Integral Methods in Science and Engineering: Analytic and Computational Procedures 3031340981, 9783031340987

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Christian Constanda Bardo E. J. Bodmann Paul J. Harris Editors

Integral Methods in Science and Engineering Analytic and Computational Procedures

Christian Constanda • Bardo E. J. Bodmann • Paul J. Harris Editors

Integral Methods in Science and Engineering Analytic and Computational Procedures

Editors Christian Constanda Department of Mathematics The University of Tulsa Tulsa, OK, USA

Bardo E. J. Bodmann Department of Mechanical Engineering Federal University of Rio Grande do Sul Porto Alegre, Rio Grande do Sul, Brazil

Paul J. Harris School of Architecture, Technology and Engineering University of Brighton Brighton, UK

ISBN 978-3-031-34098-7 ISBN 978-3-031-34099-4 https://doi.org/10.1007/978-3-031-34099-4

(eBook)

Mathematics Subject Classification: 45D05, 45Exx, 45E10, 65R20 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The international conferences on Integral Methods in Science and Engineering (IMSE) started in 1985 at the University of Texas–Arlington and continued biennially in a variety of venues around the world, bringing together specialists who employ integration techniques as essential tools in their research. These procedures exhibit generality, elegance, and efficiency, all of which are essential ingredients in the work of a wide category of practitioners. The dates and venues of the first 16 IMSE conferences are listed below. 1985, 1990: University of Texas–Arlington, TX, United States 1993: Tohoku University, Sendai, Japan 1996: University of Oulu, Finland 1998: Michigan Technological University, Houghton, MI, United States 2000: Banff, AB, Canada (organized by the University of Alberta, Edmonton) 2002: University of Saint-Étienne, France 2004: University of Central Florida, Orlando, FL, United States 2006: Niagara Falls, ON, Canada (organized by the University of Waterloo) 2008: University of Cantabria, Santander, Spain 2010: University of Brighton, United Kingdom 2012: Bento Gonçalves, Brazil (organized by the Federal University of Rio Grande do Sul) 2014: Karlsruhe Institute of Technology, Germany 2016: University of Padova, Italy 2018: University of Brighton, United Kingdom 2021: Online The online meeting in 2021 replaced the conference scheduled to be held at the Steklov Mathematical Institute in St. Petersburg, Russia, in 2020, which was ultimately cancelled due to the adverse world health conditions. The peer-reviewed chapters of this volume, arranged alphabetically by first author’s name, consist of 31 of the papers presented at the 17th IMSE conference, held online in July 2022. The editors would like to thank the reviewers for their help, Christopher Tominich at Birkhäuser–New York for his support of this project, v

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and the Springer production team for their courteous and professional handling of the publication process. The 18th IMSE conference is planned to take place in Rio de Janeiro, Brazil, in August 2024. Tulsa, OK, USA Porto Alegre, Rio Grande do Sul, Brazil Brighton, UK March 2023

Christian Constanda Bardo E. J. Bodmann Paul J. Harris

The International Steering Committee of IMSE: Christian Constanda (The University of Tulsa), Chairman; Bardo E. J. Bodmann (Federal University of Rio Grande do Sul), Co-Vice Chairman; Paul J. Harris (University of Brighton), Co-Vice Chairman; Massimo Lanza de Cristoforis (University of Padova); Sergey Mikhailov (Brunel University London); Dorina Mitrea (Baylor University); Marius Mitrea (Baylor University); Maria Perel (St. Petersburg State University); Maria Eugenia Pérez–Martínez (University of Cantabria); Adolfo Puime Pirez (State University of Northern Rio de Janeiro); Ovadia Shoham (The University of Tulsa).

Contents

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Computational Modelling Based on RIBEM Method for the Numerical Solution of Convection-Diffusion Equations . . . . . . . . . . . . . . . S. A. Al-Bayati and L. C. Wrobel 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Governing Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Boundary Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Algebraic System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Results and Discussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Conclusions and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet Problem in 2D with General Right-Hand Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. G. Ayele and M. F. Yimer 2.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Parametrix-Based Potential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Two-Operator Third Green Identity and Integral Relations . . . 2.4 Invertibility of Single Layer Potential Operator . . . . . . . . . . . . . . . . . . . . 2.5 Two-Operator BDIE Systems for Dirichlet Problem . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation of Thermal Effects in Neutron Interactions in a Physical Monte Carlo Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. G. Benvenutti, L. F. F. C. Barcellos, and B. E. J. Bodmann 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Implementation of Thermal Effects in Neutron Interactions . . . . . . . 3.3 Implementation of Thermal Effects in the Scattering Kinematics .

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3.4 Comparison to Findings From the Literature . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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On the Parameter Significance in Pandemic Modelling . . . . . . . . . . . . . . . . B. E. J. Bodmann and P. J. Harris 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 World and South Korea Statistics and a Qualitative Simulation . . . 4.3 An Analytical Solution by the Decomposition Method . . . . . . . . . . . . 4.4 Parameter Adjustment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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On a Variational Principle for Equilibrium Free Energy Functional of Simple Liquids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. S. Brikov 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Method of the Generating Functional of an Equilibrium Statistical Mechanics of Microscopic Study of Thermodynamic and Structural Parameters of Simple Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Tangent Transformation of the Generating Functional: The Functional Density Variables—The Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Introduction of a Parametrizing Set of Functions in a Form of Displaced and Averaged Gaussian Functions in 3-Dimensional Space (DAGF3Ds) to Consider Lennard-Johnson Simple Liquids in Quasicrystal Approach . . . . . . 5.5 Evaluative Calculations of Height of the 1st Structural Maximum of the Radial Distribution Function (Binary Density) by the Direct Variational Method of the Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Algorithms for Calculation of Multicenter Integrals Included in the Free Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of Variants of Seismic Signal Approximations by Proposed Sets of Functions for Statistical Structural Analysis . . . . . . . . E. S. Brikov and V. I. Dobrorodnyi 6.1 Introduction: Detection of Seismic Signals . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Modeling of the Subsystem for Detection of Seismic Signals . . . . . 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 52 57 60 61 64 65 67 67

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Topics on Space Weather: Operational Numerical Prediction for Electron Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. F. de Campos Velho, A. Petry, T. S. Klipp, G. S. Falcão, J. R. Souza, E. R. Paula, J. N. Tamoki, L. H. B. Lago, and J. V. F. Lima 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Brief SUPIM Description: Version for Operational Execution . . . . 7.3 Forecasting: Operational SUPIM-DAVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 SUMPI-DAVS: Vector Processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Ray-Tracing the Ulam Way. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 D. J. Chappell, M. Richter, G. Tanner, O. F. Bandtlow, W. Just, and J. Slipantschuk 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2 Ray-Tracing via Transfer Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.3 The Ulam Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.4 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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The Robin Boundary Value Problem for an Unbounded Plate with a Hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Constanda and D. Doty 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Exterior Robin Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 First Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Graphical Illustrations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Second Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Graphical Illustrations II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Mathematical Model of Cell Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Farmer and P. J. Harris 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A Revisit to a Double-Periodic Perforated Neumann Waveguide: Opening Spectral Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 D. Gómez, S. A. Nazarov, R. Orive-Illera, and M.-E. Pérez-Martínez 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.2 The Setting of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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11.3 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 11.4 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 12

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Spectral Homogenization Problems in Linear Elasticity: The Averaged Robin Reaction Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Gómez and M.-E. Pérez-Martínez 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Setting of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Stationary Problems and Some Preliminary Results. . . . . . . . . . . 12.4 The Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Harmonic Oscillations of a Poroelastic Body with an Application to Modelling the Spinal Cord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. J. Harris and J. Venton 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Poly-Cauchy Operator, Whitney Arrays, and Fatou Theorems for Polyanalytic Functions in Uniformly Rectifiable Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Kyeong, D. Mitrea, I. Mitrea, and M. Mitrea 14.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Influence of the Refractive Index and Absorption Coefficients in the Solution of the Radiative Conductive Transfer Equation in Cartesian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. A. Ladeia, M. Schramm, J. C. L. Fernandes, H. R. Zanetti, and A. D. Albuquerque 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Radiative Conductive Transfer Equation in Cartesian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The Discrete Ordinate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 The Decomposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 146 149 152 153 155 155 157 159 163 164

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Boundary Value Problems for Elliptic Systems on Weighted Morrey Spaces in Rough Domains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Laurel and M. Mitrea 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Calderón-Zygmund Theory on Morrey Spaces . . . . . . . . . . . . . . . . . . . . . 16.3 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recipes for Computer Implementation of a Response Matrix Spatial Spectral Nodal Method for Three-Dimensional Discrete Ordinates Neutral Particle Transport Modeling. . . . . . . . . . . . . . L. R. C. Moraes and R. C. Barros 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 The Response Matrix-Constant Nodal Method . . . . . . . . . . . . . . . . . . . . . 17.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Maximum Principles for Weak Solutions of Some Parabolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. E. Mikhailov 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Periodic Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Maximum Principles for Some Parabolic Systems . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Integral Equations Analysis of Bone Resorption Effect on Stress State Near Dental Implants . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Perelmuter 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Stresses Analysis Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Analysis of Stress State Near Dental Implants with Bone Resorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Modeling of Partially Miscible Water Alternating Gas Injection Using Geometric Thermodynamic Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Pires and B. Loza 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

191 191 196 198 203

205 205 206 215 217 218 219 219 220 221 227 229 229 230 231 237 237

239 239 240 246 253 253

xii

21

22

23

24

25

Contents

Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Ponomarev 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Philosophical Foundations of an Optimization Algorithm Inspired by Human Social Behaviour Under a Dynamical Status Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. P. L. de Oliveira and V. J. Schmidt 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 General Aspects of Optimization Algorithms. . . . . . . . . . . . . . . . . . . . . . . 22.3 An Optimization Algorithm Based on a Dynamical Social Status Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Applications of the Optimization Algorithm DySDO . . . . . . . . . . . . . . V. J. Schmidt and L. P. L. de Oliveira 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Preliminaries for the Optimization Algorithm DySDO . . . . . . . . . . . . 23.3 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Final Remarks and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Influence of the Signal to Noise Ratio on the Reconstruction of the Non-Linear Transfer Function in Signal Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Schmith, A. Schuck, B. E. J. Bodmann, and P. J. Harris 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 256 258 263 265 266

269 269 270 273 279 279 281 281 281 283 297 298

299 299 300 302 309 309

An Analytic Solution for the Transient Three-Dimensional Advection–Diffusion Equation with Non-Fickian Closure by an Integral Transform Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 V. C. Silveira, D. Buske, G. J. Weymar, and J. C. Carvalho 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 25.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Contents

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25.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 25.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 26

27

28

29

30

Failure Analysis of Composite Pipes Subjected to Bending: Classical Laminated Plate Theory vs. 3D Elasticity Solution . . . . . . . . . . T. Wang, O. Menshykov, and M. Menshykova 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Statement of the Problem and 3D Elasticity Solution . . . . . . . . . . . . . . 26.3 Laminated Plate Theory and Simplified Constitutive Equations . . . 26.4 Finite Element Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5 Numerical Results and Validation of the Models . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Analytical Formulation GILTT Applied in a Model of Contaminant Transport in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. J. Weymar, D. Buske, R. S. Quadros, I. C. Furtado, J. Konradt, and T. F. Almeida 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 Model of Contaminant Dispersion in the Porous Medium in a Solid Waste Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Existence Result for a Class of Integral Equations via Graph-Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Younis, D. Bahuguna, and D. Singh 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 Convergence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3 An Application to Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Convergence Results on the Periodic Unfolding Operator in Orlicz Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. F. Tachago, G. Gargiulo, H. Nnang, and E. Zappale 29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 Notation and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.3 The Unfolding Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Phase Flow Zero-Net Liquid Holdup in Gas-Liquid Cylindrical Cyclone (GLCC© ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Zhao, R. S. Mohan, and O. Shoham 30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 Experimental Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321 321 323 326 328 328 332 335

335 336 341 344 344 347 347 350 356 359 361 361 362 364 370 373 373 376 379

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Contents

30.4 Comparison Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 30.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 31

Error Propagation in Dynamic Iterations Applied to Linear Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Zubik-Kowal 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

387 387 389 395 399 399

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Contributors

Salam Adel Al-Bayati College of Science, Department of Mathematics and Computer Applications, University of Technology, Jadriya, Iraq Andreiky Diniz Albuquerque Engineering School, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Tamires Fonseca de Almeida Department of Mathematics and Statistics, Institute of Physics and Mathematics, Federal University of Pelotas, Capão do Leão, Brazil Tsegaye G. Ayele Addis Ababa University, Addis Ababa, Ethiopia D. Bahuguna Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India O. F. Bandtlow Queen Mary University of London, London, UK Luiz Felipe Fracasso Chaves Barcellos Nuclear Studies Group, School of Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Ricardo Carvalho de Barros University of the State of Rio de Janeiro, Polytechnic Institute, Nova Friburgo, Brazil Daniel Gustavo Benvenutti Nuclear Studies Group, School of Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Bardo Ernst Josef Bodmann Postgraduate Program in Mechanical Engineering, School of Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Evgenii Sergeevich Brikov Tyumen, Russia Daniela Buske Department of Mathematics and Statistics, Institute of Physics and Mathematics, Federal University of Pelotas, Capão do Leão, Brazil Haroldo F. de Campos Velho National Institute for Space Research, São José dos Campos, Brazil

xv

xvi

Contributors

Jonas da Costa Carvalho Faculty of Meteorology, Federal University of Pelotas, Pelotas, Rio Grande do Sul, Brazil David J. Chappell Department of Physics and Mathematics, Nottingham Trent University, Nottingham, UK Christian Constanda Department of Mathematics, The University of Tulsa, Tulsa, OK, USA Leonardo Rodrigues da Costa Moraes Polytechnic Institute, University of the State of Rio de Janeiro, Rio de Janeiro, Brazil Vladimir I. Dobrorodnyi Lugovoi village, Russia Dale Russell Doty Department of Mathematics, The University of Tulsa, Tulsa, OK, USA Gabriel Sandim Falcão National Institute for Space Research, São José dos Campos, Brazil Adam Farmer School of Architecture, Technology and Engineering, University of Brighton, Brighton, UK Júlio César Lombaldo Fernandes Institute of Mathematics and Statistics, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Igor da Cunha Furtado Science and Technology of Rio Grande do Sul, Federal Institute of Education, Pelotas, Brazil Giuliano Gargiulo University of Sannio, Benevento, Italy Delfina Gómez Departamento Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Cantabria, Spain Paul J. Harris School of Architecture, Technology and Engineering, University of Brighton, Brighton, UK W. Just Queen Mary University of London, London, UK Telmo dos Santos Klipp National Institute for Space Research, São José dos Campos, Brazil Josiane Konradt Department of Mathematics and Statistics, Institute of Physics and Mathematics, Federal University of Pelotas, Capão do Leão, Brazil Jeongsu Kyeong Temple University, Philadelphia, PA, USA Cibele Aparecida Ladeia Institute of Mathematics and Statistics, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Luiz Henrique Brock Lago Technology Center, Federal University of Santa Maria, Santa Maria, Brazil Marcus Laurel Baylor University, Waco, TX, USA

Contributors

xvii

João Vicente Ferreira Lima Technology Center, Federal University of Santa Maria, Santa Maria, Brazil Bernardo Loza State University of Norte Fluminense Darcy Ribeiro, Macaé, Brazil Oleksandr Menshykov School of Engineering, University of Aberdeen, Aberdeen, UK Marina Menshykova School of Engineering, University of Aberdeen, Aberdeen, UK Sergey E. Mikhailov Department of Mathematics, Brunel University London, Uxbridge, UK Dorina Mitrea Baylor University, Waco, TX, USA Irina Mitrea Temple University, Philadelphia, PA, USA Marius Mitrea Baylor University, Waco, TX, USA Ram S. Mohan Department of Mechanical Engineering, The University of Tulsa, Tulsa, OK, USA Sergey A. Nazarov Faculty of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia Institute of Problems of Mechanical Engineering RAS, V.O., St. Petersburg, Russia Hubert Nnang University of Yaounde I, Yaounde, Cameroon Luiz Paulo Luna de Oliveira Instituto Federal Sul Rio Grandense—IFSul, COFORGE-Charqueadas, Pelotas, Brazil Rafael Orive-Illera Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid, Spain Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Madrid, Spain Eurico R. Paula National Institute for Space Research, São José dos Campos, Brazil M. Perelmuter Ishlinsky Institute for Problems in Mechanics of RAS, Moscow, Russia Maria-Eugenia Pérez-Martínez Departamento Matemática Aplicada y Ciencias de la Computación, ETSI Caminos, Universidad de Cantabria, Santander, Spain Adriano Petry National Institute for Space Research, São José dos Campos, Brazil Adolfo Puime Pires State University of Norte Fluminense Darcy Ribeiro, Macaé, Brazil Dmitry Ponomarev Centre Inria d’Université Côte d’Azur, Sophia Antipolis, Antibes, France

xviii

Contributors

St. Petersburg Department of Steklov Mathematical Institute St. Petersburg, Russia Régis Sperotto de Quadros Department of Mathematics and Statistics, Institute of Physics and Mathematics, Federal University of Pelotas, Capão do Leão, Brazil Martin Richter The University of Nottingham, Mathematical Sciences, University Park, Nottingham, UK Vinicius José Schmidt CWI Software, São Leopoldo, Brazil Jean Schmith Polytechnic School, University of Vale do Rio dos Sinos, São Leopoldo, Brazil Marcelo Schramm Center of Engineering, Federal University of Pelotas, Pelotas, Brazil Adalberto Schuck Department of Electrical Engineering, School of Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Ovadia Shoham Department of Petroleum Engineering, The University of Tulsa, Tulsa, OK, USA Viliam Cardoso Silveira Department of Mathematics and Statistics, Institute of Physics and Mathematics, Federal University of Pelotas, Capão do Leão, Brazil D. Singh Department of Applied Science, National Institute of Technical Teachers’ Training and Research, Bhopal, India Julia Slipantschuk Mathematics Institute, The University of Warwick, Coventry, UK Jonas R. Souza National Institute for Space Research, São José dos Campos, Brazil Joel Fotso Tachago University of Bamenda, Bamenda, Cameroon J. N. Tamoki Independent HPC Analyst, São Paulo, Brazil Gregor Tanner The University of Nottingham, Mathematical Sciences, University Park, Nottingham, UK Jenny Venton The National Physical Laboratory, London, UK Tianyu Wang School of Engineering, University of Aberdeen, Aberdeen, UK Guilherme Jahnecke Weymar Department of Mathematics and Statistics, Institute of Physics and Mathematics, Federal University of Pelotas, Capão do Leão, Brazil Luiz C. Wrobel Department of Civil and Environmental Engineering, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil Markos F. Yimer Addis Ababa University, Addis Ababa, Ethiopia

Contributors

xix

Mudasir Younis Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India Heloisa Robattini Zanetti Institute of Mathematics and Statistics, Federal University of Rio Grande do Sul, Porto Alegre, Brazil Elvira Zappale Department of Basic and Applied Sciences for Engineering, Sapienza University of Rome, Rome, Italy Haoqing Zhao Department of Petroleum Engineering, The University of Tulsa, Tulsa, OK, USA Barbara Zubik-Kowal Boise State University, Boise, ID, USA

Chapter 1

Computational Modelling Based on RIBEM Method for the Numerical Solution of Convection-Diffusion Equations S. A. Al-Bayati and L. C. Wrobel

1.1 Introduction A new robust transformation technique, called the radial integration method (RIM), was developed by Gao [Ga02b], which not only can transform any complicated domain integral to the boundary without using particular solutions but can also remove various singularities appearing in the domain integrals [Ga02a]. Based on the RIM, the combined RIBEM approach was developed and applied to handle a wide range of engineering and mathematical problems, including nonhomogeneous steady-state and transient heat conduction problems, acoustics problems, diffusion problems, elastoplasticity, and other mechanical problems [Al18, CuEtAl18, QuEtAl13, GaEtAl15b]. Yang and Gao [YaGa10] proposed a new boundary element technique to handle transient heat conduction problems, for which the RIM is implemented to transform the domain integral associated with the time derivative of temperatures and the radial integral is evaluated numerically. The RIM can be applied to the combination of a power series expansion operated on the parameter plane of intrinsic coordinates or for the projection plane of global coordinates [GaEtAl15a], for which it can evaluate different types of singular boundary integrals numerically [FeEtAl16]. Recently, Feng [FeGa016] proposed a new type of single integral equation technique to solve transient heat conduction problems in multimedia with variable thermal properties. The same author has also derived an interface

S. A. Al-Bayati () Branch of Mathematics and Computer Applications, Department of Applied Sciences, University of Technology, Baghdad, Iraq e-mail: [email protected] L. C. Wrobel Department of Civil and Environmental Engineering, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_1

1

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S. A. Al-Bayati and L. C. Wrobel

integral equation method to solve general multimedia mechanical problems by considering the discontinuity of the stress-strain constitutive relationship during the transformation from elastic to plastic regions [GaEtAl16]. Feng also proposed a new BEM formulation without initial stresses for solving 2D and 3D elastoplastic problems [FeEtAl15]. Yang et al. [YaEtAl15] successfully derived a series of analytical expressions for evaluating radial integrals, which are utilized in the RIM for converting the domain integrals into equivalent boundary integrals. By using these analytical expressions, the computation time spent in the numerical calculation of radial integrals can be considerably reduced. This technique has been implemented to handle nonhomogeneous heat conduction, nonhomogeneous elasticity, and thermoelasticity problems. However, in the derivation of the analytical radial integral expressions, some special circumstances may appear, which will influence the accuracy of the results [FeEtAl16]. The RIBEM was successfully derived and implemented for the free vibration analysis of anisotropic plates [ZhEtAl16], and to thermoelasticity, elastic inclusion problems, creep damage mechanics problems, transient heat conduction problems, and viscous flow problems [AlEtAl07, FeEtAl11]. Owing to the advantages of the RIM, mainly that particular solutions are not required and various domain integrals appearing in the same integral equation can be dealt with simultaneously, RIM-based BEMs have gained considerable attention from several BEM researchers [Co02, PeEtAl13, AlWr17, IjEtAl17]. The radial integral in the RIBEM formulation is usually calculated by utilizing Gaussian quadrature, which requires it to be computed at each Gaussian point of the boundary element under consideration. Computing the radial integrals numerically, especially for a three-dimensional nonlinear and large-scale nonhomogeneous problem, is highly time-consuming, and this will lead to a reduction of the performance and the efficiency of this numerical method [FeEtAl15, FeEtAl11].

1.2 Governing Differential Equation The solution of convection-diffusion equations is a difficult task for all numerical methods because of the nature of the governing equation, which includes first-order and second-order partial derivatives in space [PaEtAl92, AlWr018, Wr02, AlWr22]. The convection-diffusion equation with source term is the basis of many physical and chemical phenomena, and its use has also spread into economics, financial forecasting, and other fields [Az19, AlWr20, YuEtAl19]. The proposed model, i.e., 2D convection-diffusion problem including constant or variable source term over a domain .Ω in .R2 limited by a boundary .Γ , for isotropic materials, is governed by the following differential equation: D∇ 2 T (x, y) − vx

.

∂T (x, y) ∂T (x, y) = P (x, y) − vy ∂y ∂x (x, y) ∈ Ω ⊂ R2

.

(1.1)

1 Computational Modelling Based on RIBEM

3

In expression (1.1), T represents the temperature of a substance, treated as a function of space; .Γ is a bounded domain in .R2 . The velocity components .vx and .vy along the x and y directions are assumed to be constant in space. Besides, D is the diffusivity coefficient, and .P (x, y) represents the source term. The boundary conditions are T

.

q=

.

over ∂T ∂n

ΓD

over

ΓN

where .ΓD and .ΓN are the Dirichlet and Neumann parts of the boundary with .Γ = ΓD ∪ΓN and .ΓD ∩ΓN = ∅. The parameter that describes the relative influence of the convective and diffusive components is called Péclet number, .Pé = |v| L/D, where ⎞1/2 ⎛ .v = vx2 + vy2 is the velocity and L is a characteristic length of the domain. For small values of .Pé, Eq. (1.1) behaves as a parabolic differential equation, while for large values, the equation becomes more like hyperbolic. These changes in the structure of the differential equation according to the value of the Péclet number have significant effects on its numerical solution.

1.3 Boundary Integral Equation Let us consider a region .Ω ⊂ R 2 bounded by a piecewise smooth boundary .Γ . The transport of T is governed by the 2D convection-diffusion Eq. (1.1). The variable T can be interpreted as temperature for heat transfer problems, concentration for dispersion problems, etc. and will be herein referred to as a temperature. For the sake of obtaining an integral equation equivalent to the partial differential equation (1.1), a fundamental solution of Eq. (1.1) is necessary. Expression (1.1) can be transformed into an equivalent integral equation by applying a weighted residual technique. Starting with the weighted residual statement: ⎞   ⎛ ∂T (x, y) ∂T (x, y) 2 ∗ D∇ T (x, y) − vx − vy T dΩ = P (x, y) T ∗ dΩ . ∂x ∂y Ω Ω

and integrating by parts twice the Laplacian and once the first-order derivatives, the following equation is obtained:  T (ξ ) = D

.

Γ

T∗

∂T dΓ − D ∂n

 T Γ

∂T ∗ dΓ − ∂n

 Γ

T T ∗ v¯n dΓ −



P (x, y)T ∗ dΩ

Ω

(1.2) where .v¯n = v.n, .n is the unit outward normal vector and the dot represents scalar product. In the above equation, .T ∗ is the fundamental solution of the convection-

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diffusion equation without source term. For 2D problems, .T ∗ is of the form T ∗ (ξ, χ ) =

.

v.r 1 e−( 2D ) K0 (μr) 2π D

where ⎛ μ=

.

v¯ 2D

⎞2

k + D

┐1

2

in which .ξ and .χ are the source and field points, respectively, and r is the modulus of r, the distance vector between the source and field points, and k represents the firstorder reaction constant or adsorption coefficient. The derivative of the fundamental solution with respect to the outward normal direction is given by: ⎛

1 ∂T ∗ − = e . ∂n 2π D

v¯ .r 2D

⎞⎡

v¯n ∂r −μ K1 (μr) − K0 (μr) ∂n 2D

┐

In the above, .K0 and .K1 are Bessel functions of second kind, of orders zero and one, respectively. The exponential term is responsible for the inclusion of the correct amount of “upwind” into the formulation [RaSk13]. Expression (1.2) is valid for source points .ξ inside the domain .Ω. A similar expression can be obtained, by a limit analysis, for source points .ξ on the boundary .Γ , in the form:  c (ξ ) T (ξ ) = D

T

.

∗ ∂T

∂n

 dΓ − D Γ

Γ

 −

∂T ∗ dΓ − T ∂n



T T ∗ v¯n dΓ

Γ

P (x, y) T ∗ dΩ

(1.3)

Ω

in which .c (ξ ) is a function of the internal angle the boundary .Γ makes at point .ξ [AlWr22].

1.4 Discretization It should be noted that the radial integral can be implemented and evaluated by direct application (analytically) when the source term is constant, whereas for variable source terms, the numerical integration will be applied. Therefore, the radial integral expression can be written as:  P (χ ) =

+1

.

−1

⎛

r (χ ) r (χ ) η+ p (χ (η)) 2 2

⎞α

r (χ ) dη 2

1 Computational Modelling Based on RIBEM

⎛ =

r (χ ) 2

5

⎞α+1 ∑ Ng

(1 + ηn )α p (χ (ηn )) wn

n=1

where r=

.

r (χ ) r (χ ) , (−1 ≤ η ≤ 1) η+ 2 2

and χi = ξi + r,i r; i = 1, 2

.

where the quantities .ξi , .r,i , and .α are constants for the radial integral equation; however, .α = 1 for the two-dimensional case. The symbol .p (χ ) means the variable p takes the values on the boundary .γ , .Ng is the number of Gaussian points, .ηn are the Gaussian point coordinates, and .wn is the associated weight. In our work, 60 Gauss points are utilized to increase the accuracy. The numerical solution of the problem, i.e., Eq. (1.3), can be represented in a discretized form, in which the integrals over the boundary are approximated by a summation of integrals over individual boundary elements: ci Ti = D

N  ∑

T∗

.

j =1 Γ

⎞ N  ⎛ ∑ ∂T ∗ vn ∗ ∂T T dΓ dΓ − D + T ∂n ∂n D j =1 Γ

j

j

.

−

N  ∑ j =1 Γ

⎡

⎤ Ng ⎛ ⎞ ∑ 2 ⎣ 1 ∂r r (1 + ξn ) p (χ (ξn )) wn ⎦ T ∗ dΓ r ∂n 2 n=1

j

where the index i stands for values at the source point .ξi , .Ng is the number of integration points, and N is the number of boundary elements. In the above expression, it can be noticed that: .

┐ ⎡ vn ∗ vn ∂T ∗ 1 ( −v.r ) ∂r + T = e 2D −μK1 (μr) + K0 (μr) ∂n D 2π D ∂n 2D

Next, the constant functions T and . ∂T ∂n within each element are approximated by their nodal values. Therefore, the following expression is obtained: ci Ti =

.

N ∑ ( ) Gij qj − Hij Tj + Bi j =1

(1.4)

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S. A. Al-Bayati and L. C. Wrobel

It should be pointed out the two influence matrices can be represented as:  Gij = D

.

T ∗ dΓ

Γj

and ⎛

 Hij = D

.

Γj

⎞ ∂T ∗ vn ∗ + T dΓ D ∂n

with  Bi =

.

⎞ Ng ⎛ r ⎞2 ∑ 1 ∂r ⎝ (1 + ξn ) p (χ (ξn )) wn ⎠ T ∗ dΓ r ∂n 2 ⎛

n=1

Γj

The above expression (1.4) involves N values of T and .q = prescribed as boundary conditions.

∂T ∂n ,

half of which are

1.5 Algebraic System of Equations In order to compute the remaining unknowns, it is necessary to generate N equations. This can be done by utilizing a simple collocation technique, i.e., by making the equation be satisfied at the N nodal points. The .ci values have been incorporated into the diagonal coefficients of matrix H . After introducing the boundary conditions, the system is reordered and solved by a direct method, for instance, least squares, Gauss elimination, or LU decomposition. The result is a system of equations of the form: H T = Gq + B

.

(1.5)

where B is the term representing the radial integral for the source term as in Eq. (1.5). Evaluation of the coefficients of matrices H and G and vector B is carried out numerically. It should be noted that the diagonal coefficients of matrix G have a weak singularity of the logarithmic type and are calculated using the self-adaptive scheme of Telles [Te87].

1.6 Results and Discussions The present section is concerned with numerical tests of the RIBEM for the solution of 2D convection-diffusion equations including constant and variable source parts. We shall examine two case studies with known analytical solutions to quantitatively

1 Computational Modelling Based on RIBEM

7

and qualitatively assess the convergence, accuracy, and robustness of the proposed formulation.

1.6.1 Example 1: Nonlinear Source Term This problem has been modeled over a square domain, .Ω = {(x, y) : 0 ≤ x, y ≤ 1}. The first tested case is a nonhomogeneous diffusion-convection equation described by: D∇ 2 T − vx

.

∂T ∂T = P (x, y) − vy ∂y ∂x

where .P (x, y) and the Dirichlet boundary condition are computed from the analytical solution as shown in Eq. (1.6): ⎛

T (x, y) = e

.

− 12 vx x

⎞

⎛

1 sin vx y 2

⎞ (1.6)

Figure 1.1 presents the solution using .vx = 1 and .D = 1. The plot shows the temperature profiles of T for 160 selected internal points along the diagonals, where the predicted results for the temperature agree perfectly well with the corresponding analytical solution. The boundary is discretized into 150 constant elements. Figure 1.2 presents the solution using .vx = −1 and .D = 1. This plot shows the temperature profiles of T for 160 selected internal nodes along the diagonals, and, once again, the predicted results for the temperature agree very well with the corresponding analytical solution.

Temperature ° C

0.1

RIBEM Analytical

0.05 0 -0.05 -0.1 20

40

60

80

100

120

140

160

Internal Points

Fig. 1.1 Computed temperature T for selected internal points with positive velocity: comparison between the analytical (solid line) and numerical (star points) solutions

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S. A. Al-Bayati and L. C. Wrobel

Temperature ° C

0.1

RIBEM Analytical

0.05 0 -0.05 -0.1 20

40

60

80

100

120

140

160

Internal Points

Fig. 1.2 Computed temperature T for selected internal points with negative velocity: comparison between the analytical (solid line) and numerical (star points) solutions

1.6.2 Example 2: Constant Source Term In this test case, a 2D transport problem with constant source term has been examined to analyze the validity of the present formulation. This problem deals with a square cross section with unit dimensions. We suppose the diffusivity .D = 1 and velocity component .vy = 0. We shall consider the cases where .P = 5, 10, 50, 100, and 500. The mixed boundary conditions are as follows: For vertical sides, i.e., .x = 0 and .x = 1, Dirichlet boundary conditions are imposed: T = 0; x = 0, 0 ≤ y ≤ 1

.

T = 1; x = 1, 0 ≤ y ≤ 1

.

and zero fluxes (Neumann boundary conditions) for the horizontal faces, i.e., .y = 0 and .y = 1: .

q=

∂T = 0; y = 0, 0 ≤ x ≤ 1 ∂n

q=

∂T = 0; y = 1, 0 ≤ x ≤ 1 ∂n

.

The analytical solution of the problem is represented by: ⎡ ⎛v Lx ⎞ ┐ x TL − T0 − (P L/vx ) P D ⎛ ⎞ e x+ .T (x, y) = T0 + −1 vx L vx e D −1 The geometry is discretized into 120 equally spaced constant elements, 30 on each side. The temperature T at boundary nodes along the sides .y = 0 and .y = 1 is investigated. Figure 1.3 displays the numerical and analytical solutions along the bottom and the top sides of the channel for .P = 5 and .vx = 10. Next, Fig. 1.4

1 Computational Modelling Based on RIBEM

9

1

Temperature ° C

RIBEM

0.8

Analytical

0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X-axis(m)

Fig. 1.3 Computed temperature T along faces .y = 0 and .y = 1 for .P = 5 and .vx = 10: comparison between the analytical (solid line) and numerical (star points) solutions 0.8

Temperature ° C

RIBEM Analytical

0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X-axis(m)

Fig. 1.4 Computed temperature T along faces .y = 0 and .y = 1 for .P = 10 and .vx = 30: comparison between the analytical (solid line) and numerical (star points) solutions 25

Temperature ° C

RIBEM

20

Analytical

15 10 5 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X-axis(m)

Fig. 1.5 Computed temperature T along faces .y = 0 and .y = 1 for .P = 500 and .vx = 20: comparison between the analytical (solid line) and numerical (star points) solutions

presents RIBEM and analytical solutions for .P = 10 and .vx = 30. Figure 1.5 shows the numerical and exact solutions for .P = 500 and .vx = 20. All figures display the expected behavior for the temperature profiles at different Péclet numbers and with various source term values, showing an excellent agreement with the analytical results. The simulation and the analytical solutions on two-dimensional refined meshes are computed with good agreement. We now solve this problem with

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S. A. Al-Bayati and L. C. Wrobel

Temperature ° C

1.8

RIBEM

1.6

Analytical

1.4 1.2 1 0.8 0.6 0.4 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X-axis (m)

Fig. 1.6 Computed temperature T along faces .y = 0 and .y = 1 for .P = 10 and .vx = −10: comparison between the analytical (solid line) and numerical (star points) solutions

Temperature ° C

2

RIBEM Analytical

1.8 1.6 1.4 1.2 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X-axis(m)

Fig. 1.7 Computed temperature T along faces .y = 0 and .y = 1 for .P = 80 and .vx = −80: comparison between the analytical (solid line) and numerical (star points) solutions 3

Temperature ° C

RIBEM Analytical

2.5 2 1.5 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X-axis(m)

Fig. 1.8 Computed temperature T along face .y = 0 and .y = 1 for .P = 100 and .vx = −50: comparison between the analytical (solid line) and numerical (star points) solutions

negative velocities to provide further validation of the proposed scheme. Figure 1.6 displays the solutions for .P = 10 and .vx = −10. In Fig. 1.7, the velocity has been increased to .vx = −80 and .P = 80. The Péclet number in this case is 80. Finally, the value of P is considered as 100 to make the velocity and the temperature profiles significantly sharp in the opposite direction. Figure 1.8 compares the BEM and analytical solutions for this case. Once again, the results show very good agreement

1 Computational Modelling Based on RIBEM

11

for a Péclet number equal to 50 in this case. Throughout this section, the figures for negative velocities show very good agreement between the numerical and the analytical solutions, for the temperature profile results at different values of the Péclet number and with different source term values. We observe that the numerical solutions are non-oscillatory and are in good agreement with the analytical solutions in all cases.

1.7 Conclusions and Discussions A computational numerical formulation of the boundary element method (BEM) combined with radial integration technique (RIM) have been implemented to solve 2D convection-diffusion problems, including constant and variable source parts. The steady-state fundamental solution of the corresponding equation is employed in this work. The numerical results presented in Sect. 1.6 show the versatility and distinct advantages of the proposed approach, which shows its ability, validity, and accuracy when solving two different types of problems with different source terms and velocity fields. It is clear that, as the velocity increases, the temperature distribution becomes steeper and more difficult to reproduce with numerical models. The computational findings produced by RIBEM provided an excellent agreement with the exact solutions when solving different numerical test cases, in which the results do not present oscillations or damping of the wave front, as may present in other computational methods. Finally, the proposed mathematical scheme is very simple, computationally fast, and easy to program; however, it can be extended to transient, temperature-dependent, and nonlinear problems with variable coefficients. This will be a key issue in our next work.

References [Al18] Al-Bayati, S.: Boundary Element Analysis for Convection-Diffusion-Reaction Problems Combining Dual Reciprocity and Radial Integration Methods, Ph.D. Thesis. Brunel University, London (2018) [AlEtAl07] Albuquerque, E.L., Sollero, P., Portilho de Paiva, W.: The radial integration method applied to dynamic problems of anisotropic plates. Commun. Numer. Methods Eng. 23, 805–818 (2007) [AlWr018] Al-Bayati, S., Wrobel, L.C.: A novel dual reciprocity boundary element formulation for two-dimensional transient convection–diffusion–reaction problems with variable velocity. Eng. Anal. Boundary Elem. 94, 60–68 (2018) [AlWr17] Al-Bayati, S., Wrobel, L.C.: DRBEM formulation for convection-diffusion-reaction problems with variable velocity, chapter in a book of proceedings. In: Chappell, D. (ed.) Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), pp. 5– 14. Nottingham Trent University Press, Nottingham (2017)

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[AlWr20] Al-Bayati, S., Wrobel, L.C.: Transient convection-diffusion-reaction problems with variable velocity field by means of DRBEM with different radial basis functions. In: Constanda, C. (ed.) Computational and Analytic Methods in Science and Engineering, pp. 21–43. Birkhd’user, Cham (2020). [AlWr22] Al-Bayati, S., Wrobel, L.C.: Numerical modelling of convection-diffusion problems with first-order chemical reaction using the dual reciprocity boundary element method. Int. J. Numer. Methods Heat Fluid Flow 32, 1793–1823 (2021) [Az19] Azis, M.I.: Standard-BEM solutions to two types of anisotropic-diffusion convection reaction equations with variable coefficients. Eng. Anal. Boundary Elem. 105, 87–93 (2019) [Co02] Constanda, C.: Direct and Indirect Boundary Integral Equation Methods. Chapman & Hall/CRC, New York (2000) [CuEtAl18] Cui, M., Xu, B.-B., Feng, W.-Z., Zhang, Y., Gao, X.-W., Peng, H.-F.: A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity. Numer. Heat Transfer, Part B: Fundamentals 73, 1–18 (2018) [FeEtAl11] Feng, Z., Gao X.-W., Liu, J., Yang, K.: Using analytical expressions in radial integration BEM for variable coefficient heat conduction problems. Eng. Anal. Boundary Elem. 35, 1085–1089 (2011) [FeEtAl15] Feng, W.-Z., Gao, X.-W., Liu, J., Yang, K.: A new BEM for solving 2D and 3D elastoplastic problems without initial stresses/strains. Eng. Anal. Boundary Elem. 61, 134–144 (2015) [FeEtAl16] Feng, W.-Z., Yang, K., Cui, M., Gao, X.-W.: Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems. Int. Commun. Heat Mass Transfer 79, 21–30 (2016) [FeGa016] Feng, W.-Z., Gao, X.-W.: An interface integral equation method for solving transient heat conduction in multi-medium materials with variable thermal properties. Int. J. Heat Mass Transf. 98, 227–239 (2016) [Ga02a] Gao, X.-W.: A boundary element method without internal cells for two-dimensional and three-dimensional elastoplastic problems. J. Appl. Mech. 69, 154–160 (2002) [Ga02b] Gao, X.-W.: The radial integration method for evaluation of domain integrals with boundary-only discretization. Eng. Anal. Boundary Elem. 26, 905–916 (2002) [GaEtAl15a] Gao, X.-W., Feng, W.-Z., Yang, K., Cui, M.: Projection plane method for evaluation of arbitrary high order singular boundary integrals. Eng. Anal. Boundary Elem. 50, 265–274 (2015) [GaEtAl15b] Gao, X.-W., Zheng, B.-J., Yang, K., Zhang, Ch.: Radial integration BEM for dynamic coupled thermoelastic analysis under thermal shock loading. Comput. Struct. 158, 140–147 (2015) [GaEtAl16] Gao, Xi,-W., Feng, W,-Z., Zheng, B,-J., Yang, K.: An interface integral equation method for solving general multi-medium mechanics problems. Int. J. Numer. Methods Eng. 50, 696–720 (2016) [IjEtAl17] Iljaž, J., Wrobel, L.C., Hriberšek, M., Marn, J.: Subdomain BEM formulations for the solution of bio-heat problems in biological tissue with melanoma lesions. Eng. Anal. Boundary Elem. 83, 25–42 (2017) [PaEtAl92] Partridge, P., Brebbia, C., Wrobel, L.C.: The dual reciprocity boundary element method. Comp. Mech. Pub., Southampton (1992) [PeEtAl13] Peng, H.-F., Bai, Y.-G., Yang, K., Gao, X.-W.: Three-step multi-domain BEM for solving transient multi-media heat conduction problems. Eng. Anal. Boundary Elem. 37, 1545–1555 (2013). [QuEtAl13] Qu, S., Li, S., Chen, H.-R., Qu, Z.: Radial integration boundary element method for acoustic eigenvalue problems. Eng. Anal. Boundary Elem. 37, 1043–1051 (2013) [RaSk13] Ravnik, J., Škerget, L.: A gradient free integral equation for diffusion–convection equation with variable coefficient and velocity. Eng. Anal. Boundary Elem. 37, 683– 690 (2013)

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[Te87] Telles, J.C.F.: A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. Int. J. Numer. Methods Eng. 24, 959–973 (1987) [Wr02] Wrobel, L.C.: The Boundary Element Method: Applications in Thermo-Fluids and Acoustics. Wiley, Chichester (2002) [YaEtAl15] Yang, K., Peng, H.-F., Cui, M., Gao, X.-W.: New analytical expressions in radial integration BEM for solving heat conduction problems with variable coefficients. Eng. Anal. Boundary Elem. 50, 224–230 (2015) [YaGa10] Yang, K., Gao, X.-W.: Radial integration BEM for transient heat conduction problems. Eng. Anal. Boundary Elem. 34, 557–563 (2010) [YuEtAl19] Yue, X., Wang, F., Hua, Q., Qiu, X.-Y.: A novel space–time meshless method for nonhomogeneous convection–diffusion equations with variable coefficients. Appl. Math. Lett. 92, 44–150 (2019) [ZhEtAl16] Zheng, B., Gao, X.-W., Zhang, C.: Radial integration BEM for vibration analysis of two-and three-dimensional elasticity structures. Appl. Math. Comput. 277, 111–126 (2016)

Chapter 2

Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet Problem in 2D with General Right-Hand Side T. G. Ayele and M. F. Yimer

2.1 Preliminaries Let .Ω be a domain in .R2 bounded by a smooth curve .∂Ω. Consider the scalar elliptic differential equation, which for sufficiently smooth function u has the following strong form, ⎞ ⎛ 2 ∑ ∂ ∂u(x) .Au(x) := A(x, ∂x )u(x) := = f˜(x), a(x) ∂xi ∂xi

x ∈ Ω,

(2.1)

i=1

where u is an unknown function and .f˜ is a given function in .Ω. We assume that a ∈ C ∞ (R2 ) and

.

0 < amin ≤ a(x) ≤ amax < ∞,

∀ x ∈ R2 .

In what follows, .D(Ω) = C0∞ (Ω), H s (Ω) = H2s (Ω), and H s (∂Ω) = H2s (∂Ω) are the Bessel potential spaces, where .s ∈ R is an arbitrary real number (see, e.g., [Lo72, McL00]). We recall that .H s coincides with the Sobolev-Slobodecki spaces s s (Ω) and .W for any nonnegative s. We also consider the spaces .H 2 s H∂Ω := {g : g ∈ H s (R2 ), supp(g) ⊂ ∂Ω}.

.

(2.2)

For .u ∈ H 2 (Ω), we denote by .Ta+ the corresponding canonical (strong) conormal derivative operator on .∂Ω in the sense of traces,

T. G. Ayele () · M. F. Yimer Addis Ababa University, Addis Ababa, Ethiopia e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_2

15

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T. G. Ayele and M. F. Yimer

Ta+ u :=

2 ∑

.

a(x)ni (x)γ +

i=1

∂u(x) ∂u(x) , = a(x)γ + ∂n(x) ∂xi

(2.3)

where .n(x) is the outward unit normal vector to .Ω at the point .x ∈ ∂Ω. However, the classical conormal derivative operator is generally not well-defined if .u ∈ H 1 (Ω); see, e.g., [Mi15, Appendix A]. For .u ∈ H 1 (Ω), the PDE Au in (2.1) is understood in the sense of distributions Ω := −Ea (u, v),

.

∀v ∈ D(Ω),

(2.4)

where  Ea (u, v) :=

a(x)∇u(x) · ∇v(x)dx

.

Ω

is a symmetric bilinear form and the duality brackets .Ω denote the value of a linear functional (distribution) g, extending the usual .L2 inner product. Since the 1 (Ω), the above formula defines a continuous operator .A : set .D(Ω) is dense in .H 1 −1 1 (Ω)]∗ H (Ω) → H (Ω) = [H Ω := −Ea (u, v),

.

1 (Ω). ∀u ∈ H 1 (Ω), ∀v ∈ H

(2.5)

−1 (Ω) = [H 1 (Ω)]∗ Let us consider also the operator, .Aˇ : H 1 (Ω) → H ˇ v>Ω : = −Ea (u, v) = − R2 = Ω , ∀u ∈ H 1 (Ω), ∀v ∈ H 1 (Ω), = Ω , ∀w ∈ H 2 (∂Ω), = ∂Ω := Ω + Ea (u, v) = ∂Ω : = Ω , ∀w ∈ H 2 (∂Ω), = ∂Ω := ∂Ω := ∂Ω + 0 for .x ∈ Ω. Since for .u ∈ H 1,0 (Ω, Δ), Au − Bu = (a − b)Δu + ∇(a − b) · ∇u ∈ L2 (Ω), we have .H 1,0 (Ω; A) = H 1,0 (Ω; B). Let

2 Two-Operator BDIEs in 2D with General Right-Hand Side .u

19

∈ H 1 (Ω) and .v ∈ H 1,0 (Ω; B). Then, we write the first Green identity for operator

B in the form  .Eb (u, v)

+ Ω

u(x)Bv(x)dx = ∂Ω ,

(2.16)

where  .Eb (u, v)

=

b(x)∇u(x) · ∇v(x)dx. Ω

−1 (Ω), then according to the definition If, in addition, .Au = f˜ in .Ω, where .f˜ ∈ H + ˜ of .Ta (f , u), in (2.8), the two-operator second Green identity can be written as .Ω

 −

 u(x)Bv(x)dx +

Ω

[a(x) − b(x)]∇u(x) · ∇v(x)dx Ω

= ∂Ω − ∂Ω .

(2.17)

Moreover, for .u, v ∈ H 1,0 (Ω; A) = H 1,0 (Ω; B) (2.17) becomes   . [v(x)Au(x) − u(x)Bv(x)]dx + [a(x) − b(x)]∇u(x) · ∇v(x)dx Ω

Ω

=

∂Ω

− ∂Ω .

2.2 Parametrix-Based Potential Operators Definition 4 We will say, a function Pb (x, y) of two variables x, y ∈ Ω is a parametrix (Levi function) for the operator B(x; ∂x ) in R2 if (see, e.g., [He77, Mi02, Mir70, Po98a, Po98b]) .B(x, ∂x )Pb (x, y)

= δ(x − y) + Rb (x, y),

(2.18)

where δ is the Dirac-delta distribution, while Rb (x, y) is a remainder possessing at most a weak singularity at x = y.

For some positive constant r0 , the parametrix and hence the corresponding remainder in 2D can be chosen as in [Mi02] .Pb (x, y)

=

Rb (x, y) =

⎛ ⎞ |x − y| 1 ln ,. 2πb(y) r0 2 ∑ i=1

xi − yi ∂b(x) , 2πb(y)|x − y|2 ∂xi

(2.19) x, y ∈ R2 .

(2.20)

20

T. G. Ayele and M. F. Yimer

The parametrix Pb (x, y) in formula (2.19) is the fundamental solution to the operator B(y, ∂x ) := b(y)Δx , with “frozen” coefficient b(x) = b(y), and .B(y, ∂x )Pb (x, y)

= δ(x − y).

(2.21)

Let b ∈ C ∞ (R2 ), and let b(x) > 0 a.e. in R2 . For some scalar function g , the parametrix-based Newtonian and the remainder volume potential operators, corresponding to the parametrix (2.19) and the remainder (2.20), are given by  .Pb g(y)

:=

R2

Pb (x, y)g(x)dx,

y ∈ R2 , .

(2.22)

Pb (x, y)g(x)dx,

y ∈ Ω, .

(2.23)

Rb (x, y)g(x)dx,

y ∈ Ω.

(2.24)

 Pb g(y) :=

Ω

 Rb g(y) :=

Ω

For g ∈ H s (R2 ), s ∈ R, (2.22) is understood as Pb g = b1 PΔ g, where the Newtonian potential operator PΔ for Laplacian Δ is well-defined in terms of the Fourier transform (i.e., as pseudodifferential operator), on any space H s (R2 ). For s (Ω) and any s ∈ R, definitions in (2.23) and (2.24) can be understood as g∈H .Pb g

=

a 1 rΩ PΔ g, Pb g = rΩ Pa g, b b

and

1 b

Rb g = − rΩ ∇ · PΔ (g∇b),

(2.25)

 , where E : while for g ∈ H s (Ω), − 12 < s < 12 , as (2.25) with g replaced by Eg s (Ω), − 1 < s < 1 , is the unique extension operator related with the H s (Ω) → H 2 2 ˚ of extension by zero (cf. [Mi11, Theorem 16]). operator E For y ∈/ ∂Ω, the single layer and the double layer surface potential operators, corresponding to the parametrix (2.19), are defined as  .Vb g(y)

:= − 

Pb (x, y)g(x)dSx , .

(2.26)

[Tb (x, n(x), ∂x )Pb (x, y)]g(x)dSx ,

(2.27)

∂Ω

Wb g(y) := − ∂Ω

where g is some scalar density function. The integrals are understood in the distributional sense if g is not integrable, while VΔ and WΔ are the single layer and double layer potentials corresponding to the Laplacian Δ. The corresponding boundary integral (pseudodifferential) operators of direct surface values to the single and the double layer potentials,Vb and Wb when y ∈ ∂Ω, are  .Vb g(y)

:= −

Pb (x, y)g(x)dSx , . ∂Ω

(2.28)

2 Two-Operator BDIEs in 2D with General Right-Hand Side

21

 Wb g(y) := −

(2.29)

Tb (x, n(x), ∂x )Pb (x, y)g(x)dSx , ∂Ω

where VΔ and WΔ are, respectively, the direct values of the single and double layer potentials corresponding to the Laplacian Δ. We can also calculate at y ∈ ∂Ω the conormal derivatives, associated with the operator A, of the single layer potential and of the double layer potential: Vb g(y) =

a(y) ± T Vb g(y), b(y) b

± Lab g(y) := Ta± Wb g(y) =

a(y) ± T Wb g(y). b(y) b

±

.Ta

.

(2.30) (2.31)

The direct value operators associated with (2.30) are ' .Wab g(y)

 := −

Wb' g(y) := −



[Ta (y, n(y), ∂y )Pb (x, y)]g(x)dSx , .

(2.32)

[Tb (y, n(y), ∂y )Pb (x, y)]g(x)dSx .

(2.33)

∂Ω

∂Ω

From Eqs. (2.22)–(2.33), we deduce representations of the parametrix-based surface potential boundary operators in terms of their counterparts ⎛ for⎞ b = 1, that is, 1 of the Laplace ln |x−y| associated with the fundamental solution PΔ = 2π r0 operator Δ. .Pa g

a Va g b a Va g b ' Wab g ± Lab g

1 PΔ g, a

1 1 1 PΔ g, Pa g = PΔ g, Pb g = PΔ g.. b a b ( ) a ⎛ bg ⎞ 1 1 = Wb g = WΔ bg , . = Vb g = VΔ g; Wa b b a b ( ) 1 a ⎛ bg ⎞ 1 = Wb g = WΔ bg , . = Vb g = VΔ g; Wa b b a b ⎡ ∂ ⎛ 1 ⎞⎤ } a⎨ ' a ' W g+ b VΔ g , . = Wb g = b b Δ ∂n b ⎡ ∂ ⎛ 1 ⎞⎤ } a⎨ a γ ± WΔ (bg) , . LΔ (bg) + b = Lb± g = b b ∂n b =

Pb g =

Lˆb g : = TΔ+ WΔ (bg) = TΔ− WΔ (bg) = LˆΔ (bg)

on

∂Ω.

It is taken into account that b and its derivatives are continuous in R2 and .LΔ (bg)

:= LΔ+ (bg) = LΔ− (bg)

by the Lyapunov-Tauber theorem. Hence,

(2.34) (2.35) (2.36) (2.37) (2.38) (2.39)

22

T. G. Ayele and M. F. Yimer .Δ(bVb g)

= 0, Δ(bWb g) = 0 .Δ(b Pb g)

=g

in

in Ω,

Ω,

∀g ∈ H s (∂Ω)

s (Ω) ∀g ∈ H

(∀s ∈ R),

(∀s ∈ R).

(2.40) (2.41)

The mapping properties of the operators (2.22)–(2.33) follow from relations (2.34)– (2.39) and are described in detail in [AM2011, Appendix A]. Particularly, we have the following jump relations: 1

1

Theorem 1 For g1 ∈ H − 2 (∂Ω), and g2 ∈ H 2 (∂Ω). Then, there hold the following relations on ∂Ω: .γ

±

V b g1 = V b g1 , .

(2.42)

1 γ ± Wb g2 = ∓ g2 + Wb g2 , . 2 1a ' Ta± Vb g1 = ± g1 . g1 + Wab 2b

(2.43) (2.44)

2.3 The Two-Operator Third Green Identity and Integral Relations Applying limiting procedures (see, e.g., [He77, S.3.8] and [Mir70]), we obtain the parametrix-based third Green identities, which proof follows in the similar way as in the corresponding proof in 3D case in [AY2021, Theorem 2]. Theorem 2 (i) If .u ∈ H 1 (Ω), then the following third Green identity holds: .u

ˇ + Zb u + Rb u + Wb γ + u = Pb Au in

Ω,

(2.45)

where the operator .Aˇ is defined in (2.7), and for .u ∈ C 1 (Ω), ˇ .Pb Au(y)

ˇ Pb (., y)>Ω = −Ea (u, Pb (., y)) : = diam(Ω), then this solution is unique and solves BDIEs (D2), while u solves the Dirichlet problem (2.67)–(2.68), and ψ satisfies (2.75). 1 (iii) If a couple (u, ψ) ∈ H 1 (Ω) × H − 2 (∂Ω) solves BDIE system (D2), then this solution is unique and solves BDIEs (D1), while u solves the Dirichlet problem (2.67)–(2.68), and ψ satisfies (2.75). Proof (i) Let u ∈ H 1 (Ω) be a solution to BVP (2.67)–(2.68). Due to Theorem 7, it is 1 unique. Setting ψ by (2.75) evidently implies ψ ∈ H − 2 (∂Ω). Theorem 3 and relations (2.64)–(2.65) follow that the couple (u, ψ) satisfies the BDIE systems (D1) and (D2), with the right-hand sides (2.71) and (2.74), respectively, which 1 completes the proof of item (i). Let now the couple (u, ψ) ∈ H 1 (Ω) × H − 2 (∂Ω) solve BDIE system (D1) or (D2). Due to Theorem 3, the hypothesis of Lemma 1 is satisfied implying that u solves PDE (2.67) in Ω, while relations in (2.53) and (2.54) also hold. 1 (ii) Let the couple (u, ψ) ∈ H 1 (Ω) × H − 2 (∂Ω) solve BDIE system (D1). Taking trace of (2.69) on ∂Ω and subtracting (2.70) from it, we obtain .γ

+

u = ϕ0

on ∂Ω,

(2.76)

that is, u satisfies the Dirichlet condition (2.68), (2.69) and Lemma 1 with Ψ = ψ, Φ = ϕ0 imply that Vb Ψ ∗ − Wb Φ ∗ = 0, in Ω, where Ψ ∗ = ψ − Ta+ (f˜, u) and Φ ∗ = ϕ0 − γ + u. Due to (2.76), Φ ∗ = 0. Then, Lemma 2(i) implies Ψ ∗ = 0, which proves condition (2.75). Thus, u obtained from the solution of BDIE system (D1) solves the Dirichlet problem, and hence, by item (i) of the theorem, (u, ψ) solve also BDIE system (D2). 1 (iii) Let now the couple (u, ψ) ∈ H 1 (Ω)×H − 2 (∂Ω) solve BDIE system (D2). Taking generalized conormal derivative of (2.72) and subtracting (2.73) from it, we get ψ = Ta+ (f˜, u) on ∂Ω. Then, substituting this in (2.54) gives Wb (ϕ0 − γ + u) = 0 in Ω, and Lemma 2(ii) then implies ϕ0 = γ + u on ∂Ω. Due to (2.71), the BDIE system (2.69)–(2.70) with zero right-hand side can be considered as obtained for (Ω) is an extension of f ∈ H −1 (Ω), that is, f˜ = 0, ϕ0 = 0, where f˜ ∈ H f = rΩ f˜, implying that its solution is given by a solution of the homogeneous problem (2.67)–(2.68), which is zero by Theorem 7. This implies uniqueness of the solution of the inhomogeneous BDIE system (2.69)–(2.70). Similar arguments work for the BDIE system (2.72)–(2.73).

30

T. G. Ayele and M. F. Yimer

2.5.3 BDIE System Operators Invertibility for the Dirichlet Problem The BDIE systems (D1) and (D2) can be written as .D

1

U D = F D1

and

D2 U D = F D2 , 1

respectively. Here .U D := (u, ψ)T ∈ H 1 (Ω) × H − 2 (∂Ω), ⎡

⎤ I + Z + R −V b b b D1 := , γ + Zb + γ + Rb −Vb . ⎡ ⎤ I + Zb + R b −Vb 2 ) ( D := , a ' Ta+ Zb + Ta+ Rb 1 − 2b I − Wab

(2.77)

while .F D1 and .F D2 are given by (2.71) and (2.74), respectively. Due to the mapping properties of the operators involved in the definitions of the operators 1 2 D1 and .F D2 (see, e.g., [AM2011, .D and .D as well as the right-hand sides .F 1 1 D1 1 ∈ H (Ω) × H 2 (∂Ω), F D2 ∈ H 1 (Ω) × H − 2 (∂Ω), Appendix A], we have .F while the operators .D

1

1

1

: H 1 (Ω) × H − 2 (∂Ω) → H 1 (Ω) × H 2 (∂Ω). 1

1

D2 : H 1 (Ω) × H − 2 (∂Ω) → H 1 (Ω) × H − 2 (∂Ω)

(2.78) (2.79)

are continuous. Due to Theorem 8(ii)–(iii), operators (2.78) and (2.79) are injective. The proof of the following result is similar to the corresponding proof of [AY2021, Lemma 3] in 3D case. 1

Lemma 3 For any couple .(F1 , F2 ) ∈ H 1 (Ω) × H − 2 (∂Ω), there exists a unique −1 (Ω) × H 12 (∂Ω) such that couple .(f˜∗∗ , Φ∗ ) ∈ H .F1

= Pb f˜∗∗ − Wb Φ∗.

˚R b f˜∗∗ , Pb f˜∗∗ ) − L + Φ∗ F2 = Ta+ (f˜∗∗ + E ∗ ab

(2.80) (2.81)

1 −1 (Ω) × Moreover, .(f˜∗∗ , Φ∗ ) = C∗∗ (F1 , F2 ) with .C∗∗ : H 1 (Ω) × H − 2 (∂Ω) → H 1 H 2 (∂Ω) a linear continuous operator given by

.f˜∗∗

ˇ F1 ) + γ ∗ (F2 + (γ + F1 )∂n b), . = Δ(b ⎞−1 ⎨ 1⎛ 1 − I + WΔ γ + − b F1 Φ∗ = b 2 ⎡ ⎛b ⎞⎤} ˇ F1 ) + γ ∗ F2 + (γ + F1 )∂n b + PΔ Δ(b a

(2.82)

(2.83)

2 Two-Operator BDIEs in 2D with General Right-Hand Side

31

ˇ ˇ F1 ) = ∇ · E∇(b where .Δ(b F1 ).

The proof of the following result is similar to the corresponding proof of [AY2021, Lemma 4] in 3D case. 1 Lemma 4 For any couple .(F˜1 , F˜2 ) ∈ H 1 (Ω) × H 2 (∂Ω), there exists a unique couple 1 −1 (Ω) × H 2 (∂Ω) such that .(f˜∗∗ , Φ∗ ) ∈ H

.F˜1

= Pb f˜∗∗ − Wb Φ∗.

(2.84)

F˜2 = γ + (Pb f˜∗∗ − Wb Φ∗ ).

(2.85)

1 −1 (Ω) × Moreover, .(f˜∗∗ , Φ∗ ) = C˜∗∗ (F˜1 , F˜2 ) with .C˜∗∗ : H 1 (Ω) × H 2 (∂Ω) → H 1 H − 2 (∂Ω) a linear continuous operator is given by

.f˜∗∗

ˇ F˜1 ) + γ ∗ (T + F˜1 + F˜2 )∂n b). = Δ(b b

Φ∗ =

(2.86)

) )−1 ( 1( 1 ˇ F˜1 ) + γ ∗ (T + F˜1 + F˜2 )∂n b)] − bF˜2 + γ + PΔ [Δ(b − I + WΔ b b 2

(2.87) ˇ ˇ F˜1 ) = ∇ · E∇(b where .Δ(b F˜1 ). Theorem 9 Let .r0 > diam(Ω). The operators (2.78) and (2.79) are continuous and continuously invertible. Proof The continuity of operators (2.78) and (2.79) is proved above. To prove the invertibility of operator (2.78), let us consider the BDIE system (D1) with arbitrary right-hand side D1

.F∗

1

D1 D1 T = (F∗1 , F∗2 ) ∈ H 1 (Ω) × H 2 (∂Ω).

D1 and .Φ = γ + F D1 − F D1 in Lemma 4, to obtain the representation Take .F˜1 = F∗1 ∗ ∗1 ∗2 D1 of .F∗ as D1

.F∗1

D1 F∗2 = γ + F˜1 − Φ∗

= F˜1

where the couple .(f˜∗ , Φ∗ )

−1 (Ω) × H 2 (∂Ω) = C˜∗∗ (F˜1 , F˜2 ) ∈ H 1

(2.88)

is unique and the operator .C˜∗∗

−1 (Ω) × H 2 (∂Ω) : H 1 (Ω) × H 2 (∂Ω) → H 1

1

(2.89)

is linear and continuous. If .r0 > diam(Ω), then, taking into account [Mi15, Remark 5.3] and applying Theorem 7 with .f = rΩ f˜ = rΩ f˜∗ , Φ∗ = ϕ0 , we obtain that BDIE system

32

T. G. Ayele and M. F. Yimer

(D1) is uniquely solvable, with solution .U1 = (A D )−1 (rΩ f˜, ϕ0 )T , .U2 = γ + U1 − ϕ0 , 1 where the inverse operator, .(A D )−1 : H −1 (Ω) × H 2 (∂Ω) → H 1 (Ω), to the left-hand 1 side operator, .A D : H 1 (Ω) → H −1 (Ω) × H 2 (∂Ω), of the Dirichlet problem (2.67)– (2.68), is continuous. Representation (2.88) and continuity of the operator (2.89) imply the invertibility of (2.78). To prove the invertibility of operator (2.79), let us consider the BDIE system (D2) with arbitrary right-hand side D2

.F∗

1

D2 D2 T = (F∗1 , F∗2 ) ∈ H 1 (Ω) × H − 2 (∂Ω).

D2 and .F = T + (F , u) = F D2 in Lemma 3 to represent .F D2 as Take .F1 = F∗1 2 1 a ∗ ∗2 D2

.F∗1

= F1

D2 F∗2 = Ta+ (F1 , u) = F2

and the couple .(f˜∗∗ , Φ∗ )

−1 (Ω) × H 2 (∂Ω) = C˜∗∗ (F1 , F2 ) ∈ H 1

is unique and the operator .C˜∗∗

−1 (Ω) × H 2 (∂Ω) : H 1 (Ω) × H − 2 (∂Ω) → H 1

1

(2.90)

is linear and continuous. Taking into account [Mi15, Remark 5.3] and applying Theorem 7 with .f˜ = f˜∗∗ , Φ∗ = ϕ0 , we obtain that BDIE system (D2) is uniquely solvable and its solution is: .U1 = (A D )−1 (rΩ f˜, ϕ0 )T , U2 = Ta+ (rΩ f˜, U1 ), where 1 the inverse operator, .(A D )−1 : H −1 (Ω) × H 2 (∂Ω) → H 1 (Ω), to the left-hand side 1 operator, .A D : H 1 (Ω) → H −1 (Ω) × H 2 (∂Ω), of the Dirichlet problem (2.67)– (2.68), is continuous. Representation (2.88) and continuity of the operator (2.90) imply the invertibility of (2.79).

2.6 Conclusion In this paper, we consider the Dirichlet problem for the linear second-order scalar elliptic differential equation with variable coefficient in a two-dimensional bounded −1 (Ω), when neither domain. The PDE right-hand side belongs to .H −1 (Ω) or .H classical nor canonical conormal derivatives of solutions are well-defined. The twooperator approach and appropriate parametrix (Levi function) are used to reduce the problem into two different systems of BDIEs. Nowadays, the theory of BDIEs in 3D is well-developed (see [CMN09, CMN11, Mi02, Mi05, CMN10]), but the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrixbased integral layer potentials and hence the unique solvability of BDIEs (see

2 Two-Operator BDIEs in 2D with General Right-Hand Side

33

[ADM2019, DM2015, ADM2017, AB2021, AD2021M, AD2021DN, AY2022]). The equivalence of the two-operator BDIE systems to the original problems, BDIE system solubility, solution uniqueness, and invertibility BDIE system are analyezed in the appropriate Sobolev spaces. Further, the two-operator BDIEs for variablecoefficient Neumann problem are to be investigated in similar way. Acknowledgments This work was supported by the International Science Program (ISP), Sweden. The second author would also like to thank the Simons Foundation based at Botswana International University of Science and Technology (BIUST) for partial support.

References [AY2022] Ayele, T.G.: Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side. J. Integral Equations and Appl. 33(4), 403–426 (2021). https://doi.org/10.1216/jie.2021.33. 403 [AB2021] Ayele, T.G., Bekele, S.T.: Two-operator boundary-domain integral equations for variable-coefficient mixed boundary value problem in 2D. Math. Meth. Appl. Sci., 1–22 2021. https://doi.org/10.1002/mma.7971 [AD2021M] Ayele, T.G., Dagnaw, M.A.: Boundary-domain integral equation systems to the mixed BVP for compressible Stokes equations with variable viscosity in 2D. Math. Meth. Appl. Sci. 44, 9899–9926 (2021). https://doi.org/10.1002/mma.7203 [AY2021] Ayele, T.G.: Analysis of two-operator boundary-domain integral equations for variable-coefficient BVPs with general data. Math. Meth. Appl. Sci. 44, 9831–9861 (2021). https://doi.org/10.1002/mma.6774 [AB2019] Ayele, T.G., Bekele, S.T.: Two-operator BDIEs for variable-coefficient Dirichlet problem in 2D. In: Constanda, C., Harris, P. (eds.) Integral Methods in Science and Engineering: Analytic Treatment and Numerical Approximations, pp. 53–66. Springer Nature, Switzerland AG (2019). ISBN 978-3-030-16077-7. https://doi.org/ 10.1007/978-3-030-16077-7_5 [ADM2017] Ayele, T.G., Dufera, T.T., Mikhailov, S.E.: Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Neumann BVP in 2D. In: Constanda, C., et al. (eds.) Integral Methods in Science and Engineering, vol. 1, pp. 21–33. Springer (Birkhäuser), Boston. Theoretical Techniques. https://doi.org/10.1007/978-3-31959384-5_3 [AD2021DN] Ayele, T.G., Dagnaw, M.A.: Boundary-domain integral equation systems to the Dirichlet and Neumann problems for compressible Stokes equations with variable viscosity in 2D. Math. Meth. Appl. Sci. 44, 9876–9898 (2021). https://doi.org/10. 1002/mma.6476 [ADM2019] Ayele, T.G., Dufera, T.T., Mikhailov, S.E.: Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Mixed BVP in 2D. In: Lindahl, K.-O., et al. (eds.) Analysis, Probability, Applications, and Computation. Trends in Mathematics, pp. 467–480. Springer Nature, Switzerland AG (2019). ISBN 9783-030-04459-6. https://doi.org/10.1007/978-3-030-04459-6_45 [AM2011] Ayele, T.G., Mikhailov, S.E.: Analysis of two-operator boundary-domain integral equations for a variable-coefficient BVP. Eurasian Math. J. 2(3), 20–41 (2011) [CMN09] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I: Equivalence and Invertibility. J. Integral Equ. Appl. 21, 499–543 (2009) [CMN10] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics. J. Integral Equations Appl. 22(1), 19–37 (2010)

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T. G. Ayele and M. F. Yimer [CMN11] Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of segregated boundarydomain integral equations for variable-coefficient problems with cracks. Numer. Methods Partial Differential Equations 27, 121–140 (2011) [CO00] Constanda, C.: Direct and Indirect Boundary Integral Equation Methods. Chapman & Hall/CRC, New York (2000) [Co88] Costabel, M.: Boundary integral operators on Lipschiz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988) [DM2015] Dufera, T.T., Mikhailov, S.E.: Analysis of boundary-domain integral equations for variable-coefficient Dirichlet BVP in 2D. In: Constanda, C., Kirsch, A. (eds.) Integral Methods in Science and Engineering: Theoretical and Computational Advances (2015), pp. 163–175. Springer (Birkhäuser), Boston. ISBN 978-3-31916727-5. https://doi.org/10.1007/978-3-319-16727-5_15 [Gr85] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston–London– Melbourne (1985) [He77] Hellwig, G.: Partial Differential Equations: An introduction. Teubner, Stuttgart (1977) [KLW15] Kohr, M.,Lanza de Cristoforis, M., Wendland, W.L.: Poisson problems for semilinear Brinkman systems on Lipschitz domains in R3 . Z. Angew. Math. Phys. 66, 833–864 (2015) [Lo72] Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972) [McL00] McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University, Cambrige (2000) [Mi18] Mikhailov, S.E.: Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains. Boundary Value Problems 2018(1), 1–52 (2018) [Mi02] Mikhailov, S.E.: Localized boundary-domain integral formulations for problems with variable coefficients. Int. J. Engineering Analysis with Boundary Elements 26, 681–690 (2002) [Mi05] Mikhailov, S.E.: Localized direct boundary-domain integro-differential formulations for scalar nonlinear BVPs with variable coefficients. J. Eng. Math. 51, 283–302 (2005) [Mi11] Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Analysis Appl. 378, 324–342 (2011) [Mi13] Mikhailov, S.E.: Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains. J. Math. Anal. Appl. 400(1), 48–67 (2013) [Mi15] Mikhailov, S.E.: Analysis of Segregated Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet and Neumann Problems with General Data, pp. 1–32 (2015). ArXiv:1509.03501 [Mir70] Miranda, C.: Partial Differential Equations of Elliptic Type, 2nd edn. Springer, Berlin (1970) [Po98a] Pomp, A.: The boundary-domain integral method for elliptic systems: with applications in shells. In: Lecture Notes in Mathematics, vol. 1683. Springer, Berlin (1998) [Po98b] Pomp, A.: Levi functions for linear elliptic systems with variable coefficients including shell equations. Comput. Mech. 22, 93–99 (1998) [St07] Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer, Berlin (2007)

Chapter 3

Implementation of Thermal Effects in Neutron Interactions in a Physical Monte Carlo Simulator D. G. Benvenutti, L. F. F. C. Barcellos, and B. E. J. Bodmann

3.1 Introduction Neutron transport problems are relevant in a wide number of applications, where in some of them thermal effects play a crucial role, which is the principal focus in the forthcoming essay. As the present discussion will show, the presence of thermal motion of the target nuclei affects the nuclear reaction rates of neutrons with target atoms, which in turn governs the properties of nuclear scenarios, especially the one in consideration. In nuclear reactors, for instance, thermal effects provide a feedback mechanism, because of an increase in temperature, and imply a negative reactivity defect, strongly related to safety issues [Re08]. Neutron transport, including temperature effects, can be implemented using either deterministic or stochastic approaches [La06], where the latter type makes commonly use of the Monte Carlo method. Generally, Monte Carlo simulators are based on a mathematical Monte Carlo paradigm, in which the equations governing the neutron transport problem are solved by stochastic means. Another Monte Carlo simulation type that has gained prominence with the development of more powerful computers, is the physical Monte Carlo method, where microscopic physical processes are “carbon-copied” at all instances in the history of each individual neutron, while interactions are simulated stochastically and the distributions from compiled nuclear data sheets [CaEtAl11] are employed.

D. G. Benvenutti · L. F. F. C. Barcellos · B. E. J. Bodmann () Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_3

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In this line, the present work reports on progress in the developments of a multi-thread physical Monte Carlo simulator, including thermal effects in the low energy regime. The physics paradigm, followed as the basis for the creation of this simulator, is to store the entire history of each neutron along all Monte Carlo steps, where interactions are generated by random events creating Markov chains. To this end, the simulator was developed using the C++ language, and it differs from others mainly by using the microscopic neutron cross sections in form of parameterized analytical functions instead of extrapolating between values of nuclear data sheets. These functions for each interaction type and energy were included in a progressively created function library so that they may be accessed by simple calls and without the need to access databases and necessary preparation procedures for the simulations. Despite the planned improvements and extensions, the simulator already provides a platform, which generates satisfactory results for shielding problems [BaEtAl21], and further its versatility allows different and novel approaches concerning aspects of neutron transport problems, e.g., obtaining pseudo-cross sections for neutron escape. These quantify escape as some sort of pseudo-interaction, which is treated in the same way an absorption reaction, is accounted for [BeEtAl22]. A current limitation is that the cross sections are parameterized for a cold temperature only (.∼0 K), and thus, the scattering kinematics used to calculate the secondary energies and directions of the neutrons in each interaction considers the target at rest hypothesis, which clearly limits the simulator for applications involving neutrons with kinetic energies beyond .∼1 eV only, because thermal motion of the target nucleus is less or not relevant. To make a virtue of a necessity, the improvement reported in the present contribution consists in the implementation and evaluation of thermal influences in simulations with the physical Monte Carlo simulator in development. In the forthcoming discussion, the influence of these effects on all neutron cross sections and also on the elastic scattering kinematics is treated, considering the free gas hypothesis. In order to maintain the philosophy of the simulator, this implementation does not involve the use of average values arising from the corrections of the cross sections, and further, no rejection technique is applied to sample the target’s velocity for the collision kinematics, as is done in other types of approaches. The motivation for this reasoning comes from the fact that in each interaction in the simulation, it is possible to sample a target’s velocity, in such a way that an effective velocity for the incident neutron is obtained after a change of the reference frame. This effective velocity is then used to calculate the effective interaction cross section. Since in the new reference frame the target is at rest, only the zero Kelvin neutron cross sections are needed, which are already parameterized and implemented in the simulator. Moreover, the same target velocity sampled to obtain the effective neutron cross section is also used in the scattering kinematics.

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3.2 Implementation of Thermal Effects in Neutron Interactions Neutron microscopic cross sections for each existing nuclide and each type of interaction are measured and reported in nuclear databases for a target nucleus at rest, and consequently, the material’s absolute temperature may be considered .0 K [He20]. This, of course, is not the situation encountered in realistic applications, since the local temperature will never be .0 K, and therefore, the targets will be in thermal motion and thus shall be taken into account. The effect of this thermal motion on the neutron cross sections is commonly addressed in the literature as nuclear Doppler broadening [La66]. To treat temperature effects in the cross sections, the more traditional Monte Carlo simulators make use of Doppler-broadened neutron cross sections, which provides the interaction probabilities already including thermal effects for a specific and predefined temperature [We17, Vi15, RoFo13]. This correction is based on the fact that the reaction rate in any reference frame shall be the same; more precisely, the reaction rate for the target at rest system and in the laboratory frame shall be the same. Considering that the thermal motion of free target nuclei, which compose the active medium, is characterized by the Maxwell-Boltzmann distribution, the corrected energy dependence of the respective cross section of interaction type x is given by a convolution integral with .T = 0 K cross sections: 1 .σ x (vn ) = vn

 B(v A , T )v rel σx (v rel ) dv A .

(3.1)

Here .vn is the neutron speed; .v A is the target velocity, all of them measured in the laboratory reference frame; and .v rel = v n − vA is the relative velocity between the neutron and the target. Finally, .σx is the microscopic cross section for interaction type x at zero Kelvin; .σ x is the corrected microscopic cross section, including the thermal effects for a medium with temperature T ; and .B(v A , T ) is the MaxwellBoltzmann distribution of the target nucleus motion. Note that the corrected cross section is evaluated from the neutron velocity in the laboratory frame. Two remarks concerning above’s procedure are in order. Recalling that for a neutron with speed .vn in the laboratory frame, Eq. (3.1) gives an average cross section, which takes into account the thermal effects at a specific temperature T [He20] and consequently results in some loss of information in the simulation results, as shown further down. Additionally, the convolution is solved numerically before the simulation process in a preprocessing step (for instance, the NJOY module [Ma16] for the MCNP simulator) for a specific temperature and then recorded in a database to be accessed during the simulation. By virtue, this implies in limitations for solving problems involving temperature gradients or even involving a large number of local temperatures, since the larger this number, the larger is the database, which has to be accessed during the simulations, and consequently the larger is the computational processing time.

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Hence, this work reports on a different proposal for implementing thermal effects, which are designed for physical Monte Carlo simulations as discussed next. The idea is to use the parameterized cross sections for .0 K already implemented in the simulator’s function library and maintain the philosophy of avoiding the use of databases during simulations. To this end, a change in the reference frame is performed in each interaction, and the needed cross sections are calculated for the respective Monte Carlo step by a sampling procedure for the target motion. Moreover, instead of using the incident neutron’s velocity in the laboratory system to evaluate the cross sections, as is usually done, these are evaluated considering the neutron’s relative velocity to the target in the rest frame of the target, which is then a sampled relative velocity from a distribution in agreement with the thermal contribution, due to target motion, and enters the interaction routine at the beginning of the Monte Carlo step. In other words, for each interaction, an effective neutron velocity is obtained changing the reference system for this instance with now target at rest and is then used to evaluate the effective cross sections (for .T = 0 K), which will then be the relevant quantity in the remainder of the corresponding simulation step. As mentioned before, we assume that the targets are free, so that the free gas model can be used, and the target’s energies follow the Maxwell-Boltzmann energy distribution: 

EA .B(EA , T ) dEA = 2 π



1 kB T

3 2

  EA dEA , exp − kB T

(3.2)

where .kB is the Boltzmann constant, i.e., an elementary entropy, and .EA is the kinetic energy of the target nucleus. Considering that an interaction involving a neutron with velocity .vn in the laboratory frame takes place in the simulation, the first step to proceed with is to sample the target’s energy following the distribution (3.2). This sampling can be done directly from a closed equation derived after applying the Box-Muller method to sample random numbers from a standard Gaussian distribution [Mo11].  π   ξ3 , EA = kB T − ln(ξ1 ) − ln(ξ2 ) cos2 2

.

(3.3)

where .ξ1 , .ξ2 and .ξ3 are sampled randomly from an unit interval. To perform now the change in the new reference frame, it is necessary to obtain the involved velocity vectors and not only the energies or speeds. Thus, with the energy of the target nucleus at hand, the next step is to find a feasible direction .Ω A of propagation. The free target is assumed to have no preferred direction and can be obtained directly from sampling an azimuthal angle .α randomly in an interval bounded by 0 and .2π and the polar angle .β following a sine distribution in an interval bounded by 0 and .π . Finally, from the known .EA and .Ω A , one computes the target velocity .v A for that specific interaction. Since .v rel is the relative velocity of the incident neutron in the rest frame of the target, it is related to the effective

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kinetic energy of the neutron, and one may determine directly the effective cross section as done in the case for .0 K. Consequently, it is consistent to use the neutron cross section functions as parameterized and implemented in the simulator without any correction procedures but with a sufficiently high sampling of the target velocity, so that a significant effect of the thermal distribution of the target motion is manifest in smooth distributions in the respective results.

3.2.1 Evaluation of Thermal Effects in Neutron Cross Sections An immediate result of the proposed methodology to consider thermal effects implemented by a collision sampling procedure is that distributional properties due to thermal effects are preserved and not averaged, differently than other simulation procedures. To illustrate this by some simulation experiment, consider a hypothetical case, where a mono-direction and monoenergetic beam of neutrons hits a homogeneous medium composed of uranium-238 with a local temperature of .300 K. In this constructed experiment, neutrons have a kinetic energy of .En = 6.67 eV in the laboratory system, and only elastic scattering interactions are considered. Figure 3.1 illustrates the microscopic cross section as a function of the neutron’s kinetic energy evaluated for all events generated in the entire simulation of the experiment, where the proposed methodology is applied (red dots in Fig. 3.1) and when the conventional Doppler-broadened cross section obtained by solving the convolution (3.1) is used (blue x in Fig. 3.1). The black solid line is an eye guide and represents the scattering cross section for a temperature of .0 K obtained from the nuclear database.

Fig. 3.1 Comparison of thermal effects treatment in cross sections to nuclear data

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Already by inspection, one observes that sampled data from our simulation follow the data from the ENDF/B-VI data base but with varying density along the curve, while the conventional approach results in one data point only, which represents one moment of a distribution, i.e., the mean value. The agreement between our simulation data and the ENDF data is due to the fact that .v rel in each interaction is essentially different and characterizes a specific microscopic cross section sampled in each Monte Carlo step from a distribution. On the other hand, the usual approach based on the cross section by convolution (3.1) relates the variation in the effective kinetic energy due to thermal effects to a single value, i.e., an average cross section. Furthermore, as a cross-check, we used the simulation data (red dots in Fig. 3.1) and determined the average cross section, which turned out to be the same as the average cross section calculated by the convolution (blue x in Fig. 3.1). This clearly shows that, in the usual approaches, a distribution is represented by one statistical moment only, while the procedure presented in this work maintains all distributional characteristics for a sufficiently high number of samples. It is noteworthy that although the above experiment considers only elastic scattering interactions, the same idea may be applied to take into account thermal effects for any other type of interaction, following the same prescription.

3.3 Implementation of Thermal Effects in the Scattering Kinematics After including thermal effects in the neutron cross sections, the next action in the simulation procedure is to consider these effects in the scattering kinematics. Currently, the simulator gives the velocity of the incident neutron after interaction by an equation derived from the target at rest hypothesis but without taking into account effects of temperature. So far, this hypothesis imposes limits on simulation scenarios, since thermal effects in the low energy region of the spectrum may result in up-scattering events, i.e., involving scattering interactions where the incident neutron gains energy, while in pure target at rest cases, these are impossible to occur. Therefore, the scattering kinematics needs to be extended, where reactions with increase of energy after collision are also considered. Based on the classical collision kinematics analysis of two rigid bodies in which both bodies have velocities different from zero, it is possible to derive a new equation, where the kinetic energy of the target nucleus is not neglected (which plays the role of thermal motion) and the velocity vector after collision of the neutron can be calculated including also up-scattering events. v 'n =

.

1 A (v n + Av A ) + ║v n − v A ║Ω  CM . A+1 A+1

(3.4)

Here A is the atomic mass of the target nucleus, .v 'n is the neutron’s velocity after collision in the laboratory system, and .Ω  CM is the neutron’s direction of motion

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in the center of mass reference frame. Note that both .v n and .v A are known, since v n is the neutron velocity from the last Monte Carlo step and .v A is sampled from the Maxwell-Boltzmann distribution in the beginning of the current step in order to evaluate the neutron cross sections before the kinematics calculation, as shown in Sect. 3.2. Finally, for simplicity, we assume that all scattering interactions are isotropic in the center of mass system, such that .cos(θ ) is sampled randomly in an interval bounded by .−1 and 1, where .θ is the scattering angle, providing .Ω 'CM , which is the last unknown of the kinematics equation. Note that the target’s velocity sampling procedure exposed in Sect. 3.2 does not involve any acception/rejection technique, as usually employed in Monte Carlo simulators which use average cross sections. In fact, Eq. (3.1) implies a probability distribution for a target velocity, such that the sampling process always accepts random values. Another difference in the proposed methodology is that in simulators, in which the averages obtained from Eq. (3.1) are used, the sampling of the target nucleus velocity is only done in the computation of the scattering kinematics, and no corrections of the microscopic cross sections are necessary. In order to evaluate now the thermal effects on the kinematics, the concept of double differential scattering cross section, sometimes called scattering kernel, will be introduced. This concept is essential in this study, since it gives the probability of a neutron having initial velocity .v n and undergoing a scattering interaction, such that its velocity .v 'n after collision lies in an infinitesimal interval .dv 'n around .v 'n , where .vn may be smaller or larger as .vn' . In fact, differently than in mathematical Monte Carlo approaches for particle transport, which have to evaluate the scattering kernel, in the physical Monte Carlo method, the procedure reduces merely to tallying microscopic interactions and tracking. Decomposing the velocity vector in an unit vector for the propagation direction and the associated kinetic energy, the double differential microscopic scattering cross section .σs (En → En' , Ω n → Ω 'n ) for a medium at .0 K is

.

