Smart Modelling for Engineering Systems: Proceedings of the International Conference on Computational Methods in Continuum Mechanics (CMCM 2021), ... Innovation, Systems and Technologies, 215) 9813346183, 9789813346185

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Smart Innovation, Systems and Technologies 215

Margarita N. Favorskaya Alena V. Favorskaya Igor B. Petrov Lakhmi C. Jain   Editors

Smart Modelling for Engineering Systems Proceedings of the International Conference on Computational Methods in Continuum Mechanics (CMCM 2021), Volume 2

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Smart Innovation, Systems and Technologies Volume 215

Series Editors Robert J. Howlett, Bournemouth University and KES International, Shoreham-by-sea, UK Lakhmi C. Jain, Faculty of Engineering and Information Technology Centre for Artificial Intelligence, University of Technology Sydney, Sydney, NSW, Australia

The Smart Innovation, Systems and Technologies book series encompasses the topics of knowledge, intelligence, innovation and sustainability. The aim of the series is to make available a platform for the publication of books on all aspects of single and multi-disciplinary research on these themes in order to make the latest results available in a readily-accessible form. Volumes on interdisciplinary research combining two or more of these areas is particularly sought. The series covers systems and paradigms that employ knowledge and intelligence in a broad sense. Its scope is systems having embedded knowledge and intelligence, which may be applied to the solution of world problems in industry, the environment and the community. It also focusses on the knowledge-transfer methodologies and innovation strategies employed to make this happen effectively. The combination of intelligent systems tools and a broad range of applications introduces a need for a synergy of disciplines from science, technology, business and the humanities. The series will include conference proceedings, edited collections, monographs, handbooks, reference books, and other relevant types of book in areas of science and technology where smart systems and technologies can offer innovative solutions. High quality content is an essential feature for all book proposals accepted for the series. It is expected that editors of all accepted volumes will ensure that contributions are subjected to an appropriate level of reviewing process and adhere to KES quality principles. Indexed by SCOPUS, EI Compendex, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST), SCImago, DBLP. All books published in the series are submitted for consideration in Web of Science.

More information about this series at http://www.springer.com/series/8767

Margarita N. Favorskaya · Alena V. Favorskaya · Igor B. Petrov · Lakhmi C. Jain Editors

Smart Modelling for Engineering Systems Proceedings of the International Conference on Computational Methods in Continuum Mechanics (CMCM 2021), Volume 2

Editors Margarita N. Favorskaya Institute of Informatics and Telecommunications Reshetnev Siberian State University of Science and Technology Krasnoyarsk, Russia Igor B. Petrov Moscow Institute of Physics and Technology National Research University Moscow, Russia

Alena V. Favorskaya Moscow Institute of Physics and Technology National Research University Moscow, Russia Lakhmi C. Jain Centre for Artificial Intelligence University of Technology Sydney Broadway, NSW, Australia KES International Shoreham-by-sea, UK Liverpool Hope University Liverpool, UK

ISSN 2190-3018 ISSN 2190-3026 (electronic) Smart Innovation, Systems and Technologies ISBN 978-981-33-4618-5 ISBN 978-981-33-4619-2 (eBook) https://doi.org/10.1007/978-981-33-4619-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 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 Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book includes the research work selected for presentation in the International Conference on Computational Methods in Continuum Mechanics CMCM2021 in Dolgoprudny, Moscow Region of the Russian Federation from 15–17 April 2021. The conference is devoted to the memory of Academician Oleg Belotserkovskii. Professor Dr. Belotserkovskii had made tremendous contribution in the field of numerical methods and the mathematical modelling of aerodynamics of spacecraft, hydrodynamics, plasma physics, mechanics of a deformable solid, thermonuclear fusion, turbulence, computational medicine and biology, and so on. The conference proceedings are arranged in two volumes including Part I “Computational Aerodynamics, Hydrodynamics and Dynamics of Plasma”, Part II “Numerical Modeling in Solid Mechanics”, Part III “Computational Modeling in Medicine and Biology”, Part IV “Numerical Methods in Continuum Mechanics”, Part V “Modern Methods in Mathematical Physics”, Part VI “Machine Learning”, and Part VII “Computer Science”. Many world-class researchers contributed in these books. Volume 2 presents Part IV “Numerical Methods in Continuum Mechanics” including Chaps. 2–12. Part V “Modern Methods in Mathematical Physics” containing Chaps. 13–14, Part VI “Machine Learning” involving Chaps. 15–18, and Part VII “Computer Science” providing Chaps. 19–23. We are confident that both theoreticians and application scientists will find this volume of interest. This volume is also directed to the postgraduate students who are looking for new research directions. We wish to express our appreciation to the authors and reviewers for their excellent contributions for making this book possible. Thanks are due to the Springer-Verlag for their assistance during the development phase of this research book. Krasnoyarsk, Russia Moscow, Russia Moscow, Russia Liverpool, United Kingdom

Margarita N. Favorskaya Alena V. Favorskaya Igor B. Petrov Lakhmi C. Jain

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Contents

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Recent Advances in Numerical Methods, Machine Learning, and Computer Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Margarita N. Favorskaya, Alena V. Favorskaya, Igor B. Petrov, and Lakhmi C. Jain

Part I 2

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4

5

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Numerical Methods in Continuum Mechanics

Correction of Boundary Conditions in Micromodels by Molecular Dynamic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viktoriia O. Podryga and Sergey V. Polyakov

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Development and Application of the SMIF Method for the Investigation of Incompressible Fluid Flows . . . . . . . . . . . . . . . Valentin A. Gushchin

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On 32nd-Order Multioperators-Based Schemes for Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrei I. Tolstykh

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Hermitian Grid-Characteristic Scheme for Linear Transport Equation and Its Dissipative-Dispersion Properties . . . . . . . . . . . . . . . Elena N. Aristova

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3-D Quasi-Conformal Mappings and Grid Generation . . . . . . . . . . . . Yuriy D. Shevelev

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Numerical Simulation of Shock-To-Detonation Transition Using One-Stage and Detailed Chemical Kinetics Mechanism . . . . . Alexander I. Lopato

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Study of the Kinetic Anomalous Transport Effects in Nonequilibrium Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vladimir V. Aristov, Anna A. Frolova, and Sergey A. Zabelok

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Contents

Different Approaches to Numerical Solution of the Boltzmann Equation with Model Collision Integral Using Tensor Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Aleksandr V. Chikitkin and Egor K. Kornev

10 Difference Scheme with a Symmetry-Analyzer for Equations of Gas Dynamics and Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . 117 Galina V. Ustyugova and Alexander V. Koldoba 11 High-Gradient Method for the Numerical Simulation of the Continuum Problems with the Strong Discontinues . . . . . . . . . 133 Vladimir V. Demchenko 12 On the Finite Difference Schemes for Burgers Equation Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Ilya V. Basharov and Aleksey I. Lobanov Part II

Modern Methods in Mathematical Physics

13 Development of Virtual Lattice Dynamics Method for Solving the Eigenvalue Problem of Three-Dimensional Elliptic Equation with a Multicenter Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Olga A. Pyrkova, Vladimir N. Pyrkov, and Petr M. Vasilets 14 Charged Particles in the Field of an Inhomogeneous Electromagnetic Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Anton A. Skubachevskii, Vladimir B. Lapshin, and Igor B. Petrov Part III Machine Learning 15 Deep Learning for Fire and Smoke Detection in Outdoor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Margarita N. Favorskaya and Lakhmi C. Jain 16 The Solution of Fractures Detection Problems by Methods of Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Maksim V. Muratov, Dmitriy I. Petrov, and Vladimir A. Biryukov 17 Solving Problems of the Strength of a Thin Thread by Machine Learning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Mykhailo Seleznov and Alexey V. Vasyukov 18 Numerical Solution of Inverse Problems of Wave Dynamics in Heterogeneous Media with Convolutional Neural Networks . . . . . 235 Andrey S. Stankevich, Igor B. Petrov, and Alexey V. Vasyukov

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Part IV Computer Science 19 A Systematic Approach to Present “Files and File Systems” in Theoretical Courses of Computer Science . . . . . . . . . . . . . . . . . . . . . 249 Vladimir E. Karpov 20 Designing Execution Models of Distributed System in Theoretical Courses on Information Technology . . . . . . . . . . . . . . . 261 Sergey L. Babichev and Konstantin A. Konkov 21 Simulation of the Drops Oscillations in the Channel . . . . . . . . . . . . . . 275 Maria S. Guskova and Lev N. Shchur 22 GPU-Accelerated Integral Equation Seismic Simulation . . . . . . . . . . 283 Alexey L. Gordov, Nikolay I. Khokhlov, and Mikhail S. Malovichko 23 On the Parallel Realization of Complex Rosenbrock ODEs Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Boris A. Polyak, Denis A. Mitroshin, Anton V. Ilin, and Maxim L. Kurkin Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

About the Editors

Dr. Margarita N. Favorskaya is Professor and Head of the Department of Informatics and Computer Techniques at Reshetnev Siberian State University of Science and Technology, Russian Federation. Professor Favorskaya is a member of KES organization since 2010, the IPC member and Chair of invited sessions of over 30 international conferences. She serves as Reviewer in international journals (Neurocomputing, Knowledge Engineering, and Soft Data Paradigms, Pattern Recognition Letters, Engineering Applications of Artificial Intelligence), Associate Editor of Intelligent Decision Technologies Journal, International Journal of KnowledgeBased and Intelligent Engineering Systems, and International Journal of Reasoningbased Intelligent Systems, Honorary Editor of the International Journal of Knowledge Engineering and Soft Data Paradigms, Reviewer, Guest Editor, and Book Editor (Springer). She is the author/co-author of 200 publications and 20 educational manuals in computer science/engineering. She co-authored/co-edited ten books for Springer recently. She supervised nine Ph.D. students and is presently supervising four Ph.D. students. Dr. Alena V. Favorskaya works as Researcher and Associated Professor at Moscow Institute of Physics and Technology, Russian Federation. Dr. Favorskaya was an advisor of more than 10 students. She is presently supervising two Ph.D. students. She is the author and the co-author of more than 100 publications in these fields. Dr. Alena Favorskaya developed smart techniques such as intelligent high-order interpolation on unstructured tetrahedral grids (2009), grid-characteristic method on unstructured tetrahedral grids (2009), and so on. Dr. Favorskaya co-edited two books for Springer recently. Dr. Alena Favorskaya was rewarded by Award of the President of the Russian Federation for Young Scientists in 2012, 2013, and 2014, by Award of the Government of the Russian Federation for Young Scientists in 2013 and 2014, by IBM PhD Fellowship Award in 2015, by the grant of the President of the Russian Federation in 2017, and by the medal of Russian Academy of Science in 2019. Prof. Dr. Igor B. Petrov is Head of the Department of Computational Physics and Full Professor at Moscow Institute of Physics and Technology, Russian Federation. xi

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He supervised more than 60 Ph.D. students. He is the author or the co-author of more than 200 publications (book chapters, research papers, and software tools) in these fields. He co-edited two books for Springer recently. Prof. Petrov developed gridcharacteristic numerical method for solving hyperbolic system of equations and for full-wave modelling of dynamical three-dimensional processes. Also, he developed a method of smooth particles for solving meteorite and asteroid protection problems. Prof. Petrov is a member of New York Academy of Sciences and a member of Advisory Board of Russian Foundation for Basic Research. He was awarded by Order for Merit of Second Degree Medal of Russian Federations in 1999. Prof. Petrov became a corresponding member of Russian Academy of Science in 2011. Dr. Lakhmi C. Jain, Ph.D., M.E., B.E. (Hons.), Fellow (Engineers Australia), is with the University of Technology Sydney, Australia, and Liverpool Hope University, UK. Professor Jain founded the KES International for providing a professional community the opportunities for publications, knowledge exchange, cooperation, and teaming. Involving around 5,000 researchers drawn from universities and companies worldwide, KES facilitates international cooperation and generates synergy in teaching and research. KES regularly provides networking opportunities for professional community through one of the largest conferences of its kind in the area of KES.

Chapter 1

Recent Advances in Numerical Methods, Machine Learning, and Computer Science Margarita N. Favorskaya, Alena V. Favorskaya , Igor B. Petrov , and Lakhmi C. Jain Abstract The chapter presents a brief description of chapters that contribute to the recent advances in numerical methods in continuum mechanics, computational physics. Also, this chapter deals with machine learning and computer science. The fourth part of the book presents novel computational methods in continuum mechanics. Modern methods in mathematical physics are presented in the fifth part of the book. The sixth part of the book deals with machine learning. Computer science is discussed in the seventh part of the book.

1.1 Introduction The fourth part of the book presents different novel modifications of numerical methods in continuum mechanics, i.e., molecular dynamic method, splitting method for an incompressible fluid, 32nd-order multioperators-based schemes, Hermitian grid-characteristic scheme, quasi-conformal mappings, and grid generation, optimize M. N. Favorskaya (B) Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31, Krasnoyarsky Rabochy ave, Krasnoyarsk 660037, Russian Federation e-mail: [email protected] A. V. Favorskaya · I. B. Petrov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] I. B. Petrov e-mail: [email protected] L. C. Jain Liverpool Hope University, Liverpool, United Kingdom e-mail: [email protected]; [email protected] University of Technology Sydney, Sydney, Australia KES International, P. O. Box 2115, Shoreham-by-Sea BN43 9AF, United Kingdom © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_1

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a low-rank tensor network with a gradient-based optimization, cross-approximation technique, Godunov-type difference schemes, high-gradient method, and finite difference schemes. Also, a one-stage and detailed chemical kinetics mechanism and kinetic anomalous transport effects are discussed in the fourth part of the book. The fifth part of the book deals with modern methods in mathematical physics, i.e., the dynamics method of the virtual lattice for solving the eigenvalue problem of a three-dimensional elliptic equation with a multicenter potential and charged particles in the field of an inhomogeneous electromagnetic wave. The sixth part of the book presents novel techniques and applications in machine learning in different areas, i.e., fire and smoke detection, geological fractures detection, the strength of a thin thread investigation, and inverse problems of wave dynamics in heterogeneous media. Computer science is discussed in the seventh part of the book. A systematic approach to the presentation of the topic “Files and file systems”, and designing execution models of distributed systems in theoretical courses on information technology are presented in this part. Also, multiphase Lattice Boltzmann method for the simulation of the drop oscillation, GPU-accelerated integral-equation seismic simulation, and the parallel realization of complex Rosenbrock ordinary differential equations solution are discussed in this part.

1.2 Chapters Included in the Book The fourth part of the book includes novel modifications of computational methods in continuum mechanics, i.e., molecular dynamics method [1], a splitting method for incompressible fluid [2], 32nd-order multioperators-based schemes [3], Hermitian grid-characteristic scheme [4], quasi-conformal mappings and grid generation [5], optimize a low-rank tensor network with a gradient-based optimization [6], cross-approximation technique, Godunov-type difference schemes [7], high-gradient method [8], and finite difference schemes [9]. One-stage and detailed chemical kinetics mechanism [10] and kinetic anomalous transport effects [11] are discussed in the fourth part of the book as well. Chapter 2 presents the capabilities of the mathematical apparatus for supercomputer modeling of gas-dynamic processes based on molecular dynamics methods. The development of splitting method for incompressible fluid flows during the last 45 years is described in Chap. 3. Multioperatorsbased schemes with the 32nd-order multioperators for fluid dynamics calculations are described in Chap. 4. Chapter 5 presents an interpolation-characteristic scheme for solving a non-stationary inhomogeneous transport equation based on Hermitian interpolation of the third order of approximation. The obtaining of the connection of the velocity components with the streamline functions for grid mapping is discussed in Chap. 6. Chapter 7 is devoted to the numerical modeling of detonation initiation in the plane channel with the profiled end-wall of the elliptic shape. Novel classes of flows caused by nonequilibrium distribution functions, in other words, on the microscopic level related to molecular velocity (or inner energy) distributions

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are introduced in Chap. 8. Two different ways of application of tensor decompositions to numerical solution of a model relaxation problem for Boltzmann equation are presented in Chap. 9. An approach using Godunov-type difference schemes to construct numerical algorithms for numerically integrating equations of ideal gas dynamics and magnetohydrodynamics is discussed in Chap. 10. The effective difference method based on characteristic directions isolation and consequent approximation of partial derivatives in pre-assigned finite-dimensional space is suggested in Chap. 11. A novel method of the difference schemes for solving the Burgers equation construction is proposed in Chap. 12. The fifth part of the book presents modern methods in mathematical physics, i.e., dynamics method of the virtual lattice for solving the eigenvalue problem of a three-dimensional elliptic equation with a multicenter potential [12] and charged particles in the field of an inhomogeneous electromagnetic wave [13]. Chapter 13 develops a method to directly solve this problem based on similarity to the problem of finding eigenfrequencies of ion oscillations in a model fragment of a solid solution. Chapter 14 generalizes an approach for simulating a wide spectrum of electromagnetic waves based on an example of inhomogeneous electromagnetic wave created by a superposition of two plane monochromatic electromagnetic waves arbitrarily directed with respect to each other. The sixth part of the book presents novel techniques and applications in machine learning in different areas, i.e., fire and smoke detection [14], geological fractures detection [15], strength of a thin thread study [16], and inverse problems of wave dynamics in heterogeneous media [17]. A novel weaved recurrent single shot detector for early smoke detection with acceptable error ratios is presented in Chap. 15. An approach to solve the inverse exploration seismology problems using methods of machine learning is presented in Chap. 16. The construction of a surrogate machine learning model for the problem of deformation and breakage of a thin thread under the action of a transverse load is considered in Chap. 17. The problem of restoring the shape of the boundary between two elastic mediums with different rheological properties is discussed in Chap. 18. Seventh part of the book deals with computer science. A systematic approach to the presentation of the topic “Files and file systems” [18], and designing execution models of distributed systems in theoretical courses on information technology are presented in this part [19]. Also, multiphase Lattice Boltzmann method for the simulation of the drop oscillation [20], GPU-accelerated integral-equation seismic simulation [21], and the parallel realization of complex Rosenbrock ordinary differential equations solution is presented in the seventh part of the book. Chapter 19 considers abstract files as mathematical objects, logical files as an implementation of abstract files in programming, and physical files as a mapping of logical files to the address space of storage media. Chapter 20 discusses the original software tools for the automated design of simulation models illustrating the various conceptions of educational courses on distributed systems and distributed algorithms. The applicability of multiphase Lattice Boltzmann method for the simulation of the drop oscillation is discussed in Chap. 21. The use of GPUs to speed-up the matrix–vector product within an iterative integral-equation solver is investigated in Chap. 22. Chapter 23

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chapter proposes research of a solver system for stiff systems of ordinary differential equations of moderate dimensionality implementing a one-stage Rosenbrock method with complex coefficients.

1.3 Conclusions The book contains the works of outstanding specialists in the field of computational methods in continuum mechanics, mathematical physics, machine learning, and computer science. Modifications of modern computational methods are presented in the book. This book will be of interest to scientists, researchers, students, graduate, and postgraduate students specializing in the scientific fields of computational physics, numerical methods, continuum mechanics, mathematical physics, parallel algorithms, fluid dynamics, GPU-acceleration, machine learning, and computer science.

References 1. Kudryashova, T., Karamzin, Yu., Podryga, V., Polyakov, S.: Two-scale computation of N2–H2 jet flow based on QGD and MMD on heterogeneous multi-core hardware. Adv. Eng. Softw. 120, 79–87 (2018) 2. Belotserkovskii, O.M., Gushchin, V.A., Shchennikov, V.V.: Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid. USSR Comput. Math. Math. Phys. 15(1), 190–200 (1975) 3. Tolstykh, A.I.: 16th and 32nd multioperators based schemes for smooth and discontinuous solutions. Commun. In Comput. Phys. 45, 33–45 (2017) 4. Aristova, E.N., Ovcharov, G.I.: Hermitian characteristic scheme for linear inhomogeneous transfer equation. Math. Models Comput. Simul. 12(6), 845–855 (2020) 5. Shevelev, Yu.D.: Application of 3-D quasi-conformal mappings for grid generation. Comput. Math. Math. Phys. 58(8), 1280–1286 (2018) 6. Chikitkin, A.V., Kornev, E.K., Titarev, V.A.: Numerical solution of the Boltzmann equation with S-model collision integral using tensor decompositions. CoRR ArXiv Preprint, arXiv: 1912.04582 (2019) 7. Koldoba, A.V., Ustyugova, G.V.: Difference scheme with a symmetry analyzer for equations of gas dynamics. Math. Models Comput. Simul. 12(2), 125–132 (2020) 8. Guskov, S.Yu., Azechi, H., Demchenko, N.N., Demchenko, V.V., Doskoch, I.Ya., Murakami, M., Nagatomo, H., Rozanov, V.B., Sakaiya, S., Stepanov, R.V.: Laser-driven acceleration of a dense matter up to ‘thermonuclear’ velocities. Plasma Phys. Control. Fusion 49(10), 1689–1706 (2007) 9. Lobanov, A.I., Mirov, FKh.: A hybrid difference scheme under generalized approximation condition in the space of undetermined coefficients. Comput. Math. Math. Phys. 58(8), 1270– 1279 (2018) 10. Lopato, A.I., Eremenko, A.G., Utkin, P.S., Gavrilov, D.A.: Numerical simulation of detonation initiation: The quest of grid resolution. In: Jain, L., Favorskaya, M., Nikitin, I., Reviznikov, D. (eds.) Advances in Theory and Practice of Computational Mechanics. Smart Innovation, Systems and Technologies, vol. 173, pp. 79–89. Springer, Singapore (2020)

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11. Aristov, V.V., Frolova, A.A., Zabelok, S.A.: Supersonic flows with nontraditional transport described by kinetic methods. Commun. Comput. Phys. 11(4), 1334–1346 (2012) 12. Pyrkova, O.A., Pyrkov, V.N., Vasilets, P.M.: Changing the geometry of the virtual lattice of the multicenter Schrodinger equation when determining eigenfunctions using the Fourier transform in time. In: XXVII International Conference on Mathematics, Economics, Education, XI Symposium Fourier Series and their Applications, Novorossiysk, Russia, (2020) (in print) 13. Lapshin, V.B., Skubachevskiy, A.A., Belinsky, A.V., Bugaev, A.S.: Emission spectrum and trajectory of a charged particle in the field of an inhomogeneous electromagnetic wave. Proc. Acad. Sci. 488(6), 1–5 (2019) 14. Favorskaya, M., Pyataeva, A., Popov, A.: Spatio-temporal smoke clustering in outdoor scenes based on boosted random forests. Procedia Comput. Sci. 96, 762–771 (2016) 15. Muratov, M.V., Biryukov, V.A., Petrov, I.B.: Solution of the fracture detection problem by machine learning methods. Doklady Math. 101(2), 169–171 (2020) 16. Vasyukov, A.V., Elovenkova, M.A., Petrov, I.B.: Modeling of thin fiber deformation and destruction under dynamic load. Matem. Mod. 32(5), 95–102 (in Russian) (2020) 17. Beklemysheva, K.A., Grigoriev, G.K., Kulberg, N.S., Petrov, I.B., Vasyukov, A.V., Vassilevski, Y.V.: Numerical simulation of aberrated medical ultrasound signals. Russian J. Numeric. Anal. Math. Model. 33(5), 277–288 (2018) 18. Karpov, V.E., Konkov, K.A.: The basics of operating systems. In: Lecture Course. 3rd edn. Fizmatkniga, Moscow (in Russian) (2019) 19. Babichev, S.L., Konkov, K.A.: Distributed System: Textbook for Universities. Urait (in Russian), Moscow (2019) 20. Guskova, M., Shchur, V., Shchur, L.: Simulation of drop oscillation using the lattice Boltzmann method. Lobachevskii J. Math. 41, 992–995 (2020) 21. Malovichko, M., Khokhlov, N., Yavich, N., Zhdanov, M.: Acoustic 3D modeling by the method of integral equations. Comput. Geosci. 111, 223–234 (2018)

Part I

Numerical Methods in Continuum Mechanics

Chapter 2

Correction of Boundary Conditions in Micromodels by Molecular Dynamic Method Viktoriia O. Podryga

and Sergey V. Polyakov

Abstract The work is devoted to presenting the capabilities of the mathematical apparatus for supercomputer modeling of gas-dynamic processes based on molecular dynamics methods. It discusses original techniques, their software implementations, as well as, some interesting molecular modeling. The relevance of the work is associated with the development of cryogenic and low-temperature nanotechnologies used in nanoelectronics, biomedicine, and other industries. One of the important tasks in this direction is the analysis of microflows of highly rarefied gases in the transonic and supersonic regions in the channels of technical systems. Due to the complexity of real-life experimentation in this area, computer and supercomputer modeling are most often used. It allows considering complex physical processes with detail down to atomic sizes. One of the possible and effective approaches in the case of atomic size detailing is molecular dynamic modeling of physical processes. This approach is free from many limitations of continuum models and allows us to take into account many important physical factors. In this chapter, we demonstrate the capabilities of the developed software package for modeling technical microsystems and calculating gas flows in their channels of different diameters, including calculating the properties of gas media and solids. The chapter gives examples of the calculation of gas-dynamic flows in conditions of violation of the medium continuity hypothesis.

V. O. Podryga (B) · S. V. Polyakov Keldysh Institute of Applied Mathematics of RAS, 4, Miusskya sq., Moscow 125047, Russian Federation e-mail: [email protected] S. V. Polyakov e-mail: [email protected] V. O. Podryga Moscow Automobile and Road Construction State Technical University, 64, Leningradsky prosp., Moscow 125319, Russian Federation S. V. Polyakov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_2

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2.1 Introduction In recent years, in connection with the massive introduction of nanotechnology in industry, there is a need to study gas-dynamic processes at the atomic-molecular level. The development of cryogenic and low-temperature nanotechnologies used in nanoelectronics, biomedicine, and other industries is relevant. One of the important tasks in this direction is the analysis of microflows of highly rarefied gases in the transonic and supersonic regions in the channels of technical systems. Due to the complexity of real-life experimentation in this area, computer and supercomputer modeling are most often used. It allows us to consider complex physical processes with detail down to atomic sizes. A similar study is carried out in cases where the theory of similarity and/or the continuity hypothesis is violated. The largest deviations from theoretical estimates are observed at the micro- and nanoscales when taking into account the specific composition of the gaseous medium and the structure of real surfaces streamlined by supersonic gas flows. Until some point, in these cases, methods for calculating free-molecular flows were successfully applied [1]. However, due to the complexity of the problems of technical gas dynamics in specific applications, methods based on the Boltzmann kinetic equation (see, for example, [2]) and molecular dynamics methods (see, for example, [3, 4]) are increasingly used. Direct molecular dynamics modeling [5–11] of physical processes at the microlevel is free from many limitations of continuous medium models and allows one to take into account many important physical factors and obtain information about the real properties of the gas medium and its interactions with the walls of microchannels. With its help, it is possible to determine the necessary properties of gases from practically the first principles. For the reasons described above, the Molecular Dynamics (MD) [5–11] approach was chosen. Based on it, effective algorithms have been developed that are implemented by the authors in the form of a set of programs for modeling technical microsystems and calculating gas flows in channels of different diameters, including calculating the properties of gaseous media and solids. The developed technology, which implements the proposed approaches and is based on mass parallel computing, allows to solve the problems of the dynamics of molecular systems of large dimension with a wide range of uncertain parameters and various conditions that imitate a physical experiment. In this work, we demonstrate the capabilities of the developed software package. The chapter gives examples of the calculation of gas-dynamic flows in conditions of violation of the continuity hypothesis of the medium. The results of studying nonlinear processes in the boundary layer are also presented. For this, a number of computational experiments were conducted on the interaction of a nitrogen flow with a nickel surface, which made it possible to formulate a macroscopic model of the boundary layer. This model can be used to calculate gas flows as a part of the approximation of continuum mechanics. In order to verify the developed technology, a full-fledged calculation of the microflow of real gas in the channel was carried out, as well as, the calculation

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of accommodation coefficients at the boundary between the gas flow and the metal wall. A system of nitrogen molecules is considered as an example of a gas-dynamic system; nickel is considered as a material of the microchannel walls. We used the results of previous studies related to the calculation of the macroparameters of individual systems of particles of nitrogen and nickel [12–17], as well as, the interaction of nitrogen with a nickel plate [17–20]. The results obtained in the calculations demonstrated the adequacy of the proposed numerical approach to modern theoretical ideas about the modeled physical process. The chapter is organized as follows. Section 2.2 provides a problem statement. Modeling stages and algorithms are developed in Sect. 2.3 Calculation results are given in Sect. 2.4. Section 2.5 concludes the chapter.

2.2 Problem Statement As a mathematical model, the molecular dynamics method [5–10] was used. In the case of studying the particles of two kinds (metal and gas), the system of equations looks as Eq. 2.1, where i is the particle number, l is the particle kind (a denotes a set of the molecules of gas, b denotes a set of atoms of metal in the nanocluster), Nl is the total number of particles of kind l, m l is the mass of particle of kind l, rl, i and vl, i are the position and the velocity vectors of the ith particle of kind l, respectively, Fl, i is the resultant force acting on the ith particle. ml

dvl,i drl,i = Fl,i = vl,i dt dt i = 1, ..., Nl l = a, b

(2.1)

The above forces are the sum of the forces of interaction of the ith particle with surrounding particles determined by the functional of potential energy and the forces of external action having the form of Eq. 2.2. Fl,i

  ∂U rl,1 , ..., rl,Nl ext =− + Fl,i i = 1, ..., Nl l = a, b ∂rl,i

(2.2)

Here, U is the total potential energy of a system of particles, Fl,exti is the interaction force of the ith particle of type l with the environment. The potential energy of the system U is represented as the sum of the partial potential energies, which are calculated according to the formulas of the selected interaction potentials. In the case of modeling the gas flow inside the channel, when considering one type of gas and one type of channel material, three types of interactions (gas–gas, gas-metal, and metal–metal) should be considered, which corresponds to three interaction potentials provided by Eqs. 2.3, where Uaa , Ubb , Uab are the potential interaction functions for gas with gas, metal with metal, and gas with metal

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particles systems, respectively. U = Uaa + Ubb + Uab Nb Na      1 ϕaa ra,i − ra, j  , Uab = ϕab ra,i − rb, j  2 i=1, j>i i=1 j=1 ⎤ ⎡ N b       ⎣ϕ1,bb rb,i + (2.3) = ϕ2,bb rb,i − rb, j  ⎦

Uaa =

Ubb

Na 

i=1

j>i

Each type of interaction is described using the corresponding potential ϕll  . For the interaction of nitrogen molecules with each other, we used the Mie potential in the form of “n – 6” [21] adapted in [22] to calculations of a mixture of hydrogen and nitrogen. For the interaction of nickel atoms with each other, the embedded atom model (EAM) potential was used [23], which takes into account not only pairwise interactions ϕ2,bb but also the influence of the environment on a specific particle ϕ1,bb . The EAM potential was taken in the form proposed in [24]. To take into account gas–metal interactions, we used the standard Lennard–Jones potential [25] with parameters calculated by the Lorentz–Berthelot formulas [26, 27]. To simulate the process of accelerating the gas flow and to reach the required temperature with gas, the Langevin thermostat was used [28], and the Berendsen thermostat was used [29] to reach only the required temperature. Both thermostats were used in the preparatory stages and were turned off during the main calculation. The initial conditions at the microlevel are determined by the equilibrium or quasiequilibrium thermodynamic state of the particle system at a given temperature, pressure, and average momentum. The boundary conditions at the molecular level depend on the modeled situation. To determine the general properties of a medium, it suffices to consider the selected three-dimensional volume with periodic boundary conditions in all coordinates. When studying microsystems of real geometry, such as a microchannel, one or several directions have a finite size, and the preservation of the shape of an object is achieved by selecting a potential or fixing the system. In this case, either the mirror boundary conditions can be used as boundary conditions when the particles interact with their mirror reflection and do not consequently leave the boundary or the condition of the particle disappearing at the boundary and its appearance in another place of the microsystem (so that the overall equilibrium in the system is not disturbed). An input stream of particles on one side of the selected volume and a free exit of particles on the other can also be specified.

2.3 Modeling Stages and Algorithms The system of equations 2.1–2.2 is solved using the velocity Verlet integration [30] provided by Eq. 2.4.

2 Correction of Boundary Conditions in Micromodels … n+1 n n Fl,i Fl,i + Fl,i (t)2 n+1 n + t vl,i = vl,i m l,i 2 2m l,i i = 1, ..., Nl l = a, b

13

n+1 n n rl,i = rl,i + vl,i t +

(2.4)

n is the force value at Here, t is the integration step, n is the step number, Fl,i the step n, F is the force calculation procedure. To achieve the desired parameters of gas and metal in a state of thermodynamic equilibrium, the temperature control procedure [28, 29] is added to Eq. 2.4. The integration step t is selected from the conditions defined by Eq. 2.5.

 n  2 F   n  v t ≤ h  l,i  (t) ≤ h i = 1, ..., Nl l = a, b l,i m  2

(2.5)

l

Here, h is a size comparable to the lattice parameters of the material from which the solid surfaces of the technical microsystem are made. In the calculations below t = 0.002 ps, h = 0.9 nm. The general algorithm certainly depends on the problem being solved. If we are talking about determining the properties of a gaseous medium or a solid wall, then we only use the well-known procedure for balancing a system of particles with given thermodynamic parameters. If you need to consider the evolution of a complex system, then at the beginning it needs to be prepared. We show this by the example of the interaction of gas with a metal plate. In this case, the algorithm consists of three main steps. The first stage involves calculating the equilibrium state of the gas volume V g we need. This can be done either completely again or by collecting the resulting large volume from unit cells with equilibrium gas. The second stage includes the calculation of the thermodynamic equilibrium of the metal plate V m . There are also two similar calculation options. The third stage consists of calculating quasiequilibrium in a gas-metal system consisting of a gas volume V g and a plate volume V m . At this stage, the volumes V g and V m are combined and joint calculations based on the Verlet scheme are performed. If the next task is to calculate the flow in the channel, then based on the results of steps 1–3, additional calculations are performed, also grouped into additional steps. At the fourth stage, the gas-metal system is replicated until the necessary channel configuration is obtained. For example, first, you can get a large system from one small gas-plate system (V g + V m ), in which the gas is bounded above and below by two large plates with a total volume of 2 * V m *nx and has a volume of 2 * V g * nx * nz (nx is the number of calibers of the new system in the x coordinate, 2 * nz is the number of calibers in the z coordinate). Then, the sidewalls of the channel can be added in a similar way (replication along the y coordinate). When such a system is formed, it should be further balanced. At the fifth stage, in the prepared channel, the gas is accelerated in the middle section. In our case, acceleration was carried out along the x coordinate. The width

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of the accelerated gas was 80% of the channel width. Acceleration lasts a short time and is carried out as a result of applying the Langevin thermostat. At the sixth stage, the Langevin thermostat is turned off and calculations are carried out to establish processes in the channel over a very long period of time. The results of the sixth stage are discussed below. In conclusion of the section, let us discuss the original software implementation of the numerical scheme based on the developed algorithm described above. This implementation initially involves the use of a cluster or supercomputer (hereinafter referred to as SC) with a central or hybrid architecture. Such a supercomputer on each node has either several multi-core Central Processing Units (CPUs) or several Vector Processing Units (VPU)/Graphic Processing Units (GPU). Parallelization of calculations is based on the principles of geometric and functional parallelism. For this, the computational domain as a whole is divided into local domains of the same power. The power of a domain is measured in the number of elementary boxes, falling into which atoms and molecules necessarily interact with each other. The domain decomposition is performed within the topology of a three-dimensional lattice. Each computational domain falls into a specific SC node. The distribution of domains by SC nodes is implemented on the base of the geometrical partition of the domain. Interaction of SC nodes during calculations is performed using the MPI communication standard. Inside the computational domain, there appears a certain number of elementary boxes grouped into a three-dimensional sublattice. This set is distributed between the trading devices of the SC node. This decision is due to the fact that the calculations inside the boxes have a higher intensity than between the elementary boxes. As a result, intermolecular interactions in the boxes are implemented in parallel mode through the use of OpenMP technology when using a CPU or VPU. When using graphics accelerators, a copy of the boxes in the GPU memory is created and the data necessary for calculating particle interactions is moved there. The implementation of such a mechanism uses CUDA technology. The results presented below were obtained using the K60 hybrid supercomputer, which has nodes with Intel Xeon CPUs and NVidia Volta V100 GPUs, as well as, a K48 cluster consisting of Intel Xeon Phi VPUs. The efficiency of parallel calculations was at least 70%.

2.4 Calculation Results Let us now consider some calculation results. As an example, the nitrogen flow inside a nickel microchannel with a length of 1–3 µm and a thickness of about 600 nm was considered. The calculated geometry contained two nickel plates (bottom and top) with layers of adsorbed nitrogen adhering to them and a free nitrogen layer between them (see Fig. 2.1). In this case, the real structure of the nickel-base of the microchannel, the effect of the initial adsorption of nitrogen molecules on the walls of the microchannel, and also the release of the energy of the walls of the

2 Correction of Boundary Conditions in Micromodels …

15

Fig. 2.1 Computational geometry

microchannel into the environment were taken into account. The calculations were carried out in the framework of the NVT model (fixed Number of particles, Volume, and Temperature). In the beginning, the gaseous medium was stationary and balanced using the Langevin thermostat with parameters corresponding to normal conditions (T = 273.15 K, P = 101,325 Pa, v = 0). Then the gas medium in the middle section was accelerated to a predetermined average velocity using the Langevin thermostat. Next, the thermostats turned off. The calculations were carried out for several widths of a nitrogen flow and several values of velocity, both exceeding the speed of sound in nitrogen and having significantly lower values. To speed up the calculations in the second direction (y), a thin layer and periodic boundary conditions were taken. The parameters of the three microsystems under consideration were the following: 1. Sizes of the lower and upper nickel plates in nanometers were 1017n×101.7×8.5. 2. Sizes of the gaseous medium between the plates in nanometers were 1017n × 101.7 × 614.5. 3. Number of nickel atoms in both plates was 162.57n millions. 4. Number of nitrogen molecules on the plates and between them was 42.38n millions (n = 1, 2, 3). The obtained calculated data showed the following. In the case of a supersonic flow during the initial evolution, the average transverse profile of the longitudinal velocity component vx  from the stepwise gradually turns into a Gaussian profile (see Fig. 2.2). In this case, small beats of velocity are observed near the channel walls associated with the nonlinear interaction of the gas not so much with the metal as with the gas adsorbed on its surface. In the case of a subsonic flow, the profile of the longitudinal velocity component vx  initially acquires a smooth shape characteristic of the Poiseuille flow.

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Fig. 2.2 Start evolution of the averaged transverse profile of the longitudinal velocity component for supersonic flow. The curves 1–6 corresponds to time moments t = 0, 0.2, 1, 2, 4, 8 ps

Long calculations showed that the final interaction of the flow with the walls of the microchannel is essentially non-linear and in some cases non-stationary in nature. For example, Figs. 2.3 and 2.4 present two-dimensional flow pictures at time t = 250 ps for subsonic and supersonic flows. Here, for comparison, in the first case, the subsonic flow velocity has changed. In the second case, the channel length has changed. The calculated flow pictures demonstrated that, in the subsonic regime, the transverse profile of the longitudinal velocity component acquires a smooth, almost parabolic shape, the center of which will shift to one or the other side relative to the channel walls. This portends over time to develop into a regime of large-scale turbulence. In a supersonic flow regime, the profile of the longitudinal velocity component acquires a sharper hyper-Gaussian shape, the center of which practically does not change its position. However, in the boundary layer, small-scale beatings of the gas flow against the channel walls are observed. Over time, this picture can develop into small-scale turbulence, however, not leaving the boundary layer. Now we discuss the behavior of the flow parameters in the boundary layer (i.e. near the channel wall). To clarify this issue, Fig. 2.5 shows the dynamics of the values of the main gas-dynamic parameters in several sections on coordinate z: – In the gas layer adsorbed on a metal (z = 11.12 nm). – In the moving layer closest to it (z = 12.18 nm).

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17

Fig. 2.3 The modulus and direction of the velocity vector for a subsonic gas flow in a microchannel, length is 1000 nm: a Mach = 0.29, b Mach = 0.58, c Mach = 0.86

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Fig. 2.4 The modulus and direction of the velocity vector for a supersonic gas flow in a microchannel, Mach = 1.15: a length is 1000 nm, b length is 2000 nm, c length is 3000 nm

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Fig. 2.5 Dynamics of flow parameters averaged over the x, y coordinates in different sections over z. Digit 1 corresponds to the layer of gas adsorbed on the metal. Digit 2 corresponds to the layer of free-moving gas nearest to it. Digit 3 corresponds to the Knudsen gas layer. Digit 4 corresponds to the central layer of gas (center of the channel). Curves 1 of density and pressure are multiplied by the value of 0.001. Dependencies of: a density, b pressure, c temperature, d x-component of velocity in time

– In the middle section of the Knudsen layer (z = 25.43 nm ≈ 1/3 of the mean free path in nitrogen under normal conditions). – In the center of the channel (z = 305.11 nm). A comparison of the above characteristics shows that a complex process is developing near the channel walls. First, the adsorbed gas does not move on the channel walls, has a very high density, and does not satisfy the usual equation of state (the pressure in this layer is negative and closes to the pressure in the metal). In this case, the gas temperature corresponds to the temperature of the metal. Second, a strong instability develops in the gas layer closest to the wall, which is associated with the interaction of the gas with the wall. Third, in the center of the Knudsen layer, a strong influence of the total gas flow is observed. Note further that the chaotic behavior of gas macroparameters near the walls is due to two factors. On the one hand, the calculation of these macroparameters is not very accurate due to the strong rarefaction of the gas. On the other hand, the presence of a gas layer adsorbed on a metal affects the rest of the gas non-linearly

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Fig. 2.6 Structure of gas flow near the solid surface

and non-stationary. These factors are taken into account below when formulating the boundary layer model. A detailed analysis of the calculated data leads us to the gas flow structure shown in Fig. 2.6. It consists of four layers, between which there is an exchange of mass, momentum, and energy. Given this structure, we can say that under the conditions of the adsorption effect, the main gas flow actually interacts not with the metal wall, but with the buffer gas and gas adsorbed on the surface. In the main gas layer, the equations of gas dynamics are valid. In the buffer gas layer, the equations of multiphase hydrodynamics should be solved. In the adsorbed gas layer, it suffices to confine to the two-dimensional convection–diffusion equation. On a solid wall, it is necessary to consider the processes of thermal conductivity and scattering of the normal component of the momentum. At the boundary of the layers, the conditions of continuity of the corresponding flows of mass, momentum, and energy are set. The formulation of these equations and boundary conditions is the subject of future analysis. Next, you need to decide which model of the boundary layer one wants to get. When it comes to the application of this model in macroscopic tasks, it is usually limited to calculating the accommodation coefficients by mass, momentum, and energy (temperature). If we are talking about multiscale modeling based on a combination of macro- and microscopic models, then the best, albeit computationally intensive, calculation option is to select boundary layers and carry out molecular calculations in them followed by matching solutions at the interface. In our works, we used both approaches [20, 31–33]. Here, we explain our approach to the calculation of accommodation coefficients. It largely corresponds to the work [34], where the coefficients of accommodation of hydrogen molecules to the surface of graphite were calculated. However, unlike [34], we proposed the calculation of not only the equilibrium accommodation coefficients (which are achieved by averaging over time) but also their instantaneous values.

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The fact is that the accommodation process (as our calculations show, see Fig. 2.5) is essentially unsteady and relatively fast proceeding. Even when calculating the thermodynamic equilibrium of a stationary gas near a metal surface, the accommodation process (adsorption) proceeds in several stages. The first stage is characterized by the accumulation of gas molecules near the surface of the metal. The second stage is characterized by the interaction of adsorbed molecules with metal atoms and molecules of overlying gas layers. With a noticeable movement of the gaseous medium along the surface, unsteady processes occur associated with the alternation of the first and second stages. In this situation, we propose the following calculation scheme. We assume that the penetration of gas molecules into the metal surface does not occur (either this process is impossible, or it has reached saturation). Then the width of the adsorption layer will weakly depend on the changing parameters of the gas in the volume and will be determined only by the interaction potential of gas molecules with metal atoms. As a result of this, at stage 3 (see Sect. 2.3), a primary adsorption layer with a fixed height h a0 is released, which is determined by the jump in the density or concentration of gas near the metal surface. In our calculations of the nitrogen is the nickel system under normal conditions, the width of the adsorption layer h a0 was approximately 1.06 nm, which corresponds to three nickel lattice lengths (a = 0.35314 nm at T 1 = 273 K, see [12]). With a change in the gas temperature within ±100 K, the value h a0 does not change, but the concentration levels in the surface layers of the gas change (increase or decrease, respectively). Let us consider in more detail the four microsystems of particles that form near the surface of metal under thermodynamic equilibrium using the same nickel-nitrogen system as an example. The first of them consists of N10 nickel atoms in the adsorption layer. The second microsystem consists of N20 gas molecules adsorbed on the surface of the metal and is located in the same layer. The third microsystem consists of N30 gas molecules in the next layer of the same height h a0 . The fourth microsystem consists of N40 gas molecules in the volume, from which we distinguish a sublayer with a height h 0λ equal to the order of the average mean free path of nitrogen under normal conditions λ ≈ 76 nm. The selected four subsystems have temperatures Tk0 and momentum pressures Pk0 (k = 1, 2, 3, 4). The physical motivation for considering these four microsystems is that the first microsystem is a metal; the second microsystem actually exhibits metallic properties that are not characteristic of gas at a given temperature. The third microsystem is unstable and has parameters that are very different from the gas parameters in the volume (fourth microsystem). An analysis of the obtained data shows that the quasiequilibrium coefficient of accommodation of gas molecules on a metal surface by concentration or density can be determined only on the basis of the MD calculation. Its true value can be determined by the parameters of the adsorption layer namely by the ratio provided by Eq. 2.6. αN ≈

N10

N20 + N20

(2.6)

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In calculations, α N ≈ 0.3402, α N ≈ 0.3306, and α N ≈ 0.3497 for T 2 = 273 K, T 2 = 200 K, and T 2 = 346 K, respectively. The values correspond to the fact that at low flow velocities (or in its absence), the accommodation of the gas with the metal surface is weak for metals with a small atomic mass. The quasiequilibrium accommodation coefficient for temperature (energy) for subsonic flows (and, therefore, in a stationary gas) in accordance with [35] should be calculated by Eq. 2.7. αT ≈

T30 − T20 T30 − T10

(2.7)

In calculations, this value is equal to α T ≈ 0.8906, α T ≈ 0.9205, and α N ≈ 0.9409 for T 2 = 273 K, T 2 = 200 K, and T 2 = 346 K, respectively. The data obtained are close to the experimental data from [35]. The calculation of the accommodation coefficients by momentum (velocity) for gas at rest is not so representative. In this situation, a pressure accommodation coefficient is usually considered. When calculating it, we note that the analyzed system has a distinguished direction (z), along which there is a small gradient of density and temperature. Therefore, the pressure accommodation coefficient is also calculated in the direction z, that is, normal to the wall (α n = α z ).The tangential components of the pressure tensor and, accordingly, the accommodation coefficients α τ are equal to zero. Calculation of normal accommodation coefficient by pressure can be done according to Eq. 2.8. αn ≈

0 0 Pz,3 − Pz,2 0 0 Pz,3 − Pz,1

(2.8)

0 Here, Pz,k is the z-component of the momentum pressure of the particle microsystem in the corresponding layer. In calculations, this value is equal to (α n ≈ 0.1106, α n ≈ 0.1023, and α n ≈ 0.1283 for T 2 = 273 K, T 2 = 200 K, and T 2 = 346 K, respectively. The obtained values correspond to the quasiequilibrium state of the system. In the case of a nonequilibrium system, these coefficients will differ at each moment of time depending on the calculation conditions and depend on the location of the particles (in which layer they are located).

2.5 Conclusions This chapter presents the technology of direct molecular dynamic supercomputer modeling of physical processes in gas-metal systems. Its application allows to calculate both the properties of gas and metal media separately and together taking into account the real conditions of the full-scale experiment. Original techniques, their software implementations, as well as some interesting molecular modeling results

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obtained by the authors were discussed. The chapter presents the calculation of gas-dynamic microflow in the channel of a technical system. Using the developed technology, nonequilibrium processes of nonlinear interaction of a nitrogen flow with the walls of a nickel channel were calculated. These calculations made it possible to form a macroscopic model of the boundary layer in several versions, depending on the objective function. The selection of specific parameters of the model shows that, with certain restrictions on the velocity and density of the gas flow, this model can be used in the framework of traditional CFD modeling. However, in the general case, a multiscale simulation technique using a combination of grid and molecular dynamics calculations should be used. Acknowledgements This work was supported by the Russian Foundation for Basic Research (grants No. 18-07-01292-a, 18-37-20062-mol_a_ved, 20-07-00790-a). The calculations were performed on the hybrid supercomputers K60 and K48 situated in the Collective Usage Centre of KIAM RAS.

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15. Podryga, V.O., Vikhrov, E.V., Polyakov, S.V.: Molecular dynamic calculation of macroparameters of technical gases by the example of argon, nitrogen, hydrogen, and methane. Math. Models Comput. Simul. 12(2), 210–220 (2020) 16. Podryga, V.O., Polyakov, S.V., Puzyrkov, D.V.: Supercomputer molecular modeling of thermodynamic equilibrium in gas-metal microsystems. Vychislitel’nye Metody I Programmirovanie 16(1), 123–138 (2015) 17. Podryga, V.O., Polyakov S.V.: Molecular Dynamic Calculation of Gas Macroparameters in the Stream and on the Boundary. KIAM Preprints 80 (2016) 18. Podryga, V., Polyakov, S.: Calculation of nitrogen flow in nickel micronozzle based on numerical approaches of gas and molecular dynamics. In: V International Conference Particle-Based Methods, Fundamentals and Applications, pp. 744–754. CIMNE, Barcelona, Spain (2017) 19. Podryga, V., Polyakov, S.: The computer simulation of nonlinear processes in gas-metal Microsystems. In: Uvarova, L.A., Nadykto, A.B., Latyshev, A.V. (eds.) Nonlinearity: Problems, Solutions and Applications, vol. 1, pp. 413–428. Nova Science Publishers Inc., New York (2017) 20. Kudryashova, T.A., Podryga, V.O., Polyakov, S.V.: Investigation of gas-dynamics processes in a boundary layer on a basis of molecular dynamics simulation. IOP Conf. Ser.: Mater. Sci. Eng. 657, 012026 (2019) 21. Mie, G.: Zur kinetischen theorie der einatomigen korper. Ann. Phys. 11(8), 657–697 (1903) 22. Fokin, L.R., Kalashnikov, A.N.: The transport properties of an N2–H2 mixture of rarefied gases in the EPIDIF database. High Temp. 47(5), 643–655 (2009) 23. Daw, M.S., Baskes, M.I.: Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B. 29(12), 6443–6645 (1984) 24. Zhou, X.W., Johnson, R.A., Wadley, H.N.G.: Misfit-energy-increasing dislocations in vapordeposited CoFe/NiFe multilayers. Phys. Rev. B. 69, 144113 (2004) 25. Lennard-Jones, J.E.: Cohesion. Proc. Phys. Soc. 43(5), 461–482 (1931) 26. Lorentz, H.A.: Uber die Anwendung des Satzes vom Virial in der kinetischen Theoric der Gase. Ann. Phys. 248, 127–136 (1881) 27. Berthelot, D.: Sur le Melange des Gaz. Comptes Rendus De L’Academie Des Sciences 126, 1703–1706 (1889) 28. Heermann, D.: Computer Simulation Methods in Theoretical Physics. Springer, Berlin, Heidelberg (1990) 29. Berendsen, H.J.C., Postma, J.P.M., van Gunsteren, W.F., DiNola, A., Haak, J.R.: Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81, 3684–3690 (1984) 30. Verlet, L.: Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159, 98–103 (1967) 31. Podryga, V.O.: Computational technology of multiscale modeling the gas flows in microchannels. IOP Conf. Ser.: Mater. Sci. Eng. 158, 012078 (2016) 32. Polyakov, S., Podryga, V.: Multiscale multilevel approach to solution of nanotechnology problems. EPJ Web Conf. 173, 01010 (2018) 33. Kudryashova, T., Karamzin, Yu., Podryga, V., Polyakov, S.: Two-scale computation of N2–H2 jet flow based on QGD and MMD on heterogeneous multi-core hardware. Adv. Eng. Softw. 120, 79–87 (2018) 34. Kovalev, V.L., Yakunchikov, A.N.: Accommodation coefficients for molecular hydrogen on a graphite surface. Fluid Dyn. 45(6), 975–981 (2010) 35. Abramovich, G.N.: Applied Gas Dynamics, 3rd edn. Nauka, Moscow (in Russian) (1973)

Chapter 3

Development and Application of the SMIF Method for the Investigation of Incompressible Fluid Flows Valentin A. Gushchin

Abstract The development of splitting method for incompressible fluid flows (SMIFs) during last 45 years is described. The hybrid explicit finite-difference scheme of SMIF method is based on modified central difference scheme (MCDS) and modified upwind difference scheme (MUDS) with special switch condition depending on the velocity sign and the signs of the first and second differences of transferred functions. Application of this method for solving some tasks is demonstrated. For example, the spatial flow around a sphere and a circular cylinder for homogeneous and stratified fluids [in a wide range of dimensionless parameters of the problem including the transitional regimes (2D-3D transition, laminar-turbulent transition in the boundary layer)], plane problem of fluid flows with a free surface, dynamics of vortex pair in a water, collapse of spots in stratified fluid, and the air, heat, and mass transfer in “clean rooms”.

3.1 Introduction Many phenomena observed in the atmosphere and ocean, as well as problems of hydromechanics, hydraulics, acoustics, circulatory physiology, and organization of processes due to moderate speeds of movement of the environment, can be studied within the model of incompressible viscous fluid. Currently, the practice puts forward to the applied scientists’ different types of tasks, the full study of which can be carried out in most cases only through a computational experiment or by a carefully set physical experiment. However, the phenomena of practical interest and technological processes either not yield to physical modeling or the costs of conducting such experiments are too high.

V. A. Gushchin (B) Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya Ul., Moscow 123056, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_3

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Tasks of practical interest tend to be characterized by multidimensionality, unsteadiness, non-linearity, free borders, and boundary layers and are described by the Navier–Stokes equations. The non-linearity of the Navier–Stokes equations and the presence of a small parameter in older derivatives (especially for large Reynolds numbers) create serious difficulties both in their analytical study (it is, in fact, only possible for model equations or private tasks) and in numerically solving these equations with the help of computers. Mathematical modeling of such complex flows as detachable flows, especially in large Reynolds numbers, the flows with the free surface, the flows stratified by the density of fluid, as well as essentially three-dimensional flows in technological devices and production facilities of special purpose, imposes a number of requirements on the methods used and developed methods of solving equations describing these flows. Such requirements include the following: a high order of approximation of finite-difference schemes—second and above, minimal schemes dissipation and dispersion, workability in wide range of parameters (Reynolds, Froude numbers, etc.), and monotonicity. The latter property is especially important when modeling currents with areas of large hydrodynamic gradients during calculating flows with a free surface, as well as flows of stratified fluid. Development of effective numerical methods with the properties listed above and calculation of their use of essentially nonlinear currents of incompressible fluid are very relevant. Currently, quite a large number of numerical methods of solving the Navier– Stokes equations describing the flows of incompressible viscous fluid are known. Most of these methods were developed for a system of equations in the form of Helmholtz stream function—vorticity. A common drawback of these methods is the use in one form or another of the boundary condition for a vortex on the solid surface of the body, which is absent in the physical setting of the task. The presence of additional iterative process associated with this boundary condition for vorticity limits the convergence rate of numerical algorithms. It is obvious that a difference scheme, allowing to calculate the flows of viscous incompressible liquid without using the boundary condition for vorticity on a solid surface, under all other equal conditions has more efficiency. The limited methods of solution stream function-vorticity systems are also associated with a significant increase in the volume of calculations when they are generalized in the case of spatial flows because in this case, in addition to the three vorticity transfer equations, three Poisson equations for the components of vector potential need to be solved. Difficulties arise in the attempts to spread these approaches for calculations of flows with a free surface, where it is necessary to set the boundary conditions for the stream function and vortex on a pre-unknown and determined in the process of solving the boundary. These circumstances explain the recent increased interest in numerical solutions to the Navier–Stokes equations recorded in natural (primitive) variables. An important step in the development of methods for solving primitive equations was the method of markers and cells (MAC) [1] developed in the Los Alamos Scientific Laboratory. The main distinguishing features of this method are the following:

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the combination of Euler and Lagrange approaches to describe the movement of fluid, which is achieved by the use of markers—weightless and non-interacting particles, which greatly facilitates the tracking of moving boundaries in the calculation process, and use the “Chess” or staggered finite-difference mesh pattern for various unknown functions. MAC method is widespread and has a number of modifications to date. It should be noted, however, that in MAC-type schemes, due to selected differencing representations, the implementation of the adhesive condition leads, with the need, to determine the boundary value of the vorticity on the solid surface of the body, satisfying Tom’s condition. The latter is known to be a condition of the first order of accuracy regarding the step of the spatial grid. Moreover, all MAC approaches proposed so far do not allow for calculations for large Reynolds numbers, and, of course, the difference schemes of these approaches do not have the property of monotonicity. As a result, new approaches have emerged that use the idea of splitting by physical processes [2, 3]. One such approach was proposed in the early 1970s—SMIF method [4]. Over the past 45 years, SMIF method has been constantly modified. To date, the method has been summarized in the case of stratified fluid flows [5], flows with a free surface [6], and spatial flows [7]. In this chapter, we briefly note these modifications and demonstrate some examples of solved problems. The chapter is organized as follows. Section 3.2 provides a description of SMIF method. One can find some examples of the challenges under consideration in Sect. 3.3. Section 3.4 concludes the chapter.

3.2 SMIF Method Section 3.2.1 provides a discussion of splitting scheme. Finite-difference scheme is represented in Sect. 3.2.2. The principle of building a finite-difference scheme is given in Sect. 3.2.3.

3.2.1 Splitting Scheme Consider the Navier–Stokes Equations in Primitive Variables ∂v 1 + (v · ∇)v = −∇ p + v, ∂t Re ∇ · v = 0, where v is the velocity vector, p is the pressure. Suppose that at a time, t n = nτ, where τ is the time step and n is the number of steps, and the fields of velocity v and pressure p are given. Then, the scheme for

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finding the unknown functions at the time t n+1 = (n + 1)•τ can be written as v˜ − vn 1 = −(vn · ∇)vn + vn , τ Re τ p = ∇ · v˜ , vn+1 − v˜ = −∇ p. τ At stage I, it is assumed that the transfer of momentum occurs only due to convection and diffusion. At stage II, given the intermediate velocity field v˜ found earlier and taking into account the solenoidity of the velocity vector vn+1 , the pressure field is found by solving the Poisson equation. At stage III, it is assumed that the transfer of momentum occurs only due to the pressure gradient (there are no convection and diffusion).

3.2.2 Finite-Difference Scheme The flow region under examination is covered by grid of cells (the grid is uniform with respect to x and y):  =

xi+1/2 = i h x h x > 0, i = 0, 1, . . . , L , Lh x = X, y j+1/2 = j h y h y > 0, j = 0, 1, . . . , M, Mh y = 2Y,

where hx and hy are the grid sizes, and L and M are the numbers of grid cells in the directions x and y, respectively (the point with the coordinates (i, j) coincides with the cell center). Here, as in the original splitting method, the staggered grid is used. This allows us to visually interpret each cell as an element of the medium volume; such an element is characterized by the pressure pi,j , density ρi,j, salinity perturbation si,j (and possibly by the temperature, energy, etc.), and the divergence Di,j (D determines the presence of a source or sink in the volume depending on its sign) computed at the cell center. The knowledge of the normal velocity component on a cell side makes it possible to directly calculate the momentum flux through this side.

