Smart Modelling For Engineering Systems: Proceedings of the International Conference on Computational Methods in Continuum Mechanics (CMCM 2021), ... Innovation, Systems and Technologies, 214) 9813347082, 9789813347083

Citation preview

Smart Innovation, Systems and Technologies 214

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 1

Smart Innovation, Systems and Technologies Volume 214

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 1

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 Informatics and Numerical Modeling Moscow Institute of Physics and Technology (National Research University) Dolgoprudny, Russia Lakhmi C. Jain Centre for Artificial Intelligence University of Technology Sydney Sydney, 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 978-981-33-4708-3 ISBN 978-981-33-4709-0 (eBook) https://doi.org/10.1007/978-981-33-4709-0 © 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, during April 15–17, 2021. The conference is devoted to the memory of Academician Oleg Belotserkovskii. Professor Dr. Belotserkovskii had made a tremendous contribution in the field of numerical methods and the mathematical modeling 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 is 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 to these books. Volume 1 presents Chap. 2 about the scientific and life path of Academician Oleg Belotserkovskii. Part I “Computational Aerodynamics, Hydrodynamics and Dynamics of Plasma” involves Chaps. 3–6. Part II “Numerical Modeling in Solid Mechanics” includes Chaps. 7–17, and Part III “Computational Modeling in Medicine and Biology” contains Chaps. 18–24. 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 Dolgoprudny, Russia Moscow, Russia Sydney, Australia

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

v

Contents

1

Recent Advances in Computational Physics . . . . . . . . . . . . . . . . . . . . . . Margarita N. Favorskaya, Alena V. Favorskaya, Igor B. Petrov, and Lakhmi C. Jain

2

The Scientific and Life Path of Academician Oleg M. Belotserkovskii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Igor B. Petrov

1

7

3

Numerical Simulation of Spatial Flows in Shear Layers . . . . . . . . . . . Svetlana V. Fortova and Elena I. Oparina

4

Modeling of Unsteady Flows in Gas Astrophysical Objects on Supercomputers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander V. Babakov

35

Modeling of the Physical Processes of the Impact of a Powerful Nuclear Explosion on an Asteroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daria S. Moiseeva, Andrey A. Motorin, and Evgenii L. Stupitsky

47

A Multidimensional Multitemperature Gas Dynamic and the Neutrino Spectrum in 2D Gravitational Collapse . . . . . . . . . Alexey G. Aksenov

61

The Airy Stress Function for Non-Euclidean Model of a Continuous Medium and Description of Residual Stresses . . . . . Mikhail A. Guzev

75

5

6

7

8

Numerical Comparison of Different Approaches for the Fractured Medium Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . Ilia S. Nikitin, Vasily I. Golubev, Yulia A. Golubeva, and Vladislav A. Miryakha

15

87

vii

viii

9

Contents

The Comparison of Two Approaches to Modeling the Seismic Waves Spread in the Heterogeneous 2D Medium with Gas Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Polina V. Stognii, Nikolay I. Khokhlov, Igor B. Petrov, and Alena V. Favorskaya

10 Mathematical Modeling of Spatial Wave Processes in Fractured Seismic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Maksim V. Muratov and Tatiana N. Derbysheva 11 Investigation of Models with Fluid- and Gas-Filled Fractures with the Help of the Grid-Characteristic Method . . . . . . . . . . . . . . . . . 125 Polina V. Stognii and Nikolay I. Khokhlov 12 Modeling Wave Responses from Thawed Permafrost Zones . . . . . . . 137 Vasily I. Golubev, Alexey V. Vasyukov, and Mikhail Churyakov 13 Modeling of Fiber-Metal Laminate Residual Strength After a Low-Velocity Impact with a Grid-Characteristic Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Katerina A. Beklemysheva 14 Modeling Movement of Train Along Bridge by Grid-Characteristic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Anton A. Kozhemyachenko, Anastasia S. Kabanova, Igor B. Petrov, and Alena V. Favorskaya 15 Seismic Evaluation of Two-Storied Unreinforced Masonry Building with Rigid Diaphragm Using Nonlinear Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Amit Sharma, Vasily I. Golubev, and Rakesh Kumar Khare 16 Numerical Study of Thin Composite Structures Vibrations for Material Parameters Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Vitalii V. Aksenov and Katerina A. Beklemysheva 17 The Study of the Physical Processes that Cause the Destruction and Fragmentation of Meteoroids in the Atmosphere . . . . . . . . . . . . . 199 Nina G. Syzranova and Viktor A. Andrushchenko 18 Personalized Geometric Modeling of a Human Knee: Data, Algorithms, Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Alexandra Yurova, Victoria Salamatova, Yuri Vassilevski, Lin Wang, Sergei Goreynov, Oleg Kosukhin, Anatoly Shipilov, and Yusuf Aliev 19 Personalization of Mathematical Models of Human Atrial Action Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Andrey V. Pikunov, Roman A. Syunyaev, Vanessa Steckmeister, Ingo Kutschka, Niels Voigt, and Igor R. Efimov

Contents

ix

20 Computational Study of the Effect of Blood Viscosity to the Coronary Blood Flow by 1D Haemodynamics Approach . . . . 237 Sergey S. Simakov and Timur M. Gamilov 21 Simulation of the Human Head Ultrasound Study by Grid-Characteristic Method on Analytically Generated Curved Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Alena V. Favorskaya 22 Reaction–Diffusion Model of Coexistence of Viruses in the Space of Genotypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Cristina Leon 23 Numerical Simulation of the Denture Prosthesis Integrity Under Typical Chewing Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Sergey D. Arutyunov, Dmitry I. Grachev, Grigoriy G. Bagdasaryan, and Alexander D. Nikitin 24 Numerical Modeling of Elastic Wave Propagation in a Human Craniocerebral Area with Discontinuous Galerkin Method . . . . . . . . 287 Katerina A. Beklemysheva and Igor B. Petrov Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

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 EngineeringSystems, 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. 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 Ph.D. 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.

xi

xii

About the Editors

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. 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., ME, BE(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 5000 researchers drawn from universities and companies world-wide, KES facilitates international cooperation and generate 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 Computational Physics 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 computational physics in different areas, i.e., computational aerodynamics, hydrodynamics, dynamics of plasma, solid mechanics, elastic and acoustic wave phenomena, seismic prospecting, seismic resistance, train movement, medicine, and biology. The first part of the book is devoted to the numerical solutions of problems of aerodynamics, hydrodynamics, and dynamics of plasma. The second part of the book deals with numerical methods and its applications in the area of solid mechanics. The third part of the book covers computational methods in medicine and biology. Keywords Computational methods · Numerical experiments · Parallel algorithms · Deformable solid bodies · Elastic and acoustic waves investigation · Gas dynamics · Aerodynamics · Fluid dynamics · Physics of plasma · Composite materials · Seismic prospecting · Seismic stability · Medicine

M. N. Favorskaya (B) Reshetnev Siberian State University of Science and Technology, Institute of Informatics and Telecommunications, 31, Krasnoyarsky Rabochy Ave, Krasnoyarsk 660037, Russia 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, Russia e-mail: [email protected] I. B. Petrov e-mail: [email protected] L. C. Jain University of Technology Sydney, Sydney, NSW, Australia 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 214, https://doi.org/10.1007/978-981-33-4709-0_1

1

2

M. N. Favorskaya et al.

1.1 Introduction This book is devoted to the development and application of numerical methods for solving problems in various areas of computational physics. At the beginning of the book, an overview of scientific way of Academician O.M. Belotserkovskii was discussed and his life path was described. In the second part of this book, numerical simulation of the spatial flows in shear layers, the unsteady flows in gas astrophysical objects, the physical processes of the impact of a powerful nuclear explosion on an asteroid, and the multidimensional multi-temperature gas dynamic and the neutrino spectrum at the 2D gravitational collapse are discussed. The second part of the book deals with different applications of the gridcharacteristic method in different areas of computational physics, i.e., elastic wave phenomena in fractured zones, seismic waves spread in the heterogeneous 2D medium with gas cavities, wave response from thawed permafrost zones, composite structure vibrations and strength, movement of train along bridge, seismic stability. Also the second part of the book discusses the airy stress function for non-Euclidean model of a continuous medium and description of residual stresses, and destruction and fragmentation of meteoroids in the atmosphere. Personalized geometric modeling of a human knee, atrial action potential, the effect of blood viscosity to the coronary blood flow, simulating the human head ultrasound, reaction–diffusion model of coexistence of viruses in the space of genotypes, and the denture prosthesis integrity under typical chewing loads are discussed in the third part of the book.

1.2 Chapters Included in the Book Chapter 2 contains a brief biography of Academician Oleg M. Belotserkovskii and describes his major achievements and areas of interest, i.e., the development of numerical methods and the mathematical modeling of aerodynamics of spacecraft [1], hydrodynamics, plasma physics, mechanics of a deformable solid, thermonuclear fusion, turbulence, computational medicine, and biology. The first part of the book deals with fluid dynamics, including turbulence [2], aerodynamics [3], physics of plasma [4], and astrophysical gas dynamics [5]. In Chap. 3, computer modeling of the phenomenon of origin and development of the vortex cascade of instability depending on various flow parameters, initial data, in the absence or presence of external force is discussed. Chapter 4 describes a numerical approach for simulation the spatial unsteady motion of matter in gas astrophysical objects. Numerical studies of the impact of the explosion at various distances from the surface of the asteroid were done in Chap. 5. The gravitational collapse of the massive star’s core with a neutrino transport is considered in Chap. 6. The second part of the book consists of numerical experiments in the area of solid mechanics, including non-Euclidean model [6], destruction processes [7], acoustic

1 Recent Advances in Computational Physics

3

and elastic wave phenomena [10] in the presence of gas cavities [11], fractured zones [12, 13], composite materials [8, 14, 15], train motion [16], and seismic resistance [17] calculated using the grid-characteristic method [10]. The non-Euclidean model of a continuous medium, for which the Saint-Venant compatibility was applied, is discussed in Chap. 7. Chapter 8 is devoted to the investigation of the dynamic loading of fractured media occurred in many applications like the seismic survey process, non-invasive material quality control, and fatigue failure of samples. Chapter 9 deals with the problem of the direct modeling of the seismic waves spread in such a medium with the presence of gas cavities and without them. The propagation of elastic waves in a fractured seismic medium is investigated in Chap. 10. The results of modeling the seismic waves spread through the homogeneous and heterogeneous media with single fractures and fracture clusters are presented in Chap. 11. The capabilities of the seismic survey for monitoring of this process of permafrost regions melting are studied in Chap. 12. Hashin failure criterion is used to model a complex loading of a GLARE fiber–metal laminate sample in Chap. 13. The problem of numerical simulation of train movement on the bridge is considered in Chap. 14. The seismic evaluation of two-storied unreinforced masonry building with rigid diaphragm when subjected to seismic lateral loading is investigated in Chap. 15. Chapter 16 deals with the vibration patterns of several thin composite samples. Chapter 17 describes the motion and destruction of three of some of the biggest meteoroids that entered the atmosphere above Russia. Computational medicine problems are discussed in the third part of the book, e.g., personalized modeling of human knee [18] and human cardiac action potential [19], coronary blood flow [20], viruses’ competition [21], denture prosthesis integrity [22] computer simulation, and human craniocerebral area [23]. The problem of patientspecific modeling of human knee joints is considered in Chap. 18. Two widely used electrophysiological models of the human atrium, i.e., the Maleckar and the Grandi models of action potential in excitation and propagation in healthy or diseased atria, are benchmarked in Chap. 19. Chapter 20 deals with how different approaches affect coronary blood flow in one-dimensional model. Solution of the direct problem of ultrasonic investigation of a human head using the grid-characteristic computational method based on the structured curved grids with a sharply changing coordinate step is discussed in Chap. 21. Persistence and evolution of two viruses in the host organism taking into account characteristic aspects of viral dynamic such as virus mutation, replication, and genotype dependent mortality, either natural or determined by an antiviral treatment are investigated in Chap. 22. The problem of stress analysis for the removable prosthesis bases of the upper and lower jaws under typical chewing loads is discussed in Chap. 23. Numerical modeling of elastic wave propagation in a human craniocerebral area with discontinuous Galerkin method is considered in Chap. 24.

4

M. N. Favorskaya et al.

1.3 Conclusions The book contains the works of outstanding specialists in the fields of computational physics and numerical methods. The achievements of recent years are in such areas of computational physics as computational aerodynamics, hydrodynamics, dynamics of plasma, solid mechanics, elastic and acoustic wave phenomena, seismic prospecting, seismic resistance, train movement, medicine, and biology. Unique numerical simulation results in these areas are presented by the authors in this book. This book will be of interest to scientists, researchers, students, graduate and postgraduate students specializing in the scientific fields of computational physics, deformable solid bodies, elastic and acoustic waves’ investigation, gas dynamics, aerodynamics, fluid dynamics, physics of plasma, composite materials, seismic prospecting, seismic stability, medicine, and biology.

References 1. Belotserkovskii, O.M.: Symmetric flow about blunt bodies in a supersonic stream of a perfect and real gas. U.S.S.R. Comput. Math. Math. Phys. 2(6), 1272–1304 (1963) 2. Fortova, S.V.: Eddy cascade of instabilities and transition to turbulence. Comput. Math. Math. Phys. 54, 553–560 (2014) 3. Babakov, A.V., Lugovsky, A.Yu., Chechetkin, V.M. Mathematical modeling of the evolution of compact astrophysical gas objects. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Lakhmi, C.J. (eds.) GCM50 2019, SIST, vol. 133, pp. 210–227. Springer, Cham (2019) 4. Andrushchenko, V.A., Moiseeva, D.S., Motorin, A.A., Stupitsky, E.L.: Modeling the physical processes of a powerful nuclear explosion on an asteroid. Comput. Res. Modeling 11(5), 861– 877 (2019) 5. Aksenov, A.G., Chechetkin, V.M.: Large-scale instability during gravitational collapse with neutrino transport and a core-collapse supernova. Astronomy Rep. 62, 251–263 (2018) 6. Guzev M.A.: Non-Euclidean Models of Elastoplastic Materials with Structure Defects. Lambert Academic Publishing (2010) 7. Syzranova, N.G., Andrushchenko, V.A.: Simulation of the motion and destruction of bodies in the Earth’s atmosphere. High Temp. 54(3), 308–315 (2016) 8. Beklemysheva, K.A., Vasyukov, A.V., Kazakov, A.O., Petrov, I.B.: Grid-characteristic numerical method for low-velocity impact testing of fiber-metal laminates. Lobachevskii J Math. 39, 874–883 (2018) 9. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Continual models of layered and block media with slippage and delamination. Proc. Structural Integrity 23, 125–130 (2019) 10. Favorskaya, A.V., Khokhlov, N.I., Petrov, I.B.: Grid-characteristic method on joint structured regular and curved grids for modeling coupled elastic and acoustic wave phenomena in objects of complex shape. Lobachevskii J. Math. 41(4), 512–525 (2020) 11. Stognii, P.V., Khokhlov, N.I.: 2D seismic prospecting of gas pockets. 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. 156–166. Springer, Cham (2019) 12. Muratov, M.V., Petrov, I.B.: Application of mathematical fracture models to simulation of exploration seismology problems by the grid-characteristic method. Comput. Res. Modeling 11(6), 1077–1082 (2019) 13. Khokhlov, N., Stognii, P.: Novel approach to modeling the seismic waves in the areas with complex fractured geological structures. Minerals 10(2), 122.1–122.17 (2020)

1 Recent Advances in Computational Physics

5

14. Beklemysheva, K.A., Vasyukov, A.V., Golubev, V.I., Zhuravlev, Y.I.: On the estimation of seismic resistance of modern composite oil pipeline elements. Doklady Math. 97(2), 184–187 (2018) 15. Aksenov, V., Vasyukov, A., Petrov, I.: Numerical modelling of thin anisotropic membrane under dynamic load. Aeronautical J. Adv. Online publication (2020). https://doi.org/10.1017/aer.202 0.61 16. Favorskaya, A., Khokhlov, N.: Modeling the impact of wheelsets with flat spots on a railway track. Proc. Comput. Sci. 126, 1100–1109 (2018) 17. Binnani, N., Khare, R.K., Golubev, V.I., Petrov, I.B.: Probabilistic seismic hazard analysis of Punasa dam site in India. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds.) Smart Modeling for Engineering Systems: Proceedings of the Conference 50 Years of the Development of Grid-Characteristic Method: Proceedings of Smart Modeling for Engineering Systems. GCM50 2018. SIST, vol. 133, pp. 105–119. Springer International Publishing AG, Cham, Switzerland (2019) 18. Salamatova, V. Y., Yurova, A. S., Vassilevski, Y. V., Wang, L.: Automatic segmentation algorithms and personalized geometric modelling for a human knee. Russian J. Numerical Anal. Math. Modelling 34(6), 361–367 (2019) 19. Smirnov, D., Pikunov, A., Syunyaev, R., Deviatiiarov, R., Gusev, O., Aras, K., Gams, A., Koppel, A., Efimov, I.R.: Genetic algorithm-based personalized models of human cardiac action potential. PLoS ONE 15(5), e0231695 (2020) 20. Gognieva, D.G., Gamilov, T.M., Pryamonosov, R.A., Vasilevsky, Y.V., Simakov, S.S., Liang, F., Ternovoy, S.K., Serova, N.S., Tebenkova, E.S., Sinitsyn, E.A., Pershina, E.S., Abugov, S.A., Mardanyan, G.V., Zakryan, N.V., Kirakosyan, V.R., Betelin, V.B., Mitina, Y.O., Gubina, A.Y., Shchekochikhin, D.Y., Syrkin, A.L., Kopylov, F.Y.: Noninvasive assessment of the fractional reserve of coronary blood flow with a one-dimensional mathematical model. Preliminary results of the pilot study. Russian J. Cardiology 24(3), 60–68 (2019) 21. Leon, C., Popov, V., Volpert, V.: Viruses competition in the genotype space. ITM Web of Conferences 31, 02002.1–02002.7 (2020) 22. Arutyunov, S.D., Chumachenko, E.N., Lebedenko, IYu., Arutyunov, A.S.: Comparative analysis of the mathematical modeling results on the stress-strain state of various designs for pin dentures. Dentistry 2, 1–41 (2001) 23. Vassilevski, Yu., Beklemysheva, K., Grigoriev, G., Kulberg, N., Petrov, I., Vasyukov, A.: Numerical simulation of aberrated medical ultrasound signals. Russian J. Numerical Anal. Math. Modelling 33(5), 277–288 (2018)

Chapter 2

The Scientific and Life Path of Academician Oleg M. Belotserkovskii Igor B. Petrov

Abstract The chapter contains a brief biography of Academician Oleg M. Belotserkovskii. The chapter describes his major achievements and areas of interest, providing a glimpse into his contribution to science and education system. Keywords Biography · Numerical methods · Hydrodynamics · Plasma physics · Mechanics of deformable solid · Thermonuclear fusion · Turbulence · Computational medicine

2.1 Introduction Oleg M. Belotserkovskii was an outstanding scientist and teacher, a full member of the world-famous Russian Academy of Sciences, a mathematician and a mechanic. He was the founder of entire scientific areas in numerical methods for continuum mechanics and mathematical modeling of physical processes. Academician Belotserkovskii served as a Rector of the Moscow Institute of Physics and Technology and a Director of the Institute for Computer Aided Design of the Russian Academy of Sciences (ICAD RAS).

I. B. Petrov (B) Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russia 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 214, https://doi.org/10.1007/978-981-33-4709-0_2

7

8

I. B. Petrov

2.2 The Scientific and Life Path Oleg M. Belotserkovskii was born in the city of Livny, Oryol region, Russia, in the family of teachers. After graduating from high school, he entered Bauman Moscow State Technical University and then continued his studies at the Department of Physics and Technology of Moscow State University. That department had been transformed into the Moscow Physics and Technology Institute (MIPT) when he graduated in 1952. After graduating from MIPT, Oleg M. Belotserkovskii was assigned to the Steklov Institute of Mathematics, USSR Academy of Sciences. In 1955, he moved to the newly founded Dorodnitsyn Computing Centre of the USSR Academy of Sciences, where he continued his research work with the full member of the USSR Academy of Sciences Anatoly A. Dorodnitsyn. Academician Dorodnitsyn was an outstanding scientist, mathematician and mechanic, well known in the field of viscous fluid mechanics. The joint work of these two great scientists continued until 1994, when Anatoly A. Dorodnitsyn passed away. Oleg M. Belotserkovskii not only continued the work of Academician Dorodnitsyn, but also created his own unique world-class scientific school. In 1972, Oleg M. Belotserkovskii was elected as a corresponding member of the USSR Academy of Sciences, and in 1979 became a full member of the USSR Academy of Sciences. In addition, he was an academician of the International Astronomical Federation, and he was the first deputy academician-secretary of the Department of Computer Science, Computer Engineering and Automation of the USSR Academy of Sciences. He received the N.E. Zhukovsky First Degree award, the gold medal “For the best work on the theory of aviation” and a number of high government awards. Scientific interests of Academician Belotserkovskii lie in the field of the development of numerical methods [1–16] and the mathematical modeling of aerodynamics of spacecraft [17–30], hydrodynamics [31–33], plasma physics, mechanics of a deformable solid [34–36], thermonuclear fusion, turbulence [37–45], computational medicine and biology [35]. Also, they included the development of approaches to solving the problem of mapping numerical methods to the architecture of multiprocessor high-performance computers [46, 47], and a number of other issues. He was particularly interested in research devoted to the direct numerical study of the complex phenomena of turbulence and hydrodynamic instability [48–52]. Oleg M. Belotserkovskii created a scientific school of computational mechanics, which is well known in the global scientific community (Fig. 2.1). He was the first one to solve one of the most important problems of aerodynamics—the supersonic flow around a blunt body with a departing shock wave [17, 24, 27, 29]. This research has a global priority, and it is an outstanding contribution to the space research, both theoretical and applied. The numerical method proposed by Oleg. M. Belotserkovskii became the basic one in aerodynamic calculations of hypersonic aircraft.

2 The Scientific and Life Path of Academician …

9

Fig. 2.1 Portrait of Academician O.M. Belotserkovskii

Academician Belotserkovskii actively participated in the development of new computational methods: “large particles” [3, 4], “conservative flow method” [5], “splitting method” [6], “grid-characteristic method” [35], “statistical particle method” [7, 8]. These methods found wide application in various research institutes and design bureaus of Russia and received the global recognition. Academician Belotserkovskii and his students obtained the fundamental theoretical results in such relevant areas of computational mechanics and physics as transonic aerodynamics, spatially unsteady flow around bodies of complex shape, dynamics of a viscous heat-conducting gas [53], modeling of statistical processes based on Boltzmann equations, etc. In the computational experiment, he together with his students boldly took up the simulation of magnetohydrodynamic instability in thermonuclear reactors. Numerical modeling of separated flows, ordered structures in turbulent flows and hydrodynamic instabilities allowed Academician Belotserkovskii to establish a number of fundamental principles that laid the foundation for a new constructive approach to numerical modeling of urgent and complex problems of nonlinear mechanics of continuous media. These works were highly appreciated by a group of scientists from the Los Alamos Laboratory in the USA under the leadership of Harlow, where he read a series of lectures and held a number of seminars in 19941995. In the following years, he dealt with the same problems, creating the Computer Modeling Center based in TSAGI (Zhukovsky, Moscow Region).

10

I. B. Petrov

Under the leadership of Academician Belotserkovskii, numerical methods are developed to solve the urgent problems of the dynamic strength of composite aircraft structures. At the same time, Oleg M. Belotserkovskii and his students began work on the numerical solution of the problems of seismic stability of nuclear facilities and aerodynamics facilities of the nuclear industry and aerodynamics together with the Russian Federal Nuclear Center in Sarov. Academician Belotserkovskii conducted a great organizational and pedagogical work. For a quarter century, he served as a Rector of the Moscow Institute of Physics and Technology. Here is the opinion of his successor, corresponding member of the RAS N.V. Karlov: “… it was Oleg M. Belotserkovskii who brilliantly used the mobility that is inherent to our institute, PhysTech (MIPT). The ability to respond to the needs of the day quickly and efficiently. In the period from 1964 to 1987, he created about 70 new graduating departments, which trained physical and mathematical engineers in a wide range of specialties in fundamental and applied science— from elementary particle physics to space research, from global ocean monitoring to microbiology, from mathematical control problems to the dynamics of large systems. Moreover, in the same period, the Far Eastern, Ural and Kiev branches of the Moscow Institute of Physics and Technology were created; students, teachers and graduating departments were completely incorporated into the system of Moscow Institute of Physics and Technology. In a word, Academician Belotserkovskii created the mighty and great in its own way empire of PhysTech.” At the Institute for Computer Aided Design, Academician Belotserkovskii launched a wide range of activities in the development of software in various scientific fields and high-performance super-computers with parallel architecture. The institute studied complex application of expert systems and mathematical modeling for solving special-purpose problems. Also it developed computer-aided design applications for various areas of the national economy. For ten years, at the initiative of Academician Belotserkovskii, ICAD RAS actively collaborated with scientific centers of India. The Izvestia newspaper wrote on December 16, 1999: “Russia and India have agreed to jointly organize a Center for Advanced Computer Research … The project will be supported by the Institute for Computer Aided Design of the Russian Academy of Sciences and the Center for High Performance Computing of the Indian Electronics Department. The RussianIndian center will be located in Moscow … According to the director of ICAD RAS Oleg M. Belotserkovskii, the center will operate on the basis of the Indian supercomputer “PARAM-10000”, capable of performing 70–100 billion operations per second. According to Academician Belotserkovskii, this is the first computer of such power in Russia.” Academician Oleg M. Belotserkovskii prepared about forty Dr Sci and dozens Ph.D. For his achievements in scientific, pedagogical and organizational activities, he received the Order of Lenin, the Order of the October Revolution, three Orders of the Red Banner of Labour and Orders “For Merit to the Fatherland” of III and IV degrees.

2 The Scientific and Life Path of Academician …

11

The most part of his books and articles are in Russian, but many of important ones were translated into English. References in this chapter contain a list of these translated articles.

2.3 Conclusions Academician Belotserkovskii was a scientist and a leader. He made a great contribution to the national and world science. His great legacy of a scientific tradition and rational approach continues in the scientific school he left behind.

References 1. Belotserkovskii, O.M., Chushkin, P.I.: A numerical method of integral relations. U.S.S.R. Comput. Math. Math. Phys. 2(5), 823–858 (1963) 2. Belotserkovskii, O.M., Bulekbaev, A., Grudnitskii, V.G.: Algorithms for numerical schemes of the method of integral relations for calculating mixed gas flows. U.S.S.R. Comput. Math. Math. Phys. 6(6), 162–184 (1966) 3. Belotserkovskii, O.M., Davydov, Yu.M.: The nonstationary method of “large particles” for gas dynamic calculations. U.S.S.R. Comput. Math. Math. Phys. 11(1), 241–271 (1971) 4. Belotserkovskii, O.M., Davydov, Yu.M.: Computation of transonic “supercritical” flows by the “coarse particle” method. U.S.S.R. Comput. Math. Math. Phys. 13(1), 187–216 (1973) 5. Belotserkovskii, O.M., Severinov, L.I.: A conservative method for “flows”, and the calculation of the circumfluence of a body of finite dimensions by a viscous heat-conducting gas. U.S.S.R. Comput. Math. Math. Phys. 13(2), 141–156 (1973) 6. 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. U.S.S.R. Comput. Math. Math. Phys. 15(1), 190–200 (1975) 7. Belotserkovskii, O.M., Yanitskii, V.E.: The statistical particles-in-cells method for solving rarefied gas dynamics problems. U.S.S.R. Comput. Math. Math. Phys. 15(5), 101–114 (1975) 8. Belotserkovskii, O.M., Yanitskii, V.E.: The statistical method of particles in cells for the solution of problems of the dynamics of a rarefied gas. II. Computational aspects of the method. U.S.S.R. Comput. Math. Math. Phys. 15(6), 184–198 (1975) 9. Belotserkovskii, O.M., Demchenko, V.V., Kosarev, V.I., Kholodov, A.S.: Numerical simulation of some problems of compression of shells by laser. U.S.S.R. Comput. Math. Math. Phys. 18(2), 117–137 (1978) 10. Belotserkovskii, O.M., Mitnitskii, V.Ya.: Some exact solutions of the problem of a magnetic dipole surrounded by an infinitely-conducting fluid. U.S.S.R. Comput. Math. Math. Phys. 19(2), 199–209 (1979) 11. Belotserkovskii, O.M., Byrkin, A.P., Mazurov, A.P., Tolstykh, A.I.: A difference method of increased accuracy for the calculation of flows of a viscous gas. U.S.S.R. Comput. Math. Math. Phys. 22(6), 206–216 (1982) 12. Belotserkovskii, O.M., Grudnitskii, V.G., Prokhorchuk, Yu.A.: A difference scheme of secondorder accuracy on a minimal pattern for hyperbolic equations. U.S.S.R. Comput. Math. Math. Phys. 23(1), 81–86 (1983) 13. Belotserkovskii, O.M., Panarin, A.I., Shchennikov, V.V.: The method of parametric correction of difference schemes. U.S.S.R. Comput. Math. Math. Phys. 24(1), 40–46 (1984)

12

I. B. Petrov

14. Belotserkovskii, O.M., Kholodov, A.S.: Majorizing schemes on unstructured grids in the space of indeterminate coefficients. Comput. Math. Math. Phys. 39(11), 1730–1747 (1999) 15. Belotserkovskii, O.M., Konyukhov, A.V.: Change of grid functions of dependent variables in finite-difference equations. Comput. Math. Math. Phys. 42(2), 224–237 (2002) 16. Belotserkovskii, O.M., Khlopkov, YuI: Monte Carlo methods in applied mathematics and computational aerodynamics. Comput. Math. Math. Phys. 46(8), 1418–1441 (2006) 17. Belotserkovskii, O.M.: Symmetric flow about blunt bodies in a supersonic stream of a perfect and real gas. U.S.S.R. Comput. Math. Math. Phys. 2(6), 1272–1304 (1963) 18. Belotserkovskii, O.M., Dushin, V.K.: The supersonic flow round blunt bodies of an unbalanced stream of gas. U.S.S.R. Comput. Math. Math. Phys. 4(1), 83–104 (1964) 19. Belotserkovskii, O.M., Golomazov, M.M., Shulishnina, N.P.: Calculation of the flow about blunt bodies with a retracted shock wave in an equilibrium dissociating gas. U.S.S.R. Comput. Math. Math. Phys. 4(2), 129–143 (1964) 20. Belotserkovskii, O.M., Sedova, E.S., Shugaev, F.V.: The supersonic flow round blunt bodies of revolution with a break in the generator. U.S.S.R. Comput. Math. Math. Phys. 6(5), 210–217 (1966) 21. Belotserkovskii, O.M., Fomin, V.N.: The calculation of flows of a radiating gas in a shock layer. U.S.S.R. Comput. Math. Math. Phys. 9(2), 188–206 (1969) 22. Belotserkovskii, O.M., Shifrin, È.G.: Transonic flows behind a detached shock wave. U.S.S.R. Comput. Math. Math. Phys. 9(4), 230–260 (1969) 23. Belotserkovskii, O.M., Popov, F.D., Tolstykh, A.I., Fomin, V.N., Kholodov, A.S.: Numerical solution of certain problems in gas dynamics. U.S.S.R. Comput. Math. Math. Phys. 10(2), 158−177 (1970) 24. Belotserkovskii, O.M., Osetrova, S.D., Fomin, V.N., Kholodov, A.S.: The hypersonic flow of a radiating gas over blunt bodies. U.S.S.R. Comput. Math. Math. Phys. 14(4), 168–179 (1974) 25. Belotserkovskii, O.M., Erofeev, A.I., Yanitskii, V.E.: A nonstationary method for direct statistical simulation of rarefied gas flows. U.S.S.R. Comput. Math. Math. Phys. 20(5), 82–112 (1980) 26. Belotserkovskii, O.M., Grudnitskii, V.G.: Study of non-stationary gas flows with complex internal structure by methods of integral relations. U.S.S.R. Comput. Math. Math. Phys. 20(6), 37–51 (1980) 27. Belotserkovskii, O.M., Golomazov, M.M., Shabalin, A.V.: Study of the effect of a strong blast of a gas from the surface on a hypersonic flow past blunt bodies. U.S.S.R. Comput. Math. Math. Phys. 21(4), 204–215 (1981) 28. Belotserkovskii, O.M., Kraginskii, L.M., Oparin, A.M.: The numerical simulation of threedimensional flows in a stratified atmosphere caused by strong large-scale disturbances. Comput. Math. Math. Phys. 43(11), 1657–1670 (2003) 29. Belotserkovskii, O.M., Tsvetkov, G.A.: The method of integral relations for computing blunt body flows with a detached shock wave. Comput. Math. Math. Phys. 51(1), 111–121 (2011) 30. Belotserkovskii, O.M., Babakov, A.V., Beloshitskiy, A.V., Gaydaenko, V.I., Dyadkin, A.A.: Numerical simulation of some problems of recovery capsule aerodynamics. Math. Models Comput. Simul. 8(5), 568–576 (2016) 31. Belotserkovskii, O.M., Belotserkovskii, S.O., Gushchin, V.A.: Numerical modeling of timedependent periodic flow of a viscous fluid in the wake of a cylinder. U.S.S.R. Comput. Math. Math. Phys. 24(4), 150–155 (1984) 32. Belotserkovskii, O.M., Gushchin, V.A., Konshin, V.N.: The splitting method for investigating flows of a stratified liquid with a free surface. U.S.S.R. Comput. Math. Math. Phys. 27(2), 181–191 (1987) 33. Belotserkovskii, O.M., Zharov, V.A., Htun, H., Khlopkov, YuI: Monte Carlo simulation of boundary layer transition. Comput. Math. Math. Phys. 49(5), 887–892 (2009) 34. Belotserkovskii, O.M., Oparin, A.M., Chechetkin, V.M.: Formation of large-scale structures in the gap between rotating cylinders (the Rayleigh–Zel’dovich problem). Comput. Math. Math. Phys. 42(11), 1661–1670 (2002)

2 The Scientific and Life Path of Academician …

13

35. Agapov, P.I., Belotserkovskii, O.M., Petrov, I.B.: Numerical simulation of the consequences of a mechanical action on a human brain under a skull injury. Comput. Math. Math. Phys. 46(9), 1629–1638 (2006) 36. Belotserkovskii, O.M., Fortova, S.V., Troshkin, O.V., Pronina, A.P., Eriklintsev, I.V., Kozlov, S.A.: Numerical simulation of the high-speed impact of two metal plates. Math. Models Comput. Simul. 8(5), 497–505 (2016) 37. Belotserkovskii, O.M.: Direct numerical modelling of free induced turbulence. U.S.S.R. Comput. Math. Math. Phys. 25(6), 166–183 (1985) 38. Belotserkovskii, O.M., Ivanov, S.A., Yanitskii, V.E.: Direct statistical simulation of some problems in turbulence theory. Comput. Math. Math. Phys. 38(3), 474–487 (1998) 39. Belotserkovskii, O.M., Oparin, A.M.: A numerical study of three-dimensional Rayleigh-Taylor instability development. Comput. Math. Math. Phys. 40(7), 1054–1059 (2000) 40. Belotserkovskii, O.M., Denisenko, V.V., Konyukhov, A.V., Oparin, A.M., Troshkin, O.V., Chechetkin, V.M.: Numerical stability analysis of the Taylor-Couette flow in the twodimensional case. Comput. Math. Math. Phys. 49(4), 729–742 (2009) 41. Belotserkovskiˇi, O.M., Fimin, N.N., Chechetkin, V.M.: Application of the Kac equation to the modeling of turbulence. Comput. Math. Math. Phys. 50(3), 549–557 (2010) 42. Belotserkovskiˇi, O.M., Fortova, S.V.: Macroparameters of three-dimensional flows in free shear turbulence. Comput. Math. Math. Phys. 50(6), 1071–1084 (2010) 43. Belotserkovskiˇi, O.M., Fimin, N.N., Chechetkin, V.M.: Coherent structures in fluid dynamics and kinetic equations. Comput. Math. Math. Phys. 50(9), 1536–1545 (2010) 44. Belotserkovskii, O.M., Konyukhov, A.V., Oparin, A.M., Troshkin, O.V., Fortova, S.V.: Structurization of chaos. Comput. Math. Math. Phys. 51(2), 222–234 (2011) 45. Belotserkovskii, O.M., Fortova, S.V.: Study of spectral characteristics of a homogeneous turbulent flow. Comput. Math. Math. Phys. 52(2), 285–291 (2012) 46. Babakov, A.V., Batsukov, O.S., Belotserkovskii, O.M., Stolyarov, L.N.: On the possibility of attaining high productivity in the solution of problems of mathematical physics using vector pipeline digital computers. U.S.S.R. Comput. Math. Math. Phys. 26(2), 171–180 (1986) 47. Belotserkovskii, O.M.: Mathematical modeling for supercomputers: Background and tendencies. Comput. Math. Math. Phys. 40(8), 1173–1187 (2000) 48. Belotserkovskii, O.M., Fimin, N.N., Chechetkin, V.M.: Possibility of explaining the existence of vortexlike hydrodynamic structures based on the theory of stationary kinetic equations. Comput. Math. Math. Phys. 52(5), 815–824 (2012) 49. Belotserkovskii, O.M., Belotserkovskaya, M.S., Denisenko, V.V., Eriklintsev, I.V., Kozlov, S.A., Oparina, E.I., Troshkin, O.V., Fortova, S.V.: On the development of a wake vortex in inviscid flow. Comput. Math. Math. Phys. 54(1), 172–176 (2014) 50. Belotserkovskii, O.M., Fimin, N.N., Chechetkin, V.M.: Statistical mechanics of vortex hydrodynamic structures. Comput. Math. Math. Phys. 55(9), 1527–1533 (2015) 51. Belotserkovskii, O.M., Fimin, N.N., Chechetkin, V.M.: Coherent hydrodynamic structures and vortex dynamics. Math. Models Comput. Simul. 8(2), 135–148 (2016) 52. Belotserkovskaya, M.S., Belotserkovskii, O.M., Denisenko, V.V., Eriklintsev, I.V., Kozlov, S.A., Oparina, E.I., Troshkin, O.V.: On the short-wave nature of Richtmyer-Meshkov instability. Comput. Math. Math. Phys. 56(6), 1075–1085 (2016) 53. Belotserkovskii, O.M., Betelin, V.B., Borisevich, V.D., Denisenko, V.V., Eriklintsev, I.V., Kozlov, S.A., Konyukhov, A.V., Oparin, A.M., Troshkin, O.V.: On the theory of countercurrent flow in a rotating viscous heat-conducting gas. Comput. Math. Math. Phys. 51(2), 208–221 (2011)

Chapter 3

Numerical Simulation of Spatial Flows in Shear Layers Svetlana V. Fortova

and Elena I. Oparina

Abstract Numerical modeling of the phenomenon of origin and development of the vortex cascade of instability depending on various flow parameters, initial data, in the absence or presence of external force is carried out. The HYPERBOLIC_SOLVER application package is used for numerical simulation of currents occurring in shear layers. To calculate the formation and evolution of large-scale formations, a series of two-dimensional and spatial computational experiments were conducted. As a mathematical model, the compressible non-viscous gas model was used, and numerical methods described in the package were used for calculations. The initial stage of development is a chaotic vortex flow in a shear layer, i.e., the process of formation of large-scale formations and statistical representation of a continuous energy flow through the vortex cascade has been studied. Direct numerical modeling shows that the transition to the developed vortex flow is carried out through a vortex cascade of instability. The conditions under which this cascade occurs are revealed. It is confirmed that it is three-dimensional staging that is essential for turbulence to occur. In the calculations, the inertial section of energy spectrum was obtained and the “Kolmogorov’s law –5/3” was confirmed with the accuracy up to 20%. Keywords Direct numerical modeling · Hydrodynamic instability · Vortex flow

3.1 Introduction Spatial vortex flows are complex nonlinear phenomena which, despite a great number of theoretical, semi-empirical and numerical studies, are still far from being fully understood [1–8]. This fully refers to the processes that connect vortex nucleation S. V. Fortova (B) · E. I. Oparina Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya Ul., Moscow 123056, Russia e-mail: [email protected] E. I. Oparina 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 214, https://doi.org/10.1007/978-981-33-4709-0_3

15

16

S. V. Fortova and E. I. Oparina

and dynamics of their development before transition to the stage of a developed vortex flow. The present chapter is devoted to the numerical analysis of the initial stage of vortex flows arising through hydrodynamic instability, vortex formation and their interaction [8–14]. The fundamental works in this direction are those of Kelvin [1], Reynolds [2], Richardson [3], Kolmogorov and Obukhov [4, 5], Belotserkovsky [6–8], Belotserkovsky [15] and others. Thanks to these works, the models that became common in the theory of turbulence were created: “Kelvin’s vortex theory,” “Richardson-Kolmogorov’s energy cascade,” “Raleigh’s stability,” “O.M. Belotserkovsky’s method of large particles,” “S.M. Belotserkovsky’s method of discrete vortexes” [1–8, 15] and a number of other significant models. Today these classical studies are an important stage of modern understanding and quantitative description of many practically important vortex modes in various objects of nature and technology. At the same time, due to the rapid development of computer technology, in recent years more and more attention has been paid to the analysis of the nature of turbulence through direct numerical simulation. In 1976, Belotserkovsky [6] pointed to the possibility of modeling free developed turbulence based on the system of the Euler equations. The most consistently proposed approach is presented in [7, 8], where the free eddy flows are numerically investigated and the determining role of large eddies in structural turbulence is justified. The present chapter is devoted to the development of this approach, to a certain extent focused on the topology and dynamics of large vortices development based on the Euler equation system. Outside the scope of our study are issues related to boundary layers, the low Reynolds numbers, and the impact of viscous effects and walls. In this chapter, we will investigate the model of vortex structures nucleation in a free shear flow [6–8]. The reader can find a discussion of the applicability of the approximation we use in the extensive literature (see [1–8]) and references there. Let us note the main points. Developed vortex movements begin at finite scales determined by velocity gradients in a flow [6–8]. Considering the initial stage with large structures, it is assumed that molecular viscosity has little influence on the development of large-scale structures [6–8]. This approach is substantiated in a large number of publications [7, 15–17] and forms the following model of vortex flow development [7, 17]. Large structures can occur if inertial terms in the equations prevail over viscosity-related stresses. In this case, forces arising from the pressure field and dynamic forces associated with the velocity field form vortex structures. The development of the eddy flow occurs by crushing large structures and generating different intervals of the energy spectrum. At the same time, it is of great interest to study conditions of appearance and dynamics of vortex flow development [6, 7, 11–13]. The model under consideration contains the following stages [7, 8, 11–13]. In a free shear flow, large vortices with the size of the order of the mixing layer width are born. Further, the interaction of large vortices with the flow gives birth to a more small-scale part of the spectrum, which subsequently fills the inertial part. The instability spectrum is determined by the physical interaction between the flow and the coarse vortex, while the development of the flow becomes a deterministic

3 Numerical Simulation of Spatial Flows in Shear Layers

17

physical phenomenon. At the same time, in accordance with the classical theory of Kolmogorov [4], the high-frequency part of the turbulence spectrum and the process of transition of the kinetic energy of turbulence into heat fall on the share of molecular viscosity. Due to the extreme complexity and nonlinearity of the phenomena under consideration, an adequate tool for their study taking into account the possibilities of modern computer technology is a numerical modeling, which is carried out by studying non-viscous and viscous flows [7, 18, 19]. A great number of works devoted to the two-dimensional numerical modeling of unsteadiness allow to interpret the natural experiment quite fully [7, 20, 21]. However, they do not explain many features of the flow. Carried out researches for two-dimensional free-shifting flows have confirmed the fact of formation of stable in time large structures [13]. At transition from calculation of two-dimensional flow to calculation of spatial flows, the loss of large vortex structures stability and transition of a flow to a developed vortex were found [9– 14]. Thus, spatial flows can be accompanied by hydrodynamic effects (loss of flow stability), which are not manifested in smaller dimensional problems. Effective numerical modeling of the mentioned hydrodynamic problems is directly connected both with the development of modern numerical methods and with the possibilities of available computing equipment. The advantage of the development of modern computing technology is the ability to make resource-intensive large-scale numerical experiments, even using personal computers. The package of applied programs HYPERBOLIC_SOLVER allows the researcher to use quickly enough a wide range of structural modules already created in computational mathematics (by numerical methods, boundary and initial conditions, various equations of substance state, etc.) and to carry out numerical simulation of a hydrodynamic task on available computational tools, e.g., personal computers. In this chapter, the HYPERBOLIC_SOLVER application package is used for numerical simulation of flows occurring in shift layers. To calculate the formation and evolution of large-scale formations, a series of two-dimensional and spatial computational experiments was carried out [6–8, 11–13]. As a mathematical model, we used the model of compressible non-viscous gas, and for calculations, we used numerical methods described in the package. Taking into account the comments made, the results of numerical modeling of the phenomenon of origin and development of the vortex cascade of instability depending on various flow parameters, initial data, in the absence or presence of external force are given below. The chapter is organized as follows. Section 3.2 provides a description of mathematical model. Vortex cascade numerical modeling is given in Sect. 3.3, while vortex cascade conditions are considered in Sect. 3.4. Section 3.5 discusses the energy characteristics of the vortex flow. Influence of flow parameters on vortex cascade structure is investigated in Sect. 3.6. Section 3.7 concludes the chapter.

18

S. V. Fortova and E. I. Oparina

3.2 Mathematical Model Compressible non-viscous gas is used as a mathematical model. To construct numerical schemes, a system of hyperbolic equations (the Euler equations) is used, which is written in divergent form in the Cartesian coordinates [7, 15]:   ∂(ρ s ) + div ρ s V = 0, ∂t ∂(ρu) ∂P + div(ρuV) = − , ∂t ∂x ∂(ρv) ∂P + div(ρvV) = − , (3.1) ∂t ∂y ∂P ∂(ρw) + div(ρwV) = − − pg, ∂t ∂z ∂(ρv) + div((ρ E + P)V) = −ρgw, ∂t  s where ρ s is the density of different components, ρ = ρ is the total gas density, V = (u, v, w) is the flow velocity, E is the total energy, ε is the internal energy, P is the pressure, g is the acceleration of gravity. The system is closed by the perfect gas state equation P = (γ − 1)ρε. To account for gravity, the right side of the equations contains additional terms that take into account the addition to momentum and total energy. Hybrid finitedifference scheme with positive operator was used for calculations [7]. The applied calculation grids contained up to 8 million nodes, which allow the use of modern personal computers.

3.3 Vortex Cascade Numerical Modeling This chapter deals with the initial stage of turbulence in the shear layer of the compressible non-viscous medium. The drift flow is formed by the presence of a finite cross section area with a nonzero speed gradient and a mixing layer height H. The evolution of the structure of a given shift flow under different boundary and initial conditions is studied numerically. The calculation area is a three-dimensional parallelepiped in a rectangular coordinate system XYZ (see Fig. 3.1) with the following dimensions: X : 0 ≤ x ≤ L x , Y : 0 ≤ y ≤ L y , Z : 0 ≤ z ≤ L z . On the upper and lower planes, the parallelogram used boundary conditions of adhesion, while on the side surfaces, conditions of periodicity are considered. Flow velocity at the start of time V 0 = –5 m/s under the mixing layer, V 0 = 5 m/s over the mixing layer. Different perturbations of the velocity components were used as initial conditions inside the mixing layer to initialize the process of large vortex

3 Numerical Simulation of Spatial Flows in Shear Layers

19

Fig. 3.1 Calculation area and basic flow parameter

structure nucleation and to model the development of instability in the shear flow. The initial conditions in the mixing layer were set as follows: Random speed disturbances. The speed of U along the X-direction is zero; speed V along Y-direction (constant speed gradient V, speed profile linear): V = V0 , 0 ≤ z ≤ bound1, V = −V0 , bound2 ≤ z ≤ L z , V = 2V0 (z − π ), bound1 < z < bound2; bound1 = π − 0.5, bound2 = π + 0.5. The velocity W along the Z-direction has a weak random perturbation inside the mixing layer bound1 ≤ z ≤ bound2 (between 1 and 20% of V 0 ). Determined speed disturbances. Single-mode speed disturbance: V = V0 , 0 ≤ z ≤ bound1, V = −V0 , bound2 ≤ z ≤ L z , V = 2V0 (z − π ), bound1 < z < bound2; bound1 = π − 0.5, bound2 = π + 0.5; U = 0.2 ∗ sin(2π ∗ y) ∗ cos(π ∗ z), W = 0.2 ∗ sin(2π ∗ y) ∗ cos(π ∗ z). Multi-mode speed disturbance:

20

S. V. Fortova and E. I. Oparina

V = V0 , 0 ≤ z ≤ bound1, V = −V0 , bound2 ≤ z ≤ L z , V = 2V0 (z − π ), bound1 < z < bound2; bound1 = π − 0.5, bound2 = π + 0.5;  ampl[i] ∗ sin(a[i]x) ∗ sin(b[i](x − c[i])), U= i

W =



ampl[i] ∗ sin(a[i]y) ∗ sin(b[i](y − c[i])).

i

The number of modes ranged from 1 to 15. The amplitude of the perturbation was ampl[i] = 2/i. The perturbation frequencies were as follows a[i] = 1/i, b[i] = 2i with shear c[i] = i. The phenomenon of the vortex cascade of instability in the shear layer is the formation of a large vortex structure during the course of the vortex and its subsequent disintegration into smaller vortex formations [7, 8, 22]. Considering various flow modes in the shear layer of the compressible unbound medium, which lead to the development of instability, we note the variety of factors affecting this development. There are the choice of the initial distribution of parameters in the perturbation layer, geometric parameters of the flow area, the influence of a constant acting force (“the Kolmogorov’s problem”), the influence of shear velocity, etc. Each of these factors has a significant influence on the formation of vortex cascades and leads to different scenarios of instability development, which are investigated in the following sections [9–14]. Numerical experiments with the HYPERBOLIC_SOLVER application package have shown [9–14] that a large vortex structure is first formed in a shear layer under certain boundary and initial conditions (see Fig. 3.2). It is a secondary flow in the form of a roll that is smoothly streamlined by an external flow [9, 10]. Then, on the surface of the vortex roll the unsteadiness in the form of vortex bundles begins to appear, forming from the streamlined external flow. These vortex tows result in the

Fig. 3.2 Formation of the vortex cascade at various points in time: a shear layer, b roll layer, c vortex bundles

3 Numerical Simulation of Spatial Flows in Shear Layers

21

destruction of a large structure and the appearance of vortexes of different scales during the flow. This phenomenon is called the vortex cascade of instability in the shear layer [9, 10, 14]. Let us pay special attention to the fact that secondary vortex rods sit on the primary vortex in transverse direction [9, 10]. The calculations show that if the secondary vortices sit in the longitudinal direction, the vortices are not crushed. This result seems to us to be obtained for the first time. A general view of the vortex cascade formation in the shear layer described above is shown in Fig. 3.2.