σs (En → En' , Ω n → Ω 'n ) = σs (En )P (En → En' , Ω n → Ω 'n ) .

.

(3.5)

Here .σs (En ) is the microscopic scattering cross section at zero Kelvin, and P (En → En' , Ω n → Ω 'n ) is the scattering probability density function, which represents for a given scattering event the distribution of the neutron’s secondary energy and secondary direction, respectively. Note the scattering kernel described in Eq. (3.5) corresponds to the case in which the medium temperature is .0 K and no thermal effects are being considered. Though in the neutron cross section treated in Sect. 3.2, temperature effects shall be included in the differential cross section to correctly simulate neutron transport problems involving different temperatures or gradients of temperature. Using an argument analog to the derivation of the Eq. (3.1) and considering only scattering reaction rates, one obtains an equation that is similar to a convolution of the .0 K differential cross section (Eq. (3.5)) with the Maxwell-Boltzmann distribution. In terms of kinetic energies, this equation is given by

.

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σ s (En → En' , Ω n → Ω 'n ) =

.

1 vn



vrel σs (vrel )P (vn , vA → En' , Ω 'n )B(vA , T ) dvA , (3.6)

where the only new term is the density function .P (vn , vA → En' , Ωn ), which is similar to .P (En → En' , Ωn → Ωn ), but now considering thermal effects. In fact, '  .P (vn , vA → En , Ωn ) gives the distribution of secondary energies and secondary directions for neutrons scattered by targets with speed .vA . In approaches in which the scattering kernel is needed to perform the simulation, the solution for the above convolution considering a specific temperature is obtained in a similar way to Eq. (3.1) but employing a preprocessing software. This solution also provides only average values for the scattering kernel including the thermal effects. Despite the fact that the scattering kernel is not necessary in the improved simulator reported in this work, some of its properties, such as the density functions, can be reconstructed and provide information on how the simulator is handling thermal effects in scattering kinematics, as will be shown in the next subsection. To evaluate now the scattering kernel introduced in the above equations by generated data from our simulations, the same idea as in the evaluation of the neutron cross section in Sect. 3.2 will be applied, i.e., a change in the reference frame in each interaction will be used, in such a way that the target at rest frame can be used to reconstruct the scattering kernel. This implies that (3.5) can be used in this reconstruction, since in each interaction the target is at rest in the frame of the analysis. Consequently, in the respective equations, the relative energy and relative direction, representing a situation in which the target is at rest and obtained using the relative velocity .v rel , is inserted instead of the neutron energy in the laboratory frame. '  '  σs (Erel → Erel , Ωrel → Ωrel ) = σs (Erel )P (Erel → Erel , Ωrel → Ωrel ). (3.7)

.

Finally, if the secondary directions of neutrons are of no interest, for instance, for isotropic scattering (in the center of mass system), then Eq. (3.7) can be simplified integrating out all directions, such that the scattering kernel depends only on the relative neutron energy, ' ' σs (Erel → Erel ) = σs (Erel )P (Erel → Erel ).

.

(3.8)

' ) is obtained for each relative In this approach, a scattering kernel .σs (Erel → Erel energy, so that for an experiment similar to the one in Sect. 3.2.1, the support of the scattering kernel is bidimensional, which means that more physical details are present that make up a simulation result as compared to approaches based on the convolution shown in Eq. (3.6). Nevertheless, for further simulation validation and comparison of the present approach with findings from the literature, in the

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remainder of this work, the convolutions for the differential scattering cross sections were computed from the generated simulation data set.

3.3.1 Evaluation of Thermal Effects on Neutron Cross Sections As a first verification, it is possible to consider a medium with .T = 0 K, which is called the asymptotic case and implies that no thermal motion affects the problem. For this specific case, it is easy to derive .P (En → En' ) analytically, so that one may compare obtained results from the simulation with the analytical ones, given by  P (En →

.

En' )

=

1 (1−α)En

0

for αEn ≤ En' ≤ En , otherwise .

(3.9)

Here .α is a factor that depends on the atomic mass of the incident neutron and of the target nucleus [He75]. Then, considering an experiment of a monoenergetic beam of neutrons with .En = 6.52 eV in the laboratory system and reaching a medium composed of uranium-238, Fig. 3.2 illustrates a comparison between the probability density function .P (En → En' ) obtained by the simulator, represented by black dots, and the same function described by Eq. (3.9), represented by the red line. By inspection, one observes that, except for small fluctuations in the simulated data, the functions are essentially the same, which shows that the simulator provides satisfactory results

Fig. 3.2 Comparison of the asymptotic probability density function .P (E → E ' ) for .T = 0 K from simulation (black dots) and analytical expression (red line)

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with respect to scattering kinematics and secondary energies, thus verifying the new implementation in this limiting case. The second verification focuses on the sensitivity of the probability density function .P (En → En' ), as the initial energy of the neutron beam varies and the medium has a finite temperature. In these simulations, a monoenergetic beam of neutrons with different energies (.En ∈ {6.67 eV, 66.7 eV, 667 eV, 6.67 MeV}) hit a target medium composed of uranium-238 at a temperature .T = 1000 K. This temperature gives origin to thermal effects and thus changes in the shape of the density function when compared to the asymptotic case, and the results are illustrated in Fig. 3.3. Figure 3.3 clearly shows that with increasing neutron beam energy, the probability density function .P (En → En' ), obtained from simulation and represented by the black dots in all sub-figures, approaches the asymptotic behavior described by Eq. (3.8), which was added to all graphs (red lines) as an eye-guide. These finding is expected, since with increasing incident neutron energy, the target energy becomes increasingly negligible even for a medium with high temperature, such as .1000 K, and the problem can be approximated by the target at rest hypothesis, i.e., considering that no thermal effects affect significantly the results; in other words, they are no longer temperature sensitive. By virtue, for incident neutrons with lower energies, this hypothesis is not appropriate, since there is an increase in up-scattering events easily identified in Fig. 3.3a as the events which have energies beyond the higher energy limit with nonzero asymptotic probability density, where the secondary energy of a some neutrons is larger than the energy of the incident neutron energy, in this case, .En = 6.67 eV in the laboratory frame. In fact, Fig. 3.3 exemplifies that with increasing neutron beam energy, the number of upscattering events decreases, which is also the expected result and thus “proves” that the simulation is taking into account the thermal effects in a consistent way. Besides the physical results, these findings also attest the improvements that have been attained in the developments so far. Without the option of the inclusion of thermal effects, only applications, such as shielding problems, could be simulated accurately, since, as shown above, for neutrons with high enough incident energies compared to the energy scale of an associated medium temperature (.∼ kB T ), the probability density function .P (En → En' ) may well be approximated by the probability density of the asymptotic case. The last verification is provided by the sensitivity of the probability density function for varying medium temperatures. Due to the fact that for lower energies the effect is more pronounced, a monoenergetic neutron beam with energy .En = 6.52 eV (laboratory reference frame) was chosen, the target medium was considered as composed by uranium-238, and the cases with different temperatures (.T ∈ {0 K, 300 K, 600 K, 100 K, 1500 K}) were simulated. The obtained probability density functions from simulation are shown in Fig. 3.4. Also in this analysis, the results are as expected, since increasing the temperature of the medium increases the width of the probability density and thus the number of neutrons gaining energy in the scattering interactions. It is noticeable that for a temperature in the range of an environmental temperature and for a neutron energy in the range of eV , temperature

3 Implementation of Thermal Effects in Neutron Interactions Fig. 3.3 Probability density function .P (En → En' ) for a neutron beam with initial energies .En ∈ {6.67 eV, 66.7 eV, 667 eV, 6.67MeV} for a medium at a temperature .T = 1000 K. (a) .En = 6.67 eV. (b) .En = 66.7 eV. (c) .En = 667 eV. (d) .En = 6.67 MeV

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effects are significant. For nuclear reactor scenarios, where the dominant part of fission reactions is induced by neutrons with low energies, these effects play a crucial role in the balance of the neutron population in the volume of the reactor vessel.

3.4 Comparison to Findings From the Literature To validate quantitatively the implementation against some results from the literature, the average up-scattering events for six different initial energies were examined. For each scenario, an experiment of a monoenergetic neutron beam with energies .En ∈ {6.52 eV, 7.20 eV, 20.20 eV, 21.50 eV, 36.25 eV, 37.20 eV} incides on a homogeneous target medium composed of uranium-238. Since the crucial aspect is up-scattering, only these interactions were considered for evaluation. While in our analysis the role of the temperature in the probability density functions was identified and is shown in Fig. 3.4, simulations reported in the literature use only average values for the cross sections; thus, we compiled comparable outcomes for three different temperatures .T ∈ {300 K, 600 K, 1000 K}. Table 3.1 shows the results obtained from the simulations and the comparisons to the findings from a deterministic and a stochastic approach from references [Gh13, WaEtAl14], which also use the free gas hypothesis. For averaged up-scattering contributions, as shown in Table 3.1, one observes that the results do not depend crucially on the chosen approach. This may be attributed mainly to the fact that whenever integration is involved in order to obtain numerical

Fig. 3.4 Probability density function .P (En → En' ) for an incident neutron beam with energy .En = 6.52 eV for different medium temperatures (.T ∈ {0 K, 300 K, 600 K, 100 K, 1500 K})

3 Implementation of Thermal Effects in Neutron Interactions

47

Table 3.1 Comparison of averaged up-scattering contributions for incident neutron energies .En ∈ {6.52 eV, 7.20 eV, 20.20 eV, 21.50 eV, 36.25 eV, 37.20 eV} and target medium temperatures .T ∈ {300 K, 600 K, 1000 K}

Energy (eV ) 6.52

7.20

20.20

21.50

36.25

37.20

Temperature (K) 300 600 1000 300 600 1000 300 600 1000 300 600 1000 300 600 1000 300 600 1000

Up-scattering (.%) Reference Simulator Deterministic [Gh13] 62.01 62.17 82.80 82.84 84.38 84.45 16.70 16.58 23.75 23.59 28.37 28.20 5.54 5.57 15.50 15.36 30.90 30.41 6.64 6.61 11.49 11.46 15.58 15.51 6.67 7.18 30.43 30.54 55.75 55.41 3.57 3.62 6.00 6.12 7.12 7.27

Stochastic [WaEtAl14] 61.26 82.82 84.29 16.23 23.47 28.17 5.50 15.67 30.65 6.44 11.42 15.39 6.87 30.66 55.26 3.51 6.04 7.19

results, this operation may stabilize the numerical figures. Here, the average relative difference between the numbers was only .1.17 %, which at least shows that the methodology and implementation of thermal effects reported here allows to obtain reliable results with the latest developments of the improved physical Monte Carlo simulator, at least for cases where the free gas hypothesis is applicable.

3.5 Conclusion In this work, temperature effects in the microscopic neutron cross section and in the scattering kinematics were implemented in a physical Monte Carlo simulator using the free gas hypothesis. As shown by examples, the novel methodology to include these effects in the neutron cross section are able to generate results with more details in comparison to established Monte Carlo method-based implementations, which commonly make use of average cross sections obtained by solving a convolution integral. The increased effort due to more details is that a larger sampling of simulated neutron histories is needed, so that the simulation provides sufficiently

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smooth distributions, which after reduction by determining the mean values provide results comparable to “consecrated” simulators along the last decades. Notwithstanding, the extensions included into the simulator’s code represent a crucial improvement, since computational issues are nowadays a minor problem, due to the fact that in the present stage of the evolution of the simulator, the latest parallelization paradigms were employed, allowing for simulations of large numbers of neutron histories (.∼105 − 106 ) even on standard personal computers, where nowadays CPUs have four cores or even more. Moreover, the philosophy of the current implementation opens pathways for trivial parallelization. Although the present discussion focused on scattering only, which is the more complex part of the interaction module, in the scattering kinematics handler the analysis of the density function .P (En → En' ) showed a good quantitative agreement with results from the literature. It is noteworthy that all other reactions are simpler, since there is no kinematics to be evaluated for an outgoing neutron. The presented results and comparisons with results from the literature nevertheless prove that the simulator is operational and accurate for scenarios where temperature effects effectively influence in the interaction treatments. One decisive feature of the improved simulator is that it is structured in a way so that only distributions from physical principles or from compilations of experimental data are used. However, most Monte Carlo simulators require preprocessing of databases previous to the simulation, which results in a reduction of information of some details, which in the present consideration occurs when the average values for the neutron microscopic cross sections are used to treat thermal effects as shown in Sect. 3.2. Furthermore, all crucial processes, which were considered so far, were implemented using one interaction structure, whether physical (scattering, capture, etc.) or of geometrical origin, such as neutrons escaping the multiplicative domain. Finally, it is important to note that other issues related to thermal effects shall be taken into account for a more complete treatment of the influence of temperature [Vi15]. For example, thermal effects on the molecular level, such as rotation and vibration, are physically significant for interactions involving low energy neutrons. In addition, for applications involving temperature gradients, such as in the nuclear fuel of thermal reactors, the presence of phonons shall be considered, which, according to established treaties in solid state physics, can be represented as pseudoparticles and interact with neutrons. Consequently these pseudo-particles also take influence on associated reaction rates [Br82]. Such effects will be implemented in future works, improving further the physical models already used in the simulator and their associated distributions, thus allowing an even more complete treatment of temperature effects and also increasing the panoply of applications to be analyzed by the use of stochastic simulations.

3 Implementation of Thermal Effects in Neutron Interactions

49

References [BaEtAl21] Barcellos, L.F.F.C., Bodmann, B.E.J., de Vilhena, M.T.: On a comparison of a neutron Monte Carlo transport simulation to a criticality benchmark experiment. Prog. Nucl. Energy 134, 103652 (2021) [BeEtAl22] Benvenutti, D.G., Barcellos, L.F.F.C., Bodmann, B.E.J.: On pseudo-cross sections for neutron escape from a domain by a physical monte carlo simulation. In: Constanda, C., Bodmann, B.E.J., Harris, P.J. (eds.) Integral Methods in Science and Engineering, pp. 19–34. Birkhäuser, Cham (2022) [Br82] Brüesch, P.: Phonons: Theory and Experiments I: Lattice Dynamics and Models. Springer,Berlin (1982) [CaEtAl11] de Camargo, D.Q., Bodmann, B.E.J., de Vilhena, M.T., de Queiroz Bogado Leite, S.: A novel method for simulating spectral nuclear reactor criticality by a spatially dependent volume size control. In: Constanda, C., Harris, P.J. (eds.) Integral Methods in Science and Engineering: Computational and Analytic Aspects, pp. 33–45. Birkhäuser, Boston (2011) [Gh13] Ghrayeb, S.Z.: Deterministic Multigroup Modeling of Thermal Effect on Neutron Scattering by Heavy Nuclides, PhD thesis. The Pennsylvania State University, Pennsylvania (2013) [He20] Hébert, A.: Applied Reactor Physics. Presses internationales Polytechnique, Québec (2020) [He75] Henry, A.F.: Nuclear-Reactor Analysis. The MIT Press, Cambridge (1975) [La66] Lamarsh, J.R.: Introduction to Nuclear Reactor Theory. Addison-Wesley Publishing Company, Boston (1966) [La06] Larsen, E.W.: An overview of neutron transport problems and simulation techniques. In: Graziani, F. (ed.) Computational Methods in Transport, pp. 513–533. Springer, Berlin (2006) [Ma16] MacFarlane, R.E.: The NJOY Nuclear Data Processing System, Version 2016. Los Alamos National Security, Los Alamos (2016) [Mo11] Mohamed, N.M.A.: Efficient algorithm for generating maxwell random variables. J. Stat. Phys. 145, 1653–1660 (2011) [Re08] Reuss, P.: Neutron Physics. EDP Sciences, Les Ulis (2008) [RoFo13] Romano, P.K., Forget, B.: The OpenMC Monte Carlo particle transport code. Ann. Nucl. Energy 51, 274–281 (2013) [Vi15] Viitanen, T.: Development of a stochastic temperature treatment technique for Monte Carlo neutron tracking, PhD thesis. Aalto University, Espoo (2015) [WaEtAl14] Walsh, J.A., Forget, B., Smith, K.S.: Accelerated sampling of the free gas resonance elastic scattering kernel. Ann. Nucl. Energy 69, 116–124 (2014) [We17] Werner, C.J.: MCNP® User’s manual. Los Alamos National Security, Los Alamos (2017)

Chapter 4

On the Parameter Significance in Pandemic Modelling B. E. J. Bodmann and P. J. Harris

4.1 Introduction During the end of the year 2019, a pandemic started by the dispersion of the Coronavirus SARS-CoV-2 in the city of Wuhan, China, in the further referred to as COVID-19 virus [Ik19, LiEtAl20, SaEtAl20]. The virus is assumed to be transmitted by close range contact between people, either via small droplets from coughing, sneezing, or even talking, or by direct contact [KuAl15, LaEtAl17, LiEtAl03, NiCh14, RiAl20, YiEtAl20]. Hence, simulations were implemented to make predictions for the time evolution of the pandemic [Al94, EaEtAl00, RaEtAl10, Wa17], which may guide political and social decisions. Evidently, translating real-life information into a mathematical equation system needs idealizations and is necessary to maintain the model tractable, such as to provide a laboratory to study possible effects of inadequate mitigation by missing discipline, a too early relaxation of restrictions, or even lack of restrictions by authorities [ChEtAl98, He00]. Some of this information may be read off the best parameter set of the model tuned to reproduce some aspects of the phenomenon, such as the reported time evolution of the infected category as published by authorities [WuEtAl20]. They also may indicate that some parameters shall be changed, thus implying adequate emergency actions [FeEtAl06, WaTe04]. Hence, the present discussion focuses on a kinetic model for simulating the time evolution of the COVID-19 disease, which differs from existing models by the use of event rates (infection, recovery, and death) instead of the traditional B. E. J. Bodmann () Federal University of Rio Grande do Sul, Porto Alegre, Brazil e-mail: [email protected] P. J. Harris University of Brighton, Brighton, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_4

51

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B. E. J. Bodmann and P. J. Harris

population fraction based models with groups of susceptible, infected, recovered, and dead fractions of the population. The event rate approach is based on a twodimensional flux like quantity, which accounts for the number of people that change their state of health (from healthy to infected, from infected to recovered, or infected to death) per unit length and time unit. Moreover, the model to be described in the further is nonlinear and deterministic, which thus provides as a result only mean values for the pandemic’s time evolution. An event rate-based flux conception in comparison to the common population fraction counting seems more adequate, since in population-based models the principal infection mechanism is represented by a term multiplying the susceptible and the infected fraction of the population, which if interpreted says that there exists a risk of infection, which depends only on the population number, while it is plausible that a population density- as well as mobility-based quantity (a flux) seems more meaningful. In order to implement an infection mechanism, one may think of fluxes of a healthy, susceptible group of people that crosses the flux of already infected ones, so that one naturally obtains the proportionality, the higher the flux of healthy and/or infected, the larger the infection rate, which is related to a probability, since contact does not necessarily imply infection. A higher (or lower) flux may have two reasons: one is simply an increase (or decrease) in the population density, and second, founded on a mobility related quantity. In a mechanical statistics picture, this would be a speed, so that the higher the speed and thus the larger the distance covered, the larger is the number of subjects a moving agent may encounter and, according to a probability, may interact, i.e., infect the subject or get infected. One of the advantages of this type of approach is that the system describing the phenomenon does not necessarily be closed or isolated; it works as well for open systems, for instance, when natural deaths, births, and other category changes are considered. Thus, the present work is an attempt to model by a nonlinear kinetics model and predict the time evolution for the pandemic, with different model parameter sets, in order to compare its related outcomes. Similar to the traditional models, four categories are considered, susceptible, infected, recovered (considered immune), and death ones. For these four groups, the model is solved and the model parameter set is interpreted in terms of some real-life properties. To this end, we adjust and compare our findings to some available statistics and related time evolution, here world and South Korea data. At the present state of the developments, the parameters are considered constant, and we leave changes in some of the parameters due to changes in social behavior (quarantine, lockdown, among other possibilities) or due to restrictions imposed by political decisions for a future work.

4.2 World and South Korea Statistics and a Qualitative Simulation As a starting point for the present modelling, available data from an open data base was employed [Wo22] (and links therein), which provides the total, active, and death numbers with a per day frequency. A comment is in order here; all data and

3x106

300000 250000

2x106 1.5x106 1x106

0

200000 150000 100000

500000 0

53

350000

Infected Recovered

2.5x106

Dead

Infected and Recovered

4 On the Parameter Significance in Pandemic Modelling

50000 20

40

60 80 t/[days]

100

0

120

0

20

40

60 80 t/[days]

100

120

10000

300

Infected Recovered

250

8000 6000

200 Dead

Infected and Recovered

Fig. 4.1 World data for the period between January 22, 2020, and June 11, 2020. Infected and recovered part of the population (left) and cumulative death cases (right)

4000

100

2000 0 0

150

50 10 20 30 40 50 60 70 80 90 100 t/[days]

0 0 10 20 30 40 50 60 70 80 90 100 t/[days]

Fig. 4.2 South Korea data for the period between February 15, 2020, and June 11, 2020. Infected and recovered part of the population (left) and cumulative death cases (right)

simulations refer to registered cases and thus may differ from the unknown real numbers. Nevertheless, we assume that the published data are similar to the real data, in the sense that the error is roughly a matter of a scale; in other words, the numbers represent the tendency reasonably well. In this work, the world data span from January 22 to June 11, the South Korea data started at February 15 and ended at June 11. From the characteristics of the infection curve (called active numbers in the aforementioned data base), one may identify two distinct scenarios for the World (Fig. 4.1) and South Korea (Fig. 4.2) statistics. While the first one shows a still strongly increasing number of the infected category, the second one is already located in a region indicating a decline of the infection wave in the considered time interval. A typical characteristic of the time evolution is that, in the initial phase, the curves for the infected, recovered, and the dead are relatively flat and then start to rise strongly. The recovered and the dead fractions tend toward an asymptotic value, where assumed immunity of the recovered drains the susceptible population fraction and is the reason for the infected curve to increase up to a maximum with subsequent decrease toward the end of the pandemic or where the end is being expected.

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For the modelling of such evolution kinetics, there are well-established approaches in physical particle systems, which consider constituents and their associated interactions manifest as reactions, for instance, in physical chemistry, statistical mechanics, and particle physics, among many others [GoKa05]. The reduction to kinetics in comparison to space-time approaches is usually employed, when spatial information is not available, not of interest, or when a larger scale information is of concern only. This reasoning in form of a systematic theoretical elaboration goes certainly back to L. Boltzmann [ReEtAl12], where a key quantity in kinetics is the reaction rate indicating changes in the state or properties of the system. Further, we use the concept of categories or groups (g), which in the present consideration represent the susceptible: .(S), the infected .(I ), the recovered .(R), and the dead .(D) constituents of the population. A simplification that is made from the beginning is that provided statistics is taken as reference, ignoring probable uncertainties or even errors in the acquisition process during the identification of either the infected ones or the true cause of death. For instance, the pathology could be compatible with the COVID-19 symptoms but could as well be due to other heart-lung diseases. The timescale, where the epidemic or pandemic is at work, is considered with a time span of the order of . 101 years. Although the proposed flux model depends explicitly on the time variable only, in order to consistently motivate its structure based on reaction rates, one shall not neglect concepts involving spatial degrees of freedom. As already mentioned in the introduction, the virus is transmitted through a short-range interaction and for simplicity is reduced to a contact type of interaction. Consequently, moving people that occasionally meet other people may transmit or receive a certain disease, such as the COVID-19 virus. Following Boltzmann’s reasoning to use distributional quantities rather than an enumerable set of particles (or subjects), one may think of a flux of people and a flux-flux interaction picture. If one considers as the spatial domain a two-dimensional one, then a flux (.φ) across a certain boundary (.Γ ), which delimits a control domain (here an area), may be given by the net number of subjects (.ΔN) that pass per time unit .Δt a line segment (.ΔΓ ) that separates two adjacent regions. If now .Δx ⊥ ΔΓ denotes a path element perpendicular to the boundary line element, then the area .ΔA = ΔΓ Δx contains .ΔN subjects that cross the line segment. Moreover, . Δx Δt is the effective speed with which the subjects cross the line segment, and the flux may be interpreted as the surface density .n = ΔN ΔA times a Δx speed .v = Δt . Note that the speed does not directly refer to locomotion but rather is a measure for an influence in the sense of causality and effect in a more or less populated target area. The flux .φ = nv has then the unit of a reciprocal time-distance dimension, for instance, one per day and kilometer .(d km)−1 . Intuitively one may state that the larger the interacting fluxes, the larger are the rates for a specific interaction. However, the reaction or interaction rate shall have the dimension of one per area and time unit .(km2 d)−1 , so that the product of fluxes shall be multiplied by a time constant and a probability. The contact of an infected with a susceptible agent has an

4 On the Parameter Significance in Pandemic Modelling

55

infection rate depending, in principle, on the contact duration. However, in real-life contact duration varies significantly, so that one should have a distribution for the rate instead of a unique parameter. For simplicity and in order to render the model into a deterministic one, the aforementioned distribution shall be represented by its mean value, which has to be determined from the real data. The equation system describing the coupled kinetics of the four groups (categories), i.e., susceptible, infected, recovered, and dead portion of the population, is then set up by the four fluxes, i.e., the susceptible flux .φS , the infected flux .φI , the flux of recovered .φR , and the flux of the dead ones .φD . Although the balance equations suggest a simple but nonlinear model, a reinterpretation of some of the terms is necessary, so that the model can be calibrated using some of the accumulated statistical data from monitoring and testing the population for a possible infection, and these shall be compared to model predictions. One of the problems is the impossibility to differentiate whether a susceptible is already infected, but infection is not manifested or not detected, so that the infection rate shall also include a self-interaction term of the type .∝ φS φS . The second aspect is related to the pertinent question of immunity of the recovered portion of the population, which may be long term or short term and thus would imply a feedback of recovered to the susceptible portion of the population. Last, also the identification of the death cause is not unambiguous; patients with, for instance, lung diseases do exist independent whether there is an ongoing epidemic or pandemic. Further, only the most crucial aspect of hidden infection is taken into account upon introducing a fraction .β of susceptible, which may give origin to infection, so that the quantitative model represents apparent changes instead of true changes. The authors are aware of the fact that several properties are reduced to simple Parameters, but we consider the present approach as an alternative to existing proposals [Al94, ChEtAl98, DeEtAl19, EaEtAl00, GuEtAl17, He00, RaEtAl10, SaEtAl20, ThEtAl19, Wa17] that make use of population kinetics rather than reaction rates. Reaction rates by their physical definition depend on densities, rather than absolute numbers, and thus are nonextensive quantities and may be used for average as well as regional or local quantities. Further, the interpretation of the interaction may be related to a probability based but deterministic model. This may be understood by the mere fact that an encounter does not necessarily lead to an infection. For the purpose of modelling, only the events, where infection in fact takes place, are relevant for the balance equation. Thus, the coupled equations that follow shall relate event rates with fluxes, which may be further translated into a representation for the susceptible, infected, recovered, and dead fraction of a population density. .

dφS = −κφI (1 − β)φS − κβφS φS. dt dφI = κφI (1 − β)φS + κβφS φS − λD φI − λR φI . dt

(4.1) (4.2)

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B. E. J. Bodmann and P. J. Harris

Fig. 4.3 Model simulation for Eq. (4.1) with parameter set .κ = 10−2 km, .β = 10−7 , .λR = 0.1 d−1 , and .λD = 10−3 d−1 and the initial conditions .φS |t=0 = 100 (d km)−1 , .φI |t=0 = 1 (d km)−1 , .φR |t=0 = 0, and .φD |t=0 = 0 with time in multiples of days

Table 4.1 Total population, population density, and area of a country or state, while for the world, the area was obtained by the sum of the six populated continents (North America, South America, Europe, Africa, Asia, Australia) [Wp19] Domain World South Korea

Population .7 713 468 000 .51 225 000

dφR = λR φI dt

Population density −2 .57 km −2 .527 km

dφD = λD φI dt

Area .136 278 000 .100 339

km

km2

2

(4.3)

A prediction of the model for the four population fluxes is shown in Fig. 4.3 as an example. For the simulation, the following parameter set was used—.κ = 10−2 km, −7 , .λ = 0.1 d−1 , and .λ = 10−3 d−1 —and the initial conditions were .β = 10 R D 2 −1 , .φ | −1 .φS |t=0 = 10 (d km) I t=0 = 1 (d km) , and .φR |t=0 = 0, .φD |t=0 = 0, while the time is given in multiples of a day. However, the present model does not provide what is being observed. More specifically data published refer to absolute numbers in a population of interest, whether world data or from South Korea are considered. To this end and recalling that the flux is defined by the population density times a quantity with the dimension of a speed, which is related to social behavior aspects and political restrictions, the link to observable numbers is established by average densities; more specifically, the numbers published in the statistics are divided by the area of the country or state of interest (see Table 4.1). The population, density, and area data were drawn from a publication of the United Nations’ Department for Economic and Social Affairs of 2019 [Wp19].

4 On the Parameter Significance in Pandemic Modelling

57

The model cast into population fluxes has effectively five independent parameters (.κ, .β, .λR , .λD , and .ΦS (0)) to be adjusted in order to reproduce figures from the published statistics. Thus, for convenience, we define the infected-susceptible coupling parameter .κ(1 − β) (.κ has the dimension of a length), the susceptiblesusceptible coupling parameter .κβ, the recovery or death rates .λR and .λD , and, finally, the unknown initial condition for the susceptible flux. Note that restrictions, like lockdown or quarantine, have the effect of reducing .κ and also .ΦS (0), while the effective recovery and death rate are determined, as well as the natural rate, by aspects like available health infrastructure such as hospitals and other centers for medical care.

4.3 An Analytical Solution by the Decomposition Method The model may be decoupled because there are no terms containing .φR and .φD on the right-hand side of the equations for .φS and .φI , so that once .φI is obtained, the fluxes .φR and .φD may be calculated by integration. Consequently, the problem to be solved is then the reduced equation system: .







−k1 −k2 k1 k2     d φR λR = φI λD dt φD d dt

φS φI



= φS

φS φI



 −

00 0λ



φS φI

 (4.4)

For convenience, we introduced the shorthand notations .k1 = βκ, .k2 = (1 − β)κ, and .λ = λR + λD , respectively. The first term on the right-hand side of the equation represents the nonlinearity, whereas the last term is a linear contribution.  Without proof, we consider the fluxes in terms of a convergent expansion .φ = i φi , which is a prerequisite for the decomposition method. By inspection of Eq. (4.4), it is noteworthy that the equation for .φS has only a nonlinear contribution, which has implications on the setup of the employed recursive scheme to solve the nonlinear equation system. The decomposition is based on the hypothesis that the original problem may be split into an infinite set of simpler (linear) problems, where the solution may be represented as a superposition of the solution of all individual recursion steps. Recalling that .φS has no linear contribution, one would like to have an indication d what type of solution a nonlinear problem like . dt ψ = −αψ 2 has. .

d ψ = −αψ 2 dt

⇒

From this, one may conclude that .φS (t)|t constant.

ψ(t) = small

Ψ (0) Ψ (0)αt + 1

(4.5)

starts its time evolution as almost a

58

B. E. J. Bodmann and P. J. Harris

Recursion Initialization The expansion hypothesis of the fluxes introduces an ambiguity with respect to the constituting equation system, which shall solve Eq. (4.4). Therefore, the nonlinearity may be split using only terms which contain the solutions of the previous recursion steps. From the findings of Eq. (4.5), one may justify the following recursion initialization.      d φS0 00 φS0 =− (4.6) . φI 0 0λ dt φI 0 The recursion initialization accounts already for the initial condition of the original problem, so that the partial initial conditions of all remaining recursion steps are homogeneous. Recursion Step 1 In the first recursion step, one may consider terms containing the solution from the recursion initialization, i.e., .φS0 and .φI 0 and which were not included in the initialization step:

.

d dt



φS1 φI 1



 + φS0

  1  k2 k1 φSi =0, −k1 −k2 + λ φI i i=0

which is a linear nonhomogeneous matrix differential equation with known solution. Recursion Step 2 In the second recursion step, one may consider terms which contain the solution from the recursion initialization and the first recursion step, i.e., .φS0 , .φS1 , .φI 0 and .φI 1 , which were not considered in the initialization and first recursion step. 

d . dt

φS2 φI 2



 + φS0

k2 k1 −k1 −k2 + λ



φS2 φI 2



 + φS1

k1 k2 −k1 −k2

 1  i=0

φSi φI i

 =0

The Generic Formula for the n-th Recursion Step Using the same schematic, one derives the generic formula for the n-th recursion step. .

d dt



φSn φI n



 + φS0 

= φSn−1

k2 k1 −k1 −k2 + λ

−k1 −k2 k1 k2

 n−1  i=0



φSi φI i

φSn φI n

 +

 (4.7) n−2  j =1

 φSj

−k1 −k2 k1 k2



φSn−1 φI n−1



Upon summing up all equations from each recursion step until infinity, one verifies that all combinations from the nonlinearity are taken into account, and thus if the

4 On the Parameter Significance in Pandemic Modelling

59

flux expansions are convergent in the infinity limit, the solution found is the exact solution. Generic Solution for Each Recursion Step Note that each recursion step is a linear nonhomogeneous equation system of the form .

d Φ = AΦ + r , dt

where the linear part of the equation has a well-known homogeneous solution and r contains known terms only, since it was constructed from the solutions of the previous recursion steps and thus represents the inhomogeneity of the differential equation in this recursion step. Independent of the step, whether initialization or recursion step, one needs the d fundamental solution of the problem . dt Ψ − AΨ = 0, so that the general solution may be constructed from the fundamental solution .Ψ .

.

 Φ = Ψ Φ(0) +

t

.

Ψ (t − τ )r(τ ) dτ

0

To this end, the characteristic equation is solved to determine the eigenvalues .γ and eigenvectors .V i of the matrix .A. For convenience, introduce the shorthand notation (.ω1 = βκφS0 and .ω2 = (1 − β)κφS0 ). Then, the eigenvalues satisfy γ 2 + (ω1 − ω2 + λ)γ + ω1 λ = 0

.

For almost all parameter choices, the distinct eigenvalues are   ω1 − ω2 + λ 4ω1 λ = −1 ± 1 − 2 (ω1 − ω2 + λ)2

γ1,2

.

and the corresponding eigenvector matrix of .A is ⎛ ⎜ V =⎝

.

ω2 ω22 +(γ1 +ω1 )2 / (γ1 +ω1 ) ω22 +(γ1 +ω1 )2

/

−

ω2 / ω22 +(γ2 +ω1 )2 / (γ2 +ω1 )

−

⎞ ⎟ ⎠ ,

ω22 +(γ2 +ω1 )2

so that finally Ψ = V eΓ t V −1

.

with

Γ = diag{γ1 , γ2 } .

The nonhomogeneous solution is determined by the convolution of .ψ with the vector .r, where .δij is the usual Kronecker symbol and

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B. E. J. Bodmann and P. J. Harris

r=

n−1 n−1   (

.

)) ( δj (n−1) + 1 − δj (n−1) δi(n−1) φSj



i=0 j =1

−k1 −k2 k1 k2



φSi φI i

 .

Upon truncating the recursion after a number terms, the neglected solutions of all the remaining recursion steps provide only a small correction when compared to the dominant terms, so that one obtains an approximate solution of the problem with an analytical representation. Nonetheless, for each specific parameter combination, the obtained solution has to be analyzed for convergence in order to justify the truncation of the recursion scheme. While the solution of the nonlinear problem is now at hand, the parameter choice remains a task still to be solved.

4.4 Parameter Adjustment As mentioned before, fluxes have to be made compatible with published statistics, where, for the first approach, we assume all the parameters including the implicit speed as a constant, so that the shape of the statistics also reflects the shape of the fluxes, except for a proportionality constant. The procedure is now to fit the data Pm with polynomial .Pm = P (t|{ci }m i=0 ) or rational functions .Rm s = Qs which, in principle, shall determine the coefficients of the respective polynomials. However, in general, the theory and data of such problems coincide only on a semiquantitative level. Nevertheless, one may use now the parameterized distribution functions from the statistical data and insert these functions into the differential equations and use these equations to define an optimization criterion. Thus, in what follows, some details on the optimization procedure to determine the five unknown parameters are presented. In the present case, polynomials of degree eight were adequate to reproduce the data for the respective time sequences (infected, recovered, and dead). Once all the polynomial coefficients are known, one may use the polynomials representing .φS and .φI in the differential equations and proceed as follows.  i Generically,let the susceptible population be . ∞ i=0 ai t , and let the infected ∞ j population be . j =0 bj t ; then, one gets the following recursion relations for the polynomial coefficients.

(i + 1)ai+1 = −(1 − β)κ

i 

(i+i mod2) 2

ak bi−k − 2βκ

k=0 .

(i + 1)bi+1 = (1 − β)κ

i 

 k=0

(i+i mod2) 2

ak bi−k + 2βκ

k=0

= −(i + 1)ai+1 − λbi .

 k=0

ak ai−k 1 + δk,i−k (4.8)

ak ai−k − λbi 1 + δk,i−k

4 On the Parameter Significance in Pandemic Modelling

61

Note that due to the initial condition .a0 , b0 ∈ R+ . One may now use the equation system and successively symbolically determine the coefficients .{ai }m i=0 , since there are no statistics available for susceptibles, so that one has to deal with an in principal unknown distribution. Thus, one may determine the expressions for .i = 0, 1 and the remaining unknown coefficients .ak in terms of the known .bj and the parameters .κ, β and .λ (which are still to be determined)  a0 =

.

1−β 4β(b1 + λb0 ) −1 1+ b0 2 2 2β κ(1 − β) b0

a1 = −b1 − λb0 .. . ak = −bk − λbk−1 . Once all coefficients for the susceptible category are determined, one may use the approximated distribution by the polynomials for the infected and susceptibles and insert them into the differential equation for the infected as one step of the optimization procedure. Thus, let .φp be the parameterized distribution of the infected or the susceptibles, where all the polynomial coefficients are known and only the parameter set of the kinetic equation system is unknown, namely, .κ, β and .λ. Note, that .a0 has already been expressed in terms of the known .bi and the aforementioned unknowns. Then the optimization problem is defined by minimizing the maximum norm of the differential equation under variation of .κ, β and .λ.

.

║ ║ ║ dφIp ║ ( ) ║ ║ − φ βκφ + λφ + (1 − β)κφ Sp Sp Ip Ip ║ ║ dt ∞

−→ κ, β, λ

min

Since all the expansion parameters are determined, the unknowns are the infection rate, the unknown fraction of infected, and the recovery and death rate, while the initial condition for the susceptible category was related to the onset of infection (dominated by .b0 and .b1 ).

4.5 Simulations The methods described in the previous section showed how one can get estimates, which semiquantitatively reflect the time evolution as evaluated from of statistical data. However, there arises the pertinent question what can be learned from a model that does not coincide with the data on a percent level. Nevertheless, a “cause and effect” analysis may still provide significant information, which may help to guide political, social, and even health strategies. Since the decomposition of .λ into .λR

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Fig. 4.4 Influence of unidentified fraction of infected .β on the time evolution of the infected (left) and recovered (right)

and .λD is a rule of the problem, no further details are addressed here, and the focus stays on the influence of the four parameters .β, κ, λ, and .φS (0) for simulating the pandemic evolution. The first parameter’s dependence to be analyzed is the influence of the unidentified fraction of infected .β on the time evolution of the infected and recovered, respectively. The results for a selection of .β between .10−7 and .100 are shown in Fig. 4.4. The .β parameter seems to have a threshold between null sampling and a sampling with a certain percentage of the population. From a minimum sampling onward the only significant difference in the evolution is the onset of the flux of infected, but the maximum height and width shows only a spurious variation. Concerning the onset, in general, it is not easy, or even possible, to determine the exact initial time (.t = 0) of the pandemic, so that more or less intense testing of the population provides no means of influencing the shape of the time line. Moreover, the model clearly states that beyond a threshold, more intensive testing does not open pathways which could be explored in favor of a more “controlled” pandemic time evolution. The next parameter is an effective infection length .κ, which is a measure for the intensity of infection transmission. Figure 4.5 show the influence of the infection length on the time evolution for the infected and the recovered categories. Even though the model is nonlinear, there are still some intuitively understandable features of the simulated kinetics. The smaller the infection length .κ, the larger is the delay of the onset of the pandemic but also the lower is the height and the larger is the width of the time evolution of the infected category. In other words, this is one of the possible attack points, which may be explored, since, besides the larger delay, also a desired effect of a flatter distribution may result. This parameter relates among others to protective measures such as the usage of masks and hygienic preventive actions. For a .κ of an order of magnitude .∼10−8 km, one would no longer talk about an epidemic or pandemic.

4 On the Parameter Significance in Pandemic Modelling

63

Fig. 4.5 Influence of the infection length .κ on the time evolution for the infected (left) and the recovered (right) categories

Fig. 4.6 Influence of the sum of the recovery and death rate .λ on the time evolution for the recovered (left) and dead (right)

The next parameter influence to be examined is the sum of the recovery and death rate, which goes with the linear term in the differential equation for the infected. The results for variations of .λ and their respective time evolutions of the infected and recovered categories are shown in Fig. 4.6. Here the range between −4 d −1 and .10−1 d −1 was scanned, so that for the largest value the situation .10 may be considered unrealistic or apocalyptic, since the pandemic for this case goes on “forever.” An already expected effect of .λ is to enlarge the distribution, but because of its linear contribution, it does not have any influence on the onset of the pandemic. The recovery rate and dead rate, respectively, do not only represent physical characteristics of the human body but also contain in parametric form information for the available infrastructure of the healthcare system. Last but not least, the role of the initial value .φS (0) for the susceptible category on the time evolution of the infected and recovered is shown. It is noteworthy that the

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Fig. 4.7 The role of the initial value .φS (0) for the susceptible category on the time evolution of the infected (left) and recovered (right)

susceptible flux is a quantity which may not be related to statistics, simply because related data does not exist. Further, personal habits, either private or professional, such as travelling or social contacts, among others, determine the magnitude of this flux, and evidently, these are almost impossible to capture in order to translate them into numbers. The results for a range of .φS (0) and its influence on the time evolution of infected and recovered are given in Fig. 4.7. Depending on the magnitude of .φS (0), there is a criticality transition, for the present parameter combination for 6 (km d)−1 . Values below this limit correspond to a rural situation, .φS (0) ≈ 10 where values above represent urban or metropolitan configurations. It is noteworthy that for the rural case .φS (0) < 10−6 (km d)−1 , there does not exist an increase in the infected category; it naturally becomes extinct. Moreover, for .φS (0) > 106 (km d)−1 , the tail of the time evolution is dominated by the recovery and death rate, so that a possible influence by recommendations or emergency actions directed by health authorities does make sense only beyond a certain threshold for .φS (0).

4.6 Conclusion The present model is a deterministic model, which at best represents average values of a distribution which were the result of a large sampling of scenarios. Discrepancies in the description of the phenomenon and their resulting simulations and the acquisition of statistical data may differ substantially, since the evaluation of real data represents only one sample of a distribution, i.e., one realization of a stochastic process, where the distributions are unknown. However, even a simplistic model, such as the one considered here, may provide insights in the influence of parameter sets and their effects as discussed in this work and thus may indicate whether certain counter measures that shall stem the spread of a disease are efficacious and thus may help to control the time evolution of a pandemic.

4 On the Parameter Significance in Pandemic Modelling

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To this end, the adopted interaction (infection) mechanism is motivated by the current interaction in analogy to the dynamical descriptions in elementary particle physics. In a future extension of the present model, mobility may be differentiated according to an age group (.α), where in an interval of a specific group of age between .[α, α + Δα], the characteristic parameter set is specified. Further, since the population density is significant for the dispersion of a virus, a natural generalization of the kinetics model shall explicitly include space degrees of freedom, i.e., a diffusion model, which may be deduced as a simplification of a transport equation. At this state of the developments, the important issue of the onset (.t = 0 d) has not been given the adequate attention. In general, this time stamp is not known right from the beginning of an epidemics or pandemics but has to be reconstructed from the available statistical data. Nevertheless, studying the available information for several countries may give some hints on the outbreak of the pandemics and the associated delay until the detection or manifestation of the disease and thus may provide a useful instrument for the decision making of the authorities.

References [Al94] Allen, L.J.S.: Some discrete-time SI, SIR, and SIS epidemic models. Math. Biosci. 124(1), 83–105 (1994) [ChEtAl98] Cha, Y., Iannelli, M., Milner, F.A.: Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model. Math. Biosci. 150(2), 177–190 (1998) [DeEtAl19] Delamater, P.L., Street, E.J., Leslie, T.F., Yang, Y.T., Jacobsen, K.H.: Complexity of the basic reproduction number (R-0). Emerg. Infect. Dis. 25(1), 1–4 (2019) [EaEtAl00] Earn, D.J.D., Rohani, P., Bolker, B.M., Grenfell, B.T.: A simple model for complex dynamical transitions in epidemics. Science 287(5453), 667–670 (2000) [FeEtAl06] Ferguson, N.M., Cummings, D.A.T. Fraser, C., Cajka, J.C., Cooley, P.C., Burke, D.S.: Strategies for mitigating an influenza pandemic. Nature 442(7101), 448–452 (2006) [GoKa05] Gorban, A.N., Karlin, I.V.: Invariant manifolds for physical and chemical kinetics. In: Lecture Notes in Physics (LNP), vol. 660. Springer, Berlin (2005). https://doi. org/10.1007/b98103. ISBN 978-3-540-22684-0 [GuEtAl17] Guerra, F.M., Bolotin, S., Lim, G., Heffernan, J., Deeks, S.L., Li, Y., Crowcroft, N.S.: The basic reproduction number (R-0) of measles: a systematic review. Lancet Infect. Dis. 17(12), E420–E428 (2017) [He00] Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653 (2000) [Ik19] Ikejezie, J.: Coronavirus Disease 2019 (covid-19): Situation Report 29 (2019) [KuAl15] Kucharski, A.J., Althaus, C.L.: The role of superspreading in Middle East respiratory syndrome coronavirus (MERS-CoV) transmission. Eurosurveillance 20(25), 14–18 (2015) [LaEtAl17] Lau, M.S.Y., Gibson, G.J., Adrakey, H., McClelland, A., Riley, S., Zelner, J., Streftaris, G., Funk, S., Metcalf, J., Dalziel, B.D., Grenfell, B.T.: A mechanistic spatio-temporal framework for modelling individual-to-individual transmissionWith an application to the 2014–2015 West Africa Ebola outbreak. PLoS Comput. Biol. 13(10), e1005798 (2017) [LiEtAl20] Li, Q., Guan, X., Wu, P., Wang, X., Zhou, L., Tong, Y., Ren, R., Leung, K.S., Lau, E.H., Wong, J.Y., Xing, X., Xiang, N., Wu, Y., Li, C., Chen, Q., Li, D., Liu, T.,

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Zhao, J., Liu, M., Tu, W., Chen, C., Jin, L., Yang, R., Wang, Q., Zhou, S., Wang, R., Liu, H., Luo, Y., Liu, Y., Shao, G., Li, H., Tao, Z., Yang, Y., Deng, Z., Liu, B., Ma, Z., Zhang, Y., Shi, G., Lam, T.T.Y., Wu, J.T., Gao, G.F., Cowling, B.J., Yang, B., Leung, G.M., Feng, Z.: Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia. N. Engl. J. Med. 382(13), 1199–1207 (2020) [LiEtAl03] Lipsitch, M., Cohen, T., Cooper, B., Robins, J.M., Ma, S., James, L., Gopalakrishna, G., Chew, S.K., Tan, C.C., Samore, M.H., Fisman, D., Murray, M.: Transmission dynamics and control of severe acute respiratory syndrome. Science 300(5627), 1966–1970 (2003) [NiCh14] Nishiura, H., Chowell, G.: Early transmission dynamics of Ebola virus disease (EVD), West Africa, March to August 2014. Eurosurveillance 19(36), 5–10 (2014) [RaEtAl10] Rakowski, F., Gruziel, M., Bieniasz-Krzywiec, L., Radomski, J.P.: Influenza epidemic spread simulation for Poland—a large scale, individual based model study. Phys A: Stat. Mech. Appl. 389(16), 3149–3165 (2010) [ReEtAl12] Reid, J.C., Evans, D.J., Searles, D.J.: Communication: beyond Boltzmann’s Htheorem: demonstration of the relaxation theorem for a non-monotonic approach to equilibrium. J. Chem. Phys. 136(2), 021101 (2012) [RiAl20] Riou, J., Althaus, C.L.: Pattern of early human-to-human transmission of Wuhan 2019 novel coronavirus (2019-nCoV), December 2019 to January 2020. Eurosurveillance 25(4), 7–11 (2020) [SaEtAl20] Sanche, S., Lin, Y.T., Xu, C., Romero-Severson, E., Hengartner, N., Ke, R.: High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus 2. Emerg. Infect. Dis. 26(7), 1470–147 (2020) [ThEtAl19] Thompson, R.N., Stockwin, J.E., van Gaalen, R.D., Polonsky, J.A., Kamvar, Z.N., Demarsh, P.A., Dahlqwist, E., Li, S., Miguel, E., Jombart, T., Lessler, J., Cauchemez, S., Cori, A.: Improved inference of time-varying reproduction numbers during infectious disease outbreaks. Epidemics 29, 100356 (2019) [WaTe04] Wallinga, J., Teunis, P.: Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Am. J. Epidemiol. 160(6), 509–516 (2004) [Wa17] Wanduku, D.: Complete global analysis of a two-scale network sirs epidemic dynamic model with distributed delay and random perturbations. Appl. Math. Comput. 294, 49–76 (2017) [Wo22] https://www.worldometers.info/coronavirus/ [Wp19] https://population.un.org/wpp/Publications/Files/WPP2019-Wallchart [WuEtAl20] Wu, J.T., Leung, K., Bushman, M., Kishore, N., Niehus, R., de Salazar, P.M., Cowling, B.J., Lipsitch, M., Leung, G.M.: Estimating clinical severity of COVID19 from the transmission dynamics in Wuhan, China. Nat. Med. 26(4), 506 (2020) [YiEtAl20] Yin, Q., Wang, Z., Xia, C., Dehmer, M., Emmert-Streib, F., Jin, Z.: A novel epidemic model considering demographics and intercity commuting on complex dynamical networks. Appl. Math. Comput. 386, 125517 (2020)

Chapter 5

On a Variational Principle for Equilibrium Free Energy Functional of Simple Liquids E. S. Brikov

5.1 Introduction Despite observed significant progress in lattice modeling and calculation with the prospect of microscopic modeling of the most frequently occurring in nature and, therefore, the most interesting nonequilibrium processes out of statistical equilibrium, algorithms of analytical statistical mechanics for calculating thermodynamic parameters in equilibrium have not become irrelevant for microscopic study of processes near equilibrium. Free energy of a statistical system depends on configurations being implemented. For illustration purposes, a rough draft of the graph demonstrating a dependence of the free energy of the statistical system on the configurations being implemented (amounted to the 1-dimensional space for simplification) is shown in Fig. 5.1. Gibbs configuration integral (the statistical sum of the system): 1 .Z = N!h3N

 exp[−βH (p1 . . . pN , x1 . . . xN )]d 3 p1 . . . d 3 pN d 3 x1 ..d 3 xN ,

where H is Hamiltonian of a statistical system; N is an amount of particles of the statistical system, .i = 1 . . . N; .xi is a coordinate and .pi is a momentum of i particle; .β = 1/kT , k is Boltzmann’s constant, T is the absolute temperature; and h is Plank’s constant. The free energy: A = −kT · ln(Z),

.

E. S. Brikov () Tyumen, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_5

67

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E. S. Brikov

Fig. 5.1 The rough draft of the graph (for illustration purposes) demonstrating the dependence of the statistical system free energy, A, on the configurations being implemented (amounted to the 1-dimensional space for simplification)

where k is Boltzmann’s constant; T is the absolute temperature; and Z is the statistical sum of a system (Gibbs configuration integral). Introduction of local density (entirely possible inhomogeneous), on which thermodynamic parameters depend functionally, is possible in case of equilibrium (possible with inhomogeneities caused by external fields) and near it, as a result of different time scales of statistical processes [Bo46, Pr62].

5.2 The Method of the Generating Functional of an Equilibrium Statistical Mechanics of Microscopic Study of Thermodynamic and Structural Parameters of Simple Liquids The Bogolyubov functional equation in case of equilibrium [Bo70, Ar04]: ρ1 (r ) = (1 + u1 (r ))

.

δW (u1 ) 1 δW =− , δu1 (r ) β δϕ1 (r )

(5.1)

where .W = ln L(u) is the generating functional of so-called additive correlation functions .gn {n}; .L(n) ≡ Z(u) is the generating functional of partial densities .ρn {n} and is the statistical sum of the system in an arbitrary additional external field: .

− βϕ1 (r ) = ln(1 + u1 (r ))

(5.2)

5 Equilibrium Free Energy Functional of Simple Liquids

69

In the simplest, binary case, .n = 2: .g2 (1, 2)

= ρ2 (1, 2) − ρ1 (1)ρ1 (2) = ρ1 (1)ρ1 (2) · (F2 (1, 2) − 1) = ρ1 (1)ρ1 (2) · M2 (1, 2),

where .ρ2 (1, 2) is a binary partial density; .F2 (1, 2) is a binary distribution function; is a binary multiplicative correlation function; .ρ1 (1) is a unary partial density; .{n} is set of n radius vectors of particles.

.M2 (1, 2)

5.3 Tangent Transformation of the Generating Functional: The Functional Density Variables—The Free Energy Functional An efficient generalization of the generating functional by tangent transformation on its extremal with a transition from the functional variable of the external field (see (5.2)) to the unary function of density (see (5.1)) was used for calculations in equilibrium properties of systems with the Leonard-Jones interaction potential [Ar04]: .W (u)

 (ρ1 ) = W (u) − −→ W

.u(1)



δW r) V δϕ1 (

−→ ρ1 (1) =

(−βϕ1 (i))d(i) + A' (ρ1 );

δW , δ(−βϕ1 (1))

where .A' (ρ1 ) is the functional, the type of which is determined for reasons of convenience and conceptual meaning. Moreover, .C2 (1, 2) function obtained by the double variation: .

 (ρ1 ) δ2 W = C2 (1, 2) δρ1 (1)δρ1 (2)

matches the Ornstein-Zernike relation [OrZe14].  (ρ1 ) may well be direct Variations in a larger number of unary densities from .W correlation functions, but the specified interpretation is not obvious.  (ρ1 (i)) (the The specified transformation introduces new generating functional .W first variation of which corresponds to an external field with addition of a logarithm of unary density: .βϕ1 (1) + ln ρ1 (1); the second one introduces a direct correlation function: .C2 (1, 2) satisfying the well-known Ornstein-Zernike relation [OrZe14], subsequent variations still need strict interpretation in conceptual meaning). In addition to determination of a new generating functional, using the potential functional theorem, the free energy functional is determined. This functional

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Fig. 5.2 Graphs of diagrams K4 and K5 included in the functional (see (5.3)) [Ar04]. Legend: edge—.M2 (r), binary multiplicative correlation function; a graph vertex of a graph—integration; .ρ is a density of a homogeneous thermodynamic system. The construction of subsequent higherorder diagram graphs requires availability of three imperative edges at each vertex of a graph [Ar04]

depends on a functional variable: a binary multiplicative correlation function (with neglect of the contribution of high order variable multiplicative correlation functions) in addition to unary density .ρ1 (i) [Ar04]. The functional of specific free energy in homogeneous case of density and accordingly with absence of an external field (in further in the text for brevity and simplification is substituted by the functional of free energy):

.M2 (1, 2)

.βA(ρ, T |M2 (r))

= ln ρ +

    1 + M2 (r) − M2 (r) r 2 dr − (1 + M2 (r)) · ln 1 + f2 (r) 0  ∞  2 2 3M  (k) 2 (k)) ρ · (2π ) M ln(1 + ρ · (2π ) 2 2 2 (k) + −M k − 4π dk − 2 ρ · (2π )3 0

+ ρ · 2π

 ∞

− K4 − K5 − · · · ,

(5.3)

2 (k) is Fourier transform of the binary correlation function; .ρ is a density of where .M a homogeneous thermodynamic system; .f2 (r) = e−βφ2 (r) − 1 is a Mayer f -function; .β = 1/kT ; .φ2 (r) is a pair potential of interactions of particles of the thermodynamic statistical system; the structures K4 and K5 are represented by graphs of diagrams (with designations: edge—.M2 (r), a binary multiplicative correlation function; the graph vertex of the graph—integration) is shown in Fig. 5.2. It should be noted that the obtained free energy functional (see (5.3)) turned out to be very convenient for a direct variational method of finding structural and thermodynamic characteristics of a thermodynamic system with natural (in calculations) temperature and volume fixation. The main problem of implementing the direct variational method by functional (see (5.3) is the introduction of convenient and easily parametrizable models of the binary multiplicative correlation function, which provide efficient calculation and interpretation of results.

5 Equilibrium Free Energy Functional of Simple Liquids

71

5.4 Introduction of a Parametrizing Set of Functions in a Form of Displaced and Averaged Gaussian Functions in 3-Dimensional Space (DAGF3Ds) to Consider Lennard-Johnson Simple Liquids in Quasicrystal Approach Introduction of multiplicative correlation functions (mentioned above) through products, which determine a multiparticle density (function) of distribution, is very promising to develop variation methods (both analytical and computational) [Ar04]: .ρN {N }|N →4

≡ ρ4 (1, 2, 3, 4) =

= ρ1 (1)ρ1 (2)ρ1 (3)ρ1 (4)

N



(1 + Mk {k}) ∼ ρ 4 ,

(5.4)

k=2 ∀{k}∈{N }

where .ρN {N } is the multi-particles density (distribution); .{N } is the set of N radius vectors of particles; .1, 2, 3, 4 is the radius vectors of particles 1,2,3,4; .Mk {k} is the multiplicative correlation function of k particles; .∀{k} ∈ {N } is all the possible options for choosing the number of k out of N particles; and .ρ is a density of a homogeneous thermodynamic system; At the same time, it should be noted that it is obvious for a proper analysis at high densities, that the proposed definition (see (5.4)) should not use the effects on introduced multiplicative correlation functions of all particles included into a tuple of multiparticle density or of a distribution function (otherwise, it is possible to obtain “absurd” equations on binary multiplicative correlation functions unrelated to the thermodynamical parameters of the system when obtaining lower multiparticle densities (distribution functions) by means of averaging of the selected more dominant coordinates). The thermodynamical parameters are determined by the multiparticle densities (the distribution functions) up to the 4th order (e.g., heat capacity, the 2nd derivative of the generating functional for temperature) in an approach using ideas about the multiparticle densities. It is common knowledge that not only important statistical information is lost, but geometry is also averaged, i.e., important geometric information about statistical behavior of structural multiparticle densities (distribution functions) is lost when obtaining the lower multiparticle densities (the distribution functions) [Zi79]. Taking into account the abovementioned, it is reasonable to consider the simplest quasicrystal model of a unary inhomogeneous density for substitution into determination of multiparticle densities (see (5.4)) with subsequent averaging, observing losses both of statistical and geometric information [Ar04, Br99, Ar07, Fr46]:

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.ρ1 (1)

≡ ρ1 (r1 ) ≡

N 

Ci · e

−

( ai −r1 )2 σi2

(5.5)

,

i=1

where N is the umber of members of decomposition of .ρ1 (1); .ai is a displacement in 3-dimensional space of the Gaussian function (i ); .σi2 is a parameter of the Gaussian function (i ) width in 3-dimensional space; .{Ci }N i=1 is a set of weight coefficients; and .r1 is a radius vector of particle “1.” Substitution of (5.5) in (5.4) in case of .ρ4 {4} (complete thermodynamic description according to a structure without loss of geometric probabilistic and statistical information) and positions .r2 , .r3 , .r4 and orientation .r12 (introduction of isotropy) averaging [Br99] under the assumption that .Mk {k} ≡ 0, for .k = 2..4 [Ar07] (despite influence of particles included in the tuple of .ρ4 {4}) [Ar04, Br99, Ar07]: .ρ2 (r12 )

≡ ρ2 (12) = 2,3,4,(n12 ) =



Cm,k · ψm,k (r12 ) ,

m.k







in general, when: Mk {k}/=0

(5.6) where displaced and averaged Gaussian functions in 3-dimensional space (DAGF3D) is non-central chi-square distribution with two freedom degrees [Pr66]: 



.ψm,k (

− → r )= e

2 → → r −− a m)  (− − σk2

− → → n− am

=

σk2 · e

−

2) r 2 +am σk2



  · sh 2 · r·a2m

2 · r · am

σk

(5.7)

and .ρ2 (r1,2 ) is a binary partial density (distribution function); .ρ1 (1) is a unary partial density (distribution function); .1, 2, 3, 4 is radius vectors of particles 1, 2, 3, 4; .n12 — is a direction of radius vector: .r12 and .|n12 | = 1; .nam —is a direction of radius vector: .a m and .| nam | = 1; .2,3,4,(n12 ) is positions .r2 , .r3 , .r4 and orientation .r12 averaging; .n am averaging; .am is a displacement in 3-dimensional space am is orientation .n of the Gaussian function (with indexes: m and k ); and .σk2 is a parameter “width” in 3-dimensional space of the Gaussian function (with indexes: m and k ). By now is scrutinized a possibility usage of non-central chi-square distributions with more freedom degrees for parametrizations of multi-particle densities (distributions). An example of evident manual fitting of binary correlation function: .M2 (r) (obtained by the molecular dynamics methods [JoZoGu93, ShJo01, GuJoLaHe19, SaJoChHe16, JoHeLaMa17]), with a sum with weights of displaced and averaged Gaussian functions in 3-dimensional space (DAGF3D), is shown in Fig. 5.3 [Br99].

5 Equilibrium Free Energy Functional of Simple Liquids

73

Fig. 5.3 An example of evident fitting .M2 (r) (which is obtained by the molecular mechanics methods [JoZoGu93, ShJo01, GuJoLaHe19, SaJoChHe16, JoHeLaMa17]) by means of the sum with weights of displaced and averaged Gaussian functions in 3-dimensional space (DAGF3D) [Br99]

5.5 Evaluative Calculations of Height of the 1st Structural Maximum of the Radial Distribution Function (Binary Density) by the Direct Variational Method of the Free Energy Functional The results of variational change in contribution to free energies at approximately similar temperatures and different densities depending on a height of the first maximum of the multiplicative correlation fiction .M2 (1, 2) with neglect of the contributions to free energy of series diagrams (see (5.3)): K4, K5, etc., i.e. in hyper-netted chain (HNC) approximation, are shown in Figs. 5.4 and 5.5 [Br99], where .ρ2 (1, 2) ≡ ρ1 (1)ρ1 (2) · F2 (1, 2) = ρ1 (1)ρ1 (2) · (M2 (1, 2) + 1) ∼ ρ 2 and in the homogeneous case, .M2 (r) + 1 = F2 (r) = (ρ2 (r))ρ 2 ). Tentative assessments of the heights of the 1st structural maximum of radial distribution functions (binary density) have been carried out by the direct variational method with its expansion in a series according to DAGF3D (quasicrystal approach) (see (5.6), (5.7)), using the criterion of minimizing the free energy functional in the hyper-netted chain (HNC) approximation [Br99].

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Fig. 5.4 The values of free energy, A, in the hyper-netted chain (HNC) approximation with a change of height of the 1st maximum .M2 (r): .hmax . .T = 2.64, .ρ = 0.85 [Br99]

Fig. 5.5 The values of free energy, A, in the hyper-netted chain (HNC) approximation with a change of the height of the 1st maximum .M2 (r): .hmax . .T = 2.68, .ρ = 0.55 [Br99]

The results of comparison of the heights of the 1st structural maximum of radial distribution functions (binary densities) obtained by the estimated direct variational method of the free energy functional in the hyper-netted chain (HNC) approximation with those obtained from the molecular dynamics experiments [JoZoGu93, ShJo01, GuJoLaHe19, SaJoChHe16, JoHeLaMa17] are presented in Table 5.1 (considering the simple connection .M2 (r) = F2 (r) − 1) [Br99]: It can be concluded that the method, based on the extreme variational properties of the free energy functional [Ar04], even with the use of a simple hyper-netted chain (HNC) approximation for estimating the heights of the first maximum of the radial distribution functions decomposed by DAGF3Ds (in quasicrystal approach), provides detectable accuracy [Ar04, Br99, Ar07]. Moreover, you can pay attention (see Table 5.1), that with a relatively higher density, the height value obtained with

5 Equilibrium Free Energy Functional of Simple Liquids

75

Table 5.1 Comparison of the heights of the 1st structural maximum of radial distribution functions, .F2 (r) (binary densities) that corresponding to .M2 (r) and obtained by the estimated direct variational method of free energy functional in the hyper-netted chain (HNC) approximation [Br99] with those obtained from the molecular dynamics experiments [JoZoGu93, ShJo01, GuJoLaHe19, SaJoChHe16, JoHeLaMa17]

No. 1.

Direct variational method using the proposed expansions (see (5.6, (5.7)) of .M2 (r) [Ar04, Br99, Ar07] .ρ .hmax T 2.64 0.85 1.21

2.

2.68

0.55

0.82

The molecular dynamics method for = M2 (r) + 1 [JoZoGu93, ShJo01, GuJoLaHe19, SaJoChHe16, JoHeLaMa17] .hmax T .ρ .rmax No. 2.22 2.75 0.835 1.038 Average 2.187 2.5 0.8 1.045 1. 2.199 2.5 0.805 1.045 2. 2.404 2.5 0.9 1.035 3. 2.117 3.0 0.8 1.035 4. 2.117 3.0 0.805 1.035 5. 2.310 3.0 0.9 1.030 6. 1.754 2.75 0.55 1.07 Average 1.754 2.5 0.5 1.075 1. 1.754 2.5 0.6 1.065 2. 1.754 3.0 0.5 1.070 3. 1.754 3.0 0.6 1.070 4. .F2 (r)

the variational method is closer to that obtained by the molecular dynamics method (which undoubtedly requires more thorough recheck and vindication) [Ar04, Br99, Ar07]. Nevertheless, no doubt that calculations by the direct variational method with a large number of expansion terms are required. It is proved [Ar04, Br99, Ar07] that the limitation on the Fourier transform of the binary correlation function (see (5.3)) is a condition of metastability region and, respectively, of instability region (following the sufficient conditions of the functional minimum). Preliminary evaluating calculations by the direct variational method show that these metastable states appear in molecular dynamics experiments at high densities and, consequently, with high first maxima in the binary distribution function [Ar04, Br99, Ar07, JoZoGu93, ShJo01, GuJoLaHe19, SaJoChHe16, JoHeLaMa17].

5.6 Algorithms for Calculation of Multicenter Integrals Included in the Free Energy Functional It is obvious that the main calculation problem of the direct variational method is the calculation of multicenter series diagrams presented in (5.3) by graphs.

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5.6.1 Calculation of the Multicenter Series Diagrams Using Expansion by DAGF3D The task of calculating the contribution of K4 (multicenter integral) to free energy (see (5.3)) in this expansion variant (see (5.6), (5.7)) leads to the accumulation of the product of expansion coefficients with a weighting coefficient, determined by a multidimensional function of 12 variables (respectively, 6 displacements and 6 widths). This function can be tabulated further. Features of this function can be determined by means of mathematical analytical methods and with use of modern computers [Br99]: .K4

=



 Ck Cl Cm Cn Cp Cq e

−ϑij6x6 · ni nj

k,l,m,n,p,q

 (2 · π )9/2 · , (detA)3 nk , nl , nm , nn , np , nq

where .{Ci }N i=1 is a set of weight coefficients (see (5.5)); .

nk , nl , nm , nn , np , nq

are orientations . nk , nl , nm , nn , np , nq averaging; detA is a function, depending on the width values of displaced and averaged Gaussian functions in 3-dimensional space (DAGF3D); and .ϑij6x6 is a matrix of functions depending on the displacement values and the width of displaced and averaged Gaussian functions in 3-dimensional space (DAGF3D). A preliminary analysis (which is not provided here) indicates the need to identify geometric features .ϑij6x6 · ni nj in the averaged multipliers [Br99]. Undoubtedly, the results of this multidimensional function analysis will allow to evaluate convenience of the proposed quasicrystal liquid model.

5.6.2 Expansion by Hermite’s Functions Another proposed method for calculation of the multicenter integrals included in free energy functional (see (5.3)): representation of binary multiplicative correlation function .M2 (r) as a sum of a series [Br99]: .M2 (r)

=

∞ 

Cn · r 2n · e−α·r , α = 0.5, 2

(5.8)

n=0

where .{Ci }N i=1 is a set of weight coefficients. For convenience of conceptual modeling .M2 (r), the series can be represented as expansion by Hermit’s functions.

5 Equilibrium Free Energy Functional of Simple Liquids

77

It should be noted that by the number of the expansion parameters, this type of expansion is comparable to the previously presented DAGF3D factorization method, but the latter one seems to be more evident in the quasicrystal explication of expansions [Br99]. The calculation of the K4 diagram is amounted to multiple differentiations of a simple and visual function of six variables .[detA]−3/2 on set of variables .{αi }6i=1 that is raised from each specimen of series of .M2 (r) (see (5.8)) [Br99]: .K4

=

ρ3 4!

∞  n1 =1,n6 =1

Cn1 Cn6

∂ n1 ∂ n6

π 9/2

, n n ∂α1 1 ∂α6 6 [detA]3/2

where    (α + α + α ) −α4 −α6 4 6   1   .detA =  . −α4 (α2 + α4 + α5 ) −α5    −α6 −α5 (α3 + α5 + α6 )

It can be assumed that due to availability of an explicit form of the differentiable function and its high symmetry, further it is entirely possible, to obtain recurrent relations and tabulated values for these multiple differentiations, which are usually sufficiently formally implemented in the form of designated algorithms on modern computer technology.

5.7 Conclusion Despite the highly convenient form for model studies when expanding according to DAGF3D (see (5.6), (5.7)), series (see (5.8)), and the rather obvious calculation algorithms presented above in the direct variational approach of analytical theory of liquid, it is obvious that a large amount of studies is still required to obtain valuable scientific results in this direction.

References [Ar04] Arinstein, E.A.: Direct variational method in the theory of liquids. Theor. Math. Phys. 141(1), 1461–1468 (2004) [Ar07] Arinstein, E.A.: A model of the liquid-crystal phase transition and the quasicrystal model of liquid. Theor. Math. Phys. 151(1), 571–585 (2007) [Bo46] Bogoliubov, N.N.: Problems of a dynamical theory in statistical physics [Problemy dinamicheskoi teorii v statisticheskoi fizike]: in russian, Gostehizdat [Gostekhizdat], Moscow (1946). In: de Boer, J., Uhlenbeck, G.E. (eds.) Studies in Statistical Mechanics, vol. 1. North-Holland, Amsterdam; Interscience (Wiley), New York (1962), Part A (or first article)

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[Bo70] Bogoliubov, N.N.: Method of functional derivatives in statistical mechanics [Metod funktsionalnykh proizvodnykh v statisticheskoi mehanike]: in russian. In: Mitropolskii, Y.A., et al. (eds.) Selected Works [Izbranye trudy], vol. 2, pp. 197– 209. Scientific thought [Naukova dumka]), Kyiv [Kiev] (1970) [Br99] Brikov, E.S.: The use of wavelet decomposition to the calculation of thermodynamic properties of simple liquids by direct variational method [Primenenie veivlet-razlozhenia k vychisleniyu termodinamicheskikh svoistv prostykh zhydkostei pryamym variatsionnym metodom]: in russian. Dis. cand. phys.-math. sci. thesis, sci. adv. prof. A. E. Arinstein. Publishing House of Tyumen State University [Izdatel’stvo Tyumenskogo gosudarstvennogo universiteta], Tyumen (1999) [Fr46] Frenkel, J.: Kinetic theory of liquids. In: Fowler, R.H., Kapitza, P., Mott, N.F. (general eds.). The international series of monographs on physics. Clarendon Press, Oxford University Press, Oxford (1946) [GuJoLaHe19] Guo, Z., Johnson, J.K., Labík, S., Henderson, D.: Test of the Duh-HaymetHenderson theory for mixtures: cavity correlation functions and excess volumes. Mol. Phys. 117(23–24), 3623–3631 (2019) [JoHeLaMa17] Johnson, J.K., Henderson, D., Labik, S., Malijevsky, A.: A comparison of the correlation functions of the Lennard-Jones fluid for the first-order Duh-HaymetHenderson closure with molecular simulations. Mol. Phys. 115(9–12), 1335–1342 (2017) [JoZoGu93] Johnson, J.K., Zollweg, J.A., Gubbins, K.E.: The Lennard-Jones equation of state revisited. Mol. Phys. 78(3), 591–618 (1993) [OrZe14] Ornstein, L.S., Zernike, F.: Accidental deviations of density and opalescence at the critical point of a single substance. Proc. RNAAS, 17(2), 793–806 (1914) [Pr62] Prigogin, I.: Nonequilibrium Statistical Mechanics. Wiley, New York (1962) [Pr66] Press, S.J.: Linear combinations of non-central chi-square variates. Ann. Math. Statist. 37(2), 480–487 (1966) [SaJoChHe16] Saeger, A.R., Johnson, J.K., Chapman, W.G., Henderson, D.: Cavity correlation and bridge functions at high density and near the critical point: a test of secondorder Percus-Yevick theory. Mol. Phys. 114(16–17), 2516–2522 (2016) [ShJo01] Shi, W., Johnson, J.K.: Histogram reweighting and finit-size scaling study of the Lennad-Jones fluids. Fluid Phase Equilib. 187–188, 171–191 (2007) [Zi79] Ziman, J.M.: Models of Disorder: the theoretical physics of homogeneously disordered systems. Cambridge University Press, New York (1979)

Chapter 6

Use of Variants of Seismic Signal Approximations by Proposed Sets of Functions for Statistical Structural Analysis E. S. Brikov and V. I. Dobrorodnyi

6.1 Introduction: Detection of Seismic Signals The task of construction of a seismic signal detector consists in creation of a device capable of not being triggered by background microseisms when detecting impacts of a concentrated type from “useful” and “harmful” objects. Signals from control objects through the seismic channel represent a broadband noise-like process. As studies and analysis of known sources have shown [Kr14, Va85, Pe03], it is advisable to use an energy detector to detect noise-like signals (see Fig. 6.1). As it is known [Kr14], the quality of signals detection in such a device is estimated, by the probability of false alarms when exposed to seismic background noise .PF A (probability of a false alarm) and the probability of a correct energy detection of signals from concentrated impacts of objects of “harmful” and “useful” classes .PEN.DET . (probability of an energetic detection). Having denoted .ρS (x) and .ρN (x)—probability densities of the seismic mixture “signal”-“noise” and “noise”, respectively, .U0 —a level of detection threshold, we can estimate .PF A and .PEN.DET . by the following expressions:  PF A =

∞

.

U0

 ρN (x)dx, PEN.DET . =

∞

ρS (x)dx.

(6.1)

U0

Mathematical relationship .PEN.DET . = f (PF A ) is a generalized characteristic of the energy threshold detector of seismic signals of a geographically distributed

E. S. Brikov () Tyumen, Russia V. I. Dobrorodnyi Lugovoi village, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_6

79

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Fig. 6.1 Energy threshold detector of seismic signals: SR—seismic receiver; LA—linear amplifier; BF—band-pass filter; QD—quadratic detector; I—integrator; TD—threshold device; AS— launch threshold adaptation scheme. .Ta —integration time from practice: is decision-making time

security system (hereinafter referred to as GDSS), which reflects patterns of a detection subsystem functioning. Since .U0 = φ(PF A ), the generalized characteristic of the detection subsystem can be represented as:  PEN.DET . =

∞

ρS (x)dx,

.

φ(PFA )

which is a mathematical model of GDSS detection subsystem, when identifying the type and characteristics of laws of seismic signals and noise distribution at the input of the threshold device [Kr14].

6.2 Modeling of the Subsystem for Detection of Seismic Signals It is proposed to consider variants for numerical smoothing of functions of probability densities .ρS (x) and .ρN (x) of the seismic signals and their approximation (realizations of fitting) where sets of functions are used: displaced and averaged Gaussian functions [Br99] in 3-dimensional space (hereinafter DAGF3D) and in 2dimensional space (hereafter, DAGF2D). In accordance with the functional blocks of “energetical conversion” of seismic signals (see Fig. 6.1), which involves stages such as filtering, amplitude quadratic detection and integration for corresponding methods of detection (filter—detector—integrator) in case when background noise and a mix of the signal and noise at the filter output have distribution functions according to not displaced Gaussian functions: 

→ ψm,k (− r )= e

.

−

(

) 

2 − → → r −− am σk2

− → → n− am

=

σk2

·e

−

(r 2 +am2 )) σk2

 · sh 2 ·

2 · r · am

 r·am σk2

− is DAGF3D,

6 Statistical Structural Analysis of Seismic Signals

 (−→r −−→a m )2  − − → σk2 .ψm,k ( r ) = e

− → → n− am

81



   2 r · am r 2 + am = exp − − is DAGF2D, ·I0 σk2 2σk2

(6.2) where .σk —a set of values of standard deviations (hereinafter SD); .am —a set of values of mathematical expectations (hereinafter ME); .nam —is a direction of radius vector: .am and .| nam | = 1; ....nam —is .nam averaging; .am is a displacement in 3-dimensional space of the Gaussian function (with indexes: m and k); .σk2 is a parameter width in 3- and 2-dimensional space of the Gaussian function (with indexes: m and k); and .I0 (x) is a modified zero-order Bessel function. Structure of a noise-like signal at the output of the integrator of the energy threshold detector of the seismic signals (see Fig. 6.1) appears to be random signal: 2 2 + Z 2 consisting of 3D components of random signals .X 2 + Y 2 + Z 2 = X + Y distributed already with displaced normal distributions [Kr14, Va85, Pe03]:

 (X + Y + Z)2

.

INT

= [X2 ]I N T + [Y 2 ]I N T + [Z 2 ]I N T =

2 + Z 2 = X2 + Y 2 + Z 2 , 2 + Y =X

(6.3)

where .[]I N T = · · ·—is integration operation in the energy threshold detector of seismic signals (see Fig. 6.1); and .· · · are random variables proportional to mean square roots of the signal components after the integrator. Obviously, normal distribution at the output and the separation of individual components of a seismic signal after the integrator is explained by fulfillment of conditions of a central limit theorem in it and also of a mutual compensation of the cross products of the signal components in the integrator due to their oscillations, which are significantly faster during the integration time. There are no fundamental changes of the energy due to a root-mean-square nature of the seismic signal components in energy detection. DAGF2D function (see (6.2)) is the most promising one for practical purposes of detection of noisy seismic signals due to insignificance of the noise component in a vertical measurement of the seismic receiver. The function of distribution of DAGF2D signals (see (6.3)) is obtained by an averaging procedure of components in the corresponding space (the mean square of the signal components is proportional to the distance in the corresponding Euclidean space), by analogy with obtaining of distributions of commonly used and known random variables (e.g., chi-squared, Rayleigh, Maxwell, etc.), by means of compositions of simple distribution functions quadratic random values, but not of central normal distributions [Pr66]. Indeed, it was revealed during field statistical studies [Kr14] that the laws of the seismic signal amplitude distribution at the energy detector input (see Fig. 6.1) and after prefiltering are normal with zero ME (.m0 = 0); after detection and integration, amplitude of the seismic background obeys generalized DAGF2D (Rice) distribution law (see (6.2)). The probability density of this law is described

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by the expression (and analogously for mixture “signal”-“noise”: .ρS (r)):     r 2 + m2N r · mN r exp − .ρN (r) = · I0 . 2σN2 σN2 σN2 It should be noted that √ the probability law of the DAGF2D (Rice) distribution has random variable .r = X2 + Y 2 , where X and Y are independent random variables having normal distribution with the same variances .σ 2 and nonzero mathematical expectations (in general, unequal). The Rice distribution is often used to describe amplitude fluctuations of a radio signal [Pe03]. Then, considering (6.1): 

∞

PF A =

.

U0



∞

PEN.DET . =

.

U0

    r 2 + m2N r · mN r exp − · I0 dr, σN2 2σN2 σN2

(6.4)

    r 2 + m2S r · mS r exp − · I0 dr, 2σS2 σS2 σS2

(6.5)

where .σN , .mN , .σS , .mS are SD and ME, respectively, of the background seismic noise of the signal and noise mixture at the threshold device input. As shown by experimental studies [Kr14], at the input of a threshold comparator (limiter), parameters of the distribution law of the amplitude of the background and signals from various objects are estimated by values other than zero, but when compensating for the constant component at the threshold limiter input .mS = 0 (and .mN = 0) (see (6.4) and (6.5)), they will be as follow:  PF A = exp −

.

U02





, PEN.DET . = exp −

2σN2

U02 2σS2



 , U0 =

2σN2 · ln

1 . PF A (6.6)

Then, considering (6.6):  PEN.DET . = exp −

.

σN2 σS2

 ln PF A

(6.7)

In case of .mN = 0, .mS = 0 .PF A it is defined similarly to (6.7). It should be noted that the estimates obtained by (6.6), (6.7) are approximate. These estimates reflect mainly conformity of processes in the detection subsystem. These estimates require experimental verification and detailing, while the value of the energy detector output signal should be centered on noise, which creates no fundamental problems.

6 Statistical Structural Analysis of Seismic Signals

83

In this case, the probability has the following form: PEN.DET .

.

1 = 2 σS





∞

r · exp −

U0

r 2 + m2S



 · I0

2σS2

r · mS σS2

 ,

(6.8)

Using the tabulated DAGF2D (Rice) function:  θ (a, b) =

∞

.

y · exp(−

0

y 2 + b2 ) · I0 (by)dy, 2

(6.9)

we get  PEN.DET . = θ

.

σN2

1

mS ; 2 2 ln PF A σS σS

 .

(6.10)

Expressions (6.6–6.10) are the basis of the mathematical model of GDSS energy detection [Kr14]. These estimates are valid for a situation of the detection subsystem single decision-making on the presence of control objects. As shown by an analysis of domestic and foreign practice of GDSS constructing, decision-making time is characterized by value: .Ta = (0, 251)c [Kr14]. If references of the seismic noise time realization at decision interval .Ta are not correlated (correlation interval .τN < Ta ) or weakly correlated (an autocorrelation coefficient at .τN = Ta interval is significantly less than 1) and the additive mixture of the signal from the control object and noise is more correlated (.Tac.s ∼ = Ta ), then this pattern can be used to increase .PEN.DET . value and decrease .PF A [Kr14]. A statistical correlation analysis of real signals showed that the seismic noises at decision-making interval .Ta are less correlated (with probability .0, 8 · RACF.N. < 0, 15, than concentrated signals from the control objects (with probability .0, 8 · RACF.S. > 0, 25 [Kr14], where, .RACF.S. is a noise and mixture “signal”-“noise” autocorrelation function respectively. Under these conditions, it is possible for the energy detector to make a decision about the presence or absence of a signal using detailed methods used in processing of noise-like signals [Va85]. Then, probability of false positives with n number of consecutive decision cycles can be approximately estimated by the following formula: PF A =

n

.

PF A,i

i=1

where .PF A,i is the probability of a false alarm on the i–th decision cycle, n - a number of the decision cycles. Then, it is not difficult to write out:

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PEN.DET . = 1 − 1 − exp −

.



n

U02 2σS2

, PF A = exp −

U02 n 2σN2

 .

(6.11)

The analysis of dependencies (6.11) shows that an increase in a number of the decision cycles can significantly increase the detector noise immunity rates, and what is more, it allows to maneuver threshold level .U0 , providing required .PF A , .PEN.DET . , .PDET . (probability of detection) as evidenced by the results of experimental studies [Kr14]. The mathematical model of the detection subsystem should provide GDSS synthesis in a form of an optimization procedure of sequential approximation with the following expressed stages [Kr14]: 1. The stage of .PF A.SDE.RE. determination based on the developed requirements (hereafter, RE.) of .PF A.RE , where SDE - seismic detection equipment. 2. The stage of determining the level of response of .U0 to ensure .PF A.SDE.RE. and .PF A.RE. for what the following is required: a. Determination of the required ratio . σUN0 for a different number of decision cycles; b. Evaluation (measurement) of seismic noise SD, determination of .U0 . 3. The stage of determination of the possibility of achieving the required level of .PEN.DET with selected .U0 and with provision of the required detection range. In this case, the threshold level is estimated by the following expression: U0 = mN + k · σN ,

.

(6.12)

where k is the proportionality coefficient that provides the required level of .PF A . When .mN = 0 from (6.11) and (6.12): k=

.

2 ln

1 PF A

.

The resulting expression allows to estimate quickly an approximate level of the proportionality coefficient relatively to the values of the seismic background. It is necessary to adapt .U0 by a change of k to conform required .PF A when conditions of seismic SE (security equipment) use change. It is necessary to do that because SD of the seismic background changes. Detailing of k should be made in the course of experimental studies (.k = 4, 04, 8). Most often, the characteristics of the detecting seismic signals (if we are talking about their graphical representation) are constructed in the form of a set of curves .PEN.DET . dependence on the signal/noise ratio. At the same time, for each dependence of the probability of energy detection on the signal/noise ratio, the value .PF A is fixed [Kr14].