3.2.3 The Principle of Building a Finite-Difference Scheme Before we begin to construct the SMIF finite-difference scheme for different coordinate systems in the case of two and three spatial variables, let us remind that it must meet the following properties: a high order of approximation—second and higher,

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minimum scheme dissipation and dispersion, workability in a wide range of Reynolds numbers, and monotony. The idea of building a heterogeneous (hybrid) scheme that possesses the above properties let us consider on the example of a one-dimensional model linear transfer equation f t + a f x = 0, a = const, where f is the desired function, a = const. Consider two-parametric family of difference schemes: n n f i+1/2 − f i−1/2 f in+1 − f in +a = 0, τ h

 n f i+1/2

=

n n α f i−1 + (1 − α − β) f in + β f i+1 a≥0

. n n α f i+2 + (1 − α − β) f i+1 + β f in a < 0 n Similar equations exist to the view of f i−1/2 . This family contains several widespread schemes. For α = 0 and β = 0, we have Godunov’s classical scheme of the first-order accuracy in the spatial variables, and for α = 0 and β = 0.5, the central difference scheme of second-order accuracy is formed, while for α = –0.5 and β = 0, we receive the upwind difference scheme of second-order accuracy. To assess the properties of the above two-parameter family of difference schemes, write a first differential approximation by expanding the grid functions into the Taylor series in a neighborhood of the point (i, n):  τ a2 h |a|(1 + 2α − 2β) − fx x . ft + a f x = 2 2 

It is known that for such equation it is impossible to build a homogeneous monotonous scheme higher than the first order of approximation. Therefore, the construction of a monotonous high-order scheme is carried out either on the basis of homogeneous second-order schemes using smoothing operators or on the basis of hybrid schemes using different criteria of switching from one scheme to another depending on the nature of the solution, possibly using smoothing. This work uses a hybrid monotonous scheme, based on a combination of modified MCDS and modified MUDS [8]. n n n n ≡ f i+1 − f in , 2 f i+1/2 =  f i+1 −  f in . Courant number Suppose that  f i+1/2   n (explicit scheme) is in interval 0 < C ≤ 1. If a ·  f · 2 f i+1/2 ≥ 0, then modified n  MUDS is used with α = –0.5 (1 – C), β = 0. Otherwise, if a ·  f · 2 f i+1/2 < 0, then modified MCDS is used with α = 0, β = 0.5 (1 – C). These schemes are schemes of the second order of approximation in both spatial and time variables and have zero scheme viscosity in the sense of equal zero ratio at the second derivative in the first differential approximation. The hybrid scheme built in this way meets the above requirements: on smooth solutions there is a second order of approximation on time and spatial variables.

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This scheme has minimal scheme viscosity and dispersion; stable when 0 < C ≤ 1, and monotonous [8]. The same idea is used in the construction of a finite-difference scheme in each direction when studying 2D and 3D tasks. For the case of two spatial variables (2D), the finite-difference scheme is given in [9].

3.3 Some Examples of the Challenges Under Consideration Hereinafter, the results of flows’ simulation around the sphere, circular cylinder, and with a free surface are represented in Sects. 3.3.1–3.3.3, respectively. Section 3.3.4 describes a process of forming a vortex pair and its dynamics. Collapse of spots in a stratified fluid is discussed in Sect. 3.3.5. Section 3.3.6 provides the calculating results of flows in clean rooms.

3.3.1 Flows Around a Sphere These flows have been investigated in details owing to the mathematical modeling on the supercomputers for the cases of homogeneous and stratified fluid in wide range of Reynolds numbers Re (1 < Re < 5·105 ) and internal Froude numbers Fr (0.005 < Fr < 103 ). The vortex structures in the sphere wake are shown in Fig. 3.1 for homogeneous fluid (Re = 350). Characteristics of calculated flows, such as time-averaged drag forces and lateral forces, angles of separation, and period are well consistent with experimental data and calculations of other authors [10]. The classification of flow regimes of the stratified viscous fluid around a sphere has been refined in [11].

Fig. 3.1 Flow around a sphere. Re = 350

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Fig. 3.2 Isosurfaces of two opposite streamwise components of the vorticity in the wake of the circular cylinder: a Re = 230, λ/d = 3.75, b Re = 320, λ/d = 0.83

3.3.2 Flows Around a Circular Cylinder These flows have been investigated in details owing to the mathematical modeling on the supercomputers for the cases of homogeneous and stratified fluid in wide range of Reynolds numbers Re (1 < Re < 5 × 105 ) and internal Froude numbers Fr (0.005 < Fr < 103 ) [12]. Let us consider the homogeneous viscous fluid flow regimes around a circular cylinder. For Re > 40, the periodical formation of vortex tubes is simulated in the wake. For Re = 191, 2D–3D transition is observed in the wake. It means that for Re > 191 there is a periodicity of the flow along the circular cylinder axis. For 191 < Re ≤ 300 and 300 ≤ Re ≤ 400, the periodicity scales are equal to 3.5d ≤ λ ≤ 4d (mode A) and 0.8d ≤ λ ≤ 1.0d (mode B), respectively (Fig. 3.2). Owing to our investigations it was found that the values of the maximum phase difference along the circular cylinder axis are approximately equal to 0.1 − 0.2Tf (for mode A) and 0.015 − 0.030Tf (for mode B), where the time Tf is the period of the separation flow [13].

3.3.3 Flows with a Free Surface. Hydraulic Jump Following the works [6, 14], we will examine the flow of the H 0 -thick fluid layer inclined on angle α to horizontal surface at the velocity U, coming on obstacle height h0 (Fig. 3.3). The coordinate system is chosen in such a way that its origin lies on the unperturbed free surface, the x-axis is directed horizontally in the direction of the incoming stream, and the y-axis is vertically upwards. The free surface η(x, t) is one of the outer boundaries of the fluid, where η is the surface deviates from the unperturbed level at the level of y = 0. The bottom is located at a depth of y = –H(x) and is a solid impenetrable surface.

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Fig. 3.3 Overall view of flow with parameters ψ = const, Fr2 = 4.8, k = 1.4, α = 1.5º

On the free surface, the kinematic and two dynamic conditions are placed. The main difficulty in numerically solving problems with a free surface is the need to calculate in areas of complex form with time-changing configuration. To overcome this difficulty, it is proposed to adopt an approach using a mobile coordinate system and allowing to display the calculated area on a rectangle in a plane (x, ξ) at any given time: x = x, ξ =

y + H (x) , t = t. η(x, t) + H (x)

The structure of the surface wave in this flow depends both on the parameters of the incoming flow (U, H 0 ) and geometric characteristics of obstacle.√The main characteristic parameters of the task are the number of Froude Fr = U/ g H0 and the relative height of the obstacle k = h0 /H 0 . It is known from the experiment that even in the modes with the rollover of the wave front, the area of rollover and the boundary layer near the obstacle are separated by a zone of quasi-potential current. This allows us to make an assumption about the weak dependence of flow modes on the type of boundary conditions on the surface of the obstacle. In the calculations below, the fluid was considered as a non-viscous fluid. The overall picture of the flow is presented in Fig. 3.3, where the streamlines obtained as a result of the numerical experiment are given. For comparison, on a free surface, the crosses mark points corresponding to experimental data [14]. Free surface profiles, obtained numerically and experimentally, are almost identical. The height of the jump is equal to 1.7 H 0 that fits well with the theoretical value 1.65•H 0 [15]. Profiles of the longitudinal component of the velocity vector in different crosssections are depicted in Fig. 3.4. AA section corresponds to the incoming flow. According to the results of the experiment, in the specified range of Froude numbers, the presence of viscosity effects near the bottom of the canal leads to the appearance of a boundary layer, which is nevertheless separated from the area of tipping by the quasi-potential flow zone. This allows the boundary layer to be neglected to roll over the rollover area, i.e., to exclude the viscosity effects near the solid surface from consideration. Measurements of the longitudinal component of the velocity vector showed the presence of reverse flows in the surface layer of the rollover zone (BB, CC, and DD sections) and their absence outside it (EE and FF sections). Therefore, it can be assumed that the flow in the tipping zone is vortex, as evidenced by the results

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Fig. 3.4 Profiles of the horizontal component of the velocity vector in different vertical crosssections. Signs—experiment [14], solid lines—calculation results [6]: Fr2 = 4.8, k = 1.4, α = 1.5º

of the numerical calculation at the above flow parameters in the same sections where the experimental measurement of velocity profiles was conducted (Fig. 3.3). Note that outside the boundary layer zone, the coincidence of the results is quite good. Similar modes are observed in numerical modeling of the flow around the underwater obstacle by the stream of stratified by the density fluid in the presence of a quasi-heterogeneous layer near the free surface [6]. There is a periodic flow with the formation of spots of mixed fluid in the vicinity of the lower boundary of the quasi-uniform layer, which move toward the incoming stream.

3.3.4 Dynamics of the Vortex Pair The task of forming a vortex pair in a homogeneous non-viscous fluid is investigated numerically. Let two vortexes of the radius r 0 and the same intensity of G, but with the opposite direction of rotation, are arranged symmetrically relative to the vertical axis. Distance between the centers of vortex is b. Due to the symmetry of the task, one can only consider the flow in the right half-plain. For certainty, let the right vortex rotate clockwise. One of the latest variants of SMIF method developed by the authors is used to solve the problem [8]. It is shown that vortex pair is formed only at relatively short distances between the centers of vortexes. As this distance increases, the pair is formed longer in time. In the future, it is planned to take into account the impact of viscosity and stratification on this formation. The process of forming a vortex pair and the beginning of its dynamics is presented in Fig. 3.5, which shows the isoclines of vorticity for the moments of time t = 1, 5, 15, 30. One can see that single vortexes set at the initial moment of time by the velocity form a pair only to t = 5. It should be noted that the distance between the centers of vortexes has decreased from 2 to 1.5. By the time of t = 15, this distance has been

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Fig. 3.5 Isoclines of vorticity. a t = 1, b t = 5, c t = 15, d t = 30

reduced to 1 and is further maintained up to a top boundary. Then the vortexes run under a top boundary in opposite directions. Note also that the induced (lower) vortex creates a pair in the lower half-plane with its antipode and, keeping the original distance between the centers, moves to the lower boundary. With the increase in b a pair is formed “in agony” and for b = 2.5 the pair is formed only to t = 25. For b = 3 the pair is formed only to t = 60.

3.3.5 Collapse of Spots in a Stratified Fluid The problem on the dynamics (collapse) of mixed fluid spots in a stratified medium is discussed. Salinity is used as the stratifying component. This problem is described by the Navier–Stokes equations in the Boussinesq approximation subject to the corresponding boundary and initial conditions. The results of comparison with analytical estimates, experimental data, and computations of other researchers have been presented in [5]. These results are in

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Fig. 3.6 Isoclines of stream function: a t = 2, b t = 8

good agreement, which confirms that the proposed model can be used for the investigation of other similar problems. The computations showed that the spot develops asymmetrically with respect to the horizontal axis, and the variation of the vertical spot size in time is not monotone and quasi-periodic. The isoclines of stream function for t = 2 and t = 8 are shown in Fig. 3.6a, b, respectively [5].

3.3.6 Calculations of Flows in Clean Rooms Other problem, which also may be considered in the frame of incompressible fluid flows, is the clean room problem. For the designing and construction of the clean rooms for aerospace, microelectronic, pharmaceutical, chemistry, and food processing industries, one can ensure minimum contamination and optimal heat conditions. The clean rooms are three-dimensional objects with complex geometry. This is the distributed systems through the air inlet and outlet vents or perforated ceiling and floor hosted by equipment with complex shapes and moving robots with the sources of particles of different sizes and various laws of their motion. The air, heat fluxes, and movement of particles are three-dimensional and turbulent in terms of gas dynamics. The air, heat, and mass transfer in the clean rooms for the pharmaceutical industry is considered. The following example demonstrates how to use the applied package Clean Room Air-dynamic Guide (CRAG) based on SMIF method at schematic design phase, i.e., when the designer wants to get answers for some questions of interest and select the optimal solution before to start designing. Let us consider an example of designing a weight room where one intends to weigh some powder substances for the pharmaceutical industry. Room configuration consisting of three compartments with the tables and operators is shown in Fig. 3.7. Air is blown into the room through

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Fig. 3.7 Example of clean room configuration and simulation: a clean room configuration, b streamlines, c the picture for concentration of powder substance

the ceiling with a speed of 0.02 m/s and is discharged through the bars with a height of 0.5 m at the bottom of the front wall of the room. Supply air temperature is 22 °C. The main question was the following: Is it possible the powder substance from a table of one of the compartments to fall into other compartments. If the temperature in the room and all objects within it (including operators) equals to the temperature of inlet air, then the situation will prove to be quite acceptable. However, if we take into account that the body temperature of the operator is equal to 36 °C, then the vertical convective flow can significantly change the situation.

3.4 Conclusions The SMIF method with explicit, hybrid finite-difference scheme (second-order approximation in the space variables, the minimum scheme viscosity and dispersion, monotonicity), which is used in this chapter for the direct numerical simulation of internal and external dimensional flow of an incompressible viscous fluid, was

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described. We consider a number of different tasks that have been resolved for the past 45 years: the spatial flow around a sphere and a circular cylinder for homogeneous and stratified fluids [in a wide range of dimensionless parameters of the problem including the transitional regimes (2D-3D transition, laminar-turbulent transition in the boundary layer)], plane problem of fluid flows with a free surface, dynamics of vortex pair in water, collapse of spots in stratified fluid, and the air, heat, and mass transfer in “clean rooms” (intended for the manufacture of microelectronics products, pharmaceuticals, medicine, and biotechnology). Many of the results of solving these problems are in a good agreement with experiments, indicating the high efficiency and reliability of SMIF method. Acknowledgements The author is very appreciated to academician O. M. Belotserkovskii, who suggested this theme for development around 50 years ago and to all the participants of these works.

References 1. Harlow, F.H., Welch, J.E.: Numerical calculation of time—dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8(12), 2182–2189 (1965) 2. Chorin, A.J.: Numerical solutions of Navier-Stokes equations. Math. Comput. 22, 745–762 (1968) 3. Fortin, M., Peyret, R., Temam, R.: Résolution numériques des equations de Navier-Stokes pour un fluide incompressible. J. De Mécanique 10(3), 357–390 (1971) 4. Belotserkovskii, O.M., Gushchin, V.A., Shchennikov, V.V.: Use of the splitting method to solve problems of the dynamics of a viscous incompressible fluid. USSR Comput. Math. Math. Phys. 15(1), 190–200 (1975) 5. Gushchin, V.A., Smirnova, I.A.: The splitting scheme for mathematical modeling of the mixed region dynamics in a stratified fluid. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics, SIST, vol. 173, pp. 11–21. Springer Nature, Singapore (2020a) 6. Belotserkovskii, O.M., Gushchin, V.A., Kon’shin, V.N.: The splitting method for investigating flows of a stratified liquid with a free surface. USSR Comput. Math. Math. Phys. 27(2), 181–191 (1987) 7. Gushchin, V.A., Narayanan, P.S., Chafle, G.: Parallel computing of industrial aerodynamics problems: clean rooms. In: Schiano, P., Ecer, A., Periaux, J., Satofuka N. (eds.) Parallel Computational Fluid Dynamics 1996: Algorithms and Results Using Advanced Computers, pp. 305–311. Elsevier Science B.V. (1997) 8. Gushchin, V.A.: Family of quasi-monotonic finite-difference schemes of the second-order of approximation. Math. Models Comput. Simul. 8(5), 487–496 (2016) 9. Gushchin, V.A., Smirnova, I.A.: Mathematical modeling of spot dynamics in a stratified medium. Comput. Math. Math. Phys. 60(5), 879–894 (2020b) 10. Gushchin, V.A., Matyushin, P.V.: Vortex formation mechanisms in the wake behind a sphere for 200 < Re < 380. Fluid Dyn. 41(5), 795–809 (2006) 11. Gushchin, V.A., Matyushin, P.V.: Numerical simulation and visualization of vortical structure transformation in the flow past a sphere at an increasing degree of stratification. Comput. Math. Math. Phys. 51(2), 251–263 (2011) 12. Gushchin, V.A., Mitkin, V.V., Rozhdestvenskaya, T.I., Chashechkin, Y.D.: Numerical and experimental study of the fine structure of a stratified fluid flow over a circular cylinder. J. Appl. Mech. Tech. Phys. 48(1), 34–43 (2007)

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13. Gushchin, V.A., Kostomarov, A.V., Matyushin, P.V.: Direct numerical simulation of the transitional separated fluid flows around a circular cylinder. Japan Soc. CFD: CFD J. 10(3), 338–343 (2001) 14. Belotserkovskii, O.M., Burynin, E.E., Gushchin, V.A., Kon’shin, V. N., Krasnikov, Yu.G., Urakov, P.Ya.: Flow of a fluid across an obstacle with breaking of the wave front. Fluid Dyn. 20, 423–426 (1985) 15. Stoker, J.J.: Water Waves. Interscience, New York (1957)

Chapter 4

On 32nd-Order Multioperators-Based Schemes for Fluid Dynamics Andrei I. Tolstykh

Abstract Multioperators-based schemes with the 32nd-order multioperators for fluid dynamics calculations are described. Their properties are illustrated by numerical examples for smooth and discontinuous solutions. An approach to their parallel implementation is outlined.

4.1 Introduction The concept of multioperators (i.e., linear combinations of basis grid operators) was introduced in [1] as an alternative to the conventional strategy of increasing approximation orders of numerical methods consisting in increasing the numbers of basis functions in the of underlying Lagrangian interpolation polynomials. In the case of finite difference schemes for partial differential equations, it means enlarging their stencils. In contrast, the multioperators techniques suggest increasing the number of basis operators without changing their stencils. It allows to circumvent many unwanted effects associated with high degrees Lagrangian polynomials. Using multioperators offers an opportunity to create numerical analysis formulas with grid operators having a desirable high approximation order. In [2], several examples of such formulas are presented. However, their main applications are in the area of grid approximations to partial differential equations. Previously, several types of the multioperators-based schemes for computational fluid dynamics (CFD) applications were constructed. Their early versions are reported in [3, 4]. They were used for high fidelity numerical simulations of flow instabilities in the cases of subsonic jets, under-expanded supersonic jets [5, 6], and vortices [7]. The complete theory of multioperators contains in [8]. Brief outlines of their main ideas and applications are presented in [2]. A. I. Tolstykh (B) Federal Research Center “Computer Science and Control” of the RAS, 40, Ul. Vavilova, Moscow 119333, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_4

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In the present chapter, the main emphasize is placed on the latest multioperators which use two-point operators only. When applied to derivatives approximations, they allow to construct arbitrary even-order schemes for convection, convection– diffusion, as well as fluid dynamics equations. The present study concerns with 32nd-order multioperators as a discretization tool for convection terms or spatial derivatives of the Euler equations. Numerical examples illustrating the properties of the resulting schemes in the cases of smooth and discontinuous solutions are presented. The main idea concerning derivatives approximations via multioperators is described in Sect. 4.2. In Sect. 4.3, the architecture of multioperators-based schemes, which serve as the basis for constructing the present 32nd-order scheme, is outlined. Section 4.4, where the numerical examples illustrating the main properties of the scheme are presented, is followed by conclusions provided in Sect. 4.5.

4.2 Family of Even-Order Multioperators for First Derivatives Starting from brief theoretical outlines, consider a uniform mesh ω_h with the mesh size h. According to the multioperators theory [2, 8], the approximations to the spatial derivatives (say, to the first x-derivative) look as Eq. 4.1, where the basis operators l(ci ) defined on ωh are generated by one-parameter family l(c) of compact approximation to the derivatives by fixing M values of the parameter c. L M (c1 , c2 , . . . , c M ) =

M 

γi l(ci )

i=1

M 

γi = 1

(4.1)

i=1

Once the parameter’s values are specified, the ci coefficients can be uniquely defined to kill M − 1 terms in the Taylor expansion series for the multioperators actions on smooth functions in terms of mesh sizes h. It gives the growth of the approximation orders proportional to either M or 2M. The multioperators family considered in the present chapter uses l(ci ) defined by two-point operators. To describe them, consider the operators pair of the form of Eq. 4.2, where − and + are the two-point left and right differences. It can be shown that the inverses Rl (c) and Rr (c) do exist if c > 1/2 + δ, δ > 0. Rl (c) = I + c+ Rr (c) = I − c−

(4.2)

It is supposed that the condition is met. Using operators Eq. 4.2, the following “left” and “right” first-order approximations to the derivative can be constructed: L l (c) =

1 Rl (c)−1 − , h

L r (c) =

1 Rr (c)−1 + . h

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To calculate them, very simple two-diagonal inversions can be performed. In the following, we consider grid operators as operators in the Hilbert space of grid functions with summable squares  defined on ωh , the inner product of two grid functions u and v being (u, v) = h ∞ j=−∞ u j v j . Skipping mathematical details, it can be shown that L l (c) > 0 and L r (c) < 0 meaning that, using the CFD slang, the operators form an upwind-down pair. Setting l(c) = L l (c) and l(c) = L r (c) in Eq. 4.1, one obtains “left” and “right” multioperators provided by Eq. 4.3 with the same γi coefficients. L M,l =

M  i=1

γi L l (ci ) L M,r =

M  i=1

γi L r (ci )

M 

γi = 1

(4.3)

i=1

  They define one-sided approximations to the derivative with O h M truncation error. It can be shown that the multioperators form an upwind-down pair provided that the input values ci are chosen to give L M,l (c1 , c2 , ..c M ) > 0. Some drawback of the multioperators is a limitation on the number M of basis functions. It is due to a steep increase of the condition number of the linear system for γi coefficients which matrix can be reduced to the Vandermonde one. Thus, the skew-symmetric operator l(c) = l0 (c) = (L l (c) + L r (c))/2, which provides the second-order approximation to the derivative, was chosen as the generator of the basis operators. In contrast to the previous case, the matrix of the systems for γi is no longer reducible to the Vandermonde one and the existence and uniqueness of the multioperators is not directly evident. However, those properties can be verified for each chosen input data c1 , c2 , . . . , c M . Once the Taylor expansion series of the action of l0 (c) contains only even powers of h, the action of the related multioperators contains them as well. It can be shown that for sufficiently smooth f (x) one has Eq. 4.4.    df = L M (c1 , c2 , . . . , c M ) f + O h 2M L M (c1 , c2 , . . . , c M ) = γi l0 (ci ) dx i=1 M

(4.4) To simplify the investigation of the multioperators properties, the ci values were chosen to be uniformly distributed between their minimal and maximal values cmin , cmax . Calculations for the various pairs did not reveal degeneration of the systems for γi . Moreover, their condition numbers were found to be of the order of unity for certain values cmin , cmax up to M = 18. Noting that difference d(c) = L l (c)− L r (c) is self-adjoint positive operator, it can be used for constructing the multioperators introducing a necessary for computational fluid dynamics dissipation mechanism. To do this, a set of the parameter’s values c1 , c2 , . . . , c M which optionally can be that for the main operator can be used. The multioperator looks as Eq. 4.5.

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D M (c1 , c2 , . . . , c M ) =

M 

M 

γˆi d(ci )

i=1

γˆi = 1

(4.5)

i=1

Its action on a sufficiently smooth function u(x) projected into mesh ωh is esti mated as O h 2M−1 . The values ci should be specified to guarantee positivity of the multioperator. It is assumed that the parameter’s values are uniformly distributed between their minimal and maximal values cmin , cmax .

4.3 Multioperators-Based Schemes Using the above multioperators for each spatial coordinate, various both explicit and implicit schemes can be constructed. The architecture of the schemes under consideration can be illustrated using model equation u t + f (u)x = 0. Their semidiscretized form with multioperators Eqs. 4.1 and 4.5 can be written using the indexfree form as Eq. 4.6. u t + L M f (u) + C D M u = 0 C = const

(4.6)

The scheme can be cast in the flux form as Eq. 4.7. q( f h ) j+1/2 − q( f h ) j−1/2 g(u h ) j+1/2 − g(u h ) j−1/2 ∂u j + +C =0 ∂t h h

(4.7)

That form follows from the multioperators representations looking as. LMu j =

q(u h ) j+1/2 − q(u h ) j−1/2 , h

DM u j =

g(u h ) j+1/2 − g(u h ) j−1/2 , h

where  1   −1 γi Rr (ci )u m+1 + Rl−1 (ci )u m , 2 i=1 M

q(u h )m+1/ 2 =

 1   −1 g(u h )m+1/ 2 = γi Rr (ci )u m+1 − Rl−1 (ci )u m . 2 i=1 M

(4.8)

Using Eq. 4.7, it is possible (if needed) to correct fluxes and control the dissipation level when dealing with discontinuous solutions. To ensure high resolution of small solutions details, parameters cmin cmax are supposed to be chosen to minimize the dispersion errors in the domain of large wave numbers supported by meshes in the case of the above model equation with f (u) = au, a = const. The minimization procedure can be accomplished by

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considering the Fourier image of L M which is a complex-valued function depending on two parameters. Once optimized, the second term in Eq. 4.6 serves as the main body of the scheme with stiffly fixed parameters. The estimates of the dispersion preserving property of the schemes are presented in [9]. Numerical tests for M = 8 and M = 16 have shown that the optimized scheme allows to preserve wave packets of high wave numbers harmonics during extremely longtime calculations. The scheme used in the present study is the particular case of Eq. 4.6 with M = 16. The third term in Eq. 4.6 with operator D M (cmin , cmax ) and constant C can be viewed as an optional one introducing O h 2M−1 truncation errors. Its role is to prevent numerical instabilities caused in particular by non-monotone properties of high-order schemes with minimal influence on physically relevant solutions components. That optimization can be achieved by choosing appropriate parameter’s values and minimal constant C. As a result, one obtains a dissipation mechanism playing role of a cut-of filter of spurious high wave number harmonics (see [9]). The semi-discretized scheme Eq. 4.6 can be readily extended to the Euler equations by constructing the multioperators for each spatial coordinate independently (see [9] for details). Approximations to the Navier–Stokes equations can be created by adding some discretization of the viscosity terms. Though the fourth- or sixth-order compact approximations were found to be appropriate for that purpose, one can use also higher-order multioperators. Starting from semi-discretized scheme Eq. 4.6, both explicit and implicit versions of the final schemes can be constructed by introducing time-stepping devices. Below the explicit scheme with the Runge–Kutta procedure is considered.

4.4 Schemes with 32nd-Order Multioperators Setting M = 16, one the 32nd-order approximations to the spatial derivatives   obtains and (optionally) O h 31 dissipation terms. The optimization procedure allows to get values cmin , cmax for which Fourier image of the multioperators is very close to the image of the derivative operator for the dimensionless wave numbers kh up to kh = 2.5 which gives formally the minimal resolvable wavelength λ = 2π /k ≈ 2.5 h. Below numerical examples illustrating the behavior of numerical solutions in both smooth and discontinuous cases are presented. Issues concerning parallel calculations are discussed. Thus, the periodic problem for the Burgers equation and verifying dispersion preserving property for large time intervals are considered in Sects. 4.4.1 and 4.4.2, respectively, while 32nd-order multioperators are presented in Sect. 4.4.3. Issues concerning the parallel calculations are discussed in Sect. 4.4.4.

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4.4.1 Smooth Solutions: Periodic Problem for the Burgers Equation To estimate the accuracy and the mesh-convergence rates of the 32th-order multioperators in the case of smooth solutions, the standard periodic problem for the inviscid Burger’s equations (see, e.g., [10]) was considered. It reads as. ∂ u2 ∂u + = 0, u(0, x) = 1 + 0.5 sin(π x), x ∈ [−1, 1]. ∂t ∂x 2

(4.9)

The exact solution up to certain value of t is smooth. It can be obtained using the iterative procedure described in [10]) with the machine precision. The calculations with the dissipation constant C = 0 were carried out using the fourth-order Runge–Kutta method. To exclude the influence of the relatively low order of the time-stepping technique, sufficiently small values of the CFL number were used. The accuracy was estimated using the discrete C-norm of the deviation of the numerical solutions from the exact one. The solution errors at t = 0.3 and the local mesh-convergence orders kc calculated using 64 bits arithmetic are gathered in Table 4.1 for different numbers N of grid points. In Table 4.1, the schemes with the 32th-order multioperators are labeled by L 32 . Additionally, the data for the scheme with the 10th-order multioperators [11] and the fifth-order compact upwind (CUD) scheme [12] are also included in Table 4.1. Approximately, the solutions obtained with the fifth-order compact schemes [10, 12], as well as the WENO scheme [13] are found to show approximately the same convergence orders for the considered problem differing in their norms by a factor O(1). Thus, they can be viewed as the reference solutions. As follows from Table 4.1, the near machine accuracy is achieved using the 32ndorder scheme in the case of N = 128 mesh, the accuracy being up to seven orders of magnitude higher than that of the fifth-order scheme. However, the real accuracy of the scheme is expected to be considerably higher. Firstly, the solution is compared with the exact one obtained iteratively with 10−14 tolerance. Secondly, additional calculations not presented here showed that the error of the L 32 solution is about 10−18 for N = 128 when using 128-bit arithmetic and 10−18 tolerance. Table 4.1 Smooth solutions of the Burgers equation; C-norms of numerical solution errors N

16

32

64

128

L 32

1.3e−3

7.9e−6

1.1e−9

1.7e−14

7.1

12.7

16.0

1.5e−3

7.6e−6

1.4e−8

6.2e−12

7.7

9.1

11.1

6.1e−3

4.3e−4

1.6e−5

5.1e−7

3.4

4.7

5.0

kc L 10 kc CUD-5 kc

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4.4.2 Smooth Solutions: Verifying Dispersion Preserving Property for Large Time Intervals To estimate high resolution of shortest wave harmonics in the case of long-term time integrations, the benchmark problem suggested in [14] was considered. It is the initial value problem which reads.    ∂u ∂u + = 0, u(0, x) = [2 + cos(βx)] exp − ln 2(x/10)2 ∂t ∂x The task is to calculate the numerical solutions at t = 400 and t = 800 for β = 1.7 using mesh size h = 1. Deviations from the exact solution which is the traveling wave packet containing very short waves harmonics allow one to estimate the resolution, dispersion, and the dissipation properties of the tested schemes. To make the test more difficult, the calculations with the dissipation-free 32ndorder scheme were carried out for β = 2.2, that is, for higher wave number content of the packet. Figure 4.1 illustrates rather good agreement between the exact and the numerical solutions at time t = 15,000 which is considerably greater than required. The test illustrates the ability of the scheme to preserve high resolution during longtime calculations. That property is important when performing direct numerical simulations in the cases of problem related to flow instabilities, sound radiation, laminar-turbulent transitions, etc. Similar calculations carried out with the 16thorder schemes showed that though the dispersion relation preserves for greater β than β = 1.7 and up to greater time than t = 800 required in the test, the solution degradation occurs at considerably lesser time values than t = 15,000. Fig. 4.1 Numerical solution at t = 15,000 obtained with 32th-order multioperators (markers) and exact solution (solid lines)

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4.4.3 The 32nd-Order Multioperators in the Case of Discontinuous Solutions The flux form Eq. 4.7 of the multioperators allows one to use the scheme for approximating conservation laws. Examples of applications of the 16th-order multioperators scheme with the same architecture to solutions with strong discontinuities are presented in [9]. The idea of using high-order schemes in the cases of discontinuous solutions is to get the solutions, which are accurately described the positions and intensities of discontinuities and provide accurate flow fields away from them where the solutions are smooth. Calculations with the linear scheme. The presence of the dissipation mechanism for C > 0 in Eq. 4.6 creates the potential for suppressing spurious numerical oscillations typical for solutions produced by high-order schemes in the cases when discontinuities are not strong (e.g., in the cases of moderate Mach number flows). Thus, stable calculations are possible in those cases with the original linear scheme. To illustrate this, calculations with the 32nd multioperators approximations written in the flux form with dissipation terms (C = 0) were carried out for the Sod problem [15]. The initial data for that problem are given by Eq. 4.10. ρ = 1, u = 0, p = 1, x < 0.5 ρ = 0.12, u = 0, p = 0.1, x > 0.5

(4.10)

Figure 4.2 shows the density distribution obtained with 400 nodes on the background of the exact solution. Good agreement between the solutions is seen in this figure. Fig. 4.2 Numerical solution of the Sod problem: density distribution (markers) and the exact solution (solid lines)

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Choosing constant C and parameters cmin , cmax , one can adjust the intensities and spectral contents of the dissipation multioperators to the problems under considerations. Using nonlinear versions. In the previous case, high-order dissipation mechanism was sufficient to suppress spurious oscillations. However, additional monotonization devices are needed in the high Mach number cases. Following the well-known ways, nonlinear schemes can be constructed. Their versions considered in [9] for the 16th-order multioperators use flux corrections and the hybridizations in the form of weighted sums of low-order monotone, as well as high-order solutions calculated at each time step (according to the procedure suggested in [16]). As in the case of the linear scheme, the aim is to obtain numerical solutions combining reasonable shocks and contacts descriptions and high accuracy and high-resolution solutions away from the discontinuities. In [9], the hybrid schemes with the 16th-order multioperators were tested against one-dimensional problems (discontinuous solutions of the Burgers equation, extremal Riemann problems). The same approach was used when constructing hybrid schemes with the 32nd-order multioperators. An example of using the 32nd-order scheme in the framework of the hybrid schemes of [9] is presented in Fig. 4.3. It is the numerical solution of the Shu-Osher shock entropy wave interaction problem [17]. Once the exact solution is not available in that case, the mesh convergence can be estimated by using successively refined meshes. The initial data for x ∈ [−5, 5] are given by ρ = 3.857143, u = 0, p = 1000, x < −4 ρ = 1 + 0.2 sin 5x, u = 2.629369, p = 10.33333, x >= −0.4.

Fig. 4.3 Shu-Osher problem: a velocity distribution, b its details. Curves 1, 2, and 3 correspond to number of nodes 200, 400, and 800, respectively

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The number of grid points N = 200 and the calculation time t = 1.8 were those of [17]. The results are shown in Fig. 4.3 with the solution for N = 800 viewed as “near exact” one. Visually, they are very close to those presented in [17].

4.4.4 Parallel Implementation The important multioperators property is the possibility to calculate their actions on known grid functions by parallel calculations of the actions of their basis operators. It can be used, for example, in the framework of MPI when performing calculations with multicore PC. In the case of massively parallel system for solving 3D CFD problems, it can be combined with domain decomposition approaches exploiting considerable amount of left and right sweeps. Their number for a line, along which the left and right sweeps are carried out, is proportional to M, the number of parameters defining the multioperators. Each sweep consists of calculating a current value with known previous one differing only in the parameter’s values and the functions which derivatives are approximated. Considering, for example, the general form of the left sweep for a grid value vi with i = 1, 2, . . . , N , the process looks as     vi = a c j vi−1 + b c j f i ,

j = 1, 2, . . . , M,

where f i , i = 1, 2, . . . , N are values of a known grid function. Thus, it is possible to use m processors by partitioning the interval i ∈ [1, N ] into m equal parts with transferring the value calculated by the kth processor to the (k + 1)th one, k = 1, 2, . . . , m − 1. The MPI realization of the idea in the 3D case of subsonic jets instabilities numerically simulated with the 16th-order scheme of the present family was described in [18]. Considerable speed-up was gained when increasing the number of the processors. In the case of the 32th-order multioperators, the technique can be directly used by changing the number of parameters from M = 8 to M = 16.

4.5 Summary and Conclusions A family of multioperators for even-order approximations to derivatives is described. Its basis operators are obtained from one-parameter compact approximations containing two-point operators only by fixing values of the parameter. That form admits very simple two-diagonal inversions for calculating their actions on grid functions and provides a good opportunity to parallelize multioperators realizations. Moreover, the coefficients for the basis operators satisfy linear systems which can be well-conditioned when increasing their dimensions. It allows to construct extremely high-order approximations. The bounds of the orders (if exist) are not clear presently; it can be said only that the 36th-order multioperator was created with great ease. The particular version of the family, that is, the 32th-order multioperators depending on 16

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free parameters were used in the present study to construct semi-discretized schemes for the Euler equations. To dump possible spurious oscillations of numerical solutions, the 31st-order self-adjoint positive multioperators are included in the resulting schemes as their dissipative components. The free parameters defining the “main” multioperator were used to decrease the phase errors in the domains of the shortest waves supported by meshes. Using the free parameters for the dissipation multioperators, the dissipation with various spectral contents can be created; in particular, it can play the role of cut-off type built-in filters. The behavior of the explicit scheme with the Runge–Kutta time stepping was investigated in the cases of smooth and discontinuous solutions. In the former case, the testing calculations apart from their very high accuracy have shown that wave packets of small wavelengths harmonics can be advected with the present scheme without degradation during extremely long times. The conservative property of the multioperators-based schemes provides the potential for using them for shock-capturing calculations, the aim being to get reasonable descriptions of discontinuities accompanied by high accuracy of the solutions in smooth parts of computational domains. It was shown that the original linear scheme with the 32nd-order approximation to convection terms with the 31st-order dissipation operator can work in the case of moderate discontinuities in the exact solutions of the Riemann problems. It was shown also that the scheme can operate in a nonlinear mode, the options being its flux correction form or blending with a monotone low-order scheme. In conclusion, the described results allow to suggest that the present schemes have the potential for being an efficient tool when performing high fidelity numerical simulations of certain classes of fluids flows. Their applications area requires high accuracy and high resolutions during longtime calculations. Examples are laminarturbulent transitions, turbulent flows, unstable flows with sound radiations, atmospheric phenomena, etc. However, the technique requires not only solutions smoothness (local or global), but also smooth meshes obtained via smooth mappings of physical domains into computational ones. Thus, it is not multipurpose one; it can be considered as an addition to the existing collection of high-order approximation methods that allow one to make a choice in accordance with the specifics of the problems to be solved. Acknowledgements Calculations related to the development of the multioperators technique were partly carried out using Lomonosov supercomputer of the Moscow State University.

References 1. Tolstykh, A.I.: Multioperator high-order compact upwind methods for CFD parallel calculations. In: Emerson, D., Fox, P., Satofuka, N., Ecer, A., Periaux, J. (eds.) Parallel Computational Fluid Dynamics: Recent Developments and Advances Using parallel Computers, pp. 383–390. Elsevier, North Holland (1998)

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2. Tolstykh, A.I.: Multioperators technique for constructing arbitrary high order approximations and schemes: Main ideas and applications to fluid dynamics equations. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds.) Smart Modeling for Engineering Systems. GCM50 2018. SIST, vol. 133, pp. 17−31. Springer, Cham (2019) 3. Tolstykh, A.I.: On multioperators principle for constructing arbitrary-order difference schemes. Appl. Numer. Math. 46, 411423 (2003) 4. Tolstykh, A.I.: Development of arbitrary-order multioperators-based schemes for parallel calculations. 1. Higher-than-fifth order approximations to convection terms. J. Comput. Phys. 225, 2333−2353 (2007) 5. Tolstykh, A.I., Lipavskii, M.V., Shirobokov, D.A.: High-order multioperators-based schemes: developments and application. Math. Comput. Simul. 139, 67–80 (2017) 6. Tolstykh, A.I.: On the use of multioperators in the construction of high-order grid approximations. Comput. Math. Math. Phys. 56(6), 932–946 (2016) 7. Tolstykh, A.I., Lipavskii, M.V.: Instability and acoustic fields of the Rankine vortex as seen from long-term calculations with the tenth-order multioperators-based scheme. Math. Comput. Simul. 147, 301–320 (2018) 8. Tolstykh, A.I.: High Accuracy Compact and Multioperators Approximations for Partial Differential Equations. Nauka, Moscow (in Russian) (2015) 9. Tolstykh, A.I.: 16th and 32nd multioperators based schemes for smooth and dis-continuous solutions. Commun. Comput. Phys. 45, 33–45 (2017) 10. Adams, N.A., Sharifi, K.: A high resolution Compact-ENO schemes for shock-turbulence interaction problems. J. Comput. Phys. 127, 27–51 (1996) 11. Lipavskii, M.V., Tolstykh, A.I.: Tenth-order accurate multioperators scheme and its application in direct numerical simulation. J. Comput. Math. Math. Phys. 53, 455−468 (2013) (transl. from Russian) 12. Tolstykh, A.I: High Accuracy Non-centered Compact Difference Schemes for Fluid Dynamics Applications. World Scientific, Singapore (1994) 13. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. NASA/CR-97–206253, ICASE Rept. No. 97–65. Hampton: Langley Research Center (1997) 14. Tam, C.K.W.: Problem 1-aliasing. In: Fourth Computational Aeroacoustics (CAA) Workshop on benchmark problems, NASA/CP-2004–2159 (2004) 15. Sod, G.A.: A survey of several finite difference schemes for hyperbolic conservation laws. J. Comput. Phys. 27, 1–31 (1978) 16. Mikhailovskaya, M., Rogov, B.: Monotone compact running schemes for systems of hyperbolic equations. Comput. Math. Math. Phys. 52, 578–600 (2012) 17. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. II. J. Comput. Phys. 83, 32–78 (1989) 18. Tolstykh, A.I., Lipavskii, M.V., Shirobokov, D.A., Chigerev, E.N.: Extremely high-order optimized multioperators-based schemes and their applications to flow instabilities and sound radiation. Commun. Comput. Inf. Sci. 965, 76–87 (2019)

Chapter 5

Hermitian Grid-Characteristic Scheme for Linear Transport Equation and Its Dissipative-Dispersion Properties Elena N. Aristova

Abstract This chapter presents an interpolation-characteristic scheme for solving a non-stationary inhomogeneous transport equation based on Hermitian interpolation of the third order of approximation. Interpolation of the solution on the lower time layer is carried out within a single cell and uses not only the nodal values of the distribution function, but also the values of its spatial derivatives. Such schemes are not new and are used by several groups of researchers. To complete the procedure of finding a solution, this scheme needs an algorithm for calculation not only the nodal values of the function, but also the nodal values of its spatial derivatives at a new time layer. For this purpose, a continued transport equation is usually used. The continued equation has the form of transport equation with respect to the spatial derivative of the function. Some algorithm for closing the resulting system of two transport equations is required. In this chapter, a variant of the Hermitian interpolation-characteristic scheme that does not require continued transport equation is proposed. The Euler– Maclaurin formula of the fourth order of approximation is used for calculation of the nodal values of spatial derivatives by calculating the integral averages over edges. This scheme might be extended for it using at non-structured tetrahedron grids. Fourier analysis of the dissipative and dispersion errors is carried out. It should be mentioned that the integral averages are responsible for the truth of conservation laws.

5.1 Introduction To solve the problems of high-temperature radiation gas dynamics, we need schemes, which allow us to build an approximation of the radiation transfer equation within a E. N. Aristova (B) Keldysh Institute of Applied Mathematics of the RAS, 4, Miusskaya sq., Moscow 125047, Russian Federation e-mail: [email protected] Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_5

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single cell that simplifies the situation near the contact discontinuities of the solution and the external boundaries. At the same time, it is desirable to construct a scheme of a higher order of approximation than the first or second. The schemes that have an approximation order higher than the trivial analysis of stencil size allows are called compact schemes. The schemes, in which a high approximation order is achieved on a two-point stencil for each dimension, are called the bicompact ones. In this chapter, conservative modification of well-known cubic-interpolated pseudo-particle (CIP) method is suggested [1–7] and compared with the bicompact Rogov’s schemes [8– 12]. Increasing the approximation order on the minimal stencil is only possible when the list of unknowns is extended. For the CIP method, the list of unknown values includes the values of its spatial derivatives in addition to the node values of the desired function. In Rogov’s schemes, this extension is carried out by introducing spatial integral averages or additional internal points in the stencil. The CIP method belongs to grid-characteristic methods based on cubic polynomic approximation [1–4]. As all grid-characteristic methods, the CIP method is not conservative. The class of Rogov’s bicompact schemes is constructed using the method of lines, and any approximation order can be reached in time (usually the third one is used). These schemes are conservative. The method of running counting or iterated factorization is used to calculate problems in multidimensional geometries. This class of methods has shown excellent results in applications to the gas dynamics problems. To the best of our knowledge, we see two drawbacks to this method. First one is in that it works on the regular rectangular spatial grids, which may not be very convenient when modeling the geometry of real technical devices. The second one is related to the peculiarities of solving the transport equation in application to radiation transfer. The transport equation contains an extinction coefficient (the sum of the absorption and scattering coefficients) so that the exact solution of the equation along each characteristic contains an exponential term. The currently developed Lebesgue averaging method on energy in applying to transport equation [13–16] has the peculiarity that it preserves the full range of changes in the absorption coefficient, which may differ by several orders of magnitude in the centers and in the wings of the lines. This means that with any reasonable choice of spatial grid in some parts of the spectrum, the cells will be optically thick. Therefore, it is important for us properties of achieved spatial approximation in the bicompact schemes. It is shown in [12] that a spatial approximation of fourth-order bicompact schemes can be obtained for the model Dahlquist equation by collocation method with the stability function R(z) =

1 + z/2 + z 2 /12 , 1 − z/2 + z 2 /12

where z is the optical dimensionless thickness of the cell. This means that this method of spatial approximation does not have “L–stability”, i.e., it does not transmit exponentially decaying solutions well. The chapter is organized as follows. Section 5.2 is devoted to the related works. In Sect. 5.3, a modification of the CIP method is described that is suitable for solving

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53

problems of radiation or neutron transport with opacities depending on space variables. In Sect. 5.4, the dissipative-dispersion properties of the CIP method are studied and compared with similar properties of the bicompact Rogov’s schemes for the Courant numbers less than one. The study of dissipative-dispersion properties of the suggested scheme for the Courant numbers more than one is given in Sect. 5.5. Section 5.6 concludes the chapter.

5.2 Related Work The CIP method was proposed 30 years ago. This method is based on a thirdorder cubic interpolation polynomial (Hermitian interpolation). This method uses the values of spatial derivatives at nodes to increase the approximation order. The need to pass the values of derivatives to the next time step leads to the use of a continued transfer equation, as in Rogov’s schemes. The difference is that in Rogov’s schemes, the continued transfer equation is integrated over the cell, while in the CIP scheme, it is not. For an inhomogeneous transport equation, the extended equation will link the original function and its spatial derivative, which is not very convenient in a numerical solution. Therefore, this chapter investigates a modification of CIP, in which the values of derivatives are restored from the Euler–Maclaurin formula (as in Rogov’s scheme). This ensures that the scheme is conservative, or, as it correctly should be said for inhomogeneous transport equation, the incoming flows are correctly redistributed between the outcoming edges of the computational cell. This is a third-order approximation scheme for both variables, in which part of the exponential dependencies is used explicitly to compute the solution along the characteristics. For this scheme, it is impossible to offer its analog for the Dahlquist equation and find its “stability function”. A description of the construction of a scheme for the inhomogeneous transport equation, numerical examples are given in [17]. For the advection equation, both the original CIP scheme and its modification [17] are equivalent to calculating the spatial derivative of the interpolant and transferring this value along the characteristic to the node. The dissipative-dispersion analysis of the CIP scheme was performed by the Fourier method in [3] for various cases of setting the initial values of the spatial derivative. It was shown that the CIP scheme reveals extreme small dispersion in the term of highest order of approximation error, and the following order has dissipation. In [10], properties of bicompact schemes of the fourth order of approximation in space and of different orders in time are investigated. For the Crank–Nicolson scheme, it is shown that this scheme is non-dissipative, but has an increased dispersion compared to the CIP scheme.

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5.3 Construction of CIP Scheme Modification To perform a dissipative-dispersion analysis of the CIP scheme, let us briefly consider its construction using the advection Eq. 5.1 as an example. ∂u(x, t) ∂u(x, t) +c =0 ∂t ∂x

(5.1)

u(x, 0) = u 0 (x), 0 ≤ x ≤ X,

(5.2)

Lu ≡ Set initial values

and boundary values in assumption that the transport velocity is positive u(0, t) = η(t), 0 ≤ t ≤ T.

(5.3)

We will use an approximation of the equation within a square cell with vertices (x m , t n ), (x m+1 , t n ), (x m , t n+1 ), (x m+1 , t n+1 ). For a positive transport velocity, c, the n+1 . Let us let unknown nodal value of the desired function in each cell will be ym+1 n+1 out the characteristics from the node (x m+1 , t ) back to the intersection with either the lower or left edge of the cell (Fig. 5.1). In the first case, the Courant number is κ = cτ/ h ≤ 1, and in the second case, the Courant number is κ ≥ 1. Let us denote the coordinate of the intersection of the backward characteristic with the cell edges either x * (κ ≤ 1) or t * (κ ≥ 1). For an exact solution of Eq. 5.1, the value from this point is transferred unchanged to the point (x m+1 , t n+1 ). Thus, the accuracy of the method is determined by the accuracy of rebuilding a non-nodal y* value. In grid-characteristic methods, some interpolation method is used to reconstruct a non-nodal value of unknown function. We will use the Hermitage cubic interpolation. When κ ≤ 1, the reverse characteristic lets out from the upper-right corner of the template intersects with the lower edge of the cell. Then this interpolation in barycentric spatial coordinates p and q on the segment has the form of Eq. 5.4. Fig. 5.1 Considered cell and backward characteristic from the (x m+1 , t n+1 ) node

5 Hermitian Grid-Characteristic Scheme for Linear Transport Equation …

55

n n P3 ( p, q) = H R ( p, q)ym+1 + H L ( p, q)ymn + G R ( p, q)dm+1 h + G L ( p, q)dmn h (5.4)

Here, H, G are the basic Hermitian functions defined by Eq. 5.5, where ymn is the nodal value of the desired function and dmn is the nodal value of spatial derivative of the desired function. p = (x ∗ − xm )/ h q = (xm+1 − x ∗ )/ h p + q = 1 h = xm+1 − xm H R = p( p + 2qp) G R = − p · qp H L = q(q + 2qp) G L = q · qp

(5.5)

For the case κ ≥ 1, where the starting point of the backward characteristic coming to the point (x m+1 , t n+1 ) belongs to the left edge of the cell, interpolation at this edge is carried out similarly taking into account the time derivatives for the desired function. If only spatial derivatives are stored in the data array, the time derivatives can be calculated directly from the transport Eq. 5.1 at the desired point. As it was mentioned above, interpolation of the form of Eq. 5.5 has been used in many works, see, for example, [1–7] and the references in them. The difference between the proposed approach and the mentioned works is in the method of determining the values dmn+1 which is necessary for continuation of calculations on the next time layer. As already mentioned, both Rogov’s bicompact schemes and the CIP scheme use a differential extension of Eq. 5.1, i.e., a differential equation of the form of Eq. 5.1 for the spatial derivative is also written, which must be solved together with the original transport equation: Lu ≡

∂u  (x, t) ∂u  (x, t) +c = 0. ∂t ∂x

(5.6)

Naturally, for the extended Eq. 5.6, it is necessary to set initial and boundary conditions for the spatial derivative u  (x, t) of type Eqs. 5.2 and 5.3. This approach has three drawbacks. First, Eq. 5.1 allows discontinuous solutions, and in this case, the derivative should be considered as a generalized function. Second, when it is considered a transport equation with absorption coefficient depending on space and time, the extended equation will contain not only the space derivative of function, but also the function itself, which complicated the scheme construction. Third, this scheme is not conservative. Under construction of a modification of the CIP method, we will not use differential extensions of Eq. 5.6. Let us assume that at the initial moment of time, we set not only the nodal values of function u 0 (xm ), but also the values of its derivative: dm0 = u 0 (xm ).

(5.7)

n+1 To close the algorithm, we need a procedure for calculation nodal values dm+1 . For a boundary node, the value of the spatial derivative in the case of boundary conditions in form of Eq. 5.3 can be calculated from Eq. 5.1 as d0n+1 = −η˙ (t n+1 )/c.

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n+1 For the advection equation, the spatial derivative dm+1 can be calculated as the n+1 interpolant derivative at the point x*: dm+1 = P3 ( p, q). However, this variant of calculating the derivative does not generalize to the inhomogeneous transfer equation with absorption:

∂ u(x, t) ∂ u(x, t) +c + σ (x, t) u(x, t) = f (x, t). ∂t ∂x

(5.8)

Here, σ (x, t) is total opacity, and f (x, t) is the source term. We will do otherwise. We calculate the integral average on the lower segment [x ∗ , xm+1 ] either exactly from the interpolant P3 , or using the Simpson formula. Due to the characteristic properties of the homogeneous Eq. 5.1, this integral averaged value over space will coincide with the integral averaged value over time for the segment [t n , t n+1 ] at the point x m+1 n on the right edge of a cell. The value of the spatial derivative dm+1 can be converted n n from Eq. 5.1 to the time derivative: gm+1 = −cdm+1 . Then on the right border of the cell, we know two nodal values, the integral averaged value and the value of the time n . This data allows one to get the value from derivative of the desired function gm+1 the Euler–Maclaurin formula using Eq. 5.9.

y m+1

1 ≡ τ

t n+1   1 n τ  n+1 1 (4) 4 n+1 n gm+1 − gm+1 u tttt τ u(xm+1 , t)dt = ym+1 + ym+1 − + 2 12 720

tn

(5.9) n+1 In turn, the value gm+1 can be converted to a spatial derivative from Eq. 5.1 as n+1 n+1 dm+1 = −gm+1 /c.

5.4 Dissipative-Dispersion Analysis Since both Simpson’s formula and the Euler–Maclaurin formula are exact on thirddegree polynomials, it is not difficult to show that for the advection equation, the method proposed in Sect. 5.2 for calculating the spatial derivative on a new layer in time is equivalent to transferring its value obtained by differentiating the interpolant at x* along the characteristic. The same method of calculating the derivative for the advection equation is used for analysis in the classical CIP scheme [1–4]. We perform a simplified (in comparison with [3]) Fourier analysis of the modified difference scheme for the investigation dissipative-dispersion properties. For an exact solution of the advection, differential equation sought in the form of Eq. 5.10. u(x, t) = eλt eikx Substituting this Fourier harmonic into Eq. 5.1 gives the connection.

(5.10)

5 Hermitian Grid-Characteristic Scheme for Linear Transport Equation …

λ = −ikc.

57

(5.11)

This means that each Fourier harmonic moves at the same velocity c, without changing in amplitude, only turning as it moves (while λ is the purely complex). For the numerical solution, we use the analog of Eq. 5.10 for the nodal values of the function and the nodal values of the spatial derivative in the form of Eq. 5.12. ymn = eλnτ eikmh dmn = βeλnτ eikmh

(5.12)

Let the Courant number be less than 1, then the characteristic comes to the lower edge of the cell. The calculation of derivatives of basic functions results in Eq. 5.13.  ∂HR 1 ∂HR = − ∂x h ∂p p(1 − 3q) ∂G R = ∂x h

   6 pq ∂ H L 6 pq ∂HR 1 ∂HL ∂HL = =− = − ∂q h ∂x h ∂p ∂q h L ∂G q(1 − 3 p) = (5.13) ∂x h

The intersection point of the backward characteristic with the lower layer x* corresponds to the value of barycentric coordinates q = κ, p = 1 − κ. Transferring the values of the function and derivative from this point to the node (t n+1 , xm+1 ) is equivalent to the system of Eq. 5.14. n+1 n n = p( p + 2 pq)ym+1 + q(q + 2 pq)ymn − p 2 qh · dm+1 + pq 2 h · dmn ym+1 n+1 n n hdm+1 = 6 pq · ym+1 − 6 pq · ymn + p(1 − 3q)h · dm+1 + q(1 − 3 p)h · dmn (5.14)

Substituting harmonics of form Eq. 5.12 into this system leads to a system for the coefficient β in the form of Eq. 5.15. eλτ = p( p + 2 pq) + q(q + 2 pq)e−ikh − p 2 qhβ + pq 2 hβe−ikh hβeλτ = 6 pq − 6 pqe−ikh + p(1 − 3q)hβ + q(1 − 3 p)hβe−ikh

(5.15)

By introducing ϕ = kh, the system of Eq. 5.15 can be rewritten as the system of Eq. 5.16.      hβ pq − p + qe−iϕ + p( p + 2 pq) + q(q + 2 pq)e−iϕ − eλτ = 0     hβ p(1 − 3q) + q(1 − 3 p)e−iϕ − eλτ + 6 pq − 6 pqe−iϕ = 0 (5.16) The compatibility condition of the system leads to a square Eq. 5.17 with respect to z = eλτ .    λτ 2 − eλτ p( p + 2 pq) + q(q + 2 pq)e−iϕ + p(1 − 3q) + q(1 − 3 p)e−iϕ e     + p( p + 2 pq) + q(q + 2 pq)e−iϕ × p(1 − 3q) + q(1 − 3 p)e−iϕ

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   − 6 p 2 q 2 1 − e−iϕ p − qe−iϕ = 0

(5.17)

This equation has two roots: The root corresponding to the dispersion ratio for this scheme corresponds to the “plus” sign before the root of the determinant. By q = κ, p = 1 − κ, we obtain Eq. 5.18. 1  ln 1 − 2κ + κ 3 + κe−iϕ (−1 + 3κ − κ 2 ) τ

+ e−iϕ κ(1 − κ) κ 2 − 4κ + 1 − 2eiϕ (κ 2 − κ − 5) + e2iϕ (κ 2 + 2κ − 2)

λ=

(5.18) The Taylor series expansion of Eq. 5.18 for long-wave harmonics ϕ = kh  1 gives Eq. 5.19.     i 1 ck(κ − 1) κ 2 − κ + 1 ϕ 3 + ck κ 2 − 1 (2κ − 1)(κ − 2)ϕ 4 72 540  2 2 5  6 1 ck(κ − 1) κ − κ + 1 ϕ + O ϕ (5.19) − 648

λ = −ick +

Equation 5.19 shows that the Hermitian characteristic scheme really has a third order of approximation, and the main error term in comparison with Eq. 5.11 is dissipative. The value of the coefficient at ϕ 3 is shown in Fig. 5.2. A negativity of coefficient means that the scheme is dissipative (and, therefore, stable). The coefficient for the dispersion term of the expansion is shown in Fig. 5.3. The long-wave harmonics phase velocity is defined by Eq. 5.20, where the dispersion coefficient is negative (for 0 < κ < 0.5), and the dispersion is outrunning because the effective velocity of wave is higher than a. Fig. 5.2 Dissipation coefficient of the CIP method as a function of the Courant number for small values of the dimensionless wave number ϕ = kh

0,000

-0,005

2

1)/7 2 -κ + )( κ

1

(κ -

-0,010

-0,015

-0,020

0,0

0,2

0,4

κ

0,6

0,8

1,0

5 Hermitian Grid-Characteristic Scheme for Linear Transport Equation … Fig. 5.3 Coefficient for the dispersion term of Eq. 5.19 at different Courant numbers for small values of the dimensionless wave number ϕ = kh

59

0,001 0,000 -0,001 2

(κ

-0,002

(2

) -1

κ-

1

)(κ

40

)/5

-2

-0,003 -0,004 -0,005

0,0

0,2

0,4

κ

0,6

  c∗ ≈ c 1 − (κ 2 − 1)(2κ − 1)(κ − 2) ϕ 4 /540

0,8

1,0

(5.20)

Conversely, where the coefficient is positive, the dispersion is lagging (for 0.5 < κ < 1).   Amplifier factor eλτ  and dependence of the ratio of the effective transport velocity c* to the exact one for the CIP method are shown in Figs. 5.4 and 5.5 as function of dimensionless wave number (0 ≤ ϕ ≤ π ) and Courant number (0 < κ < 1), respectively. Let us compare the dissipative-dispersion properties of the CIP method and Rogov’s bicompact schemes for the Courant numbers less than 1. Recall that the construction of the bicompact schemes is done using the method of lines, and the integration of resulting semi-discrete scheme over time might be carried out by using the schemes of different approximation order. Usually used for the time integration in the bicompact schemes, the third-order diagonally implicit Runge–Kutta methods are difficult for dissipative-dispersion analysis. Therefore, [11] provides an analysis only Fig. 5.4 Amplifier factor Abs(eλτ ) as a function of dimensionless wave number 0 ≤ ϕ ≤ π and Courant number 0 < κ < 1 for the Hermitian interpolation (CIP) method

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Fig. 5.5 Dependence of the ratio of the effective scheme velocity c* to the exact one c as function of dimensionless wave number 0 ≤ ϕ ≤ π and Courant number 0 < κ < 1 for the Hermitian interpolation (CIP) method

1.06 1.04 1.02 1

3

1 2

0.8 0.6 1

0.4 0.2

for an implicit Euler method of the first order of approximation and a second-order scheme of the Crank–Nicolson type. Comparing the first-order method (such as an implicit Euler method for time integration in the bicompact schemes) with the third-order CIP method does not make much sense. Therefore, we will only compare the case of using the Crank–Nicolson scheme for the time integration of semi-discrete bicompact scheme (BiC–CN). The bicompact BiC–CN scheme is non-dissipative, so its amplifier time factor is exactly equal to one. The Hermitian interpolation-characteristic scheme (equivalent to the CIP method for advection equation) has a small dissipation (Fig. 5.4). The CIP scheme has a maximum of dissipation error at the Courant numbers of about 0.5 for short-wave perturbations, and the damping of the corresponding harmonics is significant for a dimensionless wave number ϕ = kh > 2. The minimum of amplitude multiplier is equal to 0.86. Comparison of the phase error for both CIP and BiC–CN for several Courant numbers is presented at Fig. 5.6. One can see that the phase error of the Hermitian characteristic scheme is less than that the same one of the bicompact BiC–CN scheme.