3.4 Vortex Cascade Conditions Let’s consider in more details the formation of the vortex cascade depending on the size of the calculated area at random speed disturbance in the mixing layer. Figure 3.3 shows the influence of the area width (in the X-direction) on the instability development dynamics. At the beginning of time, a high-frequency random perturbation of the vertical velocity component was set within the mixing layer. The size of the area varied from 2π to π/4. The deformation of the mixing layer is as follows. At first, one large vortex is formed with the scale of the mixing layer size. The stability of this large structure depends on the width of the area in the X-direction. The critical dimensions of the vortex cascade are determined. At the size of the area within π/4 ≤ L x ≤ 2π , large structure is not sustainable. In this case, the flow passes into the developed vortex through the development of the instability cascade. For the size of the flow area 0 ≤ L x ≤ π/4, the large vortex is stable in time. This means that in a thin channel (in the X-direction) the formed secondary flow is stable, well consistent with the Batchelor’s two-dimensional theory [23]. The effect of the amplitude of random velocity disturbances in the mixing layer has also been investigated. It has been obtained that an increase in the velocity perturbation amplitude leads to a faster development of the vortex instability cascade as compared to a smaller perturbation amplitude. Figure 3.4 shows the vortex flow development at different amplitudes of velocity disturbance. At ampl = 2, the developed vortex has already formed by the time t = 7. The flow with smaller amplitude (10 times) at the same moment of time is still only entering the stage of a large structure formation. In fact, a flow with amplitude amp = 2 develops 2 times faster than a flow with amplitude amp = 0.2. Let’s consider in more detail the formation of the vortex cascade depending on the size of the calculated area in case of deterministic speed disturbance in the mixing layer. For more detailed study of the influence of initial conditions on the development of the flow, we will study the type of vertical component of velocity V (u, v, w), which will lead to the vortex cascade. For this purpose, we will introduce the function: w = a ∗ sinωy,

22

S. V. Fortova and E. I. Oparina

Fig. 3.3 Iso-surface of density in vortex cascade at various points in time t and various lengths len of the shear layer in the direction (the initial velocity perturbation in the mixing layer is chosen randomly): a t = 5, len = X − 2π, b t = 5, len = π, c t = 5, len = π /4, d t = 7, len = X–2π, e t = 7, len = π, f t = 7, len = π /4, g t = 12, len = X–2π, h t = 12, len = π, i t = 12, len = π /4

by means of which we will investigate influence of the given initial disturbances of speed on a flow. Using within the mixing layer the perturbation for the vertical component of velocity in this form, we tried to simulate a random velocity variation. However, as a result of the calculations, we have not developed a vortex cascade. The flow formed under these conditions can rather be called a “collapse” of the vortex [9]. Let us explain this phenomenon in more detail. The vortex flow is formed into

3 Numerical Simulation of Spatial Flows in Shear Layers

23

Fig. 3.4 Density iso-surface of at various points in time t and speed disturbance amplitudes amp: a t = 7, amp = 0.2, b t = 12, amp = 0.2, c t = 17, amp = 0.2, d t = 7, amp = 2, d t = 12, amp = 2, e t = 17, amp = 2

a large structure that is stable in time. This structure is a large vortex, as if flattened in transverse direction. Over time, this vortex becomes flat and disappears when it is compressed in the transverse direction. The flow becomes undisturbed. This process is shown in Fig. 3.4. Note that the change of frequency ω and the amplitude a speed disturbance affects only the vortex formation time and its disappearance: The greater

24

S. V. Fortova and E. I. Oparina

Fig. 3.5 Process of “collapse” of the vortex at various points in time: a t = 5, b t = 30, c t = 50

the amplitude and the higher the frequency, the faster the “collapse” process takes place. The explanation for this fact may be as follows. In Fig. 3.5, we can see that on the surface of the formed large structure, there is a secondary instability in the form of vortex bundles located in the longitudinal direction. This differs from the previous case where the harnesses appeared on the surface of a large structure in a transverse direction. Apparently, this is the main cause of the “collapse.” These considerations lead to the following conclusion. If secondary instability appears on the surface of a large structure in a transverse direction, the flow passes into a developed vortex. If secondary instability appears on the surface of a large structure in the longitudinal direction, such a transition does not occur, and the flow remains laminar. Perhaps, this fact can be explained by the fact that the process of development and growth of instability depends on the mechanism of vortex bundle formation on the surface of a large vortex. Vortical harnesses, being in a zone of deformation of bigger scale structure, are stretched and twisted under the action of the angular momentum conservation law. If the harnesses are located on the surface of a large structure across a shear flow, forces occur that can destroy the secondary flow and form a vortex cascade of instability. It should be emphasized that this process has a fundamentally three-dimensional nature. The secondary instability on the surface of a large structure in a transverse direction and resulting in a chaotic flow was obtained using deterministic velocity perturbations in both vertical and horizontal directions as initial data. The result of this study for a single-mode speed disturbance is presented in Fig. 3.6. The appearance of the vortex cascade and its dependence on the size of the calculated area are shown here. Similar to the variant with a random initial perturbation of the vertical component of velocity, we obtained a single large vortex with a scale of the mixing layer size [10]. The stability of this large structure also depends on the width of the area in the X-direction. The critical dimensions of this area are similar: At π/4 ≤ L x ≤ 2π we observe the collapse of a large vortex and the transition of the flow into a chaotic vortex cascade [9, 10, 14]. With area size 0 ≤ L x ≤ π/4 the large vortex is stable in time. Similar results are obtained for multi-mode speed disturbance in the mixing layer. The number of modes, amplitude, frequency and shift do not significantly change the developed picture of the flow.

3 Numerical Simulation of Spatial Flows in Shear Layers

25

Fig. 3.6 Iso-surface of density in vortex cascade at various points in time t and various lengths len of the shear layer in the direction (the initial perturbation of the vertical and horizontal velocity component is determined): a t = 5, len = X–2π, b t = 5, len = π, c t = 5, len = π /4, d t = 7, len = X – 2π, e t = 7, len = π, f t = 7, len = π /4, g t = 12, len = X – 2π, h t = 12, len = π, i t = 12, len = π /4

Summarizing the above and moving on to the conditions leading to the vortex cascade of instability, it should be noted that: The results show that both a random perturbation of the vertical velocity component and a deterministic perturbation of velocity in the vertical and transverse direction at the initial moment of time lead to a vortex cascade of instability and

26

S. V. Fortova and E. I. Oparina

the transition of the flow to a developed vortex with these cases. Secondary instability appears on the surface of a large structure in the form of vortex tows in the transverse direction, which seems to be a determining factor in the vortex cascade. In the presence of a deterministic disturbance of only the vertical component of the velocity, the secondary flow remains stable for a certain period of time, followed by a “collapse” of the vortex structure. In our opinion, the defining fact when this flow regime occurs is that the secondary instability appears on the vortex surface in the longitudinal direction. Disturbance component amplitude growth leads to vortex flow development rate increase.

3.5 Energy Characteristics of the Vortex Flow Specially introduced integral characteristics are used to describe and analyze developed vortex flows having fundamentally stochastic character [1–6]. In this section, when studying the vortex stage of a flow, we will briefly consider only some integral characteristics, which we will later use when considering the numerical results obtained. For the analysis of vortex flow development, the spectral analysis of energy characteristics is used, the founders of which are Kolmogorov [4] and Obukhov [5]. They studied and derive laws describing local properties of homogeneous and isotropic turbulence [4, 5]. Let us briefly present their main contents. At the large numbers of Reynolds’s number Re in a homogeneous turbulent flow, pulsation movements with a wide spectrum are present. In this process, the bulk of the energy of the turbulent flow contains large-scale pulsations. Small-scale pulsations are also present in motion, but contain a smaller fraction of the entire kinetic energy of the flow. Thus, from largescale movements a continuous flow of energy feeds movements with smaller scales. Eventually, on a smaller scale (due to viscosity) energy dissipation into heat occurs. The expected mechanism of energy exchange was described in detail in the Kolmogorov’s article [4]. This model has been widely recognized, confirmed by a large number of experiments and theoretical studies, and by now has become a classic one. According to [4, 5], the following scheme of formation of developed vortex motion is realized. At the large numbers of Re, the turbulent motion is divided into three intervals. On the interval of energy, which consists of large structures formation, the main part of the flow energy is generated. On the inertial interval, there is a continuous energy transfer from large vortices to small ones. And on the dissipation interval through small-scale turbulence, the kinetic energy of vortices passes into heat [4, 5]. For a wide class of physical processes at large numbers of Re at energy and inertial intervals of developed vortex motion, the influence of molecular viscosity on the general characteristics of the flow is insignificant [7, 15]. Therefore, the study of the dynamics of large vortices can be carried out on the basis of an ideal medium models, such as the Euler equations [4, 7]. In this part of the work, the main

3 Numerical Simulation of Spatial Flows in Shear Layers

27

attention is paid to the inertial interval, which plays an important role for the whole investigated process. Note the main features of the inertial interval [22]: Energy in it is not generated and does not dissipate, but is continuously transmitted in the space of wave numbers from smaller to larger. Let us give the known estimates for the energy distribution of the ripple motion E(k) of the velocity field for the inertial interval, derived from similar considerations [7]: E(k) ≈ 2/3 k −5/3 where k is the wave number,  is the rate of energy dissipation per unit mass. This law was called “Kolmogorov’s law –5/3.” The existence of inertial section is proved experimentally for different turbulent flows. The wave number corresponding to the point of transition from inertial interval to dissipation interval is called the “Kolmogorov’s” scale. For vortex flows with wave numbers smaller than this scale, the viscosity can be neglected and a numerical experiment based on the model of an ideal medium can be conducted. In the case of flows with large wave numbers comparable to the “Kolmogorov’s” scale, viscosity begins to play a significant role and affects the nature of the interaction of small vortices. At the modern level of computer technology, it is possible to carry out high-quality calculations on a dissipative section of the spectrum for flows with the Reynolds’ characteristic numbers of about 100,000. In this chapter, when numerical modeling vortex motion in shear layers, special attention is paid to the construction of kinetic energy spectral distribution of velocity field motion, determination of inertial region of spectrum and proof of the “Kolmogorov’s law –5/3.” The spectral representation of the kinetic energy will be obtained by decomposition in a two-dimensional Fourier integral. Each of the velocity components can be expanded into a series of orthogonal harmonic functions: vi (x, y) =

 kx

+ +

[vi(1) (k x , k y ) cos(k x x) cos(k y y)

ky

vi(2) (k x , k y ) cos(k x x) sin(k y y) vi(3) (k x , k y ) sin(k x x) cos(k y y)

+ vi(4) (k x , k y ) sin(k x x) sin(k y y)], i = 1, 2,

(3.2)

where vi is one of the velocity components, k x and k y are the wave vector components ( j) along OX and OY, respectively, vi (k x , k y ), (j = 1, 4) are the Fourier coefficients:   1 2π 2π E(x, y, z = const) cos(k x x) cos(k y y)dxdy, i = 1, 2, π 0 0   1 2π 2π vi(2) (k x , k y ) = E(x, y, z = const) cos(k x x) sin(k y y)dxdy, i = 1, 2, π 0 0 vi(1) (k x , k y ) =

28

S. V. Fortova and E. I. Oparina

  1 2π 2π = E(x, y, z = const) sin(k x x) cos(k y y)dxdy, i = 1, 2, π 0 0  2π  2π 1 E(x, y, z = const) sin(k x x) sin(k y y)dxdy, i = 1, 2. vi(4) (k x , k y ) = π 0 0 (3.3) vi(3) (k x , k y )

Value ε(k x , k y ) =

[v1 (k x , k y )]2 + [v2 (k x , k y )]2 , 2

where vi (k x , k y ) =

 (1) (2) (3) (4) [vi (k x , k y )]2 + [vi (k x , k y )]2 + [vi (k x , k y )]2 + [vi (k x , k y )]2 , i = 1, 2,

(3.4) will be the desired image of the kinetic energy in the space of the wave numbers. Thus, calculating Fourier coefficients using Eq. 3.3 and, further, calculating Eq. 3.4, we obtain a quantity ε(k x , k y ), depending on k x and k y . Going through all k x and k y in both directions, we obtain the energy spectrum. Let’s move on to the presentation of the results using the characteristics described above. Figures 3.7 and 3.8 show the distribution and spectrum of kinetic energy with the fulfillment of the “Kolmogorov’s law –5/3” with an accuracy of 20%. At the stage of formation of a large structure at t = 5, the flow energy is concentrated in the center of the shear layer. At t = 10, the energy gradually increases and reaches its maximum in the center of the formed large structure. From this we can conclude that in the turbulent flow large vortices not only determine the structure of the flow, but also carry its main energy [10]. Further, when the vortex cascade develops, the energy is distributed on the scale of the cascade. That is, during the vortex flow development the energy gradually passes from large vortices to smaller ones and finally dissipates into heat, which corresponds to the Kolmogorov’s theory [3, 4].

3.6 Influence of Flow Parameters on Vortex Cascade Structure Analyzing only those flow modes in which the presence of a vortex cascade is observed, let us consider in detail the evolution of the shear flow structure with a finite cross-sectional area of a constant nonzero velocity gradient with different heights of the mixing layer H. As an initial velocity perturbation within the mixing layer, we take deterministic perturbations which, as shown above, lead to the vortex

3 Numerical Simulation of Spatial Flows in Shear Layers

29

Fig. 3.7 Density iso-surfaces and kinetic energy distributions along Y-direction at various points in time: a density iso-surface, t = 5, b kinetic energy distribution, t = 5, c density iso-surface, t = 10, d kinetic energy distribution, t = 10, d density iso-surface, t = 15, e kinetic energy distribution, t = 15

30

S. V. Fortova and E. I. Oparina

Fig. 3.8 Spectrum of kinetic energy of speed pulsation at the developed stage of vortex motion at the moment of time t = 28

cascade. This part of the work aims to show possible scenarios of transition to a turbulent flow stage in a shear layer of compressible unconcerned medium. The initial stage of the vortex cascade development at the perturbation layer height H = 1 is considered in the previous paragraphs and consists of one large structure formation smoothly streamlined by the flow. In the case of small thickness of mixing layer (0.1 < H < 1), a different picture of vortex cascade formation is observed. The coarse structure first represents a bundle of two vortex coils (see Fig. 3.9). In the further development of the flow, these coils are combined, and a single large structure is also formed. Over time, the instability occurring on the surface of the coarse vortex is destroyed and a vortex cascade is formed. Figure 3.9 shows the vortex cascade development, wave number energy distribution and speed pulsation for the mixing layer height H = 0.1. Let’s consider the spectral representation of kinetic energy in the middle of the region by vertical coordinate and trace the process of formation of a stable section of the spectrum energy of ripple velocity components. Figure 3.9 shows the distribution of the speed pulsations energy    

E = u 2 + v 2 + w 2 2 on the wave numbers of longitudinal ripples with an inclination of –59° = arctg (–5/3). The initial stage of development of the vortex instability cascade at the height of disturbance layer L within 2 < H < 3 is a different picture (see Fig. 3.10). The loss of stability of the flow occurs without the stage of a large structure formation.

3 Numerical Simulation of Spatial Flows in Shear Layers

31

Fig. 3.9 Vortex instability cascade for shear layer, kinetic energy spectrum and speed pulsation at various points in time: a vortex instability, t = 0.1, b vortex instability, t = 1.6, c vortex instability, t = 2.8, d vortex instability, t = 8, e kinetic energy spectrum, f speed pulsation at H = 0.1

32

S. V. Fortova and E. I. Oparina

Fig. 3.10 Vortex instability cascade for shear layer, kinetic energy spectrum and speed pulsation at various points in time: a vortex instability, t = 0.1, b vortex instability, t = 0.7, c vortex instability, t = 2, d kinetic energy spectrum, e speed pulsation at L = 3

3 Numerical Simulation of Spatial Flows in Shear Layers

33

The instability occurs at the perturbation layer boundaries and develops in a vortex cascade by direct caving into turbulence. An analogous study of the spectral characteristics of the flow revealed the presence of an inertial region in the energy spectrum and the implementation of the “Kolmogorov’s law –5/3”. Figure 3.10 shows, in particular, the evolution of the pulsation of the longitudinal velocity component as a function of time. One can observe their gradual development along the vertical direction. This is a pulsation limit mode with zero average. Over time, these pulsations develop and fluctuate around zero. This evolution shows the establishment of a turbulent profile in flows with no viscosity or walls. These results are in good agreement with the experimental studies conducted by Cont-Bello [24]. The conclusions of this part of the work are as follows: Depending on the height of the mixing layer in the shear flow, there are different development scenarios of vortex cascades: At low layer height, the transition to the turbulent stage occurs by forming a bundle of two large structures and at high layer height this process develops without forming large structures. Regardless of the scenario, the calculations carried out in the generalized sense revealed the presence of an inertial section in the energy spectrum and implementation of the “Kolmogorov’s law –5/3” with an accuracy of up to 20%. The occurrence of turbulence in all the cases considered is a complex threedimensional phenomenon.

3.7 Conclusions In this study, we obtained the following results. The initial stage of development of a chaotic vortex flow in a shear layer, i.e., the process of formation of largescale formations and statistical representation of a continuous energy flow through the vortex cascade, has been studied. Direct numerical modeling shows that the transition to the developed vortex flow is carried out through a vortex cascade of instability. The conditions under which this cascade occurs are revealed. It has been confirmed that it is the three-dimensional production that is essential for turbulence to occur. In the calculations, the inertial section of energy spectrum was obtained, and the “Kolmogorov’s law –5/3” was confirmed with the accuracy up to 20%.

References 1. Kargon, R., Achinstein, P.: Kelvin’s Baltimore Lectures and Modern Theoretical Physics: Historical and Philosophical Perspectives. MIT Press, Cambridge, Massachusetts (1987) 2. Reynolds, O.: An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc. London 174, 935–982 (1883)

34

S. V. Fortova and E. I. Oparina

3. Richardson, L.F.: The supply of energy from and to atmospheric eddies. Proc. Roy. Soc. London 97A, 354–373 (1920) 4. Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. In: Proceedings of the Royal Society A: Mathematical and Physical Sciences 434(1890), pp. 9–13 (1991). Turbulence and stochastic processes: Kolmogorov’s ideas 50 years, Proc. R. Soc. Lond. A, pp. 9–13. Comp. a. ed. by J.C.R. Hunt et al. London: Roy. soc. (1991) 5. Obuhov, A.M.: On the distribution of energy in the spectrum of turbulent flow. Comptes Rendus de l’Académie des Sciences U.R.S.S 32 (1941) 6. Wirz, H.J., Smolderen, J.J. (eds).: Numerical Methods in Fluid Dynamics. Washington, Hemisphere Pub. Corp. (1978) 7. Belotserkovsky O.M.: Turbulence and Instabilities. Lewinstoon–Quinston–Lamper (2000) 8. Belotserkovsky, O.M., Oparin, A.M., Chechetkin, V.M.: Physical processes underlying the development of shear turbulence. J. Exp. Theor. Phys. 99(3), 504–509 (2004) 9. Belotserkovsky, O.M., Fortova, S.V.: Macroscopic parameters of three-dimensional flows in free shear turbulence. Comput. Math. Math. Phys. 50(6), 1071–1084 (2010) 10. Belotserkovsky, O.M., Fortova, S.V.: Investigation of the cascade mechanism of the turbulence development in the free-shift flow. Dokl. Phys. 57(3), 110–113 (2012) 11. Fortova, S.V.: Investigation of spectrum characteristics of the vortex cascades in shear flow. Topical Issue of the Physica Scripta, Phys. Scr. 155, 014049 (2013) 12. Fortova, S.V.: Numerical simulation of the three-dimensional Kolmogorov flow in a shear layer. Comput. Math. Math. Phys. 53(3), 311–319 (2013) 13. Fortova, S.V.: Eddy cascade of instabilities and transition to turbulence. Comput. Math. Math. Phys. 54, 553–560 (2014) 14. Fortova, S.V., Kraginskii, L.M., Chikitkin, A.V., Oparina, E.I.: Software package for solving hyperbolic-type equations. Math. Models Comput. Simul. 5(6), 607–616 (2013) 15. Belotserkovskii, O.M., Oparin, A.M., Chechetkin, V.M.: Turbulence: New Approaches. Cambridge International Science Publishing Ltd (2005) 16. Loitsyanskii, L.G.: Fluid and gas mechanics: International Series of Monographs in Aeronautics and Astronautics: Division II: Aerodynamics. Elsevier Science (2014) 17. Monin, A.S., Yaglom, A.M.: Statistical fluid mechanics: mechanics of turbulence. MIT Press, Cambridge (1971) 18. Godunov, S.K., Ryabenkii, V.S.: Difference Schemes: An Introduction to the Underlying Theory. North-Holland (1987) 19. Kulikovskii, A.G., Pogorelov, N.V., Semenov, AYu.: Mathematical aspects of numerical solution of hyperbolic systems. In: Jeltsch, R., Fey, M. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, ISNM, vol. 130, pp. 589–598. Birkhäuser, Basel (1999) 20. Rogers, M.M., Moser, R.D.: The three-dimensional evolution of a plane mixing layer: the Kelvin-Helmholtz rollup. J. Fluid Mech. 243, 183–226 (1992) 21. Moser, R.D., Rogers, M.M.: The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence. J. Fluid Mech. 247, 275–320 (1993) 22. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics (Volume 6 of Course of Theoretical Physics). Pergamon Press (1959) 23. Betchelor, G.K.: The theory of homogeneous turbulence. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1953) 24. Comte-Bellot, G.: Ecoulement turbulent entre deux parois parallèles. Publications Scientifiques et Techniques du Ministère de l’air 419 (In French) (1965)

Chapter 4

Modeling of Unsteady Flows in Gas Astrophysical Objects on Supercomputers Alexander V. Babakov

Abstract The chapter describes a numerical technique for modeling the spatial unsteady motion of matter in gas astrophysical objects. Convective processes with the formation of vortex structures in a fast-rotating star under self-gravity conditions are simulated. The modeling of spatially unsteady motion in an accretion disk at the boundary with a neutron star is also performed. The numerical technique used is based on a finite-difference analog of the conservation laws of the medium additive characteristics for a finite volume. For astrophysical objects under self-gravity conditions, the direct calculation of gravitational forces is realized by summing the interaction between all finite volumes in the integration area. Evolutionary calculations are based on the parallel algorithms implemented on the computer complexes of cluster architecture. The algorithms use parallelization in space and in physical factors. Visualized pictures of spatial non-stationary structures are presented. Keywords Astrophysics · Gas dynamics · Massive stars · Gravity · Accretion disks · Vortex structures · Mathematical simulation · Numerical methods · Conservative difference schemes · Parallel algorithms

4.1 Introduction In astrophysics, the study of the emergence and development of spatially unsteady motion of matter in gas objects is still relevant. These phenomena are typical for such compact astrophysical objects as rotating massive stars and accretion disks. In this field of research, methods of mathematical modeling based on numerical methods of continuum mechanics (in particular, aero-hydrodynamics) are quite useful [1–8]. In this case, various gas-dynamic models are used. The effectiveness of numerical A. V. Babakov (B) Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya Str., Moscow 123056, Russia 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 214, https://doi.org/10.1007/978-981-33-4709-0_4

35

36

A. V. Babakov

techniques and modeling capabilities is increased dramatically using the modern multiprocessor supercomputers [9–12]. In the present work, which is a continuation of the researches [13–15], attention will be paid to modeling unsteady motion in the accretion disk rotating around a neutron star, and vortex motion in the fast-rotating star. The chapter is organized as follows. Section 4.2 provides a general description of the numerical method used. Section 4.3 is devoted to a discussion using a direct computation of gravitational forces for compact astrophysical objects in self-gravity conditions. Section 4.4 discusses the efficiency of parallel algorithms for direct computation of the gravitational field on supercomputers of cluster architecture. Section 4.5 presents the results of simulation of instability of the stellar accretion disk area near neutron star surface. Section 4.6 is devoted to modeling the vortex structures in a rapidly rotating star. The chapter ends with conclusions in Sect. 4.7.

4.2 Numerical Technique For modeling, the conservative numerical method of flux [16, 17] implemented on multiprocessor computer complexes [18, 19] is used. The method is based on a finitedifference approximation of the conservation laws for additive characteristics of the medium, written in integral form for a finite volume. For each finite volume Ω arising from the discretization of the integration region, the conservation laws of mass M, momentum components Pk , and total energy E are written in a view of Eq. 4.1. ∂F =− ∂t



 Q F n dS + G F = SΩ

f dΩ F = (M, Pk , E) f = (ρ, ζk , ε) (4.1) Ω

Here, Q F is the vector of the flux density of the additive characteristic F, SΩ is the bounding Ω surface, f = (ρ, ζk , ε) is the corresponding density of distribution of mass, momentum components, and total energy, G reflects the contribution of the gravitational force acting on the final volume to the momentum components and total energy. Note that the equations of motion (Eq. 4.1) in the method used are always written in Cartesian coordinate system for Cartesian components of vector quantities (flux density vectors, momentum, velocity, gravitational force), regardless of the geometry of the problem and the type of computational grids used. Considering the use of a gas-dynamic model of a perfect gas for modeling the astrophysical problems, the system of equations 4.1 is supplemented by the equation of state p = (γ − 1)ρe, where p is the pressure, γ is the ratio of specific heat capacities, e is specific internal energy).

4 Modeling of Unsteady Flows in Gas Astrophysical …

37

Let the index m indicate the value of the function at the characteristic internal point of the volume Ωm , index n is the number of the time layer, τ is the step of integration over time. The center of mass of the volume Ωm is selected as the characteristic point. Then, a finite-difference analog of the conservation laws (Eq. 4.1) can be written in the form of equations 4.2. ⎛ ⎞ 3     f mn+1 − f mn =− Ωm vj gj m⎠ (Q F n dS) Si + G m G m = ⎝0, (gk )m , τ S j=1 i

(4.2) Here, gk is the Cartesian components of the gravity force acting on the volume Ωm . The flux density vectors of the additive characteristics are calculated at the characteristic points of the surfaces S i of volume Ωm . The “upwind” approximations of densities of fluxes of the medium additive characteristics are used. The choice of the approximation order densities of fluxes (first or second) is based on a local analysis of the flow structure in the vicinity of the characteristic points of the surfaces. The symmetric approximation of transfer velocities and pressure at these points are used. When using non-uniform grids, the approximation of the components of the fluxes vectors on the surfaces of a finite volume is carried out by taking into account the coefficients of interpolation and extrapolation relations. The explicit difference scheme has a first order of approximation in time. The system of equations 4.2 is supplemented by the boundary conditions, equation of state, and equation for calculating the gravitational forces and is closed with respect to the values of density ρm , velocity components (vk )m , and internal energy em at characteristic points of volumes Ωm . Most gas astrophysical objects have a fast-spin motion. When constructing a finitedifference scheme for modeling the motion of matter in such objects, the following peculiarity must be taken into account. The finite-difference model used is based on the laws of conservation of mass, momentum components, and total energy. At the same time, in the formulations considered below, the conservation laws for the angular momentum components must also be satisfied. However, these laws are not included into the defining system of equations 4.1 and their implementation when using the finite-difference analog (Eq. 4.2) is not obvious. In order to achieve a conservation of the components of angular momentum in the integration domain in evolutionary calculation in each finite volume, and at its boundaries, in addition to the general Cartesian coordinate system, the local Cartesian coordinate systems are introduced. The distribution densities of the momentum components (and transfer velocities) are approximated in these local Cartesian directions. After that, the momentum components and transfer velocities are recalculated into the general Cartesian coordinate system. Such an approach made it possible to achieve conservation of the angular momentum components with high accuracy at evolutionary calculations during integration even for long times.

38

A. V. Babakov

4.3 Direct Calculation of Gravitational Forces As mentioned above, when modeling astrophysical problems with gravitational interaction, a relation is needed that determines the gravitational forces acting on the finite volume Ωm in order to close the system of finite-difference equations 4.2. Earlier in [20] to calculate the gravitational forces, the methodology based on the solution of Poisson equation for the gravitational potential [21] was used. The method of flux is a finite volume method. Remaining within the framework of this approach for objects under the self-gravity conditions and in the gravitational field of other objects, the direct calculation of the gravitational interaction between all finite volumes which are in the field of integration is used:

     N  ρk Ωk x j m − x j k   1 ρm ≥ ρmin , δk δm = g j m = −γg ρm Ωm δm 3 0 ρm < ρmin |r | − r m k k=m (4.3) where γ g is the gravitational constant. For finite volume methods, such an approach seemed more consistent. Moreover, it is applicable to computational grids of any type. Note that in Eq. 4.3 in the calculation of gravitational forces, the finite volumes having density ρ m < ρ min are excluded. Also, calculations are not performed at the summation, if ρ k < ρ min . This approach allows to increase the performance of the algorithm without significantly losing the accuracy of calculations. In calculations, the value of ρ min was taken equal to 10−8 with respect to the characteristic density of the problem. The approach used to calculate gravitational forces is possible only when using the parallel algorithms implemented on supercomputers, since the number of operations is proportional to the square of the number of finite volumes in the integration area, which in real calculations in the spatial case are several tens of millions.

4.4 Parallel Algorithms of Numerical Technique The integration area is divided into sub-areas in two or three coordinate directions (depending on the size of the computation grid and the capabilities of the computing system). In the parallel algorithms used, in addition to parallelization over space, parallelization on physical factors is implemented. Namely, the calculation of convective motion at each integration step without taking into account gravity is implemented on some processors, while the calculation of gravitational forces in parallel on the others with their accounting in conservation laws.

4 Modeling of Unsteady Flows in Gas Astrophysical …

39

Fig. 4.1 Dependence of relative computation time of gravitational forces on the number of processors used

The calculation of the gravitational force acting on each finite volume requires the presence of the density field in entire integration area. This is accomplished by allocating a separate processor to which density fields from each sub-area are transmitted. Next, the density field of the entire integration area from this processor is transferred to processors that calculate the gravitational forces in each sub-area. It should be noted that the effectiveness of such an approach and one in which the density fields are exchanged directly between all processors may depend on the particular computing system used. The developed parallel algorithms implemented on multiprocessor computing systems provide a real opportunity for direct calculation of gravitational forces. In this case, the calculation time directly depends on the number of processors used. Figure 4.1 shows the graph of the dependence of relative computation time of gravitational forces on the number of processors used k = T (64)/T (N), obtained by calculations on the computational grid containing more than 40 million finite volumes. T (N) is the time to calculate gravitational interaction using N processors. Calculations are made for N = 64, 128, 256, 512, 1024, 2048. Calculations were performed for the problem presented in Sect. 4.6. With an increase in the number of processors used, there is no loss of efficiency due to exchange procedures (at least with N ≤ 2048). Parallel algorithms use the standardized message passing interface (MPI) message transfer system with blocking and non-block exchange procedures. Numerical simulations were carried out on a computing complex of the cluster architecture. The complex used includes 207 compute nodes with a peak performance of 521 TFlops, consisting of 2 Xeon E5-2690 (Sandy Bridge) processors (64 Gb memory) and 2 Xeon Phi 7110X (KNC) processors (16 Gb memory) connected by networks based on Infiniband FDR and Gigabit Ethernet.

40

A. V. Babakov

4.5 Simulation of Instability of the Stellar Accretion Disk Area Near Neutron Star Surface In [15], results of numerical modeling of the large-scale structures that are formed as a result of the development of hydrodynamic instability in the external area of the stellar accretion disk, which is rapidly rotating around a neutron star, are presented. The simulation of similar structures at the boundary of the accretion disk with a neutron star is investigated below. The model problem of unsteady processes in the inner region of the accretion disk of mass M a = 1.088 * 1031 g and equatorial radius R0 = 1.0 * 107 cm located in the central gravitational field of a neutron star and rapidly rotating around it is considered. The mass of a neutron star is M s = 2.7846 * 1033 g, which is ~ 1.4 solar masses, the radius of the neutron star r 0 = 1.0 * 106 cm is assumed to be 0.1 of the equatorial radius of the accretion disk R0 . When modeling movements in the accretion disk, a gas-dynamic model of a perfect, non-viscous gas with a specific heat capacity ratio of γ = 5/3 is used. The three-dimensional motion in the accretion disk is simulated in the cylindrical region (r, ϕ, z) = (r 0 ≤ r ≤ R1 )*(0 ≤ ϕ ≤ 2π )*(–z0 ≤ z ≤ z0 ): R1 = 1.5R0 , z0 = 0.3R0 . Finite volumes m are formed by splitting in constant steps along the coordinate z, the radial r and angular ϕ coordinates. At the boundary with a neutron star, the conditions are set for the normal velocity component to be equal to zero, and for the pressure to be equal to the pressure at the nearest calculated point of the accretion disk. At the external boundary of the integration region, zero values of velocity, pressure, and internal energy are set. The parameters of the integration region are chosen for the perturbation not to reach the external boundary in the process of evolutionary calculation. Recall that the system of finite-difference equations 4.2 is written in the Cartesian coordinate system for the Cartesian component vector quantities. In calculations, the computational grid including 25 million of finite volumes was used. Modeling was carried out on a cluster architecture computing complex using up to 1000 processors. For the region of the accretion disk located in the central gravitational field of a neutron star, the initial pressure and density fields at zero velocities in the r and z directions were set based on the polytropic equation of state of the gas medium, so that the entropy in the initial field of the accretion disk is constant in the integration region. This corresponds to hydrodynamic equilibrium (possibly unstable). As characteristic variables, we choose the initial equatorial radius of the accretion disk R0 , temperature T 0 , and density ρ0 at the boundary with the neutron star at the initial moment of time. For the characteristic velocity, we take a0 = a0s /γ 0.5 , where a0s is the speed of sound corresponding to T 0 . The characteristic time is t 0 = R0 /a0 , and the characteristic pressure is p0 = ρ0 a20 . In the simulation results presented below, the parameters are assigned to the specified characteristic values.

4 Modeling of Unsteady Flows in Gas Astrophysical …

41

Fig. 4.2 Initial field: a initial temperature, pressure, and density profiles (logarithmic scale) in the equatorial plane, b density field in the meridional plane (logarithmic scale)

Figure 4.2 shows the initial density profiles (curve 1, logarithmic scale), temperature (curve 2), and pressure (curve 3) in the equatorial plane z = 0 (Fig. 4.2a), and also the initial density field (logarithmic scale) in the meridional plane ϕ = 0 (Fig. 4.2b). For the values of gas-dynamic variables and the gravitational field given by analytical relations, the finite-difference equations 4.2 at the initial time is performed with a certain errors, which is a type of small disturbances. These disturbances in the unstable state of hydrodynamic equilibrium, especially in conditions of a large gravitational mass of a neutron star, can lead to a non-stationary process, which is observed in the evolutionary calculation. When integrating in time, the numerical solution on the internal boundary of the accretion disk becomes non-stationary. In this case, the accretion disk loses its axial symmetry. Figure 4.3 shows the pressure distribution in the equatorial plane (z = 0) along the coordinate ϕ at four points of the radial direction for two time instants: r = 1.0125r 0 , 1.0375r 0 , 1.1125r 0 , 1.1375r 0 . Thus, Fig. 4.3 indicates a loss of flow stability in the inner region of the accretion disk bordering the neutron star and a loss of axial flow symmetry in this region. In Fig. 4.4 for different time instants, the density field patterns in the equatorial plane are presented. The presented pictures show the development of instability in the inner region of the accretion disk bordering the neutron star.

42

A. V. Babakov

Fig. 4.3 Distribution of pressure along the circumferential coordinate in the equatorial planes of the accretion disk at various points in time: a t = 2.5, b t = 100

4.6 Vortex Structures Simulation in Fast-Rotating Star The evolutionary calculation of the structure of a fast-rotating gas object under the self-gravity conditions is considered. In the gas-dynamic model used, the compressible gas is assumed to be perfect with the ratio of specific heat capacities γ = 4/3. The parameter of the problem for the selected law of rotation is the ratio β of polar radius to equatorial (β < 1). The parameter β determines the profiles of the angular velocities of rotation and gas-dynamic variables in the initial field. In [14], the results of modeling large-scale vortex motion in rotating stars with the parameter β = 0.7, 0.6, 0.5 are presented. The following are the results of numerical modeling of the occurrence and development of large-scale vortex structures in a fast-rotating gas star at β = 0.2. Evolutionary calculation is carried out for a model of a fast-rotating star of a shape close to an ellipsoid of rotation with the indicated ratio of polar and equatorial axes β, equatorial radius R0 = 0.29R and mass M = 21.8 M with density in the center ρ 0 = 2.65 ×105 g/cm3 and temperature T 0 = 0.43*109 K (index  indicates a solar parameter). Finite-difference analogs of conservation laws are written in the Cartesian coordinate system for the Cartesian components of vector quantities. The computational grid is specified in a spherical coordinate system. The three-dimensional motion is simulated in the spherical region (r, ϕ, ψ) = (0 ≤ r ≤ R1 )*(0 ≤ ϕ ≤ 2π )*(0 ≤ ψ ≤ π ): R1 = 1.5R0 . The equilibrium initial field of variables for a fast-rotating self-gravity region is set on the basis of the technique [22] for the polytropic equation of state and angular velocity, which is a function of only the distance from the axis of rotation.

4 Modeling of Unsteady Flows in Gas Astrophysical …

43

Fig. 4.4 Density field in equatorial plane accretion disk at various times: a t = 0, b t = 2.5, c t = 50, d t = 200

For finite-difference relations, the initial configuration given in accordance with the above method is close to the equilibrium, which can be considered (similar to Sect. 4.5) as introducing small disturbances into it. Perturbations of the gravitational field and pressure gradient lead to the medium moving in the radial direction, which, in the presence of rotation, puts the numerical solution into the non-stationary mode with the appearance of large-scale vortex structures. As characteristic variables, we choose the initial equatorial radius of the star R0 , temperature T 0 , and density ρ0 in the center of star at the initial moment of time. The characteristic velocity, time, and pressure are similar to that introduced in Sect. 4.5. Below, the corresponding parameters are assigned to the specified characteristic values. In Fig. 4.5 for various time instants, instantaneous streamlines in the plane of star section passing through the axis of rotation are shown against the background of the temperature fields.

44

A. V. Babakov

Fig. 4.5 Temperature fields and instantaneous streamlines in the section of a star: a t = 0.1, b t = 0.5, c t = 1.0, d t = 2.0

The above pictures give an idea of the occurrence of vortex motion in the region of mass bulk of the star. Evolutionary calculation shows an increase in the number of torus-like vortex structures, their movement in the radial direction, and the emergence of new structures in the region of high density.

4.7 Conclusions Calculations performed on the basis of a conservative numerical technique demonstrate a possibility of modeling spatially unsteady and vortex motions in compact, fast-rotating astrophysical objects, as well as, objects under self-gravity conditions. The possibility of performing numerical calculations using direct calculation of gravitational forces based on efficient parallel algorithms implemented on modern supercomputers was also demonstrated.

4 Modeling of Unsteady Flows in Gas Astrophysical …

45

Acknowledgements Author is grateful to A.G. Aksenov for helpful discussions of the setting of the initial field for the modeling of fast-spin stellar objects. Calculations were carried out on the computational resources of the Joint Supercomputer Center of the Russian Academy of Sciences (JSCC RAS).

References 1. Belotserkovskii, O.M., Mingalev, I.V., Mingalev, V.S., Mingalev, O.V., Oparin, A.M., Chechetkin, V.M.: Formation of large-scale vortices in shear flows of the lower atmosphere of the earth in the region of tropical latitudes. Cosmic Res. 47(6), 466–479 (2009) 2. Pudritz, R.E., Norman, C.A.: Bipolar hydromagnetic winds from disks around protostellar objects. Astrophys. J. 301, 571–586 (1986) 3. Wongwathanarat, A., Muller, E., Janka, H.T.: Three-dimensional simulations of core-collapse supernovae: from shock revival to shock breakout. Astronomy Astrophys. 577, 1–20 (2015) 4. Velikhov, E.P., Sychugov, K.R., Chechetkin, V.M., Lugovskii, AYu., Koldoba, A.V.: Magnetorotational instability in the accreting envelope of a protostar and the formation of the large-scale magnetic field. Astronomy Rep. 56(2), 84–95 (2012) 5. Sawada, K., Matsuda, T., Hachisu, I.: Spiral shocks on a Roche lobe overflow in a semi-detached binary system. Monthly Not. Roy. Astron. Soc. 219, 75–88 (1986) 6. Dolence, J.C., Burrows, A., Zhang, W.: Two-dimensional core-collapse supernova models with multi-dimensional transport. Astrophys. J. 800, 10 (2015) 7. Osher, S., Solomon, F.: Upwind difference schemes for hyperbolic systems of conservations laws. Math. Comput. 38, 339–374 (1982) 8. Einfeldt, B.: On Godunov type methods for gas dynamics. SIAM J. Numer. Anal. 25, 294–318 (1988) 9. Lugovskii, AYu., Chechetkin, V.M.: The development of large-scale instability in Keplerian stellar accretion disks. Astronomy Rep. 56(2), 96–103 (2012) 10. Melson, T., Heger, A., Janka, H.T.: Supernova simulations from a 3D progenitor model—impact of perturbations and evolution of explosion properties. Monthly Notices Roy. Astronomical Soc. 472(1), 491–513 (2017) 11. Lugovsky, AYu., Mukhin, S.I., Popov, YuP, Chechetkin, V.M.: The development of large-scale instability in stellar accretion disks and its influence on the redistribution of angular momentum. Astronomy Rep. 52(10), 811–814 (2008) 12. Velikhov, Ye.P., Lugovsky, A.Yu., Mukhin, S.I., Popov, Yu.P., Chechetkin V.M.: The impact of large-scale turbulence on the redistribution of angular momentum in stellar accretion disks, Astronomy Rep. 51(2), 154–160 (2007) 13. Babakov, A.V., Popov, M.V., Chechetkin, V.M.: Mathematical simulation of a massive star evolution based on a gasdynamical model. Math. Models Comput. Simul. 10(3), 357–362 (2018) 14. Babakov, A.V., Lugovsky, A.Yu., Chechetkin, V.M.: Mathematical modeling of the evolution of compact astrophysical gas objects. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Lakhmi, C.J. (eds.) GCM50 2019, SIST, vol. 133, pp. 210–227. Springer, Cham (2019) 15. Babakov, A.V., Lugovsky, A.Yu., Chechetkin, V.M. Modelling of some astrophysical problems on supercomputers using gasdynamical model. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 23–35. Springer, Singapore (2020) 16. Belotserkovskii, O.M., Severinov, L.I.: The conservative “flow” method and the calculation of the flow of a viscous heat-conducting gas past a body of finite size. Comput. Math. Math. Phys. 13(2), 141–156 (1973)

46

A. V. Babakov

17. Belotserkovskii, O.M., Babakov, A.V.: The simulation of the coherent vortex structures in the turbulent flows. Adv. Mech. Poland. 13(3/4), 135–169 (1990) 18. Babakov, A.V.: Numerical simulation of spatially unsteady jets of compressible gas on a multiprocessor computer system. Comput. Math. Math. Phys. 51(2), 235–244 (2011) 19. Babakov, A.V.: Program package FLUX for the simulation of fundamental and applied problems of fluid dynamics. Comput. Math. Math. Phys. 56(6), 1151–1161 (2016) 20. Aksenov, A.G., Babakov, A.V., Chechetkin, V.M.: Mathematical simulation of the vortex structures in the fast rotation astrophysical objects. Comput. Math. Math. Phys. 58(8), 1287–1293 (2018) 21. Aksenov, A.G.: Numerical solution of the Poisson equation for the three-dimensional modeling of stellar evolution. Astronomy Lett. 25, 185–190 (1999) 22. Aksenov, A.G., Blinnikov, S.I.: A Newton iteration method for obtaining equilibria of rapidly rotating stars. Astronomy Astrophys. 290, 674–681 (1994)

Chapter 5

Modeling of the Physical Processes of the Impact of a Powerful Nuclear Explosion on an Asteroid Daria S. Moiseeva , Andrey A. Motorin , and Evgenii L. Stupitsky

Abstract The physical and theoretical analysis of the influence of various factors of high-altitude nuclear explosion on the asteroid in space was completed as part of the problem of the asteroid–comet hazard. In accordance with the energy and permeability of the plasma products of explosion X-ray and gamma-neutron radiation, a layered structure with a different energy density depending on angular coordinates is formed on the surface of the asteroid. As a result of numerical studies of the impact of the explosion at various distances from the surface of the asteroid, it is shown that taking into account the real geometry of the spallation layer gives the optimal height for the formation of the maximum momentum. The approximate expression is obtained for estimating the average size of fragments in the event of possible destruction of an asteroid by shock waves. Keywords Asteroid-comet hazard · High-altitude nuclear explosion · Fragmentation

D. S. Moiseeva (B) · E. L. Stupitsky Moscow Institute of Physics and Technology (National Research University), 9, Institutsky Per., Dolgoprudny, Moscow Region 141701, Russia e-mail: [email protected] E. L. Stupitsky e-mail: [email protected] A. A. Motorin · E. L. Stupitsky Institute for Computer Aided Design of the RAS, 19/18 Vtoraya Brestskaya Ul., Moscow 123056, Russia 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 214, https://doi.org/10.1007/978-981-33-4709-0_5

47

48

D. S. Moiseeva et al.

5.1 Introduction In recent decades, the scientific community of leading countries has attracted considerable attention to the problem of preventing the possibility of collision with the Earth of large cosmic bodies—asteroids and comets. Astronomical observations, geological data, information on the evolution of the biosphere and the results of space research of planets and their natural satellites showed that catastrophic collisions of these bodies with planets were not only in the distant past, but in also quite probable in the modern era. The Earth’s atmosphere as a whole completely protects us from the impact of bodies up to several meters in size. A real danger to the Earth is the collisions with large bodies—asteroids and comets several tens of meters in size. Book [1] gives a classification, the basic kinematic characteristics of meteors, asteroids and comets, as well as, a qualitative picture of the consequences of a collision with the Earth, and an assessment of model counteraction schemes using a nuclear explosion (NE). The results of the work carried out from 1996 to 2013 indicated that the creation of an Earth protection system based on modern nuclear missiles practically eliminates the risk of an asteroid collision with a planet up to 1 km in diameter. However, the initial data on the asteroids themselves are still far from enough for strictly quantitative consideration. Therefore, of considerable interest is the qualitative physical and mathematical approach to the analysis of the processes of the impact of a nuclear explosion on an asteroid. It is necessary for the formation of the conceptual appearance of a system for protecting the Earth from large cosmic bodies at a safe distance from it. This chapter is devoted to such a study of the problem under discussion. In Sect. 2, physical processes are analyzed and the results are given that were obtained in calculating various factors of a powerful nuclear explosion produced at a certain distance from the surface of an asteroid. Section 3 analyzes the fragmentation of an asteroid as a result of the nuclear explosion. Since the plasma of the nuclear explosion has a high degree of ionization, Sect. 4 evaluates the possibility of interference action of the plasma, which can occur when the second echelon of lowpower nuclear weapons is designed to destroy the largest fragments of the asteroid. Section 5 concludes the chapter.