6 Statistical Structural Analysis of Seismic Signals

85

6.3 Conclusion Thus, we can make the following conclusions: 1. The detection subsystem should represent an optimal receiver of noise-like seismic signals as a component of an amplifier, band-pass filters with passing bands (specific for detected objects), a quadratic detector, an integrator, and a threshold device. 2. Distribution functions of noise-like signals and background at the “input/output” of band-pass filters are normal with negligeable mathematical expectations. Rice’s distribution function of noise-like signals and background are at the output of the integrator (DAGF2D for 2D dimensions) and DAGF3D function for three dimensions correspondingly). 3. The mathematical model of the subsystem detection is the following: a. Mathematical description of the dependence of energy detection probability on the probability of false positives from seismic and SD noise and signals based on seismic signal approximations by DAGF2D and DAGF3D: for purposes of statistical structural analysis b. Mathematical determination of the regularities of determining .U0 of the trigger threshold for provision of required .PF A.RE. and .PEN.DET .RE. 4. Differences in the degree of correlation at the decision-making interval between seismic noise and signals of concentrated objects make it possible to use several cycles of decision-making on signal detection (at least two) to increase .PEN.DET . and decrease .PF A.RE. . At the same time, it is advisable to use special methods to make a decision [Va85]. 5. As studies of dependence of “signal/nois” ratio on distance of the control object from the seismic detection means have shown, simultaneously providing the required .PEN.DET . , .PF A.SSE. and .RDET (correlation for a detection), is a very difficult task. Provision of “signal/noise” ratio at the level of at least four is a prerequisite for implementation of the abovementioned task. The solution of this problem for SDE (seismic detection equipments) is possible using methods of adaptation of detection algorithms to conditions of an application area.

References [Br99] Brikov, E.S.: The use of wavelet decomposition to the calculation of thermodynamic properties of simple liquids by direct variational method [Primenenie veivlet-razlozhenia k vychisleniyu termodinamicheskikh svoistv prostykh zhydkostei pryamym variatsionnym metodom]: in russian. Dis. Cand. Phys.-Math. Sci. Thesis, Sci. Adv. prof. A.E. Arinstein. Publishing House of Tyumen State University [Izdatel’stvo Tyumenskogo gosudarstvennogo universiteta], Tyumen (1999) [Kr14] Kryukov I.N.: Detection of a seismic signal [Obnaruzhenie seismicheskogo signala]: in russian. In: Seismic Detection Equipment. Theory and Practice of Construction [Seis-

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E. S. Brikov and V. I. Dobrorodnyi micheskie Sredstva Obnaruzhenia. Teoria i praktika postroenia], Scientific series “Territorially distributed security systems [Territorial’no raspredelennye sistemy okhrany]”. Doc. Tech. Sci. Prof. I.N. Kryukov, pp. 80–100. Radio engineering [Radiotekhnika], Moscow (2014) [Pe03] Petrov A.I.: Statistical theory of radio engineering systems [Statisticheskaya teoriya radiotehnicheskih sistem]: in russian. Radio Engineering [Radiotekhnika], Moscow (2003) [Pr66] Press, S.J.: Linear combinations of non-central Chi-square variates. Ann. Math. Statist. 37(2), 480–487 (1996) [Va85] Varakin, L.E.: Communication systems with noise-like signals [Sistemy svyazi s shumopodobnymi signalami]: in russian. Radio and Communication [Radio I svyaz], Moscow (1985)

Chapter 7

Topics on Space Weather: Operational Numerical Prediction for Electron Content H. F. de Campos Velho, A. Petry, T. S. Klipp, G. S. Falcão, J. R. Souza, E. R. Paula, J. N. Tamoki, L. H. B. Lago, and J. V. F. Lima

7.1 Introduction Sun events are propagated by the heliosphere, producing changes in the magnetosphere, ionosphere, and the radiation on the planet Earth. The study of interactions between the Sun and Earth is called space weather. The investigation, explanation, monitoring, and possible prediction of changes for the Earth–Sun system is of interest for scientific research and applications. In fact, solar activity has strong influence on our planet; in particular, a solar electromagnetic storm can bring a huge damage on our technological infrastructure with a possible cost of more than billions of dollars. The National Institute for Space Research (INPE), Brazil, has structured several activities linked to the space weather—see INPE’s Space Weather web-page: www2.inpe.br/climaespacial/portal/en/. One of these activities is to develop, with constant innovations, an operational system for predicting the ionospheric dynamics and concentrations of ions and electrons, producing the total electron content (TEC) maps. INPE has a good background dealing with systems based on the numerical solution of partial differential equations for producing operational forecasting of weather, climate, air quality (environment), and ocean

H. F. de Campos Velho () A. Petry · T. S. Klipp · G. S. Falcão · J. R. Souza · E. R. Paula National Institute for Space Research, São José dos Campos, Brazil e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] J. N. Tamoki Independent HPC Analyst, São Paulo, Brazil e-mail: [email protected] L. H. B. Lago · J. V. F. Lima Federal University of Santa Maria, Santa Maria, Brazil e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_7

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waves. The ionospheric forecasting model is based on the Sheffield University Plasmasphere-Ionosphere Model (SUPIM) [BaSe90, BaBa96]. SUPIM is a physicsbased computational model to calculate the concentration of electrons and ions in the atmosphere. The SUPIM equations are solved using the finite difference method. The goal here is to give a description with example of the operational SUPIM development to ionospheric dynamics for daily forecasting, providing information for the TEC and concentration of ions with higher resolution over South America for 24 hours ahead. The version for the operational forecasting was named Sheffield University Plasmasphere-Ionosphere Model—Data Assimilation and Visualization System: SUPIM-DAVS.

7.2 Brief SUPIM Description: Version for Operational Execution More than three decades were necessary to develop the SUPIM. From the equations of momentum, mass, and energy conservation, the concentration and temperature + + of electrons and ions (O.+ , H.+ , N.+ , He.+ , N.+ 2 , O.2 , and NO. ) are calculated on the magnetic field lines. Several physical and chemical processes are included for simulating the ionosphere/plasmasphere dynamics. The solar radiation induces the atmospheric ion production. For computing the ionic balance, chemical reactions between ions and neutral gases, thermal diffusion, photoelectric, and friction heating need to be considered. Some inputs are necessary for the SUPIM execution: model resolution, day/time and time period for simulation, solar fluxes at different wave lengths, magnetic field, thermospheric neutral wind, and vertical drift.

7.2.1 Operational SUPIM-DAVS: Parallel Version Architectures for processing clusters is an arrangement of many processing nodes, where each node can have more than one multicore processor. The OpenMP parallelism directives can be used for shared memory multiprocessing computers. For the distributed memory multiprocessing machines, the message passing interface (MPI) is the programming environment employed. Taking advantage of the multinodes with multicore processors, a hybrid programming was applied to the parallel version of the SUPIM code. The computation in SUPIM is formulated in magnetic coordinates, where the 2D domain is employed for calculating the variables along closed magnetic field lines. Therefore, all calculations on the 2D magnetic field lines are independently performed on several processing nodes, using the MPI standard internodes, while

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OpenMP is applied for intra-node processing with a setup of four threads for some critical (greater computational effort) routines.

7.2.2 Operational SUPIM-DAVS: Data Assimilation For operational forecasting systems using time integration of differential equations, initial conditions are determined combining background data (previous predicted fields) with observations. This strategy is named data assimilation. Several methods have been developed for data assimilation. Nudging, or Newtonian relaxation [Da91, HoAn76, Ka03], is the data assimilation method employed. Measurements could have a nonuniform distribution in time and space. The scheme was thought considering a region around on the computer model point, taking into account the influence of the observations in the region following [StSe94].  Nj xa (j ) = xb (j ) + α

.

i=1 w

2 (i, j ) γ

Nj

i

[y(i) − xb (i)]

i=1 w(i, j )

Here .xa (j ) is the analysis—initial condition after data assimilation operation— at local-j ; .xb (j ) is the background (prediction from the computer model); .α is the nudging parameter determined by numerical experimentation; .γi ∈ [0, 1], indicating the observation precision degree; .y(i) is the observation; .w(i, j ) is the weighting function; and .Nj is the observation number in the considered region around the model point-i. The weighting function is given by  w(i, j ) = max 0,

.

2 R 2 − di,j 2 R 2 + di,j

 ,

where R is the influence radius and .di,j the distance between points i and j .

7.2.3 Operational SUPIM-DAVS: Decay Correction Factor Some executions have shown some instabilities. A correction factor was introduced to give an additional stability to the model. The factor is an exponential decay for an ion-neutral collisional frequency (.fin ) at high altitudes: fin = f0 e−Δs/Hi

.

where .f0 is the ion-neutral collisional frequency without correction, .Δs is a height change along a magnetic field line, and .Hi is the ion scale height.

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Fortaleza

Fig. 7.1 Electronic profile measured on March 19, 2011, by ionosondes (large dots) located at 3.88 S 38.4 W—Fortaleza (CE, Brazil) at Universal Time (UT): 07 UT (left), 13 UT (middle), 19 UT (right)

7.2.4 Operational SUPIM-DAVS: Post-Processing A software package was developed for several tasks: coordinate transformation (geographic . geomagnetic) [PeEtAl14]; temporal adjustment—there is a local time difference between different magnetic field lines and an adjustment is needed— interpolation; and map generation. The interpolation is done by the inverse distance weighting (IDW) [LiHe11]. A 3D data interpolation, based on IDW and influence explosion neighbor searching (NS) algorithm [PeEtAl16], was developed to make a map of ions and electron distributions on a homogeneous computational grid over South America [PeEtAl14].

7.3 Forecasting: Operational SUPIM-DAVS This section is dedicated to show some examples of the SUPIM execution as a package for the operational routine to predict the total electron content (TEC) and/or vertical-TEC (vTEC)—integrating the electronic concentration on the vertical coordinate, as well as ion concentrations over South America. The operational action with SUPIM was evaluated to compute the initial condition using ionosonde data from four different stations in South America during the equinox and solstice. The mentioned sensors are in Fortaleza (3.88 S 38.4 W), Jicamarca (12 S 76.8 W), Cachoeira Paulista (22.7 S 45 W), and São Luis (2.6 S 44.2 W). Figure 7.1 shows the data assimilation action with .α = 0.5 for the ionosonde located at Fortaleza (dotted black line) during equinox with the corresponding simulation output (dotted blue line) at the closest location and the simulation output after data assimilation (solid red line), while results for other locations and/or seasons can be seen in reference [PeEtAl14]. Figure 7.2 shows the vTEC by the SUPIM simulation at December 19, 2011, and the values obtained from the ground-based GPS stations. The operational SUPIM-DAVS performance was evaluated recently [KlEtAl19]. The evaluation was done with the comparison of Central and South America

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Fig. 7.2 vTEC map from the SUPIM simulation (above) and ground-based GPS stations (below) at 7, 13, and 19 h UT on December 19, 2011

with the SUPIM-DAVS forecasting, observation data, and an empirical model [JaEtAl11] developed by the German Aerospace Agency (DLR) ionospheric group—Neustrelitz (Germany). Figure 7.3 depicts the time series for the TEC values for 20 months, where an average for 24 hrs was carried out for the observations—sensors from the IGS (International GNSS—Global Navigation Satellite System)—service, and simulation by the SUPIM-DAVS. The IGS stations are spread in Central America (2 sensors) and the Caribbean (4 sensors) zones, as well as in South America (27 sensors) [KlEtAl19]. In the periods January–April (2016) and May–August (2017), there is a disagreement between the SUPIM prediction and observations. During January–April (2016), under moderate solar activity, the EUV solar flux boundary condition was estimated using the EUVAC model [RiEtAl94]—gray zone in Fig. 7.3—where the solar flux was underestimated. For the period May–August (2017), the solar flux was estimated by the Solar2000 software package [ToEtAl00], under low solar activity. Big blue dots in Fig. 7.3 are indicating values found by the SUPIM-DAVS applying the EUV flux from EUVAC (gray zone) and Solar2000 (white zone) models—in the figure, there are two big blue dot points, indicating the SUPIM-DAVS simulation with the Solar2000 (left blue dot) and employing the EUVAC (right blue dot).

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Fig. 7.3 Times series for TEC, observations (IGS), and SUPIM results Table 7.1 Routines and processing demands

Importance (1) (2) (3) (4)

Routine GLOBE7: GTS7: DENSU: SPLIN:

Demands 28.7% 21.7% 16.8% 13.1%

7.4 SUMPI-DAVS: Vector Processing A permanent effort is to investigate new computer architectures, in order to get a better performance for the SUPIM processing, since the code presents an intensive numerical computation. Therefore, an implementation for the NEC SX-Tsubasa vector computer was carried out. The first action was to evaluate the CPU time for different routines. Table 7.1 shows the processing demands for four more intensive routines. For the operational tasks, the parallel SUPIM is executed on a cluster with multicore CPUs, presenting 4730 seconds for 24 hours of simulation. The SUPIM implementation on the NEC SX-Aurora TSUBASA vector processor shows a reduction of the CPU time to 1050 seconds. The SUPIM simulation over South America for 24 hours is executed in a cluster with CPU time 01:18:50 (hours:minutes:seconds) = 4730 seconds. Looking for a reduction of the CPU time, the operational SUPIM version was implemented on a vector processor. The performance of the operational SUPIM otimized for the NEC

7 Space Weather: Prediction for Electron Content Table 7.2 Parallel optimizations and CPU-time

93 Optimazation type Without optimization Optimization directives Optimized Globe7 Optimized GTS7 Optimized DENSU Optimized SPLIN

CPU time 01:18:50 00:39:47 00:26:06 00:23:20 00:20:25 00:17:30

SX-Tsubasa computer is the goal of this paper, where all optimization steps are shown in Table 7.2.

7.5 Final Remarks This contribution is a brief report of main developments of an operational forecasting system for the electron and ion distribution in the atmosphere. One important product is the prediction of the total electron content (TEC) mapping. The SUPIM vector version was .∼4.5 faster than the sequential version. The SUPIM team is investigating other developments, such as ensemble prediction, an automatic system to select the boundary condition (solar flux), new schemes for data assimilation (3D-Var and neural networks), and evaluation of new computing environments (cloud computing and hybrid computing, CPU + coprocessors).

References [BaSe90] Bailey, G.J., Sellek, R.: A mathematical model of the Earth’s plasmasphere and its application on a study of He+ at L = 3. Ann. Geophys. 8, 171–190 (1990) [BaBa96] Bailey, G.J., Balan, N.: A low-latitude ionosphere-plasmasphere model. In: Schunk, R.W. (ed.) STEP Handbook on Ionospheric Models, pp. 173–206. Utah State University, USA (1996) [Da91] Daley, R.: Atmospherics Data Analysis. Cambridge University Press, New York (1991) [HoAn76] Hoke, J., Anthes, R.: The initialization of numerical models by a dynamic relaxation technique. Mon. Weather Rev. 104, 1551–1556 (1976) [JaEtAl11] Jakowsky, N., Mayer, C. Hoque, M.M., Wilken, V.: Total electron content models and their use in ionosphere monitoring. Radio Sci. 46, RS0D18 (2011) [Ka03] Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge (2003) [KlEtAl19] Klipp, T.S., Petry, A., Souza, J.R., Falcão, G.S., Campos Velho, H.F., Paula, E.R., Antreich, F., Hoque, M., Kriegel, M., Berdermann, J., JoKowisky, N., FernandezGomez, I., Borries, C., Sto, H., Wilken, V.: Evaluation of ionospheric models for Central and South Americas. Adv. Space Res. 64, 2521–2136 (2019)

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[LiHe11] Li, J., Heap, A.D.: A review of comparative studies of spatial interpolation methods in environmental sciences: performance and impact factors. Eco. Inform. 6, 228–241 (2011) [PeEtAl14] [SUPIM-2014] [SUPIM-2014] Petry, A., Souza, J.R., Campos Velho, H.F., Pereira, A.G., Bailey, G.J.: First results of operational ionospheric dynamics prediction for the Brazilian Space Weather program. Adv. Space Res. 54, 22–36 (2015) [PeEtAl16] Petry, A., Pereira, A.G., Souza, J.R.: Approximate nearest neighbors searching algorithm for low-dimensional grid locations. Earth Sci. Inf. 10, 1–14 (2016) [RiEtAl94] Richards, P.G., Fennelly, J.A., Torr, D.G.: EUVAC: A solar EUV flux model for aeronomic calculations. J. Geophys. Res. Space Phys. 99, 8981–8992 (1994) [StSe94] Stauffer, D.R., Seman, N.L.: Multiscale four-dimensional data assimilation. J. Appl. Meteorol. 33, 416–434 (1994) [ToEtAl00] Tobiska, W., Woods, T., Eparvier, F., Viereck, R., Floyd, L., Bouwer, D., Rottman, G., White, O.: The SOLAR2000 empirical solar irradiance model and forecast tool. J. Atmos. Sol. Terr. Phys. 62, 1233–1250 (2000)

Chapter 8

Ray-Tracing the Ulam Way D. J. Chappell, M. Richter, G. Tanner, O. F. Bandtlow, W. Just, and J. Slipantschuk

8.1 Introduction Ray-tracing is a well-established approach for modelling wave propagation at high frequencies, in which the ray trajectories are defined by a Hamiltonian system of ODEs [Ce01]. An approximation of the wave amplitude is then derived from estimating the density of rays in the neighborhood of a given evaluation point. An alternative approach is to formulate the ray-tracing model directly in terms of the ray density in phase-space using the Liouville equation. The solutions may then be expressed in integral form using the Frobenius-Perron (F-P) operator, which is a transfer operator transporting the ray density along the trajectories [CvEtAl20]. The classical approach for discretizing such operators dates back to 1960 and the work of Stanislaw Ulam [Ul64]. The convergence of the Ulam method has been established in some cases, typically in low-dimensional settings with continuous densities and hyperbolic dynamics (see, e.g., [Li76, Fr99, BoMu01]). In this chapter, we outline some recent work investigating the convergence of the Ulam method for ray tracing in triangular billiards, where the dynamics are parabolic and the flow map contains jump discontinuities. This study builds upon recent work on ray tracing in circular billiards [SlEtAl20], where it was found that a spectral Fourier Galerkin approximation of the F-P operator gave faster

D. J. Chappell () Nottingham Trent University, Nottingham, UK M. Richter · G. Tanner University of Nottingham, Nottingham, UK O. F. Bandtlow · W. Just Queen Mary University of London, London, UK J. Slipantschuk University of Warwick, Coventry, UK © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_8

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convergence rates than would be possible using the Ulam method, and the precise rate depends critically on the regularity of the boundary data driving the problem. However, the rigorous study of polygonal billiards, such as the triangle, is innately more challenging, owing to the presence of vertices. In particular, the momentum component of the boundary flow map, which defines the phase-space coordinates of the ray trajectory at discrete times corresponding to boundary collisions, is discontinuous at the vertices. The presence of these discontinuities necessitates the use of function spaces, including discontinuous functions, such as Sobolev spaces of low regularity or spaces of bounded variation [Ke85, Sa00]. A Galerkin projection using a Fourier basis as proposed for circular billiards in [SlEtAl20] and analyzed within a Sobolev function space setting is therefore less suited to polygonal billiards, since there is no smoothness of the boundary flow map to exploit. Instead, Ulam-type methods appear to provide a more natural fit, owing to their use of discontinuous basis functions that better reflect the properties of the F-P operator here. In the remainder of this chapter, we will first outline a mathematical model for propagating ray densities using transfer operators. We then describe the discretization of this model using the Ulam method for triangular domains, as well as giving some pointers toward a convergence analysis for this discretization. Finally, we draw some conclusions from our findings and discuss some related studies, where faster convergence rates are plausible.

8.2 Ray-Tracing via Transfer Operators The transport of densities along a ray trajectory flow map .ϕ τ through time .τ and space .Rd can be formulated in terms of the F-P operator (see, e.g., [CvEtAl20]). The action of this operator on a density f may be expressed as  L τ f (X) =

.

δ(X − ϕ τ (Y ))f (Y ) dY,

where X and Y are phase-space coordinates in .R2d . Solving such problems when d > 1 and for physically relevant systems is often considered intractable due to both high dimensionality and potentially complex geometries [SiEtAl07]. In this work, we reformulate the F-P operator as a phase-space boundary integral operator, which is discretized using the Ulam method. We restrict our attention to modelling the propagation of a density f through a convex polygonal domain 2 .Ω ⊂ R . Let us assume that the ray trajectory flow is governed by the Hamiltonian .H (r, p) = |p| = 1 in .Ω, where .r ∈ Ω and .p is the momentum coordinate. Let the phase-space P on the boundary of .Ω be written .P = ∂Ω × (−1, 1). Then, the associated coordinates are given by .X = [s, p] ∈ P with .s ∈ [0, L) parameterizing .∂Ω, where L is the total length of the boundary, and .p ∈ (−1, 1) parameterising the component of the inward unit vector .p tangential to .∂Ω. Next, .

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we define .ϕ : P → P to be the boundary flow map, which takes a vector in P and maps it along the Hamiltonian flow defined by H to a vector in P . The propagation of the density f along the map .ϕ is given by a modified F-P operator acting on this map as follows:  L f (X) =

μ(Y )δ(X − ϕ(Y ))f (Y )dY.

.

(8.1)

P

The operator .L describes the propagation of f along a trajectory between two points on the boundary of .Ω, together with a specular reflection at the arrival point. The term .μ : P → (0, 1) is incorporated to model energy losses at boundary reflections. The stationary density .ρ on P due to an initial boundary distribution .ρ0 on P is the density accumulated in the long time (many iterate) limit. That is, ρ(X) =

∞ 

.

L n ρ0 (X),

(8.2)

n=0

where .L n is the nth iterate of the operator (8.1). Note that the incorporation of the dissipative term .μ within (8.1) is necessary for the sum (8.2) to converge. The stationary density .ρ may then be obtained as the solution of the following integral equation: (I − L )ρ = ρ0

.

(8.3)

via the standard Neumann series result for (8.2).

8.3 The Ulam Method In this section, we restrict our attention to the case when .Ω is a triangle. We first outline the implementation of the Ulam method to discretize the integral equation (8.3), before briefly discussing some of the building blocks that we believe will lead to a rigorous convergence analysis of the resulting numerical approximation of the stationary density .ρ.

8.3.1 Implementation of the Ulam Method In order to apply the Ulam method to discretize (8.3), we first subdivide the boundary phase-space .P = [0, L) × (−1, 1) into MN rectangles .Rm,n , with .m = 1, 2, . . . , M and .n = 1, 2, . . . , N. We do this by performing an equi-spaced

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subdivision along the p coordinate with step-size .Δp = 2/N, and along each edge of the triangle .e = 1, 2, 3 we perform an equi-spaced subdivision with step-size .Δse = Le /Me , where .M = M1 + M2 + M3 and .Le is the length of edge e. The number of subintervals on edge e, .Me is defined by first choosing a target number of subdivisions for the whole boundary of length .L = L1 + L2 + L3 as .M ∗ and then taking .Me = round(M ∗ Le /L), with round denoting rounding to the nearest integer. The stationary density .ρ is now approximated by its projection .PMN ρ onto a finite-dimensional space of piecewise constant functions of the form ρ(s, p) ≈ (PMN ρ) (s, p) =

N M  

.

m=1 n=1

ρm,n χm (s)χn (p), Δse(m) Δp

(8.4)

where .χn (p) = 1 if .2(n − 1)/N < p < 2n/N and is zero otherwise. Likewise, χm (s) = 1 is an indicator for the mth subdivision of the s-coordinate. The index .e(m) = 1 for .m = 1, 2, . . . , M1 , .e(m) = 2 for .m = M1 + 1, M1 + 2, . . . , M1 + M2 , and .e(m) = 3 for .m = M1 + M2 + 1, M1 + M2 + 2, . . . , M. A Galerkin projection of Eq. (8.3) onto the finite-dimensional basis (8.4) may be written in the form .

(I − T )ρ = ρ0 ,

(8.5)

.

where I is the .NM × NM identity matrix and T is a matrix with entries   Tα,α  = μ

.

1









δ(X − ϕ(Y ))χm (s )χn (p )dY χm (s)χn (p)dX, Δse(m ) Δp P  μ χm (ϕs (s  , p ))χn (ϕp (s  , p ))χm (s  )χn (p )dY, = Δse(m ) Δp P P

=μ

Area(Rm ,n ∩ ϕ −1 (Rm,n )) . Area(Rm ,n )

In the second line, we note that the boundary map .ϕ has been split into its position and momentum components, which are denoted .ϕs and .ϕp , respectively. In addition,   .Area(Rm ,n ) = Δse(m ) Δp denotes the area of .Rm ,n , .Y = [s , p ], .α = m + (n −    1)M and correspondingly .α = m + (n − 1)M. The vector .ρ = [ρα ]α=1,2,...,MN contains the coefficients .ρm,n = ρα from the basis expansion (8.4). The vector .ρ 0 contains the corresponding coefficients for the expansion of the initial density .ρ0 in the form (8.4). Furthermore, we have assumed that the damping factor .μ is constant for simplicity. In the next section, we outline a theoretical setting that we believe will lead to rigorous analysis for the convergence of the approximation (8.4) obtained by solving (8.5).

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8.3.2 A Route Toward Rigorous Analysis The theory underpinning our analysis stems from the work of Keller in the 1980s [Ke85]. We conjecture that a function space setting that remains invariant under the action of .L can be devised using spaces of functions of generalized bounded variation .Vφ for a particular choice of the map .φ. In fact, .Vφ are Banach spaces when .φ : [0, η0 ] → [0, ∞) is a monotonically increasing map with .limη↓0 φ(η) = 0. To define the norm on .Vφ , we first introduce the oscillation of a function f in a set 2 .A ⊂ R as osc(f, A) = ess sup(x,y)∈A×A |f (x) − f (y)|.

.

The norm on .Vφ may then be written as  f φ := f L1 + sup

.

η∈(0,η0 ]

R2

osc(f, Bη (x))dx , φ(η)

where . · L1 denotes the standard .L1 norm and .Bη (x) ⊂ R2 is an open ball of radius .η and center x. Assuming that .L f ∈ Vφ for every .f ∈ Vφ , then Lemma 1.11 from [Ke85] provides f − PMN f L1  φ(h)f φ ,

.

 where .h = Δs 2 + Δp2 and .Δs = maxe {Δse }. That is, the convergence rate depends critically on .φ, as does the validity of the assumption .L f ∈ Vφ . Evidence from numerical experiments typically shows first-order convergence, corresponding to .φ(h) ∝ h. However, the assumption of .L f ∈ Vφ would break down in this case owing to the discontinuities in the boundary map .ϕ. We conjecture that instead choosing φ(h) = |ln (h)|−β

.

with fixed .β ∈ N will provide the necessary assumption of .L f ∈ Vφ for every .f ∈ Vφ . The price that we pay for this choice is a rather slower logarithmic convergence rate. We remark that sublinear convergence of the Ulam method has previously been proved rigorously for piecewise expanding interval maps [BoMu01] and appears to be a realistic starting point for our analysis here.

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8.4 Conclusions and Outlook We have introduced a ray-tracing model for the propagation of phase-space densities through convex polygonal domains, which was expressed in terms of a damped F-P operator .L for the boundary flow map .ϕ. The stationary density .ρ may be obtained from the solution of a second-kind Fredholm equation (8.5). We discussed the discretization of (8.5), and consequently the damped F-P operator .L , using the Ulam method. Some ideas were then proposed regarding how one may prove the convergence of the solution obtained via this discretization to the solution of the original problem. We only expect to obtain sublinear convergence rates, and the reason for this slow convergence can be directly linked to the lack of regularity of the boundary map .ϕ for the triangle, or indeed any convex polygon. We note that there are a number of settings in which one can realistically hope for faster convergence of discretization schemes for estimating .ρ, such as for domains with smooth boundaries. This was explored further in [SlEtAl20], where rigorous convergence estimates were established for circular billiards using a Fourier basis approximation of .ρ and more general smooth domains were investigated numerically. A further case that may yield faster convergence rates is that of stochastically smoothed F-P operators, which describe uncertain ray dynamics [ChTa14]. A significant advantage in this case is that the transfer operator is compact, meaning one can draw upon a considerably wider body of supporting theory as discussed in [BaCh18].

References [BaCh18] Bajars, J., Chappell, D.J.: A boundary integral method for modelling vibroacoustic energy distributions in uncertain built up structures. J. Comp. Phys. 373, 130–147 (2018) [BoMu01] Bose, C., Murray, R.: The exact rate of approximation in Ulam’s method. Discrete Contin. Dynam. Systems 7(1), 219–235 (2001) ˇ V.: Seismic ray theory. Cambridge University Press, Cambridge (2001) [Ce01] Cervený, [ChTa14] Chappell, D.J., Tanner, G.: A boundary integral formalism for stochastic ray tracing in billiards. Chaos 24(4), 043147 (2014) [CvEtAl20] Cvitanovi´c, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and Quantum. Niels Bohr Institute, Copenhagen, Denmark (2012). ChaosBook.org [Fr99] Froyland, G.: Ulam’s method for random interval maps. Nonlinearity 12, 1029–1052 (1999) [Ke85] Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrscheinlichkeitstheorie verw Gebiete 69(3), 461–478 (1985) [Li76] Li, T.-Y.: Finite approximation for the Frobenius–Perron operator. A solution to Ulam’s conjecture. J. Approx. Theory 17, 177–186 (1976) [SlEtAl20] Slipantschuk, J., Richter, M., Chappell, D.J., Tanner, G., Just, W., Bandtlow, O.F.: Transfer operator approach to ray-tracing in circular domains. Nonlinearity 33, 5773– 5790 (2020) [Sa00] Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Isr. J. Math. 116, 223–248 (2000)

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[SiEtAl07] Siltanen, S., Lokki, T., Kiminki, S., Savioja, L.: The room acoustic rendering equation. J. Acoust. Soc. Am. 122, 1624–1635 (2007) [Ul64] Ulam, S.: Problems in Modern Mathematics. Wiley, New York (1964)

Chapter 9

The Robin Boundary Value Problem for an Unbounded Plate with a Hole C. Constanda and D. Doty

9.1 Introduction In this chapter, we construct a method for approximating the solution of bending of a load-free, unbounded elastic plate with a hole, under Robin-type conditions prescribed on the boundary of the hole and a given far-field behavior. The procedure is implemented by means of a generalized Fourier series method that makes use of a complete set of functions spanning the space of the solution. The members of this set are constructed from elements intrinsically tied to the analytic structure of the mathematical model. Similar problems have been considered for finite plates with Dirichlet, Neumann, and Robin boundary conditions and for infinite plates with Dirichlet and Neumann data on the boundary, in [CoDo17a, CoDo17b, CoDo18, CoDo19a, CoDo19b, CoDo19c, CoDo20] and [CoDo22].

9.2 The Mathematical Model In the sequel, .S + is a finite domain in .R2 bounded by a simple, closed, .C 2 –curve − = R2 \(S + ∪∂S), .x(x , x ) and .y(y , y ) are generic points in .S + , .S − , or on .∂S, .S 1 2 1 2 .∂S, and .|x − y| is the distance between x and y in the Cartesian metric. For a matrix M, we denote by .M (i) and .M(i) its columns and rows, and by .M T its transpose. Additionally, .C 0,α (∂S) and .C 1,α (∂S), .α ∈ (0, 1) are, respectively, the spaces of

C. Constanda () · D. Doty The University of Tulsa, Tulsa, OK, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_9

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Hölder continuous and Hölder continuously differentiable functions on .∂S, .· , · and .  ·  are the inner product and norm on .L2 (∂S), and I is the identity operator. We assume that the three-dimensional region .(S − ∪ ∂S) × [−h0 /2, h0 /2], where .h0 = const, is occupied by a (homogeneous and isotropic) material with Lamé constants .λ and .μ. The model of bending of plates with transverse shear deformation consists of the following mathematical elements (see [Co16]): (i) The displacements are characterized by a vector of the form .u = (u1 , u2 , u3 )T , whose components are functions of .x1 and .x2 . (ii) The columns .F (i) of the matrix ⎛

⎞ 1 0 0 .F = ⎝ 0 1 0⎠ −x1 −x2 1 form a basis for the space of rigid displacements. (iii) The system of partial differential equations governing the state of equilibrium when the body forces are negligible is written as A(∂1 , ∂2 )u(x) = 0,

.

where A(∂1 , ∂2 )

.

⎞ ⎛ 2 h 2 (λ + μ)∂1 ∂2 −μ∂1 h μΔ + h 2 (λ + μ)∂12 − μ ⎟ ⎜ =⎝ h 2 (λ + μ)∂1 ∂2 h 2 μΔ + h 2 (λ + μ)∂22 − μ −μ∂2 ⎠, μ∂1

μ∂2

μΔ

∂α = ∂/∂xα , .α = 1, 2, .D(x, y) is a matrix of fundamental solutions, .h2 = h20 /12, and .Δ = ∂12 + ∂22 is the Laplacian. (iv) An associated matrix of singular solutions that plays an important role in the study of the model is defined by .

 T P (x, y) = T (∂y )D(y, x) ,

.

(9.1)

where T (∂) = T (∂1 , ∂2 )

.

⎛ ⎜ =⎝

h 2 (λ + 2μ)n1 ∂1 + h 2 μn2 ∂2

h 2 μn2 ∂1 + h 2 λn1 ∂2

0

h 2 λn2 ∂1 + h 2 μn1 ∂2

h 2 μn1 ∂1 + h 2 (λ + 2μ)n2 ∂2

0

μn1

μn2

μnα ∂α

⎞ ⎟ ⎠

9 Robin Boundary Value Problem

105

is the boundary moment-force operator, .n(n1 , n2 ) is the unit normal to .∂S pointing outside .S − , and .nα ∂α = n1 ∂1 + n2 ∂2 .

9.3 Exterior Robin Problem Let .A be the class of vector functions in .S − with far-field expansion, as .r → ∞,

 u1 (r, θ ) = r −1 m0 sin θ + 2m1 cos θ − m0 sin(3θ ) + (m2 − m1 ) cos(3θ )

+ r −2 (2m3 + m4 ) sin(2θ ) + m5 cos(2θ ) − 2m3 sin(4θ )  + 2m6 cos(4θ )

+ r −3 2m7 sin(3θ ) + 2m8 cos(3θ ) + 3(m9 − m7 ) sin(5θ )  + 3(m10 − m8 ) cos(5θ ) + O(r −4 ),

.



u2 (r, θ ) = r −1 2m2 sin θ + m0 cos θ + (m2 − m1 ) sin(3θ ) + m0 cos(3θ )

+ r −2 (2m6 + m5 ) sin(2θ ) − m4 cos(2θ ) + 2m6 sin(4θ )  + 2m3 cos(4θ )

+ r −3 2m10 sin(3θ ) − 2m9 cos(3θ ) + 3(m10 − m8 ) sin(5θ )  + 3(m7 − m9 ) cos(5θ ) + O(r −4 ),

 u3 (r, θ ) = −(m1 + m2 ) ln r − m1 + m2 + m0 sin(2θ ) + (m1 − m2 ) cos(2θ )

+ r −1 (m3 + m4 ) sin θ + (m5 + m6 ) cos θ − m3 sin(3θ )  + m6 cos(3θ )

+ r −2 m11 sin(2θ ) + m12 cos(2θ ) + (m9 − m7 ) sin(4θ )  + (m10 − m8 ) cos(4θ ) + O(r −3 ), where .m0 , . . . , m12 are constants. We make the decomposition D = DA + D∞,

.

where (D A )(i) ∈ A , AD A = AD ∞ = 0 in S + ∪ S − .

.

At the same time, for the P matrix we have

106

C. Constanda and D. Doty

P ∞ = 0, P = P A , (P A )(i) ∈ A .

.

The exterior Robin problem consists in finding .u ∈ C 2 (S − ) ∩ C 1 (S¯ − ) that satisfies Au = 0 in S − , .

(T − σ )u = R on ∂S,

(9.2)

u∈A, where the symmetric, positive definite, continuous .3 × 3 matrix function .σ , and the 3 × 1 vector function .R ∈ C 0,α (∂S), .α ∈ (0, 1), are prescribed on .∂S.

.

Theorem 1 Problem (9.2) has a unique solution u for any .R ∈ C 0,α (∂S). The fact that .u ∈ A ∗ means that u = uA + F c,

.

where .uA ∈ A and F c is a rigid displacement. Since .AF (i) = 0 and .T F (i) = 0, .i = 1, 2, 3, we see that problem (9.2) becomes AuA = 0 in S − , .

(T − σ )uA = R + σ F c

(9.3) on ∂S.

As a solution of (9.3), .uA admits the representation formulas uA (x) = −



.

D(x, y)T uA (y) ds(y) +

∂S

P (x, y)uA (y) ds(y),

x ∈ S−,

P (x, y)uA (y) ds(y),

x ∈ S+,

∂S

0=−





A

D(x, y)T u (y) ds(y) +

∂S

∂S

which, with .T uA = R + σ uA + σ F c on .∂S, become A .u (x) = − D(x, y)(R + σ uA + σ F c)(y) ds(y) ∂S

+ ∂S

P (x, y)uA (y) ds(y),

x ∈ S−, .

(9.4)

9 Robin Boundary Value Problem



107

D(x, y)(R + σ uA + σ F c)(y) ds(y)

0=− ∂S

+

P (x, y)uA (y) ds(y),

x ∈ S+.

(9.5)

∂S

Since .T uA is the Neumann boundary data of the solution .uA of (9.3), it follows that [CoDo22] .T uA , F (i)  = 0, .i = 1, 2, 3, translated in our case as R + σ uA + σ F c, F (i)  = 0,

i = 1, 2, 3.

.

It can be shown that D and P can be replaced in (9.4) and (9.5) by .D A and .P A , respectively. Recalling that .P = P A , we arrive at

A

u (x) = −

.

 A D(x, y) (σ uA )(y) ds(y) +

∂S

−

0=−

 A D(x, y) (R + σ F c)(y) ds(y),

∂S

−

P (x, y)uA (y) ds(y)

∂S

 A D(x, y) (σ uA )(y) ds(y) +

∂S





x ∈ S−, .

(9.6)

P (x, y)uA (y) ds(y)

∂S

 A D(x, y) (R + σ F c)(y) ds(y),

x ∈ S+.

(9.7)

∂S

Equalities (9.6) and (9.7) can also be written for the solution u of (9.2). If we replace .uA = u − F c in them, a simple reorganization of the terms yields u(x) =

.



  P (x, y) − D(x, y))A σ (y) u(y) ds(y)

∂S

−

0=

 A D(x, y) R(y) ds(y) + (F c)(x),

x ∈ S−, .

(9.8)

x ∈ S+.

(9.9)

∂S

  P (x, y) − D(x, y))A σ (y) u(y) ds(y)

∂S

− ∂S

 A D(x, y) R(y) ds(y) + (F c)(x),

108

C. Constanda and D. Doty

Let .∂S∗ be a simple, closed, .C 2 –curve lying strictly inside .S + , and let .{x (k) }∞ k=1 be a set of points densely distributed on .∂S∗ . We consider the sequence of vector functions on .∂S defined by

  G = σ F (i) , i = 1, 2, 3 ∪ ϕ (j k) , j = 1, 2, 3, k = 1, 2, . . . ,

.

where   A (j ) ϕ (j k) (x) = T − σ (x) D(x, x (k) ) .

.

(9.10)

There are two useful alternative expressions for .ϕ (j k) , in terms of the rows of matrix .D A and those of matrix P , derived from (9.10):   A (j ) ϕ (j k) (x) = T − σ (x) D(x, x (k) )   A T = T − σ (x) D(x (k) , x) , (j )

.

j = 1, 2, 3, k = 1, 2, . . . , (9.11)

and  T ϕ (j k) (x) = ( P (x (k) , x) (j ) ) −

.





(( D(x (k) , x) A )(j ) σ (x)) , T

j = 1, 2, 3, k = 1, 2, . . . .

(9.12)

Theorem 2 The set .G is linearly independent on .∂S and complete in .L2 (∂S). We order the elements of .G as the sequence σ F (1) , σ F (2) , σ F (3) , ϕ (11) , ϕ (21) , ϕ (31) , ϕ (12) , ϕ (22) , ϕ (32) , . . .

.

and re-index them, noticing that for each .i = 4, 5, . . . , there is a unique pair .{j, k}, j = 1, 2, 3, . k = 1, 2, . . . , such that .i = j + 3k. Then,

.

 G = ϕ (1) , ϕ (2) , ϕ (3) , ϕ (4) , ϕ (5) , ϕ (6) , ϕ (7) , ϕ (8) , ϕ (9) , . . . ,

.

(9.13)

where ϕ (i) = σ F (i) ,

.

i = 1, 2, 3,

(9.14)

and ϕ (i) = ϕ (j k) ,

.

j = 1, 2, 3, k = 1, 2, . . . , i = j + 3k = 4, 5, . . . .

(9.15)

Our intention is to construct a nonsingular linear system of finitely many algebraic equations to approximate the solution u of (9.2).

9 Robin Boundary Value Problem

109

Rewriting the representation formulas (9.8) and (9.9) as   A  u(x) = P (x, ·) − D(x, ·) σ, u

.

  A − D(x, ·) , R + (F c)(x),

x ∈ S−, .

(9.16)

  A  P (x, ·) − D(x, ·) σ, u + (F c)(x)   A = D(x, ·) , R ,

x ∈ S+,

(9.17)

we would normally use (9.17) to approximate .u|∂S , and then approximate u in .S − by means of (9.16). However, this operation cannot be performed since (9.17) contains the rigid displacement Fc =

3 

.

ci F (i) ,

i=1

which may not be known a priori and needs to be determined as part of the computation process. We set ψ = u|∂S .

.

Since the .x (k) are points in .S + , from (9.17) it follows that   A  P (x (k) , ·) − D(x (k) , ·) σ, ψ + (F c)(x (k) )

.

A   = D(x (k) , ·) , R ,

j = 1, 2, 3, k = 1, 2, . . . ,

from which 







( P (x (k) , ·) (j ) )T − (( D(x (k) , ·) A σ )(j ) ) , ψ  + (F c)j (x (k) ) T

.

=





(( D(x (k) , ·) A )(j ) ) , R , T

j = 1, 2, 3, k = 1, 2, . . . .

(9.18)

We have  A  . D(x, ·) σ, ψ =



 A  A  D(x, y) σ (y)ψ(y) ds(y) = D(x, ·) , σ ψ ,

∂S

so, by the argument developed in [CoDo19c],

110

C. Constanda and D. Doty

 A T   A T  D(x (k) , ·) σ (j ) , ψ = D(x (k) , ·) ,σψ (j )

.

A T    ,ψ = σ D(x (k) , ·) (j ) 

=



=

D(x (k) , ·) D(x (k) , ·)

A (j )

A (j )

T  σT ,ψ σ

T

 ,ψ .

Replacing in (9.18) and using (9.12) and (9.15), we deduce that ψ, ϕ (i)  = ψ, ϕ (j k)   T = ( P (x (k) , ·) (j ) ) −

.











T





T

(( D(x (k) , ·) A )(j ) σ ) , ψ 

=

( P (x (k) , ·) (j ) )T − (( D(x (k) , ·) A σ )(j ) ) , ψ 

=

(( D(x, ·) A )(j ) ) , R  − (F c)j (x (k) ),





T

j = 1, 2, 3, k = 1, 2, . . . , i = j + 3k = 4, 5, . . . .

(9.19)

At the same time, since .T u = R + σ ψ is the Neumann boundary data of the solution, we have [CoDo22] T u, F (i)  = 0,

i = 1, 2, 3,

.

so σ ψ, F (i)  = −R, F (i) ,

i = 1, 2, 3,

.

which (see [CoDo19c]) yields ψ, σ F (i)  = σ ψ, F (i)  = −R, F (i) ,

.

i = 1, 2, 3.

(9.20)

In view of Theorem 2, there is a unique expansion ψ=

∞ 

.

ch ϕ (h) ,

h=1

so we can seek an approximation for .ψ of the form ψ

.

(n)

= (u|∂S )

(n)

=

n  h=1

ch ϕ (h) .

(9.21)

9 Robin Boundary Value Problem

111

For symmetry and ease of computation, in the construction of the approximation ψ (n) we use the subsequence with .n = 3N + 3, where N is the number of points (k) chosen on .∂S . This choice inserts in the computation, for each value of .k = .x ∗ 1, 2, . . . , N, all three columns/rows of .D A and P involved in the definition (9.10) and its alternatives (9.11) and (9.12) of the vector functions .ϕ (i) , .i = 4, 5, . . . , n = 3N + 3, with the re-indexing (9.15). The combinations of (9.21) and (9.20), and then of (9.21) and (9.19), yield the approximate equalities .

3N +3  .

ch σ F (i) , ϕ (h)  = −F (i) , R ,

i = 1, 2, 3, .

(9.22)

h=1 3N +3 

ch ϕ (i) , ϕ (h)  +

h=1

3 

(l)

cl fj (x (k) ) =





(( D(x (k) , ·) A )(j ) ) , R , T

l=1

j = 1, 2, 3, k = 1, . . . , N, i = j + 3k = 4, 5, . . . , 3N + 3. (9.23) Equations (9.22) and (9.23) form our system for determining the .3N + 3 coefficients .ci . Finally, we construct the function   A  u(n) (x) = P (x, ·) − D(x, ·) σ, ψ (n)

.

  A − D(x, ·) , R + (F c)(n) ,

x ∈ S−.

Theorem 3 The vector function .u(n) is an approximation of the solution u of problem (9.2) in the sense that .u(n) → u uniformly on any closed and bounded subdomain of .S − .

9.4 First Numerical Example Let .S + be the disk of radius 1 centered at origin, let the plate parameters (after rescaling and non-dimensionalization) be h = 0.5,

.

λ = μ = 1,

and let the positive definite matrix .σ defined on .∂S be

112

C. Constanda and D. Doty

⎛

⎞ 4 2 −2 .σ (θ ) = ⎝ 2 5 −1 ⎠ . −2 −1 5 We consider the boundary condition function (in polar coordinates on .∂S) ⎛

⎞

2 − 32 cos θ + 12 sin θ − 38 cos(2θ ) + 15 sin(2θ ) + 13 cos(3θ )

⎜ ⎟ ⎜ − 8 sin(3θ ) − 41 cos(4θ ) + 38 sin(4θ ) + 24 cos(5θ ) + 3 sin(5θ ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 + 4 cos θ + 39 sin θ − 27 cos(2θ ) + 8 sin(2θ ) + 12 cos(3θ ) ⎟ ⎜ ⎟. 2 R(x) = ⎜ ⎟ ⎜ + 28 sin(3θ ) − 34 cos(4θ ) + 27 sin(4θ ) + 24 cos(5θ ) + 18 sin(5θ ) ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −5 + 2 cos θ − 3 sin θ + 62 cos(2θ ) − 22 sin(2θ ) + 2 cos(3θ ) ⎟ ⎝ ⎠ + 21 sin(3θ ) + 28 cos(4θ ) − 24 sin(4θ ) − 15 cos(5θ ) Direct verification shows that the exact solution of problem (9.2) generated by this function is ⎛

(x12 + x22 )−1 [2x1 − 2x2 ]

⎞

⎟ ⎜ ⎟ ⎜ + (x12 + x22 )−2 [x12 + 2x1 x2 − x22 − 3x13 + 6x12 x2 + 9x1 x22 − 2x23 ] ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ + (x12 + x22 )−3 [−4x13 − 9x12 x2 + 12x1 x22 + 3x23 − 2x14 − 16x13 x2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3 2 2 4 ⎟ ⎜ + 12x1 x2 + 16x1 x2 − 2x2 ] ⎟ ⎜ ⎟ ⎜ 5 3 3 5 2 2 4 2 2 4 −4 ⎟ ⎜ + x ) [−6x + 15x x + 60x x − 30x x − 30x x + 3x ] + (x 2 1 1 1 2 2 ⎟ 1 2 1 1 2 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (x 2 + x 2 )−1 [−2x − 4x ] 1 2 ⎟ ⎜ 1 2 ⎟ ⎜ ⎟ ⎜ 3 3 2 2 2 2 2 2 −2 + (x1 + x2 ) [3x1 − 2x1 x2 − 3x2 − 2x1 − 9x1 x2 + 6x1 x2 + 3x2 ] ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3 3 3 2 2 2 2 4 −3 + (x1 + x2 ) [x1 − 24x1 x2 − 3x1 x2 + 8x2 + 4x1 − 8x1 x2 ⎟ ⎜ ⎟, u(x) = ⎜ ⎟ ⎜ ⎟ ⎜ − 24x12 x22 + 8x1 x23 + 4x24 ] ⎟ ⎜ ⎜ 2 + x 2 )−4 [−3x 5 − 30x 4 x + 30x 3 x 2 + 60x 2 x 3 − 15x x 4 − 6x 5 ] ⎟ ⎟ ⎜ + (x 2 1 1 2 1 1 2 2 1 1 2 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 + 1 ln(x 2 + x 2 ) ⎟ ⎜ 1 2 2 ⎟ ⎜ ⎟ ⎜ 2 2 2 2 −1 + (x1 + x2 ) [−x2 − 3x1 + 4x1 x2 + 3x2 ] ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 3 3 2 2 2 2 2 2 −2 + (x1 + x2 ) [−12x1 + 10x1 x2 + 12x2 − x1 − 6x1 x2 + 3x1 x2 + 2x2 ] ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ + (x12 + x22 )−3 [−6x13 − 36x12 x2 + 18x1 x22 + 12x23 − 2x14 + 4x13 x2 ⎟ ⎜ ⎟ ⎜ 2 x 2 − 4x x 3 − 2x 4 ] ⎟ ⎜ + 12x 1 2 1 2 2 ⎠ ⎝ +(x12 + x22 )−4 [−18x14 + 36x13 x2 + 108x12 x22 − 36x1 x23 − 18x24 ]

and that the class .A coefficients of this solution are

9 Robin Boundary Value Problem

113

m0 = −2, m1 = 1, m2 = −2, m3 = 2, m4 = −3, m5 = 1, m6 − 1,

.

3 1 m7 = − , m8 = −2, m9 = − , m10 = −4, m11 = 5, m12 = −12. 2 2 We take the auxiliary curve .∂S∗ to be the circle of radius 1/2 centered at the origin. This seems a reasonable choice, since having .∂S∗ too far from .∂S makes the set .G “less linearly independent”, whereas positioning it too close to .∂S increases the sensitivity of .G to the singularities of matrices D and P . This solution contains no rigid displacement other than the vertical translation supplied by the structure of class .A . It is obvious that

the accuracy of the approximation depends on the selection of the set of points . x (k) on .∂S∗ . For the sake of symmetry, we make a uniformly distributed choice; specifically, for .N = 1, 2, . . . ,

(k) N   . x = k=1 Cartesian



1 2π k , 2 N

N    

.

k=1 Polar

The numerical computation method used in this example is row reduction.

9.5 Graphical Illustrations I The graphs of the three components of .(uA )(93) computed from .ψ (93) (with .N = 30 points .x (k) on .∂S∗ ) for r ≥ 1.01, 0 ≤ θ < 2π,

.

together with the graph of .ψ (93) , are shown in Fig. 9.1. The influence of the singularities of .D(x, y) and .P (x, y) for .x ∈ S − very close to .y ∈ ∂S is mitigated by increasing the floating-point accuracy in the vicinity of

Fig. 9.1 Graphs of the components of .(uA )(93) and .ψ (93) for .r ≥ 1.01, . 0 ≤ θ < 2π

114

C. Constanda and D. Doty

Fig. 9.2 Graphs of the components of .(uA )(93) for .1.01 < r < 100, . 0 ≤ θ < 2π

Fig. 9.3 Graphs of the components of the error .(uA )(93) − uA

∂S but is never completely eliminated. The gap between the computed subdomain and .∂S is filled by appropriate interpolation. The graphs of the components of .(uA )(93) constructed from .ψ (93) for

.

1.01 < r ≤ 100, 0 ≤ θ < 2π,

.

which illustrate the class .A behavior of the solution away from the boundary, are displayed in Fig. 9.2. Figure 9.3 contains the graphs of the components of the error .(uA )(93) − uA . The approximation is 1–2 digits of accuracy near .∂S but improves significantly away from the boundary. The behavior of the relative error .

ψ (3N +3) − uA |∂S  uA |∂S 

as a function of N reflects the efficiency and accuracy of our computational procedure. The logarithmic plot of the size of the relative error in terms of N can be seen in Fig. 9.4. This plot strongly suggests that the relative error improves exponentially as N increases. Fitting a linear curve to the logarithmic data produces the model 8.4039 − 0.29722 N.

.

9 Robin Boundary Value Problem

115

Fig. 9.4 Logarithmic plot of the relative error as a function of N

The relative error may be modeled by

.

ψ (3N +3) − uA |∂S  = 2441.3 × 0.506304N . uA |∂S 

(9.24)

System (9.22), (9.23) also generates the numerical approximation F (93) = −0.014480 F (1) + 0.00020337 F (2) + 0.0063486 F (3)

.

of the rigid displacement .F = 0.

9.6 Second Numerical Example The first example, where the exact solution of the problem was known, validated the efficiency of our approximation method. We now solve the boundary value problem (9.2) with a data function .R for which the exact solution is not known. Specifically, for the same domain .S − but with ⎛ σ (θ ) =

.

7 + cos(2θ ) −3 − cos(2θ ) 0

3 ⎜ ⎜ −3 − cos(2θ ) 7 + cos(2θ ) 2 ⎝ 0 0

⎞

⎟ 0⎟ ⎠,

8

we choose ⎛

−6 + 70 cos θ − 50 sin θ − 51 cos(2θ ) − 55 sin(2θ )

⎜ ⎜ + 60 cos(3θ ) + 18 sin(3θ ) + 114 cos(4θ ) − 60 sin(4θ ) ⎜ ⎜ + 6 cos(5θ ) + 9 cos(6θ ) − 9 sin(6θ ) ⎜ ⎜ 1 ⎜ ⎜ 6 − 68 cos θ + 52 sin θ + 37 cos(2θ ) + 69 sin(2θ ) .R(x) = 2 ⎜ ⎜ ⎜ − 66 cos(3θ ) + 24 sin(3θ ) − 48 cos(4θ ) + 126 sin(4θ ) ⎜ ⎜ ⎜ − 6 cos(5θ ) − 9 cos(6θ ) + 9 sin(6θ ) ⎝ −48 − 24 sin θ + 24 cos(2θ ) + 168 sin(2θ ) + 396 cos(3θ )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

116

C. Constanda and D. Doty

The approximation is computed with the same auxiliary curve .∂S∗ and points x (k) , and the same parameters as in Sect. 9.5, by means of the row reduction method.

.

9.7 Graphical Illustrations II The graphs of the components of .u(93) , generated with 30 points .x (k) on .∂S∗ , for r ≥ 1.01, 0 ≤ θ < 2π,

.

are shown in Fig. 9.5. The graphs of the components of .u(93) for 1.01 ≤ r ≤ 100, 0 ≤ θ < 2π,

.

which indicate the class .A behavior of the solution away from the origin, are displayed in Fig. 9.6. Since .uA ∂S is not known in this case, we cannot use (9.24) to estimate the error. Instead, we design a roundabout procedure that uses .ψ (93) to construct the solution  (93) .u ˜ of an exterior Dirichlet problem, and then compute . (T − σ )u˜ by the method  (93) described in [CoDo20]. As can be seen in Fig. 9.7, . (T − σ )u˜ is close to .R, which confirms that our technique is efficient.

Fig. 9.5 Graphs of the components of .(uA )(93) and .ψ (93) for .r ≥ 1.01, . 0 ≤ θ < 2π

Fig. 9.6 Graphs of the components of .(uA )(93) for .1.01 < r < 100, . 0 ≤ θ < 2π

9 Robin Boundary Value Problem

117

Fig. 9.7 Components of  (93) − R in polar (T − σ )u˜ coordinates

.

A far smaller relative error is obtained if we take 200 points .x (k) on .∂S∗ ; then,    (T − σ )u˜ (603) − R  = 5.8739 × 10−25 . . R The computational algorithm also produces the rigid displacement approximation F (93) = −2.4603 × 10−10 F (1) − 8.3284 × 10−11 F (2) − 2.0779 × 10−11 F (3) ,

.

which suggests that .F = 0.

References [Co16] Constanda, C.: Mathematical Methods for Elastic Plates. Springer, London (2016) [CoDo17a] Constanda, C., Doty, D.: Bending of elastic plates: generalized Fourier series method. In: Integral Methods in Science and Engineering: Theoretical Techniques. Birkhäuser, New York, pp. 71–81 (2017) [CoDo17b] Constanda, C., Doty, D.: The Neumann problem for bending of elastic plates. In Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 2017, vol. II. Cádiz, Spain, pp. 619– 622 (2017) [CoDo18] Constanda, C., Doty, D.: Bending of elastic plates with transverse shear deformation: the Neumann problem. Math. Methods Appl. Sci. 41, 7130–7140 (2018). https://doi. org/10.1002/mma.4704 [CoDo19a] Constanda, C., Doty, D.: The Robin problem for bending of elastic plates. Math. Methods Appl. Sci. 42, 5639–5648 (2019). https://doi.org/10.1002/mma.5286 [CoDo19b] Constanda, C., Doty, D.: Bending of plates with transverse shear deformation: The Robin problem. Comput. Math. Methods 1, e1015 (2019). https://doi.org/10.1002/ cmm4.1015 [CoDo19c] Constanda, C., Doty, D.: Bending of elastic plates: generalized Fourier series method for the Robin problem. In: Integral Methods in Science and Engineering: Analytic Treatment and Numerical Approximations, pp. 97–110. Birkhäuser, New York (2019) [CoDo20] Constanda, C., Doty, D.: Analytic and numerical solutions in the theory of elastic plates. Complex Var. Elliptic Equ. 65, 40–56 (2020). https://doi.org/10.1080/ 17476933.2019.1636789 [CoDo22] Constanda, C., Doty, D.: The traction boundary value problem for thin elastic structures. In: Integral Methods in Science and Engineering: Applications in Theoretical and Practical Research, pp. 49–62. Birkhäuser, New York (2022)

Chapter 10

A Mathematical Model of Cell Clustering A. Farmer and P. J. Harris

10.1 Introduction In this paper, we have developed a mathematical model of how biological cells migrate through the medium in which they are immersed and combine to form clusters of cells. In many cases, the cells migrate in response to chemical signals by moving in the direction in which the concentration of the chemical signal is increasing, a process called chemotaxis [LaHo96]. Often (one or more) cells will secrete the chemical in order to attract other nearby cells and collide with the other cells to form clusters. The cells under consideration here typically consist of an outer membrane, which encloses the cytoplasm that makes up the majority of the interior of the cell. Each cell will have a nucleus that is located within the cytoplasm and is usually a lot denser than the cytoplasm [HeEtAl19]. Previous models of cell motion have included the particle-based models, where the cells are treated as rigid particles with a simple fixed geometry, such as circles in two-dimensional models or spheres in three-dimensional models (see, e.g., [ChEtAl18b, Ha17, Ve15]). The rigid body method has been extended to cells of different shapes moving through a viscous fluid by Harris [Ha18]. The main disadvantage of these methods is that they do not simulate the changes to the shapes of the cells that occur as they move and collide. Indeed, the method presented in Harris [Ha18] terminates when two cells collide, as the method cannot simulate what happens to the cells after they have collided. The alternative is to use a cell-based model (which are called agent-based models in some work), where the cells is treated as a deformable structure, which moves by

A. Farmer · P. J. Harris () The University of Brighton, Brighton, UK e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_10

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changing shape [ChEtAl18a, MoDo15]. As receptors in the cells, membranes that are on the side closest to the source of the chemical signal react to and deform the cell membrane in the direction in which the concentration of the chemical signal is increasing. One approach is to represent the deformations of the different parts of the cell as a system of springs, as described in Chen [ChEtAl18a]. These models have been used to simulate cells in isolation or for cells moving through domains with what are effectively rigid obstructions (see, e.g., [SaWe20]). However, the methods used in Chen [ChEtAl18a] have not been used to simulate what happens when two (or more) cells collide and stick together to form clusters. As the results in this paper show, these spring-based models can be used to effectively model the interactions of cells, as they collide and show the shapes that individual cells take as they stick together to form clusters.

10.2 Mathematical Model In the mathematical model presented here, we will assume that when each cell is in its equilibrium state, it forms a circle with a circular nucleus located at the center of the cell. The deformations of the cytoplasm and nucleus are represented by a system of springs connected together at .n nodal points on both the membrane and nucleus boundary, and the cells, center of mass as shown in Fig. 10.1. Let .mi be the position of the .ith nodal point on the cell membrane, .ni be the position of the .i th nodal point on the nucleus, and .c be the position of the center of mass. Let .xi be the original position vector of .ni relative to .mi (i.e., .xi = ni − mi when .t = 0), and similarly, let .xˆ i be the original position vector of .ni relative to the center of mass .c. In the model presented here, the nodes are consecutively numbered around the membrane and the boundary of the nucleus in an anticlockwise direction starting at the positive x-axis. It is noted that both the cell membrane and the boundary of its nucleus are closed curves, and therefore, the first and last nodes on both curves will Fig. 10.1 An illustration of the nodal points on the cell boundary membrane and surface of the nucleus. The cytoskeleton is represented as a collection of springs

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be adjacent to each other. This means that if there are .n nodes on the cell membrane, then, when deriving the equations for the first and last nodes on the membrane, we have .m−1 = mn and .mn+1 = m1 . Similarly, if there are .n nodes on the boundary of the nucleus, then .n−1 = nn and .nn+1 = n1 . This needs to be taken into account when substituting .i = 1 and .i = n into the general equations derived in the sections below but will not be explicitly mentioned again. First, we shall consider the forces acting on a node on the membrane of the cell. Each node on the membrane is connected by a spring, with stiffness .α to a corresponding node of the boundary of the cell’s nucleus, which gives a contribution in the form α(ni (t) − mi (t) − xi ).

.

In addition, the node is connected to the two adjacent nodes on the membrane by springs with stiffness .δ, which gives the contribution δ [yi − (mi−1 − mi ) − yi+1 + (mi − mi+1 )]

.

where .yi is the initial location of .mi relative to .mi−1 . The final contribution to the force acting on a node on the membrane is that due to the gradient in the concentration of the chemical signal, which can be expressed in the form β∇c(t, mi (t))

.

where .β a parameter, which controls how sensitive the cells are to the chemical signal. The exact form of the chemical signal .c will be discussed below. Summing the contributions gives the total force acting on the .ith node on the membrane as

.

dmi (t) = β∇c(t, mi (t)) dt +α(ni (t) − mi (t) − xi ) dt +δ [yi − (mi−1 − mi ) − yi+1 + (mi − mi+1 )] dt.

(10.1)

We now consider the forces acting on each node of the boundary of the nucleus. Each node on the boundary of the nucleus is connected to the corresponding node on the membrane, the node at the center of the cell, and the two nodes on the boundary on the nucleus that are adjacent current node. As .ni is the location of the .ith node on the boundary on the nucleus and .c is the location of the center of the nucleus, then following a similar analysis to that carried out above for nodes on the membrane gives

.

dni (t) = αn (c(t) − ni (t) − xˆ i )dt −α(ni (t) − mi (t) − xi )dt   +δn yˆ i − (ni−1 − ni ) − yˆ i+1 + (ni − ni+1 ) dt.

(10.2)

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where .αn and .δn are the internal nucleus relaxation coefficient and nucleus boundary relaxation coefficient, respectively; .xˆ i is the position vector of .ni relative to .c when ˆ i is the position vector of .ni relative to .ni−1 when .t = 0. .t = 0; and .y Finally, to be able to accurately model the motion of the cell, we need to account for the movement of the center of mass of the cell. If each node on the membrane is connected to the adjacent nodes by straight lines and each node on the boundary of the nucleus is connected to the adjacent nodes by straight lines, then the membrane and boundary of the nucleus form a pair of concentric polygons. We note that provided a sufficiently large number of nodes is used on both the membrane and nucleus, then the discrepancies between the polygons and the exact shapes of the membrane and nucleus are negligible. If .ρ1 is the density of the cytoplasm and .ρ2 is the density of the nucleus, then it can be shown that the coordinates of the center of mass of the cell .(x, ˆ y) ˆ are given by xˆ =

.

Mx , A

yˆ =

My A

(10.3)

where n ρ1  [(xi − xi+1 )[(yi + 2yi+1 )xi+1 + (2yi + yi+1 )xi ]] 6 i=1 n ρ2 − ρ1  + [(xˆi − xˆi+1 )[(yˆi + 2yˆi+1 )xˆi+1 + (2yˆi + yˆi+1 )xˆi ]] 6 i=1 n ρ1  2 [(xi − xi+1 )(yi2 + yi yi+1 + yi+1 )] . My = 6 i=1 n ρ2 − ρ1  2 + [(xˆi − xˆi+1 )(yˆi2 + yˆi yˆi+1 + yˆi+1 )] 6 i=1 n n ρ1  ρ2 − ρ1  A = (yi + yi+1 )(xi − xi+1 ) + (yˆi + yˆi+1 )(xˆi − xˆi+1 ). 2 2

Mx =

i=1

i=1

and recall that .n is the number nodes on both the membrane and the boundary of the nucleus. Assume that spreading of the concentration .c(t, x) of the signaling chemical can be modeled using the linear diffusion equation .

  2 ∂ c ∂c ∂ 2c =μ + ∂t ∂x 2 ∂y 2

(10.4)

where .μ > 0 is the diffusion constant. It is straightforward to show that the solution to the differential equation (10.4) is c(t, xi (t)) =

.

  A (x(t) − xs )2 + (y(t) − ys )2 exp − μ(t + t ) 4μ(t + t )

(10.5)

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where .A is the source density and .t = 1 × 10−6 is a small parameter to avoid computational problems when .t = 0. Here .(xs , ys ) is the location of the source point for the chemical signal, and since it is the cells themselves that are emitting the chemical signal, then the source point will be the center of mass of the emitting cell. For computational convenience in this work, it will be the first cell which is emitting the signal. The gradient of the concentrations, as required in (10.1), can be found by simply differentiating (10.5) to give   A(x − xs ) ∂c (x(t) − xs )2 + (y(t) − ys )2 =− 2 exp − 4μ(t + t ) ∂x 2μ (t + t )2 .

  ∂c (x(t) − xs )2 + (y(t) − ys )2 A(y − ys ) exp − =− 2 4μ(t + t ) ∂y 2μ (t + t )2

The equations above model the motion of cells toward the cell, which is emitting the chemical signal. If the chemical signal lasts long and/or is strong enough, then the other cells will eventually collide with the emitting cell, and possibly each other as well. In the model presented here, when two cells collide, the nodes on the membrane of one cell become attached to the nodes on the membrane of the second cell. To model cell collisions, we first need to detect when a node on the membrane of cell A has moved inside the membrane of cell B (or vice versa). To detect a collision, let .(xA , yA ) be a node on the membrane of cell A, let .(x, ˆ y) ˆ be the center of mass of cell .B, and then consider a pair of adjacent membrane nodes .mj and .mj +1 , which has coordinates .(xj , yj ) and .(xj +1 , yj +1 ), respectively. Form the equations xA = (1 − u − v)xˆ + uxj + vxj +1 .

yA = (1 − u − v)yˆ + uyj + vyj +1 which can be solved for .u and .v. If the three conditions .u ≥ 0, .v ≥ 0, and .u + v ≤ 1 are all true for one pair of adjacent membrane nodes on cell B, then .(xA , yA ) is inside cell B, and so cells A and B must have collided. This is repeated for each node on the membrane of cell A. If a collision is detected for a particular pair of adjacent nodes on the membrane of cell B, then the nodal point on cell A will combine with to the closest of either .(xj , yj ) or .(xj +1 , yj +1 ). If n is small, then this change in location of .(xA , yA ) may be noticeable, as the distance between two nodal points is visible, but for larger the values of n, this becomes more negligible. Suppose that the .ith node from cell A needs to be combined with the .j th node from cell B. The single combined equation for both nodes is (where subscripts i and j refer to nodes on the membrane of the appropriate cell and the superscripts denote

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which cell the nodes are on) A A A dmi (t) = β∇c(t, mA i (t)) dt + α(ni (t) − mi (t) − xi ) dt A A A A A −δ(mi (t) − mi+1 (t) − yi ) dt + δ(mi−1 (t) − mA i (t) − yi−1 ) dt . B B B B B +α(nB j (t) − mj (t) − xj ) dt − δ(mj (t) − mj +1 (t) − yj ) dt B B B +δ(mj −1 (t) − mj (t) − yj −1 ) dt (10.6)

When two cells collide, they form a common boundary or edge between them, which is no longer in direct contact with the surrounding fluid. This means that the rate at which the cell reacts to the chemical signal or its gradient will be reduced. This is incorporated into the model presented here by reducing the value of .β at all the nodes on the cells membrane by a factor of .k nc,i , where .k ∈ (0, 1] is a constant, which we will refer to as the reaction damping factor, and .nc,i is the number of nodes on the membrane of the .ith cell that are attached to the nodes of a different cell. The differential equations (10.1) (or (10.6) if two or more cells have collided), for the nodes on the membranes of the cells, and (10.2), for nodes on the boundaries of the cell nuclei, form a system of ordinary differential equations, which we can solve for the locations of the nodes on the cell membranes and nuclei. However, the equations are too complex to solve analytically and so have to be solved numerically. In this paper, the following algorithm, where .h denotes the time-step, has been used. 1. If this is the first time-step, then update the locations of all the nodes on the membrane and boundary of the nucleus of each cell in turn using the Euler’s scheme [At89] .

mi (h) = mi (0) + h dmi (0) ni (h) = ni (0) + h dni (0)

and then go to step 3. 2. Update the locations of all the nodes on the membrane and boundary of the nucleus of each cell in turn using the multi-step scheme [At89] 4 1 2h mi (t) − mi (t − h) + dmi (t) 3 3 3 . 4 1 2h ni (t + h) = ni (t) − ni (t − h) + dni (t) 3 3 3 mi (t + h) =

3. For each cell in turn, use the values of .{mi (t + h)} and .{ni (t + h)} to compute the new location of the center of mass .c using (10.3). 4. Check to see if there have been any new collisions and that nodes on different cells become connected. If so, modify the equations to use (10.6) for the newly connected node rather than (10.1) for the original individual nodes.

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5. Repeat from step 2 until the required end time for the simulation has been reached.

10.3 Numerical Results For all of results presented in this paper, we use the parameters given in Table 10.1, unless stated otherwise. In all the figures below, the cell emitting the signal is colored green, the other cells are colored red, and the nuclei for all cells are colored blue. Figure 10.2 shows the cells placed equidistant at a distance of 6.0 cell radii away from the center of the source cell, and in a symmetrical pattern, along the x and y axes. In this example, we did not introduce a damping factor and so .k = 1. As expected, in this example, the motion of the four outer cells has the same symmetry as the original cells, and that the central cell does not move. The results presented in Fig. 10.2 show that the motion of the four cells do have the expected symmetry, and this gives us some confidence that our numerical method is performing as expected. In addition, the motion of the central cell is smaller than the expected error in the numerical method used and so can be considered to be zero to with the precision of the numerical methods being used. The methods described above can be used to simulate the motion of a large number of cells. For example, Fig. 10.3 show the results of computing the motion and interactions of 100 cells. In this example, a reaction damping factor of .k = 0.94 was used. Here, more than one cell is able to secrete the chemical signal (as indicated by the number of cells colored green). As expected, this shows that in most cases, Table 10.1 The values of the parameters used in the calculations

Number of nodal points (n) Radius of membrane Radius of nucleus Internal cell membranes deformation relaxation (.α) External cell membranes deformation relaxation (.δ) Internal nucleus’ deformation relaxation (.αn ) Cell’s response to external signals (.β) Density of cytoplasm (.ρ1 ) Density of nucleus (.ρ2 ) Diffusion constant (.μ) Source intensity (a) Reaction damping factor (k) Number of time-steps Maximum time

500 1.0 0.25 0.5 30.0 100.0 2.5 1.2 1.4 2.0 15.0 Variable 10,000 20 s

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(a)

(b)

(c)

(d)

(e)

(f)

Fig. 10.2 The motion of four cells placed equidistant and symmetrically about a central cell, which is emitting the signal. (a) .t = 0. (b) .t = 1. (c) .t = 4.5. (d) .t = 7.5. (e) .t = 10. (f) .t = 20

the cells are attracted toward the nearest cell that is emitting the chemical signal. However, there are some cells, which become elongated as they are attracted toward two (or more) emitting cells with approximately the same force. In such cases, the cell does not move very much from its original location as the forces from the emitting cells essentially cancel each other.

10.4 Discussion and Conclusions This paper has presented a new mathematical model of how individual cells can come together and join to form clusters. The results given in Sect. 10.3 show that the model is behaving as expected, as demonstrated in Fig. 10.2, which shows that when cells are placed equidistant from and symmetrically about the source cell, the symmetry in their locations is maintained. The results in Fig. 10.3 show that the methods here can be used to simulate what happens with a realistic number of cells. In theory, there is no limit on the number of cells that can be used with this model, but practical considerations, such as available computer resources, are usually the limiting factor on the size of problem which can be considered. The results in Fig. 10.3 show that it is possible to simulate what happens when more than one cell secretes the chemical signal. These results also show how cells can be stretched if they are attracted to more than one source. A future development is to allow different cells to start secreting the chemical signal at different times.

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(a)

(b)

(c)

(d)

(d)

(e)

Fig. 10.3 The motion of four groups of 25 randomly placed cells (a) .t = 0. (b) .t = 1. (c) .t = 4.5. (d) .t = 7.5. (e) .t = 10. (f) .t = 20

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In experiments, cells have been observed to leave small particles of membrane and cytoplasm behind as the rear of the cells detach from the surface over which they are moving. In addition, cells can also absorb any such particles that they encounter as they move. The formation and absorption of these particles as the cells move will be incorporated in a future model. Acknowledgments The authors would like to thank Matteo Santin from the Brighton Centre for Regenerative Medicine and Devices for his help and advice with the biological aspects of this work. They would also like to thank Manolia Andredaki for her help and advice with this research.

References [At89] Atkinson, K.E.: An Introduction to Numerical Analysis (2nd ed.) John Wiley and Sons, New York (1989) [ChEtAl18a] Chen, J., Weihs, D., Van Dijk, M., Vermolen, F.J.: A phenomenological model for cell and nucleus deformation during cancer metastasis. Biomech. Model. Mechanobiol. 17(5), 1429–1450 (2018) [ChEtAl18b] Chen, J., Weihs, D., Vermolen, F.J.: A model for cell migration in non-isotropic fibrin networks with an application to pancreatic tumor islets. Biomech. Model. Mechanobiol. 17, 367–386 (2018) [Ha17] Harris, P.J.: A simple mathematical model of cell clustering by chemotaxis. Math. Biosci. 294(May), 62–70 (2017) [Ha18] Harris, P.J.: Modelling the motion of clusters of cells in a viscous fluid using the boundary integral method. Math. Biosci. 306, 145–151 (2018) [HeEtAl19] Hervas-Raluy, S., Garcia-Aznar, J.M., Gomez-Benito, M.J: Modelling actin polymerization: the effect on confined cell migration. Biomech. Model. Mechanobiol. 18, 1177–1187 (2019) [LaHo96] Lauffenburger, D.A., Horwitz, A.F.: Cell migration: a physically integrated molecular process. Cell 84, 359–369 (1996) [MoDo15] Mousavi, S.J., Doweidar, M.H.: Three-dimensional numerical model of cell morphology during migration in multi-signaling substrates. PLoS ONE 10, 1–33 (2015) [SaWe20] Saeed, M., Weihs, D.: Finite element analysis reveals an important role for cell morphology in response to mechanical compression. Biomech. Model. Mechanobiol. 19, 1155–1164 (2020) [Ve15] Vermolen, F.J.: Particle methods to solve modelling problems in wound healing and tumor growth. Comput. Part. Mech. 2, 381–399 (2015)

Chapter 11

A Revisit to a Double-Periodic Perforated Neumann Waveguide: Opening Spectral Gaps D. Gómez, S. A. Nazarov, R. Orive-Illera, and M.-E. Pérez-Martínez

11.1 Introduction In this paper, we revisit the problem of opening gaps in the essential spectrum of a Neumann waveguide considered first in [GoEtAl23a]. We provide a new perspective for proofs, which are possible, thanks to the novel results in [GoEtAl23b]. In particular, we examine the bandgap structure of the spectrum of the Neumann problem for the Laplace operator in a strip with periodic dense transversal perforation by identical holes of a small diameter .ε > 0. The periodicity cell itself contains a string of holes at a distance .O(ε) between them. Under certain assumptions on the symmetry of the holes and of the width of the strip, we provide asymptotic formulas for the endpoints of the spectral bands in the low frequency range of the spectrum as .ε → 0. We prove that, for .ε small enough, some spectral gaps are open. The position and size of the opened gaps depend on the strip width, the perforation period, and certain integral characteristics of the holes. To derive the asymptotic formulas for the bandgap structure, we use a result based on uniform bounds for discrepancies between the dispersion curves .{Λεp (η) : η ∈ [−π, π ]} and .{Λ0p (η) : η ∈ [−π, π ]} for different values of p (cf. [GoEtAl23b]), along with the thorough analysis of these dispersion curves near the band edges

D. Gómez · M.-E. Pérez-Martínez () Universidad de Cantabria, Santander, Spain e-mail: [email protected]; [email protected] S. A. Nazarov Institute for Problems in Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg, Russia e-mail: [email protected] R. Orive-Illera Universidad Autónoma de Madrid, Madrid, Spain e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_11

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performed in [GoEtAl23a]. This analysis near the band edges involves both new “fast Floquet variables” and boundary layers in the vicinity of the perforation string of holes. The global uniform bounds for discrepancies for the low frequencies allow us to skip the technique used in [GoEtAl23a] based on the control of the total number of perturbed eigenvalues inside certain boxes of the band .[−π, π ] × [0, ∞) in the coordinate axis .(η, Λ). In Sect. 11.2, we introduce the spectral problem under consideration and provide some background, which relates this problem with the so-called model problem in the unit cell, namely, a parametric family of homogenization problems, with the Floquet parameter arising on the boundary conditions (see Sect. 11.2.1) and .ε the perturbation parameter. It allows the bandgap structure of the spectrum of the original perturbation problem to be described. In Sect. 11.2.2, we also introduce the parametric family of homogenized problems. The aim of the paper being to show the opening of gaps in the bandgap structure, we conclude Sect. 11.2.3 with the description of the paper structure: in short, Sect. 11.3 contains some preliminary results, and Sect. 11.4 contains the main results of the paper and their proofs. Finally, it should be emphasized that the method here developed can be applied to many problems arising in waveguide theory (cf. [GoEtAl21, GoEtAl23a, GoEtAl23b] for a large list of references).

11.2 The Setting of the Problem Let Π = {x = (x1 , x2 ) ∈ R2 : x1 ∈ R, x2 ∈ (0, H )}

.

(11.1)

be an open strip of width .H > 0, and let .ω be a domain in the plane .R2 , which is bounded by a smooth simple closed curve .∂ω and has the compact closure .ω = ω ∪∂ω inside .Π . Let .ε = N −1 , where N is a large natural number. We introduce the strip .Π ε (see Fig. 11.1(a)), obtained from .Π via perforation by the family of holes ωε (j, k) = {x : ε−1 (x1 − j, x2 − εkH ) ∈ ω},

.

j ∈ Z, k = 0, 1, . . . , N − 1, (11.2)

distributed periodically along line segments parallel to the ordinate .x2 -axis. Each hole is homothetic to .ω of ratio .ε and translation of .εω = ωε (0, 0). Namely,

Fig. 11.1 (a) The perforated strip .Π ε . (b) The periodicity cell . ε

11 Spectral Gaps in a Double-Periodic Perforated Neumann Waveguide

Πε = Π \ Ωε

.

where

Ωε =

−1  N

131

ωε (j, k).

j ∈Z k=0

The period of perforation along the abscissa .x1 -axis in the domain .Π ε is made equal to one by rescaling, which also fixes the dimensionless width .H > 0. The period along the .x2 -axis is .εH with .ε  1. We consider the spectral Neumann problem .

−Δx uε (x) = λε uε (x), x ∈ Π ε , .

(11.3)

∂ν u (x) = 0, x ∈ ∂Π ,

(11.4)

ε

ε

where .∂ν is the directional derivative along the outward normal while .∂ν = ±∂/∂x2 at the lateral sides .Υ± = {x : x1 ∈ R, x2 = (H ± H )/2} of the strip (11.1).

11.2.1 The Model Problem in the Periodicity Cell The Floquet–Bloch–Gelfand transform (FBG transform in short) (see [Ge50, ReSi78])

.

1  −ipη ε uε (x) → U ε (x; η) = √ e u (x1 + p, x2 ) 2π p∈Z

(11.5)

converts problem (11.3)–(11.4) in the infinite waveguide .Π ε into problem .

−Δx U ε (x; η) = Λε (η)U ε (x; η), x ∈ ε , . 1  1   U ε , x2 ; η = eiη U ε − , x2 ; η , x2 ∈ (0, H ), . 2 2     ε ε 1 ∂U ∂U 1 , x2 ; η = eiη − , x2 ; η , x2 ∈ (0, H ), . ∂x1 2 ∂x1 2 ∂ν U ε (x; η) = 0,

x ∈ {x ∈ ∂Π ε : |x1 | < 1/2},

(11.6) (11.7) (11.8) (11.9)

where . ε (see Fig. 11.1(b)) is defined by .

ε = {x ∈ Π ε : |x1 | < 1/2},

(11.10)

and (11.7)–(11.8) are the so-called quasiperiodicity conditions. .η ∈ [−π, π ] is the Floquet parameter, while .Λε (η) and .U ε (·; η), respectively, are the new notations for the eigenvalues and eigenfunctions in the model problem. Notice that .x ∈ Π ε on the left of (11.5) but .x ∈ ε on the right. The basic properties of the FBG transform can be found in the above-cited publications; see also [GoEtAl23a, GoEtAl23b].

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The problem (11.6)–(11.9) has a standard variational formulation in 1,η the Sobolev space .Hper ( ε ) of .H 1 functions satisfying (11.7)–(11.8) (cf. [GoEtAl23a, GoEtAl23b]), with a discrete spectrum constituting the unbounded monotone sequence of eigenvalues .

0 ≤ Λε1 (η) ≤ Λε2 (η) ≤ · · · ≤ Λεp (η) ≤ · · · → +∞,

as p → +∞,

(11.11)

where their multiplicities are taken into account. Furthermore, the functions [−π, π ] η → Λεp (η),

.

p ∈ N,

(11.12)

are continuous and .2π -periodic (see [Ge50, ReSi78]). Hence, the spectral bands βpε = {Λεp (η) : η ∈ [−π, π ]} ⊂ R+

.

are closed, connected, and finite segments. It is well-known in the framework of the Floquet–Bloch–Gelfand theory that the spectrum of the operator associated to the boundary-value problem (11.3)–(11.4) is given by .

σε =

 p∈N

βpε .

(11.13)

ε Also, the spectral bands .βpε and .βp+1 may intersect each other but can also be disjointed, so that the spectral gap

.

γpε =



max Λεp (η) ,

η∈[−π,π ]

min

η∈[−π,π ]

 Λεp+1 (η) ,

p ∈ N,

(11.14)

becomes open between them. Recall that an open spectral gap is recognized as a nontrivial open interval in .R+ , which is free of the essential spectrum but has both ε endpoints in it. If .βpε ∩ βp+1

= ∅, then the gap .γpε is closed. See, for instance, Fig. 11.3, where the open spectral gaps correspond with the projections of the shaded bands on the .Λ-axis, to show these openings in (11.13) constitute the aim of the paper.

11.2.2 The Limit Problem and the Limit Dispersion Curves For each .η ∈ [−π, π ], the boundary-value problem (11.6)–(11.9) constitutes a homogenization problem. Using standard techniques in homogenization, in

11 Spectral Gaps in a Double-Periodic Perforated Neumann Waveguide

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[GoEtAl23a], we have obtained its homogenized problem: .

−Δx U 0 (x; η) = Λ0 (η)U 0 (x; η), x ∈ 0 , . ∂U 0

(11.15) 

∂U 0

1 1 − , ,. 2 2

(x1 , 0; η) = (x1 , H ; η) = 0, x1 ∈ ∂x2 ∂x2 1  1   U 0 , x2 ; η = eiη U 0 − , x2 ; η , x2 ∈ (0, H ), . 2 2     0 0 ∂U 1 ∂U 1 , x2 ; η = eiη − , x2 ; η , x2 ∈ (0, H ). ∂x1 2 ∂x1 2

(11.16) (11.17) (11.18)

In (11.15), . 0 is the rectangle 0 = {x : |x1 | < 1/2, x2 ∈ (0, H )}

.

obtained from the periodicity cell (11.10) by filling all voids, namely, the perforation string .ωε (0, 0), . . . , ωε (0, N − 1) ⊂ 0 (cf. (11.2)). Problem (11.15)–(11.18) has the following explicit eigenvalues and eigenfunctions k2 , H2 .  x  2 , Uj0k (x; η) = ei(η+2πj )x1 cos π k H Λ0j k (η) = (η + 2πj )2 + π 2

j ∈ Z, k ∈ N0 = N ∪ {0}.

(11.19)

The renumeration in (11.19) is needed to compose the monotone sequence of eigenvalues .

0 ≤ Λ01 (η) ≤ Λ02 (η) ≤ · · · ≤ Λ0p (η) ≤ · · · → +∞,

as p → +∞,

(11.20)

which repeat according to their multiplicities. In [GoEtAl23a], the convergence result between entries of the sequence (11.11) and those of the sequence (11.20) has been proved and, for .p ∈ N, the following convergence holds Λεpk (ηk ) → Λ0p ( η),

.

as k → +∞,

(11.21)

where .{(εk , ηk )}∞ η ∈ [−π, π ] as k=1 is any sequence such that .εk → 0 and .ηk →  k → +∞ (see Sections 2–3 in [GoEtAl23a] for the proof). Note that it involves a strong stability of both problems on the parameter .η. Note that formulas (11.19) are of great interest for drawing the limit dispersion curves for different values of H . Recall that these curves are the graphs of .Λ0p (η), for .η ∈ [−π, π ]. Figure 11.2(a) shows the great variety of possibilities of behaviors of such curves depending on the value of H . Also, along with Fig. 11.3, it gives an

.