5.5 Dissipative-Dispersion Analysis of the Scheme at the Courant Number More Than One In this case, interpolation will be performed along the left edge of the cell. Similarly to Eq. 5.4, we can write Eq. 5.21. P3 ( p, q) = H U ( p, q)ymn+1 + H D ( p, q)ymn + G U ( p, q)gmn+1 τ + G D ( p, q)gmn τ (5.21) Here, H, G are the basic Hermitian functions provided by Eq. 5.22.

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Fig. 5.6 Comparison of the phase velocity c*/c of the Hermitian characteristic scheme (solid lines) and bicompact BiC–CN scheme (dashed lines) depending on the phase ϕ for several Courant numbers: a κ = 0.1, b κ = 0.2, c κ = 0.3, d κ = 0.4

    p = t ∗ − t n /τ q = t n+1 − t ∗ /τ

p + q = 1 τ = t n+1 − t n

H U = p( p + 2qp) G U = − p · qp H D = q(q + 2qp) G D = q · qp (5.22) In this case, the scheme for the advection equation will also be equivalent to transferring from the point t* not only the value of the function, but also the value of its time derivative. Similar Fourier analysis of type Eq. 5.15 for the function and its time derivative gives the following equations: eikh = p( p + 2qp) + e−λτ q(q + 2qp) + τβ(− p 2 q) + τβe−λτ pq 2 ,

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τβeikh = 6 pq + e−λτ (−6 pq) + τβ · p(1 − 3q) + τβe−λτ q(1 − 3 p). Taking into account the connection p + q = 1, this leads to the square equation:      −λτ 2 2 q + 2 qe−λτ 1 − 2q 2 + q 3 − eiϕ 1 − 3q + q 2 qe   + e2iϕ + (1 − q)4 − 2eiϕ 1 − 2q + q 3 = 0. The point t* of intersection backward characteristic with the left edge of the cell = κ −1 , and then the roots of this square are defined will have coordinates q = h/c τ by Eq. 5.23.

1 e−λτ = 3 1 − 2q 2 + q 3 − eiϕ 1 − 3q + q 2 q 

  2    − q 2 e2iϕ + (1 − q)4 − 2eiϕ 1 − 2q + q 3 ± 1 − 2q 2 + q 3 − eiϕ 1 − 3q + q 2

(5.23) Expressed in terms of the Courant number, the physically meaningful root corresponds to the dispersion relation with the plus sign before the root. Taylor series expansion for long-wave harmonics ϕ = kh  1 gives the dispersion relation in the form of Eq. 5.24.    ick  2 ck (κ − 1) κ 2 − κ + 1 ϕ 3 − κ κ − 1 (κ − 2)ϕ 4 72 540  2   ck (κ − 1) κ 2 − κ + 1 ϕ 5 + O ϕ 6 (5.24) + 648

λ = −ick −

This scheme is dissipative at κ > 1, i.e., for the case under consideration, the scheme is stable. For long-wave harmonics, we have Eq. 5.25.   c∗ ≈ c 1 + κ(κ 2 − 1)(κ − 2) ϕ 4 /540

(5.25)

It means that for κ > 2, dispersion is outrunning, while in case 1 < κ < 2, the dispersion is lagging. Comparing Eqs.5.20 and 5.25, it can be seen that the dispersion corrections turn to zero at mutually inverse Courant numbers κ. Both roots in Eq. 5.23 are less than one, and each of them contains a branch-tobranch transition when calculating the root of a complex number. It is quite difficult to draw these dependencies.

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5.6 Conclusions The chapter presents a Hermitian grid-characteristic scheme, which is a modification of the CIP method. In [17], a set of numerical solutions of problems for an inhomogeneous transport equation with decreasing smoothness were considered on the basis of suggested scheme. In the numerical calculations, it was shown that the scheme has a small dissipation and ultra-small dispersion. In this chapter, the Fourier analysis of this scheme is performed for the advection equation and compared with the Fourier analysis of Rogov’s bicompact scheme BiC–CN. The bicompact comparison scheme BiC–CN, which has the fourth order of approximation in spatial variables and the second in time, is non-dissipative due to the complete symmetry of the construction of all space–time approximations. The proposed version of the CIP method has a third order of approximation both space and time. Comparison of dispersion properties shows that the CIP method has an extra small dispersion. For the radiation transfer problems, which are characterized by both exponentially decaying solution and discontinuous solutions, the small dispersion is a very attractive property of the constructed scheme. The built CIP modification is conservative. The closure of the built scheme does not base on the solution of the continued equation, but based on the calculation of the integral averages and the Euler–Maclaurin formula. This fact references modification of the CIP method to the family of Rogov’s bicompact schemes. Acknowledgements The work is done under financial support of the RFBR, grant no. 18-01-00857.

References 1. Yabe, T., Xiao, F., Utsumi, T.: The constrained interpolation profile method for multiphase analysis. J. Comput. Phys. 169(2), 556–593 (2001) 2. Yabe, T., Aoki, T., Sakaguchi, G., Wang, P.Y., Ishikawa, T.: The compact CIP (cubicinterpolated pseudo-particle) method as a general hyperbolic solver. Comput. Fluids 19(3/4), 421–431 (1991) 3. Utsumi, T., Kunugi, T., Aoki, T.: Stability and accuracy of the cubic interpolated propagation. Comput. Phys. Commun. 101, 9–20 (1997) 4. Tsai, T.L., Chiang, S.W., Yang, J.G.: Characteristics method with cubic–spline interpolation for open channel flow computation. Int. J. Numer. Methods Fluids 46, 663–683 (2004) 5. Golubev, V.I., Petrov, I.B., Khokhlov, N.I.: Compact grid-characteristic schemes of higher orders for 3D linear transport equation. Math. Models Comput. Simul. 8(5), 577–584 (2016) 6. Favorskaya, A.V., Petrov, I.B.: Numerical modeling of wave processes in rocks by gridcharacteristic method. Math. Models Comput. Simul. 10(5), 639–647 (2018) 7. Petrov, I.B., Favorskaya, A.V., Khokhlov, N.I.: Grid-characteristic method on embedded hierarchical grids and its application in the study of seismic wave. Comput. Math. Math. Phys. 57(11), 1771–1777 (2017) 8. Rogov, B.V., Mikhailovskaya, M.N.: Fourth order accurate bicompact schemes for hyperbolic equations. Doklady Math. 81(1), 146–150 (2010) 9. Aristova, E.N., Rogov, B.V.: Bicompact scheme for the multidimensional stationary linear transport equation. Appl. Numer. Math. 93, 3–14 (2015)

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10. Rogov, B.V.: Dispersive and dissipative properties of the fully discrete bicompact schemes of the fourth order of spatial approximation for hyperbolic equations. Appl. Numer. Math. 139, 136–155 (2019) 11. Chikitkin, A.V., Rogov, B.V.: Family of central bicompact schemes with spectral resolution property for hyperbolic equations. Appl. Numer. Math. 142, 151–170 (2019) 12. Aristova, E.N., Chikitkin, A.V., Rogov, B.V.: Optimal monotonization of a high-order accurate bicompact scheme for the nonstationary multidimensional transport equation. Comput. Math. Math. Phys. 56(6), 962–976 (2016) 13. Shilkov, A.V., Gerthev, M.N.: Verification of the Lebesgue averaging method. Math. Models Comput. Simul. 8(2), 93–107 (2016) 14. Mozheiko, S.V., Tsvetkova, I.L., Shilkov, A.V.: Calculation of radiative transfer in hot air. Mat. Model. 4(1), 65–82 (1992) 15. Shilkov, A.V., Tsvetkova, I.L., Shilkova, S.V.: System ATRAD for calculations of atmospheric radiation: Lebesgue averaging of radiation spectra and cross sections. Math. Model. 9(6), 3–24 (1997) 16. Aristova, E.N., Gertsev, M.N., Shilkov, A.V.: Lebesgue averaging method in serial computations of atmospheric radiation. Comput. Math. Math. Phys. 57, 1022–1035 (2017) 17. Aristova, E.N., Ovcharov, G.I.: Hermitian characteristic scheme for linear inhomogeneous transfer equation. Math. Models Comput. Simul. 12(6), 845–855 (2020)

Chapter 6

3-D Quasi-Conformal Mappings and Grid Generation Yuriy D. Shevelev

Abstract The difficulties of creating three-dimensional (3-D) analogs of conformal mappings are related to the topological and analytical features of 3-D space. The most interesting quasi-conformal mappings of 3-D regions are obtained with help of hydrodynamic analogy. For the steady irrotational flow of incompressible inviscid fluid together with the potential of velocities, two streamline functions are introduced. Any solenoid vector can be represented as a vector product of the gradients of two streamline functions. We obtain the connection of the velocity components with the streamline functions. These transformations are the basis for Lavrentiev’s type of harmonic mappings. On the other hand, these conditions can be considered as a generalization of the Cauchy-Riemann conditions in the 3-D case. As shown in our work, generalized 3-D Cauchy-Riemann conditions for harmonic mappings are reduced to Cauchy-Riemann conditions for two functions of a usual complex variable. An analog of 3-D quasi-conformal mappings is obtained as combination of two ordinary functions of a plane complex variable. Examples of grid generation obtained by the theory of 3-D quasi-conformal mappings are given. The best proof of these results is their visualization.

6.1 Introduction For many years, attempts have been made to extend the methods of 2-D conformal maps to the 3-D case. Two-dimensional conformal mappings are a powerful and elegant tool for solving many mathematical and physical problems. Conformal maps are used for calculating and visualizing harmonic vector fields in solving problems of incompressible fluid hydrodynamics, elasticity theory, gravitation theory, electromagnetism, etc. Y. D. Shevelev (B) Institute of Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya Ul., Moscow 123056, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_6

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In general, the properties of plane conformal maps are not generalized for 3D problems. It is proved that conformal mappings of 3-D domains are narrower in comparison with the class of such mappings of 2-D regions, and special cases of conformal transformations of 3-D space are the only transformation of motion, similarities, inversions and orthogonal transformations. The application of conformal mappings is related to the solution of boundary value problems for the Laplace equation to which many steady problems of mathematical physics reduces. Harmonic functions describe steady flows of an incompressible inviscid fluid, wave propagation, the theory of gravitation, diffusion processes, heat propagation, etc. The importance of harmonic functions in relation to 2-D complex analysis is that the real and imaginary parts of any analytical function are harmonic functions. The theory of functions of many complex variables is covered in many monographs [1–4]. A good theory is strong by its applications as it is evident from the two-dimensional theory of the function of one complex variable. For theories of functions of many complex variables, the use of a powerful mathematical apparatus has led just to most significant complications. The main issues of the theory of 3-D conformal mappings have not been studied enough, and, most importantly, the theory of multidimensional conformal mappings has not found sufficient practical applications in real applications. If we refuse from some restrictions, some properties of 2-D conformal maps can be generalized to the 3-D case [5]. The consideration of Lavrentiev’s type of harmonic mappings presents the great interest from the point of view of the extension of Riemann’s theorem on the existence of 3-D quasi-conformal mappings and other properties of conformal maps of plane region. This chapter is organized as follows. The mathematical background of Lavrentiev’s type of harmonic mappings is introduced in Sect. 6.2. In Sect. 6.3, the equations of motion are written in a curvilinear coordinate system connected with the introduced streamlines and solution in simple form obtained. Section 6.4 provides 3-D quasi-conformal mappings as combination of two ordinary functions of a plane complex variable. The numerical results and visualization are displayed in Sect. 6.5. Conclusions are presented in Sect. 6.6.

6.2 Lavrentiev’s Type of Harmonic Mapping Let us consider the steady irrotational flow of an incompressible inviscid fluid. The equations of motion in the Cartesian coordinate system have the form: ∂v ∂w ∂u + + = 0, ∂x ∂y ∂z

∂w ∂v − = 0, ∂y ∂z

∂u ∂w − = 0, ∂z ∂x

∂v ∂u − = 0. (6.1) ∂x ∂y

The vector field u(u, v, w) is potential and solenoid. In contrast to the 2-D system, Eq. 6.1 is redefined. We have four equations with respect to three variables. The theory

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of redefined systems of differential equations with constant coefficients is essentially based on the theory of functions of many complex variables. Due to equality r ot u = 0, there is a function ξ(x, y, z) in the steady flow region for stationary motion or a function of coordinates and time ξ(x, y, z, t) for unsteady flow such that u = ξx , v = ξ y , w = ξz . We will call the function ξ the potential of velocities and assume that function is continuous together with its first two derivatives coordinates. The potential of velocities is determined up to the additive constant. Mappings that meet conditions u = ξx , v = ξ y , w = ξz or conditions div u = 0, r ot u = 0 are called harmonic mappings. By substitution the relations u = ξx , v = ξ y , w = ξx into the equation of continuity, we obtain that ξ(x, y, z) satisfy the Laplace equation: ∂ 2ξ ∂ 2ξ ∂ 2ξ + + = 0. ∂x2 ∂ y2 ∂z 2 Consider the potential flow of an incompressible fluid past body of finite dimension. The harmonic function is searched for by the value of its normal derivative set at the boundary of the domain. We find the potential ξ(x, y, z) as a solution of the Laplace equation that satisfies the condition to an impervious surface ∂ξ/∂n = 0 and the conditions on the boundary for infinitely distant points: ∂ξ/∂ x = U, ∂ξ/∂ y = 0, ∂ξ/∂z = 0. The problem of determining a function in some domain D that satisfies the Laplace equation from the known values of the normal derivative of the function ξ on the surface S is called the Neumann problem. If the flow region contains an infinite point, then we will require the existence of a limit gradξ —the velocity at infinity u∞ at x 2 + y 2 + z 2 → ∞ and assume this vector to be given. For regions D with a sufficiently smooth boundary, the harmonic function ξ , which satisfies the boundary condition ∂ξ/∂n = 0 and condition at infinity if D contains an infinite point, always exists and is defined up to the real constant. Different areas of mathematical physics are related to theory of potential. Fluid flows that have a potential of velocities are called potential flows. The concept of potential coincides with the definition of potential of forces, with the only difference that the gradient of the potential forces is equal to the tension force field, whereas the gradient of potential is equal to the velocity of flow. The same type has a gravitational potential formed by the distribution of weighty particles, an electrostatic potential from a set of point charges, similarly leads an electric current in a homogeneous conductor. With minor changes, the theory applies to magnetism as well. As a rule in real problems and applications, there are harmonic functions of three variables. Together with the potential of velocities ξ(x, y, z), we introduce two stream functions and such that the surfaces ζ (x, y, z) = const and η(x, y, z) = const intersect along the stream function line. For a vortex-free flow, ∇ × u = 0.

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Each solenoid vector can be represented as a vector product of the gradients of two functions: u = ∇ζ × ∇η. The velocity vector is tangent to the two families of surfaces of the stream function. The value of the velocity vector is represented as: u = ui + vj + wk = ∇ζ × ∇η. Recall that ⎤ i j k     ∇ζ × ∇η = ⎣ ζx ζ y ζz ⎦ = i ζ y ηz − ζz η y + j(ζz ηx − ζx ηz ) + k ζx η y − ζ y ηx . η x η y ηz (6.2) ⎡

Let us write out the relationship between the new variables in the form of Eq. 6.3. ξ x = ζ y ηz − ζz η y ξ y = ζz η x − ζ x ηz ξz = ζ x η y − ζ y η x

(6.3)

The flow problem is reduced to mapping the flow area to the area in the space of the velocity potential and two streamline functions. Equation 6.3 can be considered as a generalization of the Cauchy-Riemann conditions in the three-dimensional case, from which three-dimensional quasi-conformal maps follow. Equation 6.3 are the basis of Lavrentiev’s harmonic mappings. The velocity vector is tangent to two families of streamline surfaces, which is equivalent to the equations: u · ∇ζ = 0, u · ∇η = 0 or uζx + vζ y + wζz = 0, uηx + vη y + wηz = 0.

(6.4)

As it follows from Eq. 6.4, the functions are related by the relations: ξx ζx + ξ y ζ y + ξz ζz = 0, ξx ηx + ξ y η y + ξz ηz = 0. A family of velocity potential surfaces ξ = const and two families of stream functions ζ (x, y, z), η(x, y, z) form a coordinate system in Cartesian space x, y, z. Consider the surfaces of streamlines ζ (x, y, z), η(x, y, z). On the surface ζ (x, y, z) = const, the coordinates ξ ,η form an orthogonal coordinate system. Similarly, on the surface η = const, coordinates ξ , η form also an orthogonal coordinate system. In general, the coordinate system formed by an intersection is not an orthogonal coordinate system: Surfaces ζ = const and η = const can intersect at some angle.

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Consider the inverse transformation. In Eq. 6.3, we will make variables x, y, z dependent and variables ξ, η, ζ independent. To do this, we substitute the formulas for converting derivatives when transfer from the coordinate system ξ, η, ζ to the coordinates x, y, z in Eq. 6.3. As a result, we get Eq. 6.5. xξ = yζ z η − z ζ yη yξ = z ζ xη − xζ z η z ξ = xζ yη − yζ xη

(6.5)

These relations connect hydrodynamic and geometric values.

6.3 Basic Equations for Arbitrary Curvilinear Coordinate System Connected with Streamlines Let us write down Eq. 6.1 in an arbitrary curvilinear coordinate system connected with the introduced streamlines. Mass conservation of an incompressible fluid is vanishing divergence of the velocity field. For arbitrary curvilinear coordinate system, this equation has the form of Eq. 6.6. 1 ∂ √ i  gu = 0 √ g ∂xi

(6.6)

The absence of vortex (∇ × u = 0) in an arbitrary curvilinear coordinate system leads to three equations: ∂u 2 ∂u 3 ∂u 3 ∂u 1 ∂u 1 ∂u 2 − 2 = 0, − 3 = 0, − 1 = 0. ∂x3 ∂x ∂x1 ∂x ∂x2 ∂x

(6.7)

Recall that any vector u(R) can be represented as a decomposition in various coordinate systems in the form u = ui + vj + wk =

3  k=1

u k Rk =

3 

u k Rk , R = xi + yj + zk, (k = 1, 2, 3).

k=1

Here, u, v, w are the components of velocities in cartesian coordinates, and u k , u k (k = 1, 2, 3) are the contravariant and covariant components in the coordinate system x 1 , x 2 , x 3 (u k = u · Rk ,u k = u · Rk (k = 1, 2, 3)). The √ g is the Jacobian of  the transformation for the right coordinate system  value √ g = ∂(x, y, z)/∂ x 1 , x 2 , x 3 . The local basis is a triple of vectors Rk ≡ ∂R/∂ x k directed along coordinate lines or a triple of vectors Rk directed perpendicular to coordinate surfaces. In the coordinate system associated with the streamlines, we have:

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u = 1

∂ξ ∂x

2

+

∂ξ ∂y

2

+

∂ξ ∂z

2 = u 2 + v2 + w2 = U 2 , u 2 = 0, u 3 = 0.

The relations between the physical components u(k), contravariant u k , covariant u k and Cartesian components of the velocity vector in a system of curvilinear coordinates have the form of Eq. 6.8. ∂xk ∂xk ∂xk u+ v+ w ∂x ∂y ∂z ∂x ∂y ∂z u k = k u + k v + k w (k = 1, 2, 3) ∂x ∂x ∂x

u(k) =

√

gkk u k u k =

(6.8)

In our case, the physical components of the velocity are related to the contravariant components by the following relations: u(1) =

√

g11 u 1 ,

√

g11 = U =

 u 2 + v2 + w2 , u(2) = 0, u(3) = 0.

As a result, we √ obtained: u(1) = U . Along the streamline, we have component of the velocity U = u 2 + v2 + w2 other than zero. The value of velocity is specified by the value of the boundary conditions. If we consider a uniform flow at infinity, then the components of the velocity along the streamline change, but the velocity preserves its value. The covariant components in the coordinate system connected with the streamlines are the following: u 1 = 1, u 2 = 0, u 3 = 0. In the coordinate system associated with streamlines, the condition is satisfied automatically. Let us find the Jacobian of the transformation J = ∂(ξ, η, ζ )/∂(x, y, z). Thus, we obtain Eq. 6.9.     J = ξx ζ y ηz − ζz η y + ξ y (ζz ηx − ζx ηz ) + ξz ζx η y − ζ y ηx

(6.9)

All flow features are related to the velocity field and the points where the total velocity vanishes. In order for the mapping to be one-to-one, the Jacobian of the transformation must be finite and non-zero. Using Eq. 6.3, we get J = ξx2 + ξ y2 + ξz2 = u 2 + v2 + w2 = U 2 = g 11 . On the other hand, Jacobian’s transformation can be represented as: 2 2   2  J = ζ y ηz − ζz η y + (ζz ηx − ζx ηz )2 + ζx η y − ζ y ηx = g 22 g 33 − g 23 . In the coordinate system associated with stream functions, we find the values of the contravariant metric tensor. As a result, we get g 11 = ξx2 + ξ y2 + ξz2 , g 22 = ζx2 + ζ y2 + ζz2 , g 33 = ηx2 + η2y + ηz2 , g 12 = g 21 = 0, g 13 = g 31 = 0, g 23 = g 32 = ζx ηx + ζ y η y + ζz ηz .

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If metric coefficient g 23 = ζx ηx + ζ y η y + ζz ηz = 0, then we get a 3-D orthogonal coordinate system. In steady flow, the potential of velocities permits to compute the streamlines that it corresponds identically with the particle trajectories used for flow visualization. Along any streamline, the steady flow vector is parallel to each streamline. Thus, any solid wall is a streamline. In the coordinate system connected with the streamlines, the solution of the boundary value problem is defined along each streamline. Equation 6.1 is invariant with respect to the transformation of the flow direction. Thus, in some problem, it is possible to take one direction of the velocity at the input in the same streamline tubes and in output—another direction. Since the flow is in the absence of friction, this approach makes it possible to design solutions in complex systems that satisfy the laws of mass conservation.

6.4 Quasi-Conformal Mappings of 3-D Domain The consideration of Lavrentiev’s type of harmonic mappings presents the great interest from the point of view of the extension of the Riemann’s theorem on the existence of 3-D quasi-conformal mappings and other properties of conformal maps of plane region. (a) Let us return to tshe generalized Cauchy-Riemann conditions (Eq. 6.3). If the streamline function η(x, y, z) depends on two variables, i.e., −ηz = 0, Eq. 6.3 is reduced to the form of Eq. 6.10.

ξx = −ζz η y ξ y = ζz ηx ξz = ζx η y − ζ y ηx

(6.10)

Let us show that the solution of Eq. 6.10 can be represented using two independent functions of the complex variable ζ1 = f 1 (z 1 ) and ζ2 = f 2 (z 2 ) defined in different domains. Denote C 2 by the space of two independent complex variables z = (z 1 , z 2 ). The points of this space z = (z 1 , z 2 ) are the sets of complex numbers z 1 = x + i y,z 2 = z + it. A space C 2 can be identified with a Euclidean space R 4 , whose points are sets (x, y, z, t) of real numbers. When changing from R 4 to C 2 , coordinates are divided into pairs that act as complexes z 1 = x + i y, z 2 = z + it. The space C 2 differs from a four-dimensional space R 4 by some asymmetry. The region D of the variable space C 2 is the set of points z 1 , z 2 that is obtained if z 1 ∈ D1 and z 2 ∈ D2 . Here, D1 and D2 are simply connected plane areas. For each complex number z 1 = x + i y from a certain domain C 2 , we will match the complex number ζ1 = τ + iη. Let a single-leaf analytical function ζ1 = f 1 (z 1 ) be set in the region C 2 . Similarly, the function is a single-leaf analytical function ζ2 = f 2 (z 2 ) of a complex variable z 2 = z + it. Each complex number z 2 = z + it from a certain domain C 2 is assigned a complex number ζ2 = ξ + iζ .

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The basic facts of the theory of analytic (holomorphic) functions of a single plane variable, sometimes in modified form, can be applied to analytical (holomorphic) functions of two variables. In order for the functions f 1 (z 1 ), f 2 (z 2 ) to be analytical in a domain, Cauchy-Riemann conditions must be satisfied in this domain (Eq. 6.11). ξz = ζt ξt = −ζz τx = η y τ y = −ηx

(6.11)

Let us search for a solution of Eq. 6.10 in the next form: ξ = ξ (z, τ (x, y)), ζ = ζ (z, τ (x, y)), η(x, y). Here, we will select a function t as τ (x, y). Then from Eqs. 6.10, 6.12 follows. ξτ τx = −ζz η y ξτ τ y = ζz ηx ξz = ζx η y − ζ y ηx

(6.12)

Since Eq. 6.11 is satisfied, the first and second equations in Eq. 6.12 have the form of Eq. 6.13. ξτ = −ζz ξτ = −ζz ξz = ζx η y − ζ y ηx

(6.13)

The third equation of Eq. 6.13 is reduced to Eq. 6.14.   ξz = ζτ τx η y − τ y ηx = ζτ J1

(6.14)

2 Here, value J1 = τx η y − τ y ηx = τx2 + τ y2 = ηx2 + η2y = f 1 (z 1 ) . If Jacobian J1 = 1, then we obtain ξz = ζτ . Remembering Eq. 6.12, we get Eq. 6.15. ξz = ζτ

ξτ = −ζz

(6.15)

Thus, in the case τ = t, Eq. 6.15 takes the same form as Eq. 6.11. However, we have a different system of Eq. 6.16. ξz = ζτ J1 ξτ = −ζz

(6.16)

∗ √ That ∗it has √a more common case than Eq. 6.15. By replacing the variables z = J1 z, ζ = J1 ζ , Eq. 6.16 can be reduced to the usual Cauchy-Riemann conditions for the system of Eq. 6.17.

ξz ∗ = ζτ∗ ξτ = −ζz∗∗

(6.17)

  And solution of Eq. 6.17 can be represented as ζ2∗ = f 2 z 2∗ . Thus, in the domain of real space R 4 , you can set four regular harmonic functions ξ, ζ ∗ , η, τ from which you can combine the relations:     ξ = ξ z ∗ , t , ζ ∗ = ζ ∗ z ∗ , t , η = η(x, y), t = τ (x, y)

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that are the solution of the generalized Cauchy-Riemann system (Eq. 6.10). If we omit the index “*”, we get a quasi-conformal mapping of the space in transformed variables with Jacobian equal to one. Let us recall that by virtue of the construction, the following relations are fulfilled: τx x + τ yy = 0, ηx x + η yy = 0 and ξz ∗ z ∗ + ξtt = 0, ζz∗∗ z ∗ + ζtt∗ = 0.

(6.18)

(b) Let us consider the inverse transformation. When we study the case z η = 0, then Eq. 6.5 takes the form of Eq. 6.19. xξ = −yη z ζ yξ = xη z ζ z ξ = yη xζ − xη yζ

(6.19)

Let us show that the solution of Eq. 6.19 can be represented using two analytical functions of the complex variable z 1 = F(ζ1 ) and z 2 = F(ζ2 ). Here, z 1 = x + i y and ζ1 = τ + iη. Similarly, z 2 = z + it and ζ2 = ξ + iζ . Each complex number ζ1 = τ + iη from a certain region G 1 (G 1 ⊂ C 2 ) is assigned a complex number z 1 = x + i y. In other words, a single-leaf analytical function is defined in the region G1. Similarly, the function z 2 = F(ζ2 ) is a single-leaf analytical function of a complex variable ζ2 = ξ + iζ . Each complex number ζ2 = ξ + iζ from some region G 2 (G 2 ⊂ C 2 ) is corresponded the complex number z 2 = z + it. Since the solution is searched in the form of two independent functions of a complex variable, the Cauchy-Riemann conditions are satisfied for each of them: xτ = yη , xη = −yτ for the function z 1 = F1 (ζ1 ) and the conditions of integrability z ξ = tζ , z ζ = −tξ for the function z 2 = F2 (ζ2 ). Consider the first and second equations of Eq. 6.19. We will try to find a solution in the form x = x(t(ξ, ζ ), η), y = y(t(ξ, ζ ), η), z = z(ξ, ζ ). We will choose τ as a function t (ξ, ζ ). Then the first and second equations of Eq. 6.19 take the form of Eq. 6.20. xt tξ = −yη z ζ yt tξ = xη z ζ

(6.20)

Thus, the relation z ζ = tξ holds, and then from Eq. 6.20, we get Eq. 6.21. xt = yη xη = −yt

(6.21)

Recall that the Cauchy-Riemann conditions for the function z 1 = f 1 (ζ1 ) coincide with Eq. 6.21 and have the same form (at t = τ ). The third equation of Eq. 6.19 takes the form of Eq. 6.22.   z ξ = tζ xt yη − xη yt = tζ J2

(6.22)

Thus, the generalized Cauchy-Riemann (Eq. 6.19) conditions when J2 = xτ yη − yτ xη , z η = 0 are reduced to Eq. 6.23.

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xτ = yη , xη = −yτ and z ξ = tζ J2 , z ζ = −tξ

(6.23)

The first two equations are the usual Cauchy-Riemann conditions, and the last two equations of Eq. 6.23 can be reduced to the form of Eq. 6.24. z ξ∗ = tζ ∗ z ζ∗∗ = −tξ

(6.24)

Equation 6.24 is Cauchy-Riemann conditions for a complex function provided by Eq. 6.25.     z 2∗ = F2 ζ2∗ z ∗ = z/ J2 ζ ∗ = ζ / J2

(6.25)

Similarly to what was considered above, the solution of Eq. 6.19 is obtained   by using arbitrary single-leaf analytical functions z 1 = F1 (ζ1 ) and z 2∗ = F2 ζ2∗ . The solution of Eq. 6.19 can be presented as    x = x t ξ, ζ ∗ , η ,

       y = y t ξ, ζ ∗ , η , z ∗ = z ∗ ξ, ζ ∗ , t = t ξ, ζ ∗ = τ.

If you omit the index “*”, then solution is the same as the case when x = x(t(ξ, ζ ), η),

y = y(t(ξ, ζ ), η), z = z(ξ, ζ ), t = t(ξ, ζ ) = τ.

It corresponds to the case when J2 = 1. Recall that the following relations are performed xτ τ + xηη = 0,

yτ τ + yηη = 0 and z ξ∗ξ + z ζ∗∗ ζ ∗ = 0, tξ ξ + tζ ∗ ζ ∗ = 0.

(c) Let us show that the system of functions—the potential of velocities and two stream functions—forms a 3-D orthogonal coordinate system. Let us find the components of a contravariant metric tensor whose values are given by the formula gi j =

∂xi ∂x j ∂xi ∂x j ∂xi ∂x j + + , ∂x ∂x ∂y ∂y ∂z ∂z

where i, j = 1, 2, 3, (x 1 = ξ ,x 2 = ζ ,x 3 = η). For variables z ∗ , ζ ∗ , the values of the components of the contravariant metric tensor have the form    g ∗11 = τx2 + τ y2 ξt2 + ξz2∗ , g ∗22 = ζt∗2 + ζz∗2 = ξt2 + ξz2∗ , g ∗33 = τx2 + τ y2 = ηx2 + η2y , g ∗13 = g ∗23 = 0.

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Taking into account Eq. 6.17, we obtain ξτ ζτ∗ + ξ y ζ y∗∗ = 0, i.e., g ∗12 = 0. All non-diagonal components of the contravariant metric tensor are equal to zero g ∗12 = g ∗13 = g ∗23 = 0. The resulting mappings are a family of pairwise orthogonal surfaces. Note that—g ∗11 = g ∗22 · g ∗33 . Let us find covariant components of the metric tensor for variables z ∗ , ζ ∗ . Since the equations z ξ∗ = tζ ∗ , z ζ∗∗ = −tξ are fulfilled, the diagonal components of the metric tensor have the following form:    ∗ ∗ ∗ = xt2 + yt2 tξ2 + z ξ∗2 , g22 = tζ2∗ + z ζ2 ∗ , g33 = xt2 + yt2 . g11 ∗ requires particular consideration. When changing for The metric coefficient g12 √ √ variables z ∗ = z/ J2 , ζ ∗ = ζ / J2 , we get

    2   ∗ xt + yt2 . = tξ tζ xt2 + yt2 + yξ yζ = tξ tζ ∗ + z ξ∗ z ζ∗∗ g12 ∗ Since Eq. 6.24 is fulfilled, we obtain a 3-D orthogonal coordinate system g12 = ∗ ∗ ∗ ∗ = g23 = 0. Note that—g11 = g22 g33 . This result is evident from the connection of contravariant and covariant components of the metric tensor. ∗ g13

(d) Let us find Jacobian’s transformation J = ∂(ξ, ζ, η)/∂(x, y, z) in the case when ηz = 0. We use Cauchy-Riemann’s type conditions in the form of Eq. 6.10. As a result, we get: 2   2     = f 1 (z 1 ) f 2 z 2∗ . J = τx2 + τ y2 ζx∗2 + ζz∗2 ∗ Note that J = g ∗11 = g ∗22 g ∗33 . The theory of two-dimensional conformal maps is fully described by the theory of one-leaf analytic functions of a single complex variables z 1 = x +i y,z 1 = x −i y and z 2 = z+it,z 2 = z−it. Hence, quasi-conformal mappings ζ1 = f 1 (z 1 ), ζ2 = f 2 (z 2 ) are performed by holomorphic functions of a complex variable or by antiholomorphic functions. It should be noted that in 3-D case, a number of possible variants increase. In addition to the above case, it is possible to represent the solution of Eq. 6.10 in the form of antiholomorphic functions. Besides, Eq. 6.10 admits other cases of complex variables. In order for the mapping performed by analytical functions to be quasi-conformal, it must be one-to-one, i.e., the functions F1 (ζ1 ), F2 (ζ2 ) must be one-leafed in the domain. Recall that in the two-dimensional case for an analytical function, the necessary condition for one-leafed in the domain is the condition F  (ζ ) = 0, i.e., the derivative must be everywhere different from zero in this domain if we exclude significantly special points from consideration. The transformation is quasi-conformal and continuous everywhere except in domain, where the derivatives F1 (ζ1 ), F2 (ζ2 ) or 1/F1 (ζ1 ), 1/F2 (ζ2 ) do not exist. A mapping that is the reverse of a quasi-conformal mapping is also quasi-conformal.

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In other words, if the functions z 1 = F1 (ζ1 ),z 2 = F2 (ζ2 ) that quasi-conformal map the area D to the area G, then the inverse function quasi-conformal maps the area G to the area D. The local homeomorphism of the map can only be violated at points, where the Jacobian of the transformation vanishes. The question of the zeros of the Jacobian map is reduced to the study of the properties of critical points. We will require that 2   2 the Jacobian of the transformation J = f 1 (z 1 ) f 2 z 2∗ to be greater than zero,   2 2 which corresponds to the usual requirements f 1 (z 1 ) > 0 and f 2 z 2∗ > 0. This guarantees a one-to-one mapping of the parametric parallelepiped to a given area of physical space. Non-degenerated simply connected plane regions are finite regions, infinite regions with a finite boundary, curved angular regions (curved half-planes), curved bands and curved bands with sleeves. Simply connected regions of a simple type (the unit circle, the exterior of the unit circle, the upper half-plane, the horizontal band, etc.) are called canonical regions. By analogy with plane regions, we can consider 3D canonical regions (the sphere, the exterior of the sphere, and bicylindrical regions formed by the product of plane regions, rotationally symmetric regions formed by the rotation of the upper half-plane, etc.). The bicylindrical region D of the variable space C 2 is the set of points z 1 , z 2 that is obtained if z 1 ∈ D1 and z 2 ∈ D2 . Here, D1 and D2 are simply connected plane areas. On the basis of Riemann’s theorem for plane regions, the interior of a bicylindrical region can be topologically and analytically mapped to the interior of a single bicylindrical domain.

6.5 Results of Numerical Calculations Quasi-conformal mappings z 1 = F1 (ζ1 ) and z 2 = F2 (ζ2 ) that display a threedimensional region ξ, ζ, η to a physical region can be visualized if 2-D analogs are known. Simple codes in Fortran and C++ have been developed to generate 3-D quasiconformal mappings and visualize them. The standard libraries of these programs which include sets of functions designed for working with complex numbers are used. (a) Let us consider the case z 1 = ζ1 , z 2 = F2 (ζ2 ). Hence, it should be clear that x = τ, y = η, z = z(ξ, ζ ), t = t (ξ, ζ ) = τ, J2 = 1. Equation 6.5 is reduced to the form xξ = −z ζ , xξ = xζ , x = x(ξ, ζ ), z = z(ξ, ζ ). It is an ordinary 2-D conformal mapping of the plane ξ, ζ onto the plane x, z. In this case, the domain under consideration has the form of infinite cylinders with a plane section, in which 2-D conformal maps can be used. (b) Consider the rotational-symmetric coordinates [6]. Coordinates are obtained from the corresponding plane coordinates by rotating on an angle η around the axis of symmetry. Coordinates ξ, ζ change in the half-plane. Let us choose z 1 = F1 (ζ1 ), z 1 = x + i y, ζ1 = τ eiη , z 2 = F2 (ζ2 ), z 2 = z + it, ζ2 = ξ + iζ . As a result, we obtain x = τ cos η, y = τ sin η, z = z(ξ, ζ ) for τ = t (ξ, ζ ).

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(c) Let us functions y r = y r (ξ, ζ, η), r = 1, 2, 3 (x = y 1 ,y = y 2 ,z = y 3 ) are known. Solution is obtained as x = x(t (ξ, ζ ), η), y = y(t (ξ, ζ ), η), z = z(ξ, ζ ), t = t (ξ, ζ ) = τ . It displays a one-to-one continuous mapping of the computational region to the physical domain. The computational region will be a grid with coordinates yi,r j,k , r = 1, 2, 3 of nodes. Choose Cartesian variables or polar coordinates as variables ξ, ζ, η (ξmin ≤ ξ ≤ ξmax ,ζmin ≤ ζ ≤ ζmax ,ηmin ≤ η ≤ ηmax ). The computational grid is defined by a set of nodes ξi = i h 1 , ζ j = j h 2 , ηk = kh 3 (i = 1, 2, ..., N , j = 1, 2, ..., M, k = 1, 2, ..., K ). The coordinates of nodes are given by a set of integers i, j, k. The values N , M, K are the number of nodes in the directions (ξ, ζ, η), respectively. In Fig. 6.1, the examples of quasi-conformal mappings are given. The presented visual examples of 3-D quasi-conformal mappings correspond to mappings: (1) (2) (3) (4) (5) (6)

z1 z1 z1 z1 z1 z1

= ia sin ζ1 , z 2 = ib sin ζ2 , a, b > 0 (Mapping 1). = ζ12 , z 2 = i sin ζ2 (Mapping 2). = ζ12 , z 2 = sin ζ2 (Mapping 3). √ 2 = ζ12 , z 2 = ζ2 + i (Mapping 4). = ζ12 , z 2 = i(t + 1/t − 2), t = 1 − iζ2 , ζ2 = ξ + iζ (Mapping 5). = sin ζ1 , z 2 = sin ζ2 (Mapping 6).

The grid generation using quasi-conformal mappings for solving 3-D problems in the theory of hydrodynamics, filtration, thermal conductivity, gravity, electromagnetic field theory, etc. is just beginning. Problems differ from the point of view of

Fig. 6.1 Examples of quasi-conformal mappings: a Mapping 1, b Mapping 2, c Mapping 3, d Mapping 4, e Mapping 5, f Mapping 6

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topology of considered regions in setting the initial and boundary conditions. Each problem requires separate consideration in terms of using 3-D conformal mappings.

6.6 Conclusions Thus, in 3-D case, a wide class of mappings is found that preserves all the basic properties of conformal maps of plane regions. Conformal mappings are used for calculating and visualizing harmonic vector fields in hydrodynamics, elasticity theory, filtration, electromagnetism, etc. The quasi-conformal mappings discussed in this chapter allow us to generalize the application of two-dimensional conformal mappings to the 3-D case by a natural way. The best proof of obtained results is their visualization. Acknowledgements The work was carried out within the framework of the state contract of the Institute for Computer Aided Design of the RAS.

References 1. 2. 3. 4. 5.

Bochner, S., Martin, W.T.: Several Complex Variables. Prinston (1948) Scheidemann, V.: Introduction to Complex Analysis in Several Variables. Birkhäuser (2005) Gunning, R., Rossi, H.: Analytical Functions of Several Complex Variables: Prentice-Hall (1965) Hermander, L.: An Introduction to Complex Analysis in Several Variables. Prinston (1966) Shevelev, Yu.D.: Application of 3-D quasi-conformal mappings for grid generation. Comput. Math. Math. Phys. 58(8), 1280–1286 (2018) 6. Madelung, E.: Die mathematischen hilfsmittel des physikers. Springer, Berlin (1957)

Chapter 7

Numerical Simulation of Shock-To-Detonation Transition Using One-Stage and Detailed Chemical Kinetics Mechanism Alexander I. Lopato Abstract The chapter is devoted to the numerical modeling of detonation initiation in the plane channel with the profiled end wall of the elliptic shape. Mathematical model is based on two-dimensional Euler equations supplemented by the chemical kinetics model to describe the combustion of the hydrogen–oxygen mixture under the low pressure. Both the global kinetic model and the detailed Petersen-Hanson model are applied. The computations are performed on unstructured triangular grids using the numerical method of the second approximation order. The mechanism of detonation initiation by the incident shock wave with the Mach number 2.7 is described. The results of initiation of the detonation wave using two models of chemical kinetics are compared with each other with the identification of similarities and differences in the process.

7.1 Introduction Many researchers investigated the problem of gaseous detonation initiation as a result of relatively weak shock wave (SW) interaction with the profiled end wall of the channel. In the experimental study [1], the reflectors in the form of twodimensional wedges, semi-cylinder, and parabola were considered. The peculiarities of mild and strong ignition inside the reflector cavity were visualized. It was shown that the mild ignition inside the reflector cavity can lead to detonation initiation outside the cavity. The system of elliptic reflectors of different numbers and relative position was considered in [2] and named “multifocused system”. The multifocused system follows the concept of minimization of detonation initiation energy due to the time-spatial factors that is due to the usage of spatially distributed initiators A. I. Lopato (B) Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya Ul., Moscow 123056, Russian Federation e-mail: [email protected] Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_7

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which ignite non-simultaneously. In the numerical study of detonation initiation using such profiled end walls, the particular problem is the choice of the type of the computational grid. As noted in [3], many computations on detonation initiation in the channel have been conducted on the structured and Cartesian grids. The reason is that the detailed analysis of the detonation phenomena requires higher-order accuracy especially for capturing the detonation cell structures. In turn, the unstructured grid method has a greater numerical dissipation than the structured grid scheme. For example, in the work [3], the rectangular channel with detonation initiation and propagation in the hydrogen-air mixture was considered, and it was shown that to capture some key structures of the flow, the unstructured grid requires about five times resolution in comparison with the structured one. However, the unstructured grid has the advantages of facilitating grid refinement and flexibility for complicated domains with curved boundaries that is important in the case of channels with the profiled end wall. The choice of the chemical kinetics model in the study of detonation initiation also plays an important role. Thus, in [4], the authors consider the influence of chemical kinetics on the modeling of detonation initiated by a temperature gradient in the reactive mixtures. Temperature gradient selected by the authors provides the formation of a hot spot and leads to a spontaneous wave capable to initiate detonation. It is shown that the temporal evolution of the spontaneous wave calculated using the complex chemical kinetic model differs significantly from the results obtained using the one-stage model. The difference in the steepness of temperature gradients, and correspondingly, in the hot spot size capable of producing detonation for considered two types of kinetics model authors explain by two main reasons. First, the induction times for a one-stage model calibrated in such a way that the model more or less correctly reproduces the speed and width of the laminar flame are several orders of magnitude smaller than the induction times in the case of detailed kinetics and measured in experiments. Second, the one-stage kinetics model is exothermic for all temperatures, while chain branching reactions in detailed kinetics start with endothermic induction stage representing chain initiation and branching. As a consequence, for one-stage models, the hot spots are much smaller and the temperature gradients initiating a detonation much steeper than those calculated using detailed chemical models. Despite the differences in the results of the two types of kinetic models, the authors of the work [4] noted that detailed kinetics may be unnecessary in problems when the gas dynamics model has the leading role and chemistry is required to get an amount of energy suitable for simulations. The goal of this work is the investigation of the mechanisms of detonation initiation in the plane channel with the profiled end wall of the elliptic form using the numerical technology based on fully unstructured computational grids and comparison of results of detonation wave (DW) initiation in the case of one-stage and detailed kinetics. The chapter is organized as follows. Section 7.2 provides the statement of the problem. The mathematical model is considered in Sect. 7.3. The computational

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Fig. 7.1 Schematic statement of the problem. All sizes are in millimeters

algorithm is described in Sect. 7.4. Section 7.5 presents the results of verification and numerical experiments. Section 7.6 gives the conclusions.

7.2 Problem Statement Consider the plane channel filled with the quiescent stoichiometric hydrogen–oxygen mixture under the initial pressure 0.04 atm and temperature 298 K. The geometry of the channel is following. The height of the channel is 60 mm, and the length is 37.1 mm. The end wall is the semi-elliptical curve with the semi-axes 10 and 7.1 mm, see Fig. 7.1. The incident SW location is x = 28 mm. The SW Mach number is taken equal to 2.7. At the initial time moment in the area x ≤ 28 mm, the parameters behind the SW are set. Slip conditions are set at the bottom boundary, the inflow conditions at the left boundary, and the symmetry conditions at the upper boundary. For the sake of computational cost, diminishing the computational domain corresponds to the one half of the channel. Note that a geometry of the considered computational domain is in accordance with a geometry from the experimental work of Vasil’ev [2]. The work is devoted to the experimental study of detonation initiation in hydrogen–oxygen mixture, and results of the work can be used for further numerical research.

7.3 Mathematical Model Mathematical model is based on two-dimensional system of Euler equations written in the Cartesian frame (x, y) for the multicomponent media and supplemented by

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the complex chemical kinetics model:

⎡

⎤ ρ ⎢ ρu ⎥ ⎢ ⎥ ⎢ ⎥ U = ⎢ ρv ⎥, ⎢ ⎥ ⎣ e ⎦ ρYs

∂U ∂F ∂G + + = S, ∂t ∂x ∂y ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ρv 0 ρu ⎢ ρvu ⎥ ⎢ 0 ⎥ ⎢ ρu 2 + p ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ 2 ⎢ ⎢ ⎥ ⎥ ⎥ F = ⎢ ρuv ⎥, G = ⎢ ρv + p ⎥, S = ⎢ 0 ⎥, ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎣ (e + p)v ⎦ ⎣ 0 ⎦ ⎣ (e + p)u ⎦ ρωs ρYs v ρYs u

NS 

ρYs ρ 2 u + v2 , p = RT , μ 2 μs s s=1 s=1

 NS β j  NR  NS NS      

ρYi γ js γ js  γ js − γ js K f j ωs = μs αl j cl ci − K bj ci , ci = . μi j=1 l=1 i=1 i=1

e=

NS  ρYs

h s (T )− p +

(7.1) Here, ρ is the total mixture density, u and v are the velocities in the x and y directions, p is the pressure, e is the total energy density, R is the universal gas constant, Ys is the mass fraction of the mixture component s, ωs is the production rate, h s is the molar enthalpy, μs is the molecular weight, ci is the molar concentration, αl j is the third body coefficient, γ js and γ js are the stoichiometric coefficients, K f j is the forward rate constant,K bj is the backward rate constant, NS is the total number of components, and NR is the total number of reactions. The molar enthalpy is calculated as:  a3s 2 a4s 3 a5s 4 a6s  a2s T+ T + T + T + , h s (T ) = RT a1s + 2 3 4 5 T where the coefficients a1s , ..., a6s are taken from [5]. The specific heat ratio of the multicomponent mixture in such mathematical model depends on the temperature as: NS Ys /μs γ (T ) = 1 + R NS s=1 ,

s=1 Ys C ps (T ) − R /μs where Cps is the molar heat capacity at constant pressure of the component s defined by:

Cps (T ) = R a1s + a2s T + a3s T 2 + a4s T 3 + a5s T 4 . The sound velocity is calculated using the formula:

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   NS  s=1 c = RT NS

Ys Cps (T ) NS Ys s=1 μs μ s

Ys C −R (T ) ps s=1 μs

In this work, the Petersen and Hanson (PH) model [6] is applied to describe combustion and detonation in the stoichiometric hydrogen–oxygen mixture. The model takes into account NS = 9 components (H2 , O2 , H, O, OH, HO2 , H2 O2 , H2 O, and N2 ) and NR = 18 elementary reactions. The stoichiometric coefficients γ js and γ js , rate constants K f j and K bj , and third body coefficients αl j can be found in [6]. The applicability and efficiency of detailed kinetics for numerical modeling of chemical reactions in hydrogen-air and hydrogen–oxygen mixtures are confirmed by a number of works (see, e.g., Ref. [3]).

7.4 Numerical Algorithm The main feature of the computational technique is the use of fully unstructured computational grids with triangular cells. The Delaunay triangulation based on the arrangements of the computational grid nodes to satisfy the Delaunay condition is carried out to construct the grid. The computational grids are built using software SALOME [7]. The example of construction of the grid for the computational domain corresponding to the problem statement of the work is shown in Fig. 7.2. The curved end wall of the channel is fairly well approximated by a polyline containing the edges of triangular cells. The green line corresponds to the internal boundary that coincides with the initial position of SW front. The computational algorithm is based on the Strang splitting principle in terms of physical processes [8]. When passing from one time layer to another one, one first Fig. 7.2 Example of grid construction using SALOME. The green line marks the internal boundary corresponding to the position of the shock wave front

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integrates the gas dynamics equations without considering the chemical reactions (S = 0 in Eq. 7.1) and, thereby, performs the first stage of the splitting procedure. The first stage is described in [9] for the case of the two-component mixture (reagent and product) and constant value of the specific heat ratio. In this work, the procedure was extended for the case of the multicomponent mixture with considered dependence γ (T ). The spatial part of Eq. 7.1 is discretized using the finite volume method. Time integration is carried out using the explicit Runge–Kutta method of the second approximation order. The time step is chosen dynamically to satisfy the stability condition [10]: ⎛

⎞

⎜ ⎟   Sk ⎜ ⎟ t = CFL · min⎜ 3 ⎟, Ak,σ = u k n k,σ x + vk n k,σ y + ck , ⎠ k ⎝  lk Ak,σ σ =1

where the index k goes through all cells, Sk is the cell area, lk is the length of the edge σ , n k,σ x , n k,σ y are the components of the outer unit normal to the edge σ , and ck is the sound velocity. The flux is calculated using the Advection Upstream Splitting Method (AUSM) scheme [11] extended to the case of a multicomponent mixture. Note that the use in the calculations AUSM scheme is not an obligatory requirement. Thus, in the paper [12], numerical modeling of combustion and detonation waves is carried out using the Courant-Isaacson-Rees flux scheme. On the second stage, the chemical reactions are taken into account without considering the convection (the second stage of splitting). The stage involves solving the system of ordinary differential equations, which describes the chemical reaction model for the molar concentrations and temperature in each computational grid cell:

 NS β j  NR NS NS       

dcs γ js γ js  = αl j cl ci − K bj ci , s = 1, 2, ..., NS, γ js − γ js K f j dt j=1 l=1 i=1 i=1 dT = dt

RT

NS  s=1

dcs dt

−

NS  s=1

h s (T ) dcdts

NS

 cs Cps (T ) − R

.

s=1

The system is integrated on the time step t. According to the splitting method, initial conditions of the system are taken from the solution of the first gas-dynamic stage. The system is solved with the use of the implicit Euler method with Newton linearization.

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7.5 Verification and Results The estimation of the practical approximation order of the algorithm on the problem of isentropic vortex evolution [13] gave the value near 2. The results of using the algorithm realized in the code for the case of the two-component mixture can be found in the works [9, 14]. To verify the realized PH chemical reaction model, the 0D homogeneous ignition simulation for the stoichiometric hydrogen-air mixture was carried out. Figure 7.3 demonstrates the time dependence of the mass fractions of the components at the initial pressure 1 atm and temperature 1000 K. The obtained dependencies are similar to the results from [3]. Relatively small quantitative differences (the obtained value of the ignition time is 210 µs, while the value in [3] is about 225 µs) can be explained by the fact that the polynomial coefficients a1s , a2s , ... in [3] that are used in the calculation of heat capacities and other thermodynamic quantities were taken from another database. The author of the chapter did not try to achieve a complete match in the results. It was important to establish that the chemical kinetics model is realized correctly and can be used for modeling of complex problems, including combustion and detonation in hydrogen-air and hydrogen–oxygen mixtures. Let us characterize the process of detonation initiation and DW propagation in the plane channel with the profiled end wall of the elliptic form for the computation on the grid with the average edge size x = 35 µm and the total number of cells N = 1,500,000. Some stages of the process are shown in Fig. 7.4, where the temperature distributions and numerical schlieren images are illustrated. The schlieren technique is one of the well-known methods used to visualize density gradients in compressible flows (see references in [15], for example). So, numerical schlieren images represent Fig. 7.3 Dependences of mass fractions of the mixture components on time for initial pressure p = 1 atm and temperature T = 1000 K

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Fig. 7.4 Predicted distributions at the successive time moments in the case of the detailed kinetics model and Mach number 2.7: a numerical schlieren for 8.5 µs, b numerical schlieren for 14 µs, c numerical schlieren for 15.5 µs, d temperature distribution. The temperature scale is in Kelvin degrees. The coordinate axes are in millimeters

density gradient fields and can be used for visualization of the considered problem data, in particular, shock and detonation waves. The incident SW with the Mach number 2.7 enters the reflector cavity at first microseconds with the formation of some typical structures including the triple point near the elliptical surface, the Mach stem, the reflected wave, the shear layer (see Fig. 7.4a). The incident SW reaches the end point of the reflector at 6.5 µs. At the time of about 9 µs, two Mach stems collide with increasing pressure and temperature in the area near the symmetry axis. Increasing the parameter values leads to the ignition of the mixture in two zones on the axis of symmetry of the reflector (see Fig. 7.4b). The first zone represents the shear layer and the second one includes a mushroomshaped pair of vortices. Gradually, the combustion area is being extended, occupying the region between these two structures. The combustion front propagates from the axis of symmetry of the channel (see Fig. 7.4c). Detonation initiation occurs after 17 µs. The DW front propagates away from the reflector cavity. The author also performed a computation in the case of one-stage kinetics with parameters from the database of Shepard [16], which more or less correspond to the pressures and temperatures of the considered problem. In this case, the observed pattern of detonation initiation has the qualitative similarities with the results obtained using the detailed kinetics model. Detonation initiation occurs almost at a time of about 13 µs according to the mechanisms noted above. Combustion takes place after the collisions of the waves in the same zones. But the zone with the shear layer is situated at some distance from the symmetry axis. The difference in detonation initiation moment can be explained by the factors given in the introduction of this work. The hot spots are more localized in space, and temperature gradients have

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Fig. 7.5 Predicted distributions at different time moments in the case of the one-stage kinetics model and Mach number 2.6: a numerical schlieren for 16 µs, b numerical schlieren for 16.5 µs, c temperature distribution. The temperature scale is in Kelvin degrees. The coordinate axes are in millimeters

higher values in comparison with the values obtained by the use of the complex kinetics model. Note also that the detonation initiation pattern for the detailed kinetics model and Mach number 2.7 qualitatively correlates with the results obtained in the case of the one-stage kinetics model and Mach 2.6. Indeed, in the computation on the grid with cells number of about N = 6,000,000, one can see that the combustion zones are formed in a similar manner near the symmetry axis of the channel that is shown in Fig. 7.5a. Detonation initiation occurs at the time of 16.5 µs (see Fig. 7.5b). Thus, one can see that the global kinetics in this work gives the quantitative differences in the value of the ignition delay of about 20% in the case of the Mach number 2.7. Taking into account relatively small quantitative differences in ignition delays and qualitative similarities in the initiation processes, we suppose that the applied global model may be suitable for calculations and makes it possible to obtain some results that are qualitatively close to the pictures associated with the use of complex kinetics. Calculations for the lower Mach number 2.6 using the global kinetics model show that it is possible to reduce the quantitative differences in the value of the ignition delay to the value obtained using the detailed model.

7.6 Concluding Remarks The work demonstrates the possibility of the numerical modeling of initiation and propagation of gaseous detonation waves in the two-dimensional statement with the detailed chemical kinetics model on fully unstructured computational grids with triangular cells. The computational algorithm based on the scheme of the second approximation order is described. The computational algorithm was tested in terms of realization of the Petersen and Hanson chemical kinetics model used in the work.

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The work also demonstrates the efficiency of the suggested computational algorithm for numerical modeling of detonation waves’ propagation in the plane channel with the profiled end wall of the elliptic form. The mechanism of detonation initiation by the incident shock wave is described. The features of the detonation initiation process are noted. The obtained patterns of detonation wave initiation in the case of the detailed and one-stage kinetics models are compared. The kinetic models are shown to give different ignition delays at the same Mach number of the incident wave. Qualitative similarities in the processes of detonation initiation are discussed. Acknowledgements This work is carried out under the state task of the ICAD RAS.

References 1. Gelfand, B.E., Khomik, S.V., Bartenev, A.M., Medvedev, S.P., Gronig, H., Olivier, H.: Detonation and deflagration initiation at the focusing of shock waves in combustible gaseous mixture. Shock Waves 10, 197–204 (2000) 2. Vasil’ev, A.A.: Cellular structures of a multifront detonation wave and initiation (Review). Combus. Exp. Shock Waves 51(1), 1–20 (2015) 3. Togashi, F., Lohner, R., Tsuboi, N.: Numerical simulation of H2 /air detonation using unstructured mesh. Shock Waves 19, 151–162 (2009) 4. Liberman, M., Wang, C., Qian, C., Liu, J.: Influence of chemical kinetics on spontaneous waves and detonation initiation in highly reactive and low reactive mixtures. Comb. Theory Model. 23(3), 467–495 (2019) 5. Burcat thermochemical data. https://burcat.technion.ac.il, last accessed 08.06.2020 6. Petersen, E.L., Hanson, R.K.: Reduced kinetics mechanisms for RAM accelerator combustion. J. Prop. Power 15(4), 591–600 (1999) 7. Salome. The open source integration platform for numerical simulation. https://www.salomeplatform.org, last accessed 2020/06/14 8. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd edn. Springer, Berlin (2009) 9. Lopato, A.I., Utkin, P.S.: Numerical study of detonation wave propagation in the variable cross-section channel using unstructured computational grids. J. Comb. 3635797, 1–8 (2018) 10. Mavriplis, D.J.: Accurate multigrid solution of the Euler equations on unstructured and adaptive meshes. AIAA J. 28, 213–221 (1990) 11. Liou, M.S., Steffen, C.J., Jr.: A new flux splitting scheme. J. Comput. Phys. 107, 23–39 (1993) 12. Lopato, A.I., Utkin, P.S.: The usage of grid-characteristic method for the simulation of flows with detonation waves. In: Petrov, I., Favorskaya, A., Favorskaya, M., Simakov, S., Jain, L. (eds.) Smart Modeling for Engineering Systems. GCM50 2018. Smart Innovation, Systems and Technologies, vol. 133, pp. 281–290. Springer, Cham (2018) 13. Hu, C., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 1, 97–127 (1999) 14. Lopato, A.I., Eremenko, A.G., Utkin, P.S., Gavrilov, D.A.: Numerical simulation of detonation initiation: The quest of grid resolution. In: Jain, L., Favorskaya, M., Nikitin, I., Reviznikov, D. (eds.) Advances in Theory and Practice of Computational Mechanics. Smart Innovation, Systems and Technologies, vol. 173, pp. 79–89. Springer, Singapore (2020) 15. Hadjadj, A., Kudryavtsev, A.: Computation and flow visualization in high-speed aerodynamics. J. Turbul. 6(16), 1–25 (2005) 16. Schultz, E., Shepherd, J.: Validation of detailed reaction mechanisms for detonation simulation. CalTech Explosion Dynamics Lab, Report No. FM99–5 (2000)

Chapter 8

Study of the Kinetic Anomalous Transport Effects in Nonequilibrium Flows Vladimir V. Aristov , Anna A. Frolova , and Sergey A. Zabelok

Abstract The second law of thermodynamics is one of the basic physical principals. The expression of the law in the macroscopic thermodynamics is the well-known Fourier equation. However, the question can be asked whether the validity of this equation for microscopic scales is confirmed by the kinetic description by means of the Boltzmann and other equations. We introduce the new classes of flows caused by nonequilibrium distribution functions, in other words, on the microscopic level related to molecular velocity (or inner energy) distributions. These nonequilibrium flows are realized by introducing new types of boundary conditions. For these flows, the Fourier equation for the thermal conduction is invalid (the Newton-Stokes relationships for the stress transport are also invalid for some physical situations). The anomalous nonclassical transport effects, in which in fact negative viscosity and thermal conductivity coefficients appear, are observed in computer simulations on the basis of direct solutions of the Boltzmann kinetic equation. The validity of results is verified by comparison with the simulations by means of a popular Direct Simulation Monte Carlo (DSMC) method, and the results are found to be in good agreement. The second law of thermodynamics is treated in terms of the H-theorem, and it is valid. Although Fourier equation in the scales of the mean free path can be invalid, the Clausius formulation of the second law can be true—as heat is transferred from a cooler region of the flow to a warmer one, the nonequilibrium boundary conditions must be maintained that requires permanent energy flux and provides the total increase of entropy. Possible experiments for testing the effects are also discussed.