5.2 Physics of the Impact of a Nuclear Explosion There are two possible physical mechanisms for actively countering the space threat using a nuclear explosion: Destruction of an object and its fragmentation into parts no larger than a few meters. Removal of the body from a dangerous orbit as a result of the transmission of a momentum to it from the plasma flow of the explosion, penetrating radiation, and

5 Modeling of the Physical Processes of the Impact …

49

also as a result of the reactive action of the mass evaporating from the surface of the body. The nature and results of the impact of a nuclear explosion on an asteroid depend primarily on the position of the explosion center relative to the asteroid’s surface. This chapter considers the solution of the problem in two stages using a rocket with two carrier nuclear modules at a great distance from the Earth. The first module places a nuclear weapon with a power of ~1 ÷ 5 megatons, which explodes at an optimal distance from the surface of the asteroid (40–100 m). In this case, the asteroid is affected by: X-radiation (X-ray), γ-radiation, neutron fluxes and Plasma Products of Explosion (PPE). As a result of the influence of these factors, a certain mass of the heated substance is carried away from the perturbed layer of the surface facing the explosion, and shock waves propagate inside the asteroid. Thus, the asteroid receives a momentum that displaces it from the trajectory, and fragmentation is possible. The second multi-purpose module houses low-power charges (1–10 kt), and a radar system for pointing them at large fragments formed from the first powerful explosion is applied in conditions of plasma-dust interference in order to destroy the fragments formed. The main parameters of the explosion that determine the momentum of the asteroid are the energy E, the mass of the explosion products Mp and the height of the explosion h above the surface of the asteroid. Figure 5.1 shows a geometric scheme of the effect of NE damaging factors on the asteroid and angular characteristics.

Fig. 5.1 Geometric scheme of the nuclear explosion above the surface of asteroid

50

D. S. Moiseeva et al.

The geometric parameters in Fig. 5.1 are defined according to the following formulae: dS⊥ = dS sin γ = dS cos(α + β), γ = π2 − (α + β),  = sin γ = cos(α + β), dS = Rdβ ∗ 2π sin β. λ The general connection of the integral values of mass Mc , momentum Pc , and energy E c − Pc2 /2M = E c gives Eq. 5.1. Pc =



2π R 2 ρE ·



(1 − cos βk )(1 − cos αk )

(5.1)

√ (R+h)2 −R 2 r R Since cos αk = R+h = and cos βk = R+h differentiating Eq. 5.1 by R+h h and equating the derivative to zero, we obtain the approximate optimal height for the nuclear explosion: hm =

√  2 − 1 R.

(5.2)

Since the physical content of the processes that determine the effect of the above damaging factors of NE is different, it is of interest to consider in more detail the process of formation of the asteroid momentum, assuming the dependence (β). This is also important because the shock waves that arise in this case have spherical geometry, and this can affect the fragmentation process. The following parameters of NE were adopted for specification: E = 4.2 × 1022 erg, M P = 106 g, A = 27(Al) is the molecular weight of PPE, N P = 2.2 × 1028 is the number of ions of PPE. For instantaneous radiation of high-energy neutrons and γ-quanta, the average value of 4 meV was assumed, and their numbers Nn = 1.5×1026 1/Mt, Nγ = 6×1025 1/Mt [2]. For the accepted value of the specific energy E/M P = 4.2 × 1012 J/kg, the average initial temperature of PPE is T ≈ 3.6 keV [2]. The following energy parts were taken for these factors in: Neutrons δn = 0.023. γ-quanta δγ = 0.09. X-ray δν = 0.87. PPE δ p = 0.1. For the accepted energy values, the neutron velocity was υn = 2.7 × 109 cm/s, the front velocity of√PPE was υ f r = 1.2 × 108 cm/s and, accordingly, the average velocity was υ P = 3/5υ f r = 0.93 × 108 cm/s. If X-ray is assumed to be Planck’s radiation, then the maximum of its spectrum ˙ will fall on the energy of quanta: hνmax = 2.82T ∼ = 10 keV (1.24 A). The following values were taken for the calculation: ρ = 2 g/cm3 , A = 25 g/mol. The radius of the asteroid was set equal to 100 m. Based on the above parameters of the nuclear explosion and asteroid material, the nature of their interaction is analyzed. Fast neutrons experience elastic and diffraction

5 Modeling of the Physical Processes of the Impact …

51

scattering by atomic nuclei. The total scattering cross section is: σn = 2π R 2N , where R N = 1.5 × 10−13 A1/3 . Then mean free path of the fast neutron will be: λn =

A 1 A1/3 = = 11.8 (cm). σn n σn ρ N A ρ

We consider the stone with parameters: A = 25 g/mol, ρ = 2 g/cm3 , λn = 17.3 cm (for iron: A = 56 g/mol, ρ = 7.9 g/cm3 , λn = 5.7 cm). The mean free path of γ-radiation is λγ ≈ 16 cm. The mean free path of βelectrons is λβ ≈ 0.2 cm. Since the maximum of X-ray emission falls on quanta with an energy of ≈10 keV, the absorption of such quanta is determined mainly by the ionization of electrons from the internal K-shells of atoms. For z = 13 and hν = 2300 eV σν ≈ 5 × 10−22 cm2 , the mean free path is λν = 4.2 × 10−2 cm. As shown in [3], based on detailed calculations in the molecular dynamics approximation, ions of PPE penetrate the substance S i to a depth of ∼ = 1 microns, that is, this is a significant surface effect. Next, the characteristic time of the energy release processes and their time sequence are estimated in relation to the conditions of the explosion at a height close to the optimum (h ≈ 40 m) and for the epicentral region on the surface of the asteroid (r ≈ h). The first momentum comes in ≈ 10−7 s of X-ray and γ-radiation. X-ray radiation due to the photoelectric effect from the K-shell creates electrons with an energy of ~100–1000 eV; γ-quanta as a result of the Compton scattering creates MeV-electrons. Afterward, the ionization of substance by these fast electrons occurs, and at the same time the energy from electrons transfers to ions that is the substance is heated. Since X-ray radiation carries away the main part of the explosion energy, and the thickness of its absorption layer is small, high temperatures (≈100 eV) are reached in this layer. At the electron energy εe ≈ 100 ÷ 150 eV, the electron ionization cross section is maximum and is σi ≈ 10−16 cm2 , then the characteristic time of energy exchange between electrons and ions of different multiplicity will be τei = 1.6 × 10−11 ÷ 1.6 × 10−13 s. Thus, τg  τi , τei in the X-ray absorption layer, and it can be assumed that the transformation of X-ray energy into ionization and heat will occur long before the layer has time to noticeably expand. The propagation time of the X-ray momentum is r/c ≈ 1.3 × 10−7 s; then, after a time of the order of 10−8 s, the expansion of the heated X-ray layer of the substance and the formation of the recoil momentum to the asteroid begin. After a moment of time r/υn ≈ 1.5 × 10−6 s, a neutron momentum arrives. The scattering of neutron energy occurs on atoms that is it immediately turns into heat. The total neutron flux energy is much smaller than the X-ray energy, and the depth of their energy dissipation layer is n  ν . Therefore, as similar estimates and the

52

D. S. Moiseeva et al.

calculation results below show, the temperature in the layer n is much lower than in the layer ν . Moreover, it can be assumed that during the time of neutron energy release, the substance in the layer n does not have time to significantly expand: τg = n /υ >> τn = n /υn as υ ∼ 106 cm/s, υn = 2.7 × 109 cm/s. Thus, the second momentum occurs due to the momentum given by the neutrons themselves and the recoil momentum of the layer  √n . The total neutron momentum Pn = 2m n Nn Eδn and its value directed along the OO1 are determined by:

Pn||

1 = Pn 2

βk  2 R sin β cos(α + β) cos αdβ. r 0

Similar expressions determine the flux density σ P|| and the total momentum PP|| created√by the plasma of the explosion products, for which the total momentum is PP = 2m P N P Eδ P (Fig. 5.2). Since the average mean free path of neutrons and γ-radiation is approximately the same λn ≈ λγ , the thickness of the layers in which energy is released depends on the angle β: nγ = λnγ cos(α + β), ν = λν cos(α + β). The density of energy released in these layers is determined by the expressions: ρ Enγ =

E(δn + δγ ) Eδν , ρ Eν = , 4πr 2 ρλnγ 4πr 2 ρλν

Fig. 5.2 Projection of plasma pulse flux density σ P|| and neutrons σn|| created by nuclear explosion products on the surface of an asteroid depending on the angle β

5 Modeling of the Physical Processes of the Impact …

53

Fig. 5.3 Density of energy released in the gamma-neutron absorption layer ρ Enγ and X-ray radiation ρ Eν depending on the angle β

Figure 5.3 shows the energy density depending on the angle in the gamma-neutron and X-ray absorption layers. In the X-ray layer, it is much higher than in the gamma-neutron. The total mass and energy in these layers were determined from the expressions: βk M0nγ = 2π R ρλn

sin β cos(α + β)dβ,

2

0

E nγ

1 = δn + δγ 2

βk  2 R sin β cos(α + β)dβ. r 0

Similar expressions are defined for parameters M0ν , E ν , and E P . It should be noted that the plasma of the explosion products arrives after a time equal to 4 × 10−5 c, when the neutron layer has flown off already by a distance of the order of a meter, and the X-ray layer has flown off even further. The plasma flow is stretched in time ∼ t13 , and transmitting momentum to the asteroid, its effect on the surface is apparently erosive. The energy density determines the temperature increase in both layers and, accordingly, their thermal expansion rate and recoil momentum: 1 Tν = (ρ Eν − Q − Q i ) , C where Q is a specific fracture energy of the structure of the asteroid material. Fusion heat of aluminum is 3.8 × 109 erg/g and iron is 2.7 × 109 erg/g. In amorphous bodies,

54

D. S. Moiseeva et al.

which, apparently, are mainly asteroids, there is no clear value of the particle binding energy in the equilibrium position, but according to approximate representations its average value is of the order of 1 eV [3]. Then, the specific fracture energy for a substance with A = 25 g/mol is approximately 3 × 1010 erg/g. Evaporation heat is close to the value 9 × 1010 erg/g, Qi is a specific ionization energy. For Qi , an approximation expression was obtained depending on the degree of ionization of the

substance α = n e n, C is a specific heat of the substance—C = 5 × 106 (1 + α) erg/g degree. The average speed is determined by the expression:  υν = 3.2 × 103 T0 + Tν , (cm/s). Similar expressions are used for calculation parameters Tnγ and υnγ . Figure 5.4 shows the angular distribution of temperatures and velocities.

Fig. 5.4 Temperatures T and velocities υ in the absorption layers of radiation depending on the angle β for: a neutron, b X-ray

5 Modeling of the Physical Processes of the Impact …

55

Table 5.1 Integral by angle values of mass, momentum and energy Pn|| (g cm/s)

Pp|| (g cm/s)

M 0n (g)

M 0v (g)

P0n|| (g cm/s)

P0v|| (g cm/s)

8.873 × 1010

1.156 × 1013

2.053 × 109

4.928 × 106

7.462 × 1014

5.036 × 1013

E nγ (erg)

E v (erg)

E p (erg)

P0Σ (g cm/s)

u0 (cm/s)

E Σ (erg)

2.030 × 1020

5.484 × 1021

6.164 × 1020

8.082 × 1014

96

6.303 × 1021

These values are used to calculate the projection of the recoil momentum from neutrons and X-rays in the direction OO1 : βk P0ν|| = 2π R ρλν

υν (β) sin β cos(α + β) cos βdβ,

2

0

βk P0n|| = 2π R ρλn

υnγ (β) sin β cos(α + β) cos βdβ.

2

0

Thus, the total momentum that the asteroid receives from the explosion is calculated using Eq. 5.3. P0 = Pn|| + Pp|| + P0n|| + P0ν||

(5.3)

The asteroid speed acquired from this pulse is as follows: u = P0 /M. Table 5.1 shows the main integral values obtained in the calculations. Calculations were also performed for various explosion heights. The results show that a correct consideration of the geometry of the spallation layers of an asteroid will lead to a significant change in the optimal explosion height (Fig. 5.5).

5.3 Research of the Asteroid Fragmentation. Theoretical Analysis At present, there are no any well-founded ideas about the structure of the material, of which stone asteroids consist [4, 5]. It can be assumed that as a result of prolonged exposure to solar radiation, their stone structures contain a large number of cracks of various sizes. For that reason, compression and rarefaction waves formed inside the asteroid by neutron, X-ray and plasma flows can lead to its fragmentation [6]. A brief analysis of the work on assessing the possible fragmentation of a large asteroid due to a nuclear explosion is given in [7]. From this review, it becomes clear that a

56

D. S. Moiseeva et al.

Fig. 5.5 Total momentum, determined by the Eq. 5.3, which the asteroid receives, depending on the height of the nuclear explosion at a distance h

significant number of fragments formed of several tens of meters in size that cannot be burned in the atmosphere poses an additional problem on how to eliminate them. In [7], based on physical analysis, an approximate expression is obtained for the average fragment size: r=

2π R 4 σm2 , E Eg

(5.4)

where E is the Young’s modulus; E g is the energy released on the asteroid, which passes into the limiting energy of elastic deformation. Since the material of the asteroid is not known definitely enough to use an adequate value of σm and E, and also about the value of E g can only be talked about as an approximate, the structure of Eq. 5.4 was tested on a simple experiment, to a certain extent corresponding to the logic of its obtaining. From a certain height h of several meters, a body (brick) fell onto an absolutely solid surface and split into fragments of various sizes. Since in the reference editions there is a significant variation in the values σm and E, in accordance with the available data, their average values were used in this work: σ m = 1.87 × 107 Pa i E = 2.75 × 1010 Pa. Assuming that all potential energy mgh goes into E g , then for the values of m = 3.3 kt and h = 3 m accepted in the experiment, it follows from Eq. 5.4 that r = 2.65 cm, the value corresponds rather well to the average value of r obtained in the experiment. This fact, apparently, indicates that Eq. 5.4, although very approximately, generally correctly describes the dependence of the average fragment size on the parameters. The distribution by the number of fragments of this size also approximately corresponds to a power-law dependence, which is typical for the fragmentation process [8]. The above calculation results show that with the adopted initial parameters, the total energy transferred to the absorbed layer of the asteroid is ≈ 15% from E 0 that is ≈ 6.3 × 1021 erg. As a result of the momentum transfer, the velocity of the asteroid in the direction OO1 (Fig. 5.1) will be u ∼ = 100 cm/s, and, accordingly, the kinetic energy of its movement will be Mu 2 /2 ∼ = 4.2 × 1016 erg. Based on general

5 Modeling of the Physical Processes of the Impact …

57

physical considerations (Virial theorem), it can be assumed that approximately the same energy will be spent on elastic deformation, which will lead to fragmentation. The decisive contribution to the transfer of energy to the asteroid is made by shock waves generated by X-ray and neutron momentum, as well as, by products of a nuclear explosion. However, it is clear from the geometry (Fig. 5.1) that these sufficiently powerful perturbations release their energy in a converging direction, approximately in a cone is with  a solid angle  2π (1 − cos βk ). In the case when the optimal explosion  height √  √ ◦ 2 − 1 R, the limit angle is βk = 45 , so the solid angle is π 2 − 2 /2, h=  √  which is a part 2 − 2 /8 of the total volume of the asteroid. equivalent spherical radius of this sector volume is estimated as re f = The   √  √  R 3 2 − 2 /8 = 0.42R = 42[3] 2 − 2 /8 = 0.42R = 42. The radius r for the average size of an asteroid fragment can be estimated from Eq. 5.4, assuming its effective radius re f = 42 m as the size of the body, the energy that is released in it—4.2 × 1016 erg = 4.2 × 109 J, same values for σ m , and E, as for the brick. Substituting these values in Eq. 5.4, the resulting radius is r = 58 m, i.e., this radius corresponds to re f in order of magnitude. Thus, the analysis of geometric and physical representations shows that the resulting Eq. 5.4, apparently, gives a physically correct idea of the size of the asteroid. Since the explosion power of 1 Mt relative to the size of the asteroid D = 200 m is relatively small, the asteroid can split into sufficiently large fragments, the number  3 of which is of the order of N ∼ = R r ≈ 8. Then, to eliminate them, the second echelon of nuclear explosions of relatively low power can be used (~1 kt).

5.4 On the Radiophysical Situation in the Field of Explosion and the Possibility of Using Space Rocket Defense in Solving the Problem of Asteroid–Comet Hazard A certain difficulty exists in the process of inducing nuclear explosive on the fragments of the asteroid, since the radiophysical situation after the first powerful nuclear explosion is complicated by the formed ionized cloud (IC). The expanding products of the nuclear explosion, to which the escaped matter of the asteroid is added, determine the degree of ionization. The main mass of the cloud is associated with the addition of neutrals M0ν = 5 × 106 g, but its degree of ionization is small. The expanding thin layer formed by X-ray is substantially ionized. Its mass M0ν = 5 × 106 g is comparable to the mass of PPE M P = 106 . According to calculations [9], as a result of rapid expansion, the charge composition is quenched. At specific energy ~4.2 × 1012 J/kt, the average degree of ionization of an expanding plasma is α ∼ 1. However, the plasma cloud of the explosion expands at a speed of ~108 cm/s and, accordingly, the electron concentration decreases on average as:

58

D. S. Moiseeva et al.

Fig. 5.6 Dependence of the delay time t on the frequency of an electromagnetic wave f when a radio signal passes through an ionized region

ne =

N 3 4 π 5 (υt)

After the electron concentration satisfies the inequality n e < n ∗e (where n ∗e is the critical electron concentration, which is proportional to the square of the frequency of the electromagnetic wave ~ f 2 ), electromagnetic waves can with a certain absorption pass through the ionized region. Figure 5.6 shows the change in the critical electron concentration n ∗e and the time t(s), through which the passage of the wave through IC becomes possible. But during this time, all the fragments of the asteroid will have time to shift to a distance S from the place of impact, when the first powerful nuclear explosion occurred. For f = 1 GHz, this displacement is approximately 200 m at an initial velocity of ~30 km/s. Thus, the kinematic problem arises the possibility of directing low-power nuclear explosive to large fragments of an asteroid. In this chapter, only some physical questions of the problem of eliminating the asteroid hazard are discussed, since the goal was not to discuss the kinematic issues of the time–space orientation of the objects in question during their dynamics. It is clear that even with sufficiently reliable and detailed calculation data on the spatiotemporal behavior of the ionization parameters of the plasma region of the explosion produced at outside magnetospheric distances, the development of a kinematic algorithm for the operation of rocket and space technology is a serious independent problem.

5.5 Conclusions In the present work, the issue of the transformation of the energy in a nuclear explosion produced at a distance from the surface of the asteroid was studied in sufficient

5 Modeling of the Physical Processes of the Impact …

59

detail. It showed that the main contribution to the momentum comes from the heated expanding mass formed by the neutron flux of 10 dyne·s. The asteroid velocity acquired from this pulse is ~10 cm/s. Based on the obtained approximate relationship for the average fragment size, it is shown that at a blast power of ~1 Mt, it is likely that only a few large fragments can form. The two-module rocket system is proposed, and the possibility of destroying asteroid fragments with an additional set of low-power nuclear charges is estimated. Acknowledgements The work was performed within the state task of the ICAD RAS.

References 1. Shustova, B.M., Ryhlovoj, L.V. (eds.): Asteroid-Comet Hazard: Yesterday, Today, Tomorrow. Fizmatlit, Moscow (2010).(in Russian) 2. Physics of Nuclear Explosion. vol. 1, Fizmatlit, Moscow (in Russian) (2009) 3. Smirnov, E.V., Stupitskii, E.L.: Numerical simulation of the effect of rarefied plasma flow on the solid surface. J. Surf. Investigation: X-ray Synchrotron Neutron Tech. 4(4), 965–975 (2010) 4. Ghiorso, M.S., Nevins, D., Cutler, I., Spera, F.J.: Molecular dynamics studies of CaAl2Si2O8 liquid II. Equation of state and a thermodynamic model. Geochim. Cosmochim Acta 73(22), 6937–6951 (2009) 5. Chopelas, A.: Thermal properties of forsterite at mantle pressures derived from vibrational spectroscopy. Phys. Chem. Minerals 17(2), 149–156 (1990) 6. Solem, J.C.: Interception of comets and asteroids on collision course with Earth. J. Spacecraft Rockets 30(2), 222–228 (1993) 7. Andrushchenko, V.A., Moiseeva, D.S., Motorin, A.A., Stupitsky, E.L.: Modeling the physical processes of a powerful nuclear explosion on an asteroid. Comput. Res. Modeling 11(5), 861–877 (2019) 8. Pilyugin, N.N., Vinogradov, Yu.A., Ermolaev, I.K.: On the modeling of the disruption of cosmic bodies at high-speed impacts. Sol. Syst. Res. 35(2), 141–150 (2001) 9. Stupitsky, E.L., Kholodov, A.S.: Physical Research and Mathematical Modeling of Large-Scale Geophysical Experiments. Intellekt, Dolgoprudny (2019).(in Russian)

Chapter 6

A Multidimensional Multitemperature Gas Dynamic and the Neutrino Spectrum in 2D Gravitational Collapse Alexey G. Aksenov

Abstract A multitemperature code intended for the numerical solution of the multicomponent gas dynamics equations in problems with a high energy density in matter is described. The velocities of all components with “nonzero” masses are assumed to be identical. The gas dynamic part is based on the Godunov’s scheme and an efficient Riemann problem solver with an approximate local equation of state. As an example of the code application, the gravitational collapse of the massive star’s core with a neutrino transport is considered. A self-consistent formulation of the gravitational collapse is solved using 2D gas dynamics, taking into account the spectral transport of neutrinos in the framework of neutrino flux-limited diffusion. Large-scale convection leads to an increase in the mean energy of the neutrinos from 10 to 15 MeV, which is important for explaining supernovae, as well as, for designing experiments on detecting high-energy neutrinos from supernovae. Keywords Neutrino · Supernovae · Multitemperature gas dynamic

6.1 Introduction A special interest in physics is concerned with problems for a multicomponent gas of different substances α described by a set of densities ρα (r, t) ≡ cα (r, t)ρ(r, t), where cα are concentrations, ρεα (r, t) are internal energy densities, εα is specific energy. All massive particles (atoms, ions, and nucleons) have identical velocities v(r, t) and temperatures, while the “massless” fast particles from viewpoint of the total density ρ (likeelectrons, photons) have their own temperatures. The equations of state are P = α P(ρ, εα ), εα = εα (ρ, Tα ). The components can exchange energy, can transfer energy by heat conduction not associated with the velocity of the massive particles, and can participate in reactions. Such problems arise in inertial thermonuclear fusion [1], laser ablation experiments [2, 3], and astrophysics [4]. This A. G. Aksenov (B) Institute for Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow, Russian Federation 123056 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 214, https://doi.org/10.1007/978-981-33-4709-0_6

61

62

A. G. Aksenov

is an intermediate case between the description based on the Boltzmann equations for the one particle distribution function provided by Eq. 6.1, where r is the radius vector and p is the momentum, and classical single component gas dynamics. ∂ fα ∂ fα ∂ f α (r, p, t) +v +F = ∂t ∂r ∂p



df dt

 (6.1) coll

The basic mathematical problem concerns the gas dynamic part because of the discontinuities in the solution. Efficient Riemann problem solvers for such mixture of gases are constructed; see e.g. [5, 6] for some special case of Equation Of State (EOS). Below an original method based on the Riemann problem solver for the multitemperature non-equilibrium gas [7–9] is briefly described. The method was applied within the plasma physics for the inertial heavy ion fusion [1], for the laser ablation [7], and useful in astrophysical tasks with hydrodynamic and the radiation transfer [10, 11]. Each component in the mixture has its own density and specific internal energy. In the local model for EOS proposed, it is assumed that the entropy variations in neighboring mesh cells are small at the evaluation of the dimensionless coefficients EOS from the pressure jump across the discontinuity. In the case of an arbitrarily large pressure jump, the model yields physically reasonable results. However, the algorithm for solving the Riemann problem should be tested for correctness. During long time in our computations [1], it was assumed that the Riemann solver deals with small pressure jumps on a shock wave (SW), but in real cases the pressure jumps are not small in nearest cells of the computational grid. The samples’ multitemperature shock wave structure in plasmas was considered in [12]. This solution is a suitable test for the technique developed. An alternative similar code in the literature is CASTRO [13], which uses a different approximation for the Riemann problem solver. Other numerous gas-dynamical models for mixtures with the separate descriptions of the gas dynamics of the matter and fast particles are very limited by the computational power in the explicit multidimensional tasks. In an optically dense, opaque region, the number of time steps for the computations of the energy exchange between fast particles of various energies will be determined by the opacity for the fastest reaction. In a joint treatment of the matter and fast particles, the transport only weakly affects the thermal distribution, and reactions computed using an implicit scheme in individual steps do not influence the number of time steps required for the gas-dynamical transport. The chapter is organized as follows. Section 6.2 includes a discussion about the multidimensional multitemperature gas dynamic code with the approximate Riemann problem solver. The application of the code to a 2D gravitational collapse of a rotating core with the neutrino spectrum is given in Sect. 6.3. Section 6.4 concludes the chapter.

6 A Multidimensional Multitemperature Gas Dynamic …

63

6.2 The Multidimensional Multitemperature Gas Dynamic Code with the Approximate Riemann Problem Solver The system in the fixed Euler coordinates is: the mass transfer equations for the components ∂ρα + divρα v = ρ c˙α , ∂t

(6.2)

the momentum conservation law ∂ρv + Div = 0, ∂t

(6.3)

and the energy density equations ∂ρ E α + div(ρ E α + Pα )v + v(cα gradP − gradPα ) = div(κgradTα ) + ρqα , (6.4) ∂t where the energy densities E α = εα + cα v2 /2, the tensor i j = ρvi v j + Pδi j , and the equation of state P = α Pα (ρ, c, εα ) with specific energies εα (ρ, c, Tα ). The kinetic coefficients c˙α , κα , and qα depend on ρ, c, and T. The problem is computed by applying dimensional splitting. The heat conduction equations are solved using central difference approximations. As a result, the system of partial differential equations (PDEs) is reduced to an ordinary differential equations (ODEs) system for ε˙ α, i . The ODEs system is solved by applying the implicit Gear’s method [14]. To describe the kinetics of reactions, the ODEs system for ρ˙α and ε˙ α is solved in each grid cell also by the Gear’s method. The hydrodynamic part of the code is based on a high-order explicit Godunov scheme for single-temperature single-component gas dynamics [7, 15]. A local model for EOS simplifies the solution of the Riemann problem and makes it possible to obtain fluxes and partial pressures of the components in any flow region with discontinuities. Following [16], such model for a multicomponent gas is constructed so as it holds strictly in the case of weak discontinuities. The increment of the specific entropy s across SW is a quantity of the third order of smallness with respect to the pressure jump: O([P]3 ). Neglecting the entropy variation behind SW, one computes the dimensionless coefficients γα ≡

Pα τ + 1, εα

(6.5)

as functions of the state ahead of SW and the total pressure behind SW. The local model for EOS is used to solve the Riemann problem with constant values specified initially to the left and to the right of x/t = 0. On the left and right of the contact discontinuity, the concentrations remain constant, EOS is independent of cα . Using the relation dεα (sα , τ ) = Tα dsα − Pα dτ and the assumption dsα = 0, one can

64

A. G. Aksenov

write Eq. 6.6, where α ≡ Cα2 τ/Pα , i.e., one obtains an explicit dependence of the increment of γα on the increment of the pressure of a component.   τ Pα dτ Pα τ dεα dτ γα 1 dγα = + − 2 = (γα − 1) 1 − . dPα εα εα dPα εα dτ d Pα α Pα

(6.6)

The partial pressure increment depends explicitly on the total pressure increment provided by Eq. 6.7, where the squared Lagrangian speed of sound of a component 2 is Cα ≡ −d Pα /dτ = (∂ Pα /∂εα )Pα − ∂ Pα /∂τ , and total pressure is P(ε, τ ) = α Pα (εα , τ ). d Pα =

Cα2 dP C2

(6.7)

In the computations, it is convenient to use the fraction of the specific energy of a component γαε ≡ εα /ε: εdεα − εα dε (ε Pα − εα P)d P (ε Pα − εα P)τ d P = = ε2 ε2 C 2  Pε2 d P γ − γ α . (6.8) = γαε  P   The increment of the dimensionless variable γ ≡ α γα εα / α εα can be evaluated [7] dγαε =

 γ d P , dγ = (γ − 1) 1 −  P

(6.9)

as expected for a single component in Eq. 6.6. A key point of the method and the main difference of the proposed gas dynamic approach for multicomponent multitemperature gases from [15, 16] lie in Eq. 6.8 and in the derivation of Eq. 6.9 for a multicomponent gas. The assumption that the entropy variation is negligibly small is used to compute only the variations in the dimensionless coefficients. This assumption is not used in the computation of other parameters behind SW. Firstly, the local model for EOS proposed resolves the uncertainty occurring when the specific internal energy and the pressure of a mixture component behind SW (rarefaction wave) are computed from known values of γαε behind the wave. Secondly, the local EOS with the a priory known dimensionless coefficients as the functions of the full pressure behind the wave reduces the Riemann problem solver to the case of one temperature gas with EOS of the ideal gas [7, 8, 16]. When the energy equations for all components  are added, the resulting difference scheme is conservative if the normalization α Pα∗ = P∗ is used.

6 A Multidimensional Multitemperature Gas Dynamic …

65

As a test, an SW structure in hydrogen arising in hydrogen plasma in a tube at rest with a piston moving with the constant velocity into the gas is considered. If the hydrogen is completely ionized, the system involves protons and electrons. The “massless” electrons are transfer heat by conduction and exchange energy with the protons. The temperatures of protons and electrons near SW are different. The kinetic coefficients are√specified as by Shafranov [12] with the √ relaxation time τe = 3 m e (kTe )3/2 /(4 2π λq 4 n e ), where the Coulomb logarithm 3  e Ti λ = 21 ln TkTe +T /(q 6 n e ) and the thermal conductivity i κe = 3.16n e kB Te τe /n e

(6.10)

are used. An ideal monoatomic gas for protons and electrons with γi, e = 5/3 is used in EOS of ideal gas. The initial hydrogen pressure is selected approximately equal to the atmospheric pressure. The initial temperature is chosen so as to achieve the ionization. The hydrogen is half ionized when the initial temperature is 104 K, and the density is 10−6 g cm−3 . Shafranov [12] operated with relations on discontinuities and solved a system of ODEs from both sides of the discontinuity. However, due to the piecewise-smooth temperature of electrons (−∂ Te /∂ x)1 > (−∂ Te /∂ x)2 (Fig. 6.1), the heat fluxes are different on SW because the independence of the conduction coefficient κe from the concentration. Only PDEs can operate with the different heat fluxes. The easiest way is to solve the system of PDEs in the Lagrangian coordinates for density, momentum, and specific energies of protons and electrons on fine computational grid; see details in [8]. To exclude discontinuities, one need introduce the viscosity protons passing

a

b

Fig. 6.1 Numerical solution of the problem: a density profiles (circles), b proton temperatures profiles (squares), electron temperature profiles (triangles) and comparison with exact solution of Shafranov problem (solid curves) for strong SW for M = 8 at time moment 7.2 × 10−7 s. Numbers near SW on the plots show relative accuracy of solution

66

A. G. Aksenov

from the hyperbolic system of equations to a parabolic one. In the present calculations, we are not interested by the fine structure of SW due to finite protons viscosity and artificially reduced the physical proton viscosity. The velocity of the gas injected at the right boundary (the velocity a moving piston) is chosen so as to obtain a strong stationary SW on which the density jump near to ρ2 /ρ1 = 4. In present calculations, we prefer to select not huge velocity of the piston −2×107 cm s−1 to have the strongly correct physical task about the steady SW stricture in hydrogen plasma. Behind SW, the gas velocity is −1 × 107 cm s−1 , the pressure is 1.37 × 108 din cm−2 , the temperature is 2.17 × 105 K, and the black body radiation can be disregarded. The density behind SW is 3.83 × 10−6 g cm−3 and its Mach number is 8. The numerical solution of the problem is shown in Fig. 6.1. In the direction of motion of the gas, a stationary SW is formed, propagating relative to the unperturbed gas. The profiles of all quantities near the SW shift at a constant velocity and remain unchanged. The plasma is in a non-equilibrium state near SW, but equilibrium is established at some distance behind the SW front. Jumps in the proton density and temperature are observed on SW. Due to electron heat conduction, the electron temperature Te is continuous and piecewise smooth. Figure 6.1 demonstrates good agreement of the numerical solution with the “exact” solution obtained in the Lagrangian coordinates.

6.3 The Application of the Code to a 2D Gravitational Collapse of a Rotating Core with the Neutrino Spectrum Type II supernovae (SN II), so-called core-collapse supernovae, are associated with the evolution of stars with masses more than 10M , in whose centers an iron core with a mass ∼1.4M forms as a result of thermonuclear burning. The total energy of a Supernova (SN) explosion can reach 1054 erg and is carried primarily by neutrinos. A small fraction of this energy, ∼1051 erg, is absorbed in the stellar shell. During the collapse, a core with a mass of 1.4M transfers into a neutron star with a radius of ∼10 km, ∼100 MeV per nucleon is emitted in the form of neutrinos, while thermonuclear energy can provide only 8 MeV per nucleon. A neutrino mechanism for such explosions related to the absorption of the energy from the neutrinos in the envelope of an initial star is most promising [17]. The mathematical model of a gravitational collapse includes the kinetic Boltzmann equations for distribution function of neutrinos of various types (Eq. 6.1) with the reactions of weak interaction described by [17]. In addition to the dependence on the spatial variables and time, the neutrino distribution functions also depend on the momentum (two angles and the particle energy). The solution is complicated by the presence of regions that are transparent and opaque to the neutrinos. Due to the different characteristic time scales for the processes involved, different spatial

6 A Multidimensional Multitemperature Gas Dynamic …

67

scales in the full physical problem can be treated numerically only in the spherically symmetrical case, in the framework of an implicit kinetic code [18]. 1D spherically symmetrical task yields not sufficient energy deposited by neutrino in the shell. It cannot explain SN II explosion. The problem is multidimensional due to convective instabilities [19, 20]. The diffusion model with so-called flux limiters first applied in [4] for a spherically symmetrical collapse is attractive in the required multidimensional case. In this model, there is no dependence of the distribution functions on the angles, and the model contains a parameter for joining the solutions for the fluxes in the transparent and opaque regions, which is undefined at the boundary of these regions. The fluxes themselves are proportional to the gradients of the spectral energy densities. The model can easily be extended to the multidimensional case, if the problem of resolving the multicomponent gas dynamics on a fixed Euler grid can be solved. A probable reason for unsuccessful attempts to construct models for a collapse SN II is the carrying out multidimensional calculations with a reach and a difficult input physics. One region of convective instability forms at the center of the collapsing stellar core, which is extended and exists for ∼ 10 ms, while another unstable region in the vicinity of the region of accretion is thin, but exists throughout the collapse. Convection at the center can have two important effects: Apart from increasing the neutrino energy flux, the center contains more high-energy neutrinos than the region near the neutrinosphere [21]. 2D [13] and 3D [22] models demonstrate the large-scale character of convection and lead to the SN explosion. At the same time, improved 3D computations cast doubt on large-scale convection and the explosion [23]. The 2D modeling of the collapse with neutrino transport, neutronization, and strong degeneracy in real EOS without taking into account the neutrino spectrum [24, 25] confirmed the development of large-scale convection at the center over a time ∼10 ms. Due to the loss of some fraction of the neutrinos, neutronization is non-equilibrium and irreversible, a scenario for the instability close to the Schwarzschild convective instability criterion for the pure gas dynamic without the chemistry ((∇s)g > 0) is realized. The absorption of neutrinos in the envelope of the collapsing stellar core with a constant neutrino flux was considered in [26]. A specified mean energy for the neutrinos 30 MeV provides enough kinetic energy for the envelope of 1.5 × 1051 erg to explain SN II. This is a consequence of the proportionality between the scattering cross section and the fraction of the mean neutrino energy transmitted to the matter [18]. The next key question is the influence of convection on the spectrum of the emergent neutrinos. If we are interested only in the neutrino spectrum, but not in the interaction of the neutrinos with the envelope, we can apply a simple model for the interaction of the neutrinos with matter. The multicomponent hydrodynamical contains independent variables: the concentrations n i = ρ/m p and the internal energies εi of the components i. One has a system of the Euler equations that can be used to describe the gravitational collapse. The equation for the number density of baryons is

68

A. G. Aksenov

∂ρ/m p + div((ρ/m p )v) = 0, ∂t

(6.11)

and the equation for the difference between the concentrations of electrons and positrons is ∂n e + div(n e v) = Y˙e ρ/m p . ∂t

(6.12)

The velocities of all components with nonzero mass were taken to be equal, with energy transport occurring for massless particles. The conservation of momentum for the matter can be written as ∂ρv j + ∇i imj = ρg j + ρ f ν , ∂t

(6.13)

and the equation for the energy density of the matter is defined as ∂ρ E m + div(E m ρ + Pm )v = ρvg + ρqm . ∂t

(6.14)

The term ρqm represents heating of the matter by neutrinos. The gravity acceleration is the gradient of the potential g = −∇, obtained from the Poisson equation. In the non-relativistic assumption v  c the distribution functions in the laboratory frame and a frame comoving with the matter are the same. The neutrino spectrum in thermal equilibrium described by equilibrium temperature and the neutrino chemical potential equals to the chemical potential of electrons μν = μe ,   ε−μν −1 eq 2 f ν = (2πc) . We must introduce a grid for the neutrino energy 3 1 + exp kT eq and use the spectral energy density of the neutrinos and antineutrinos, ρεν, ω ≡ εω+1/2 3 2 εω−1/2 dεεν (4π c εν f ν ) in each grid interval (εω−1/2 , εω+1/2 ). The equation of transport for the spectral energy density of neutrino is defined by Eq. 6.15, where the flux is determined by the gradient of the zeroth moment in 1 max thick the opaque case, Fν, ω = − 3χ gradUν, ω , and in the transparent case, Fν, ω = cUν, ω .   ∂ρεν, ω + v∇ ρεν, ω = divFν, ω − ρqm, ω ∂t

(6.15)

In the arbitrary case, we can use the interpolation (the flux limiter) in the form of Eq. 6.16. thick Fν, ω Fν, ω = thick

F /F max + 1 ν, ω

ν, ω

(6.16)

6 A Multidimensional Multitemperature Gas Dynamic …

69

The constraint on the flux refers to the introduction of nonlinear thermal conductivity and a certain arbitrary fit of the fluxes in the intermediate case. The nonlinear diffusion flux transfers a (parabolic) equation in the opaque region into hyperbolic transport equation in the transparent region for the spectral energy densities. The exchange of energy between neutrinos by the relax  and matter is described ation to a thermal distribution, ρqm = cχ Uν, ω − Uν,th ω . The relaxation rate was chosen to be proportional to the concentration of free nucleons, χ ≈ σ0 n n , with the constant cross section of weak interactions σ0 = 1.7 × 10−40 cm2 . We adopted the relaxation time for the difference of the numbers of electrons and positrons per nucleon (below “number of electrons,” Ye ≡ n e m p /ρ) τ0 = 3 12 −3  eq 10−3 s 10 gρcm 5 × 1010 K/T eq and the relaxation rate Y˙e = −(Ye − Ye )/τ0 toward the beta-equilibrium distribution from [27], where only two reactions were ˜ with the free escape of the considered: e− + p → n + ν and e+ + n → p + ν, neutrinos. Such beta-equilibrium is correct only near the neutrinosphere, where it is important to take the energy flux of the neutrinos into account. In a deeply opaque region and the simple beta-equilibrium model is quantitatively incorrect, however, the neutrino energy flux is small in the opaque region. EOS of the matter, Pm = Pm (ρ, εm , Ye ), contains the free nucleons and the nuclei in statistical equilibrium. Also matter contains degenerate ultrarelativistic pair and photons. As initial data, we adopted the relationship between the pressure and density for a polytrope, P ∝ ρ 1+1/n , with the polytropic index n = 3 corresponding to ultrarelativistic electrons. The rotation law is a constant ratio of the centrifugal and gravitational forces in the equatorial plane (it is a generalization of rigid-body rotation for an incompressible fluid) [28]. We chose with modest initial rotation with the ratio of the polar radius to the equatorial radius: req was taken to be 0.9 and the rotational energy was 1.25% of the gravitational energy. The polytropic initial model contains three independent physical parameters: the gravitational constant G, central density ρc , and equatorial radius req . For the adopted for calculations central density ρc = 2 × 1012 g/cm3 and the mass of the collapsing core of 1.4M , the polytropic model gives the equatorial radius req = 2.68×107 cm, gravitational energy E gr = 2.93 × 1052 . For the specified density and pressure profiles, we must calculate the parameters T and Ye = Y eq cm, but the initial stage of the neutrino energy losses continues for several seconds, appreciably exceeding the gas-dynamical time scale (Gρ)−1/2 , even for a low initial central density ∼ 109 g × cm−3 [18]. As a result of reconstructing the solution in 2D problem, we obtained an entropy profile that falls off toward the center, ds/dr < 0. This type of profile arises due to neutronization, which lowers Ye , so that the specific energy from the electron component goes into nucleons (Fig. 6.2). In 2D computations in spherical coordinates (r, 0 < θ < π/2), we used a 60 × 30 grid in the computational domain restricted to a region r ≤ 0.6req assuming axial symmetry, with the plane of symmetry θ = π/2. For the neutrino spectrum, we are using a logarithmic grid with 15 intervals up to 40 MeV. The density contours shown in Fig. 6.3 illustrate the development of largescale convection over the gas-dynamical time scale (G ρ) ¯ −1/2 = 4 ms for the

70

A. G. Aksenov

Fig. 6.2 Initial model in the equatorial plane in 2D computations. The radial dependence of the density ρ, temperature T , entropy per nucleon s/(kB T ), electron number Ye , and chemical potential of the electrons μe are shown

Fig. 6.3 Density contours in the y = 0 plane for the collapse of a mass 1.4M taking into account neutrino transport and the neutrino spectrum at time t = 15.77 ms (lg ρmin = 7, lg ρmax = 13,  lg ρ = 0.1) that demonstrates a development of large-scale convection

mean density ρ ≈ 1011 g/cm3 . This is corresponded to the initial unstable region ds/dr < 0 (Fig. 6.2). Neutrinos are lost here, and a convection condition close to the Schwarzschild condition arises due to non-equilibrium neutronization. Computations taking into account rotation demonstrate the development of longer-wavelength perturbations than in the case without rotation [24, 25]. An important result of the computations is the spectral flux of neutrinos near the boundary of the computational domain. Figure 6.4 shows the spectral neutrino luminosity during the collapse, d L/dε, and the spectral particle flux, dn/dε, at time t = 15.8 ms for 2D computations and in a spherically symmetrical formulation using

6 A Multidimensional Multitemperature Gas Dynamic …

71

Fig. 6.4 Spectral neutrino luminosity during the collapse, d L/dε (solid), and spectral particle flux dn/dε (dashed) at time moment 15.77 ms. The energy ε is in MeV, and the total luminosity is in erg/s. 2D computations demonstrate an enhancement of the mean energy in neutrinos compared to 1D case

the same code and grid. 1D computations yield the mean energy 10 MeV, as spherically symmetric computations with an exact allowance for the main weak-interaction reactions [18]. In 2D computations, the escaping neutrino spectrum becomes harder due to convection in the central region with high-energy neutrinos that are trapped by optically dense matter in 1D computations. The energy of the emergent neutrinos is approximately equal to the maximum electron chemical potential 20 MeV in Fig. 6.2. The mean energies are equal to the ratio of the luminosity and particle flux, (ε1D ) = 10.5 MeV and (ε1D ) = 15 MeV in the 1D and 2D cases. The growth in the energy of the emergent neutrinos we have obtained is in agreement with old simulations [29].

6.4 Conclusions In this research, we tested multidimensional multitemperature code on the SW structure in hydrogen plasma. We have considered the development of large-scale convection in a collapsing stellar core with weak rotation taking into account spectral neutrino transport. Convection arises in the central region over the gas-dynamical time scale, ∼10 ms, due to non-equilibrium neutronization with the loss of some fraction of the neutrinos. The mean energy of escaping neutrino of 15 MeV is a factor of 1.5 higher than the energy obtained in spherically symmetrical computations. This effect is important for explaining SN explosions.

72

A. G. Aksenov

There is no doubt that observations of neutrinos from SN are important for confirming models for the collapse and explosion. Only 20 neutrino events were registered for close SN1987A. The first publications about this event indicate high energies: 20–40 MeV IMB [30], 9–35 MeV Kamiokande-II [31], and 20 MeV Baksan-LSD [32], closer to the large-scale convection model than to expectations for a spherically symmetrical collapse. Taking into account the interaction of neutrinos and the envelope is the next step in the explanation of the supernovae explosion energy. The interactions of the neutrinos with the envelope of the stellar core will require to calculate the integrals of the collisions of the neutrinos with the matter and their annihilation. The number of computations increases at least as the square of the number of intervals used to represent the neutrino energy.

References 1. Basko, M.M., Churazov, M.D., Aksenov, A.G.: Prospects of heavy ion fusion in cylindrical geometry. Laser Part. Beams 20, 411–414 (2002) 2. Anisimov, S.I., Zhakhovskii, V.V., Inogamov, N.A., Nishihara, K., Petrov, Y.V., Khokhlov, V.A.: Ablated matter expansion and crater formation under the action of ultrashort laser pulse. JETP 103, 183–197 (2006) 3. Fortov, V.E., Hoffmann, D.H., Sharkov, B.Y.: Intense ion beams for generating extreme states of matter. Adv. Phys. Sci. 178(2), 113–138 (2008). (in Russian) 4. Bruenn, S.W.: Stellar core collapse: numerical model and infall epoch. Astrophys. J. Suppl. 58, 771–841 (1985) 5. Miller, G.H., Puckett, E.G.: A high-order Godunov method for multiple condensed phases. J. Comput. Phys. 128, 134–164 (1996) 6. Pelanti, M., Shyue, K.-M.: A mixture-energy-consistent six-equation two-phasenumerical model for fluids with interfaces, cavitation and evaporation waves. J. Comput. Phys. 259, 331–357 (2014) 7. Aksenov, A.G.: Computation of shock waves in plasma. Comput. Math. Math. Phys. 55, 1752– 1769 (2015) 8. Aksenov, A.G., Chechetkin, V.M., Tishkin, V.F.: Godunov type method and the shafranov’s task for multi-temperature plasma. Math. Models Comput. Simul. 11, 360–373 (2019) 9. Vereshchagin, G.V., Aksenov, A.G.: Relativistic kinetic theory with applications in astrophysics and cosmology. Cambridge University Press (2017) 10. Aksenov, A.G., Chechetkin, V.M.: Supernova explosion mechanism with the neutrinos and the collapse of the rotation core. Astron. Rep. 62, 834–839 (2018) 11. Aksenov, A.G., Chechetkin, V.M.: Large-scale instability during gravitational collapse and the escaping neutrino spectrum during a supernova explosion. Astron. Rep. 63, 900–909 (2019) 12. Shafranov, V.D.: The structure of shock waves in a plasma. Sov. Phys. JETP 5, 1183–1188 (1957). (in Russian) 13. Dolence, J.C., Burrows, A., Zhang, W.: Two-dimensional core-collapse supernova models with multi-dimensional transport. Astroph. J. 800, 10.1–10.14 (2015) 14. Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. PrenticeHall, Inc., Englewood Cliffs (1971) 15. Colella, P., Woodward, P.R.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201 (1984) 16. Colella, P., Glaz, H.M.: Efficient solution algorithms for the Riemann problem for real gases. J. Comput. Phys. 59, 264–289 (1985)

6 A Multidimensional Multitemperature Gas Dynamic …

73

17. Fowler, W.A., Hoyle, F.: Neutrino processes and pair formation in massive stars and supernovae. Astrophys. J. Suppl. 9, 201–319 (1964) 18. Aksenov, A.G., Chechetkin, V.M.: Computations of the collapse of a stellar iron core allowing for the absorption, emission, and scattering of electron neutrinos and anti-neutrinos. Astron. Rep. 56, 193–206 (2012) 19. Bethe, H.A.: Supernova mechanisms. Rev. Mod. Phys. 62, 801–866 (1990) 20. Imshennik, V.S., Nadezhin, D.K.: Supernova 1987A in the large magellanic cloud: observations and theory. Astrophys. Space Phys. Rev. 8, 1–147 (1989) 21. Herant, M., Benz, W., Hix, W.R., Fryer, C.L., Colgate, S.A.: Inside the supernova: a powerful convective engine. ApJ 435, 339–361 (1994) 22. Wongwathanarat, A., Muller, E., Janka, H.-T.: Three-dimensional simulations of core-collapse supernovae: From shock revival to shock breakout. Astronony Astroph. 577, A48.1–A48.20 (2015) 23. Couch, S.M., Ott, C.D.: The role of turbulence in neutrino-driven core-collapse supernova explosions. Astrophys. J. 799, 5.1–5.12 (2015) 24. Chechetkin, V.M., Aksenov, A.G.: Supernova-explosion mechanism involving neutrinos. Phys. At. Nucl. 81, 128–138 (2018) 25. Aksenov, A.G., Chechetkin, V.M.: Large-scale instability during gravitational collapse with neutrino transport and a core-collapse supernova. Astron. Rep. 62, 251–263 (2018) 26. Baikov, I.V., Suslin, V.M., Chechetkin, V.M., Bychkov, V., Stenflo, L.: Radiation of a neutrino mechanism for type II supernovae. Astron. Rep. 51, 274–281 (2007) 27. Bisnovatyi-Kogan, G.S.: Analytic solution for kinetic equilibrium with respect to betaprocesses in nucleon plasmas with relativistic pairs. Astrophysics 55, 387–396 (2012) 28. Aksenov, A.G., Blinnikov, S.I.: A Newton iteration method for obtaining equilibria of rapidly rotating stars. Astron. Astrophys. 290, 674–681 (1994) 29. Burrows, A.: Convection and the mechanism of Type II supernovae. ApJ Lett. 318, L57–L61 (1987) 30. Bionta, R.M., Blewitt, G., Bratton, C.B., Casper, D., Ciocio, A.: Observation of a neutrino burst in coincidence with supernova 1987A in the large magellanic cloud. Phys. Rev. Lett. 58, 1494–1496 (1987) 31. Hirata, K., Kajita, T., Koshiba, M., Nakahata, M., Oyama, Y.: Observation of a neutrino burst from the supernova SN1987A. Phys. Rev. Lett. 58, 1490–1493 (1987) 32. Schaeffer, R., Declais, Y., Jullian, S.: The neutrino emission of SN1987A. Nature 330, 142–144 (1987)

Chapter 7

The Airy Stress Function for Non-Euclidean Model of a Continuous Medium and Description of Residual Stresses Mikhail A. Guzev Abstract In the frame of the non-Euclidean model of a continuous medium for which the Saint-Venant compatibility condition for deformations is not fulfilled, an equation for the Airy stress function was derived. A representation was constructed for the field of internal stresses, and it was shown to consist of the classical field of elastic stresses and the stress field parameterized through the incompatibility function. The obtained relations of the non-Euclidean continuum model were used to describe the internal residual stresses in the samples. The phenomenological parameters of the model were determined using the experimental data of the residual stresses measurement. Keywords Incompatible deformation · Non-Euclidean model · Biharmonic equation · Residual stress

7.1 Introduction Residual stresses σi j are those stresses that exist inside a material or body when there are no external forces acting on it [1]. It means that in the absence of mass forces inside the volume V of the body, the Cauchy equations of equilibrium are valid: ∂σi j = 0. ∂x j

(7.1)

Residual stresses are also called “self-balanced,” since areas of tensile and compressive stresses exist within the material provided that the effects of tension and compression must be balanced. It leads to the total zero resultant forces and moments [2] and the integral equilibrium conditions in the form of Eq. 7.2, where ∂ V is the boundary of the body, n i are the direction cosines of the outer normal to M. A. Guzev (B) Institute for Applied Mathematics FEB of the RAS, 7 Radio ul., Vladivostok, Russian Federation 690041 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 214, https://doi.org/10.1007/978-981-33-4709-0_7

75

76

M. A. Guzev

the boundary: 

 σi j n j dS = 0 ∂V

(σil xk − σkl xi )n l dS = 0.