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Fig. 11.2 (a) The first limit dispersion curves for different values of H . The first one is .Λ = η2 . The parabolic curve translations of this one along the .Λ-axis are .Λ = η2 + (π/H )2 depending on H . The other pieces of parabola are the translations along the .η-axis, namely, .Λ = (η ± 2πj )2 , while .j = 1, 2. (b) The first limit dispersion curves for a certain value of .H < 1/3

Fig. 11.3 The dispersion curves of the limit problem √ and the perturbed problem with the mirror symmetry of the hole .ω in the case where .H ∈ (1/ 8, 1/2). Spectral gaps are the projections of the shaded rectangles on the ordinate .Λ-axis

idea of what we can expect for the behavior of the perturbed dispersion curves; see, for instance, the second graphic in Fig. 11.3. It is worth mentioning that the explicit formulas (11.19) allow us to obtain the precise multiplicity of each eigenvalue .Λ0p (η) for each p and for each .η ∈ [−π, π ]. This proves to be essential for Theorem 1, which provides the precise number of eigenvalues of the perturbed problem at a distance .O(ε) of the eigenvalues of the homogenized problem (cf. Fig. 11.5). It becomes particularly complicated near the points .η0,p , in which the limit eigenvalues .Λ0p (η) change the multiplicity from 1 to 2 or 3. Note the different behavior of the limit dispersion curves depending on H ; see Figs. 11.3 and 11.2 to realize the difference for the following cases:

11 Spectral Gaps in a Double-Periodic Perforated Neumann Waveguide

 1  a) H ∈ 0, √ , 8

.

 1 1 b) H ∈ √ , , 8 2

c) H ∈

1 2

 ,1 ,

135

d) H ∈ [1, +∞).

As a matter of fact, limit eigenvalues of multiplicity 3 appear first for larger values of H , the first one being .Λ = 4π 2 when .H = 1/2. This is the main reason why our final results, which rely on the obtained uniform bounds for discrepancies in different intervals, depend strongly on H , and we focus our attention on the first eigenvalues.

11.2.3 The Aim and the Structure of the Paper As in [GoEtAl23a], the main goal of our paper is to show that, under certain restrictions on the width H and the perforation shape, the problem (11.3)–(11.4) can have at least one open gap in its spectrum. Also, we aim to derive asymptotic formulas for the position and geometric characteristics of several bands and gaps in the low frequency range of the spectrum. It should be mentioned that the traditional homogenization procedure in the problem (11.3)–(11.4) is not enough to detect open gaps. The crucial role is played by the uniform estimates for discrepancies of the dispersion curves and the boundary layer phenomenon. The width of the gaps is expressed in terms of certain integral characteristics of the Neumann hole .ω of unit size in the strip .Π with the periodicity conditions at its lateral sides (cf. (11.36) and (11.39)). For a technical reason (cf. (11.28)) and for simplification of asymptotic structures, we make the assumption ω = {ξ = (ξ1 , ξ2 ) ∈ R2 : (ξ1 , H − ξ2 ) ∈ ω}.

.

(11.22)

which means that the holes possess the mirror symmetry (see Fig. 11.4). Also, for simplicity, we assume that the boundary of .ω is of class .C ∞ . The structure of the rest of the paper is as follows: Sect. 11.3 contains the necessary previous results obtained in [GoEtAl23a] and [GoEtAl23b] to derive the asymptotic formulas for the end points of the bands .βpε while .p = 1, 2, 3. The problem being multiscale in both spatial and Floquet variables, we gather these preliminary results into three sections related to global approximation results over the interval .[−π, π ], the boundary layer problem in the periodicity unbounded

Fig. 11.4 The strip .Ξ with two different possible geometries for the hole .ω

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unit cell corresponding to the homogenization model problem, and the fast Floquet variable enlarging neighborhoods of the nodes of trusses of the dispersion curves. However, we emphasize that the proof of these results in [GoEtAl23a] and [GoEtAl23b] are not independent of each other. Section 11.4 contains the main results, the abovementioned asymptotic formulas, and the opening of the gaps, depending on H and the characteristic of the holes.

11.3 Preliminary Results For convenience for proofs in the next section, here we gather some important results obtained in [GoEtAl23a] and [GoEtAl23b].

11.3.1 The Uniform Approach for the Spectrum of the Model Problem The following theorem establishes that in certain .ε-neighborhoods of the first limit dispersion curves, there are dispersion curves of the perturbation problem; in addition, the number of these curves is preserved by perturbation. They become essential to open gaps in Sect. 11.4. Theorem 1 (i) There exist .ε0 > 0, C0 > 0 independent of .ε and .η ∈ [−π, π ] such that for .m = 1√and .H > 0, for .m = 2 and .H ∈ (0, 1/2), and for .m = 3 and .H ∈ (0, 1/ 8), we have |Λεm (η) − Λ0m (η)| < C0 ε,

.

for ε ≤ ε0 , η ∈ [−π, π ].

(11.23)

Above .Λ0m (η) means the m-th eigenvalue of (11.15)–(11.18) in the sequence (11.20), while .Λεm (η) mean the m-th eigenvalue of (11.6)–(11.9) in the sequence (11.11). (ii) For .H ∈ [1/2, 1), and .η ∈ [−π, a] or .η ∈ [b, π ], with .−π < a < 0 < b < π and such that the above intervals do not contain abscises of the intersection points of the limit dispersion curves (cf. (11.19)) with one of the corresponding eigenfunctions depending on .x2 , then (11.23) holds for .m = 2 and .η ∈ [−π, a] √ ∪ [b, π ]. (iii) For .H ∈ [1/ 8, 1/2), and .η ∈ [a, b], with .−π < a < 0 < b < π and such that the above intervals do not contain abscises of the intersection points of the limit dispersion curves (cf. (11.19)) with one of the corresponding eigenfunctions depending on .x2 , then (11.23) holds for .m = 3 and .η ∈ [a, b].

11 Spectral Gaps in a Double-Periodic Perforated Neumann Waveguide

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See Theorem 5.1 in [GoEtAl23b] for the proof of the first statement, while see Theorem 4.2 and third step in the proof of Theorem 5.1 in [GoEtAl23b] for the statement of the second and third statement in Theorem 1. The proofs are based on results of spectral perturbation theory (cf. [ViLj62]). The following results provide regions of the band .(−π, π ) × (0, +∞) free of certain perturbed eigenvalues. They have been proved in Section 3 of [GoEtAl23a] using (11.21). Proposition 1 Let .H ∈ (0, 1). Let .δ1 > 0 (and .< π ). Then, there exists a positive constant .ε1 = ε1 (H, δ1 ) such that for .ε < ε1 the entries .Λε2 (η) of the eigenvalue sequence (11.11) meet the estimates ε .Λ2 (η)



π 2 (1 − H 2 ) > π +min 2π δ1 , 2H 2 2

for η ∈ [−π +δ1 , π −δ1 ].

(11.24)

Proposition 2 Let .H ∈ (0, 1/2). Let .δ3 > 0 (and .< π ). Then, there exists a positive constant .ε1 = ε1 (H, δ3 ) such that for .ε < ε1 the entries .Λε3 (η) of the eigenvalue sequence (11.11) meet the estimates ε .Λ3 (η)

π 2 (1 − 4H 2 ) > 4π + min 4π δ3 , H2 2

for η ∈ [−π, −δ3 ] ∪ [δ3 , π ]. (11.25)

11.3.2 The Boundary Layer Problem Near the perforation string .ωε (0, 0), . . . , ωε (0, N − 1) ⊂ 0 , there appears a boundary layer, which is described in the stretched coordinates ξ = (ξ1 , ξ2 ) = ε−1 (x1 , x2 − εkH ).

.

Using these auxiliary coordinates which transform, for each .k = 0, · · · , N − 1, ωε (0, k) into .ω (cf. (11.2)), below we introduce a boundary layer problem in the unbounded perforated strip .Ξ := {x ∈ R × (0, H )} \ ω (cf. Fig. 11.4). Taking into account the mirror symmetry (11.22), we consider .W01 a .ξ2 -periodic function, even in the .ξ2 − H variable, .W01 to be the solution of

.

.

−Δξ W01 (ξ ) = 0, ξ ∈ Ξ,.

(11.26)

W01 (ξ1 , H ) = W01 (ξ1 , 0), ξ1 ∈ R, .

(11.27)

∂W01

(11.28)

∂ξ2

(ξ1 , H ) =

∂W01 ∂ξ2

(ξ1 , 0), ξ1 ∈ R, .

∂ν(ξ ) W01 (ξ ) = −ν1 (ξ ), ξ ∈ ∂ω,

(11.29)

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where .ν(ξ ) = (ν1 (ξ ), ν2 (ξ )) denote the outward normal vector on .∂ω. The existence and uniqueness (up to an additive constant) of solution of (11.26)–(11.29) in a suitable weighted Sobolev space has been proved in Section 4 of [GoEtAl23a]. In particular, it implies .∇ξ W01 ∈ L2 (Ξ ). Let us mention Lemma 4.3 in [GoEtAl23a] in connection with the symmetry (11.22) and the Neumann condition (11.28). Also, connected with .W01 , we introduce a so-called integral characteristic, namely, the quantity .m1 (Ξ ) given by .

m1 (Ξ ) =

 1  ∇ξ W01 ; L2 (Ξ )2 + |ω| > 0, 2H

(11.30)

where .|ω| = mes2 ω (cf. Section 4.2 of [GoEtAl23a]).

11.3.3 Asymptotic Analysis Near Nodes The results in this section deal with the points, where the limit eigenvalue under consideration is always multiple and gives rise to a node of the dispersion curves in Figs. 11.3 and 11.5. Examining the splitting of the band edges and the opening of spectral gaps requires obtaining more precise asymptotic formulas for the eigenvalues in (11.11), which are valid in a neighborhood of a certain value of the Floquet parameter .η. The asymptotic analysis is somehow double, since it takes into account the small parameter and the small neighborhood of the nodes 2 2 .(η◦ , Λ◦ ) = (0, 4π ) and .(η , Λ ) = (±π, π ). 11.3.3.1

The Node (η◦ , Λ◦ ) = (0, 4π 2 ) for H ∈ (0, 1/2)

This node marked with .◦ occurs in Fig. 11.3 under the assumption .H ∈ (0, 1/2) as the intersection point of the two (plus and minus) limit dispersion curves .(η ± 2π )2 with .η ∈ [−π, π ]. Thus, the problem (11.15)–(11.18) with .η = 0 has the eigenvalue 0 0 2 .Λ◦ := Λ (0) = Λ (0) = 4π of multiplicity 2. 2 3 Fig. 11.5 Graphics of the limit dispersion curves .Λ01 (η) and .Λ02 (η) when .H ∈ (0, 1/2). Associated .ε-neighborhoods are the regions between the surrounding red lines

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To investigate the perturbed dispersion curves (11.12) with .p = 2, 3 near the point .(η◦ , Λ◦ ) = (0, 4π 2 ), we consider the rapid Floquet variable .ψ = η/ε in a neighborhood of .η = 0. We formulate the result on the splitting edges of the second and third limit spectral bands (cf. Section 7 of [GoEtAl23a] for the proof): Theorem 2 Let .H ∈ (0, 1/2) and .ψ0 > 0. Then, there exist positive .ε0 = ε0 (H , ψ0 ) and .C = C(H , ψ0 ) such that, for .ε ∈ (0, ε0 ] and .|η| ≤ εψ0 , we obtain the estimates     ε 2  η  . Λ2 (η) − 4π − εΛ−  ≤ Cε2 , ε   η    ε  ≤ Cε2 , Λ3 (η) − 4π 2 − εΛ+ ε where the functions .Λ± (·) are given by  . Λ± (ψ)

= 4π 2π

 |ω| 2H



− m1 (Ξ ) ±

4π 2

  |ω| 2 2 m1 (Ξ )+ +ψ . 2H (11.31)

From (11.31) and (11.30), we have that Λ− (ψ) ≤ −16π 2 m1 (Ξ ) < 0

.

and

Λ+ (ψ) ≥ 8π 2

|ω| > 0. H

(11.32)

In this way, combining Theorem 2 with Theorem 1 and Proposition 2 will likely give rise to the open gap .γ2ε (cf. Fig. 11.3). We show this in Sect. 11.4.2. The Node (η , Λ ) = (±π, π 2 ) for H ∈ (0, 1)

11.3.3.2

Following the scheme in Sect. 11.3.3.1 for the node (η◦ , Λ◦ ) = (0, 4π 2 ), we consider the node .(η , Λ ) = (±π, π 2 ) under the assumption .H ∈ (0, 1) (cf. Fig. 11.2a). For the sake of brevity, here we only outline the main changes. To investigate the perturbed dispersion curves (11.12) with .p = 1, 2 near the point .(±π, π 2 ), we consider the rapid Floquet variable .ψ = (η ∓ π )/ε in a neighborhood of .η = ±π . We formulate the result on the splitting edges of the first and second limit spectral bands: Theorem 3 Let .H ∈ (0, 1) and .ψ1 > 0. Then, there exist positive .ε0 = ε0 (H, ψ1 ) and .C = C(H , ψ1 ) such that for .ε ∈ (0, ε0 ] we obtain the estimates: .

  η − π    ε  ≤ Cε2 Λ1 (η) − π 2 − εΛ− ε   η − π    ε  ≤ Cε2 Λ2 (η) − π 2 − εΛ+ ε

for |η − π | ≤ εψ1 , for |η − π | ≤ εψ1 ,

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for the node .(π, π 2 ), and similarly for the node .(−π, π 2 ): .

  π + η    ε  ≤ Cε2 Λ1 (η) − π 2 − εΛ− ε   π + η    ε  ≤ Cε2 Λ2 (η) − π 2 − εΛ+ ε

for |π + η| ≤ εψ1 , for |π + η| ≤ εψ1 .

The functions .Λ± (·) are given by . Λ (ψ) ±

     |ω| |ω| 2 − m1 (Ξ ) ± π 2 m1 (Ξ )+ = 2π π + ψ2 . 2H 2H

(11.33)

From (11.33) and (11.30), we have that Λ− (ψ) ≤ −4π 2 m1 (Ξ ) < 0 and Λ+ (ψ) ≥ 2π 2

.

|ω| > 0. H

(11.34)

In this way, combining Theorem 3 with Theorem 1 and Proposition 1 will likely give rise to the open gap .γ1ε (cf. Fig. 11.3). We show this in Sect. 11.4.1.

11.4 The Main Results In this section, we show that, under the mirror symmetry condition of the holes (11.22), there are open spectral gaps for the spectrum (11.13) of the problem (11.3)– (11.4). Further specifying, when the width .H ∈ (0, 1), we get at least one open gap, while for .H ∈ (0, 1/2), there are at least two open gaps. We provide asymptotic formulas for their localization and width (cf. (11.35), (11.36), (11.38), and (11.39)). In Sects. 11.4.1 and 11.4.2, respectively, we broach the cases where .H ∈ (0, 1) and .H ∈ (0, 1/2).

11.4.1 Opening Spectral Gap Near the Node (η , Λ ) Recall .(η , Λ ) = (±π, π 2 ). Based on the preliminary results (see Sect. 11.3), we prove the following result for the opening of spectral gaps: Theorem 4 Let .H ∈ (0, 1). Then, there exists a positive constant .ε0 = ε0 (H ) such that, for .ε ∈ (0, ε0 ], the asymptotic formulas max Λε1 (η) ≤ π 2 − 4π 2 εm1 (Ξ ) + O(ε2 ),

η∈[−π,π ] .

min

η∈[−π,π ]

Λε2 (η) ≥ π 2 + 2π 2 ε

|ω| + O(ε2 ) H

(11.35)

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141

hold. Moreover, the gap .γ1ε , defined by (11.14), has positive length .

 |ω|  + O(ε2 ), |γ1ε | ≥ 4π 2 ε m1 (Ξ ) + 2H

(11.36)

where .m1 (Ξ ) is the integral characteristic given by (11.30). Proof According to the estimates for the node .(−π, π 2 ) in Theorem 3 and the bound (11.34), inequalities (11.35) hold for .η ∈ [−π, −π + εψ1 ] and .ε ≤ ε0∗ , ∗ .ε , depending on H and .ψ1 , for arbitrary positive .ψ1 . By the symmetry, one gets 0 the same inequality for .η ∈ [π − εψ1 , π ] Now, according to (11.23) for .m = 1, we get Λε1 (η) ≤ (−π + εψ1 )2 + C0 ε

.

for ε ≤ ε0 , η ∈ [−π + εψ1 , π − εψ1 ].

It suffices to fix a .ψ1 such that .ψ1 ≥ (2π )−1 C0 + 2π m1 (Ξ ), to guarantee that (−π + εψ1 )2 + C0 ε ≤ π 2 − 4π 2 εm1 (Ξ ) + O(ε2 ).

.

Thus, we get the uniform estimate (11.35) for the first dispersion curve .Λε1 (η). Let us proceed with the estimate for .Λε2 (η). Let us first consider the case where .H ∈ (0, 1/2). According to statement i) in Theorem 1, for .m = 2, we get Λε2 (η) ≥ (π + εψ1 )2 − C0 ε

.

for ε ≤ ε0 , η ∈ [−π + εψ1 , π − εψ1 ]. (11.37)

It suffices to fix a .ψ1 such that .ψ1 ≥ (2π )−1 C0 + π |ω|H −1 , to guarantee that (εψ1 + π )2 − C0 ε ≥ π 2 + 2π 2 ε|ω|H −1 + O(ε2 ).

.

Thus, we get the uniform estimate (11.35) for the second dispersion curve .Λε2 (η). Fixing   ψ1 = (2π )−1 max 4π 2 m1 (Ξ ) + C0 , C0 + 2π 2 |ω|H −1

.

for .ε small enough, we have the estimates for the two dispersion curves. In this way, the spectral gap .γ1ε stays open and has the width (11.36). In the case where .H ∈ [1/2, 1), we apply statement ii) in Theorem 1, and we get the estimate (11.37) in subintervals .η ∈ [−π + εψ1 , a] or .η ∈ [b, π − εψ1 ], for .a and .b as outlined in the statement. Then, using (11.24) for a suitable .δ1 and .ε small enough, we derive the above estimates for the two dispersion curves and for the width of .γ1ε .

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11.4.2 Opening Spectral Gap Near the Node (η◦ , Λ◦ ) Recall .(η◦ , Λ◦ ) = (0, 4π 2 ). With the idea of Sect. 11.4.1 and on the basis of Theorems 1 and 2, we prove the result on opening spectral gap .γ2ε (see Fig. 11.3). Theorem 5 Let .H ∈ (0, 1/2). Then, there exists a positive constant .ε0 = ε0 (H ) such that, for .ε ∈ (0, ε0 ], the asymptotic formulas max Λε2 (η) ≤ 4π 2 − 16π 2 εm1 (Ξ ) + O(ε2 ),

η∈[−π,π ] .

Λε3 (η) ≥ 4π 2 + 8π 2 ε

min

η∈[−π,π ]

|ω| + O(ε2 ) H

(11.38)

hold. Moreover, the gap .γ2ε , defined by (11.14), has positive length .

 |ω|  + O(ε2 ), |γ2ε | ≥ 16π 2 ε m1 (Ξ ) + 2H

(11.39)

where .m1 (Ξ ) is the integral characteristic given by (11.30). Proof According to the estimates for the node .(0, 4π 2 ) in Theorem 2 and the bound (11.32), inequalities (11.38) hold for .η ∈ [−εψ0 , εψ0 ] and .ε ≤ ε0 , .ε0 depending on H and .ψ0 , for arbitrary positive .ψ0 . Now, according to (11.23) for .m = 2, we get for .η ∈ [−π, −εψ0 ] ∪ [εψ0 , π ] Λε2 (η) ≤ (2π − εψ0 )2 + C0 ε

.

for

ε ≤ ε0 .

It suffices to fix a .ψ0 such that .ψ0 ≥ (4π )−1 C0 + 4π m1 (Ξ ), to guarantee that (2π − εψ0 )2 + C0 ε ≤ 4π 2 − 16π 2 εm1 (Ξ ) + O(ε2 ).

.

Thus, we get the uniform estimate (11.38) for the second dispersion curve .Λε2 (η). Let us proceed with the estimate for .Λε3 (η). √ First, we consider the case where .H ∈ (0, 1/ 8). According to statement i) in Theorem 1, for .m = 3, we get for .η ∈ [−π, −εψ0 ] ∪ [εψ0 , π ]: Λε3 (η) ≥ (2π + εψ0 )2 − C0 ε

.

for ε ≤ ε0 .

(11.40)

It suffices to fix a .ψ0 such that .ψ0 ≥ (4π )−1 C0 + 2π |ω|H −1 , to guarantee that (2π + εψ0 )2 − C0 ε ≥ 4π 2 + 8π 2 ε|ω|H −1 + O(ε2 ).

.

Thus, we get the uniform estimate (11.38) for the third dispersion curve .Λε3 (η).   Now, fixing .ψ0 = (4π )−1 max 16π 2 m1 (Ξ ) + C0 , C0 + 8π 2 |ω|H −1 , for .ε small enough, we have the estimates for the two dispersion curves. In this way, the spectral gap .γ2ε stays open and has the width (11.39).

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√ In the case where .H ∈ [1/ 8, 1/2), we use statement (iii) in Theorem 1 and get (11.40) for .η ∈ [a, −εψ0 ]∪[εψ0 , b] along with estimate (11.25) to get the estimates above for the dispersion curves and the width of .γ2ε . Remark 1 Since .m1 (Ξ ) + |ω|(2H )−1 > 0 (cf. (11.30)), the estimates (11.36) and (11.39) ensure that the widths of the gaps .γ1ε and .γ2ε are of order .O(ε) when .0 < H < 1 and .0 < H < 1/2, respectively. Acknowledgments The work has been partially supported by MICINN through PGC2018098178-B-I00, PID2020-114703GB-I00, and Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S).

References [Ge50] Gelfand, I.M.: Expansions in characteristic functions of an equation with periodic coefficients [in Russian]. Doklady Akad. Nauk. SSSR 73, 1117–1120 (1950) [GoEtAl21] Gómez, D., Nazarov, S.A., Orive-Illera, R., Pérez-Martínez, M.-E.: Remark on justification of asymptotics of spectra of cylindrical waveguides with periodic singular perturbations of boundary and coefficients. J. Math. Sci. 257, 597–623 (2021) [GoEtAl23a] Gómez, D., Nazarov, S.A., Orive-Illera, R., Pérez-Martínez, M.-E.: Spectral gaps in a double-periodic perforated Neumann waveguide. Asymptot. Anal., 131, 385–441 (2023) [GoEtAl23b] Gómez, D., Nazarov, S.A., Orive-Illera, R., Pérez-Martínez, M.-E.: Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide. Math. Nachr., (2023), 23 p. https://doi.org/ 10.1002/mana.202100589 [ReSi78] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV: Analysis of Operators. Academic Press, New York (1978) [ViLj62] Vishik, M.I., Ljusternik, L.A.: Regular degeneration and boundary layer for linear differential equations with small parameter. Am. Math. Soc. Transl. 20, 239–364 (1962)

Chapter 12

Spectral Homogenization Problems in Linear Elasticity: The Averaged Robin Reaction Matrix D. Gómez and M.-E. Pérez-Martínez

12.1 Introduction In this paper, we study a spectral problem associated to the vibrations of an elastic body, which has large surface reaction terms concentrated in small regions. We assume that the elastic material fills the domain .Ω of the the upper half space .R3+ , and a part .Σ of its surface lies on the plane .{x3 = 0} and contains small regions .T ε of size .rε , at a distance .ε between them (cf. Fig. 12.1). The boundary conditions are of Winkler-Robin type on .T ε . Outside, the surface .Σ is traction-free, while the rest of the surface .∂Ω \ Σ is clamped to a rigid support. Here .ε and .rε are two small parameters, .ε  1 and .rε  ε or .rε = O(ε). As is well-known, from the mechanical viewpoint, the small regions behave as “springs,” and the elastic coefficients of these springs are defined through the socalled Robin reaction matrix, which we denote by .β(ε)M(x). .β(ε) is a positive parameter, which is referred to as the Robin/Winkler coefficient of reaction (the reaction parameter, in short). Let us introduce parameters .r0 , .β 0 , and .β ∗ , which play an important role in the description of the homogenized problems. They are defined by three limits: .

rε = r0 , ε→0 ε 2 lim

lim rε β(ε) = β 0 ,

ε→0

and

β(ε)rε2 = β ∗. ε→0 ε2 lim

(12.1)

An analysis of all the possible relations between the three parameters of the problem, .ε, .rε , and .β(ε), involving all the possible limits in (12.1), has been carried out in [GoNaPe20a] by using asymptotic expansions. The justification scheme for the critical relations in which .r0 > 0 and .β 0 > 0 or .r0 > 0 and .β 0 = +∞ has been D. Gómez · M.-E. Pérez-Martínez () Universidad de Cantabria, Santander, Spain e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_12

145

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Fig. 12.1 Geometrical configuration of the problem

developed in [GoNaPe20b]. In both cases, a strange term arises in the homogenized problem which contains a so-called capacity matrix, which is a function of the macroscopic variable. It has been obtained from the solutions of the parametric family of the associated microscopic problems, and it depends on .M(x) only in the most critical situation when .r0 > 0 and .β 0 > 0 (the extended capacity matrix). In this paper, we address the convergence of the spectrum for a different critical relation between parameters, which imply a different homogenized problem. More specifically, we deal with the case when .β ∗ > 0 and .r0 = +∞, which provides an “averaged spring-type condition” on .Σ, where the averaged Robin reaction matrix is .M(x) multiplied by the constant .β ∗ |T | (cf. (12.10)). It should be emphasized that, in this case, to each reaction parameter .β(ε) corresponds a critical size of the reaction regions .rε = O(εβ(ε)−1/2 ), while to each size .rε corresponds a critical reaction parameter .β(ε) = O(ε2 rε−2 ) (see [GoNaPe20a] for details). The solutions of the parametric family of associated microscopic problems obtained in [GoNaPe20a] are constants, and, consequently, the technique for justifications differs very much from that in [GoNaPe20b]. Also, it should be noted that when .rε = O(ε), we have the same geometrical configuration considered in [GoNaPe18] for a different spectral problem in linear elasticity with Winkler-Steklov boundary conditions. Finally, the structure of the paper is as follows. Section 12.2 contains the setting of the spectral homogenization problem as well as the corresponding homogenized problem. Section 12.3 describes the associated stationary problems and some preliminary results, which allow us to prove the spectral convergence. In Sect. 12.4, we show the convergence of solutions and spectrum for .r0 = +∞ and .β ∗ > 0.

12.2 The Setting of the Problem Let .Ω be an open bounded domain of .R3 situated in the upper half-space .R3+ = {x ∈ R3 : x3 > 0}, with a Lipschitz boundary .∂Ω. Let .Σ be the part of the boundary in contact with the plane .{x3 = 0}, which is assumed to be non-empty, and let .ΓΩ be the rest of the boundary of .Ω: .∂Ω = Γ Ω ∪ Σ. Let T denote an open bounded domain of the plane .{x3 = 0} with a Lipschitz boundary. Without any restriction, we can assume that both .Σ and T contain the origin of coordinates.

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Let .ε be a small parameter .ε  1. Let .rε be an order function such that .rε  ε or rε = O(ε). For .k = (k1 , k2 ) ∈ Z2 , we denote by . xkε the point of the plane .{x3 = 0} ε ε of coordinates . xk = (k1 ε, k2 ε, 0), and by .T xk , the homothetic domain of T of ratio ε =  ε .rε after translation to the point . xkε : .T x xk k + rε T . If there is no ambiguity, we ε ε ε shall write . xk instead of . xk , and .T instead of .T xk . In this way, for a fixed .ε, we have constructed a grid of squares in the plane ε ε ε .{x3 = 0}, whose vertices are the points . xkε ∈ T xk . Let .J denote .J = {k ∈ ε 2 ε Z : T of .J : .Nε  |Σ|/ε2 = xk ⊂ Σ}, while .Nε denotes the number of elements  O(ε−2 ). Finally, if no confusion arises, we denote by . T ε the union of all the .T ε contained in .Σ. Also, in what follows, .x = (x1 , x2 , x3 ) denotes the usual cartesian coordinates, while by .xˆ = (x1 , x2 ), we refer to the two first components of .x ∈ R3 . Under the basis that the domain .Ω is filled by an elastic material, for .i, j, k, l = 1, 2, 3, we denote by .aij kl = aij kl (x) the elastic coefficients of the material, which are assumed to be continuous functions defined in .Ω and satisfy the standard symmetry and coercivity properties (cf., e.g., [SaSa89] and [OlEtAl92]): .

aij kl = aj ikl = aklij , .

i, j, k, l = 1, 2, 3,

∃α1 > 0 : aij kl ξij ξkl ≥ α1 ξij ξij ,

∀ξ 3 × 3 − matrix : ξij = ξj i .

(12.2)

Also, for a given displacement vector .u(x) = (u1 (x), u2 (x), u3 (x)), we use the standard notations for stress and strain tensors .σ (u) and .e(u); namely, their components are related by the Hooke’s law: σij (u) = aij kl ekl (u),

for i, j = 1, 2, 3,

.

where ekl (u) =

.

∂ul  1  ∂uk , + 2 ∂xl ∂xk

for k, l = 1, 2, 3.

Above, and in what follows, we use the convention of summation over repeated indexes. In connection with the elastic coefficients on boundary conditions, let us introduce a symmetric and positive definite .3 × 3-matrix, .Mij ∈ C(Σ): ∃α2 > 0 : Mij (x1 , x2 , 0)ξi ξj ≥ α2 ξi2 ,

.

∀ξ ∈ R3 ,

∀(x1 , x2 , 0) ∈ Σ.

(12.3)

Let us consider the spectral problem ⎧ ε ⎪ − ∂σij (u ) = λε uε ⎪ ⎪ i ⎪ ∂xj ⎪ ⎨ ε u =0 . ⎪ ⎪ σ ε nj = 0 ⎪ ij ⎪ ⎪ ⎩ ε σij nj + β(ε)Mij uεj = 0

in Ω , on ΓΩ ,  on Σ \ T ε ,  ε on T ,

i = 1, 2, 3,

(12.4)

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where .λε denotes the spectral parameter and .uε = (uε1 , uε2 , uε3 ) the corresponding eigenvector. .n stands for the unit outer normal to .Ω along .Σ, namely, .n = (0, 0, −1). .β(ε) > 0 can be very large or of order 1. For fixed .ε > 0, the weak formulation of problem (12.4) reads: find .λε ∈ R, ε ε .u ∈ V, and .u = 0, satisfying



σij (u )eij (v) dx + β(ε) ε

.



Ω

Mij uεi vj

d xˆ = λ

Tε

uεi vi dx, ∀v ∈ V,

ε

(12.5)

Ω

where .V denotes the space .{v ∈ (H 1 (Ω))3 : v = 0 on ΓΩ } with the norm generated by the scalar product

.

(u, v)V =

eij (u)eij (v) dx .

(12.6)

Ω

On account of (12.2) and (12.3), the left-hand side of (12.5) defines a bilinear, symmetric continuous and coercive form on .V ⊂ (L2 (Ω))3 . Consequently (see, e.g., [SaSa89]), (12.5) has the discrete spectrum: n→∞

0 < λε1 ≤ λε2 ≤ · · · ≤ λεn ≤ · · · −−−−→ + ∞,

.

(12.7)

where we have adopted the convention of repeated eigenvalues according to their multiplicities. The corresponding eigenfunctions form a basis in .V and .(L2 (Ω))3 , and we assume that they are subject to the orthonormalization condition  n,ε n,ε  u , u (L2 (Ω))3 = δn,m .

.

(12.8)

The minimax principle gives the following bounds for the eigenvalues of (12.4) (cf. [GoNaPe20a] for details): for each fixed .n ∈ N, there exist .C and .Cn constants independent of .ε such that 0 < C ≤ λεn ≤ Cn ,

.

∀ε > 0.

(12.9)

In this paper, we address the asymptotic behavior of .(λε , uε ) as .ε → 0, for the critical relation where .β ∗ > 0 with .r0 = +∞. Using the technique of matched asymptotic expansions, in [GoNaPe20a], we have obtained the homogenized problem ⎧ ⎪ ∂σij (u0 ) ⎪ ⎪ = λ0 u0i in Ω ⎨− ∂xj . , u0 = 0 on ΓΩ ⎪ ⎪ ⎪ ⎩ σ (u0 )n + β ∗ |T |M u0 = 0 on Σ ij j ij j

i = 1, 2, 3.

(12.10)

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The Robin reaction matrix .β ∗ |T |M, which appears in the boundary condition on .Σ, is referred to as averaged Robin reaction matrix. The discreteness of the spectrum of problem (12.10) follows as that of (12.4), with minor modifications. Let us denote by n→∞

0 < λ01 ≤ λ02 ≤ · · · ≤ λ0n ≤ · · · −−−−→ + ∞,

.

(12.11)

the increasing sequence of eigenvalues repeated according to their multiplicities. 2 3 The corresponding eigenfunctions .{un,0}∞ n=1 forms an orthonormal basis in .(L (Ω)) . ε ε The aim of this paper is to prove the convergence of the eigenpairs .(λ , u ), as .ε → 0, toward those of (12.10).

12.3 The Stationary Problems and Some Preliminary Results The proof of the convergence for the eigenpairs of (12.5) (cf. Theorem 2) is based on a general result on spectral perturbation theory (cf. Section III.1 of [OlEtAl92]). In order to be self-contained, we introduce below a simplified version of such a result: Lemma 1 Let .H be a separable Hilbert space with the norm .·. Let .L (H) denote ε 0 the space of continuous linear operators  on .H. Let .A , A ∈ L (H), and .W be a subspace of .H such that .I m A 0 = {v  v = A 0 u : u ∈ H} ⊂ W . We assume that the following properties are satisfied: (i1) .A ε and .A 0 are positive, compact, and self-adjoint operators on .H and ε .A L (H) .≤ c, where .c denotes a constant independent of .ε. (i2) For any .f ∈ W , .A ε f − A 0 f  → 0 as ε → 0. (i3) The family of operators .A ε is uniformly compact, that is, for any sequence ε ε ε satisfying .f ∈ H such that . supε f  ≤ c, we can extract a subsequence .f ε ε − w 0   → 0, as .ε  → 0, for a certain .w 0 ∈ W . .A f ε 0 ∞ ε 0 Let .{μεi }∞ i=1 (.{μi }i=1 , respectively) be the sequence of the eigenvalues of .A (.A , ε ∞ respectively) with the usual convention of repeated eigenvalues. Let .{wi }i=1 and (.{wi0 }∞ i=1 , respectively) be the corresponding eigenvectors, which are assumed to form an orthonormal basis in .H. Then, for each fixed k, .μεk → μ0k , as .ε → 0 . In addition, for each sequence, still   denoted by .ε, we can extract a subsequence .ε → 0 such that .A ε wkε − wk∗  → 0 as .ε → 0 , where .wk∗ is an eigenvector of .A 0 corresponding to .μ0k and the set ∗ ∞ .{w } i i=1 forms an orthogonal basis of .H.

On account of this result, (12.8) and (12.9), proving the convergence for the eigenpairs of (12.5) amounts to showing the convergence of solutions of the associated stationary problems. Hence, it proves useful to introduce here the stationary homogenization problem: Find .uε ∈ V satisfying

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σij (uε )eij (v) dx+β(ε)

.

Ω



Tε

Mij uεi vj d xˆ =

fi vi dx

∀v ∈ V,

(12.12)

Ω

where .f = (f1 , f2 , f3 ) ∈ (L2 (Ω))3 represent the given forces acting on the body. Because of the Korn and Poincaré inequalities, (12.2) and (12.3), the unique solution of (12.12) satisfies uε V ≤ C,

(12.13)

.

with C a constant independent of .ε (cf. (12.6)). Therefore, for each sequence of {uε }ε>0 , we can extract a subsequence, still denoted by .ε, such that

.

ε→0

uε −−−−→u0

weakly in (H 1 (Ω))3 ,

.

(12.14)

for some .u0 ∈ V ⊂ {v ∈ (H 1 (Ω))3 : v = 0 on ΓΩ }. We aim to identify .u0 with the solution of a homogenized problem, and this is performed in Theorem 1. When .β ∗ > 0 and .r0 = +∞, let us introduce the homogenized problem which, similarly to (12.10), has been obtained using asymptotic expansions (cf. [GoNaPe20a] for details): Find .u0 ∈ V satisfying

∗



σij (u )eij (v) dx + β |T | 0

.

Ω

Mij u0i vj

Σ

d xˆ =

fi vi dx

∀v ∈ V.

(12.15)

Ω

The existence and uniqueness of solution of (12.15) holds as that of (12.12). For the sake of completeness, we introduce here certain results that we use for the proof of Theorem 1. See Theorem 4 in [KoOl88] for the proof of Lemma 2. We refer to Lemma 2.4 in Section II.3 of [MaKh74] for the proof of Lemma 3. See Theorem 1.2 in Section I.1 of [OlEtAl92] in connection with Lemma 4. Lemma 2 Let Y be a bounded domain in .R3 with a Lipschitz boundary. Let V be a linear space, weakly closed in .(H 1 (Y ))3 such that .V ∩ R = {0}, where .R denotes the set of rigid motions, namely, .R := {Aξ + b : A skew-symmetric 3 × 3 − matrix, b ∈ R3 }. Then, the following Korn inequality holds:



|∇ξ wi |2 dξ ≤ C

.

Y

|eij (w)|2 dξ

∀w ∈ V .

Y

Lemma 3 For .w ∈ H 1 (Ω), wL2 (Ω∩{0 0 in (12.1). For each k, .k = 1, 2, 3 . . ., ε ε → λ0 as .ε → 0, where .{λ0 }∞ are 0 .λ in (12.7) and .λ in (12.11) satisfy .λ k k k k=1 k k the eigenvalues of (12.10). In addition, for each infinitesimal sequence .ε, we can extract a subsequence, still denoted by .ε, such that the corresponding eigenfunctions ε,k converge toward .u0,k in .(L2 (Ω))3 , where .u0,k is an eigenfunction of (12.10) .u 2 3 corresponding to .λ0k , and the set .{u0,k }∞ k=1 forms an orthogonal basis of .(L (Ω)) . Proof Let us introduce the operators .A ε , A 0 : (L2 (Ω))3 → (L2 (Ω))3 . For .f ∈ (L2 (Ω))3 , we set .A ε f = uε , where .uε ∈ V is the unique solution of (12.12). 0 0 Similarly, we set .A 0 f = u .u ∈ V is the unique solution of (12.15). Thus,   , εwhere ε , uε,k )}∞ , the eigenpairs of ε −1 ε,k the eigenpairs of .A are .{ (λk ) , u }∞ k k=1 with .{(λ k=1   0 0 0,k ∞ (12.5), and the eigenpairs of .A 0 are .{ (λk )−1 , u0,k }∞ k=1 with .{(λk , u )}k=1 , the eigenpairs of (12.10). We define .W = V, and considering Theorem 1, properties (i1) and (i2) in Lemma 1 become self-evident. To prove property (i3), we consider .f ε ∈ (L2 (Ω))3 uniformly bounded in .(L2 (Ω))3 , and hence, we find a subsequence .ε → 0 and a  certain .f ∈ (L2 (Ω))3 such that .f ε → f weakly in .(L2 (Ω))3 . We replace f by  ε in (12.12), and since (12.13) also holds, we rewrite the proof of Theorem 1 with .f  minor modifications, to show the convergence of solutions .uε toward .u0 weakly in 1 3  .(H (Ω)) , as .ε → 0, and property (i3) is also verified. Consequently, the convergence of the eigenvalues and the corresponding eigenfunctions in the statement of the theorem holds from Lemma 1. Note that all the results hold when .β ∗ = 0 replacing the Robin boundary condition in (12.10) by the Neumann one. Acknowledgments Work supported by grant PGC2018-098178-B-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”.

References [GoNaPe18] Gómez, D., Nazarov, S.A., Pérez, E.: Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity. Z. Angew. Math. Phys. 69, 35, 23 p. (2018) [GoNaPe20a] Gómez, D., Nazarov, S.A., Pérez, E.: Spectral homogenization problems in linear elasticity with large reaction terms concentrated in small regions of the boundary. In: Constanda, C. (ed.) Computational and Analytic Methods in Science and Engineering, pp. 127–150. Springer Nature, Switzerland (2020) [GoNaPe20b] Gómez, D., Nazarov, S.A., Pérez, E.: Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. J. Elast. 142, 89–120 (2020) [GoPeSh12] Gómez, D., Pérez, E., Shaposhnikova, T.A.: On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems. Asymptot. Anal. 80, 289–322 (2012)

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[KoOl88] Kondratiev, V.A., Oleinik, O.A.: Boundary value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities. Russ. Math. Surv. 43, 65–119 (1988) [LoEtAl97] Lobo, M., Oleinik, O.A., Pérez, M.E., Shaposhnikova T.A.: On homogenization of solutions of boundary value problems in domains, perforated along manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4e série 25, 611–629 (1997) [MaKh74] Marchenko, V.A., Khruslov E.Ya: Boundary Value Problems in Domains with a Fine-Grained Boundary. Izdat. Naukova Dumka, Kiev (1974). In Russian [OlEtAl92] Oleinik, O.A., Shamaev, A.S., Yosifian G.A.: Mathematical Problems in Elasticity and Homogenization. North-Holland, London (1992) [PeSh12] Pérez, E., Shaposhnikova T.A.: Homogenization of a variational inequality with nonlinear restrictions specified on subsets located on part of the domain boundary. Dokl. Math. 85, 198–203 (2012) [SaSa89] Sanchez-Hubert, J., Sanchez-Palencia, E.: Vibration and Coupling of Continuous Systems. Asymptotic Methods. Springer, Heidelberg (1989)

Chapter 13

Time-Harmonic Oscillations of a Poroelastic Body with an Application to Modelling the Spinal Cord P. J. Harris and J. Venton

13.1 Introduction The medical condition syringomyelia is characterised by fluid-filled cavities (called syrinxes in the medical literature) which can form in the spinal cord. Cerebrospinal fluid (CSF) fills the spinal canal and surrounds the spinal cord, and extracellular fluid (ECF) surrounds the cells within the cord tissue itself. As such the cord tissue is often modelled as a porous medium. Fluid in the cavities typically has the same composition as CSF and ECF, although the exact mechanism causing the cavities to form is unknown. It is commonly thought that they are formed by a mechanical process within the spinal canal, as no viral, bacteriological or chemical causes have been detected. Most cases of syringomyelia occur in patients where the flow of the cerebrospinal fluid is restricted either by traumatic damage to their spine or by a malformation of the bones at the base of the skull called a Chiari malformation [Bo14]. If the flow of the CSF is restricted, then the speed of the fluid will have to be greater than if there was no restriction in order for the same volume of fluid to pass through the restriction in the same time. The Venturi principle states that the pressure in a liquid will decrease as its velocity increases and so the pressure drop at a restriction will be greater than when there is an unrestricted flow. It has been hypothesised that this larger pressure drop is one possible mechanism for causing the damage to the spinal cord where the flow of the cerebrospinal is restricted [Wi76, Wi80].

P. J. Harris () The University of Brighton, Brighton, UK e-mail: [email protected] J. Venton The National Physical Laboratory, London, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_13

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Fig. 13.1 A cross section of the spinal cord showing the white matter (shown in blue), grey matter (shown in green) and possible location of the region of oedema (shown in red). This figure also shows the finite element mesh used for the results presented in this paper. The mesh has 5072 quadratic triangular elements and 10293 nodes

The cord consists of two types of tissue, referred to a white and grey matter in the medical literature, and the geometrical shapes of the different parts of the spinal cord are shown in Fig. 13.1. The cord may also have a region of oedema, as shown in Fig. 13.1, where there is an accumulation of ECF. The region of oedema is usually not present in a healthy spinal cord. In cases where it does occur, it is thought that a region of oedema might be a precursor to the formation of the cavities that are characteristic of syringomyelia. A number of mathematical models for simulating the motion of the spinal cord have been proposed. Many of the models treat the spinal cord as an axisymmetric structure [BeEtAl05, BeEtAl06, CiEtAl18, CaCa21, HaHa06, HaHa10, HeBe16]. However, the cross section of the spinal cord is not circular, and so such models are just an approximation to the true geometry of the cord. Venton [Ve18] employs a two-dimensional model that has a more accurate representation of the geometrical shape of the spinal cord. Many of the models [CiEtAl18, HaHa06, HaHa10, HeBe16, Ve18] make use of the finite element method to solve the governing differential equations as it can easily include effects such as the poroelastic properties of the spinal cord into the models. The frequency domain model presented here is the first step in developing a model based on a Fourier series representation of the pressure in the fluid phase and the displacements in the solid phase. Most of existing finite element models of the motion of the spinal cord are based in the time domain [BeEtAl05, HaHa06, HaHa10, HeBe16, Ve18] and make use of time-stepping schemes to integrate the solution from one time-step to the next. However, as observed by Venton [Ve18], it is often necessary to use a large number of time-steps to get a stable and accurate numerical solution. An alternative is to use a Fourier series approach which requires us to solve a number of time-harmonic problems at different frequencies to construct

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a time-domain solution, although the solution it will give will be periodic rather than truly transient. The development of the Fourier series solution will be the subject of a future paper.

13.2 Mathematical Modelling In the model presented here, we will assume that the spinal cord consists of a poroelastic material where the pores within the cord are filled with ECF. Additionally, we are going to assume that properties of the ECF are not significantly different from those of water. We will present a two-dimensional model representing a cross section of the spinal cord where the elastic deformations of the cord are modelled using a planestrain elastic models which are applicable when the perpendicular length of the body is large compared to the diameter of the cross section. Since the length of the spinal cord in a healthy adult person is around 42 cm and its maximum diameter is approximately 1.3 cm (e.g. see [MaKe17]), such models are applicable to this studying the pressure and deformations inside the spinal cord in a plane which is halfway along the length of the cord. The time-dependent deformations of an elastic solid will satisfy the differential equation ∇ · σ = ρs

.

∂ 2u ∂t 2

(13.1)

where .σ denotes the symmetric stress tensor in the solid phase, .u is the vector-valued displacement and .ρs is the density of the elastic solid. Note that here underlining will be used to denote geometrical vectors (such as the vector of the displacements at a point) and a bold type face will be used to denote more general vectors such that that will occur in the finite element method. For a poroelastic material equation (13.1) is modified as  ∂ 2u  ∇ · [(1 − η)σ − ηpI ] = (1 − η)ρs + ηρf ∂t 2

.

(13.2)

where p denotes the excess pressure of the fluid in the pores, .η is the void-fraction of the porous solid, .ρf is the density of the fluid and .I is the identity tensor. From Darcy’s law, it is possible to show that the pressure of the fluid phase in the porous body satisfies [LeSc98] .

∂jj ∂p = Mb ∇ · (κ∇p) = Mb μ∇ 2 p + Mb α ∂t ∂t

(13.3)

where . denotes the symmetric strain tensor for the solid phase, and the usual convention of summing over repeated subscripts has been used. The constants

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Mb , .α and .μ are the Biot modulus, Biot-Willis coefficient and diffusion constant respectively, where we have assumed that the permeability of the spinal cord is the same in all directions. A full description of these constants can be found in [Ve18]. The boundary condition on the coupled system of differential equations (13.2) and (13.3) is that the pressure is known on the outer boundary of the body. For the spinal cord problem under consideration here, the excess pressure on the boundary will be the excess pressure in the surrounding cerebrospinal fluid. The system of differential equations (13.2) and (13.3) can be solved using an appropriate numerical method. It is straightforward to discretise the equations in space using a domain method such as the finite element method. However, many time-stepping schemes for integrating these equations through time suffer from stability and accuracy issues. For problems where the external loading is periodic, a Fourier series type method can be used to express the solution as linear combination of the solutions to a sequence of time-harmonic problems. In this paper, we will discuss the method for solving an individual time-harmonic problem, and the use of Fourier series to solve more general periodic problems will be discussed in a future paper. Assume that the system is undergoing time-harmonic motion with angular frequency .ω and that the pressure, displacements and stresses can be expressed in the form

.

p = pω eiωt

u = uω eiωt

.

σ = σω eiωt

(13.4)

where .pω , .uω and .σω are the frequency dependant amplitudes of the pressure, displacement and stress oscillations, respectively, and .i denotes the unit imaginary number. We note that in general the amplitudes will be complex valued. Substituting (13.4) into (13.2) and (13.3) yields   ∇ · [(1 − η)σω − ηpω I ] = −ω2 (1 − η)ρs + ηρf uω

.

iωp + iωMb αjj,ω = Mb μ∇ 2 pω

.

(13.5) (13.6)

Equations (13.5) and (13.6) can be discretised and solved numerically using the finite element method. The full details of the finite element method are not given here but can be found in one of the many texts on the subject, such as [ZiTa89]. Applying the finite element method to discretise (13.5) and (13.6) gives .

Kuω − ω2 Muω − Qpω − Lpω = 0 iωSpω + iωRuω − H pω = qω

(13.7)

where .K and .M are the usual structural stiffness and mass matrices, .H and .S are stiffness and mass matrices for scalar Laplace type problems, .Q and .R are coupling matrices and .L is matrix which gives the loading on the solid phase due to the pressure in the external fluid. Here the vectors .pω and .uω are the vectors of the

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159

nodal values of the fluid phase pressure and solid-phase displacements, and .qω is the vector nodal values of the known pressure on the surface of the poroelastic body. The full definitions of these matrices are given in [Ve18] and are not repeated here. The matrices appearing in the linear system of Eqs. (13.7) are sparse matrices, and so the system can be efficiently solved using an iterative method, such as GMRES. In many implementations, the iterations for solving .Ax = b are stopped when .

Axn − b ε

1 f (ζ ) dζ = lim ζ −z ε→0+ 2π

ζ ∈∂∗ Ω |ζ −z|>ε

f (ζ ) ν(ζ ) dσ (ζ ), ζ −z

for .σ -a.e. point .z ∈ ∂∗ Ω. Note that the integrals in (14.2)  are absolutely convergent,  σ (ζ ) 1 so .C f is well defined for each .f ∈ L ∂∗ Ω, 1+|ζ | and, in fact, .∂ (C f ) ≡ 0 in .Ω. Thus, if .O(Ω) denotes the space of holomorphic functions in .Ω, we have   σ (ζ ) C f ∈ O(Ω) for each f ∈ L1 ∂∗ Ω, 1+|ζ | .

.

Since .∂∗ Ω is countably rectifiable (according to a classical result of De GiorgiFederer), it turns out that the boundary-to-boundary Cauchy operator is also

14 The Poly-Cauchy Operator, Whitney Arrays, and Fatou Theorems

167

well defined when acting on functions from the weighted Lebesgue space  σ (ζ ) . See [MiEtAl22] for details, where more information on the L1 ∂∗ Ω, 1+|ζ | size, regularity, and boundary behavior of those objects may be found. To state a concrete result to this effect, we introduce the nontangential approach regions of aperture .κ ∈ (0, ∞) as

.

 Γκ (z0 ) := z ∈ Ω : |z − z0 | < (1 + κ) dist(z, ∂Ω) for each z0 ∈ ∂Ω.

.

Then the nontangential maximal operator .Nκ acts on each .L 2 -measurable function .u : Ω → C according to   Nκ u (z) := u L∞ (Γκ (z), L 2 ) for each z ∈ ∂Ω.

.

 κ−n.t.  In addition, we agree to denote by . u ∂Ω (z0 ) the .κ-nontangential trace of a given 2 .L -measurable function .u : Ω → C at the point .z0 ∈ ∂Ω, defined as the complex number (which is unique, if it exists) .w ∈ C with the property that for every ε > 0 there exists some r > 0 such that |u(z) − w| < ε forL 2 -a.e. z ∈ Γκ (z0 ) ∩ B(z0 , r).

.

Following G. David and S. Semmes (cf., e.g., the discussion in [MiEtAl22]), we shall call .Σ ⊆ R2 a uniformly rectifiable set (or UR set, for short) if .Σ is a closed Ahlfors regular set with the property that .ε, M ∈ (0, ∞) such  there exist constants  that for each .z ∈ Σ and each .R ∈ 0, 2 diam (Σ) , it is possible to find some Lipschitz map .ϕ : [0, R] → R2 with . ϕ L∞ ≤ M such that   H 1 Σ ∩ B(z, R) ∩ ϕ([0, R]) ≥ εR.

.

Next, we shall say that .Ω ⊆ R2 is a UR domain (cf. [HoEtAl10]) if .Ω is an open set for which .∂Ω is a UR set and .H 1 (∂Ω \ ∂∗ Ω) = 0. Fix a UR domain .Ω ⊆ R2 . Define the tangential derivative .∂τ ϕ of any given function .ϕ ∈ C 1 (R2 ) as   



.∂τ ϕ := ν1 ∂y ϕ − ν2 ∂x ϕ . ∂Ω

∂Ω

More generally, we shall say that the tangential derivative of some .f ∈ Lp (∂Ω, σ ) with .1 < p < ∞ exists and belongs to the space .Lp (∂Ω, σ ) if one can find a function .∂τ f ∈ Lp (∂Ω, σ ) with the property that the following formula (mimicking integration by parts on the boundary) holds:



f (∂τ ϕ) dσ = −

.

∂Ω

∂Ω

(∂τ f )ϕ dσ for each ϕ ∈ Cc1 (R2 ).

(14.3)

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Finally, for each .p ∈ (1, ∞) define the boundary Sobolev space (cf. [HoEtAl10], [MiEtAl22], [MiEtAl13])   p L1 (∂Ω, σ ) := f ∈ Lp (∂Ω, σ ) : ∂τ f ∈ Lp (∂Ω, σ ) ,

.

(14.4)

and equip it with the natural norm . f Lp (∂Ω,σ ) := f Lp (∂Ω,σ ) + ∂τ f Lp (∂Ω,σ ) . 1 The result below, found in [MiEtAl22] (cf. also [MiEtAl20]), deals with the nontangential behavior of the Cauchy operator and related matters. Theorem 1 Assume .Ω ⊆ R2 is a UR domain. Fix a parameter .κ ∈ (0, ∞) and an integrability exponent .p ∈ (1, ∞). Then the following properties hold: (1) [Fatou Theorem and Integral Representation Formula] For each ∈ O(Ω)  . uκ−n.t. p

with .Nκ u ∈ L (∂Ω, σ ), the .κ-nontangential boundary trace . u ∂Ω (z) is

κ−n.t. meaningfully defined at .σ -a.e. point .z ∈ ∂Ω, the function .u ∂Ω belongs to p .L (∂Ω, σ ), and (also assuming .u(z) = o(1) as .z → ∞ if .Ω is an exterior domain) one has  κ−n.t.  in Ω. u = C u ∂Ω

.

(2) [Jump Relation] If I denotes the identity operator then, for each function .f ∈ Lp (∂Ω, σ ),

κ−n.t.   (C f ) ∂Ω = 12 I + C f at σ -a.e. point on ∂Ω.

.

(3) [Size Estimate] For each .f ∈ Lp (∂Ω, σ ) one has   Nκ (C f ) p  f Lp (∂Ω,σ ) . L (∂Ω,σ )

.

(4) [Boundedness and Involution Property] The operator C is bounded from the space .Lp (∂Ω, σ ) into itself, i.e., for each given function .f ∈ Lp (∂Ω, σ ) one has .Cf ∈ Lp (∂Ω, σ ) and

Cf Lp (∂Ω,σ )  f Lp (∂Ω,σ ) .

.

In addition, C 2 = 14 I as operators on Lp (∂Ω, σ ).

.

p

(5) [Regularity] For each .f ∈ L1 (∂Ω, σ ) one has   Nκ (∇C f ) p  f Lp (∂Ω,σ ) . L (∂Ω,σ )

.

1

14 The Poly-Cauchy Operator, Whitney Arrays, and Fatou Theorems

169 p

Also, the operator C is continuous from the boundary Sobolev space .L1 (∂Ω, σ ) p p into itself, i.e., for each .f ∈ L1 (∂Ω, σ ) one has .Cf ∈ L1 (∂Ω, σ ) and

Cf Lp (∂Ω,σ )  f Lp (∂Ω,σ ) .

.

1

1

The goal here is to devise a higher-order analogue of this result, in which the m salient role of the Cauchy-Riemann operator .∂ is now played by .∂ for some arbitrary fixed integer .m ∈ N.

14.2 Main Results The first order of business is to figure out what the correct higher-order analogues of the Lebesgue and Sobolev spaces considered above should be. Let .Ω ⊆ R2 be a UR domain with compact boundary. Fix .m ∈ N and .p ∈ (1, ∞). We shall replace the standard Lebesgue space and the boundary Sobolev space from (14.4) with families   g˙ := g(a,b) : a, b ∈ N0 with a + b ≤ m − 1

.

of functions in .Lp (∂Ω, σ ) satisfying certain compatibility conditions (henceforth abbreviated as CC), namely,  g˙ ∈ CC ⇐⇒

.

∂τ g(a,b) = iνg(a+1,b) − i νg(a,b+1) σ -a.e. on ∂Ω whenever a, b ∈ N0 are such that a + b ≤ m − 2.

(14.5)

Concretely, as the Lebesgue-based complex Whitney array space (of order m) take  . g˙ = {g(a,b) }

CWAm−1 [Lp (∂Ω, σ )] := a,b∈N0 a+b≤m−1

∈ CC : g(a,b) ∈ Lp (∂Ω, σ ) if a + b ≤ m − 1 ,

equipped with the norm . g ˙ CWAm−1 [Lp (∂Ω,σ )] :=



g(a,b) Lp (∂Ω,σ ) . In the

a,b∈N0 a+b≤m−1

higher-order theory we envision, this space is going to play the role of the ordinary Lebesgue space .Lp (∂Ω, σ ). Similarly, we define the Sobolev-based complex Whitney array space (of order m) as p

 . g˙ = {g(a,b) }

CWAm−1 [L1 (∂Ω, σ )] := a,b∈N0 a+b≤m−1

p ∈ CC : g(a,b) ∈ L1 (∂Ω, σ ) if a + b ≤ m − 1 ,

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and equip it with the norm . g ˙ CWAm−1 [Lp (∂Ω,σ )] :=



1

g(a,b) Lp (∂Ω,σ ) . In 1

a,b∈N0 a+b≤m−1

the context of the higher-order theory we aim to develop, this space is going to play p the role of the boundary Sobolev space .L1 (∂Ω, σ ) from (14.4). We then use the space of (Lebesgue-based) complex Whitney arrays to define the poly-Cauchy operator as follows. Definition 1 Let .Ω ⊆ R2 be a UR domain with compact boundary. Fix .m ∈ N and .p ∈ (1, ∞). Define the action of the boundary-to-domain poly-Cauchy operator (of   order m) on each array .g˙ = g(a,b) a,b∈N0 ∈ CWAm−1 [Lp (∂Ω, σ )] as a+b≤m−1

m−1  1   C˙m−1 g˙ (z) := 2π i



.

k=0

∂Ω

(z − ζ )k g(0,k) (ζ ) dζ for all z ∈ Ω. k!(ζ − z)

When .m = 1, our boundary-to-domain poly-Cauchy operator reduces precisely to the standard boundary-to-domain Cauchy operator recalled earlier. The boundaryto-domain poly-Cauchy operator from Definition 1 satisfies a number of remarkable properties, such as the following Fatou result and integral representation formula. Theorem 2 Let .Ω ⊆ R2 be an arbitrary UR domain, and fix an arbitrary integer .m ∈ N. Pick some aperture parameter .κ ∈ (0, ∞) along with some integrability exponent .p ∈ (1, ∞). Let u be a polyanalytic function of order m m in .Ω, i.e., a function .u ∈ C ∞ (Ω) satisfying .∂ u = 0 in .Ω. Associate with this polyanalytic function the family of auxiliary functions .{uj }0≤j ≤m−1 defined, for each .j ∈ {0, 1, . . . , m − 1}, as uj (z) :=

m−1−j 

.

=0

(−1) z¯  j + (∂ u)(z) for each z ∈ Ω. !

(14.6)

Make the assumption that Nκ uj ∈ Lp (∂Ω, σ ) for each j ∈ {0, 1, . . . , m − 1},

.

(14.7)

and if .Ω is an exterior domain also assume that uj (z) = o(1) as z → ∞, for each j ∈ {0, 1, . . . , m − 1}.

.

(14.8)

Then for each . ∈ {0, 1, . . . , m − 1}, it follows that the .κ-nontangential trace

κ−n.t.  (∂ u) exists at σ -a.e. point on ∂Ω.

.

∂Ω

(14.9)

14 The Poly-Cauchy Operator, Whitney Arrays, and Fatou Theorems

171

m−1 (z−ζ )k k κ−n.t.

Also, for each .z ∈ Ω the function .∂Ω  ζ → k=0 k!(ζ −z) (∂ u) ∂Ω (ζ ) is absolutely integrable on .∂Ω (with respect to the measure .σ ) and 1 .u(z) = 2π i

m−1  (z − ζ )k k κ−n.t. (∂ u) ∂Ω (ζ ) dζ. k!(ζ − z)

(14.10)

∂Ω k=0

Note that Theorem 2 is the most natural higher-order generalization of item (1) in Theorem 1 (to which this reduces when .m := 1). Here is the proof of Theorem 2. Proof We proceed by induction on m. The case .m = 1 is contained in Theorem 1. To carry out the induction step, assume .m ≥ 2 and that all claims in the statement are valid for polyanalytic functions of order .m − 1. Decompose the given u as u(z) = w(z) + z¯ m−1 ω(z) for each z ∈ Ω, .

where w(z) := u(z) − ω(z) :=

1 (m−1)! (∂

m−1

m−1 z¯ m−1 u)(z) (m−1)! (∂

and

(14.11)

u)(z) for each z ∈ Ω.

1 um−1 , we conclude from (14.7)–(14.8) Then .ω ∈ O(Ω) and since .ω = (m−1)! (corresponding to .j := m − 1) that .Nκ ω ∈ Lp (∂Ω, σ ) and .ω vanishes at infinity if .Ω is an exterior domain. As such, all conclusions in item (1) of Theorem 1 apply to .ω. In terms of u, these imply that

(∂

m−1

.

κ−n.t.

u) exists at σ -a.e. point on ∂Ω ∂Ω

(14.12)

and for each .z ∈ Ω we have (∂

.

m−1

u)(z) =

1 2π i

∂Ω

1 m−1

κ−n.t. (∂ u) ∂Ω (ζ ) dζ. ζ −z

(14.13)

m−1

w = 0 (hence w is a In addition, (14.11) implies that .w ∈ C ∞ (Ω) satisfies .∂ polyanalytic function of order .m−1 in .Ω) and, for each .j,  ∈ N0 with .j + ≤ m−2, (∂

.

j +

w)(z) = (∂

j +

u)(z) −

z¯ m−1−j − m−1 (∂ u)(z) for each z ∈ Ω. (m − j −  − 1)! (14.14)

This may be used to compute the auxiliary functions associated with w as in (14.6). Specifically, for each .j ∈ {0, 1, . . . , m − 2}, we have

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wj (z) =

m−2−j 

.

=0

=

m−2−j  =0

−

(−1) z¯  j + (∂ w)(z) ! (−1) z¯  j + (∂ u)(z) !

 m−2−j  =0

=

m−2−j  =0

−

=



 (−1) m−1 z¯ m−1−j (∂ u)(z) !(m − j −  − 1)!

(−1) z¯  j + (∂ u)(z) !

 m−1−j (−1)m−1−j  m−1−j m−1 1 · (−1)+1 z¯ u)(z) − (∂ (m−1−j )! (m − 1 − j )!

m−1−j  =0

(−1) z¯  j + (∂ u)(z)=uj (z) for each z ∈ Ω. !

In concert with (14.7)–(14.8), this guarantees that, for each .j ∈ {0, 1, . . . , m−2}, we have .Nκ wj ∈ Lp (∂Ω, σ ) and .wj vanishes at infinity if .Ω is an exterior domain. As such, by the induction hypothesis, (14.12), and (14.14) with .j := 0, it follows that for each . ∈ {0, 1, . . . , m − 2} the .κ-nontangential trace

κ−n.t.  exists at σ -a.e. point on ∂Ω (∂ u)

.

∂Ω

(which, together with (14.12), takes care of (14.9)), and for each .z ∈ Ω, we have 1 .w(z) = 2π i

=

1 2π i

m−2  (z − ζ )k k κ−n.t. (∂ w) ∂Ω (ζ ) dζ k!(ζ − z)

m−2   (z−ζ )k  k κ−n.t. ζ¯ m−1−k m−1

κ−n.t. (∂ u) ∂Ω (ζ )− (∂ u) ∂Ω (ζ ) dζ. k!(ζ −z) (m − k − 1)!

∂Ω k=0

Hence, using (14.11), (14.13), and (14.15), we may write u(z) = w(z) +

.

(14.15)

∂Ω k=0

z¯ m−1 m−1 (∂ u)(z) (m − 1)!

14 The Poly-Cauchy Operator, Whitney Arrays, and Fatou Theorems

1 = 2π i

173

m−2   (z − ζ )k  k κ−n.t. ζ¯ m−1−k m−1

κ−n.t. (∂ u) ∂Ω (ζ ) − (∂ u) ∂Ω (ζ ) dζ k!(ζ − z) (m − k − 1)!

∂Ω k=0

1 + 2π i



z¯ m−1 m−1

κ−n.t. (∂ u) ∂Ω (ζ ) dζ, (m − 1)!(ζ − z)

∂Ω

(14.16)

for each .z ∈ Ω. Observe that for each z and .ζ , we have m−1  m−1  (z − ζ )k ζ¯ m−1−k (z − ζ ) + ζ¯ z¯ m−1 = = . . (m − 1)! (m − 1)! k!(m − 1 − k)! k=0

Plugging this back in (14.16) and canceling like-terms yields (14.10). This completes the proof of Theorem 2. The point of the next theorem is that our boundary-to-domain poly-Cauchy operator acting on complex Whitney array spaces can absorb .m − 1 derivatives without becoming hyper-singular. Theorem 3 Let .Ω ⊆ R2 be a UR domain with compact boundary. Fix an arbitrary integer  .m ∈N and select an integrability exponent .p ∈ (1, ∞). Then for any array .g ˙ = g(a,b) a,b∈N0 ∈ CWAm−1 [Lp (∂Ω, σ )], it follows that a+b≤m−1

m ˙ = 0 in Ω, C˙m−1 g˙ ∈ C ∞ (Ω) and ∂ (C˙m−1 g)

.

hence .C˙m−1 g˙ is a well-defined polyanalytic function of order m in .Ω. Furthermore, given any .r, s ∈ N0 with .r + s ≤ m − 1, for each .z ∈ Ω, one has ∂s∂

.

r

m−1−s−r    1 C˙m−1 g˙ (z) = 2π i k=0

−

∂Ω

(z − ζ )k g(s,k+r) (ζ ) dζ k!(ζ − z)

s−1  1 j! × 2π i ∂Ω (m − s − r + j )! j =0

 × ∂τ (ζ )

(z − ζ )m−s−r+j (ζ − z)j +1

 g(s−1−j,m−s+j ) (ζ ) dσ (ζ ).

As a corollary of this and the integration  by parts on the boundary formula in (14.3),  it follows that whenever .g˙ = g(a,b) a,b∈N0 actually belongs to the Soboleva+b≤m−1

p

based complex Whitney array space .CWAm−1 [L1 (∂Ω, σ )], then for any pair of numbers .r, s ∈ N0 with .r + s ≤ m − 1 any point .z ∈ Ω, one has

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m−1−s−r    1 C˙m−1 g˙ (z) = .∂ ∂ 2π i s r

k=0

∂Ω

(z − ζ )k g(s,k+r) (ζ ) dζ k!(ζ − z)

s−1  j! 1 + × 2π i ∂Ω (m − s − r + j )! j =0

×

 (z − ζ )m−s−r+j  ∂τ g(s−1−j,m−s+j ) (ζ ) dσ (ζ ). j +1 (ζ − z)

This is proved by induction on the number of derivatives, using the integration by parts on the boundary formula (14.3) and the compatibility conditions from (14.5). We also have higher-order nontangential maximal function estimates for our boundary-to-domain poly-Cauchy operator acting on complex Whitney array spaces, of the sort described below. Theorem 4 Let .Ω ⊆ R2 be a UR domain with compact boundary, and fix some integer .m ∈ N. Pick an aperture parameter .κ ∈ (0, ∞) and an exponent .p ∈ (1, ∞). Then (with the convention that, if .Ω is an exterior domain, the corresponding nontangential maximal operator is truncated) m−1  .

    Nκ ∇ C˙m−1 g˙  p  g ˙ CWAm−1 [Lp (∂Ω,σ )] L (∂Ω,σ )

=0

  uniformly in the array .g˙ = g(a,b)

.

a,b∈N0 a+b≤m−1

m      Nκ ∇ C˙m−1 g˙ 

Lp (∂Ω,σ )

∈ CWAm−1 [Lp (∂Ω, σ )], and also

 g ˙ CWAm−1 [Lp (∂Ω,σ )] 1

=0

  uniformly in the array .g˙ = g(a,b)

a,b∈N0 a+b≤m−1

p

∈ CWAm−1 [L1 (∂Ω, σ )].

Theorem 4 is proved based on the formulas for successive derivatives of the poly-Cauchy operator .C˙m−1 from Theorem 3 and the Calderón-Zygmund theory developed in [MiEtAl22]. The above theorem provides a satisfactory higherorder analogue of the nontangential maximal function estimates from items (3), (5) in Theorem 1. The boundary-to-domain poly-Cauchy operator turns out to be particularly useful in establishing the following higher-order Fatou type theorem for polyanalytic functions in UR domains, involving arbitrary combinations of derivatives. Theorem 5 Let .Ω ⊆ R2 be a UR domain with compact boundary. Fix an integer m ∈ N, an integrability exponent .p ∈ (1, ∞), and an aperture parameter .κ in ∞ (Ω) .(0, ∞). Suppose u is a polyanalytic function of order m in .Ω, i.e., .u ∈ C .

14 The Poly-Cauchy Operator, Whitney Arrays, and Fatou Theorems

175

m

satisfies .∂ u = 0 in .Ω, and assume that Nκ (∇  u) ∈ Lp (∂Ω, σ ) for each  ∈ {0, 1, . . . , m − 1}.

.

Then for each . ∈ {0, 1, . . . , m − 1}, the .κ-nontangential limit   

κ−n.t. ∇ u exists at σ -a.e. point on ∂Ω.

.

∂Ω

The above theorem suggests making the following definition. Definition 2 Given a UR domain with compact boundary .Ω ⊆ R2 , along with an integer .m ∈ N, an integrability exponent .p ∈ (1, ∞), and some aperture parameter p,m (Ω) (of order m in .Ω) as .κ ∈ (0, ∞), define the higher-order Hardy space .H m ∞ the collection of all functions .u ∈ C (Ω) with .∂ u = 0 in .Ω, satisfying Nκ (∇  u) ∈ Lp (∂Ω, σ ) for all  ∈ {0, 1, . . . , m − 1}.

(14.17)

.

If .Ω is an exterior domain, it is assumed that the nontangential maximal operator is truncated and one also asks that the auxiliary functions .{uj }0≤j ≤m−1 associated with u as in (14.6) vanish at infinity. Equip this higher-order Hardy space with H p,m (Ω)  u −  → u H p,m (Ω) :=

m−1 

.

  Nκ (∇  u)

Lp (∂Ω,σ )

.

=0 p,m

Finally, define the regular higher-order Hardy space .H1 (Ω) (of order m in .Ω) in an analogous fashion, now replacing (14.17) by the stronger condition Nκ (∇  u) ∈ Lp (∂Ω, σ ) for all  ∈ {0, 1, . . . , m},

.

p,m

and equipping .H1

p,m .H (Ω) 1

(Ω) with the norm  u −→ u

p,m H1 (Ω)

m    Nκ (∇  u) p := . L (∂Ω,σ ) =0

We may then refine our earlier higher-order Fatou type theorem as follows. Theorem 6 Let .Ω ⊆ R2 be a UR domain with compact boundary. Fix an integer .m ∈ N, an integrability exponent .p ∈ (1, ∞), and an aperture parameter .κ in .(0, ∞). Then the higher-order complex nontangential trace operator  p,m TrC (Ω) −→ CWAm−1 Lp (∂Ω, σ ) m−1 : H

.

defined as

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J. Kyeong et al.

κ−n.t.  b  ∂a∂ u

TrC m−1 (u) :=

.

a,b∈N0 a+b≤m−1

∂Ω

for each u ∈ H p,m (Ω),

(14.18)

is meaningful, linear, and bounded. In addition, TrC m−1 : H1

p,m

.

 p (Ω) −→ CWAm−1 L1 (∂Ω, σ ) p,m

defined as in (14.18) above for each .u ∈ H1 (Ω) ⊆ H p,m (Ω) is well defined, linear, and bounded. Moreover, the boundary-to-domain poly-Cauchy operator introduced in Definition 1 induces well-defined, linear, and bounded mappings

.

C˙m−1 : CWAm−1 [Lp (∂Ω, σ )] −→ H p,m (Ω),  p p,m (Ω). C˙m−1 : CWAm−1 L (∂Ω, σ ) −→ H 1

1

Finally, one has the following poly-Cauchy reproducing formula:   p,m u = C˙m−1 TrC (Ω). m−1 (u) for each u ∈ H

.

Proceeding forward, we define a boundary-to-boundary poly-Cauchy operator as follows. Given a UR domain .Ω ⊆ R2 with boundary and an integrability  compact  exponent .p ∈ (1, ∞), for any array .g˙ = g(a,b) a,b∈N0 ∈ CWAm−1 [Lp (∂Ω, σ )], a+b≤m−1   ˙ ˙ we define .Cm−1 g˙ := Cm−1 g˙ (a,b) a,b∈N0 by setting, for each .a, b ∈ N0 with a+b≤m−1

a + b ≤ m − 1,

.

.

m−1−a−b    C˙ m−1 g˙ (a,b) (z) := lim k=0

−

a−1  j =0

ε→0+

lim

ε→0+

1 2π i

1 2π i

× ∂τ (ζ )

ζ ∈∂Ω |ζ −z|>ε

ζ ∈∂Ω |ζ −z|>ε

(z − ζ )k g(a,k+b) (ζ ) dζ k!(ζ − z)

j! × (m − a − b + j )!

 (z − ζ )m−a−b+j  (ζ − z)j +1

g(a−1−j,m−a+j ) (ζ ) dσ (ζ )

at .σ -a.e. point .z ∈ ∂Ω. As indicated in our next theorem, this operator acts naturally between our complex Whitney array spaces (compare with [HoEtAl21]). Theorem 7 Let .Ω ⊆ R2 be a UR domain with compact boundary, and fix an arbitrary integer .m ∈ N. Also, pick an integrability exponent .p ∈ (1, ∞). Then the boundary-to-boundary poly-Cauchy operator .C˙ m−1 yields well-defined, linear, and bounded mappings both on the Lebesgue-based complex Whitney array

14 The Poly-Cauchy Operator, Whitney Arrays, and Fatou Theorems

177

space .CWAm−1 [Lp (∂Ω, σ )] and on the Sobolev-based complex Whitney array p space .CWAm−1 [L1 (∂Ω, σ )].   As such, for each array .g˙ = g(a,b) a,b∈N0 ∈ CWAm−1 [Lp (∂Ω, σ )], one has a+b≤m−1

  C˙ m−1 g˙ 

.

CWAm−1 [Lp (∂Ω,σ )]

  and for each array .g˙ = g(a,b)   C˙ m−1 g˙ 

.

a,b∈N0 a+b≤m−1 p

 g ˙ CWAm−1 [Lp (∂Ω,σ )] , p

∈ CWAm−1 [L1 (∂Ω, σ )], one has

CWAm−1 [L1 (∂Ω,σ )]

 g ˙ CWAm−1 [Lp (∂Ω,σ )] , 1

Remarkably, the boundary-to-boundary poly-Cauchy operator .C˙ m−1 is tied up with its boundary-to-domain version .C˙m−1 via the jump-formula described below. Theorem 8 Let .Ω ⊆ R2 be a UR domain with compact boundary. Fix an integer .m ∈ N along with an integrability exponent .p ∈ (1, ∞). Then for each given array .g ˙ ∈ CWAm−1 [Lp (∂Ω, σ )], one has    1 ˙ ˙ ˙ TrC m−1 Cm−1 g˙ = 2 I + Cm−1 g.

.

(14.19)

As a corollary of Theorem 6 and (14.19), one obtains (see item (4) in Theorem 1) .

  2 C˙ m−1 = 14 I as operators on CWAm−1 Lp (∂Ω, σ ) .

Acknowledgments The authors gratefully acknowledge partial support from the Simons Foundation (grants .# 958374, .# 637481), and NSF (grant .# 1900938).

References [HoEtAl10] Hofmann, S., Mitrea, M., Taylor, M.: Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains. Int. Math. Res. Not. IMRN 2010(14), 2567–2865 (2010) [HoEtAl21] Hoepfner, G., Liboni, P., Mitrea, D., Mitrea, I., Mitrea, M.: Multi-layer potentials for higher order systems in rough domains. Anal. PDE 14(4), 1233–1308 (2021) [Mi18] Mitrea, D.: Distributions, Partial Differential Equations, and Harmonic Analysis (2nd ed.). Springer Nature, Switzerland (2018) [MiEtAl13] Mitrea, I., Mitrea, M.: Multi-Layer Potentials and Boundary Problems for HigherOrder Elliptic Systems in Lipschitz Domains. Lecture Notes in Mathematics, vol. 2063. Springer, Berlin (2013) [MiEtAl20] Mitrea, D., Mitrea, I., Mitrea, M.: A sharp divergence theorem with nontangential traces. Notices Amer. Math. Soc. 67(9), 1295–1305 (2020) [MiEtAl22] Mitrea, D., Mitrea, I., Mitrea, M.: Geometric Harmonic Analysis I–V, Developments in Mathematics, vol. 72–76. Springer Nature, Switzerland (2022)

Chapter 15

The Influence of the Refractive Index and Absorption Coefficients in the Solution of the Radiative Conductive Transfer Equation in Cartesian Geometry C. A. Ladeia, M. Schramm, J. C. L. Fernandes, H. R. Zanetti, and A. D. Albuquerque

15.1 Introduction During the last decades, general interest boomed in research on the radiative conductive transfer equation in a semi-transparent medium [AbDe00, FaEtAl22, HoEtAl20, LaEtAl02, LiEtAl04, LuEtAl10, YiEtAl12]. Some examples of applications are furnaces, ceramics, optical flame production, glass production, coating systems, and atmospheric problems [MiEtAl18], among many others. Following this trend, we are interested in the influence of the refractive index and absorption coefficients in the solution of the radiative conductive transfer problem in Cartesian geometry. In this context, numerical solutions of the nonlinear .SN problem [Ch50, LeMi84] are addressed, where the solution is constructed using the finite difference [OzEtAl17] and the decomposition method [Ad88]. Further, some particularities of the combination of the mentioned methods are discussed. Thus, the objective of this chapter is to show the results obtained for a semitransparent medium bounded by two black surfaces with prescribed temperatures for several constant absorption coefficients. In Sect. 15.2, we present the model, i.e., the radiative conductive transfer equation in Cartesian geometry. Next, Sects. 15.3 and 15.4 are dedicated to the discrete ordinate method in the angular variable and the spatial discretization using the backward and central difference schemes. Then,

C. A. Ladeia () · J. C. L. Fernandes Institute of Mathematics, Federal University of Rio Grande do Sul, Porto Alegre, Brazil e-mail: [email protected]; [email protected] M. Schramm Graduate Program in Mathematical Modeling, Federal University of Pelotas, Pelotas, Brazil e-mail: [email protected] H. R. Zanetti · A. D. Albuquerque Engineering School, Federal University of Rio Grande do Sul, Porto Alegre, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_15

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in Sect. 15.5 the decomposition method for the treatment of the nonlinearity is outlined, and numerical results for a set of physical parameters for a semi-transparent medium are presented in Sect. 15.6. Finally, Sect. 15.7 contains concluding remarks and some future perspectives.

15.2 The Radiative Conductive Transfer Equation in Cartesian Geometry As a starting point, we consider the radiative conductive transfer equation in Cartesian geometry though without using dimensionless quantities as in reference [Oz73]. σ n2 (x) 4 σs ∂I (x, μ) T (x) + + βI (x, μ) = κ .μ π 2 ∂x



1

−1

    P μ, μ I x, μ dμ , (15.1)

Here .x ∈ [0, L] is the geometric optical depth in units of m, .μ ∈ [−1, 1] is the direction cosine .μ = cos θ , .θ is the direction polar angle in radian, .μ is the direction cosine of the incident rays, .σ = 5.6699×10−8 W m−2 K −4 is the Stefan-Boltzmann constant, .n(x) is the refractive index, and .κ, .σs , and .β are the absorption, scattering, and extinction coefficients (.m−1 ), which we assume to be constant. The radiation intensity .I (x, μ) (.W m−2 ) and the temperature .T (x) (K) are the unknowns. Further, P is the phase function and may be approximated as a truncated series of Legendre polynomial [Ch50] L      P μ, μ = α P (μ) P μ ,

.

=0

where .P is the Legendre polynomial of degree ., .L is the truncation order, and .α are constant coefficients. The heat conduction equation with radiation input is k

.

d2 T (x) dqr (x) , = 2 dx dx

(15.2)

where k is the thermal conductivity (.W m−1 K −1 ) and the radiative flux .qr (x) (.W m−2 is computed by  qr (x) = 2π

1

.

−1

μI (x, μ) dμ .

The boundary conditions of equation (15.1) are

(15.3)

15 The Influence of Refractive Index and Absorption Coefficients

I (0, μ) = 1

.

σ n20 4 T (0) − 2ρ1 π



0 −1

181

I (0, μ )μ dμ

(15.4)

I (L, μ )μ dμ

(15.5)

for .μ > 0 and I (L, μ) = 2

.

σ n2d 4 T (L) + 2ρ2 π



1 0

for .μ < 0. Here, . 1 and . 2 are the emissivities of the surfaces at .x = 0 and .x = L, respectively. Further, .n0 and .nd are refractive indexes, and .ρ1 and .ρ2 are the diffusive reflectivities on the surfaces at .x = 0 and .x = L, respectively. The boundary conditions of equation (15.2) are .

T (0) = TO , .

(15.6)

T (L) = TL ,

(15.7)

where .TO and .TL are prescribed temperatures. Note that further, the focus is put on the behavior of .T (x) admitting variations in .n(x).

15.3 The Discrete Ordinate Method One of the approaches which may be found in the literature for the transport equation is the so-called .SM equation [Ch50, LeMi84] for the discretized variable .μ. This method is based on choosing M values for .μ, .{μm }M 0 , and their corresponding weights in a quadrature rule, .{wm }M , so that equations to be evaluated are (15.1), 0 (15.4), and (15.5) for all .μm . For convenience one may define .Im (x) = I (x, μm ), so that the .SM equations are M dIm (x) σs  σ n2 (x) 4 T (x) + + βIm (x) = κ wm P (μm , μm )Im (x) .μm π 2  dx m =1 (15.8) for .m = 1, 2, . . . , M

Im (0) = 1

.

M/2  σ n20 4 T (0) − 2ρ1 μm wm Im (0) , π 

(15.9)

m =1

for .m = M/2 + 1, M/2 + 2, . . . , M and Im (L) = 2

.

σ n2d 4 T (L) + 2ρ2 π

M  m =M/2+1

μm wm Im (L)

(15.10)

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for .m = 1, 2, . . . , M/2. The radiative flux in equation (15.3) is then computed using the quadrature rule qr (x) = 2π

M 

.

wm μm Im (x) .

(15.11)

m=1

15.4 Spatial Discretization  N The spatial variable x is then discretized in .N + 1 equally spaced nodes, . x i 0 , and the shorthand notations .Imi = Im (x i ), .T i = T (x i ), and .qri = qr (x i ) are introduced. Now, from equation (15.11), .qri is computed qri = 2π

M 

.

wm μm Imi

m=1

for .i = 0, 1, . . . , N . We use the backward difference scheme oriented by the sign of .μm in the derivatives with respect to x in (15.8). This means that for .m = 1, 2, . . . , M/2, we evaluate equation (15.8) in .xi for .i = 0, 1, . . . , N − 1, leading to μm

.

Imi+1 − Imi i + βImi = ψm . Δx

(15.12)

Similarly, for .m = M/2 + 1, M/2 + 2, . . . , M, we evaluate equation (15.8) in .xi for .i = 1, 2, . . . , N, yielding μm

.

Imi − Imi−1 i + βImi = ψm , Δx

(15.13)

where .Δx = L/N is the distance between two consecutive nodes. In equations (15.12) and (15.13), i .ψm

M σ n2 (x i )  i 4 σs  T + wm P (μm , μm )Imi  . =κ π 2 

(15.14)

m =1

The intensities at the boundaries (15.9) and (15.10) may be written as 0 .Im

M/2  σ n20  0 4 T = 1 − 2ρ1 μm wm Im0  π  m =1

(15.15)

15 The Influence of Refractive Index and Absorption Coefficients

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for .m = M/2 + 1, M/2 + 2, . . . , M, and as ImN = 2

.

σ n2d  N 4 T + 2ρ2 π

M  m =M/2+1

μm wm ImN

(15.16)

for .m = 1, 2, . . . , M/2. Solving now equations (15.12) and (15.13) for .Imi yields Imi =

.

i − μ I i+1 Δxψm m m Δxβ − μm

(15.17)

for .m = 1, 2, . . . , M/2 and .i = 0, 1, . . . , N − 1 Imi =

.

i + μ I i−1 Δxψm m m Δxβ + μm

(15.18)

for .m = M/2 + 1, M/2 + 2, . . . , M and .i = 1, 2, . . . , N. Then, evaluating equation (15.2) in .x i for .i = 1, 2, . . . , N − 1 using the central finite difference scheme in the derivatives, one gets k

.

qri+1 − qri−1 T i+1 − 2T i + T i−1 , = 2Δx Δx 2

(15.19)

where the solution for .T i is Ti =

.

Δx  1  i+1 T qri+1 − qri−1 . + T i−1 − 2 4k

(15.20)

Last, equations (15.6) and (15.7) may be cast in index notation T 0 = TO

.

and

T N = TL .

15.5 The Decomposition Method We use the Adomian decomposition method [Ad88] to solve the emerging set of nonlinear equations. Generally, the decomposition method is based on the expansion of the unknowns in an infinite series and upon substituting the infinite series in the governing equation of the problem allows to set up a recursive system for each term, due to the fact that initially one differential equation may be decomposed into an infinite set of differential equations, where the nonlinearity is composed from known solutions of previous recursion steps and thus they are known. To this end, we expand .Imi , .T i and .qri in infinite series

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Imi =

.