V. V. Aristov (B) · A. A. Frolova · S. A. Zabelok Federal Research Center “Computer Science and Control” of the RAS, 40, Ul. Vavilova, Moscow 119333, Russian Federation e-mail: [email protected] A. A. Frolova e-mail: [email protected] S. A. Zabelok e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_8

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8.1 Introduction Besides numerous traditional and actual problems of the kinetic theory and rarefied gas dynamics such as aerodynamics of high-altitude aviation and space-aircrafts, vacuum technology, and micro- and nanotechnologies, there is an increased interest for studying specific kinetic effects including thermal transpiration, which are used as a basis in new technologies, for example, Knudsen pumps (compressors). These phenomena such as thermodiffusion, thermostress, thermal transpiration, and Knudsen force [1–9] are caused as a rule by gradients of the macroscopic parameters, mainly temperature. On the other hand, the kinetic theory can also be a basis for describing multiscale phenomena in nonequilibrium open systems or (in terms introduced by Prigogine) dissipative structures. Flows with strong nonequilibrium can lead to phenomena that seem paradoxical at first sight. It is well known that in the thermodynamic systems, the transport of heat from a cold region to a hot one is possible if an appropriate work is performed (an ordinary refrigerator is a source of heat into a room). Now we investigate theoretically a possibility of nonequilibrium flows with nonclassical transport properties with negative coefficients of viscosity and thermal conduction. Transport processes are as a rule interpreted as nonequilibrium processes but caused by gradients of quantities between states at local equilibrium. Thus, in thermodynamics, the transport of mass, momentum, and heat between close equilibrium states is considered, and the well-known relations between thermodynamic forces and fluxes are written. This leads to the ordinary transport. Kinetics can deal with possible local nonequilibrium states, so the transfer processes can be nonclassical. It should be emphasized that in the kinetic description, heat flux is an appropriate moment of the distribution function, but in a general case, it cannot be expressed through other macroparameters. The Chapman-Enskog distribution is valid in a special case only. A simple example for two one-dimensional nonequilibrium velocity distributions is considered in Fig. 8.1. We deal with the distribution functions, which are constant within 2 intervals and are equal to 0 beyond them. We do not use any kinetic equation now. It is assumed only that two nonequilibrium distributions are prepared. Let the gas to be at rest and b > a. Then for the left distribution, we find the following mean velocity u and numerical density n, temperature (for 1 degree of freedom) T, and heat

Fig. 8.1 Example of the anomalous heat transfer

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flux q: 

 a2 1 2 f b b − f a a 2 = 0, f b = f a 2 , 2n b  a  n= f dξx = f b b + f a a = f a a +1 , b   a2 1 3 1 fb b + fa a3 = T = ξx2 f dξx = f a (b + a), n 3n 3n   a2 1 4 1 fb b − fa a4 = f a (b + a)(b − a) > 0. q= ξx3 f dξx = 2 8 8 1 u= n

ξx f dξx =

Let for the right distribution, c > b and taking into account that the gas is at rest u1 =

 1  2 f c c − f a a 2 = 0, 2n 1

fc = fa

a2 , c2

so one can derive a set of simple inequalities: n1 = fc c + fa a = fa a T1 =

a c

 + 1 < n,

   1 3 1 3 1  3 fc c + fa a3 > f b b + f a a 3 = T, fc c + fa a3 > 3n 1 3n 3n  a2 1 4 fc c − fa a4 = f a (c + a)(c − a) 8 8 a2 > f a (b + a)(b − a) = q > 0. 8

q1 =

Hence, grad T > 0 and q > 0. Of course, it is desirable to describe this anomalous transport by the kinetic equations. These effects intrinsic to nonequilibrium processes can be obtained for the spatial scales of the order of the mean free path. A question arises concerning simulation of gas (and other media) flows: Under which conditions this anomalous transport can occur? The next step in the development of this approach is experimental validation. Our chapter consists of the following sections. In Sect. 8.2, we consider the formulation and basic properties of 1D nonuniform relaxation problem. In Sect. 8.3, the methods of solution and results of simulations for this problem are presented. Section 8.4 provides the results of simulations for problems with membrane-like nonequilibrium boundary conditions. In Sect. 8.5, we discuss the considered results, and Sect. 8.6 presents conclusions.

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8.2 Nonuniform Relaxation Problem The solutions of boundary problems with nonequilibrium boundary conditions result in the appearance of effects of nonclassical transport. Initially, the effect of anomalous transfer of viscous stress and heat flux was obtained in 1D nonequilibrium relaxation problem (NRP) [10, 11], which gave rise to the study of a new class of nonequilibrium flows corresponding to processes occurring in open systems. The description of such phenomena is based on the kinetic formalism, namely the Boltzmann equation, which has the form of Eq. 8.1. ∂f + ∇r · (ξ f ) = I (t, r, ξ ). ∂t

(8.1)

Here, f (r, ξ , t) is the velocity distribution function (VDF) that depends on the   coordinate vector r = (x, y, z) in physical space, velocity vector ξ = ξx , ξ y , ξz , and time t. Collision integral I (t, r, ξ ) in the case of binary collisions represents local integral operator in physical space. Macroscopic variables, namely the particle number density n, the velocity vector u, temperature T, the heat flux vector q, pressure tensor P, and pressure p, are defined by VDF as follows:   (n, nu, 3k B nT, q, P) = ∫ 1, ξ , mc2 , mcc2 /2, mc ◦ c f dξ ,

p = nk B T,

where c = ξ − u, m, and k B are the vector of intrinsic molecular velocity, molecular mass, and Boltzmann constant, respectively. The formulation of NRP in 1D geometry consists of defining a stationary solution of Eq. 8.1 on the half-axis (x > 0) with a given nonequilibrium distribution function f (x = 0, ξ ) which causes supersonic incoming fluxes. And it should be noted that there are two possible equilibrium solutions downstream, namely corresponding supersonic and subsonic branches. Density, velocity, and pressure at downstream infinity can be determined from the moment relations of the stationary Boltzmann equation. In the case of supersonic equilibrium at (x = +∞) and localization of the boundary distribution function in a narrow range of positive longitudinal velocities (large Mach number), the distribution function for negative velocities at infinity can be set equal to zero. For a high supersonic flow, one can construct an analytical expansion for the Boltzmann equation in powers of a small parameter α = (ξx −u)/u [10]. The following anomalous relations (for Maxwellian molecules or the model BGK collision integral) are obtained for the first-order approximation σx x = μU

du x dT , qx = λU . dx dx

Here, σx x = Px x − p, u x , qx are the components of a viscous stress tensor, mean velocity, and heat flux, respectively. For the case of Maxwell molecules, μU =

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u 20 m(m/8K )1/2 /(6A), λU = 3k B u 20 (m/8K )1/2 /(8A), where u0 is the mean velocity in the boundary, and A and K are certain positive constants. These relationships obviously contradict the known Stokes and Fourier laws. Recently, Ilyin in [12] has confirmed analytically the anomalous character of heat transfer in NRP. For this and more complex problems, the numerical methods need to be used. We apply the software package Unified Flow Solver (UFS) [13] developed with our colleagues from the USA. Below we present numerical solutions of the Boltzmann equation for some variants of boundary distribution functions and the molecular model of hard spheres.

8.3 Methods of Solution and Results of Calculations Here and after, all variables and functions are non-dimensional and are normalized to reference values. Consider NRP, for a boundary function representing the composition of two Maxwellians f M (n, u, T ), F(0, ξ ) = FM (1.5, 2.5, 1.0) + FM (0.5, 6.0, 1.0). The VDFs at the left and at the right boundaries obtained by numerical solution are presented in Fig. 8.2a, b. Numerically, NRP was solved using the direct method of solving the Boltzmann equation in the framework of UFS, and these results are verified with a variant DSMC method (the SMILE (Statistical Modeling in Low-Density Environment) software [14]). The profiles of macroscopic parameters for stationary solution are presented in Fig. 8.2c–f. The solutions by two du σx x > 0, and different methods are in a good agreement. It can be seen that dx dT q > 0. dx x The solutions of NRP with the boundary function being the composition of two Maxwellian F(0, ξ ) = 0.7FM (1.0, 4.24, 1.0) + 0.3FM (1.0, 2.12, 6.0) with the supersonic and subsonic conditions at infinity are presented in Fig. 8.3a, b. It is interesting to note that the solution with a subsonic boundary condition at infinity provides a more complex shock transition. Firstly, relaxation to equilibrium occurs with supersonic parameters and with anomalous heat transfer, and then a shock transition to subsonic flow with ordinary heat transfer takes place. For the supersonic boundary condition, the anomalous heat transport is observed for all x. For the subsonic branch, the anomalous heat transport is seen in the region near the left boundary, and it is related to the transition to the supersonic branch of the solution downstream, the next region corresponds to the transition to the subsonic branch of the solution downstream, and for this region, the transport (as in the shock wave structure) is ordinary. Anomalous transport effects were also revealed for the problem of nonequilibrium relaxation in the case of flows for mixtures of simple gases, for molecular gases with internal degrees of freedom, and also for mixtures of gases with chemical reactions [11]. When generalizing 1D NRP to 2D and 3D geometry, the problem of a jet expansion with nonequilibrium boundary conditions at the entrance to a half-space with the gas at rest can be considered.

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Fig. 8.2 Nonclassical anomalous transfer in nonequilibrium flows for NRP: a boundary distribution function at the left end of the boundary, b boundary distribution function at the right end of the boundary, c profiles of macroscopic parameters: the component of viscous stress tensor σ xx , d mean velocity ux , e heat flux qx , f temperature T. Solid red lines are the results obtained with direct solving the Boltzmann equation by UFS, and circles are solutions by Bondar and Kokhanchik (DSMC)

8.4 Boundary Problems with Nonequilibrium Conditions This class of problems allows us to describe the occurrence of anomalous transport in regions of different geometries with complex gas-surface interactions. Let us regard the well-known problem of thermal transfer between two parallel differently heated plates located at x = –1/2, x = 1/2. The classical approach considers the diffuse

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Fig. 8.3 Different nonequilibrium flows with demonstration of the anomalous heat transfer: a profile of heat flux, b profile of temperature for 1D NRP with the supersonic nonequilibrium left boundary condition and the equilibrium supersonic and subsonic right boundary conditions

reflection of molecules reaching the plates. The solutions for this problem [15, 16] at any Knudsen number do not show a distinction from the qualitative relations between the heat flux and the temperature gradient. However, the law of gas-surface interactions may be more complex because it is based on the surface structure and the interaction potential of incoming and surface particles. One of the methods to construct gas-surface condition is to choose the reflection function in some special form with free parameters or accommodation coefficients matched with experimental data. For example, the velocity-dependent reflection model with accommodation coefficients in the form of ellipsoidal functions was considered in [17], and reflection with different accommodation coefficients for tangential and normal energies was presented and analyzed in [18]. According to theory of gas-surface interaction [18], the scattering kernels must satisfy the conditions of positivity, normalization, and reciprocity. The last condition is valid for the quasi-equilibrium state of the surface atoms but may be violated in the nonequilibrium state of the solid. Let us consider some special cases of the boundary function modeling nonequilibrium scattering. In the first example, the model of reflection is assumed by ellipsoidal law in the form of Eq. 8.2, where Txl ,Tnl are the reflection parameters and have the different values, and N − is determined by the condition of impermeability. f (−1/2, ξ , ξx > 0) = N −



1 2π Txl

1/2



 1 exp −ξx2 /2Txl − ξ y2 + ξz2 /2Tnl . 2π Tnl (8.2)

Another nonequilibrium boundary condition represents the distribution of reflected molecules as a superposition of Maxwellians with different parameters provided by Eq. 8.3.

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f (−1/2, ξ , ξx > 0) = N − (α FM (u 0 , T0 ) + (1 − α)FM (u 1 , T1 ))

(8.3)

The reflection of gas atoms from the right plate at x = 1/2 is assumed to be diffuse with full accommodation. It is possible to estimate the range of Txl , Tnl for Eq. 8.2 and u, T for Eq. 8.3, for which there will be abnormal transport effect and solutions confirming this have been obtained in [19]. Note that in the so-called inverted-temperature gradient effect [20, 21], the heat flux from the cold plane condensed phase to the hot plane is transferred, but in that problem due to a special condensation, the temperature of a gas near the cold plane is greater than the gas temperature near the hot plane; thus, the thermal transport in a gas is realized by the ordinary manner. Another example is a problem in the finite region with the “membrane-like” boundary conditions. VDFs of particles ejected from the left and right boundaries are nonequilibrium in a general case, and they do not depend on VDFs of molecules entering the surface. One can suppose that molecules leaving the region under consideration are captured by the membrane; thus, these particles do not interact with particles entering the region. Even for equilibrium boundary VDFs of molecules entering the region, the nonequilibrium steady state in the region is obtained. Conditions for satisfying the true formulation could be as follows: lp  λ, where, l p is the size of a pore, and λ is the mean free path of the gas molecule. Hence, Knp = λ/l p  1, one can expect that this boundary condition is valid if even Knp ~ 1. In this case, the boundary conditions can be generalized taking into account that part of outcoming gas molecules can be reflected from the molecules of a pore with accommodation, which is characterized by a parameter a. This parameter is linked to coefficient of the permeability P (a = 1 – P) introduced in [22]. The full permeability P = 1 corresponds to the null reflection a = 0. These conditions may be related to recently manufactured membranes with micro- and nanopores [23]. This problem for 1D case is studied for mixtures of gases with chemical reactions, and a gas of four components Ai with masses of molecules m i (i = 1, …, 4) and the energies of chemical bonds E i is considered. The regime of slow chemical reactions is considered, i.e., it is assumed that the relaxation time due to elastic collisions is much lesser than the relaxation time during chemical interaction. The gas components are involved in a reversible bimolecular reaction, and the change in the internal chemical energy E of the mixture is assumed to be positive A1 + A2 ⇔ A3 + A4 , E = −

4 

λi E i = E 4 + E 3 − E 2 − E 1 > 0,

i=1

where λi is the stoichiometric coefficients, respectively (1, 1, –1, –1). The formulation of a problem with membrane-type boundary conditions for 1D flow of a gas mixture of four components is similar to that for a single-component

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gas. Also, as above, it is assumed that the outgoing gas does not interact with the incoming gas. The solution of the problem is determined by the system of Boltzmann equations, but the direct solution of the system of Boltzmann equations is extremely complicated; therefore, the use of model equations in the case of a chemically reacting gas is reasonable. The approximating system of BGK type equations is proposed in [8, 9], in which the integral terms are replaced by one relaxation operator for each component of the mixture     ∂fi ∂fi + ξ, = Q i ( f ) ≡ νi f Mi − f i , i = 1, . . . , 4, ∂t ∂x

(8.4)

where  f M,i (ξ ) = n i

mi 2π kTi

 23

  m i (ξ − ui )2 , i = 1, . . . , 4. exp − 2kTi

The free five parameters of the equilibrium function for each reacting component, namely the particle number density n i , the three components of velocity ui , and temperature Ti , are determined from the equality of momentum, energy and density exchanges for each component in Eq. 8.4, and the system of complete Boltzmann equations. The boundary conditions are analogous to the mentioned above but now should be set for each component. Some results are shown in Fig. 8.4. Here, solutions for the boundary conditions set to be Maxwell functions with parameter n L = {1.0, 1.0, 0.01, 0.01}, u L = {1.0, 1.0, 1.0, 1.0}, n R = {0.5, 1.0, 0.01, 0.01}, u R = {−0.5, −0.5, −0.5, −0.5}, and temperature on both sides T = 1 are presented (for E = 1). One can see that

Fig. 8.4 Profiles of heat flux qx and temperature T for gas mixtures with chemical reactions for different Knudsen number with the left boundary parameters nL = {1.0, 1.0, 0.01, 0.01}, uL = {1.0, 1.0, 1.0, 1.0} and nR = {0.5, 1.0, 0.01, 0.01}, uR = {–0.5, –0.5, –0.5, –0.5} and T = 1 for different Knudsen numbers (Kn): a Kn = 0.1, b Kn = 1

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Fig. 8.5 Solutions of 2D problem with nonequilibrium boundary conditions for different Kn and for a = 0. Distributions of cos(q, grad T ): a Kn = 0.1, b Kn = 0.25, c Kn = 1.0, d Kn = 10

in this case, the change of temperature is small; nevertheless, the anomalous zone is significant (marked by a yellow color). The results of solutions of the problem with “membrane-like” boundary conditions for 2D geometry are presented in Figs. 8.5 and 8.6. In this case, we consider one component gas flow in the square region. At the borders of the region, the following boundary conditions are set: • Boundary I (x = –0.5, –0.5 < y < 0.5): f = f M (n1 , 0.16, 0.1) + f M (n1 , 0.04, 0.1), n1 = 0.1–0.09(y + 0.05). • Boundary II (–0.5 < x < 0.5, y = 0.5): f = f M (0.1, –0.3, 0.1) + f M (0.1, –0.6, 0.1). • Boundary III (x = 0.5 < x < 0.5, y = 0.5): f = f M (n3 , –0.12, 1) + f M (n3 , –0.04, 0.1), n3 = 0.1–0.09(x + 0.05). • Boundary IV (–0.5 < x < 0.5, y = –0.5): f = f M (0.1, 0.8, 1) + f M (0.1, 0.03, 1). • Here, f M (n, u,T) are Maxwellians with density n, mean velocity u, and temperature T. The cosine of the angle between heat flux vector and gradient of temperature is presented in Figs. 8.5 and 8.6. For traditional heat transfer, this value equals to –1 (the directions of heat flux and gradient of temperature are opposite). The manifestation of the anomalous heat transfer is given. This problem is studied for different value of the permeability coefficient and for a wide range of Knudsen numbers.

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Fig. 8.6 Solutions of 2D problem with nonequilibrium boundary conditions for different coefficient of reflection a (connected with the coefficient of permeability). Knudsen number Kn = 1. Distributions of cos(q, grad T ): a a = 0.0, b a = 0.4, c a = 0.8, d a = 1.0

Results confirm that for different parameters of the problem, the anomalous heat transfer is appeared, and the anomalous region is smaller for smaller Knudsen numbers. Results in Fig. 8.5 demonstrate specific features of an anomalous thermal transfer for different rarefaction parameters. For small Knudsen numbers, this region is in the narrow layer near the corner of the square where boundary condition sharply changes. For larger rarefication, these zones of the anomalous transfer are larger and spread to the whole region. Possibilities of control anomalous processes by the parameters of the problem under consideration are demonstrated in Fig. 8.6. Namely, the coefficient a of the reflection in the membrane-like boundary (connected with the permeability coefficient as mentioned above) of one of the boundaries (x = –0.5) changes from 0 to 1. Results of calculations show that the maximum of the anomalous properties corresponds to the minimum a, i.e., to the maximum permeability.

8.5 Discussion One of the most important questions concerns the connection of the mentioned kinetic effects with the second law of thermodynamics. Indeed, at first sight, it seems that the unconventional transport phenomena revealed so far (theoretically) contradict the

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existing ideas on thermodynamic processes. In classical problems (with the exception, of local disturbances in a shock wave), there is no known case where the Fourier and Stokes laws are qualitatively violated. The theory of the Boltzmann equation has a unique character because it actually combines the first and the second principles of thermodynamics: the first principle is in the form of conservation laws, and the second principle is in the form of the H-theorem. There is a well-known Eq. 8.5, where dS is the change of entropy in the system, de S is the entropy flux through boundaries, and di S is the entropy production (non-negative). dS = de S + di S

(8.5)

This equation is derived from the Boltzmann equation. For a closed system, this leads to the H-theorem. In the steady case when dS = 0, we obtain the analogue of the H-theorem. For 1D NRP, this means the entropy flux downstream has to increase. In NRP with an anomalous transport, the validity of the H-theorem was verified in [24]. However, the behavior of entropy S (or reduced entropy S/n, where n is density) can differ, namely with nonequilibrium distributions, local maximum of entropy and reduced entropy have been noticed in [24]. In 2D problems, the situation is more complicated, for example, in the case of slow chemical reactions, the H-theorem is not analytically proved, but in our calculations, the corresponding relations have always been confirmed, e.g., these properties were confirmed for 2D nonequilibrium flow for a mixture with chemical reactions [11]. The thermodynamics of irreversible processes refers to local equilibrium states; in this macroscopic description, the Fourier and Stokes relationships are valid also as Eq. 8.5. But they do not have to be satisfied on a microscale, i.e., on the scales of the mean free path, where the Chapman-Enskog method is not applicable. One might say that the second principle in Clausius’s formulation “heat cannot by itself pass from a body that is less heated to the body more heated” is violated, although an important stipulation should be made. In the problems under consideration, such transport occurs just “not by itself”: For the flow of heat from the cold zone to the hot zone, maintaining of the nonequilibrium distribution is required, which means constant energy expenditure (and the increase in the total entropy). Thomson’s formulation that it is impossible to transfer heat from a colder body to a warmer one without compensation, i.e., without changing the system itself and the surrounding bodies, is actually equivalent to the above. Thus, with the mentioned stipulation, we can say that the second principle is also satisfied for more complex nonequilibrium systems. Notice that some models of Maxwell’s demon have recently been realized, see [25]. For experimental tests, it is necessary to create special conditions under which external factors would maintain the nonequilibrium distribution function at the boundary. Although as we saw in the “membrane-like” problem the nonequilibrium flows with anomalous transport properties can exist with equilibrium functions at the boundaries, the main issue is concerned with the realizability of these hypothetical boundary conditions. Modern admissible techniques for creating the nonequilibrium

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states are as follows: optical lattices [26], magnetic trapping [27], and molecules emitted from a surface with the nonequilibrium distribution function (here, the technique of the laser ablation can be useful, see [28]). With the recent advances in microfabrication technique, mixed cellulose ester membranes were elaborated with 100 nm (or lesser) spore-size [23]. Thus, Knp ~ 1 at atmospheric pressure, and the “membrane-like” boundary conditions could be realized. The numerical calculations for creating nonequilibrium distribution was performed in [24] with the aim to simulate results by the optical lattice. Then the appropriate nonequilibrium function was obtained, and the anomalous stress and heat transfer have been confirmed for the total region downstream. The upstream nonequilibrium distribution similar to the distribution obtained with the optical lattices technique (the crossed laser beams provide the nonequilibrium state due to interference and polarization) is formed and maintained. For measuring the distribution function, an electron beam fluorescence can be applied. This diagnostic tool has been used by Muntz et al. for measuring the distribution function in the shock wave structure [29]. A simple method without the diagnostics could be proposed for testing these effects. Namely, one can measure the temperature and heat flux (and also velocity and viscous stress tensor) in the inlet point as appropriate moments of the distribution function, the form of which is determined by the character of the electromagnetic field in the optical lattice. Downstream in the equilibrium zone, the heat flux (as the appropriate viscous stress tensor component) equals to zero, and the temperature (and velocity) can be measured by ordinary devices, e.g., a thermometer. If the difference between the temperatures downstream and upstream has the same sign as the nonequilibrium heat flux (upstream), then the effect of anomalous heat transfer is confirmed because there is a spatial zone with anomalous thermal transfer. It is also worth mentioning an interesting issue about possible natural referents with nonequilibrium nonclassical transfer. The nonequilibrium states must be maintained by “pumping” energy and negentropy, under normal conditions a spontaneously arising nonequilibrium distributions relax to equilibrium in a very short time (in air, the relaxation time is ~10–9 s). However, it is possible that nonequilibrium states can be self-maintained also in astrophysical high-energy states like masers, and lasers have been found in astrophysical objects. Note also that under terrestrial conditions, nonequilibrium states, and currents can be observed in biosystems.

8.6 Conclusion A new class of nonequilibrium flows with the nonclassical transport properties, which can be interesting from the theoretical and possibly from the practical point of views, has been considered. Some problems with different nonequilibrium boundary conditions have been formulated and solved. Analytical and direct numerical methods for solving the Boltzmann and other kinetic equations were used for investigation of these processes. Problems of spatial relaxation for supersonic and subsonic conditions have demonstrated regions of the anomalous transfer with the same signs of the

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temperature gradient and heat flux and also of the longitudinal velocity gradient and the appropriate component of the stress tensor. Comparison of the obtained solutions with DSMC method has shown good agreement. The existence of nonclassical transfer zones has been confirmed by numerical calculations for the heat transfer problem with nonequilibrium boundary conditions, as well as with membrane-like boundary conditions for 1D case for a chemically reacting mixture and two-dimensional case of simple gas. The perspectives of experimental testing these effects have been considered. The relation of these phenomena to the second law of thermodynamics has been discussed. Acknowledgements The work of Aristov and Zabelok has been supported Russian Foundation for Basic Research (grants No. 18-07-01500 and No. 18-01-00899). The computations have been partly performed on computer MVS-10P of Joint Supercomputer Center of Russian Academy of Sciences.

References 1. Maxwell, J.C.: VII. On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. London 170, 231–256 (1879) 2. Reynolds, O.: XVIII. On certain dimensional properties of matter in the gaseous state. Part I. Experimental researches on thermal transpiration of gases through porous plates and on the laws of transpiration and impulsion, including an experimental proof that gas is not a continuous plenum. Part II. On an extension of the dynamical theory of gas, which includes the stresses, tangential and normal, caused by a varying condition of gas, and affords an explanation of the phenomena of transpiration and impulsion. Phil. Trans. R. Soc. London 170, 727–845 (1879) 3. Knudsen, M.: Eine revision der gleichgewichtsbedingung der gase. thermische molekularstromung. Ann. Phys. (Leipzig) 336(1), 205–229 (1909) 4. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases. Cambridge University Press, Cambridge (1952) 5. Kogan, M.N., Galkin, V.S., Fridlender, O.G.: Stresses produced in gasses by temperature and concentration inhomogeneities. New types of free convection. Sov. Phys. Usp. 19, 420–428 (in Russian) (1976) 6. Vargo, S.E., Muntz, E.P.: Initial results from the first MEMS fabricated thermal transpirationdriven vacuum pump. In: Bartel, T.J., Gallis, M.A. (eds.) Rarefied Gas Dynamics: 22nd International Symposium, AIP Conference Proceedings, vol. 585, pp. 502–509. Melville, New York (2001) 7. Passian, A., Warmack, R.J., Ferrell, T.L., Thundat, T.: Thermal transpiration at the microscale: a Crookes cantilever. Phys. Rev. Lett.90 (12), 124503.1–124503.20 (2003) 8. Sista, S.V., Bhattacharya, E.: Knudsen’s force based MEMS structures. J. Micromech. Microeng. 24(4), 045003 (2014) 9. Strongrich, A., Pikus, A., Sebastião, I.B., Alexeenko, A.: Microscale in-plane Knudsen radiometric actuator: design, characterization, and performance modeling. J. Microelectromech. Syst. 26(3), 528–538 (2017) 10. Aristov, V.V.: A steady state, supersonic flow solution of the Boltzmann equation. Phys. Lett. A 250, 354–359 (1998) 11. Aristov, V.V., Frolova, A.A., Zabelok, S.A.: Nonequilibrium kinetic processes with chemical reactions and complex structures in open systems. EPL (Europhys. Lett.)106 (2), 20002.1– 20002.6 (2014)

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12. Ilyin, O.V.: Anomalous heat transfer for an open non-equilibrium gaseous system. J. Stat. Mech. 2017, 053201 (2017) 13. Kolobov, V.I., Arslanbekov, R.R., Aristov, V.V., Frolova, A.A., Zabelok, S.A.: Unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement. J. Comp. Phys. 223, 58–608 (2007) 14. Ivanov, M.S., Markelov, G.N., Gimelshein, S.F.: Statistical simulation of reactive rarefied flows. In: Numerical Approach and Applications. AIAA Paper 98-2669 (1998) 15. Aristov, V.V., Ivanov, M.S., Theremissine, F.G.: Two methods for solving the problem of heat transfer in a rarefied gas. USSR J. Comp. Math. Math. Phys. 30(2), 193–195 (1990) 16. Ohwada, T.: Heat flow and temperature and density distributions in a rarefied gas between parallel plates with different temperatures. Finite-difference analysis of the nonlinear Boltzmann equation for hard-sphere molecules. Phys. Fluids 8(8), 2153–2160 (1996) 17. Epstein, M.: A model of the wall boundary condition in kinetic theory. AIAA J. 5(10), 1797– 1800 (1967) 18. Cercignani, C., Lampis, M.: On the recovery factor in free-molecular flow. J. Appl. Math. Phys. 27, 733–738 (1976) 19. Aristov, V.V., Frolova, A.A., Zabelok, S.A.: The possibility of anomalous heat transfer in flows with nonequilibrium boundary conditions. Dokl. Phys. 62, 149–153 (2017) 20. Aoki, K., Cercignani, C.: Evaporation and condensation on two parallel plates at finite Reynolds numbers. Phys. Fluids 26(5), 1163–1164 (1983) 21. Cercignani, C., Fiszdon, W., Frezzotti, A.: The paradox of the inverted temperature profiles between an evaporating and a condensing surface. Phys. Fluids 28(11), 3237–3240 (1985) 22. Erofeev, A.I., Kogan, M.N., Fridlender, O.G.: Quasiequilibrium Knudsen boundary layer on a nonisothermal porous body. Fluid Dyn. 45, 134–146 (2010) 23. Gupta, N.K., Gianchandani, Y.B.: Thermal transpiration in mixed cellulose ester membranes: enabling miniature, motionless gas pumps. Microporous Mesoporous Mater. 142, 535–541 (2011) 24. Aristov, V.V., Frolova, A.A., Zabelok, S.A.: Supersonic flows with nontraditional transport described by kinetic methods. Commun. Comput. Phys. 11(4), 1334–1346 (2012) 25. Koski, J.V., Kutvonen, A., Khaymovich, I. M., Ala-Nissila, T., Pekolaet, J. P.: On-chip Maxwell’s demon as an information-powered refrigerator. Phys. Rev. Lett. 115, 260602.1– 260602.11 (2015) 26. Fulton, R., Bishop, A.I., Shneider, M.N., Barker, P.F.: Controlling the motion of cold molecules with deep periodic optical potentials. Nat. Phys. 2, 465–468 (2006) 27. Lobser, D.S., Barentine, A.E.S., Cornell, E.A., Lewandowski, H.J.: Observation of a persistent non-equilibrium state in cold atoms. Nat. Phys. 11, 1009–1012 (2015) 28. Morozov, A.A., Evtushenko, A.B., Bulgakov, A.V.: Gas-dynamic acceleration of laser-ablation plumes: hyperthermal particle energies under thermal vaporization. Appl. Phys. Lett. 106(5), 054107.1–054107.11 (2015) 29. Pham-Van-Diep, G., Erwin, D., Muntz, E.P.: Nonequilibrium molecular motion in a hypersonic shock wave. Science 245(4918), 624–626 (1989)

Chapter 9

Different Approaches to Numerical Solution of the Boltzmann Equation with Model Collision Integral Using Tensor Decompositions Aleksandr V. Chikitkin

and Egor K. Kornev

Abstract We consider two different ways of application of tensor decompositions to the numerical solution of a model relaxation problem for the Boltzmann equation with the BGK collision operator. The first approach is to optimize a low-rank tensor network with a gradient-based optimization for a loss functional related to time step of a numerical method. The second approach uses cross-approximation technique. Numerical experiments show that the second approach is more robust and allows to obtain a significant reduction in amount of memory and computations required by the numerical method.

9.1 Introduction The Boltzmann kinetic equation is the main mathematical model of the theory of rarefied gases. There are several simplified models based on a simplification of the collision operator. The most straightforward model is the Bhatnagar, Gross, Krook (BGK) model [1], and a more accurate approximation is given by the Shakhov model [2, 3]. The common property of all simplified models is that for the collision integral computation, only a certain number of macroparameters is needed. In turn, computation of macroparameters is performed via integration in three-dimensional velocity space, which requires significantly less operations compared to the exact five-dimensional collision integral. As a result, numerical solution of model equations in three-dimensional space domains becomes feasible. Nevertheless, since distribution function depends on seven parameters, all discrete velocity methods still suffer from large number of degrees of freedom. A. V. Chikitkin (B) · E. K. Kornev Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] E. K. Kornev e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_9

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Several techniques have been developed to overcome this problem. One of the most obvious approaches is to use an adaptive mesh in the velocity space [4–8]. However, the use of adaptive unstructured meshes significantly complicates the algorithm and often requires some a priori information about the problem. The simplest computational algorithm can be obtained with the use of structured Cartesian meshes in the velocity space. Therefore, it is an urgent problem to develop algorithms with reduced number of parameters while retaining simplicity provided by the structured velocity mesh. Values of distribution function on a structured velocity mesh naturally form a three-dimensional array which will be called tensor throughout the chapter. For such function-related tensors, low-rank approximations exist [9, 10]. Therefore, the number of degrees of freedom in the discrete velocity method can be reduced by replacing full tensors with their low-parametric approximations. In [11], this idea has been applied to the three-dimensional (in space) discrete velocity method for the Shakhov model. All operations with full tensors were replaced with their counterparts for the Tensor-Train format [12]. In this chapter, we test two other approaches on a simple spatially homogeneous relaxation problem for the BGK model. In the first approach, the time step of the numerical method is formulated as a minimization problem for a loss functional. This functional is minimized using gradient descent method. Gradient with respect to all parameters of any tensor format is easily computed using automatic differentiation, since all tensor formats or tensor networks use simple multilinear operations. The second approach employs the cross-approximation technique [13], which allows to approximate a tensor defined by a black-box function for computation of its elements. The present chapter is organized as follows. In Sect. 9.2, we briefly describe equations of the BGK model. The model problem along with the baseline numerical method is described in Sect. 9.3. Then we briefly present all necessary theory regarding tensor decompositions in Sect. 9.4. Section 9.5 is devoted to the description of modifications of the baseline method using tensor formats. We present results of numerical experiments in Sect. 9.6. Section 9.7 concludes the chapter.

9.2 Governing Equations We consider a time evolution of the distribution function of a monoatomic gas in spatially isotropic case, when all derivatives with respect to spatial coordinates equal zero. The Boltzmann equation with a model collision integral has the following form: ∂ f (t, v) = I ( f ), ∂t

(9.1)

where f (t, v) is the distribution function, v = (v1 , v2 , v3 ) is the velocity vector, and I ( f ) is the collision operator. Macroparameters are defined via moments of

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the distribution function provided by Eq. 9.2. 

  1 1 3 f dv, nu = v f dv, mn Rg T + mnu 2 = m v2 f dv 2 2 2  1 q = m ww2 f dv, w = v − u, ρ = mn, p = ρ Rg T (9.2) 2 n=

Here, n is the numerical density, u = (u 1 , u 2 , u 3 ) is the velocity vector, T is the temperature, m is the molecular weight (weight of one particle), p is the pressure, and Rg is the gas constant. For the sake of brevity, we consider the simplest collision model—the BGK model [1]. In the BGK model, the collision integral is defined by Eq. 9.3. I ( f ) = q( f M − f ), q =

p μ

  2 n fM =  3/2 exp −c , c = v/ 2Rg T 2π Rg T

(9.3)

Here, μ = μ(T ) is the dynamic viscosity, and f M is the equilibrium distribution function.

9.3 Model Problem We consider the following problem: ∂ f (t, v) = I ( f ), ∂t

f (0, v) =

L 

  f M v; p j .

(9.4)

j=1

Initial condition  of several Maxwell distributions for different macropa is the sum rameters p j = n j , v j , T j . If centers v j of Maxwell functions (Gaussians) are located far from each other in the velocity space, then in order to resolve them with high accuracy on a uniform Cartesian mesh, one has to use a large mesh with small mesh step. This leads to large number n v of nodes (or cells) in one dimension and large number of unknowns: n 3v . We have tested different numbers L of Maxwell distributions, but present results only for L = 2 since for that case it is possible to visualize results. Besides, in external aerodynamics problem, the distribution function is also some intermediate state between two Maxwellians corresponding to the free-stream conditions and boundary conditions on the surface. The following macroparameters were used (hereafter, we use International System of Units):   n 1 = 3 × 1020 m−3 , v1 = (2000, 100, 10000) [m/s], T1 = 3000 [K],

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  n 2 = 3 × 1020 m−3 , v2 = (−5000, −50, −8000) [m/s], T2 = 5000 [K]. (9.5) Computational domain in the velocity space is (−vmax , vmax )3 with vmax ≈ 1.44 × 10 , n v = 200, v = 1.44 × 102 . Mesh parameters are chosen in such a way that the mesh is sufficient to reconstruct (by numerical integration) macroparameters of each Maxwellian distribution with relative accuracy 10−11 . As a baseline method, we consider an explicit scheme for the problem (Eq. 9.2) discretized on the uniform structured mesh: 4

  vi1 i2 i3 = v1,i1 , v2,i2 , v3,i3 , vk,i = −vmax + iv, i = 1, . . . , n v .

(9.6)

Hereafter, we will denote by f (t) the tensor formed by the values of the distribution function on the mesh:   f (t)(i 1 , i 2 , i 3 ) = f t, vi1 i2 i3

(9.7)

All integrals involved in the computation of macroparameters and collision integral are computed by the simple second-order quadrature rule, for example:  n=

f (v) dv ≈ v3

nv 

f (i 1 , i 2 , i 3 )

(9.8)

i 1 i 2 i 3 =1

The time step of the resulting scheme is:   f n+1 = f n + t I f n

(9.9)

We use t ≈ 0.01 · (2/q) for the collision frequency q computed from one of the initial Maxwell distributions to satisfy stability restriction of the explicit scheme.  The storage and number of operations required for one time step are O n 3v . As it mentioned, it is too much in the case when we use a uniform mesh with small step in a large cubic domain in the velocity space, since n v is large itself. To circumvent this problem, tensor formats can be used.

9.4 Tensor Formats Tensor decompositions extend the idea of separation of variables to multidimensional arrays. In the two-dimensional case, for any matrix of rank r , the singular value decomposition (SVD) exists:

9 Different Approaches to Numerical Solution of the Boltzmann Equation …

A = U V T ,

A(i 1 , i 2 ) =

r 

σk u k (i 1 )vk (i 2 ),

109

(9.10)

k=1

where U, V are the unitary matrices, and  is the diagonal matrix with singular values σk on the main diagonal. A direct generalization of SVD to higher dimensions leads to the canonical decomposition (CANDECOMP, PARAFAC) [14] of the form: A(i 1 , i 2 , ..., i d ) =

r 

u 1k (i 1 )...u dk (i d ).

(9.11)

k=1 j

where r is the canonical rank, and u k is one dimensional vectors. Direct employment of the canonical decomposition in numerical methods is difficult due to instability of the basic algorithms for its computation. On the other hand, there are theoretical estimates, which show that tensors generated by the values of a smooth function on structured meshes can be approximated with high accuracy by tensor with small canonical ranks [10]. In the three-dimensional case, the Tucker decomposition is often used: A(i 1 , i 2 , i 3 ) =

r1 ,r2 ,r3

G(k1 , k2 , k3 )U1 (i 1 , k1 )U2 (i 2 , k2 )U3 (i 3 , k3 )

(9.12)

k1 ,k2 ,k3 =1

Tensor G is called core, and matrices U1 , U2 , U3 are called factors. This representation allows to employ robust SVD-based procedures for fast linear algebra operations for tensors in this format. Obviously, the Tucker decomposition does not circumvent the “curse of dimensionality”, since r d elements are needed to store the core G for d dimensions. However, in many problems, the ranks are very small and total number of parameters in the Tucker tensor r 3 + 3nr is much less than n 3 elements in the corresponding full tensor (here, n is the number of elements in one dimension, not numerical density). There are two tensor decompositions applicable to arbitrary number of dimensions, which generalize idea of Tucker format: the Hierarchical-Tucker (HT) format [15] and the Tensor-Train (TT) format [12]. Both formats are based on a dimensionality reduction tree and use SVD of auxiliary matrices for a low-rank approximation of an arbitrary tensor. In the TT-format, a tensor is represented as: A(i 1 , ..., i d ) =



G 1 (i 1 , k1 )G 2 (k1 , i 2 , k2 )...G d (kd−1 , i d ), k j = 1, ..., r j

k1 ,...,kd−1

(9.13) where G k are called TT-cores. Two cores, the first and the last, are matrices whereas all the rest are 3D tensors. The numbers r j are called TT-ranks.

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There are relatively simple algorithms for basic algebraic operations on tensors in the Tucker format [16] and the TT-format. Below we list the main operations with tensors in the TT-format along with its complexities. Complexities for the Tucker format are very similar (linear in n). Details can be found in the mentioned papers. For simplicity in all estimates, we assume that n 1 = n 2 = n 3 = n and r1 = r2 = r3 = r . 1. Construction of a rank-one tensor from its canonical factors. If canonical ranks of some tensor equals 1, i.e., A(i 1 , i 2 , i 3 ) = u 1 (i 1 )u 2 (i 2 )u 3 (i 3 ), then TT-ranks also equal to 1 and TT-cores equal to canonical factors. An example of a rank-one tensor is the tensor of values of Maxwell distribution on the grid, which is used in the model collision integral. 2. Element-wise summation. Given two tensors A1 , A2 in the TT-format one can explicitly construct TT-cores of their sum A3 = A2 + A1 . However, TT-ranks of A3 are equal to the sum of ranks of summands. 3. Convolution with one-dimensional vectors w1 , w2 , ..., wd . In the considered problem vectors, w1 , w2 , ..., wd may arise from weights of the quadrature on a structured non-uniform (in general) mesh: S=



A(i 1 , ..., i d )w1 (i 1 )w2 (i 2 )...wd (i d )

(9.14)

i 1 ...i d

  Convolution can be computed in O dnr 2 operations.

  n 2 3 4. Frobenius norm  A F = i 1 i 2 i 3 =1 Ai 1 i 2 i 3 can be computed in O dnr operations. 5 Rounding. For a tensor A, one can find tensor B with lower ranks such that: A − B F ≤ εA F . The algorithm is a sequence of SVD  and QR decompositions of auxiliary unfolding matrices and has complexity O dnr 3 . Using listed operation, the baseline method (Eq. 9.8–9.9) can be easily modified to handle with tensors in any tensor format. The key modifications are: 1. Integrals are computed using convolution. 2. Element-wise sum is computed using proper algorithm for the tensor format used. 3. Intermediate rounding is used to prevent rank growth due to summation. Such a straightforward approach was successfully used in [11]. However, it can lead to a large amount of computations, since the rounding algorithm costs O dnr 3 and ranks at intermediate steps can be rather high. Therefore, it may be more efficient to use approaches described below. 1. Gradient-based optimization of tensor network.

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Many problems involving low-rank tensors in some tensor format can be formulated as a minimization of some functional depending on tensor parameters (cores, factors, etc.). Advantage of such approach is that it is possible to eliminate direct computation of intermediate tensors with high ranks, arising, for example, from element-wise sum of several tensors. 2. Cross-approximation. Cross-approximation method [13] is applicable for all stable SVD-based formats (Tucker, TT, HT). This algorithm reconstructs a low-rank tensor using only a few of its elements. Therefore, one may only define a tensor as a function of several integer arguments. Then just a small number of elements (proportional to the number of parameters in a low-rank representation) of this tensor should be computed using this function. In Sect. 9.5, we describe how the baseline method can be modified using these two approaches.

9.5 Numerical Method Based on Tensor Formats Hereinafter, optimization-based method and cross-approximation are discussed in Sects. 9.5.1 and 9.5.2, respectively.

9.5.1 Optimization-Based Method The straightforward application of the tensor decompositions to the model problem is to find values of the distribution function on the velocity grid among low-rank tensors (e.g., in the Tucker format) with fixed ranks r : f n+1 (i 1 , i 2 , i 3 ) =

r 

G(k1 , k2 , k3 )U1 (i 1 , k1 )U2 (i 2 , k2 )U3 (i 3 , k3 )

(9.15)

k1 ,k2 ,k3 =1

Then one can define a loss function as the norm of the residual of equations of the explicit scheme (Eq. 9.9):    R = f n+1 − f n − tq f M − f n , Loss = R2F = R 2 (i 1 , i 2 , i 3 ) (9.16) i 1 ,i 2 ,i 3

With this loss function, the time step is formulated as a minimization problem of the form: min

G,U1 ,U2 ,U3

Loss

(9.17)

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Natural initial guess is f n+1 = f n . After that, one can compute gradient of the loss function with respect to elements of the core and factors of the tensor f n+1 using automatic differentiation (backpropagation) and apply a gradient descent method. Such approach is typical for machine learning problems. The key advantage of this approach is that there is no need to store explicitly all elements of large tensors. Residual (Eq. 9.15) is the sum of a few low-rank tensors; thus, it also has low rank. Then norm of the residual is computed in O(nr 3 ) operations. Another way is to compute parts (partial sums) of loss function using small subset of indices (batches) and minimize loss using some stochastic optimization method. It allows to circumvent the computation of the element-wise sum in the residual which leads to rank growth. Disadvantage of this method is that convergence of gradient-based methods is much slower than for cross-approximation methods, which utilize special structure of the tensor decomposition. However, optimization-based approach is more universal and allows to impose any additional constraints on the solution. For example, we can provide conservation of some quantity such as the numerical density, which can vary due to approximation errors. In fact, the proposed algorithm is very similar to the training of an artificial neural network with the difference that we use multilinear neural units instead of combination of fully connected dense layers and nonlinear activation functions which are more typical for machine learning problems. In our case, such an architecture is more convenient because it provides fast computation of 3D integrals (convolutions at the discrete level), which is the core of many kinetic models. For implementation of the proposed method, we used tntorch library [17], which is based on the popular machine learning library PyTorch [18].

9.5.2 Cross-Approximation In order to use cross-approximation, we define the function of two arguments: Fun( f, f M ) = f + t q( f M − f ).

(9.18)

At each time step, we apply cross-approximation algorithm to this function substituting into it tensor f n and tensor f M . Macroparameters and collision frequency q are preliminarily computed using algorithm for convolution. Ranks of the resulting tensor are fixed. Therefore, there is no need for rounding. Evaluation of each element since it is one rank by construction. Evaluation of the f n of the tensor f M is cheap,   2 3 element costs O r for the TT-format and O r for the Tucker format. Consequently, to outperform the baseline method, the number of evaluations must be  n 3V /r 2 or  n 3V /r 3 correspondingly. Method based on the cross-approximation has been implemented using tntorch library also. Implementation is available in [19].

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9.6 Computational Results The comparative numerical results of optimization-based method and method based on the cross-approximation are presented in Sects. 9.6.1 and 9.6.2, respectively.

9.6.1 Optimization-Based Method Numerical experiments reveal that this method is not suitable for the considered problem due to the effect of vanishing gradients, which is well known in the field of artificial neural networks. The problem is that tensors generated by values of distribution functions are very specific: most of their elements are essentially zero. Consequently, many elements of cores and factors of the underlying tensor format are near zero also. To demonstrate that, let us consider the following derivative for the Tucker format: ∂ Ai 1 i 2 i 3 = Ui11 k1 Ui22 k2 Ui33 k3 ∂G k1 k2 k3

(9.19)

For many index sets, it will be near zero. Consequently, G k1 k2 k3 will not be updated during gradient descent. This effect is visualized in Fig. 9.1. It shows cross-section of the initial condition along with solutions after several time steps obtained by the baseline method and the optimization-based method with tensors in the TT-format of ranks 4. It is clear that due to vanishing gradients, solution between two peaks remains zero as in the initial closure.

Fig. 9.1 Numerical solution by optimization-based method

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9.6.2 Method Based on Cross-Approximation Figure 9.2 shows 2D cross-section of the numerical solutions obtained using the baseline method and the method with cross-approximation after 15 time steps. Absolute value of difference between two solutions is shown in Fig. 9.3. Table 9.1 shows how the relative error in the Frobenius norm and the number of evaluations in the cross-approximation algorithm depend on the ranks of tensor. For small ranks, the number of evaluations is very high because iterations in crossapproximation do not converge to prescribed tolerance (1E−6 for relative error).

Fig. 9.2 Cross-section of numerical solutions obtained using: a baseline method, b method based on cross-approximation

Fig. 9.3 Absolute value of difference between two solutions

9 Different Approaches to Numerical Solution of the Boltzmann Equation … Table 9.1 Dependence of the relative error on maximum rank

Max. rank

Error

Number of evaluation

1

8.85e+00

1000

2

1.07e+00

70,000

3

1.78e−03

135,000

4

6.57e−07

17,600

5

1.62e−10

13,000

6

3.73e−11

18,000

7

3.56e−13

23,800

8

1.74e−14

30,400

115

Starting from rank 4, the number of evaluations increases slowly with rank. It is also clear that rank 4 is sufficient to obtain a very accurate solution, and for rank 8 the limit connected with machine precision is achieved. For rank 4, the TT tensor contains 4800 parameters; thus, tensorized method requires 1000 times less memory than the baseline method. The number of elementary operations during cross-approximation is approximately 2 × 17600 × 42 = 563200, which is only 14 times less than for the baseline method.

9.7 Conclusions Two approaches based on tensor decompositions have been applied to a model relaxation problem for the BGK equation. Numerical experiments show that the approach utilizing gradient optimization of a tensor network suffers from vanishing gradient effect. The second approach based on the cross-approximation technique is more promising and allows to reduce significantly both amount of memory and amount of computations required by the numerical method. Acknowledgements This study was supported by Grant of President of Russian Federation MK2855.2019.1.

References 1. Betahatnagar, P., Gross, E., Krook, M.: A model for collision processes in gases. Phys. Rev 94, 511–525 (1954) 2. Shakhov, E.: Generalization of the Krook kinetic relaxation equation. Fluid Dyn. 3(5), 95–96 (1968) 3. Rykov, V.: A model kinetic equation for a gas with rotational degrees of freedom. Fluid Dyn. 10(6), 959–966 (1975) 4. Arslanbekov, R., Kolobov, V., Frolova, A.: Kinetic solvers with adaptive mesh in phase space. Phys. Rev. E 88(6), art. no. 063301 (2013)

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5. Baranger, C., Claudel, J., H´erouard, N., Mieussens, L.: Locally refined discrete velocity grids for stationary rarefied flow simulations. J. Comput. Phys. 257(PA), 572–593 (2014) 6. Titarev, V., Utyuzhnikov, S., Chikitkin, A.: OpenMP + MPI parallel implementation of a numerical method for solving a kinetic equation. Comput. Math. Math. Phys. 56(11), 1919– 1928 (2016) 7. Guo, W., Cheng, Y.: A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations. SIAM J. Sci. Comput. 38(6), A3381–A3409 (2016) 8. Titarev, V.: Application of model kinetic equations to hypersonic rarefied gas flows. Comput. Fluids 169, 62–70 (2018) 9. Tyrtyshnikov, E.: Kronecker-product approximations for some function-related matrices. Linear Algebra Appl. 379(1–3 SPEC. ISS), 423–437 (2004) 10. Tyrtyshnikov, E.: Tensor approximations of matrices generated by asymptotically smooth functions. Sbornik Math. 194(5–6), 941–954 (2003) 11. Chikitkin, A.V., Kornev, E.K., Titarev, V.A.: Numerical solution of the Boltzmann equation with S-model collision integral using tensor decompositions. CoRR ArXiv Preprint, arXiv: 1912.04582 (2019) 12. Oseledets, I.: Tensor-train decomposition. CSIAM J. Sci. Comput. 33(5), 2295–2317 (2011) 13. Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010) 14. De Lathauwer, L., De Moor, B., Vandewalle, J.: On the best rank-1 and rank-(r1, r2,…, rn) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000) 15. Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31(4), 2029–2054 (2009) 16. Oseledets, I.V., Savostyanov, D.V., Tyrtyshnikov, E.E.: Linear algebra for tensor problems. Computing 85(3), 169–188 (2009) 17. tntorch: Tensor Network Learning with PyTorch. https://github.com/rballester/tntorch, last accessed 2020/05/03 18. PyTorch: An open source machine learning framework. https://pytorch.org/, last accessed 2020/07/23 19. Implementation of described algorithms. https://colab.research.google.com/drive/1i8XmdJIvkIma5HeTRO9sVVPqRJqHmFb?usp=sharing, last accessed 2020/07/23

Chapter 10

Difference Scheme with a Symmetry-Analyzer for Equations of Gas Dynamics and Magnetohydrodynamics Galina V. Ustyugova

and Alexander V. Koldoba

Abstract The chapter describes an approach using Godunov-type difference schemes to construct numerical algorithms for numerically integrating equations of ideal gas dynamics and magnetohydrodynamics. This approach allows calculating the flows of conservative variables through the interface between the computational cells. The Riemann problem is solved (exactly or approximately) on the decay of the discontinuity between the states that are formed using some procedures for reconstructing grid data to this interface. During reconstructing vector quantities, the question arises: what components of vector fields should be preferred during the reconstruction? To automate this choice, we propose a symmetry-analyzer as an element of the computational algorithm for the numerical solution of two-dimensional equations of ideal gas dynamics and magnetohydrodynamics. A symmetry-analyzer is an algorithm that allows using grid data to give a preference to a component (in the present work, Cartesian or polar) of a vector field for its reconstruction on the cell interfaces of a computational grid. A computational algorithm is constructed using a polar-type computational grid with a symmetry-analyzer. The algorithm is easily transferred to a three-dimensional cylindrical type of computational grids.

10.1 Introduction The complication of computational fluid dynamics problems has led to the emergence of new and improved computational algorithms for numerical solutions of equations, in particular, ideal gas dynamics and magnetohydrodynamics [1]. Even if we restrict ourselves to finite-volume type schemes [2], we can indicate different approaches for G. V. Ustyugova Keldysh Institute of Applied Mathematics, 4, Miusskaya pl., Moscow 125047, Russian Federation e-mail: [email protected] A. V. Koldoba (B) Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_10

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the spatial and temporal approximations [3–8], different algorithms for the approximate solution of the Riemann problem [9–14], and methods for excluding magnetic charges [15–17]. Certainty, these approaches are not limited only to the problems of gas dynamics and magnetohydrodynamics; the range of their application is much wider. Each of these approaches alone does not lead to an ideal computational algorithm and, in some cases, requires adaptation to the specifics of the class of problems under consideration. Many numerical magnetohydrodynamic codes have been developed for modeling plasma flows in astrophysics. Some of the most well-known codes are ZEUS [18], FLASH [19], PLUTO [20], ATHENA [21, 22]. In some astrophysical hydrodynamic problems, the main flow is the movement in circular orbits around the gravitating center, on which additional perturbations are superimposed and have axial symmetry. In relation to the indicated astrophysical problems, this circumstance can be associated, for example, with a mismatch between the axis of rotation of the system and the magnetic axis of the star, the presence of planets or other sources in the system that violate the axial symmetry of the main flow, the development of instabilities, etc. (e.g., see [23–25]). Often, polar (in the two-dimensional case) or cylindrical (in the three-dimensional case) coordinates are used to solve such problems. When modeling flows in polar or cylindrical coordinates, a number of problems arise related to the approximation of the hydrodynamics equations in the vicinity of the axis, which generates a geometric feature. If the flow has axial symmetry, the use of polar or cylindrical coordinates is natural (we do not address the issue of reducing the dimension of the problem in these cases). If the flow does not have axial symmetry, the use of such coordinates can lead to significant errors. In addition, the use of a detailed grid along the azimuthal angle leads to strongly elongated spatial grid cells in the vicinity of the axis and a strong restriction on the time step of integration in the case of using explicit difference schemes. If the flow has translational symmetry in any direction, it is natural to use Cartesian coordinates to approximate the equations of gas dynamics. The choice of the computational grid on which differential equations are approximated (in this case, gas dynamics), and the choice of the method of representing vector quantities (in this case, velocities), Cartesian, polar, or other components, are not interconnected. Although in many cases, the choice of the coordinate system determines both the representation of vector quantities and the type of a computational grid. Apparently, the gas-dynamic variables themselves and their flows can be recalculated from polar (cylindrical) coordinates to Cartesian and vice versa. However, to calculate the flows between the computational cells in the Godunov approach, the gas-dynamic quantities should be set on both sides of the face separating these cells. The vector quantities can be transferred from the cells to the faces in various ways. Depending on the chosen method, different data for solving the Riemann problem or its approximations can be obtained. In this chapter, we propose a computational algorithm for the numerical integration of gas dynamics equations containing a flow symmetry-analyzer. The computational algorithm is based on a finite-volume Godunov type scheme with an approximate solution of the Riemann problem [26], which, however, is not fundamental. Instead

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of the energy balance equation, the entropy balance equation is used. This circumstance is also not fundamental for the proposed algorithm but limits its applicability to continuous solutions. In this work, the equations of gas dynamics are approximated on a polar type grid, and Cartesian components represent the velocity field. The computational grid consists of several circular blocks (concentric rings) with different resolutions in the azimuthal angle. In the indoor unit, the azimuthal direction is allowed by six cells. In each subsequent block, the number of cells in the azimuthal direction is doubled. In all the blocks, the grid is uniform in terms of the azimuthal angle. On the one hand, the use of such a grid avoids artificial (geometryrelated) restrictions on the time integration step. On the other hand, in the vicinity of the grid axis, the azimuthal resolution is coarse, while the difference between the plane and axisymmetric vector fields is most pronounced. On such a grid, the use of a symmetry-analyzer produces the greatest positive effect. If we use a computational grid with a fixed (in all the blocks) azimuthal resolution to approximate the equations, using a symmetry-analyzer will not lead to a significant improvement in the quality of the numerical solution. In this case, computational problems are caused by other factors. In the present work, the Cartesian coordinate system is used to approximate the momentum equations, i.e., the momentum density and the corresponding flows are represented by the Cartesian components. This approach eliminates the need to approximate the terms due to the curvature of the coordinate system. It is assumed that the flows are either predominantly homogeneous (plane-symmetric) or predominantly axisymmetric (relative to the axis of the grid). To reconstruct vector variables, a symmetry-analyzer is used, which allows locally attributing the flow to one of the indicated types. Depending on what type the vector field will be assigned to, the corresponding components of the vectors are used for its reconstruction. This approach makes it possible to more accurately reproduce the flow features on a coarse grid in the vicinity of the axis. Section 10.2 describes the Magnetic HydroDynamics (MHD) model considered in this work and presents the proposed algorithm for analyzing the flow symmetry. Section 10.3 is devoted to testing the computational algorithm, including the proposed symmetry analyzer, and comparing it with the “standard” algorithm for the numerical integration of two-dimensional equations of an ideal MHD model. Section 10.4 concludes the chapter.