(7.2)

∂V

In the classical theory of elasticity at mechanical equilibrium of a rigid body in the absence of bulk forces and loads at the boundary of the body, the stress components are assumed to be zero everywhere, as inside the body and on its surface. However, it is well known that under equilibrium Eqs. 7.1 and 7.2 stresses inside the solid are nonzero. Welds can serve as an example. Experimental study [1] shows that stresses in welds are comparable with those arising under external actions. In engineering, various technological procedures are used making it possible to reduce the level of internal stresses or, on the contrary, to elevate it. As such procedures, we can indicate various types of thermal treatments: annealing, quenching, etc. In this case, a complete description of the physicomechanical properties of materials necessitates the development of new theoretical models of their behavior under various conditions. On the other hand, the study of the microcharacteristics of various materials by physical methods led to the introduction of concepts such as dislocation, disclination, vacancy, and other characteristics of the crystal structure defects into the materials science literature, e.g., [3]. As a rule, the explanation of results of the action of such technological procedures to a material is associated with a redistribution or disappearance of defects in the internal structure of the material. Many works appeared, in which various characteristics of the internal structure of materials were introduced, including scalar and tensor characteristics of “damage,” “defectiveness,” etc. From the viewpoint of classical continuum mechanics and nonequilibrium thermodynamics, the characteristics introduced by various researchers to describe the internal structure of materials explicitly require an extension of the kinematic foundations of the classical theory. As early as the fifties in the last century, Kondo [4] and Bilby et al. [5] came to the conclusion that it is necessary to use nonEuclidean geometric objects, forbidden in the classical theory of elasticity, in their description. Thus, experimental studies clearly indicate the existence of nonzero internal selfbalanced stresses. To describe them on the basis of physical models of defects in the internal structure of the material, it is necessary to use mathematical objects that do not fit into the framework of the Euclidean geometric description of the deformation properties of an elastic continuum. Analysis of this problem for various decades can be found in [6–10]. Common to the approaches of all authors is the use of geometric non-Euclidean objects (torsion, non-metricity and curvature tensors) as variables characterizing the geometric structure of the incompatibility of the medium material elements. It should be noted that non-Euclidean characteristics are used in the construction of material models at its various scale levels. From the view of continuum mechanics, the introduction of incompatibility at the macroscale and the construction of a non-Euclidean model are associated with the rejection of the kinematic conditions for the compatibility of

7 The Airy Stress Function for Non-Euclidean Model …

77

Saint-Venant deformations [11]. Therefore, it seems natural to apply the methods of classical theory for the analysis of the non-Euclidean model. The most important tool for this is the stress tensor function. In the classical theory of elasticity, the Airy stress function method is well known for investigating the plane deformation problems [12]. Obtaining the equation for the stress function involves using the Saint-Venant compatibility condition. Therefore, the natural extension of the classical theory consists in rejecting the compatibility condition and introducing the incompatibility function [7], i.e., transition to a non-Euclidean model. This chapter is organized as follows. In Sect. 7.2, the stress function in a nonEuclidean model is shown to satisfy the inhomogeneous biharmonic equation, the right-hand side of which coincides with the incompatibility function. In Sect. 7.3, the complete stress function is calculated through the classical and the incompatibility functions. The equation for the incompatibility function is presented, and it is shown that the field of internal stresses consists of the classical field of elastic stresses and the non-Euclidean stress field. The last one is constructed through the incompatibility function. In Sect. 7.4, as an application of the obtained results, a solution is constructed for the incompatibility function in the Cartesian coordinate system, and a representation is obtained for the longitudinal and transverse stresses. The theoretical results obtained are used to analyze the experimental data and to select the phenomenological parameters of the non-Euclidean model. Section 7.5 concludes the chapter.

7.2 Transition from the Classical Model to the Non-Euclidean One For plane problems, the equations of mechanical equilibrium Eq. 7.1 have the form of Eq. 7.3. ∂σ12 ∂σ21 ∂σ22 ∂σ11 + =0 + = 0. ∂x1 ∂x2 ∂x1 ∂x2

(7.3)

Equation 7.3 are identically satisfied if we introduce the stress function  according to Eq. 7.4. σ11 =

∂ 2 ∂ 2 ∂ 2 σ = σ = − 22 12 ∂ x 2∂ x 2 ∂ x 1∂ x 1 ∂ x 1∂ x 2

(7.4)

The relationship between the components of the stress and strain field (Hooke’s law) is linear: σi j = λδi j εkk + 2μεi j ,

(7.5)

78

M. A. Guzev

where λ, μ are Lamé’s parameters of materials. Functions εi j are easy to write through σi j :   λσ 1 σi j − δi j , σ = σkk . εi j = 2μ 3λ + 2μ

(7.6)

In classical theory, a further definition of the stress function is based on the additional assumption that material strains εi j are compatible. This equation of compatibility Saint-Venant is given by Eq. 7.7. ∂ 2 ε22 ∂ 2 ε12 ∂ 2 ε11 + − 2 =0 ∂ x 2∂ x 2 ∂ x 1∂ x 1 ∂ x 1∂ x 2

(7.7)

Substituting Eq. 7.6 into Eq. 7.7 and using Eqs. 7.3 and 7.4, we obtain a homogeneous biharmonic equation for the classical stress function clas : 2 clas = 0.

(7.8)

Considering the various possibilities of expanding the classical model, we retain some physical hypotheses, in particular, the fulfillment of the equations of mechanical equilibrium in the form Eq. 7.3 and the linear relation Eq. 7.4 [7]. This allows us to present the solution in the form of Eq. 7.4 with some stress function. However, the compatibility condition Eq. 7.3 may not be satisfied in the general case. We introduce the function R by setting ∂ 2 ε11 ∂ 2 ε22 ∂ 2 ε12 R = 2 2 + 1 1 −2 1 2. 2 ∂x ∂x ∂x ∂x ∂x ∂x

(7.9)

From the mathematical view, the fulfillment of compatibility conditions (Eq. 7.9), i.e., R ≡ 0, means that εi j = (∂u i /∂ x j + ∂u j /∂ xi )/2 [11] and we have the classical model. However, the conditions for the material deformation can lead to the formation of configurations having an internal structure for which R = 0. As an example, thin plates studied in [13] should be mentioned. In this case, R coincides with the Gaussian curvature of the surface obtained after deformation of the plate. We gave an example of the incompatibility appearance due to the mechanical state. However, incompatibilities can arise due to temperature, magnetization, and other factors. With the expansion of the classical model, the functions εi j are determined through the stress field Eq. 7.6 [7]. If we substitute Eq. 7.6 into Eq. 7.9 and use Eqs. 7.3 and 7.4, then we obtain Eq. 7.10 for the stress function: 2  =

λ μ R, ν = . 1−ν 2(λ + μ)

(7.10)

7 The Airy Stress Function for Non-Euclidean Model …

79

A comparison of Eq. 7.10 with Eq. 7.8 shows that the transition from the classical model to the non-Euclidean one leads to the inhomogeneous biharmonic equation for the stress function:

7.3 Structure of the Stress Function Equation 7.10 is linear, then its solution can be represented as the sum of the classical stress function clas and the additional contribution of non-Eucl :  = clas + non-Eucl .

(7.11)

Since clas satisfies the homogeneous biharmonic Eq. 7.8, then non-Eucl is a particular solution of Eq. 7.10. Performing a renormalization for R in Eq. 7.10, i.e., assuming μR → (1 − ν)R, we rewrite Eq. 7.10 in the form of Eq. 7.12. 2 non-Eucl = R.

(7.12)

The function R is unknown in the general case. The equation for R was obtained in [14] assuming a quadratic dependence of the internal energy of the medium on thermodynamic variables and has the form of Eq. 7.13. 2 R = γ R, γ = 0.

(7.13)

When analyzing the phenomenon of zonal disintegration [15] in the stationary case, an equation was obtained in the form Eq. 7.13. From Eqs. 7.12 and 7.13, we obtain   R = 0. 2 non-Eucl − γ

(7.14)

Since we assigned the solutions of the homogeneous biharmonic equation to the function clas Eq. 7.8, it follows from Eq. 7.14: non-Eucl =

R . γ

(7.15)

After substituting Eq. 7.15 into Eq. 7.11, we obtain the stress components according to Eq. 7.4: ∂ 2 (clas + non-Eucl ) 1 ∂2 R = σ + σ , σ = , 11,clas 11,non-Eucl 11,non-Eucl ∂ x 2∂ x 2 γ ∂ x 2∂ x 2 ∂ 2 (clas + non-Eucl ) 1 ∂2 R = = σ + σ , σ = , 22,clas 22,non-Eucl 22,non-Eucl ∂ x 1∂ x 1 γ ∂ x 1∂ x 1

σ11 = σ22

80

σ12 = −

M. A. Guzev

∂ 2 (clas + non-Eucl ) 1 ∂2 R = σ + σ , σ = − . 12,clas 12,non-Eucl 12,non-Eucl ∂ x 1∂ x 2 γ ∂ x 1∂ x 2 (7.16)

It can be seen from Eq. 7.16 that the field of internal stresses consists of the classical field of elastic stresses and the stress field created by defects. The contribution of the latter to the stress distribution is parameterized via the incompatibility function R. It is easy to show that Airy’s two-dimensional solution in terms of a stress function (Eqs. 7.4 and 7.16) is self-equilibrated, i.e., Eq. 7.2 are valid.

7.4 Description of Residual Stresses To illustrate the capabilities of the formulas obtained when self-balanced stress fields were modeled, we consider their application in the analysis of residual stresses in metal samples. Rectangular samples of length 100 mm (in coordinate x 1 ) and width 20 mm (in coordinate x 2 ), in which the longitudinal ends were rounded with a radius of ~10 mm, were used in the study [16]. The researchers measured the longitudinal residual and transverse residual stresses on the center line x 2 = 0 depending on the variable x 1 . Square dots in Fig. 7.1 correspond to the experimentally measured longitudinal stress, round dots in Fig. 7.2 correspond to the transverse stress [16]. Let’s represent R as R = R+ + R− , where R+ , R− satisfy the equations:   √  √   + γ R+ = 0,  − γ R− = 0.

(7.17)

Longitudinal residual stress in MPa

600 Experimental

400 Theoretical

200

0 0

10

20

30

-200

-400

-600

Distance to centre in mm

Fig. 7.1 Theoretical and experimental longitudinal residual stresses

40

7 The Airy Stress Function for Non-Euclidean Model …

81

Transverse residual stress in MPa

600 Experimental

400

TheoreƟcal

200

0 0

10

20

30

40

-200

-400

-600

Distance to centre in mm

Fig. 7.2 Theoretical and experimental transverse residual stresses

When γ > 0, we obtain the classical Helmholtz equation for R+ , where the √ parameter γ has the meaning of a wave number. If γ < 0, then we have the Helmholtz equation with the imaginary wave number. In Cartesian coordinates, it is possible to construct solutions of Eq. 7.17 using the variable separation method. Then, considering R+ = R+,1 R+,2 ,

R+,1 = R+,1 (x 1 ), R+,2 = R+,2 (x 2 ),

R− = R−,1 R−,2 ,

R−,1 = R−,1 (x 1 ), R−,2 = R−,2 (x 2 ),

we obtain from Eq. 7.17:  R+,1

R+,1

+

 R+,2

R+,2

+

√ γ = 0,

 R−,1

R−,1

+

 R−,2

R−,2

−

√

γ = 0,

(7.18)

where two dashes denote differentiation with respect to the corresponding argument of the function. Introducing constant divisions ω12 = −

    R+,1 R+,2 R−,1 R−,2 , ω22 = − , 21 = − , 22 = − , R+,1 R+,2 R−,1 R−,2

we write Eq. 7.18 as ω12 + ω22 =

√

√ γ , 21 + 22 = − γ .

(7.19)

Further specification of the obtained relations is based on the additional simplifications. Since the second condition in Eq. 7.19 follows from the first when replacing ω1 → i 1 , ω2 → i 2 , where i is the imaginary unit, it suffices to consider the

82

M. A. Guzev

solution for R+ , setting R− = 0. Let’s represent R+ as R+ = aγ cos ω1 x 1 cos ω2 x 2 . From here and from Eq. 7.15, we obtain non-Eucl = a cos ω1 x 1 cos ω2 x 2 and stress components (Eq. 7.16): σ11 = σ11,clas − aω22 cos ω1 x 1 cos ω2 x 2 , σ22 = σ22,clas − aω12 cos ω1 x 1 cos ω2 x 2 , σ12 = σ12,clas − aω1 ω2 sin ω1 x 1 sin ω2 x 2 .

(7.20)

From Eq. 7.20, it follows that on the x 2 = 0 axis the components of the stresses under study are given by the relations: σ11 (x 1 ) = σ11,clas − a

√

γ − ω12 cos ω1 x 1 , σ22 (x 1 ) = σ22,clas − aω12 cos ω1 x 1 (7.21)

and we used the first condition in Eq. 7.19, when recording them. The parameters σ11,clas , σ22,clas , ω1 , a, γ should be determined basing on experimental data. In accordance with the experimental results, the values of σ11 , σ22 in the discrete set of points xi1 = x0 + T (i − 1), 1 ≤ i ≤ n are known. On the other hand, the values of σ11 (x 1 ), σ22 (x 1 ) are parameterized by means of Eq. 7.21, then using the procedure of averaging over xi1 , we can determine σ11,clas , σ22,clas . In particular, setting n 

cos ω1 xi1 =

i=1

n 

cos ω1 (x0 + T (i − 1)) = 0,

(7.22)

i=1

we obtain n σ11,clas =

i=1

σ11 (xi1 ) , σ22,clas = n

n i=1

σ22 (xi1 ) . n

For Figs. 7.1 and 7.2, we have n = 8, then σ11,clas = 189 MPa, σ22,clas = 15 MPa. Formulated condition (Eq. 7.22) limits the choice of the parameter ω1 . To verify this, we perform the summation in Eq. 7.22. Since cos ω1 [x0 + T (i − 1)] = cos ω1 x0 cos ω1 T (i − 1) − sin ω1 x0 sin ω1 T (i − 1), n 

cos kx = cos

k=0 n  k=1

sin kx = sin

(n + 1)x 1 nx sin , 2 2 sin x2

(n + 1)x nx 1 , sin 2 2 sin x2

7 The Airy Stress Function for Non-Euclidean Model …

83

then n  i=1 n 

cos ω1 T (i − 1) = cos

(n − 1)ω1 T sin nω21 T , 2 sin ω21 T

sin ω1 T (i − 1) = sin

(n − 1)ω1 T sin nω21 T . 2 sin ω21 T

i=1

The combination of the obtained relations gives n 

cos ω1 xi1 =

i=1

nω1 T 2 sin ω21 T

sin

  (n − 1)ω1 T . cos ω1 x0 + 2

From here and from Eq. 7.22, we obtain, that ω1 is parametrized as follows: nω1 T π (n − 1)ω1 T = π N or ω1 x0 + = + π M, 2 2 2 where N, M are non-negative integers. In accordance with Figs. 7.1 and 7.2, we have n = 8, T = 5 mm, x 0 = 2.5 mm and get two series of sets for ω1 :   1 πN π (1/mm), ω1 = M+ (1/mm). ω1 = 20 20 2

(7.23)

The corresponding period is equal to T = 2π/ω1 , then substituting here (Eq. 7.23) we obtain: T =

40 40 (mm) or T = (mm). N M + 21

(7.24)

From the point arrangement in Figs. 7.1 and 7.2, it can be seen that the length of the period lies between the values of 20 mm and 25 mm. By direct calculation, it is easy to verify that N = 2 gives the only such value of the period in Eq. 7.24. Thus, we finally obtain: ω1 =

π (mm). 10

To determine a, γ , we use the experimental data on the magnitude of the stresses in the center of the sample. Let σ1,0 = σ11 |x 1 →0 be the longitudinal stress and σ2,0 = σ22 |x 1 →0 be the transverse stress at the center of the sample. We believe that they satisfy to Eq. 7.20: σ1,0 = σ11 |x 1 →0 = σ11,clas − a

√

γ − ω12 , σ2,0 = σ22 |x 1 →0 = σ22,clas − aω12 . (7.25)

84

M. A. Guzev

The last relation allows us to find a=

σ22,clas − σ2,0 . ω12

For the experiment under study, we obtain a = 861 MPa · (mm)2 . Note that the product MPa · (mm)2 has the dimension of force. Substituting the  obtained  value of √ a into the first condition (Eq. 7.25), we determine γ = 0.198 1/mm2 . Graphs for stresses constructed in accordance with formula (Eq. 7.21) using the values of the parameters obtained are presented in Figs. 7.1 and 7.2 by a continuous line corresponds to a longitudinal stress, and a dashed line corresponds to a transverse one.

7.5 Conclusions In this chapter, the use of a non-Euclidean model to describe the residual stresses in a material led to the solution of an important problem for the plane-deformed state of a material in equilibrium associated with the construction of a stress function when the Saint-Venant compatibility condition is not fulfilled. The stress function in the non-Euclidean model is shown to be the sum of the classical stress function and the additional contribution determined through the incompatibility function. The parameterization of the incompatibility function is given through the solution of the Helmholtz equation, for which a representation in the Cartesian coordinate system is obtained as applied to the material sample under study. The phenomenological parameters of the proposed model were determined using experimental data for the longitudinal and transverse stresses of metal samples subjected to laser processing. A good qualitative agreement between the analytical and experimental results provides optimism for the further application of the proposed theoretical approach. The task, which is a natural continuation of this study, is connected with the study of the stress distribution over the sample height. For this case, the authors of [16] also presented experimental data; however, comparing the simulation results with them will require analysis of the complete equation system of a continuous medium equilibrium, in contrast to the equations used (Eq. 7.3). Acknowledgements The study was carried in the frame of RSF, project № 19-19-00408.

7 The Airy Stress Function for Non-Euclidean Model …

85

References 1. Schajer, G.S., Whitehead, P.S.: Hole-drilling method for measuring residual stresses. Synth. SEM Lect. Exp. Mech. 1(1), 1–186 (2018) 2. Rossini, N.S., Dassisti, M., Benyounis, K.Y., Olabi, A.G.: Methods of measuring residual stresses in components. Mater. Des. 35, 572–588 (2012) 3. Sih, G.C., Tang, X.S.: Screw dislocations generated from crack tip of self-consistent and selfequilibrated systems of residual stresses: atomic, meso and micro. Theoret. Appl. Fract. Mech. 43(3), 261–307 (2005) 4. Kondo, K.: On the geometrical and physical foundations of the theory of yielding. In: Proceedings of the Second Japan National Congress for Applied Mechanics, vol. 2, pp. 41–47. University of Tokyo Press, Tokyo (1953) 5. Bilby, B.A., Bullough, R., Smith, E.: Continuous distributions of dislocations: a new application of the methods of non-Reimannian geometry. Proc. R. Soc. A 231, 263–273 (1955) 6. Stojanovic, R.: Equilibrium conditions for internal stresses in non-Euclidian continua and stress space. Int. J. Eng. Sci. 1, 323–327 (1963) 7. Kröner, E.: Incompatibility, defects, and stress functions in the mechanics of generalized continua. Int. J. Solids Struct. 21(7), 747–756 (1985) 8. Myasnikov, V.P., Guzev, M.A.: Thermo-mechanical model of elastic-plastic materials with defect structures. Theoret. Appl. Fract. Mech. 33(3), 165–171 (2000) 9. Guzev, M.A.: Non-Euclidean Models of Elastoplastic Materials with Structure Defects. Lambert Academic Publishing (2010) 10. Yavari, A., Goriely, A.: Riemann-Cartan geometry of nonlinear dislocation mechanics. Arch. Ration. Mech. Anal. 205(1), 59–118 (2012) 11. Godunov, S.K., Romensky, E.: Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic Publication, Dordrecht (2003) 12. Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Linear Theories of Elasticity and Thermoelasticity. Springer, Berlin (1973) 13. Klein, Y., Efrati, E., Sharon, E.: Shaping of elastic sheets by prescription of non-Euclidean Metrics. Science 315(5815), 1116–1120 (2007) 14. Guzev, M.A.: Structure of kinematic and force fields in the Riemannian continuum model. J. Appl. Mech. Tech. Phys. 52(5), 709–716 (2011) 15. Guzev, M.A., Paroshin, A.A.: Non-Euclidean model of the zonal disintegration of rocks around an underground working. J. Appl. Mech. Tech. Phys. 42(1), 131–139 (2001) 16. Kromm A., Cabeza S., Mishurova T., Nadammal N., Thiede T., Bruno G.: Residual stresses in selective laser melted samples of a nickel based superalloy. Mater. Res. Proc. 6, 259–264 (2018)

Chapter 8

Numerical Comparison of Different Approaches for the Fractured Medium Simulation Ilia S. Nikitin , Vasily I. Golubev , Yulia A. Golubeva , and Vladislav A. Miryakha Abstract This chapter is devoted to the investigation of the dynamic loading of fractured media occurred in many applications like the seismic survey process, non-invasive material quality control, and fatigue failure of samples. Two different approaches were described. The first one is based on the continual model of solid media with a discrete set of slip planes and with nonlinear-type slip conditions at them. The constitutive equations of the resulting system of equations contain a small viscosity parameter in the denominator of nonlinear free terms. For a stable numerical solution of a system of differential equations, an explicit–implicit method is used with an explicit approximation of motion equations and an implicit approximation of constitutive relations containing a small parameter. The second one is based on the explicit crack positioning inside the computational grid. With the help of correct contact conditions, unfilled and fluid-filled fractures are described. To solve numerically the elastic system of equations, the grid-characteristic method on structured grids is used. It allows to set precisely all necessary boundary conditions. Both approaches were successfully applied for the simulation of the seismic survey problem in fractured media. Full-wave numerical solutions were obtained and compared. The striker–target interaction problem was simulated using Lagrange grids. Based on the analysis of the dynamic stress state, the process of the new crack initiation was described. The influence of the material strength on it was investigated. Positions of damages were compared with the delamination region. I. S. Nikitin · Y. A. Golubeva · V. A. Miryakha Institute of Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow, Russian Federation 123056 e-mail: [email protected] Y. A. Golubeva e-mail: [email protected] V. A. Miryakha e-mail: [email protected] V. I. Golubev (B) · Y. A. Golubeva · V. A. Miryakha Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per., Dolgoprudny, Moscow Region, Russian Federation 141701 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 214, https://doi.org/10.1007/978-981-33-4709-0_8

87

88

I. S. Nikitin et al.

Keywords Fractured media · Layered media · Slip conditions · Explicit-implicit method · Grid-characteristic method · Numerical simulation

8.1 Introduction The dynamic problem of fractured media occurs in many applications such as the seismic survey process [1], non-invasive material quality control [2, 3], and fatigue failure of samples. With a development of modern high-performance computing systems, the ability of the usage of more complicated and realistic mathematical models is created. One of the promising approaches is connected with continual models [4–6]. Models of deformable solid media with a discrete set of slip planes (layered, block media) and with nonlinear (dry friction or viscous-plastic types) slip conditions at contact boundaries can be obtained with the discrete variant of the slippage theory [7] or with the method of the asymptotic homogenization [8]. In all of these cases, in the constitutive system of equations with the nonlinear free term and small stress relaxation time are included. For the stable numerical solution of differential equations, the explicit–implicit method with the explicit approximation of motion equations and the implicit approximation of constitutive equations containing small parameter in the denominator of the free term was proposed [9]. Different numerical methods are intensively used for problems of the seismic survey. Most popular of them are the finite difference method, Galerkin method, finite element method, and some hybrid methods [10–12]. It should be noticed that the solution of any hyperbolic system can be naturally calculated by the grid-characteristic method [13–16]. Some efforts were recently applied for increasing the simulation precision. For example, the compact schemes [17] were adopted for various range of models and phenomena: supersonic flows [18], lubricant transfer from media to head during heat-assisted magnetic recording [19], elliptic problems [20], incompressible Navier–Stokes equations [21], and acoustic problems with discontinuous coefficients [22]. This chapter is the extension of our continuum model development. Recently, nonstationary two-dimensional [23] and three-dimensional [24] problems of generating a response from an oriented layered or block cracked cluster located in a homogeneous medium were successfully solved using a high-performance computing system. In current research, we compare two different approaches taking into account the medium fracturing. This chapter is organized as follows. Section 8.2 presents two different mathematical models and numerical grid-characteristic method. Simulation results of the initially fractured medium and dynamic damaging process are discussed in Sect. 8.3. Section 8.4 concludes the chapter.

8 Numerical Comparison of Different Approaches for the Fractured …

89

8.2 Mathematical Model In this work, two different approaches for the simulation of dynamic processes in fractured media are used. In Sect. 8.2.1, a continuum model with an implicit fracturing mechanism is formulated. In Sect. 8.2.2, an approach for explicit fracturing is described. A high precision method for solving hyperbolic systems of equations is proposed in Sect. 8.2.3.

8.2.1 Implicit Fracturing Model Consider a material consisting of periodically alternating elastic layers and gluing layers between them. If the tensile strength is exceeded, the slip and delamination planes may occur. Their orientation can be specified by the unit normal vector n. Shear stress at such a boundary is τ = σ · n − (n · σ · n)n, and tensile stress is σn = n · σ · n. Let the distance between possible slip planes be the same and equal to . We assume that the thickness of gluing layers is h, and the relation h   is valid. Then we can remove the explicit dependence on h and introduce the corresponding contact conditions. Initially, we are dealing with an elastic material. However, due to local exceeding the tensile strength of the material, microcracking zones will be formed. In this work, we consider a strength condition in the following form. For |τ | < τ0 and |σn | < σ0 , the material is described by the elastic model, but if at least one of the conditions is violated, then new area of damage appears. Introducing the scalar damage function , which is initially zero in all points of the material, we can write the dependencies expressed by Eqs. 8.1 and 8.2.  σ0 () =  τ0 () =

σ0 if  = 0 0 else

(8.1)

τ0 if  = 0 0 else

(8.2)

Here  = 1 − H − (σn − σ0 )H − (|τ | − τ0 ) and H − (y) = 1 − H (y), where H (y) is the Heaviside function. To describe the behavior of the damaged material under the dynamic load, we can divide this process into the elastic and inelastic parts. Sliding along the boundary of elastic layers can be specified by the tangent velocity vector γ and delamination process by the normal velocity vector ω. Their connection with the jump in the particle velocity vector of the medium at the contact boundary is γ = Vτ / and ω = Vn /. In this work, we use a combination of Coulomb sliding friction conditions with a small viscous additive on pressed contacts and weak delamination on unloaded planes [7]. If σn < 0, then

90

I. S. Nikitin et al.

 τ = q|σn |

 γ + ηγ , ω = 0, |γ |

(8.3)

else γ =

1 ∂τ , kγ ∂t

(8.4)

ω=

1 ∂σn , kω ∂t

(8.5)

kγ , kω  1. μ

(8.6)

Final expressions for velocities γ and ω are of the form: γ =

  |τ | 1 τ 1 ∂τ H (σn − σ0 ), − 1 H − (σn − σ0 ) + η |τ | q|σn | kγ ∂t

(8.7)

1 ∂σ n H (σn − σ0 ). kω ∂t

(8.8)

ω=

Formulating a continuum model of a medium containing a system of slipdelamination planes, we consider the unknown γ and ω as the continuous functions of coordinates and time that have the meaning of distributed sliding and delamination velocities. Relations of the slip theory, which was used by many authors to construct models of inelastic media with a continuous distribution of slip planes, were applied. The inelastic part of the stress tensor consists of two parts in a view of Eqs. 8.9 and 8.10. eγ = (n ⊗ γ + γ ⊗ n)/2 n · γ = 0

(8.9)

eω = (n ⊗ ω + ω ⊗ n)/2 = ωn ⊗ n ω = ω

(8.10)

The full strain rate tensor e is obtained by summing elastic and inelastic components:   ∇v + ∇v T . e=e +e +e = 2 γ

ω

e

(8.11)

Constituent layers are described as a linear isotropic material. The govern system of equations contains motion equations and rheological relations: ρ

∂v = ∇ · σ, ∂t

(8.12)

8 Numerical Comparison of Different Approaches for the Fractured …

∂σ = λ(∇ · v)I + 2μe, ∂t

91

(8.13)

where ρ is the density, v is the velocity vector, σ is the stress tensor, λ and μ are the Lame parameters. Based on the described theory, the explicit formulas were obtained that take a form of corrections after performing the elastic [9].

8.2.2 Explicit Fracturing Model The other approach can be used to describe a presence of some amount of fractures inside the elastic medium [25]. Relying on rectangle grids, it provides the mechanism of describing arbitrarily oriented cracks. The real crack is replaced by a set of tiny fractures crossing nearest nodes. They are split into two independent sub-nodes, and the appropriate contact conditions were used based on the known normal direction. For example, in two-dimensional case for the fluid-saturated crack we can write: v 1 · n = v 2 · n,

(8.14)

(σ 1 · n) · n = −(σ 2 · n) · n,

(8.15)

(σ 1 · n) · l = 0,

(8.16)

(σ 2 · n) · l = 0,

(8.17)

where different subscripts mean different borders of the crack, n is the normal vector, l is the tangent vector. This method was previously verified by the comparison with the discontinuous Galerkin method on the unstructured grids [25]. To describe a process of the fracture occurrence in each nodes, the main stress is compared with the material stiffness. In the case of exceeding, a new fracture with the plain of orthogonal direction is created in the grid. Zero-force conditions are used at the both sides to simulate the unfilled crack provided by Eq. 8.18. σ 1,2 · n = 0

(8.18)

8.2.3 Grid-Characteristic Method The elastic part of equations was solved numerically with the grid-characteristic method, which uses hyperbolic properties of a system of partial differential equations.

92

I. S. Nikitin et al.

Initially, all equations were rewritten in the canonical form: q t + Ax q x + A y q y = f ,

(8.19)

where Ax , A y are the square matrices defined by material parameters with a full set of eigenvalues and eigenvectors, the vector f is determined by the right side. Let’s consider the homogenous equation for simplicity, so f = 0, and q is the vector of T  unknown functions. It equals to q = vx , v y , σx x , σx y , σ yy . To solve the initial system, we split it by two orthogonal directions and solve the one-dimensional for each one consecutively. For the most general case of curvilinear grids, we have to perform the splitting by an arbitrary direction Oξ given by a unit directional vector n. Thus, we obtain a simplified system: q t + Aξ q ξ = 0.

(8.20)

The matrix Aξ can be singularly decomposed into the product of matrices Aξ = −1 , where  is the diagonal matrix of eigenvalues. Now we can introduce the Riemann invariants ω = q, in terms of which the system of PDEs becomes a system of independent transport equations. The resulting transport equations are solved iteratively. During each time step, the value on the next time step is calculated using the interpolation of the necessary order using Eq. 8.21. ωi (t + t, ξ, η) = ωi (t, ξ − λi t, η)

(8.21)

The return procedure is described by q = −1 ω. Using initial values, we obtain a direct time-stepping algorithm. The interpolation procedure requires the knowledge of values ωi in the neighboring nodes. For nodes on the boundary, the positive or negative characteristics go out from the computational domain. Therefore, additional conditions are necessary to determine the corresponding Riemann invariants ωi . These relations are given as a system of linear equations from which the unknown correction values are derived, and then the correction is applied. It should be noticed that with this approach the contact condition may be implemented as a consequently application of boundary conditions for both sub-domains.

8.3 Simulation Results Hereinafter, the problem of dynamic loading of the fractured medium is investigated. The cases of unfilled fractures and fluid-filled fractures are considered in Sects. 8.3.1 and 8.3.2, respectively.

8 Numerical Comparison of Different Approaches for the Fractured …

93

8.3.1 Unfilled Fractures The numerical simulation of the striker collision with a plate was carried out. A two-dimensional statement of the problem was considered. The computational grid contained about five thousand nodes. The dimensionless parameters of the task are as follows: The striker velocity equals 0.02, the material density equals 10, the Pwave velocity inside the material equals 1, and the S-wave velocity inside the material equals 0.7. The orientation of possible slippage planes is horizontal (layered composite model). The process of the interaction between the striker and target was calculated on moving Lagrange grids, which made it possible to take into account the deformation of the top surface of the target, as well as, the reflection of the striker. Initially, the compression and tension zones were identified during the numerical simulation with the switched off damage mechanics. In Fig. 8.1, the spatial distributions of component σ yy are depicted. Immediately after the collision moment, the compression wave propagates to the revert side of the target. The local zone of the intensive tension is formed after the P-wave reflection from the free border. It should lead to the material destruction. Further, the mechanism of the explicit fracturing was switched on. For the visualization of destruction zones, the tiny white line normal to the main stress was placed in the appropriate grid node. In Fig. 8.2, the simulation results are presented. They correspond to the gradual grows of the material stiffness. It is clearly seen that this process leads to the generation of a single destruction zone on the opposite side of the plate. Finally, the simulation was done in the frame of the continual model of solid media with a discrete set of slip planes. Figure 8.3 shows the distribution of the calculated

Fig. 8.1 Component σ yy of the elastic stress tensor. Blue color corresponds to the compression zones, and red color corresponds to the tension zones. There are consecutive points in time: a 100 time steps, b 110 time steps, c 120 time steps, d 130 time steps

94

I. S. Nikitin et al.

Fig. 8.2 Modulus of velocity in the color scale (red is the maximum). Initiated cracks are depicted with the white color. Normalized stiffness is equaled: a 0.6, b 0.8, c 1.0, d 1.2

Fig. 8.3 Absolute  values at delamination planes after the interaction process. Nonzero values correspond to destruction zones (red color)

8 Numerical Comparison of Different Approaches for the Fractured …

95

Fig. 8.4 Computational domain was the rectangle with the point source (star) and the fractured cluster (grey rectangle) placed in it

delamination zone after the stabilization of wave processes. Due to the fact that at the initial moment the compression wave penetrates into the material, its upper part remains intact. Reflecting from a free revert surface, this wave transforms into a tensile wave, which leads to the delamination of elastic layers and the occurrence of the internal damage.

8.3.2 Fluid-Filled Fractures The numerical simulation of seismic waves’ propagation in initially fractured media was carried out. The problem was considered in two-dimensional case. The rectangular medium was elastic and isotropic with the density ρ = 2500 kg/m3 , the P-wave velocity c p = 4500 m/s, and the S-wave velocity cs = 2250 m/s. The total area of 10 km × 3 km was covered with the square mesh 1 m × 1 m, more than 30 million nodes were generated. The initially damaged zone had length of 3 km and the width of 100 m. The computational domain with the source position is represented in Fig. 8.4. Initially, the dynamic wave patterns calculated with both models were compared on vertically oriented fractures. In the case of explicit cracks identification, 151 objects (with free glide conditions) were injected in the model to compare its influence. For continuum model, the parameters q = 0, δ = μ/η t = 0.01, α = kγ /μ = 0.1, β = kω /(λ + 2μ) = 0.1 were used. The analysis of obtained results proved that the continuum model successfully reproduces the smooth seismic signal but doesn’t capture high-frequency oscillations (see Fig. 8.5). Also, the amplitude of the response is higher because of the significantly larger volume of scatters. Further, the case of non-vertical cracks was investigated. The fractured region was filled with cracks rotated on 45° from the vertical. Obtained wave patterns are presented in Fig. 8.6. The non-symmetry of the model led to the visible anisotropy of seismic signals. The main amplitude difference is detected on transmitted waves. It may be explained by the different behavior of models in the tension regime. The explicit crack model provides the discontinuity for the normal component of the

96

I. S. Nikitin et al.

Fig. 8.5 Velocity modulus for: a explicit fractures inclusion, b continuum model. The crack orientation is vertical

velocity vector. Thus, the crack borders cannot become farther or closer. By contrast, the continuum model allows a crack borders’ movement in different directions. That may be a reason of the generation of different wave patterns.

8.4 Conclusions In this chapter, a process of the seismic waves’ propagation in heterogeneous fractured media was investigated. To formulate the govern system of equations, two different approaches were used. The first one is relied on the continual model of solid media with a discrete set of slip planes and nonlinear-type slip conditions at them. Relations of the slip theory, which was used by many authors to construct models of inelastic media with a continuous distribution of slip planes, were applied. The numerical algorithm contains the explicit correction formulas after performing the elastic step. The second approach is based on the explicit crack description as a

8 Numerical Comparison of Different Approaches for the Fractured …

97

Fig. 8.6 Velocity modulus for: a explicit fractures inclusion, b continuum model. The crack orientation is 45°

set of grid nodes with the predefined normal vector. They are split into two independent sub-nodes, and appropriate contact conditions were used. To describe a process of fracture occurrence in each node, the main stress is compared with the material stiffness. Two fundamentally different problems were numerically solved. The striker collision with a plate was simulated. It requited a use of the movable grids for the adequate description of the surface deformation. The first model demonstrated the creation of the damaged zone on the revert side of the plate. The second model confirmed this statement. The influence of the material stiffness on the size of the damaged zone was demonstrated. The second problem was connected with the propagation of low intensity waves in initially fractured media. The comparison of seismic responses for both approaches in two cases (vertical and inclined cracks) was done. They successfully reproduced the smooth wave pattern. The observed difference in amplitudes may be explained by the different behavior of models in the tension regime. Further development of the continuum model may be connected with the extension on the more complicated rheological models like VTI, HTI, and other anisotropies.

98

I. S. Nikitin et al.

Acknowledgements This work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC “Kurchatov Institute”, http://ckp.nrcki.ru/. This work was carried out with the financial support of the Russian Science Foundation, project no. 19-71-10060.

References 1. Golubev, V.I., Muratov, M.V., Petrov, I.B.: Different approaches for solving inverse seismic problems in fractured media. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 199–212 (2020) 2. Beklemysheva, K.A., Kazakov, A.O., Petrov, I.B.: Numerical modeling of ultrasound phased array for non-destructive testing of composites. Lobachevskii J. Math. 40(4), 415–424 (2019) 3. Beklemysheva, K.A., Golubev, V.I., Vasyukov, A.V., Petrov, I.B.: Numerical modeling of the seismic influence on an underwater composite oil pipeline. Math. Models Comput. Simul. 11(5), 715–721 (2019) 4. Sadovskii, V.M., Sadovskaya, O.V.: Numerical algorithm based on implicit finite-difference schemes for analysis of dynamic processes in blocky media. Russian J. Numer. Anal. Math. Model. 33(2), 111–121 (2018) 5. Chentsov, E.P., Sadovskii, V.M., Sadovskaya, O.V.: Modeling of wave processes in a blocky medium with fluid-saturated porous interlayers. AIP Conf. Proc. 1895, 080002 (2017) 6. Sadovskii, V.M., Sadovskaya, O.V., Lukyanov, A.A.: Modeling of wave processes in blocky media with porous and fluid-saturated interlayers. J. Comput. Phys. 345, 834–855 (2017) 7. Nikitin, I.S.: Dynamic models of layered and block media with slip, friction and separation. Mech. Solids 43(4), 652–661 (2008) 8. Burago, N.G., Zhuravlev, A.B., Nikitin, I.S.: Continuum model and method of calculating for dynamics of inelastic layered medium. Math. Models Comput. Simul. 11(3), 59–74 (2019) 9. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Mathematical modeling of the dynamics of layered and block media with nonlinear contact conditions on supercomputers. J. Phys.: Conf. Ser. 1392(1), 012057 (2019) 10. Virieux, J., Calandra, H., Plessix, R.E.: A review of the spectral, pseudo-spectral, finitedifference and finite-element modelling techniques for geophysical imaging. Geophys. Prospect. 59(5), 794–813 (2011) 11. Carcione, J.M., Herman, C.G., Kroode, P.E.: Y2K Review article: seismic modeling. Rev. Lit. Arts Am. 67(4), 1304–1325 (2002) 12. Lisitsa, V., Tcheverda, V., Botter, C.: Combination of the discontinuous Galerkin method with finite differences for simulation of seismic wave propagation. J. Comput. Phys. 311, 142–157 (2016) 13. Massau, J.: Memoire sur L’integration Graphique des E´quations aux Deriv´ess Partielles. F. Meyer-van Loo, Ghent (1899) 14. Zhukov, A.I.: Using the method of characteristics for the numerical solution of one-dimensional problems of gas dynamics. Tr. Mat. Inst. Akad. Nauk SSSR 58, 4–150 (1960) 15. Butler, D.S.: The numerical solution of hyperbolic systems of partial differential equations of three independent variables. Proc. R. Soc. London Ser. A 255(1281), 232–241 (1960) 16. Golubev, V.I., Shevchenko, A.V., Petrov, I.B.: Taking into account fluid saturation of bottom sediments in marine seismic survey. Doklady Math. 100(2), 488–490 (2019) 17. 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, 421–431 (1991)

8 Numerical Comparison of Different Approaches for the Fractured …

99

18. Azarova, O.A.: Complex conservative difference schemes for computing supersonic flows past simple aerodynamic forms. Comput. Math. Math. Phys. 55, 2025–2049 (2015) 19. Sakhalkar, S.V., Bogy, D.B.: A model for lubricant transfer from media to head during HeatAssisted Magnetic Recording (HAMR) writing. Tribol. Lett. 65, art. no. 166 (2017) 20. Mittal, H.V.R., Ray, R.K.: Solving immersed interface problems using a new interfacial pointsbased finite difference approach. SIAM J. Sci. Comput. 40(3), A1860–A1883 (2018) 21. Mittal, H.V.R., Kalita, J.C., Ray, R.K.: A class of finite-difference schemes for interface problems with an HOC approach. Int. J. Numer. Methods Fluids 82(9), 567–606 (2016) 22. Golubev, V.I., Khokhlov, N.I., Nikitin, I.S., Churyakov, M.A.: Application of compact gridcharacteristic schemes for acoustic problems. J. Phys.: Conf. Ser. 1479, 012058.1–012058.11 (2020) 23. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Continual models of layered and block media with slippage and delamination. Procedia Struct. Integr. 23, 125–130 (2019) 24. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Methods for calculating the dynamics of layered and block media with nonlinear contact conditions. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 171–183 (2020) 25. Khokhlov, N., Stognii, P.: Novel approach to modeling the seismic waves in the areas with complex fractured geological structures. Minerals 10, 122–139 (2020)

Chapter 9

The Comparison of Two Approaches to Modeling the Seismic Waves Spread in the Heterogeneous 2D Medium with Gas Cavities Polina V. Stognii , Nikolay I. Khokhlov , Igor B. Petrov , and Alena V. Favorskaya Abstract The problem of detecting the hydrocarbon deposits in the heterogeneous media is very important and difficult for solving. We suppose the grid-characteristic method of the third order of accuracy for solving the problem of the direct modeling of the seismic waves spread in such a medium with the presence of gas cavities and without them. The numerical method used in all the computations is described in detail. We present the results of modeling, the wave fields of the normal component of the seismic velocity, and the seismograms for the models of the heterogeneous media with several gas cavities and without them. The results demonstrate the possibility of detecting the seismic reflections from the geological layers and gas cavities. In the previous work, we solved a problem of modeling the seismic waves spread through the heterogeneous media with the use of the transparent method, which also showed the correct results. In this work, we carry out the comparative analysis of the previous results with the new ones. The grid-characteristic method of the third order of accuracy and the transparent method are both appropriate for solving the described problem in general. However, the results, which were obtained using the grid-characteristic method of the third order of accuracy under consideration the contact conditions between the geological layers, demonstrate the clearer seismic reflections and the more accurate velocity meanings near the contact boundaries

P. V. Stognii (B) · N. I. Khokhlov · I. B. Petrov · A. V. Favorskaya Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] N. I. Khokhlov e-mail: [email protected] I. B. Petrov e-mail: [email protected] A. V. Favorskaya 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 214, https://doi.org/10.1007/978-981-33-4709-0_9

101

102

P. V. Stognii et al.

between different media respect to the transparent method, which does not consider any contact conditions between the geological layers. Keywords Numerical modeling · Computer simulation · Gas cavities · Grid-characteristic method

9.1 Introduction The numerical solution of the problem of modeling heterogeneous media is very important as most hydrocarbon deposits are located in such media. The hydrocarbon deposits are often located in such inhomogeneities as fractures [1, 2], oil layers, and gas cavities [3]. The last of them, gas cavities, are widely spread in the Arctic region [4, 5], where the geological medium is very nonuniform. In addition, the gas cavities are situated quite far from the Earth surface, at distance of 1000–1500 m from the surface. Thus, modeling the heterogeneous media is a high computational cost problem. The geological layers are described by different parameters (density, wave velocities), which should be considered in the model. One of the simplest ways of solving the problem of modeling the seismic waves spread through the heterogeneous media is the application of the optional numerical scheme to solving the problem. In the previous work, we solved the discussed problem using the method of transparent computations [6, 7]. This means that the geological layers are described by their characteristics, i.e., the longitudinal and transverse wave velocities and the density of the geological layer. This method does not take into account the specific contact conditions between the layers, so the results lose the useful information about the seismic waves spread through the heterogeneous media. However, even that simplification of the problem solution let us detect the differences between the models containing different number of gas cavities. The wave fields and seismograms demonstrated the similar results, presenting the rising wave reflections from gas-containing layers in time. In this chapter, we introduce the approach of modeling the seismic waves spread in the heterogeneous medium with gas cavities, taking into account the certain contact conditions between the different geological layers. On the contact boundary between the two geological layers, the numerical solution is corrected by the appropriate conditions. And beyond the contacts between the layers, we used the grid-characteristic method [8, 9] of the third order of accuracy [10]. The full adhesion contact conditions are estimated between all the geological layers. From a mathematical view point, it is more correct that computation the models using the transparent method. We present the results of modeling: The wave fields and seismograms for the four-year period of gas spread through the heterogeneous medium. Then, we carry out the comparative analysis between the results of computed models with and without contact conditions between the geological layers.

9 The Comparison of Two Approaches to Modeling the Seismic Waves …

103

The chapter is organized as follows. Section 9.1 introduces the general description of the problem of investigation of different regions with the gas cavities. In Sects. 9.2 and 9.3, we describe the numerical method and the contact conditions between the geological layers, respectively. Section 9.4 contains the results of the numerical modeling of the seismic waves spread through the heterogeneous medium with gas cavities. In Sect. 9.5, we carry out the analysis of the computational results obtained by two different methods: the transparent method and grid-characteristic method of the third order of accuracy. Section 9.6 concludes the chapter.