∞   Imi , j =0

Ti =

∞   T i ,. j =0

qri =

j

(15.21)

j

∞   qri , j =0

(15.22)

j

where the terms are computed in recursive fashion as detailed next. Upon substituting the decomposition of .Imi in equation (15.21) in the boundary conditions (15.15) and (15.16), we build the recursive equation system ⎧ ⎨ σ n2d T N 4 ,

 2 π N . Im =  N  ⎩2ρ2 M j m =M/2+1 μm wm Im j −1 ,

j =0 j ≥1

for .m = 1, 2, . . . , M/2 and ⎧ ⎨ σ n20 T 0 4 ,

 j =0 1 π 0 . Im =  0 M/2 ⎩−2ρ1 j   m =1 μm wm Im j , j ≥ 1 for .m = M/2+1, M/2+2, . . . , M. Further, substituting the expansion in equations (15.17) and (15.18) leads to  i  

 Δx ψm − μm Imi+1 j j i . Im = j Δxβ − μm for .m = 1, 2, . . . , M/2 and .i = 0, 1, . . . , N − 1 and  i  

 Δx ψm + μm Imi−1 j j i . Im = j Δxβ + μm for .m = M/2 + 1, M/2 + 2, . . . , M and .i = 1, 2, . . . , N in all recursive steps, where

.

i ψm

 j

m−1



 σs  = Aˆ i + wm P (μm , μm ) Imi  j j 2  m =1

+

M

 σs  wm P (μm , μm ) Imi  , j −1 2  m =m

15 The Influence of Refractive Index and Absorption Coefficients

185

 from equation (15.14). Here . Aˆ i is the nonlinear term in equation (15.14) in the j

j -th recursion, such that σ n2 (x i )  i 4  ˆ i  T A = . j π ∞

κ

.

j =0

 We compute . Aˆ i as j

.

Aˆ i

 j

=κ

 2  2  σ n2 (x i ) i   i , T Tj + Tji+1 Tji + Tji+1 j π

where .Tji is the partial solution .T i in equation (15.21) added up to the j -th recursion or Tji =

.

j   Ti j  =0

j

.

Substituting the expansions for equations .[T i ]j and .[qri ]j in equation (15.20),   we build the recursive equations with a relaxation in the .qri input to compute . T i j  Z 

    



Δx  i+1  1 i+1  i−1 i i−1 − qr T . T = + T − qr j j j −z j −z j 2 4kZ z=1 (15.23) for .i = 1, 2, . . . , N − 1, where .Z  = min(j, Z). The relaxation is the sum in z and introduced in the computer code to avoid overflow. Here, Z is a relaxation   parameter, i.e., it is the quantity of delayed effects on the input of . qri j in the . T i j equation. Note that adding up the equations (15.23) for .j  =  0, 1, . . . and using the expansions in equations (15.21), we get (15.20), where . qri j /Z is added up Z times in this sum for all j . Moreover, 

 TO , 0 . T = j 0,

j =0 j ≥0

and

T

N

 j

 =

TL , j = 0 0,

j ≥0

.

15.6 Numerical Results In this section, we present cases to test the consistency of the proposed method for applications. Although the presented computational model is adequate to simulate

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Fig. 15.1 The temperature field for cases 1 and 2, .κ = 1m−1 . (a) Case 1, .k = 0.1 2, .k = 1 mWK

W mK .

(b) Case

Fig. 15.2 The temperature field for cases 3 and 4, .κ = 10 m−1 . (a) Case 3, .k = 0.1 4, .k = 1 mWK

W mK .

(b) Case

Fig. 15.3 The temperature field for cases 5 and 6, .κ = 100 m−1 . (a) Case 5, .k = 0.1 Case 6, .k = 1 mWK

W mK .

(b)

several configurations, in this discussion, we consider isotropic scattering (.L = 0) only, with . 1 = 2 = 1, .ρ1 = ρ2 = 0, .n0 = nd = n(x) = 1.5, .L = 0, 01 m, .TO = 1000 K, and .TL = 1500 L for all cases. In Figs. 15.1, 15.2, 15.3, 15.4, we

15 The Influence of Refractive Index and Absorption Coefficients

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Fig. 15.4 The temperature field for cases 7 and 8, .κ = 1000 m−1 . (a) Case 7, .k = 0.1 Case 8, .k = 1 mWK Table 15.1 Parameter sets for the eight cases

 Cases 1 2 3 4 5 6 7 8

.κ .

1 m

1 1 10 10 100 100 1000 1000

  k . mWK 0.1 1 0.1 1 0.1 1 0.1 1

M 2 2 20 10 20 10 20 10

N 10 2 30 10 40 20 60 40

W mK .

(b)

Z 1 1 10 1 50 10 50 10

present the influence of different values for the absorption coefficient .κ and thermal conductivity k (listed in Table 15.1) on the temperature field. These parameters were obtained from [AbDe00], and the present results agreed reasonably well with the findings from this reference. M We used a Gauss-Legendre quadrature in .{μm }M 1 and .{wm }0 . To test stability of the implemented algorithm, simulation with higher values of M and N were executed, however the solutions for the convergent cases did not present meaningful differences in the temperature values. Also, the Z values presented in Table 15.1 are the lowest simulated values that were sufficient for convergence.

15.7 Conclusions In the present work, the radiative conductive transfer equation in Cartesian geometry was solved by a numerical scheme using a spatial, angular mesh and the decomposition method, where the nonlinear contributions appear as a source terms in a recursive scheme. This way, the original nonlinear problem was decomposed in a system of linear recursive equations. Differently than the original method,

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we introduced a modified decomposition procedure using a Z parameter, which ultimately resulted in convergent series, equations (15.21). We obtained results for the temperature field for all eight cases, where different values for the thermal conductivity and the absorption coefficient were considered. The found solutions are in agreement with expectations for semi-transparent media, which in turn may be used as an indication that the computational implementations are consistent. In our simulations, we presented the results for the eight cases from reference [AbDe00], where all of them converged for an appropriate choice of Z; however, our results were slightly different from the reference due to the considered space dependence on n in this work. However, this issue is still a subject of investigation in a future work. The computational time depended only weakly on the choice of M, but noticeably on the choice of N and depended strongly on the choice of Z. Cases with low Z values were computed in a few seconds, while the cases where higher values of Z were required also higher computational times of the order of hours were needed until the convergence criterion was satisfied. All the cases were simulated on a standard personal computer. As already outlined in Sect. 15.6, the Z values presented in Table 15.1 were the lowest implemented values that resulted in convergence. Although these values are not the lowest possible choices for Z, they represent reference values, since considering higher values for Z implies a longer computational time. Note that the present work is the first one into a new direction, where our next steps are to investigate the behavior of the algorithm in cases with space-dependent refractive index, anisotropy, reflective boundary conditions for I , and convective boundary conditions for T . Further, we plan to introduce changes in the formulation and allow for a space dependence in the thermal conductivity and in the scattering, absorption, and extinction coefficients, considering a heterogeneous medium. Evidently, also pertinent operational issues will be analyzed such as convergence and stability criteria of the methodology. Moreover, we will also focus on physics and engineering applications like radiotherapy dose distributions, thermal comfort in buildings, or research in the solar energy sector.

References [AbDe00] Abdallah, P.B., Dez, V.L.: Radiative flux field inside an absorbing-emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling. J. Quant. Spectrosc. Radiat. Transfer 67, 125–137 (2000) [Ad88] Adomian, G.A.: A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 145, 501–544 (1988) [Ch50] Chandrasekhar, S.: The Radiative Transfer. Oxford University Press, New York (1950) [FaEtAl22] Fan, C., Xia, X.L., Du, W., Sun, C., Li Y.: Numerical investigations of the coupled conductive-radiative heat transfer in alumina ceramics. Int. Commun. Heat Mass Transf., 135, 1–11 (2022) [HoEtAl20] Howell, J.R., Menguç, M.P., Daun, K., Siegel, R.: Thermal Radiation Heat Transfer. CRC Press, Boca Raton (2020)

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[LaEtAl02] Lacroix, D., Parent, G., Asllanaj, F., Jeandel, G.: Coupled radiative and conductive heat transfer in a non-grey absorbing and emitting semitransparent media under collimated radiation. J. Quant. Spectrosc. Radiat. Transfer 75, 589–609 (2002) [LeMi84] Lewis, E.E., Miller, W.F.: Computational Methods of Neutron Transport. Wiley, New York (1984) [LiEtAl04] Liu, L., Zhang, H., Tan, H.P.: Discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index. J. Quant. Spectrosc. Radiat. Transfer 83, 223–228 (2004) [LuEtAl10] Luo, J.F., Yi, H.L., Chang, S.L., Tan, H.P.: Radiation and conduction in an isotropic scattering rectangular medium with one semitransparent and diffusely reflecting boundary. J. Heat Transfer 132, 1127011–11270111 (2010) [MiEtAl18] Mironov, R.A., Zabezhailov, M.O., Cherepanov, V.V., Rusin, M.Yu.: Transient radiative-conductive heat transfer modeling in constructional semitransparent silica ceramics. Int. J. Heat Mass Transf. 127, 1230–1238 (2018) [Oz73] Ozisik, M.N.: Radiative Transfer and Interaction with Conduction and Convection. Wiley, New York (1973) [OzEtAl17] Ozisik, M.N., Orlande, H.R.B., Colaço, M.J., Cotta, R.M.: Finite Difference Methods in Heat Transfer. CRC Press, Boca Raton (2017) [YiEtAl12] Yi, H.L., Wang, C.H., Tan, H.P., Zhou, Y.: Radiative heat transfer in semitransparent solidifying slab considering space-time dependent refractive index. Int. J. Heat Mass Transf. 55, 1724–1731 (2012)

Chapter 16

Boundary Value Problems for Elliptic Systems on Weighted Morrey Spaces in Rough Domains M. Laurel and M. Mitrea

16.1 Introduction The main result in this article is a well-posedness result for the Dirichlet boundary value problem, formulated in a rough domain and with boundary data in a Muckenhoupt weighted Morrey space. We state the theorem now with the promise that the various mathematical objects appearing in the statement will be explained later. Theorem 1 Let .Ω ⊆ Rn be an AR domain and define .σ := H n−1 ∂Ω. Fix two integrability exponents .p, q ∈ (1, ∞), togetherwith a Muckenhoupt  weight .w ∈ Ap (∂Ω, σ )∩RHq (∂Ω, σ ), and a threshold .λ ∈ 0, q(n−1)/(q −1) . Suppose L is a second-order, weakly elliptic, homogeneous, constant complex coefficient, .M × M system in .Rn , and consider the following Dirichlet boundary value problem: ⎧ M  ⎪ u ∈ C ∞ (Ω) , Lu = 0 in Ω, ⎪ ⎪ ⎨ Nκ u ∈ M p,λ (∂Ω, w), . ⎪ ⎪ ⎪  M ⎩

κ−n.t. u ∂Ω = f ∈ M p,λ (∂Ω, w) .

(16.1)

If .Ω is a .δ-AR domain for .δ ∈ (0, 1) sufficiently small, and the system L along with its transpose .L have distinguished coefficient tensors, then (16.1) is uniquely solvable and the solution satisfies Nκ uM p,λ (∂Ω,w) ≤ Cf [M p,λ (∂Ω,w)]M ,

.

M. Laurel · M. Mitrea () Baylor University, Waco, TX, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_16

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for some constant .C ∈ (0, ∞), depending only on the AR constants of .∂Ω, n, p, 1 q, .λ, .[w]Ap , .[w]RHq , and A. Moreover, if .A ∈ Adis L , then the operator . 2 I + KA (where .KA is the boundary-to-boundary double layer associated with A, cf. (16.4)) is invertible on .[M p,λ (∂Ω, w)]M , and the unique solution u of (16.1) is given by  u := DA ( 12 I + KA )−1 f , in Ω,

.

(16.2)

where .DA is the boundary-to-domain double layer associated with A (cf. (16.3)). Throughout, .H n−1 stands for the .(n − 1)-dimensional Hausdorff measure in We say that .Σ ⊆ Rn is Ahlfors-David regular (ADR for short) if .Σ is a closed set and there exist two constants .cΣ , CΣ ∈ (0, ∞) so that, for any “surfaceball” .Δ(x, R) := B(x, R) ∩ Σ centered at .x ∈ Σ and of radius .R ∈ 0, 2diam(Σ) , one has n .R .

  cΣ R n−1 ≤ H n−1 Δ(x, R) ≤ CΣ R n−1 .

.

Unless otherwise stated, it is agreed that .Σ denotes a closed Ahlfors-David regular subset of .Rn , and we denote the corresponding “surface measure” by the symbol n−1 Σ. .σ := H An Ahlfors regular domain (AR domain, for short) is an open set .Ω ⊆ Rn whose topological boundary .∂Ω is ADR and .H n−1 (∂Ω \ ∂∗ Ω) = 0. Here, .∂∗ Ω denotes the geometric measure theoretic boundary of .Ω, defined by    L n B(x, r) ∩ Ω .∂∗ Ω := x ∈ ∂Ω : lim sup > 0, rn r→0+ lim sup r→0+

  L n B(x, r) \ Ω > 0 , rn

where .L n stands for n-dimensional Lebesgue measure. Also appearing in the statement of the Dirichlet problem (16.1) is a size condition given in terms of the nontangential maximal operator. To define this, we first introduce the nontangential approach region .Γκ (x). Let .Ω ⊂ Rn be an open set, and fix an aperture parameter .κ ∈ (0, ∞), then

 Γκ (x) := y ∈ Ω : |y − x| < (1 + κ) dist(y, ∂Ω) for each x ∈ ∂Ω.

.

For example, in the case when .Ω := Rn+ , the nontangential approach regions are upright, infinite, circular cones, with vertices at points on .∂Rn+ , and aperture angle determined by .κ. The (.κ-)nontangential maximal operator, .Nκ , acts on any given n .L -measurable function .u : Ω → R according to   Nκ u (x) := uL∞ (Γκ (x), L n ) for each x ∈ ∂Ω,

.

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193

  with the convention that . Nκ u (x) = 0 whenever .Γκ (x) = ∅. In our formulation of the Dirichlet problem in (16.1), because we do not require continuity of solutions up to and including the boundary, we must describe the boundary trace in a nontangential sense. The nontangential limit of u at x from within .Γκ (x) exists, and its value is .a ∈ R, provided for every ε > 0 there exists some r > 0 such that

.

|u(y) − a| < ε for L n -a.e. point y ∈ Γκ (x) ∩ B(x, r).  κ−n.t.  When this limit exists, we denote it by . u ∂Ω (x). In our work, we consider domains which are sufficiently “flat,” a property quantified in terms of the size of the BMO seminorm of their measure theoretic outward unit normals. The John-Nirenberg space of bounded mean oscillation, 1 (Σ, σ ) such that .BMO(Σ, σ ), is the collection of functions .f ∈ L loc f BMO(Σ,σ ) := sup

.

Δ⊆Σ

Δ



f −

Δ



f dσ dσ < +∞,

where the supremum is taken over all surface balls .Δ ⊆ Σ. Given some .δ ∈ (0, 1), we say .Ω ⊆ Rn is a .δ-AR domain provided .Ω is an AR domain with the additional property that .νBMO(∂Ω,σ ) < δ, where .ν denotes the measure theoretic outward unit normal to .Ω. The existence of .ν at .σ -a.e. point on .∂Ω is guaranteed by the fact that .H n−1 (∂Ω \ ∂∗ Ω) = 0, a defining trait of AR domains. A set .Σ ⊆ Rn is called uniformly rectifiable (UR set, for short) if .Σ is ADR, and there exist  constants .ε,  M ∈ (0, ∞) such that for each location .x ∈ Σ and each scale .R ∈ 0, 2diam(Σ) , it is possible to find a Lipschitz map .ϕ : BRn−1 → Rn (where .BRn−1 denotes a .(n − 1)-dimensional ball) with Lipschitz constant .≤ M such that   H n−1 Σ ∩ B(x, R) ∩ ϕ(BRn−1 ) ≥ εR n−1 .

.

We shall often assume that .Ω is a UR domain, i.e., an open subset of .Rn whose topological boundary is a UR set with .H n−1 (∂Ω \ ∂∗ Ω) = 0. This is the most inclusive geometric setting in which we can study singular integral operators of Calderón-Zygmund type. Let L be a second-order, homogeneous, constant complex coefficient, .M × M αβ system in .Rn . Hence, .L = (aj k ∂j ∂k )1≤α,β≤M (here and elsewhere we employ the αβ

convention of summation over repeated indices). The coefficient .aj k belongs to .C for each .j, k ∈ {1, . . . , n} and .α, β ∈ {1, . . . , M}, and the collection of coefficients  αβ  .A := aj k 1≤j,k≤n will be referred to as a coefficient tensor. The choice of 1≤α,β≤M

coefficient tensor is not unique, and the collection of all possible coefficient tensors for a given system L will be denoted by .AL . In fact, .A, B ∈ AL if and only if .A − B

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is antisymmetric in the lower indices. The system .L = (aj k ∂j ∂k )1≤α,β≤M acts on a vector-valued function .u := (uβ )1≤β≤M according to   αβ Lu = aj k ∂j ∂k uβ

.

1≤α≤M

.

Such a system L will be called weakly elliptic provided the characteristic matrix L(ξ ) := −

.

   αβ aj k ξj ξk 1≤α,β≤M for each ξ = (ξi )1≤i≤n ∈ Rn ,

satisfies .det[L(ξ )] = 0 for each .ξ ∈ Rn \ {0}. Note that the characteristic matrix is independent of the choice of coefficient tensor. From here on out, let .E = (Eαβ )1≤α,β≤M be the fundamental solution to L in the sense of [Mit18]. Given a UR domain .Ω ⊆ Rn , a weakly elliptic, second-order, homogeneous, .M × M system L, and a coefficient tensor .A ∈ AL , the boundary  σ (x)  M to-domain double-layer potential, .DA , acts on functions .f ∈ L1 ∂Ω, 1+|x| n−1 as  ˆ   βα  .DA f (x) := − νk (y)aj k ∂j Eγβ (x − y)fα (y) dσ (y) , (16.3) 1≤γ ≤M

∂Ω

at each point .x ∈ Ω. Here, .ν denotes the geometric measure theoretic outward unit normal to .Ω (in the sense of De Giorgi and Federer), which exists .σ -a.e. on .∂Ω given that .Ω is a UR domain. The boundary-to-domain double-layer potential is a mechanism for producing many null solutions for the weakly elliptic system L. That is, .L(DA f ) = 0 for any choice of coefficient tensor .A ∈ AL , and any choice   σ (x)  M (cf. [M+ 22]). A closely related operator of function .f ∈ L1 ∂Ω, 1+|x| n−1 is the boundary-to-boundary double-layer potential, denoted .KA , whose action on   σ (x)  M each function .f ∈ L1 ∂Ω, 1+|x| at .σ -a.e. point .x ∈ ∂Ω is n−1  KA f (x) :=

.

ˆ − lim

ε→0+ y∈∂Ω |x−y|>ε

 βα  νk (y)aj k ∂j Eγβ (x − y)fα (y) dσ (y)

 . 1≤γ ≤M

(16.4) Unlike the characteristic matrix and the fundamental solution, the double-layer potentials strongly depend on the choice of coefficient tensor. Sometimes there exists what we perceive as a particularly “good” choice of coefficient tensor. Specifically, we say a coefficient tensor .A ∈ AL is distinguished if there is a matrix M×M , which is even, and positive homogeneous valued function .k ∈ C ∞ (Rn \{0}) of degree .−n such that the integral kernels appearing in the double layers associated with A in (16.3) and (16.4) have the form .ν(y), x − yk(x − y). The set of all distinguished coefficient tensors will be denoted .Adis L .

16 Boundary Value Problems for Elliptic Systems

195

For example, the Laplacian, .Δ := ∂j ∂j , and the Lamé system of elasticity, Lμ,λ := μΔu + (μ + λ)∇div u (where .μ, λ ∈ C are such that .μ = 0, .2μ + λ = 0, and .3μ + λ = 0), are examples of weakly elliptic systems which have distinguished coefficient tensors. However, the system .LD := Δu − 2∇div u is weakly elliptic, but does not have a distinguished coefficient tensor, and this will result in a failure of both existence and uniqueness of solutions to (16.1) when .L := LD (cf. [M+ 22]). The Muckenhoupt class .Ap (Σ, σ ) is defined as the collection of weights (i.e., .σ -measurable, positive, and finite .σ -a.e. functions) .

  Ap (Σ, σ ) := w : Σ → [0, ∞] : w is a weight on Σ and [w]Ap < ∞ ,

.

where for .p, p ∈ (1, ∞) with .1/p + 1/p  = 1, the .Ap characteristic .[w]Ap of w is  [w]Ap := sup



Δ⊆Σ



p−1

w 1−p dσ

w dσ

.

Δ

,

Δ

with the supremum taken over all surface balls. We will often identify the measure wσ with the weight function w. Closely related to the Muckenhoupt class is the reverse Hölder class, .RHq (Σ, σ ), with .q ∈ (1, ∞). This is the collection of all weights .w : Σ → [0, ∞] for which there exists a constant .C ∈ (0, ∞) such that .



1/q w(x)q dσ (x) ≤C

.

Δ

(16.5)

w(x) dσ (x), Δ

for all surface balls .Δ ⊆ Σ, with the smallest such constant C denoted by .[w]RHq . It is a fact that for any .w ∈ Ap (Σ, σ ) with .p ∈ (1, ∞) arbitrary, there exist .C ∈ (0, ∞) and .q ∈ (1, ∞), both of which depend on n, p, .[w]Ap , and the Ahlfors regularity constants of .Σ such that (16.5) holds (hence .w ∈ RHq (Σ, σ )). Given an Ahlfors regular set .Σ, an integrability exponent .p ∈ (1, ∞), a Muckenhoupt weight .w ∈ Ap (Σ, σ ), and a threshold .λ ∈ [0, ∞), the Muckenhoupt-weighted Morrey space is defined by   M p,λ (Σ, w) := f : Σ → R : f is σ -measurable and f M p,λ (Σ,w) < +∞ ,

.

where  f M p,λ (Σ,w) :=

.

sup x∈Σ 0ε

ˆ Tf (x) := lim

.

ε→0+ y∈Σ |x−y|>ε

Then .T∗ f is meaningfully defined in (16.8) as a .σ -measurable   function on .Σ for σ (x)  any given .f ∈ L1 Σ, 1+|x| . Additionally, whenever . λ ∈ 0, (n − 1)/q , one n−1 has    .T∗ f M p,λ (Σ,w) ≤ C sup |∂ γ k| f M p,λ (Σ,w) , n−1 |γ |≤N S

where .C ∈ (0, ∞) depends only on the UR constants of .Σ, n, p, .λ, and .[w]Ap in such a way that C stays bounded whenever .[w]Ap stays bounded.  σ (x)  Furthermore, for each .f ∈ L1 Σ, 1+|x| n−1 , the limit defining .Tf (x) in (16.9)   exists .σ -a.e. on .Σ, and for each .λ ∈ 0, (n − 1)/q  the operator T : M p,λ (Σ, w) → M p,λ (Σ, w)

.

is well defined, linear, and bounded. Next, let .Ω ⊆ Rn be a UR domain. Pick an aperture parameter .κ ∈ (0, ∞). With integral kernel k as before, define for each .f ∈ M p,λ (∂Ω, w) the operator ˆ T f (x) :=

k(x − y)f (y) dσ (y) for all x ∈ Ω.

.

∂Ω

  Then, assuming .λ ∈ 0, (n − 1)/q  , there exists .C ∈ (0, ∞), which depends only on the UR constants of .Σ, n, p, .λ, q, .[w]RHq , and .[w]Ap in such a way that C stays bounded whenever .[w]Ap stays bounded, with the property that for each given function .f ∈ M p,λ (∂Ω, w), one has Nκ (T f )M p,λ (∂Ω,w) ≤ C ·

.

 

 sup |∂ γ k| · f M p,λ (∂Ω,w) .

(16.10)

n−1 |γ |≤N S

Moreover, for each .f ∈ M p,λ (∂Ω, w), one has the jump relation  .

κ−n.t.    1 T f ∂Ω (x) = √  k ν(x) f (x) + (Tf )(x) at σ -a.e. x ∈ ∂Ω, 2 −1

(16.11)

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where  .k denotes the Fourier transform of k and .ν denotes the geometric measure theoretic (GMT) outward unit normal to .Ω. If the integral kernel in the definition of the principal value operator in (16.10) exhibits a specific algebraic structure, referred to in [M+ 22] as being of chord-dotnormal type, then it is possible to obtain finer estimates on the operator norm, now involving the BMO seminorm of the GMT outward unit normal. Theorem 5 Let .Ω ⊆ Rn be a UR domain and set .σ := H n−1 ∂Ω. Consider a sufficiently large number .N = N(n) ∈ N, and suppose .k ∈ C N (Rn \ {0}) is a complex-valued function whichis even and positive homogeneous of degree .−n. Let σ (x)  T act on each function .f ∈ L1 ∂Ω, 1+|x| according to n−1 ˆ (Tf )(x) := lim

.

ε→0+

ν(y), x − yk(x − y)f (y) dσ (y)

y∈∂Ω |x−y|>ε

for .σ -a.e. point .x ∈ ∂Ω.  Next, fix .p, q ∈ (1, ∞), .w ∈ Ap (∂Ω, σ ) ∩ RHq (∂Ω, σ ), and .λ ∈ 0, (n − 1)/q  , where .q  := q/(q − 1) denotes the Hölder conjugate of q. Then there exists .C ∈ (0, ∞), depending only on the UR constants of .∂Ω, n, p, q, .λ, .[w]RHq , and .[w]Ap such that  T M p,λ (∂Ω,w)→M p,λ (∂Ω,w) ≤ C · νBMO(∂Ω,σ ) · ln

.

e νBMO(∂Ω,σ )

 .

(16.12)

Note that the right-hand side of (16.12) vanishes as .νBMO(∂Ω,σ ) goes to 0.

16.3 Boundary Value Problems 16.3.1 Existence To establish the solvability of (16.1), recall the embedding in Theorem 2. Then what we know about double layers ensures that for any .g ∈ [M p,λ (∂Ω, w)]M and any choice of coefficient tensor .A ∈ AL , we have .DA g ∈ [C ∞ (Ω)]M and .L(DA g) = 0 in .Ω. The Calderón-Zygmund theory outlined in Theorems 4 and 5 is applicable to the double-layer potentials introduced in (16.3) and (16.4). In fact, each double layer is a sum of operators of the types considered in Theorem 4, and [M+ 22] ensures that the integral kernels have the right smoothness, homogeneity, and parity. As such, (16.10) of Theorem 4 implies that the size condition in (16.1) is satisfied. Also, one can show that (16.11) implies that the nontangential boundary trace of .DA g is given by

16 Boundary Value Problems for Elliptic Systems

199

κ−n.t. (DA g) ∂Ω = ( 12 I + KA )g at σ -a.e. point on ∂Ω.

.

We now must find a suitable .g ∈ [M p,λ (∂Ω, w)]M so that .( 12 I + KA )g = f . Because the space .[M p,λ (∂Ω, w)]M is Banach, the operator . 21 I + KA is invertible on .[M p,λ (∂Ω, w)]M via a Neumann series provided KA [M p,λ (∂Ω,w)]M →[M p,λ (∂Ω,w)]M < 12 .

.

(16.13)

If .νBMO(∂Ω,σ ) is small enough, i.e., if .Ω is a .δ-AR domain for sufficiently small δ ∈ (0, 1), then we can guarantee (16.13) via Theorem 5, provided .A ∈ Adis L (a condition ensuring that the integral kernel of .KA is of chord-dot-normal type). As noted in [M++ 22], if .δ ∈ (0, 1) is small enough, then .Ω is in fact a UR domain; hence Theorems 4 and 5 are indeed relevant. This allows us to conclude that a solution to (16.1) is given by (16.2), as stated. We also obtain from (16.10) of Theorem 4 the naturally accompanying estimate

.

Nκ uM p,λ (∂Ω,w) ≤ Cf [M p,λ (∂Ω,w)]M ,

.

for some constant .C ∈ (0, ∞), independent of f .

16.3.2 Uniqueness The problem of uniqueness presents us with a different set of challenges, which we address by considering a brand of spaces whose dual are Muckenhoupt weighted Morrey spaces. To elaborate, call .b : Σ → C a .(q, λ, ω)-block if .b ∈ Lq (Σ, ω) and there exists .x0 ∈ Σ and .R ∈ (0, 2diam(Σ)) (both of which may depend on b) such that supp b ⊆ Δ(x0 , R) and bLq (Σ,ω) ≤ R λ(1/q−1) .

.

Fix .q ∈ (1, ∞) and .ω ∈ Aq (Σ, σ ). Let .ε ∈ (0, q −1) be such that .ω ∈ Aq−ε (Σ, σ ), the existence of such weights.  an .ε being guaranteed  by properties of Muckenhoupt ∞ Also, fix .λ ∈ 0, ε(n − 1)/(q − 1) . Suppose .g = λ b for some j =1 j j 1 .{λj }j ∈N ∈  and .{bj }j ∈N a sequence of .(q, λ, ω)-blocks, where convergence of  σ (x)  the series is in .L1 Σ, 1+|x| n−1 . Call such a series a block-representation of g. Define the Muckenhoupt weighted block space .B q,λ (Σ, ω) as the collection of  σ (x)  1 all functions belonging to .L Σ, 1+|x|n−1 which have a block representation. The

space .B q,λ (Σ, ω) is a Banach space when equipped with the norm

200

M. Laurel and M. Mitrea

gB q,λ (Σ,ω) := inf

∞



.

j =1

|λj | : g =

∞ 

 λj bj .

j =1

We also have a similar embedding as in (16.6), encapsulated in the following theorem. Theorem 6 Let .Σ ⊆ Rn be a closed Ahlfors regular set with .σ := H n−1 Σ. Fix .q ∈ (1, ∞) and .ω ∈ Aq (Σ, σ ). Let .ε ∈ (0, q − 1) be such that .ω ∈ Aq−ε (Σ, σ ).   Then with .λ ∈ 0, ε(n − 1)/(q − 1) ,  B q,λ (Σ, ω) → L1 Σ,

.

σ (x)  continuously. 1 + |x|n−1

Our next theorem establishes the duality pairing of Morrey and block spaces and provides a very useful Hölder-type inequality. Theorem 7 Fix .p, p  ∈ (1, ∞) with .1/p + 1/p  = 1 and some .w ∈ Ap (Σ, σ )  with its conjugate weight .w  := w −p /p ∈ Ap  (Σ, σ ). Let .ε ∈ (0, p  − 1) be such  that .w ∈ Ap  −ε (Σ, σ ) and, lastly, fix .λ ∈ 0, ε(n − 1)/(p  − 1) .  Then the space .B p ,λ (Σ, w  ) is the predual of .M p,λ (Σ, w). Moreover, for every p,λ (Σ, w) and .g ∈ B p  ,λ (Σ, w  ), one has .f ∈ M fgL1 (Σ,σ ) ≤ f M p,λ (Σ,w) gB p  ,λ (Σ,w ) .

.

We have seen two different ranges for the parameter .λ present themselves in Theorem 3 and Theorem 7. Both ranges for .λ are reconciled in the following theorem, which points to the fact that each range for the parameter .λ is actually natural. Theorem 8 Fix integrability exponents .p, p  , q, q  ∈ (1, ∞) with .1/p +1/p  = 1 and .1/q + 1/q  = 1. Fix a Muckenhoupt weight .w ∈ Ap (Σ, σ ) ∩ RHq (Σ, σ ) and  denote its conjugate by .w  := w −p /p ∈ Ap  (Σ, σ ). Then .w  ∈ Ap  −ε (Σ, σ ) if   .ε := (p − 1)/q . With a solid functional analytic understanding of Muckenhoupt weighted block spaces, we are now tasked with developing (from the ground up) a CalderónZygmund theory for block spaces. We obtain results akin to Theorems 4 and 5, but suitably formulated for Muckenhoupt weighted block spaces. The reason for exerting so much effort to study the preduals of Morrey spaces stems from our method of proving uniqueness, which relies on a result from [M+ 22]. Namely, under the hypotheses of Theorem 1, for every .x0 ∈ Ω, the value of any function u as in the first three lines of (16.1) at .x0 may be expressed via the following Poisson integral representation formula:

16 Boundary Value Problems for Elliptic Systems

ˆ uβ (x0 ) =

.

∂Ω

 κ−n.t.      u ∂Ω (y) , ∂νA Gx. 0β (y) dσ (y),

201

∀ β ∈ {1, . . . , M}, (16.14) A .∂ν

(Gxαβ0 )1≤α,β≤M

where = is a suitable Green function and stands for  the conormal derivative associated with .A acting on the columns of the matrixx0 x0 valued function   .Gx0 . By “suitable” we mean .G must have certain properties. For one thing, . L G . β α = −δx0 δαβ in the distributional sense. For another, there

κ −n.t. x exists an aperture parameter  .κ such that .G 0

= 0 at .σ -a.e. point on .∂Ω, ∂Ω

 κ −n.t. and .(∇Gx0 ) ∂Ω exists .σ -a.e. on .∂Ω. Finally, we must have the following absolute integrability condition: x .G 0

ˆ .

∂Ω

Ω\K

Nκ u · Nκ

(∇Gx0 ) dσ < +∞

(16.15)

Ω\K

where, for a compact neighborhood K of .x0 , the symbol .Nκ denotes the version of the nontangential maximal operator in which the essential norm is taken only the nontangential approach region intersected with .Ω \ K. Since .Nκ u ∈ M p,λ (∂Ω, w), the requirement on .Gx0 in (16.15) serves as motivation for considering a space of Ω\K functions to which .Nκ (∇Gx0 ) should belong so that the above integral pairing is guaranteed to be finite. This also highlights the significance of Theorem 7 and the inequality therein. In view of (16.14), the issue of uniqueness for (16.1) therefore hinges on the existence of such a Green function. To construct said Green function with pole at .x0 ∈ Ω proceed as follows. Let E denote the fundamental solution of L as in [Mit18] and fix an additional point .x1 ∈ Rn \ Ω. Define for each .x ∈ Ω \ {x0 } the matrix-valued function .Gx0 (x) := E  (x − x0 ) − E  (x − x1 ) − η(x), where .η = (ηαβ )1≤α,β≤M is a “correction,” satisfying ⎧ ⎪ ηαβ ∈ C ∞ (Ω), L (ηγβ )1≤γ ≤M = 0 in Ω, ⎪ ⎪ ⎨  Nκ ηαβ , Nκ (∇ηαβ ) ∈ B p ,λ (∂Ω, w  ), . ⎪ ⎪

κ −n.t.  ⎪ 

⎩ ηαβ ∂Ω = Eβα (· − x0 ) − Eβα (· − x1 ) ∂Ω ,

(16.16)

for each .α, β ∈ {1, . . . , M}. In a broader perspective, this leads us to consider the solvability of the regularity boundary value problem for the transposed system .L on Muckenhoupt weighted block spaces which, after adjusting notation, reads ⎧  ⎪ u ∈ C ∞ (Ω) M , L u = 0 in Ω, ⎪ ⎪ ⎨ Nκ u, Nκ (∇u) ∈ B q,λ (∂Ω, ω), . ⎪ ⎪ ⎪  q,λ M ⎩

κ−n.t. u ∂Ω = h ∈ B1 (∂Ω, ω) .

(16.17)

202

M. Laurel and M. Mitrea q,λ

Here .q ∈ (1, ∞), .ω ∈ Aq (∂Ω, σ ), and .B1 (∂Ω, ω) denotes a block-based Sobolev space of order one, defined as the subspace of .B q,λ (∂Ω, ω) having weak tangential derivatives (mimicking integration by parts on the boundary) in q,λ (∂Ω, ω). .B In order to prove the existence of .η as in (16.16), and hence uniqueness of a solution to (16.1), we first demonstrate the existence of solutions to (16.17). We go about this in an analogous way to the proof of solvability for (16.1), but now using our newly developed Calderón-Zygmund theory for block spaces. In the process, we again require .νBMO(∂Ω,σ ) to be sufficiently small. Additionally, for (16.17) to be solvable, we need .L to have a distinguished coefficient tensor, something not necessarily guaranteed even if L itself has a distinguished coefficient tensor. There is one more subtle point to make regarding the solvability of (16.16). Namely, for the solvability of (16.17) of (16.16), we must  to imply the solvability 

actually have that the function .h := Eβα (· − x0 ) − Eβα (· − x1 ) ∂Ω indeed belongs q,λ

to .B1 (∂Ω, ω) for each .α, β ∈ {1, . . . , M}. The justification relies on the fact that n−1 )−1 ∈ B q,λ (∂Ω, ω) and .(1 + | · | −1 −1   |h(x)|  1 + |x|n−1 and |∇h(x)|  1 + |x|n−1 ,

.

q,λ

uniformly in .x ∈ ∂Ω, which is enough to conclude that .h ∈ B1 (∂Ω, ω). Lastly, we note that the entire argument continues to hold with the roles of Morrey and block spaces switched. Therefore, we can obtain a result in the spirit of Theorem 1, now suitably formulated for Muckenhoupt weighted block spaces.

16.3.3 Examples in the Upper-Half Space Thanks to one of the defining properties of distinguished coefficient tensors, .KA = 0 1 n −1 = 2I in this geometric setting. We if .A ∈ Adis L and .Ω := R+ ; hence .( 2 I + KA ) state two immediate corollaries of Theorem 1 corresponding to the Laplacian and the Lamé system in the upper-half space. Both results are new and nontrivial. Corollary 1 Fix two integrability exponents .p, q ∈ (1, ∞) along with an arbitrary Muckenhoupt weight .w ∈ Ap (Rn−1 , L n−1 ) ∩ RHq (Rn−1 , L n−1 ), and some number .λ ∈ 0, (n − 1)/q  , where .q  := q/(q − 1). Identify .∂Rn+ ≡ Rn−1 , and pick an aperture parameter .κ ∈ (0, ∞). Then the following boundary value problem is well posed: ⎧ u ∈ C ∞ (Rn+ ), Δu = 0 in Rn+ , ⎪ ⎪ ⎪ ⎨ Nκ u ∈ M p,λ (Rn−1 , w), . ⎪ ⎪ ⎪

κ−n.t. ⎩ u ∂Rn = f ∈ M p,λ (Rn−1 , w), +

16 Boundary Value Problems for Elliptic Systems

203

and if .ωn−1 is the area of the unit sphere in .Rn , then its unique solution is given by u(x  , t) :=

.

2 ωn−1

ˆ Rn−1

t    n  n/2 f (y ) dy for all (x , t) ∈ R+ . |x  − y  |2 + t 2

n−1 n−1 ) ∩ RH (Rn−1 , L n−1 ) with .p, q ∈ (1, ∞), Corollary 2 Let q  .w ∈ Ap (R   , L and pick .λ ∈ 0, (n − 1)/q where .q  := q/(q − 1). Also, pick an aperture parameter .κ ∈ (0, ∞) and choose .μ, λ ∈ C with .μ = 0 and .2μ + λ = 0. Finally, consider ⎧  n u ∈ C ∞ (Rn+ ) , μΔu + (μ + λ)∇div u = 0 in Rn+ , ⎪ ⎪ ⎪ ⎨ Nκ u ∈ M p,λ (Rn−1 , w), . (16.18) ⎪ ⎪

κ−n.t.  ⎪ n ⎩ u n = f = (f ) p,λ (Rn−1 , w) . α 1≤α≤n ∈ M ∂R +

Then, under the assumption that .3μ+λ = 0, the boundary value problem (16.18) is well posed, and the unique solution .u = (uα )1≤α≤n is given at any .(x  , t) ∈ Rn+ by uα (x  , t) =

.

1 4μ 3μ + λ ωn−1 +

ˆ

t n

(|x  − y  |2 + t 2 ) 2

Rn−1

μ + λ 2n 3μ + λ ωn−1

ˆ Rn−1

fα (y  ) dy 

t (x  − y  , t)α (x  − y  , t)β (|x 

− y  |2

+ t 2)

n+2 2

fβ (y  ) dy  .

Acknowledgments Work supported by the Simons Foundation grant .# 637481.

References [DR20] Duoandikoetxea, J., Rosenthal, M.: Boundedness of operators on certain weighted Morrey spaces beyond the muckenhoupt range. Potential Anal. 53, 1255–1268 (2020) [Lau22] Laurel, M.: Boundary Value Problems in Muckenhoupt Weighted Morrey Spaces. Ph.D. Thesis, M. Mitrea advisor, Baylor University (2022) [M++ 22] Marín, J., Martell, J., Mitrea, D., Mitrea, I., Mitrea, M.: Singular Integral Operators, Quantitative Flatness, and Boundary Value Problems. Progress in Mathematica, vol. 344. Birkhäuser, Basel (2022) [Mit18] Mitrea, D.: Distributions, Partial Differential Equation, and Harmonic Analysis. Springer International Publishing, New York (2018) [M+ 22] Mitrea, D., Mitrea, I., Mitrea, M.: Geometric Harmonic Analysis I-V. Springer, New York (2022)

Chapter 17

Recipes for Computer Implementation of a Response Matrix Spatial Spectral Nodal Method for Three-Dimensional Discrete Ordinates Neutral Particle Transport Modeling L. R. C. Moraes and R. C. Barros

17.1 Introduction Deterministic neutral particle transport computational modeling has shown great potential as a viable design and analysis tool for realistic nuclear engineering applications, such as the optimization of neutron beams for tumor therapy [KoGi08] and the determination of neutron sources driving prescribed power generations in subcritical systems [LeEtAl21, LeEtAl22], among other peaceful applications of nuclear energy [VaEtAl20, LeEtAl20, BaSi99]. In the 1980s, analytical nodal methods such as the linear nodal (LN) method [WaDe81] and the linear-linear Nodal (LLN) method [Az88] were developed to generate numerical results for multidimensional neutron/photon transport problems in the discrete ordinates (SN ) formulation [LeMi93]. These analytical nodal methods are based on transverse integrations of the SN transport equations within the spatial nodes that compose the multidimensional spatial discretization grid, using low-order polynomial approximations for both the scattering source terms and the transverse leakage terms. The spectral nodal methods [Ba90, BaLa90] came to the fore in the 1990s. Similarly to the LN and LLN methods, they are also based on the transverse integrations of the SN transport equations within the spatial nodes. However, the low-order polynomial approximation is used only for the transverse leakage terms. The scattering source terms are treated analytically.

L. R. C. Moraes The Ohio State University, Columbus, OH, USA e-mail: [email protected] R. C. Barros () Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_17

205

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The spectral nodal methods seek, at first, to generate numerical solutions for SN transport problems, by determining the local homogeneous and particular solutions of the transverse-integrated SN transport equations in each discretization node of the domain (local general solution). Regarding the local homogeneous solution, it is assumed an exponential behavior for the transverse-integrated particle angular fluxes. This assumption leads to the generation of a local eigenvalue problem defined by the discrete directions and weights of the angular quadrature considered in the SN formulation and the material parameters of the spatial node. The process of building and solving the local eigenvalue problems, in conjunction with the analysis of the generated eigenvalues and eigenfunctions, is referred to as spectral analysis. The response matrix-constant nodal (RM-CN) method is a spectral nodal method with constant approximation for the transverse leakage terms. In the RM-CN method, the arbitrary constants that appear in the local general solution, as the result of a linear combination of the local homogeneous solutions, are not obtained explicitly. Instead, a matrix relation is constructed for these variables making use of boundary and/or continuity conditions and the particular solutions. The matrix relation associated with the arbitrary constants is used with the local particular solutions to generate sweeping matrices that can be used to implement iterative numerical schemes. As the computer calculation of eigenvalue problems has generally high cost, and the human workload for the development of the sweeping matrices of the RMCN method is significantly heavy, the main goal of this chapter (Sect. 17.2) is to describe efficient recipes to solve only one eigenvalue problem for each material zone in one fixed spatial coordinate direction (say x-direction) and then obtaining the eigenvalue solution for the other two coordinate directions (y and z-directions) by rotation transformations of the projection axes of the particle direction of motion. Also described here is a simple technique to build the sweeping matrices for the partial one-node block inversion (NBI) iterative scheme used in the offered RM-CN method. Numerical results are given in Sect. 17.3, and we close this chapter with some concluding remarks in Sect. 17.4.

17.2 The Response Matrix-Constant Nodal Method We begin considering the steady-state and monoenergetic version of the threedimensional neutral particle transport equation in the SN formulation with isotropic scattering. That is [LeMi93], μm .

∂Ψ (x, y, z) ∂Ψ (x, y, z) ∂Ψm (x, y, z) + ηm m + ξm m + σT (x, y, z)Ψm (x, y, z) = ∂x ∂y ∂z σS (x, y, z)  Ψn (x, y, z)ωn + Q(x, y, z). 8 n=1 (17.1) M

17 A Response Matrix-Constant Nodal method for 3-D Transport Calculations

207

Fig. 17.1 Three-dimensional spatial domain with I J K contiguous regions Ri,j,k

In Eq. (17.1) Ψm (x, y, z) is the particle angular flux at a spatial point of a three 3 dimensional domain D = (x, y, z) ∈ R | 0 ≤ x ≤ X; 0 ≤ y ≤ Y ; 0 ≤ z ≤ Z , migrating in the discrete direction Ωm = (μm , ηm , ξm ), with m = 1, 2, . . . , M and M = N(N + 2) where N is the order of the angular quadrature set considered. In this case, the level symmetric angular quadrature (LQN ) [LeMi93]. In addition, σT (x, y, z) and σS (x, y, z) are the total and scattering macroscopic cross sections, and Q(x, y, z) is an isotropic source term. Functions σT (x, y, z), σS (x, y, z), and Q(x, y, z) are considered piecewise constant functions with respect to the spatial variables. They are spatially uniform inside each region Ri,j,k that compose the domain (viz., Fig. 17.1), i.e., σT (x, y, z) = σTi,j,k , σS (x, y, z) = σSi,j,k and Q(x, y, z) = Qi,j,k for (x, y, z) ∈ Ri,j,k . In essence, the RM-CN method is composed of three steps: (1) Transverse integrations of Eq. (17.1) over xy, xz and yz-faces, approximating by constants the leakage terms from each region (Sect. 17.2.1); (2) Determination of the analytical general solution of the transverse-integrated SN transport equations in each region Ri,j,k by a spectral analysis (Sect. 17.2.2); (3) Construction of the discretized equations of the RM-CN method (Sect. 17.2.3).

17.2.1 The Transverse Integration of the Three-Dimensional SN Transport Equations Let us focus on a region Ri,j,k of the domain represented in Fig. 17.1. Region Ri,j,k is a rectangular prism with length equal to hxi = xi+1/2 − xi−1/2 , width hyj = yj +1/2 −yj −1/2 , and height hzk = zk+1/2 −zk−1/2 . To transverse-integrate Eq. (17.1) over region Ri,j,k xy-face, we apply the operator

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1 . hxi hyj



yj +1/2



xi+1/2

(·)dxdy yj −1/2

xi−1/2

on Eq. (17.1), obtaining xy

ξm .

−

dΨm,i,j (z) dz

xy

+ σTi,j,k Ψm,i,j (z) =

M σSi,j,k 

8

xy

Ψn,i,j (z)ωn + Qi,j,k

n=1

  η  μm  y y x x Ψm,i+1/2,j (z) − Ψm,i−1/2,j (z) − m Ψm,i,j +1/2 (z) − Ψm,i,j −1/2 (z) . hyj hxi (17.2) y

In Eq. (17.2) Ψm,i±1/2,j (z) is the region Ri,j,k angular flux averaged along its y coordinate direction  yj +1/2 1 y Ψm (xi±1/2 , y, z)dy, .Ψ (z) = m,i±1/2,j hyj yj −1/2 x Ψm,i,j ±1/2 (z) is the region Ri,j,k angular flux averaged along its x coordinate direction  xi+1/2 1 x .Ψm,i,j ±1/2 (z) = Ψm (x, yj ±1/2 , z)dx, hxi xi−1/2 xy

and Ψm,i,j (z) is the region Ri,j,k angular flux averaged over its xy-face xy

Ψm,i,j (z) =

.

1 hxi hyj



yj +1/2



yj −1/2

xi+1/2

xi−1/2

Ψm (x, y, z)dxdy.

Now, we approximate the leakage terms in Eq. (17.2) by constants, i.e., y

yz

Ψm,i±1/2,j (z) ≈ Ψm,i±1/2,j,k =

.

1 hzk



zk+1/2 zk−1/2

y

Ψm,i±1/2,j (z)dz,

(17.3a)

x Ψm,i,j ±1/2 (z)dz,

(17.3b)

and x .Ψm,i,j ±1/2 (z)

≈

xz Ψm,i,j ±1/2,k

1 = hzk



zk+1/2 zk−1/2

and then we substitute these approximations (Eqs. 17.3) into Eq. (17.2), thus obtaining

17 A Response Matrix-Constant Nodal method for 3-D Transport Calculations xy

ξm

dΨm,i,j (z) dz

xy

+ σTi,j,k Ψm,i,j (z) =

M σSi,j,k 

8

.

209

xy

Ψn,i,j (z)ωn + Qi,j,k

n=1

(17.4a)

μ η yz xz − m ΔΨm,i±1/2,j,k − m ΔΨm,i,j ±1/2,k . hxi hyj yz

xz In Eqs. (17.3) and (17.4), Ψm,i±1/2,j,k and Ψm,i,j ±1/2,k are the region Ri,j,k angular yz fluxes averaged over its yz and xz-faces, respectively. Moreover, ΔΨm,i±1/2,j,k = yz yz xz xz xz Ψm,i+1/2,j,k − Ψm,i−1/2,j,k and ΔΨm,i,j ±1/2,k = Ψm,i,j +1/2,k − Ψm,i,j −1/2,k . Following analogous procedures, we can obtain the equations

ηm

xz dΨm,i,k (y)

dy

xz + σTi,j,k Ψm,i,k (y)

=

M σSi,j,k 

8

.

xz Ψn,i,k (y)ωn + Qi,j,k

n=1

(17.4b)

μ ξ xy yz − m ΔΨm,i±1/2,j,k − m ΔΨm,i,j,k±1/2 , hzk hxi and yz

μm

dΨm,j,k (x) dx

yz

+ σTi,j,k Ψm,j,k (x) =

M σSi,j,k 

8

.

yz

Ψn,j,k (x)ωn + Qi,j,k

n=1

(17.4c)

ξm η xy xz ΔΨm,i,j,k±1/2 . − m ΔΨm,i,j ±1/2,k − hzk hyj Equations (17.4) are the transverse-integrated SN constant nodal transport equations. The point of transverse-integrating Eq. (17.1) over regions xy, yz, and xz-faces and then approximate the leakage terms by constants is that we seek to generate results for three-dimensional transport problems through the solution of a coupled system of “one-dimensional” ordinary differential equations (observe that Eqs. (17.4) are centered in only one spatial variable) instead of the solution of a coupled system of three-dimensional partial differential equations.

17.2.2 The RM-CN Spectral Analysis As stated in the previous section, we seek to generate numerical results for threedimensional transport problems by solving Eqs. (17.4). Thus, we write the analytical general solution of Eq. (17.4a) as xy

xy

h

xy

p

Ψm,i,j (z) = Ψm,i,j (z) + Ψm,i,j,k ,

.

(17.5)

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where the superscripts h and p stand for the homogeneous and particular components of the general solution, respectively. xy

p

The particular component Ψm,i,j,k is considered spatially uniform due to the fact that the source term is constant inside each region that composes the domain. The particular component can be obtained through the solution of the following matrix system  p −1 Ψ xy = Υ i,j,k Qi,j,k − I i,j,k

.

μ hxi

ΔΨ yz −I i±1/2,j,k xy

η hyj

, ΔΨ xz i,j ±1/2,k

(17.6)

p

which is generated by the substitution of Ψm,i,j,k into Eq. (17.4a) and the variation of m in the resulting equation. In Eq. (17.6) Υ i,j,k is a square matrix of order M whose entries depend on the material parameters; I μ and I η are diagonal matrices of order M whose non-zero entries are

μ hxi

hxi

and

η hyj

hyj

, respectively; ΔΨ yz and i±1/2,j,k

are M-dimensional column vectors whose entries are the region Ri,j,k ΔΨ xz i,j ±1/2,k angular fluxes averaged over its yz and xz-faces; and Qi,j,k is an M-dimensional column vector composed by the source term. To determine the homogeneous component, we consider the ansatz [BaSi99, OdEtAl20, LeEtAl22] h

xy Ψm,i,j (z) = am (ϑ)e−

.

(z−γ ) ϑ

,

γ =

zk−1/2 , ϑ > 0 . zk+1/2 , ϑ < 0

(17.7)

Substituting Eq. (17.7) into the homogeneous version of Eq. (17.4a), we obtain the relation  M σSi,j,k ωn δm,n σTi,j,k  1 . (17.8) − an (ϑ) = am (ϑ), 8ξm ϑ ξm n=1

which defines an eigenvalue problem of order M (varying m). Once this eigenvalue problem is solved, we can write the homogeneous component as a linear combination of the solutions proposed in Eq. (17.7) and the analytical general solution of Eq. (17.4a) as xy Ψm,i,j (z) =

M 

.

β am (ϑ )e

) − (z−γ ϑ

p

xy + Ψm,i,j,k ,

(17.9a)

=1

or in matrix form as .Ψ



(z−γ )  − ϑ −1

(z) = A diag +Υ e β i,j,k i,j,k i,j,k Qi,j,k −I i,j xy

μ hxi

ΔΨ yz −I i±1/2,j,k

η hyj

ΔΨ xz , i,j ±1/2,k

(17.9b)

17 A Response Matrix-Constant Nodal method for 3-D Transport Calculations

211

where  a square matrix of order M whose entries are the eigenvectors am (ϑ);

Ai,j,k is diag e

) − (z−γ ϑ

is a diagonal matrix of order M whose non-zero entries are e−

(z−γ ) ϑ

;

and β i,j,k is an M-dimensional vector composed by the arbitrary constants. For convenience, we consider that the eigenvalues are ordered in a decreasing fashion. At this point we remark that an analogous procedure should be done in order to generate the analytical general solution of Eq. (17.4b) and (17.4c). However, we can avoid this by considering three different mappings for the discrete directions of motion, each one to be applied in one set of ODEs, i.e., Eq. (17.4a) to (17.4c). We term “mapping” as a different organization of the discrete directions of motion in each set of equations. We work with three different mappings for Eq. (17.4a) to (17.4c) in order to use the symmetry properties of the LQN angular quadrature. As the LQN angular quadrature satisfies reflective conditions across all three orthogonal surfaces, and it requires that all weights must be equal for points obtained by permuting the direction cosines [LeMi93], depending on the way one maps the discrete directions of motion in Eq. (17.4a) to (17.4c), it is possible to generate the same eigenvalue problems (hence matrices Ai,j,k and diag ) and particular solutions (with the exception of vectors ΔΨ yz , ΔΨ xz , and ΔΨ xy i,j,k±1/2 ) for these sets of i±1/2,j,k i,j ±1/2,k equations. This means that one can directly use the analytical general solution for one set of equations, for example, Eqs. (17.9), to generate the analytical general solution for the other two set of equations, hence reducing the effort in order to generate the analytical general solution of the transverse-integrated SN transport equations. One configuration for the three different mappings that generate analytical general solutions of Eqs. (17.4) with the same structure can be obtained by (I) defining the mapping for one set of equations as the standard (ξ, η, μ) and (II) applying separately a clockwise (η, μ, ξ ) and counterclockwise (μ, ξ, η) rotation in the axis of the mapping considered as standard. These “rotations” automatically define the mappings for the other two sets of equations. To visualize this procedure, let us consider the mapping for Eq. (17.4a) as the standard. Figure 17.2 displays, for the case of N = 8, the ordering of Ωm in the first octant (μm > 0, ηm > 0, ξm > 0) of discrete directions in Eq. (17.4a) (standard mapping). Applying separately a clockwise and counterclockwise rotation with respect to the standard mapping (Fig. 17.2), we generate the mappings for Eq. (17.4b) and (17.4c), as illustrated in Fig. 17.3. To conclude the mapping of the discrete direction of motion, it is necessary to organize differently the way that the signs of the components of Ωm change. Again, we consider the mapping of Eq. (17.4a) as the standard, and then we modify the order that the sign of the components of Ωm change according to the clockwise and counterclockwise rotations with respect to the standard mapping. Figure 17.4 illustrates these procedures. By using these mappings, we can write the analytical general solution of Eq. (17.4b) and (17.4c) making use of Eq. (17.9). Thus, we have

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a LQ angular quadrature. The components of Ω are defined such that μ = η = ξ , where μ = m n n n 8 1 0.2182179, μ2 = 0.5773503, μ3 = 0.7867958 and μ4 = 0.9511897. The weights associated with Ωm are ω1 = 0.1209877, ω2 = 0.0907407, and ω3 = 0.0925926 [LeMi93].

Fig. 17.2 Mapping of the discrete directions of motion in the first octant (μm > 0, ηm > 0, ξm > 0) in Eq. (17.4a)

a The correspondent discrete direction of motion in the standard mapping.

Fig. 17.3 Mapping of the discrete directions in the first octant (μm > 0, ηm > 0, ξm > 0) in Eq. (17.4b) and (17.4c)

.Ψ

(y−γ )  − ϑ −1

e α Qi,j,k −I (y) = A diag +Υ i,j,k i,j,k i,j,k m,i,k xz

μ h xi

ΔΨ yz −I i±1/2,j,k

ξ hzk

, ΔΨ xy i,j,k±1/2

(17.10a)

and .Ψ

(x−γ )  − ϑ −1

e ζ (x) = A diag +Υ i,j,k i,j,k i,j,k Qi,j,k −I m,j,k yz

η h yj

ΔΨ xz −I i,j ±1/2,k

ξ hzk

, ΔΨ xy i,j,k±1/2

(17.10b)

where α i,j,k and ζ i,j,k are M -dimensional column vectors composed by arbitrary constants.

17 A Response Matrix-Constant Nodal method for 3-D Transport Calculations

213

Fig. 17.4 Mapping of the discrete directions in all octants in Eq. (17.4)

17.2.3 The Discretized Equations of the RM-CN Method To determine the region average angular fluxes, it is necessary to calculate the arbitrary constants in Eqs. (17.9) and (17.10). Therefore, let us write Eq. (17.9b) in a convenient form

(z−γ )  p − ϑ xy xy

+ Ψ . (17.11) .Ψ (z) = A diag e β i,j,k i,j,k i,j,k m,i,j Now, we consider in Eq. (17.11) z = zk−1/2 for the discrete directions of motion that compose octants 1, 2, 3, and 4 and zk+1/2 for the discrete directions of motion that compose octants 5, 6, 7, and 8, viz., Fig. 17.4. Thus, we obtain xy Ψ i,j

.

(in)

(in)

p

xy = Ti,j,k β i,j,k + Ψ i,j,k ,

(17.12)

xy where Ψ i,j is an M-dimensional column vector composed of the region Ri,j,k angular fluxes averaged over its xy-faces in the incoming directions and

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Ti,j,k =

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ A2,1 ⎣

hz  ⎤ − k A1,2 diag e |ϑ | ⎥  ⎥  ⎥ ⎥

=(M/2+1):M

hz  ⎥, ⎥ − ϑk

⎥ A2,2 diag e ⎦    A1,1

=1:M/2

where A1,1 , A1,2 , A2,1 , and A2,2 are square sub-matrices of order M/2 of matrix Ai,j,k . From Eq. (17.12) we obtain a matrix relation for the arbitrary constants, i.e.,  p −1 (in) xy . β i,j,k = T i,j,k Ψ xy − Ψ i,j i,j,k

(17.13)

.

Substituting Eq. (17.13) into Eq. (17.11), and then setting in the resulting equation zk+1/2 for the discrete directions that compose octants 1, 2, 3, and 4, and zk−1/2 for the discrete directions that compose octants 5, 6, 7, and 8, we obtain Ψ xy i,j

.

(out)

x,y (in)

= R i,j,k Ψ i,j

p

+ H i,j,k Ψ xy , i,j,k

(17.14)

(out)

where Ψ xy is an M-dimensional column vector composed of the region Ri,j,k i,j angular fluxes averaged over its xy-faces in the outgoing directions

hz  ⎤ − k A1,1 diag e ϑ A1,2 ⎥ ⎢ ⎥ ⎢    ⎥ −1 ⎢

=1:M/2 ⎢

hz  ⎥ , ⎥T ⎢ − |ϑ k| ⎥ i,j,k ⎢ ⎢ A2,1 A2,2 diag e ⎥ ⎦ ⎣    ⎡

R i,j,k =

=(M/2+1):M

and H i,j,k = I − R i,j,k , where I is the identity matrix of order M. Substituting Eq. (17.6) into Eq. (17.14), we obtain .Ψ

xy (out) i,j

= R i,j,k Ψ xy i,j

(in)

 + H i,j,k Υ −1 i,j,k Qi,j,k − I

μ hxi

ΔΨ yz −I i±1/2,j,k

η hyj

ΔΨ xz . i,j ±1/2,k

(17.15a) Analyzing carefully Eq. (17.15a), we can observe that entries of matrices R i,j,k and H i,j,k depend on the eigenvalues and eigenvectors obtained through the solution of the eigenproblem defined in Eq. (17.8) and the height hzk of region Ri,j,k . Considering the fact that the eigenvalues and eigenvectors are the same in Eqs. (17.9) and (17.10), if one considers a spatial discretization grid composed of cubic nodes, i.e., hxi = hyj = hzk = h, Eq. (17.15a) can be used to also generate the relations for the region Ri,j,k angular fluxes averaged over its xz and yz-faces in the outgoing directions. That is,

17 A Response Matrix-Constant Nodal method for 3-D Transport Calculations

.Ψ

xz(out) i,k

= R i,j,k Ψ xz i,k

(in)

215

  xy yz μ + H i,j,k Υ −1 i,j,k Qi,j,k − I ΔΨ i±1/2,j,k − I ξ ΔΨ i,j,k±1/2 , h

h

(17.15b)

and .Ψ

yz(out) j,k

yz = R i,j,k Ψ j,k

(in)

  xy xz η + H i,j,k Υ −1 i,j,k Qi,j,k − I ΔΨ i,j ±1/2,k − I ξ ΔΨ i,j,k±1/2 . h

h

(17.15c)

Equations (17.15) are the discretized equations of the RM-CN method. They are used to implement an iterative numerical scheme to converge numerical results for the node angular fluxes averaged over its xy, xz, and yz-faces. In this work, we use the partial one-node block inversion (partial NBI) iterative scheme that uses the boundary conditions and the most recent estimates available for the node angular fluxes averaged over its xy, xz, and yz-faces in the incoming directions to evaluate the outgoing fluxes that constitute the incoming fluxes for the adjacent nodes in the directions of the transport sweeps. The partial NBI iterative scheme can be found in more detail in the literature [OdEtAl20].

17.3 Numerical Results To illustrate that the RM-CN method can, in conjunction with the partial NBI iterative scheme, produce accurate results in three-dimensional calculations, we consider a test problem composed of four regions (I–IV) with two different 3 materials, where a source Q = 1 neutron/cm s is embedded into region I. In this test problem, reflective boundary conditions apply on the top and at the bottom of the cube and at the xz-face at y = 0 cm and at the yz-face at x = 0 cm. Vacuum boundary conditions apply at the xz-face at y = 10 cm and at the yz-face at x = 10 cm. This test problem is illustrated in Fig. 17.5. In this test problem, we seek to generate numerical results for the region average scalar fluxes, i.e., xyz .Φ = i,j,k h

 xi+1/2  yj +1/2  zk+1/2  M M  1 xyz Ψm (x, y, z)ωm dzdydx= Ψm,i,j,k ωm . h h xi yj zk xi−1/2 yj −1/2 zk−1/2 m=1 m=1 (17.16a) xyz

The region average angular fluxes Ψm,i,j,k can be obtained through the solution of the following equation: μm η ξm yz xy xyz xz ΔΨm,i±1/2,j,k + m ΔΨm,i,j ±1/2,k + h ΔΨm,i,j,k±1/2 + σTi,j,k Ψm,i,j,k = hx i hy j zk .

M σSi,j,k 

8

xyz

Ψn,i,j,k ωn + Qi,j,k ,

n=1

(17.16b)

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L. R. C. Moraes and R. C. Barros

Fig. 17.5 Test problem

which is obtained after the application of the operator .

 xi+1/2  yj +1/2  zk+1/2 1 (·)dzdydx hxi hyj hzk xi−1/2 yj −1/2 zk−1/2

on Eq. (17.1). The values of the region angular fluxes averaged over its xy, xz, and yz-faces are generated making use of the discretized equations of the RMCN method (Eqs. 17.15) in conjunction with the partial NBI iterative scheme. The stopping criterion adopted for the partial NBI iterative scheme required that the relative deviations between two consecutive estimates of the node average scalar fluxes be less than or equal to 10−5 . As in the RM-CN method, we consider a constant approximation for the leakage terms, it is necessary to set up a spatial discretization grid over the domain in order to produce accurate results. We consider in this case a spatial discretization grid composed of cubic nodes with edges of 0.25 cm on the three-dimensional domain. Moreover, we consider the LQ4 angular quadrature. To compare the results generated by the RM-CN computer program, we also solve this problem making use of the classical fine-mesh diamond difference (DD) method in conjunction with the source iteration scheme [LeMi93]. For the DD solution, we set up a spatial discretization grid composed of cubic nodes with edges of 0.05 cm. Table 17.1 presents the region average scalar fluxes in regions I, II, and IV, generated by the computer programs created to implement the RM-CN and DD methods, as well as the computer running time of these simulations. To measure the computer running time of the simulations, we use the routine high-resolution clock from the standard C++ Chrono library. As can be noted in Table 17.1, the

17 A Response Matrix-Constant Nodal method for 3-D Transport Calculations Table 17.1 Numerical results of the test problem

Region I II IV Time (s)

Average scalar flux RM-CN DD 1.67378 1.67373 0.04005 0.04006 0.00189 0.00188 11.7782 45.1564

217

Relative deviation (%) 0.003 0.022 0.360

RM-CN method generated accurate results in three-dimensional calculations as the maximum value of the percentage relative deviation with respect to the DD solution was only 0.36%. However, the computer program created to implement the DD method took approximately 3.83 more time to generate the numerical results compared with the program created to implement the RM-CN method. This can be explained by the fact that similarly to the LN and LLN methods, the DD method also approximates both the transverse leakage terms and the scattering source term. As the RM-CN method approximates only the transverse leakage terms, the spatial discretization grid necessary to generate accurate results may be composed of fewer nodes for the RM-CN method in comparison to the one considered for the DD method.

17.4 Concluding Remarks Presented here is a spectral nodal method, namely, Response Matrix-Constant Nodal method, applied to numerically solve three-dimensional Cartesian geometry neutral particle transport problems in the discrete ordinates formulation. The RM-CN method is described with a special focus on an efficient scheme for the eigenvalue problems related to the local homogeneous solutions associated with the transverseintegrated SN constant-nodal equations. Each set of transverse-integrated equations is centered in only one spatial variable, viz., Eq. (17.4). We adopt three different mappings for the discrete directions of motion, each one applied in one set of equations, such that the eigenvalue problems associated with these sets become the same. Thus, we reduce the effort in the calculation of the homogeneous solutions of the transverse-integrated SN constant-nodal equations, since once the eigenvalue problems are the same, we do not need to calculate them multiple times. In addition, if one also considers cubic spatial nodes (hxi = hyj = hzk = h, viz., Fig. 17.1), the discretized equations of the RM-CN method have the same structure. This means that the whole process of building the discretized equations can be done with respect to only one set of transverse-integrated SN transport equations (e.g., Eq. (17.15a)). The other two discretized equations (Eqs. 17.15b and 17.15c) can be easily generated through matrix procedures with respect to the already built discretized equation (Eq. 17.15a). At this point we remark that all the discretized equations of the RM-CN method must have the same mapping in order

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to implement an iterative numerical scheme to converge numerical results. In this case, if one considers the different mappings given in Sect. 17.2 to generate the discretized equations of the RM-CN method (Eqs. 17.15), extra matrix procedures must be applied to write Eqs. (17.15b) and (17.15c) in the same mapping of the one considered for Eq. (17.15a). We conclude this chapter by pointing out that although we provide an efficient scheme for the RM-CN method, the improvement of the RM-CN computer program is of interest in order to deal with more complex three-dimensional neutral particle transport problems. One way to improve the RM-CN method is to reduce the size of the eigenvalue problems. This can be done by the use of symmetric quadrature schemes defined in the semi-interval, which generate eigenvalue problems whose order is half of the number of discrete directions [BaSi99]. Another way to improve the efficiency of the RM-CN solution is to develop iterative schemes that improve the efficiency of the RM-CN computer program.

References [Az88] Azmy, Y.Y.: Comparison of three approximations to the linear-linear nodal transport method in weighted diamond-difference form. Nucl. Sci. Eng. 100, 190–200 (1988) [Ba90] Badruzzaman, A.: Nodal methods in transport theory. Adv. Nucl. Sci. Tech. 21, (1990) [BaSi99] Barichello, L.B., Siewert, C.E.: A discrete-ordinates solution for pouiseuille flow in a plane channel. J. Appl. Math. Phys. (ZAMP) 50, 972–981 (1999) [BaLa90] Barros, R.C., Larsen, E.W.: A numerical method for one-group slab-geometry discrete ordinates problems with no spatial truncation error. Nucl. Sci. Eng. 104, 199–208 (1990) [KoGi08] Korobeynikov, A., Ginkin, V.: Computing analysis and optimization of neutron beam for tumor therapy. Transp. Theory Stat. Phys. 37, 65–108 (2008) [LeMi93] Lewis, E.E., Miller, W.F.: Computational Methods of Neutron Transport. American Nuclear Society, Illinois, USA (1993) [LeEtAl20] Moraes, L.R.C., Alves Filho, H., Barros, R.C.: Estimation of neutron sources driving prescribed power generations in subcritical systems using one-speed two-dimensional discrete ordinates formulations. Ann. Nucl. Energy 136, 107053 (2020) [LeEtAl21] Moraes, L.R.C., Mansur, R.S., Moura, C.A., Curbelo, J.P., Alves Filho, H., Barros, R.C.: A response matrix method for slab-geometry discrete ordinates adjoint calculations in energy-dependent neutral particle transport. J. Comput. Theor. Transp. 50, 159–179 (2021) [LeEtAl22] Moraes, L.R.C., Alves Filho, H., Barros, R.C.: On the calculation of neutron sources generating steady prescribed power distributions in subcritical systems using multigroup X,Y-geometry discrete ordinates models. Ann. Nucl. Energy 168, 108854 (2022) [OdEtAl20] Silva, O.P., Guida, M.R., Alves Filho, H., Barros, R.C.: A response matrix spectral nodal method for energy multigroup X,Y-geometry discrete ordinates problems in non-multiplying media. Prog. Nucl. Energy 125, 103288 (2020) [VaEtAl20] Vasques, R., Moraes, L.R.C., Barros, R.C., Slaybaugh, R.N.: A spectral approach for solving the nonclassical transport equation. J. Comput. Phys. 402, 109078 (2020) [WaDe81] Walters, W.F., O’Dell, R.D.: Nodal methods for discrete-ordinates transport problems in (x,y) geometry. Proc. Int. Top. Meet. Ad. Math. Methods Sol. Nucl. Eng. Probl. Munich Am. Nucl. Soc. 1, 115–129 (1981)

Chapter 18

On Maximum Principles for Weak Solutions of Some Parabolic Systems S. E. Mikhailov

18.1 Introduction Maximum principles for solutions parabolic equations constitute a traditional part of PDE analysis. It is well developed for classical solutions of scalar parabolic equations with constant coefficients, and these results also generalized to weak solutions of scalar elliptic and parabolic equations with variable coefficients; see, e.g., [LaSoUr67, Chapter III, Theorem 7.2]. The estimates of the essential maximum of weak solutions of parabolic systems are also available although with a constant depending on the system coefficients, cf., e.g., [LaSoUr67, Chapter VII, Theorem 2.1]. In this paper, by employing special test functions, sharper versions of the maximum principle for weak solutions of several linear parabolic variable-coefficient systems have been proved. The considered systems include non-stationary convection-reaction-diffusion systems as well as the Stokes and Brinkman systems. The obtained maximum principles for weak solutions can be employed to prove global existence of solutions of some nonlinear parabolic systems, cf. [PoRo16], where a maximum principle for strong solutions of the Burgers system has been used for this. We presented here maximum principles for the spatially periodic solutions, i.e., solutions on the n-dimensional flat torus; similar results can be re-stated also for solutions on bounded domains.

S. E. Mikhailov () Brunel University London, Uxbridge, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_18

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18.2 Periodic Function Spaces We will employ some function spaces on torus and periodic function spaces (see, e.g., [Ag65, p.26], [Ag15], [Mc91], [RuTu10, Chapter 3], [RoRoSa16, Section 1.7.1] for more details). Let .n ≥ 1 be an integer and .T be the n-dimensional flat torus that can be parametrized as the semi-open cube .T = [0, 1)n ⊂ Rn , cf. [Zy02, p. 312]. Let n .Z denote the set of integers and .ξ ∈ Z denote the n-dimensional vector with integer components. The Lebesgue space on the torus .Lp (T), .1 ≤ p ≤ ∞, can be identified with the periodic Lebesgue space .Lp# = Lp# (Rn ) that consists of functions .φ ∈ Lp,loc (Rn ), which satisfy the periodicity condition φ(x + ξ ) = φ(x)

.

∀ ξ ∈ Zn .

for a.e. .x ∈ Rn . For .s ∈ R, let .H#s := H#s (Rn ) := H s (T) denote the .L2 -based periodic/toroidal Sobolev (Bessel-potential) spaces, cf. [RuTu10, Definition 3.2.2, Proposition 3.2.6]. For any .s ∈ R, the space .H#−s is adjoint (dual) to .H#s , i.e., .H#−s = (H#s )∗ . Note that the torus/periodic norms on .H#s are equivalent to the corresponding standard (non-periodic) Bessel potential norms on .T as a cubic domain, see, e.g., [Ag15, Section 13.8.1]. For any .s ∈ R, let us also introduce the space H˙ #s := {g ∈ H#s : g, 1 T = 0}.

(18.1)

.

with the same norm as .H#s . Definition (18.1) and the Riesz theorem also imply that the space adjoin to .H˙ #s can be expressed as .(H˙ #s )∗ = H˙ #−s . The corresponding spaces of n-component vector functions/distributions are denoted as .Lq# := (Lq# )n , .Hs# := (H#s )n , etc. Let us also define the Sobolev spaces of divergence-free functions and distributions   ˙ s : div w = 0 , ˙ s := w ∈ H H #σ #

.

s ∈ R,

endowed with the same norm as .Hs# . Similarly, .Lq#σ denote the subspaces of divergence-free vector-functions from .Lq# , etc. Some more details about the periodic Sobolev spaces used here are available in [Mi22, Section 16.3]. For the evolution problems, we will use the spaces of Banach-valued functions s .Lq (0, T ; H ), .s ∈ R, .1 ≤ q ≤ ∞, .0 < T < ∞, which consists of functions that # map .t ∈ (0, T ) to a function or distribution from .H#s . For .1 ≤ q < ∞, the space s .Lq (0, T ; H ) is endowed with the norm #  hLq (0,T ;H#s ) =

T

.

0

1/q q h(·, t)H s dt #

3 GPa. At lower values of the spongy bone elasticity modulus (.0.5 ≤ Es ≤ 3GPa), the value of the maximum stresses changes slightly and practically does not depend on the cortical bone elasticity modulus.

19.3.2 Stress State of the Screw Joint of the Implant and Bones with Resorption The analysis of bone tissue resorption effect on the stress state near the implant was also performed on a model of a screw joint between the implant and bone tissues; it is the second stage of computations. The outer part of the implant structure (crown and abutment) was cut off, and the load was applied directly to the implant root. It was assumed that the hollows in the spongy bone which had formed in this bone after the implant penetration, are conformed to the screw thread on the implant root. It was assumed also that there is formation of the full material joint (osseointegration) on the boundary line between the implant and bones. The cross-

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Fig. 19.3 Maximum stresses .σi in the spongy bone, inclined load; resorption at 1/3 of the implant length

sectional shape of the implant root screw connection and the shape of the hollows in the spongy bone were modeled as an equilateral triangles with a side .a = 0.93 mm (with 12 teeth along the length .L = 14 mm of the implant root). The rounding radius r of the implant teeth tops and the hollows between the screw carving were taken equal to half of the circle radius R inscribed in an equilateral triangle with a side a: ◦ .R = 0.5a · tan(30 ), .r = 0.5R. Computations were made for the normal and inclined loads. The model consisted of three sub-domains (see Fig. 19.4). Boundary conditions (zones of zero displacements) are shown in the figure by a thin dashed strip, as well as in Fig. 19.1. The inclined load was applied at an angle to the plane of the implant root section, assuming that the load applied to the top of the crown was transferred to the implant section. Due to the asymmetry of the crown, a bending moment also occurs as the load was transferred. Taking into account the increase in the load due to the bending moment, a normal load as the traction .p = 10 MPa or an inclined load in the form of two equal components acting in the directions of the coordinate axes was applied to the implant section. Computations were made for three variants for the bone tissue resorption cavity depth—at .h/L = 1/3 of the implant length (see Fig. 19.4), half of the implant length (.h/L = 1/2), and .h/L = 0.6. The computation results when taking into account the shape of the screw joint of the implant and bone tissue showed that, as in the case of the full implant structure without taking into account the shape of the screw joint, there is the significant stresses redistribution near the implant with bone resorption. With the increase in the resorption cavity depth, the stresses in the cortical bone tissue decrease, whereas in the sponge bone tissue, the stresses increase, both under normal and inclined

19 Boundary Integral Equations Analysis of Bone Resorption

235

Fig. 19.4 Bone resorption cavity depth at .1/3 of the implant height, boundary elements discretization with screw joint of the implant and surrounded bone tissues under inclined load application, 3 subregions, total 1740 nodes: (a) the whole model, 1 is cortical bone; 2 is sponge bone; 3 is implant root; (b) enlarged resorption zone

Fig. 19.5 Screw joint, maximum stresses in the bone tissues; resorption depth .h/L variation

loadings (see Fig. 19.5). Stress distributions along the spongy bone contour, with accounting the screw joint the implant and bone, are shown in Fig. 19.6. Under the normal loading, the maximum stresses are observed in the contact zone of the spongy bone and the implant at the base of the cavity formed during bone resorption. Significant stresses also occur in the area of the contact between the spongy bone and the lower part of the implant (Fig. 19.6a). With the inclined loading, due to the

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Fig. 19.6 Intensity of stresses .σi along the spongy bone contour, .Ec = 18 GPa, .Es = 0.5 GPa, (a) normal load on the implant; (b) inclined load on the implant

Fig. 19.7 Intensity of stresses .σi along the implant contour, .Ec = 18 GPa, .Es = 0.5 GPa, (a) normal load on the implant; (b) inclined load on the implant

bending, the stress intensity increases significantly. The maximum stresses are also observed at the base of the cavity formed during bone resorption, in the contact zone between the spongy bone and the implant on the side of compressive stresses. The stresses at the base of the resorption cavity on the tension side are less than the maximum values of the stresses on the compression side by 3–5% (see Fig. 19.6b). In the contact zone between the spongy bone and the lower part of the implant (Fig. 19.6b), the stresses are significantly lower than the maximum values. The bone tissue resorption leads to a significant redistribution of stresses also in the implant. Stresses along the implant model contour under normal load, taking into account the bone resorption, are shown in Fig. 19.7a. As the resorption result, the position of the maximum stresses in the implant is moved from the implant upper part (near its neck; see [Pe18a]) and is observed at the base of the resorption cavity. The distribution of stresses along the implant model contour under inclined loading and with accounting bone resorption is shown in Fig. 19.7b. Under the inclined load action, the maximum stresses in the implant are also observed at the base of the resorption cavity (on the compression side), and the magnitude of these maximum stresses is much higher than in the resorption absence. With the resorption cavity depth increasing, the maximum stresses increase significantly.

19 Boundary Integral Equations Analysis of Bone Resorption

237

19.4 Conclusion It was found that the bone tissue resorption causes significant redistribution of stresses in the bones near the implant. With the increase in resorption depth cavity, the maximum stress intensity in the cortical bone tissue decreases; in the spongy bone tissue it increases. The stress increasing in the spongy bone depends on the degree of bone tissue resorption and, with significant resorption, can achieve of this bone tissue ultimate strength. The bone resorption effect on the stress state near the implant is the most significant at an inclined load. In this case, even for the resorption cavity depth not exceeding a third of the implant length may result in destruction of the spongy bone tissue. High stresses after the implant installation in the cortical bone in the contact zone of this bone with the implant (neck zone) could be one of the causes of bone tissue resorption onset. With significant inclined loads, one-sided resorption of the bone is possible. Bone resorption leads to the redistribution of stresses not only in the bone tissues, but also in the implant. With significant inclined loads, the maximum equivalent stresses in the implant increase significantly, which can lead to the decreasing of the implant service life.

References [AyEtAl22] Aydin, K., Okten, K., Ugur, L.: An analytical and numerical approach to the determination of thermal necrosis in cortical bone drilling Int. J. Numer. Meth. Biomed. Eng. 38, 3640–3644 (2022) [BrEtAl84] Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.: Boundary Element Techniques - Theory and Applications in Engineering. Springer, Berlin (1984) [BuEtAl22] Büyük, F.N., Savran, E., Karpat, F.: Review on finite element analysis of dental implants. J. Dent. Implant Res. 41, 50–63 (2022) [ChEtAl18] Chang, Y., Tambe, A.A., Maeda, Y. et al: Finite element analysis of dental implants with validation: to what extent can we expect the model to predict biological phenomena? A literature review and proposal for classification of a validation process. Int. J. Implant Dent. 4, 1–14 (2018) [CiEtAl12] Citarella, R., Armentani, E., Caputo, F., Lepore, M.: Stress analysis of an endosseus dental implant by BEM and FEM. Open Mech. Eng. J. 6, 115–124 (2012) [GePe10] Genna, F., Perelmuter, M.: Speeding-up Finite Element analyses by replacing the linear equation solver with a Boundary Element code. Part 1: 2D linear elasticity. Comput. Struct. 88, 845–858 (2010) [HaCi20] Hashim, D., Cionca, N.: A comprehensive review of peri-implantitis risk factors. Curr. Oral Health Rep. 7, 262–273 (2020) [KiEtAl05] Kitamura, E., Stegaroiu, R., Nomura, S., Miyakawa, O.: Influence of marginal bone resorption on stress around an implant - a three-dimensional finite element analysis. J. Oral Rehabil. 32, 279–286 (2005)

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[LaEtAl21] Kowalski, J., Lapinska, B., Nissan, J., Lukomska-Szymanska, M.: Factors influencing marginal bone loss around dental implants: a narrative review. Coatings 11, 865–877 (2021) [LiEtAl17] Linetskiy, I, Demenko, V., Linetska, L., Yefremov, O.: Impact of annual bone loss and different bone quality on dental implant success - a finite element study. Comput. Biol. Med. 91, 318–325 (2017) [NiEtAl21] Nimbalkar, S., Dhatrak, P., Gherde, C., Joshi, S.: A review article on factors affecting bone loss in dental implants. Mater. Today Proc. 43, 970–976 (2021) [Pe13] Perelmuter, M.: Boundary element analysis of structures with bridged interfacial cracks. Comput. Mech. 51, 523–534 (2013) [Pe18a] Perelmuter, M.N.: Analysis of stress-strain state of dental implants by the boundary integral equations method. PNRPU Mech. Bull. (in Russian) 2, 83–95 (2018) [Pe18b] Perelmuter, M.: Analysis of interaction of bridged cracks and weak interfaces. Int. J. Mech. Sci. 149, 349–360 (2018) [PeEtAl15] Perrella, M., Franciosa, P., Gerbino, S.: FEM and BEM stress analysis of mandibular bone surrounding a dental implant. Open Mech. Eng. J. 9, 282–292 (2015) [Wo93] Wolfe, L.A.: Stress analysis of endosseous implants using the Boundary Integral Equation (BIE) method. J. Biomed. Eng. 15, 319–323 (1993) [WoEtAl4] Wolff, J., Narra, N., Antalainen, A-K., Valasek, J., Kaiser, J., Sandor, G. K., Marcian, P.: Finite element analysis of bone loss around failing implants. Mater. Des. 61, 177– 184 (2014)

Chapter 20

Mathematical Modeling of Partially Miscible Water Alternating Gas Injection Using Geometric Thermodynamic Variables A. Pires and B. Loza

20.1 Introduction The water alternating gas (WAG) injection was proposed as an effort to combine the mobility control of waterflooding and the more efficient technique to improve the oil recovery in unswept zones through gasflooding [CaDy58]. In EOR projects, water and gas are injected separately by alternating slugs [ChEtAl11]. Physically WAG injection results in a compositional flow through the porous medium. Mathematically it can be represented by systems of hyperbolic partial differential equations that express the mass conservation of each component considering its mass transfer between flowing phases [Or07]. Once the phase equilibrium behavior and relative permeability curves are defined, the analytical solution for one-dimensional multicomponent flow can be found by the analysis of the fractional flow curve [PiBe05]. A number of researchers have obtained analytical solutions for miscible and immiscible WAG injection [Be93, MaPl01]. In both cases, a .2×2 hyperbolic system of mass conservation laws describes the one-dimensional displacement of oil by gas and water, assuming system volume conservation. There are few works published in literature regarding the development of analytical solutions of the process of partially miscible WAG injection. Recently [LaJo05] presented analytical solutions for the simultaneous water and gas (SWAG) injection. These solutions consider the partitioning of components between aqueous and hydrocarbon phases. In this contribution, we extend the two-phase equilibrium parametrization presented in [Wa64] to analyze the Riemann problem for partially miscible WAG

A. Pires () · B. Loza Universidade Estadual do Norte Fluminense Darcy Ribeiro, Rio de Janeiro, Brazil e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_20

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injection. The cases evaluated consider constant injection composition and variable initial compositions. In order to update the initial condition, we define the new reservoir component concentration as the average of the last composition profile. We show that most of the composition paths during the gas or water injection are defined by the initial tie line.