10.2 Computational Algorithm with a Symmetry-Analyzer We describe an algorithm with a symmetry-analyzer for the numerical solution of two-dimensional equations of ideal MHD. As already noted, momentum balance equations are considered in the Cartesian coordinate system, and instead of the energy balance equation, the entropy balance equation is used: ∂ρv y ∂ρvx ∂ρ + + = 0, ∂t ∂x ∂y

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    B y2 − Bx2 Bx B y ∂ ∂ρvx ∂ 2 + ρvx v y − = 0, ρvx + P + + ∂t ∂x 8π ∂y 4π     Bx2 − B y2 ∂ρv y Bx B y ∂ ∂ 2 + ρv y vx − + ρv y + P + = 0, (10.1) ∂t ∂x 4π ∂y 8π  ∂ Bx ∂  v y Bx − vx B y = 0, + ∂t ∂y  ∂ By ∂  vx B y − v y Bx = 0, + ∂t ∂x ∂ρ Sv y ∂ρ S ∂ρ Svx + + = 0. ∂t ∂x ∂y Here, x, y are the Cartesian coordinates, t is the time, ρ, P, S are the density, pressure, and specific entropy, respectively, vx , v y are the Cartesian components of the velocity, Bx , B y are the Cartesian components of the magnetic field. The computational grid consists of several blocks (concentric rings), in each of which the resolution in the azimuthal direction is different. When moving from the previous (internal) block to the next (external) block, the resolution detail is doubled. The usage of such grids allows us to make the computational cells approximately equilateral and avoid fragmentation of the integration step over time. This is due to the geometry of the computational grid in the vicinity of the axis (Fig. 10.1). Fig. 10.1 Fragment of the computational grid

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For the numerical solution of Eq. 10.1, an explicit difference scheme of the Godunov type of the third-order approximation in time and space is used with an approximate HLLD (Harten-Lax-van Leer-Discontinuities) solution of the Riemann problem [13, 26] and an anti-diffusion limiter [1, 11]. It is understood that the reconstruction of variables on the verge of the computational grid is carried out according to an algorithm that, in the one-dimensional case and on sufficiently smooth solutions, provides a third order of approximation on spatial variables. Reconstruction is performed for pressure, specific entropy, and components of the velocity vector and the magnetic field, selected by a symmetry-analyzer. The anti-diffusion limiter has the form: ⎧ ⎪ ⎪ ⎨

0 (2/δ + 1)x/6 α(x, y) = ⎪ x/6 + y/3 ⎪ ⎩ y

if if if if

xy xy xy xy

0 |x| < δ|y| > 0 δ|y| < |x| < 4|y| >0 |x| > 4|y|

with parameter δ = 0.5. Concerning the reconstruction of pressure in the radial direction, the anti-diffusion corrections yield the following values on the opposite faces of the cell (i, j) in a block l: d1 = Pi, j,l − Pi−1, j,l , d2 = Pi+1, j,l − Pi, j,l and Pi,Lj,l = Pi, j,l − α(d2 , d1 ), Pi,Rj,l = Pi, j,l + α(d1 , d2 ). The directly calculated quantities are the mass density ρ, the entropy density ρ S, and the Cartesian components of the momentum flux density vector ρvx , ρv y and the magnetic field Bx , B y , assigned to the cells of the computational grid. These variables are used to calculate other variables necessary for reconstructing the functions and solving the Riemann problem on the faces between the grid cells. To solve the Riemann problem, two sets of variables containing density, pressure, and polar components of the velocity vector and the magnetic field that have the meaning of MHD states on both sides of the face are fed to the boundary between the cells. The use of polar components in constructing a solution to the Riemann problem (on the computational grids used in this chapter) is natural since they are normal and tangential components of vectors relative to the faces of the computational grid. An algorithm for calculating these values will be presented below. As a result of an approximate solution of the Riemann problem, the flows of mass, entropy, and momentum vector (specified by the polar components) are calculated through the face. The momentum flow vectors on the faces are converted to Cartesian coordinate system to calculate the change in the momentum vector in the cells specified by the Cartesian components. For the numerical integration of time equations, the three-stage Runge–Kutta method is used, or more specifically, the third-order approximation Heun method [27]. Time in the intermediate stages is calculated according to the Runge rule. To calculate the flows of conservative variables between the calculation cells (e.g., A and B, see Fig. 10.2) in the Godunov approach, it is necessary to solve (exactly or approximately) the Riemann problem of the decay of the discontinuity between

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Fig. 10.2 Transfer of vector b from point B to point A while preserving the Cartesian components—b and preserving the polar components—b

the states on two sides of the cell separating cells A and B. The MHD-quantities describing these states are obtained using a procedure for reconstructing functions (densities, components of the velocity vector, etc.) from the grid data. The following question arises concerning the reconstruction of vector quantities: which components of the vector fields are preferable: Cartesian or polar (or some others)? To answer this question, we use a symmetry-analyzer as this algorithm allows us to give preference to the Cartesian or polar (other options are not considered) components of a vector field for the grid data to reconstruct it on the edge of the computational grid. The analysis is carried out separately for each face so that the vectors’ Cartesian components can be used on parts of the faces for reconstruction, and the polar ones on the other part [28]. To analyze the type of a vector field, the following algorithm was adopted. Two two-dimensional vectors are considered: a and b in the (x, y) planes defined at two points A and B, e.g., the velocity vectors at the centers of two adjacent cells marked with the same letters), differing only in the azimuthal coordinate. The goal of the symmetry analysis is to classify this vector field (consisting of two vectors) as plane-symmetric or axisymmetric (azimuthally symmetric). The symmetry analysis establishes the proximity of these vectors when transferring them to one common point, e.g., to point A (Fig. 10.2). Two vector transfer methods are considered to transfer vector b from point B to point A: vector b is transferred while maintaining its Cartesian components bx , b y and vector b is transferred with the preservation of its polar components br , bφ . As a result of these operations at point A, in addition to vector a, two vectors are defined, b and b , which are obtained as a result of applying the two indicated vector transfer methods. In both cases, the lengths of these vectors are equal to the length of vector b. The proximity of vectors a and b , b is estimated by the angle between them. Since the lengths of the vectors did not the

during

change transfer, the moduli of the Z–component of the vector products a, b z and a, b z are compared. If |[a, b ]z | ≤ |[a, b ]z |, then it is assumed that the vector field is represented by vectors a and b is approximately plane-symmetric. Otherwise, the

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field is considered approximately axisymmetric. Under the assignment of the field to one of the indicated classes, the vector field is interpolated on the edge of the grid cells. In the first case, the Cartesian components of the vectors are used for interpolation, and in the second case, the polar ones. As noted, for the numerical integration of the momentum balance equations, the Cartesian components of the vectors are used. If at points A and B only the Cartesian components of the vectors are given, then the following holds: 1. The polar components of vectors a and b are calculated:

ar = ax cos φ A + a y sin φ A , aφ = −ax sin φ A + a y cos φ A , br (A) = br (B) = bx cos φ B + b y sin φ B , bφ (A) = bφ (B) = −bx sin φ B + b y cos φ B , where φ A and φ B are the azimuthal coordinates of points A and B. 2. The z-components of the vector products are calculated: [a, b ]z = ax b y − a y bx ,

[a, b ]z = ar bφ − aφ br = (ax b y − a y bx ) cos φ + (ax bx + a y b y ) sin φ, where φ = φ A − φ B is the azimuthal angle between the directions to points A and B (grid step in azimuth in the corresponding block).

3. Based on a comparison of values a, b z and a, b z , the velocity (or magnetic) field is reconstructed in the corresponding face using the Cartesian or polar components of the velocity (or magnetic) field. Note that the symmetry analysis algorithm is applied locally, i.e., for some pairs of the cells, the vectors specified in them are classified as belonging to a plane-symmetric type, and for others, to an axisymmetric type. After the variables are reconstructed on the face of the computational grid, the HLLD algorithm for the approximate solution of the Riemann problem reconstructed on the sides of this face is used to calculate the flows of conservative variables through a certain face. It is assumed that the solution contains five discontinuities: fast shock waves and Alfvenic waves propagating to the left and right (rarefaction maybe) and a tangential discontinuity separating the areas of constant flow. The shock wave speeds were taken to be D = 1.1 cf , where cf are the fast magnetosonic speeds before the shock waves. The speed of matter between the shock waves is determined from the laws of conservation of momentum, taking into account the fact that at

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a tangential discontinuity, the velocity normal to the front is continuous, and the tangential component of the momentum flow is zero. Specific entropy is transferred through shock waves without changes.

10.3 Testing the Algorithm To evaluate the effectiveness of the proposed algorithm, test calculations were performed. In all the calculations, a computational grid consisting of five radial blocks was used. The number of radial layers in the blocks was 4 (in the indoor block) and 4, 8, 16, and 32 (in the outdoor block), giving a total of 64 layers. The azimuth angle resolution in the blocks was as follows: six cells (in the indoor block); and 12, 24, 48, and 96 cells (in the outdoor block). The time integration step was automatically selected from the following condition: the Courant number is equal to 0.4. Solutions of the test problems of plane parallel-flow and axisymmetric flow are discussed in Sects. 10.3.1 and 10.3.2, respectively, while rapid rotation of the cylinder in a stationary medium with a uniform magnetic field is considered in Sect. 10.3.3.

10.3.1 Plane-Parallel Flow A flow was considered against a background of constant pressure, in which the components of the velocity vector and density depend on the variable ξ = r sin(φ − α) − v1 t: ρ = ρ0 exp(−2ξ 2 ), p = p0 , vx = v0 cos α cos 2ξ − v1 sin α, v y = v0 sin α cos 2ξ + v1 cos α . Figure 10.3 shows the results of modeling this flow with the following flow parameters: α = 30◦ ,ρ0 = 1, p0 = 1,v0 = 1,v1 = 1 for two options for the reconstruction of vector variables. In Fig. 10.3a, for the reconstruction of the velocity field, its polar components are used, and in Fig. 10.3b, the components selected by a symmetryanalyzer depending on the type of flow, which in the case of this flow coincide with the choice of the Cartesian components for the reconstruction of the velocity field, are used.

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Fig. 10.3 Solution of the test problem of plane parallel-flow (all the points of the computational grid are shown): a obtained during the reconstruction of the velocity field by a symmetry-analyzer, b obtained during the reconstruction of the velocity field by polar components. The velocity vector components and density are shown

10.3.2 Axisymmetric Flow As a second test for the proposed algorithm, we consider the Bondi cylindrical flow— accretion onto the gravitating center of mass. This one-dimensional axisymmetric stationary isoentropic flow is described by a system of Eqs. 10.2–10.3 as follows: vr

1 dP M dvr l2 + = − 2 + 3, dr ρ dr r r

(10.2)

ρvr dρ dvr + vr = , dr dr r

(10.3)

ρ

taking into account the conservation of entropy P/ρ γ = const and specific angular momentum l = vφr = const. Combining Eqs. 10.2–10.3, taking into account d p =  c2 dρ, where c = γ p ρ is the sound speed, we obtain Eq. 10.4. vr c2 − M/r + l 2 /r 2 dvr = dr r vr2 − c2

(10.4)

At a sound point, the conditions for the conservation of entropy and mass flow lead to the relation c2 = c∗2 (c∗r∗ /vr r )γ −1 . To construct the solution describing stationary accretion onto the gravitating center, we set the “mass” of the gravitating center M = 1, sound point position r∗ = 1, and specific angular momentum l = 0.25, while the sound speed at the sound point was c∗ = 0.968. Dependence vr (r ) was

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Fig. 10.4 Solution of the axisymmetric accretion test problem to the gravitating center (all the points of the computational grid are shown): a obtained during the reconstruction of the velocity field by a symmetry-analyzer, b obtained during the reconstruction by Cartesian components

obtained by integrating Eq. 10.4 from the sound point in both directions (to the inner radius R1 = 0.12 and external radius R2 = 3.32 of the computational domain) on a sufficiently detailed grid. Figure 10.4 shows the exact solution obtained by integrating Eq. 10.4 and the numerical solutions obtained by the integrating system Eq. 10.1 according to the third-order approximation scheme, depending on the radius. Figure 10.4a shows the azimuthal and radial components of the velocity vector and the speed of sound obtained by reconstructing the velocity field from its Cartesian components, and Fig. 10.4b shows the solution obtained using a symmetry-analyzer. The solid line shows the exact solution, and the dots indicate the numerical solutions in all the cells of the two-dimensional computational grid. The grid step along the radius, as in all the blocks, was r = 0.05.

10.3.3 Rapid Rotation of the Cylinder in a Stationary Medium with a Uniform Magnetic Field (Rotor Problem) The problem was proposed in [6] and is often used in the study of numerical schemes for MHD equations. The main purpose of this test is to check for negative pressures in the center of the computational domain. At the initial moment of time, a uniform magnetic field B y0 , constant pressure P0 , and the distribution of density and azimuthal velocity are set in a circle of radius R. When a substance with a density solidly rotates at a constant angular velocity ω0 , vφ = ωr . When a substance with a density is at rest vφ = 0 . In the transition zone, linear interpolation of quantities is used:

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Fig. 10.5 Profiles along the axis of the components of the speed of matter vr , vφ , and the sound speed c: a reconstruction by a symmetry-analyzer, b reconstruction by polar components of vectors

 Fig. 10.6 Profiles along the axis of gas P and magnetic B 2 8π pressures: a reconstruction by a symmetry-analyzer, b reconstruction by polar components of vectors

ω = ω0

r1 − r r1 − r , ρ = ρ1 + (ρ0 − ρ1 ) . r1 − r0 r1 − r0

In such a problem, it is obvious that the velocity field should be reconstructed in its polar components and the magnetic field in the Cartesian ones. Moreover, the situation when both fields are reconstructed using their polar components looks much worse than the reconstruction of vectors from Cartesian components. When reconstructing both fields using Cartesian components, the maximum difference with the variables obtained using a symmetry-analyzer is achieved on the axis and is about 2%.

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Fig. 10.7 Profiles along the axis of the Alfven velocity components ar , aφ : a reconstruction with a symmetry-analyzer, b reconstruction by polar components of vectors

Figures 10.5, 10.6, 10.7 and 10.8 present the results of the calculation of this problem for the following parameters: the boundaries of the domains R = 1, r0 = 0.1, r1 = 0.115, the initial data parameters B y0 = 2.5, P0 = 0.5, ω0 = 1, ρ0 = 10, ρ1 = 1 In the calculations, a grid consisting of six radial blocks was used. The number of radial layers in the blocks was: 2 (in the indoor block), 4, 8, 16, 32, 130 (in the outdoor block), a total of 192 layers. Resolution in the azimuthal angle in the blocks: 6 cells (in the indoor block), 12, 24, 48, 96, 192 (in the outdoor block). The grid step along the radius, the same in all the blocks, was r = 0.005. The time integration step was automatically selected from the condition: Courant number is equal to 0.4. The calculation results are presented at a point in time t = 0.295. For comparison, a calculation was carried out on a Cartesian grid of 800 × 800 squared in domain (−0.8, 0.8)×(−0.8, 0.8). The results of this calculation are taken as “reference.” Fig. 10.5 shows the profiles alongthe axis of the components of the  velocity of matter vr , vφ and the sound speed c = γ P ρ. The “reference” solution is shown by a solid line, the markers in Fig. 10.5a show the variables obtained using a symmetry-analyzer, and Fig. 10.5b shows the reconstruction of vector fields by polar components.  Figure 10.6 shows gas pressure P and magnetic pressure B 2 8π profiles. The “reference” solution is shown by a solid line, the markers in Fig. 10.6a show the variables obtained using a symmetry-analyzer, and in Fig. 10.6b, by the polar components. Figure 10.7 shows the profiles of the x- and y-components of the Alfven velocity. Figure 10.8 shows the level lines of the modulus of the magnetic field induction vector and the magnetic field lines for the solution in the internal part of the computational domain, obtained using a symmetry-analyzer (Fig. 10.8a), and by reconstructing the velocity and magnetic fields from the polar components (Fig. 10.8b). Figure 10.8c shows the corresponding fields in the “reference” solution.

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Fig. 10.8 Distributions some MHD variables: a pressure distribution, b magnetic pressure distribution, c velocity module distribution, d azimuthal velocity distribution

A comparison of the results of the test calculations shows that for different types of flows, a symmetry-analyzer selects the most suitable velocity components for its reconstruction at the boundaries of the calculated cells. The numerical solution obtained for the plane-parallel flow using a reconstruction of the polar velocity components has a large error due to the loss of approximation of the equations in the vicinity of the axis of the computational grid. Similarly, the numerical solution obtained for the axisymmetric flow by reconstructing the Cartesian velocity components has a large error. At the same time, a symmetry-analyzer in all the cases selects the appropriate components for reconstructing the velocity field, which leads to a good coincidence of the numerical and exact solutions.

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Thus, we can conclude that the use of a symmetry-analyzer can improve the quality of numerical solutions of problems in which the type of the predominant flow symmetry in the vicinity of the axis of the computational grid can change.

10.4 Conclusions A computational algorithm is proposed for the numerical integration of the gas dynamics equations containing a flow symmetry-analyzer. The computational algorithm is based on a finite-volume Godunov type scheme with an approximate solution of the Riemann problem: three-wave HLLC (Harten-Lax-van Leer-Contact [11]) for gas-dynamic or five-wave HLLD [13] for magnetohydrodynamic equations. The performed test calculations demonstrated the feasibility of the proposed approach in relation to some problems of gas dynamics and magnetohydrodynamic.

References 1. Kulikovskii, A.G., Pogorelov, N.V., Semenov, AYu.: Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman and Hall, Boca Raton (2001) 2. Godunov, S.K. (ed.): Numerical Solution of Multidimensional Problems of Gas Dynamics, Nauka, Moscow (1976). (in Russian) 3. Brio, M., Wu, C.C.: An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comp. Phys. 75(2), 400–422 (1988) 4. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems. J. Comp. Phys. 84(1), 90–113 (1989) 5. Dai, W., Woodward, P.R.: An approximate Riemann solver for ideal magnetohydrodynamics. Comp. Phys. 111(2), 354–372 (1994) 6. Dai, W., Woodward, P.R.: Extension of the piecewise parabolic method to multidimensional ideal magnetohydrodynamics. Comp. Phys. 115(2), 485–514 (1994) 7. Balsara, D.S., Spicer, D.S.: A staggered mesh algorithm using high order godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comp. Phys. 149(2), 270–292 (1999) 8. Ustyugov, S.D., Popov, M.V., Kritsuk, A.G., Norman, M.L.: Piecewise parabolic method on a local stencil for magnetized supersonic turbulence simulation. J. Comp. Phys. 228(20), 7614– 7633 (2009) 9. Harten, A., Lax, P.D., van Leer, B.: Upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61 (1983) 10. Roe, P.L.: Characteristic-based schemes for the Euler equations. Ann. Rev. Fluid Mech. 18, 337–365 (1986) 11. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A practical introduction. Springer, Berlin (1997) 12. Gardiner, T.A., Stone, J.M.: An unsplit Godunov method for ideal MHD via constrained transport. J. Comp. Phys. 205(2), 509–539 (2005) 13. Miyoshi, T., Kusano, K.: A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208(1), 315–344 (2005)

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14. Miyoshi, T., Terada, N., Matsumoto, Y., Fukazawa, K., Umeda, T., Kusano, K.: The HLLD approximate Riemann solver for magnetospheric simulation. IEEE Trans. Plasma Sci. 38(9), 2236–2242 (2010) 15. Balsara, D.S., Spicer, D.S.: A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magneto-hydrodynamic simulations. J. Comp. Phys. 149(2), 270–292 (1999) 16. Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I., De Zeeuw, D.L.: A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comp. Phys. 154(2), 284–309 (1999) 17. Toth, G.: The constraint in shock-capturing magnetohydrodynamics codes. J. Comp. Phys. 161(2), 605–652 (2000) 18. Stone, J.M., Norman, M.L.: ZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. I. The hydrodynamic algorithms and tests. Astrophys. J. Suppl. Ser. 80, 753–790 (1992) 19. Fryxell, B., Olson, K., Ricker, P., Timmes, F.X., Zingale, M., Lamb, D.Q., MacNeice, P., Rosner, R., Truran, J.W., Tufo, H.: FLASH: an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. Astrophys. J. Suppl. Ser. 131(1), 273–334 (2000) 20. Mignone, A., Bodo, G., Massaglia, S., Matsakos, T., Tesileanu, O., Zanni, C., Ferrari, A.: PLUTO: a numerical code for computational astrophysics. Astrophys. J. Suppl. Ser. 170(1), 228–242 (2007) 21. Stone, J.M., Gardiner, T.S., Tauben, P., Hawley, J.F., Simon, J.B.: Athena: a new code for astrophysical MHD. Astrophys. J. Suppl. Ser. 178(1), 137–177 (2008) 22. Skinner, M.A., Ostriker, E.C.: The Athena astrophysical magnetohydrodynamics code in cylindrical geometry. Astrophys. J. Suppl. Ser. 188(1), 290–311 (2010) 23. Romanova, M.M., Blinova, A.A., Ustyugova, G.V., Koldoba, A.V., Lovelace, R.V.E.: Properties of strong and weak propellers from MHD simulations. New Astron. 62, 94–114 (2019) 24. Romanova, M.M., Lii, P.S., Ustyugova, G.V., Koldoba, A.V., Blinova, A.A., Lovelace, R.V.E.: 3D simulations of planet trapping at disc–cavity boundaries. Mon. Not. R. Astron. Soc. 485(2), 2666–2680 (2019) 25. Blinova, A.A., Romanova, M.M., Lii, P.S., Ustyugova, G.V., Koldoba, A.V., Lovelace, R.V.E.: Comparisons of MHD propeller model with observations of cataclysmic variable AE Aqr. Mon. Not. R. Astron. Soc. 487(2), 1754–1763 (2019) 26. Koldoba, A.V., Ustyugova, G.V., Lii, P.S., Comins, M.L., Dyda, S., Romanova, M.M., Lovelace, R.V.E.: Numerical MHD codes for modeling astrophysical flows. New Astron. 45, 60–76 (2016) 27. Petrov, I.B., Lobanov, A.I.: Lectures in Computational Mathematics. Internet-Univ. Inform. Tekhnol, Moscow (in Russian) (2006) 28. Koldoba, A.V., Ustyugova, G.V.: Difference scheme with a symmetry analyzer for equations of gas dynamics. Math. Models Comput. Simul. 12(2), 125–132 (2020)

Chapter 11

High-Gradient Method for the Numerical Simulation of the Continuum Problems with the Strong Discontinues Vladimir V. Demchenko

Abstract The effective difference method based on characteristic directions isolation and consequent approximation of partial derivatives in pre-assigned finitedimensional space is suggested for the numerical simulation of physical processes with the strong discontinues in mechanics of continua and plasma physics.

11.1 Introduction The chapter deals with the problems of numerical modeling of some physical processes relevant to modern physics such as Inertial Fusion (IF), high-speed shock, and the development of hydrodynamic instabilities [1–3, 7]. The history of laser fusion goes back several decades. In this area, encouraging results have been obtained: mass velocities of the substance at the level of 1000 km/s and pressure in the laser-plasma of about 50 Gbar have been achieved. Section 11.2 presents a numerical method. Section 11.3 discusses the results of solving the model problem of high-speed impact and the convergence of the obtained numerical data to the analytical solution. An analysis of the possibility of using one of the analytical solutions to the Cauchy problem of the Euler equation system in IF is presented in Sect. 11.4. Section 11.5 contains data from a multidimensional numerical simulation of one of the shock ignition problems in IF. Section 11.6 concludes the chapter.

V. V. Demchenko (B) Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_11

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11.2 Numerical Method The realization of three main conservation laws: mass, linear momentum, and energy are necessary for the right description of the continuum behavior. The mathematical formulation of these laws leads to the quasi-linear system of first-order hyperbolic type partial differential equations [1, 2]. This equations system is represented for vector–matrix form in three-dimensional Cartesian space as follows:     ∂W ∂W ∂W ∂W +A +B +C = 0, ∂t ∂x ∂y ∂z

(11.1)

 = {ρ, ρu, ρv, ρw, ρ E}T , W ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎫ ⎪ 0 1 0 0 0 ⎪ ⎪ ⎪  2     ⎪ −u + Pρ 2u + Pρu Pρv Pρw Pρ E ⎬ , A= −uv v u 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −uw w 0 u 0 ⎪ ⎪ ⎪ ⎩ −u(E + P ρ + P  ) E + P ρ + u P  u P  u P  u(1 + P  ) ⎪ ⎭ ρ ρu ρv ρw ρE ⎧ ⎫ ⎪ ⎪ 0 0 1 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −uv v u 0 0 ⎨ ⎬  2     −v + P P 2v + P P P B= , ρ ρu ρv ρw ρE ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −vw 0 w v 0 ⎪ ⎪ ⎪ ⎩ −v(E + P ρ + P  ) v P  E + P ρ + v P  v P  v(1 + P  ) ⎪ ⎭ ρ ρu ρv ρw ρE ⎧ ⎫ ⎪ ⎪ 0 0 0 1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −uw w 0 u 0 ⎨ ⎬ −vw 0 w v 0 C= , ⎪ ⎪  2     ⎪ ⎪ ⎪ ⎪ −w + P P P 2w + P P ⎪ ⎪ ρ ρu ρv ρw ρE ⎪ ⎩ −w(E + P ρ + P  ) w P  w P  E + P ρ + w P  w(1 + P  ) ⎪ ⎭ ρ ρu ρv ρw ρE where x, y, z, t are the independent variables, ρ is density, u, v, w are the velocity   is the pressure. components, E is the total specific energy, P = P W Let’s introduce the diagonal matrices consisting of eigenvalues for matrices A T , B T , C T and their absolute values  A = diag{u + c; u; u; u; u − c}, | A | = diag{|u + c|; |u|; |u|; |u|; |u − c|},  B = diag{v + c; v; v; v; v − c}, | B | = diag{|v + c|; |v|; |v|; |v|; |v − c|},

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C = diag{w + c; w; w; w; w − c}, |C | = diag{|w + c|; |w|; |w|; |w|; |w − c|}, where c is the sound velocity and as well matrices  A ,  B , C consisting of linear independent eigenvectors for matrices A T , B T , C T . Then the initial Eq. 11.1 is multiplied by the matrix TA from the left and is used the following correlations: TA A = (A T  A )T = ( A  A )T =  A TA . The system of transformed equations receives in the form of Eq. 11.2. TA

    ∂W ∂W ∂W ∂W +  A TA + TA B + TA C =0 ∂t ∂x ∂y ∂z

(11.2)

Let’s divide the matrix  A into two ones: nonnegative +A = ( A + | A |)/2 and nonpositive −A = ( A − | A |)/2 and introduce the two-point approximation of the first derivatives with respect to x taking into account the inclination of characteristics. Then transformed equation is multiplied from the left by the matrix (TA )−1 , which exists always, because the matrix TA consists of linearly independent eigenvectors of a matrix A T and, consequently, is nonsingular and provided by Eq. 11.3, where h x is x-difference mesh width. n n  l,m,k  l+1,m,k  −W W ∂W + (TA )−1 −A TA ∂t hx n n  l−1,m,k  l,m,k − W W + (TA )−1 +A TA hx   ∂W ∂W +C =0 +B ∂y ∂z

(11.3)

Comparing the equations in the forms Eqs. 11.2–11.3, it’s seen that equation members, which don’t depend on the x-derivative, don’t differ from each other. Therefore, those transformations that are fulfilled for x-first-order derivative approximation may be repeated for y, z is the first-order derivative approximations if we define eigenvalues and eigenvectors of matrices B T and C T . Introducing the difference approximation of the derivative with respect to time between the nearest time layers, we shall obtain the final calculating formula for the transition from n-time layer to n + 1-time layer in the form of Eq. 11.4, where h y , h z , and τ are the difference mesh widths on y, z, and t, respectively. n+1 n n n  l,m,k  l,m,k  l+1,m,k  l,m,k W =W + (TA )−1 −A TA (W −W )τ/ h x n n  l,m,k  l−1,m,k + (TA )−1 +A TA (W −W )τ/ h x T −1 − T  n n  l,m,k )τ/ h y + ( B )  B  B (Wl,m+1,k − W n n  l,m,k  l,m−1,k + (TB )−1 +B TB (W −W )τ/ h y

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V. V. Demchenko n n  l,m,k+1  l,m,k + (CT )−1 C− CT (W −W )τ/ h z T −1 + T  n n  l,m,k−1 )τ/ h z + (C ) C C (Wl,m,k − W

(11.4)

11.3 The Strong Arbitrary Discontinues Decay Problems in the Continuum Mechanics The test calculation series was carried out for the properties study of some developed variant of the finite difference method (Eq. 11.4) in order to provide the possibility of the comparison with already well-known analytical solutions. The mathematical model of this task includes the three main conservation laws in the continuum mechanics: mass, momentum, energy, the equation of ideal gas state with the isentropic exponent γ = 5/3, the initial and boundary conditions: ∂ρu ∂ρ + = 0, ∂t ∂x

∂ ρu 2 + P ∂ρu + = 0, ∂t ∂x ∂ρε ∂ρuε ∂u + +P = 0, ∂t ∂x ∂x  P = (γ − 1)ρε, γ = 5 3, x0 ≤ x ≤ x1 , t0 ≤ t ≤ T, ρ|t=t0 = ρ(t0 , x),

ε|t=t0 = ε|(t0 , x),

u|t=t0 = u(t0 , x),

ρ|x=x0 = ρ(t, x0 ),

ε|x=x0 = ε|(t, x0 ),

u|x=x0 = u(t, x0 ),

ρ|x=x1 = ρ(t, x1 ),

ε|x=x1 = ε|(t, x1 ),

u|x=x1 = u(t, x1 ).

Let us suppose, there are four infinite planar layers made from the same matter. All layers have 15 μ thickness. At the first moment of time, their temperature is the same and equal to 250°K. The two central layers density is equal 2.7 g/cm3 , they are surrounded by the two layers with a density 10–5 g/cm3 . Let one from the central layer collides with other central layers at the relative speed of 80 km/s. In initial momentum of time the model task contains three planes of the discontinuity into the integration domain: (1) on the boundary between vapor—rest layer the density undergoes the jump, (2) on the boundary of two layers the velocity undergoes the jump, (3) in the region of the contact moving layer-vapor the density and velocity are changed by jump. In these cases, the discontinuity jumps of gas dynamics

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functions achieve six-seven orders of scale units in centimeter-gram-second. The discontinuity decay on the boundary of two layers is accompanied by two shock waves formation propagating in the opposite directions and on the boundary moving layer—rest vapor—two central rarefaction waves arise. The scheme of the strong arbitrary discontinues decay problems in the continuum mechanics at the initial time is        ρ = 10−5 g cm3  ρ = 2.7 g cm3  ρ = 2.7g cm3  ρ = 10−5 g cm3 u = 0   u = 0   u = −80 km s  u=0  10 2 2  ε = 10 cm s  ε = 1010 cm2 s2  ε = 1010 cm2 s2  ε = 1010 cm2 s2  ← 15 μ →  ← 15 μ →  ← 15 μ → ← 15 μ → The numerical calculation results are presented in three next Figs. 11.1, 11.2 and 11.3 that were obtained on sequentially doubling grids for these cases of the discontinuity decay. The slowest convergence of density values is observed in the region of existence of centered rarefaction waves, where the minimum density values approach values of the order 10–13 g/cm3 . Here and in the third graph, a semi-logarithmic scale is used to represent the results. The different curves correspond to various calculations with some variant of scheme (Eq. 11.4) at the differing grid steps along with space coordinates. In all cases, a uniform space grid is used, but it’s doubled from one variant to another. The corresponding curves are noted by dotted lines with different markers. Note that in the area of the existence of two centered rarefaction waves, the use of the Euler equation system is not fully justified, since the Knudsen number is

Fig. 11.1 The graphs of density at 150 ps after the beginning of the discontinuity decays at the boundaries of both the two layers and the moving layer-vapor calculated on the sequentially difference grids

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Fig. 11.2 The graphs of velocity at 150 ps after the beginning of the discontinuity decays at the boundaries of both the two layers and the moving layer-vapor calculated on the sequentially difference grids

Fig. 11.3 The graphs of temperature at 150 ps after the beginning of the discontinuity decays at the boundaries of both the two layers and the moving layer-vapor calculated on the sequentially difference grids

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approximately equal ~1015 and the free molecular flow regime is realized. However, despite this discrepancy, it is possible to observe the convergence of the main gasdynamic values obtained by numerical calculations to a physically justified result. This raises questions about the possibility of using a system of Euler equations to describe flows with Knudsen numbers ≥10−3 , and under what conditions this approach is acceptable.

11.4 Collapsing Mode of Compression of the Substance One of the most important problems of modern physics is the implementation of controlled thermonuclear fusion. Among the possible approaches to solving this problem, we consider inertial fusion, when such densities and temperatures are created in a limited volume that the energy of thermal motion is sufficient to overcome the Coulomb forces of interaction between the plasma nuclei of thermonuclear fuel. Preliminary theoretical estimates show that this requires the formation of a region with the following parameters: ρ R ∼ 1 g/cm2 and Ti ∼ several keV. Here, ρ density Ti is the ionic temperature, R is the linear size of the region. To describe the behavior of a substance under such conditions, the basic laws of conservation are used: mass, amount of motion, and energy, along with other physical processes [2, 3, 5–8]. The mathematical model corresponding to physical laws includes a system of the Euler equations with the equation of state of an ideal gas. Let’s focus on one of the analytical solutions of this system, which allows us to satisfy the above requirements [5, 6], and formulate the Cauchy problem for the one-dimensional case in the Cartesian coordinate system: ∂ρu ∂ρ + = 0, ∂t ∂x

∂ ρu 2 + P ∂ρu + = 0, ∂t ∂x ∂ρε ∂ρuε ∂u + +P = 0, ∂t ∂x ∂x P = (γ − 1)ρε,

 P = C0 ρ γ , γ = 5 3,

   3 2 3 C0 = 2ε0 (3ρ0 / ), ρ|t=t0 = ρ0 x x0  ,   2  ε|t=t0 = ε0 x x0 , u 

t=t0

 = u 0 x x0 .

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Problem solution: density ρ = ρ0 (x/x0 )3 ( f )6 , x ≥ 0, velocity u = (x/x0 )χ f , specific internal energy ε = ε0 (x/x0 )2 ( f )4 , defined from initial conditions  for ρ and u: f = 1/[cos(2α/3) + 1/2], t0 = 0, u 0 = 0, χ = (80tε0 sin(2α/3)) (x0 sin(2α)),

√ cos α = − t 160ε0 /(2x0 ). In the practical implementation of this solution in experimental conditions, it is difficult to achieve a linear velocity distribution at the initial moment along the length of the target, so the initial condition is zero velocity. Let’s assume that it is possible to simulate the cubic density distribution along the length of the target by layers with different densities up to the maximum. For material from DT (D is deuterium, T is tritium), this corresponds to ρ0 = 0.2 g/cm3 . We will also assume that a quadratic dependence of temperature on the spatial coordinate with a maximum value of 1 eV is created, and assume that the specific heat capacity at a constant volume is CVe = 5.8 × 1014cm2 /(s2 keV). Let the target have the initial size x0 = 100 μ, ε0 = 5.8 × 1011 cm2 s2 (Fig. 11.4).

Fig. 11.4 Graphs: density, velocity, and temperature as a function of the spatial coordinate at the initial time

If the analytical solution conditions are maintained at the boundary of the target, x = Pm =  50 then there will come a point in time when 0 m ρd x ≈ 1 and P(tm, xm ) √ 160ε0 , Gbar. For option u0 = 0 by t 0 = 0, time of collapse is tk = 2x0  5  3 pressure is P = P0 x x0 ( f )10 , ρm = ρ0 xm x0 ( f m )6 , the time of reach Pm on the border x m is defined as follows:

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Fig. 11.5 Graphs: density, velocity modulus, temperature at a time t m = 1.95 ns, Pm = 50 Gbar, x m = 65 μ

    √   3 P0 1/ 10 1 2x0 tm 160ε0 · cos arccos tm = − √ − , cos αm = − 2 Pm 2 2x0 160ε0     

 f m = 1 cos 2αm 3 + 1 2 ; u m = 80tm ε0 sin 2αm 3 (x0 sin(2αm )) f m . Let’s assume that it was possible to form the central core of the target in accordance with the analytical solution, and, thus, the external crown of the target was formed symmetrical to the central part (Fig. 11.5). For further consideration of the development of physical processes, we use the numerical method (Eq. 11.4) and obtain distributions of the main physical quantities: density, temperature, velocity after the moment of collapse, and the propagation of a shock wave (“thermonuclear fusion waves”) by the substance of the target, which is shown in Fig. 11.6 at time t = 2.2 ns. The obtained results show that, in general, it is possible to achieve the conditions of thermonuclear ignition, although a number of significant physical processes were not taken into account, for example, energy release due to thermonuclear fusion reactions. At the same time, the rating of the density of the number of thermonuclear neutrons gives an order of magnitude Q = 1018 n/cm2 .

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Fig. 11.6 Graphs: density, velocity modulus, temperature at a time t = 2.2 ns

11.5 Numerical Experiment on Shock Ignition in Inertial Fusion In a number of experimental studies, results were obtained on the acceleration of layers of matter to speeds of about 1000 km/s, and a proposal was made to use this fact for thermonuclear ignition [4]. The results of numerical simulation of one of these problems are presented below. The original system of equations written in the Euler variables (r, θ ) of the spherical coordinate system has the following form: 1 ∂(r 2 ρU ) 1 ∂(sin θρV ) ∂ρ + 2 + = 0, ∂t r ∂r r sin θ ∂θ ∂ρU 1 ∂[r 2 (ρU 2 + P)] 1 ∂(sin θρU V ) 2P + 2 + = , ∂t r ∂r r sin θ ∂θ r ∂ρV 1 ∂r 2 (ρU V ) 1 ∂[sin θ (ρV 2 + P)] cos θ P + 2 + = , ∂t r ∂r r sin θ ∂θ r sin θ ∂ρ E 1 ∂[r 2 U (ρ E + P)] 1 ∂[sin θ V (ρ E + P)] + 2 + ∂t r ∂r r sin θ ∂θ 1 ∂[sin θ (We )θ ] 1 ∂[r 2 (W e)r ] + , = Q laz + 2 r ∂r r sin θ ∂θ

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1 ∂(r 2 ρU εi ) 1 ∂(sin θρV εi ) ∂ρεi + 2 + ∂t r ∂r r sin θ ∂θ   1 ∂(r 2 U ) 1 ∂(sin θ V ) = Q ei (Te − Ti ), + Pi 2 + r ∂r r sin θ ∂θ where ρ is the density, U is the radial r-component of velocity, V is the tangential θ component of velocity, εi = CVi Ti is the ionic specific internal energy, εe = CVe Te is the electronic specific internal energy, T e , T i are the electronic and ionic

temperature; CVe = 7.2391014 cm2 / s2 keV , CVi = 7.239 × 1014 cm2 / s2 keV is the specific electron and ionic heat capacity at a constant volume, E = εe +εi +(U 2 + V 2 )/2 + I is the total specific energy, I = 8.345 × 1013 cm2 /s2 is the total ionization energy of carbon and deuterium atoms, which was assumed to be constant, P = Pe + Pi is the pressure, Pe = (γ − 1)[ρ E − ρεi − ρ(U 2 + V 2 )/2 − ρ I ] is the electronic pressure, γ = 5/3 is the adiabatic index, Pi = (γ −1)ρεi is the ionic pressure,  W e = ke grad Te is the heat flow due to electronic thermal conductivity, ke = coefficient of electron-ion exchange, k1 =

k12 + k22 , Q ei is the

5 2  erg  1.31 × 1020 Te / , ln(1 + 2.87 × 104 Te2 /ρ + 3.427ρ 1/ 3 ) cm s keV

k2 =

 erg  9.097 × 1016 ρTe , 1 3 ln(5.738 + 41.957Ti /ρ / ) cm s keV  [ρ] = g cm3 , [Te ] = [Ti ] =keV,

Q ei = 5.498 × 1024

ρ 2 ln(1 + 2.87 × 104 Te2 /ρ + 3.427ρ 1/ 3 )  erg  . 3 2 cm3 s keV Te / + 6.575 × 10−4 ρ

The propagation and absorption of laser radiation in the target corresponded back to the braking mechanism and was found from the solution of the equation 

 qL   · ∇ r q L = −kL q L q L 

where  2   ρ ρcr [ln(5.27Te ) + 4.5] 1  , kL = 136.4  3 2 cm Te / 10 · (1 − ρ ρcr ) ⎛ r ⎞ " −2 3 ρcr = 1.35 × 10 g/cm , qL (r, t) = q0 (t) exp⎝ −kL dr ⎠ R

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max q0 (t) = 8.36 × 1020

 erg  , cm2 s

 where [ρ] = g cm3 , [Te ] = keV, R is the external radius of the integration area. The dependence of the laser radiation intensity on time and angular coordinates was determined by the target configuration and parameters of the laser, whose total energy was 1 kJ and was divided into two beams of 500 J each. The graph of the change in the power of the laser pulse over time corresponded to the shape of an isosceles trapezoid that is, during the first 0.89 ns there was a linear increase in power, then it remained constant up to 1.3 ns, and in the following 0.89 ns, up to 2.19 ns, its linear decrease to zero was observed. Further, the initial statement of the problem is discussed in Sect. 11.5.1, while Sect. 11.5.2 provides the results of numerical simulation.

11.5.1 The Initial Statement of the Problem Figure 11.7 shows a cross-section of the target from CD (C-carbon, D-deuterium) with a plane passing through the symmetry axis θ = 0° (Z-axis) and bounded by the symmetry plane θ = 90° (XY ). In the central part, there is a ball with a radius of 100 microns and a density of 200 g/cm3 . A conical cavity with an internal volume of 0° < θ < 45°, 100 μ < r < 340 μ and a wall of 45° < θ < 54° closed on the outside by a segment 10 microns thick is attached to it. The density of the substance of the cone and segment is 1 g/cm3 , and the gas filling the inner cavity and the space surrounding the cone is 0.001 g/cm3 . At the initial moment, the target is at rest, the electron and ionic temperatures in the entire integration region of 1 eV. The energy of the laser pulse is 500 J, the power change over time corresponds to an isosceles trapezoid with a duration of 2.19 ns and linear sections of 0.89 ns, the radiation flux density is 8.36 × 1013 W/cm2 .

11.5.2 The Results of Numerical Simulation Figure 11.8 shows the density isolines at the time of 1.8 ns when the accelerated segment decelerates near the surface of the ball. Figure 11.9 shows the isotherms of the ion temperature at the time of 1.8 ns when the accelerated segment decelerates near the surface of the ball. Two shock waves are formed: one propagates to the center of the target along with the ball, and the other along the conical cavity in the plasma corona region. The density graph along the symmetry axis between the shock waves at a time of 1.8 ns is shown above (Fig. 11.10). It can be seen that the first shock wave interacting with the rarefaction wave from the surface of the ball compresses the substance, heating it up to several eV.

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Fig. 11.7 The initial statement of the problem

The graph of ionic temperature along the symmetry axis between shock waves at a time of 1.8 ns is shown in Fig. 11.11. In the second shock wave, the density is several hundredths of g/cm3 , and the ionic temperature is about 1 keV. Figure 11.12 shows the isolines of the thermonuclear neutron generation power in the integration region at the time of 1.8 ns. Thus, the following results have been obtained: • A finite-difference method for numerical simulation of physical processes in thermonuclear targets “impact ignition” was developed and implemented as a software package. • A computational experiment was performed to study one of the variants of a shock ignition target. • Results were obtained on the conversion of laser radiation energy over time into the energy of the accelerated segment, the walls of the conical cavity, and the scattering plasma. • The estimation of the number of produced thermonuclear neutrons is given, the spatial regions of their generation are specified, and the temporal and spatial changes in the sizes of these regions are determined.

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Fig. 11.8 The density isolines at the time of 1.8 ns

11.6 Conclusions The research carried out on the basis of an improved high-gradient method has shown the possibility of numerical simulation of complex processes in actual problems of modern physics. It is necessary to develop different methods of a higher order of convergence to increase the efficiency and accuracy of numerical calculations.

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Fig. 11.9 The isotherms of the ion temperature at the time of 1.8 ns

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Fig. 11.10 The density graph along the symmetry axis at the time of 1.8 ns

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Fig. 11.11 The graph of ionic temperature along the symmetry axis at a time of 1.8 ns

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Fig. 11.12 The isolines of the thermonuclear neutron generation power at a time of 1.8 ns

References 1. Belotserkovskii, O.M., Demchenko, V.V., Kosarev, V.I., Kholodov, A.S.: Numerical simulation of some problems of the laser compression of shells. Zh. Vychisl. Mat. Mat. Fiz. 18(2), 420–444 (in Russian) (1978) 2. Kidder, R.E.: Theory of homogeneous isentropic compression and its application to laser fusion. Nucl. Fusion 14, 53–60 (1974) 3. Ievlev, V.M., Son E.E.: Hydrodynamic description of high-temperature media. MIPT, Dolgoprudny (1977) 4. Demchenko, V.V.: An arbitrary gas dynamical discontinuity decay. MIPT, Moscow (1998) 5. Demchenko, V.V.: On a certain exact solution of the Euler equation system taking into account nonlinear thermal conductivity. In: Proceedings of the XXIII scientific conference of MIPT 1977, series “Aerophysics and applied mathematics”, Dolgoprudny, (1978) 6. Demchenko, V.V.: A comparative investigation of some hydrodynamic compression processes. USSR Comput. Math. Math. Phys. 19(2), 286–292 (1979) 7. Sedov, L. I.: On integration of equations of one-dimensional gas motion. Rep. USSR Acad. Sci. 90(5), 735 (1955) 8. Kidder, R.E.: Theory of homogeneous isentropic compression and its application to laser fusion. Nucl. Fusion 14, 53–60 (1974) 9. Azechi, H., Guskov, S.Yu., Demchenko, V.V., Demchenko, N.N., Doskoch, I.Ya., Murakami, M., Nagatomo, H., Rozanov, V.B., Sakaiya, S., Stepanov, R.V.: Laser-driven acceleration of a dense matter up to ‘thermonuclear’ velocities. Plasma Phys. Control. Fusion 49(10), 1689–1706 (2007)

Chapter 12

On the Finite Difference Schemes for Burgers Equation Solution Ilya V. Basharov

and Aleksey I. Lobanov

Abstract The new method of the difference schemes for solving the Burgers equation constructing is discovered. The method is based on the two different divergent forms for the Hopf equation. To search for the optimal difference schemes in this family, an analysis in the space of insufficient coefficients was applied using the technique of self-dual problems of linear programming solving. The multiple before the third derivate of exact solution grid mapping is used as a target functional. It is necessary and sufficient that the complementary slackness conditions be fulfilled. Based on complementary slackness conditions analysis, a new version of the Lax–Wendroff scheme is built. The new hybrid schemes with maximum anti-dispersion are also constructed. Some numerical results demonstrate the properties of the new difference schemes. Such a consideration opens up a way to build the optimal hybrid schemes with a successful choice of the target functional linear by the scheme insufficient coefficients.

12.1 Introduction The simplest non-linear partial differential equation describes a continuous media motion called a Burgers equation. Hopf was the first who investigated this equation [1]. The solution of the Burgers equation can be reduced to the solution of the linear heat transfer equation using the transformation of Cole and Hopf. In the case where the viscosity coefficient is zero, the equation turns to the Hopf equation. The first order Hopf equation has the hyperbolic type. The solutions of the Hopf equation can be a discontinuous function even with smooth initial data. Many features of the first-order quasi-linear equations are significant even if the viscosity coefficient is different from zero. Galerkin’s discontinuous methods for the Burgers equation solving are successfully used for this reason. The discontinuous I. V. Basharov · A. I. Lobanov (B) Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_12

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Galerkin method is often applied to construct the discontinuous solutions of hyperbolic equations and equations with dissipative terms. The method’s application of such class for solving the Burgers equation is described, for instance, in [2]. The Burgers equation has the same type of non-linearity as the Navier–Stokes equations. Thus, the Burgers equation is a good test problem to construct the difference schemes of new types and test the well-known difference schemes [3–5]. Splitting methods often use the specifics of the Burgers equation. This is a balance between dissipation and weak non-linearity. Some methods deal with either fast Fourier transform [3] or the Cole–Hopf transformation [4]. This chapter does not consider the schemes obtained by using integral transformations. If we take into account only the families of finite-difference schemes to solve the Burgers equation, we can observe the method of high (fourth) approximation order [5] based on the ideas of “compact difference schemes” [6]. In this chapter, the finite-difference schemes based on several other principles are described. The way of finite difference schemes constructions using analysis in the insufficient coefficients space was proposed by Magomedov and Kholodov in [7]. The chapter is organized as follows. Section 12.2 describes the problem statement of the analysis of difference schemes for the Burgers equation numerical solution. Search of insufficient coefficients and optimization problem is studied in Sect. 12.3. Some numerical results obtained by the new difference schemes usage are given in Sect. 12.4. Section 12.5 comprises short conclusions.

12.2 The Problem Statement Herein after, we formulate the differential Cauchy problem in Sect. 12.2.1. Section 12.2.2 comprises the description of a new family of finite difference schemes based on the two different divergent forms for the Hopf equation usage.

12.2.1 Differential Problem Burgers equation arises while researching the continuum mechanics tasks. Thus, from the Navier–Stokes equations for the velocity of one-dimensional isothermal flows of viscous incompressible fluids, Eq. 12.1 follows. ∂u 1 ∂ 2u ∂u +u = ∂t ∂x Re ∂ x 2

(12.1)

Here, u is the dimensionless velocity, Re is the Reynolds number. Equation 12.1 is a significant model to test numerical methods and develop the difference schemes for the Navier–Stokes equations solution. We will use the following Eq. 12.2.

12 On the Finite Difference Schemes for Burgers Equation Solution

∂u ∂ 2u ∂u +u =D 2 ∂t ∂x ∂x

153

(12.2)

As a rule, we set D  1. This is a case of the convective term dominance. We consider the divergence form of the equation except for the characteristic form Eq. 12.2 with Cauchy-like data in the form of Eq. 12.3. ∂ ∂u + ∂t ∂x



u2 ∂u −D 2 ∂x

 =0

(12.3)

Using the Cole–Hopf transformation ⎛ 1 η(x, t) = exp⎝ D

⎞ +∞ u(y, t)dy ⎠,

(12.4)

x

every solution of the Cauchy problem for Eq. 12.2 must correspond to the linear heat transfer equation solution. Continuously differentiable functions are important solution properties of Eq. 12.2, differ from the Hopf equation: ∂u ∂u +u = 0. ∂t ∂x

(12.5)

The solutions of Eq. 12.5 possibly have discontinuities even under continuous initial conditions. They are defined as weak solutions.

12.2.2 Finite Difference Schemes Equation 12.1 will be passed into the Hopf equation Eq. 12.5 at D = 0. If we multiply both the right and the left parts of Eq. 12.5 by u, we can get another divergence form in a view of Eq. 12.6. ∂ 2u 3 ∂u 2 + =0 ∂t ∂x 3

(12.6)

Let us apply the finite difference approach. We introduce the grid with constant steps τ by time and h by space. The simplest conservative explicit scheme is obtained by replacing the first derivatives with the upwind differences of the first order and by replacing the second derivative with the central difference. In this case, we get an analog of the Courant–Isaacson–Reese scheme for convective terms provided by Eq. 12.7.

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n 2 n 2 n − ym−1 y y n − 2ymn + ym−1 ymn+1 − ymn + m = D m+1 τ 2h h2

(12.7)

Here, we use traditional designations. The grid function y corresponds to the velocity. The upper index corresponds to the time layer number. The subscript corresponds to the node number by spatial coordinate. The scheme has the first-order approximation on both variables. The main term of the approximation error of Eq. 12.7 on the exact solution of Eq. 12.2 will be 2 2 τ ∂2u − h4 ∂∂ xu2 . To increase the order of approximation up to the second, the main term 2 ∂t 2 of the approximation error must be subtracted from the left side of the difference Eq. 12.7. When considering the domination of the convective terms case for the evaluation of the main error term, we will use the differential continuation of the Hopf equation. Let us assume that the contribution of dissipative terms is insignificant. It follows from Eq. 12.5: 1 ∂ ∂u 2 ∂ 2u = − . ∂t 2 2 ∂ x ∂t Then using Eq. 12.6, one can write: 1 ∂ 2u3 ∂ 2u = . ∂t 2 3 ∂x2 The second approximation order by space and the first-order by time difference scheme is written in the following forms: n 2 n 2 n 3 3 n 3 ym − ym−1 − 2 ymn + ym−1 ymn+1 − ymn τ ym+1 + = τ 2h 6 h2 2 n 2 n 2 h ym+1 − 2 ymn + ym−1 − 4 h2 n n n y − 2ym + ym−1 + D m+1 h2 or 3 n 3 n 2 n 2 n 3 − 2 ymn + ym−1 ym+1 − ym−1 τ ym+1 ymn+1 − ymn + = τ 4h 6 h2 n n n y − 2ym + ym−1 + D m+1 . (12.8) h2 This is an analog of well-known Lax–Wendroff difference scheme. It should be noted that this is different from the traditional Burgers equation scheme recording.

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155

12.3 Insufficient Coefficients and Optimization Problem The description of the difference schemes family in the insufficient coefficients space is given in Sect. 12.3.1. Section 12.3.2 provides the pair of self-dual linear programming problems. It also contains the auxiliary materials dealing with the dual problems of linear programming. A system of the complementary slackness conditions is given. The solutions of this system are differential schemes that are optimal in their class.

12.3.1 Family of Difference Schemes Let us consider a five-point stencil with 4 points on the lower time layer (Fig. 12.1) as the examination of the linear transfer equation in [7]. A family of difference schemes can be written as follows: 1 

u n+1 = m

αμ u nm+μ +

μ=−2

1 



βμ (u nm+μ )2 − (u nm )2 .

μ=−2 μ=0

necessary for the first-order 1The following 1conditions are 1 approximation 1 τ κ 2 α = 1, μα = 0, μ α = 2D , and μ μ μ 2 μ=−2 μ=−2 μ=−2 μ=−2 μβμ = − 2 , h τ here κ = h . It is not possible to build high-order approximation schemes in this way. Thus, if we need to achieve a second-order approximation only by spatial variable, we can additionally write 1μ=−2 μ2 βμ = κ4 and 1μ=−2 μ3 αμ = 0. When we use the second derivative of the solution third power as a grid dissipation term, we can achieve the second order of approximation. This was done for an analog of the Lax–Wendroff scheme [7]. Then the family of difference schemes can be written as Eq. 12.9. = u n+1 m

1 

αμ u nm+μ +

μ=−2

1  μ=−2 μ=0

1



βμ (u nm+μ )2 − (u nm )2 + γμ (u nm+μ )3 − (u nm )3 μ=−2 μ=0

(12.9) Fig. 12.1 Five-point stencil

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Four conditions provided by Eq. 12.10 must be satisfied for the first order of approximation. 1 

αμ = 1

μ=−2 1 

1 

μαμ = 0

μ=−2

μβμ = −

μ=−2

κ 2

1 

μ2 αμ = 2

μ=−2 1 

Dτ h2

μγμ = 0

(12.10)

μ=−2

If we additionally write 1μ=−2 μ3 αμ = 0, 1μ=−2 μ2 βμ = κ4 , 1μ=−2 μ3 βμ = 2 − 3Dτ κ, 1μ=−2 μ2 γμ = κ3 , as a result the second-order on spatial variables scheme 4h 2 is obtained. The coefficients α, β are fully defined for the Burgers equation solving is the difference scheme. On introducing the dimensionless parameter σ = Dτ h2 Courant number analog for dissipative terms, we have α−2 = 0, α−1 = σ, α0 = 1 − 2σ, α1 = σ. Now let’s analyze a family of difference schemes in the form of Eq. 12.11. 1 

u n+1 = u nm + m

αμ (u nm+μ − u nm ) +

μ=−2 μ=0

+

1 

1 



βμ (u nm+μ )2 − (u nm )2

μ=−2 μ=0



γμ (u nm+μ )3 − (u nm )3

(12.11)

μ=−2 μ=0

This way of representation is more convenient for further consideration. The Lax–Wendroff scheme analog (Eq. 12.8) belongs to this family. We will study smooth (infinitely continuously differentiable) solutions of the differential problem and write out the approximation conditions. The requirements of zero-order approximation for this type of record are fulfilled automatically. Using the standard technique of decomposition into the Taylor series of the exact solution projection on the uniform grid, we obtain the following order’s conditions. The first order of approximation requires the following conditions: 1  μ=−2 μ=0 1 

1 

μαμ = 0,

μ=−2 μ=0

κ μβμ = − , 2 μ=−2 μ=0

μ2 αμ = 2

1 

τD = 2σ, h2

μγμ = 0

μ=−2 μ=0

Let us notice then if we used Eq. 12.11, the insufficient coefficients α 0 , β 0 , γ 0 are not included in the sum on the right-hand side. For the second-order difference

12 On the Finite Difference Schemes for Burgers Equation Solution

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schemes by a spatial variable not lower than the first-order approximation by time constructing, we need 2 additional conditions on the insufficient coefficients 1  μ=−2 μ=0

μ βμ = 0, 2

1  μ=−2 μ=0

μ2 γμ =

κ2 . 3

We have a set of 9 coefficients under 6 connection conditions. Three coefficients in the set can be left free. Let us formulate the conditions of the grid solution monotonicity. For this purpose, one can rewrite the family of difference schemes Eq. 12.11 in the equivalent form provided by Eq. 12.12. u n+1 = u nm m +

1 



αμ + βμ u nm+μ + u nm + γμ (u nm+μ )2 + u nm+μ u nm + (u nm )2 μ=−2 μ=0

(u nm+μ − u nm )

(12.12)

The monotonic difference scheme transfers the monotonic (non-increase or nondecrease) grid function into monotonic one. We will require the inequalities to be performed in order to maintain the monotony of the grid function profile in the form of Eqs. 12.13–12.14.



αμ + βμ u nm+μ + u nm + γμ (u nm+μ )2 + u nm+μ u nm + (u nm )2 ≥ 0 ∀μ 1−

(12.13)

1 



αμ + βμ u nm+μ + u nm + γμ (u nm+μ )2 + u nm+μ u nm + (u nm )2 ≥ 0 μ=−2 μ=0

(12.14) Now we use the coupling equations (the approximation order conditions) to save only one from each set α, β, γ. We have equalities described earlier 1 three variables – 1 2 μα = 0 and μ μ=−2 μ=−2 μ αμ = 2σ . Then using the notation α−2 = α we will μ=0

μ=0

obtain α1 = σ −α, α−1 = σ −3α. Similarly, β−2 = β, β1 = − κ4 −β, β−1 = κ4 −3β, 2 2 γ−2 = γ , γ1 = κ6 − γ , γ−1 = κ6 − 3γ . Let us introduce the target functional. It must be linear by the insufficient coefficients of the scheme and somehow characterize the quality of the difference scheme. For instance, we choose the coefficient for the third derivative of the solution over space in the first differential approximation of the difference scheme as such a functional. It will characterize the dispersion part of the approximation error. Let us introduce the target function by Eq. 12.15.

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⎛

=

1  μ=−2 μ=0

⎞

1 9 ⎟ n 2  3 μ3 βμ + κσ ⎟ μ γμ → min (12.15) + u m 2 ⎠ μ=−2 μ=−2

⎜ μ3 αμ + u nm ⎜ ⎝

1 

μ=0

μ=0

This choice of target functionality reduces the positive dispersion but the difference schemes with anti-dispersion, which will be considered close to optimal.