9.2 Numerical Method We used the linear-elastic system of equations [11] for describing the dynamic behavior of seismic waves: ρ

∂v = (∇ · σ)T , ∂t

  ∂σ = λ(∇ · v)I + μ (∇ × v) + (∇ × v)T , ∂t

(9.1) (9.2)

where v is the velocity of the seismic waves in the medium, σ is the Cauchy stress tensor, λ and μ are the Lame parameters, ρ is the density of the medium. The finite-volume [11], discontinuous Galerkin [12–14], spectral element [15, 16], finite differences [17, 18], and staged grid [19] methods might be used to solve Eqs. 9.1 and 9.2 in order to model the elastic wave phenomena in geological media. We have used the grid-characteristic method [8, 9]. Recently, this method was modified [8, 20, 21] and used to solve the inverse [22] and direct problems of seismic prospecting [21, 23], ultrasound diagnostic [23] including medicine [24], icebergs explosions [25], seismic resistance [26, 27], and shock waves [28] calculation. The grid-characteristic method for 2D case was used for solving the system of Eqs. 9.1 and 9.2. We can rewrite the system of Eqs. 9.1 and 9.2 in the form: ∂q ∂q ∂q +Ax +A y = 0, ∂t ∂x ∂y

(9.3)

  where the vector q looks as q = σx x , σx y , σ yy , vx , v y and the matrixes Ax , A y are constructed out of the coefficients of the system of Eqs. 9.1 and 9.2. After splitting Eq. 9.3 in space coordinates, we obtain two 1D systems of equations: ∂q ∂q +Ai = 0, i ∈ {x, y}. ∂t ∂i

(9.4)

104

P. V. Stognii et al.

Now, we work with Eq. 9.4 for the case, when i = x (the case, when i = y is analogical): ∂q ∂q +Ax = 0. ∂t ∂x

(9.5)

The system of Eq. 9.5 is hyperbolic, then it can be represented as ∂q ∂q + x x −1 = 0, x ∂t ∂x

(9.6)

where x is the matrix, made of the  matrix Ax , x is the diag eigenvectors of the onal matrix with the eigenvalues −c p , c p , −cs , cs , 0 on the diagonal, c p is the longitudinal wave velocity, cs is the transverse wave velocity: cp =

 (λ + 2μ)/ρ,

cs =

 μ/ρ.

(9.7)

Then, we make the variable change p = −1 x q, and the system of Eq. 9.6 will transfer to ∂p ∂p +x = 0. ∂t ∂x

(9.8)

The system of Eq. 9.8 consists of five independent equations, each of which is solved using the Rusanov scheme of the third order of accuracy [10] provided by Eqs. 9.9 and 9.10. 6( pi )nm−1 − 3( pi )nm − 2( pi )nm+1 − ( pi )nm−2 6h (λi τ )2 ( pi )nm+1 − 2( pi )nm + ( pi )nm−1 + 2 h2 n 3 (λi τ ) ( pi )m−2 − 3( pi )nm−1 + 3( pi )nm − ( pi )nm+1 + 6 h3

= ( pi )nm + λi τ ( pi )n+1 m

6( pi )nm+1 − 3( pi )nm − 2( pi )nm−1 − ( pi )nm+2 6h (λi τ )2 ( pi )nm−1 − 2( pi )nm + ( pi )nm+1 + 2 h2 n 3 (λi τ ) ( pi )m+2 − 3( pi )nm+1 + 3( pi )nm − ( pi )nm−1 − 6 h3

(9.9)

= ( pi )nm − λi τ ( pi )n+1 m

(9.10)

In Eqs. 9.9 and 9.10, τ is the time step, h is the coordinate step along considered direction. Equation 9.9 is used for positive λi , and Eq. 9.10 is applied for negative = ( pi )nm . λi . For zero λi ( pi )n+1 m

9 The Comparison of Two Approaches to Modeling the Seismic Waves …

105

9.3 Contact Conditions The full adhesion contact condition between two geological layers (determined by indexes l and r) described by the system of the linear-elastic equations has a view: vl = vr = V, σl nl = σr nr , nl = −nr

(9.11)

    σ n+1 = σ n+1(ins) − ρ n · gn+1 c p − 2cs − c3 (n × n) + c3 I   +cs gn+1 × n + n × gn+1 ,

(9.12)

λ2 where c3 = ρ(λ+2μ) , gn + 1 =vn + 1(ins) − V, nl and nr are the normal vectors toward the contact between the layers. The system of Eqs. 9.9 and 9.10 is used for all contacts between the geological layers.

9.4 The Results of Numerical Modeling of the Seismic Waves Spread Through the Heterogeneous Media We carried out the numerical modeling of the seismic waves spread through the heterogeneous media with the presence of gas cavities and without them. The basic model without any gas cavities consists of nine geological layers, from top to bottom: water, silt, water-saturated sand, clay, water-saturated sand, clay, water-saturated sand, clay, and oil. The second model contains one gas cavity, the third model includes two gas cavities, and the fourth model consists of three gas cavities. Four static models with different number of gas cavities were applied for modeling the dynamic spread of gas through the heterogeneous medium with time. Gas stays only in the water-saturated sand because the density of clay is too large. The geological layers were characterized by the following parameters. The longitudinal wave velocity in water was 1500 m/s, in the silt layer was 1600 m/s, in the water-saturated sand was 1600 m/s, in clay was 2000 m/s, in the oil layer was 1400 m/s, and in gas was 1000 m/s. The transverse wave velocity in water was 10 m/s, in the silt layer was 1131 m/s, in the water-saturated sand was 1131 m/s, in clay was 1414 m/s, in the oil layer was 1000 m/s, and in gas was 707 m/s. The water layer was modeled as a linear-elastic layer with very small transverse wave velocity. The density of water was 1000 kg/m3 , of silt was 1600 kg/m3 , of water-saturated sand was 1200 kg/m3 , of clay was 2500 kg/m3 , of oil was 850 kg/m3 , and of gas was 900 kg/m3 . We computed the Reiker impulse spread through the heterogeneous medium during 2 s. The time step was equal to 0.0001 s, the overall number of steps was 20,000. Two hundred seismic receivers were established on the surface of water in the center at distance of 10 m between them.

106

P. V. Stognii et al.

Fig. 9.1 Wave fields at time 0.64 s for the models with different number of gas cavities: a no gas cavities, b one gas cavity, c two gas cavities, d three gas cavities

The result wave field for the model of the first computational year is presented in Fig. 9.1a. The figure depicts the normal component of the velocity in the heterogeneous medium without gas cavities at time 0.64 s. The wave fields at the same time moment for the models of the second, third, and fourth computational years are presented in Fig. 9.1b–d, respectively. The black rectangles in Fig. 9.1b–d depict the gas cavities. The wave reflections from the geological layers above the gas cavities look very similar. The differences occur at the gas cavity location and lower. The seismograms for the model without gas cavities, with one gas cavity, with two gas cavities, and with three gas cavities are presented in Fig. 9.2a–d, respectively. We analyze the seismograms only for the y-component of the velocity (Vy) as they are more representative and contain more differences for the different computational methods than the seismograms for the x-component of the velocity (Vx). The differences in the signals are noticeable in the centers of the seismograms. The extra short horizontal seismic lines make it possible to differ the models with the presence of gas cavities and without them. Moreover, the number of short horizontal seismic lines increases with the increase of the number of gas cavities. It is connected with the seismic waves spreading between the gas cavities.

9 The Comparison of Two Approaches to Modeling the Seismic Waves …

107

Fig. 9.2 Seismograms of the y-components of the velocity (Vy) for the models with different number of gas cavities: a no gas cavities, b one gas cavity, c two gas cavities, d three gas cavities

9.5 The Comparison of the Models Computed by Two Different Methods In this section, we carry out the analysis of the computational results obtained by the two different methods: the transparent method and numerical method with contact conditions between the geological layers. Earlier, we investigated the results of computing 2D and 3D models with/without gas cavities using the transparent method [6, 7]. The results were adequate, but the method accuracy was near the first order on the contact boundaries between the geological layers. In this chapter, we present the results of computing 2D models with taken into account contact conditions between the layers. The results of modeling were described in the previous section. Now, we are going to compare the amplitudes of the seismic reflections on the contact boundaries of the models computed by different methods. The seismograms for the models of the first computational year (without gas cavities) for the 0.5 s period of time are presented in Fig. 9.3a, b. The seismogram based on the transparent method (Fig. 9.3b) brings in extra reflections into the figure if compared with the same time-period seismogram based on the computational method taking into account contact conditions (Fig. 9.3a). These reflections start at the time, when the seismic wave approaches the first contact between the geological layers, the water–silt contact (0.03 s). The seismograms for the 0.5–1.0 s period are shown in Fig. 9.3c, d. The seismogram based on the transparent method (Fig. 9.3d) brings

108

P. V. Stognii et al.

Fig. 9.3 Seismograms of the y-components of the velocity (Vy) for the models computed with different modifications of the grid-characteristic method: a the grid-characteristic method considering contact conditions (seismograms present the 0.0–0.5 s record), b the transparent method, which does not consider the contact conditions (seismograms present the 0.0–0.5 s record), c the gridcharacteristic method considering contact conditions (seismograms present the 0.5–1.0 s record), d the transparent method, which does not consider the contact conditions (seismograms present the 0.5–1.0 s record)

a lots of extra reflections into the figure respect to the same time-period seismogram based on the computational method considering the contact conditions (Fig. 9.3c). The extra reflections in the seismogram of Fig. 9.3b allow to interpret the information about the seismic waves, coming from one contact between the layers. But when the extra reflections interfere with each other with time, and, moreover, more extra reflections appear from other contacts between the geological layers (as in Fig. 9.3d), it becomes too difficult to analyze the seismic reflections coming from different geological layers. Then, we analyze the wave fields for the models computed by different numerical methods. The wave fields for the models of the first computational year are presented in Fig. 9.4. The wave fields in Fig. 9.4 show the normal component of the velocity in the water and silt layers. The wave fields were made using the same scale, where the maximum meaning, presented by the red color, corresponds to 0.000167 m/s. The wave field in Fig. 9.4a presents the seismic velocities at time 0.16 s in the model computed by the method with taken contact conditions into account, and the wave field in Fig. 9.4b presents the seismic velocities at time 0.16 s in the model computed by the transparent method.

9 The Comparison of Two Approaches to Modeling the Seismic Waves …

109

Fig. 9.4 Wave fields at time 0.16 s for the models, computed with different modifications of the grid-characteristic method: a the grid-characteristic method considering the contact conditions, b the transparent method, which does not consider the contact conditions

The wave fields for the both models look similar. The corresponding graphs of the meanings of the normal components of the seismic velocities along the central vertical axis are shown in Fig. 9.5a, b. The differences between the meanings are less than 0.1%, so we can neglect them. The corresponding graphs of the relative

Fig. 9.5 Graphs at time 0.16 s for the models: a the relative meanings of the normal components of the seismic velocities v-norm (the grid-characteristic method considering contact conditions), b the relative meanings of the normal components of the seismic velocities v-norm (the transparent method, which does not consider contact conditions), c relative meanings of the vertical components of the seismic velocities Vy (the grid-characteristic method considering contact conditions), d relative meanings of the vertical components of the seismic velocities Vy (the transparent method, which does not consider contact conditions)

110

P. V. Stognii et al.

meanings of the vertical components of the seismic velocities (Vy) along the central vertical axis are shown in Fig. 9.5c, d. The differences between the meanings of the Vy-components of the velocities are less than 0.1%, so we can also neglect them. We obtain more interesting results, if we gather the meanings of the seismic velocities along the X-axis in the center of the model near the contact water–silt. Graphs in Fig. 9.6a, b present the meanings of the normal components of the velocity in the water layer at distance of 15 m from the contact for the model computed by the method with taken contact conditions into account (Fig. 9.6a), and for the model computed by the transparent method (Fig. 9.6b). Graphs in Fig. 9.6c, d present the meanings of the normal components of the velocity in the silt layer at distance of 15 m from the contact for the model computed by the method with taken contact conditions into account (Fig. 9.6c) and for the model computed by the transparent method (Fig. 9.6d). In both cases, in the graphs in the water layer and in the graphs in the silt layer, we obtain the smooth graphs only for the model computed by the method with taken contact conditions into account (Fig. 9.6a, c). The graphs for the model computed by the transparent method possess several gap points that is

Fig. 9.6 Graphs at time 0.16 s for the models: a the relative meanings of the normal components of the seismic velocities v-norm along the x-axis in water (the grid-characteristic method considering contact conditions), b the relative meanings of the normal components of the seismic velocities vnorm along the x-axis in water (the transparent method, which does not consider contact conditions), c the relative meanings of the vertical components of the seismic velocities v-norm along the x-axis in silt (the grid-characteristic method considering contact conditions), d the relative meanings of the vertical components of the seismic velocities v-norm along the x-axis in silt (the transparent method, which does not consider contact conditions)

9 The Comparison of Two Approaches to Modeling the Seismic Waves …

111

Fig. 9.7 Time moment 0.56 s, the relative meanings of the normal components of the seismic velocities v-norm: a the wave fields, which are built using the grid-characteristic method considering contact conditions, b the wave fields, which are built using the transparent method without considering the contact conditions, c the graphs provided by the grid-characteristic method considering contact conditions, d the graphs provided by the transparent method without consideration the contact conditions

incorrect according to the initial formulation of the problem. However, the overall form of the graphs for the models computed by different methods is similar. At last, we analyze the wave fields for the models of the second computational year computed by different numerical methods nearby the gas cavity. Figure 9.7 presents the wave fields above the water-saturated sand, where the gas cavity is located. The normal component of the seismic velocity is shown in Fig. 9.7a for the model computed with taken contact conditions into account, and Fig. 9.7b depicts the same wave field for the model computed by the transparent method. The wave fields in Fig. 9.7 present the 0.56 s time moment. The wave fields look similar except for the extra reflections in Fig. 9.7b, which appear as a result without consideration the contact conditions between the geological layers. We obtained the same result in the seismograms (Fig. 9.3). The corresponding graphs of the normal components of the velocities for the models of the second computational year with one gas cavity are shown in Fig. 9.7c, d. The graph is smooth in Fig. 9.7c, and the graph in Fig. 9.7d contains a few gap points, especially in the point with x = 1900 and x = 2100 that is critical, because the reflections from the gas cavity are just located in that area. But the overall form of the graph for the model computed by the transparent method (Fig. 9.7d) is similar to the graph for the model computed with taken contact conditions between the layers into account (Fig. 9.7c). Then, arising from the results in Figs. 9.6 and 9.7, we can conclude that if we need the overall solution of the problem of contact interference between the geological layers with the presence of gas cavities, the

112

P. V. Stognii et al.

transparent method can be applied as a numerical method of its solution. However, it is better to use the numerical method considering the contact conditions between the geological layers, when we need the accurate meanings of the seismic velocities in all the geological layers.

9.6 Conclusions In this chapter, we presented the results of the numerical modeling of the seismic waves spread through the heterogeneous media with gas cavities using the gridcharacteristic method of the third order of accuracy. The contact conditions between the different media were set as the full adhesion conditions. The wave fields of the normal component of the velocity and the seismograms demonstrated the correctness of applying the grid-characteristic method of the third order of accuracy to solving the problem of modeling the seismic waves spread in the heterogeneous media with gas cavities. The wave fields, as well as, the seismograms, show the possibility of detecting the seismic reflections coming from the contacts between the geological layers and gas cavities. The results of modeling the heterogeneous media with several gas cavities show the availability to detect the seismic reflections from each gas cavity that can be used further in determining the number of gas cavities in the geological area. The computations were carried out for the 2D case of the problem formulation as the correctness of replacing 3D models by 2D models in the problems of the seismic wave spread through the heterogeneous media with gas-containing layers. We carried out the comparative analysis of the results obtained from modeling the seismic waves spread through the heterogeneous medium using the gridcharacteristic method of the third order of accuracy and considering the contact conditions between the layers, and the results of computing the same models applying the transparent method, which does not consider the contact conditions between the layers. The analysis demonstrated the correct application of these methods for solving the problem in general. However, the grid-characteristic method of the third order of accuracy considering the contact conditions between the layers demonstrated the clearer seismograms and graphs of the velocity meanings near the contact boundaries between the geological layers than the transparent method, which did not consider the contact conditions between the layers. Acknowledgements The reported study was funded by RFBR according to the research project № 19-07-00366. This work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC “Kurchatov Institute”, http://ckp.nrcki.ru/.

9 The Comparison of Two Approaches to Modeling the Seismic Waves …

113

References 1. Zhan, Q., Sun, Q., Ren, Q., Fang, Y., Wang, H., Liu, Q.H.: A discontinuous Galerkin method for simulating the effects of arbitrary discrete fractures on elastic wavepropagation. Geophys. J. Int. 210(2), 1219–1230 (2017) 2. Khokhlov, N., Stognii P.: Novel approach to modelling the seismic waves in the areas with complex fractured geological structures. Minerals 10(2), 122.1–122.17 (2020) 3. Castroa, J., Cashman, K., Joslin, N., Olmstedc, B.: Structural origin of large gas cavities in the Big Obsidian Flow, Newberry Volcano. J. Volcanol. Geotherm. Res. 114(3), 313–330 (2002) 4. Reshetova, G., Cheverda, V., Lisitsa, V., Khaidykov, V.: A parallel algorithm for studying the ice cover impact onto seismic waves propagation in the shallow arctic waters. Commun. Comput. Inf. Sci. 965, 3–14 (2019) 5. Marchenko, A., Eik, K.: Iceberg towing in open water: mathematical modeling and analysis of model tests. Cold Reg. Sci. Technol. 73, 12–31 (2011) 6. Stognii, P.V., Khokhlov, N.I.: 2D seismic prospecting of gas pockets. 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. 156–166. Springer, Cham (2019) 7. Stognii, P., Khokhlov, N., Petrov, I.: Numerical modeling of wave processes in multilayered media with gas-containing layers: comparison of 2D and 3D models. Doklady Math. 100(3), 586–588 (2019) 8. Favorskaya, A.V., Khokhlov, N.I., Petrov, I.B.: Grid-characteristic method on joint structured regular and curved grids for modeling coupled elastic and acoustic wave phenomena in objects of complex shape. Lobachevskii J. Math. 41(4), 512–525 (2020) 9. Magomedov, K., Kholodov, A.: Grid Characteristic Methods. Nauka, Moscow (1988) 10. Kholodov, A.S., Kholodov Y.A.: Monotonicity criteria for difference schemes designed for hyperbolic equations. Comput. Math. Math. Phys. 46(9), 1560–1588 (2006) 11. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002) 12. De Basabe, J.D., Sen, M.K., Wheeler, M.F.: The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion. Geophys. J. Int. 175(1), 83–93 (2008) 13. Dumbser, M., Käser, M.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes—II. The three-dimensional isotropic case. Geophys. J. Int. 167(1), 319–336 (2006) 14. Wilcox, L.C., Stadler, G., Burstedde, C., Ghattas, O.: A high-order discontinuous Galerkin method for wave propagation through coupled elastic–acoustic media. J. Comput. Phys. 229(24), 9373–9396 (2010) 15. Komatitsch, D., Tromp, J.: Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139(3), 806–822 (1999) 16. Capdeville, Y., Chaljub, E., Montagner, J.P.: Coupling the spectral element method with a modal solution for elastic wave propagation in global earth models. Geophys. J. Int. 152(1), 34–67 (2003) 17. Moczo, P., Robertsson, J.O., Eisner, L.: The finite-difference time-domain method for modeling of seismic wave propagation. Adv. Geophys. 48, 421–516 (2007) 18. Wang, T., Tang, X.: Finite-difference modeling of elastic wave propagation: a nonsplitting perfectly matched layer approach. Geophysics 68(5), 1749–1755 (2003) 19. Graves, R.W.: Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences. Bull. Seismol. Soc. Am. 86(4), 1091–1106 (1996) 20. Khokhlov, N., Golubev, V.: On the class of compact grid-characteristic schemes. 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. 64–77. Springer, Cham (2019) 21. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Methods for calculating the dynamics of layered and block media with nonlinear contact conditions. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) CMMASS 2019, SIST, vol. 173, pp. 171–183. Springer, Singapore (2019)

114

P. V. Stognii et al.

22. Golubev, V.: The usage of grid-characteristic method in seismic migration problems. 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. 143–155. Springer, Cham (2019) 23. Favorskaya, A.V.: Elastic wave scattering on a gas-filled fracture perpendicular to plane P-wave front. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) CMMASS 2019, SIST, vol. 173, pp. 213–224. Springer, Singapore (2019) 24. 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) 25. Favorskaya, A., Khokhlov, N.: Icebergs explosions for prevention of offshore collision: computer simulation and analysis. In: Czarnowski, I., Howlett, R.J., Jain, L.C. (eds.) KES-IDT 2020, SIST, vol. 193, pp. 201–210. Springer, Singapore (2020) 26. Favorskaya, A., Golubev, V.: Study the elastic waves propagation in multistory buildings, taking into account dynamic destruction. In: Czarnowski, I., Howlett, R., Jain, L.C. (eds.) KES-IDT 2020, SIST, vol. 193, pp. 189–199. Springer, Singapore (2020) 27. Favorskaya, A.: Computation the bridges earthquake resistance by the grid-characteristic method. In: Czarnowski, I., Howlett, R.J., Jain, L.C. (eds.) KES-IDT 2020, SIST, vol. 193, pp. 179–187. Springer, Singapore (2020) 28. Lopato, A., Utkin, P.: The usage of grid-characteristic method for the simulation of flows with detonation waves. 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. 281–290. Springer, Cham (2019)

Chapter 10

Mathematical Modeling of Spatial Wave Processes in Fractured Seismic Media Maksim V. Muratov

and Tatiana N. Derbysheva

Abstract This chapter is devoted to the study of the propagation of elastic waves in a fractured seismic medium using methods of mathematical modeling. The obtained results are compared with the results of physical modeling on similar models. For mathematical modeling, a grid-characteristic method is used with 1-3-order hybrid schemes with approximation on unstructured triangular meshes (2D case) and tetrahedral meshes (3D case). Such meshes make it possible to specify inhomogeneities (fractures) of various complex shapes and spatial orientations. There is a description of developed mathematical models of fractures, which can be used for numerical solution of exploration seismology problems. The base of developed models is the concept of infinitely thin fracture, which aperture does not influence on wave processes in fracture area. These fractures are represented by boundaries and contact boundaries with different conditions on the contact and boundary surfaces. Such approach significantly reduces the computational costs due to the absence of the mesh definition inside the fracture. On the other side, it lets to state the fractures discretely in integration domain. Therefore, one can observe qualitative new effects such as diffractive waves forming and multiphase wave front due to multiple reflections between surfaces of neighbor fractures. These effects are not available to observe using effective models of fractures, actively applied in computational seismology. Keywords Grid-characteristic method · Exploration seismology problems · Mathematical modeling · Mathematical models of fractures · Mesofractures · Physical modeling

M. V. Muratov (B) · T. N. Derbysheva Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] T. N. Derbysheva 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 214, https://doi.org/10.1007/978-981-33-4709-0_10

115

116

M. V. Muratov and T. N. Derbysheva

10.1 Introduction The problem of seismic waves’ propagation through fractured geological medium simulation is quite popular. Fractured reservoirs are the potential hydrocarbon collectors. The papers [1–4] are devoted to different approaches of fracture modeling. According to Leviant et al. [5], the geologists distinguished several types of fractures in dependence of dimensions: microfractures, which aperture is about tens of microns and extension is about several centimeters, mesofractures, which aperture is about hundreds of microns, and extension is up to several meters, and macrofractures, which aperture is about several millimeters and more and extension is from tens to hundreds of meters. At the same time, for description of microfractures, as well as mesofratures in the scale of geological medium for regional exploration, the use of popular effective models [6, 7] is the most optimal for performance. For modeling mesofractures and macrofractures, the modern high-performance computational systems let us to use the discrete models of fractures (Fig. 10.1), which significantly increase the possibilities of explorations. For different problems, one can used discrete models with different degree of accuracy: fluid in fracture with use of viscous liquid model, model of ideal liquid, model with state of elastic characteristics of liquid, etc. In the case of wave response observing in macroscale of entire exploration seismology experiment, the most optimal approach is setting of fractures as boundary and contact boundary conditions—model of infinitely thin fracture. The applicability of this model was shown in [8–10]. This chapter is dedicated to selecting of the most suitable conditions for different cases to solve the exploration seismology problems using grid-characteristic method [11, 12] on unstructured meshes [13, 14]. Let us notice that the most often used in practice cases fractures are the gas-saturated, fluid-filed, and glued. The range of models was developed for these cases and their combinations. The chapter is organized as follows. Section 10.2 is about a mathematical model. In Sect. 10.3, we consider the fracture models based on a concept of infinitely thin fracture. Section 10.4 describes the results of mathematical and physical modeling and their comparison. Section 10.5 concludes the chapter.

Fig. 10.1 Examples of fractures discrete models

10 Mathematical Modeling of Spatial Wave Processes in Fractured …

117

10.2 Mathematical Model For mathematical modeling, a linear-elastic medium model is used. During 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 [15]. According to this model, the state of an infinitely small volume of 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 the following expressions: ∂ T ji ∂ Vi = , ∂t ∂x j      ∂ Vk ∂Vj ∂ Ti j ∂ Vi , =λ + Ii j + μ ∂t ∂ xk ∂x j ∂ xi k

ρ

where Ti j are the components of stress tensor, Vk are the components of velocity, λ and μ are the Lame coefficients, and Ii j is the unit tensor. 3D system of such equations can be represented in the form:  ∂ u ∂ u + Ai = 0, ∂t ∂ξ i i=1,2,3 where ξi take values as x, y, and z, and u is vector of values that can be represented as   u = Vx , Vy , Vz , Tx x , Tyy , Tzz , Tx y , Tx z , Tyz . Such system can be solved by grid-characteristic method [11, 16]. 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 vector is transferred to the time layer n + 1 according to the formula:



wkn+1 ξi = wkn ξi − ωk τ , where τ is the timestep.

118

M. V. Muratov and T. N. Derbysheva

After all the values are transferred, there is a reverse transition to the vector of the desired values u.

10.3 Fracture Models This section is about developed models of fractures based on a model of infinitely thin fracture. We consider the most popular cases such as gas-saturated (Sect. 10.3.1), fluid-filled (Sect. 10.3.2), glued (Sect. 10.3.3), and partially glued (Sect. 10.3.4) fractures. For each case, the special model was developed [17].

10.3.1 Gas-Saturated (Empty) Fracture The model of gas-saturated (empty) fracture describes well a behavior of fractures saturated by air or gas on a little depth up to 100–150 m. In more depths, the fractures with air are closed by action of large pressure and gas acquires properties of liquid. The fracture can be set as boundary condition of full reflection on the surfaces of fracture: T n = 0. Such model is applicable for describing situation. Velocity of longitudinal elastic wave propagation in geological medium (1500–7000 m/s) is significantly more the wave velocity in air (330 m/s) and natural gas (430 m/s) in little pressures. Velocity of shear elastic wave propagation in air is equal to zero. The same situation is with densities (1000–3000 kg/m3 vs. 1.2 kg/m3 ). Therefore, the reflection coefficient is closed to zero. Thus, a fracture saturated by gas in pressure closed to normal can be considered as empty and set by free boundary condition, which gives a full reflection of falling wave.

10.3.2 Fluid-Filled Fracture In the most of practical problems, fractures are filled by fluid: water, oil, liquefied gas, etc. Thus, it is expediently to develop a model allowing to describe such situation. Fluid-filled fracture is set as contact boundary with free slide condition: va · n = vb · n,

fna = − fnb , fτa = fτb = 0.

10 Mathematical Modeling of Spatial Wave Processes in Fractured …

119

Such contact boundary fully passes any longitudinal vibrations without reflection and fully reflects shear waves. This situation corresponds to 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 closed to zero.

10.3.3 Glued Fracture In the big depth under pressure action, the surfaces of fracture can be closed to the elastic waves almost fully propagate through the fracture. In such case, the most optimal is to use the contact condition of gluing: va = vb ,

fa = − fb ,

where v are the velocities of closing boundary point and f are the forces acting on boundary. Let a be the first surface and b be the second surface of fracture.

10.3.4 Partially Glued Fracture In real exploration seismology, we take a deal mostly with partially gluing fractures, where one part of sides’ surface is gluing and another part is separated by fluid or gas. Such fractures give partial propagation elastic wave front that influences on the amplitudes of response wave of seismograms. We developed such model of fracture. In this model, in different points of surfaces conditions gas-saturation (fluid-filling) and full gluing are set randomly. Number of point of each type is regulated by weight coefficient–coefficient of gluing. Such model gives us a possibility to set the gas-saturated and fluid-filling fractures with fracture of gluing points from 0 to 100%. As the fracture reflects the wave front in some points and propagates it in others, the superposition of diffracted waves formed by a wave interaction with all points is a response of gas-saturated (fluid-filled) fracture with less amplitude.

10.4 Results of Mathematical and Physical Modeling In this section, we consider the mathematical and physical modeling for problem with geological model described in Sect. 10.4.1. The results of mathematical modeling are represented in Sect. 10.4.2. Section 10.4.3 is devoted to the results of physical modeling and their comparison with the results of mathematical modeling.

120

M. V. Muratov and T. N. Derbysheva

10.4.1 Geological Medium Model We study a wave responses from a system of uniformly oriented mesofractures located at a depth of 1640 m. The horizontal length of the formation is equaled to 2800 m, while the vertical length equals 120 m. The fractures are evenly distributed in the formation. The height of the fractures equals 12 m with the distance between them equaled to 12 m and angle of inclination equaled to 5°. A point of perturbation is used (located on the day surface), and its horizontal position coincides with the center of the fractured layer. Frequency of the excited seismic impulse is equal to 10 Hz. Three observation profiles, on which wave responses in the form of seismograms have been recorded, are used: • First profile. Receiving points with sensors ch1–ch49 are located on the day surface with an interval of 40 m to the right and left of the source, 24 sensors each. Sensors ch1–ch49 are located from left to right. Sensor ch25 combined with the point of excitation. • Second profile. Receiving points with sensors ch1–ch49 are located at a depth of 400 m with an interval of 40 m to the right and left of the source with 24 sensors. Sensors ch1–ch49 are located from left to right. Sensor ch25 is under the point of excitation. • Third profile. Receiving points with sensors ch1–ch49 are located at a depth of 2000 m with an interval of 40 m to the right and left of the source, 24 sensors each. Sensors ch1–ch49 are located from left to right. Sensor ch25 is under the point of excitation. The scheme of this observation is represented in Fig. 10.2.

Fig. 10.2 Scheme of observation

10 Mathematical Modeling of Spatial Wave Processes in Fractured …

121

10.4.2 Results of Mathematical Modeling Mathematical modeling for the task was carried out using the above methodology. The results in the form of wave patterns of velocity are represented on Fig. 10.3. In Figs. 10.4, 10.5 and 10.6, visualization in the form of synthetical seismograms is represented for all three observation profiles.

Fig. 10.3 Visualization of results of mathematical modeling in form of velocity wave patterns: a is at time when propagating wave front has not yet reached the fractures, b is at time after wave front propagation through the fractures. On b we can observe the wave response from fractured layer

Fig. 10.4 Results of mathematical modeling in form of synthetical seismograms on observation profile on the surface: a horizontal component of velocity (X), b vertical (Z)

Fig. 10.5 Results of mathematical modeling in form of synthetical seismograms on observation profile on the depth of 400 m: a horizontal component of velocity (X), b vertical (Z)

122

M. V. Muratov and T. N. Derbysheva

Fig. 10.6 Results of mathematical modeling in form of synthetical seismograms on observation profile on the depth of 2000 m: a horizontal component of velocity (X), b vertical (Z)

10.4.3 Comparison with Results of Physical Modeling Technology of physical modeling. The physical model is made of a plexiglass sheet with dimensions of 1640 × 800 × 4 mm and simulates a homogeneous medium, in which the studied object is placed in a given shape and with specified physical parameters. With a coefficient of geometric similarity K = 4000, a geological section of 6560 × 3200 m in size can be simulated. Comparison of results of mathematical and physical modeling. Figure 10.7 shows a comparison of the records of the vertical component of the seismograms obtained

Fig. 10.7 Comparing the records of reflections of the mesofracture layer of: a is mathematical modeling data, b is physical modeling data

10 Mathematical Modeling of Spatial Wave Processes in Fractured …

123

by the mathematical and physical modeling. In both cases, the longitudinal reflected wave was traced on the records with a smooth decrease in intensity, and the regular reflected waves close in form of recording.

10.5 Conclusions The developed technique based on the grid-characteristic method with interpolation on unstructured meshes allows a mathematical modeling of exploration seismology in layers of uniformly oriented mesofractures. Several models of fractures based on the concept of infinitely thin fracture were developed for modeling the heterogeneities with various types of saturation: gas-saturated, fluid-filled, glued and partially glued gas-saturated, and fluid-filled fractures. Numerical simulation is performed for the problem of exploration seismology of a mesofracture formation that is close to real data with the registration of wave responses in three profiles of receivers. A similar problem is solved using physical modeling, the solution of which verified the results of mathematical modeling. Acknowledgements This work was supported by the Russian Foundation of Basic Research, project no. 19-01-00432.

References 1. Khokhlov, N., Stognii, P.: Novel approach to modeling the seismic waves in the areas with complex fractured geological structures. Minerals 10(2), 122.1–122.17 (2020) 2. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Methods for calculating the dynamics of layered and block media with nonlinear contact conditions. 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. 171–183. Springer, Singapore (2020) 3. Cui, X., Lines, L.R., Krebes, E.S.: Seismic modelling for geological fractures. Geophys. Prospect. 66, 157–168 (2018) 4. Chentsov, E.P., Sadovskii, V.M., Sadovskaya, O.V.: Modeling of wave processes in a blocky medium with fluid-saturated porous interlayers. AIP Conf. Proc. 1895, 080002.1–080002.10 (2017) 5. Leviant, V.B., Petrov, I.B., Kvasov, I.E.: Numerical modeling of seismic response from subvertical macrofractures as possible fluid conduits. Seismic Technol. 4, 41–61 (2011) 6. Bakulin, A., Grechka, V., Karaev, N., Anisimov, A., Kozlov, E.: Physical modeling and theoretical studies of seismic reflections from a fault zone. SEG Technical Program Expanded Abstracts, pp. 1674–1677 (2004) 7. Willis, M.E., Burns, D.R., Rao, R., Minsley, B., Toksöz, M.N., Vetri, L.: Spatial orientation and distribution of reservoir fractures from scattered seismic energy. Geophysics 71(5), O43–O51 (2006) 8. Leviant, V.B., Petrov, I.B., Muratov, M.V.: Numerical simulation of wave responses from subvertical macrofractures system. Seismic Technol. 1, 5–21 (2012)

124

M. V. Muratov and T. N. Derbysheva

9. Biryukov, V.A., Muratov, M.V., Petrov, I.B., Sannikov, A.V., Favorskaya, A.V.: Application of the grid-characteristic method on unstructured tetrahedral meshes to the solution of direct problems in seismic exploration of fractured layers. Comput. Math. Math. Phys. 55(10), 1733– 1742 (2015) 10. Petrov, I.B., Muratov, M.V.: Application of the grid-characteristic method to the solution of direct problems in the seismic exploration of fractured formations (review). Math. Models Comput. Simul. 11, 924–939 (2019) 11. Magomedov, K.M., Kholodov, A.S.: Grid-Characteristic Numerical Methods. Nauka, Moscow (1988). (in Russian) 12. Petrov, I.B., Kholodov, A.S.: Numerical study of some dynamic problems of the mechanics of a deformable rigid body by the mesh-characteristic method. Comput. Math. Math. Phys. 24(3), 61–73 (1984) 13. Aurenhammer, F.: Voronoi diagrams—a study of fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991) 14. Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge University Press (2006) 15. Novatskii, V.K.: Elasticity theory. Izd. Mir, Moscow (1975). (in Russian) 16. 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. SIST, vol. 133, pp. 167–181. Springer, Cham (2019) 17. 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)

Chapter 11

Investigation of Models with Fluidand Gas-Filled Fractures with the Help of the Grid-Characteristic Method Polina V. Stognii

and Nikolay I. Khokhlov

Abstract Heterogeneous media with fractured geological structures are widely spread, and therefore, they need a very careful exploration. In this work, we present the results of modeling the seismic waves spread through the homogeneous and heterogeneous media with single fractures and fracture clusters using the gridcharacteristic method. We analyze the seismic reflections from inclined fractured structures described by the model of a two shore extremely thin fracture. The models with little differing and significantly differing characteristics for single fluid- and gas-filled fractures and fracture clusters are analyzed. The results demonstrate the dependence of the normal components of the velocity distribution from the media parameters. Keywords Fractures · Numerical modeling · Grid-characteristic method

11.1 Introduction Seismic prospecting is one of the most popular methods of searching the geological areas for hydrocarbon deposits. During this process, the source of impulse is established on the surface, and the receivers record the seismic velocities, which are resulted from seismic waves spread and reflected from media with different characteristics. The hydrocarbons are often kept in fractures. Then, they are more difficult to detect because the fractures can be fluid- or gas-filled and with different length and width. Fractures reflect the seismic waves in different ways, and it should also be taken into account.

P. V. Stognii (B) · N. I. Khokhlov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] N. I. Khokhlov 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 214, https://doi.org/10.1007/978-981-33-4709-0_11

125

126

P. V. Stognii et al.

The numerical modeling of the seismic waves spread through heterogeneous media is usually carried out together with the real seismic survey of the geological areas. It allows to build many models with different media characteristics and different hydrocarbons locations including fractures. Fluid-filled and gas-filled fractures differ by the way of seismic waves’ reflection. Thus, the result wave fields for these models are different. These differences are taken into account during the numerical modeling of geological areas with fractures, and the result wave fields and seismograms are further compared with the real ones in order to specify the information about the explored geological areas. There are different fracture models. One of them is the linear slip model of fracture [1], which is often used in mathematical modeling. However, it allows to model only vertical fractures. The fracture model of Hudson [2] and explicit–implicit method [3] are often used for fractures’ modeling. There are many explorations about the geological media models with fractures, for example, [4, 5]. In this chapter, we use a model of an extremely thin fracture [6]. This model allows to build the fluid- and gas-filled fractures correctly. We explore the influence of the medium characteristics on the wave fields from single fractures and clusters of fractures filled with fluid and gas. We present the main equations of the gridcharacteristic method [7] used in all calculations. Then, we demonstrate the result wave fields for the models with single fractures and clusters of fractures, fluid- or gas-filled located in homogeneous media and heterogeneous media with different characteristics. The chapter is organized as follows. In Sect. 11.2, we describe the numerical method used in all computations. We present the results of the numerical modeling of the seismic waves spread in the medium with single fractures and clusters of fractures in Sects. 11.3 and 11.4, respectively. We conclude the chapter by Sect. 11.5.

11.2 Numerical Method The system of equations for the linear-elastic medium was used for describing the dynamic behavior of the seismic wave spread in the medium [8]: ρ

∂v = (∇ · σ)T , ∂t

  ∂σ = λ(∇ · v)I + μ (∇ × v) + (∇ × v)T . ∂t

(11.1) (11.2)

In Eqs. 11.1 and 11.2, σ is the Cauchy stress tensor, v is the speed of the seismic waves, t is the time, ρ is the medium density, and λ μ are the Lame parameters. The grid-characteristic method of the third order of accuracy was used for solving the system of Eqs. 11.1 and 11.2 numerically. For this, we rewrite the system of Eqs. 11.1 and 11.2 as:

11 Investigation of Models with Fluid- and Gas-Filled …

∂q ∂q ∂q +Ax +A y = 0. ∂t ∂x ∂y

127

(11.3)

  In Eq. 11.3, the vector q is q = σx x , σx y , σ yy , vx , v y . The matrixes Ax and A y consist of the coefficients of the system provided by Eqs. 11.1 and 11.2. Then, we split Eq. 11.3 in coordinates and obtain two 1D systems of equations: ∂q ∂q +Ax =0 ∂t ∂x

(11.4)

Then, if we examine the system of Eq. 11.4 for the coordinate x taking into account the fact that it is hyperbolic, we will be able to rewrite it as: ∂q ∂q + x x −1 = 0. x ∂t ∂x

(11.5)

In Eq. 11.5, the matrix x is made of the eigenvectors of the matrix Ax , and the matrix x is made of the eigenvalues of the matrix Ax , which√are equal to −c p , c√ p , −cs , cs , 0 . Here, c p and cs can be found from c p = (λ + 2μ)/ρ, cs = μ/ρ. Afterward, we make the variable change p = −1 x q, and the system of Eq. 11.5 will transfer to: ∂p ∂p +x = 0. ∂t ∂x

(11.6)

The system of Eq. 11.6, which consists of five independent equations, was solved using the Rusanov scheme of the third order of accuracy [9]. The following conditions were established on the boundaries of fractures. For describing the fluid-filled fracture, we applied the free slip condition [10]: υan = υbn , f an = − f bn , f aτ = f bτ = 0.

(11.7)

For describing the gas-filled fracture, we applied the condition of free reflection [11]: σ × q = 0.

(11.8)

The model of two shore extremely thin fracture [6] was used in all calculations.

128

P. V. Stognii et al.

11.3 Models with Single Fractures in Homogeneous and Heterogeneous Media In this section, we investigate the influence of the medium characteristics on the wave fields from the single fluid- and gas-saturated fractures. Firstly, we computer simulated the seismic waves spread in the homogeneous medium with one fracture. The characteristic parameters of the medium were the following. The longitudinal sound velocity was equal to 5000 m/s, the transverse sound velocity was 3000 m/s, and the medium density was 2500 m/s. The fracture was inclined at the angle of 5° to the right. The single fracture had the length of 5 m (small fracture) and was located in the center of the medium. In all the experiments, the initial impulse was characterized by the Reiker impulse of 30 Hz. The result seismic reflections from the fluid- and gas-filled single fractures are presented in Fig. 11.1a, b, respectively. The wave fields for the normal component of the velocity in Fig. 11.1 and other figures, which will be presented in the chapter, are made in the same color scale at the same moment of time, and thus, it is correct to compare them. The reflections from the gas-filled fracture (Fig. 11.1b) are more intensive if compared them with the reflections from the fluid-filled fracture (Fig. 11.1b). If we analyze the very first reflection wave (marked as number 1), the maximum minus meaning is displaced from the center to the left side. The displacement of the minus maximum meaning is more noticeable in Fig. 11.1b. It is connected with the inclination of the fracture at 5° to the right. The first wave reflection from the vertical

Fig. 11.1 Wave fields from single fractures in the homogeneous medium: a the wave filed from the fluid-filled fracture, b the wave filed from the gas-filled fracture

11 Investigation of Models with Fluid- and Gas-Filled …

129

fracture has the maximum meaning in the center [12]. The 5° angle is very small for such a large displacement of the maximum, and it can be explained by the significant increase of the reflected surface. Each node along the fracture starts reflecting the seismic waves if we rotate the fracture at any angle, even very small. Secondly, we computer simulated the seismic waves spread in the heterogeneous medium with one fracture close to the contact border between the layers. The characteristic parameters of the first medium were the same to the first experiment. The characteristics of the second medium, which contained the fracture (fluid- or gasfilled) were the following. The longitudinal sound velocity was equal to 5500 m/s, the transverse sound velocity was 3500 m/s, and the medium density was 2500 m/s. The sound velocities of the second medium were a little larger than the same characteristics of the first medium. The fracture was inclined at the angle of 5° to the right. The single fracture had the length of 50 m (big fracture) and was located in the center of the second medium at distance of 3 m from the border between different layers. The result seismic reflections from the fluid- and gas-filled single fractures are presented in Fig. 11.2a, b, respectively. The contact border between the layers does not prevent us from collecting the seismic reflections from the fractures. The first wave (marked as number 1) in Fig. 11.2 depicts the reflection from the contact border between the layers. The first reflection wave from the fracture (marked as number 2) coming through the contact between the different media has the maximum minus meaning displaced from the center to the left side. We observed the same

Fig. 11.2 Wave fields from single fractures in the heterogeneous media: a the wave filed from the fluid-filled fracture, b the wave filed from the gas-filled fracture

130

P. V. Stognii et al.

displacement in the seismic reflections from the small fracture in the homogeneous medium (Fig. 11.1). However, in [13], the seismic reflections from the single big fracture inclined at 5° are symmetric. The differences can be explained by the use of the unstructured grids in [13] and the lower order of accuracy of the scheme.

11.4 Models with Clusters of Fractures in Heterogeneous Media In this section, we analyze the influence of the media characteristics on the wave fields in the models with clusters of fluid- and gas-filled fractures. Firstly, we calculated the wave field from the clusters of fluid- and gas-filled fractures in the homogeneous medium. The characteristic parameters of the medium were the same as in Sect. 11.3 for the fractures investigation in the homogeneous medium. The fractures were inclined at the angle of 5° to the right. The fractures had the length of 5 m (small fracture) and were located in the center of the medium at distance of 10 m from each other. The cluster consisted of several rows of fractures. The result seismic reflections from the cluster of fluid- and gas-filled fractures are presented in Fig. 11.3a, b, respectively. If we analyze the first wave reflection, it contains the maximum positive meanings on the left and right endings of the wave, and the minimum positive meaning is displaced to the left side. We got the same

Fig. 11.3 Wave fields from fracture clusters in the homogeneous medium: a the wave filed from the fluid-filled fractures, b the wave filed from the gas-filled fractures

11 Investigation of Models with Fluid- and Gas-Filled …

131

result for the seismic reflection from the single fracture (Figs. 11.1 and 11.2). The displacement to the left is more clearly seen for the model with gas-filled fractures (Fig. 11.3b). The next reflected waves from the cluster of inclined fractures, not only the first one, contain the minimum positive meaning displacement to the left side. Secondly, we computer simulated the seismic waves spread in the heterogeneous medium with the cluster of fractures close to the contact border between the layers. The characteristic parameters of the media were the same to the second experiment with one single fracture in Sect. 11.3. The sound velocities of the second medium were a little larger than the same characteristics of the first medium. The fractures were inclined at the angle of 5° to the right. The fractures were 50 m long and were located in the center of the second medium at distance of 2 m from the contact border between the layers. In addition, we carried out the same experiment with the heterogeneous media containing the clusters of fluid- and gas-filled fractures, but the characteristic parameters of the second medium were the following. The longitudinal sound velocity was equal to 6500 m/s, the transverse sound velocity was 3500 m/s, and the medium density was 2500 m/s. The sound velocities of the second medium were noticeably larger than the same characteristics of the first medium (6500 m/s in the second medium against 5000 m/s in the first medium). The result wave fields in the models with little differing parameters are presented in Fig. 11.4a, b with fluid- and gas-filled fractures, respectively. If we analyze the wave fields for the model with the little differing media characteristics (Fig. 11.4),

Fig. 11.4 Wave fields from fracture clusters in the heterogeneous media with little differing parameters, larger for the second medium: a the wave filed from the fluid-filled fractures, b the wave filed from the gas-filled fractures

132

P. V. Stognii et al.

the first reflected wave has the minimum positive meaning displaced to the right side (white and blue part). Then, the overall meanings along the first wave are larger for the fluid-filled fractures cluster (Fig. 11.4a) if compare with the meanings along the first wave for the gas-filled fractures cluster (Fig. 11.4b). The first experiment with the single fracture in Sect. 11.3 (Fig. 11.1) demonstrated the more intensive reflections from the gas-filled fracture. The same results were observed in the experiment with the single fracture in the model with two media (Fig. 11.2). In this case, the contact border influences on the reflections from the cluster of fractures significantly, which leads to the weakness of the reflections from the gas-filled fractures if compare with the reflections from the fluid-filled fractures. The wave fields in the models with significantly differing parameters are depicted in Fig. 11.5a, b with fluid- and gas-filled fractures, respectively. If we analyze the wave fields for the model with the significantly differing media characteristics (Fig. 11.5), the minimum positive meaning of the first reflected wave is not displaced to the right side. The first wave reflection is almost symmetric, which was not so in the model with the slightly differing media parameters (Fig. 11.4). However, the overall meanings of the reflections in the model with the cluster of gas-filled fractures (Fig. 11.5a) are less in comparison with the reflections in the model with the cluster of fluid-filled fractures (Fig. 11.5b). Then, in case of the significantly differing media characteristics, the first wave reflection from the cluster of fractures is almost symmetric, if compare with the model for the slightly differing parameters, where the minimum positive meaning is noticeably displaced to the right side. However, in both models. The

Fig. 11.5 Wave fields from fracture clusters in the heterogeneous media with significantly differing parameters, larger for the second medium, for: a the fluid-filled fractures, b the gas-filled fractures

11 Investigation of Models with Fluid- and Gas-Filled …

133

seismic reflections from the clusters of fluid- and gas-filled fractures demonstrate the larger meanings for the models with fluid-filled fractures. Thirdly, we computer simulated the seismic waves spread in the heterogeneous medium with the cluster of fractures close to the contact border between the layers. Now, the characteristic parameters of the second medium, which contained fractures, were smaller than the parameters of the first medium. The characteristic parameters of the first medium were the same to the first experiment with one single fracture in Sect. 11.3. The second medium was characterized by the following parameters. The longitudinal sound velocity was equal to 4500 m/s, the transverse sound velocity was 3500 m/s, and the medium density was 2500 m/s. The sound velocities of the second medium were a little smaller than the same characteristics of the first medium (4500 m/s in the second medium against 5000 m/s in the first medium). The fractures were inclined at the angle of 5° to the right. The fractures were 50 m long and were located in the center of the second medium at distance of 2 m from the contact border between the two layers. In addition, we carried out the same experiment with the heterogeneous media containing the clusters of fluid- and gas-filled fractures, but the characteristic parameters of the second medium were the following. The longitudinal sound velocity was equal to 3500 m/s, the transverse sound velocity was 3500 m/s, and the medium density was 2500 m/s. The sound velocities of the second medium were noticeably smaller than the same characteristics of the first medium (3500 m/s in the second medium against 5000 m/s in the first medium). The result wave fields in the models with little differing parameters (smaller for the second medium) are presented in Fig. 11.6a, b with fluid- and gas-filled fractures,

Fig. 11.6 Wave fields from fracture clusters in the heterogeneous media with little differing parameters, larger for the first medium, for: a the fluid-filled fractures, b the gas-filled fractures

134

P. V. Stognii et al.

Fig. 11.7 Wave fields from fracture clusters in the heterogeneous media with significantly differing parameters, larger for the first medium: a the wave filed from the fluid-filled fractures, b the wave filed from the gas-filled fractures

respectively, and the result wave fields in the models with significantly differing parameters (smaller for the second medium) are presented in Fig. 11.7a, b with fluidand gas-filled fractures, respectively. If we compare the wave fields for the model with the little differing media characteristics (Fig. 11.6) and for the model with the significantly differing media characteristics (Fig. 11.7), the first reflected wave is almost symmetric. Then, in case the seismic wave spreads from the medium with the larger sound velocities to the medium with the smaller sound velocities, containing the cluster of fractures, the symmetricity of the wave does not change, though the fractures are inclined at the 5° angle. However, in this case, the overall meanings of the seismic velocities are larger in the models with gas-filled fractures (Figs. 11.6b and 11.7b), than in the models with fluid-filled fractures (Figs. 11.6a and 11.7a). In [14], the authors obtained the symmetric wave reflections from the cluster of the fractures inclined at 5° in the homogeneous medium. The differences in the result reflections from the single fracture and fractured clusters are connected with the use of the unstructured grids in [13, 14] and structured grids in our research. The mathematical equations describing the fluid- and gas-filled fractures are the same in both methods. On the one hand, the 5° angle is small for the significant displacement of the maximum. However, the area of the reflected surface significantly increases. If we rotate the fracture from the vertical position to the position of a very small angle inclination, each node along the fracture starts reflecting the seismic waves, and we observe the meanings of the extremum points’ displacement.