20.2 Mathematical Model The following assumptions are adopted: • • • • • • • • • •

One-dimensional isothermal multiphase flow; Incompressible system; Homogeneous porous media; Instantaneous thermodynamic equilibrium; Small pressure gradient; Negligible capillary and gravitational effects; No diffusion; No chemical reaction or adsorption effects; Pure component density is the same in both phases; No volume change from mixing.

Under these assumptions, the mass conservation equation for the i-th component is given by .

∂Fi ∂Ci + = 0, ∂T ∂X

i = 1, . . . , nc

(20.1)

where .nc is the number of components, X is the dimensionless distance, and T is the dimensionless time, equivalent to the number of pore volumes injected (PVI). The overall volumetric fraction of the i-th component, .Ci , and the overall fractional flow of the i-th component, .Fi , are defined as Ci =

np 

.

cij Sj

j =1

Fi =

np 

.

cij fj

j =1

where .np is the number of phases, .cij is the volumetric fraction of the i-th component in phase j , and .Sj and .fj are the saturation and the fractional  cflow of phase j , respectively. As the overall concentration and flow are given by . ni=1 Ci = c Fi = 1, it is necessary to solve .nc − 1 equations. For ternary systems, 1 and . ni=1

20 Mathematical Modeling of Partially Miscible Water Alternating Gas. . .

241

Fig. 20.1 Phase equilibrium diagram for ternary systems

we introduce a thermodynamic parametrization in Eq. 20.1 that allows us to solve the eigenvalue problem using a previously developed theory.

20.2.1 Two-Phase Formulation The detailed derivation of the compositional model for two-phase flow can be found in [PiEtAl05]. The two-phase behavior for ternary systems is shown in Fig. 20.1, which displays the lower L and upper U phase behavior at a fixed pressure and temperature. The phase equilibrium compositions are connected by tie lines, which can be represented by the following geometric variables: α=

.

c3u − c3l c1u − c1l

β = c3l − αc1l

.

where .α and .β are the slope and .C3 -intercept of the tie line, respectively. By the introduction of these new variables, we can write the overall composition of component .C3 as a function of the component .C1 as C3 = αC1 + β .

(20.2)

.

The two-phase three-component problem, after substitution of Eq. 20.2 into Eq. 20.1, is rewritten in matrix form as ⎡ .

⎣

C β

⎤

⎡

⎦ +⎢ ⎣ T

FC

Fβ

0

αβ F +1 αβ C+1

⎤⎡ ⎥⎣ ⎦

C β

⎡ ⎤ 0 ⎦ =⎣ ⎦ ⎤

X

0

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where C and F are the overall composition and flow of the third component; .FC and .Fβ are the derivatives of F with respect to C and .β, respectively; and .αβ is the derivative of .α with respect to .β. The eigenvalues and eigenvectors of this system are λ C = FC ,

.

rC =

.

 1 , 0

λβ =

αβ F + 1 , αβ C + 1

rβ =

Fβ λ β − FC

 .

The corresponding Rankine-Hugoniot conditions are defined by σC =

.

σβ =

.

F− + C− +

[F ] , [C]

[β] [α] [β] [α]

=

F+ + C+ +

[β] [α] [β] [α]

,

where .[A] = A+ − A− represents the jump of A from the left state (.−) to the right state (.+) and .σi is the shock velocity along the i-th characteristic curve. For the sake of simplicity, the following nomenclature will be adopted from this point and beyond: for water-gas system .C ≡ CCO2 and .S ≡ Sw , for water-oil system .C ≡ CC10 and .S ≡ Sw , and for oil-gas system .C ≡ CCO2 and .S ≡ Sg .

20.2.2 Three-Phase Formulation According to Gibbs phase rule, there is only one equilibrium state for a threecomponent three-phase system at fixed pressure and temperature [Sa06]. Thus, a three-component three-phase system can be represented by the following equations [FaSc92, LaJo05]:

.

S1 S3



+

T

f11 f13 f31 f33



S1 S3

 = X

 0 0

where .fij is the derivative of .fi with respect to .Sj . The eigenvalues and eigenvectors of this system are λk =

.



1 (f11 + f33 ) + (−1)k (f11 − f33 )2 + 4f13 f31 , 2

k = 1, 2 ,

20 Mathematical Modeling of Partially Miscible Water Alternating Gas. . .

243

Table 20.1 Critical properties, accentric factors, and binary interaction coefficients for PR-1976 equation of state Comp. H.2 O C.10 CO.2

[MPa] 22.0483 2.4196 7.3765

.pc

[K] 647.13 626 304.7

.Tc

[kg/kmol] 18.015 134 44.01

.MW



f13 .rk = λk − f11



[m3/kmol] 0.0560 0.5340 0.0940

.vc

 λk − f33 = , f31

.ω

0.345 0.385 0.225

BI C H.2 O 0.0

C.10 0.0 0.0

CO.2 0.0 0.1 0.0



k = 1, 2 ,

where the slow path is represented by the subscript .k = 1 and the fast path by the subscript .k = 2. The corresponding Rankine-Hugoniot conditions are calculated by solving the following system of nonlinear equations: .

[S1 ] σk = [f1 (S1 , S3 )] , [S3 ] σk = [f3 (S1 , S3 )]

k = 1, 2

In this work .S1 ≡ Sw is water phase saturation, .S3 ≡ Sg is gas phase saturation, and the oil phase saturation is given by .So ≡ 1 − S1 − S3 .

20.2.3 Thermodynamic and Hydrodynamic Modeling In this section, we present the basic parameters and functions used for the solution of this problem. Thermodynamic equilibrium conditions are computed by the  software at a pressure of .12 MPa and a temperature of .345 K. Pure .W inprop component and mixture properties are calculated using the equation of state PR1976 [PeRo76] at system pressure and temperature [Da98]. The other parameters for the equation of state PR-1976 are listed in Table 20.1. The results of phase equilibrium calculations are found in molar fractions. Thus, volumetric fractions are obtained using Amagat’s law cij =

.

xij vi , nc  xkj vk

i = 1, . . . , nc

j = 1, . . . , np

k=1

where .xij is the molar fraction of i-th component in phase j and .vi is the molar volume of pure component i. Figure 20.2 shows the thermodynamic relation between the tie triangle (three-phase region) and the corresponding binary tie lines (two-phase regions). For water-gas and oil-gas two-phase systems, the flux

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Fig. 20.2 Geometric thermodynamic variables for H.2 O-C.10 -CO.2 ternary system

function .α(β) is linear while for water-oil two-phase system, it is a second-degree polynomial. The viscosity of hydrocarbon phases is calculated through the [LoEtAl64] correlation (LBC), .μj . We consider pure water for the determination of water phase viscosity and calculate .μw using the correlation from reference [Ya09], given by μw = 10(−11.6225+1.949·10

.

3 /T +2.1641·10−2 T −1.599·10−5 T 2 )

,

where .μw is the water phase viscosity [cp] and T is the temperature [K]. The fractional flow function fj =

.

krj /μj n p 

krk /μk

k=1

is calculated using the Corey-type relative permeability model ⎞nj

⎛

⎟ ⎜ o ⎜ Sj − Srj ⎟ krj = krj ⎟ ⎜ np ⎠ ⎝  1− Srj

.

k=1

20 Mathematical Modeling of Partially Miscible Water Alternating Gas. . . Table 20.2 Relative permeability parameters

245

Property .Srj o .krj

Value Water .0.12 1

Oil .0.1 1

Gas .0.05 1

.nj

2

2

2

Fig. 20.3 Phase behavior and integral curves of the eigenvectors (paths) for a H.2 O-CO.2 -C.10 ternary system. Fast (blue) and slow (red) paths are shown within the three-phase region (red tie triangle); dotted lines are phase residuals. Nontie-line paths (blue) are shown in the two-phase regions

o is the endpoint of the relative permeability of phase j , .S is the residual where .krj rj saturation of phase j , and .nj is the Corey exponent of phase j . The relative permeability parameters used in this study are shown in Table 20.2. The phase behavior of the ternary system H.2 O-CO.2 -C.10 using the parameters listed in Table 20.1 is presented in Fig. 20.3. In this figure, also the paths of the solutions for the two- and three-phase regions are shown. These paths are the integral curves of the eigenvectors and are calculated by simple numerical integration using the Runge-Kutta technique.

246

A. Pires and B. Loza

20.3 Analytical Solutions In this section, the H.2 O-C.10 -CO.2 system is used to model the phases composition during water alternating gas injection (WAG) process. We evaluate the injection of several slugs of H.2 O and CO.2 into an oil reservoir containing liquid hydrocarbon and mobile water. The first initial oil composition .I1 is the average composition calculated at a previous waterflooding breakthrough. The solution path for ternary systems during partially miscible WAG process is constructed via the following steps: 1. Identify the tie lines that extend through the injection conditions. Since injection fluids are either pure water or pure gas, both conditions can be represented by the H.2 O-C.10 and the H.2 O-CO.2 tie line, respectively. Additionally, for all twophase systems analyzed in this work, the common value for the tie lines that represent binary systems is .β = 0. 2. Identify the tie lines that extend through the initial conditions. At the beginning of the WAG process, the initial condition belongs to the H.2 O-C.10 tie line. As water and gas slugs are injected, C.10 concentration decreases, and consequently different equilibrium states define the initial condition. 3. Construct the semi-shock from the injection gas composition to determine the tangent shock point in the corresponding H.2 O-CO.2 tie line. 4. Identify the equal-eigenvalue point on the initial tie line to start the construction of the nontie-line path to the injection tie line. 5. Starting at the initial tie line for gas injection, draw the tangent shock between two- and three-phase regions. Transitional phase shocks are usually tangent to one of the paths within the three-phase region. 6. Build a spreading wave in the three-phase region up to the tangent point. At this point, the shock velocity between three and two-phase regions is equal to the spreading wave eigenvalue. 7. Construct a rarefaction along the gas injection tie line that connects the tangent shock point (step 3) with the intersection point of the transitional phase shock (step 6). 8. After gas injection, update the initial composition and repeat the step 4 for water injection. 9. At the new initial tie line, draw a semi-shock for which the nontie-line shock velocity is equal to the tie line eigenvalue, .σβ = λC . 10. Construct a rarefaction along the water injection tie line that connects the water injection condition with the intersection point of the nontie-line path. 11. Determine whether a rarefaction or a rarefaction-shock connects the equaleigenvalue point on the initial tie line with the updated initial composition. 12. After water injection, update the initial composition and go to step 2 for the next cycle.

20 Mathematical Modeling of Partially Miscible Water Alternating Gas. . .

247

Fig. 20.4 Composition paths for water and gas flooding. Shocks are denoted by dashed lines and spreading waves by solid lines

20.3.1 Water and Gas Flooding We consider the WAG process beginning at the water breakthrough of an oil reservoir waterflooding. At this point, we update the initial reservoir composition. Based on this assumption, we solve the Riemann problem for variable injection conditions during the WAG process. Pure gas is also injected into the same oil reservoir to evaluate the efficiency of the WAG process. As injection of pure water (H.2 O) or pure gas (CO.2 ) into the original oil reservoir (represented by C.10 ) is considered, the problem consists of a two-phase binary displacement where the complete solution takes place on a single tie line, .β (J ) = β (I ) [Or07]. The composition routes and profiles for initial water injection and continuous gas injection are shown in Figs. 20.4 and 20.5. For initial water injection, there is a spreading wave, .C-rarefaction, with continuous variation of the compositions (.C, β) that connect the injection composition .Iw to point a. Next, there is a shock from point a to the initial condition .I0 . The shock (J ) .C-shock is tangent to the curve .F (C, β w ) at a.

248

A. Pires and B. Loza

Fig. 20.5 Composition profiles for water and gas flooding

Continuous gas injection .Ig results in a route with a tangent shock to the curve F (C, β) at b, followed by a spreading wave along the .C10 -CO2 tie line to c. From c there is a shock (tangent to the tie-line path at c) to the initial condition .I0 .

.

20.3.2 Single-Cycle WAG Injection Most of the composition paths during WAG processes have the same fundamental wave structure. The solution path for a single-cycle WAG model (Fig. 20.6) is defined as a combination of spreading waves, shock waves, and constant states. In the case of a first gas slug injection, we have the following wave groups: 1. A constant state at the injection composition .Jg . 2. A shock wave (or .C-shock) from the injection composition .Jg to state a such that (J →a)

3.

4. 5. 6.

the shock velocity is tangent to the curve .F (C (a) , β (Jg ) ), .σC g = λ(a) C . A rarefaction wave (or .C-rarefaction) from state a to b. At point b, the solution path switches to the nontie-line path. This switch creates a constant state zone because the tie-line path wave velocity is smaller than the nontie-line path wave (b) velocity, .λ(b) C < λβ . A transitional shock wave (or .τ -shock) from two-phase region at b to three-phase (b→c) (c) region at c. This shock has the same velocity as the fast wave at c, .στ = λ2 . A short spreading wave (or .2-rarefaction) in the three-phase region at the fast path to d. Another transitional shock .τ -shock from d to e. In this case, the transitional wave starts in the three-phase region and ends in the two-phase region. This shock has

20 Mathematical Modeling of Partially Miscible Water Alternating Gas. . .

249

Fig. 20.6 Composition paths for water alternating gas injection. Single-cycle WAG with .P V I = 1 and ratio .1 : 1

the same velocity as the fast path wave at d and as the tie-line path wave at e, (d) (d→e) (e) λ2 = στ = λC , resulting in a degenerate shock. Both transitional shocks, step 4 and 6, satisfy the following Rankine-Hugoniot conditions:

.

(b)

− Fi

(b)

− Ci

(d)

− Fi

(d)

− Ci

στ(b→c) =

Fi

στ(d→e) =

Fi

.

.

Ci

Ci

(c)

(c)

, i = 1, 3

(e)

(e)

, i = 1, 3

7. A continuous variation along the initial tie line .C-rarefaction from e to f. (f ) (f →I1 ) (I ) 8. A semi-shock .C-shock from f to the initial condition .I1 , .λC = σC > λC 1 . As the initial composition is further enriched with .CO2 , .β value increases defining new equilibrium states. The next composition path is the injection of the first water slug. Because both initial and injection states are in the same two-phase system, the composition route does not enter the three-phase region. Then, the composition route results in a twophase partially miscible displacement with the following wave groups: 1. A constant state at composition g. 2. A shock wave (or .β-shock) from state g to h such that the shock velocity is (g→h) (h) = λC . tangent to the curve .F (C (h) , β (I2 ) ), .σβ 3. A rarefaction wave (or .C-rarefaction) from state h to i.

250

A. Pires and B. Loza

Fig. 20.7 Composition profiles for injection of 1st-gas slug and 1st-water slug. Single-cycle WAG model with .P V I = 1 and ratio .1 : 1 (i)

(i→I2 )

4. A semishock .C-shock from i to the initial condition .I2 , .λC = σC

(I )

> λC 2 .

The composition profiles for a single-cycle WAG injection are shown in Fig. 20.7.

20.3.3 Five-Cycle WAG Injection In this case, the wave structure for each cycle is similar to the structure for the single-cycle WAG. The solution path for each cycle is shown in Fig. 20.8. These results show the evolution of the initial composition at each cycle. In all the solution paths during gas slug injection, three-phase flow occurs, and the wave solution is always defined by the initial tie line. The composition profiles for each gas and water slug injection are shown in Figs. 20.9 and 20.10. In the case of gas slug injection, an intermediate spreading wave (.2-rarefaction) increases as the number of cycles increases. Then, three-phase flow effects are more pronounced as the initial composition is further enriched with .CO2 .

20.3.4 Comparison of Solutions In this section, we evaluate the efficiency of the WAG process through the cumulative oil (.C10 ) recovery up to .P V = P V Iwater breakthrough + P V IW AG +

20 Mathematical Modeling of Partially Miscible Water Alternating Gas. . .

251

Fig. 20.8 Composition paths for water alternating gas injection. Five-cycle WAG model with = 1 and ratio .1 : 1

.P V I

Fig. 20.9 Composition profiles for injection of gas slugs. Five-cycle WAG model with .P V I = 1 and ratio .1 : 1

P V Ichase water . Figure 20.11 shows the cumulative oil (.C10 ) recovery for waterflooding, continuous gas injection, single- and five-cycle WAG injection. The behavior of the recovery factor is the same for all the cases up to the breakthrough. As the number of slugs increases (and the size of the slugs decreases), the WAG solutions present better recovery at early times. After a long period of WAG

252

A. Pires and B. Loza

Fig. 20.10 Composition profiles for injection of water slugs. Five-cycle WAG model with .P V I = 1 and ratio .1 : 1

Fig. 20.11 Recovery of .C10 . Comparison between water injection, continuous gas injection, single- and five-cycle WAG with .P V I = 1 and ratio .1 : 1

injection, both single- and five-cycle model solutions tend to the continuous gas injection solution. This results show that the effective recovery factor can be obtained by gas mobility control, and consequently less amount of gas is necessary. This is a very important result for fields where the source of gas is limited.

20 Mathematical Modeling of Partially Miscible Water Alternating Gas. . .

253

20.4 Summary and Conclusions The key result of this work is the introduction of geometric thermodynamic variables to construct the transition waves between two- and three-phase regions in oil recovery from WAG injection process. We analyze the eigenvalue problem for ternary systems in which we have one-dimensional two- and three-phase displacements in porous medium. Under the assumptions adopted, the compositions for the partially miscible WAG injection are unique. The eigenvalue problem can be parameterized in the twophase region, and the behavior of the tie lines and two-phase boundaries can be characterized by geometric variables. Within the two-phase region, the nontie-line eigenvalue can be found using the derivative of the tie line slope function, .αβ , and the equal-eigenvalue point in the two-phase region is a key point to construct shocks between two- and three-phase regions. Most of composition paths during WAG injection are defined by the initial tie line. The Hugoniot locus of shocks between two- and three-phase compositions is a series of curves in space that intersect at points where the nontie-triangle eigenvalue is equal to the shock velocity. The results of this work can be useful in the development of WAG injection strategies. Further, our methodology to solve the Riemann problem for WAG injection considering instantaneous phases redistribution is ready for applications.

References [Be93] Bedrikovetsky, P.G.: Mathematical Theory of Oil and Gas Recovery. Kluwer Academic Publishers, London, UK (1993) [CaDy58] Caudle, B.H., Dyes, A.B.: Improving Miscible Displacement by Gas-Water Injection. In: 32nd Annual Fall Meeting of Society of Petroleum Engineers, vol. SPE-911-G, pp. 281–283 (1958) [ChEtAl11] Christensen, J.R., Stenby, E.H., Skauge, A.: Review of WAG field experience. SPE Reserv. Eval. Eng. 4, 97–106 (2001) [Da98] Danesh, A.: PVT and Phase Behavior of Petroleum Reservoir Fluids. Elsevier, Amsterdam, The Netherlands (1998) [FaSc92] Falls, A.H., Schulte, W.M.: Theory of three-component, three-phase displacement in porous media. SPE Reserv. Eng. 7, 377–384 (1992) [LaJo05] LaForce, T.C., Johns, R.T.: Composition routes for three-phase partially miscible flow in ternary systems. SPE J. 10, 161–174 (2005) [LoEtAl64] Lohrenz, J., Bray, B.G., Clark, C.R.: Calculating viscosities of reservoir fluids from their compositions. J. Pet. Technol. 16, 1171–1176 (1964) [MaPl01] Marchesin, D., Plohr, B.: Wave structure in WAG recovery. SPE J. 6, 209–219 (2001) [Or07] Orr Jr., F.M.: Theory of Gas Injection Processes. Tie-Line Publications, Copenhagem, Denmark (2007) [PeRo76] Peng, D.Y., Robinson, D.B.: A new two-constant equation of state. Ind. Eng. Chem. Fundam. 15, 59–64 (1976) [PiBe05] Pires, A.P., Bedrikovetsky, P.G.: Analytical modeling of 1-D nc component miscible displacement of ideal fluids. In: SPE Latin American and Caribbean Petroleum Engineering Conference, vol. SPE-94855 (2005)

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[PiEtAl05] Pires, A.P., Bedrikovetsky, P.G., Shapiro, A.A.: A splitting technique for analytical modelling of two-phase multicomponent flow in porous media. J. Pet. Sci. Eng. 51, 54–67 (2005) [Sa06] Sandler, S.I.: Chemical, Biochemical, and Engineering Thermodynamics. Wiley, NJ, USA (2006) [Wa64] Wachman, C.: A mathematical theory for the displacement of oil and water by alcohol. SPE J. 4, 250–266 (1964) [Ya09] Yaws, C.L.: Transport Properties of Chemicals and Hydrocarbons. Elsevier, Amsterdam, The Netherlands (2009)

Chapter 21

Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load D. Ponomarev

21.1 Introduction Due to its practical importance, contact problems with wear has been an area of active research for decades. A problem when indented wearable punch slides with a constant speed on an elastic layer or a half-space is a classical setting (see, e.g. [AlKu69, AlKu70]). Numerous empirical laws of wear were proposed to fit predictions of different models [ArCh19, ArCh20, ArFa11, FeEtAl16, Go98, Ko79, ZhEtAl07] to experiments such as a pin on a rotating disk. Recently, a RiemannLiouville relation, generalising the classical simple integral relation between worn volume and pressure, was motivated in [Ar22]. Mathematical analysis of that model and its further extension was given in [Po22]. Namely, in addition to the fractional order integration [GoMa08], another generalisation of the model has been incorporated aiming to account for possible relaxation effects [YeEtAl96]. The long-time behaviour of the pressure profile has been investigated in the set-ups where the exterior load was either constant or eventually constant (i.e. the so-called transitional load describing a smooth switch between two values over some finite time interval). In the present work, we are concerned with analysis of the long-time behaviour of the solution of the generalised model in the situation where the exterior load is time-harmonic. The applied analysis also automatically applies to the classical model (with worn volume and pressure related through the basic Archard’s law [Ar53, Ga76]) as a particular case.

D. Ponomarev () FACTAS team, Centre Inria d’Université Côte d’Azur, Sophia Antipolis, France St. Petersburg Department of Steklov Mathematical Institute, Petersburg, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_21

255

256

D. Ponomarev

The structure of the paper is as follows. In Sect. 21.2, we briefly recall the previously proposed model as well as some auxiliary functions and their properties that are going to be essential for the present work. Section 21.3 is dedicated to the analysis of the solution of the model: we will identify time-dependent stationary state for the pressure profile and estimate the convergence rate depending on the value of the model parameter .α. We illustrate the obtained results numerically in Sect. 21.4 and conclude with their discussion in Sect. 21.5.

21.2 Model According to [AlKo84, ArCh19, ArCh20, ArFa11, Ga76, Ko85, VoEtAl74], the pressure under the punch satisfies the following equation for displacements:  ηp (x, t) +

a

.

−a

K (x − ξ ) p (ξ, t) dξ

= δ (t) − w [p] (x, t) − Δ (x) , x ∈ (−a, a) , t ≥ 0,

(21.1)

and the force equilibrium condition 

a

.

−a

p (x, t) dx = P (t) ,

t ≥ 0.

(21.2)

The contact load is denoted .P (t), and, in the present work, we study its particular form, namely, P (t) = P0 − PΔ + PΔ cos (ωt)

.

(21.3)

for some given constants .P0 , .PΔ , .ω > 0. The interval .(−a, a) corresponds to the contact area under the punch; .K (x) is a kernel function of the “pressure-to-displacement” operator. Such an operator stems from the Green’s function pertinent to a given geometry. In particular, we are interested in an elastic half-space problem which is the limiting case of the thicklayer problem. In this case, we have K (x) := − log |x| + CK ,

.

(21.4)

for some constant .CK > log a. The illustration of the problem geometry is given in Fig. 21.1. The first term on the left-hand side of (21.1) accounts for the additional deformation due to the presence of a coating or to model surface roughness [AlKo82, Ko79]. The strength of this effect is measured by the constant .η > 0.

21 Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load

257

Fig. 21.1 Illustration of the geometry of the problem

The initial punch profile .Δ (x) is a known function whereas the punch indentation δ (t) is a function of only time. Its initial value .δ (0) can be found from solving

.

 ηp (x, 0) +

a

.

−a

K (x − ξ ) p (ξ, 0) dξ = δ (0) − Δ (x) ,

x ∈ (−a, a) ,

(21.5)

a and requiring that . −a p (x, 0) dx = P0 , as follows from (21.1) and (21.2), respectively, evaluated at .t = 0. Finally, .w [p] (x, t) is the wear term which is an operator acting on the contact pressure .p (x, t). Following the discussion in [Po22, Sec. 3], we take it as 

t

w [p] (x, t) = −νμ1/α−1

.

  Eα μ1/α (t − τ ) p (x, τ ) dτ,

(21.6)

0

where .μ > 0 is a constant and the special function .Eα can be defined as ∞ α  (−1)k kx αk , .Eα (x) := x Γ (αk + 1)

x > 0,

α > 0,

(21.7)

k=1

with .Γ being the Gamma function. Note that, in particular case, when .α = 1, we have .E1 (x) = − exp (−x). We will make use of the following asymptotics Eα (x) = −

.

1 α +O Γ (1 + α) x 1−α



1 x 1−2α

 ,

|x|  1,

(21.8)

258

D. Ponomarev

⎧   1 1 ⎨− α , α ∈ (0, 1) ∪ (1, 2) , + O Γ (1−α) x α+1 x 2α+1 .Eα (x) = ⎩− exp (−x) , α = 1,

x  1, (21.9)

as well as the integral relation  .

0

x0

    Eα λ1/α x dx = λ−1/α Eα −λx0α − 1 ,

x0 > 0,

α > 0,

λ > 0, (21.10)

where .Eα is the Mittag-Leffler function defined as Eα (z) :=

∞ 

.

k=0

zk , Γ (αk + 1)

z ∈ C,

α > 0.

(21.11)

In particular, we have .E1 (z) = exp z, .z ∈ C, and .Eα (0) = 1, .α > 0. Moreover, the following useful asymptotic holds true: Eα (−x) =

.

⎧ ⎨

+O

1 1 Γ (1−α) x

⎩exp (−x) ,



1 x2



, α ∈ (0, 1) ∪ (1, 2) , α = 1,

x  1.

(21.12)

References for the above-mentioned results can be found in [Po22, Appendix A]. Finally, we recall from [Po22, Sec. 3] that when .α = 1 and .μ = 0 (or more precisely, in the limit of .μ  0), the relation (21.6) reduces to what is consistent with the classical Archard’s law [Ar53]:  w [p] (x, t) = −ν

t

p (x, τ ) dτ.

.

0

21.3 Analysis The model (21.1)–(21.2) has been rigorously analysed in [Po22, Sec. 4]. We adapt here a general theory [Po22, Thm 6] using the result of [Po22, Prop. 14] valid for the particular form of the kernel function (21.4). Namely, we have the following theorem. Theorem 1 Assume that .μ ≥ 0, .η, .ν > 0, .α ∈ (0, 2), .a = 2, and that 2 p  a(·, 0) ∈ L (−a, a) solves (21.5) with K given by (21.4) and .δ (0) such that . p 0) dx = P0 for some .P0 > 0. Suppose .P (t) is as in (21.3) and .w [p] (x, −a

 is defined in (21.6). Then, the unique solution .p ∈ Cb R+ ; L2 (−a, a) of (21.1) satisfying (21.2) is given by .

21 Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load

259

∞

p (x, t) =

.

P (t)  + dk (t) φk (x) , 2a

x ∈ (−a, a) ,

t ≥ 0,

(21.13)

k=1

where dk (t)

.

:=dk0

 1+

ν μ (η + σk ) + ν

   Eα − μ +

ν η + σk



 t

α

 −1

(21.14)

 1/α−1 ν νlk lk μ+ − (P (t) − P (0)) − 2a (η + σk ) η + σk 2a (η + σk )2   1/α  t ν μ+ × Eα (t − τ ) [P (τ ) − P (0)] dτ, k ≥ 1, η + σk 0  dk0 :=

a

.

−a

p (ξ, 0) φk (ξ ) dξ,

lk :=

1 2a



a

−a



a

−a

K (ζ − ξ ) φk (ξ ) dζ dξ,

k ≥ 1.

Here, .Eα and .Eα are as in (21.7) and (21.11), respectively, whereas .φk ∈ L20 (−a, a), .σk > 0, .k ≥ 1, are normalised eigenfunctions and eigenvalues of the compact self-adjoint operator    a 1 K (x − ξ ) − (K (ζ − x) + K (ζ − ξ )) dζ φ (ξ ) dξ 2a −a −a

 K2 [φ] (x) :=

.

a

defined on the functional space 2 .L0 (−a, a)

  2 := f ∈ L (−a, a) :

a

−a

 f (x) dx = 0 .

Proof The statement of the theorem is merely a rephrasement of several results from [Po22]. First of all, it is straightforward (see also [Po22, Prop. 11]) to verify that the kernel function (21.4) and the exterior load (21.3) satisfy all assumptions of [Po22, Thm 6]. Then, thanks to [Po22, Prop. 14], we observe that a form of the solution given by [Po22, Thm 6] simplifies since .Ker K2 , the kernel space of the auxiliary operator .K2 , is empty. This last part calls for the additional assumption .a = 2 appearing in the formulation. Finally, the fact that .σk > 0 for .k ≥ 1 follows from [Po22, Prop. 4] and another use of [Po22, Prop. 14]. We now proceed with the main goal of the paper. We identify the stationary state and perform long-time behaviour analysis to study qualitative character and speed of the convergence of the solution to this stationary state.

Proposition 1 Under assumptions of Theorem 1, there exists .δp ∈ C R+ ; L2  . (−a, a)) with .δp (·, t) 2 −→ 0 as .t → +∞ such that the solution .p (x, t) L (−a,a) to the model (21.1)–(21.2) can be written as

260

D. Ponomarev

p (x, t) = p ∞ (x) − W0 (x) cos (ωt − ψ (x)) + δp (x, t) =: p∞ (x, t) + δp (x, t) , (21.15) where .

.

 ∞   μ (η + σk ) 1 PΔ μlk dk0 + φk (x) , (P0 − PΔ )+ 2a μ (η + σk ) + ν 2a [μ (η + σk ) + ν] k=1 (21.16) W 1 (x) 1/2 , .W0 (x) := [W1 (x) + W2 (x)] , ψ (x) := sign (W2 (x)) arccos W0 (x)

p ∞ (x) :=

  ∞ ν PΔ  lk 1−α c,ω .W1 (x) := −1 + β Ck + 1 φk (x) , 2a η + σk η + σk k k=1

W2 (x) :=

.

 Ckc,ω :=

.

∞

∞ νlk PΔ  β 1−α Cks,ω φk (x) , 2 k 2a + σ (η ) k k=1

 Eα (βk τ ) cos (ωτ ) dτ,

0

 .βk := μ+

Cks,ω :=

ν η + σk

∞

(21.17)

Eα (βk τ ) sin (ωτ ) dτ,

0

(21.18)

1/α ,

k ≥ 1.

(21.19)

Moreover, we have .

  δp (·, t)

L2 (−a,a)

   = O exp − μ +

  . δp (·, t) 2 =O L (−a,a)



1 tα

ν η + σ1

  t ,

t  1,

α = 1,

 ,

t  1,

α ∈ (0, 1) ∪ (1, 2) .

Proof Plugging (21.3) into (21.14) and employing (21.10), we obtain PΔ lk [cos (ωt) − 1] 2a (η + σk )   α α  PΔ lk ν Eα −βk t − 1 dk0 + + 2a (η + σk ) μ (η + σk ) + ν   t νlk PΔ 1−α cos − β Eα (βk τ ) cos (ωτ ) dτ (ωt) 2a (η + σk )2 k 0   t − sin (ωt) Eα (βk τ ) sin (ωτ ) dτ ,

dk (t) =dk0 −

.

0

(21.20) (21.21)

21 Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load

261

where we used definition (21.19). Note that, due to (21.9), the integrals here are converging even when .t → +∞. Hence, using the definitions in (21.18), we can write  t  ∞ c,ω . Eα (βk τ ) cos (ωτ ) dτ = Ck − Eα (βk τ ) cos (ωτ ) dτ, 0



t

t

.

0

 Eα (βk τ ) sin (ωτ ) dτ = Cks,ω −

∞

Eα (βk τ ) sin (ωτ ) dτ.

t

Consequently, we arrive at lk PΔ μ (η + σk ) μ (η + σk ) + μ (η + σk ) + ν 2a μ (η + σk ) + ν

 ν lk PΔ βk1−α Ckc,ω + 1 cos (ωt) − 2 2a (η + σk )

dk (t) =dk0

.

−

(21.22)

ν lk PΔ β 1−α Cks,ω sin (ωt) + rk (t) , 2a (η + σk )2 k

where ν .rk (t) := μ (η + σk ) + ν +

 dk0 +

ν lk PΔ 2a (η + σk )



 PΔ lk Eα −βkα t α 2a (η + σk )  ∞ 1−α β Eα (βk τ ) cos (ω (t − τ )) dτ. 2 k

(21.23)

t

Inserting (21.22) into (21.13) and taking into account (21.16)–(21.17) yields (21.15) with δp (x, t) :=

∞ 

.

rk (t) φk (x) .

k=1

It now remains to deduce (21.20)–(21.21). To this effect, we first note that, due to the mutual orthogonality of functions .{φk }∞ k=1 , we have ∞ 1/2    |rk (t)|2 . δp (·, t) 2 = . L (−a,a)

(21.24)

k=1

Since .0 < σk ≤ σk−1 for .k > 1, we have βkα ≥ β1α ,

.

1 1 , ≤ μ (η + σk ) + ν μη + ν

k ≥ 1.

262

D. Ponomarev

Hence, for sufficiently large t, we can estimate ∞   1/2 

α α  0 2 ν   .  μ (η + σ ) + ν Eα −βk t dk  k

(21.25)

k=1

≤

 α α  ν Eα −β t  p0 (·, t) 2 L (−a,a) , 1 μη + ν

∞   1/2 

α α  2 PΔ ν   .  2a (η + σ ) [μ (η + σ ) + ν] Eα −βk t lk  k k

(21.26)

k=1

≤

 α α  PΔ ν Eα −β t  K1 2 L (−a,a) , 1 2aη (μη + ν)

with  K1 (x) :=

a

.

−a

K (ζ − x) dζ

= 2a (1 + CK ) − (a + x) log (a + x) − (a − x) log (a − x) . Here, we employed the Parseval’s identity

.

∞     0 2 dk  = p (·, 0) 2L2 (−a,a) ,

∞ 

k=1

k=1

|lk |2 = K1 2L2 (−a,a) ,

and the fact that .|Eα (−t)| is a monotonically decreasing function for sufficiently large values of t which follows from (21.9). Then, thanks to the triangle inequality for the Euclidean .l 2 norm, we estimate (21.24) using (21.23) and (21.25)–(21.26) .

  δp (·, t) 2 L (−a,a)     ν PΔ K1 L2 (−a,a) Eα −β1α t α  p (·, 0) L2 (−a,a) + (21.27) ≤ 2aη μη + ν  ∞    PΔ ν 1−α  . K + sup β E τ cos − τ dτ ) (ω (t )) (β 2 1 L (−a,a) α k k   2 2aη t k≥1

In case .α = 1, the functions .Eα and .Eα in (21.27) reduce to exponential functions. Employing the simple result 

∞

.

t

e−βk τ cos (ω (t − τ )) dτ =

βk e−βk t , βk2 + ω2

21 Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load

263

we arrive at .

  sup βk1−α  k≥1

∞ t

  β 2−α e−βk t ≤ e−β1 t , Eα (βk τ ) cos (ω (t − τ )) dτ  ≤ sup k 2 2 β + ω k≥1 k

and thus deduce (21.20). In case .α ∈ (0, 1) ∪ (1, 2), we first use (21.12) in the first line of (21.27) to deduce the .O (1/t α ) decay of the corresponding term. Next, we estimate   βk1−α 

.

t

∞

  Cα 1 Eα (βk τ ) cos (ω (t − τ )) dτ  ≤ , α tα

which is possible due to the finiteness of the constant .Cα := supτ >0 τ 1+α |Eα (τ )| entailed by the continuity of .Eα (away from .t = 0) and asymptotics (21.8)–(21.9). Therefore, the decay result (21.21) follows.

21.4 Numerical Illustrations We fix the following set of parameters .a = 1, .ν = 2, .η = 1, .μ = 1.2, .CK = log 5 1.61. We take the oscillatory load profile (21.3) with .P0 = 6, .PΔ = 0.5, .ω = 1.5. For simplicity, we assume that the punch profile .Δ (x) is such that p (x, 0) =

.

2P0  2 a − x2, a2π

x ∈ (−a, a) ,

(21.28)

which is a reasonable initial pressure form. All computations are performed using 60 terms in the expansion (21.13). We illustrate the results for three different values of the parameter .α (.α ∈ {0.8, 1.0, 1.8}) with the parameter .μ = 1.2 and again with .μ = 0. In the latter case, the model is purely of a fractional order (no relaxation effect). Also, recall that when .α = 1, the model reduces to that which does not involve fractional calculus (with or without relaxation, depending on .μ). In Fig. 21.2, we plot the stationary state pressure profile (or, more precisely, a collection of curves .p∞ (·, t) evaluated for .t ∈ [0, 2π/ω]) and investigate the ∞ ± W0 ) on the choice of model parameters dependence of its envelope (given by .p .α and .μ. Evidently, for .μ = 1.2, the impact of the parameter .α on the stationary state is almost undetectable, which is not the case when .μ = 0. Figure 21.3 shows the character and the speed of convergence  of the solu tion to the stationary state .p∞ measured by the quantity .δp (·, t)L∞ (−a,a) := p (·, t) − p∞ (·, t) L∞ (−a,a) . Since the initial profile (21.28) is bounded, the

264 Fig. 21.2 Stationary state (x, t) for multiple values of t over the period .T 4.19 with .μ = 1.2 (above) and its envelope for three different values of .α (top) with .μ = 1.2 (middle) and .μ = 0 (bottom)

D. Ponomarev 3.5

.p∞

3

p

2.5 2 1.5 1 -1

-0.5

0 x

0.5

1

0.5

1

0.5

1

3.5 3 2.5

p

=0.8 =1.0 =1.8

2 1.5 1 -1

-0.5

0

x

p

3

2.5

2 -1

=0.8 =1.0 =1.8 -0.5

0

x

pressure values at larger times are expected to remain bounded too. This is why we replaced the .L2 norm with the .L∞ norm in visualising the solution convergence. Clearly, the results are in direct correspondence to the analytical prediction given by (21.20)–(21.21).

21 Generalised Model of Wear in Contact Problems: The Case of Oscillatory Load

=0.8 =1.0 =1.8

1.2 1 || p ( ,t)||

265

0.8 0.6 0.4 0.2 0

0

10

20

3

=0.8 =1.0 =1.8

2.5 2 || p ( ,t)||

30

t

1.5 1 0.5 0

0

10

20

30

t Fig. 21.3 Convergence of the solution to the stationary state for three different values of .α with = 1.2 (top) and .μ = 0 (bottom)

.μ

21.5 Discussion and Conclusion We have revisited the classical sliding punch problem with a recently proposed generalised model of wear. In particular, we have investigated long-time evolution of the pressure profile under a practically important case of exterior time-harmonic load. We have derived an explicit form of the stationary pressure distribution in terms of eigenfunctions of an auxiliary integral operator. Moreover, we have analysed a speed of the convergence of the model solution to this distribution. We note that, in contrast to previous results when the load was constant (or eventually constant; see [Po22, Sec. 5]), here the stationary distribution .p∞ is a function of

266

D. Ponomarev

both space and time. Its time dependence is, nevertheless, clear and structurally simple: it is harmonic with the same frequency as the exterior load but with a phase shift that depends on the spatial variable (see (21.15)). Numerical simulations have been performed to illustrate the obtained results. In particular, the focus was on the dependence of the mentioned results on the parameters .α, .μ which are characteristic for the present model. It is remarkable that the dependence of the stationary state on the model order .α is insignificant when .μ = 0. The parameter .α, however, has an essential impact on the speed of the convergence towards the stationary state: the convergence is exponential for .α = 1, whereas for .α ∈ (0, 1) ∪ (1, 2), it is algebraic but its rate grows with the increase of .α. Moreover, when .α ∈ (1, 2) the convergence happens in a nonmonotone fashion. The similar effect of .α on the convergence rate was observed in the previous work [Po22] when the load was constant or eventually constant. We thus confirm here the previous observation that the model parameters .μ and .α affect essentially the stationary state profile and the speed of convergence, respectively. These statements would constitute important guidelines when trying to fit the new model to experimental data. Such a fit would be essential for a practical validation of the model. Acknowledgments The author is grateful for the support of Arbeitsmarktservice (AMS) Österreich during the period of working on this manuscript.

References [AlKu69] Alblas, J.B., Kuipers, M.: Contact problems of a rectangular block on an elastic layer of finite thickness: the thin layer. Acta Mech. 8(3), 133–145 (1969) [AlKu70] Alblas, J.B., Kuipers, M.: Contact problems of a rectangular block on an elastic layer of finite thickness: the thick layer. Acta Mech. 9(1), 1–12 (1970) [AlKo84] Aleksandrov, V.M., Kovalenko, E.V.: Mathematical methods in problems with wear (in Russian). In: K.V. Frolov (ed.) Nonlinear Models and Problems of Mechanics of Solids, pp. 77–89. Nauka, Moscow (1984) [AlKo82] Aleksandrov, V.M., Kovalenko, E.V.: On the theory of contact problems in the presence of nonlinear wear (in Russian). Mech. Solids 4, 98–108 (1982) [Ar53] Archard, J.F.: Contact and rubbing of flat surfaces. J. Appl. Phys. 24, 981–988 (1953) [Ar22] Argatov, I.I.: A fractional time-derivative model for severe wear: hypothesis and implications. Front. Mech. Eng. 8, (2022). https://doi.org/10.3389/fmech.2022.905026 [ArCh19] Argatov, I.I., Chai, Y.S.: Effective wear coefficient and wearing-in period for a functionally graded wear-resisting punch. Acta Mech. 230, 2295–2307 (2019) [ArCh20] Argatov, I.I., Chai, Y.S.: Wear contact problem with friction: Steady-state regime and wearing-in period. Int. J. Sol. Struct. 193–194, 213–221 (2020) [ArFa11] Argatov, I.I., Fadin, Yu.A.: A macro-scale approximation for the running-in period. Tribol. Lett. 42, 311–317 (2011) [FeEtAl16] Feppon, F., Sidebottom, M.A., Michailidis, G., Krick, B.A., Vermaak, N.: Efficient steady-state computation for wear of multimaterial composites. J. Tribol. 138(3), 031602 (2016) [Ga76] Galin, L.A.: Contact problems of the theory of elasticity in the presence of wear. J. Appl. Math. Mech. 40(6), 931–936 (1976)

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[GoMa08] Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order, 56 p. (2008). arXiv:0805.3823 [Go98] Goryacheva, I.G.: Contact Mechanics in Tribology. Kluwer Academic Publishers, Dordrecht–Boston–London (1998) [Ko79] Kovalenko, E.V.: On an efficient method of solving contact problems with linearly deformable base with a reinforcing coating (in Russian). Mech. Proc. Natl. Acad. Sci. Armenia 32(2), 76–82 (1979) [Ko85] Komogortsev, V.F.: Contact between a moving stamp and an elastic half-plane when there is wear. J. Appl. Math. Mech. 49, 243–246 (1985) [Po22] Ponomarev, D.: A generalised time-evolution model for contact problems with wear and its analysis. MEMOCS 10(3), 279–319 (2022) [SaEtAl93] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach Science Publishers, Philadelphia (1993) [VoEtAl74] Vorovich, I.I, Aleksandrov, V.M., Babeshko, V.A.: Non-Classical Mixed Contact Problems in Elasticity Theory (in Russian). Nauka, Moscow (1974) [YeEtAl96] Yevtushenko, A.A., Pyr’yev, Yu.A.: The applicability of a hereditary model of wear with an exponential kernel in the one-dimensional contact problem taking frictional heat generation into account. J. Appl. Math. Mech. 63(5), 795–801 (1996) [ZhEtAl07] Zhu, D., Martini, A., Wang, W., Hu, Y., Lisowsky, B., Wang, Q.J.: Simulation of sliding wear in mixed lubrication. ASME. J. Tribol. 129(3), 544–552 (2007)

Chapter 22

On the Philosophical Foundations of an Optimization Algorithm Inspired by Human Social Behaviour Under a Dynamical Status Distribution L. P. L. de Oliveira and V. J. Schmidt

22.1 Introduction Metaheuristics have emerged as a powerful and robust option compared to classical optimization methods, which have some limitations concerning successful applications, such as requirements of differentiability and continuity of the objective function, and also a proper choice of a starting point when facing large-size problem instances. Metaheuristics do not require any assumption for the objective function and work essentially using stochastic mechanisms to ensure an effective search of the optimum solution in the search space. Since the 1960s, physical and biological concepts have inspired remarkable metaheuristic techniques. Methods like evolutionary strategy, evolutionary programming, genetic algorithms and simulated annealing are just some examples of precursor metaheuristics which showed the applicability and advantages of this class of algorithms [Go89, OsLa96, Ta09] and [Ya10]. Aiming to solve real-life problems more efficiently and effectively, many metaheuristics are still being proposed. The ones inspired by nature are some of the most known and used ones nowadays [FiEtAl13, PaSa15] and [WaEtAl15]. Metaheuristics based on biological factors consist in simulating some behaviour of real animal life aspects or in some cases genetics itself. However, so far those algorithms which are focused on animal behaviour patterns are based on agents and are driven only by their instincts, in other words, by non-social behaviour. Some examples of such metaheuristics are particle swarm optimization [KeEb95], ant colony optimization [CoEtAl91], bat algorithm [Ya10a], firefly algorithm

L. P. L. de Oliveira () Instituto Federal Sul Rio Grandense—IFSul, COFORGE-Charqueadas, Pelotas, Brazil V. J. Schmidt CWI Software, São Leopoldo, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_22

269

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[IzEtAl13], hunting animals algorithm [OfEtAl10], fruit fly algorithm [Pa12], spider algorithm [YuLi15] and artificial bee colony [KaBa07]. When using methods in the spirit of the aforementioned ideas, some important aspects of human intelligence are lost: our learning capacity and how we are able to teach and learn from our peers through social interactions. Also, we are able to learn via our own experiences and environmental interference. Therefore, our knowledge is built from three major mechanisms: individual behaviour, social interaction and environmental influence [Ba71, Ba77, Ba86] and [Ba01]. Species can evolve under a Darwinian mechanism but not the individuals, while under social evolution, individuals can evolve dealing with their intrinsic (genetic) limitations via a social status. Thus, we think that these paradigms have their associated and specific time scales each with its advantages. To be more specific, the evolutionary one tends to be more effective but demanding more time, while the social one tends to be more efficient but more susceptible to stagnation. Of course, these are general tendencies, and a number of adaptations can be implemented to work around one or another factor. In this contribution, we propose a Dynamic Status Distribution Optimization (DySDO) algorithm, an agent-based model (ABM) algorithm based on (mainly human) social behaviour. The driving force is the social status differences between the individuals for searching the optimum solution in continuous domain problems. To this end, we were motivated by the fact that humans learn from their own experiences and also from others in order to find better solutions for a living as comfortable as possible. In fact, it is common to see least well-positioned individuals to follow or imitate other more successful examples. Also, the individual search for good social positions are not neglected, and after experimenting new solutions, an individual keeps the best encountered ones so far. Therefore, our metaheuristics is a combination of two such search strategies, i.e. personal and social. The DySDO algorithm is based on the fact that in a society, everyone tries to achieve a better position under the diverse aspects, such as financial advantages, educational perspectives, etc. Because of its relative nature, social status tends to push the society to get better in terms of an average position as a whole. In this line, we believe that a combination of personal-like and social-like strategies can generate better solutions for each agent and moreover a greater chance for the whole population of agents to achieve higher developmental degrees, which in turn rises the chances for each agent to find even better solutions and forming a virtuous cycle of progress. The latter can drive in average the population of agents to a global optimum.

22.2 General Aspects of Optimization Algorithms An optimization problem is represented by the following scheme: find a global maximum (or minimum) of the function .fo : Γ ⊂ RD −→ R in a search domain .Γ . In the specialized jargon, each .x ∈ Γ is a solution (a candidate to be the optimum) and .fo is referred as the optimization function. Note that a

22 On the Philosophical Foundations of an Optimization Algorithm Inspired. . .

271

Fig. 22.1 Easom’s function in 2D, showing lots of local minima around the global one at .(π, π ). Left: Graph in the search domain .Γ = [−100, 100] × [−100, 100]. Right: Zoom of the graph for a subset of .Γ around the global minimum

minimum problem can be converted into a maximum one by change in sign making fo → −fo and vice versa. An additional issue to the above-mentioned definition is the complexity of the objective function .fo . For example, consider a 2D objective function whose graph is similar to the relief of a mountain and the problem is to find its largest value at some coordinates .(x ∗ , y ∗ ), corresponding to the latitude and longitude of the maximum height. In general an objective function .fo is a fairly complex one, with a number of local maxima and possible plateaus, both making it difficult to solve the problem. Such a challenge becomes even more problematic for larger dimensions. It is obvious that testing a number of randomly chosen solutions .xi ∈ Γ, i = 1, . . . N, does not define an efficient strategy. For example, a set of solutions chosen at random would hardly give a good result for the minimum optimum in a situation like the one shown in Fig. 22.1, the graph of Easom’s function, a benchmark frequently used to test metaheuristics. A better idea would be to apply some analytical strategies based on the partial derivatives of .fo to restrict the tests to its critical points, where .fo ’s gradient is the null vector or does not exist. However, in some practical situations, the analytical strategies are not feasible or efficient. To work around these issues, well-defined algorithms are designed for leading an initial set A of not optimized solutions towards a better set B of solutions. These types of algorithms are called heuristics. As an improvement B should be a set with at least one element with a higher value for .fo compared to the solutions in A, i.e. there exists .x∗ ∈ B such that .f (x∗ ) > f (x) for any .x ∈ A in the case of a maximum problem. Generally speaking, the heuristics are strategies for searching better solutions (set B) from a previously determined set (A) with a less qualified solutions. Roughly, the heuristics can be classified as local and global ones, where the first types are those which improve solutions searching better solutions in small neighbourhoods compared to the size of the search domain .Γ , while the global ones extend the search to the whole of .Γ . A local heuristics results in the intensification of the search, while a global one results in the exploration (or diversification) in the search process.

.

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L. P. L. de Oliveira and V. J. Schmidt

Fig. 22.2 Rastrigin’s function in 2D, showing local minima around the global one at .(0, 0). On the left, search domain .Γ = [−6, 6] × [−6, 6] and on the right a zoom into a subset of .Γ around the global minimum

In most situations, two or more heuristics are combined to solve an optimization problem with at least one of each kind—local and global. The reason for using a local heuristics is that it leads to a refinement of solutions, which is in accordance with the objective of finding an optimum. The use of global heuristics is less evident and is related to the possible existence of local optima. Indeed, each local optimum can trap a local heuristics into its basin of attraction, possibly far away from the global optimum and thus impeding the algorithm of finding it, so that there is need for an exploratory (global) heuristics to open a path and find other optima, where the global optimum is hopefully one of them. See, for example, the Rastrigin’s function with a number of “traps” (local minima) in Fig. 22.2, which is another benchmark function frequently used to test metaheuristics (the ways how these heuristics are linked together). There are a number of ways to construct a metaheuristics algorithms, and some of the more frequently used implementations were referred to in Sect. 22.1. The proposed algorithm (DySDO) is an ABM algorithm, where each agent carries two positions (or addresses) in .Γ , its present location and its memory (its best position found so far). The algorithm puts the agents to move in such a way that their memories evolve towards an optimum of the objective function .fo (intensification), but changes in their locations maintain a uniform distribution in .Γ (exploration). Figure 22.3 shows these characteristics of DySDO applied to Rastrigin’s objective function, where details are given in Sect. 22.3. Here, exploration comes mainly from an individual searching not involving agent-agent interactions. On the other hand, the memory evolves according to agent-agent interactions led by a social status parameter which emulates the corresponding concept in social science. Social status is greatly associated with social intelligence, according to the social cognitive theory (see Sect. 22.3).

22 On the Philosophical Foundations of an Optimization Algorithm Inspired. . .

273

Fig. 22.3 The agents’ real locations in .Γ (left) and their memory positions (right) resulting from DySDO applied to Rastrigin’s function with search domain .Γ = [−6, 6] × [−6, 6]

22.3 An Optimization Algorithm Based on a Dynamical Social Status Distribution There are two basic search mechanisms an agent can use for getting a higher social status, individual experimentation and learning from others. In the further, we call these Mechanism 1 and Mechanism 2, related to learning-by-direct-experience and to learning-through-model modes of driving individual behaviour in a society according to the social learning theory [Ba71]. The first one consists in the agent evaluating the current solution, trying to find a better one in a sort of trial-anderror strategy with some degree of self-learning from that experience. However, it is a common practice that a person gets better solutions (higher status) by learning from those more successful examples (Mechanism 2). Indeed, people tend to imitate successful individuals by trying to follow their formula and appreciating their knowledge because of their higher status. Because of that, the more successful, the more influence individuals usually have on others, when compared to the less successful ones. The tendency of a society under Mechanisms 1 and 2 is to develop as a whole and thus to improve solutions in the context defined by the considered objective function. Note that the role of Mechanism 2 in this process is driven by the degree of social inequality among the individuals in terms of status and a

274

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collaborative attitude of the successful individuals to influence positively the less successful ones. From now on, we will use the word agent to refer to each individual or person of a society and apply the concept of agents as usually done in ABMs, where details may be found in reference [BaEtAl12]. Let .fo : Γ ⊂ RD → R be the objective function whose global minimum is to be found. For simplicity we suppose that the search domain .Γ is a Ddimensional parallelepiped, the result of a Cartesian product of D closed intervals, .Γ = [α1 , β1 ] × [α2 , β2 ] × . . . × [αD , βD ]. The adaptation for the global maximum finding problem can be done considering .fo ← −fo . Following the above-described social aspects, we consider a set of .Na agents, which are moved around in the search space from an initial instant .t = 0 to a final instant .t = T . We consider the time interval .[0, T ] divided into time steps .t = 0, 1, 2, . . . , T . From each time t on until .t + 1 each agent .ai moves from .xi (t) ∈ Γ to a new position .xi (t + 1) ∈ Γ searching for better solutions. This procedure represents the Mechanism 1 mentioned above. Also, each agent .ai interacts with those who are next to them, trying to learn from the more successful ones, which represents Mechanism 2, and both are more detailed below. In our model, the parameters considered for each agent .ai are: • Location .xi (t) = (xi1 (t), xi2 (t), . . . , xiD (t)): a point in .Γ , where .ai is located at instant t. For each .ai , .xi = xi (t) varies with time describing a quasi-random walk trajectory. • Memory .x∗i (t): the best solution found by .ai up to an instant t by means of Mechanisms 1 or 2. Thus, there are two kinds of locations associated with each agent .ai at t, its current location .xi (t) and the location .x∗i (t), where it found the smallest value for .fo up to an instant t. • Social status .Si (t): taking values in a closed interval .[SMin , SMax ] and representing the social status of agent .ai at t as a function of .fo (x(t)∗i ), as described by Eq. (22.3). The choices of values for minimal status .SMin and maximum status .SMax are user defined, where for convenience we choose .SMin = 0 and .SMax = 1; further the instantaneous values of .Si = Si (t) vary with time. • Influence radius .Ri (t): the radius of influence of agent .ai , where other agents .aj are influenced if .Sj (t) < Si (t) and .|xj (t) − xi (t)| < Ri (t) and the Euclidean metric is used. Thus, this parameter relates to the communicability (popularity) degree of an individual .ai at t. .Ri (t) is assumed to be an increasing function of .ai ’s social status with respect to the context defined in the objective function. Therefore, it is an increasing function of the quality of .ai ’s solution .x∗i (t) in comparison to the ones found by the others in the society up to instant t. For each agent .ai at each time t, the values .fo (x∗i (t)), .Si (t) and .Ri (t) are derived from two basic parameters, its current location .xi (t) and its memory location .x∗i (t). Then, we can completely represent an agent at instant t as .ai (t) = ai (xi (t), x∗i (t)) and denote the population of .Na agents as .P(t) = {ai (xi (t), x∗i (t)); i = 1, . . . Na }. Now, a simulation starts at .t = 0 with an initialized population ∗ .P(0) = {ai (xi (0), x (0)); i = 1, . . . Na } of .Na agents randomly distributed in i the D-dimensional parallelepiped shaped search space .Γ . The initialization starts

22 On the Philosophical Foundations of an Optimization Algorithm Inspired. . .

275

setting .x∗i (0) = xi (0), and the other parameters .Si (0) and .Ri (0) are derived from ∗ ∗ .xi (0) and .x (0) as described below. At .t + 1 the memory .x (t) may change to i i a different vector when .fo is evaluated at new test solutions at .t + 1 and in case that .ai gives a smaller result compared to the previous value of .fo (x∗i (t)). The new j test solutions are .xi (t + 1) and .xi (t + 1) and result from the two Mechanisms. At .t + 1, each agent .ai updates memory with .x∗i (t + 1), i.e. the position of j ∗ .min{fo (x (t)), fo (xi (t + 1)), fo (x (t + 1))}. i i Mechanism 1: Each agent explores by a random walk in .Γ , where the location change is generated by an individual step vector .Ei (t) with random direction. The test solution obtained from Mechanism 1 is the location .xi (t +1) = (xi1 (t +1), xi2 (t +1), . . . , xiD (t +1)) obtained from the present location .xi (t) = (xi1 (t), xi2 (t), . . . , xiD (t)). xid (t + 1) = xid (t) + Eid (t + 1)vid (t + 1) ,

.

(22.1)

where .vi (t + 1) = (vi1 (t + 1), vi2 (t + 1), . . . , viD (t + 1)) is a unitary vector (.|vi (t + 1)| = 1 and .t = 1, 2, . . . , T ). Except for .t = 1, .vi (t + 1) is randomly chosen, while for .t ≥ 2, .vi (t + 1) is generated according to the following rule: fo (xi (t)) < fo (xi (t − 1)) ⇒ vi (t + 1) = vi (t) ,

.

(22.2)

otherwise, .vi (t + 1) is randomly chosen. According to (22.2) we keep the direction followed by .ai to go from .xi (t − 1) to .xi (t) also to go from .xi (t) to .xi (t + 1), if .ai results in a better location at .xi (t) when compared to .xi (t − 1). This is a sort of individual intensification and plays the role of a reinforcement effect in the learningby-direct-experience mode of social learning [Ba71]. Further, in (22.1) each .Eid (t + 1) is chosen in a semi-closed interval .[0, Emax ), where .Emax can  be defined in a

number of valid ways. Valid choices are, for example, .Emax = L21 + . . . + L2D , the diagonal length of the search space .Γ and .Emax = (L1 + L2 + · · · + LD )/D. Before describing Mechanism 2, it is instructive to define a social status and an influence radius for each agent. We assign the status .Si (t) to agent .ai as a function of the best solution .x∗i (t) found until an instant t Si (t) =

.

SMax − SMin SMax M(t) − SMin m(t) fo (x∗i (t)) − , m(t) − M(t) m(t) − M(t)

(22.3)

where .m(t) is the least value for .fo found by the population .P(t), while .M(t) is the largest one, that is, .m(t) = min{fo (xi∗ (t)), i = 1, 2, . . . , Na } and .M(t) = max{fo (xi∗ (t)), i = 1, 2, . . . , Na } and thus Eq. (22.3) is nothing but an affine relation between .Si (t) and .fo (x∗i (t)). More precisely, for each time t, the relation describes a straight line defined by the points .(m(t), SMin ) and .(M(t), SMax ). The choice of such an affine relation was motivated by its simplicity, but other relations could be adopted. The consequences of the variation of this choice is out of the scope

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of the present work; however, it defines some sort of social knowledge, achieved by the whole society trying to evolve according to the problem of finding the global minimum of the objective function .fo . Also, it guaranties that there is always a degree of social inequality among the individuals in terms of status which is crucial for .m(t) to converge to the global minimum. The influence radius .Ri (t) is defined by a Gaussian-like function depending on .Si (t) centred at the maximum value .SMax , with parameters A and s to be set by the user for more or less search intensification.   (Si (t) − SMax )2 (22.4) .Ri (t) = A L exp − , 2s 2 Here A is usually taken such that .A ≤ 1, and we call it an attenuator, and s is a measure for a (standard) deviation around .SMax , where .Ri (t) assumes its maximum. The parameter L can be chosen in a number of valid ways, for instance, .L = min{L1 , . . . , LD }, .L = max{L1 , . . . , LD }, or even choosing different values for each dimension. Mechanism 2: The dominant influence for intensification occurs, when some agent .ai gets inside the influence radius .Rj (t) of a more successful agent .aj , that is, when |xi (t) − xi (t)| < Rj (t)

and

.

Sj (t) > Si (t) ⇒ aj influences ai .

(22.5)

Then, as we shall show, there is a chance of raising .ai ’s status by approaching the one of .aj . When .ai is influenced by another agent .aj at t, that is, when conditions (22.5) are satisfied, the influenced agent .ai visits the region from .aj ’s memory .x∗j (t). More precisely, .ai incorporates a candidate solution for .t + 1, j

j

j

j

xi (t + 1) = (xi1 (t + 1), xi2 (t + 1), . . . , xiD (t + 1)) according to

.

j .x (t id

+ 1) =

xj∗d (t) +



Sj (t) − SMax + Si (t) − SMax



∗ (t) xj∗d (t) − xid

|x∗j (t) − x∗i (t)|

Iid ,

(22.6)

for .d = 1, 2 . . . , D, where .0 <  Si (t). 

(Si (t) − SMax )2 .Ri (t) = A L exp − 2s 2



Here, the attenuation factor .0 < A ≤ 1 and s is in the spirit of the standard deviation in Gaussian distributions around .SMax , last not least .L = (L1 + L2 + . . . + LD )/D. Further, Si (t) =

.

SMax − SMin SMax M(t) − SMin m(t) fo (x∗i (t)) − , m(t) − M(t) m(t) − M(t)

with .m(t) = min{fo (xi∗ (t)), i = 1, 2, . . . , Na } as found by the agent population P(t) and .M(t) = max{fo (xi∗ (t)), i = 1, 2, . . . , Na }. For convenience in the present computations, .[SMin , SMax ] was set to .[0, 1]. Now, when .ai is influenced by an agent .aj with higher social rank, the influenced agent .ai copies the location from .aj ’s memory .x∗j (t) and thus the better solution for .t + 1.

.

j .x (t id

+ 1) =

xj∗d (t) +



Sj (t) − SMax + Si (t) − SMax



∗ (t) xj∗d (t) − xid

|x∗j (t) − x∗i (t)|

Eid

Here .d = 1, 2 . . . , D and .0 <  2, Eq. (23.2) may be considered a challenge, due to a large number of local minima, so that finding the global one is not trivial at all. Now, upon applying the proposed metaheuristics, we solved the L-J potential problem with .N = 10 atoms (the CEC 2011 case). After the aforementioned dimensional reduction, the degrees of freedom of the optimization problem had .D = 24 dimensions. Some of the best obtained results and the respective configurations are shown in Table 23.4. The best findings for each number of .fo evaluations are shown in bold face. The following restrictions were applied, .Na = 20 D .A ∈ {0.25, 0.5} and .s ∈ {0.1, 0.5}. Note, that the least found solution .−28.422532 was found for .Na = 20 D, .A = 0.25 and .s = 0.1, with .5 × 106 .fo evaluations after about one minute CPU time in the used setup. Limiting the number of .fo evaluations to .1.5 × 105 and comparing our findings to the CEC 2011 competitors, our results show that the proposed metaheuristics is competitive with the reported ones in [Su11]. In fact, we got better mean and better best results for 13 out of 15 benchmark cases from [Su11] (see also [Su11a]). In the next problem, we consider a frequency-modulated (FM) sound wave, where the reconstruction of the parameter vector .x = (A1 , ω1 , A2 , ω2 , A3 , ω3 ) is the goal, subject to the restriction .Ai ∈ [−6.4, 6.35]. The reconstructed signal has the form .

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Table 23.4 Best L-J results

.

.

.

.

Eval. 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06

Na = 20 D A = 0.25 s = 0.1 Na = 20 D A = 0.25 s = 0.5 Na = 20 D A = 0.5 s = 0.1 Na = 20 D A = 0.5 s = 0.5

Mean .−11.856704 .−18.260470 .−20.497768 .−26.186434 .−11.616691 .−16.673864 .−19.052565 .−26.665731 .−4.409733 .−4.572617 .−4.623519 .−20.142541 .−3.842851 .−4.124038 .−4.287330 .−19.222096

Std. dev. 2.012857 2.837316 3.292360 2.889595 1.586002 2.337389 2.743481 2.976702 0.884385 0.768808 0.735623 2.737729 0.574447 0.538364 0.550927 3.200937

Best .−14.845718 .−22.702906 .−24.688865 .−28.422532 .−13.774743 .−19.362227 .−22.406634 .−28.422523 .−5.898848 .−5.898848 .−5.898848 .−22.785522 .−4.880621 .−4.897711 .−5.191709 .−24.299630

y(t) = A1 sin(ω1 θ t + A2 sin(ω2 θ t + A3 sin(ω3 θ t))) ,

.

Worst .−9.353071 .−13.961607 .−15.307041 .−18.304759 .−9.103748 .−12.379061 .−14.562728 .−18.304508 .−3.378917 .−3.902195 .−3.905302 .−14.385681 .−3.062138 .−3.155255 .−3.453572 .−14.088976

(23.3)

with .θ = 2π/100, while the target function (original signal) is yT (t) = sin(5.0θ t − 1.5 sin(4.8θ t + 2 sin(4.9θ t))) .

.

(23.4)

The objective function is the sum over the square error between the estimated wave and the target function. fo (x) =

.

100  [y(t) − yT (t)]2 .

(23.5)

t=0

By comparing (23.3) and (23.4), one identifies the global minimum of (23.5) for fo (x∗ ) = 0 with .x∗ = (1, 5, 1.5, 4.8, 2, 4.9). It is noteworthy that finding this solution by conventional numerical means is a tedious if not impossible task because the signal represents a highly complex multimodal problem. Table 23.5 shows the best results (cast in bold face) we obtained depending on the number of .fo evaluations. For these runs, we considered .Na = 20 D .A ∈ {0.25, 0.5} and .s ∈ {0.1, 0.5}. A global minimum was found for two configurations with .5 × 106 .fo evaluations after about 3 min CPU time with .A = 0.25, s = 0.1 and 5 .A = 0.25, s = 0.5. Also, with .1.5 × 10 .fo evaluations, we approached the 14th CEC2011 competitor for the found mean solution and got a superior best solution compared to the 11th competitor (see [Su11] and [Su11a]). .

23 On Applications of the Optimization Algorithm DySDO

297

Table 23.5 Best frequency modulation results

.

Na = 20 D A = 0.25 s = 0.1

.

Na = 20 D A = 0.25 s = 0.5

.

Na = 20 D A = 0.5 s = 0.1

.

Na = 20 D A = 0.5 s = 0.5

Eval 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06 5.0E.+04 1.0E.+05 1.5E.+05 5.0E.+06

Mean 1.08E.+01 1.08E.+01 1.08E.+01 9.41E.+00 1.07E+01 1.05E+01 1.04E+01 6.23E+00 1.85E.+01 1.64E.+01 1.62E.+01 1.33E.+01 2.15E.+01 1.81E.+01 1.68E.+01 1.60E.+01

Std. dev. 6.43E.+00 6.43E.+00 6.43E.+00 7.03E.+00 5.62E.+00 5.80E.+00 5.80E.+00 6.63E.+00 3.91E.+00 3.92E.+00 4.11E.+00 2.11E+00 1.89E.+00 2.87E.+00 3.74E.+00 4.43E.+00

Best 8.56E-05 1.04E-08 6.38E-12 0.00E+00 3.05E.−01 1.45E.−04 7.68E.−08 0.00E+00 1.13E.+01 1.12E.+01 1.12E.+01 1.12E.+01 1.78E.+01 1.49E.+01 1.26E.+01 1.02E.+01

Worst 1.94E.+01 1.94E.+01 1.94E.+01 1.94E.+01 1.74E+01 1.74E+01 1.74E+01 1.42E+01 2.37E.+01 2.33E.+01 2.33E.+01 1.76E.+01 2.33E.+01 2.16E.+01 2.14E.+01 2.13E.+01

23.4 Final Remarks and Conclusion In this contribution, we applied the DySDO algorithm to a selection of pathological and benchmark problems. The experiments showed that the proposed metaheuristics is quite robust with respect to the necessary choices for parameters and initial values and in most cases the runs achieved good approximations for the global optimum. When applied to the benchmark problems considered in reference [WaEtAl15], DySDO showed to be superior in most of the situations. When applied to a LennardJones potential problem, the proposed metaheuristics was successful and found the best global minimum which is referenced in the literature. The frequencymodulated problem was also considered, and the global minimum was also found. Both minima were found within .5.0 × 106 evaluations of .fo with 1 min CPUtime for L-J and 3 min for FM on a standard computer. In terms of performance measured by the number of .fo evaluations, we obtained results better than some of the CEC2011 competitors for those two problems (L-J and FM). However, we point out that the number of .fo evaluations is not always a fair comparison criterion for performance, since depending on the metaheuristics a significant number of arithmetic and logical operations for the .fo evaluations may be necessary which increase the need for computational power. Nevertheless, comparisons with other established algorithms showed that the proposed algorithm is competitive. The presented DySDO algorithm is in a basic (canonical) version and highlights the role of the considered social behaviour of pursuing higher social status as a paradigm in optimization problems. However, the reported results and comparisons with other

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competitive algorithms showed already that DySDO is a promising proposition, so that we believe that the present version can still be improved in a number of ways, but such explorations are left for future works.

References [DaSu10] Das, S., Suganthan, P.: Problem definitions and evaluation criteria for CEC 2011 competition on testing evolutionary algorithms on real world optimization problems. Technical Report (2010) [MoAl05] Moloi, N., Ali, M.: An iterative global optimization algorithm for potential energy minimization. J. Comput. Optim. Appl. 30, 119–132 (2005) [Su11] Suganthan, P.: Testing evolutionary algorithms on real-world numerical optimization problems. http://www3.ntu.edu.sg/home/epnsugan/index_files /CEC11-RWP/CEC2011_ranking.pdf (2011) [Su11a] Suganthan, P.: Ranking. www3.ntu.edu.sg/home/epnsugan/index_files /CEC11-RWP/ranking.xlsx (2011) [SuEtAl05] Suganthan, P., Hansen, N., Liang, J., Deb, K., Chen, Y., Auger, A., Tiwari, S.: Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization. Nanyang Technological University, Singapore (2005) [WaEtAl14] Wang, L., Ni, H., Yang, R., Fei, M., Ye, W.: A simple human learning optimization algorithm. Comput. Intell. Netw. Syst. Their Appl. 462, 56–65 (2014) [WaEtAl15] Wang, L., Ni, H., Yang, R., Pardalos, P., Du, X., Fei, M.: An adaptive simplified human learning optimization algorithm. Inf. Sci. 320, 126–139 (2015)

Chapter 24

On the Influence of the Signal to Noise Ratio on the Reconstruction of the Non-linear Transfer Function in Signal Amplification J. Schmith, A. Schuck, B. E. J. Bodmann, and P. J. Harris

24.1 Introduction During the 1970s, pioneer digital audio technologies made the first steps in a direction to mimic analog devices, whereas already in the 1980s, these developments resulted in the first generation of commercial digital audio products. Since then, the evolution of increasingly sophisticated digital equipment and methods to simulate classical analog instrument sound effects with high fidelity [DuEtAl20, Zo11, ZoEi18] is well established, but in turn opened also novel unprecedented possibilities of instrument sound manipulation. As a matter of fact, high-fidelity simulation of guitar amplifiers or effect pedals faces limits in digital hardware performance because simulations have to be implemented with low latency while still providing expected results in real time. Employed algorithms [EiEtAl17, EiZo16, MoEtAl15] generally are of numerical nature, so that efficient computational codes are needed to solve the inverse problem [Og07, SjSc12]. Therefore, analytical models seem to be a good choice for a variety of real-time audio equipment simulations including also parallel sound effect chains. As already reported in the literature [ScEtAl22], the basis for the analytical method superimposes the input and output signals to produce a Lissajous-like curve but in the complex plane, which besides filter properties manifest in phase

J. Schmith University of the Vale do Rio dos Sinos, São Leopoldo, Brazil e-mail: [email protected] A. Schuck · B. E. J. Bodmann () Federal University of Rio Grande do Sul, Porto Alegre, Brazil e-mail: [email protected] P. J. Harris University of Brighton, Brighton, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_24

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shifts also represent the non-linearity of a Wiener-Hammerstein type of model [EiZo18, RoEtAl14, ScEtAl14]. While in a previous work [ScEtAl22] an idealised system with the absence of any noise was considered, in the present discussion, a selection of signal to noise ratios were introduced, and the analytical method for the reconstruction of the non-linearity was adapted in order to suppress the stochastic component in the signal and extract the desired output, a necessity for reconstructing the transfer function in the spirit of [ScEtAl22]. In this reference, focus was put on the identification of different types of non-linear transfer functions for idealised noiseless cases. In general, the principal challenge in using the Wiener-Hammerstein model is to separate the filter influence before and after the amplification component from the part which causes the non-linear response. Here, the first filter introduces a frequency-dependent phase shift in the input signal, the non-linear amplification step creates harmonic contributions to the original signal, and the second filter causes additional frequency-dependent phase shifts besides changes in the respective amplitudes. Thus, the created harmonics are partially cut off by the frequency response of the second filter, so that parts of the amplification characteristics are suppressed and consequently difficult the reconstruction of the non-linearity. Is the method still applicable if noise is added to the original signal? In order to get an affirmative answer, a modification of the original method was introduced, and the worst-case noise—white noise—was superimposed on the input signal to evaluate the effectiveness of the modified procedure. In the idealised case, one Lissajous cycle was sufficient to extract the non-linearity from sampled data, whereas, if besides the signal also noise is present, a number of cycles are necessary so that uncorrelated fluctuations cancel out at least to a significant extent. Thus, the performance of the analytical method depends on the relation between the signal to noise ratio and the number of cycles which allows for a unique determination of the non-linearity.

24.2 Method The Wiener-Hammerstein model depicted in Fig. 24.1 is a widespread schematics used to represent signal amplification. Here, the components .h1 (t) and .h2 (t) are the input and output filters, and these are interconnected by a non-linear transfer function. The principal changes in the original signal are the aforementioned phase shifts and frequency multiplications by the non-linear block. The reconstruction of the non-linear transfer function as an analytical expression starts with the combined input-output signal in form of a Lissajous-like curve in the complex plane [ScEtAl22]. To this end, an input signal with a known frequency together with white noise and the related output signal of the Wiener Hammerstein system was used to generate the complex curve. From this curve, the amplitudes and phases of the harmonics were calculated to rebuild the non-linear function by setting all phases in the expression for the output signal to zero. For a noiseless

24 On the Influence of the Signal to Noise Ratio on the Reconstruction of the. . .

301

Fig. 24.1 The Wiener-Hammerstein model diagram

signal, only one Lissajous cycle was sufficient to extract the non-linearity, while due to the stochasticity of uncorrelated fluctuations in the signal, an adequate number of cycles are necessary so that the influence of noise is sufficiently suppressed and the output signal can be presented as a unique sum of terms specified by amplitude, frequency and related phase shifts. Let .I (ω|t) + BI be the time dependent input signal with the angular frequency .ω = 2πf and the noise contribution .BI , which is a random variable compatible with a white noise distribution. In the same way, the output signal is defined as K Ω({kω}K k=1 , {φk }k=1 |t) + BΩ ,

.

where k represents the number of the harmonics and .φk the corresponding phase. Here, .BΩ represents the noise manifestation in the output signal. The time dependence of the Lissajous-like curve with noise is then defined by Z({kω}k , {φk }k |t) = I (ω|t) + ıΩ({kω}k , {φk }k |t) + BI + ıBΩ ,

.

where both random variables have vanishing expectation values .E[BI +ıBΩ ] = 0. The deterministic parts of the input and output signals are given by I (ω|t) = A0 eıωt

Ω({kω}k , {φk }k |t) =

and

.

K 

Ak eı(kωt+φk ) .

k=1

For any given input and output signal from laboratory experiments, the amplitudes (.Ak ) and phases (.φk ) shall be determined making use of the vanishing correlation time scale of the stochastic part of the signal. Theoretically, on average noise cancels out, which is especially true for a noisy periodic input signal where the number of sampled periods tends to infinity. For practical implementations, one has to find a suitable truncation for that number such that the difference between signal and noise is effectively several tenth dB. An infinitesimal line element on the noisy Lissajous curve is then given by  dZ =

.

dΩ dI +ı dt dt

 dt + δB ,

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where .δB is a complex random variable. Now, in order to separate specific output modes with its associated phases, one applies an integral transform to . dZ dt . 

T

.

0

dZ −ımωt dt = 2π ıA0 δm 1 − 2π mAm eıφm + E[B] e dt

The amplitudes (Eq. 24.1) and phases (Eq. 24.2) are then computed using the sum over the N sample points along the closed Lissajous curve and for C cycles, where .B is the complex sampled distribution due to the presence of noise. 1 .Am ≈ C

 CN −1  2 In cos(2π n/N ) + Ωn sin(2π n/N) δ1m A20 − A0 N



n=0

⎞1 2 CN −1 CN −1 m(n−p) 1   † 2⎠ ı2π N . + 2 Zn Zp e + E[B] N n=0

(24.1)

p=0

 ⎞ CN −1 ı2π mn N A0 δm1 +  Nı + E[BI + BΩ ] n=0 Zn e ⎠ (24.2)  = arctan ⎝ CN −1 ı ı2π mn N + E[B]  N n=0 Zn e ⎛

(inv) φm

Here, the number of cycles (C) has to be chosen such that .E[B] ≈ 0.

24.3 Numerical Results and Discussion A set of numerical experiments provided a test of the method and showed the influence of noise in the reconstruction of the non-linear transfer function. To this end, a synthetic pair of input and output signals (Eq. 24.3) was considered. The frequency of the input signal was .f = 1 kHz, and the sampling frequency was .100 kHz, while the output signal contained the third, fifth and seventh harmonics. I (t) = sin(ωt) (24.3)        1 π 1 π 1 π π + sin 3ωt − + sin 5ωt − + sin 7ωt − Ω(t) = sin ωt − 2 2 2 4 4 8 6 .

A .10% white noise level was added to the input signal, and Fig. 24.2 shows the Lissajous-like curve of the signals. For the evaluation of the amplitude and phases (Eqs. 24.1 and 24.2), ten cycles were taken into account, and a noisy input signal together with the output signal with all the phase shifts set to zero is shown in Fig. 24.3 (left), while the comparison of the original output signal together with the reconstructed one is shown on the right. Visually the reconstructed signal presented a good agreement with the original

24 On the Influence of the Signal to Noise Ratio on the Reconstruction of the. . .

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Fig. 24.2 Lissajous-like curve for a synthetic input/output signal with .10% white noise level

Fig. 24.3 Signal reconstruction for synthetic noisy input/output signals, using 10 cycles

Fig. 24.4 Amplitude (left) and phase reconstruction (right) for noisy signals using 10 cycles

output. The comparison of the spectra of the output signals (see Fig. 24.4 (left)) with and without noise shows only a small difference, whereas the phases of the reconstructed output signal with and without noise (right) presented significant differences for higher modes. By inspection one notes that the harmonics for the output signal were correctly reproduced and the phases of the original output harmonics were reasonably

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Fig. 24.5 Reconstruction of the Lissajous curve and the non-linear transfer function for 10 cycles using a synthetic noisy signal Fig. 24.6 The influence of a noise level in the input on the output signal

.100%

identified; the noise contribution gave rise to additional random phases for small amplitude contributions, which influenced the shape of the reconstructed Lissajouslike curve only marginally (see Fig. 24.5 (left)). The same figure (right) shows the reconstructed non-linear transfer function compared to the version without the addition of noise. As a measure for the reconstruction error, the Euclidean distance of the reconstructed and original non-linear transfer function was used, which for the synthetic signal (10 cycles and .10% noise) was .20.35%. In a subsequent step, the Wiener-Hammerstein model with two identical bandpass filters .h1 (t) and .h2 (t) was analysed. The time invariant filters were set up with low- and high-frequency cuts at .20 Hz and .20 kHz, respectively, which corresponds to the audible range. In all cases the input signal had a frequency of .1 kHz, and the sampling rate was .100 kHz. The parameters of the filters was chosen to have only a moderate influence on the output and on the noise characteristics in the audible range. As an example Fig. 24.6 shows the .1 kHz input signal with an addition of .100% white noise and the output signal from the Wiener-Hammerstein model. The robustness of the algorithm was evaluated adding a selection of white noise amplitudes to the input signals. Experiments with a pure signal (.f = 1 kHz) and the addition of .5%, .10%, .20% and .50% white noise were performed with different

24 On the Influence of the Signal to Noise Ratio on the Reconstruction of the. . .

(a) 5 % noise level.

(b) 10 % noise level.

(c) 20 % noise level.

(d) 50 % noise level.

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Fig. 24.7 Influence of the noise level (a) .5%, (b) .10% (c) .20% and (d) .50% on the reconstructed non-linear transfer function using 100 cycles

numbers of cycles. It is noteworthy that industrial standards and product manuals for audio amplifiers prescribe as low noise a typical limit less than .5%. Here, the non-linear transfer function chosen for the non-linear Wiener-Hammerstein block was the hyperbolic tangent function shown in Fig. 24.7a–d for the four percentages of noise and 100 cycles. Although the signals matched visually to a fairly good degree, the discrepancies in the amplitudes and phases of the signals with and without noise are shown in Fig. 24.8. In general, the amplitudes of the harmonics coincided with the original ones, while in the experiment with .50% noise the amplitudes for the harmonics five and seven are slightly smaller for the reconstructed output signal. The phase information for the experiments presented apparent discrepancies though predominantly for the signal amplitudes of the harmonics which were comparable to the noise intensity. Note that the larger amplitudes are located in the low odd harmonics range, where, for instance, for .m < 15 odd in most cases the reconstructed phases are close to the original ones. Thus, the major discrepancies are associated with higher harmonics with their associated low amplitudes, which are also more susceptible to the influence of noise. As was to be expected, with increasing noise intensity, more deviations occur, either in the Lissajous-like input/output signal representation or in the reconstructed non-linear transfer function (see Fig. 24.9). The quality of the signal reconstruction may be improved upon increasing the number of cycles due to the fact that the expectation value of the noise contribution approaches zero. The presented results

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(b) 10 % noise level.

(c) 20 % noise level.

(d) 50 % noise level.

Fig. 24.8 Amplitudes and phases for (a) 5%, (b) 10% (c) 20% and (d) 50% noise level. For comparison the amplitudes and phases without noise are also shown

24 On the Influence of the Signal to Noise Ratio on the Reconstruction of the. . .

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(a) 5% noise level.

(b) 10% noise level.

(c) 20% noise level.

(d) 50% noise level.

Fig. 24.9 Lissajous curves of the noisy input versus output and reconstructed output signals (left) and the corresponding non-linear transfer functions (right). (a) 5% noise level. (b) 10% noise level. (c) 20% noise level. (d) 50% noise level

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Fig. 24.10 Error dependence on the number of cycles

were obtained using a number of 100 cycles. Thus, Fig. 24.9 shows the closed curve constructed from the averages over the total number of cycles. Even for a relatively small number of cycles, there is a fairly good agreement between the theoretical and the identified non-linear transfer function, even for an unrealistic case with .50% noise level. Therefore, for applications the developed method is efficient in separating the linear filter blocks from the non-linear transfer function in the WienerHammerstein model. As a quality evaluation for a hundred cycles, the reconstruction of the non-linear transfer function already showed a good agreement with the original one, which may be quantified using as a measure the Euclidian distance of the reconstructed and the original transfer function. The errors for the respective noise levels of .5%, .10%, .20% and .50% amounted to .1.62%, .1.71%, .1.93% and .3.11%, respectively. Evidently, as the signal to noise ratio increases, the error also increases if the same number of Lissajous like cycles are used for data evaluation. In the experiments the noise from the lower percentage to the higher ratio had an increment of one order in magnitude, while the error roughly doubled only. This shows the robustness of the algorithm to reconstruct the non-linear transfer function even for the case with severe noise. Last not least, the influence of the number of cycles on the reconstruction error of the non-linear transfer function is shown in Fig. 24.10, where the four experiments with different signal to noise ratios for eight cases of cycle numbers are presented (.C ∈ {1, 5, 10, 15, 20, 30, 50, 100}). For only one cycle, all the errors were above .3%, while for the cases up to .20% approached the .2% level, only the .50% signal to noise ratio experiment did not attain an error level below the .3% mark. Although we presented experiments for up to hundred cycles, which was sufficient to “prove” that the error decreases as the number of cycles increases, however, the specific choice of the number of cycles is up to user’s definition and criteria for error tolerance. Nevertheless, the quantity of samples does certainly not impose any computational limitation, since for a signal of .1 kHz, only one second of signal acquisition is needed for 1000 cycles of the Lissajous-like signal representation.

24 On the Influence of the Signal to Noise Ratio on the Reconstruction of the. . .

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24.4 Conclusion In the present contribution, we implemented the necessary modifications so that the modified method presented in reference [ScEtAl22] works also for the case where the non-linear transfer function has to be reconstructed from a case with an input signal superimposed by white noise fluctuations. The fact that the expectation value for a random signal with vanishing correlation time is zero indicates that for practical applications a sufficient large number of cycles shall be sufficient so that the expectation value becomes orders of magnitude smaller than the deterministic part of the signal. The robustness of the algorithm was validated considering signal to noise ratios from .5% to .50%. The algorithm presented good results with an error of .3.11% for the extreme case of .50% white noise added to the input signal, and the reconstruction was based on 100 closed cycles. Moreover, the fact that the transfer function is given in form of an analytical expression, computational limits are unlikely to impose an implementation limit, which is not obvious for numerical approaches. An excessive example (a .100% signal to noise ratio) shows that .∼ 103 cycles are sufficient, which for data acquisition with a signal of .1 kHz corresponds to one second. Moreover, once the non-linear transfer function is known, the output signal can be computed directly in contrast to commonly used numerical methods, which may show computational limitations especially in cases where a chain of simulated devices is large or where parallel chains are employed to create novel sound impressions. Thus, an analytical method as the one proposed allows for more complex setups with its signal processing building blocks, where latency is not an issue even for limited computational setups. Due to the perspectives that arise from the proposed approach, in future works we will employ real sound input signals, which generally have a rich spectrum and by nature are transients, so that a natural continuation of the developments is to consider more realistic scenarios. Apart from the original idea of instrument signal processing, another scope of application may be explored where the principal challenge is to match input and output signals for different types of non-linearities [OlSc13, ScOl21], as, for instance, the computation of hysteresis curves for transformers where the core is in saturation, among many others. Concluding, the proposed method should in principle be applicable to problems where a unique link between a cause and its effect is determined by a non-linear transfer function.