12.3.2 Dual Problem, Complementary Slackness Conditions To construct the difference schemes with high approximation order by spatial variable with error dissipative component minimum and minimum positive dispersion the optimization problem Eq. 12.15 with the inequality constraints Eqs. 12.13 and 12.14 should be solved. The technique of dual linear programming problems to analyze the difference schemes properties in the space of insufficient coefficients was used previously [8]. Based on this approach to the linear problem solution, the new difference schemes were constructed in [9]. So far these schemes have not been found in computational practice. More interesting results in the linear case are obtained for the difference schemes taking into account the dissipative effects, in particular linear drain [10]. Presumably, this technique can be used to analyze the difference schemes in a nonlinear case. For linear functional in Eq. 12.15 with constraints provided by Eqs. 12.13–12.14, the minimization problem can be represented as  2 

= −6 α + u nm β + u nm γ

− l−2 α + β u nm−2 + u nm

+ γ (u nm−2 )2 + u nm−2 u nm + (u nm )2   κ

− l−1 σ − 3α + − 3β u nm−1 + u nm 4    2

κ − − + 3γ (u nm−1 )2 + u nm−1 u nm + (u nm )2 6    κ

− l1 σ − α + − − β u nm+1 + u nm 4    2 n

κ 2 n n n 2 − γ (u m+1 ) + u m+1 u m + (u m ) + 6    κ

− 3β u nm−1 + u nm − l0 1 − σ − 3α + 4    2 n

κ 2 n n n 2 − 3γ (u m−1 ) + u m−1 u m + (u m ) + 6

12 On the Finite Difference Schemes for Burgers Equation Solution

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− α + β u nm−2 + u nm + γ (u nm−2 )2 + u nm−2 u nm + (u nm )2   κ 

− σ − α + − − β u nm+1 + u nm + 4  2   n

κ 2 n n n 2 − γ (u m+1 ) + u m+1 u m + (u m ) → min 6 under the given distribution of grid function u. The conditions of the Lagrange multipliers l–2 , l –1 , l 0 , l 1 non-negativity will be satisfied. The Linear Programming (LP) problem is the minimization or maximization problem of a linear functional subject to linear constraints. The general LP (the LP subject to mixed constraints) is to maximize the linear functional c1 x1 + c2 x2 → max with respect to the variables x1 ∈ R n 1 , x2 ∈ R n 2 subject to the linear constraints A11 x1 + A12 x2 ≤ b1 , A21 x1 + A22 x2 = b2 , x1 ≥ 0. Here, b1 ∈ R m 1 , b2 ∈ R m 2 , b1 ∈ R m 1 , c1 ∈ R n 1 , c2 ∈ R n 2 , and the matrices A11 , A12 , A21 , A22 have the size (m 1 × n 1 ), (m 1 × n 2 ), (m 2 × n 1 ), (m 2 × n 2 ). The dual problem of LP is the minimization problem of the linear functional b1 u1 + b2 u2 → min with respect to the variables u1 ∈ R m 1 , u2 ∈ R m 2 subject to the linear constraints AT11 u1 + AT21 u2 ≥ c1 , AT12 u1 + AT22 u2 = c2 , u1 ≥ 0. Here, AT is the transpose of matrix A. The problem dual to the dual LP is the original LP. These problems form a pair of self-dual LPs. The well-known theorem (optimality criterion) is the following. Let (x 1 , x 2 ) satisfy the constraints of the general LP and (u1 , u2 ) satisfy the constraints of the dual problem. For (x 1 , x 2 ) to be a solution to LP and (u1 , u2 ) to be a solution to the dual problem, it is necessary and sufficient that the following complementary slackness

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conditions be fulfilled: u1 (A11 x1 + A12 x2 − b1 ) = 0,

x1 AT11 u1 + AT21 u2 − c1 = 0. For the difference schemes for Burgers equation optimization, the system of complementary slackness conditions will be   κ

− 3β u nm−1 + u nm l−1 σ − 3α + 4    2 n

κ 2 n n n 2 − 3γ (u m−1 ) + u m−1 u m + (u m ) = 0, + 6



l−2 α + β u nm−2 + u nm + γ (u nm−2 )2 + u nm−2 u nm + (u nm )2 = 0,    κ

− 3β u nm−1 + u nm l0 1 − σ − 3α + 4   2 

n κ + − 3γ (u m−1 )2 + u nm−1 u nm + (u nm )2 6

− α + β u nm−2 + u nm

− +γ (u nm−2 )2 + u nm−2 u nm + (u nm )2   κ 

− σ − α + − − β u nm+1 + u nm 4    2

κ − γ (u nm+1 )2 + u nm+1 u nm + (u nm )2 = 0, + 6    2   κ  n



κ n n 2 n n n 2 − γ (u m+1 ) + u m+1 u m + (u m ) l1 σ − α + − − β u m+1 + u m + = 0, 4 6 α(−6 − l−2 + 3l−1 − 3l0 + l1 ) = 0,





β −6u nm − l−2 u nm−2 + u nm + 3l−1 u nm−1 + u nm + l1 u nm+1 + u nm

−l0 (−u nm−2 + 3u nm−1 + 3u nm + u nm+1 ) = 0,



2 γ −6(u nm ) − l−2 (u nm−2 )2 + u nm−2 u nm + (u nm )2 + 3l−1 (u nm−1 )2 + u nm−1 u nm + (u nm )2



− l0 (u nm+1 )2 + u nm+1 u nm + (u nm )2 + l0 (u nm−2 )2 + u nm−2 u nm + (u nm )2



−3l0 (u nm−1 )2 + u nm−1 u nm + (u nm )2 + l1 (u nm+1 )2 + u nm+1 u nm + (u nm )2 = 0. In total, the system of complementary slackness equations defines 128 points where a minimum can be reached. Many solutions to this system do not satisfy

12 On the Finite Difference Schemes for Burgers Equation Solution

161

the non-negativity of the Lagrange coefficients conditions or the non-negativity of the difference schemes coefficients [11]. We will not impose any restrictions on the grid function of the solution. We believe that only if the initial conditions are non-negative, the exact solution should also be non-negative at any time. Only those points (solutions of the complementary slackness equations) that are of interest for further research should be considered. One family of the points is constructed as follows: From the second condition, if l –2 is different from 0, all other Lagrange multipliers must be zero. The remaining physical cases assume zero l–2 . Then the non-zero value takes either l–1 or l 1 . In this class of solutions, only one Lagrange multiplier is non-zero. There are 9 possible solutions in this case. We have a scheme Eq. 12.8 when α = β = γ = 0. Under the condition σ+

κ

κ2 n (u m+1 )2 + u nm+1 u nm + (u nm )2 ≥ u nm+1 + u nm 6 4

(12.16)

we have tree solutions of the complementary slackness system (the first one



2 under conditions α = σ − κ4 u nm+1 + u nm + κ6 (u nm+1 )2 + u nm+1 u nm + (u nm )2 , β = γ = 0, the second one under conditions α = γ = 0, β = 3 u n σ +u n + ( m−1 m )  2  n 2 n n n 2  (u m−1 ) +u m−1 u m +(u m ) κ κ , and the third one under conditions α = β = 0, + 18 12 (u nm−1 +u nm ) 2 κ u nm+1 +u nm ) γ = (u n )2 +u nσ u n +(u n )2 − 4 (u n )(2 +u + κ6 ). All of them led to the n n n 2 ( m+1 ( m+1 m ) m+1 m m+1 u m +(u m ) ) following difference scheme

= u nm + σ u nm−2 − 2u nm−1 + u nm u n+1 m

κ − (u nm+1 )2 − (u nm−1 )2 4

κ2 n (u m+1 )3 − 2(u nm )3 + (u nm−1 )3 + 6  

κ2 n

κ (u m+1 )2 + u nm+1 u nm + (u nm )2 + − u nm+1 + u nm + 4 6 n n n (u m−2 − 3u m−1 + 3u m − u nm+1 ) (12.17) Another form of Eq. 12.17 will be written as Eq. 12.18 or Eq. 12.19.

= u nm + σ u nm−2 − 2u nm−1 + u nm u n+1 m

κ − 3(u nm+1 )2 − 4(u nm−1 )2 + (u nm−2 )2 4

2

2  2 κ  n 2 + u m−2 − 3 u nm−1 + 3 u nm − u nm+1 4



− u nm+1 + u nm u nm−2 − 3u nm−1 + 3u nm − u nm+1

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κ2 n (u m+1 )3 − 2(u nm )3 + (u nm−1 )3 6

κ2 n (u m+1 )2 + u nm+1 u nm + (u nm )2 + 6 (u nm−2 − 3u nm−1 + 3u nm − u nm+1 ) +

u n+1 = u nm + σ u nm−2 − 2u nm−1 + u nm m

κ − 3(u nm+1 )2 − 4(u nm−1 )2 + (u nm−2 )2 4

κ n n − u m−2 u m+1 + u nm − u nm−2 4



− 3u nm−1 u nm+1 + u nm − u nm−1 + 2u nm u nm+1

κ 2  n (u m+1 )2 + u nm+1 u nm + (u nm )2 + 6  (u nm−2 − 3u nm−1 + 2u nm ) + (u nm−1 )3 − (u nm )3

(12.18)

(12.19)

The terms (u nm+1 )2 and (u nm+1 )3 were excluded in Eq. 12.19. The difference scheme can be interpreted as a second-order approximation upwind scheme (an analog of the Beam and Warming difference scheme). The last two items in the right-hand side Eq. 12.18 define a grid dispersion (or negative dispersion) with the coefficient depending on the numerical solution itself. The condition in Eq. 12.16 acts as a switch in a hybrid difference scheme. If it is violated, either a first-order scheme with an anti-flow difference or an analog of the Lax–Wendroff scheme (Eq. 12.8) is applied. The other solutions of this family are found to be insignificant. For them, either the conditions of non-negativity of coefficients are violated, or the value of the target functional becomes deliberately large. The difference schemes of other families (in which two or three Lagrange multipliers differ from zero) require separate detailed analysis.

12.4 Some Numerical Results Problem 1 One of the traditional tests for difference schemes can be described in the following way. Let us consider the problem for Eq. 12.2 with D = 1 on x ∈ [0, 1] with initial condition u(x, 0) = sin(π x). The boundary conditions are u(0, t) = 0, u(1, t) = 0. The numerical results on the test grid consist of 11 nodes are given in Table 12.1. We used scheme Eq. 12.8 and hybrid scheme Eq. 12.17 when constraint Eq. 12.16 is fulfilled and Eq. 12.7 when constraint Eq. 12.16 is violated. The final time was equal to 1. Both schemes on the rough grid are in good agreement with the exact solution. When the step by time increases, the scheme Eq. 12.8 becomes unstable. The hybrid scheme remains stable when the Von Neumann condition is violated.

12 On the Finite Difference Schemes for Burgers Equation Solution

163

Fig. 12.2 Numerical solution of Problem 2. The initial stage of calculation. Solid line corresponds to the scheme Eq. 12.8, dashed line corresponds to the Lax–Wendroff scheme Eq. 12.20, dash-dotted line corresponds to the hybrid scheme with main scheme Eq. 12.17 and upwind scheme Eq. 12.7 when constraint Eq. 12.16 is violated, and the dotted line corresponds to the hybrid scheme with main scheme Eq. 12.17 and Lax–Wendroff scheme Eq. 12.8 when constraint Eq. 12.16 is violated. Only part of calculation domain is shown

Problem 2 Equation 12.2 with D = 0.01 on [–10, 10] is considered with the periodic boundary conditions. The initial disturbance is given as the non-continuous function  1, |x| ≤ 1, u(x, 0) = ϕ(x), where ϕ(x) = . The grid parameters are: step by 0, |x| > 1, space is equal h = 0.04, step by time is equal τ = 0.02, κ = 1/2. The numerical solutions are shown in Figs. 12.2, 12.3 and 12.4. The numerical solution with the maximal oscillation corresponding to the smalltime is shown in Fig. 12.2. The continuous line corresponds to scheme Eq. 12.8. The dashed line corresponds to the ordinary variant of the Lax–Wendroff scheme provided by Eq. 12.20. n 2 n 2 n − ym y ) y˜m+1/2 − 0.5 (ymn + ym+1  =0 + m+1 2h τ 2 n 2 n 2 n − ym−1 y y˜m−1/2 − 0.5 (ymn + ym−1 )  =0 + m 2h τ 2

2

2 n y˜m+1/2 − y˜m−1/2 y n − 2ymn + ym−1 ymn+1 − ymn + = D m+1 2 τ 2h h

(12.20)

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Fig. 12.3 Numerical solution of Problem 2. One period, 6100 steps by time. Solid line corresponds to the scheme Eq. 12.8, dash-dotted line corresponds to the hybrid scheme with the main scheme Eq. 12.17 and the upwind scheme Eq. 12.7 when the constraint Eq. 12.16 is violated, and the dotted line corresponds to the hybrid scheme with main scheme Eq. 12.17 and the Lax–Wendroff scheme Eq. 12.8 when the constraint Eq. 12.16 is violated

Fig. 12.4 Numerical solution of Problem 2. Three periods, 26,800 steps by time. Solid line corresponds to the Lax–Wendroff scheme Eq. 12.8, dash-dotted line corresponds to the hybrid scheme with the main scheme Eq. 12.17 and the upwind scheme Eq. 12.7 when the constraint Eq. 12.16 is violated

12 On the Finite Difference Schemes for Burgers Equation Solution

165

Fig. 12.5 Numerical solution of Problem 2. Rough grid, one period, 726 steps by time. Solid line corresponds to the scheme Eq. 12.8, dash-dotted line corresponds to the scheme Eq. 12.17 with scheme Eq. 12.7 when constraint Eq. 12.16 is violated, and the dotted line corresponds to the scheme Eq. 12.17 with scheme Eq. 12.8 when Eq. 12.16 is violated

The dash-dotted line corresponds to scheme Eq. 12.17 with the scheme Eq. 12.7 when the constrain Eq. 12.16 is violated. The dotted line corresponds scheme Eq. 12.17 with scheme Eq. 12.8 when the constrain Eq. 12.16 is violated. One can see that the best results are given by the scheme Eq. 12.17 with scheme Eq. 12.7. The grid parameters are as follows: space step is h = 0.04, time step is τ = 0.02. Figure 12.2 corresponds to time 0.2 (10-time steps). The oscillations, which occurred in the vicinity of the break function of the initial condition, are small and almost invisible in the scale. Further, we investigate the properties of schemes under long-time calculations. The graphs of the numerical solution when the initial disturbance has moved for one period (time 122, which corresponds to 6100 steps by time) are shown in Fig. 12.3. The numerical solutions using all the difference schemes differ slightly for this time. The solid line corresponds to the scheme Eq. 12.8. The dashed line corresponds to the hybrid difference scheme, which consists of the main scheme Eq. 12.20 when the condition Eq. 12.16 is satisfied and the upwind scheme Eq. 12.7 when the constrain Eq. 12.16 is violated. The points denote the solution according to another hybrid scheme with main scheme Eq. 12.17 and the Lax–Wendroff scheme Eq. 12.8 when the constrain Eq. 12.16 is violated. This curve practically coincides with the solid line. Similar properties of the difference schemes are preserved with more time steps. Figure 12.4 corresponds to the case when the initial disturbance has passed 3 periods in space (26,800 steps by time). The hybrid difference scheme with the main scheme

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Eq. 12.17 and the Lax–Wendroff one Eq. 12.8 in case of the condition Eq. 12.16 violation is not shown since the solution has a great amplitude decrease due to the non-conservativeness of the scheme. The properties of schemes can be illustrated better with a rough grid calculation provided. The results for the grid with h = 0.2 and τ = 0.1 (one period, 726 steps by time) are shown in Fig. 12.5 The Lax–Wendroff scheme Eq. 12.8 oscillations did not decay. The hybrid scheme with main scheme Eq. 12.17 and the scheme Eq. 12.8 gave a decrease solution with a significant anti-dispersion. Another hybrid scheme with the main scheme Eq. 12.17 and the upwind scheme Eq. 12.7 in violation Eq. 12.16 constraint is the best for Problem 2.

12.5 Conclusions The new method of the difference schemes for solving the Burgers equation constructing is discovered. The method is based on the two divergent forms for the Hopf equation. To search for the optimal difference schemes in this family, an analysis in the space of insufficient coefficients was applied. This approach was developed earlier by Magomedov and Kholodov [7] for linear problems. The technique of self-dual problems of linear programming solving is used. Based on this analysis, a new version of the Lax–Wendroff scheme was built. The new variant constructed here has slight oscillations on the discontinuous solutions and in the areas of large gradients, while on the smooth solutions both the schemes (new ant usual forms) give the same result. The new hybrid scheme with maximum anti-dispersion also was built. The constructed scheme turned out to be monotonous and formally has an order of approximation above the first. The difference scheme reproduces the solution well in short time, but a significant dissipative error occurs with a large calculating time. However, such a consideration opens up a way to build the optimal hybrid schemes with a successful choice of the target functional.

12 On the Finite Difference Schemes for Burgers Equation Solution

167

Table 12.1 Problem 1: numerical solution x

Hybrid scheme τ = 0.005

Scheme Eq. 12.8 τ = 0.005

Hybrid scheme τ = 0.001

Scheme Eq. 12.8 τ = 0.001

Exact solution

0.1

0.11180

0.11313

0.10978

0.11104

0.10954

0.2

0.21410

0.21670

0.21013

0.21266

0.20979

0.3

0.29734

0.30156

0.29181

0.29586

0.2919

0.4

0.35401

0.35954

0.34721

0.35260

0.34792

0.5

0.37782

0.38411

0.37045

0.37651

0.37158

0.6

0.36534

0.37128

0.35797

0.36374

0.35905

0.7

0.31583

0.32057

0.30935

0.31390

0.30991

0.8

0.23283

0.23572

0.22793

0.23072

0.22782

0.9

0.12374

0.12490

0.12111

0.12221

0.12069

References 1. Hopf, E.: The partial differential equation. Commun. Pure Apply Math. 3(3), 201–230 (1950) 2. Zhao, G.-Z., Yu, X.-J., Wu, D.: Numerical solution of the Burgers’ equation by local discontinuous Galerkin method. Appl. Math. and Comput. 216(12), 3671–3679. 3. Seydao˘glu, M., Erdo˘gan, U., Ozi¸s, T.: Numerical solution of Burgers’ equation with high order splitting methods. J. Comput. Appl. Math. 291(1), 410–422 (2016) 4. Pandy, K., Verma, L.: A note on Crank-Nicolson scheme for Burgers’ equation. Appl. Math. 2(7), 883–886 (2011) 5. Hassanien, I.A., Salama, A.A., Hosham, H.A.: Fourth-order finite difference method for solving Burger’s equation. Appl. Math. Comput. 170(2), 781–800 (2005) 6. Tolstykh, A.I.: High accuracy non-centered compact difference schemes for fluid dynamics applications. Word Scientist Publ. (1994). 7. Magomedov, K.M., Kholodov, A.S.: Grid-and-characteristics numerical methods. Nauka, Moscow (in Russian) (1988) 8. Lobanov, A.I.: Finite difference schemes for linear advection equation solving under generalized approximation condition. Comput. Res. Model. 10(2), 181–193 (In Russian) (2018) 9. Lobanov, A.I., Mirov, FKh.: A hybrid difference scheme under generalized approximation condition in the space of undetermined coefficients. Comput. Math. Math. Phys. 58(8), 1270– 1279 (2018) 10. Lobanov, AI.,Mirov, F.Kh.: Difference schemes for drain transfer equation based on space of insufficient coefficients analysis. Math. Model. 32(9), 53–72 (in Russian) (2020) 11. Pandy, K., Verma, L., Verma, A.: Du Fort-Frankel finite difference scheme for Burgers equation. Arab. J. Math. 2, 91–101 (2013)

Part II

Modern Methods in Mathematical Physics

Chapter 13

Development of Virtual Lattice Dynamics Method for Solving the Eigenvalue Problem of Three-Dimensional Elliptic Equation with a Multicenter Potential Olga A. Pyrkova , Vladimir N. Pyrkov , and Petr M. Vasilets Abstract Investigation of thermal conductivity and electron transfer processes often requires accurate calculations of a spectrum of ion vibrations in semiconductor solid solutions. One of the main problems in obtaining the above information is to determine the symmetry of the ion interactions based on quantum chemical calculations. An important step here is solving a three-dimensional elliptic eigenvalue problem for the Schrödinger equation with multicenter potential. Until now, a common approach to problems involving many ions has been to seek a solution in the form of a linear combination of plane waves or Gaussians. In our previous work, we proposed a method to directly solve this problem based on similarity to the problem of finding eigenfrequencies of ion oscillations in a model fragment of a solid solution. To do that we used virtual lattice dynamics and Fourier transform. However, many time steps were required to separate closely located eigenvalues. The present study develops this method. We show how selecting a uniform potential offset can efficiently reduce the number of iterations of the virtual lattice motion. We also demonstrate examples of solving this problem involving several ions. Finally, we discuss further steps for implementing the virtual lattice method.

O. A. Pyrkova (B) Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] V. N. Pyrkov · P. M. Vasilets Space Research Institute of the RAS, 84/32, Profsoyuznaya Str, Moscow 117997, Russian Federation e-mail: [email protected] P. M. Vasilets e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_13

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13.1 Introduction Accurate calculation of vibration spectra in semiconductor materials is very important to account for the processes of thermal conductivity and electron scattering in microelectronics. As it is exemplified in [1], many factors should be taken into account to determine the maximum of thermal conductivity, as well as, to determine the output to the 1/T asymptotic. These factors include the dispersion of phonons, the behavior of matrix elements of the oscillation interaction, and geometric factors. The important role of the oscillation polarization was further highlighted in [2, 3], as well as, the influence of geometric factors such as phonon focusing on silicon nanofilms [4, 5]. Similarly, the investigation of electron transport requires an accurate calculation of phonon dispersion and the matrix elements of electron–phonon interactions, including optical phonons [1]. Thus, to study the thermal conductivity and electron transport, it is necessary to determine phonon eigenfrequencies and eigenvectors. In turn, this requires first identify the force constants of ions interactions. Hence, we are faced with the need to solve the quantum-chemical problem of ions interactions in a solid solution. In quantum-chemical problems, the main attention is paid to solving the stationary non-relativistic Schrödinger equation: Hˆ (r1 , s1 , . . . , r N , s N ) = E(r1 , s1 , . . . , r N , s N ),

(13.1)

where H is the Hamilton operator representing the total energy of a molecular system consisting of M nuclei and N electrons in the absence of external magnetic or electric fields: 2

h Hˆ = − 2m

 i

∇i2 +

 i, j

z e2 1 e2  j +  . r j − R j  2 r j − r j  i, j

(13.2)

Here, J is the index of M nuclei, while i and j refer to a system of N electrons. The first term describes the kinetic energy of electrons. The second and third terms describe the electrons attraction to ions and the mutual repulsion of electrons. Investigation of large molecules and condensed matter always relies on the approximation of the many-electron function Eq. 13.1 by single-electron functions. This can be done using, for example, Slater determinant. In the case of one-electron functions, one needs to solve a three-dimensional elliptic eigenvalue problem with a multicenter potential to account for the attraction of electrons to positively charged ions. Next, we use the Hartree-Fock approximation or one of the approximation options of the density-functional theory approach. Each approach has its advantages and disadvantage. Properties of condensed matter are most commonly simulated by a combined approach Generalized Gradient Approximation (GGA) [6]. This drives us to the Kohn–Sham equations:

13 Development of Virtual Lattice Dynamics Method …

173

    h2 −  + Veff r φi r = εi φi r , 2m

(13.3)

→ − →  −    n r d r 2  −  + Vxc r , Veff r = eVext r + e →   r − r 

(13.4)

N        2 n r = φi r  .

(13.5)

i=1

  In the above equations, φi r and εi are the one-electron eigenfunctions and their  corresponding energies in a self-consistent effective potential Veff r , respectively. The latter includes the potential of attraction to ions (external potential for the electron subsystem), the mutual repulsion of electrons (the second term in Eq. 13.4), and the exchange–correlation contribution represented by the last term in Eqs. 13.4 and 13.6. Equation 13.3 as well as, Eq. 13.1 is the Schrödinger equation. The total energy of the system within the framework of the Kohn–Sham formalism is equal to:   − →  − →  n  r n r d r d r  − E= →  r − r  i           + n r εxc n r − μxc n r dr. 

e2 εi − 2

(13.6)

The efficacy of this approximation is determined by the prevalent role of the ion attractive potential. This potential is multicenter in the case of several ions and, therefore, V eff is also an essentially multicenter potential. There exist well-proven direct numerical methods for solving the one-dimensional Eq. 13.3 with the central external field U(r) provided by Eqs. 13.7–13.8.  −

 h2 d 2 l(l + 1) + + U − ε (r ) nl Pnl = 0 2m 2r 2 2mr 2  φi r = Pnl (r )Ylm (θ, ϕ)

(13.7) (13.8)

Here, functions in Eq. 13.8 are referred to as atom-like functions. Linear combinations of functions are used to solve the Schrödinger equations for a multicenter potential in a three-dimensional case. This is followed by the energy minimization of the total energy Eq. 13.6 with the coefficients of the basic functions being optimized. The above atom-like orbitals in the form of a linear combination of Gaussians and plane waves in combination with the pseudopotential method are used as the basis functions.

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Since 2015, software package PHONOPY [7] has been a popular tool for determining the force constant of ion interactions, eigenfrequencies of phonons, and eigenvectors. It implements the finite displacement method as the main method for determining force constants. It is also possible to define force constants using density functional perturbation theory [8]. Namely, PHONOPY suggests an option to use VASP-Vienna package [9] which implements an ab initio method for solving Eqs. 13.3–13.6. GGA approximation is used to account for exchange–correlation effects. It is also possible to use QE (pw) and TURBOMOLE instead of VASP. All three packages use the basic functions in the form of plane waves along with the pseudopotential method. Software package CRYSTAL can also be used to represent atom-like orbitals as a linear combination of Gaussian type orbital as a set of basic functions. It is worth noting here earlier works [10, 11], which describe calculation of the properties of vibrations from the first principles. These studies also implement plane waves were as basic functions. Thermodynamic parameters obtained using VASP and ABINIT were compared in [11]. In certain cases, the discrepancy exceeded as much as 15%. All three papers [7, 10, 11] consider the role of the macroscopic electric field that emerges when ions are displaced along wave vectors close to the center of the Brillouin zone. As can be seen from this brief review of articles, all modern approaches to solving such problems necessarily rely on basic functions in the form of plane waves or atomic orbitals. Note that a set of plane waves is used together with the pseudopotential method. Also, double sets of valence functions are often used [12]. The abovementioned valence functions are a linear combination of atom-like orbitals. Atomlike orbitals, in turn, are linear combinations of Gaussians. Here, the terminology itself—the pseudopotential method and the double set of valence functions—suggests that the methods used in quantum chemical calculations are looking forward to refinement and further development. In the present chapter, we address these problems and suggest approaches to solving this problem, and to complement the methods already used in practice. This study extends the development of the previously proposed [13] method of virtual lattice dynamic for directly solving the problem of finding eigenvalues and eigenfunctions of the multicenter one-electron three-dimensional Schrödinger equation without a predefined set of basic functions. Our method is based on the similarity of the problem of determining the eigenfrequencies of ion vibrations for a solid solution fragment [14] to the problem of determining the eigenvalues of the Schrödinger equation. The chapter is organized as follows. Virtual lattice method [13] and switch of the virtual lattice geometry [15] previously proposed by the authors are described in detail in Sect. 13.2. Particularly, we highlight the need for many time step iterations to delineate close energy levels. To solve this problem, we suggest a method of homogeneous potential displacement. This method is described in detail in Sect. 13.3. We also describe results obtained using the proposed method. In Sect. 13.4, we outline the possible directions for implementing the virtual lattice method.

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13.2 Calculation Method Using Virtual Lattice and Switching the Geometry of the Virtual Lattice The method proposed in [13, 14] is based on a discrete representation of the Schrödinger equation provided by Eqs. 13.9–13.10. −

ND ∂ 2 (x) h2  + (U (x) − E) · (x) = 0 2m i=1 ∂ xi2

(13.9)

∂(x) 1  = 2 · (x1 , x2 , . . . , xi−1 xi + δi , xi+1 , . . . , x D ) 2 ∂ xi δi + (x1 , x2 , . . . , xi−1 xi − δi , xi+1 , . . . , x D ) 

−2 · (x1 , x2 , . . . , xi−1 xi , xi+1 , . . . , x D ) + O δi2 1  ≈ 2 · (x1 , x2 , . . . , xi−1 xi + δi , xi+1 , . . . , x D ) δi + (x1 , x2 , . . . , xi−1 xi − δi , xi+1 , . . . , x D )  −2 · (x1 , x2 , . . . , xi−1 xi , xi+1 , . . . , x D )

(13.10)

Let xi = i • δ, then we obtain Eqs. 13.11–13.12, where K = 2 / 2mδ 2 . i, j,k = (i · δ, j · δ, k · δ) Ui, j,k = U (i · δ, j · δ, k · δ) − E ∗ i jk + K ∗ [6∗ i jk − i+1 jk − i j+1k − i jk+1  − i−1 jk − i j−1k − i jk−1 + Ui jk ∗ i jk = 0

(13.11)

(13.12)

In order to solve Eq. 13.12, it is necessary to diagonalize the matrix of size M × M, where M = N D , where N is the number of steps in each dimension, and D is the dimension of the problem—so far, we have put D = 3. For N = 100 and D = 3 (M × M = 1,000,000 × 1,000,000) and the task of diagonalization already becomes practically impossible. The problem can be greatly simplified by using time√ dependent equations. Let us write i jk (t) = i jk ∗ e−i Et . And rewrite Eq. 13.12 in the form of Eq. 13.13. d2 i jk (t)/dt 2 + K ∗ [6∗ i jk − i+1 jk − i j+1k − i jk+1  −i−1 jk − i j−1k − i jk−1 + Ui jk ∗ i jk = 0

(13.13)

Thus, we obtain a system of linear ordinary differential equations, which describe harmonic vibrations of a virtual lattice with particles of the same mass, connected to the nearest neighbors by springs of the same stiffness K and related to the equilibrium position by springs of stiffness Ui jk . The eigenfrequencies of Eq. 13.13 are then related to the eigenenergies of Eq. 13.12 as follows: ω2 = E.

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The eigenfrequencies of Eq. 13.13 can be found by spectral analysis, for example, by extracting Fourier transform peaks from the time-varying readings of a sensor in a virtual lattice during its arbitrary excitation. Figure 13.1 shows an example of a Fourier transform graph obtained by solving Eq. 13.13. The typical number of time steps in these calculations was 32,768 and 131,072. After the Fourier transform of the response, the position of the peaks was identified to get the eigenvalues of the equations. We tested several approaches for determining the eigenfunction of Eqs. 13.3–13.6 and found that the most stable was the following one. Excitation was applied at a frequency corresponding to one selected eigenvalue near the equilibrium position of the function f . At the first iteration, we used the form of excitation corresponding to that used to find the eigenvalues. After a time interval of motion equal to n ∗ (2π/ω) + δ, the response form was recorded for further use as the excitation form at the next iteration. The first term of the above value determines the integer number of periods at the selected frequency, and the second term adds a phase shift. The need to use an additional phase shift (approximately 0.3 of the full period) is illustrated in Fig. 13.2. Namely, the response is growing in a time when the phase shift is applied, while without that shift it would not. We carried out calculation tests and obtained results in good accordance with theoretical solutions for problems that can be solved analytically: hydrogen-like ions, various rectangular potential wells, and oscillatory potentials. We obtained reliable values for the affinity of a carbon atom to an electron in the Hartree self-consistent framework and taking into account the exchange energy. However, we encountered the calculations difficulty even with the number of electrons exceeding eight for one ion (e.g. a fluorine atom). At the grid step of 0.1 a.u (atomic units) the rank of the 2s and 2p levels was as expected, i.e. 2p > 2s. While a grid step of 0.15 a.u in a self-consistent simulation yielded a qualitatively wrong result 2p < 2s. Fig. 13.1 Example of Fast Fourier Transform (FFT) plot obtained by solving Eq. 13.13. The peaks at FFT wavenumbers of 42, 277, 301, 309 correspond to the sequence of energies of a hydrogen-like ion

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Fig. 13.2 Illustration of the need to add a phase shift during calculating the eigenfunction when excited at a predefined eigenfrequency. Alternating blue and green segments indicate complete periods of oscillation

Decreasing grid step is computationally costly. In one-dimensional case, it is straightforward to reduce the grid step in one-dimensional Eq. 13.7 without a substantial increase in the number of nodes of the virtual lattice. Indeed, when the grid step decreases, the increase in the number of steps in the radial case is linear. Things change dramatically in the three-dimensional case. Here, a decrease in the spatial grid step by a factor of two will lead to an increase in the calculation time by 8 = 23 times due to the need to increase the corresponding number of steps in each dimension. Taking this into account, we now set up the task as follows. It is necessary to find a method that will be able to: • Solve Eq. 13.13 with a grid step, which is not too small in order to afford reasonable computation times for a large number of particles in the virtual lattice. • Take into account the physically-relevant features of the deep potential in the vicinity of the ion. • Avoid artifacts introduced by the commonly used implementation of pseudopotential. In our previous work [15] we showed that the above problems can be solved by switching the geometry of grid of the Eq. 13.13 in the vicinity of ions. We suggested the assumption of the three-dimensional potential U (x) being dependent only on the distance to an ion in its vicinity. wave function of the electron can therefore be The represented as in Eq. 13.8 φ r = lm Plm (r )Ylm (θ, ϕ) and satisfies the equation depending on one variable r in the vicinity of the ion, see Eq. 13.7. Outside of the ion neighborhood, Eq. 13.9 for the spatial coordinates x, y, z can be used. Figure 13.3 illustrates the schematic diagram of switching the geometry of a virtual lattice. Thus, the continuity of the function and its normal derivative at the boundary of each ion neighborhood are determined by Eqs. 13.14–13.15.

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Fig. 13.3 Schematic diagram of switching the geometry of a virtual lattice. The rectangular grid changes to a radial grid near the ions. (r) and φ(r) represent wave function in the rectangle part and radial part of the grid, respectively

φ(t, r) = (t, r)

(13.14)

  ∂φ(t, r) ∂n = ∂(t, r) ∂n

(13.15)

Application of the described assumptions on switching in the geometry of grid of Eq. 13.13, yielded reliable results for the electron affinity of a sulfur atom with 16 electrons. The equilibrium positions of the ions obtained for MgS molecule also demonstrated good correspondence with the reference data values [16]: 2.07 < 2.142 < 2.19 Å. Below we describe in detail the examples of mapping of one-electron selfconsistent functions for molecule MgS to a two-center potential calculated using the modified virtual lattice geometry: s2 function for S ion (Fig. 13.4), s2 function for Mg ion (Fig. 13.5), s3 function for S ion (Fig. 13.6), px function for S ion (Fig. 13.7), and px 2 for S ion (Fig. 13.8). These figures represent the  ijk function at the level of 0.5 of grid step above the position of ions along axis z. The x-axis is marked by the grid ticks. Functions  ijk were calculated using the values of the φ(r) in the vicinity of ions, where necessary. We used the ionic orbital as the initial excitation form for calculating the molecular function as is indicated by the notation for the molecular function  ijk . The wave functions were virtually indistinguishable from the functions of an individual sulfur atom for the s2 function of the sulfur ion S (Fig. 13.4) and px function of the sulfur ion S (Fig. 13.7) as can be seen by their sufficiently deep levels.

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Fig. 13.4 The s2 function of sulfur ion S

Fig. 13.5 The s2 function of magnesium ion Mg

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Fig. 13.6 The s3 function of the sulfur ion S

Fig. 13.7 The px function of the sulfur ion S

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Fig. 13.8 The px 2 function of the sulfur ion S (binding orbital)

On the contrary, the levels were not deep for the s2 function for Mg ion (Fig. 13.5), s3 function for S ion (Fig. 13.6), and function px 2 for S ion (Fig. 13.8). This indicates that the wave functions are fairly well distinguishable from that of an individual sulfur atom. Function px 2 for S ion is the binding orbital for molecule MgS.

13.3 Calculation Technique Using a Virtual Lattice and Uniform Potential Offset More complex three-dimensional problems, however, require an increasing number of time steps to separate close eigenfrequencies. To address this problem, we take into account the fact that the energy E and the potential U are both defined relative to an arbitrary constant. Upon a uniform displacement of the potential U by a constant value, the energy changes by the same amount. The spatial part of the wave functions , meanwhile, does not change. If the displacement of U is such that E becomes √ negative, the function i jk (t) = i jk ∗e−i Et will grow exponentially with time. The wave function corresponding to the deepest energy level will experience the greatest growth. Thus, for a small number of time steps, the spatial part of the wave function i jk (t) will be represented by the spatial part of the wave function with the deepest energy i0jk . The energy E 0 for the resulting spatial wave function can be found using Eq. 13.2 by substitution of the function i0jk into it. To obtain the spatial part of the next deepest level function i1jk we need to perform its orthogonalization to i0jk at

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Fig. 13.9 The py function of the carbon ion in molecule CH4

each iteration. Similarly, in order to obtain the following wave functions with the next deepest energy, we need to perform orthogonalization to all previously defined wave functions. In case of a degenerate energy level, several orthogonal spatial functions i jk will correspond to this level. We carried out a test of the practical utility of this approach for a methane molecule CH4 . Using grid step of 0.15 a.u., we obtained a good agreement for the equilibrium distances between atoms to the reference average values [17]: 0.97 < 1.087 < 1.1 Å. Figure 13.9 illustrates the example of mapping of one-electron self-consistent py function for CH4 molecule. An additional peak is seen behind the peak in the foreground (which corresponds to a high electron density in the vicinity of the carbon ion). This additional peak corresponds to an increase in the electron density near the hydrogen nucleus.

13.4 Discussion of Results and Conclusions We proposed an iterative method for the direct solution of the multicenter Schrödinger equation in the framework of the Hartree and Kohn–Sham models. Our method is based on the previously proposed spectral analysis of virtual lattice vibrations using an additional uniform potential offset.

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We demonstrated the efficacy of the method for multi-electron molecules in combination with a switch of grid geometry of the virtual lattice. Yet, when determining the force constants, we faced significant computational costs for calculating the matrix elements of the interaction of the obtained eigenfunctions for the elliptic three-dimensional eigenvalue problem in a multicenter potential. The results of the corresponding calculations will be reported in further studies. Since all the proposed methods for solving the Schrödinger equation are iterative, we plan to determine the rate of convergence of the error [18] in our future work. We also plan to use the proposed method of uniform displacement of the potential to find eigenvectors when studying the dynamics of the model lattice of a semiconductor solid solution under the influence of an external macroscopic electric field. Acknowledgements Authors express deep gratitude to V. Burlakov for fruitful discussions and useful comments and suggestions during the work preparation and B. Mainin for fruitful discussions during the work preparation. The work was carried out using the technologies and data of the collective use center of the IKI Monitoring Center [19]. This work was supported by the Ministry of Education and Science of the Russian Federation (the topic “Monitoring”, state registration No. 01.20.0.2.00164)

References 1. Ziman, J.H.: Electrons and Phonons. Clarendon Press, Oxford (1960) 2. Asen-Palmer, M., Bartkowski, K., Gmelin, E., Cardona, M., Zhernov, A.P., Inyushkin, A.V., Taldenkov, A., Ozhogin, V.I., Itoh, K.M., Haller, E.E.: Thermal conductivity of germanium crystals with different isotopic compositions. Phys. Rev. B 56, 9431–9447 (1997) 3. Kuleyev, I.G., Kuleyev, I.I.: Normal phonon-phonon scattering processes and the thermal conductivity of germanium crystals with isotope disorder. J. Exp. Theor. Phys. 93, 568–578 (2001) 4. Cahill, D.G., Braun, P.V., Chen, G.: Nanoscale thermal transport II. Appl. Phys. Rev. 1, 2003– 2012 (2014) 5. Kuleyev, I.I., Bakharev, S.M., Kuleyev, I.G., Ustinov, V.V.: Phonon focusing and temperature dependences of thermal conductivity of silicon nanofilms. J. Exp. Theor. Phys. 120, 638–650 (2015) 6. Perdew, J.P., Burke, K., Ernzerhof, M.: Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996) 7. Togo, A., Tanaka, T.: First principles phonon calculations in materials science. Scr. Mater. 108, 1–5 (2015) 8. Giannozzi, P., de Gironcoli, S., Pavone, P., Baroni, S.: Ab initio calculation of phonon dispersions in semiconductors. Phys. Rev. B 43, 7231–7242 (1991) 9. Kresse, G., Furthmuller, J.: Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996) 10. Alfe, D.: PHON: a program to calculate phonons using the small displacement method. Comput. Phys. Commun. 180, 2622–2633 (2009) 11. Cardona, M., Kremer, R.K., Lauck, R., Siegle, G., Munoz, A., Romero, A.H., Schindler, A.: Electronic, vibrational, and thermodynamic properties of ZnS with zinc-blende and rocksalt structure. Phys. Rev. B 81, 075207 (2010) 12. Oliveira, D.V., Laun, J., Peintinger, M.F., Bredow, T.: BSSE correction scheme for consistent gaussian basis sets of double and triple zeta valence with polarization quality for solid state calculations. J. Comput. Chem. 40(27), 2364–2376 (2019)

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13. Pyrkov, V.N.: Investigation of the possibility of using a conservative difference scheme of the wave equation to determine the symmetry of the force coefficients of interaction of neighboring ions in solid solutions A II B VI. In: Reznichenko, G.Y., Rubina, A.B. (eds.) 23th International Conference on Mathematics, Computer, Education, Symposium with International Participation “Biophysics of Complex Systems, Molecular Modeling, Systems Biology”, Dubna 2016, p. 29. Moscow, Izhevsk (in Russian) (2016) 14. Pyrkov, V.N., Kozyrev, S.P., Vodopyanov, L.K.: Model calculation of optically active lattice vibrations for an alloy A(1–x) B (x)C (for example, Hg (1–x)Cd (x)Te). Phys. Solid State 35, 2479–2489 (1993) 15. Pyrkova, O.A., Pyrkov, V.N., Vasilets, P.M.: Changing the geometry of the virtual lattice of the multicenter Schrodinger equation when determining eigenfunctions using the Fourier transform in time. In: XXVII International Conference on Mathematics, Economics, Education, XI Symposium Fourier Series and their Applications, Novorossiysk, Russia (2020) (in print) 16. Grigoriev, I.S., Meilikhov, E.Z.: Handbook of Physical quantities. Energoatomizdat, Moscow (in Russian) (1991) 17. Nikolsky, B.P.: Chemist’s Handbook. Khimiya, Leningrad (in Russian) (1971) 18. Grabowski, Paul E., Chernoff, D.F.: Pseudospectral calculation of the wave function of helium and the negative hydrogen ion. Phys. Rev. A81, 032508 (2010) 19. Lupyan, E.A., Proshin, A.A., Burtsev, M.A., Balashov, I.V., Bartalev, S.A., Efremov, VYu., Kashnitsky, A.V., Mazurov, A.A., Matveev, A.M., Sudneva, O.A., Sichugov, I.G., Tolpin, V.A., Uvarov, I.A.: Center for collective use of systems for archiving, processing and analysis of satellite data of ICI RAS for solving problems of studying and monitoring the environment. Mod. Prob. Remote Sens. Earth Space 12(5), 263–284 (2015)

Chapter 14

Charged Particles in the Field of an Inhomogeneous Electromagnetic Wave Anton A. Skubachevskii , Vladimir B. Lapshin , and Igor B. Petrov

Abstract The chapter generalizes an approach for simulating a wide spectrum of electromagnetic waves based on an example of an inhomogeneous electromagnetic wave created by a superposition of two plane monochromatic electromagnetic waves arbitrarily directed with respect to each other. The complete system of equations that describe the motion of an electron in an inhomogeneous electromagnetic field is numerically solved. The features of the trajectory and spectrum of the electron radiation were discovered and investigated.

14.1 Introduction The electron trajectory in a plane electromagnetic wave was first determined in [1], while the electron trajectory in a plane non-polarized electromagnetic wave was obtained by solving the Dirac equation in a semi-classical formulation: the electromagnetic field in the classical formulation and the electron in the quantum–mechanical one. The Dirac equation is applicable to describe the interaction of particles by generalizing the methods of classical and quantum theory only to particles with halfinteger spins. Volkov’s solution made a significant contribution to the study of the behavior of electrons in powerful electromagnetic fields. However, the results of [1] are not applicable for inhomogeneous electromagnetic waves because the equations of motion of the electron, in this case, are nonlinear. A. A. Skubachevskii (B) · I. B. Petrov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] I. B. Petrov e-mail: [email protected] V. B. Lapshin Fedorov Institute of Applied Geophysics, 9, Rostokinskaya ul., Moscow 129128, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_14

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The use of the Dirac equation is justified when the scale of the phenomenon is comparable to the Compton wavelength. In this regard, for a large class of physical phenomena, when studying the interaction of charged particles with an electromagnetic field, it is permissible to use the equations of classical electrodynamics. In [2–8], the motion of a charged particle in a uniform electromagnetic wave was considered. For a special case of a charged particle in an inhomogeneous wave, namely, a transverse electric and a transverse magnetic one, approximate solutions were obtained by the perturbation method in [2]. The rest of the chapter is organized as follows. Section 14.2 includes a problem statement. Section 14.3 provides the results and discussion of numerical simulations. Concluding remarks are given in Sect. 14.4.

14.2 Formulation of the Problem In this chapter, as an example of an inhomogeneous electromagnetic wave created by a superposition of two plane monochromatic electromagnetic waves arbitrarily directed relative to each other, an approach is proposed that allows to simulate of a wide range of interference structures. In the classical formulation, the trajectory of a charged particle in an arbitrary electromagnetic wave is found as a solution to a system of nonlinear differential equations with variable coefficients in Eq. 14.1. It is convenient to write the equation of particle motion for a numerical solution in the form: q y (E x + Hz − mγ c  q z (E y + Hx − y  = mγ c q x (E z + Hy − z  = mγ c x  =

z 1 Hy − 2 (x 2 E x + x  y  E y + x  z  E z ), c c x 1 Hz − 2 (x  y  E x + y 2 E y + y  z  E z ), c c y 1 Hx − 2 (x  z  E x + y  z  E y + z 2 E z ), c c

(14.1)

where x, y, z, x  , y  , z  , x  , y  , z  are the particle coordinates, velocities, and accel− → − → erations, m is the mass of the particle, q is the charge of the particle, E , H are the electric and magnetic field of inhomogeneous wave,  E x , E y , E z , Hx , Hy , Hz are the electric and magnetic field components, γ = 1/ 1 − v 2 /c2 , c is the speed of light, t is the time. The position of plane monochromatic waves is defined by three angles: ϕ, θ, ψ, where ϕ is the angle between the direction of OX axis and the projection of the wave vector onto the plane (x, y), θ is the angle between the direction of the wave vector and the projection of this vector onto the plane (x, y), ψ is the angle between the direction of the electric field vector and the vector product of the wave vector and the normal to the plane (x, y).

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We write the superposition of two arbitrarily directed plane waves:   (1) sin ω0 t − A x k0 x − A y k0 y − A z k0 z + 1 E x = E x,0   (2) + E x,0 sin ω0 t − Bx k0 x − B y k0 y − Bz k0 z + 2 ,   (1) E y = E y,0 sin ω0 t − A x k0 x − A y k0 y − A z k0 z + 1   (2) + E y,0 sin ω0 t − Bx k0 x − B y k0 y − Bz k0 z + 2 ,   (1) E z = E z,0 sin ω0 t − A x k0 x − A y k0 y − A z k0 z + 1   (2) + E z,0 sin ω0 t − Bx k0 x − B y k0 y − Bz k0 z + 2 ,   (1) Hx = Hx,0 sin ω0 t − A x k0 x − A y k0 y − A z k0 z + 1   (2) + Hx,0 sin ω0 t − Bx k0 x − B y k0 y − Bz k0 z + 2 ,   (1) Hy = Hy,0 sin ω0 t − A x k0 x − A y k0 y − A z k0 z + 1   (2) + Hy,0 sin ω0 t − Bx k0 x − B y k0 y − Bz k0 z + 2 ,   (1) sin ω0 t − A x k0 x − A y k0 y − A z k0 z + 1 Hz = Hz,0   (2) + Hz,0 sin ω0 t − Bx k0 x − B y k0 y − Bz k0 z + 2 , where A x , A y , A z , Bx , B y , Bz are projections of normalized wave vectors of waves on the coordinate axis, the index “0” indicates the components of amplitudes, is the phase,ω0 is the wave frequency, A x = cos ϕ1 cos θ1 , A y = sin ϕ1 cos θ1 , A z = sin θ1 , Bx = cos ϕ2 cos θ2 , B y = sin ϕ2 cos θ2 , Bz = sin θ2 , E x,0 = E 0 (cos ψ1 sin ϕ1 − sin ψ1 sin θ1 cos ϕ1 ) + E 0 (cos ψ2 sin ϕ2 − sin ψ2 sin θ2 cos ϕ2 ), E y,0 = E 0 (− cos ψ1 cos ϕ1 − sin ψ1 sin θ1 sin ϕ1 ) + E 0 (− cos ψ2 cos ϕ2 − sin ψ2 sin θ2 sin ϕ2 ), E z,0 = E 0 sin ψ1 cos θ1 + E 0 sin ψ2 cos θ2 , Hx,0 = H0 (sin ψ1 sin ϕ1 + cos ψ1 sin θ1 cos ϕ1 ) + H0 (sin ψ2 sin ϕ2 + cos ψ2 sin θ2 cos ϕ2 ), Hy,0 = H0 (− sin ψ1 cos ϕ1 + cos ψ1 sin θ1 sin ϕ1 ) + H0 (− sin ψ2 cos ϕ2 + cos ψ2 sin θ2 sin ϕ2 ), Hz,0 = −H0 cos ψ1 cos θ1 − H0 cos ψ2 cos θ2 ,

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where E 0 and H0 are the amplitudes of the inhomogeneous wave. The Runge–Kutta scheme with an adaptive stepsize known as the Dorman–Prince scheme was used to solve this system of equations. As a numerical value for the next step, we took the result obtained according to the fifth-order accuracy scheme. In this method, a fourth-order scheme is also used to control the integration stepsize. The above schemes differ by the bottom line of the coefficients in the Butcher table. This scheme differs from most schemes with a variable integration step in that as a solution we take the result obtained according to a higher-order scheme, and the time step is set using a lower-order scheme. In experiments below ϕ1 = ϕ2 = π2 ,θ2 = π − θ1 ,ψ2 = 0,1 = 2 = 0. In this case, the electric field is parallel to OX axis, while the wave vector of inhomogeneous electromagnetic wave is parallel to OZ axis [2]. The result of interaction of such  waves is interference structure with the distance between maxima d = λ/2 sin π2 − θ1 . There are also planes of zero electric field with coordinates    yn = λ/2 sin π2 − θ1 21 + n [2]. The trajectory and particle drift velocities in the plane-wave field calculated numerically coincide with those calculated analytically for different values of the initial parameters [2].

14.3 Results of Numerical Experiments To calculate the emission spectrum of a particle, an inhomogeneous wave can be represented as a Fourier integral. Using Parseval’s equality and Larmor’s formula for the total radiation power, we calculate the spectral radiation density of a charged particle. The spectrum of charge radiation in an inhomogeneous electromagnetic wave generally contains a set of frequencies due to the complex interference structure, in the general case non-stationary, which determines the particle acceleration. The value of the frequency of the modulating signal and side frequencies ω0 ± of the modulated signal ω0 are set by angles ϕ, θ, ψ. Figure 14.1 shows the dependence of the frequency of the modulating signal on angles θ, ψ. Figure 14.2 shows the calculated dependence of the particle velocity on the electric field. Dependences of the carrier frequency of the wave and the angles θ, ψ are depicted in Fig. 14.2a, b respectively. At the initial moment of time t = 0, the  particle is at a point with coordinates x(0) = z(0) = 0, y(0) = λ/2 sin π2 − θ1 = 0.3 m and has zero initial velocity. The trajectory of a charged particle located between the interference maxima is shown  14.3. The particle is injected with the initial coordinates y(0) =  in Fig. λ/4 sin π2 − θ1 = 0.3 m, x(0) = z(0) = 0 m. The calculated trajectory (Fig. 14.3) is an exact numerical solution of the complete system of Eq. 14.1, in contrast to the solutions in [2, 5], which are approximate and obtained by the perturbation method for particular cases of a transverse electric wave and is valid only in locality of plane of zero electric fields.

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Fig. 14.1 Dependence of modulating frequency of inhomogeneous wave on angles θ1 and ψ1 (ϕ1 = ϕ2 = π2 , θ2 = π − θ1 , ψ2 = 0, 1 = 2 = 0) with amplitude of electric and magnetic fields 3 · 105 V /m, carrier frequency ω0 9 2π = 10 H z, initial coordinates x0 = z 0 = 0 m, y0 = 0.3 m, and initial velocity v0 = 5 · 105 m/s.

Fig. 14.2 Dependences: a of the particle velocity on the amplitude of the electric field and the carrier frequency of the wave, where ϕ1 = ϕ2 = π2 , θ1 = π3 , θ2 = 2π 3 , ψ1 = ψ2 = 0, b on the angles ψ1 , θ1 , where ϕ1 = ϕ2 = π2 , θ2 = π − θ1 , ψ2 = 0. The initial coordinates x0 = z 0 = 0 m, y0 = 0.3 m, initial velocity v0 = 5 · 105 m/s. Electric and magnetic fields amplitude 3 · 105 V /m, ω0 carrier frequency 2π = 109 H z,1 = 2 = 0.

When solving Eq. 14.1, the following must be taken into account. In a nonuniform oscillating electromagnetic field, nonlinear phenomena occur accompanied by the appearance of gradient forces. The inhomogeneity  of the wavefield leads to the  appearance of a nonlinear force proportional to ∇ E 2 an oscillating electromagnetic field, first introduced in 1958 in [4]. This force is also considered in [5, 9, 10]. In the approximation, when the electron motion can be represented as a combination of a → slowly changing component − r1 (t) and a component that oscillates rapidly with the

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Fig. 14.3 The trajectory of a particle in a wave created by two plane electromagnetic waves: (ϕ1 = ϕ2 = π2 ,ψ1 = ψ2 = 0, θ1 = π3 , θ2 = 2π 3 , 1 = 2 = 0). The initial coordinates x0 = z 0 = 0 m, y0 = 0.3 m, the initial speed is directed along OY, v0 = 5 · 105 m/s, frequency ω0 9 5 2π = 10 H z,E 0 = H0 = 6 · 10 V /m.

→ carrier frequency of the wave − r2 (t), the expression for the gradient force will have the following form [5]: ..

− → r1 = −(

 e 2 ) ∇ E2 2mω0

(14.2)

The limiting value of the initial velocity v0 is defined by Eq. 14.3. v0 = √

eE 2mω0

(14.3)

Let us compare the value of the threshold initial velocity (Eq. 14.3) obtained in this way with the results of numerical simulation. These numerical experiments will help to understand the essence of the gradient force better. We inject an electron into a wave with a carrier frequency ω0 = π · 109rad/s, the amplitude of the electric field of plane waves E = 10 CGS units, with initial velocity along the axis OY −2.3 · 109 cm/s, in the point with coordinates (0;30;0)cm. The eE ≈ 2.37 · 109 cm/s. That is, the electron is injected threshold velocity is v0 = √2mω 0 into the region of zero electric fields with a speed slightly lower than the threshold. The electron trajectory is depicted in Fig. 14.4. The electron trajectory is between the maxima of the interference pattern. Now let’s do the same experiment but with an initial speed, −2.5 · 109 cm/s. The electron trajectory is shown in Fig. 14.5. The electron overcoming the gradient force crosses the interference maximum and continues to move on.

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Fig. 14.4 Electron trajectory in field of an inhomogeneuos electromagnetic wave: a in 3D space ω0 with parameters (ϕ1 , θ1 , ψ1 ) = (π/2; π/3; 0),(ϕ2 , θ2 , ψ2 ) = (π/2; 2π/3; 0),1 = 2 = 0, 2π = 2π c 8 1.25 · 10 H z, E 0 = 10 CGS units. λ = ω0 = 60 cm. The initial coordinates of an electron x0 = z 0 = 0 cm, y0 = 3 · 101 cm, initial velocity v0x = v0z = 0 cm/s, v0y = −2.3 · 109 cm/s. Time step dt = 2π/100ω0 . Simulation time T = 1500dt, b the projection of the same trajectory onto XY plane

Fig. 14.5 Electron trajectory in field of an inhomogeneuos electromagnetic wave: a in 3D space ω0 with parameters (ϕ1 , θ1 , ψ1 ) = (π/2; π/3; 0),(ϕ2 , θ2 , ψ2 ) = (π/2; 2π/3; 0),1 = 2 = 0, 2π = 2π c 8 1.25 · 10 H z, E 0 = 10 CGS units. λ = ω0 = 60 cm. The initial coordinates of an electron x0 = z 0 = 0 cm, y0 = 3 · 101 cm, initial velocity v0x = v0z = 0 cm/s,v0y = −2.5 · 109 cm/s. Time step dt = 2π/100ω0 . Simulation time T = 1500 dt, b the projection of the same trajectory onto XY plane

14.4 Conclusions The presented results allow us to draw the following conclusions. Frequency value

modulating signal and side frequencies of the modulated signal ω0 ± depends on

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the angles ϕ, θ, ψ defining the spatial directions of plane waves forming an inhomogeneous electromagnetic wave. Thus, it is possible to control lateral frequencies by changing the values ϕ, θ, ψ. Particle velocity with its trajectory is determined by parameters θ, ψ, ω0 , E 0 . The particle velocity in X direction is mainly determined by the electric field of the wave, and in Y direction, by the so-called gradient force proportional to the gradient of the square of the electric field. The influence of the gradient force determines the maximum speed at which the particle oscillates between interference maxima. The value of this speed calculated theoretically is completely consistent with the numerical experiment. The results can be used in various technical applications.

References 1. Wolkov, D.M.: Electron in the field of a plane unpolarized electromagnetic wave from the point of view of the Dirac equations. Z. Phys. 94, 250–260 (1935) 2. Bolotovskii, B.M., Serov, A.V.: Special features of motion of particles in an electromagnetic wave. Physics-Uspekhi. 46(6), 645–655 (2003) 3. Lapshin, V.B., Skubachevskiy, A.A., Belinsky, A.V., Bugaev, A.S.: Emission spectrum and trajectory of a charged particle in the field of an inhomogeneous electromagnetic wave. Proceed. Acad. Sci. 488(6), 1–5 (2019) 4. Gaponov, A.V., Miller, M.A.: Potential wells for charged particles in a high-frequency electromagnetic field. JETP 7(1), 168–169 (1958) 5. Serov, A.V.: Ponderomotive force, proportional to, acting on a charged particle travelling across an inhomogeneous electromagnetic wave. Quantum Electron. 25(3), 197–200 (1998) 6. Andreev, S.N., Makarov, V.P., Rukhadze, A.A.: On the motion of a charged particle in a plane monochromatic electromagnetic wave. Quantum Electron. 39(1), 68–72 (2009) 7. Zhu L.-W., Sheng Z.-M., Yu M. Y. Direct acceleration of electrons by a circular polarized laser pulse with phase modulation. Phys. Plasmas. U.S.: American Phys. Soc. 20(11), 113112 (2013) 8. Kopytov, G.F., Martynov, A.A., Akintsov N.S.:Motion of a charged particle in the field of a circularly polarized amplitude-modulated electromagnetic wave in the presence of a constant magnetic field. Russian Phys. J. U.S.: Springer New York Consultants Bureau 58(4), 508–516 (2015) 9. Aseev, S.A., Mironov, B.N., Minogin, V.G., Chekalin, S.V.: Visualization of the spationtemporal structure of a pulsed photoelectron beam formed by femtosecond laser radiation. JETP 112, 780 (2011) 10. Castillo, A.J., Milant’ev, V.P.: Relativistic ponderomotive forces in the field of intense laser radiation. Tech. Phys. 59(9), 1261–1266 (2014)

Part III

Machine Learning

Chapter 15

Deep Learning for Fire and Smoke Detection in Outdoor Spaces Margarita N. Favorskaya

and Lakhmi C. Jain

Abstract Recent investigations in deep learning provide a new approach for fire and smoke detection in outdoor spaces. Fire and smoke detection is an additional function in smart urban surveillance, as well as, in wildlife monitoring using stationary cameras or UAV cameras. Smoke detection plays an important role in a fire alarm. The algorithms based on the traditional machine learning techniques show high values of errors that cause additional economic costs. Deep learning techniques solve this problem partly but raise new challenges. In this chapter, we analyze deep learning models applicable for this task and present a Weaved Recurrent Single Shot Detector (WRSSD) for early smoke detection with acceptable error ratios.

15.1 Introduction Wildfires have serious ecological, environmental, and economic impacts including the loss of human life and damage to property all over the world. Since the 1980s, a number of different approaches are reported in this area, depending on the available visual data. The historically first approach connects with using the satellite data. Sensors in active fire/smoke detection are constantly improved. Among the advanced sensors, we can mention the very high-resolution radiometer, moderate resolution imaging spectroradiometer, sea and land surface temperature radiometer, and visible infrared imaging radiometer suite [1]. Different satellite datasets have been created for comparing a variety of active fire/smoke detection algorithms and M. N. Favorskaya (B) Reshetnev Siberian State University of Science and Technology, 31, Krasnoyarsky Rabochy Prosp., Krasnoyarsk 660037, Russian Federation e-mail: [email protected] L. C. Jain Liverpool Hope University, Hope Park, Liverpool L16 9JD, UK e-mail: [email protected] University of Technology Sydney, Broadway, PO Box 123, Sydney, NSW 2007, Australia © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_15

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products. The obvious advantage of this approach is the large territory range, especially remote territories. Limited resolution of satellite images, durable time series (in days), cloudiness, areas with small fire incidents, and so on are the main disadvantages of satellite inspection. Thus, for the prediction of the current state, nearly 20 days of previous observations are required. However, the long-term forecast can be utilized using a series of satellite images. Another level of observation using the optic and infrared cameras is based on using unmanned aerial vehicles, helicopters, and so on. The resolution of images becomes higher but the territory coverage is lesser than the satellite images. More often, video sequences become non-stationary that requires additional complex processing. The third level of observation is a terrestrial shooting. It may be as the additional function of urban video surveillance or observation towers in non-far wilds near cities. Both these types provide the operating control of territories and are available for early fire/smoke detection in outdoor spaces. For all levels of observations, the most perspective algorithms on fire/smoke detection combine the multi-temporal data and include the time series analysis. Often smoke appears before a flame that is why early smoke detection is an important direction of investigations. After this, we pay attention to methods for processing video sequences with/without the features of wildfires and urban fires in the outdoor spaces. The rest of the chapter is organized as follows. Section 15.2 provides a brief description of traditional and deep learning techniques for fire/smoke detection. Section 15.3 includes a problem statement. The proposed deep network architecture is presented in Sect. 15.4. Some experimental results are reported in Sect. 15.5. Concluding remarks are given in Sect. 15.6.