11 Investigation of Models with Fluid- and Gas-Filled …

135

11.5 Conclusions In this work, we present the results of exploration the models with fractures in heterogeneous media. We gave a brief overview of the problem of fractures detection and the importance of investigating heterogeneous media with fractured formations. We describe the grid-characteristic method used in all computations and the main equations applied on boundaries of fluid- and gas-filled fractures. Then, we present the wave fields for the models with single inclined fractures in homogeneous and heterogeneous media and the differences between them. The models demonstrate the significant displacement of the maximum values from the wave center, though the angle inclination of fractures was only 5°. It does not correspond to the results obtained by modeling the vertical fractures. In addition, we carried out the same computations in the homogeneous and heterogeneous media with fracture clusters. The result wave fields demonstrated the significant displacement of the maximum values from the wave center for the models with little differing characteristics. However, the models with significantly differing parameters show the symmetric image of the wave reflection from the fractures. This result is unexpected for such models and demands further explorations. This chapter is the first one in the description of the problem of comparing the results on structured/unstructured grids for the fractured zones. We aimed to demonstrate the results for the reflections from the fractures depending on the size filling parameters of fractures and media characteristics. The further research will refer to the explanation of differences between the results obtained from computations on structured/unstructured grids. Acknowledgements The reported study was funded by RFBR according to the research project № 18-01-00710 A.

References 1. Schoenberg, M.: Elastic wave behavior across linear slip interfaces. J. Acoust. Soc. Am. 68(5), 1516–1521 (1980) 2. Hudson, J.A.: Overall properties of a cracked solid. Math Proc Cambridge Philos. Soc. 88, 371–384 (1980) 3. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Mathematical modeling of the dynamics of layered and block media with nonlinear contact conditions on supercomputers. J. Phys. Conf. Ser. 1392, 012057.1–012057.6 (2019) 4. Muratov, M.V., Petrov, I.B.: Application of mathematical fracture models to simulation of exploration seismology problems by the grid-characteristic method. Compu. Res. Model. 11(6), 1077–1082 (2019) 5. Zhan, Q., Sun, Q., Ren, Q.: A discontinuous Galerkin method for simulating the effects of arbitrary discrete fractures on elastic wave propagation. Geophys. J. Int. 210(2), 1219–1230 (2017) 6. Khokhlov, N., Stognii, P.: Novel approach to modeling the seismic waves in the areas with complex fractured geological structures. Minerals 10(2), 122.1–122.17 (2020)

136

P. V. Stognii et al.

7. Ivanov, A.M., Khokhlov, N.I.: Efficient inter-process communication in parallel implementation of grid-characteristic method. In: Petrov, I., Favorskaya, A. (eds.) Smart Modeling for Engineering Systems. GCM50 2018. Smart Innovation, SIST, vol. 133, pp. 91–102. Springer, Cham (2019) 8. Savin, G.N., Rushchitskii, Y.Y., Novatskii, V.: The theory of elasticity. Sov. Appl. Math. 7, 808–811 (in Russian) (1971) 9. Ivanov, A.M., Khokhlov, N.I.: Parallel implementation of the grid-characteristic method in the case of explicit contact boundaries. Comput. Res. Model. 10(5), 667–678 (2018) 10. Cerquaglia, M.L., Deliege, G., Boman, R., Terrapon, V., Ponthot, J.-P.: Free-slip boundary conditions for simulating free-surface incompressible flows through the particle finite element method. Int. J. Numer. Meth. Eng. 110(10), 921–946 (2017) 11. Huang, L.-J., Mora, P., Fehler, M.C.: Absorbing boundary and free-surface conditions in the phononic lattice solid by interpolation. Geophys. J. Int. 140(1), 147–157 (2000) 12. Muratov, M., Petrov, I., Leviant, V.: Grid-characteristic method as optimal tool of fracture formations research. In: European Association of Geoscientists & Engineers. Conference Proceedings, vol. 2018, pp. 1–6. Saint Petersburg (2018) 13. 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) 14. Muratov, M.V., Petrov, I.B., Levyant, V.B.: The development of fracture mathematical models for numerical solution of exploration seismology problems with the use of grid-characteristic method. Comput. Res. Model. 8(6), 911–925 (2016)

Chapter 12

Modeling Wave Responses from Thawed Permafrost Zones Vasily I. Golubev , Alexey V. Vasyukov , and Mikhail Churyakov

Abstract The presence of technological subsurface heat sources in permafrost regions leads to the melting process. It significantly decreases the strength of the geological massif and may lead to the subsidence of the day surface. In this chapter, we investigated the capabilities of the seismic survey for monitoring of this process. To describe the dynamic behavior of thawed zones, the model of the porous fluidfilled medium was used. Its numerical solution in two-dimensional case was obtained with the grid-characteristic method on rectangular grids. It allows to set physically correct contact conditions between elastic and porous media. To increase the simulation precision, the compact scheme for one-dimensional case was proposed and successfully tested. It is based on the extension of the original system with differential consequences. This approach provided the third order of the accuracy on the threepoint spatial stencil. Based on a set of computer experiments, the seismic response was analyzed quantitatively and qualitatively. Keywords Permafrost · Melting · Seismic survey · Computer simulation · Grid-characteristic method · Porous medium

V. I. Golubev (B) · M. Churyakov Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] M. Churyakov e-mail: [email protected] A. V. Vasyukov Keldysch Institute of Applied Mathematics of the RAS, 4, Miusskaya sq., Moscow 125047, 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 214, https://doi.org/10.1007/978-981-33-4709-0_12

137

138

V. I. Golubev et al.

12.1 Introduction Currently, a lot of oil and gas fields are being developed in the northern latitudes on permafrost. The extraction of deposits in such conditions leads to a significant increase of damage risks for expensive drilling equipment associated with the thawing of permafrost. In this research, we estimated the capabilities of the seismic survey for monitoring of this process. A mathematical approach has been developed that allows to model the propagation of seismic waves taking into account the thawing of hydrates. It is based on the assumption, that the medium is porous and consists of a solid skeleton and a fluid saturating it. Due to the different time scales of the thermal and wave problems, the latter can be solved with respect to the currently fixed saturation distribution pattern. The simplest way of taking into account the medium saturation is the Gassmann model [1]. With the development of modern high-performance computing systems, the Biot model was widely implemented [2, 3]. It was successfully applied for describing waves in complex media, for example, in [4, 5]. A method of determining the Biot’s coefficient, which is one of the key poro-elastic parameters of a rock, was presented in [6] and tested on shale samples. To describe the dynamic behavior of a porous medium, we used the Dorovsky model [7]. Unlike the Biot model, it is described by three elastic parameters instead of four. In [8], simulation results were compared with the Biot–Johnson theory. Both models showed excellent agreement with each other. However, in order to accurately depict the dependence of Stoneley waves on a frequency that can be measured experimentally, special corrections may be required. In general, this porous model is actively used in seismic modeling. For example, modeling of a well with a clay crust between the borehole fluid and the porous surrounding formation is described in [9]. This chapter is organized as follows: Sect. 12.2 presents the used mathematical model and numerical method [10]. The new compact scheme for one-dimensional case is discussed in Sect. 12.3. Numerical solutions for two-dimensional problems are presented in Sect. 12.4. Section 12.5 concludes the chapter.

12.2 Mathematical Model and Numerical Method Hereinafter, a mathematical model is given in Sect. 12.2.1, while application of grid-characteristic method for solving the problem is discussed in Sect. 12.2.2.

12 Modeling Wave Responses from Thawed Permafrost Zones

139

12.2.1 Mathematical Model Two-continuum approach assumes that in every small volume of the medium there is both a skeleton and a fluid filling pores, which allows us to consider the following unknown functions in the entire simulation domain: the skeleton velocity, fluid velocity, stress matrix in the skeleton, and fluid pressure. The equations describing the dynamic behavior of such medium can be written in the form of Eqs. 12.1–12.4 [7, 11], where h and u are the minus skeleton stress tensor and skeleton velocity vector, respectively, p and v are the pressure and fluid velocity vector, respectively, ρ0 = ρs + ρ f , ρ f and ρs are the specific densities of the fluid and skeleton, respectively, K , μ, λ, and α are the elastic characteristics of a saturated porous medium. ut +

1 1 (∇ · h)T + ∇ p = F ρs ρ0

(12.1)

1 ∇p= F (12.2) ρ0      ρs ρs ht + μ ∇ ⊗ u + (∇ ⊗ u)T + λ − K (∇ · u) − K (∇ · v) I = 0 ρ0 ρ0 (12.3) vt +

pt − (K − αρ0 ρs )(∇ · u) + αρ0 ρ f (∇ · v) = 0

(12.4)

The main feature of this model is the propagation of three different waves (fast and slow longitudinal and one transverse). Moreover, Eqs. 12.5–12.7 are true, where c p1 , c p2 , and cs are these velocities. μ = ρs c2s 8ρ f 2 ρ0 ρs 2 K = (c + c2p2 − c − 2ρ f p1 3ρ0 s 8ρ f 2 1 2 (c + c2p2 − c + α3 = 3ρ0 s 2ρ02 p1



(12.5)

(c2p1 − c2p2 )2 −

64ρ f ρs 4 c ) 9ρ02 s

(12.6)

(c2p1 − c2p2 )2 −

64ρ f ρs 4 c ) 9ρ02 s

(12.7)



12.2.2 Grid-Characteristic Method The original system (Eqs. 12.1–12.4) is hyperbolic, and for zero, F can be rewritten in the canonical form of Eq. 12.8, where the vector q contains all unknown functions.

140

V. I. Golubev et al.

qt + A x qx + A y qy = 0

(12.8)

This type of systems can be effectively solved with the grid-characteristic method. It was successfully applied for direct [12, 13] and inverse [14] seismic survey problems, global seismicity problems [15] and destruction problems [16–18]. The initial system can be initially split by coordinate directions into two one-dimensional systems with matrices A x and A y . The consequence solutions of them lead to the solution of the initial system. Let us consider the X step. The matrix A x has a full set of eigenvalues Λx and eigenvectors Ωx and can be represented as A x = Ωx−1 Λx Ωx . With the introduction of the Riemann invariants and the consideration of w  = Ωx q, we obtain a set of independent transport equations in the form of Eq. 12.9. x = 0 w  t + Λx w

(12.9)

Obviously, the solution of each equation of Eq. 12.8 can be found analytically as Eq. 12.10. wi (t + dt, x) = wi (t, x − λi dt)i = 1, 6

(12.10)

Using different space stencils and interpolation procedures, we can obtain numerical schemes with wide ranges of approximation orders and stability conditions. In this chapter, for two-dimensional problems, the Rusanov scheme was applied  providing third order of accuracy. The inverse transformation is trivial: q = Ωx−1 w.

12.3 High-Order One-Dimensional Scheme Let us look closer to the one-dimensional case of the system of Eqs. 12.1–12.4 with zero F: ⎡ ⎤⎛ ⎛ ⎞ ⎞ 0 0 1/ρs 1/ρ0 vs vs ⎢ ⎜ vf ⎟ 0 0 0 1/ρ0 ⎥ vf ⎟ ⎢ ⎥⎜ ⎜ ⎟ ⎟ = 0. (12.11) ⎥⎜ ρf ρf 4μ ⎝ hxx ⎠ + ⎢ ⎝ 0 ⎦ hxx ⎠ ⎣ 3 + K ρ0 −K ρ0 0 p t p x −K + αρ0 ρs αρ0 ρ f 0 0 Introducing additional notations       τ = 3Kρ0 ρ f − ρs ρs + ρ02 ρs 4μ + 3αρs ρ f + ρs /6ρ02 ρs2 ,     ψ = (−12ρ f ρs −3K 2 + αρ0 4μρ0 + 3K ρ f + ρs ,      2  + 3K ρ f − ρs + ρ0 4μ + +3αρs ρ f + ρs /6ρ02 ρs2

(12.12)

(12.13)

12 Modeling Wave Responses from Thawed Permafrost Zones

141

we obtain the matrix of eigenvalues ⎡  ⎤ √ − τ− ψ 0 0 0 √ ⎢ ⎥ 0 τ− ψ  0 0 ⎢ ⎥ Λ=⎢ √ ⎥. ⎣ ⎦ 0 0 − τ+ ψ 0 √ 0 0 0 τ+ ψ

(12.14)

To obtain the matrix of eigenvectors with the short form, we introduce more notations in the form of Eqs. 12.15–12.20.      2   2 β = ρ02 ρs2 9K 2 ρ f + ρs + ρ02 9α 2 ρ 2f ρs2 + 6αρ f ρs −4μ + 3αρs2 + 4μ + 3αρs2    −6Kρ0 4μρs + 3αρ 2f ρs + 3αρs3 + ρ f −4μ + 6αρs2 (12.15)

      γ = ρ0 ρs 3K ρ f − ρs + ρ0 4μ + 3αρs ρ f + ρs

(12.16)

      ω = ρ0 ρs 3K ρ f + ρs + ρ0 4μ − 3αρs ρ f + ρs

(12.17)

     δ = 6K 2 ρs − 3K αρ0 ρs ρ f + 3ρs + αρ02 ρs −4μ + 3αρs ρ f + ρs

(12.18)

= 6ρ02 ρs (−K + αρ0 ρs )

(12.19)

√ 6(K − αρ0 ρs )/ρ0

(12.20)

φ=

The final form of both matrices is defined by Eqs. 12.21 and 12.22. ⎡ √

√ √ √ γ − β −φ γ − β √ √ ⎢ (δ−α (δ−α √ β) √ β) ⎢ √ 6ρ√s ⎢ − √ 6ρ√s ⎢ γ− √ β ⎢ ω−γ√−β β ω− β ⎣

φ

Ω=

1 ⎡√

Ω −1

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

√ − β+γ (−α 2 β+δ 2 ) √ 4 β(αγ −δ)φ √ √ − β+γ (α 2 β−δ 2 ) √ √√4 β(αγ 2−δ)φ2 β+γ (α β−δ ) √ 4 β(αγ −δ)φ √√ β+γ (−α 2 β+δ 2 ) √ 4 β(αγ −δ)φ

1 √

√ √ √ γ + β −φ γ + β √ √ (δ+α (δ+α √ β) √ β) √ 6ρ√s − √ 6ρ√s γ√+ β γ+ √ β ω+ β ω+ β

φ

√

1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(12.21)

1

√ √ √ − β+γ ( β+γ )(α β−δ ) √ √ − − 4√ β s √ √ 4 6√ β(αγ −δ)ρ √ − β+γ ( β+γ )(α β−δ ) √ √ − 4√ β 4 √ 6 β(αγ −δ)ρs √ √ √ ( β−γ√) √ β+γ (α β+δ)

√ 4 β 4 6 √ β(αγ −δ)ρs √ √ √ β+γ (α β+δ ) β−γ

√ − ( 4√)6√β(αγ −δ)ρ 4 β s

 1+  1 1+ 4 1 4

√ω β √ω β

1 4

−

ω √ 4 β

1 4

−

ω √ 4 β

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (12.22)

142

V. I. Golubev et al.

One of the ways for increasing the scheme approximation order without widening the spatial stencil is the using the differential sequences [19]. These schemes are called compact. This approach is rapidly developed in recently years [20, 21]. For acoustic media, compact schemes were described in [22–25]. In this chapter, we constructed the compact scheme for one-dimensional Dorovsky model. Due to the continuity of parameters, we can directly differentiate the q vector. We introduce Eq. 12.23. ν=

∂q ∂ω = R−1 ∂x ∂x

(12.23)

Based on two function values and its derivatives, we can reconstruct the third-order polynomial interpolant. For the left side, we obtain Eqs. 12.24–12.30. ω+n (x) = ax 3 + bx 2 + kx + l

(12.24)

v +n (x) = 3ax 2 + 2bx + k

(12.25)

a= b=

+n +n νm+n + νm−1 ωm+n − ωm−1 − 2 h2 h3

(12.26)

+n ω+n − ω+n 2νm+n + νm−1 − 3 m 2 m−1 h h

(12.27)

k = νm+n , l = ωm+n

(12.28)

ωm+n+1 = ω+n (xm − cτ )

(12.29)

νm+n+1 = v +n (xm − cτ )

(12.30)

For the right side, we obtain Eqs. 12.31–12.37. ω−n (x) = ax 3 + bx 2 + kx + l

(12.31)

v −n (x) = 3ax 2 + 2bx + k

(12.32)

a=

−n −n νm+1 ωm+1 + νm−n − ωm−n − 2 2 h h3

(12.33)

−n ω−n − ω−n νm+1 + 2νm−n + 3 m+1 2 m h h

(12.34)

b=−

12 Modeling Wave Responses from Thawed Permafrost Zones

143

k = νm−n , l = ωm−n

(12.35)

ωm−n+1 = ω−n (xm + cτ )

(12.36)

νm−n+1 = v −n (xm + cτ )

(12.37)

The constructed scheme was tested for the approximation order. One-dimensional domain x ∈ [−10, 10] filled with the material of μ = 2.2815 × 109 N/m2 , α = 2507.905873 m5 /kg/s2 , K = 2.642306867 × 109 N/m2 , ρs = 1350 kg/m3 , ρ f = 100 kg/m3 was simulated. These parameters correspond to c1 = 2000 m/s and c2 = 450 m/s. Initial conditions were smoothly set as p(x) = 106 ∗ e−4x . 2

(12.38)

The total time was 4 ms. The Courant number was fixed to 0.1. A set of grids containing N = 101, 201, 401, 1601, 3201 nodes were used. Two different norms |xi | × h and L ∞ = max|xi |, where xi = u i − u itheor were calculated: L 1 = i

is the difference between analytical and numerical solutions. The results for the Courant–Isakson–Rees (CIR) and compact cubic interpolated pseudoparticle (CIP) schemes were presented in Table 12.1. The comparison of simulated pressure profiles is presented in Fig. 12.1. The spatial step was 0.05, and the time step was 2.5 µs. Table 12.1 Errors and approximation orders of CIR and CIP schemes (see Fig. 12.2) Scheme title

N

L1

Order L 1

L∞

Order L ∞

CIR

101

726,816.0

–

152,323.0

–

201

538,112.0

0.43

130,633.0

0.22

401

368,768.0

0.55

104,490.0

0.32

801

234,041.0

0.66

76,357.5

0.45

1601

138,099.0

0.76

50,399.7

0.60

3201

76,670.1

0.85

30,240.2

0.74

101

45,238.6

–

15,370.5

–

201

6752.5

2.74

2740.7

2.49

401

880.0

2.94

372.9

2.88

801

111.0

2.99

47.3

2.98

1601

13.9

3.00

5.9

3.00

3201

1.7

3.00

0.7

3.00

CIP

144

V. I. Golubev et al.

Fig. 12.1 Spatial distribution of the pressure. The first-order scheme (dotted line) and the third-order scheme (solid line) were used

12.4 Simulation Results In this work, the process of the seismic wave interaction with the thawed permafrost zone was simulated. Full-wave two-dimensional case was investigated. The total computational domain covered 2000 m × 1000 m. The squared grid with the spatial step of 1 m was constructed. The geological massif had such properties: 2000 m/s of P-wave velocity, 1300 m/s of S-wave velocity, and 1350 kg/m3 of density. At the depth of 120 m, the thawed zone was places with sizes 20 m × 20 m. To describe the dynamical behavior of it, the Dorovsky model was used with such parameters: μ = 2.2815 ∗ 109 N/m2 , K = 2.642306867 ∗ 109 N/m2 , α = 2507.905873 m5 /kg/s2 , ρs = 1350 kg/m3 , and ρ f = 100 kg/m3 . To take into account the amount of the melted material, the porosity of the model changed in the range from 5 to 25%. The specified above parameters with the 10% porosity correspond to c p1 = 2000 m/s, c p2 = 450 m/s, and cs = 1300 m/s. To obtain the realistic wavefield, the point source buried at the depth of 3 m with the Ricker time function was used. The main frequency was 30 Hz. To analyze the internal structure of seismic responses, the velocity modulus was visualized. The third-order Rusanov scheme was used for solving independent linear transport equations (Fig. 12.2). The initial moment of the seismic wave excitation is represented in Fig. 12.3. It is well known that in the elastic half space, two volumetric waves (fast P-wave and slow S-wave) and one surface Rayleigh wave are propagated. All of them are clearly seen in Fig. 12.4. Due to the presence of the inhomogeneity in the geological massif the seismic response was generated. It consists of quasi-spherical waves because the melted zone size is compared with the wavelength (see Fig. 12.5). The Rayleigh wave does not feel this heterogeneity because of its exponential attenuation with the depth (see Fig. 12.6). Of the particular interest is the answer to the question of how much the seismic response depends on the volume of the thawed material. A series

12 Modeling Wave Responses from Thawed Permafrost Zones

145

Fig. 12.2 Error drop with the nodes increase for: a L 1 , b L ∞ . The first-order scheme (plus line) and the third-order scheme (cross line) were used

Fig. 12.3 Spatial distribution of the velocity modulus. The moment right after the emitting of seismic waves. The thawed permafrost zone is depicted with the white rectangle

of calculations was carried out with a change in the proportion of saturated material. Its analysis showed that the response intensity is directly proportional to the fraction of the thawed medium.

146

V. I. Golubev et al.

Fig. 12.4 Spatial distribution of the velocity modulus. The moment of the P-wave interaction with the melted zone. P-wave, S-wave, and Rayleigh waves are easily identified

Fig. 12.5 Spatial distribution of the velocity modulus. The quasi-spherical reflected waves are registered

12.5 Conclusions In this work, the process of the seismic wave interaction with the thawed permafrost zone was investigated. To describe the dynamic behavior of the geological massif, the elastic system of the linear elasticity was used. The melted region was described by the Dorovsky model applied for porous fluid-filled media. The numerical solution of 2D problems was obtained by the grid-characteristic method on rectangular grids.

12 Modeling Wave Responses from Thawed Permafrost Zones

147

Fig. 12.6 Spatial distribution of the velocity modulus. Exponential attenuation with the depth for Rayleigh wave leads to the inhomogeneity insensitivity

The compact scheme for one-dimensional case was proposed and tested. The explicit contact conditions between elastic and porous media were set with the help of the Riemann invariants. A set of calculations was carried out with the different amount of the melted material. The linear dependency of the response amplitude on the fraction of thawed zone was observed. Acknowledgements This work was supported by the Russian Science Foundation, grant no. 1611-00100.

References 1. Gassmann, F.: Elasticity of porous media. Vierteljahrsschrder Naturforschenden Gesselschaft 96, 1–23 (1951) 2. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Lowfrequency range. J. Acoust. Soc. Am. 28(2), 168–178 (1956) 3. Winkler, K.W., Liu, H.L., Johnson, D.L.: Permeability and borehole Stoneley waves: comparison between experiment and theory. Geophysics 54(1), 66–75 (1989) 4. Sidler, R.: A porosity-based Biot model for acoustic waves in snow. J. Glaciol. 61(228), 789–798 (2015) 5. Capelli, A., Kapil, J.C., Reiweger, I., Or, D., Schweizer, J.: Speed and attenuation of acoustic waves in snow: Laboratory experiments and modeling with Biot’s theory. Cold Reg. Sci. Technol. 125, 1–11 (2016) 6. He, J., Rui, Z., Ling, K.: A new method to determine Biot’s coefficients of Bakken samples. J. Nat. Gas Sci. Eng. 35(A), 259–264 (2016) 7. Dorovsky, V.N.: Continual theory of filtration. Russ. Geol. Geophys. (Geol. i Geofiz.) 30(7), 39–45 (1989)

148

V. I. Golubev et al.

8. Dorovsky, V.N., Perepechko, YuV, Fedorov, A.I.: Stoneley waves in the Biot-Johnson and continuum filtration theories. Russ. Geol. Geophysics 53(5), 475–483 (2012) 9. Sinev, A.V., Romensky, E.I., Dorovsky, V.N.: Effects of a mudcake on Stoneley waves in a fluid-filled porous formation around a borehole. Russ. Geol. Geophys. 53(8), 823–828 (2012) 10. Favorskaya, A., Golubev, V., Khokhlov, N.: Two approaches to the calculation of air subdomains: theoretical estimation and practical results. Proced. Comput. Sci. 126, 1082–1090 (2018) 11. Golubev, V.I., Shevchenko, A.V., Petrov, I.B.: Taking into account fluid saturation of bottom sediments in marine seismic survey. Dokl. Math. 100(2), 488–490 (2019) 12. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Methods for calculating the dynamics of layered and block media with nonlinear contact conditions. Smart Innovation Syst. Technol. 173, 171–183 (2020) 13. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Mathematical modeling of the dynamics of layered and block media with nonlinear contact conditions on supercomputers. J. Phys: Conf. Ser. 1392(1), 012057 (2019) 14. Golubev, V.I., Muratov, M.V., Petrov, I.B.: Different approaches for solving inverse seismic problems in fractured media. Smart Innovation Syst. Technol. 173, 199–212 (2020) 15. Golubev, V.I., Golubeva, YuA: Full-wave simulation of the earthquake initiation process. CEUR Workshop Proc. 2267, 346–350 (2018) 16. Breus, A., Favorskaya, A., Golubev, V., Kozhemyachenko, A., Petrov, I.: Investigation of seismic stability of high-rising buildings using grid-characteristic method. Procedia Comput. Sci. 154, 305–310 (2019) 17. Beklemysheva, K.A., Golubev, V.I., Vasyukov, A.V., Petrov, I.B.: Numerical modeling of the seismic influence on an underwater composite oil pipeline. Mathe. Mod. Comput. Simul. 11(5), 715–721 (2019) 18. Beklemysheva, K.A., Vasyukov, A.V., Golubev, V.I., Zhuravlev, Y.I.: On the estimation of seismic resistance of modern composite oil pipeline elements. Dokl. Math. 97(2), 184–187 (2018) 19. 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) 20. Garanzha, V.A., Konshin, V.N.: Numerical algorithms for viscous fluid flows based on highorder accurate conservative compact schemes. Comput. Math. Math. Phys. 39(8), 1321–1334 (1999) 21. 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) 22. Oh, S., Okubo, K., Tsuchiya, T., Takeuchi N.: Two-dimensional numerical analysis of acoustic field using the constrained interpolation profile method. Acoust. Imag. 29 (2008) 23. Tachioka, Y., Yasuda, Y., Sakuma, T.: Application of the constrained interpolation profile method to room acoustic problems: examination of boundary modeling and spatial/time discretization. Acoust. Sci. Technol. 33, 21–32 (2012) 24. Yamashita, O., Tsuchiya, T., Iwaya, Y., Otani, M., Inoguchi, Y.: Reflective boundary condition with arbitrary boundary shape for compact-explicit finite-difference time-domain method. Japan. J. Appl. Phys. 54 (2015) 25. Golubev, V.I., Khokhlov, N.I., Nikitin, I.S., Churyakov, M.A.: Application of compact gridcharacteristic schemes for acoustic problems. J. Phys.: Conf. Ser. 1479, 012058 (2020)

Chapter 13

Modeling of Fiber-Metal Laminate Residual Strength After a Low-Velocity Impact with a Grid-Characteristic Numerical Method Katerina A. Beklemysheva Abstract Fiber-metal laminates are very perspective materials, but their behavior under low-velocity impacts and their residual strength require extensive research. In this chapter, a hybrid grid-characteristic numerical method on regular grids and Hashin failure criterion are used to model a complex loading of a GLARE fibermetal laminate sample. After an initial low-velocity impact, the sample is subjected to a quasi-static in-plane compression loading in the same calculation. Full threedimensional dynamic wavefront patterns for samples with different ply laying, striker angle, and friction coefficient are obtained and presented. Keywords Numerical modeling · Composites · Low-Velocity impact · Grid-Characteristic method · Residual strength · GLARE

13.1 Introduction Barely visible impact damage (BVID) is a well-known problem in composite research [1]. It is an inner damage during exploitation of a composite part caused by weak impacts like hail, gravel, or bird strikes [2]. It is called barely visible because this type of damage can be discovered only during a thorough ultrasound examination. That effect makes composites either dangerous or expensive. BVID usually consists of multiple microscopic matrix cracks and extensive delaminations between plies [3–5]. It leads to a loss in material strength, especially under in-plane compression loading [6, 7]. A research on residual strength of a fiber-metal laminate is given in [8]. In [9], numerical modeling by the finite element method is employed for two successive tests of one plate: low-speed impact loading and the subsequent compression.

K. A. Beklemysheva (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 214, https://doi.org/10.1007/978-981-33-4709-0_13

149

150

K. A. Beklemysheva

The fiber-metal laminate modeled in this research is GLAss-REinforced aluminum laminate (GLARE)an impact resistant material used in aerospace structures [10, 11]. It consists of alternating 2024-T3 aluminum and S2-glass/FM94-epoxy prepreg. Aluminum layers protect the brittle glass-epoxy fiber laminate from impacts and scratches during exploitation. To model the low-velocity impact, a small aluminum striker was considered. Usually, this type of loading is reproduced in experiment by standard equipment with a large steel striker [1, 12]. To correspond to the strike energy of a hailstone moving with a high velocity, the standard striker is set to a low velocity, which explains the origin of the term. In several researches on numerical modeling, it is modeled by a weak static loading [13, 14]. This approach ignores the dynamic pattern and is unable to reproduce inherently dynamic effects like spalling [15, 16]. To model the dynamic loading provided by both low-velocity impact and quasistatic in-plane compression, the grid-characteristic method on regular cubic grids [17, 18] was used. This method is close to finite difference methods, and it is faster than other grid methods [19]. Also, it allows to implement various border and contact conditions, including friction contact, and complex destruction models for anisotropic materials [20]. It should be noticed that only Hashin destruction criterion was used in [21]. Full three-dimensional dynamic wavefront patterns for samples with different ply laying, striker angle, and friction coefficient are obtained and presented. This chapter is organized as follows. Section 13.2 contains a description of the material model for the composite and the used numerical method. Section 13.3 is devoted to the numerical results presenting the distributions of velocity vector modulus and stress tensor components. Section 13.4 contains conclusions.

13.2 Material Model and Numerical Method The composite material model used in this research is the homogenous orthotropic elastic material with the Hashin destruction criterion and destructible contact [20]. The damaged node is marked and calculated according to the Prandtl–Reuss material model with a zero plasticity limit. It allows to model effectively a damaged material despite of this model requires an improvement to distinguish different composite damage types that are provided by the Hashin criterion. It is necessary to improve this algorithm by modeling the ultrasound testing of a composite. The full description of a general anisotropic material and the algorithm for the grid-characteristic method are given in [22]. The main idea behind the grid-characteristic method is that the governing system of equations for an anisotropic elastic material is linear and hyperbolic. After the transition to Riemann invariants, this system falls into a set of independent linear equations. To solve each of these equations, we can find the value of a corresponding Riemann invariant on the previous time layer. After all the invariants are gathered, the original set of variables can be restored.

13 Modeling of Fiber-Metal Laminate Residual Strength …

151

To apply this method to a three-dimensional statement, a spatial splitting is used [16]. On this stage of the algorithm, a splitting by physical processes can be applied to implement viscosity, plasticity, and damage models.

13.3 Numerical Results Hereinafter, we consider results of the numerical modeling. Section 13.3.1 describes the general problem statement. Section 13.3.2 contains the results for a typical normal low-velocity impact. Section 13.3.3 shows the difference in the wavefront pattern caused by different layings of the composite. Section 13.3.4 shows the dependency of wavefront patterns on friction effects during an angled strike. Section 13.3.5 shows the different destruction modes in the calculation area before and after a quasi-static loading.

13.3.1 Problem Statement GLARE consists of alternating 2024-T3 aluminum and S2-glass/FM94-epoxy prepreg. In this research, a single aluminum layer with 0.5 mm thickness was located above two layers of Carbon Fiber Reinforced Plastics (GFRP), 0.5 mm thickness each. The laying of plies is different in different statements. Material parameters are given in [9]. The sample size is 20 × 20 mm. The cubic aluminum striker modeling a breakaway piece of cover has the size of 1 × 1 × 1 mm and velocity 1.4 mm/s. The contact between plies had a strength of 40 MPa, which approximately corresponds to matrix strength, and the contact between the top ply and the aluminum layer had a lower strength of 30 MPa. The plies are located in the XZ plane, and Y-axis is perpendicular to them (Fig. 13.1). The static loading is applied along X-axis: it grows linearly for 12 s to the maximum value and then becomes a constant loading. A Courant time step was used. It was measured, in terms of mechanics, by the fastest elastic wave. For all the given calculations, this value was 0.0037 s.

13.3.2 Wavefronts During Low-Velocity Impacts Firstly, let us consider that both plies are directed along Z-axis, the friction coefficient between the striker and the composite is zero, and the striker velocity is strictly along Y-axis. In this statement, striker velocity is extremely low, and no damage occurs.

152

K. A. Beklemysheva

Fig. 13.1 General view of the calculation area

The dynamic velocity pattern on YZ slice through the sample center is given on Fig. 13.2. The striker reflects from the composite, and elastic waves propagate in the sample. The pattern on XZ slice is different due to material anisotropy but is less informative on this scale than the following patterns. To illustrate the propagation of elastic waves in a layered material, Figs. 13.3, 13.4 and 13.5 show the dynamic distribution of velocity modulus on top surface of all the layers. On these figures Z axis in directed upward. A complex anisotropic pattern is visible. For the further analysis, the distributions of stress deviator and tension on contact surfaces are given in Figs. 13.6, 13.7, 13.8 and 13.9.

Fig. 13.2 Velocity vector and modulus on YZ slice during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102

13 Modeling of Fiber-Metal Laminate Residual Strength …

153

Fig. 13.3 Velocity modulus on top of the aluminum layer during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102, g t = 122, h t = 142

Fig. 13.4 Velocity modulus on top of the upper GFRP ply during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102, g t = 122, h t = 142

154

K. A. Beklemysheva

Fig. 13.5 Velocity modulus on top of the lower GFRP ply during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102, g t = 122, h t = 142

Fig. 13.6 Tension on top of the upper GFRP ply during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102, g t = 122, h t = 142

13.3.3 Wavefronts at Different Layings of Composite Plies In this statement, the direction of plies is changed. The rest of the statement stays the same as in the previous part. The friction coefficient between the striker and

13 Modeling of Fiber-Metal Laminate Residual Strength …

155

Fig. 13.7 Deviator on top of the upper GFRP ply during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102, g t = 122, h t = 142

Fig. 13.8 Tension on top of the lower GFRP ply during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102, g t = 122, h t = 142

composite is zero, and the striker velocity is strictly along Y-axis. Striker velocity is extremely low, and no damage occurs. Figures 13.10 and 13.11 represent the stress pattern on contact surfaces. The laying of plies influences these patterns on the qualitative and quantitative levels.

156

K. A. Beklemysheva

Fig. 13.9 Deviator on top of the lower GFRP ply during the collision, striker velocity is aligned with Y-axis, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102, g t = 122, h t = 142

Fig. 13.10 Tension on contacts during the collision, striker velocity is aligned with Y-axis, time step equals 100: a contact between GFRP and aluminum in ZZ plane, b contact between GFRP and aluminum in XZ plane, c contact between GFRP and aluminum in ZX plane, d contact between plies in ZZ plane, e contact between plies in XZ plane, f contact between plies in ZX plane

13 Modeling of Fiber-Metal Laminate Residual Strength …

157

Fig. 13.11 Deviator on contacts during the collision, striker velocity is aligned with Y-axis, time step equals 100: a contact between GFRP and aluminum in ZZ plane, b contact between GFRP and aluminum in XZ plane, c contact between GFRP and aluminum in ZX plane, d contact between plies in ZZ plane, e contact between plies in XZ plane, f contact between plies in ZX plane

13.3.4 Angled Impacts with Different Friction Coefficients To demonstrate the friction coefficient influence on calculation, let us firstly consider a direct strike, when the striker velocity is strictly along Y-axis. Stress patterns with zero friction coefficients can be observed in Figs. 13.10 and 13.11. In the case of the direct strike, the stress patterns with 0.75 friction coefficient are also similar. After that, let us consider a strike at an angle of 45°. Stress patterns with zero friction coefficients qualitatively coincide with patterns for the direct strike and can be observed on Fig. 13.10. Stress patterns with 0.75 friction coefficient are given in Figs. 13.11 and 13.12. A dependency on plies laying is clearly visible, and it is qualitatively different than in case of the direct strike. To illustrate the friction contact conditions, the velocity vector and modulus for an angled strike with and without friction are given in Figs. 13.13 and 13.14.

13.3.5 Quasi-Static Loading A statement with an angled strike for different friction coefficients and loading amplitude was considered. The striker velocity is at 45° respect to Y-axis. Figure 13.15 shows the dynamic pattern of contact destruction during the quasi-static loading. The

158

K. A. Beklemysheva

Fig. 13.12 Tension on contacts during the collision, striker velocity is at 45° respect to Y-axis, friction coefficient is 0.75, time step equals 100: a contact between GFRP and aluminum in ZZ plane, b contact between GFRP and aluminum in XZ plane, c contact between GFRP and aluminum in ZX plane, d contact between plies in ZZ plane, e contact between plies in XZ plane, f contact between plies in ZX plane

Fig. 13.13 Deviator on contacts during the collision, striker velocity is at 45° respect to Y-axis, friction coefficient is 0.75, time step equals 100: a contact between GFRP and aluminum in ZZ plane, b contact between GFRP and aluminum in XZ plane, c contact between GFRP and aluminum in ZX plane, d contact between plies in ZZ plane, e contact between plies in XZ plane, f contact between plies in ZX plane

13 Modeling of Fiber-Metal Laminate Residual Strength …

159

Fig. 13.14 Velocity vector and modulus on YZ slice during the collision, striker velocity is at 45° respect to Y-axis, friction coefficient is zero, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102

Fig. 13.15 Velocity vector and modulus on a YZ slice during the collision, striker velocity is at 45° respect to Y-axis, friction coefficient is 0.75, plies laying is ZZ. Consecutive time steps t are: a t = 2, b t = 22, c t = 42, d t = 62, e t = 82, f t = 102

whole low-velocity impact is occurring before the first depicted frame. Figure 13.16 shows the volume destruction calculated by the Hashin criterion. For comparison, the same calculation results without any low-velocity impact are given in Fig. 13.17. In presence of BVID, cracks start to grow earlier, and the resulting damage is larger. An interesting pattern can be observed in Fig. 13.18. The tension distribution qualitatively coincides with characteristic destruction patterns that are often observed after low-velocity impacts [4]. After the quasi-static loading, amplitude of 750 MPa of all the GFRP material is damaged. The plies direction slightly alters the general destruction pattern, but does not influence the outcome.

160

K. A. Beklemysheva

Fig. 13.16 Destructed contact during the quasi-static loading, amplitude is 750 MPa, friction coefficient is zero, plies laying is ZZ, consecutive time steps t are: a contact between GFRP and aluminum, t = 500, b contact between GFRP and aluminum, t = 1000, c contact between GFRP and aluminum, t = 1500, d contact between GFRP and aluminum, t = 20,000, e contact between GFRP and aluminum, t = 2500, f contact between GFRP and aluminum, t = 3000, g contact between plies, t = 500, h contact between plies, t = 1000, i contact between plies, t = 1500, j contact between plies, t = 20,000, k contact between plies, t = 2500, l contact between plies, t = 3000

Fig. 13.17 Failed nodes during the quasi-static loading, amplitude is 750 MPa, friction coefficient is zero, plies laying is ZZ, consecutive time steps t are: a the upper ply, t = 1500, b the upper ply, t = 2000, c the upper ply, t = 2500, d the upper ply, t = 30,000, e the upper ply, t = 3500, f the upper ply, t = 4000, g the bottom ply, t = 1500, h the bottom ply, t = 2000, i the bottom ply, t = 2500, j the bottom ply, t = 30,000, k the bottom ply, t = 3500, l the bottom ply, t = 4000

After the quasi-static loading, the amplitude of 300 MPa and the zero friction coefficient of the damage in material are limited to the damage after the low-velocity impacts. In the same statement but with a 0.75 friction coefficient, the pattern changes. The low-velocity impact damages the contact between the aluminum layer and GFRP, and the following quasi-static loading completely destroys this contact. At the same time, the bottom GFRP ply stays intact, and the volume damage to the top ply is very low (Figs. 13.19 and 13.20).

13 Modeling of Fiber-Metal Laminate Residual Strength …

161

Fig. 13.18 Failed nodes during the quasi-static loading, amplitude is 750 MPa, friction coefficient is zero, plies laying is ZZ, no low-velocity impact, consecutive time steps t are: a the upper ply, t = 1500, b the upper ply, t = 2000, c the upper ply, t = 2500, d the upper ply, t = 30,000, e the upper ply, t = 3500, f the upper ply, t = 4000, g the bottom ply, t = 1500, h the bottom ply, t = 2000, i the bottom ply, t = 2500, j the bottom ply, t = 30,000, k the bottom ply, t = 3500, l the bottom ply, t = 4000

Fig. 13.19 Tension on top surfaces during the quasi-static loading, amplitude is 750 MPa, friction coefficient is zero, plies laying is ZZ, consecutive time steps t are: a the upper ply, t = 100, b the upper ply, t = 500, c the upper ply, t = 900, d the upper ply, t = 1300, e the upper ply, t = 1700, f the upper ply, t = 2100, g the bottom ply, t = 100, h the bottom ply, t = 500, i the bottom ply, t = 900, j the bottom ply, t = 1300, k the bottom ply, t = 1700, l the bottom ply, t = 2100

Fig. 13.20 Failed nodes after the quasi-static loading, amplitude is 300 MPa, friction coefficient is 0.75, plies laying is along: a ZZ, b XX, c XZ, d ZX

162

K. A. Beklemysheva

13.4 Conclusions Composite parts are designed with a large margin of safety and require a full replacement after a serious damage unlike traditional aircraft cover materials. A stable and reliable numerical method can help to investigate effects, occurring after a lowvelocity impact and during a subsequent quasi-static loading. The modern level of technology inevitably requires numerical modeling as a part of the industrial design process, but modeling of composites still requires an extensive research to be safe and reliable for engineering uses. This chapter as a proof of concept provides an illustration to capabilities of the grid-characteristic method and its applicability to modeling of residual strength of fiber-metal laminates. The anisotropic elastoplastic material model with the Hashin failure criterion and destructible contacts allows modeling the barely visible impact damage and its influence on material strength during quasi-static compressive inplane loading. The implemented friction contact model allows to model the angled strikes and destructed contacts correctly. The grid-characteristic method on regular grids used in this chapter proves to be stable during long-scale calculations and provides the numerical results that qualitatively coincide with experimental data. Acknowledgements This work was carried out with the financial support of the Russian Science Foundation, project no. 19-71-00147.

References 1. Abrate, F.: Impact on laminated composites: recent advances. Appl. Mech. Rev. 11(47), 517– 544 (1994) 2. Liu, J., Li, Y., Yu, X., Gao, X., Liu, Z.: Design of aircraft structures against threat of bird strikes. Chin. J. Aeronaut. 31(7), 1535–1558 (2018) 3. Lopresto, V., Caprino, G.: Damage mechanisms and energy absorption in composite laminates under low velocity impact loads. In: Abrate, S., Castanie, B., Rajapakse, Y. (eds.) Dynamic Failure of Composite and Sandwich Structures. Solid Mechanics and Its Applications, SMIA, vol. 192, pp. 209–289. Springer Science + Business Media, Dordrecht (2013) 4. Richardson, M.O.W., Wisheart, M.J.: Review of low-velocity impact properties of compo-site materials. Compos. A Appl. Sci. Manuf. 12(29), 1123–1131 (1996) 5. Beklemysheva, K.A., Petrov, I.B.: Damage modeling in hybrid composites subject to low-speed impact. Math. Models Comput. Simul. 11(3), 469–478 (2019) 6. Siow, Y.P., Shim, P.W.: An experimental study of low velocity impact damage in woven fiber composites. J. Compos. Mater. 32, 1178–1202 (1998) 7. Nettles, A.T., Hodge, A.J.: Compression-after-impact testing of thin composite materials. In: The 23rd International SAMPE Technical Conference (Soc. Advancement of Mater. Proc. Eng.), pp. 177–183. Covina, CA, USA (1991) 8. Sun, C.T., Dicken, A., Wu, H.F.: Characterization of impact damage in ARALL laminates. Comp. Sci. Technol. 49, 139–144 (1993) 9. Moriniere, F.D., Alderliesten, R.C., Sadighi, M., Benedictus, R.: An integrated study on the low-velocity impact response of the GLARE fibre-metal laminate. Compos. Struct. 100, 89–103 (2013)

13 Modeling of Fiber-Metal Laminate Residual Strength …

163

10. Gonzalez, E.V., Maimi, P., Camanho, P.P., Turon, A., Mayugo, J.A.: Simulation of drop-weight impact and compression after impact tests on composite laminates. Compos. Struct. 94, 3364– 3378 (2012) 11. Vlot A., Gunnink J.W.: Fibre Metal Laminates—An Introduction. Kluywer Academic Publisher (2001) 12. Hu, N., Zemba, Y., Okabe, T., Yan, C., Fukunaga, H., Elmarakbi, A.: A new cohesive model for simulating delamination propagation in composite laminates under transverse loads. Mech. Mater. 40, 920–935 (2008) 13. Sjoblom, P.O., Hartness, J.T., Cordell, T.M.: On low-velocity impact testing of composite materials. J. Compos. Mater. 22, 30–52 (1988) 14. Shivakumar, K.N., Elber, W., Illg, W.: Prediction of low-velocity impact damage in thin circular laminates. AIAA J. 23, 442–449 (1985) 15. Kanel, G.I., Razorenov, S.V., Utkin, A.V., Fortov, V.E.: Shock Wave Phenomena in Condensed Matter. Yanus-K, Moscow (in Russian) (1996) 16. Ivanov, V.D., Kondaurov, V.I., Petrov, I.B., Holodov, A.S.: Calculation of dynamic deformation and destruction of elastoplastic bodies by grid-characteristic methods. Mat. Model. 2(11), 10–29 (1990) 17. Ivanov, A.M.; Khokhlov, N.I.: Efficient inter-process communication in parallel implementation of grid-characteristic method. 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. 91–102. Springer, Cham (2019) 18. Beklemysheva, K.A., Danilov, A.A., Petrov, I.B., Salamatova, V.Yu., Vassilevski, Yu.V., Vasyukov, A.V: Virtual blunt injury of human thorax: age-dependent response of vascular system. Russian J. Numer. Anal. Math. Modell. 30(5), 259–268 (2015) 19. Biryukov, V.A., Miryaha, V.A., Petrov, I.B., Khokhlov, N.I.: Simulation of elastic wave propagation in geological media: intercomparison of three numerical methods. Comput. Mathe. Math. Phys. 56(6), 1086–1095 (2016) 20. Beklemysheva, K.A., Ermakov, A.S., Petrov, I.B., Vasyukov, A.V.: Numerical simulation of the failure of composite materials by using the grid-characteristic method. Math. Models Comput. Simul. 5(8), 557–567 (2016) 21. Hashin, Z.: Failure criteria for unidirectional fiber composites. J. Appl. Mech. 47, 329–334 (1980) 22. Beklemysheva, K.A., Vasyukov, A.V., Kazakov, A.O., Petrov, I.B.: Grid-characteristic numerical method for low-velocity impact testing of fiber-metal laminates. Lobachevskii J Math. 39, 874–883 (2018)

Chapter 14

Modeling Movement of Train Along Bridge by Grid-Characteristic Method Anton A. Kozhemyachenko , Anastasia S. Kabanova , Igor B. Petrov , and Alena V. Favorskaya

Abstract The aim of the study is the application of the grid-characteristic method on structured grids in problems related to railway transport safety. The main objective of the study is to consider the problem of numerical simulation of train movement on the bridge. We consider two types of bridges: ballast bridge and nonballast bridge. The model of interaction, based on analytical approach between rail and train is proposed. The grid-characteristic method solves the system of equations describing the propagation of wave processes in the bridge model. As a result of numerical simulation, various patterns of the dynamic distribution of the vertical component of the Cauchy stress tensor are obtained for different types of bridges. The embankment significantly reduces the load transmitted to the ballast bridge structure. The wave front spreads faster in nonballast bridges. According to the obtained results, the dynamic load from train movement to the nonballast bridge structure is about ten times greater than the similar load on the ballast bridge structure. Keywords Wheel–rail contact · Railway bridge · Railway safety

A. A. Kozhemyachenko (B) · A. S. Kabanova · I. B. Petrov · A. V. Favorskaya Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] A. S. Kabanova e-mail: [email protected] I. B. Petrov e-mail: [email protected] A. V. Favorskaya e-mail: [email protected] I. B. Petrov · A. V. Favorskaya Scientific Research Institute for System Analysis of the RAS, 36/1, Nahimovskij Props., Moscow 117218, 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 214, https://doi.org/10.1007/978-981-33-4709-0_14

165

166

A. A. Kozhemyachenko et al.

14.1 Introduction Safety in heavy and high-speed railway traffic directly depends on the condition of the railway track during long-term operation. It is important to determine the most dangerous sections of the train movement, as well as to know the causes of defects in the railway track. Thus, numerical simulation is used in the safety problems of railway transport. In addition, it is important to use different methods for the numerical solution of these problems to expand the capabilities of computer modeling and increase the applicability of the method itself. We use the grid-characteristic method for calculating wave processes in a railway track presented in the form of a multilayer heterogeneous linear-elastic medium. The grid-characteristic method is used in problems of railway safety [1], seismic problems [2, 3], and destruction of objects under dynamic loads [4]. Scientists solve the class of problems of numerical simulation related to the calculation of wave processes in railway rails by different calculation methods: the finite element method [5], the discontinuous Galerkin method [6], finite-difference methods and their modifications [7], semi-analytic finite element method [8], and commercial closed-source software ANSYS [9]. Based on the analytical approach [10], we apply the developed special boundary condition to set the pressure in the wheel–rail system. The chapter considers the structure of the bridge of two types: the rail track on the ballast layer or on the concrete slabs (nonballast). The ballast railway along the entire length of the track is traditional and has a low cost of laying the rail track. In turn, the nonballast track with concrete slabs reduces maintenance costs due to the preservation of geometric parameters in long-term operation. Slabs are the extremely promising design solution in connection with operational wear resistance and durability of the upper part of the track. In [11], the dynamic reactions of multi-span viaducts were analyzed during train movement. The coupling effect between the bridge and rail in the presence of a ballast layer was considered. On several ballast bridges in Austria, dynamic tests were conducted with short and medium span lengths [12]. Various ballast paths were studied in [13]. It is concluded that the structure of the track is a filter for high-frequency vibrations of ground-based railway bridges. We calculate full-wave patterns for different types of bridges in conditions of high-speed train movement. The effect of increasing the speed of the train on the transmitted load is investigated. The rest of chapter is the following. Section 14.2 discusses the main features of the applied grid-characteristic method and the initial and boundary conditions in the area of integration of the problem. Section 14.3 contains the results of numerical simulation for the ballast and nonballast bridges. Section 14.4 presents the conclusions on the numerical simulation using the grid-characteristic method.