References [DuEtAl20] Düvel, N., Reinhard, K., Anna, W., Weihe, P.: Confusingly similar: discerning between hardware guitar amplifier sounds and simulations with the kemper profiling amp. Music Sci. 3, 1–16 (2020) [EiEtAl17] Eixas, F., Möller, S., Zölzer, U.: Block-oriented gray box modeling of guitar amplifier. In: Proc. of the 20th Int. Conference on Digital Audio Effects (DAFx-17), Edinburg, UK, pp. 184–191 (2016)

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[EiZo16] Eixas, F., Zölzer, U.: Black-box modeling of distortion circuits with block-oriented models. In: Proc. of the 19th Int. Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, pp. 39–45 (2016) [EiZo18] Eichas, F., Zölzer, U.: Virtual analog modeling of guitar amplifiers with WienerHammerstein models. In: 44th Annual Convention on Acoustics, DAGA 2018 (2018) [MoEtAl15] Möller, S., Eichas, F., Zölzer, U.: Block-oriented modeling of distortion audio effects using iterative minimization. In: Proc. of the 18th Int. Conference on Digital Audio Effects (DAFx-15), Trondheim, Norway, pp. 1–6 (2015) [Og07] Ogunfunmi, T.: Adaptive Non-linear System Identification: The Volterra and Wiener Model Approaches. Springer Science and Business, LCC (2007) [OlSc13] De Oliveira, L.P.L., Schmith, J.: Ponderomotive force as a cause for chaos in loudspeakers. Appl. Math. Comput. 219, 6449–6456 (2013) [RoEtAl14] Rolain, Y., Schoukens, M., Vandersteen, G., Ferranti, F.: Fast identification of WienerHammerstein systems using discrete optimisation. Electron. Lett. 50(25), 1942–1944 (2014) [ScEtAl22] Schmith, J., Schuck Jr., A., Bodmann, B.E.J., Harris, P.J.: Analytical reconstruction of the nonlinear transfer function for a Wiener-Hammerstein model. In: Ch. Constanda, B.E.J. Bodmann, P.J. Harris (eds.) Integral Methods in Science and Engineering: Applications in Theoretical and Practical Research, pp. 307–322. Springer Nature Switzerland AG, Basel (2022) [ScOl21] Schmith, J., de Oliveira, L.P.L.: Dimensioning sealed enclosures for suppressing nonlinear distortions in woofers. Appl. Acoust. 178, 107975 (2021) [ScEtAl14] Schoukens, M., Zhang, E., Schoukens, J.: Structure detection of Wiener-Hammerstein systems with process noise. IEEE Trans. Instrum. Meas. 66(3), 569–576 (2014) [SjSc12] Sjöberg, J., Schoukens, J.: Initializing Wiener-Hammerstein models based on partitioning of the best linear approximation. Automatica 48(1), 353–359 (2012) [Zo11] Zölzer, U.: DAFX: Digital Audio Effects, Vol. 2. Wiley Online Library (2011) [ZoEi18] Zölzer, U., Eichas, F.: Gray-box modeling of guitar amplifiers. J. Audio Eng. Soc. 66(12), 1006–1015 (2018)

Chapter 25

An Analytic Solution for the Transient Three-Dimensional Advection–Diffusion Equation with Non-Fickian Closure by an Integral Transform Technique V. C. Silveira, D. Buske, G. J. Weymar, and J. C. Carvalho

25.1 Introduction Air pollution is a recurring phenomenon in our daily lives and can be caused by natural effects (e.g. methane gas emissions) or anthropogenic ones (e.g. by burning fossil fuels). The pollutants emitted in a given place can cause several problems to the environment and reach regions far from where they were emitted, justifying the importance of studying the pollutants behaviour in the most diverse situations that can occur in the atmosphere. Anthropogenic activity brings numerous problems with the gas emissions, generating an ecological imbalance. The gases and dust released in the atmosphere can cause problems near the sources, reducing air quality in urban regions, causing acid rain at medium and long distances and holes in the ozone layer on a global scale. Atmospheric motions are classified according to their horizontal dimensions into three broad categories: macroscale (horizontal scale in order of 1000 km), mesoscale (horizontal scale in order of 100 km) and microscale (horizontal scale in order of 10 km or less) [Ar99], where the present model applies for the latter. In the literature one finds two mathematical models to numerically simulate the pollutant concentrations in the atmosphere: Lagrangian and Eulerian. The Lagrangian model follows the instantaneous velocity of the fluid, and the Eulerian is fixed with respect to the earth [An05]. The main scheme of Eulerian dispersion models is the solution of the advection-diffusion equation, which is expressed through the parameterization of turbulent flows. Under specific conditions, expressions for the concentration field are obtained that are functions of pollutant emission, meteorological variables and plume dispersion parameters.

V. C. Silveira · D. Buske () · G. J. Weymar · J. C. Carvalho Federal University of Pelotas, Pelotas, Rio Grande do Sul, Brazil e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_25

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There are several works available in the literature that focus on the planetary boundary layer (PBL) dynamics study, in which several models are considered to close the turbulent flow equations, for example, first-order models or K theory, second-order or higher. The gradient transport hypothesis (or K-theory) is one of the most used hypothesis to solve the problem of closing the advection-diffusion equation that assumes that the turbulent flow concentration is proportional to the magnitude of the average concentration gradient. However, the gradient transport hypothesis is not valid in the upper part of the convective boundary layer (CBL), because in this region a countergradient flow is present [DeWi75]. It was observed that in the upper part of the CBL, the potential temperature flow is contrary to the potential temperature profile gradient of the medium [De66]. The potential temperature gradient of the medium and the flow change sign at different levels affect the first-order turbulence closure, because it does not take into account the inhomogeneous character of the CBL turbulence. Considering a non-local closure for turbulence, the asymmetry in the process of pollutant dispersion in the atmosphere shall be taken into account. To this end, a generic equation for turbulent diffusion can be used, considering that the flux plus its derivative are proportional to the average gradient causing an additional term to appear in the equation that is associated with the countergradient term [WyWe91, VaVe01]. Several studies have already been developed to study pollutant dispersion in the atmosphere considering the advection-diffusion equation under different atmospheric stability conditions and using the GILTT (Generalized Integral Laplace Transform Technique). The innovation of this work is to consider the advectiondiffusion equation in its most complete form, considering a three-dimensional model and the non-local turbulence closure term.

25.2 Methodology Atmospheric advection and diffusion can be modelled by applying the mass conservation equation [SePa97]

.

∂u c ∂v  c ∂w  c ∂c ∂c ∂c ∂c +v +u +w =− − − , ∂x ∂t ∂y ∂z ∂x ∂y ∂z

(25.1)

where .c = c (x, y, z, t) denotes the average concentration of a passive contaminant, u, .v and .w are the Cartesian components of the mean wind in units of .(m/s) and 2 .u c  and .v  c  and .w  c  represent the turbulent contaminant flows in units of .(g/sm ) in the longitudinal, lateral and vertical directions. Equation (25.1) has four unknowns (turbulent flows and the concentration) and therefore cannot be uniquely solved leading to the so-called turbulence closure problem [St88]. One of the common ways to solve the closure problem of the advection-diffusion equation (25.1) is based on the gradient transport hypothesis

.

25 Advection–Diffusion Equation with Non-Fickian Closure

313

(or K-theory or first-order closure) which assumes that the turbulent concentration flow is proportional to the magnitude of the mean concentration gradient [SePa97]. Following this reasoning, one obtains the following advection-diffusion equation [Bl97]: .

∂ ∂c ∂c ∂c ∂c +u = +w +v ∂t ∂z ∂y ∂x ∂x

    ∂ ∂w  c ∂c ∂c Kx + Ky − . ∂x ∂y ∂y ∂z

However, the gradient transport hypothesis is inconsistent with the turbulent diffusion characteristics in the upper part of the mixed layer for convective cases where a countergradient material flow is present [DeWi75]. To describe and characterize the diffusion in this region, [Er42, De66] and [De72] proposed to modify the usual application of gradient flow in the K-theory approach as follows:  w  c = −Kz

.

∂c −γ ∂z



where .γ represents the countergradient term in form of a constant parameter. Many schemes and parameterizations for the countergradient term have been developed, and in this work we use the parameterization proposed by [VaVe01], which are based on the work of [WyWe91]. The turbulence closure problem in the advection-diffusion equation was modified considering a generic equation for turbulent diffusion so that the turbulent vertical flow of concentration and the derivative is proportional to the mean gradient.   ∂w  c 1 ∂c ∂w  c +τ + w  c . =− . Kz β ∂z ∂t ∂z Using the Cauchy-Euler scheme, one can write the transient advection-diffusion equation in three dimensions that considers the aforementioned nonlocal turbulence closure       ∂ ∂ ∂c ∂ ∂c ∂c ∂c ∂c ∂c ∂c Kx + Ky − β + = +w +v +u ∂x ∂y ∂y ∂z ∂t ∂t  ∂x  ∂y  ∂z ∂x      ∂ ∂ ∂ ∂ ∂ ∂c ∂c ∂c ∂c β Kx + + βw − βv − βu − ∂z ∂x ∂x ∂z ∂z ∂y ∂z ∂x ∂z        . ∂ ∂ ∂ 2c ∂ ∂c ∂c ∂ ∂c + + Ky + Kz −τ 2 − τu β ∂x ∂z ∂z ∂t ∂z  ∂y  ∂y  ∂t    ∂ ∂ ∂ ∂ ∂ ∂c ∂c ∂ ∂c ∂c τ Kx + τ Ky , + τw − τv − ∂t ∂x ∂x ∂t ∂y ∂y ∂z ∂t ∂y ∂t (25.2) where .Kx , .Ky and .Kz represent the eddy diffusivities in the longitudinal, lateral and vertical directions, respectively. Since the domain is finite, Eq. (25.2) is subject to

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the following boundary conditions: Kx

.

∂c(x, Ly , z, t) ∂c(x, 0, z, t) ∂c(Lx , y, z, t) = Ky = Ky =0 ∂x ∂y ∂y

Kz

.

∂c(x, y, 0, t) ∂c(x, y, h, t) = Kz =0 ∂z ∂z

(25.2a)

(25.2b)

and the source condition u c(0, y, z, t) = Qδ(y − yo )δ(z − Hs )

.

(25.2c)

where Q is the source intensity in .(g/s), h is the planetary boundary layer height in .(m), .Hs is the source height in .(m) and .Lx and .Ly define the domain boundaries in .(m), which shall be sufficiently far from the source in the x and y directions, respectively. Last, .δ is the Dirac delta functional. The mean wind components .u and .v are given, respectively, by .u = V sin(θ ) and .v = V cos(θ ), where V is the wind speed and .θ is the wind direction. Applying now in Eq. (25.2) the Laplace transform in the time variable (t), we obtain the following pseudo-stationary problem. u

∂ 2C ∂C ∂ 2C ∂C ∂C ∂C ∂C +v +w = Kx 2 + Kx + Ky 2 − β r + ∂x ∂y ∂z ∂x ∂z ∂x ∂y ∂C ∂C ∂ 2C ∂ 2C ∂ 2C − (βu) − βv − (βv) − βw 2 + ∂z∂x ∂x ∂z∂y ∂y ∂z 3 2 2   ∂ C ∂ C ∂ C ∂C  ∂C + βKx + βKx + + (βKx ) + βKx − (βw) ∂z ∂z∂x ∂x ∂z∂x 2 ∂x 2  ∂ 2 C  ∂ 3C ∂ 2C ∂C ∂C − τ r2 C − τ u r + +βKy + βK + K + Kz y z 2 2 2 ∂z ∂x ∂z∂y ∂y ∂z

− (β r) C − βu .

∂ 2C ∂C ∂ 2C ∂C ∂C − τw r + τ r Kx + τ r Ky + τ r Kx − r C. 2 ∂y ∂z ∂x ∂x ∂y 2 (25.3) In order to apply the integral transform technique in the y variable, the pollutant concentration is expanded in a series [BuEtAl12] −τ v r

C(x, y, z, r) =

.

N  cn (x, z, r)ζn (y) 1

n=0

,

(25.4)

Nn2

L where .Nn is given by .Nn = 0 y ζn2 (y)dy, considering .cn = cn (x, z, r), then substituting the Eq. (25.4) in Eq. (25.3) and projecting out moments, in other words

25 Advection–Diffusion Equation with Non-Fickian Closure

applying the integral operator equation for each projection: α. n,m u

.

1 1 Nm2

Ly 0

315

(.)ζm (y)dy, one arrives at the following

∂ 2 cn ∂cn ∂cn ∂cn = αn,m Kx 2 + αn,m Kx + γn,m v cn + αn,m w ∂z ∂x ∂x ∂x

− αn,m λ2n Ky cn − αn,m β r

∂cn ∂ 2 cn − αn,m (β r) cn − αn,m βu ∂z ∂z∂x

− αn,m (βu)

∂cn ∂ 2 cn ∂cn − γn,m (βv) cn − αn,m βw 2 − γn,m βv ∂z ∂x ∂z

− αn,m (βw)

∂ 3 cn ∂ 2 cn ∂ 2 cn ∂cn + αn,m βKx + αn,m (βKx ) + αn,m βKx 2 2 ∂z ∂z∂x ∂z∂x ∂x

(25.5)

    ∂cn ∂cn ∂ 2 cn − αn,m λ2n βKy − αn,m λ2n βKy cn + αn,m Kz 2 + αn,m βKx ∂x ∂z ∂z ∂c ∂c ∂cn n n − αn,m τ r 2 cn − αn,m τ u r − γn,m τ v rcn − αn,m τ w r + αn,m Kz ∂z ∂z ∂x + αn,m τ r Kx

∂cn ∂ 2 cn − αn,m λ2n τ r Ky cn − αn,m rcn + αn,m τ r Kx 2 ∂x ∂x

To solve the problem given in Eq. (25.5), we followed the methodology described in [Si17].

25.2.1 Turbulence Parameterizations In this work were used the turbulence parameterizations proposed by [DeEtAl01] and [DeEtAl02]. These parameterizations take into account a memory effect in the pollutant plume 0.583w∗ zi ci ψ 2/3 (z/zi )4/3 X∗ [0.55(z/zi )2/3 + 1.03ci ψ 1/3 (fm∗ )i X∗ ] 1/2

Kα =

.

1/3

[0.55(z/zi )2/3 (fm∗ )i

2/3

1/2

+ 2.06ci ψ 1/3 (fm∗ )i X∗ ]2

where .α refers to the directions x, y and z, .w∗ is the convective velocity scale, .zi is the convective boundary layer height, .ci (cu = 0.3; cv,w = 0.36) is a constant and .ψ is the dissipation function as given in reference [Ho82]  ψ

.

1/3

=

z 1− zi

2 

z −L

−2/3

1/2 + 0.75

,

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where z is the height above ground, L is the Monin-Obukhov length and .X∗ is the dimensionless distance given by the following equation: xw∗ uzi

X∗ =

.

The normalized spectral peak frequency [DeEtAl00]:  ∗ fm i =

.

 ∗   fm i is given by Degrazia et al.

.

z (λm )i

where .(λm )i is the peak wavelength of the turbulent velocity spectrum, with (λm )u = (λm )v = 1, 5h and

.

    4z 8z − 0.0003 exp .(λm )w = 1, 8zi 1 − exp − zi zi The longitudinal component is given by .(fm∗ )u = 0.67 [OlEtAl84], and the dissipation function used here is the average value .ψ = 0.4 as reported in reference [Ca82].

25.2.2 Wind Profile The wind speed profile is described by a power law, justified in reference [PaDu84] V .

V1

 =

z z1

α ,

where .V and .V 1 are the average horizontal wind speeds at heights z and .z1 and .α is a constant, which under unstable conditions is .α = 0.1.

25.2.3 Experimental Data To validate the model under unstable atmospheric conditions, low wind data were used as reported in the IIT Delhi experiment [ShEtAl96, ShEtAl96a, ShEtAl02]. These experiments measured the ground-level concentrations of a sulphur hexafluoride (.SF6 ) tracer in defined distances from the source. The pollutant was released without buoyancy from a height of 1 m, and the concentrations were observed close to the ground (.0.5) m. The sampling period of .SF6 was 30 min at a rate of 30 to −1 .50 ml min . The meteorological variables, wind and temperature, were obtained at

25 Advection–Diffusion Equation with Non-Fickian Closure Table 25.1 Meteorological parameters observed in the IIT Delhi experiment

V Run 1 2 6 7 8 11 12 13

.(ms

.θ

−1 )

1.36 0.74 1.40 1.54 0.89 1.07 1.55 1.08

.(

◦)

343 291 286 284 301 320 334 331

.w∗ .(ms

317

−1 )

2.37 2.26 2.04 2.28 1.09 1.83 2.32 1.72

.u∗ .(ms

−1 )

0.34 0.21 0.34 0.37 0.25 0.25 0.35 0.21

L

h

.(m)

.(m)

−32.78 −8.70 −33.25 −33.54 −56.63 −19.53 −25.64 −8.11

1570 1240 1070 1240 943 1070 1325 1070

4 heights (.2, 4, 15 and 30) m from a micrometeorological tower of 30 m height. The samplers were placed in arcs of 50 and 100 m radius. In a total eight experiments were performed, and the following micrometeorological parameters were measured, wind speed (V ), wind direction (.θ ), convective speed (.w∗ ), friction speed ( .u∗ ), Monin-Obukhov length (L) and atmospheric boundary layer height (h). Table 25.1 shows the meteorological data from the IIT Delhi experiment reported in references [ShEtAl96, ShEtAl96a] and [ShEtAl02].

25.2.4 Statistical Indices Some statistical indices are used to evaluate the model performance compared to the observed data. These statistical indices were established by reference [Ha89], which recommends to evaluate the normalized mean square error (NMSE), the Correlation Coefficient (COR), the Factor of two cut (F A2), the Fractional Bias (F B) and the Standard Fractional Bias (F S).

25.3 Results For the briefly described experiment, the concentrations of the pollutant dispersion process were computed using the solution of the advection-diffusion model closed by the countergradient term. Therefore, micrometeorological data from the IIT Delhi experiment and the turbulence parameterizations proposed by Degrazia were used. To analyse the influence of the countergradient term in the turbulent transport simulation, two simulations were performed one with .Sk = 0 and another one with .Sk = 0.6. The statistical evaluation of the observed data and the present model predictions with .Sk = 0 and .Sk = 0.6, respectively, and the eddy diffusivities proposed by Degrazia is presented in Table 25.2. The comparison of the experimental data and the predictions by the model simulation are shown in Fig. 25.1 in form of scattering diagrams. On the left side, the

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Table 25.2 Statistical performance of the model prediction for the IIT Delhi experiment, (a) for = 0 and (b) for .Sk = 0.6

.Sk

.Sk

0 .0.6

N MSE 0.28 0.18

COR 0.88 0.84

F AT 2 0.88 0.94

FB −0.17 0.18

FS −0.44 0.18

Fig. 25.1 Scatter diagram of observed (.co ) and predicted (.cp ) concentrations by the 3D-GILTT method for the IIT Delhi experiment and eddy diffusivities proposed by Degrazia: (a) .Sk = 0 and (b) .Sk = 0.6

model calculation used an asymmetry term with .Sk = 0 and on the right side with Sk = 0.6, while for both computations the eddy diffusivities proposed by Degrazia [DeEtAl00, DeEtAl02] were employed. When the asymmetry term is considered in the model, better results are obtained when compared with the experimental data, and as a consequence, the error in the simulations is smaller. In general, the model satisfactorily simulates the observed concentrations.

.

25.4 Conclusions The pollutant concentrations were satisfactorily simulated considering a non-local closure for turbulence, which is different to approaches in the literature. It is noteworthy that this solution is the most complete one obtained by the GILTT method. In the solution derivation, the countergradient and the longitudinal diffusion terms are incorporated, and further this model is able to analyse transients. The obtained results were computed considering .Sk = 0.6 for the asymmetry term. In future works, more light will be shed on the influence of the asymmetry on the solution.

25 Advection–Diffusion Equation with Non-Fickian Closure

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References [An05] Anfossi, D.: Lagrangian Dispersion in the Planetary Boundary Layer. Editora da UFRGS, Porto Alegre (2005) [Ar99] Arya, S.P.: Air Pollution Meteorology and Dispersion. Oxford University Press, New York (1999) [Bl97] Blackadar, A.K.: Turbulence and Diffusion in the Atmosphere: Lectures in Environmental Sciences. Springer, Heidelberg (1997) [BuEtAl12] Buske, D., Vilhena, M.T., Tirabassi, T., Quadros, R.S., Bodmann, B.: A closed form solution for pollutant dispersion in atmosphere considering nonlocal closure of the turbulent diffusion. WIT Trans. Ecol. Environ. 1, 59–69 (2012) [Ca82] Caughey, S.J.: Diffusion in the convective boundary layer. In: Nieuwstadt, F.T.M., Van Dop, H., Reidel (eds.) Atmospheric Turbulence and Air Pollution Modelling, Boston (1982) [De66] Deardoff, J.W.: The countergradient heat flux in the lower atmosphere and in the laboratory. J. Atmos. Sci. 23, 503–506 (1968) [De72] Deardoff, J.M.: Numerical investigation of neutral and unstable planetary boundary layers. J. Atmos. Sci. 29, 91–115 (1972) [DeEtAl00] Degrazia, G., Anfossi, D., Carvalho, J., Mangia, C., Tirabassi, T., Velho, H.C.: Turbulence parameterisation for PBL dispersion models in all stability conditions. Atmos. Environ. 34, 3575–3583 (2000) [DeEtAl01] Degrazia, G.A., Moreira, D.M., Vilhena, M.T.: Derivation of an eddy diffusivity depending on source distance for vertically inhomogeneous turbulence in a convective boundary layer. J. Appl. Meteorol. 40(7), 1233–1240 (2001) [DeEtAl02] Degrazia, G.A., Moreira, D.M., Campos, C.R.J., Carvalho, J.C., Vilhena, M.T.: Comparison between an integral and algebraic formulation for the eddy diffusivity using the Copenhagen experimental dataset. Il Nuovo Cimento 25C, 207–218 (2002) [DeWi75] Deardoff, J.W., Willis, G.E.: A parameterization of diffusion into the mixed layer. J. Appl. Meteorol. 14, 1451–1458 (1975) [Er42] Ertel, H.: Der vertikale Turbulenz-Wärmestrom in der Atmosphäre. Meteorologische Zeitschrift 59, 250–253 (1942) [Ha89] Hanna, S.R.: Confidence limit for air quality models as estimated by bootstrap and jacknife resampling methods. Atmos. Environ. 23, 1385–1395 (1989) [Ho82] Hojstrup, J.: Velocity spectra in the unstable boundary layer. J. Amos. Sci. 39, 2239– 2248 (1982) [OlEtAl84] Olesen, H.R., Larsen, S.E., Hojstrup, J.: Modelling velocity spectra in the lower part of the planetary boundary layer. Boundary-Layer Meteorol. 29, 285–312 (1984) [PaDu84] Panofsky, H.A., Dutton, J.A.: Atmospheric Turbulence. John Wiley & Sons, New York (1984) [SePa97] Seinfeld, J.H., Pandis, S.N.: Atmospheric Chemistry and Physics of Air Pollution. John Wiley & Sons, New York (1997) [ShEtAl02] Sharan, M., Yadav, A.K., Modani, M.: Simulation of short-range diffusion experiment in low wind convective conditions. Atmos. Environ. 36, 1901–1906 (2002) [ShEtAl96] Sharan, M., Singh, M.P., Yadav, A.K.: A mathematical model for the atmospheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmos. Environ. 30, 1137–1145 (1996) [ShEtAl96a] Sharan, M., Singh, M.P., Yadav, A.K., Agarwal, P., Nigam, S.: A mathematical model for dispersion of air pollutants in low winds conditions. Atmos. Environ. 30, 1209– 1220 (1996) [Si17] Silveira, V.C.: Simulação tridimensional da dispersão de poluentes em um modelo Euleriano levando em conta o efeito do meandro do vento. Tese de Doutorado, UFSM, Santa Maria (2017)

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[St88] Stull, R.B.: An Introduction to Boundary Layer Meteorology. Kluwer Academic Publishers, Dordrecht (1988) [VaVe01] Van Dop, H., Verver, G.S.: Countergradient transport revisited. J. Atmos. Sci. 58, 2240–2247 (2001) [WyWe91] Wyngaard, J.C., Weil, J.C.: Transport asymmetry in skewed turbulence. Phys. Fluids A 3, 155–162 (1991)

Chapter 26

Failure Analysis of Composite Pipes Subjected to Bending: Classical Laminated Plate Theory vs. 3D Elasticity Solution T. Wang, O. Menshykov, and M. Menshykova

26.1 Introduction Interest in products made from fibre-reinforced polymer composites is increasing in the oil and gas industry due to their high specific strength, as well as their overall weight reduction and resistance to various liquids. Taking into account that oil and gas pipelines are working in complex and hazardous condition, filamentwound composite pipes have been identified as the best alternative to conventional oil pipelines in the future due to their ability to be tailored for specific purposes. Due to this, it is important to study the properties of fibre-reinforced polymer composite pipe in service condition. Improving performance to achieve better structural quality is of paramount importance. Xia et al. [XiEtAl01] presented research on thin-walled composite pipes exposed to internal pressure. The appropriate solution for deformations and stresses was computed using a three-dimensional elasticity solution, and three distinct angle ply pipes were studied. Later, they expanded their research and proved the validity of the theoretical approach using experimental methods and illustrated that theoretical and experimental results are consistent within the elastic state [TaEtAl02]. Because pipes may also be subjected to thermomechanical stress, Bakaiyan et al. [BaEtAl09] presented a study of composite pipe subjected to a temperature gradient and internal pressure. Hastie et al. used the finite element model to evaluate the stress states of thermoplastic pipe with liners that was subjected to a combination of pressure, tension and temperature gradient loadings. The modified Tsai-Hill failure criterion was used to predict failure, and the detailed failure coefficients distribution was presented [HaEtAl19, HaEtAl19]. The time-dependent stress, strain

T. Wang · O. Menshykov () · M. Menshykova School of Engineering, University of Aberdeen, Aberdeen, UK e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_26

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and deformation distributions for multi-layered composite pipe exposed to cyclic temperature and internal pressure loadings were studied using three-dimensional anisotropic elasticity [AnEtAl10]. Guz et al. examined the differences between Tsai-Hill and modified Tsai-Hill failure coefficients for pipes with various winding angles [GuEtAL17]. By utilizing a three-dimensional elasticity solution, Cox et al. predicted the failure mechanisms of composite pipes subjected to pressure, axial and bending load [CoEtAl19]. Meanwhile, numerous studies examined composite pipe subjected to asymmetric loading. The stresses and deformations of coaxial hollow cylinders were studied using a general analytical elasticity solution with two distinct interface boundary conditions: no sliding and no friction [JoCa94]. Zhang et al. established an elasticity solution to analyse composite pipes with winding angles of .0◦ and .90◦ that were exposed to pure bending, as well as numerical analysis of pipe with arbitrary winding angles between .0◦ .90◦ [ZhEtAl14]. An efficient method based on homogenization theory was presented to conduct the stress analysis of thick or moderated thick-walled pipe with varied winding angles subjected to pressure, tension and bending [SuEtAl14]. The bending behaviour of thick-walled sandwich pipes composed of a resin core layer and alternated ply layers was studied in [XiEtAl02]. The classical laminated-plate theory was used to solve for stresses, strains and cross-sectional deformations of composite pipes under bending load. Natsuki and his colleagues determined the bending strength of filament wrapped composite pipe [NaEtAl03]. The stresses were theoretically calculated, and the maximum stress failure criterion was employed to estimate the failure mode of composite pipe. In [MeGu14] two types of spoolable filament wound composite pipes, one with the metal internal layer and the other without, were investigated, obtaining the stress distribution through the wall thickness. Other methods based on three-dimensional laminate theory were presented in [ShEtAl11] and [AhHo16] for studying composite pipes. The researchers calculated the strain distribution using the equivalent flexural stiffness of pipe and determined the relationship between bending and axial strain. The layer-wise method [Re93] was proposed to analytically determine the stress, and the results were verified by comparing with the FEM results and experimental data. The thick-walled composite pipe subjected to combined loadings was studied using a high-order analytical approach based on layer-wise theory [SaHo16]. Additionally, a beam theory was used to compute the stiffness of a thin-walled composite pipe exposed to combined bending and torsion loads, and the findings were confirmed by FE analysis using the shell element [JoEtAl15]. In [ZhEtAl20], a novel method was employed to compute the stress distribution of multi-layer composite cantilever tubes by combining bending- and shear-like analytical formulas. In this study layered composite pipes subjected to pure bending were considered. The efficient three-dimensional elasticity solution (3D-ES) was proposed to predict the stress state of pipe, and comparison with the results obtained using the laminated plate theory (LPT) was presented.

26 Failure Analysis of Composite Pipes Under Bending

323

26.2 Statement of the Problem and 3D Elasticity Solution Let us consider a composite pipe consisting of orthotropic layers under pure bending moment M; please see Fig. 26.1. Considering the low ductility of composite materials compared to conventional metal, we assume that the pipe is subjected to a minor strain condition. The three-dimensional elasticity constitutive equation of composite in cylindrical coordinate system can be presented as [He98] and [JoCa94]: ⎤ ⎡ S11 εr ⎢ ε ⎥ ⎢S ⎢ θ ⎥ ⎢ 12 ⎢ ⎥ ⎢ ⎢ εz ⎥ ⎢S13 .⎢ ⎥=⎢ ⎢γθz ⎥ ⎢S14 ⎢ ⎥ ⎢ ⎣ γrz ⎦ ⎣ 0 γrθ 0 ⎡

S12 S22 S23 S24 0 0

S13 S23 S33 S34 0 0

S14 S24 S34 S44 0 0

0 0 0 0 S55 S56

⎤⎡ ⎤ σr 0 ⎢σ ⎥ 0 ⎥ ⎥⎢ θ ⎥ ⎥⎢ ⎥ 0 ⎥ ⎢ σz ⎥ ⎥⎢ ⎥, 0 ⎥ ⎢τθz ⎥ ⎥⎢ ⎥ S56 ⎦ ⎣ τrz ⎦ S66 τrθ

(26.1)

where .Sij are transformed compliances and .σr , .σθ , .σz and .τ ; .εr , .εθ , .εz and .γ are the radial, hoop, axial and shear stresses and strains, respectively. The stresses could be expressed in terms of Lekhnitskii’s stress functions, .F (r, θ ) and .Ψ (r, θ ), as [Le81] and [JoCa94]: σr =

.

∂ 2F ∂F ∂ 2F + 2 2 , σθ = , r∂r r ∂θ ∂r 2

∂ 2F ∂F , − r∂r∂θ r 2 ∂θ

τrθ =

.

σz =

.

τrz =

∂Ψ , r∂θ

τθz = −

(26.2) ∂Ψ , ∂r

1 (Ar sin θ − S13 σr − S23 σθ − S34 τθz ) . S33

(26.3) (26.4)

Substituting Eqs. (26.2)–(26.4) into the constitutive Eq. (26.1), strains could be represented as: Fig. 26.1 Composite layered pipe under bending

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εr = β11 σr + β12 σθ + S13 Ar sin θ + β14 τθz ,

.

εθ = β12 σr + β22 σθ + S23 Ar sin θ + β24 τθz ,

.

γθz = β14 σr + β24 σθ + S34 Ar sin θ + β44 τθz ,

.

γrθ = β56 τrz + β66 τrθ ,

.

γrz = β55 τrz + β56 τrθ ,

.

S S

where .βij = Sij − i3S333j are the reduced compliances and the related reduced compliances matrix is: ⎡

β11 ⎢β ⎢ 12 ⎢ ⎢ 0 .[β ij ] = ⎢ ⎢β14 ⎢ ⎣ 0 0

β12 β22 0 β24 0 0

0 0 0 0 0 0

β14 β24 0 β44 0 0

0 0 0 0 β55 β56

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ β56 ⎦ β66

Assuming that the pipe is bent in one direction and solving the CauchyEuler strain-compatibility equations by the separation of variables, the general Lekhnitskii’s stress functions could be represented as: F (r, θ ) = A sin θ

.

4  αi i=1

χi

μ1 3 r , 2

(26.5)

β56 + μ2 r 2 . β66

(26.6)

r χi +1 +α5 r + α6 r ln r +

Ψ (r, θ ) = A sin θ

4 

.

αi gi r χi + α6

i=1

The four roots of the characteristic equations could be computed as follows: χ1,2 =

.

−b ±

√ b2 − 4ac , 2a

2 a = β22 β44 − β24 ,

.

χ3,4 = −

−b ±

√

b2 − 4ac , 2a

2 c = β55 (β11 + 2β12 + β22 + β66 ) − β56 ,

2 b = β24 (2β14 + β24 + 2β56 ) − β44 (β11 + 2β12 + β22 + β66 ) − β22 β55 + β14 ,

.

and the other constants are:

26 Failure Analysis of Composite Pipes Under Bending

325

β24 χi3 + (β14 + β24 )χi2 − β56 χi

gi =

.

(i = 1, . . . , 4),

β44 χi2 − β55





−1 1 2S34 μ1 λ1 λ2 = , . μ2 λ3 λ4 S33 S13 − S23 λ1 = −2β14 − 6β24 + β56

λ2 = 4β44 − β55 ,

.

λ3 = −β11 − 2β12 + 3β22 − β66 ,

λ4 = 2β14 − 2β24 + β56 ,

.

where .α1 -.α4 are unknown constants that vary by layer and could be determined from the boundary conditions and continuity equations. The remaining constants .α5 and .α6 are equal to zero since the corresponding stresses dissipate throughout the derivation process. Substituting the general Lekhnitskii’s stress functions Eqs. (26.5) and (26.6) into the Eqs. (26.2)–(26.4), the stresses could be rewritten as: 4

 χi −1 .σr = A sin θ αi r + μ1 r , i=1

σθ = A sin θ

4 

.

αi (χi + 1) r

χi −1

+ 3μ1 r ,

i=1

τrθ = A cos θ −

4 

.

αi r

χi −1

− μ1 r ,

i=1

τrz = A cos θ

4 

.

αi r

χi −1

+ μ2 r ,

i=1

τθz = A sin θ −

4 

.

g i χi α i r

χi −1

− 2μ2 r ,

i=1

σz = A sin θ

.

4  gi χi S34 − (S13 + S23 + χi S23 ) i=1

.

S33

 1 + 2μ2 S34 − μ1 (S13 + 3S23 ) r . S33

αi r χi −1 +

(26.7)

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Boundary conditions and continuity conditions are applied to determine the unknown constants .αi .(i = 1, 2, 3, 4) for each layer. The interfaces between layers are assumed to be perfectly bonded, and as a consequence, the radial, hoop and axial displacements and radial and shear stresses are continuous from layer to layer. Meanwhile, as no internal pressure was applied, the radial and shear (.τrθ , .τrz ) stresses at the interior and exterior faces of the pipe are equal to zero. The constant A is derived from equilibrium equation of composite pipe subjected to bending moment applying at the ends; its expression is presented as [MeGu14]: M=

N  

2π



r(k+1)

.

k=0 0

σz r 2 sin θdθ dr.

(26.8)

rk

Substituting the expression of axial stress Eq. (26.7) into the equilibrium equation (26.8), the constant A can be computed as follows: 4 (k) (k)   N (k) (k)   η α χi +2 χi +2 i i + r .M = Aπ − r (k) (k−1) (k) k=1 i=1 χi + 2 (k) (k)

.

(k)

1 1 + 2μ2 S34 − μ1 (k) 4 S33

  (k) (k) S13 + 3S23 

⎞  4 4 ⎠, r(k) − r(k−1)

where ηi =

.

gi χi S34 − (S13 + S23 + χi S23 ) , S33

i = 1, 4.

26.3 Laminated Plate Theory and Simplified Constitutive Equations The detailed equations of the laminated plate theory for a coaxial multi-layered composite pipe under pure bending were presented in [Le81, XiEtAl02] and [MeGu14]. Thus, the constitutive equation can be represented as: ⎡

⎤ ⎡ εr S11 ⎢ εθ ⎥ ⎢S12 ⎥ ⎢ .⎢ ⎣ εz ⎦ = ⎣S13 γrθ 0

S12 S22 S23 0

and the reduced compliances matrix is:

S13 S23 S33 0

⎤⎡ ⎤ 0 σr ⎢ σθ ⎥ 0 ⎥ ⎥⎢ ⎥, 0 ⎦ ⎣ σz ⎦ τrθ S44

(26.9)

26 Failure Analysis of Composite Pipes Under Bending

⎡

β11 ⎢β12 .[β ij ] = ⎢ ⎣ 0 0

β12 β22 0 0

0 0 0 0

327

⎤ 0 0 ⎥ ⎥. 0 ⎦ β44

Applying the continuity and boundary conditions provided above, the components of the stress can be represented as:   σr = a1 r −1+ρ − a2 r −1−ρ + Awr sin θ,

.

 σθ = a1 (1 + ρ)r −1+ρ − a2 (1 − ρ)r −1−ρ + 3Awr] sin θ

.

  τrθ = − a1 r −1+ρ − a2 r −1−ρ + Awr cos θ,

.

  σz = s1 a1 r −1+ρ + s2 a2 r −1−ρ + s3 Ar sin θ,

.

(26.10)

where s1,2 = ∓

.

ρ=

.

1+

S13 + S23 (1 ± ρ) , S33

β11 + 2β12 + β66 , β22

s3 = 1 −

w=

(S13 + 3S 23 ) w , S33

S23 − S13 . β11 + 2β12 + β66 − 3β22

The parameters .β, .ρ, w, p and q can be computed from the layers’ material constants, and unknown constants A, .a1 and .a2 are to be computed from the boundary and continuity conditions and equilibrium equation (26.8) with the substitution of the appropriate stress component (26.10). It shall be noted that both approaches (3D-ES and LPT) are analytical methods based on the elasticity solutions derived from Lekhnitskii’s functions, but under different assumptions. In particular, considering a single-layered composite pipe, Lekhnitskii assumed that all the planes normal to the axis and passing through the axis are planes of elastic symmetry. Thus, for the case of bending, this assumption leads in a simplified compatibility expression [Le81] and [He98]: β22

.

.

∂ 4F ∂ 4F ∂ 3F ∂ 4F + + β + β + 2β − (2β ) 12 44 11 22 ∂r 4 r 2 ∂r 2 ∂θ 2 r 4 ∂θ 4 r∂r 3

(2β12 + β44 )

∂ 2F ∂ 2F ∂ 3F + + 2β + β + − β (2β ) 11 12 44 11 r 2 ∂r 2 r 4 ∂θ 2 r 3 ∂r∂θ 2

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.

+ β11

∂F sin θ . = 2 (S13 − S23 ) A r r 3 ∂r

(26.11)

Here the stress function F (analogous to Eq. (26.5)) has been changed appropriately, and the function .Ψ (Eq. (26.6)) has been eliminated, so the reduced compliances .β14 , .β24 , .β66 ,. β55 , .β56 that relate to the shear coupling have been eliminated too, and the constitutive equation Eq. (26.1) was simplified to Eq. (26.9). Thus, as long as shear coupling is not taken into account, the in-plane shear stress cannot be calculated when the problem is solved using LPT approach.

26.4 Finite Element Modelling Three-dimensional FE model is developed to compute the distribution of stresses through the thickness in a section of composite pipe under pure bending loading. Mechanical loads are applied in static steps. As the bending load cannot be applied directly at the end of the pipe, we symmetrically applied forces to the reference points at the ends of the pipe on both sides, which are coupled to the end crosssectional faces. The solid element (C3D20R) is prudently used in the model to avoid the hourglass effects and improve the accuracy in computing the curved structure. To minimize the effect of concentrated force at the ends on the results, the length to outer radius ratio is more than 4 times; results are analyzed at the middle of the pipe. In addition, the mesh density (element global dimension/elements in one layer) is determined as 4/3 to keep convergence of results. A comprehensive set of convergence results was presented in [WaEtAl23].

26.5 Numerical Results and Validation of the Models To validate the feasibility of the models, the results obtained by both analytical methods will be compared with FEM results and results presented by Zhang et al. [ZhEtAl14]. A composite pipe with winding angles (.90◦ /+45◦ /−45◦ /0◦ ) subjected to 1.51 kNm is considered. The inner radius is 25 mm and the thickness of each layer is 3.5 mm. The mechanical properties of the material (T300/LY5052) are as follows: .E1 = 135 GPa, .E2 = 8 GPa, .G12 = 3.8 GPa, .ν12 = 0.27 GPa, .ν23 = 0.49 GPa. Radial, hoop, axial and shear stresses through the thickness of the pipe are presented in Figs. 26.2, 26.3, 26.4, and 26.5. One can notice that three out of four solutions fully coincide, with the LPT results following the same rising and falling trends, but being quite different quantitatively. Please also note (see Fig. 26.5) that shear stress cannot be calculated using LPT approach as it was mentioned above.

Fig. 26.2 Axial stresses through the thickness of the pipe, .M = 1.51 kNm

100

Stress (MPa)

80 60

Axial Stress, [90 /+45 /-45 /0 ] 3D-ES LPT FEM Zhang

40 20 0 25

27

29

31

33

35

37

39

R (mm) Fig. 26.3 Hoop stresses through the thickness of the pipe, .M = 1.51 kNm

10

Hoop Stress, [90 /+45 /-45 /0 ]

Stress (MPa)

5 0 -5 -10

3D-ES LPT FEM Zhang

-15 -20 -25 25

27

29

31

33

35

37

39

R (mm) Fig. 26.4 Radial stresses through the thickness of the pipe, .M = 1.51 kNm

0.5

Radial Stress, [90 /+45 /-45 /0 ]

Stress (MPa)

0 -0.5 -1

3D-ES LPT FEM Zhang

-1.5 -2 25

27

29

31

33

R (mm)

35

37

39

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Fig. 26.5 Shear stresses through the thickness of the pipe, .M = 1.51 kNm

10

Shear Stress, [90 /+45 /-45 /0 ]

Stress (MPa)

5

0 3D-ES LPT FEM Zhang

-5

-10 25

27

29

31

33

35

37

39

R (mm) Fig. 26.6 Axial stresses through the thickness of the pipe, .M = 10 kNm

60

Stress (MPa)

50

Axial Stress, [0 /+30 /-30 /0 ] 3D-ES FEM Menshykova

40 30 20 10 0 2.5

3.5

4.5

5.5

6.5

7.5

R (mm)

Similar results have been obtained for the problem considered by Menshykova and Guz in [MeGu14]; see Figs. 26.6, 26.7, 26.8 and 26.9. As above, the LPT solution (red dashed line) does not coincide with 3D-ES and FEM results. It could be concluded that the developed 3D elasticity solution model taking the extensional shear couplings into account shall be used for failure analysis and optimal design of composite pipes under combined loading that includes bending. Meanwhile, it is worth to add that the preliminary analysis shows the applicability of the LPT approach for thick- and thin-walled composite pipes with .0◦ and ◦ .90 winding angles only. The detailed parametric analysis will be the next step of the current studies followed by the failure analysis and optimal design based on the failure coefficients and safety zones, e.g. see [WaEtAl20] and [WaEtAl21].

3

Stress (MPa)

2

Hoop Stress, [0 /+30 /-30 /0 ] 3D-ES FEM Menshykova

1 0 -1 -2 2.5

3.5

4.5

5.5

6.5

7.5

R (mm) Fig. 26.7 Hoop stresses through the thickness of the pipe, .M = 10 kNm

0.4

Radial Stress, [0 /+30 /-30 /0 ]

Stress (MPa)

0.2 0 -0.2 3D-ES FEM Menshykova

-0.4 -0.6 2.5

3.5

4.5

5.5

6.5

7.5

R (mm) Fig. 26.8 Radial stresses through the thickness of the pipe, .M = 10 kNm

10

Shear Stress, [0 /+30 /-30 /0 ] 3D-ES FEM Menshykova

Stress (MPa)

5

0

-5

-10 2.5

3.5

4.5

5.5

6.5

R (mm) Fig. 26.9 Shear stresses through the thickness of the pipe, .M = 10 kNm

7.5

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References [AhHo16] Ahmad, M.G., Hoa, S.: Flexural stiffness of thick walled composite tubes. Compos. Struct. 149, 125–133 (2016) [AnEtAl10] Ansari, R., Alisafaei, F., Ghaedi, P.: Dynamic analysis of multi-layered filamentwound composite pipes subjected to cyclic internal pressure and cyclic temperature. Compos. Struct. 92(5), 1100–1109 (2010) [BaEtAl09] Bakaiyan, H., Hosseini, H., Ameri, E.: Analysis of multi-layered filament-wound composite pipes under combined internal pressure and thermomechanical loading with thermal variations. Compos. Struct. 88(4), 532–541 (2009) [CoEtAl19] Cox, K., Menshykova, M., Menshykov, O., Guz, I.: Analysis of flexible composites for coiled tubing applications. Compos. Struct. 225, 111118 (2019) [GuEtAL17] Guz, I.A., Menshykova, M., Paik, J.K.: Thick-walled composite tubes for offshore applications: an example of stress and failure analysis for filament-wound multilayered pipes. Ships Offshore Struct. 12(3), 304–322 (2017) [HaEtAl19] Hastie, J.C., Guz, I.A., Kashtalyan, M.: Effects of thermal gradient on failure of a thermoplastic composite pipe (TCP) riser leg. Int. J. Press. Vessels Pip. 172, 90–99 (2019) [HaEtAl19] Hastie, J.C., Kashtalyan, M., Guz, I.A.: Failure analysis of thermoplastic composite pipe (TCP) under combined pressure, tension and thermal gradient for an offshore riser application. Int. J. Press. Vessels Pip. 178, 103998 (2019) [He98] Herakovich, C.T.: Mechanics of Fibrous Composites. John Wiley and Sons, Hoboken (1998) [JoCa94] Jolicoeur, C., Cardou, A.: Analytical solution for bending of coaxial orthotropic cylinders. J. Eng. Mech. 120(12), 2556–2574 (1994) [JoEtAl15] Jonnalagadda, A., Sawant, A., Rohde, S., Sankar, B., Ifju, P.: An analytical model for composite tubes with bend–twist coupling. Compos. Struct. 131, 578–584 (2015) [Le81] Lekhnitskii, S.: Theory of Elasticity of an Anisotropic Body. MIR Publishers, Moscow (1981) [MeGu14] Menshykova, M., Guz, I.: Stress analysis of layered thick-walled composite pipes subjected to bending loading. Int. J. Mech. Sci. 88, 289–299 (2014) [NaEtAl03] Natsuki, T., Takayanagi, H., Tsuda, H., Kemmochi, K.: Prediction of bending strength for filament-wound composite pipes. J. Reinf. Plast. Compos. 22(8), 695– 710 (2003) [Re93] Reddy, J.: An evaluation of equivalent-single-layer and layerwise theories of composite laminates. Compos. Struct. 25(1–4), 21–35 (1993) [SaHo16] Sarvestani, H.Y., Hojjati, M.: A high-order analytical method for thick composite tubes. Steel Compos. Struct. 21(4), 755–773 (2016) [ShEtAl11] Shadmehri, F., Derisi, B., Hoa, S.: On bending stiffness of composite tubes. Compos. Struct. 93(9), 2173–2179 (2011) [SuEtAl14] Sun, X., Tan, V., Chen, Y., Tan, L., Jaiman, R., and Tay, T.: Stress analysis of multi-layered hollow anisotropic composite cylindrical structures using the homogenization method. Acta Mech. 225(6), 1649–1672 (2014) [TaEtAl02] Takayanagi, H., Xia, M., Kemmochi, K.: Stiffness and strength of filament-wound fiber-reinforced composite pipes under internal pressure. Adv. Compos. Mat. 11(2), 137–149 (2002) [WaEtAl20] Wang, T., Menshykov, O., Menshykova, M., Guz, I.: Modelling and optimal design of thick-walled composite pipes under in-service conditions. IOP Conf. Ser. Mat. Sci. Eng. 936, 012046 (2020) [WaEtAl21] Wang, T.Y., Menshykova, M., Menshykov, O., Guz, I.: Failure analysis of multilayered thick-walled composite pipes subjected to torsion loading. Mat. Sci. Forum 1047, 25–30 (2021)

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[WaEtAl23] Wang, T., Menshykova, M., Menshykov, O., Guz, I.A., Bokedal N.K.: Mechanical analysis of thick-walled filament wound composite pipes under pure torsion load: safety zones and optimal design. Appl. Compos. Mat. (2023). https://doi.org/10. 1007/s10443-022-10088-3 [XiEtAl01] Xia, M., Takayanagi, H., Kemmochi, K.: Analysis of multi-layered filament-wound composite pipes under internal pressure. Compos. Struct. 53(4), 483–491 (2001) [XiEtAl02] Xia, M., Takayanagi, H., Kemmochi, K.: Bending behavior of filament-wound fiberreinforced sandwich pipes. Compos. Struct. 56(2), 201–210 (2002) [ZhEtAl14] Zhang, C., Hoa, S.V., Liu, P.: A method to analyze the pure bending of tubes of cylindrically anisotropic layers with arbitrary angles including 0◦ or 90◦ . Compos. Struct. 109, 57–67 (2014) [ZhEtAl20] Zhang, C., Chang, R., Li, A.: Novel two-level strategy to exactly solve multilayer composite cantilever tubes of cylindrically orthotropic materials. Compos. Struct. 237, 111866 (2020)

Chapter 27

The Analytical Formulation GILTT Applied in a Model of Contaminant Transport in Porous Media G. J. Weymar, D. Buske, R. S. Quadros, I. C. Furtado, J. Konradt, and T. F. Almeida

27.1 Introduction Water is one of the essential natural resources for the existence of living beings; however, this resource is finite, so its preservation and conscious use are essential. According to Beckie [Be13], about 50% of the total drinking water available on the planet is provided by groundwater. Groundwater is very important in the hydrological cycle as it is connected and exchanges with surface water bodies and the atmosphere. Groundwater discharge feeds springs, lakes, and rivers and in the end drains into the ocean. A large part of groundwater contamination originates from the earth’s surface, where the soil serves as some sort of groundwater protection, i.e. functioning as a natural filter capable of retaining a good part of the impurities that infiltrate. However, a part can reach the water table modifying its quality, so that studies which aim to investigate the dispersion of contaminants in soils are of real importance, since depending on how substances are dispersed, they can reach the water table. Thus, the present work intends to carry out a study of the one-dimensional mathematical model of the transport of contaminants in a sanitary landfill. To this end, the dimensionless form of the contaminant dispersion model in a porous medium will be considered, together with the hypothesis of a continuous and uniform leakage of pollutant from a storage cell of urban solid waste into the soil.

G. J. Weymar · D. Buske () · R. S. Quadros · J. Konradt · T. F. Almeida Federal University of Pelotas, Pelotas, Rio Grande do Sul, Brazil I. C. Furtado Federal Institute of Education, Science and Technology of Rio Grande do Sul, Pelotas, Rio Grande do Sul, Brazil © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_27

335

336

G. J. Weymar et al.

27.2 Model of Contaminant Dispersion in the Porous Medium in a Solid Waste Cell The equation that describes the transport of non-reactive contaminants in a saturated porous medium considering the process of adsorption is represented by: .

      ρ ∂S ∂C ∂ 2C ∂ 2C ∂ 2C ∂C ∂C ∂C 1+ = Dx 2 +Dy 2 +Dz 2 − vx +vy +vz , ∂x ∂y ∂z θ ∂C ∂t ∂x ∂y ∂z (27.1)

where .ρ is the soil density, S is the concentration of the contaminant in the solid phase, .θ represents porosity, C is the contaminant concentration in the liquid phase, t is the time, and .Di and .vi are the diffusivity coefficient and the advection velocity, respectively, in the directions i with .i = x, y, and z. In this work, the following fundamental hypotheses were used [Al18]: • • • •

Soil is considered a homogeneous saturated porous medium. Contaminant flow through the soil is transient and one-dimensional. The release of the contaminant is continuous. The physical properties of the contaminating medium and the soil are considered constant. • The mass flow at y is much greater than at x. • The diffusion process in the soil is negligible when compared to the diffusion in the contaminant. • The velocity profile for contaminant flow is considered uniform. Equation (27.1) results in the expression: R

.

∂C ∂ 2C ∂C = Dy 2 − vy ∂y ∂t ∂y

(27.2)

∂S is known as the soil retardation factor. where .R = 1 + ρθ ∂C The boundary and initial conditions are:

C(0, t) = C0 ,

.

.

− Dy

∂C (Ly , t) = hm (C − C∞ ) , ∂y C(y, 0) = Ci ,

.

(27.3) (27.4) (27.5)

where .C0 represents the concentration at the landfill-soil interface, .Ly is the distance from the leachate to the water table, .hm is the mass transfer coefficient, .C∞ is the contaminant concentration at the soil-water table interface and .Ci is the initial concentration in the solid waste storage cell.

27 The Analytical Formulation GILTT Applied in a Model of Contaminant Transport

337

The problem can be rewritten in the dimensionless form; thus, for the dimenD sionless process, the following relationships are considered: .Y = Lyy , .τ = L2y t, y

Ci −C∞ ∗ ∞ V = vy = 1, .Bi = mDy y , .P e = Dyy , .C ∗ = C−C ΔC and .C0 = ΔC . In the expressions Y represents the dimensionless soil depth, .τ is the dimensionless time, V is the dimensionless average fluid velocity, .V is the average speed, Bi is the Biot number, P e is the Péclet number, .C ∗ is the contaminant concentration in the dimensionless liquid phase, .ΔC = C0 − C∞ and .C0∗ is the concentration at the dimensionless landfill-soil interface. Equation (27.2) in dimensionless form becomes:

.

v

h L

vL

R

.

∂C ∗ ∂C ∗ ∂ 2C∗ − Pe . = 2 ∂Y ∂τ ∂Y

(27.6)

The dimensionless boundary and initial conditions (Eqs. 27.3–27.5) are: C ∗ (0, τ ) = 1 ,

(27.7)

∂C ∗ (1, τ ) + BiC ∗ (1, τ ) = 0 , ∂Y

(27.8)

C ∗ (Y, 0) = C0∗ .

(27.9)

.

.

.

27.2.1 Solution of the Problem With the dimensionless equations, represented by (27.6)–(27.9), in a first step, the superposition method [Oz93] is used to solve the linear problem. ˜ C ∗ (Y, τ ) = C(Y, τ ) + CF (Y ) ,

(27.10)

.

where .C˜ is an auxiliary function that is subject to homogeneous boundary conditions and .CF is the solution to the steady-state problem and subject to the nonhomogeneous boundary conditions of the problem. Then, substituting equation (27.10) in (27.6), ∂ 2 C˜ ∂ C˜ d 2 CF = .R + − Pe ∂τ ∂Y 2 dY 2



∂ C˜ dCF + ∂Y dY

 .

(27.11)

From Eq. (27.11), there are two differential equations, one ordinary differential equation (ODE) and another partial differential equation (PDE). The stationary problem is obtained by solving the following boundary value problem:

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.

d 2 CF dCF =0, − Pe 2 dY dY

(27.12)

with boundary conditions CF (0) = 1 ,

.

.

dCF (1) + BiCF (1) = 0 . dY

The analytical solution of the ODE (27.12) is: CF (Y ) =

.

eP e (P e + Bi) − BieP eY . eP e (P e + Bi) − Bi

The transient problem with homogeneous boundary conditions is represented by the following PDE: R

.

∂ 2 C˜ ∂ C˜ ∂ C˜ = , − Pe 2 ∂τ ∂Y ∂Y

(27.13)

The associated boundary conditions are given by the following equations: ˜ C(0, τ) = 0 ,

.

.

∂ C˜ ˜ (1, τ ) + Bi C(1, τ) = 0 , ∂Y

And the initial condition is: ˜ C(Y, 0) = C0∗ − CF (Y ) .

.

(27.14)

To obtain the solution of Eq. (27.13), the GILTT method was used [BuEtAl12]. To this end, the Sturm-Liouville auxiliary problem is solved considering the Y direction. .

d 2ϕ + λ2 ϕ = 0 , dY 2 ϕ(0) = 0 ,

dϕ(1) + Biϕ(1) = 0 . dY The differential equation has as solution the eigenfunctions: ϕn (Y ) = sin(λn Y ) ,

.

(27.15)

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where λn cot(λn ) = −Bi ,

(27.16)

.

the eigenvalues .λn are the roots of the transcendental equation (27.16) and were calculated by the Newton-Raphson method. In the sequence, the solution of the PDE (27.13) is written expanding the concentration as a series in terms of the eigenfunctions (27.15): ˜ C(Y, τ) =

.

N 

lim

N →+∞

n=1

N0 

≈ ϕn (Y )C n (τ ) 

ϕn (Y )C n (τ ) ,

(27.17)

truncation n=1

where .C n (τ ) are terms to be determined and .N0 is to be determined such as to satisfy a precision criterion. The substitution of solution (27.17) into the Eq. (27.13) and the application of

1 the integral operator . 0 (.)ϕm (Y )dY provide the components of the expansion: N0  .

 R 0

n=1

1

0 ∂C n (τ )  ϕn (Y )ϕm (Y )dY = ∂τ

N



1

n=1 0

−P e

N0   n=1 0

1

ϕn (Y )ϕm (Y )dY C n (τ )

ϕn (Y )ϕm (Y )dY C n (τ ) .

(27.18)

Defining .Y (τ ) = {Cn (τ )}, .A = {am,n } and .B = {bm,n } with 

1

am,n = R

ϕn (Y )ϕm (Y )dY

.

0

bm,n = −λ2n



1

 ϕn (Y )ϕm (Y )dY − P e

0

0

1

ϕn (Y )ϕm (Y )dY .

Then, Eq. (27.18) may be cast in matrix form with .F = −A−1 B. Y  (τ ) + F Y (τ ) = 0 .

.

Applying the same procedures performed in the PDE also in the initial condition (Eq. 27.14), one obtains Y (0) = RA−1 E ,

.

1 where .em = 0 (C0∗ − CF (Y ))ϕm (Y )dY . The next step of the GILTT method is to apply the Laplace transform (.L {(.); τ → s}) to solve the matrix ODE.

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sL {Y (τ ), τ → s} − Y (0) + F L {Y (τ ), τ → s} = 0 .

.

Considering that the matrix F may be diagonalized, F = XDX−1 ,

.

where X is the matrix with the columns containing linearly independent eigenvectors of F and D is the diagonal matrix whose elements are the eigenvalues of F . After some manipulations, we get L {Y (τ ), τ → s} = X(sI + D)−1 X−1 Y (0) .

.

To determine .Y (τ ) = {Cn (τ )}, we apply the inverse Laplace transform Y (τ ) = XL −1 {(sI + D)−1 , s → τ }X−1 Y (0) .

.

⎛ (sI + D)

.

−1

⎜ ⎜ =⎜ ⎜ ⎝

1 s+d0

0 .. . 0

0 1 s+d1

.. . 0

··· ··· .. . ···

0 0 .. .

⎞ ⎟ ⎟ ⎟ , ⎟ ⎠

1 s+dN

where .d0 , d1 , · · · , dN are the eigenvalues of the matrix F . Upon performing the inversion of the Laplace transform element by element in the matrix, ⎛

e−d0 τ 0 ⎜ 0 e−d1 τ ⎜ −1 .L {(sI + D)−1 , s → τ } = ⎜ . .. ⎝ .. . 0 0

··· ··· .. .

0 0 .. .

⎞ ⎟ ⎟ ⎟ = G(τ ) . ⎠

(27.19)

· · · e−dN τ

With the matrix .G(τ ) defined in the Eq. (27.19), we conclude that the solution of the matrix ODE is Y (τ ) = XG(τ )X−1 Y (0) .

.

Therefore, the solution of the one-dimensional model of pollutant dispersion in porous media, represented by Eqs. (27.6)–(27.9), is given by ˜ C ∗ (y, τ ) = C(Y, τ ) + CF (Y ) =

N0 

.

n=1

ϕn (Y )C n (τ ) + CF (y) .

(27.20)

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27.3 Numerical Results The solution (27.20) from the previous section was now used to generate numerical results, where for all numerical implementations we used the online software Google Collaboratory (Python language). The simulations presented here considered some fixed parameters besides the variables, so that the same problem presented in [Al18] could be addressed. Figure 27.1 shows the dimensionless concentration development as a function of time; for the simulation we considered .P e = 2, .Y = 0.5 and the variable soil retardation factor (R). The results show that as the value of R increases, the dimensionless concentration will be lower, that is, the concentration of contaminant is retained by the porosity of the soil. For example, considering the values of .R = 0.1 and .R = 1 and for .τ > 0.60, one observes that the dimensionless concentration reaches its maximum; however at different times, because for a low soil retardation factor (for instance, .R = 0.1), the maximum concentration is reached in a time of .0.05, while for .R = 1, the maximum concentration is reached in a time of .0.60. The graph presented in Fig. 27.2 shows the concentration distribution with dimensionless soil depth, where in this simulation .P e = 2, .τ = 0.5 was considered and the soil retardation factor was varied again. Here, one observes from the concentration distribution profiles that the higher the soil retardation factor, the more abrupt is the variation of the concentration. This behaviour shows some kind of an obstruction effect on the contaminant’s infiltration. Figures 27.3 and 27.4 show the dimensionless concentration as a function of time and depth, respectively. In these simulations, .R = 1 was considered and the P e was varied.

Fig. 27.1 Evolution of the dimensionless concentration with R varying, considering fixed .P e = 2 and .Y = 0.5

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Fig. 27.2 Dimensionless concentration for soil depths with varying R and considering fixed .P e = 2 and .τ = 0.5

Fig. 27.3 Evolution of the dimensionless concentration with P e varying, considering fixed .R = 1 and .Y = 0.5

Note that as the value of P e increases, the concentration also increases, so that the soil will be contaminated more quickly by the pollutant. This occurs because the contaminant flows more quickly and contaminates the soil less; however, the pollutant reaches the groundwater considerably faster and with more intensity, i.e. the advection mechanism is dominant. Figure 27.5 shows the graph of the dimensionless concentration as a function of time for different depths (.Y = 0.2, .Y = 0.4, .Y = 0.6 and .Y = 0.8), considering the values of .R = 1 and .P e = 2 fixed. Since P e is a low value, one observes in Fig. 27.5 that the pollutant flow is slow and for results with .τ > 0.40 the concentration practically does not change (equilibrium concentration). Moreover,

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Fig. 27.4 Dimensionless concentration distribution with P e varying, considering fixed .R = 1 and = 0.5

.τ

Fig. 27.5 Evolution of the dimensionless concentration for different depths (.Y = 0.2, .Y = 0.4, = 0.6, and .Y = 0.8), considering fixed .P e = 2 and .R = 1

.Y

considering a value of Y closer to the surface and consequently to the landfill, one verifies by inspection that the dimensionless concentration gets closer to unity, which corresponds to the maximum concentration, mainly in comparison to the value of .Y = 0.8, which is a depth more adjacent to the groundwater and the concentration of the contaminant is lower. Figure 27.6 shows an illustration of a waste storage cell and a graph of the concentration profile as a function of soil depth for some time instants, considering .P e = 2 and .R = 1. With the profiles presented, one observes that the concentrations in each position with increasing ground depth for instants (.τ ) greater than .0.4 is already in a range of (pseudo-)equilibrium.

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Fig. 27.6 Dimensionless concentration distribution for different instants (.τ = 0.2, .τ = 0.4, .τ = 0.6 and .τ = 0.8), considering fixed .P e = 2 and .R = 1

27.4 Conclusions The results presented are consistent with expectations for the infiltration problem. The analysis of the influences of the parameters in the response of the model, with respect to the Péclet number indicated, the higher its value, the higher the concentration, shows that the soil will be contaminated more quickly by the pollutant and thus the substances reach the water table in a more intensive way. Concerning the soil retardation factor, simulations showed that soil contamination occurs more slowly for greater R values, which shows consistency, because in the physical sense a lower value in the retardation factor means an increase in soil porosity or equivalently an increase in the number of voids in the soil. From the adopted formalism point of view, we may confirm that the GILTT technique proved to be a useful tool to work with this type of problem, in addition to presenting an analytical solution for the dispersion process of pollutants in porous media.

References [Al18] Albuquerque, F.A.: Contaminants Spread of Study in Landfill Sanitary via GITT. Thesis (PhD in Mechanical Engineering) – Federal University of Paraiba, João Pessoa (2018) [Be13] Beckie, R.D.: Reference Module in Earth Systems and Environmental Sciences: Groundwater. Elsevier, Amsterdam (2013)

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[BuEtAl12] Buske, D., Vilhena, M.T., Tirabassi, T., Quadros, R.S., Bodmann, B.: A closed form solution for pollutant dispersion in atmosphere considering nonlocal closure of the turbulent diffusion. In: WIT Transactions on Ecology and The Environment, Lyndhurst, pp. 59–69 (2012) [Oz93] Özisik, M.N.: Heat Conduction. John Wiley Sons, New York (1993)

Chapter 28

An Existence Result for a Class of Integral Equations via Graph-Contractions M. Younis, D. Bahuguna, and D. Singh

28.1 Introduction Nonlinear functional analysis includes the essential and comprehensive field of fixed point theory, which is used as a method to solve numerous nonlinear problems in the science and engineering. Since a couple years ago, experts in fixed point theory have been focusing on how to apply their theories to a variety of physical-related engineering challenges. Fixed point theory has entered a new phase that is inextricably linked to measurements, abstract language, space analysis, and the mining of empirical studies in engineering. By incorporating the metric fixed point theory into a plethora of literature from the fields of computational engineering, quantum dynamics, and medical research, this was frequently maintained. In the analysis of metric spaces, fixed point theory has briefly been mentioned as independent literature while referencing numerous other mathematical groups. The definition and generalization of the various metric spaces and the concept of contractions are common applications of metric fixed point theory. The expected outcome of these extensions is also a deeper understanding of the geometric characteristics of Banach spaces, set theory, and inexpensive mappings. Some recent applications of fixed point theory lie in the following noteworthy papers.

M. Younis () · D. Bahuguna Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India e-mail: [email protected]; [email protected] D. Singh Department of Applied Science, National Institute of Technical Teachers’ Training and Research, Bhopal, India © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_28

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• Radenovi´c at al. [RaEtAl16] utilized cyclic contractions to find the unique solution of an integral equation for an unknown function within the context of b-metric spaces. • By using the fixed point technique, Dvorsky et al. [DvEtAl11] were able to construct a nonlinear potential problem for the pertinent boundary temperatures, fully taking into account the heat transfer coefficient’s temperature dependency. • Younis et al. [YoEtAl22a] focused their work on graph-Kannan contractions with an emphasis on the applications for the existence of the solution of different models associated with engineering problems, such as fourth-order two-point boundary value problems specifying deformations of an elastic beam, ascending motion of a rocket, etc. • In [YoEtAl22b] Younis et al. proposed an application to dynamic programming connected with the multistage process by adopting easier proofs of several wellknown fixed point findings in rectangular metric spaces. • Kuznetsov [Ku97] presented some results on solvability and qualitative properties of solutions of the abstract Cauchy problem for a new class of nonlinear equations of evolution in Banach spaces and applied fixed point theorems to nonlinear mathematical problems of nuclear reactor dynamics. More literature on this subject can be found in [NaEtAl17, SiEtAl17, YoEtAl19, YoEtAl22c]. Researchers, on the other hand, are now concentrating on graph-theoretical fixed point theory. Because of the importance and applications of graph theory, academics combine it with other disciplines of mathematics, particularly engineering mathematics. In this regard, Jachymski’s [Ja08] approach has altered the trajectory of metric fixed point theory and opened the door to a wide variety of applications. This subject sculpted a wide range of mappings in several disciplines, including internet network nodes, data flow networks, denotational semantics, and so on, based on its application. The contraction on underlying mappings is critical for locating the fixed point. Several academics have enhanced and extended the Banach’s contraction based on this basic premise. Wardowski [Wa12] proposed a new contraction, called F -contraction, and enunciated some novel fixed point results based on this new contraction. Later on in [Pi14] and [Se13], authors refined the concept given in [Wa12] by launching some weaker conditions on .F and on the self mapping involved. In our subsequent discussion, we drop the assumption .(F 2 ). Thus we utilize the functions .F : R+ → R which satisfy .(F 1) and .(F 3 ). The class of all such functions satisfying .(F 1) and .(F 3 ) is denoted by .ΞF . Let .X be the set of functions .χ : [0, ∞) → [0, ∞) such that: 1. .χ is monotonic increasing, i.e., .l1 ≤ l2 ⇒ χ (l1 ) ≤ χ (l2 ) 2. .χ is continuous and .χ (l) < l for each .l > 0 In this work, we introduce a graphical .χ -contraction within the context of bdislocated metric spaces and establish certain fixed point results for such mappings. Our aim is to establish the sufficient conditions for the existence and uniqueness of

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the solution for a class of integral equation utilizing the .χ -contraction in the sense of graph. An example is also enunciated to uphold the given results. Before presenting our main findings, some basic definitions and notations are worth mentioning. Alghamdi et al. [AlEtAl13] launched dislocated b-metric spaces by amalgamating the notions given in [Cz93] and [Ha12]. These spaces present a bond between metric spaces and logic programming semantics. Due to the applicative nature of these spaces, many authors worked on these spaces and presented some quality papers with applications (see, e.g., [SiEtAl18, Ve16]). For the rest of the hypothesis, unless otherwise specified, we use the standard notations of .N, .R, and .R+ .The axiomatic definition of a dislocated .b−metric space is as under. Definition 1 ( [AlEtAl13]) A nonempty set .S equipped with .Db : S × S → R+ is said to be a b-dislocated metric space denoted by the pair .(S , Db ), if for some .1 ≤ s, the following axioms hold: .(Db 1) .Db (υ, ϑ) = 0 implies .υ = ϑ. .(Db 2) .Db (υ, ϑ) = Db (ϑ, υ), .∀ .υ, ϑ ∈ S . .(Db 3) .Db (υ, ϑ) ≤ s[Db (υ, ς ) + Db (ς, ϑ)], .∀ .υ, ϑ, ς ∈ S . If .s = 1, then the resulting space is called dislocated metric space. Example 1 ( [AlEtAl13]) Let .S = R+ and the mapping .Db : S × S → R+ be defined by Db (υ, ϑ) = [max{υ, ϑ}]2 ,

.

for all .υ, ϑ ∈ S . Then .(S , Db ) is a b-dislocated metric space with the coefficient s = 2 > 1, but it is neither a b-metric nor a dislocated metric space.

.

Remark 1 The class of b-dislocated metric space .(S , Db ) is effectively larger than the class of dislocated metric space, since a dislocated metric space is a special case of b-dislocated metric space .(S , pb ) when .s = 1. Also, the class of b-dislocated metric space .(S , Db ) is effectively larger than the class of b-metric space, since a b-metric space is a special case of a b-dislocated metric space .(S , Db ) when the self distance .Db (υ, υ) = 0. Each b-metric-like .Db on .S generalizes a topology .τDb on .S whose base is the family of open .Db -balls .B − Db (υ, ) = {ϑ ∈ S : |Db (υ, ϑ) − Db (υ, υ)| < } for all .υ ∈ S and . > 0. Definition 2 ( [AlEtAl13]) Suppose that .(S , Db ) is a b-dislocated metric space. A mapping .A : S → S is said to be continuous at .υ ∈ S , if for every . > 0, there exists .δ > 0 such that .A (BDb (υ, δ)) ⊂ BDb (A υ, ). We say that .A is continuous on .S if .A is continuous at all .υ ∈ S . Lemma 1 ( [AlEtAl13]) Let .{ϑn } be a sequence in a b-metric-like space .(S , Db ) such that for some .λ, 0 < λ < 1s and each .n ∈ N. Then .{ϑn } is a Cauchy sequence in .S and. lim Db (ϑn , ϑm ) = 0. n,m→∞

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Remark 2 ( [AlEtAl13]) Let .(S , Db ) be a b-dislocated metric space with constant s ≥ 1. Then it is clear that

.

Dbs (υ, ϑ) = |2Db (υ, ϑ) − Db (υ, υ) − Db (ϑ, ϑ)|

.

satisfies .Dbs (υ, υ) = 0, for all .υ ∈ S . So it is considered to be a b-metric induced by b-dislocated metric spaces. Remark 3 ( [ChEtAl15]) Let .(S , Db ) be a b-dislocated metric space, and let .A : S → S be a continuous mapping. Then .

lim Db (υn , υ) = Db (υ, υ) ⇒ lim Db (f x, f υn ) = Db (υ, υ).

n→∞

n→∞

One may refer to [AlEtAl13] for additional terms and notations such as .Db completeness, continuity, and topology as well as the noteworthy observations in the corresponding space. Let .S be a non-void set; a graph .K = (U(K ), E(K )) is said to be associated with .S , whenever the set of vertices of the graph .K is equal to the set .S , that is, .S = U(K ). Let .Δ denotes the diagonal of .S × S , for a nonempty set .S . The set of edges of the underlying graph .K contains all the self loops on each of the vertex of .K , that is, .E(K ) ⊇ Δ. If the direction of edges of .K is reversed, then we denote the it by .K −1 . More investigation on this theme accompanied by notable observations and remarks can be seen in [YoEtAl21, YoEtAl22c].

28.2 Convergence Results The primary definition of this chapter is as follows. Definition 3 Let .K be a graph endowing a dislocated b-metric space .(S , Db ). A self mapping .A : S → S is said to be a graphical .χ -contraction in dislocated b-metric spaces if the following assumptions hold: .(i) There exists .υ ∈ S such that .(υ, A υ) ∈ E(K ). .(ii) .A preserves edges of .K . .(iii) For all .υ, ϑ ∈ S , where .a, b, c ∈ [0, 1] (not all zero simultaneously), such that .a + b + c ≤ 1, .χ ∈ X , Db (A υ, A ϑ) > 0 ⇒

.

   b c τ + F(Db (A υ, A ϑ)) ≤ F χ aDb (υ, ϑ) + Db (υ, A ϑ) + Db (ϑ, A υ) . 2s 2s (28.1) 

The unique fixed point result using graphical .χ -contraction is as follows.

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Theorem 1 Let .(S , Db ) be a complete b-dislocated metric space and .A a continuous graphical .χ -contraction. If .Db (A υ, A υ) ≤ Db (υ, υ), then .A has a unique fixed point in .S . Proof To start with, since .A is a graphical .χ -contraction, there exists .υ0 ∈ S such that .(υ0 , A υ0 ) ∈ E(K ). Define a sequence .{υn } ∈ S by the following υn+1 = A υn ,

.

n ∈ N.

(28.2)

If there exists .n0 ∈ N such that Db (υn0 , υn0 +1 ) = 0,

.

then .υn0 is the required fixed point of .A ; hence the proof is complete in this case. Consequently, we suppose .Db (υn , υn+1 ) > 0, for all .n ∈ N. Since .A preserves the edges of .K , there exists .υn , υn+1 ∈ S such that 0 < Db (υn , υn+1 ) = Db (A υn−1 , A υn ),

.

∀n ∈ N,

(28.3)

and by (28.1) we obtain τ + F(Db (A υn−1 , A υn ))   b c ≤ F χ (aDb (υn−1 , υn ) + Db (υn−1 , A υn ) + Db (υn , A υn−1 ) . 2s 2s (28.4)

.

Now, we show that Db (υn , A υn ) < Db (υn−1 , A υn−1 ), ∀n ∈ N.

.

(28.5)

Assume to the contrary that there exists .n0 ∈ N, such that Db (υn0 , A υn0 ) ≥ Db (υn0 −1 , A υn0 −1 ).

.

Thanks to (28.1), we obtain τ + F(Db (υn0 , A υn0 )) = τ + F(Db (A υn0 −1 , A υn0 ))   b c ≤ F χ (aDb (υn0 −1 , υn0 ) + Db (υn0 −1 , A υn0 ) + Db (υn0 , A υn0 −1 ) 2s 2s  b ≤ F χ (aDb (υn0 −1 , υn0 ) + (sDb (υn0 −1 , υn0 ) + sDb (υn0 , A υn0 )) 2s  c + (sDb (υn0 , υn0 −1 ) + sDb (υn0 −1 , A υn0 −1 )) 2s

.

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 b = F χ (aDb (υn0 −1 , A υn0 −1 ) + (Db (υn0 −1 , A υn0 −1 ) + Db (υn0 , A υn0 )) 2  c + (Db (A υn0 −1 , υn0 −1 ) + Db (υn0 −1 , A υn0 −1 )) 2   ≤ F χ (aDb (υn0 −1 , A υn0 −1 ) + bDb (υn0 , A υn0 ) + cDb (υn0 −1 , A υn0 −1 )) . Taking into account the properties of .X and .(F 1), we infer D(υn0 , A υn0 ) < aDb (υn0 −1 , A υn0 −1 )+bDb (υn0 , A υn0 )+cDb (υn0 −1 , A υn0 −1 ); (28.6) this implies that

.

Db (υn0 , A υn0 )
0. Using the same technique as previously mentioned, we infer that τ + F(Db (A υn−1 , A υn ))   ≤ F (aDb (υn−1 , A υn−1 ) + bDb (υn−1 , A υn−1 ) + cDb (υn−1 , A υn−1 ) .

.

Letting .n → ∞ and utilizing .(F 3 ), we have τ + F(l) ≤ F((a + b + c)l).

.

This is a contradiction, in view of the properties of .(F 1) and the fact that .a+b+c ≤ 1. So, we must have .l = 0, i.e., .

lim Db (υn , A υn ) = 0.

n→∞

(28.7)

In order to prove .{υn } is a Cauchy sequence, we shall prove that .

lim Db (υn , υm ) = 0.

n,m→∞

(28.8)

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On the contrary, suppose that there exists . > 0 and sequences .{pn } and .{qn } on natural number such that .l(n) > q(n) > n, Db (υl(n) , υq(n) ) ≥  and Db (υl(n)−1 , υq(n) ) < , ∀n ∈ N.

(28.9)

.

From the property of b-dislocated metric spaces, we obtain Db (υl(n)−1 , υq(n)−1 ) < sDb (υl(n)−1 , υq(n) ) + Db (υq(n) , υq(n)−1 ) < sDb (υq(n) , υq(n)−1 ) + s

.

(28.10)

= sDb (A υq(n)−1 , υq(n)−1 ) + s, ∀n ∈ N. From (28.7), there exist .N1 ∈ N, such that Db (υl(n)−1 , A υl(n)−1 ) < , Db (υq(n)−1 , A υq(n)−1 ) < , ∀n ∈ N1 .

.

(28.11)

From (28.10) and (28.11), we get Db (υl(n)−1 , υq(n)−1 ) < 2s, ∀ n ∈ N1 ;

.

(28.12)

consequently, we acquire F(Db (υl(n)−1 , υq(n)−1 )) < F(2s), ∀ n ∈ N1 .

.

(28.13)

Now, from (28.9), we have  ≤ Db (υl(n) , υq(n) ) = Db (A υl(n)−1 , A υq(n)−1 ).

.

Utilizing (28.1), one can obtain τ + F(Db (A υl(n)−1 , A υq(n)−1 ))  b ≤ F χ (aDb (υl(n)−1 , υq(n)−1 ) + Db (υl(n)−1 , A υq(n)−1 ) 2s  c + Db (υq(n)−1 , A υl(n)−1 )) 2s  b Db (υl(n)−1 , υq(n)−1 ) ≤ F χ (aDb (υl(n)−1 , υq(n)−1 ) + 2 +Db (υq(n)−1 , A υq(n)−1 )  c D(υq(n)−1 , υl(n)−1 ) + Db (υl(n)−1 , A υl(n)−1 ) , ∀n ∈ N1 . + 2

.

Taking into consideration (28.11), (28.12), (28.13), and .χ , we obtain

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c b τ + F(Db (A υl(n−1) , A υq(n−1) )) < F[a(2s) + (2s + ) + (2s + )] 2 2 . b c ⇒ Db (A υl(n−1) , A υq(n−1) ) < a(2s) + (2s + ) + (2s + ). 2 2 This asserts that . lim Db (A υl(n)−1 , A υq(n)−1 ) = 0. Hence . lim Db (υl(n) , υq(n) ) n→∞ n→∞ .= 0, which is a contradiction with (28.9). Thus the validation of (28.8) is acquired. Therefore .{υn } is a Cauchy sequence in .S . Since .(S , Db ) is a complete, there exists .w ∈ S , such that Db (w, w) = lim Db (υn , w) =

.

n→∞

lim Db (υn , υm ) = 0.

n,m→∞

A being continuous implies

.

Db (A w, A w) = lim Db (A υn , A w) = lim Db (υn+1 , A w) = 0.

.

n→∞

n→∞

Since .D(A w, A w) ≤ D(w, w), we have . lim Db (υn , A w) = 0. n→∞

Also, .Db (w, A w) ≤ s [Db (w, υn ) + Db (υn , A w)]; letting .n → ∞, we obtain that .Db (w, A w) = 0. Thus .w = A w and so .A has a fixed point. In order to show the uniqueness of fixed point, suppose .μ be another fixed point such that .w = μ. Then we have Db (A w, A u) > 0 ⇒

.

F(Db (w, μ)) =F(Db (A w, A μ)) 

 Db (μ, A w)  Db (w, A μ) +c −τ ≤ F χ aDb (w, μ) + b . 2s 2s

 c b + )Db (w, μ) − τ ≤ F (a + 2s 2s This is contradiction, in view of F 1. Thus we have .w = μ. Hence .A has a unique fixed point. This completes the proof. In order to illustrate our result, we present the following example. Example 2 Let .S = [0, ∞), and let the function .Db : S × S → [0, ∞) be defined by .Db (υ, ϑ) = |υ − ϑ|2 + υ 2 + ϑ 2 , for all .υ, ϑ ∈ S . It is obvious that .(S , Db ) is a complete b-dislocated metric space with .s = 2. Let the mapping .A : S → S be defined by υ2

Aυ =

.

e 2+υ υ2

e 2+υ + 1

.

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Define .K = (U(K ), E(K )) by .U(K ) = S and E(K ) = {(υ, ϑ) ∈ S × S : υ ≥ ϑ}.

.

3 3 and c = 26 ; then .a + b + c ≤ 1 We verify the condition (28.1) with .a = 34 , b = 25 + and .F(l) = l + log l, for all .l ∈ R . It is not hard to see that .F ∈ ΞF and .Db (A υ, A ϑ) > 0, for all υ, ϑ ∈ S . Consider .χ : [0, ∞) → [0, ∞) given by .χ (l) = 4l . The following are the calculations for the various terms involved in (28.1):

2 ⎛ ⎞2 ⎞2 ⎛ ϑ2 υ2 ϑ2 υ2 2+ϑ 2+υ 2+ϑ e 2+υ e +⎝ e ⎠ ; ⎠ +⎝ e Db (A υ, A ϑ) = υ 2 − ϑ2 ϑ2 υ2 e 2+υ + 1 e 2+ϑ + 1 e 2+ϑ + 1 e 2+υ + 1

.

Db (υ, ϑ) =|υ − ϑ|2 + υ 2 + ϑ 2 ; 2 ⎛ ⎞2 ϑ2 ϑ2 2+ϑ 2+ϑ e e 2 +υ +⎝ ⎠ ; Db (υ, A ϑ) = υ − ϑ 2 ϑ2 2+ϑ 2+ϑ e +1 e +1 ⎛ ⎞2 2 υ2 υ2 2+υ 2+υ e e + ϑ2 + ⎝ ⎠ . Db (ϑ, A υ) = ϑ − υ 2 υ2 2+υ 2+υ e +1 e +1

Making use of the above calculations, L.H.S. of (28.1) is as follows: 2 ⎛ ⎞2 ⎞2 ⎛ ϑ2 υ2 ϑ2 υ2 2+ϑ 2+υ 2+ϑ e 2+υ e e e +⎝ ⎠ ⎠ +⎝ τ + υ 2 − ϑ2 ϑ2 υ2 e 2+υ + 1 e 2+ϑ e 2+ϑ + 1 e 2+υ + 1 + 1 ⎡ . 2 ⎛ ⎞2 ⎤ ⎞2 ⎛ υ2 ϑ2 υ2 ϑ2 2+ϑ 2+υ 2+ϑ 2+υ e ⎢ e +⎝ e ⎠ ⎥ ⎠ +⎝ e + log ⎣ υ 2 − ϑ2 ⎦; ϑ2 υ2 e 2+υ + 1 e 2+ϑ + 1 e 2+ϑ + 1 e 2+υ + 1 and the R.H.S. is acquired as ⎧ ⎪ 1⎨

⎛

2 ⎞ ϑ2 e 2+ϑ ⎟ b⎜ 2 ⎝ e 2 2 2 ⎠ + υ − ϑ 2 . a(υ + ϑ + |υ − ϑ| ) + ⎝υ + ⎠ ϑ2 4⎪ 4 ⎩ e 2+ϑ + 1 e 2+ϑ + 1 ⎛ 2 ⎞⎫ ⎛ ⎞2 ⎪ υ2 υ2 e 2+υ ⎟⎬ c ⎜ 2 ⎝ e 2+υ ⎠ + ϑ − υ 2 + ⎝ϑ + ⎠⎪ υ2 4 e 2+υ + 1 e 2+υ + 1 ⎭ ⎛

ϑ2 2+ϑ

⎞2

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Fig. 28.1 Domination of R.H.S. over L.H.S. of (28.1)

⎛ ⎡ ⎧ 2 ⎞2 ⎞ ⎛ ⎪ ϑ2 ϑ2 ⎨ 2+ϑ 2+ϑ e b⎜ ⎢1 +υ 2 + ⎝ e ⎠ ⎟ a(|υ − ϑ|2 +υ 2 +ϑ 2 )+ ⎝ υ − ϑ 2 + log ⎣ ⎠ ϑ2 4⎪ 4 ⎩ e 2+ϑ + 1 e 2+ϑ + 1 ⎛ 2 ⎛ ⎞2 ⎞⎫⎤ ⎪ υ2 υ2 e 2+υ e 2+υ ⎠ ⎟⎬⎥ c ⎜ 2 ⎝ + ⎝ ϑ − υ 2 ⎠ ⎦. +ϑ + υ2 ⎪ 4 ⎭ e 2+υ + 1 e 2+υ + 1 Figure 28.1 shows clearly that the surface symbolizing right-side function is dominating over the surface symbolizing left-side function. As a result, condition (28.1) is confirmed. Furthermore, .A is continuous, and we also have that .Db (A υ, A υ) ≤ Db (υ, υ) for all .υ ∈ S . As a result, all of the assumptions of Theorem 1 are contended, and the mapping .A has a unique fixed point .υ = 0.527495. Taking .b = c = 0 in Theorem 1, the following corollary is obtained as a consequence in the setting of b-dislocated metric spaces. Corollary 1 Let .(S , Db ) be a b-dislocated metric space and .A : S → S be a continuous self mapping. If there exists .τ > 0 and .F ∈ ΞF such that for all .υ, ϑ ∈ S , Db (A υ, A ϑ) > 0 ⇒

.