15.2 Traditional and Deep Learning Techniques in Fire/Smoke Detection Recently, a number of fire/smoke detection methods for different observation conditions in the outdoor spaces have been developed. However, the main source of original data remains visual features of the object of interest provided by cameras, optical and infrared. For close smoke detection, it is reasonable to use the optical camera, and for far smoke detection, sometimes laser scanners are additionally applied to distinguish smoke from, for example, a frosty haze over the city. Infrared cameras for fire detection have a restricted application due to their cost and limited distance parameters. Usually, fire visualization includes the color, dynamic features, and flicker, while smoke representation involves a richer set of features such as dynamic features, color, texture, turbulence, and flicker. Section 15.2.1 presents the methods for fire detection. Section 15.2.2 describes the methods for smoke detection. Methods for simultaneous fire/smoke detection are presented in Sect. 15.2.3.

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15.2.1 Fire Detection Methods Color analysis is very important in fire detection, especially for single images. Different color spaces are utilized in fire pixels detection, for example, RGB, YIQ, HSV, YCbCr, and CIE L * a * b* color spaces [2, 3]. However, some researchers extend the color analysis by specific color spaces using machine learning methods. Thus, in [4] a conversion-based target-oriented color space using particle swarm optimization and K-medoids clustering algorithms was introduced. This is reasonable for fire detection in industrial enterprises, when the flame is blue or has a structure of nested rings of colors, changing from white in the core to yellow, orange, and red in the border. Researchers reported a scheme [4] for forest fire detection with the flame colors in the ranges of red and yellow. As it was shown [5], wavelet transform is useful for detection of fire and flamecolored moving regions in videos, as well as, for flicker estimation in flame borders. These authors proved that turbulent flames flicker with a flicker frequency of around 10 Hz is independent of the burning material and the burner. Different clustering and classification methods had been deployed by researchers to better differentiate between fire and non-fire components. Thus, Fuzzy C-Means clustering was used to choose the candidate of fire areas [6], Support Vector Machine (SVM) was applied to classify fire and non-fire pixels even in night time [7]. In [8], a hierarchical detection system was built allowing to detect the fire pixels using a range of features from low-level pixel-based features to high-level video-based semantic features based on some rules. Elmas and Sönmez [9] designed a multi-agent decision support system for a forest fire. The main aim was to estimate the fire danger rating depending on the meteorological state and environmental observations. In this project, artificial neural networks, Naive Bayes classifier, fuzzy switching, and image processing were utilized. Convolutional Neural Networks (CNNs) also generated an interest in their exploration for fire detection. The architecture for early fire detection during surveillance based on CNN and the Internet of multimedia things elaborating the probabilities for both classes of fire and non-fire was explored in [10]. Sharma et al. [11] investigated Deep CNNs, VGG16, and ResNet-50 architectures on the imbalanced fire dataset. Wildfire detection using transfer learning on augmented datasets was discussed in [12]. Three networks, Inception-V3, ResNet-50, and Xception, were tested for fire detection at the very early stage in [13]. YOLO network was used to detect directly the fire regions. Li and Zhao [14] tested several deep architectures for the detection of fire regions and feature extraction. Faster-RCNN, R-FCN, SSD, and YOLO v.3 were compared for detecting the fire regions. The best result was provided by YOLO v.3 with 83.7% accuracy. Two architectures, Inception Resnet V2 and Darknet-53, were yielded as feature extraction networks having 235 and 53 convolutional layers, respectively.

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15.2.2 Smoke Detection Methods Smoke has physical, visual, and dynamic properties but only the visual and dynamic properties are captured by cameras in outdoor spaces. Early smoke detection remains a challenging problem for computer vision-based detection systems. The traditional smoke detection algorithms used image preprocessing, feature extraction, and feature classification in order to detect the smoke regions in the consecutive frames [15, 16]. SVM, AdaBoost, and Random Forest were the most popular machine learning classifiers, and the precision results were obtained in the cases of smoke appearance. The worst cases deal with the atmospheric haze, fog, clouds, and semi-transparent smoke. The motion, color, and contour features are the basic data required for smoke detection [17, 18]. In recent years, deep learning contributed significantly to smoke detection, beginning from typical CNN to special deep networks. Yin et al. [19] suggested a deep normalization and convolutional neural network with 14 layers, in which the convolutional layers were replaced with normalization and convolutional layers. This improvement allowed us to accelerate the training process and boost the performance of smoke detection. The authors obtained detection rates above 96.37% and false alarm rates below 0.60% on the training smoke dataset. Wildfire smoke dilated DenseNet was developed in [20] with the possibility for multi-scale feature extraction by combining the dilated convolutions with dense block. Accuracy rate and false acceptance rate achieved values of 99.20% and 0.24%, respectively. Yin et al. [21] proposed a video-based smoke detection method via Recurrent Neural Network (RNN). Generally speaking, RNNs achieved a higher accuracy of smoke detection in comparison to CNNs due to their ability to analyze a motion carefully but incur a longer detection time. The goal of such complicated deep networks is to provide invariance for large variations in luminance and weather conditions. In [22], a conditional GAN combined with U-net detected the foreground target (generative model) and calculated the higher dimensional differences between the generated smoke distribution and real data distribution. Video smoke detection method based on deep saliency network was proposed in [23]. This network included two branches: the master branch (pixel-level CNN) and partner branch for existence prediction. The final smoke saliency map as a result of fusion of the pixel-level and object-level saliency maps was combined with the feature map for smoke prediction. It is worth to notice a family of SqueezeNet neural networks adopted for smoke detection in [24]. These authors integrated the manual features and deep learning features using the Gaussian mixture model algorithm for detecting the motion blocks and depthwise separable convolution instead of a standard convolution to optimize the architecture of SqueezeNet. The best-achieved precision value equated to 97.95% in real-time implementation.

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15.2.3 Simultaneous Fire and Smoke Detection Methods Simultaneous fire/smoke detection cannot be implemented in early fire-alarm methods because in most cases smoke appears before flame. Flame usually denotes a middle phase of the fire, and changes of dynamic parameters of fire are important for the prediction of fire development. Chen et al. [25] suggested extracting the fire pixels and smoke-pixels in RGB color space using chromatic and disorder measurements. There are few attempts of simultaneous fire and smoke detection using deep learning methods. Namozov and Cho [26] modified the architecture of the original VGG-net with 12 layers using the adaptive piecewise linear activation function instead of using traditional ReLU functions in convolutional layers and tuning the filters. Zhang et al. [27] studied the training process faster Region-based CNN (RCNN) using synthetic smoke images. Two strategies were applied to insert the real smoke and synthetic smoke into the forest background creating RF dataset and SF dataset, respectively. The most of deep learning methods for fire is oriented on fine-tuning full CNNs like VGG16, ResNet, GoogleNet, SqueezeNet, and MobileNetV2. Jadon et al. [28] trained the light CNN, which can be implemented on low-cost hardware like a Raspberry Pi for real-time fire detection. They called their network FireNet, which contains a total of 14 layers (including pooling, dropout, and “Softmax” output layer), has consequently a small number of trainable parameters (646,818) and small size on disk memory (7.45 MB). The accuracy was claimed as 93.91% and higher. As intermediate conclusions, it is evident that at present the investigations are shifted to the deep learning models. The systems for separate fire and smoke detection are more popular than simultaneous fire and smoke detection due to their complexity.

15.3 The Problem Statement The problem statements for fire, smoke, or fire/smoke detection are very similar. Let S = {F i }, i = 1, k, where k is the number of frames F i , be a video sequence. The output data Outi = {Rij }, j = 1, n j , where {Rij } is the set of regions bounded in rectangles, nj is the number of fire, smoke, or fire/smoke regions in each frame F i , provide the labeled regions of interest. Thus, the input data S represented by the consecutive frames F i are transformed into a set of {OUTi } through the preprocessing, feature extraction, and feature classifying steps. The limited task is to use a still image with possible fire and/or smoke presence. Such a statement makes the task poor, with worse results because the dynamic features are rejected from a feature set. The list of fire features includes the following items: • Color dependently from the burning material. • Flickering. • Motion in dynamics.

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The list of smoke features is expended by additional properties: • • • • •

Color dependently from the stage of fire. Semi-transparence in dynamics. Turbulence. Flickering at the edges. Motion in dynamics.

From these two lists, it is clear that dynamic properties play an important role in detection. For traditional machine learning methods, these features are hand-crafted [29, 30], while the use of deep learning methods frees us from construction of feature set for different environmental conditions. It should be noted that a problem statement can be viewed as the recognition task or segmentation task. The recognition task means that we label each frame F i as fire/non-fire or smoke/non-smoke. The segmentation task is the most complex and computationally expensive, where the exact boundaries of the fire or smoke regions should be determined in each frame.

15.4 The Proposed Deep Network Architecture There is a trend to use complicated deep networks for fire/smoke detection under the environment of the unpredictable motion process of non-rigid fire and smoke shapes. To the best of our knowledge, the specific motion features of these dynamic textures are more significant for reliable detection than the static features, mostly color. Therefore, the architecture of deep networks ought to take into consideration both spatial and temporal domains. RNNs are traditionally specialized in modeling the temporal dependencies. Convolutional recurrent networks, which are able to analyze the spatial and temporal information simultaneously, are widely used in many applications. Thus, a convolutional encoder-decoder and Long Short-Term Memory (LSTM) as a key unit of RNN were used for monaural (single-microphone) speech enhancement in real-time as noise- and speaker-independent system [31]. Chen et al. [32] suggested an ensemble application of CNN and RNN to capture both the global and local textual semantics and to model high-order label correlations into text. The recurrent neural network layers were embedded into fully CNNs for developing an end-to-end network for human skin detection [33]. A generic deep framework for activity recognition based on the convolutional and LSTM recurrent units were developed in [34]. For smoke detection, it is reasonable to follow the same approach and construct CNN architecture with LSTM units as a part of RNN. In RNN, the connections between units form a directed cycle for processing the arbitrary sequences of inputs. The internal states are such that generate internal “memory” and simulate the temporal properties. The main problem of RNN training is learning the long-term dependencies that may cause the gradient vanishing problem. Use of LSTM units is a wide-yielded way to avoid this problem.

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The main components of LSTM unit are three gates including the input gate (input modulation gate), forget gate, and output gate and peephole connections, which allow the gates to “spy” on the state of LSTM unit. Three gates are trained to protect the linear unit from false signals. An input gate protects the unit from non-existent input signals. The forget gate allows the unit to forget the previous state of memory cell. The output gate opens (or does not open) the contents of the memory cell at the output of LSTM unit. The output of LSTM unit is recurrently connected to its input and all gates of the unit. In all three gates of LSTM unit, sigmoidal activation functions are used (bringing the argument to the range [–1, 1]). The tanh function is usually used as an activation function at the input and output of a unit. An activation value equaled to 1 means “remember everything,” and a value equaled to 0 means “forget everything.” Usually, the bias of forget gate is initialized with a large value so that it learns the long-term dependencies (a large default value equals 1). The functioning of LSTM unit is implemented using three steps. Step 1. The process of identifying information that is not required is executed by the sigmoid function that takes the output of the previous LSTM unit ht–1 at time t – 1 and the current input X t at time t. Additionally, the forget gate determines, which part from the old output should be eliminated. The function of forget gate f t is a vector with values ranging from 0 to 1, corresponding to each number in the cell state, C t–1 . This function is defined by Eq. 15.1, where σ is the sigmoid function, W f and bf are the weight matrices and bias, respectively, of the forget gate.     f t = σ W f h t−1 , X t + b f

(15.1)

Step 2. The process of storing information on the new input X t , as well as, to update the unit state is carried out. First, the new information is updated or ignored using the sigmoid layer it (0 or 1) and, second, the tanh function N t weights the values which passed by ranging in [–1, 1]. Then the unit state C t at time t is updated considering the previous state of LSTM unit C t–1 at time t – 1. Equations 15.2–15.4 describing this step are given below, where W and b are the weight matrices and bias of the sigmoid layer it and tanh layer N t , respectively.     i t = σ Wi h t−1 , X t + bi

(15.2)

    Nt = tanh Wn h t−1 , X t + bn

(15.3)

Ct = Ct−1 f t + Nt i t

(15.4)

Step 3. The output values ht are formed by the output unit state Ot . These values are filtered by the sigmoid layer. Then the output of the sigmoid gate value Ot is multiplied by the unit state C t , with a value ranging in [–1, 1]. Equations 15.5–15.6 describe this process, where Wo and bo are the weight matrices and bias of the output gate, respectively.

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Fig. 15.1 Basic SSD architecture

    Ot = σ Wo h t−1 , X t + bo

(15.5)

h t = Ot tanh(Ct )

(15.6)

Our Weaved Recurrent Single Shot Detector (WRSSD) architecture is CNN architecture with the incorporated LSTM units. For smoke detection in videos, SSD architecture depicted in Fig. 15.1 was chosen. MobileNet architecture [35] as one of SSD types was used as the basic architecture due to its capability to work in real-time. In order to provide high precision of smoke detection, a place of LSTM units in SSD architecture was defined during experiments. Input frames reduced to the size 300 × 300 pixels enter at the first convolutional layer through three color channels and then on a sequence of MobileNet convolutional layers. The specific feature of MobileNet is that the standard convolutional filters are replaced by two layers under the assumption of signal factorization: the depthwise convolution and a 1 × 1 convolution called pointwise convolution. Such depthwise separable convolution provides a separate layer for filtering and a separate layer for combining. The factorization reduces the computation cost and model size. It should be noted that each color channel is processed separately. After each convolutional layer, batch normalization is executed in order to increase the productivity and reliability of the network. Due to batch normalization, data values are in the range [0, 1]. The obtained data are entered into the input of the activation ReLU function having a standard view of Eq. 15.7, where x is the input value. f (x) = max(0, x)

(15.7)

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Fig. 15.2 SSD network operation scheme with LSTM units

During its work, SSD forms a set of bounding rectangles (anchors) with different sizes and locations placed uniformly in a frame. After the prediction of rectangles’ placement, the non-maximum suppression procedure is applied in order to determine the best rectangles. The network output includes the set of coordinates of the bounded rectangles, set of object classes, and probabilities of belonging objects to certain classes. An object can be assigned to one of four classes: smoke with low density (semi-transparent smoke), smoke with medium density, dense smoke, and no smoke. The architecture of WRSSD contains 34 layers. LSTM units have been added to the network structure in order to consider the temporal features. LSTM units modify the feature maps considering the feature maps obtained by previous frames’ processing. SSD network operation scheme with LSTM units is depicted in Fig. 15.2. Two feature maps with sizes W × H × M obtained after a set of convolutional layers and with sizes W × H × N obtained from the output of LSTM unit at the previous step enter to LSTM unit. These feature maps are joint through the channels forming a feature map W × H × (M + N) incoming to the inputs of the forget, input, and output gates. The output of LSTM unit is the feature maps that are transmitted to the next convolutional layers, as well as, to LSTM unit at the next step. Creation of feature map using LSTM unit is a time-consuming task. It is reasonable to place LSTM units after low-resolution feature maps. Also, it is required to select the optimal number of LSTM units that provide the highest values of precision. During experiments, several architectures were tested: • LSTM unit was located after the 13th convolutional layer. • LSTM units were located after the 12th and 13th convolutional layers. • LSTM units were located after the 6th, 12th, and 13th convolutional layers. Neural network was trained using the gradient descent algorithm. The correspondences between the detected rectangles and sample rectangles were estimated. If the p intersection area of these rectangles exceeds 50%, then the output z i j = 1, otherwise

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p

z i j = 0 [36]. The following step was to estimate the error function. Then the network parameters were changed in order to minimize the error function. Error function of rectangles localization is calculated using Smooth L1-loss as a combination of L1-loss and L2-loss provided by Eq. 15.8, where l is the detected rectangle, g is the sample rectangle, d is the anchor, w and h are the width and height of rectangle, respectively, cx and cy are the shifts of the anchor origin with coordinates p (x, y), Pos is the set of rectangles, for which z i j = 1, N is a correspondence number of rectangles. L loc (z, l, g) =

N 



  p z i j Smooth L1 lim − gˆ mj

i∈Pos m∈{cx,cy,w,h} cy cy − g j − di cy = g ˆ = gˆ cx j j diw dih

 w g hj gj h gˆ j = log h gˆ j = log w di di

g cx j

dicx

(15.8)

Error function of classification is calculated using Eq. 15.9, where c is the probp ability of classification, Neg is the set of rectangles, for which z i j = 0. Class “0” is associated with background objects. L conf (z, c) = −

N  i∈Pos

 p    p z i j log cˆi − log cˆi0

(15.9)

i∈Neg

The resulting loss function is calculated using Eq. 15.10, where α is the weight coefficient. L(z, c, l, g) =

1 (L conf (z, c) + αL loc (z, l, g)) N

(15.10)

In order to avoid overfitting, WRSSD was initially trained without recurrent elements. Then in the pre-trained WRSSD, LSTM units have been added, and the training continued. The paths to images and videos, coordinates of bounded rectangles, and their classes are written in XML-files, which further are transformed into a format available for neural network training using open machine learning library TensorFlow.

15.5 Experimental Results During experiments, the main attention was paid to early smoke recognition as the most promising approach for fire alarm problem. Dataset including around 1500

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images and short videos with smoke and around 1300 images without smoke were selected from datasets: • Smoke detection dataset [37]. This dataset is one of the last, formed in July 2012 at the University of Salerno. It was composed using 149 videos, each of approximately 15 min, with a total duration of more than 35 h. This dataset is available both for testing smoke detectors and fire detectors since containing red houses in a wide valley, mountains at sunset, sun reflections in the camera, several smokes, and clouds. Videos have good quality and applicable to experiments with deep learning models. • Bilkent [38]. Database of Bilkent University is a directory with sample videos containing smoke and smoke/fire flows in open or large spaces. • DynTex [39]. The total contents of DynTex database are more than 650 sequences but a small part of them is concerned with videos with smoke. This database includes seven types of dynamic textures, such as waving/oscillating motion, directed motion, turbulent/irregular motion, oscillating motions, directed motions, irregular motions, and direct appearance change. Smoke and fire are concerned with medium categories. • Free accessed Internet resources. The public dataset was extended by augmentation involving color changing and affine transformations (translation, rotation, scaling, flipping, and shearing) with the following parameters [40]: • • • •

Random rotation in the range of [–45°, +45°]. Random flip of image with following horizontal alignment. Random resizing in the range [0.75, 1.25] or scaling up to 25% zooming in/out. Random cropping an image.

The extended dataset was divided into the training, validation, and testing samples with proportions 60%, 20%, and 20%, respectively. Training samples for WRSSD with LSTM units involved the batches consisting of ten consecutive frames. Six models including one model with hand-crafted features and five deep neural models were implemented in order to obtain the estimates and compare the results: • • • • •

Model-based on conventional spatio-temporal features (Model 1). SSD model without motion verification (Model 2). SSD model with motion verification (Model 3). WRSSD model with one LSTM unit located after 13th layer (Model 4). WRSSD model with two LSTM units located after 12th and 13th layers (Model 5). • WRSSD model with three LSTM units located after 6th, 12th, and 13th layers (Model 6). The experiments were conducted using PC with Intel Core i5-7300HQ, 2.5 GHz, NVIDIA GeForce GTX 1050, RAM 6 Gb. The obtained results with True Recognition (TR), False Rejection Rate (FRR), and False Acceptance Rate (FAR) are shown in Table 15.1.

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Table 15.1 Average results for early smoke detection using Model 1–Model 6 Model

Number of analyzed frames per sec.

Averaged TR (%)

Averaged FRR (%)

Averaged FAR (%)

Model 1

2

82.7

20.3

16.5

Model 2

8

88.0

16.2

8.1

Model 3

8

89.1

15.1

8.1

Model 4

7

92.0

12.1

6.3

Model 5

7

92.5

12.0

5.9

Model 6

5

94.5

10.2

3.7

The time costs increase when LSTM units are added to neural network architecture. The precision of smoke detection using a neural network is higher than the precision of smoke detection based on image processing methods. Checking a movement after application of the neural network allows reducing FRR values. Extraction of temporal features using LSTM units decreases FAR values and improves the precision of smoke detection. The precision of smoke detection based on WRSSD with two LSTM units is not much higher than the precision of smoke detection by WRSSD with one LSTM unit. The use of WRSSD with three LSTM units improves smoke detection, but it is time-consuming. The output result of WRSSD is a list of object classes with probabilities. If a probability value exceeds the predetermined threshold, then it means that classification is carried out correctly. A convenient way to visualize the precision values at different thresholds is to build ROC-curves. The percentage of FRR and FAR depends on the given threshold value. The optimal threshold value in interval [0…1] can be selected for minimizing FRR or FAR. For each threshold value, the true positive ratio and false-positive ratio are plotted on OX and OY axes, respectively. ROC-curves depicted in Fig. 15.3 show the results of SSD and WRSSD classification. The conducted experiments show that smoke with different densities in video scenes with variable depth can be detected successfully. Smoke with high optical density can be detected easily using the algorithms based on the spatio-temporal Fig. 15.3 ROC-curves for SSD and WRSSD models

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features, extracted manually or automatically. Precision of semi-transparent smoke detection is worse. Thus, the precision results are changed in time due to the variations in smoke density. Obviously, deep learning models require the optimization of parameters more carefully than conventional machine learning techniques.

15.6 Conclusions Visual-based wildfires detection remains a crucial task since the 1980s. Presently, the investigations are shifted from the traditional machine learning methods to the deep learning models. The systems for separate fire and smoke detection are more common than simultaneous fire and smoke detection. A set of smoke features is richer than a set of fire features. Both fire and smoke have the spatial and temporal aspects that warrant a necessity to construct the combined deep architecture. During experiments, the main attention was paid to early smoke recognition as the most promising approach for fire alarm problem. The proposed WRSSD architecture is CNN architecture with the incorporated LSTM units. Six models including one model with hand-crafted features and five deep neural models were implemented in order to obtain the estimates and to compare the results. The best results show WRSSD with three LSTM units with averaged TR, FRR, and FAR estimates equaled to 94.5%, 10.2%, and 3.7%, respectively, but this network is the most timeconsuming. Smoke with high optical density is easily detected using all methods. Precision of semi-transparent smoke detection is worse due to the smoke density variations in time. Future research is directed at flame segmentation problems using video sequences in order to estimate a degree of wildfire.

References 1. Lin, Z., Chen, F., Niu, Z., Li, B., Yu, B., Jia, H., Zhang, M.: An active fire detection algorithm based on multi-temporal FengYun-3C VIRR data. Remote Sens. Environ. 211, 376–387 (2018) 2. Celik, T.: Fast and efficient method for fire detection using image processing. ETRI J. 32(6), 881–890 (2010) 3. Celik, T., Demirel, H.: Fire detection in video sequences using a generic color model. Fire Saf. J. 44(2), 147–158 (2009) 4. Khatami, A., Mirghasemi, S., Khosravi, A., Lim, C.P., Nahavandi, S.: A new PSO-based approach to fire flame detection using K-Medoids clustering. Expert Syst. Appl. 68, 69–80 (2017) 5. Töreyin, B.U., Dedeoglu, Y., Güdükbay, U., Cetin, A.E.: Computer vision based method for real-time fire and flame detection. Pattern Recogn. Lett. 27(1), 49–58 (2006) 6. Truong, T.X., Kim, J.M.: Fire flame detection in video sequences using multi-stage pattern recognition techniques. Eng. Appl. Artif. Intell. 25(7), 1365–1372 (2012) 7. Ho, C.-C.: Nighttime fire/smoke detection system based on a support vector machine. Math. Probl. Eng. ID 428545.1–ID 428545.7 (2013)

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8. Zhao, Y., Tang, G., Xu, M.: Hierarchical detection of wildfire flame video from pixel level to semantic level. Expert Syst. Appl. 42(8), 4097–4104 (2015) 9. Elmas, C., Sönmez, Y.: A data fusion framework with novel hybrid algorithm for multi-agent decision support system for forest fire. Expert Syst. Appl. 38(8), 9225–9236 (2011) 10. Muhammad, K., Ahmad, J., Baik, S.W.: Early fire detection using convolutional neural networks during surveillance for effective disaster management. Neurocomputing 288, 30–42 (2018) 11. Sharma J., Granmo OC., Goodwin M., Fidje J.T.: Deep convolutional neural networks for fire detection in images. In: Boracchi G., Iliadis L., Jayne C., Likas A. (eds.) Engineering Applications of Neural Networks. EANN 2017. CCIS, vol. 744, pp. 183–193. Springer, Cham (2017) 12. Sousa, M.J., Moutinho, A., Almeida, M.: Wildfire detection using transfer learning on augmented datasets. Expert Syst. Appl. 142, 112975.1–112975.14 (2020) 13. Wu, H., Wu, D., Jinsong Zhao, J.: An intelligent fire detection approach through cameras based on computer vision methods. Process Saf. Environ. Prot. 127, 245–256 (2019) 14. Li, P., Zhao, W.: Image fire detection algorithms based on convolutional neural networks. Case Stud. Thermal Eng. (2020) (in print) 15. Calderara, S., Piccinini, P., Cucchiara, R.: Vision based smoke detection system using image energy and color information. Mach. Vis. Appl. 22, 705–719 (2011) 16. Favorskaya, M., Pyataeva, A., Popov, A.: Spatio-temporal smoke clustering in outdoor scenes based on boosted random forests. Procedia Comput. Sci. 96, 762–771 (2016) 17. Toreyin, B.U., Dedeoglu, Y., Cetin, A.E.: Contour based smoke detection in video using wavelets. In: 14th European Signal Processing Conference, Florence, Italy, pp. 1–5 (2006) 18. Lin, G., Zhang, Y., Zhang, Q., Zhang, J., Jia, Y., Xu, G., Wang, J.: Smoke detection in video sequences based on dynamic texture using volume local binary patterns. KSII Trans. Internet Inf. Syst. 11(11), 5522–5536 (2017) 19. Yin, Z., Wan, B., Yuan, F., Xia, X., Shi, J.: A deep normalization and convolutional neural network for image smoke detection. IEEE Access 5, 18429–18438 (2017) 20. Li, T., Zhao, E., Zhang, J., Hu, C.: Detection of wildfire smoke images based on a densely dilated convolutional network. Electronics 8(10), 1131.1–1131.12 (2019) 21. Yin, M., Lang, C., Li, Z., Feng, S., Wang, T.: Recurrent convolutional network for video-based smoke detection. Multimedia Tools Appl. 8, 1–20 (2018) 22. Jia, Y., Du, H., Wang, H., Yu, R., Fan, L., Xu, G., Zhang, Q.: Automatic early smoke segmentation based on conditional generative adversarial networks. Optik Int. J. Light Electron. Opt. 193, 162879.1–162879.13 (2019) 23. Xu, G., Zhang, Y., Zhang, Q., Lin, G., Wang, Z., Jia, Y., Wang, J.: Video smoke detection based on deep saliency network. Fire Saf. J. 105, 277–285 (2019) 24. Peng, Y., Wang, Y.: Real-time forest smoke detection using hand-designed features and deep learning. Comput. Electron. Agric. 167, 105029.1–105029.18 (2019) 25. Chen, T.-H., Wu, P.-H., Chiou, Y.-C.: An early fire-detection method based on image processing. In: 2004 International Conference on Image Processing, vol. 3, pp. 1707–1710 IEEE (2004) 26. Namozov, A., Cho, Y.I.: An efficient deep learning algorithm for fire and smoke detection with limited data. Adv. Electr. Comput. Eng. 18(4), 121–128 (2018) 27. Zhang, Q., Lin, G., Zhang, Y., Xu, G., Wang, J.: Wildland forest fire smoke detection based on faster R-CNN using synthetic smoke images. Procedia Eng. 211, 441–446 (2018) 28. Jadon, A., Omama, M., Varshney, A., Ansari, M.S., Sharma, R.: FireNet: a specialized lightweight fire and smoke detection model for real-time IoT applications. CoRR ArXiv Preprint, arXiv:1905.11922v2 [cs.CV] (2019) 29. Favorskaya, M., Levtin, K.: Early video-based smoke detection in outdoor spaces by spatiotemporal clustering. Int. J. Reason.-Based Intell. Syst. 5(2), 133–144 (2013) 30. Favorskaya, M., Pyataeva, A., Popov, A.: Verification of smoke detection in video sequences based on spatio-temporal local binary patterns. Procedia Comput. Sci. 60, 671–680 (2015) 31. Tan, K., Wang, D.: A convolutional recurrent neural network for real-time speech enhancement. Interspeech 1405, 3229–3233 (2018)

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32. Chen, G., Ye, D., Xing, Z-C., Chen, J., Cambria, E.: Ensemble application of convolutional and recurrent neural networks for multi-label text categorization. In: International Joint Conference on Neural Networks, pp. 2377–2383 (2017) 33. Zuo, H., Fan, H., Blasch, E., Ling, H.: Combining convolutional and recurrent neural networks for human skin detection. IEEE Signal Process. Lett. 24(3), 289–293 (2017) 34. Ordóñez, F.J., Roggen D.: Deep convolutional and LSTM recurrent neural networks for multimodal wearable activity recognition. Sensors 16, 115.1–115.25 (2016) 35. Howard, A.G., Zhu, M., Chen, B., Kalenichenko, D., Wang, W., Weyand, T., Andreetto, M., Adam, H.: MobileNets: efficient convolutional neural networks for mobile vision applications. CoRR arXiv preprint, arXiv:1704.04861 (2017) 36. Liu, W., Anguelov, D., Erhan, D., Szegedy, C., Reed, S., Fu, C.Y., Berg, A.C.: SSD: Single shot multibox detector. In: Leibe, B., Matas, J., Sebe, N., Welling, M. (eds.) Computer Vision— ECCV 2016, LNCS, vol. 9905, pp. 21–37. Springer, Cham (2016) 37. Smoke Detection Dataset. https://mivia.unisa.it/datasets/video-analysis-datasets/smoke-detect ion-dataset/. Last accessed 03 May 202 38. Database of Bilkent University. https://signal.ee.bilkent.edu.tr/VisiFire/Demo/SmokeClips/. Last accessed 03 May 2020 39. DynTex. https://projects.cwi.nl/dyntex/. Last accessed 03 May 2020 40. Favorskaya, M., Pakhirka, A.: Animal species recognition in the wildlife based on muzzle and shape features using joint CNN. Procedia Comput. Sci. 159, 933–942 (2019)

Chapter 16

The Solution of Fractures Detection Problems by Methods of Machine Learning Maksim V. Muratov, Dmitriy I. Petrov, and Vladimir A. Biryukov

Abstract The aim of this chapter is to represent approach to solve the inverse exploration seismology problems using methods of machine learning. The twodimensional problem of fracture size, placement, and spatial orientation detection is considered in this chapter. The geological medium including the fracture has the determined and initially known elastic characteristics. To solve this problem, the neural network is used. The training of network was produced by the direct exploration of seismology problems with different sizes, placement, and orientation of fracture solutions using mathematical modeling by grid-characteristic method on regular meshes. The use of such a numerical method takes into consideration the characteristic physical properties of describing processes and gives us a possibility to construct correct algorithms on boundaries and contact boundaries in the integrational domain. The presented approach has given good accuracy in determining the parameters of a fracture in a medium with known characteristics. Quite good accuracy was achieved.

16.1 Introduction Recently, machine learning techniques and, in particular, deep neural networks have shown impressive results in many areas, such as computer vision, speech recognition, and machine translation. For example, in the field of computer vision, it was possible to solve many problems, such as the classification problem [1], recognition problem [2], and the problem of image generation [3]. One of the significant advantages of M. V. Muratov (B) · D. I. Petrov · V. A. Biryukov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] D. I. Petrov e-mail: [email protected] V. A. Biryukov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_16

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deep learning methods is that these methods can be transferred to many other areas related to the processing of large amounts of data. One such area is the exploration seismology problems. Several works in this field have already been carried out. In [4], the problem of fault detection in 2D was solved using a deep convolutional neural network. Synthetic data obtained by solving large direct problems were used for training the neural network. In [5], a similar problem was solved in three dimensions. The great advantage that these papers draw attention to the input data for deep learning algorithms, which do not require special processing and, therefore, such methods can be simpler to use than standard exploration seismology methods. Flexibility and relative simplicity make such methods effective for solving practical problems. Thus, in [6], deep neural networks are used to detect CO2 emissions, and these methods are used to detect and classify defects in composite materials in [7]. The results show that the use of machine learning methods in the exploration seismology is an important topic for research. The chapter is organized as follows. Section 16.2 is devoted to problem formulation. Section 16.3 is about problem solution technique, method of direct problem solution, and neural network structure. In Sect. 16.4, the results of the solution are represented. Section 16.5 concludes the chapter.

16.2 The Problem Formulation Two-dimensional problem is considered, in which it is proposed to find the size, the spatial position, and angle of inclination of a single fracture using seismic data. The fracture is in a homogeneous elastic medium with the following elastic characteristics: Cp = 4500 m/s, Cs = 2500 m/s, ρ = 2500 kg/m3 . The size of the computational domain is 2 km × 2 km (Fig. 16.1). The position of the fracture varies in the range of 1000 m vertically and horizontally. The angle of inclination is in the range of ± 15° (subvertical fracture). The size of the fracture is in the range from 50 to 200 m. In the middle of the border of the study area, a sinusoidal elastic pulse consisting of five periods (a wavelength of 100 m) is excited. The values of the vertical component of the velocity of the reflected waves on seismic receivers uniformly located on the excitation surface of the wave pulse (65 receivers in total) are recorded—seismic data.

16.3 Solution Technique The process of solving the problem of recognizing the spatial position of a fracture in an elastic medium consists of two stages: training of the neural network and recognizing a control sample of seismic data. To create a training sample, direct problems are solved with different fracture parameters [8].

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Fig. 16.1 The scheme of fracture placement: fracture can be placed in any point of fracture domain, angle of fracture is in range of ± 15°, size of fracture is in range from 50 to 200 m

In Sect. 16.3.1, we consider the direct problem solution, mathematical model, numerical method, boundary conditions, and condition on fracture. Section 16.3.2 is devoted to inverse problem solution technic, structure of the used convolutional neural network, and their types of layers description.

16.3.1 Direct Problem Solution Mathematical model. For mathematical modeling, a linear-elastic medium model is used. In the computation, the grid-characteristic method with a second-order hybrid scheme is used. Wave processes in elastic geological medium are described on the basis of the governing equations of the theory of a linearly elastic medium. According to this model, the state of an infinitely small volume of a medium obeys a system of two equations: the local equation of motion and rheological relation that relates stress and strain in the medium. They can be reduced to Eqs. 16.1 and 16.2. ∂ T ji ∂ Vi = ∂t ∂x j      ∂ Vk ∂ Ti j ∂Vj ∂ Vi + Ii j + μ =λ ∂t ∂ xk ∂x j ∂ xi k ρ

(16.1)

(16.2)

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3D-system of such equations can be represented as. → →  ∂− u ∂− u + Ai = 0, ∂t ∂ξi i=1,2,3

(16.3)

→ u is the vector of values represented as: where ξi take values as x, y, and z, and −   u = Vx , Vy , Vz , Tx x , Tyy , Tzz , Tx y , Tx z , Tyz . Such a system can be solved by the grid-characteristic method [9, 10]. Numerical method. By changing variables, each of them can be reduced to a set of independent scalar transport equations in Riemann invariants: ∂w  ∂w  + i  = 0, i = 1, 2, 3, ∂t ∂ξi where i is the diagonal matrix of eigenvalues. For each transfer equation, all nodes of the computational mesh are bypassed, and characteristics are omitted for each node. From the time layer n, the corresponding component of the vector is transferred to the time layer n + 1 according to the formula: 



wkn+1 ξi = wkn ξi − ωk τ , where τ is timestep. After all the values are transferred, there is a reverse transition to the vector of → the desired values − u . For the interpolation on the irregular triangle (in 2D-case) and tetrahedral (in 3D-case) meshes, using values → we define the values in each point− → → p i jkl − r as in supporting points of mesh − w − r i jkl and weights of these points → follows: − → − →  → → − → p i jkl − r → w − r i jkl . w − r = i, j,k,l

Boundary conditions and conditions on the fracture. The boundary condition can be written in the general form as: − → → D− u (ξ1 , ξ2 , t + τ ) = d . − → → where D is the matrix 5 × 2, d is the vector, − u is the value of the required velocity values and components of the stress tensor at the boundary point at the next time step.

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At the boundaries of the computational domain, the condition of a free boundary has a view: → T− n (ξ1 , ξ2 , t + τ ) = 0. To specify the fracture, an infinitely thin fracture model with the condition of fluidfilled fracture was used [11, 12]. Such a fracture is defined as a contact boundary with the condition of free sliding: − →a − →b − →a − →b → → → − → n =− v b•− n , f n = f n , f τ = f τ = 0. v a •− Such contact boundary fully passes any longitudinal vibrations without reflection and fully reflects shear waves. This situation corresponds to the real one: values of longitudinal waves propagation in fluids velocities and densities are comparable with velocities and densities of real geological media. At the same time velocities of shear vibrations in fluids are close to zero.

16.3.2 Neural Network Structure The Keras deep learning library based on the Tensorflow library and the CUDA parallel computing architecture was used [13]. The Keras library was chosen because of its simplicity of use and possibilities sufficient to solve the problem. The training set consisted of pairs (X, y). X is a seismogram (a matrix of real numbers 94 × 94 in size). y is a set of parameters defining the position of the fracture. In the case under consideration, y was given by four real numbers: the coordinates of the ends of the fractures (the height of the fracture is remained constant). The convolutional network differs from other types of neural networks by the presence of convolutional and pooling layers. These layers can significantly reduce the number of network parameters and accelerate the speed of learning. The following types of layers were used in the current task: convolutional layer, max-pooling layer, and fully connected layer. Experiments were carried out with the addition of Dropout layers, but their use impaired the accuracy of the predictions. Here is a brief description of each of the layers used. Fully connected layer. The fully connected layer is a classic for most types of neural networks. In this layer, each neuron from the previous layer is connected to the neuron of the next layer. Layers of this group are used in many types of tasks: their advantage is that they take into account the maximum amount of information and connections between neurons. The disadvantage is a large number of parameters, which is equal to the number of edges in conjunction with the number of output neurons. Another drawback is the fact that a large number of parameters can degrade the convergence of the optimized function.

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Convolutional layer. Convolutional layers are a characteristic feature of the convolutional neural network. Their feature is that instead of pairwise combining of outgoing and incoming neurons, only certain local neurons are combined. The next feature is the joint use of the same scales for different ribs, which speeds up and simplifies training. For clarity, the convolutional layer scheme can be represented in the form of a filter sliding along the selected axes (there can be 1, 2, 3 or more). One filter passes through all possible points and forms the next level of neurons by scalar multiplication of the filter values by the values of the object. The number of parameters is equal to the product of the number of filters and the filter size. During the current experiments, two-dimensional convolutional layers were used, in which one axis corresponded to the measurement time, and the second axis corresponded to the position of the sensor. Thus, a two-dimensional map of signals is created. As the advantages of this type of layers, it should be noted a small number of parameters that are limited by the size of the filter and sharing of weights. These factors accelerate the training of the neural network and find specific features of objects along the indicated axes. These features make convolutional neural networks very popular for solving pattern recognition tasks (2 axes—width and height of the image), text analysis (1 axis—letter/word position in the text), or audio (one axis— time or two axes—time and sound frequency). Among the disadvantages, the use of this type of layer is limited to certain types of tasks. Max-pooling layers. Max-pooling (also called average-pooling) layers are typical for convolutional neural networks. These layers are nonparametric, and their work is to select the maximum (or average) value inside the given window and transfer this value to the neuron of the next layer. Pooling layers allow us to reduce the number of neurons in the next layer 4 (9, 16, etc.) times, thereby reducing the number of weights in the next layers. In practice, these types of layers are almost universally used in the training of convolutional neural networks to improve their convergence. Activation function. In neural networks, output signals from neurons pass through a nonlinear activation function. Examples include hyperbolic tangent, sigmoid, Restricted Linear Unit (ReLU), function of the form y = max (0, x). Without activation functions, a neural network (or a subset of its layers) would turn into a simple linear function, thus the presence of nonlinear activations is an essential component of a neural network. The choice of the activation function is left to the discretion of the researcher. In this work, we used the ReLU activation function, the graph of which is shown in Fig. 16.2. Used neural network. In this chapter, a two-dimensional inverse problem was solved, but the proposed method can be extended to the case of 3D. To solve the problem, a neural network consisting of three convolutional (Conv), 3 max-pooling (MP), and two fully connected (FC) layers was proposed. The structure of neural network is represented in Fig. 16.3. The network layer parameters are as follows: the first convolutional layer: (94, 94, 64), the second convolutional layer: (44, 44, 64), the third convolutional layer: (20, 20, 64), the first fully connected layer: (6400, 64), the second fully connected layer: (64, 4). Between convolutional layers, max-pooling

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Fig. 16.2 The graph of activation function ReLU

Fig. 16.3 The structure of neural network

layers are placed. The dimensions of all filters in the convolutional layers are 3 × 3, the activation function is ReLU. The total number of network parameters: 2.7 × 106. As the optimizer of the neural network, Adam was chosen with a learning rate of 0.001.

16.4 Simulating Results In this section, the process of recognition (Sect. 16.4.1) and the results of recognition (Sect. 16.4.2) are represented.

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Fig. 16.4 The graph of functional J dependence on period of study

16.4.1 Recognition Process The recognition of a control sample is the process of minimizing the functional [13]:

J=



pr ed 2 L 2 .

yir eal − yi

i

Its graph is represented in Fig. 16.4. With each new era in learning, the value of the function decreases and tends to a certain value. Therefore, the method can be used to solve this class of problems. Functional decreases were minimized to value of 0.07. As the value of maxy L 2 − miny L 2 ~10, so this is quite a good result.

16.4.2 Results of Recognition For the described problem with a variation in size, the spatial placement and orientation of single fracture training were performed on a set of 3000 solutions of the direct problem and then validation was performed on a set of 200 control samples. Figure 16.5 shows several results of recognition (blue line is real fracture placement and red line is its recognized image). In Fig. 16.5a, the seismograms of control samples are represented, and in Fig. 16.5b, the results on recognition for these control samples represented in the form of two-dimensional pictures of fractures are depicted. One can see good qualitative conformity in results.

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Fig. 16.5 Examples of recognition with qualitative comparison: a seismogram of control samples, b the results on recognition for these control samples

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16.5 Conclusions The approach to solve the inverse exploration seismology of fracture detection problems with the use of convolutional neural networks was developed. Training set in this problem was formed by direct problem solutions by mathematical modeling with the use of grid-characteristic method. Neural network consists of three convolutional layers connected by max-pooling layers and two full-connected layers. As activation function ReLU function was used. Recognition gave an excellent result. But in more complex problems with more number of variables and different interferences due to heterogeneity of medium, the error will be larger, and their solution will require the revision of neural network. Acknowledgements This work was supported by the Russian Foundation of Basic Research, project no. 20-01-00572.

References 1. Krizhevsky, A., Sustskever, I., Hinton, G.E.: Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems (NIPS’12), Lake Tahoe, Nevada, USA, pp. 1097–1105 (2012) 2. Szegedy, C., Toshev, A., Erhan, D.: Deep neural networks for object detection. Advances in neural information processing systems (NIPS’13), Lake Tahoe, Nevada, USA, pp. 2553–2561 (2013) 3. Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial nets. Advances in neural information processing systems (NIPS’14), Montréal, Canada, pp. 2672–2680 (2014) 4. Zhang, C., Frogner, C., Araya-Polo, M., Hohl, D.: Machine-learning based automated fault detection in seismic traces. In: 76th EAGE Conf. and Exhibition, Amsterdam, The Netherlands, pp. 1–5 (2014) 5. Araya-Polo, M., Dahlke, T., Frogner, C., Zhang, C., Poggio, T., Hohl, D.: Automated fault detection without seismic processing. Lead. Edge 36(3), 194–280 (2017) 6. Wu, Yu., Lin, Yo., Zhou, Zh., Delorey, A.: Seismic-net: A deep densely connected neural network to detect seismic events. CoRR ArXiv Preprint, arXiv:1802.02241 (2018) 7. Menga, M., Chua, Y.J., Woutersonb, E., Ong, C.P.K.: Ultrasonic signal classification and imaging system for composite materials via deep convolutional neural networks. Neurocomputing 257, 128–135 (2017) 8. Petrov, I.B., Muratov, M.V.: The application of grid-characteristic method in solution of fractured formations exploration seismology direct problems (review article). Matem. Mod. 31(4), 33–56 (2019) 9. Magomedov, K.M., Kholodov, A.S.: Grid-characteristic numerical methods. Nauka, Moscow (in Russian) (1988) 10. Favorskaya, A.V., Breus, A.V., Galitskii, B.V.: Application of the grid-characteristic method to the seismic isolation model. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds) Smart Modeling for Engineering Systems. GCM50 2018. SISIT, vol. 133, pp. 167–181. Springer, Cham (2019) 11. Muratov, M.V., Petrov, I.B.: Application of mathematical fracture models to simulation of exploration seismology problems by the grid-characteristic method. Comput. Res. Model. 11(6), 1077–1082 (2019)

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12. Favorskaya, A., Petrov, I., Grinevskiy, A.: Numerical simulation of fracturing in geological medium. Procedia Comput. Sci. 112, 1216–1224 (2017) 13. Muratov, M.V., Biryukov, V.A., Petrov, I.B.: Solution of the fracture detection problem by machine learning methods. Doklady Math. 101(2), 169–171 (2020)

Chapter 17

Solving Problems of the Strength of a Thin Thread by Machine Learning Methods Mykhailo Seleznov

and Alexey V. Vasyukov

Abstract The chapter considers the construction of a surrogate machine learning model for the problem of deformation and breakage of a thin thread under the action of a transverse load. This dynamic physical problem is computationally simple, which makes it convenient for testing various approaches, but at the same time, it contains a significant range of effects, which allows us to expect how the methods worked out on it can be further applied to more complex problems. The chapter proposes an algorithm for the formation of a small training set, which will still provide a reasonable quality of the surrogate model. It was found that the machine learning model can provide a reasonable quality of prediction for a dynamic physical problem even with a small training set: an F-score of above 0.8 is obtained when training on 500 samples. Quantitative characteristics for other tasks can vary significantly; some issues regarding it are also discussed in the present work.

17.1 Introduction Designing engineering structures and protective shells made of composite materials often requires to solve the problem of optimizing their parameters. For instance, it can be minimization of the mass of the structure for a given static bearing capacity or minimization of the mass of the protective shell while maintaining the ability to withstand a given spectrum of shock loads. Traditionally, mathematical modeling methods are used to solve optimization problems. There are different optimization algorithms, but any algorithm requires solutions to a large number of direct problems. For instance, to optimize the parameters of a shock-proof composite protective screen, it will be necessary to iterate M. Seleznov (B) · A. V. Vasyukov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] A. V. Vasyukov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_17

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over the possible parameters of the screen such as the elastic and strength properties of the material, as well as, the thickness and geometry of the screen. And for each set of screen parameters, it will be necessary to calculate the consequences of impacts by particles of different mass and shape moving at different speeds and from different angles. Such calculations are possible, but extremely resource-intensive for a dynamic three-dimensional problem or the problem statement can be extremely complicated and the calculation time of one statement can take many hours or even days [1]. To accelerate the solution of such optimization problems, one can use surrogate models built using Machine Learning (ML) methods [2]. In this approach, a traditional numerical solver for the direct problem is used to generate data on which ML model is trained. The resulting ML model gives answers “instantly” in comparison with the time of direct numerical simulation (fractions of a second for the ML model compared to the hours of direct calculation). After that, ML model is used by the optimization algorithm when solving the multi-parameter optimization problem. ML model prediction is used as an answer for a specific set of parameters, not the direct calculation results. Of course, due to the nature of machine learning, the response of ML model may be incorrect in some cases. However, if in most cases it gives a reasonable answer, then its application allows one to estimate the promising ranges of parameters by orders of magnitude faster than using a direct solver. If necessary, these ranges of parameters can be further checked and updated by direct solvers. A separate question, in this case, is the size of the training set, which will be sufficient to build a surrogate model of reasonable quality to describe a dynamic physical problem. Obviously, for practical tasks, it is desirable to be able to use not too large set, since for each sample in this set it is still required to calculate a complete physical task using hours of machine time. Therefore, in this chapter, we consider a computationally simple, but physically quite a complex dynamic problem—deformation and breakage of a thin thread under the action of a transverse load impulse. It is possible to calculate for this problem a large number of statements in a reasonable time. A surrogate ML model is trained after that using different train sets, and the quality of ML model is evaluated with respect to the size of the training set. The chapter is organized as follows. Section 17.2 presents the models, methods, and related works on which this study is based. Section 17.3 describes the considered ranges of parameters and also describes the algorithms for the formation of training and test sets, which are an important part of the present work. Section 17.4 is devoted to the consideration of the results. It presents the dependencies from the direct calculations and the quality metrics of ML models. Section 17.5 gives the final remarks and concludes the chapter.

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17.2 Background In this chapter, we consider the deformation and breakage of a thin thread under the action of a transverse load. The thread is described by the model from [3], and this chapter uses numerical implementation from [4]. In this model, the following system of equations is used: d2x ∂(T cos φ) = + ρ0 P sin φ, dt 2 ∂s0 d2 y ∂(T sin φ) ρ0 2 = + ρ0 P cos φ, dt ∂s0

ρ0

T = ε · E. Here ρ 0 is the density of the thread material, s0 is the distance along the thread, T is the tension force of the thread, ϕ is the angle between the OX axis and the tangent vector to the thread, P is the external force acting in the normal direction to the thread. The tension force acts along the thread and is associated with deformation ε in a linear manner. A thread breaks at a point when a strain limit εk is reached at this point. The transverse load is defined as a short pressure pulse described by three parameters – the width, amplitude, and duration of the pulse. Thus, the problem statement is described by eight parameters: the length of the thread, the diameter of the thread, Young’s modulus, the density of the material of the thread, the strain limit, the duration of the pressure pulse, the radius of the pulse, and the amplitude of the pulse. The result of the calculation in the framework of this work is a binary answer: the thread is broken or not. The parameters and calculation results are input to the training of the surrogate model. The trained model should give an answer according to the specified parameters: this thread remains intact when exposed to this load, or it breaks. This problem becomes actually a binary classification problem for a sample of eight parameters. To build ML model, in this chapter, the following standard methods and algorithms are used: XGBoost tree boosting [5], logistic regression [6], Support Vector Classification (SVC) [7], Stochastic Gradient Descent (SGD) [8], naive Bayes [9], random forest classifier [10], multilayer perceptron [11], and k-nearest neighbors [12]. The implementation uses the Scikit-Learn library.

17.3 Data Sets As noted above, the problem has eight parameters. In this chapter, we use two main sets of statements. Data set #1 considers 10 values for each of the eight parameters (a total of 100 million problem statements). The parameters in data set #1 vary in the following

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ranges: length 10–100 cm, diameter 0.01–0.5 mm, the Young modulus 60–300 GPa, density 1000–2000 kg/m3 , strain limit 0.2–10.0%, pressure time 0–100 ms, pressure radius 0–5 cm, pressure amplitude 0–200 MPa. The selected ten values for each parameter are evenly distributed over the specified range for this parameter. For example, values for length are: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. Data set #2 considers 9 values for each parameter (a total of 43 million problem statements). The values in the set #2 are located between the values in the set #1. For example, for length values in the second set are: 15, 25, 35, 45, 55, 65, 75, 85, 95. Due to the fact that the problem under consideration is computationally simple, it is possible to perform direct numerical modeling of the dynamics of the thread for all samples from sets #1 and #2. The optimized solver of the direct problem requires for each sample 40–500 ms of CPU time on 1 core of the Intel Xeon processor depending on the required level of approximation. The calculations use one central node (Intel Core i5, 8 GB RAM, SSD hard drive) that coordinates workers and stores the results and 100 worker nodes that do computations. Total computation time for 100 million statements using 100 working nodes varies from half a day to 1 week depending on the required level of approximation. The calculation of 43 million statements takes proportionally less time. In compressed form, the input parameters and binary results for 100 million statements require about 1 MB. In this case, of course, the complete spatio-temporal picture of the thread is not stored. One of the goals of this work is to build ML model of reasonable quality using a small training set. Therefore, in addition to the two indicated large-sized data sets, a small-sized set #3 was created. The set #3 should include samples that cover substantially different problem statements. Each new sample is supposed to provide essentially new information for training the model. At the same time, we do not use the full information of direct calculations from the previous large sets when building the set #3. We do it this way to reflect the situation when direct calculation requires a large amount of computational resources. The set #3 is built using the following algorithm: (1) Two initial samples are taken. The first sample is a thread with large values of length, diameter, Young modulus, density, strain limit, and small values of pressure time, pressure radius, and pressure amplitude. There is no breakage of the thread in this statement. The second sample is a thread with small values of length, diameter, Young modulus, density, strain limit, and large values of pressure time, pressure radius, and pressure amplitude. There is a breakage in this statement. These two samples differ in parameters as much as possible. (2) An ensemble of seven models is trained on all current samples of the set #3: XGBoost, logistic regression, SVC, linear SVC, SGD, naive Bayes, random forest. Of course, at the initial stages, this training using a few samples does not provide any quality, but it is formally possible. (3) These seven models are used to predict 100 million statements of set #1. The actual answers from the direct solver are not used at this stage.

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(4) The samples are selected on which the models unanimously coincide with each other in predicting the absence of a break. If there is no unanimous decision, then the samples predicted to have no break by the maximum number of models are selected. (5) For each sample from stage 4, the Euclidean distance in the parameter space is calculated for all samples without a break in the current set. A sample with the maximum given distance is selected (or several samples if their distances coincide). In other words, one tries to select the sample without a break (based on the ensemble prediction), which is the farthest from existing samples in the set. (6) Similarly, the distances are calculated from the selected samples from stage 5 to the samples with a break in the current set. The sample with the maximum distance is selected. If there are two or more samples with equal distance, then one of them is selected arbitrarily. This sample is considered the best candidate for inclusion in the training set. (7) For the sample from stage 6, the calculation is performed by the direct solver (potentially slow and resource-intensive operation) and the actual (rather than predicted) answer is determined. This sample is added to set #3. (8) Stages 2–7 are repeated until set #3 has a required number of samples (from 100 to 500 in this work). Further, we will evaluate the quality of the prediction if one uses this set #3 for training.

17.4 Computational Results This section includes a discussion of the obtained results of direct calculations (Sect. 17.4.1) and the results of ML models (Sect. 17.4.2).

17.4.1 Results of Direct Calculations For a general assessment of data, Fig. 17.1 visualizes the correlations of the parameters with the result of the calculation (whether or not there is a break in the thread). Figure 17.1 uses the samples from set #1. This figure shows the results for one fixed load, the values of load parameters are: pressure time is 33.33 ms, pressure radius is 1.67 cm, and pressure amplitude is 66.67 MPa. All samples for a given load are selected, pairwise diagrams of the influence of the parameters of the thread on the result are presented. The values of all parameters are linearly scaled in the range from 0 to 10 for the convenience of visualization. On the off-diagonal graphs, the axes show the values of the corresponding pair of parameters, and the isolines show the percentage levels—the proportion of outcomes

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Fig. 17.1 Heatmaps of parameters’ correlations with the fiber breakage probability. The parameter numbers in this diagram are the following: 0 means thread length, 1 means thread diameter, 2 means Young modulus, 3 means material density, 4 means strain limit

without breaking the thread for given values of the two selected parameters and the variation of the other three parameters. Thus, for example, graph 4–2 shows the correlations for the strain limit of the thread (parameter 4) and Young’s modulus (parameter 2). It can be seen from the isolines that the outcomes without a break in the thread are localized in the corner of the diagram, in which the threads with high values of the strain limit and Young’s modulus are located. It is important to note that there are samples without the break on the entire plane of the graph 4–2, but they are not displayed in the lower-left corner, since there their share is small. For some pairs, the influence of both parameters is visible, for some, one parameter from the pair clearly prevails, for the pair 0–3 (length and density), a weak influence of both parameters is visible.

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Diagonal graphs show the proportion of outcomes without breaking the thread when changing one corresponding parameter. If the parameter “strengthens the thread” increases, then the graph increases. Oscillations on the graphs are caused by the graphic library used and do not carry a semantic load. Thus, for instance, in graph 1–1, it is clear that with an increase in the diameter of the thread, the proportion of outcomes without a break increases rapidly. It should be noted that these diagrams are valid within the selected ranges of parameters. Dependencies can change if one selects different ranges.

17.4.2 Results of ML Models For a typical training using a large amount of samples, data set #1 was used. This set is balanced. A training set with 100 to 1 million samples was randomly selected from all the samples in set #1. The samples of the set #1 remained the test set. The algorithms XGBoost, logistic regression, k-neighbors, and linear SVC were used. The training does not cause any problems when using large train sets. For example, XGBoost reaches 97% accuracy with 10,000 samples in the training set. Next, a variation of the same problem was considered with an unbalanced set. To do this, all samples with zero values of three parameters related to the pressure were removed from the set #1. All these samples refer to the case when the break of the thread does not occur. Thus, an unbalanced set #1.2 with 72.9 million samples was formed, in which 87% of the statements lead to a break in the thread against 13% without a break. The accuracy metric poorly estimates the quality of training for this set. Therefore, ROC AUC and PR AUC metrics were used. Figures 17.2, 17.3 show the graphs of ROC AUC and PR AUC depending on the size of the training set. Each figure shows the results for the algorithms listed above. It can be seen that with a sufficiently large training set (again, about 10,000 samples), all models show good results. The values obtained with the training set with 1 million samples can be used as a baseline, which results are achievable for the surrogate model in this problem. To assess what results are achievable using a small training set we used set #3, formed according to the algorithm described above. The size of this training set varied from 100 to 500 samples. For testing, we used set #1.2, which contains the parameter values used for training, and set #2, which contains a completely different grid for parameters. Nine machine learning algorithms were used: XGBoost, logistic regression, SVC, linear SVC, k-neighbors, SGD, naive Bayes, random forest, multilayer perceptron. Figures 17.4, 17.5 and 17.6 show the results of testing using set #1.2, and Figs. 17.7, 17.8 and 17.9 provide the results of testing using set #2. The main indicator of the quality of the prediction in this problem is the F-score metric for the minor class, since other metrics may be over-optimistic based on the results of the prediction of the dominant class. All graphs show only those algorithms that showed the best values and whose results were stable when varying the size of the training set. It can be seen that the transition to set #2 for testing, which contains a different

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Fig. 17.2 ROC AUC dependency from train set size when using random samples for training and test set #1.2

Fig. 17.3 PR AUC dependency from train set size when using random samples for training and test set #1.2

grid for parameters, led to a noticeable decrease in quality metrics for all algorithms. It is also seen that set #3 gives significantly better results when compared with a training set chosen randomly—this is seen when comparing Fig. 17.2 with Fig. 17.4 and Fig. 17.3 with Fig. 17.5.

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Fig. 17.4 ROC AUC dependency from train set size when set #3 for training and test set #1.2

Fig. 17.5 PR AUC dependency from train set size when set #3 for training and test set #1.2

17.5 Conclusions It was demonstrated that ML models can provide a reasonable prediction quality for a dynamic physical problem even with a small training set. An algorithm for the formation of such a small training set is proposed and tested. F-score of above 0.8 is obtained when training on just 500 samples. These numbers are implicitly based

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Fig. 17.6 F-score dependency from train set size when set #3 for training and test set #1.2

Fig. 17.7 ROC AUC dependency from train set size when set #3 for training and test set #2

on the fact that the dependence of the physical problem calculation result on the input parameters is described by fairly smooth functions. The results are obtained using the problem of deformation and breakage of a thin thread under the action of a transverse load impulse as an example. The application of a similar approach to more complex physical problems is the topic of subsequent work because the quantitative characteristics for other problems can vary significantly.

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Fig. 17.8 PR AUC dependency from train set size when set #3 for training and test set #2

Fig. 17.9 F-score dependency from train set size when set #3 for training and test set #2

It should be noted that the algorithm for generating a small training set described in this chapter is not the only possible one. For other physical problems with different dependencies of the result on the parameters, other algorithms may be more effective. The discussion of possible other algorithms is beyond the scope of this chapter. Acknowledgements The work was supported by RFBR project 18-29-17027.