14 Modeling Movement of Train Along Bridge …

167

14.2 Methods Hereinafter, Sect. 14.2.1 explains the features of applying the grid-characteristic method to a system describing a linear-elastic medium. Section 14.2.2 formulates the initial and boundary conditions for the specified integration area.

14.2.1 Grid-Characteristic Method The components of the propagation velocity of the disturbance V and symmetric Cauchy tensor σ in the linearly elastic medium were described by the following system of equations: ρVt = (∇ · σ)T ,

(14.1)

σt = λ(∇ · V)I + μ(∇ ⊗ V + (∇ ⊗ V)T ),

(14.2)

where λ, μ are the Lame parameters that determine the properties of an elastic material, ρ is the density of the material, and I is the unit tensor of the second rank. Equation 14.1 is a local equation of motion. We derived Eq. 14.2 from the law of Hook. We applied the grid-characteristic method for the numerical solution of systems of Eqs. 14.1 and 14.2. Therefore, method allowed to construct correct numerical algorithms to calculate the boundary points and points of contacting media with different Lame and/or density parameters. The vector of unknowns in the case of an isotropic linearly elastic medium in two-dimensional case is provided by Eq. 14.3. q = (V, σ)T = (V1 , V2 , σ11 , σ22 , σ12 )T .

(14.3)

System of Eqs. 14.1 and 14.2 in two-dimensional case can be represented in the form of Eq. 14.4. qt + A1 qx + A2 q y = 0

(14.4)

Cleavage is performed in two directions, and we obtained the further expression: qt + A1 qx = 0.

(14.5)

The matrix A1 is the hyperbolic, and it had a set of eigenvectors:  −1 A1 = 1 1 1 .

(14.6)

Where the matrix 1 is composed of eigenvectors, and the eigenvalues of the matrix A1 are the elements of the diagonal matrix 1 .

168

A. A. Kozhemyachenko et al.

Thus, we revised Eq. 14.5 in the form of Eq. 14.7.  −1 qt + 1 1 1 qx = 0

(14.7)

Further calculations were divided into three stages. At the first stage, multiplication by the matrix (1 )−1 and the transition to new variables is performed (Eq. 14.8). At the second stage, one-dimensional transfer equations are solved using the method of characteristics or conventional finite-difference schemes. At the third stage, the inverse change is made (Eq. 14.9).  −1 ω(x, y, t) = 1 q(x, y, t)

(14.8)

q(x, y, t + τ ) = 1 ω(x, y, t + τ )

(14.9)

We calculated a wavefront propagation with the Rusanov scheme of the third order of accuracy in the multilayer medium. This scheme was stable for Courant numbers (Eq. 14.8) not more than one: C = cτ/ h,

(14.10)

where τ is the time step, h is the space step, and c is the eigenvalue of the matrix 1 corresponding to the transfer equation.

14.2.2 Initial and Border Conditions The model of the railway track includes different elements marked with colors in Fig. 14.1: rail, railway sleepers, railway embankment, and sedimentary cover. We selected the parameters of each layer based on a typical railway structure in Russia or Europe (Tables 14.1, 14.2 and 14.3). We consider the model in Fig. 14.1 as an integration area for the numerical simulation problem. For each part of railway track, non-reflecting boundary conditions are satisfied on the left and right boundaries of the integration area. This implied that the difference of values along the characteristics (Eq. 14.8) leaving the boundaries of the integration area are equal to zero. We set glue contacts (Eq. 14.11) between the layers of model. We set the conditions of the detached boundary (Eq. 14.12) for the upper and lower boundary of the railway bridge and along the borders of all rectangular areas between the rail, sleepers, and the embankment or concrete slab, except for the pressure setting area in the wheel–rail contact: va = vb = V, f a = −f b ,

(14.11)

14 Modeling Movement of Train Along Bridge …

169

Fig. 14.1 Railway track model: a ballast track, b nonballast track Table 14.1 Parameters of the elasticity of the medium in the composition of the track Part of the track Material

Velocity P-wave (m/s)

Velocity S-wave (m/s)

Density (kg/m3 )

Rail

Steel

5740

3092

7800

Railway sleepers

Reinforced concrete

4200

2200

2500

Railway embankment

Gravel

500

300

1400

Concrete slab

Reinforced concrete

4200

2200

2500

6000

3000

7650

Bridge structure Steel

Table 14.2 Dimensions of the components of the ballast bridge Part of the track

Length (m)

height (m)

Rail

25

0.18

Railway sleepers

0.25

0.19

Railway embankment

25

0.3

Bridge structure

25

0.34

170

A. A. Kozhemyachenko et al.

Table 14.3 Dimensions of the components of the nonballast bridge Part of the track

Length (m)

height (m)

Rail

25

0.18

Railway sleepers

0.25

0.19

Concrete slab

25

0.44

Bridge structure

25

0.5

f = 0,

(14.12)

where V is the velocity of the contact boundary, f = σ·n is the density of external forces, and n is the external normal to the surface of the layer under consideration. We considered different speed conditions of the train: 120, 180, 240, 300, and 350 km/h. The time τ step is equal 8.5 × 10−7 s., the space steps hx and hz were equal to 10−2 and 5 × 10−3 , respectively. The mass of the train moving along the railway was 90 t. Time of train movement is 0.068 s. We set a wheel pressure in wheel–rail contact by the boundary condition from [1]. The calculations were performed with the package developed at the Laboratory of Applied and Computational Geophysics of Moscow Institute of Physics and Technology. The ParaView package was applied to visualize the simulation results.

14.3 Results Hereinafter, Sects. 14.3.1 and 14.3.2 contain the results of numerical simulation in the cases of ballast railway track and nonballast railway track, respectively. Results of numerical simulation are presented in the form of wave field patterns of the vertical component of the Cauchy stress tensor and characteristic values of the vertical component of the Cauchy stress tensor in some sleepers.

14.3.1 Ballast Railway Track Figure 14.2 shows the range of values of the vertical component of the stress tensor in Pa. Figure 14.3 shows the wave propagation patterns of the vertical component of the

Fig. 14.2 Value range

14 Modeling Movement of Train Along Bridge …

171

Fig. 14.3 Ballast railway track: a 120 km/h, b 180 km/h, c 240 km/h, d 300 km/h, e 350 km/h

Table 14.4 Dynamic loads for ballast track Train velocity (km/h)

Vertical component of cauchy stress tensor (MPa) Sleeper №12

Sleeper №25

Sleeper №30

120

+34 ÷ −51

+17 ÷ −6.0

+8.5 ÷ −15

180

+25 ÷ −43

+13 ÷ −6.2

+7.8 ÷ −19

240

+15 ÷ −42

+5.7 ÷ −8.2

+8.0 ÷ −37

300

+12 ÷ −38

+7.1 ÷ −41.0

+12 ÷ −47

350

+14 ÷ −38

35.0 ÷ −49.0

+14 ÷ −40

stress tensor in conditions of different train velocity for the finite time 0.068 s on the ballast track. Blue color corresponds to negative values of the vertical component (these components are the compression waves), and orange color corresponds to positive values of the vertical component (these components are the tensile waves). The load in the supporting structure of the bridge increases slightly due to gravel ballast. The calculated load transferred to the bridge structure fluctuates around a value of 5 MPa. Table 14.4 shows the results of the distribution of dynamic load in some sleepers for each speed mode. Values in Table 14.4 are indicated as a range from maximum tensile pressure to maximum compression pressure. These results show that at any train speed, the residual stresses in the sleepers, which remain after the train has passed, are preserved. The compression loads in the sleepers located between the wheelsets increase several times.

14.3.2 Nonballast Railway Track Figure 14.4 shows the range of values of the vertical component of the stress tensor in Pa. Figure 14.5 shows the wave propagation patterns of the vertical component of the stress tensor in conditions of different train velocity for the finite time 0.068 s on the

172

A. A. Kozhemyachenko et al.

Fig. 14.4 Value range

Fig. 14.5 Nonballast railway track: a 120 km/h, b 180 km/h, 240 km/h, d 300 km/h, e 350 km/h

nonballast track. Blue color corresponds to negative values of the vertical component (these components are the compression waves), and orange color corresponds to positive values of the vertical component (these components are the tensile waves). The wave front penetrates deeper into the structure of the bridge due to the absence of a gravel layer. The calculated load transferred to the bridge structure fluctuates around a value of 50 MPa. Table 14.5 shows the results of the distribution of dynamic load in some sleepers for each speed mode. Values in Table 14.5 are indicated as a range from maximum tensile pressure to maximum compression pressure. Table 14.5 Dynamic loads for nonballast track Train velocity (km/h)

Vertical component of Cauchy stress tensor (MPa) Sleeper №12

Sleeper №25

Sleeper №30

120

+68 ÷ −67

+6.3 ÷ −3.2

+16 ÷ −15

180

+68 ÷ −67

+14 ÷ −11

+13 ÷ −12

240

+62 ÷ −62

+18 ÷ −15

+7.9 ÷ −7.3

300

+59 ÷ −58

+21 ÷ −60

+5.5 ÷ −4.2

350

+54 ÷ −55

+31 ÷ −30

+9.1 ÷ −4.7

14 Modeling Movement of Train Along Bridge …

173

The load at the rail–sleeper contact increases. However, when the wheelset approaches the №30 sleeper, the value of the vertical component of the stress tensor does not increase, in contrast to the case with the ballast bridge. On the other hand, the pressure in the sleeper №12 does not decrease with time. A similar result leads to the fact that the applied method does not give an accurate representation of the distribution of the load in the entire calculation area.

14.4 Conclusions During simulating a movement of the train, we obtained patterns of the distribution of the vertical component of the stress tensor. The fundamental difference in the formulation of the problem with ballast and nonballast bridges is in the penetrating ability of the wave front, which emanates from the contacts of the wheels and rail. The load transferred to the nonballast bridge structure is about ten times greater than the load on the ballast bridge structure. The resulting model shows a plausible picture of the distribution of dynamic loads in areas between two wheelsets. Nonetheless, there are a number of conflicting results in our study. First of all, the ambiguous relation causes a preservation of residual stresses when the train passes from the previously covered section of the track. Probably, the explicit scheme gives an unstable solution for these sections of the path, and additional research is required. We have applied the grid-characteristic method to simulate the movement of a train along a section of a bridge. Despite some shortcomings, this method gives a representation of the wave phenomena that occur when the train moves on the bridge. Acknowledgements The publication was carried out as part of the state assignment of Federal State Institution “Scientific Research Institute for System Analysis of the Russian Academy of Sciences” (fundamental scientific research GP 47) on the topic No. 0065-2019-0005 “Mathematical modeling of dynamic processes in deformable and responsive environments using multiprocessor computing systems” (No. AAAA-A19-119011590092-6).

References 1. Favorskaya, A., Khokhlov, N.: Modeling the impact of wheelsets with flat spots on a railway track. Proc. Comput. Sci. 126, 1100–1109 (2018) 2. Favorskaya, A., Petrov, I., Khokhlov, N.: Numerical modeling of wave processes during shelf seismic exploration. Proc. Comput. Sci. 96, 920–929 (2016) 3. Favorskaya, A., Petrov, I.: The use of full-wave numerical simulation for the investigation of fractured zones. Math. Mod. Comput. Simul. 11, 518–530 (2019) 4. Breus, A., Favorskaya, A., Golubev, V., Kozhemyachenko, A., Petrov, I.: Investigation of seismic stability of high-rising buildings using grid-characteristic method. Proc. Comput. Sci. 154, 305–310 (2019) 5. Nejad, R.: Using three-dimensional finite element analysis for simulation of residual stresses in railway wheels. Eng. Fail. Anal. 45, 449–455 (2014)

174

A. A. Kozhemyachenko et al.

6. Petrov, I., Favorskaya, A., Khokhlov, N., Miryakha, V., Sannikov, A., Golubev, V.: Monitoring the state of the moving train by use of high performance systems and modern computation methods. Math. Mod. Comput. Simul. 7, 51–61 (2015) 7. Feng, D., Feng, M.: Model updating of railway bridge using in situ dynamic displacement measurement under trainloads. J. Bridge Eng. 20(12), 04015019 (2015) 8. Bartoli, I., Marzani, A., di Scalea, F., Viola, E.: Modeling wave propagation in damped waveguides of arbitrary cross-section. J. Sound Vib. 295(3–5), 685–707 (2006) 9. Zumpano, G., Meo, M.: A new damage detection technique based on wave propagation for rails. Int. J. solids and structures 43(5), 1023–1046 (2006) 10. Loktev, A., Sychev, V., Buchkin, V., Bykov, Y., Andreichicov, A., Stepanov, R.: Determination of the pressure between the wheel of the moving railcar and rails subject to the defects. In: Proceedings of the 2017 International Conference Quality Management, Transport and Information Security, Information Technologies, pp. 748–751. St. Petersburg, Russia (2017) 11. Liu, K., Lombaert, G., De Roeck, G.: Dynamic analysis of multi-span viaducts under the passage of the train using a substructure approach. Bridge Eng. 19(1), 83–90 (2014) 12. Rebelo, C., Da Silva, L., Rigueiro, C., Pircher, M.: Dynamic behavior of twin single-span ballasted railway viaducts. Eng. Struct. 30(9), 2460–2469 (2008) 13. Rigueiro, C., Rebelo, C., Da Silva, L.: Influence of ballast models in the dynamic response of railway viaducts. Sound Vibr. 329(15), 3030–3040 (2010)

Chapter 15

Seismic Evaluation of Two-Storied Unreinforced Masonry Building with Rigid Diaphragm Using Nonlinear Static Analysis Amit Sharma , Vasily I. Golubev , and Rakesh Kumar Khare Abstract The present work investigates the seismic evaluation of two-storied unreinforced masonry building with rigid diaphragm when subjected to seismic lateral loading. The work consists of three-dimensional modeling of two-storied masonry building, which forms a common building stock in India. Nonlinear analysis is performed using SAP 2000 software. The building is designed for Bhuj (Zone V). After nonlinear analysis, retrofitting of masonry building is done by providing bond beam designed as per IS 4326:1993. The demands are calculated as per IS 1893:2002 and FEMA-356 (2000) and compared with the capacity of building obtained from nonlinear static curves. The main objective of the work is to perform pushover analysis on the masonry building in order to assess its performance for Bhuj (Zone V). Keywords Seismic evaluation · Unreinforced masonry · Modeling · SAP-2000 · Bhuj · Zone V · Retrofitting · Bond beam

A. Sharma (B) IPS Academy, Institute of Engineering and Science, Rajendra Nagar, Indore, India e-mail: [email protected] V. I. Golubev Moscow Institute of Physics and Technology (National Research University), 9, Institutsky per Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] R. K. Khare Shri G.S. Institute of Technology and Science, Indore, India 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 214, https://doi.org/10.1007/978-981-33-4709-0_15

175

176

A. Sharma et al.

15.1 Introduction Earthquakes lead to the major amount of destructions and deaths each year. Due to the development of the modern computing systems, precise physical simulations can be used by scientists and engineers [1–3]. A nonlinear static (pushover) analysis approach as per FEMA-356 (2000) [4] is used in the present work to estimate the seismic performance of unreinforced masonry (URM) building with rigid diaphragm. The analysis of building without bond beams and with bond beams is performed. Thus, in the present work, initially, a two-storied URM building [5] with rigid diaphragm representing a common building stock in India is considered. The building is designed for Bhuj (Zone V) as per IS 1893:2002 [6]. Main objective of the work is to obtain the seismic performance of two-storied URM building with rigid diaphragm design for Bhuj (Zone V). Two types of buildings considered are (i) without bond beams and (ii) retrofitted building with bond beams. Nonlinear static (pushover) analysis is used for seismic performance assessment. Performance assessment is done using SAP 2000. The chapter is organized as follows. Section 15.2 provides a description of sample existing building. Section 15.3 presents a linear static analysis, while nonlinear static (pushover) analysis is given in Sect. 15.4. Section 15.5 contains the results and conclusions of this research.

15.2 Description of Sample Existing Building Plan of the building considered for analysis is shown in Fig. 15.1. The building has 250 mm thick walls that act as the main lateral force resisting elements. It has a rigid roof diaphragm, which represents one of the typical building types in India. In the present study, three cases for buildings are considered depending upon the number of shear walls available for bearing the lateral load. As per IS 4326:1993 [7], the building designed in Zone V must be provided with reinforced cement concrete (RCC) bands (bond beams) at different levels. Thus, all cases mentioned below are also checked by providing RCC bond beams. The positions of shear walls in three different cases considered are shown in Fig. 15.2. Properties of masonry taken for analysis are given in Table 15.1. Let us consider three cases mentioned below: • Case I. In this case, three shear walls are provided in the building to carry the lateral load so that the middle shear wall takes the half of total lateral load acting on the building. • Case II. In this case, four shear walls are provided in the building to carry lateral load so that two middle shear walls take one-third of total lateral load acting on the building.

15 Seismic Evaluation of Two-Storied Unreinforced Masonry …

Fig. 15.1 Plan of building

Fig. 15.2 Plan outline with shear wall positions for: a case I, b case II, c case III Table 15.1 Properties of masonry taken for analysis Parameter

Condition

Masonry condition

Fair

Wall thickness Compressive strength



f m



Elastic modulus of compression (E m )

250 mm 2.5 MPa 1375 MPa

Modulus of rigidity (Gm )

550 MPa

Density of masonry

20 kN/m3

Poisson’s ratio (µ)

0.13

Type of material

Isotropic

Coefficient of thermal expansion (A)

1.1 × 10−6

177

178

A. Sharma et al.

Table 15.2 Linear static analysis parameter for building at DBE and MCE hazard level as per IS 1893:2002 for URM building without bond beams Case

Maximum seismic weight (W) shared by wall (kN)

DBE (kN)

MCE (kN)

Case I Case II

2717.59

815.277

1630.6

3032.36

909.71

Case III

1819.4

3327.16

998.148

1996.3

Table 15.3 Linear static analysis parameter for building at DBE and MCE hazard level as per IS 1893:2002 for URM building with bond beams Case

Maximum seismic weight (W) shared by wall (kN)

DBE (kN)

Case I

2755.08

495.91

MCE (kN)

Case II

3144.85

566.07

1132.1

Case III

3469.46

624.54

1249.1

991.83

• Case III. In this case, five shear walls are provided in the building to carry lateral load so that three middle shear walls take one-fourth of total lateral load acting on the building.

15.3 Linear Static Analysis Lateral load acting on buildings for three different cases are calculated. Loads are calculated as per IS 1893:2002 and FEMA 356 (2000). The details are given in Sects. 15.3.1 and 15.3.2.

15.3.1 As Per IS 1893:2002 URM building used for the analysis is situated in Bhuj, which is in Zone V as per Indian seismic code. The building type considered is residential having rigid diaphragm with different combinations of shear walls. Tables 15.2 and 15.3 show the values of base shear for different cases.

15.3.2 As Per FEMA-356 (2000) For calculation of base shear using FEMA 356 (2000), seismic hazard of Bhuj is estimated using United States Geological Survey (USGS) maps [8, 9]. The response

15 Seismic Evaluation of Two-Storied Unreinforced Masonry …

179

Fig. 15.3 Response spectrum curve for Bhuj (Zone V)

Table 15.4 Linear static analysis parameters for the building at DBE and MCE hazard levels as per FEMA-356 (2000) (URM building without bond beams) Case

Maximum seismic weight (W) shared by wall (kN)

DBE (kN)

MCE (kN)

Case I Case II

2717.59

2747.01

3797.20

3032.36

3065.19

4237.02

Case III

3327.16

3363.19

4648.93

Table 15.5 Linear static analysis parameters for the building at DBE and MCE hazard levels as per FEMA-356 (2000) (URM with bond beams) Case

Maximum seismic weight (W) shared by wall (kN)

DBE (kN)

MCE (kN)

Case I Case II

2755.08

2784.91

3849.59

3144.85

3178.90

Case III

4394.20

3469.46

3507.03

4874.77

spectrum graphs of Bhuj for design basis earthquake (DBE) and maximum considered earthquake (MCE) hazard levels are shown in Fig. 15.3. Tables 15.4 and 15.5 show the values of base shear for different cases.

15.4 Nonlinear Static (Pushover) Analysis The nonlinear static pushover analysis of the building considered is carried out using SAP 2000 [10]. In the present study, for DBE and MCE hazard levels, immediate occupancy and collapse prevention performance levels are considered. For immediate occupancy (IO) and collapse prevention (CP) performance levels, the drift percentage limits for unreinforced masonry building (non-infill) are 0.3% and 1%, respectively (FEMA-356, 2000), while for reinforced masonry buildings, the drift limits are 0.2% and 1.5%, respectively. Table 15.6 shows the allowable displacement values for URM and reinforced masonry buildings considered for analysis.

180

A. Sharma et al.

Table 15.6 Limiting displacement values for DBE and MCE hazard levels Building

Displacement limit (mm) to achieve the predefined performance level DBE (IO performance level)

MCE (CP performance level)

URM (non-infill)

24

80

Reinforced masonry

16

120

Hereinafter, modeling of masonry building is discussed in Sect. 15.4.1. Nonlinear static analysis of URM building without bond beams and with bond beams are presented in Sects. 15.4.2 and 15.4.3, respectively.

15.4.1 Modeling of Masonry Building A homogeneous modeling approach is applied assuming masonry units, mortar elements to be smeared and considered isotropic. In the homogeneous modeling approach, the test results and analytical curve suggested in research papers [11, 12] are adopted. For the pushover analysis, a three-dimensional model of building is made with 0.5 m mesh size. The stress values are taken up to 0.25 f m [13]. The strain values are taken up to 0.003 levels. Table 15.1 shows the property of masonry taken in analysis. As per IS 1905:1987, the tensile stress is taken up to 68 kN/m2 and the thickness of wall taken as 250 mm. The internal wall thickness is 125 mm. The live load considered on floor is 120 kN and no live load is considered on roof. The considered mesh size models without bond beams (or bands) and with bond beams are shown in Figs. 15.4 and 15.5, respectively. The seismic force is calculated as per IS 1893:2002 and IS 4326:1993. The nonlinear static analysis (pushover analysis) is performed for URM building by modeling masonry wall as shown in Fig. 15.4 with RCC slab acting as rigid diaphragms. For URM building, the masonry walls and RCC slabs are modeled using shell element in SAP 2000. In URM with bond beams as shown in Fig. 15.5, bond beams are modeled as the line elements. The nonlinearity parameters for stress–strain curve used in bond beams and RCC slabs are adopted from Mander model.

15.4.2 Nonlinear Static Analysis of URM Building Without Bond Beams The nonlinear static analysis (pushover analysis) is performed for URM building by modeling masonry wall as shown in Fig. 15.4 with RCC slab acting as rigid diaphragms. The results for three different cases considered are given below.

15 Seismic Evaluation of Two-Storied Unreinforced Masonry …

Fig. 15.4 URM building model without bond beams

Fig. 15.5 URM building with bond beams model

181

182

A. Sharma et al.

Case I URM with three shear walls The pushover analysis results are presented in Table 15.7 and pushover curve of building is plotted in Fig. 15.6. Base shear value of building at yield is 731.13 kN. The target displacement for different hazard levels and corresponding target base shears is calculated. It is observed that for DBE and MCE hazard levels, the target displacements (demand) calculated as per FEMA-356 (2000) for the building are 10 mm and 61.84 mm, respectively. The corresponding target base shear at DBE is 651.675 kN. The target base shear corresponding to MCE is not obtained. For DBE hazard level, 80% of base shear at yield is 584.904 kN. The building will reach the target displacement level for DBE. The acceptable drift (as shown in Table 15.6) corresponding to 0.3% for immediate occupancy performance level (DBE) and 1% for collapse prevention level (MCE) is 24 mm and 80 mm, respectively. Additionally, FEMA-356 (2000) specifies that the base shear at target displacement (651.675 kN) must be greater than 80% of the effective yield strength of the building (584.906 kN). For DBE hazard level, both criteria are within acceptable limits. For MCE, hazard level criteria are not fulfilling, and thus building needs to retrofit for MCE hazard level. Table 15.7 Linear static analysis parameters for the building at DBE and MCE hazard levels as per FEMA-356 (2000) (URM without bond beams) Parameters

Bhuj Zone V case

Bhuj Zone V case II

Bhuj Zone V case II

DBE

MCE

DBE

MCE

DBE

MCE

V y (kN)

731.133

NA

992.60

1006.98

1269.58

1293.82

0.8 V y (kN)

584.904

NA

794.08

805.58

1015.66

1035.06

K i (kN/mm)

185.228

NA

317.34

317.34

370.81

370.81

K e (kN/mm)

194.49

NA

298.07

297.36

357.79

357.79

T i (s)

0.1944

0.1868

0.1695

0.1695

0.1488

0.1488

T e (s) PL

0.1897 IO

0.1915 CP

0.1749 IO

0.1751 CP

0.1514

0.1514

IO

CP

Calculated target displacement C0

0.7

1.15

0.998

1.011

1.1833

1.1582

C1

1.407

2.55

1.422

1.422

1.145

1.447

C2

1.0

1.0

1.0

1.0

1.0

1.0

C3

1.0

2.25

1.0

1.0

1.0

1.0

Sa

0.8085

1.049

0.8085

1.049

0.8085

1.049

S t (mm)

10

61.84

8.65

11

7.834

9.96

1053.04

1064.68

1362.51

1376.05

Base shear at target displacement V t (kN)

651.675

NA

15 Seismic Evaluation of Two-Storied Unreinforced Masonry …

183

Fig. 15.6 Pushover curves for URM building without bond beams (three cases)

Case II URM with four shear walls The pushover analysis results are presented in Table 15.7 and pushover curve of building is shown in Fig. 15.6. Base shear value of building at yield for DBE and MCE are 992.603 kN and 1006.98 kN, respectively. The target displacement (demand) calculated for both the hazard levels as per FEMA-356 (2000) is 8.65 mm and 11 mm, respectively. The corresponding target base shear at DBE and MCE hazard levels are 1053.04 kN and 1064.68 kN, respectively. For DBE hazard level, 80% of base shear at yield is 794.8 kN. For MCE hazard level, 80% of base shear at yield is 805.58 kN. The building will reach the target displacement level for both the hazard levels. The acceptable drift (as shown in Table 15.6) corresponding to target performance levels are calculated. It is observed that for DBE and MCE corresponding to 0.3% for immediate occupancy performance level (DBE) and 1% for collapse prevention level (MCE) are 24 mm and 80 mm, respectively. Additionally, FEMA-356 (2000) specifies that the base shear at target displacement (992.603 kN for DBE and 1006.98 kN for MCE) must be greater than 80% of the effective yield strength of the building (794.8 kN for DBE and 805.58 kN for MCE). Thus, the building meets the acceptance criteria for both the hazard levels. Case III URM with five shear walls The pushover analysis results are presented in Table 15.7 and pushover curve of building is plotted in Fig. 15.6. Base shear value of building at yield for DBE and MCE are 1269.58 kN and 1293.82 kN, respectively. It is observed that for DBE and MCE hazard levels, the target displacements (demand) calculated as per FEMA-356 (2000) for the building are 7.83 mm and 9.96 mm, respectively. The corresponding target base shear at DBE and MCE hazard levels are 1362.51 kN and 1376.05 kN, respectively. For DBE and MCE hazard level, 80% of base shear at yield are 1015.66 kN and 1035.06 kN. The building will reach the target displacement level for both the hazard levels. Additionally, FEMA-356 (2000) specifies that the base shear at target displacement (1362.51 kN for DBE and 1376.05 kN for MCE) must be greater than 80% of the effective yield strength of the building (1015.66 kN for DBE and 1035.06 kN for MCE). Thus, the building meets the acceptance criteria for both the hazard levels.

184

A. Sharma et al.

15.4.3 Nonlinear Static Analysis with Bond Beams The nonlinear static analysis (pushover analysis) is performed for URM building by modeling masonry wall as shown in Fig. 15.5 with RC slab acting as rigid diaphragms. The building is provided with RCC elements designed as per IS 4326-1993. The details are given in Table 15.8. The results for three different cases considered are presented below. Case I URM building with bond beams with three shear walls The pushover analysis results are presented in Table 15.9 and pushover curve of building is plotted in Fig. 15.7. Base shear value of building at yield for DBE and MCE hazard levels are 693.11 kN and 709.90 kN, respectively. The target displacement for different hazard levels and corresponding target base shears is calculated. It is observed that for DBE and MCE hazard levels, the target displacement (demand) calculated as per FEMA-356 (2000) is 12 mm and 15 mm, respectively. The corresponding target base shear at DBE and MCE hazard levels are 746.96 kN and 748.03, respectively. For DBE hazard level, 80% of base shear at yield is 554.49 kN, while for MCE hazard level is 567.9 kN. The acceptable drift (as shown in Table 15.6) corresponding to 0.3% for immediate occupancy performance level (DBE) and 1% for collapse prevention level (MCE) is 24 mm and 80 mm, respectively. Additionally, FEMA-356 (2000) specifies that the base shear at target displacement (746.96 kN for DBE and 748.03 for MCE) must be greater than 80% of the effective yield strength of the building (554.49 kN for DBE and 567.92 kN for MCE). For DBE and MCE hazard levels, both criteria are within acceptable limits. Case II URM with bond beams and four shear walls The pushover analysis results are presented in Table 15.9 and pushover curve of building is plotted in Fig. 15.7. Base shear value of building at yield for DBE and MCE are 1009.72 kN and 1034.15 kN, respectively. The target displacement for different hazard levels and corresponding target base shears is calculated. It is observed that for DBE and MCE hazard levels, the target displacements (demand) calculated as per FEMA-356 (2000) for the building are 8.42 mm and 11 mm, respectively. The corresponding target base shear at DBE and MCE hazard levels are 1074.8 kN and 1074.8 kN, respectively. For DBE hazard level, 80% of base shear at Table 15.8 Reinforcement detailing and member sizes are used in retrofitting Levels

Member

Size

Reinforcement detailing

Diaphragm

Concrete beam (M 25 )

250 mm × 250 mm

12 mm # 2 nos at top and bottom (Fe415 )

Lintel

Concrete beam (M 25 )

250 mm × 250 mm

12 mm # 2 nos at top and bottom (Fe415 )

Plinth

Concrete beam (M 25 )

250 mm × 250 mm

12 mm # 2 nos at top and bottom (Fe415 )

15 Seismic Evaluation of Two-Storied Unreinforced Masonry …

185

Table 15.9 Performance parameters for Bhuj (Zone V) at MCE and DBE hazard levels (URM building with bond beams) Parameters

Bhuj Zone V case I

Bhuj Zone V case II

Bhuj Zone V case II

DBE

MCE

DBE

MCE

DBE

MCE

V y (kN)

693.11

709.90

1009.72

1034.15

1290.94

1316.07

0.8V y (kN)

554.49

567.92

807.78

827.32

1032.75

1052.86

K i (kN/mm)

207.97

207.96

328.84

328.84

383.86

383.86

K e (kN/mm)

197.10

196.7

312.08

311.24

373.06

372.71

T i (s)

0.1934

0.1934

0.1669

0.1669

0.1481

0.1481

T e (s) PL

0.1906 IO

0.1988 CP

0.1713

0.1715

0.1503

0.1503

IO

CP

IO

CP

Calculated target displacement C0

1.0651

1.0593

1.010

1.017

1.1831

1.1576

C1

1.3978

1.3982

1.426

1.426

1.448

1.448

C2

1.0

1.0

1.0

1.0

1.0

1.0

C3

1.0

1.0

1.0

1.0

1.0

1.0

Sa

0.8085

1.049

0.8085

1.049

0.8085

1.049

S t (mm)

12

15

8.42

11

7.72

9.81

1074.88

1074.80

1385.95

1398.06

Base shear at target displacement V t (kN)

746.96

748.03

Fig. 15.7 Pushover curves for URM building with bond beams (three cases)

yield is 807.78 kN. For MCE hazard level, 80% of base shear at yield is 827.32 kN. The building will reach the target displacement level for both the hazard levels. The acceptable drift (as shown in Table 15.6) corresponding to 0.3% for immediate occupancy performance level (DBE) and 1% for collapse prevention level (MCE) is 24 mm and 80 mm, respectively. Additionally, FEMA-356 (2000) specifies that the base shear at target displacement (1074.8 kN for DBE and 1074.80 kN for MCE) must be greater than 80% of the effective yield strength of the building (807.78 kN for DBE and 827.32 kN for MCE). Thus, the building meets the acceptance criteria for both the hazard levels.

186

A. Sharma et al.

Case III URM with bond beams and five shear walls The pushover analysis results are presented in Table 15.9 and pushover curve of building is plotted in Fig. 15.7. Base shear value of building at yield for DBE and MCE are 1290.94 kN and 1316.07 kN, respectively. The target displacement for different hazard levels and corresponding target base shears is calculated. It is observed that for DBE and MCE hazard levels, the target displacements (demand) calculated as per FEMA-356 (2000) for the building are 7.72 mm and 9.81 mm, respectively. The corresponding target base shear at DBE and MCE hazard levels are 1385.95 kN and 1398.06 kN, respectively. For DBE hazard level, 80% of base shear at yield is 1032.75 kN. For MCE hazard level, 80% of base shear at yield is 1052.86 kN. The building will reach the target displacement level for both the hazard levels. The acceptable drift (as shown in Table 15.6) corresponding to 0.3% for immediate occupancy performance level (DBE) and 1% for collapse prevention level (MCE) is 24 mm and 80 mm, respectively. Additionally, FEMA-356 (2000) specifies that the base shear at target displacement (1385.95 kN for DBE and 1398.06 kN for MCE) must be greater than 80% of the effective yield strength of the building (1032.75 kN for DBE and 1052.86 kN for MCE). Thus, the building meets the acceptance criteria for both the hazard levels.

15.5 Results and Conclusions From the present study, it can be interpreted that there is uncertainty in calculation of base shear in linear approach while calculating it from two approaches, viz., IS 1893:2002 and FEMA-356 (2000). For two-storied unreinforced masonry building with rigid diaphragm without bond beams on comparing capacities obtained from pushover curve with demands calculated using IS 1893:2002, it is observed that in case I (building with three shear walls), building is unsafe at both the hazard levels, while in case II (building with four shear walls) and case III (building with five shear walls), building is unsafe at MCE hazard level. For two-storied unreinforced masonry building with rigid diaphragm without bond beams on comparing capacities obtained from pushover curve with nonlinear demands calculated using FEMA-356 (2000), the building is unsafe for case I at MCE hazard level. Thus, the building fails in achieving performance point as target displacement calculated as per FEMA-356 (2000) (61.84 mm) is beyond the capacity curve of building. For case II and case III, building meets the acceptance criteria for both the hazard levels. For two-storied unreinforced masonry building with rigid diaphragm with bond beams on comparing capacities obtained from pushover curve with demands calculated using IS 1893:2002, it is observed that the building is unsafe at MCE level for case I and case II. For remaining cases, the building is safe. For two-storied unreinforced masonry building with rigid diaphragm with bond beams on comparing capacities obtained from pushover curve with nonlinear

15 Seismic Evaluation of Two-Storied Unreinforced Masonry …

187

demands calculated using FEMA-356 (2000), the building is safe for all the three cases at both the hazard levels. Also, the building has achieved both the performance criteria. From the present study, it is observed that unreinforced masonry building with bond beams showed a better response in nonlinear stage. Also, it is observed that sufficient shear walls should be provided in unreinforced masonry buildings to resist lateral forces satisfying the demands required. It is proposed to incorporate certain provisions from FEMA-356 (2000) in IS 1893:2002 along with inclusion of performance-based design of buildings built in India.

References 1. Golubev, V.I., Muratov, M.V., Petrov, I.B.: Different approaches for solving inverse seismic problems in fractured media. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 199–212. Springer, Singapore (2020) 2. Nikitin, I.S., Burago, N.G., Golubev, V.I., Nikitin, A.D.: Methods for calculating the dynamics of layered and block media with nonlinear contact conditions. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds.) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 171–183. Springer, Singapore (2020) 3. Bagaev, R.A., Golubev, V.I., Golubeva, YuA: Full-wave 3D earthquake simulation using the double-couple model and the grid-characteristic method. Comput. Res. Mod. 11(6), 1061–1067 (2019) 4. FEMA-356 Prestandard and commentary for the seismic rehabilitation of buildings. Washington DC (2000) 5. Agarwal, P., Shrikhande, M.: Earthquake resistant design of structures. PHI Learning Private Limited, New Delhi (2009) 6. IS-1893 Indian standard criteria for earthquake resistant design of structures, Fifth Revision Part-1, Bureau of Indian Standards, New Delhi (2016) 7. IS-4326 Code of Practice for Earthquake Resistant Design and Construction of Building, Second Revision (1993) 8. USGS Homepage. https://www.usgs.gov/products/maps/topo-map. Last accessed 21 May 2020 9. Binnani, N., Khare, R.K., Golubev, V.I., Petrov, I.B.: Probabilistic seismic hazard analysis of Punasa dam site in India. In: Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C. (eds.) Smart Modeling for Engineering Systems: Proceedings of the Conference 50 Years of the Development of Grid-Characteristic Method: Proceedings of Smart Modeling for Engineering Systems. GCM50 2018. SIST, vol. 133, pp. 105–119. Springer International Publishing AG, Cham, Switzerland (2019) 10. SAP2000 Integrated Software for Structural Analysis and Design. Version 14.0. Computers & Structures, Inc., Berkeley, California (2014) 11. Kaushik, H.B., Rai, D.C., Jain, S.K.: Uniaxial compressive stress-strain model for clay brick masonry. Curr. Sci. 92(4), 497–501 (2007) 12. Kaushik, H.B., Rai, D.C., Jain, S.K.: Stress-Strain characteristics of clay brickmasonry under uniaxial compression. J. Mater Civil Eng 19(9), 728–739 (2007) 13. IS-1905 Indian standard criteria for structural use of unreinforced masonry. Third Revision. Bureau of Indian Standards, New Delhi (1987)

Chapter 16

Numerical Study of Thin Composite Structures Vibrations for Material Parameters Identification Vitalii V. Aksenov

and Katerina A. Beklemysheva

Abstract The chapter considers the numerical modeling of the vibration patterns of several thin composite samples. The work utilizes the finite element model of thin composite structures, as well as the model and numerical method support complex geometries along with arbitrary anisotropy of the material. The chapter presents the results of modeling the vibration spectra of solid and perforated thin composite structures. The role of damping in the experimental installation is also discussed. The possibility to substitute densely perforated parts with solid parts with effective compliance tensor with the same frequency properties is considered. The solution of this direct problem targets the transition to the inverse problem of restoring the properties of the sample material from the results of its vibration tests. Keywords Numerical modeling · Thin membrane · Composite materials · Vibrations · Finite element method

16.1 Introduction This work considers the problem of material properties identification for a carbon– carbon composite material based on its non-destructive test results. Composite materials are used in various fields of industry, and over the years of their operation, standards and regulations have been developed that can be reliably used in the construction of buildings, sea and river vessels, and cars. The destruction of composites is an open topic in modern science. Many teams around the world study it by experimental and numerical methods [1]. One of the serious problems is modeling the destruction of the composite under the influence V. V. Aksenov (B) · K. A. Beklemysheva Moscow Institute of Physics and Technology, National Research University, 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] K. A. Beklemysheva 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 214, https://doi.org/10.1007/978-981-33-4709-0_16

189

190

V. V. Aksenov and K. A. Beklemysheva

of weak impacts that do not cause visible damage to the composite, but significantly reduce its residual strength. Such impacts can happen as single events or can occur multiple times during vibrations of composite parts. A comprehensive review of shock loading on composite materials is given in [2]. The vibrational load on products made of carbon–carbon composite material is considered in [3, 4]. In these works, the effect of random vibration on µ20 engine components with 10 g loading in all directions was studied. The increasingly active use of composite materials in critical elements of hightech products leads to the fact that new materials are constantly being created. These materials have improved characteristics, but their experimental status means unsteady production technology and the lack of reliable data from the manufacturer about the properties. The matrix of the carbon–carbon composite material, which is a mixture of coke residue and pyrocarbon, has a large variation in the elastic and strength parameters of the material. There is also a risk of pores, latent defects, and small cracks. The uneven stacking of the reinforcing carbon filaments further increases this dispersion of properties. Some studies of these problems are described in [5, 6]. In addition, the transient technology for the production of experimental material can lead to the fact that the properties of different samples vary significantly. For the practical application of such a material, it is necessary to determine the actual properties of the sample using a small number of non-destructive tests. Also, during the production processes, composite material may change its elastic properties that are important for the resulting detail. Thus, it is desirable to have the technique to obtain the elastic parameters of the material of the final details having complex geometry. In this chapter, a direct problem is considered—the numerical modeling of the vibration patterns of a thin composite sample. The solution of this problem is the starting point for the transition to the inverse problem of restoring the properties of the sample material from the results of its vibration tests. To simulate dynamics, collisions, and non-destructive testing of composite materials, various numerical methods can be used. Many problems in this area are successfully studied using the grid-characteristic method [7, 8]. However, for modeling vibration tests of a thin structure, it seems more efficient to use a model of a thin anisotropic membrane and a finite element method [9]. The chapter is organized as follows. In Sect. 16.2, the mathematical model and numerical method, which are used in this work to simulate thin composite structure vibrations, are discussed. Section 16.3 presents the numerical results obtained for perforated and non-perforated structures and discusses them. Section 16.4 concludes the chapter.

16 Numerical Study of Thin Composite Structures Vibrations …

191

16.2 Mathematical Model and Numerical Method Thin composite structure is described as 2D anisotropic membrane that moves in 3D space [9]. This approach allows to reduce the number of required computations significantly still having arbitrary geometry, load, and material anisotropy. Compared with the original model, this work adds damping, since it is an inevitable part of experimental setup during vibration tests of the samples. The system of equations of motion takes the form: M a¨ + C a˙ + K a = f (t). Here, following [9], a is the displacement vector, M is the mass matrix, K is the stiffness matrix, and f is the load vector. The new component here is the matrix C, which describes the damping, i.e., energy dissipation, occurring in the sample. The choice of the correct damping model for the vibration testing problem is an open question itself. In this work, we consider a damping force of a viscous type, which results in a damping matrix C =γM with γ being an unknown damping factor. For the problem of vibration testing, the load vector f (t) is taken in the form f eiωt with ω being the cyclic frequency of the external load. Here, f is the amplitude vector, which has nonzero components only for those vertices where the load is applied. The solution of the system then has the form: a(t) = a p eiωt +

3N nodes 

Ci ai eλi t

i=1

In the equation above, λi and ai are the eigenvalues and eigenvectors of the equation without the right part, and the coefficients C i are chosen to satisfy the initial conditions. It can be easily shown that due to introduction of the damping, the real parts of λi will be negative, and thus, the magnitude of these components will decrease exponentially with time. Thus, these components can be neglected to describe the behavior of the sample after a sufficiently long time. The partial solution ap is found from the linear system:   2 −ω M + iωC + K a p = f. The solution is in general complex, with |[a]i | denoting the amplitude of the vibrations, and arg[a]i the phase shift relative to the external load.

192

V. V. Aksenov and K. A. Beklemysheva

16.3 Numerical Results The calculations were performed for a series of problem statements. The baseline problem statement was the perforated structure. The geometry of this statement is presented in Fig. 16.1. One can see that in this statement, all holes are taken into account explicitly. An unstructured mesh of triangles is used that provides a reasonable spatial resolution of the geometry. The length of the specimen is 16 cm, and the width is 4 cm. Baseline material parameters are: ρ = 900 kg/m3 , E = 100 GPa, and ν = 0.45. The border of the large hole in the left part of the sample is fixed, and the load is applied to the area in the right part of the sample that is symmetrical to this large hole. Figures 16.2, 16.3, 16.4 and 16.5 shows examples of the vibration modes of this perforated structure at different frequencies. Color shows the displacement at a given point when oscillating at a given frequency. These data can be used for direct comparison with an experiment, in which plate motion is directly measured at a specific point using a high-precision accelerometer or rangefinder.

Fig. 16.1 Geometry of the computation domain for the baseline problem statement

Fig. 16.2 Vibration mode for 500 Hz

Fig. 16.3 Vibration mode for 1000 Hz

16 Numerical Study of Thin Composite Structures Vibrations …

193

Fig. 16.4 Vibration mode for 1500 Hz

Fig. 16.5 Vibration mode for 2000 Hz

It worth noting that vibration modes are significantly different for the perforated and non-perforated structures. Figures 16.6 and 16.7 demonstrate this difference— the mode for 1600 Hz for perforated and non-perforated samples from the same material. One can see that patterns of vibration differs completely, and the perforated structure has more complex modes with more local maxima.

Fig. 16.6 Vibration mode for 1600 Hz, perforated sample

Fig. 16.7 Vibration mode for 1600 Hz, non-perforated sample

194

V. V. Aksenov and K. A. Beklemysheva

Fig. 16.8 Amplitude–frequency characteristics of all structures

Figure 16.8 shows the amplitude–frequency characteristics of the structure, which can also be directly compared with the experimental data. Calculations of the amplitude–frequency characteristics are performed for three statements. In the first statement, the geometry of the entire perforated structure is completely specified. On the graphs, the data of these calculations are indicated as “perf.” In the second statement, perforation is not specified; oscillations of a continuous plate of the same material are considered. On the graphs, the data of these calculations are indicated as “no_perf_same.” In the third statement, perforation is not specified, and the density of the material is adjusted, so that it coincides with the effective density of the real perforated sample (rho = 706.78 kg/m3 ). On the graphs, the data of these calculations are indicated as “no_perf_smaller.” Figure 16.8 shows that the high-frequency part of the spectrum is very difficult to analyze—the structure of the spectrum in this region is quite complex, and the spectra of different samples overlap. Figure 16.9 shows separately the low-frequency part of the spectrum. One can see a significant difference in the spectra of different samples, which can be used to determine the properties of the material. It should be separately noted that the damping parameter introduced in the model, of course, significantly affects the exact form of the obtained spectra. Figure 16.10 shows the frequency response of the same sample, performed using the damping parameter 1, 100, and 1000 m3 /s. The damping parameter is not known a priori for real materials. Apparently, it should be determined together with the major material parameters by comparing the numerical and experimental data. Figures 16.11 and 16.12 show the phase-frequency characteristics for different samples. Figure 16.11 shows the dependence of the phase-frequency characteristic on the damping parameter, the same values of 1, 100, and 1000 m3 /s were used

16 Numerical Study of Thin Composite Structures Vibrations …

195

Fig. 16.9 Low-frequency part of the spectra

Fig. 16.10 Spectra for the perforated structure calculated using different damping parameters

here. Figure 16.12 shows the phase-frequency characteristic for different samples— perforated structure, solid structure of the same material, and solid structure with the effective density.