   τ + F(Db (A υ, A ϑ)) ≤ F χ aDb (υ, ϑ) ,

(28.14)

where .a ∈]0, 1] and .χ ∈ X . Then .A has a unique fixed point.

28.3 An Application to Integral Equations In this section, we obtain the solution of the subsequent integral equation for an unknown function .κ:

28 An Existence Result for a Class of Integral Equations b

κ(l) = p(l) +

357

R(l, ς )Υ (ς, κ(ς ))dς,

.

l ∈ [a, b],

(28.15)

a

where .κ : R+ → R is a non-decreasing function, .Υ : [a, b] × R+ → R is a non-increasing continuous function, and .R : [a, b] × [a, b] → [0, ∞) is a nondecreasing continuous function. Let .p : [a, b] → R be given continuous function. Furthermore, let the graph .K = (U(K ), E(K )) be such that U(K ) = S and E(K ) = {(κ, μ) ∈ S × S : μ ≥ κ}.

.

Let .S be the set .C[a, b] of real continuous functions on .[a, b], and let .Db : S × S → [0, ∞) be given by Db (κ, μ) = max |κ(l) − μ(l)|2 ,

.

a≤l≤b

(28.16)

One can easily see that .(S , Db ) is a complete b-dislocated metric space. Let the mapping .A : S → S is defined by b

A κ(l) = p(l) +

.

R(l, ς )Υ (ς, κ(ς ))dς,

l ∈ [a, b],

(28.17)

a

then .κ(l) is a solution of (28.15) if and only if it is a fixed point of .A . To demonstrate the existence of an integral equation solution, we prove the following theorem. Theorem 2 Assume that the following statements are true: (1) b .

max

a≤l≤b a

|R(l, ς )|2 dς ≤

1 ; b−a

(2) For all .υ, ϑ ∈ R+ , the following inequality holds: |Υ (ς, υ) − Υ (ς, ϑ)|2 ≤

.

1 |υ − ϑ|2 e−τ . 2

Then the integral equation (28.15) has a solution. Proof Taking into account the integral equation (28.15) and the conditions (1) and (2), we have

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Db (A κ1 , A κ2 )

.

= max |A κ1 (l) − A κ2 (l)|2 a≤l≤b

= max p(l)+ a≤l≤b

! = max

b

a≤l≤b

≤ max

a≤l≤b

= ≤ ≤

b

a

1 b−a

 2 R(l, ς )Υ (ς, κ2 (ς ))dς

2 " Υ (ς, κ1 (ς )) − Υ (ς, κ2 (ς )) dς

"! |R(l, ς )|2 dς .

b a

"! .

1 2(b − a)

b a

a≤l≤b a

a≤l≤b

a

b

  2 " R(l, ς )Υ (ς, κ1 (ς )) − R(l, ς )Υ (ς, κ2 (ς )) dς

|R(l, ς )|2 dς.

b

max

!

 R(l, ς )Υ (ς, κ1 (ς ))dς − p(l)+

a

!

= max =

a

!

b

$ " 1# (|κ1 (ς ) − κ2 (ς )|2 )e−τ dς 2

b a b

2 " Υ (ς, κ1 (ς )) − Υ (ς, κ2 (ς )) dς

max [|κ1 (l) − κ2 (l)|2 e−τ ]dς

a a≤l≤b

1 |κ1 (l) − κ2 (l)|2 e−τ 2

1 Db (κ1 , κ2 )e−τ . 2

Hence, we acquire Db (A κ1 , A κ2 ) ≤

.

Db (κ1 , κ2 )e−τ . 2

As a result, by switching to logarithms, we obtain  Db (κ1 , κ2 )  . 2     Consequently, we obtain .τ + F Db (A κ1 , A κ2 ) ≤ F χ (Db (κ1 , κ2 )) , for .F(l) = log l, l > 0, .a = 1 and .χ (l) = 0.5. Thus, all the conditions of Corollary 1 are satisfied. Hence, we conclude that .A has a unique fixed point .κ ∗ in .S . Which yields, integral equation (28.15) has a unique solution which belongs to .S = C[a, b]. τ + log(Db (A κ1 , A κ2 )) ≤ log

.

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References [AlEtAl13] Alghamdi, M.A., Hussain, N., Salimi, P.: Fixed point and coupled fixed point theorems on b-metric-like spaces. J. Inequal. Appl. 402, 1–25 (2013). https://doi. org/10.1186/1029-242X-2013-402 [ChEtAl15] Chen, C.F., Dong, J., Zhu, C.X.: Some fixed point theorems in b-metric-like spaces. Fixed Point Theory Appl. 2015, 1–10 (2015). https://doi.org/10.1186/s13663-0150369-3 [Cz93] Czerwik, S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1(1), 5–11 (1993) [DvEtAl11] Dvorsky, K., Gwinner, J., Liess, H.D.: A fixed point approach to stationary heat transfer in electric cables. Math. Model. Anal. 16(2), 286–303 (2011) [Ha12] Harandi, A.A.: Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory Appl. 2012, 1–10 (2012). https://doi.org/10.1186/1687-1812-2012-204 [Ja08] Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Amer. Math. Soc. 136, 1359–1373 (2008) [Ku97] Kuznetsov, Y.A.: Mathematical modeling, nonlinear equations of evolution, and the dynamics of nuclear reactors. Comput. Math. Model. 8(1), 49–61 (1997) [NaEtAl17] Nashine, H.K., Agarwal, R.P., Shukla, S., Gupta, A.: Some fixed point theorems for almost (GF ; δb )-contractions and application. Fasciculi Mathematici. 58(1), 123–43 (2017) [Pi14] Piri, H., Kumam, P.: Some fixed point theorems concerning F -contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 210 (2014) [RaEtAl16] Radenovi´c, S., DoŠenovi´c, T., Lampert, T.A., Goluboví´c, Z.: A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations. Appl. Math. Comput. 273, 155–164 (2016) [Se13] Secelean, N.A.: Iterated function system consisting of F -contractions. Fixed Point Theory Appl. 2013, 277 (2013). https://doi.org/10.1186/1687-1812-2013-277 [SiEtAl17] Singh, D., Chauhan, V., Altun I.:, Common fixed point of a power graphic (F − ψ)contraction pair on partial b-metric spaces with application. Nonlinear Anal. Model. Control. 22(5), 662–678 (2017) [SiEtAl18] Singh, D., Chauhan, V., Kumam, P., Joshi. V.: Some applications of fixed point results for generalized two classes of Boyd-Wong’s F -contraction in partial b-metric spaces. Math. Sci. 12(2), 111–127 (2018) [Ve16] Vetro, F: F -contractions of Hardy-Rogers type and application to multistage decision processes. Nonlinear Anal. Model. Control 21(4), 531–546 (2016) [Wa12] Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012) [YoEtAl19] Younis, M., Singh, D., Goyal, A.: Solving existence problems via F -reich contraction. In: Constanda, C., Harris, P. (eds.) Integral Methods in Science and Engineering, pp. 451–463. Birkhäuser, Cham (2019) [YoEtAl21] Younis, M., Singh, D., Altun, I., Chauhan, V.: Graphical structure of extended bmetric spaces: an application to the transverse oscillations of a homogeneous bar. Int. J. Nonlinear Sci. Numer. Simul. (2021). https://doi.org/10.1515/ijnsns-2020-0126 [YoEtAl22a] Younis, M., Singh, D., Chen, L., Metwali, M.: A study on the solutions of notable engineering models. Math. Model. Anal. 27(3), 492–509 (2022) [YoEtAl22b] Younis, M., Stretenovi´c, A., Radenovi´c, S.: Some critical remarks on “some new fixed point results in rectangular metric spaces with an application to fractional-order functional differential equations”. Nonlinear Anal. Model. Control 27(1), 163–178 (2022) [YoEtAl22c] Younis, M., Singh, D., Shi, L.: Revisiting graphical rectangular b-metric spaces. Asian-Eur. J. Math. 15(04), 2250072 (2022)

Chapter 29

Some Convergence Results on the Periodic Unfolding Operator in Orlicz Setting J. F. Tachago, G. Gargiulo, H. Nnang, and E. Zappale

29.1 Introduction Homogenization of periodic structures via two-scale convergence in Orlicz setting was introduced in [FoEtAl12] and later expanded in [FoEtAl19, FoEtAl21, FoEtAl21] in order to deal with convex integral functionals, multiscale problems, and differential operators. With the aim of making two-scale convergence a convergence in a function space, the unfolding method has been introduced in [CiEtAl02, CiEtAl08, CiEtAl09, CiEtAl18]. The scope of this note is to introduce the unfolding operator in [CiEtAl02, CiEtAl08] to the Orlicz setting, having in mind that these spaces generalize classical p .L spaces and capture more information than those available in the latter setting. Let .m, d be positive integers, .Y = ]0, 1[d , and let .A0 be the class of all bounded open subsets of .Rd with Lipschitz boundary, and let B be an .N −function satisfying .∇2 and .2 conditions (cf. Sect. 29.2.1 for definitions). We intend for each .Ω ∈ A0 (Rs ), i.e., .Ω bounded open subset of .Rd , with Lipschitz boundary, and for every sequence .{ε} ⊆ ]0, +∞[ converging to .0, to

J. F. Tachago University of Bamenda, Bamenda, Cameroon G. Gargiulo University of Sannio, Benevento, Italy e-mail: [email protected] H. Nnang University of Yaounde I, Yaounde, Cameroon E. Zappale () Sapienza University of Rome, Rome, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_29

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define the unfolding operator in .LB (Ω; Rm ) and make a parallel with the analogous notion and convergence in .Lp (Ω; Rm ). Since we can argue in components, we will assume in the sequel, without loss of generality that .m = 1. To this end, in Sect. 29.2 we deal with preliminaries on Orlicz spaces, while the remaining of the paper is devoted to define the unfolding method in this framework and prove some convergence results, establishing parallels with the strong and weak convergence in .LB (Ω × Y ) and .LB (Ω) between unfolded sequences and their original counterparts, thus extending to the Orlicz setting the result proved by Cioranescu et al. [CiEtAl02, CiEtAl08, CiEtAl09, CiEtAl18], leaving the parallel with the notion of two-scale convergence in the standard and Orlicz setting (cf. [Ng89, Al92, LuEtAl02, V06, FoEtAl12]), as well as the applications and the extension to more scales and to higher-order derivatives as in [AlEtAl96, FoEtAl03, LuEtAl09, FoEtAl19, FoEtAl21] and [FoEtAl22] for a forthcoming paper. Indeed our main theorem is the following: Theorem 1 Let B be an N-function satisfying .∇2 and .Δ2 conditions. The following results hold: (i) For .w ∈ LB (Ω × Y ) , Tε (w) → w strongly in .LB (Ω × Y ). (ii) Let .{wε }ε be a sequence in .LB (Ω) such that .wε → w strongly in .LB (Ω) and then .Tε (wε ) → w strongly in .LB (Ω × Y ) . (iii) For every relatively weakly compact sequence .{wε }ε in .LB (Ω) the corresponding .{Tε (wε )}ε is relatively weakly compact in .LB (Ω × Y ) . Furthermore, if .Tε (wε )  w  weakly in .LB (Ω × Y ) , then .wε  MY ( w ) weakly in B .L (Ω). (iv) If .Tε (wε )  w  weakly in .LB (Ω × Y ) , then .

 wLB (Ω×Y ) ≤ lim inf (1 + |Y |) wε LB (Ω) . ε→0

29.2 Notation and Preliminaries In what follows X and V denote a locally compact space and a Banach space, p respectively, and the spaces .Lp (X; V ) and .Lloc (X; V ) (X provided with a positive p Radon measure) are denoted by .Lp (X) and .Lloc (X), respectively, when .V = R. We refer to [FoEtAl07] for integration theory and more details. In the sequel we denote by Y the cube .]0, 1[d . The family of open subsets in .Rd will be denoted by .A (Rd ), and its subsets constituted by bounded sets with Lipschitz boundary are denoted by .A0 (Rd ).

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For any .N ∈ N, and any subset E of .RN , by .E, we denote its closure in the  its interior, and by .∂E its topological boundary. For any set relative topology, by .E  1 in E, . E, its characteristic function .χE is defined as . 0 otherwise. For every .x ∈ Rd , we denote by .[x] its integer part, namely, the vector in .Zd , which has as components the integer parts of the components of x. By .L d we denote the Lebesgue measure in .Rd ; hence for any .E ⊂ Rd , .L d (E) denotes its measure, but for the sake of shortening the notation, when .E = Y , we will adopt the notation .|Y | instead of .L d (Y ).

29.2.1 Orlicz Spaces Let .B : [0, +∞[ → [0, +∞[ be an .N−function (see [Ad75]), i.e., B is continuous, B(t) convex, with .B (t) > 0 for .t > 0, B(t) t → 0 as .t → 0, and . t → ∞ as .t → ∞. t Equivalently, B is of the form .B (t) = 0 b (τ ) dτ, where .b : [0, +∞[ → [0, +∞[ is nondecreasing, right continuous, with .b (0) = 0, b (t) > 0 if .t > 0 and .b (t) → +∞ if .t → +∞.  the complementary .N−function of B defined by We denote by .B,  = sup {st − B (s) , t ≥ 0} . B(t)

.

s≥0

It follows that .

tb(t) ≥ 1 (or > if b is strictly increasing), B(t)  B(b(t)) ≤ tb(t) ≤ B(2t) for all t > 0.

.

An .N-function B is of class .2 near .∞ (denoted by .B ∈ 2 ) if there are .α > 0 and t0 ≥ 0 such that

.

B (2t) ≤ αB (t)

(29.1)

.

for all .t ≥ t0 .  satisfy the .2 condition; In what follows every N-function B and its conjugate .B this latter property can be equivalently stated saying that B satisfies the .∇2 condition. In the sequel c refers to a positive constant, which, in principle, may vary from line to line. Let .Ω be a bounded open set in .Rd ; the Orlicz space 



L (Ω) = u : Ω → R measurable, lim

.

B

δ→0+ Ω

 B (δ |u (x)|) dx = 0

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is a Banach space with respect to the Luxemburg norm: .

    |u (x)| uB,Ω = inf k > 0 : dx ≤ 1 . B k Ω

It follows that .D (Ω) is dense in .LB (Ω), .LB (Ω) is separable and reflexive, the   dual of .LB (Ω) is identified with .LB (Ω) , and the norm on .LB (Ω) is equivalent B to .·B,Ω  . We will denote the norm of elements in .L (Ω), both by . · LB (Ω) and with . · B,Ω , the latter symbol being useful when we want emphasize the domain .Ω. Furthermore, it is also convenient to recall that:

  (i) . Ω u (x) v (x) dx ≤ 2 uB,Ω vB,Ω for .u ∈ LB (Ω) and .v ∈ LB (Ω)  

(ii) Given .v ∈ LB (Ω) the linear functional .Lv on .LB (Ω) defined by    B B .Lv (u) := ≤  Ω u (x) v (x) dx, u ∈ L (Ω) belongs to .L (Ω) with .vB,Ω Lv [LB (Ω)] ≤ 2 vB,Ω  1 B 1 (iii) The property .limt→+∞ B(t) t = +∞ implies .L (Ω) ⊂ L (Ω) ⊂ Lloc (Ω) ⊂ D (Ω) , each embedding being continuous.

For the sake of notations, given any .m ∈ N, when .u : Ω → Rm , we mean that each component .(ui ), of u, .(1 ≤ i ≤ m), lies in .LB (Ω), and we will denote the norm of u with the symbol .uLB (Ω)d := di=1 ui B,Ω . Given a function space S defined in Y , the subscript .Sper means that its elements are periodic in Y . In particular .Cper (Y ) denotes the space of periodic functions in d d d ∞ .C(R ), i.e., that verify .w(y + k) for .y ∈ R and .k ∈ Z . .Cper (Y ) := Cper (Y ) ∩ p p ∞ d C (R ). For every .p ≥ 1, .Lper (Y ) is the space of Y -periodic function in .Lloc (Rd ). B B d Analogously .Lper (Y ) is the space of Y -periodic functions in .Lloc (R ).

29.3 The Unfolding Operator Here and in the sequel, taking into account the above notation, we will consider Ω ∈ A0 (Rd ) and .Y :=]0, 1[d . Following [CiEtAl08], for .z ∈ Rd , [z]Y is the vector part with components .[zi ] where .[zi ] is theinteger   of .zi . It follows that .z − [z]Y = {z}Y ∈ Y. Then for each .x ∈ Rd , .x = xε Y + xε Y . Define   d .Ξε := ξ ∈ Z , ε (ξ + Y ) ⊂ Ω , ⎫ ⎧ ⎨  ⎬ ε := int Ω , (29.2) ε ξ +Y ⎩ ⎭

.

ξ ∈Ξε

ε . Λε := Ω\Ω

29 Unfolding in Orlicz Setting

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ε is the largest union of cells .ε(ξ + Y ) (with .ξ ∈ Zd ) included in .Ω, The set .Ω while .Λε is the subset of .Ω containing the parts from cells .ε(ξ + Y ) intersecting the boundary .∂Ω. We are in position of introducing the unfolding operator, i.e., we recall [CiEtAl08, Definition 2.1], since it acts on Lebesgue measurable functions. Definition 1 For .φ Lebesgue measurable on .Ω, the unfolding operator .Tε is defined as   x  ε × Y φ ε ε Y + εy a.e for (x, y) ∈ Ω .Tε (φ) (x, y) = 0 a.e for (x, y) ∈ Λε × Y. ε . Clearly .Tε (φ) is Lebesgue measurable in .Ω × Y and is 0 whenever x is outside .Ω Moreover, for every .v, w Lebesgue-measurable, Tε (vw) = Tε (v) Tε (w) .

.

Proposition 1 Let .f ∈ L1per (Y ) define the sequence .{fε }ε by .fε (x) := f for .x ∈ Rn ; then  Tε (fε |Ω ) (x, y) =

.

x ε

a.e

ε × Y, f (y) a.e for (x, y) ∈ Ω 0 a.e for (x, y) ∈ Λε × Y.

B If .f ∈ LB per (Y ), .Tε (fε |Ω ) → f strongly in .L (Ω × Y ) .    Proof For any .k > 0 such that . Y B |fk | (y) dy ≤ 1, observe that    ε × Y, 0 a.e for (x, y) ∈ Ω |Tε (fε |Ω )−f | y) = .B (x, |f (y)| k a.e for (x, y) ∈ Λε × Y. k Hence     |f | |Tε (fε |Ω ) − f | B . B (x, y) dxdy = (x, y) dxdy = k k Λε ×Y Ω×Y   |f | d L (Λε ) B (y) dy. k Y

Since .∂Ω is Lipschitz and bounded, .L d (Λε ) → 0 as .ε → 0, and recalling the definition of norm in .LB spaces, .Tε (fε |Ω ) → f strongly in .LB (Ω × Y ) . Proposition 2 For every N-function B, the operator .Tε is linear and continuous from .LB (Ω) to .LB (Ω × Y ). It results that    (i) . |Y1 | Ω×Y B(Tε (w))(x, y)dxdy = Ω B(w(x))dx − Λε B(w(x))dx  .=  B(w(x))dx. Ωε   1 (ii) . |Y | Ω×Y B(Tε (w))(x, y)dxdy ≤ Ω B(w(x))dx.

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(iii)

.

1 |Y | |

 Ω×Y

B(Tε (w))(x, y)dxdy −

 Ω

B(w(x))dx| ≤

 Λε

|B(w(x))|dx.

(iv) .Tε (w)LB (Ω×Y ) = wχΩε LB (Ω) .  ε , 1 in Ω with .χΩε = as in Sect. 29.2.1. In particular, for every .ε, 0 otherwise, .

Tε (w)LB (Ω×Y ) ≤ (1 + |Y |) wLB (Ω) .

(29.3)

Proof Arguing as in [CiEtAl08, (i) in Proposition 2.5], it results that 1 . |Y |







B(Tε (w))(x, y)dxdy = Ω×Y

B(w(x))dx − 

=

Ω

ε Ω

B(w(x))dx Λε

B(w(x))dx.

Indeed, by the definitions of unfolding operator .Tε and .Ωε , it results .

1 |Y |

 1 B(Tε (w))(x, y)dxdy = |Y | Ωε ×Y  1  B(Tε (w))(x, y)dxdy. |Y | {εξ +εY }×Y



B(Tε (w))(x, y)dxdy = Ω×Y

ξ ∈Ξε

(29.4) In particular .Tε (w)(x, y) = w(εξ + εy) is constant in x on each .{εξ + εY } × Y . Consequently, each summand in the above equality becomes 

 B(Tε (w))(x, y)dxdy = L (εξ + εY )

B(w(εξ + εy))dy

d

.

{εξ +εY }×Y

Y

 = ε |Y | d

εξ +εY



B(w)(εξ + εy)dy = |Y |

B(w)(x)dx. Y

(29.5) By summing over .ξ ∈ Ξε in (29.4) and exploiting (29.5), we obtain 1 . |Y |



 B(Tε (w))(x, y)dxdy = Ω×Y

ε Ω

B(w)(x)dx.

(29.6)

   Thus, using the same argument, for every .k > 0 such that . Ω B |w| (x) dx ≤ 1, k

29 Unfolding in Orlicz Setting



 B

.

Ω×Y

367

|Tε (w)| (1 + |Y |) k 

(x, y) dxdy

 |Tε (w)| 1 B (x, y) dxdy k (1 + |Y |) Ω×Y     |Tε (w)| |w| 1 B B ≤ (x, y) dxdy ≤ (x) dx. |Y | Ω×Y k k Ω

≤

Therefore .Tε (w)LB (Ω×Y ) ≤ (1 + |Y |) wLB (Ω) . Observe that (i), (ii), (iii), and (iv) follow by (29.6). From this result, in particular from (ii), it is possible to provide, as in the standard Lp setting (cf. [CiEtAl08, Proposition 2.6]), an unfolding criterion for integrals in the Orlicz setting. For the sake of a more complete parallel with the standard p setting, we recall the unfolding criterion for integrals, u.c.i. for shortness, as .L introduced in [CiEtAl08].  Proposition 3 If .{wε }ε is a sequence in .L1 (Ω) satisfying . Λε |wε | dx → 0 as .ε → 0, then   1 |wε | dx − Tε (wε ) dxdy → 0 . |Y | Ω×Y Ω .

as .ε → 0. This result justifies the following notation for integrals of unfolding operators. Indeed, if .{wε }ε is a sequence satisfying .u.c.i, we write  .

Tε

1 |wε | dx  |Y | Ω

 Tε (wε ) dxdy. Ω×Y

B Proposition  4 (u.c.i. in the Orlicz Setting) If .{wε }ε is a sequence in .L (Ω) satisfying . Λε B(wε )dx → 0 as .ε → 0, then

 B(wε )dx −

.

Ω

1 |Y |

 B (Tε (wε )) dxdy → 0 Ω×Y

as .ε → 0. Proof The result is a consequence of (iii) in Proposition 2. 

Proposition 5 Let .{uε }ε be a bounded sequence in .LB (Ω) and .v ∈ LB (Ω); then 

Tε

uε vdx 

.

Ω

1 |Y |

 Tε (uε ) Tε (v) dxdy. Ω×Y

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 Proof To obtain the stated result, it is enough to prove that . Λε |uε v| dx → 0 as .ε → 0. By Hoelder’s inequality in Orlicz setting (i.e., (i) in Sect. 29.2.1), it results that  

  

   

χΛ uε v dx ≤ 2 uε  B  |uε v| dx = . L (Ω) χΛε v LB(Ω) ≤ χΛε v LB(Ω) . ε Λε

Ω

   |χΛε v | = 0 a.e. in Since . lim χΛε (x) = 0, for a.e. .x ∈ Rd . Thus .∀k > 0, . lim B k ε→0    ε→0    | χΛε v | |v|    |v| dx ≤ 1. ≤ B .Ω. Moreover .∃k1 > 0 such that .B and . B Ω k1 k1 k1     |χΛε v | (x) dx = 0, By Lebesgue’s dominated convergence theorem . lim Ω B k 1 ε→0   that is . lim χΛε v LB(Ω) = 0. ε→0

As in the classical Lebesgue setting, we can define the mean value operator acting on .LB spaces. Definition 2 The mean value operator .MY : LB (Ω × Y ) → LB (Ω) is defined as follows:  1 w (x, y) dy .MY (w) (x) := |Y | Y for a.e. .x ∈ Ω and for every .w ∈ LB (Ω × Y ). Remark 1 As a consequence .MY (w)LB (Ω) ≤ |Y |−1 wLB (Ω×Y ) , for every B .w ∈ L (Ω × Y ) . Indeed, using Jensen’s inequality, we get for .k > 0,           w(x,y) w(x,y) |Y |MY (w) dxdy. dx = dy dx ≤ B B . B Ω Y Ω Y Ω k k k Hence .

MY (w)LB (Ω) ≤ |Y |−1 wLB (Ω×Y ) .

Theorem 1, which we restate here for the reader’s convenience, extends to the Orlicz setting the correspective one in classical .Lp spaces (cf. [CiEtAl08, Proposition 2.9]). To this end, we recall the convergence properties related to the unfolding operator when .ε → 0, i.e., for w uniformly continuous on .Ω, with modulus of continuity .mw , it is easy to see that .

sup

ε ,y∈Y x∈Ω

|Tε (w)(x, y) − w(x)| ≤ mw (ε).

Theorem 2 Let B be an N-function satisfying .∇2 and .Δ2 conditions. The following results hold: (i) For .w ∈ LB (Ω × Y ) , Tε (w) → w strongly in .LB (Ω × Y ). (ii) Let .{wε }ε be a sequence in .LB (Ω) such that .wε → w strongly in .LB (Ω) and then .Tε (wε ) → w strongly in .LB (Ω × Y ) .

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(iii) For every relatively weakly compact sequence .{wε }ε in .LB (Ω) the corresponding .{Tε (wε )}ε is relatively weakly compact in .LB (Ω × Y ) . Furthermore, if .Tε (wε )  w  weakly in .LB (Ω × Y ) , then .wε  MY ( w ) weakly in B .L (Ω). (iv) If .Tε (wε )  w  weakly in .LB (Ω × Y ) , then .

 w LB (Ω×Y ) ≤ lim inf (1 + |Y |) wε LB (Ω) .

(29.7)

ε→0

Proof (i) We start proving the result when .ϕ ∈ D(Ω). Thus it exists a compact set K such that .suppϕ ⊂ K, and there exists .ε0 > 0, .0 < ε < ε0 such that ε . Let .k > 0, and let us prove that .K ⊂⊂ Ω 

 B

.

Ω×Y

|Tε (ϕ (x)) (x, y) − ϕ (x)| dxdy → 0, k

as .ε → 0. For .0 < ε < ε0 , for every .x ∈ Λε we have that .ϕ (x) = 0,; hence |Tε (ϕ (x)) (x, y) − ϕ (x)| dxdy k Ω×Y   |Tε (ϕ (x) (x, y)) − ϕ (x)| = dxdy B k ε ×Y Ω    |ϕ (εξ + εy) − ϕ (x)| = dxdy B k {εξ +εY }×Y 



B

.

ξ ∈Ξε

≤

  {εξ +εY }×Y

ξ ∈Ξε

B



!

mϕ (ε)

k

dxdy.

Since .mϕ1 (ε) → 0, as .ε → 0, there exists .0 < ε1 , ε < ε1 → mϕ1 (ε) < 1. Therefore, for .0 < ε < min (ε0 , ε1 ) , |Tε (ϕ (x)) (x, y) − ϕ (x)| dxdy B k Ω×Y    1 ≤ mϕ (ε) dxdy = B k {εξ +εY }×Y

 .

 mϕ (ε)

ε Ω



ξ ∈Ξε

 Y

   1 1 dy ≤ mϕ (ε) L d (Ω) B dy B k k Y ≤ L d (Ω) Cmϕ (ε) → 0,

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   as .ε → 0, since clearly k can be chosen such that . Y B k1 dy < ∞. The general case follows from the density of .D(Ω) in .LB (Ω) and from the estimates Tε (w) − wLB (Ω×Y ) = Tε (w) − Tε (φ) + Tε (φ) − φ + φ − wLB (Ω×Y )

.

≤ Tε (φ) − φLB (Ω×Y ) − C(1 − |Y |)φ − wLB (Ω×Y ) where .φ ∈ D(Ω). Observe that the inequality is a consequence of the linearity of .Tε and (29.3) in Proposition 2. Indeed, by the previous step, we have .

lim sup Tε (w) − w ≤ C(1 + |Y |)φ − wLB (Ω×Y ) , ε→0

which concludes the proof of (i). (ii) It is straightforward by the linearity of .Tε and (29.3) in Proposition 2. (iii) By (29.3) boundedness of .{Tε (wε )}ε is preserved. Moreover, taking into ˜ account that .wε  wˆ in .LB (Ω), for every .v ∈ LB (Ω), by Proposition 5, 

Tε

uε (x)v(x)dx 

.

Ω

1 |Y |

 Tε (uε ) (x, y)Tε (v) (x, y)dxdy. Ω×Y

Hence, passing to the limit as .ε → 0, one has  

 .

lim

ε→0 Ω

wε (x)v(x)dx = Ω

1 |Y |



 w(x, ˆ y)dy v(x)dx,

Y

which, by Definition 2, concludes the proof of (iii). (iv) For what concerns the proof of (29.7), it results that it is a consequence of the lower semicontinuity of the norm with respect to the weak convergence and (ii) in Proposition 2. Hence the proof is concluded.

References [Ad75] Adams, R.: Sobolev Spaces. Academic Press, New York (1975) [Al92] Allaire, G.: Homogenization and two scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992) [AlEtAl96] Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenization. Proc. Royal Soc. Edin. 126, 297–342 (1996) [CiEtAl02] Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335(1), 99–104 (2002)

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[CiEtAl08] Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008) [CiEtAl09] Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. Multiple scales problems in biomathematics, mechanics, physicsand numerics. GAKUTO Internat. Ser. Math. Sci. Appl. 31, 1–35 (2009) [CiEtAl18] Cioranescu, D., Damlamian, A., Griso, G.: The Periodic Unfolding Method. Series in Contemporary Mathematics, vol. 3. Theory and Applications to Partial Differential Problems, p. xv+513. Springer, Singapore (2018) [FoEtAl03] Fonseca, I., Zappale, E.: Multiscale relaxation of convex functionals. J. Convex Anal. 10(2), 325–350 (2003) [FoEtAl07] Fonseca, I., Leoni, G.: Modern methods in the calculus of variations: Lp spaces. Springer Monographs in Mathematics, p. xiv+599. Springer, New York (2007) [FoEtAl12] Fotso Tachago, J., Nnang, H.: Two-scale convergence of Integral functionals with convex, periodic and Nonstandard Growth Integrands. Acta Appl. Math, 121, 175– 196 (2012) [FoEtAl19] Fotso Tachago, J., Nnang, H., Zappale, E.: Relaxation of periodic and nonstandard growth integrals by means of two-scale convergence. In: Integral Methods in Science and Engineering, pp. 123–131. Birkhäuser/Springer, Cham (2019) [FoEtAl21] Fotso Tachago, J., Nnang, H., Zappale, E.: Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands. Opuscula Mathematica 41(1), 113–143 (2021) [FoEtAl21] Fotso Tachago, J., Gargiulo, G., Nnang, H., Zappale, E.: Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evol. Equations Control Theory 10(2), 297–320 (2021) [FoEtAl22] Fotso Tachago, J., Nnang, H., Zappale, E.: Reiterated homogenization of nonlinear degenerate elliptic operators with nonstandard growth. Differ. Integral Equ. (to appear). https://arxiv.org/abs/2110.02882 [LuEtAl02] Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2(1), 33-81 (2002) [LuEtAl09] Lukkassen, D., Nguetseng, G., Nnang, H., Wall, P.: Reiterated homogenization of nonlinear monotone operators in a general deterministic setting. J. Funct. Spaces Appl. 7(2), 121–152 (2009) [Ng89] Nguetseng, G.: A general convergent result for functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–623 (1989) [V06] Visintin, A.: Towards a two-scale calculus. ESAIM Control Optim. Calc. Var. 12(3), 371–397 (2006)

Chapter 30

Three-Phase Flow Zero-Net Liquid Holdup in Gas-Liquid Cylindrical Cyclone (GLCC© ) H. Zhao, R. S. Mohan, and O. Shoham

30.1 Introduction The petroleum industry has relied in the past on conventional separators to process production of oil, water, and gas. Conventional separators are gravity based and, thus, require a large retention time for separation of the production fluids. As a result, conventional separators are large, heavy, and expensive. An alternative to the conventional separator has always been considered by the industry in the form of compact separators. Development and application of compact multiphase cyclonic separators have started in the 1990s, owing to their smaller size, smaller footprint, lower weight and ease of construction, simplicity, and last but not least their economic advantage and lower cost. The Tulsa University Separation Technology Projects (TUSTP) research consortium has developed and implemented compact multiphase cyclonic separation technology for gas-oil-water-sand flow for over 28 years. Among several developed compact separators, the Gas-Liquid Cylindrical Cyclone (GLCC©1 ) is the most commonly used compact separator by the industry, which is utilized for a variety of applications. These include multiphase flow metering loop, offshore platform separation, full separation, partial separation, control of gas liquid ratio upstream of multiphase flow meters and pumps, equal splitting of multiphase flow upstream of separation trains, portable well test system, pre-separation upstream of conventional separators, and slug mitigation. To date, there are about 8600 GLCC compact separators deployed in the field in the USA and around the world. A schematic of the GLCC© is presented in Fig. 30.1. As can be seen, GLCC© body is simply a piece of pipe, mounted vertically with a downward inclined

H. Zhao · R. S. Mohan · O. Shoham () The University of Tulsa, Tulsa, OK, USA e-mail: [email protected]; [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_30

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Fig. 30.1 Schematic of the GLCC© compact separator

tangential inlet. The top of the GLCC© is connected to the gas leg, and the bottom part of the GLCC is connected to the liquid leg. The downward inclined inlet (at approximately .27◦ to horizontal) promotes stratification of the gas-liquid flow and, thus, helps with pre-separation of the flow prior to entering the GLCC. A nozzle, which has a cross-sectional area of 25% of the full-bore inlet pipe, is installed at the end of the inlet tangentially to the GLCC© wall to accelerate the incoming mixture. Thus, the mixture flow from the nozzle forming a swirling motion around the GLCC© inlet region wall. The swirling flow generates centrifugal forces that act on the gas and the liquid, whereby the heavier liquid phase is forced radially to the wall of the cylinder and moves downward due to gravity, while the gas is forced radially inward toward the center of the cyclone and flows upward toward its top. Finally, the liquid exits from the bottom of the GLCC© through the liquid leg, and the gas flows out through the gas leg. For efficient separation, the liquid level is maintained at about 6” below the inlet. The GLCC© operation is limited by two physical phenomena, namely, liquid carry-over (LCO) and gas carry-under (GCU). LCO occurs when some liquid is carried over into the gas leg by the gas. On the other hand, some gas can be carried under by the liquid into the liquid leg, which is designated as GCU. Both LCO and GCU could cause problems in downstream separation facilities, such as failure of pumps and compressors downstream of the GLCC© . Also, in the GLCC© multiphase flow metering loop, LCO and GCU can increase the uncertainty of gas and liquid meter measurements, respectively. When operating the GLCC© at or below the OPEN for LCO, some liquid is held in the GLCC© upper part in the form of Churn flow, while no liquid is produced from the top into the gas leg, as shown in Fig. 30.1. This phenomenon is designated as Zero-Net Liquid Flow (ZNLF), whereby the gas phase velocity is sufficient to entrain and lift some liquid into the upper part of the GLCC© , but is insufficient to push it into the gas leg. The associated liquid holdup (quantified measure of liquid entrainment in the gas) in the upper part of the GLCC© in the ZNLF Churn region

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is termed as Zero-Net Liquid Holdup (ZNLH). The maximum ZNLH occurs at the OPEN for LCO flow conditions where the Churn region is extended to the top of the GLCC, as presented in Fig. 30.1 (but no liquid is produced into the gas leg). The ZNLH plays an important role in the GLCC© design, as it controls the pressure balance in the GLCC© . Dynamic ZNLH experimental data for flow conditions at the GLCC Operational Envelope for Liquid Carry-Over (OPEN LCO) were acquired and reported by Karpurapu [Ka18a] and Karpurapu et al. [Ka18b]. For these operational conditions, the ZNLH is the maximum possible liquid entrainment as the Churn flow region extends all the way to the top of the GLCC, but no liquid is produced into the gas leg. For these conditions, Karpurapu [Ka18a] also developed a correlation for predicting the ZNLH for flow conditions at the GLCC OPEN for LCO based on the “Flooding and Flow Reversal” correlation by Wallis [Wa61] for pipe flow. The prediction of the developed correlation shows a good agreement with the acquired experimental data. An experimental and theoretical investigation on the variation of the ZNLH and the associated Churn region height in the GLCC for both air-oil and air-water was carried out by Shah [Sh18] and Shah et al. [Sh21]. Experimental data were acquired for air-oil and air-water flow at normal operating conditions below the GLCC OPEN for LCO. The correlation developed by Karpurapu et al. [Ka18b] was extended for predicting the variation of the ZNLH and the Churn region height for normal operating conditions below the GLCC OPEN for LCO. The correlation developed by Karpurapu [Ka18a] for predicting the ZNLH at the GLCC OPEN for LCO was extended for both gas-oil and gas-water flows, enabling predictions of the variation of the ZNLH and associated Churn region height along the upper part of the GLCC, for normal operating conditions below the OPEN for LCO. The studies of Karpurapu [Ka18a] Shah [Sh18] preceded the current study, whereby the same facility has been utilized in all the three studies. As can be seen, the previous GLCC ZNLH investigations were limited to gas-oil and gas-water twophase flow. No studies were conducted for three-phase gas-oil-water flow; this is the gap that the current study aims to bridge. Following are the objectives of the current study: 1. Acquire experimental data for three-phase gas-oil-water flow at the GLCC© OPEN for LCO operating conditions. The acquired data include the ZNLH, as well as the oil fraction in the zero-net liquid phase. 2. Extend the Karpurapu [Ka18a] correlation that was developed for the prediction of two-phase gas-liquid flow ZNLH to enable the prediction of the ZNLH and the oil fraction in the zero-net liquid phase in three-phase gas-oil-water flow. 3. Compare the extended ZNLH correlation with the acquired experimental data.

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30.2 Experimental Program The TUSTP experimental facility, which is located at The University of Tulsa North Campus, is a state-of-the-art fully instrumented four-phase flow loop, enabling experimental data acquisition for gas, oil, water, and solid particles. The following sections provide details of the flow loop and the GLCC test section.

30.2.1 GLCC© TUSTP Flow Loop TUSTP Flow Loop: A schematic of the TUSTP flow loop is presented in Fig. 30.2. The oil and water are stored in their respective 400-gallon tanks, and air is supplied by a compressor with flow capacity of 250 CFM and discharge pressure of 100 psi. Either the oil or the water is pumped from its respective tank into the flow loop, where the flows are controlled by control valves and metered by respective Coriolis mass flow meters. The three phases are mixed at a mixing-Tee, and the multiphase mixture flows into the GLCC test section.

30.2.2 GLCC© Test Section A schematic and a photograph of the GLCC test section are presented in Figs. 30.3 and 30.4, respectively. It is fabricated with schedule 80 acrylic transparent pipe. The GLCC body is a 3” ID 10-foot-tall vertical pipe, extending 65 in. above the inlet (GLCC upper part) and 55 in. below the inlet (GLCC lower part). The GLCC inlet (also 3” ID) is inclined downward at .−27◦ to horizontal to promote stratification and pre-separation. A nozzle is installed at the end of the inlet section with a 25% area reduction (of the inlet cross-sectional area), which diverts the flow tangentially to the GLCC wall. A 2” ID liquid leg and a 2” ID gas leg are installed at the bottom and top of the GLCC body, respectively. As shown in Figs. 30.3 and 30.4, a 0.75” ID transparent acrylic pipe, which is installed vertically in parallel to GLCC body, is utilized as a liquid level indicator. The level in the GLCC is also measured with a differential pressure transducer, Fig. 30.2 Schematic of experimental flow loop

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Fig. 30.3 Schematic of GLCC test facility

Fig. 30.4 Photograph of GLCC test section with ZNLH trap

which measures the pressure difference between the top and the bottom of the GLCC.

30.2.3 Trap Section for ZNLH Measurement The trap section for the ZNLH measurement is shown in Fig. 30.4 in the GLCC photograph on the right-hand side of the figure. As can be seen, an open/close trap valve is installed 15” above the GLCC inlet, which makes the entire upper part of the GLCC (above the valve) a trap section. The trap valve is shown in the expanded photograph on the right-hand side of Fig. 30.4. Once steady-state conditions are reached, where the ZNLF reaches the top of the GLCC, namely, the flow conditions corresponding to the onset of LCO (OPEN), the trap valve is closed, and the inlet flow is diverted through a bypass. Thus, the ZNLF in the GLCC upper part is trapped, whereby the liquid phase is drained downward and accumulated above the trap valve, while the gas moves upward and stays above the liquid phase. The total height, as well as the heights of the oil and liquid phases, is measured, enabling determination of the ZNLH and the volume fraction of the oil in the zero-net liquid phase.

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Test Matrix

Experiments are conducted for ZNLF in the GLCC upper part at the OPEN for LCO flow conditions. The acquired data for these conditions include the ZNLH and the oil fraction in the zero-net liquid phase as a function of the gas and liquid superficial velocities and the WC. Data are acquired at WCs of 0, 25, 50, 75, and 100% at the OPEN for LCO flow conditions for each WC, with superficial liquid velocities of 0.5, 1.75, 2, 2.25, and 2.5 ft/s, with respective values of superficial gas velocities at the OPEN for LCO. The working fluids utilized in this study are air, tap water, and Tulco Tech 80 mineral oil.

30.2.4 Repeatability Runs Data are acquired and compared with Karpurapu [Ka18a] and Shah [Sh18] data that were taken at the operational envelope just before Liquid Carry-Over (LCO) occurs. Comparisons are presented for both the OPEN for LCO and the ZNLH at these conditions. Figures 30.5 and 30.6 show, respectively, the repeatability comparisons for the OPEN for LCO, as well as the ZNLH for air-oil flow, showing a good agreement.

Fig. 30.5 Repeatability runs for air-oil flow (OPEN for LCO)

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Fig. 30.6 Repeatability runs for air-oil flow ZNLH

30.3 Experimental Results This section presents the acquired experimental data for three-phase ZNLF for the different WCs. These include the OPEN for LCO, ZNLH, and the in situ volume fraction of the water in the ZNLH liquid phase, as given below.

30.3.1 Operational Envelope The operational envelopes for 0% WC (air-oil), 50%, 75%, and 100% WC (airwater) are presented in Fig. 30.7. In reality, the figure represents the experimental test matrix as it shows the combinations of gas and liquid superficial velocities under which the ZNLH data are acquired. The operational envelope for 100% WC is the highest, namely, occurring at the highest superficial gas velocity for a given superficial liquid velocity. The opposite occurs for 0% WC, for which the operational envelope is the lowest. This is owing to the difference in densities. As the water is heavier, it takes a higher superficial gas velocity to bring the liquid to the top of the GLCC, for a given superficial liquid velocity. The operational envelope for the 75% WC is very close to the 100% WC owing to the high-water fraction in the liquid phase, while the 50% WC operational envelope is in between the 0% and 100% WCs.

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Fig. 30.7 Operational envelope for various water cuts

Fig. 30.8 Zero-net liquid holdup for various water cuts

30.3.2 Zero-Net Liquid Holdup (ZNLH) Figure 30.8 provides the most important experimental results, namely, the acquired ZNLH data for different WCs of 0% (air-oil), 50%, 75%, and 100% (air-water). This data exhibits an opposite trend to the operational envelope results. For the ZNLH case, the results for the 0% WC are the highest, while the results are the lowest for the 100% case, at the same superficial gas velocity. This is owing to the higher oil viscosity, which results in a higher drag force with the oil phase that results in

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Fig. 30.9 Water volume fraction in ZNLH liquid phase

retaining more oil in the upper part of the GLCC. As can be seen in Fig. 30.8, at constant WC, the ZNLH increases as the gas superficial velocity decreases, since the data are acquired at the operational envelope flow conditions, where the liquid superficial velocity increases as the gas superficial velocity decreases, along the OPEN.

30.3.3 Water Volume Fraction in ZNLH Liquid Phase The in situ volume fractions of the oil and water in the zero-net liquid phase are very close to the corresponding GLCC inlet values, as shown in Fig. 30.9. This is owing to the flow pattern, occurring in the upper part of the GLCC, namely, Churn flow. The chaotic and oscillatory flow behavior in Churn flow keeps the oil and water well mixed, maintaining the inlet volume fractions in the ZNLF.

30.4 Comparison Study 30.4.1 Three-Phase Flow Zero-Net Liquid Holdup Correlation A new correlation is proposed for the prediction of the three-phase flow ZNLH, HZNL , as well as the oil fraction in the zero-net liquid phase, .fOZ , at the GLCC OPEN for LCO flow conditions. The Karpurapu [Ka18a] correlation for the prediction of ZNLH is based on the Wallis [Wa61] flooding and flow reversal model. The original model was modified by Karpurapu [Ka18a] to predict the

.

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GLCC© ZNLH at the OPEN for LCO flow conditions. The modified correlation was developed for gas-oil flow and gas-water flow and was validated with acquired data for these conditions. In this study, the Karpurapu [Ka18a] correlation is extended and validated for gas-water-oil three-phase ZNLF in the GLCC. The set of input inlet flow conditions includes the pipe diameter; superficial velocities at the OPEN for LCO, namely, .νSG , .νSO , and .νSW , where .νSL = νSO + νSW ; and the fluid properties of the gas, oil, and water. For the given operational flow conditions, the liquid phase properties are averaged based on no-slip conditions between oil and water, as follows: .

ρL = ρW fW C + ρO (1 − fW C )

(30.1)

μL = μW fW C + μO (1 − fW C )

(30.2)

σL = σW fW C + σO (1 − fW C )

(30.3)

.

.

where fW C =

.

νSW νSO + νSW

(30.4)

Following Karpurapu [Ka18a], the proposed extended correlation is given by .

 ∗ 0.5  ∗ 0.5 + νSG =1 νSL

(30.5)

∗ and .ν ∗ are defined, respectively, by The dimensionless parameters .νSG SL ∗ 0.5 νSG = νSG ρG [gd (ρL − ρG )]−0.5

(30.6)

∗ 2n νSL = νSLZ ρLn [gd (ρL − ρG )]−n

(30.7)

.

.

where .νSLZ is the associated flooding superficial oil velocity for ZNLF conditions and the exponent n is given as  n=

.

√ 0.25 25μL gd σL

(30.8)

Finally, the zero-net liquid holdup at the OPEN for LCO flow conditions is calculated by HZN L =

.

νSLZ νSLZ + νSG

(30.9)

30 Three-Phase Flow Zero-Net Liquid Holdup

383

30.4.2 Calculation Procedure For a given set of input flow conditions, the following procedure is followed to solve for the three-phase flow ZNLH at OPEN for LCO conditions: 1. 2. 3. 4. 5.

∗ from (30.6) for a given inlet .ν . Determine .νSG SG ∗ . Substitute into (30.5) and solve for .νSL Solve for .νSLZ from (30.7) using n from (30.4). Determine .HZN L from (30.9). The in situ volume fractions of oil and water in the ZNLH are the respective inlet fractions of the phases (no-slip conditions in the liquid phase).

30.4.3 Comparison Between Correlation Prediction and Experimental Data The predictions of the developed three-phase flow ZNLH correlation are tested against the acquired experimental data at the OPEN flow conditions. Figure 30.10 is a plot presenting an overall comparison between the predictions of the extended correlation and the experimental data. As can be seen, the predictions follow closely the experimental data trend for all the WCs, namely, that the ZNLH increases as the gas superficial velocity deceases. This is owing to the fact that the data are acquired at the OPEN flow conditions, where the liquid superficial velocity increases as the gas superficial velocity decreases along the OPEN. On the other hand, the extended correlation also follows the trend when decreasing the WC from 100% to 0%, at the same superficial gas velocity. As the WC is reduced (at the same superficial gas velocity), the ZNLH increases owing to the higher oil viscosity, which results in a higher drag force with the oil phase and is the highest for 0% WC.

30.5 Summary and Conclusions Summary and conclusions of the current study are given below. • Novel experimental data for Zero-Net Liquid Flow (ZNLF) in the upper part of the GLCC are acquired for three-phase flow, with water cuts of 0%, 50%, 75%, and 100%. The data include the Zero-Net Liquid Holdup (ZNLH) and the corresponding volume fractions of the oil and water in the liquid phase. The data are acquired at the GLCC Operational Envelope (OPEN) for liquid carry-over (LCO) flow conditions. • At a constant WC, the ZNLH increases as the gas superficial velocity decreases, since the data are acquired at the OPEN flow conditions, where the liquid

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Fig. 30.10 Comparison between extended correlation predictions and experimental data

•

•

•

•

•

•

superficial velocity increases as the gas superficial velocity decreases, along the OPEN. The ZNLH increases as the WC decreases, at the same superficial gas velocity, owing to the higher oil viscosity, which results in a higher drag force with the oil phase, and is the highest for 0% WC. The volume fractions of the oil and water in the zero-net liquid phase are very close to the corresponding GLCC inlet values. This is owing to the Churn flow pattern occurring in the upper part of the GLCC, which keeps the oil and water well mixed, maintaining the inlet volume fractions in the ZNLF. The correlation developed by Karpurapu [Ka18a] for prediction of ZNLH for two-phase flow is extended to three-phase flow, by utilizing average liquid phase properties based on the WC. The discrepancies between the extended correlation predictions and experiment data are very small, demonstrating the excellent performance of the correlation. The discrepancies are 2%, 1%, 0.7%, and 2.8%, respectively, for 0%, 50%, 75%, and 100% WC. The discrepancies for 0% and 100% WCs, respectively, for air-oil and air-water flows, are higher than those of the three-phase flow cases with WC of 50% and 75%. The discrepancies for 100% WC are the highest and they are the lowest for the 75%. Uncertainty analysis for the ZNLF gas and liquid superficial velocities shows 2% to 25% variations of the measured experimental values.

30 Three-Phase Flow Zero-Net Liquid Holdup

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References [Ka18a] Karpurapu, N.V.M.P.: Dynamic Zero-Net Liquid Holdup in a Gas-Liquid Cylindrical Cyclone (GLCC© ) Separator. M.S. Thesis, The University of Tulsa, Tulsa (2018) [Ka18b] Karpurapu, N.V.M.P., Kolla S.S, Mohan R.S, Shoham O.: Dynamic Zero-Net Liquid Holdup in Gas-Liquid Cylindrical Cyclones[conference presentation] ASME 2018 5th Joint US-European Fluids Engineering Summer Conference, Montreal, Canada, (2018) [Sh18] Shah, M. : Variation of Zero-Net Liquid Holdup in Gas-Liquid Cylindrical Cyclone (GLCC© ). M.S. Thesis, The University of Tulsa, Tulsa (2018) [Sh21] Shah, M., Zhao, H., Mohan R., Shoham O.: Variation of Zero-Net Liquid Holdup in Gas-Liquid Cylindrical Cyclone (GLCC© ). Integral Methods for Scientists and Engineers (IMSE) (2021) [Wa61] Wallis, G.B.: Flooding Velocities for Air and Water in Vertical Tubes. In: The United Kingdom Atomic Energy Authority Report AEEW-R 123 (1961)

Chapter 31

Error Propagation in Dynamic Iterations Applied to Linear Systems of Differential Equations B. Zubik-Kowal

31.1 Introduction Consider the system of ordinary differential equations .

d x(t) = Ax(t) + f (t), dt

(31.1)

where .A = [aij ]ni,j =1 is a given real matrix, .f (t) = (f1 (t), . . . , fn (t))T is a given real-valued vector function, and .t > 0. System (31.1) is supplemented by the initial condtion x(0) = x0 ,

.

(31.2)

where .x0 ∈ Rn is a given initial vector. Let A = L + D + U,

.

where D is the diagonal matrix consisting of the diagonal entries of A and L, U are the lower and the upper diagonal matrices of A, respectively. For (31.1), consider the dynamic iteration scheme .

d (k+1) x (t) = (L + D)x (k+1) (t) + U x (k) (t) + f (t), dt

(31.3)

B. Zubik-Kowal () Boise State University, 1910 University Drive, Boise, ID, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4_31

387

388

B. Zubik-Kowal

where .k = 1, 2, 3, . . . and the initial iterate .x 0 (t) is an arbitrary continuous realvalued vector function such that .x (0) (0) = x0 . Using (31.2), we require that the successive iterates .x (k) (t), .k = 1, 2, 3, . . . satisfy the initial condition x (k) (0) = x0 .

.

The dynamic iterations (31.3) generate the errors  T (k) e(k) (t) = e1 (t), . . . , en(k) (t) ,

.

where ei(k) (t) = xi(k) (t) − xi (t).

.

(31.4)

In this chapter, we focus on the influence of the magnitudes of the inherent physical parameters on the propagation of the errors (31.4) as k increases. Scheme (31.3) is also called Gauss-Seidel waveform relaxation. Note that since .L + D involves only zeros in the upper diagonals, the application of implicit time integration methods to (31.3) is straightforward and does not require solving any algebraic system in each time step. Therefore, the time integration of system (31.3) is simple and allows the use of larger step sizes, which is not possible when explicit methods are directly applied to (31.1). Implicit methods are frequently used to solve and simulate mathematical models written in terms of differential systems applied to a great number of disciplines. The parameters of the physical systems frequently have different orders of magnitude, ranging from very small to very large values as is common in practical applications and emphatically demonstrated, for example, by Graedel and Crutzen [GrCr95]. One of the models from atmospheric chemistry is described in [GrCr95] as a system of differential equations involving constant and time-dependent coefficients ranging from .10−40 to .105 in magnitude. As demonstrated in [Zu17, Zu19, Zu20], such contrast in the magnitudes of the given physical parameters may be efficiently used to speed up the convergence of dynamic iterations. Dynamic iterations are more than a 100 years old, and a subclass of them, known as Picard iterations, were introduced by Charles Émile Picard in 1893 to prove the existence of solutions of differential problems. Upon the arrival of computers, dynamic iterations have been introduced as efficient techniques for solving large systems of differential equations in parallel computing environments [LeEtAl82]. They have been investigated not only for systems of ordinary differential equations, e.g., in [Bu95] and [MiNe96], but also for systems of delay differential equations [Bj94, Bj95] and for more general systems of functional differential equations, e.g., in [ZuVa99, Zu00]. Recently, the role of given physical parameters in the error propagation of dynamic iteration schemes has been investigated in [Zu17, Zu19, Zu20], and permutations of equations in a given linear differential system have been proposed as

31 Errors of Dynamic Iterations

389

an acceleration technique to speed up the convergence of the schemes. Acceleration properties of permutations have been also revealed for nonlinear differential systems applied in population dynamics [Zu22] when .n = 2. The recurrence relations derived in [Zu17, Zu19, Zu20] for the errors of dynamic iterations inform how the parameters of given differential systems influence their convergence. However, these results apply only to systems of dimension .n ≤ 4. In this chapter, we extend these results to arbitrarily large systems, valid under the assumption that .n ≥ 4, and derive recurrence relations for the errors (31.4). We then employ these relations to conclude that the smaller (or larger) the magnitudes of the parameters multiplied by the previous iterates, the faster (or slower) the convergence of the applied dynamic iterations. The rest of the chapter is organized as follows. In the next section, we derive the recurrence relations for the errors (31.4). In Sect. 31.3, we present results of numerical experiments illustrating conclusions following from the recurrence relations. Finally, in Sect. 31.4, we finish with concluding remarks and sketch future work.

31.2 Error Propagation The following result is an extension of those of [Zu17, Zu19, Zu20] to general n-dimensional linear systems, where .n ≥ 4 is any positive integer. Theorem 1 Let .n ≥ 4. Define (k) ωml =

k−1 

.

k−1−j j all ,

(31.5)

amm

j =0

Ωml (t) =

∞ k  t k=1

k!

(k)

ωml ,

(k) + am+1,m am+2,m+1 γm(k) = am+2,m ωm,m+2

k−2 

k−2−i (i+1) amm ωm+1,m+2 ,

i=0

Γm (t) =

∞ k  t k=1

k!

γm(k) .

for .m = 1, . . . , n − 2. Then, (k+1) .e (t) 1



t

= 0

(k)

ea11 (t−s) L1 (s)ds, .

(31.6)

390

B. Zubik-Kowal

(k+1) e2 (t) (k+1)

e3

  t (k) a22 (t−s) (k) a21 Ω12 (t −s)L1 (s) + e = L2 (s) ds, .

(t) =

(31.7)

0

 t (k) (k) Γ1 (t −s)L1 (s) + a32 Ω23 (t −s)L2 (s).

(31.8)

0

 (k) +ea33 (t−s) L3 (s) ds, (k+1) (t) = em

 t  m−3  0

(k)

(k)

αm,i (t −s)Li (s) + Γm−2 (t −s)Lm−2 (s).

(31.9)

i=1

 (k) +am,m−1 Ωm−1,m (t −s)Lm−1 (s) + eamm (t−s) L(k) (s) ds, m for m = 4, 5, . . . , n − 1,  t  n−3 (k) (k) en(k+1) (t) = αn,i (t −s)Li (s) + Γn−2 (t − s)Ln−2 (s) 0

(31.10)

i=1

 (k) +an,n−1 Ωn−1,n (t −s)Ln−1 (s) ds, where L(k) i (t) =

n 

ai,j ej(k) (t),

.

i = 1, . . . , n,

(31.11)

j =i+1

for .k = 0, 1, . . . and the coefficients .αm,i (t −s), .m = 4, 5, . . . , n, .i = 1, 2, . . . , m− 3 depend solely on the strictly lower diagonal entries of A. Proof We want to show that  (k) n (L + D)k = βij

.

i,j =1

,

where (0)

.

and

βi,i = 1,

(k) βi,j

= 0,

(0)

βi,j = 0,

i = j,

i < j, i, j = 1, 2, . . . , n,

(31.12)

31 Errors of Dynamic Iterations

391

(k)

k , βi,i = ai,i i = 1, . . . n, (k) (k) .β = a ω i = 2, . . . n, i,i−1 i−1,i , i,i−1 (k) (k) βi,i−2 = γi−2 , i = 3, . . . , n

(31.13)

for all .k = 1, 2, 3, . . . (k) Note that the main diagonal entries of the matrix .(L + D)k are .βi,i , where .k = 1, 2, 3, . . . . We now want to determine the .(i, i −1) and .(i, i −2) entries of .(L+D)k . For .k = 1, we get 0 0 ai,i−1 = ai,i−1 ai−1,i−1 ai,i = ai,i−1

1−1 

.

1−1−j

j

(1) ai−1,i−1 ai,i = ai,i−1 ωi−1,i = βi,i−1 ,

j =0

and 0 0 =a ai,i−2 = ai,i−2 ai−2,i−2 ai,i i,i−2

1−1 

1−1−j

j =0 1−2 

.

j

(1)

ai−2,i−2 ai,i = ai,i−2 ωi−2,i

(1) = ai,i−2 ωi−2,i + ai−1,i−2 ai,i−1

1−2−j

(j +1)

(1) ai−2,i−2 ωi−1,i = γi−2 = βi,i−2 .

j =0 (k)

(k)

We now assume that .βi,i−1 and .βi,i−2 are defined by (31.13) and derive the entries (k+1)

(k+1)

βi,i−1 and .βi,i−2 for .(L + D)k+1 . From (31.12) and (31.13), we get

.

(k+1)

(k)

k =a βi,i−1 = ai−1,i−1 βi,i−1 + ai,i−1 ai,i i−1,i−1 ai,i−1 .

= ai,i−1

k−1 

k−1 

k−1−j

j =0 k−j

j

k ai−1,i−1 ai,i + ai,i−1 ai,i

j

0 k ai−1,i−1 ai,i + ai,i−1 ai−1,i−1 ai,i = ai,i−1

j =0 (k+1)

k 

k−j

j

ai−1,i−1 ai,i

j =0

= ai,i−1 ωi−1,i , which shows that the second equation in (31.13) is fulfilled. We now show the third equation in (31.13). From (31.12) and (31.13), we get (k+1)

(k)

k k βi,i−2 = ai−2,i−2 βi,i−2 + ai−1,i−2 βi,i−1 + ai,i−2 ai,i ⎛ ⎞ k−2  k−2−j (j +1) (k) . = ai−2,i−2 ⎝ai,i−2 ωi−2,i + ai−1,i−2 ai,i−1 ai−2,i−2 ωi−1,i ⎠ j =0 (k)

k . + ai−1,i−2 ai,i−1 ωi−1,i + ai,i−2 ai,i

We now use (31.5) and get

392

B. Zubik-Kowal

(k+1) βi,i−2 = ai−2,i−2 ai,i−2

k−1 

k−1−j

j

ai−2,i−2 ai,i

j =0

+ai−2,i−2 ai−1,i−2 ai,i−1 +ai−1,i−2 ai,i−1

.

= ai,i−2 ⎛

k 

k−1 

k−2 

k−2−j

ai−2,i−2

j =0

j 

j −s

s ai−1,i−1 ai,i

s=0

k−1−j

j

k−k k ai−1,i−1 ai,i + ai,i−2 ai−2,i−2 ai,i

j =0 k−j

j

ai−2,i−2 ai,i + ai−1,i−2 ai,i−1

j =0

× ⎝ai−2,i−2

k−2 

k−2−j

ai−2,i−2

j =0

j 

j −s

s ai−1,i−1 ai,i +

k−1 

⎞ ai−1,i−1 ai,i ⎠ . k−1−j

j

j =0

s=0

So, by rearranging the terms, we get (k+1)

(k+1)

βi,i−2 = ai,i−2 ωi−2,i + ai−1,i−2 ai,i−1 .

k−1  k−1−(k−1)

+ ai−2,i−2

 k−2



k−1−j

ai−2,i−2

j =0

j 

j −s

s ai−1,i−1 ai,i

s=0

k−1−s s ai−1,i−1 ai,i

s=0 (k+1)

= ai,i−2 ωi−2,i + ai−1,i−2 ai,i−1

k−1 

k−1−j

ai−2,i−2

j =0

j 

j −s

s ai−1,i−1 ai,i

s=0

and, from (31.5),

.

(k+1) (k+1) βi,i−2 = ai,i−2 ωi−2,i + ai−1,i−2 ai,i−1

k−1 

k−1−j

(j +1)

(k+1) ai−2,i−2 ωi−1,i = γi−2 .

j =0

We now introduce the following notation ∞ n ∞ k  t k (k)  n  t k (L + D) = β . αi,j (t) = exp (t (L + D)) = i,j =1 k! k! i,j k=0

k=0

.

i,j =1

(31.14)

Then, αi,i (t) =

∞ k  t

.

k=0

k!

(k)

βi,i =

∞  (ai,i t)k k=0

k!

= eai,i t ,

i = 1, 2, . . . , n.

(31.15)

31 Errors of Dynamic Iterations

393

From (31.13), we get

αi,i−1 (t) =

∞ k  t

=

∞ k  t

k!

ai,i−1

k−1 

k−1−j

j

ai−1,i−1 ai,i

k=0 j =0 ∞ k ∞ k  k−1   t (k) t k−1−j j ω ai,i−1 ai−1,i−1 ai,i = ai,i−1 k! k! i−1,i k=0 j =0 k=0 k=0

.

k!

(k)

βi,i−1 =

= ai,i−1 Ωi−1,i (t),

(31.16)

i = 2, 3, . . . , n,

and

.

αi,i−2 (t) =

∞ k  t k=0

k!

(k)

βi,i−2 =

∞ k  t k=1

k!

(k)

γi−2 = Γi−2 (t),

i = 3, 4, . . . , n.

(31.17)

From (31.1), (31.3), and (31.4), we get

.

d (k+1) e (t) = (L + D)e(k+1) (t) + U e(k) (t) dt

and  e

.

(k+1)

(t) =

t

  exp (t − s)(L + D) U e(k) (s)ds.

0

From (31.11), we get ⎡

n 

⎤

⎡ ⎤ L(k) ⎢ ⎥ 1 (s) ⎢ j =2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ n ⎢  ⎥ ⎢ L(k) (s) ⎥ ⎢ ⎥ ⎢ 2 ⎥ (k) ⎢ ⎥ a2,j ej (s) ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ (k) ⎢ ⎢ ⎥ ⎥ j =3 .U e (s) = ⎢ ⎥ = ⎢ ... ⎥, ⎢ ⎥ ⎢ ⎥ ... ⎢ n ⎥ ⎢ (k) ⎥ ⎢ ⎢ ⎥ Ln−1 (s) ⎥ (k) ⎢ ⎢ ⎥ ⎥ a e (s) n−1,j j ⎢ ⎥ ⎣ ⎦ ⎣ j =n ⎦ 0 0 (k) a1,j ej (s)

(31.18)

394

B. Zubik-Kowal

and, further from (31.14), we get ⎡

⎤

1 

(k) − s)Lj (s)

α1,j (t ⎢ ⎥ ⎢ ⎥ j =1 ⎢ ⎥ ⎢ ⎥ ... ⎢ ⎥ ⎢  ⎥ m ⎢ ⎥ (k) ⎢ αm,j (t − s)Lj (s) ⎥ ⎢ ⎥   ⎢ j =1 ⎥ (k) ⎥, . exp (t − s)(L + D) U e (s) = ⎢ ⎢ ⎥ ... ⎢ ⎥ ⎢ ⎥ n−1 ⎢ ⎥ (k) ⎢ ⎥ α (t − s)L (s) n−1,j j ⎢ ⎥ ⎢ j =1 ⎥ ⎢ n−1 ⎥ ⎢  ⎥ (k) ⎣ α (t − s)L (s) ⎦ n,j

j

j =1

where .m = 1, . . . , n − 1. Therefore, from (31.18), we get (k+1) .e (t) 1



t

= 0

(k)

α1,1 (t − s)L1 (s)ds,

which, by (31.15), implies (31.6). Since the second component in (31.18) is written in the form  t (k+1) (k) (k) .e (t) = α2,1 (t − s)L1 (s) + α2,2 (t − s)L2 (s)ds, 2 0

from (31.16) and (31.15), we get (31.7). Then, for the third component in (31.18), we get (k+1) .e (t) 3

 = 0

t

(k) (k) α3,1 (t − s)L(k) 1 (s) + α3,2 (t − s)L2 (s) + α3,3 (t − s)L3 (s)ds,

which, by (31.17), (31.16), and (31.15), implies (31.8). For .m = 4, 5, . . . , n − 1, from (31.18), we get (k+1)

em .

 (t) =

t m−3  0 j =1

(k)

(k)

αm,j (t − s)Lj (s) + αm,m−2 (t − s)Lm−2 (s)

+ αm,m−1 (t

(k) − s)Lm−1 (s) + αm,m (t

(31.19)

(k) − s)Lm (s)ds.

We now use (31.17), (31.16), and (31.15) for the factors .αm,m−2 (t − s), .αm,m−1 (t − s), and .αm,m (t − s) in (31.19), respectively, and get (31.9).

31 Errors of Dynamic Iterations

395

Finally, for the third component in (31.18), we get en(k+1) (t)

.

=

 t n−3 0 j =1

(k)

(k)

(k)

αn,j (t − s)Lj (s) + αn,n−2 (t − s)Ln−2 (s) + αn,n−1 (t − s)Ln−1 (s)ds

and use (31.17), (31.16) for the factors .αn,n−2 (t − s), .αn,n−1 (t − s), respectively, which implies (31.10) and finishes the proof of the theorem. By Theorem 1 combined with the results in [Zu17, Zu19, Zu20], we conclude that for any given n by n linear system of differential equations, the smaller the magnitudes of the coefficients multiplied by the previous iterates, the faster the convergence of the dynamic iterations applied to the system. In the next section, we illustrate this conclusion using numerical experiments.

31.3 Numerical Experiments Consider three systems of differential equations written in the form (31.1), where the matrix A is defined by ⎡

−1.5 ⎢ 0.04 ⎢ ⎢ ⎢ −0.3 .Aa = ⎢ ⎢−0.001 ⎢ ⎣ 0.002 −0.002

a −1.1 0.5 −0.4 −0.003 0.001

0.05 −0.06 −1.3 0.6 −0.5 0.003

−0.03 −0.02 0.04 −1.0 0.6 0.5

0.02 0.01 −0.03 0.05 −1.4 −0.7

⎤ −0.01 −0.03⎥ ⎥ ⎥ 0.02 ⎥ ⎥ −0.03⎥ ⎥ −0.02⎦ −1.2

(31.20)

and .a = 0.05 for the first system, .a = 0.5 for the second system, and .a = 5 for the third one. The non-homogeneous function .f (t) is defined in such a way that the solution of (31.1) is .xi (t) = sin(t), for .i = 1, . . . , 6. For the initial condition (31.2), we use the vector .x0 = (0, 0, 0, 0, 0, 0)T . We apply the dynamic iteration scheme (31.3) initiated from the constant initial iterate .x (0) (t) ≡ (0, 0, 0, 0, 0, 0)T , for any t, and supplemented by the initial condition .x (k) (0) = (0, 0, 0, 0, 0, 0)T . Then, the successive iterates .x (k) (t) are defined recursively by (31.3) from the initial iterate .x (0) (t), meaning that for each k the present iterate .x (k+1) (t) is defined by the previous iterate .x (k) (t). By Theorem 1, we conclude that the errors (31.4) are smallest when .a = 0.05, and they are larger if .a = 0.5. The errors are largest when .a = 5. To illustrate this conclusion numerically, we integrate system (31.3) in time and get discrete approximations to the successive iterates .x (k) (t), which are continuous functions.

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Note that the application of implicit time integration methods to (31.1) requires solving an algebraic system in each time step. However, if we instead apply implicit time integration methods to (31.3), we get straightforward recursive algorithms, and no algebraic systems have to be solved. This is due to the structure of the lower diagonal matrix in (31.3) multiplied by the present iterate .x (k+1) (t), that is, the previous iterates .x (k) (t) are only multiplied by the strictly upper diagonals of the matrix A. To illustrate the advantage of the dynamic iteration scheme (31.3), as an example, we apply the BDF3 method to (31.3) and describe the entire resulting recursive (k+1) algorithm as follows. The successive iterates .x1,n+3 for the first solution .x1 are (k)

(k)

(k)

computed from the previous iterates .x2,n+3 , x3,n+3 , . . . , x6,n+3 as follows: −1  18 (k+1) 6 9 (k+1) 2 (k+1) − x + x = 1 − ha1,1 x 11 1,n+2 11 1,n+1 11 1,n 11  6   6 (k) h a1,m xm,n+3 + f1 (tn+3 ) , + 11 

(k+1) x1,n+3 .

(31.21)

m=2

where h is the step size of the integration in time, .tn = nh are temporal grid-points, (k) (k) and .xm,n are approximations to .xm (tn ), .m = 1, 2, . . . , 6. (k+1) Then, once the approximations .x1,n+3 are already computed from (31.21), they (k) (k) can now be used together with the previous iterates .x3,n+3 , . . . , x6,n+3 to compute (k+1) the iterates .x2,n+3 for the second solution .x2 from

−1   18 (k+1) 6 9 (k+1) 2 (k+1) (k+1) x2,n+2 − x2,n+1 + x2,n x2,n+3 = 1 − ha2,2 11 11 11 11   . 6  6 (k+1) (k) h a2,1 x1,n+3 + a2,m xm,n+3 + f2 (tn+3 ) . + 11

(31.22)

m=3

For any .j

(k+1)

(k+1)

= 3, . . . 5, the approximations .x1,n+3 and .x2,n+3 computed (k+1)

from (31.21) and (31.22), respectively, together with the approximations .xi,n+3 , (k)

(k)

for .i < j , can now be used together with the previous iterates .xj +1,n+3 , . . . , x6,n+3

(k+1) to compute the present iterates .xj,n+3 for the j -th solution .xj from

−1   18 (k+1) 9 (k+1) 2 (k+1) 6 (k+1) xj,n+2 − xj,n+1 + xj,n xj,n+3 = 1 − haj,j 11 11 11 11 ⎛ ⎞ .  j −1 6  6 ⎝ (k+1) (k) ⎠ , h aj,m xm,n+3 + aj,m xm,n+3 + fj (tn+3 ) + 11 m=1

m=j +1

(31.23)

31 Errors of Dynamic Iterations (k+1)

397

(k+1)

where .x1,n+3 , . . . , xj −1,n+3 have been computed from (31.21), (31.22), and (31.23) (k)

(k)

for .i < j , respectively, and only .6 − j previous iterates, .xj +1,n+3 , . . . x6,n+3 , are used. (k+1) Finally, the present iterates .x6,n+3 for the sixth solution .x6 can be computed from −1   18 (k+1) 9 (k+1) 2 (k+1) 6 (k+1) x6,n+2 − x6,n+1 + x6,n x6,n+3 = 1 − ha6,6 11 11 11 11   5 .  6 (k+1) a6,m xm,n+3 + f6 (tn+3 ) , + h 11

(31.24)

m=1

(k+1)

(k+1)

(k+1)

(k+1)

(k+1)

where .x1,n+3 , .x2,n+3 , .x3,n+3 , .x4,n+3 , .x5,n+3 have been computed from (31.21)– (k+1)

(31.23), respectively. Note that no previous iterates are used in (31.24) and .x6,n+3 are computed only from present iterates. The recurrence relations (31.21)–(31.24) have become possible through the application of the dynamic iterations (31.3) to (31.1). Note also that (31.3) brings two kinds of advantages. The first advantage is that (31.21)–(31.24) allow to apply larger values of h than those to be used for explicit methods applied directly to solve system (31.1). Another advantage of the application of (31.3) is that the direct formulas (31.21)–(31.24) allow to avoid solving systems of algebraic equations at each time step, which would be a necessity if implicit methods were to be applied directly to (31.1). The recursive scheme (31.21)–(31.24) generates the sequence of approximations  (k)  . xm,n to the exact solution .xm (tn ) at the grid-points .tn . Since     (k) (k) (k) (k) xm,n − xm (tn ) = xm,n − xm (tn ) + xm (tn ) − xm (tn ) ,

.

the errors of the approximations are composed of two errors: the first one is the (k) error of the time integration of system (31.3), and the second one is .em (tn ) defined by (31.4). For graphical illustration, we define the maximum errors (k) Em =

.

max

n=0,1,2,...,N

(k) |xm,n − xm (tn )|,

(31.25)

where .k = 1, 2, 3, . . . , .m = 1, 2, . . . , 6, .tn = nh, N is a positive integer such (k) that .N h = T and .[0, T ] is the interval over which the numerical solutions .xm,n are computed. Numerical errors (31.25), where .T = 10, .h = 0.01, and .N = 1000, are presented in Fig. 31.1 for the three systems (31.1) involving .Aa defined by (31.20), where .a = 0.05, .a = 0.5, and .a = 5. Each of the six subplots in Fig. 31.1 presents the errors (31.25) for all three systems, and the m-th subplot (counting from the left (k) to the right) presents the errors of the m-th approximations .xm,n to the exact solution .xm , where .m = 1, 2, . . . , 6.

398

B. Zubik-Kowal

a=0.05 a=0.5 a=5

10-2 10-4 10-6 10-8

0

5

k

10

error e 4(k)

error e 3(k)

10-4

0

5

k

10

15

a=0.05 a=0.5 a=5

100

10-6

10-2 10-4 10-6

0

5

k

10

10-8

15

a=0.05 a=0.5 a=5

100 10-2 10-4

0

5

k

10

10-6

15

a=0.05 a=0.5 a=5

100 error e 6(k)

error e 5(k)

10-4

10-8

15

a=0.05 a=0.5 a=5

10-2

10-8

10-2

10-6

100

10-8

a=0.05 a=0.5 a=5

100 error e (k) 2

error e (k) 1

100

10-2 10-4 10-6

0

5

k

10

15

10-8

0

5

k

10

15

Fig. 31.1 Maximum errors (31.25) as functions of k

Figure 31.1 illustrates that the larger the magnitudes of the coefficients multiplied by the previous iterates, the slower the convergence of the applied dynamic iterations. From 31.1, we observe that the convergence of dynamic iterations is slightly slower for .a = 0.5 than for .a = 0.05, but it is similar for both systems. Only for the first three exact solutions .x1 , .x2 , and .x3 , the convergence for .a = 0.05 is faster by one iteration than the convergence for .a = 0.5. Namely, the iterations for .x1 , .x2 , and .x3 converge in .k = 6 iterations when .a = 0.05, and in .k = 7 iterations when .a = 0.5. For the next three exact solutions .x4 , .x5 , and .x6 , convergence occurs in .k = 6 iterations for both .a = 0.05 and .a = 0.5. Moreover, for both .a = 0.05 and .a = 0.5, and for all six exact solutions .xm , .m = 1, . . . , 6, the convergence of the

31 Errors of Dynamic Iterations

399

dynamic iterations is faster than for .a = 5. Therefore, if we order the systems based on the convergence rate of the applied dynamic iterations, from fastest to slowest, then the order would be .a = 0.05 (fastest convergence), .a = 0.5, and finally .a = 5 (slowest convergence). As demonstrated in Fig. 31.1, we conclude that changes in the magnitude of even just one parameter value affects the rate of convergence of dynamic iterations. Moreover, the smaller the magnitudes of the parameters multiplied by the previous dynamic iterates, the faster the convergence of the applied dynamic iteration scheme. These conclusions are confirmed by Theorem 1 for systems of linear differential equations of arbitrary dimension.

31.4 Conclusions We investigated dynamic iterations applied to n-dimensional linear systems of differential equations and addressed the question of how the parameters inherent to the systems influence the convergence of the iterations. We derived recursive formulas for the errors of the iterations in terms of the given parameters and expose their roles in the propagation of the errors. We conclude that the smaller the magnitudes of the parameters multiplied by the previous iterates, the faster the convergence of the applied dynamic iterations. We also conclude that a change in even a single parameter may lead to significant changes in the errors and number of required iterations. We presented numerical experiments demonstrating these conclusions. Our future work will address the roles of the magnitudes of model parameters on the convergence of dynamic iterations applied to n-dimensional nonlinear systems of differential equations.

References [Bj94] Bjorhus, M.: On dynamic iteration for delay differential equations. BIT 43, 325–336 (1994) [Bj95] Bjorhus, M.: A note on the convergence of discretized dynamic iteration. BIT 35, 291–296 (1995) [Bu95] Burrage, K.: Parallel and Sequential Methods for Ordinary Differential Equations. Oxford University Press, Oxford (1995) [GrCr95] Graedel, T.E., Crutzen, P.J.: Atmosphere, Climate, and Change (Scientific American Library), Freeman and Company, New York (1995) [LeEtAl82] Lelarasmee, E., Ruehli, A., Sangiovanni-Vincentelli, A.: The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Trans. CAD 1, 131–145 (1982) [MiNe96] Miekkala, U., Nevanlinna, O.: Iterative solution of systems of linear differential equations. Acta Numer. 5, 259–307 (1996)

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[ZuVa99] Zubik-Kowal, B., Vandewalle, S.: Waveform relaxation for functional differential equations. SIAM J. Sci. Comput. 21, 207–226 (1999) [Zu00] Zubik-Kowal, B.: Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations. Appl. Numer. Math. 34, 309–328 (2000) [Zu17] Zubik-Kowal, B.: Propagation of errors in dynamic iterative schemes. In: Mikula, K., Sevcovic, D., Urban, J. (eds.) Proceedings of EQUADIFF 2017, Published by Slovak University of Technology, SPEKTRUM STU Publishing, pp. 97–106 (2017) [Zu19] Zubik-Kowal, B.: On the convergence of dynamic iterations in terms of model parameters. In: Constanda, C., Harris, P. (eds.) Integral Methods in Science and Engineering, Analytic Treatment and Numerical Approximations, pp. 219–230. Birkhäuser, New York (2019) [Zu20] Zubik-Kowal, B.: Error analysis and the role of permutation in dynamic iteration schemes. In: Constanda, C. (ed.) Computational and Analytic Methods in Science and Engineering, pp. 239–256. Birkhäuser, New York (2020) [Zu22] Zubik-Kowal, B.: Dynamic iterations for nonlinear systems applied in population dynamics. In: Ehrhardt, M., Günther, M. (eds.) Progress in Industrial Mathematics, ECMI 2021. The European Consortium for Mathematics in Industry, vol. 39. Springer, Berlin (2022)

Index

A Absorption coefficient, 179 Advection-diffusion equation, 311 Ahlfors regular set, 167 Analytical formulation, contaminant transport, porous media, 335 Analytic solution, 311

B Block norm, 199 Bones resorption, 229 Boundary geometric measure theoretic (GMT), 165 Boundary integral equations, 229 Boundary Sobolev space, 168 Boundary-to-boundary Cauchy operator, 166 Boundary-to-domain Cauchy operator, 166 Boundary value problems, 191

C Cauchy-Riemann operator, 166 Cell clustering, 119 Complex Whitney array space, 169 Composite pipes, 321 Contact problem, 255 Contaminant transport, 335 Convergence of dynamic iterations, 387 Correlation functions, 67

D Direct correlation function, 67 variational method, 67 Dynamic iterations, 387 E Elasticity solution, 321 Elastic structures, 103 Electron content, 87 Enhanced oil recovery, thermodynamics, 239 Equilibrium statistical mechanics, 67 F Failure analysis, 321 Finite elements, porous media, 119, 155 Fixed point, 347 Formula jump, 197 Free energy, 67 functional, 67 Function polyanalytic, 170

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 C. Constanda et al. (eds.), Integral Methods in Science and Engineering, https://doi.org/10.1007/978-3-031-34099-4

401

402 G Gauss-Seidel, Jacobi, waveform relaxation, convergence analysis, 387 Generating functional, 67 Geometric measure theoretic (GMT) boundary, 165 Gibbs configuration integral, 67 distributions, 67 GILTT method, 335 Graph-contractions, 347 H Hermit’s functions, 67 Higher-order complex nontangential trace, 175 Higher-order Hardy space, 175 I Integral equations, nonuniqueness, 95 equations, uniqueness, 347 Ionospheric dynamics, 87 J John-Nirenberg space, 193 Jump formula, 197 L Laminated plate theory, 321 Layer potential operator boundary-to-boundary double, 194 boundary-to-domain double, 194

M Maximum principle, 219 Metaheuristics, 269, 281 Mittag-Leffler function, 258 Morrey norm, 195 Muckenhoupt weight, 195 Muckenhoupt-weighted block space, 199 Muckenhoupt-weighted Morrey space, 195 Multiplicative correlation function, 67

Index N Non-linear transfer function, 299 Nontangential approach region, 192 Nontangential maximal operator, 167, 192 Norm block, 199 Morrey, 195

O Operator boundary-to-boundary Cauchy, 166 boundary-to-domain Cauchy, 166 Cauchy-Riemann operator, 166 nontangential maximal, 167, 192 poly-Cauchy, 170 Optimization algorithm, 269, 281 Orlicz setting, 361

P Pandemic modelling, kinetics equation, SIRD-model, 51 Partial densities, 67 distribution functions, 67 Periodic solutions, 219 Periodic Sobolev spaces, 219 Plane deformation state, 229 elasticity, 95 Pointwise nontangential trace, 167 Polyanalytic function, 170 Poly-Cauchy operator, 170 Porous media, 335

Q Quasicrystal model approach, 67 Quasicrystal model calculations, 67

R Radiative conductive transfer equation, 179 Radiative conductive transfer equation, refractive index, absorption coefficients, 179 Ray-tracing, 95 Refractive index, 179

Index Response matrix spectral nodal method, discrete ordinates, neutral particle transport, 205 Rigid displacement, 106 Robin boundary value problem, 103

S Set Ahlfors regular, 167 uniformly rectifiable (UR), 167 Signal amplification, 299 Sobolev-based complex Whitney array space, 169 Solution regularity, 219 Space boundary Sobolev, 168 complex Whitney array, 169 higher-order Hardy, 175 John-Nirenberg, 193 Muckenhoupt-weighted block, 199 Muckenhoupt-weighted Morrey, 195 Sobolev-based complex Whitney array, 169 Space weather, 87 Statistical structural analysis of seismic signals, 79 Stress state near dental implants, 229 Syringomyelia, spinal cord, 119, 155

403 T Thermal effects physical Monte Carlo simulator, 35 Three-phase flow zero-net liquid holdup, 373 Time harmonic porous-elasticity, 155 Trace higher-order complex nontangential, 175 pointwise nontangential, 167

U Ulam method, 95 Unfolding method, 361 Uniformly rectifiable (UR) set, 167 V Variational principle, 67 W Water alternating gas, 239 Water alternating gas injection, 239 Wear model, 255 Wear model for contact problems, 255 Weight Muckenhoupt, 195