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References 1. Beklemysheva, K.A., Petrov, I.B.: Damage modeling in hybrid composites subject to low-speed impact. Mathematical Models and Computer Simulations 11, 469–478 (2019) 2. Kim, S.H., Boukouvala, F.: Machine learning-based surrogate modeling for data-driven optimization: a comparison of subset selection for regression techniques. Optimization Lett. 14, 989–1010 (2020) 3. Rakhmatulin, Kh.A., Demianov, Yu.A.: Strength under high transient loads. Daniel Davey, New York (1966) 4. Vasyukov, A.V., Elovenkova, M.A., Petrov, I.B.: Modeling of thin fiber deformation and destruction under dynamic load. Matem. Mod. 32(5), 95–102 (2020) 5. Chen, T., Guestrin, C.: XGBoost: A scalable tree boosting system. The 22nd ACM SIGKDD Int. Conf. Knowledge Discovery and Data Mining, pp. 785–794. San Francisco, California, USA (2016). 6. Yu, H.F., Huang, F.L., Lin, C.J.: Dual coordinate descent methods for logistic regression and maximum entropy models. Machine Learning 85, 41–75 (2011) 7. Platt, J.: Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods. Advances in Large Margin Classifiers, pp. 61–74 (1999) 8. Bottou, L.: Large-scale machine learning with stochastic gradient descent. The 19th Int. Conf. Computational Statistics, pp. 177–187. Paris, France (2010) 9. McCallum, A., Nigam, K.: A comparison of event models for naive Bayes text classification. AAAI/ICML-98 Workshop on Learning for Text Categorization, pp. 41–48 (1998) 10. Breiman, L.: Random forests. Machine Learn. 45, 5–32 (2001) 11. Rumelhart, D.E., Hinton, G.E., Williams, R.J.: Learning internal representations by error propagation. In: Rumelhart, D.E., McClelland, J.L. PDP Research Group (eds) Parallel distributed processing: Explorations in the microstructure of cognition, vol. 1: Foundations, pp. 318–362. MIT Press (1986) 12. Altman, N.S.: An introduction to kernel and nearest-neighbor nonparametric regression. Am. Stat. 46(3), 175–185 (1992)

Chapter 18

Numerical Solution of Inverse Problems of Wave Dynamics in Heterogeneous Media with Convolutional Neural Networks Andrey S. Stankevich, Igor B. Petrov, and Alexey V. Vasyukov Abstract The chapter considers the problem of restoring the shape of the boundary between two elastic mediums with different rheological properties. The problem has many variations. This chapter concentrates on the application to the medical ultrasound aberrations. The problem statement includes two acoustically contrast materials: the curved layer of acoustically rigid material on the surface and the main volume of acoustically soft material. The direct problem of the propagation of the ultrasonic pulse in the medium is solved numerically using the discontinuous Galerkin method. The inverse problem is solved using neural networks. The proposed method is based on the assumption that the continuous observations of wave amplitude and velocity on the surface of the medium could also be treated as images, and the initial problem can be reduced to the optimization problem. The neural networks are trained using the data from multiple numerical solutions. During the testing phase, the networks restored the shape of the boundary using the data from the surface. Three different neural network architectures were tested and compared.

18.1 Introduction The chapter considers the problem of determining the boundary between two materials with different elastic properties using the data of non-destructive scanning. This problem has many applications and variations. In seismic exploration, it applies to studying the structure of geological layers and the identification of underground A. S. Stankevich (B) · I. B. Petrov · A. V. Vasyukov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] I. B. Petrov e-mail: [email protected] A. V. Vasyukov e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_18

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cracks and reservoirs based on seismic data from the surface. In medical technology, a similar problem arises when interpreting data from an ultrasound scan of a human body. Depending on the subject area, different approaches to this problem can be used due to different features of the medium being studied, limitations of the equipment used, and requirements for the resulting solution. The present work concentrates on the problem related to transcranial ultrasound studies. Despite all the development of modern medical equipment, research of this type is still difficult and requires sophisticated equipment. The problem is caused by the fact that the layer of the bone tissue of the skull acts as an aberrator, distorts, and noises the image constructed from the signal reflected from the internal structures of the brain [1]. The shape of the skull is quite complex, its external and internal surfaces are not flat. Because of this, the correction of aberrations turns into a difficult task. This chapter considers the problem of determining the location and shape of the boundary between two acoustically contrast materials a layer of acoustically rigid material on the surface (models the wall of the skull) and the main volume of acoustically soft material (models inner soft tissues). The scanning pulse and the recorded response in this chapter are close to the ones used in a typical modern ultrasonic device with a linear phased array. A direct calculation of the propagation of the entire spectrum of elastic waves in the medium is used. This approach is not the only possible one, for example, the need to take into account the full spectrum of waves when calculating medical ultrasound is the subject of separate studies [2]. Many works are successfully limited to the acoustic model and ray tracing methods. Nevertheless, in this work, a complete model of elasticity is used since this will potentially make it possible to further generalize the results to other mediums and problems. Machine learning methods are used for the inverse problem of determining the boundary. Nowadays, machine learning algorithms and deep neural networks in particular are used widely for the numerical solution of different problems that are incorrectly posed from a mathematical point of view. A significant advantage of neural network algorithms is the ability to work with data without preprocessing, which allows them to be successfully used in the fields of modeling natural languages and computer vision. A similar approach can be successfully applied for the numerical solution of seismic exploration tasks. Thus, for example, in [3] a technique is proposed for solving the problem of determining the spatial location of a fracture in a geological environment using a convolution network. Neural network algorithms are also successfully used to detect defects in materials using ultrasound testing data as input [4, 5]. Neural network training requires a large amount of data, which was obtained as a result of mathematical modeling of the response from aberrations of various forms. The neural network trained on this data allows one to solve the target inverse problem and restore the boundary from the response registered on the surface. The chapter is organized as follows. In Sect. 18.2, the geometry and major problem parameters are presented. Section 18.3 describes the neural network used in this work. The numerical results containing the samples for both direct and inverse problems

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are presented in Sect. 18.4. Section 18.5 concludes the chapter and gives some final remarks.

18.2 Problem Statement In this research, a two-dimensional problem is considered. It is required to determine the shape of the interface between two media with different physical parameters. A computational domain is a square with sizes 10 × 10 cm. In the upper part of the domain, there is a 5 cm long aberrator plate, the boundary of which is 3–6 sine wave modes. The geometry of the computational domain is presented in Fig. 18.1. A linear elastic medium model is used for direct numerical simulation since it was proofed for ultrasound imaging using a direct comparison of the modeling with experimental data [6]. The phased array was calculated explicitly following [7]: at the upper boundary of the computational domain, there are 50 equidistant sensors recording the vertical components of the medium velocity, and an ultrasound pulse in the form of a Gaussian is excited in the middle of the plate. Hereinafter, Sect. 18.2.1 discusses the system of equation for the direct problem and gives brief comments on the numerical method used for the direct problem. Section 18.2.2 provides a description of the inverse problem statement. Fig. 18.1 Geometry of the computational domain

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18.2.1 Direct Problem The equations for a linear elastic isotropic medium at constant temperature include the equations of motion, the generalized Hooke’s law, and the equations of state. They have the following form [8]: 2 i

ρ ddtu2 = ∇ j σ i j + f i ,   σ i j = λI1 εˆ g i j + 2μεi j , εi j = 21 ∇i u j + ∇ j u i . In these equations, ui is the displacement vector component, ρ is the medium density, σ ij is the stress tensor components, εij is the strain tensor components, gij is the metric tensor, I 1 (ε) = gij εij . The rheology of the medium is described by the Lame coefficients λ and μ. The equations under consideration must be supplemented with initial and boundary conditions. At the upper boundary of the computational domain, the condition of the free boundary was specified. For all other boundaries, the absorbing boundary condition was set. The numerical solution of the problem under consideration is found using approximation on an irregular triangular grid by the discontinuous Galerkin method. The family of the Galerkin methods is based on the following concepts: • In each finite element, the exact solution is approximated with a finite linear combination of basis functions. • The residual obtained by approximation is considered to be orthogonal to the basis functions in the sense of L 2 scalar product. The variation of the method chosen in this study allows the sought function to undergo discontinuities at the boundaries of finite elements. For numerical integration, it uses the solution of a one-dimensional Riemann problem. A detailed description of the discontinuous Galerkin method and its convergence analysis can be found in [9], and features of the implementation used in this work are provided in [10]. This method was previously successfully used for modeling biomedical ultrasound [11].

18.2.2 Inverse Problem Let the computational domain G = [0, L] × [0, L] have two different subdomains G1 and G2 . The geometry of the medium under consideration is defined using a regular rectangular grid ωi j =



  h i , h j , i, j ∈ 0, N , N h = L .

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As thetarget variable, one chooses the matrix. 1, if ωi j ∈ G 1 yi j = . 0, else Thus, the problem under consideration can be reduced to the problem of simultaneous binary classification of each node of the regular grid according to the data from the sensors. There is a data set (X, Y), consisting of N pairs (X, Y), where Y is the matrix described above of size 128 × 128, and this matrix describes the geometry of the medium, X is a seismogram represented by a matrix of size 50 × 100, obtained by direct numerical simulation. Neural network can be described as a parametric family of functions.  F(θ ) = f : X × θ → Y; f ∈ C 1 {X × θ} . Among these functions, one needs to find a function that delivers a minimum of error functional L f = arg min L( f (X, θ ), Y) θ

In this chapter, a combination of weighted binary cross-entropy and generalized Dice overlap [12] is used as the error functional: L(X, Y) = BC E(X, Y, W) + G D O(X, Y, W), N 128      i  i 1 i i BC E(X, Y, W) = N1 −y , ω log x − 1 − y log 1 − x jk jk jk jk jk 128×128 i=1

G D O(X, Y, W) = 1 −

j.k=1 128 i i i j,k=1 2x jk y jk w jk 1 . 128  i N i i x w +y j,k=1 jk jk jk i=1 N 

Here, similar to [13] W was introduced as the matrix of weights for each pixel in the image calculated bythe formula:   128×128 i i i y W jk =1+  + J ω ∈ δG 128 i j 1 , jk yi j.k=1

jk

where J(x) is the indicator function, δG1 is the boundary of the projection G1 onto the introduced uniform grid. Its purpose is to balance the number of pixels in the image according to their type and make the neural network focus on the border of the area. The metric of the quality of the solution was the Jaccard Similarity Index, which is widely used in semantic segmentation problems.

18.3 Neural Network Architecture Section 18.3.1 provides a short discussion about the convolutional neural networks, autoencoders, and self-attention mechanism used in this work. Section 18.3.2 describes the resulting architecture of the network.

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18.3.1 Convolutional Neural Networks Convolutional neural networks [14] is a special neural network architecture proposed by Yann LeCun in 1988 and designed to work with images. The basis of the architecture is the convolutional layers exploiting the ideas of spatial correlation between image pixels and shared parameterization. Such neural networks are widely used in a variety of computer vision tasks. Autoencoder is a special type of neural network whose purpose is to translate data into a new, more informative presentation [15]. Autoencoders are trained to restore their own input. The usage of autoencoders allows reducing effectively the dimension of raw data. The architecture consists of two independent blocks, trained together: the encoder translates the data into a new representation, and the decoder tries to restore the original data from the output of the encoder. Originally introduced for neural machine translation tasks, the self-attention mechanism [16] is a way to discover the informative representation of the sequencelike data by relating different positions inside the sequence. Neural networks based on the attention mechanism show state-of-the-art performance in the problems of natural language processing and time series analysis.

18.3.2 Resulting Architecture To solve this problem, two different types of architectures are proposed. The first one is based on SegNet architecture for semantic segmentation [17], which can be trained end-to-end with a backpropagation algorithm. The diagram of this architecture is presented in Fig. 18.2. Training pipeline for the second architecture consists of two steps. (1) On an independently generated set of (Y, Y ) target variables, an autoencoder is trained. (2) On the set (X, Y ), a new neural network that consists of a parser and decoder obtained in the previous step, the parameters of which are fixed, and are trained. The design of a parser may vary depending on the representation of a seismogram. For two-dimensional matrices, we offer two possible approaches: LeNet-like parser proposed in [3] and parser based on the simplified self-attention mechanism. Autoencoder for this architecture is also based on SegNet containing a bottleneck layer with a single channel between encoder and decoder blocks. The diagram of this architecture is presented in Fig. 18.3. All models are implemented using PyTorch framework. Models were trained on the single Tesla K80 GPU with the Adam optimization algorithm.

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Fig. 18.2 The diagram of SegNet autoencoder. The bottleneck-in (-out) layer is a convolutional layer with the number of channels at the input and output (N, 1) ((1, N)), respectively

Fig. 18.3 The diagram of the network with LeNet parser is used to predict the target variable

18.4 Numerical Results Figure 18.4 shows an example of the direct problem solution: sequential wave patterns formed by a short ultrasonic pulse in the center of the upper boundary of the calculation domain. For numerical simulation, the basis functions were the Legendre polynomials of the fourth-order. The resulting finite-difference scheme was integrated in

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Fig. 18.4 Wave propagation in the medium: a initial stage, step 1, b initial stage, step 2, c initial stage, step 3, d middle stage, step 4, e middle stage, step 5, f middle stage, step 6, g stage of active multiple reflections, step 7, h stage of active multiple reflections, step 8, i stage of active multiple reflections, step 9

time with fourth-order Runge–Kutta method. The calculations were carried out on two machines, each with a six-core Intel Core i7-5820 K CPU and 64 GB of RAM. Considering the approximation order of the numerical method, the estimated time for the solution of a single problem on the triangular grid of about 2000 finite elements with the hardware given is around 2 min. Figure 18.4a–c corresponds to the initial stage of the process. One can see the primary formation of a pulse emitted by one element of a phased array. It can be seen that the wavefront is not spherical. This is due to the fact that a complete system of elasticity equations is used. In this case, the longitudinal P-wave, pair of shear S-waves, and set of surface waves are formed in the medium after the short pulse on the surface. These waves propagate at their speeds, which are not equal to each other. Figure 18.4d–f corresponds to the stage of the process, at which the formed packet propagates inside an acoustically rigid material and begins to interact with the boundary between the materials. The structure of waves is most clearly visible from these images. Figure 18.4g–i corresponds to the stage of active multiple reflections of waves from the boundary of the computational domain and boundary between the materials. The overall wave picture is already difficult to analyze. It should be noted that shear waves decay relatively slowly in bone tissue. Therefore, it should be expected that at this stage of the process the pattern formed by elastic waves is more correct than the pattern that could be obtained using a simpler model of acoustics. The neural network will solve the inverse problem and restore the shape of the boundary based on the information recorded by the sensors on the

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Fig. 18.5 Scaled phased array data

surface at this initial stage of signal propagation. This fact additionally illustrates why the system of elasticity equations is used in this work. The data recorded by the phased array elements on the surface is presented in Fig. 18.5. This representation is typical for many areas. Seismograms in geology and B-scans in ultrasound testing use this format with only minor differences. The neural network uses these data as input. The training and validation datasets were obtained by running the numerical simulation for 5000 computational domains with varied thicknesses of an abberator, the number of sine wave modes between the media, and the amplitude of the sinusoid. With that said, creating a dataset takes about 160 h. All models described before were implemented using PyTorch framework for deep learning. Models were trained for approximately 2 h on the single Tesla K80 GPU with the Adam optimization algorithm. For training, we used an exponentially decaying learning rate. The testing was performed using the data that was not previously seen by the model. Figures 18.6, 18.7 and 18.8 show the results. New sample geometry was created, and the ultrasound response from this boundary was calculated using the same direct problem solver. The response from the phased array was passed to the neural network that predicted the shape of the boundary. Figures 18.6, 18.7 and 18.8 show the real geometry and prediction by the different types of networks. Table 18.1 summarizes the metrics for different architectures used in this work.

Fig. 18.6 Predictions of neural networks for a smooth aberrator with few waves: a original geometry, b SegNet, c LeNet parser + decoder, d attention parser + decoder

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Fig. 18.7 Predictions for a relatively sharp aberrator with large wave amplitude: a original geometry, b SegNet, c LeNet parser + decoder, d attention parser + decoder

Fig. 18.8 Predictions for a relatively sharp aberrator with small wave amplitude: a original geometry, b SegNet, c LeNet parser + decoder, d attention parser + decoder

18.5 Conclusions The results show the potential of using convolutional neural networks to reduce aberrations and improve the quality of ultrasound medical imaging. It should be noted that in this work a rather large number of restrictions and assumptions were used. However, the current results for the synthetic problem show that the chosen approach is reasonable. It will be necessary to verify the stability of the results with respect to the variations of the problem statement. This is a part of future work. The major areas are the following: recheck the results for significantly different forms of the aberrator, study the dependence on elastic properties of materials, and evaluate if the noise of the input signal influences the results. The possibility of transferring the result to engineering practice requires additional research. Typical ultrasound equipment does not have the ability to emit a pulse using a single element of the phased array. There are limitations to the sampling frequency. The signal from the near zone that was used in this work is often interpreted as noise and filtered out. Nevertheless, the next logical step of this work may be validation according to the data of laboratory experiments.

18 Numerical Solution of Inverse Problems of Wave Dynamics in Heterogeneous … Table 18.1 Comparison of performance of different neural networks architectures on test dataset

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Model

Number of parameters

Jaccard index

Jaccard index for border pixels

SegNet

0.7 × 106

0.94

0.87

LeNet parser + decoder

17 × 106

0.94

0.88

Attention parser + decoder

3.4 × 106

0.97

0.91

Acknowledgements The work was supported by RFBR project 18-29-02127. The authors are grateful to Dr. Beklemysheva for important and valuable remarks to the problem statement.

References 1. Beklemysheva, K.A., Grigoriev, G.K., Kulberg, N.S., Petrov, I.B., Vasyukov, A.V., Vassilevski, Y.V.: Numerical simulation of aberrated medical ultrasound signals. Russian J. Numerical Anal. Math. Model. 33(5), 277–288 (2018) 2. Jiang, C., Li, D., Xu, F., Li, Y., Liu, C., Ta, D.: Numerical evaluation of the influence of skull heterogeneity on transcranial ultrasonic focusing. Frontiers Neurosci. 14, artc. no. 317 (2020) 3. Muratov, M.V., Biryukov, V.A., Petrov, I.B.: The solution of fractures detection problem by methods of machine learning. Doklady Math. 491(1), 107–110 (2020) 4. Tripathi, G., Anowarul, H., Agarwal, K., Prasad, D.K.: Classification of micro-damage in piezoelectric ceramics using machine learning of ultrasound signals. Sensors 19(19), 4216.1– 4216.13 (2019) 5. Ye, J., Ito, S., Toyama, N.: Computerized ultrasonic imaging inspection: From shallow to deep learning. Sensors 18(11), 3820.1–3820.17 (2018) 6. Vassilevski, Y.V., Beklemysheva, K.A., Grigoriev, G.K., Kulberg, N.S., Petrov, I.B., Vasyukov, A.V.: Numerical modelling of medical ultrasound: Phantom-based verification. Russian J. Numerical Anal. Math. Model. 32(5), 339–346 (2017) 7. Beklemysheva, K.A., Vasyukov, A.V., Ermakov, A.S., Favorskaya, A.V.: Numerical modeling of ultrasound beam forming in elastic medium. Procedia Comput. Sci. 112, 1488–1496 (2017) 8. Novatsky, V.: Theory of elasticity. Mir, Moscow (in Russian) (1975) 9. Hong, Q., Wang, F., Wu, S., Xu, J.: A unified study of continuous and discontinuous Galerkin methods. Sci. China Math. 62, 1–32 (2019) 10. Miryaha, V.A., Sannikov, A.V., Petrov, I.B.: Discontinuous Galerkin method for numerical simulation of dynamic processes in solids. Math. Models Comput. Simul. 7, 446–455 (2015) 11. Beklemysheva, K.A., Biryukov, V.A., Kazakov, A.O.: Numerical methods for modeling focused ultrasound in biomedical problems. Procedia Comput. Sci. 156, 79–86 (2019) 12. Sudre, C.H., Li, W., Vercauteren, T., Ourselin, S., Cardoso, M.J.: Generalised Dice overlap as a deep learning loss function for highly unbalanced segmentations. In: Cardoso, M.J., Arbel, T., Carneiro, G., Syeda-Mahmood, T., Tavares, J.M.R.S., Moradi, M., Bradley, A., Greenspan, H., Papa, J.P., Madabhushi, A., Nascimento, J.C., Cardoso, J.S., Belagiannis, V., Lu, Z. (eds.) Deep learning in medical image analysis and multimodal learning for clinical decision support. LNCS, vol. 10553, pp. 240–248. Springer, Cham (2017)

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13. Ronneberger, O., Fischer, P., Brox, T.: U-Net: Convolutional networks for biomedical image segmentation. In: Navab, N., Hornegger, J., Wells, W., Frangi, A. (eds.) Medical image computing and computer-assisted intervention, LNCS, vol. 9351, pp. 234–241. Springer, Cham (2015) 14. Li, Z., Yang, W., Peng, S., Liu, F.: A survey of convolutional neural networks: Analysis, applications, and prospects. CoRR ArXiv Preprint, arXiv, abs/2004.02806 (2020) 15. Bank, D., Koenigstein, N., Giryes, R.: Autoencoders. CoRR ArXiv Preprint, arXiv, abs/2003.05991 (2020) 16. Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., Polosukhin, I.: Attention is All You Need. CoRR ArXiv Preprint, arXiv, abs/1706.03762 (2017) 17. Badrinarayanan, V., Kendall, A., Cipolla, R.: SegNet: A deep convolutional encoder-decoder architecture for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 39(12), 2481– 2495 (2017)

Part IV

Computer Science

Chapter 19

A Systematic Approach to Present “Files and File Systems” in Theoretical Courses of Computer Science Vladimir E. Karpov

Abstract The chapter considers a systematic approach to present “Files and file systems” in theoretical courses. The approach is based on different levels of abstraction for the concept introduced. We consider abstract files as mathematical objects, logical files as an implementation of abstract files in programming, and physical files as a mapping of logical files to the address space of storage media. Different levels of abstraction for file systems are introduced in the same way: logical file systems as collections of logical files, physical file systems as a mapping of logical file systems to storage media, and file systems as software tools that serve physical file systems. The abstraction hierarchy allows one to naturally introduce a hierarchy of operations on files and file systems and explain the functions performed by the file subsystems of operating systems. Logical structuring of the topic material facilitates the listeners’ perception of the topic, distinguishing essential general ideas from a set of secondary details.

19.1 Introduction The topic of “Files and file systems” for many lecturers is nontrivial to explain to the students of non-core specialties. The abundance of heterogeneous facts, a large amount of material, and discrepancies in various textbooks and monographs did not allow us to build a logically coherent presentation of the topic. For example, in books [1–3], there is no agreement about the set of basic operations on files and the types of files that may occur in operating systems. The definitions of the term “file” vary almost everywhere (e.g. [1–5]). This is probably due to the fact that the concept of file itself can be considered at different levels of abstraction. One can talk about a file as a set of data, where the primary focus is on the contents of this set. Or one can talk about the file as a certain set of blocks on the disk, where the main question is V. E. Karpov (B) Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per, Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_19

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what blocks the file occupies. It should also be taken into account that by the time of lectures, most students have already formed their own (unfortunately, not always correct) idea of what a file is and what a file system is. A clear distinction between the terms at different levels of abstraction allows to present the course material logically. For example, it is not possible to explain the memory management mechanism in modern computing systems if concepts of physical address space (the set of all available RAM addresses) and logical address space (the set of all addresses that the processor can work with) are not introduced [6]. The rest of the chapter is structured as follows. In Sect. 19.2, we provide an example of integer data explaining different levels of abstraction depending on its representation. Section 19.3 describes the various levels of abstraction for files and Sect. 19.4 describes them for file systems. Section 19.5 concludes the chapter.

19.2 Different Levels of Abstraction for Integers The abstraction levels given above exist for most data types used in programming. Let’s have a look at an example of integers to understand these levels more accurately. Section 19.2.1 gives an idea of integer data as mathematical objects. Section 19.2.2 discusses logical integer data. Section 19.2.3 is dedicated to the physical integer data description.

19.2.1 Integers as Mathematical Objects We can use integers in ordinary practice, not related to computing systems at all. With this approach, we often forget that these abstract integers are objects studied in theoretical mathematics. For us, it does not matter that they are introduced as the closure of a set of natural numbers, and even further, it does not matter how the concept of a natural number was introduced—through Peano’s axioms or through the Bourbaki approach. It is essential for us that: (1) The set of valid integers is in the range between −∞ to +∞. (2) This set is a countable set. (3) A certain set of basic operations on integers can be performed: addition, subtraction, multiplication, and division (integer division or division with a remainder). This level of abstraction is sufficient for use in everyday life or when setting any discrete mathematical modeling problems.

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19.2.2 Integer Data as Abstract Objects in Programming—Logical Integer Data In computer science, we can consider integer data as objects in high-level programming languages that are not tied to the architectures of specific computing systems. Now, integer data is understood to be a sequence of bits of a limited size interpreted as some record of a mathematical integer in a binary number system. What changes during the transition from theoretical to computational mathematics? The answer to this question is the following: (1) Limiting the size of a bit sequence interpreted as an integer reduces the set of acceptable values for logical integer data. In computer mathematics, signed integer data can take values from −2n to 2n –1, where n is the number of bits allocated to represent an integer. (2) The set of valid values for logical integer data is a finite set. (3) The basic operations on integer data remain the same: addition, subtraction, multiplication, and division (integer division or division with a remainder). However, when performing operations one must consider the outcome to be possibly out of the range of acceptable values. (4) Since the logical integer given is nothing but a sequence of bits, new bitwise operations are possible on the integer data: bitwise “and”, bitwise “or”, bitwise shifts, etc. And in some programming languages, it is even possible to treat integer data as logical values and, therefore, they can participate in logical expressions. Some bitwise operations can be used instead of mathematical operations (e.g. to gain better performance). Changing the set of operations on data when changing the abstraction level is quite normal. This level of abstraction is sufficient for writing mathematical modeling programs in high-level languages.

19.2.3 Integer Data as Physical Data in the Memory of a Computing System Finally, we can study the representation of integer data in the physical memory of a particular computing system and the operations on such physical data. What is new at this level of abstraction? The answer to this question is the following: (1) The interpretation of a bit sequence in memory as a value of a mathematical integer refers precisely to this level of abstraction. The highest bit in the record is considered to be a sign bit. The value of bit 0 corresponds to a positive number, and the value of bit 1 corresponds to a negative number. Negative numbers are usually represented in two’s complement. (2) When the result of an operation is out of the range of possible values, the higher bits of the result are often simply discarded.

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(3) The number of bits of a logical integer representation in memory is usually a multiple of the number of bits in a byte. (4) Each integer that occupies more than one byte is located in RAM in a continuous sequence of bytes, starting with a specific physical address. For many architectures, the start address of a multibyte integer record must be a multiple of a specific value commonly equal to the number of bytes in the integer. This value is called alignment. (5) The architecture of the computer determines the representation of multibyte integer data in memory. In some architectures, the highest byte is located at the lowest address (big ending order), and in others the highest byte is situated at the highest address (little ending order). Understanding this level of abstraction is useful for low-level programming and transferring numerical data between computers of different architectures.

19.3 Different Levels of Abstraction for Files The presence of different abstraction levels for numerical data has been historically determined. By the time electronic computers appeared, theoretical foundations had already been developed for working with integers, real and complex numbers. These foundations constituted the mathematical level of abstraction. Representation of numbers in the memory of computing systems and their processing during low-level programming led to the appearance of a physical level of data abstraction. The need for portability of written programs and the emergence of high-level programming languages led to the emergence of a logical level of data abstraction. For files, things were different. Before the advent of computers, the concept of a file as a certain set of information, on which certain operations can be performed, was almost completely absent. The concept of the “file” came to computer science from the archival (library) service, and the first objects in electronic computing systems called “files” were not information sets, but devices or storage media where such sets could be stored [7]. Later files started to be referred to as named datasets stored on some external storage medium. In fact, these are physical files, the physical level of abstraction. Initially, access to information stored in physical files required working with them at a low-level. This knowledge could change as the user moved from one computer system to another. Therefore, with the advent of high-level programming languages, a new level of abstraction appeared called the logical files or files as objects in programming languages. There is often no clear distinction between the logical and physical files in computer science courses. One of the few people who try to separate the logical and physical levels of abstraction is Niklaus Wirth [8]. Mathematical files are usually not considered at all. Perhaps the consideration of all three levels of abstraction for files is very thread that we can pull to build logical, harmonious teaching of “Files and File Systems”.

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Section 19.3.1 discusses files as mathematical objects. Section 19.3.2 is devoted to the description of the logical files. Section 19.3.3 gives an idea of files at the physical level of abstraction.

19.3.1 File as a Mathematical Object It is quite natural to consider the concept of a file as a mathematical object without any connection with computer systems. A file as a mathematical object is an ordered set of information in a digital (for example, bitwise) representation, which has the beginning and the end, provided with an additional attribute—a pointer to the current position. The current position pointer can be set at the beginning of the file, at the end of the file, or between elements in the set. One can perform several basic operations on this mathematical object as follows: (1) Read operation. There is no input data for this operation. If the current position pointer is not set at the end of the file, the result of the read operation is a set of information containing the bit directly following the current position pointer in the file. The pointer itself will move one bit towards the end of the file. If the current position pointer is at the end of the file, the result is an empty set, and the position of the pointer does not change. (2) Write operation. The input data for this operation is a data set containing a single bit. As a result of this operation, the file content changes. If the current position pointer is not set at the end of the file, the bit immediately following the pointer is replaced with a bit from the input data set. If the current position pointer is located at the end of the file, the bit from the input set is added to the end of the file. The current position pointer is always set after the recorded bit. (3) Truncate operation. The operation has no input data. As a result of this operation, the last bit is deleted from the file. If the current position pointer was not located at the end of the file, its position does not change. If the current position pointer was at the end of the file, it will be set to the end of the modified file after the operation. (4) In addition to these operations associated with obtaining data from a file or modifying the contents of a file, there is an operation that allows moving the pointer to a new current position. Depending on the method of changing the current position pointer, two types of abstract mathematical files are considered: sequential access files and direct access files. For sequential access files, the operation to change the current position pointer (rewind) has no input data. As a result, the pointer moves to the beginning of the file. For direct access files, the operation to change the current position pointer (seek) allows setting it to any valid position, the exact value of which is determined by the input data for this operation. There are no other file operations at this level of abstraction. However, this does not end the consideration of files as abstract mathematical objects.

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In mathematical theory, after introducing the integer data concept, we can study different subsets of integers (even, odd, prime, etc.). Within such subsets, the basic operations on integers can change their meaning. For example, if we work only with odd integers, the meaning of the addition operation changes. The sum of 3 + 7 cannot equal 10, it must be either 9 or 11. Some new operations may appear. For example, for a set of primes, take the next largest prime number. Similarly, after defining abstract direct and sequential access files, we can consider some subsets of them, in which the files have some internal structure. For example, we can assume that a set of information in a file is structured as records. Each record is something whole. We can only read the entire record, not some parts of it. Everything that we write to a file with a single write operation (not necessarily just one bit, but maybe more) will be a record. The current position pointer can no longer be located anywhere in the file. It can only point to the beginning of the file, the end of the file, or be set on the border between records. A set of basic operations for a sequential access file with records can look like this: read record, add record, delete record, and rewind. Any introduction of a structure in a file at the level of mathematical abstraction leads to a change in the meaning of basic operations or to a change in their set.

19.3.2 Logical Files—Files in Programming When we start talking about files in relation to computing systems as the objects used in programming, we move to a new level of abstraction. What are the features of logical files in modern computers? The answer to this question is the following: (1) A logical file must be located somewhere in the memory of the computing system. Most often, logical files are located in secondary memory, but sometimes in RAM (for example, some temporary files). How such placement is made is a question that is solved at the physical level of abstraction. (2) Logical files usually do not cease to exist after the completion of the process that used the file or after the shutdown of the operating system. (3) Placing a file on a storage medium having a limited size leads to restrictions on the size of the logical file. (4) For mathematical files, the minimum unit of information for read and write operations is a bit. For logical files, such a minimum unit of information is a byte. We can assume that unstructured logical files are mathematical files containing byte records. (5) Modern computing systems as a rule are multi-program and multi-user. Therefore, a logical file can potentially be used by different processes of different users, sometimes simultaneously. These features of logical files cause them to have new attributes and change the set of operations on files:

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(1) Placing a logical file in the secondary or the main memory is associated with the allocation of computing system resources. These resources should naturally be released when the file becomes unnecessary. The create file and delete file operations appear to allocate and release resources. (2) The lifetime of a logical file can exceed the lifetime of individual processes that use the file and the operating system. Therefore, a logical file can have time attributes: the time when the file was created, the time when the file was last modified, and so on. (3) A multi-user computer operation requires protection of information stored in a logical file from unauthorized access. This means that a logical file can have attributes that contain the ID of the file owner and some restrictions on access to it by various users. (4) The presence of a large number of attributes in a logical file leads to new operations on the file that allows finding out and setting attribute values—getting attributes and setting attributes. (5) If multiple processes can work with a logical file at the same time, operations are required to avoid conflict situations such as locking and unlocking the file.

19.3.3 Physical Files as Logical Files Mapped to the Address Space of a Computer Considering the mapping of a logical file to the address space of a computing system (usually the address space of an external storage medium) takes us to the next level of abstraction—to the level of physical files. A physical file is an object that exists in reality regardless of the operation of the operating system and processes. In modern computing systems, not a single file is usually stored on external media, but a certain set (or collection) of files. Therefore, the consideration of the organization of a physical file is naturally related to the consideration of the physical organization of a file collection storage, or in other words, to the consideration of a physical file system concept (Sect. 19.4.2). Moreover, a physical level for file essentially cannot be considered out of concept of a physical file system, while, of course, there are some properties that can be discussed separately: nonlinear storage of logically linear data, inability to allocate memory for particular file even if there is logically available memory, access speed, etc.

19.4 Different Levels of Abstraction for the File System For the concept of the file system in IT courses mixing of different levels of abstraction is often observed. It is similar to mixing different levels of abstraction for the concept of the file.

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It is one thing to talk about a file system as a way to organize a collection of logical files. Then we deal with a logical file system. A logical file system is a kind of analogous systematization of a book collection, a stamp collection, in general, a set of any objects of value to the collector. Another thing is when we talk about the file system as a logical file system mapping to the address space of secondary memory. Then we learn how files (i.e. the information contained in files), their attributes, and the structure of the collection are stored on the storage medium. In this case, we deal with a physical file system. Finally, it is entirely different when we refer to the file system as a part of the operating system that serves the physical file system. Here, it would be better to use the term “file management system”. This chapter is not a summary of a lecture on “Files and File systems” but is methodological in nature. Therefore, in the following sections, we will not go into substantive details of the topic, but only outline the dotted line of presentation. Section 19.4.1 discusses the concept of a logical file system. Section 19.4.2 explains the concept of a physical file system. Section 19.4.3 discusses the file system as a part of the computer’s operating system.

19.4.1 Logical File System When considering the logical level of abstraction for a file, it was tacitly assumed that we were dealt with a single file type object. If we have several logical files, then we deal with a collection of logical files. Any collection needs some organization. The easiest way to organize is to have no organization at all. Let’s keep all the logical files in a pile. But even in this pile, we must be able to distinguish files from each other, i.e., files must have names. The rules for building file names are already the beginning of an organization. No two files in our pile should have the same names. When adding a file to the collection, it should get a name that has not yet been encountered in the collection. To avoid possible conflicts, it is convenient to have a list of names of all files in the collection. It is also desirable that this list contains information about how to access the data stored in the file and its attributes. Since our collection of files can be stored on an external media for a long time, this list should also be stored on an external media along with the files. We will call this list a folder or directory. We are not interested in the internal structure of the list (directory) yet. What matters is that it consists of records. Each record corresponds to a logical file in the collection and contains instructions on how to access the file’s attributes and data. It turns out that our pile is actually a single-level directory. A new object appears in the logical file system that is different from the files in the collection. It can be considered as a service file. For the simplest way to organize a collection, we consider two types of files: the directory and all other logical files, i.e. regular files. The meaning of file creation and deletion is changing. The file creation operation involves not only allocating resources to place the file but also adding a new record to the directory. Deleting a file involves not only freeing up resources

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but also deleting the corresponding record from the directory. It should be noted that any external structuring of the file collection leads to new operations on files or changes in the meaning of existing operations. The same effect was observed for mathematical files with the appearance of the internal structure in them. It is easy to move from a single-level directory to a two-level directory and a directory tree. This transition assumes that the set of files directly located in a certain directory may include not only regular files but also other subdirectories. Moreover, part of the subdirectories may not contain any files; they may be empty. The complication of the logical file system structure leads to another new operation to create an empty directory (make directory). In the future, the word “file” will mean both regular files and directories if the difference in types does not matter. In a tree-based logical file system, not all files must have unique names. Uniqueness is only required for file names that directly belong to the same directory. Files in different directories may have the same names. To uniquely identify files, the concepts of full and relative file names are naturally introduced. Further complicating the structure of the logical file system leads us to an acyclic graph and a graph of arbitrary form. New operations appear: create a hard link and create a symbolic (soft) link. Once again, the meaning of the file deletion operation is changed. There is a third type of files, i.e. files that have symbolic links. These three file types: regular, directory, and symbolic links are sufficient to describe the complete structure of the logical file system. All other file types are not related to the organization of the logical file system. Their appearance is associated either with the functioning of other parts of the operating system or with the operation of any applications. The pinnacle of considering logical file systems is the ability to combine different file systems into a single whole and discuss the operations of mounting and unmounting file systems.

19.4.2 Physical File System A physical file system is a mapping of a logical file system to an external storage medium. This part of the topic is related to the study of the structure of the external media address space and methods for allocating blocks for placing files: continuous block allocation, linked list, FAT, direct indexing, multi-level indexing, index nodes method, etc. As a rule, all these objects are presented in a uniform manner in different IT courses. The presentation details are not related to the purpose of this chapter, and we will not dwell on them in detail.

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19.4.3 File System as Part of the Operating System A file system as part of an operating system, i.e., a file management system, is software that serves a physical file system. The file management system performs the following main functions: (1) Allocation of external memory between files. Accounting for the occupied and free space of external memory. (2) File identification. Associating a file name with its location on an external device. (3) Providing operations on files and logical file systems including the ability to share files. (4) Protection of information in files from unauthorized access. (5) Ensuring the reliability and fault tolerance of the physical file system. When discussing the implementation of operations, we can also talk about new trends in this area related to the multicore nature of modern computing systems and the wide use of threads. It should be emphasized that it is here that the file open and close operations (open and close) appear, which is necessary for the effective operation of the file management system. At this level of abstraction, in addition to a detailed analysis of the execution of these functions, it turns out to be appropriate to consider the concept of files displayed in memory. It is appropriate to consider the concept of memory-mapped files.

19.5 Conclusions This chapter attempts to offer a systematic approach to the presentation of the topic “Files and file systems” in theoretical courses in computer science. The main idea is a consistent presentation of the material based on the relationships between different levels of abstraction of the concepts “file” and “file system”. This approach has been applied by the author over the past two years when giving lectures to students of Moscow Institute of Physics and Technology and Higher School of Economics. The results of the control activities showed that the students began to learn much better the topic in question.

References 1. 2. 3. 4.

Silberschatz, A., Gagne, G., Galvin, P.B.: Operating system concepts. 10th edn. Wiley (2018) Stallings, W.: Operating systems: Internals and design principles. 8th edn. Pearson (2015) Tanenbaum, A.S., Bos, H.: Modern operating systems. 4th edn. Pearson (2015) Hailperin, M.: Operating systems and middleware: Supporting controlled interaction. 1.2.1 edn. Gustavus Adolphus College, St. Peter, Minnesota (2016) 5. Giampaolo, D.: Practical file system design with the be file system. Morgan Kaufmann Publishers Inc., San Francisco (1999)

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6. Karpov, V.E., Konkov, K.A.: The basics of operating systems. Lecture course. 3rd edn. Fizmatkniga, Moscow (in Russian) (2019) 7. Weik, M.H.: A third survey of domestic electronic digital computing systems. Ballistic Research Laboratories Report #1115, 314–331 (1961) 8. Wirth, N.: The programming language Pascal. Acta Informatica 1(1), 35–63 (1971)

Chapter 20

Designing Execution Models of Distributed System in Theoretical Courses on Information Technology Sergey L. Babichev

and Konstantin A. Konkov

Abstract This chapter is concerned to simulation of distributed systems. There is a large number of specialized software systems designed to build a simulation model of distributed systems and conduct a simulation experiment. The chapter discusses the original software tools for the automated design of simulation models illustrating the various conceptions of educational courses on distributed systems and distributed algorithms.

20.1 Introduction Technological limitations, such as speed of light and atomic size, hinder the increase in the clock frequency of modern processors and then slow down their evolution. Therefore, the increase in productivity of modern computing systems in recent years is largely due to the widespread introduction of parallel architectures with distributed memory, both local and distributed, where interacting processes are forced to using messaging passing by the communication subsystem. Effective messaging is of interest to shared memory systems since a messaging system can be modeled on a shared memory system, and vice versa, that is, two paradigms are equivalent. Despite the fact that the idea of the interaction of independent processes or computational threads using messages has been known for a long time (one example is the Inmos transputer systems of 1980), computer science still suffers from a shortage of algorithms that can be effectively parallelized without using shared memory. In modern developing languages, such as Rust and Go, it is the message mechanism, called channels in them, that is the preferred way of cross-threading interaction even for local programs. If for distributed systems messaging was always natural, then S. L. Babichev · K. A. Konkov (B) Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] S. L. Babichev e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. N. Favorskaya et al. (eds.), Smart Modelling for Engineering Systems, Smart Innovation, Systems and Technologies 215, https://doi.org/10.1007/978-981-33-4619-2_20

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modern local multi-threaded systems began to use the same principle to increase productivity: message passing. If there are appropriate algorithms, this method of interaction allows the developers to scale the performance of the system just by adding computing cores (for local systems) and nodes (for distributed). Thus, the shortage of high-performance scalable algorithms based on messaging can be overcome only if the developing and debugging tools for such algorithms exist. The simulation modeling discussed in this chapter is one of the effective ways to study the features of the communication algorithms. It is well known that modeling allows to explore the essence of complex processes and phenomena by experimenting with a model of a real system, and not with the real system itself. Simulation is often the only practical way to obtain information about the behavior of a complex system. Currently, simulation has found wide application in areas such as logistics, manufacturing, and business processes. Its fundamental capabilities are very extensive and allow it to explore systems of varying complexity and affiliation. The object of this work is to create the discrete-event simulation models and tools to support the design of distributed systems based on some knowledge about the elements of the system. Such properties of simulation models as the observability and controllability (the possibility of a flexible effect on the target functions of the model) can reduce the uncertainty of knowledge about the characteristics and behavior of the system and can be used as a means of studying the most important algorithmic and structural problems of the functioning of distributed systems. Currently, there is a large number of specialized software systems designed to build a simulation model of distributed systems and conduct the simulation experiments. Examples of such systems are given below. For simulation models, a quality requirement is imposed (qualimetry of the simulation model), which, in particular, assumes that a change in the structure and algorithms of the model’s behavior will not entail a deterioration in the characteristics of the model, which determine its ability to satisfy the set or expected needs of the user. Algorithms on distributed systems must have the properties of scalability (stability to increase computing power) and fault tolerance (invariance of distributed execution in the event of failure of individual components). This means that the simulation model of a distributed information system when performing operations to add or remove its components associated with expanding or out of order computing power should retain its characteristics and provide users with reliable results [1]. Of great interest from the point of view of modeling is the set of widely known distributed algorithms (wave algorithms, election algorithms, protocols for making consistent decisions, etc.), the description of which can be found, for example, in classical monographs [2, 3]. To perform such algorithms, it is necessary to maintain data structures (arrays, structures), the sizes of which depend on the number of interacting processes operating on different nodes. When changing the topology of a distributed system, the data structures of the algorithms that are executed in this distributed environment must change accordingly. Therefore, the model should provide the opportunity to change the scenario of the simulation model or to change

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the algorithms and corresponding data structures that describe the behavior of the simulated objects. Subsequently, the chapter is presented in the following sequence. Section 20.2 provides a brief overview of the existing tools for simulation of distributed systems, and Sect. 20.3 formulates the requirements for a distributed system simulator that could be used for learning purposes. Section 20.4 describes the model of the original distributed system simulator and the most important features of its implementation. The conclusions are given in Sect. 20.5.

20.2 Tools for Simulation of Distributed Systems The process of constructing a simulation tool that adequately displays a distributed system is a non-trivial challenge. Therefore, there is a need for automation tools for this process that will maximize the computer technology potential and ensure the construction of a complex technological chain: from a mathematician who formalizes the subject area of a distributed system to a programmer who creates a simulation model and means of access to this model and its application. According to reviews published on the Internet today, several hundred software products focused on simulation modeling appear on the information technology market. Simulation systems that provide an opportunity to study various aspects of the functioning of distributed systems can be divided into general-purpose systems and highly specialized systems. Existing simulation systems and a list of requirements for them are described below, following mainly the review [4]. General-purpose simulation systems are typically difficult to learn and use and require qualified model designers. The most popular general-purpose systems are the following: AnyLogic [5], ExtendSim [6], GPSSWorld [7], MicroSaint [8], Simplex3 [9], Arena [10], Simulink [11], and others. They are universal and can be used when modeling almost any subject area. Such systems support various modeling paradigms: methods of system dynamics, discrete-event and agent modeling, as well as, a hybrid approach to model building. They have the opportunity by varying the parameters of various types of numerical experiments and the implementation of animation including three-dimensional cases that allows to observe a simulation process in real time. The construction of model objects is performed using high-level programming languages: Java, C++ , as well as, using built-in object-oriented ones. The dominant concepts can also include a powerful graphical interface that allows the designing process diagrams, code debugging tools, and monitoring objects in the modeling process. It includes the ability to import and export data. In some systems, it is allowed to connect external specially designed modules. Among the shortcomings of universal systems, the usually noted are the following: run mainly under Windows-based operating systems, the need for a number of additional functionalities for effective modeling of specific distributed systems, etc. General-purpose systems do not allow fully taking into account the specifics features

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of the subject area, which requires the development of special tools to enable the study of the effectiveness and reliability of the domain-specific distributed systems functioning. Highly specialized simulation systems are focused on solving a specific class of problems or are used to model specific subject areas. For example, QualNet system is the network simulation software that acts as a planning, testing, and training tool which mimics the behavior of a physical communications network. The interface of such programs is designed for professional in the simulated domain. Examples of such systems are SIMSCRIPT, NetSim, Riverbed Modeler, ns-3, and EXata. To simulate the distributed systems depending on the way the functioning processes are reproduced, the emulators and simulators can be used. The emulator allows to reproduce the behavior of a real device or program in real time, when the simulator allows to simulate the real systems displaying some of the real phenomena and properties in a virtual environment. The following systems can be classified as distributed media simulators: OptorSim, GridSim, Bricks, CloudSim, WorkflowSim, SimGrid, GridMe, SIRIUS, and others. These systems allow to study the most important system characteristics, for example, the channel capacity, probability of data loss, behavior of users, planning policies, load balancing, and other properties.

20.3 Simulator Components and Simulator Requirements The crucial disadvantage of the simulation model, which appears in the computer implementation of the simulation method, is that a solution is always partial and corresponds to fixed elements of the structure, behavior algorithms, and system parameter values. This complicates the application of the standard tools and solutions discussed above for modeling domain-specific distributed systems. Additional disadvantages are mentioned below: • Lack of support for conceptual modeling and, as a consequence, the inability to take into account the features of specific problem areas of the tasks being solved. • Primary focus on modeling the operation of communication systems and limited capabilities for describing the computational process in the network nodes themselves. • The necessary detailing of the system structure is not provided. • Hiding the details of the functioning algorithms of a number of commercial systems. • Limited simulated architectures of distributed systems. • The need to learn complex software packages and special programming languages and specific data formats. • The inability to use multi-platform for the development and use of models. • High cost of license to use packages.

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Below, we discuss the software package used for educational purposes, which partially solves some of the above limitations of simulation. The main elements and requirements for such a package should be: • Relatively small size of program code to make it possible of learning the package for a few of hours in the educational process. • Ability to simulate synchronous and asynchronous systems. • Ability of modeling systems with failures of various types, including Byzantine ones. • Availability of graphic design tools for the fragments of the model, analysis of input data, and assessing the degree of their suitability in modeling systems. • Ability to monitor the simulation process in real time. • Availability of software to optimize a simulation experiment. • The ability to account for dynamic changes in the structure of the simulated system. • Adaptability of software for simulators of distributed information systems to include new devices and new algorithms in the simulation model. • Specification of the performance features of modeling algorithms in one of the high-level languages. • Availability of code debugging facilities. • Ability to simulate the behavior of real communication subsystems and modeling computer networks. • Use of developed two- and three-dimensional animation in real time. Particular attention should be paid to modeling the behavior of senders and recipients of messages (On/Off model). During distributed execution, a certain number of messages are generated, the sources of which can be in two states: active (On), in which they generate a message, and passive (Off), in which they wait. These periods are alternate. Sources generate the queries with a certain intensity, which can be determined, or, in the case of asynchronous systems, be a random value.

20.4 Model of an Asynchronous Distributed System The main difficulty in the study of distributed system algorithms is that there is a set of processes that make up a distributed system that are executed simultaneously and independently of each other. It is important that the behavior of processes is determined exclusively by received messages. Each of the processes can interact with any connected process by receiving messages from it and by sending messages to it. Since the proposed software package (hereinafter referred to as the simulator) is primarily intended for participation in the educational process, it was required to provide a low threshold for students to enter into its understanding and development of components. For this, it was divided into two parts. The base part (kernel) was written in portable C++ for executing on all modern operating systems, such as

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Windows, Linux, and MacOS. The graphic part allows visualizing the process of creating the model and the simulation itself, including the passage of messages within the system. The graphic part uses the kernel components to build a model and monitor the modeling process. In this paper, we focus on the features of the implementation of the base part. A small amount of program code (the basic part takes about 400 lines in C++), used to implement the simulator, and an object model that maps the simulated concepts to objects and methods of the language means that the student can develop his/her own distributed algorithm in the first lesson, further expanding his/her knowledge of the system’s capabilities. Let us consider the basic part in detail. Hereinafter, we describe the simulator model and its composition (Sects. 20.4.1 and 20.4.2), implementation features of a specific distributed algorithm (Sect. 20.4.3). Section 20.4.4 provides an example of such an implementation. Synchronous algorithms are discussed in Sect. 20.4.5.

20.4.1 Model In order for the model reflects adequately the real distributed systems, all the entities of such systems were implemented as C++ programming language classes. The behaviour of computing nodes are simulated. On each node an arbitrary number of processes that implement the algorithms is executed. Nodes can send and receive the typed messages, each of which can be processed by one or more algorithms. The main unit of the message handler is the operational function of the algorithm. It is unique for each of the implemented algorithms. The operational function is that which implements the reaction of the distributed process to external influences, that is, to received messages. The operational function should deal with external influences sending messages to processes and should also inform us about the events that take place with the process. For a student, the development of a specific algorithm consists in writing an operational function that implements the algorithm. To determine the modeling system, a configuration file is used. It describes the composition of the nodes of the system, metrics of links between nodes, methods of interaction, the intensity and types of failures of communications and individual nodes, and more. The default operating mode of the system is asynchronous. In asynchronous mode, there is no concept of global time, but each message sent by one process to another is delivered at an interval determined by the latency of link. To simulate synchronous systems, it is possible to run a special synchronization process, which will send messages containing global time to the participating nodes. When modeling, it is possible to introduce errors in the transmission of messages, for example, a message with some probability may not be delivered to the addressee. It is possible to simulate a temporary or permanent failure of a process including Byzantine one.

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The graphic part of the simulator makes it possible to determine interactively the composition of the simulated system, nodes, and connections between nodes and visualize the modeling process. It is allowed to use the configuration file of the base model or export the created or edited system configuration as a new configuration file. The format of configuration file was designed especially for automated generation. The sample of the graph of distributed processed and its configuration file is described below. ;file config.data processes 0 6 bidirected 1 link from 0 to 1 latency 6 link from 0 to 3 latency 2 errorRate 0.02 link from 0 to 5 latency 6 link from 0 to 6 latency 10 link from 1 to 2 latency 2 link from 1 to 4 latency 7 errorRate 0.1 link from 2 to 6 latency 6 link from 3 to 5 latency 6 link from 5 to 6 latency 4 setprocesses 0 6 CM transmitionErrorRate 0.02 launch timer 1 send from −1 to 0 CM_INIT wait 40 send from −1 to 1 CM_INIT wait 40

The given configuration file describes the following network configuration presented in Fig. 20.1.

6

P0

P1

3 2

P3 2

7

10 P2

6

P4 6 P5

Fig. 20.1 Sample network

4

P6

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20.4.2 Base Model Objects At least one object of World class is required to run the simulator. It manages the nodes, models of distributed processes, links between them, sets the mode of functioning of the system, and registers the operational functions. The created processes may fail, which is specified in the configuration file. Some messages that were supposed to be delivered to the failed process are destroyed; some messages are accumulated in the delay queue. When resuming a failed process, messages from the delay queue are transferred to the input queue of this process. The NetworkLayer class is responsible for communication between processes and for message delivery between processes. NetworkLayer class simulates the asynchronous modes of sending messages between processes, which corresponds to User Datagram Protocol (UDP) of the IP protocol family. Just as in UDP, messages may not be delivered to the recipient. The failure rate is set by the configuration file. The Process class models the distributed process itself. Messages can come from other processes and from the system. In particular, it is possible to initiate the start of any algorithm by sending a corresponding message on behalf of the system. The computational thread that processes incoming messages cycles through the following phases: waiting, receiving, analyzing, scheduling, and executing. Its main state is waiting for an event—receiving a message. When a message appears in the input queue, a flag of the Event class is set and the computational thread wakes up. This splitting allows more flexible processing of incoming messages. If the algorithm can process several different classes of messages, then it is possible to discover the class of the received message only on the analysis phase. Messages of different classes implement different algorithms. For example, messages related to routing (sending messages to a process that is not a direct neighbor through direct neighbor) can be processed by routing-aware executor, and messages related to the election of the leader of a group of processes can be processed by an executor designed specifically for implementing such algorithm. Each message for transmission over communication channels is just a set of bits, but from the point of view of the application, it is necessary to transmit the contents of objects, such as variables, structures, and instances of classes. The Message class allows serializing many objects into a single object, which is then transferred as a bit stream to another process and provides methods for deserializing these parameters on the receiver side. Representation of distributed system objects by simulator objects is in Table 20.1.

20.4.3 Operational Function The response to an arrived message is implemented in an operational function, which is active only while the message is being processed. As soon as the operational function is completed, the process goes into a wait state. All local variables used

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Table 20.1 Representation of distributed system objects by simulator objects Distributed system

Simulator

Distributed process

Process::workerThread

Creation of distributed process

World::createProcess

Process’ context

Process::context_common

Algorithm’s ALG context

Process::context_ALG

Message executor for algorithm ALG

opFunction_ALG

Network infrastructure

class NetworkLayer

Message sending

NetworkLayer::send

Message receiving

NetworkLayer::receive

Process’ received messaged queue

Process::workerMessageQueue

Metrics of link between processes P1 and P2

NetworkLayer::networkMap[p1][p2]

by the operational function are inaccessible between calls. The data required by the algorithm are stored in a data structure called the context of the algorithm. Two arguments are passed to the operational function—the context of the Process class and a copy of the received message. Each algorithm can add its context to the Process class, which the operational function will use. Processing of incoming messages by an operational function includes: analyzing of the message content; analyzing and changing the context of the algorithm along with the global context of the process, and sending messages to other processes. The simulator does not use the client/server model, and the process that sent the message continues its activity without waiting for the delivery of this message to the recipient. The development of systems based on such interaction of processes is often a stumbling block for students. The use of only such asynchronous interaction serves as a motivating force for the development of asynchronous algorithms.

20.4.4 Example: Asynchronous Algorithm Implementation The proposed simulator allows implementation of almost any distributed algorithms, both asynchronous and synchronous. The book [12] gives examples of the implementation of algorithms and operational functions for them. There are some of them: • • • • • • •

Physical time synchronization: Christian and Berkley algorithms. Logical time synchronization: Lamport time stamps and vector time stamps. ABD synchronizers. Minimal spanning tree algorithms. Variables distribution: FLOOD algorithms. Shortest path search: Toueg, Chandy-Misra, NETCHANGE algorithms. Group reliable messaging: BROADCAST and CONVERGECAST algorithms.

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Distributed search depth or breadth priority algorithms. Elections: BULLY and Le Lann-Chang-Roberts algorithms. Detection of completion: Rana algorithm. Mutual exclusion algorithms. Consensus algorithms.

In this chapter, a distributed algorithm for determining the shortest paths in a graph is chosen—the Chandy-Misra algorithm [13]. The Chandy-Misra algorithm is one of the variants of the Ford-Bellman algorithm designed for distributed systems. This algorithm calculates the shortest paths from the node specified by the initial message to all reachable nodes storing the number of the neighboring node, through which this shortest path is reachable. The Chandy-Misra algorithm belongs to the class of diffuse ones. This means that the algorithm is triggered by a certain message, which starts the execution of the algorithm in one process and, as the algorithm executes, other processes are connected to it. For algorithms that are not initially diffuse ones, it is necessary to create their diffuse form, which complicates their implementation. The ChandyMisra algorithm was originally developed as diffuse, so it should be expected that its implementation will be straightforward. An algorithm context, thus data structures which are unique to a given algorithm belonging to a particular process, is created for each node, and, upon completion of the algorithm, it will contain two mappings in each node. In mapping D, each achievable process will correspond to the length of the shortest path from the current process. In mapping Nb, it will correspond to the number of the neighboring process through which this best path passes. The Boolean variable inited determines whether the execution of the algorithm on a given node has begun. struct context_CM { bool inited = false; bool isRoot = false; map D; // best weights map map Nb; // best neighbors map };

The algorithm is described as follows: (1) At the beginning of the algorithm, all processes fill mapping D with the value INF, and mapping Nb with the value UNDEF. To do this, it is possible to use another distributed algorithm: the distribution of variable values. (2) The initial INIT message is sent by the system to the root node R from which the shortest paths will be built. The root node and later on the other nodes that received the INIT message send it to neighboring nodes on their own behalf. After receiving this message, the inited context variable is set to true. If the process has already received an INIT message, diffuse propagation decays. (3) The root node starts diffusion by sending a MYDIST message with two arguments: the number of the root node and the best distance to the sending process message. If relaxation occurs (a shorter path is found), then the calculated DR

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value changes according to the formula DR = DP + DistP,node , where DP is the distance transmitted by the message and DistP,node is the metric of the path to the current process. NbR in this case is set to P. (4) After the algorithm terminates, the mappings D and Nb in the context of the algorithm will contain the calculated values. The operational function implements the algorithm in a fairly straightforward manner. The DR and NbR mappings are filled in a “lazy” way only after receiving the first MYDIST message containing R as the root. 00 int opFunction_CM(Process *dp, Message m) { 01 string s = m.getString(); 02 auto lc = &dp->context_CM; 03 const int UNDEF = 2000000001; 04 const int INF = 1000000000; 05 auto nl = dp->networkLayer; 06 int node = dp->node; 07 auto neibs = dp->neibs(); 08 // (1) INIT message handling 09 if (s == "CM_INIT") { 10 if (lc->inited) return true; 11 lc->D.clear(); 12 lc->Nb.clear(); 13 if (m.from == -1) lc->isRoot = true; 14 inform("CM[%d]: INIT from %d\n", node, m.from); 15 for (auto n : neibs) { 16 nl->send(node, n, Message("CM_INIT")); 17 } 18 lc->inited = true; 19 // Init wave is launched. Launch MYDIST wave. 20 if (lc->isRoot) { 21 for (auto n : neibs) { 22 nl->send(node,n,Message("CM_MYDIST", node, 0)); 23 } 24 } 25 inform("CM[%d]: INIT ENDED\n", node); 26 } else if (s == "CM_MYDIST") { 27 int v0 = m.getInt(); 28 if (lc->D.find(v0) == lc->D.end()) { // First 29 lc->D[v0] = INF; 30 lc->Nb[v0] = UNDEF; 31 } 32 int d = m.getInt(); 33 inform(“CM[%d]: MYDIST[v0 = %d,d = %d] came from %d\n”,33a m.from); 34 int omega = nl- > getLink(m.from, node); // dist to neib. 35 if (d + omega < lc->D[v0]) { 36 lc->D[v0] = d + omega; 37 lc->Nb[v0] = m.from; 38 for (auto n : neibs) { 39 nl->send(node, n, 40 Message("CM_MYDIST", v0, lc->D[v0]) 41 );

node, v0, d,

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42 } 43 } 44 stringstream out; 45 out