16.4 Conclusions The results presented in this chapter are considered by the authors as the starting point for the transition to the inverse problem of restoring the properties of the sample material from the results of its vibration tests. The model and the method

196

V. V. Aksenov and K. A. Beklemysheva

Fig. 16.11 Phase-frequency characteristic calculated using different damping parameters

Fig. 16.12 Phase-frequency characteristic for different sample

allow to calculate amplitude–frequency characteristics for geometrically complex structures using arbitrary anisotropy of the material. The vibration modes obtained for solid and perforated structures can be compared directly with the plate motion from the experimental data. The amplitude–frequency characteristics can also be directly compared with the experimental data.

16 Numerical Study of Thin Composite Structures Vibrations …

197

Calculation of model parameters is likely to require special attention. As stated above, the damping parameter should be determined together with the major material parameters by comparing the numerical and experimental data. Moreover, the damping may depend not only on the material, but on experimental setup also. However, the damping is described using single scalar value, and it makes the authors think that the problem of determining this parameter can be solved. The results also demonstrate clearly that substitution of densely perforated parts with solid parts with effective compliance tensor is not a trivial problem. It is still desirable, since this procedure will allow for usage of coarser computational grids, thus speeding up the practical computations. However, effective compliance tensor becomes in this case one more complex parameter of the model that should be determined. Acknowledgements The work was supported by RSF project 19-71-00147.

References 1. Hinton, M.J., Kaddour, A.S.: Maturity of 3D failure criteria for fibre-reinforced composites: comparison between theories and experiments: part B of WWFE-II. J. Compos. Mater. 7, 925– 966 (2013) 2. Abrate, S.: Impact on laminated composites: recent advances. Appl. Mech. Rev. 47(11), 517–544 (1994) 3. Nishiyama, K., Shimizu, Y., Kuninaka, H., Miyamoto, T., Fukuda, M., Nakai, T.: Research and development status of microwave discharge ion thruster µ20. In: IEPC-2005-055. 29th International Electric Propulsion Conference, pp. 1–9. Princeton University (2005) 4. Hosoda, S., Nishiyama, K., Toyoda, Y., Kuninaka, H.: Intermediate report of MU-20 microwave discharge ion thruster development. In: IEPC-2009-155, 31st International Electric Propulsion Conference, pp. 1–7. University of Michigan, Ann Arbor, Michigan, USA (2009) 5. Madeev, S.V., Lovtsov, A.S., Laptev, I.N., Sitnikov, N.N.: Determination of the operational characteristics of structural materials from a carbon-carbon composite for electrodes of ion engines. Compos. Nanostruct. 8(2), 141–150 (in Russian) (2016) 6. Madeev, S.V., Selivanov, M.Y., Shagayda, A.A., Lovtsov, A.S.: Experimental study of ion thruster ID-200 with non-circular apertures. In: 7th Russian-German Conference on Electric Propulsion and Their Application “Electric Propulsion—New Challenges”, p. 3. Castle of Rauischholzhausen, Germany (2018) 7. Beklemysheva, K.A., Vasyukov, A.V., Kazakov, A.O., Petrov, I.B.: Grid-characteristic numerical method for low-velocity impact testing of fiber-metal laminates. Lobachevskii. J. Math. 39(7), 874–883 (2018) 8. Beklemysheva, K.A., Vasyukov, A.V., Kazakov, A.O., Ermakov, A.S.: Numerical modelling of composite delamination and non-destructive testing. In: Favorskaya, A.V., Petrov, I.B. (eds.) Innovations in Wave Processes Modelling and Decision Making: Grid-Characteristic Method and Applications, SIST, pp. 161–185. Springer, Berlin (2018) 9. Aksenov, V., Vasyukov, A., Petrov, I.: Numerical modelling of thin anisotropic membrane under dynamic load. Aeronaut. J. Advance online publication (2020). https://doi.org/10.1017/aer.202 0.61

Chapter 17

The Study of the Physical Processes that Cause the Destruction and Fragmentation of Meteoroids in the Atmosphere Nina G. Syzranova

and Viktor A. Andrushchenko

Abstract We study the interaction of celestial bodies with the atmosphere of the Earth using the expanded physical theory of meteors. We describe the motion and destruction of three of some of the biggest meteoroids that entered the atmosphere above Russia. Previously, the phenomena accompanying such entries have not been fully understood. We investigate the effects of aerodynamic loading and heat fluxes on those bodies leading to an intense loss of mass and the destruction of meteoroids. Our studies involve various ranges of kinematic and physical parameters of the meteoroids. Keywords Meteoroid · Strength · Destruction · Heat flux · Fragmentation

17.1 Introduction Every year, dozens of space bodies of meter and decameter size invade the Earth’s atmosphere with very high hypersonic speeds ranging from 11 to 72 km/s [1]. Processes occurring during the motion of large meteoric bodies in the atmosphere at a hypersonic velocity are associated with extremely high temperatures arising in the shock layer near the body being flown around, which leads to intense evaporation of the meteorite material and to other ablation processes (peeling, melting, and blowing away the melt film from the surface). In addition, the pressure in the shock layer at a certain altitude of the flight may become comparable with the strength of the meteorite material, and the body fragmentation, often being multiple, occurs.

N. G. Syzranova (B) · V. A. Andrushchenko Institute of Computer Aided Design of the RAS, 19/18, Vtoraya Brestskaya ul., Moscow 123056, Russian Federation e-mail: [email protected] V. A. Andrushchenko 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 214, https://doi.org/10.1007/978-981-33-4709-0_17

199

200

N. G. Syzranova and V. A. Andrushchenko

The study of the movement and destruction of each meteoroid is a purely independent task, since it is impossible to build a strict theory of the movement and destruction of bodies of arbitrary shape and structure like the aeromechanics of aircraft for various purposes of a certain configuration. Therefore, it is advisable in this situation to use a fairly simple, but nevertheless multifunctional mathematical model, the theory of meteor physics, which expands it for each individual object taking into account additional determining factors. The purpose of this work is to study both general regularities in the fall and interaction with the atmosphere of space bodies of different scales and to identify previously unexplored features for each of these bodies, which are manifested due to differences in their kinematic parameters and physical properties of the material. Here, we analyze the movement and destruction in the atmosphere of three specific meteor bodies: the Tungusky bolide (1908), the Vitimsky bolide (2002), and the Chelyabinsk bolide (2013) [2–4]. The variety of unique events associated with the flight of these meteoroids indicates the importance of a comprehensive study of the problem of comet-asteroid danger for each of them and effective monitoring of nearearth space for the purpose of early detection of such celestial bodies from the meter and decameter range. Section 17.2 discusses the problem of determining dynamics of a meteoroid prior to fragmentation of meteoroids, Sect. 17.3 presents a model of meteor body fragmentation, and Sect. 17.4 analyzes the movement and destruction of three specific meteor bodies in the atmosphere. Conclusions are given in Sect. 17.5.

17.2 Dynamics of a Meteoroid Prior to Fragmentation We consider a model describing the meteorite fall in the Earth’s atmosphere. One of the important aspects of this model is to determine a law of motion of the meteorite center-of-mass. Another important aspect is to study the parameters of a flow around a body with allowance for the effects of ablation, heat transfer, radiation, and mechanical destruction. Variations in the meteorite velocity V , mass M, and the angle of inclination of the velocity vector to the horizon θ are described by Eqs. 17.1–17.4 known from the physical theory of meteorites [5]. M

MV

ρV 2 dV = Mg sin θ − CD S dt 2

(17.1)

dθ M V 2 cos θ ρV 2 = Mg cos θ − − CN S dt RE + z 2

(17.2)

dM ρV 3 = −CH S dt 2

(17.3)

Heff

17 The Study of the Physical Processes that Cause the Destruction …

dz = −V sin θ dt

201

(17.4)

Here, CD , CN , and CH are the coefficients of drag, lift, and heat transfer to the body surface, respectively, S is the area of the body midsection, RE is the Earth’s radius, Heff is the effective enthalpy of evaporation of the meteorite material, and z is the altitude of the meteoric body above the Earth’s surface. The change in the air density ρ with altitude z is given by Eq. 17.5, where ρ0 is the atmospheric density with z = 0 and h is the characteristic scale of altitude. ρ = ρ0 exp(−z/ h)

(17.5)

The area of midsection S in the general case is a variable quantity since the mass of the meteoric body changes with altitude provided by Eq. 17.6. Se = S



Me M

μ (17.6)

The index e in Eq. 17.6 corresponds to the parameters of atmospheric entry of the body. The parameter μ characterizes the influence of variation in the body shape  due to its ablation. With μ = 2 3, ablation occurs uniformly over the entire surface, and the shape factor of the body is maintained. A necessary condition for this is the fast and chaotic rotation of the meteoric body which ensures homogeneous ablation from the entire surface. In the other limiting case, when a meteoroid moves without rotation, the maximum heat flow is observed in the vicinity of the stagnation point. This case is equivalent to the assumption about the constancy of the midsection, i.e., S = const, μ = 0. One of the most important characteristics in meteor physics is the loss of kinetic energy per length depending on the flight altitude. As a rule, the segment of the trajectory at which a rapid loss of kinetic energy occurs with its transition to the kinetic and internal energy of the surrounding gas is called in meteor physics the “explosion” of a meteoric body in flight. Taking into account Eqs. 17.1 and 17.3, we can determine the energy released in the atmosphere assuming it to be equal to the loss of the kinetic energy E of the body:   SρV 2 V2 dE . = CD + CH dz 2 sin θ 2Heff

(17.7)

Two summands in brackets in the resulting expression are responsible for the kinetic energy release due to deceleration and ablation. The colossal speed of meteor bodies leads to the appearance of pressures and temperatures around them of the order of tens of thousands of atmospheres and tens of thousands of degrees. Under these conditions, there are two mechanisms of heat

202

N. G. Syzranova and V. A. Andrushchenko

transfer from the gas to the surface of the body: convective heat transfer and heat transfer by radiation. The main kind of heat transfer during the hypersonic motion of a meteoric body is a radiative heat transfer. At present time, there are a number of numerical solutions of the flow problem in the context of radiative gas dynamics. In applied problems, as a rule, empirical formulas are used for estimating the radiative flux to the body surface. For the heat-transfer coefficient at the stagnation point, Eq. 17.8 is offered in [6]. CH = f · e A1 ρ A2 +A3 V −1 R A4 +A5 V +A6 V V A7 +A8 V +A9 V 2

2

−3

(17.8)

Here, Ai are the numerical coefficients obtained experimentally. Equation 17.8 is valid for a certain density, velocity of the free stream, and parameters of the body being flown around.

17.3 Fragmentation of Meteoroid Statistics of falls of meteoroids shows that most of them fell onto the Earth as fragmented pieces. Therefore, calculation of the meteoric-body ablation requires us to take into account its fragmentation. In this study, a process of meteoroid fragmentation is considered in the framework of a model of sequential fracturing, taking into account the influence of the scale factor on the ultimate strength of the object. We used a model of sequential crushing of a body based on the statistical theory of strength [7], when fragmentation occurred due to defects and cracks that are inherent in such structurally inhomogeneous bodies as meteoroids. As a result, the fragmentation proceeds as the elimination of defects under increasing load by the destruction of a body along these defects, so that the resulting fragments are stronger than the original body. In this regard, a fragmentation process was completed when the speed pressure began to decrease. This model of fragmentation is presented in detail in [8, 9]. In this case, the problem was solved at three stages. At the first stage, the motion of the single body is considered from the altitude of atmospheric entry to the altitude of the start of fragmentation. At the second stage, the motion of the swarm of fragments from the altitude of the beginning of fragmentation to the altitude of the maximum velocity head is considered. It is assumed that the fragments have the same size. At the third stage, the motion of a single fragment is traced. According to the model under consideration, the fragment strength is written using Eq. 17.9, where σe and Me are the ultimate strength and mass of the meteoroid at the extra-atmospheric leg of the trajectory, respectively, σf∗ and Mf are the strength characteristic and the mass of the fragment, respectively.   α σf∗ = σe Me Mf

(17.9)

17 The Study of the Physical Processes that Cause the Destruction …

203

The parameter α is an indicator of the degree of heterogeneity of the material (for large, the heterogeneity is higher). The value of α is variable for different meteoroids and conditions. However, values between 0.1 and 0.5 are usually used for stony asteroids [10]. The condition for the beginning of destruction will be written in the form: ρ∗ V∗2 = σ ∗ ,

(17.10)

where σ ∗ is one of the strength characteristics of meteoroid. The height of the beginning of fragmentation z ∗ in an exponential atmosphere was determined under the assumption that by this moment the body had not time to slow down, and its speed was equal to the initial velocity of entry into the atmosphere Ve :    z ∗ = h ln ρVe2 σ ∗ .

(17.11)

Beginning from this altitude, a swarm of splitting fragments will move with their number growing, i.e., in the model under consideration, the number of meteorite fragments N at each instant is added to the set of characteristic variables.    It is assumed that the created pieces have the same spherical shape and mass Mf Mf = M N . From Eqs. 17.9–17.11, their number N was obtained depending on the current values of the velocity head and the total mass of all fragments: N=

 1/α M ρV 2 . M∗ σ ∗

A variable area of the midsection was defined by the formula (it was suggested that the pieces created do not overlap): S = S∗

 1/3α M ρV 2 . M∗ ρ∗ V∗2

17.4 The Results of the Calculation Hereinafter, Sects. 17.4.1–17.4.3 analyze the movement and destruction of the Tunguska bolide (1908), the Vitim meteoroid (2002), and the Chelyabinsk meteorite (2013), respectively.

204

N. G. Syzranova and V. A. Andrushchenko

17.4.1 The Tunguska Bolide Despite the more than a century-old period of the Tunguska event, the nature of this phenomenon is still not definitively established. Some researchers, such as the authors [11], believe that this phenomenon was due to the interaction of the comet’s core with the Earth’s atmosphere, while others hold the opinion that the Tunguska celestial body was not a comet, but a large bolide that entered the Earth’s atmosphere, did not fall on its surface, but flew back into space [12, 13]. For many decades, the Tunguska meteorite has been the subject of research, as a result of which many hypotheses have been expressed. However, a generally accepted explanation for this phenomenon has not yet been found. Several significant fragments of this celestial body were neither found, nor were there a crater formed by the fall of a large body or any of its fragments. In this chapter, we considered the options for the fall of the Tunguska body, which consisted of various substances that corresponded to the comet, stone, and iron composition. It was assumed that the body had a spherical shape, its initial mass was Me = 1 × 106 t, and the angle of entry into the atmosphere was equal to θ = 30◦ . Table 17.1 shows the body density ρb , effective enthalpy of destruction H eff , and strength characteristics σ * for three types of substances. Figure 17.1 shows the calculated values of the mass entrainment of the Tunguska body depending on the altitude of flight for three types of this body, consisting of comet, stone, and iron substances. These calculations were performed within the framework of a single body model (without taking into account fragmentation). From Fig. 17.1, it can be seen that the “comet” body was completely burned up in the atmosphere, while about a quarter of the original mass remained of the stone body and half of the iron body. This fact is explained by the relatively low enthalpy of destruction of matter in the comet body, as well as the difference in the density of the bodies under consideration (the initial mass of all the bodies is the same, and the size and, consequently, the radiation flows were different). Figure 17.2 demonstrates a mass entrainment depending on the altitude of flight for three types of substance of crushed meteoroids. It can be seen that the mass entrainment values for this model had higher values compared to the corresponding data in Fig. 17.1 due to an increase in the evaporating surface in the flow. When taking into account the fragmentation process, the index of the degree of heterogeneity of the material was assumed to be 0.25. The calculations showed different values of the heights of the beginning and end of fragmentation and the number of fragments formed for the considered types of meteoroids. Table 17.1 Parameters of meteoroids Type of meteoroid

ρ b , kg/m3

H eff , J/kg

σ *, N/m2

Comet

1000

2.5 × 106

105

Stone

3300

8.0 ×

106

107

Iron

7800

8.0 × 106

108

17 The Study of the Physical Processes that Cause the Destruction …

205

Fig. 17.1 Changes of the mass of the Tunguska body without taking into account its fragmentation in dependence on the altitude of flight for three types of meteor matter: 1—comet; 2—stone; 3—iron

Fig. 17.2 Changes of the mass of the Tunguska body taking into account its fragmentation in dependence on the altitude of flight for three types of meteor matter: 1—comet; 2—stone; 3—iron composition

Thus, the comet body began to collapse at an altitude z = 67 km, the fragmentation process ended at an altitude of ~35 km with the formation of ~1.5 × 107 small fragments that almost instantly evaporated in the atmosphere. The stone body began to collapse at an altitude of 35 km, fragmentation ended at an altitude of 12 km, the

206

N. G. Syzranova and V. A. Andrushchenko

maximum number of fragments was 4.2 × 104 , and the final mass of the fallen fragments was calculated to be equal to one tenth of the original mass of the asteroid. The iron body began to fragment at low altitudes of ~19 km, and this process continued almost to the surface of the Earth (up to 2 km). The number of fragments formed was about 2000, and the final fallen mass was one-third of the original mass of the meteoroid. The obtained results showed that under for the considered initial data and the used calculation models, the comet body would have lost all its mass by the height of 30 km, and the final mass of the fallen fragments of the stone and iron meteoroids would have been 1 × 105 t, and 3 × 105 t, respectively. In the last two cases, this would lead to catastrophic consequences, as well as to the appearance of craters and the fall of many fragments on the Earth’s surface. Figure 17.3 shows the values of kinetic energy loss per unit length for the Tunguska body consisting of different substances. It can be seen that the maximum loss of kinetic energy that is, the so-called explosion by definition [14] for a comet body, was at a height of 35 km, for a stone body—12 km, for an iron body—2 km. In literature on the Tunguska meteorite, for example in [15], it was found that its explosion occurred at an altitude of 7–10 km. In [15], we solved the inverse problem of selecting parameters of a celestial body that gave a satisfactory coincidence of the calculated size of the forest fallout area during an explosion with the real fallout area. As the calculations showed, the flight path of the meteor body strongly depends on the angle of the velocity vector to the horizon θ . At sufficiently small angles of inclination of the trajectory, it might not fall to the Earth at all, but rather pierce the atmosphere and exit it into interplanetary space. Calculated data in Fig. 17.4

Fig. 17.3 Loss of the kinetic energy per length depending on the altitude of the Tunguska body for three types of meteor matter: 1—comet; 2—stone; 3—iron composition

17 The Study of the Physical Processes that Cause the Destruction …

207

Fig. 17.4 Dependence of the flight altitude on the flight time in the case of the Tunguska stone body

demonstrate how the height of the asteroid’s flight changed depending on the flight time for different angles of entry into the atmosphere θe . Data were obtained for the stone body without taking into account fragmentation. From the above-calculated results, it can be seen that when θe > 9◦ the meteorite fell to the Earth, and when θe ≤ 9◦ starting from a certain height, its trajectory became ascending. Similar data were obtained for the iron body. The results obtained made it possible to explain many manifestations of the Tunguska phenomenon in 1908. So, if the Tunguska meteorite intruded into the atmosphere at a low angle to the horizon, then it could be a flyby, which would not exclude its fragmentation with the explosion of some of its fragments in the atmosphere that caused the fall forest. The remaining large enough fragments could fall far from the epicenter of the explosion, or go into outer space if their residual velocity exceeded of the second space speed. This assumption is also confirmed by the estimates of other authors, for example, given in [12].

17.4.2 The Vitim Bolide On September 25, 2002, a bright bolide was observed in the vicinity of Bodaibo, Irkutsk region. The U.S. air force recorded the appearance of a luminous object at an altitude of 62 km. It was escorted to a height of 30 km, at which it exploded. The power of the explosion was estimated by experts at 200 tons of trinitrotoluene. None of the three expedition teams were able to find traces of the fallout of the body or its fragments, as well as craters on the surface along the flight path were

208

N. G. Syzranova and V. A. Andrushchenko

found only areas of broken and fallen trees. Samples of snow cover material along the flight path revealed microparticles presumably associated with cosmogenic material from the meteorite ablation trace [16]. This meteoroid got its name from the name of the river and the nearest village “Vitimsky.” In [3], its kinetic energy, mass, and diameter were estimated using the magnitude of the glow of a bolide. According to the data given in [3], its initial mass was about 50 t, diameter 3.2 m, the velocity of entry into the atmosphere was 20 km/s, and the angle of entry was 30°. Figure 17.5 presents the calculated data on the change in the mass of the Vitim meteorite with and without fragmentation. It should be noted that during the fragmentation of the Vitim bolide, the ablation of its mass initially increased due to an increase in the area of the evaporating surface in the stream, which led to an increase in the braking of the crushing body, i.e., to a decrease in the speed of the meteoroid and its fragments, and, in turn, decreased the heat flux to the surface and slowed down the process of ablation of the crushing body. Naturally, this process was also influenced by the initial parameters of entry into the atmosphere and size of the meteorite. As a result, the final mass taking into account fragmentation exceeded the final mass without taking it into account. As calculations showed, the body began to fragment at an altitude of 39 km, and, taking into account the progressive fragmentation model at an altitude of 30 km, it decomposed into many fragments of the order of 10,000 with a radius of ~7 cm. At the final stage of the movement, the process of destruction of these fragments could continue due to temperature stresses [17]. Thermal stresses do not play a significant

Fig. 17.5 Change of the mass of the Vitim meteorite without taking into account (curve 1) and taking into account fragmentation (curve 2)

17 The Study of the Physical Processes that Cause the Destruction …

209

Fig. 17.6 Loss of the kinetic energy per length depending on the altitude of the Vitim meteoroid

role for large meteoroids; however, if the fragment size reaches a few centimeters, the arising temperature gradients can later destroy this fragment into small pieces or even into large dust particles, which rapidly melts and evaporates in the high-temperature air. Figure 17.6 shows the change in kinetic energy depending on the flight altitude. It can be seen that the maximum value of this characteristic occurred at a flight altitude of 30 km. Thus, apparently, the explosion of the Vitim bolide at a relatively high altitude led to the formation of many small fragments that evaporated in the air.

17.4.3 The Chelyabinsk Bolide The following observational data for the Chelyabinsk meteorite are given in [4]: the body entered the Earth’s atmosphere at an angle of ~18° to the horizon at a speed of ~19.2 km/s; the meteorite size was ~19.8 ± 4.6 m. The final explosion occurred at an altitude of about 23 km. The collected fragments had an ordinary chondritic structure; the effective enthalpy of evaporation of such a material was assumed to be Heff = 8 kJ/g. The body was destroyed in several stages: the destruction began at an altitude of 45 km, and the final explosion occurred at an altitude of about 23 km. The value of the strength parameter corresponding to the height of the onset of crushing in this case was σ ∗ = 106 N/m2 . If we assume that the meteorite was a ball of radius

210

N. G. Syzranova and V. A. Andrushchenko

R = 9.8 m, then taking into account the density characteristic of stone chondrite 3.3 g/cm3 , its mass was about 13,000 tons. These data were used in this work to simulate the flight and destruction of the Chelyabinsk meteorite in the atmosphere. The modeling of the fall of the Chelyabinsk meteorite was considered in detail, for example, in [9, 17]. Changes in the total mass of the meteorite depending on the flight altitude within the framework of the model of a single body and for a crushing meteorite are shown in Fig. 17.7. The data in Fig. 17.7 are presented for different values α of the parameter characterizing the degree of heterogeneity of the material. It can be seen that the total mass of meteorite fragments in a wide range of parameter variations α at the Earth’s surface was about 2000 t, while the mass of the found fragments of the Chelyabinsk meteorite did not exceed one ton. The question arose as to how to explain the insignificant mass of the found fragments, with such a huge initial mass of the Chelyabinsk celestial body. For example, in [18], this was explained by the low initial strength of the meteoroid material, as a result of which, when an explosion at an altitude of about 23 km, the meteoroid broke up mainly into very small fragments. Some of these fragments were further crushed due to ablation and remained in the stratosphere, while others were burned under the influence of heat fluxes in the lower layers of the troposphere, or were scattered as dust particles over very long distances from the trajectory of the fall. Only the largest fragments of centimeter or rarely meter sizes were collected during the search. This question was also discussed in [19], which proposed its own original hypothesis for explaining the small mass of fragments found. The authors [19] believed that the cause of the meteorite explosion should be sought among the gas-detonation mechanisms of the formation of a supersonic shock wave front. They suggested

Fig. 17.7 Change of the total mass of the Chelyabinsk meteorite depending on altitude for different values of parameter α

17 The Study of the Physical Processes that Cause the Destruction …

211

considering the vapor–gas detonation concept of a bolide explosion. If a solid space body entered the dense layers of the atmosphere at high speeds, then a hot boundary layer formed on its surface adiabatically compressed to high pressures. The object overheated much higher than the boiling point of the substance that formed it, and as the bolide decelerated and the compressing pressure decreased, the body mass boiled over in an ultra-short period of time. Converted to a gas–vapor state and still compressed to high pressures, the substance explosively disintegrated, that is, there was a “volumetric steam explosion,” which formed a shock wave with consequences. The shock wave scattered a gas–vapor cloud of matter from an exploding meteorite in the atmosphere over a large area, and, therefore, it was not possible to collect a more or less significant volume of its fragments from the Earth’s surface. Unfortunately, as noted in [19], the theoretical foundations explaining the phenomenon of a steam explosion have not yet been created, and its mathematical models have not been constructed. To this end, it is necessary to develop theoretical ideas about the nature of steam explosions, to create models of processes taking place in real conditions.

17.5 Conclusions Our numerical results explain main effects of the motion and destruction of the Tunguska, the Vitimsky, and the Chelyabinsk meteoroids in the atmosphere. Firstly, our results justify the assumption of a stone or an iron composition of the Tunguska meteoroid. This meteoroid fragmented when entering the atmosphere at a small angle. Some of its shards exploded above the taiga and knocked over the trees. The other shards departed from the center of the explosion and either fell far away or returned to space. This explains the lack of craters and of the material of the meteoroid, which has been looked for by numerous expeditions. Secondly, we establish the explosion of the Vitimsky meteoroid at a finite altitude. Its destruction started at an altitude of 40 km. We describe its decomposition into small fragments at a height of 30 km taking progressive fragmentation into account. These fragments melted and evaporated due to thermal stress before reaching the Earth’s surface. This explains, on the one hand, the lack of traces from the fragments on the surface and, on the other hand, damages to the forest caused by the weak shock waves from the explosions and discovered by several expeditions. Thirdly, we explain the lack of the material of the stone Chelyabinsk meteorite relative to its estimated mass using the assumption of vapor–gas explosion. This assumption implies that the fragments of the meteorite were scattered over a large area.

212

N. G. Syzranova and V. A. Andrushchenko

References 1. Dudorov, A.E., Eretnova, O.V.: Periodicity of meteorite falls. Vestn. Chelyabinsk. Gos. Univ. Fiz. 19(1), 58–67 (in Russian) (2014) 2. Vasilyev, N.V.: The Tunguska meteorite problem today. Planet. Space Sci. 46, 129–150 (1998) 3. Chernogor, L.F.: Oscillations of the geomagnetic field caused by the flight of Vitim bolide on September 24, 2002. Geomag. Aeron. 51(1), 116–130 (2011) 4. Emel’yanenko, V.V., Popova, O.P., Chugai, N.N., Shelyakov, M.A., Pakhomov, Yu.V., Shustov, B.M., Shuvalov, V.V., Biryukov, E.E., Rybnov, Yu.S., Marov, M.Ya., Rykhlova, L.V., Naroenkov, S.A., Kartashova, A.P., Kharlamov, V.A., Trubetskaya, I.A.: Astronomical and physical aspects of the Chelyabinsk event. Solar Syst. Res. 47(4), 240–254 (2013) 5. Levin, B.Yu.: Physical Theory of Meteors and Meteoric Matter in the Solar System. Akademii Nauka, SSSR, Moscow (in Russian) (1956) 6. ReVelle, D.O.: Dynamics and thermodynamics of large meteor entry: a quasi-simple ablation model. Herzberg Institute of Astrophysics, National Research Council, Canada (1976) 7. Weibull, W.: A statistical theory of the strength of materials. Proc. Roy Swedish Inst. Eng. Res. 151, 1–45 (1939) 8. Syzranova, N.G., Andrushchenko, V.A.: Simulation of the motion and destruction of bodies in the Earth’s atmosphere. High Temp. 54(3), 308–315 (2016) 9. Tirskiy, G.A., Khanukaeva, D.Y.: Ballistics of a fragmenting meteor body with allowance made for ablation in the non-isothermal atmosphere. Cosm. Res. 46(2), 120–132 (2008) 10. Svetsov, V.V., Nemtchinov, E.V., Teterev, A.V.: Disintegration of large meteoroids in Earth’s atmosphere: theoretical models. Icarus 116, 131–153 (1995) 11. Whipple, F.J.W.: The great Siberian meteor and the waves, seismic and aerial, which it produced. Q. J. Roy. Meteorol. Soc. 56(236), 287–301 (1930) 12. Murzinov, I.V.: The problem of the century: where the Tunguska meteorite flew. Cosmonautics Rocket Sci. 83(4), 65–72 (2015) 13. Khrennikov, D.E., Titov, A.K., Ershov, A.E., Pariev, V.I., Karpov, S.V.: On the possibility of through passage of asteroid bodies across the Earth’s atmosphere. Mon. Not. R. Astron. Soc. 493(1), 1344–1351 (2020) 14. Grigorian, S.S.: On the movement and destruction of meteorites in the atmospheres of the planets. Kosm. Issled. 17, 875–893 (in Russian) (1979) 15. Korobeinikov, V.P., Shurshalov, L.V., Vlasov, V.I., Semenov, I.V.: Complex modeling of the Tunguska catastrophe. Planet. Space Sci. 46(2/3), 231–244 (1998) 16. Antipin, V.S., Yazev, S.A., Perepelov, A.B., Efremov, S.V., Mitichkin, M.A., Ivanov, A.V., Pavlova, L.A., Karmanov, N., Ushchapovskaya, Z.F.: The 25 September 2002 Vitim meteorite: results of complex research. Geol. Geofiz. 46(10), 1050–1064 (in Russian) (2005) 17. Andrushchenko, V.A., Syzranova, N.G., Shevelev, Yu.D., Goloveshkin, V.A.: Destruction mechanisms of meteoroids and heat transfer to their surfaces. Math. Models Comput. Simul. 8(5), 506–512 (2016) 18. Fortov, V.E., Sultanov, V.G., Shutov, A.V.: Chelyabinsk superbolide explosion in the Earth’s atmosphere: a common phenomenon or unique coincidence? Geochem. Int. 51, 549–567 (2013) 19. Barelko, V., Kuznetsov, M., Bykov, L.: To the question about the role of “steam explosion mechanism in the natural catastrophes”. Explosions of meteorites and volcanic eruptions. Civ. Secur. Technol. 12(2), 60–83 (in Russian) (2015)

Chapter 18

Personalized Geometric Modeling of a Human Knee: Data, Algorithms, Outcomes Alexandra Yurova , Victoria Salamatova , Yuri Vassilevski , Lin Wang , Sergei Goreynov , Oleg Kosukhin , Anatoly Shipilov , and Yusuf Aliev Abstract We address the problem of patient-specific modeling of human knee joints. We describe the algorithm of finding ligaments attachment points based on a segmentation procedure and anatomical landmarks. The obtained coordinates of ligaments’ origin and insertion points are implemented into a simple OpenSim knee joint model. A. Yurova · V. Salamatova (B) · Y. Vassilevski · S. Goreynov · O. Kosukhin · A. Shipilov · Y. Aliev Sechenov University, 8-2, Trubetskaya str., Moscow 119991, Russian Federation e-mail: [email protected] A. Yurova e-mail: [email protected] Y. Vassilevski e-mail: [email protected] S. Goreynov e-mail: [email protected] O. Kosukhin e-mail: [email protected] A. Shipilov e-mail: [email protected] Y. Aliev e-mail: [email protected] V. Salamatova · Y. Vassilevski Moscow Institute of Physics and Technology, 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russian Federation Y. Vassilevski · S. Goreynov Institute of Numerical Mathematics of the RAS, 8 Gubkina str., Moscow 119333, Russian Federation L. Wang SIAT, 1068 Xueyuan Avenue, Shenzhen, China e-mail: [email protected] O. Kosukhin · Y. Aliev Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, 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 214, https://doi.org/10.1007/978-981-33-4709-0_18

213

214

A. Yurova et al.

The model is used to study knee passive flexion. The obtained ligaments forces and strains are in the range of physiological values. Keywords Knee joint · OpenSim · Segmentation · Ligaments

18.1 Introduction One of the most frequently injured joints is the human knee [1]. Personalized mathematical models of the knee joint allow us to estimate ligaments laxity and muscles forces under certain types of movement. Biomechanical simulation of the knee joint in normal and pathological conditions helps to predict surgery outcomes and to assess effectiveness of rehabilitation. One can find knee models of different complexity in the literature: from elementary hinge point models to complex finite element models based on nonlinear tissue mechanics [2, 3]. Discrete element models offer a compromise between model complexity and fast simulation. They are appealing in the patient-specific study how leg geometry and leg movement affect tension of ligaments and cartilage contact load. OpenSim is an extensible and user-friendly open-source software package which assists in personalized biomechanical simulations based on discrete elements [4]. Personalization of OpenSim-based models requires tedious anthropometric scaling of once developed dynamic model of the human knee. We aim to develop a workflow for automatic construction of an OpenSim-based patient-specific dynamic model for the human knee joint. In the present chapter, we address ligament tensions under passive flexions of the knee joint and thus we restrict our model to knee bones and ligaments. It is well-known that bones geometry and ligament attachments points influence greatly on the ligament tensions [5]. That is why patient-specific geometric models of the knee joint are inevitable. We extract geometric features of bones (via segmentation) and positions of ligaments origins and insertions (via geometric analysis) from computed tomography (CT) images of patient’s lower limbs. Although MRI is commonly used in clinical examination of knees due to better visualization of ligaments, tendons, cartilages, and muscles, intensity-based segmentation algorithms are not applicable to sustainable segmentation of MRI images in contrast to CT-images [6]. The rest of the chapter is organized as follows. In Sect. 18.2, we present our approach for automated generation of personalized geometric models of human knee joints. In Sect. 18.3, we address a simple biomechanical model describing knee joint dynamics. In Sect. 18.4, we consider an application of the personalized biomechanical model of a knee joint. Section 18.5 concludes the chapter.

A. Shipilov Moscow Center of Advanced Sports Technologies, 6, Soviet Army ul., Moscow 129272, Russian Federation

18 Personalized Geometric Modeling of a Human Knee …

215

18.2 Personalized Geometric Model of a Human Knee The simplest personalized geometric model of a knee is composed of three connected rigid segments representing femur, patella, coupled tibia and fibula, and five ligaments supporting the knee joint (Figs. 18.1 and 18.3a). The femur and the tibia/fibula are connected at the knee joint. The femur is usually attached to the pelvis at the hip joint where it can rotate within defined degrees of freedom. Each rigid segment is characterized by mass, length, position of its barycenter, and its inertia tensor. The knee joint is characterized by limits for degrees of freedom (three translations and three rotations). Each ligament is defined by its stiffness and length (measured in the upright position) as well as insertion and origin points whose positions are fixed with respect to the host bone. All the above listed geometric data give the basis for the simplest dynamic model of a knee joint to be discussed below. The data are retrieved from CT-images as follows. Section 18.2.1 presents a geometric model of knee bones, while detection of attachment sites of all five knee ligaments is given in Sect. 18.2.2.

18.2.1 Knee Bones Geometric model of each knee bone results from segmentation of CT-image of the patient’s knee joint. The algorithm is thoroughly described in [6]. The main difficulty of bone segmentation stems from the anatomical structure of the bone.

Fig. 18.1 Bones and ligaments of the knee joint: a lateral view, b medial view (Modified Fig. 6.74 (A, B) from [7])

216

A. Yurova et al.

Fig. 18.2 Transversal slice of CT scan and cortical bone segmentation using intensity range [200; 2000] HU for thresholding: a original image, b continuous contour, c discontinuous contour

The outer layer (cortical bone) admits automatic segmentation with the use of known threshold values [8, 9]. However, its thickness may be lower than CT resolution and the outer surface of the bone is not a closed surface, see Fig. 18.2. The intensity range of the inner part (cancellous bone) intersect with intensity ranges of other tissues surrounding the bone [9] and its threshold-based segmentation is useless. Due to discontinuities of the cortical bone surface, the 3D hole-filling algorithm [10] for the cancellous bone segmentation is not applicable. The operation of cortical contour closing requires patient-specific parameters and results in manual segmentation which is time-consuming. An automated algorithm for knee bones segmentation suggested in [6] alleviates the described above obstacles. On input, the algorithm takes the threshold-based initial segmentation of the knee and extracts the four largest components of the cortical bone for femur, tibia, fibula, and patella. For each component, one iterates over all 2D slices in transversal (orthogonal to z-axis), sagittal (orthogonal to x-axis), and coronal (orthogonal to y-axis) planes and examines the background of each slice. If the background contains several connected components, the other than the largest components belong to the cancellous part of the appropriate bone. If the background contains only one connected component, then the contours of the cortical bones are open in this 2D slice and it is not processed. Voxel representation of each bone provides the ground for a simplified geometric model to be used in the simulation. In OpenSim framework, we consider femur, patella, and coupled tibia/fibula as rigid segments connected to each other through joints. Geometric and mechanical characteristics of each segment are recovered from voxel representation of the appropriate bone. The segment barycenter coordinates x¯ are computed by averaging voxel i coordinates xi with weight wi , corresponding to cortical and cancellous bone. Although here we set wi = 1, in future works the weights will adjust according to the bones densities that correlate with intensity. The inertia tensor is defined by: T =

  |wi x i − x¯ |2 I3 − (wi x i − x¯ )(wi x i − x¯ )T . i

18 Personalized Geometric Modeling of a Human Knee …

217

Fig. 18.3 Cruciate ligaments of the knee joint: a Fig. 6.75 from [7], b detection of ACL and PCL attachments to tibia

18.2.2 Knee Ligaments Instead of ligament segmentation, we detect attachment sites of ligaments. This is sufficient for simple personalized biomechanical models of the human knee. The attachment sites (origin and insertion) can be found based on anatomical arrangements and geometric features of bones. Examples of such features are tubercles (eminences on a bone surface), depressions on a bone, epicondyles, and condyles. We developed automated algorithms for the detection of attachment sites of all five knee ligaments: patellar ligament, medial (tibial) collateral ligament (MCL), lateral (fibular) collateral ligament (LCL), anterior and posterior cruciate ligaments (Figs. 18.1 and 18.3a). An algorithm for detection of patellar ligament attachment sites is given in [6]. It exploits automated analysis of bone tuberosity. Here, we present an algorithm for detection of attachment sites for anterior cruciate ligament (ACL) and posterior cruciate ligaments (PCL) which are known to be torn frequently. ACL and PCL connect femur and tibia (Fig. 18.3a). To detect their attachments to the tibia, we take advantage of the anatomy: ACL and PCL attach to the anterior and posterior part of the intercondylar area of the tibia, correspondingly. The intercondylar area is an eminence separating the medial and lateral condyle on the upper extremity of the tibia and is divided by intercondyloid eminence into two parts: anterior and posterior area. We exploit the fact that a plane parallel to z-axis and passing through the mid-edge connecting medial and lateral intercondylar tubercles and tibial tuberosity, passes through the intercondyloid eminence (Fig. 18.3b).

218

A. Yurova et al.

The algorithm for tibia attachments detection is as follows: 1. Compute the curvature of the tibia. 2. Select points with maximum curvature values in the neighborhood of radius r = 5. 3. Select two points with maximal value of z-coordinate (corresponding to the medial and lateral intercondylar tubercles) and find their mean Pm = (x m , ym , zm ). 4. Consider projection of tibial tuberosity T = (x t , yt , zt ) to the plane z = zm and define a vector v = (x t − x m , yt − ym , 0). 5. Compute points I ACL = Pm + v/2 and I PCL = Pm − v/2, corresponding to insertions of ACL and PCL to the tibia. As far as concerns ACL and PCL attachment to femur, we exploit anatomy atlas once again: ACL attaches to the back of the lateral wall of the femur intercondylar fossa, whereas PCL attaches to the medial wall of the femur intercondylar fossa. Both attachments are located on the femoral condyles, and after distinguishing the medial and the lateral condyles, we find their lateral and medial walls as voxels with maximum (minimum) x-coordinate in the condyle. Automated algorithms for detection of origins and insertions of all five knee ligaments will be presented in detail elsewhere.

18.3 Biomechanical Model of a Human Knee We consider the simplest biomechanical model of a knee targeting the estimation of ligaments load under passive knee flexion. The objective of the study allows us to ignore otherwise important issues of musculoskeletal dynamics and cartilage loads. In OpenSim framework, bones are represented as rigid bodies connecting to each other through joints. The generalized coordinates q are joint degrees of freedom (joint angles and positions). According to the second Newton law, the multibody dynamics are described by the following equations [11]: ˙ + G(q) ˙ + F, M(q)q¨ = τ + C(q, q) where M(q) is the mass matrix, q¨ , q˙ are coordinates accelerations and velocities, ˙ is Coriolis and centrifugal forces, G(q) ˙ respectively, τ are joint torques, C(q, q) is gravity force, and F is other forces applied to the model. The result of forward dynamics is generalized coordinates of kinematic joints q(t) and corresponding velocities and accelerations. We consider the patellofemoral joint with one translational degree of freedom [12] and the knee (tibiofemoral) joint as the simple hinge joint with one degree of freedom (knee flexion), so we have only two degrees of freedom. Coordinate axes for joints are the same as in [12] and are shown in Fig. 18.4. Global coordinates of joint frames origins are found during post-processing of bones segmentation results. We ignore the

18 Personalized Geometric Modeling of a Human Knee …

219

Fig. 18.4 Joints frames for the simple knee joint model in OpenSim. The blue surfaces are contact surfaces for detecting tibia-femur contact. The green cylinders are ligaments

Coriolis and centrifugal forces and account contact forces in the tibiofemoral joint and elastic forces due to ligaments tension. The contact and ligament forces are calculated according to [12]. Namely, the ligaments are considered as nonlinear springs with force–strain relationship defined as a quadratic toe region and subsequent linear portion after 6% strain (ε):  Fligament =

ε2 /0.12 ε − 0.03

ε ≤ 0.06 . ε > 0.06

The contact forces are calculated according to a linear elastic foundation model implemented in OpenSim. The contacts between ligaments and bones are assumed to be pointwise, rather than pathwise. Generalization to more realistic contact patches will be done in the future.

18.4 Example of Personalized Simulation The proposed geometric algorithms and biomechanical models are applied to the construction in OpenSim of simplified patient-specific model of the right knee joint. Figure 18.5 presents the attachments points for patellar ligament, medial (tibial) collateral ligament, lateral (fibular) collateral ligament, anterior and posterior cruciate ligaments for a right lower limb of a patient. The coordinates of attachment points, the origins for the joints frame, and the position of the contact surfaces define the simplified patient-specific geometric knee model. The forward dynamics simulation

220

A. Yurova et al.

Fig. 18.5 Attachment points of ligaments (subscript ‘o’ for origin, ‘i’ for insertion): patellar ligament (1o-1i), medial collateral ligament (2o-2i), lateral collateral ligament (3o-3i), anterior cruciate ligament (4o-4i), posterior cruciate ligament (5o-5i). The point 6t is the attachment point of quadriceps femoris tendon (not used in this model)

in OpenSim produces ligaments forces and ligaments strains under passive knee flexion as shown in Fig. 18.6. The obtained forces and ligament’s elongation are in the physiologically reasonable ranges [12]. Generally, the simulation results are sensitive to the stiffness and other model parameters [12]. In the present study, we focus on the construction of geometrical personalized model, and do not vary the model parameters to obtain better agreement with literature. The spike for MCL at 33 degrees is due to nonmonotonicity of numerical scheme (fifth-order Runge–Kutta– Feldberg method).

18.5 Conclusions We address the problem of patient-specific modeling of human knee joints. The base of personalized modeling is the segmentation of medical images. In the study, we present the algorithm for detection of ligaments attachment points. The algorithm exploits anatomical landmarks in post-processing of the obtained segmentation. The same approach can be utilized for detection of tendons attachment points. In order to study knee passive flexion, coordinates of ligaments origin and insertion points

18 Personalized Geometric Modeling of a Human Knee …

221

Fig. 18.6 Ligament forces and strains during the knee passive flexion obtained by OpenSim forward dynamics for the patient-specific right knee model

and bones mechanical characteristics are incorporated into the OpenSim knee joint model as a hinge joint. The obtained ligaments forces and strains are in the range of physiological values. The presented simplified knee joint model will be developed further: more degrees of freedom and active muscles will be added to the knee joint, ligaments, and muscles will be treated as bundles of fibers attaching to bones. Acknowledgements The work was supported in part by Moscow Center for Fundamental and Applied Mathematics (agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2019-1624), and the 2019 International Collaboration Special Plan, Chinese Academy of Sciences, grants of Guangdong Province, China, 2018A030313065 and Shenzhen, China, JCYJ20170818163445670.

References 1. Gage, B., McIlvain, N., Collins, C., Fields, S., Dawn, C.: Epidemiology of 6.6 million knee injuries presenting to United States emergency departments from 1999 through 2008. Acad. Emerg. Med. 19(4), 378–385 (2012) 2. Xu, H., Bloswick, D., Merryweather, A.: An improved OpenSim gait model with multiple degrees of freedom knee joint and knee ligaments. Comput. Methods Biomech. Biomed. Eng. 18(11), 1217–1224 (2014) 3. Pena, E., Calvo, B., Martinez, M.A., Doblare, M.: A three-dimensional finite element analysis of the combined behavior of ligaments and menisci in the healthy human knee joint. J. Biomech. 39(9), 1686–1701 (2006) 4. OpenSim homepage. https://opensim.stanford.edu/. Last accessed 2020/06/28

222

A. Yurova et al.

5. Cullen, M.: Sensitivity study of knee ligament properties in a computer simulation of a total knee arthroplasty. Doctoral dissertation, The Ohio State University (2014) 6. Salamatova, V.Y., Yurova, A.S., Vassilevski, Y.V., Wang, L.: Automatic segmentation algorithms and personalized geometric modelling for a human knee. Russ. J. Numer. Anal. Math. Model. 34(6), 361–367 (2019). 7. Drake, R., Vogl, A.W., Mitchell, A.W.: Gray’s Anatomy for Students, 3rd edn. Elsevier Churchill Livingstone, London (2015) 8. de Carvalho Felinto, J., Poloni, K.M., de Lima Freire, P.G., Aily, J.B., de Almeida, A.C., Pedroso, M.G., Mattiello, S.M., Ferrari, R.J.: Automatic segmentation and quantification of thigh tissues in CT images. In: Gervasi, O., Murgante, B., Misra, S., Stankova, E., Torre, C.M., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O., Tarantino, E., Ryu, Y. (eds.) Computational Science and Its Applications—ICCSA 2018, LNCS, vol. 10960, pp. 261–276. Springer, Cham (2018) 9. Hofer, M.: CT Teaching Manual: A Systematic Approach to CT Reading. Thieme, New York (2010) 10. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 4th edn. Prentice Hall, Pearson (2018) 11. OpenSim documentation. https://simtk-confluence.stanford.edu:8443/display/OpenSim/ How+Forward+Dynamics+Works. Last accessed 2020/06/28 12. Schmitz, A., Piovesan, D.: Development of an open-source, discrete element knee model. IEEE Trans. Biomed. Eng. 63(10), 2056–2067 (2016)

Chapter 19

Personalization of Mathematical Models of Human Atrial Action Potential Andrey V. Pikunov , Roman A. Syunyaev , Vanessa Steckmeister , Ingo Kutschka , Niels Voigt , and Igor R. Efimov

Abstract Atrial cardiomyocytes demonstrate a wide spectrum of patient-specific, tissue-specific, and pathology-specific Action potential (AP) phenotypes due to differences in protein expression and posttranslational modifications. Accurate simulation of the AP excitation and propagation in healthy or diseased atria requires a mathematical model capable of reproducing all the differences by parameter A. V. Pikunov · R. A. Syunyaev (B) Moscow Institute of Physics and Technology, National Research University, 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] A. V. Pikunov e-mail: [email protected] R. A. Syunyaev Sechenov First Moscow State Medical University, 8/2, Trubetskaya ul., Moscow 119991, Russian Federation R. A. Syunyaev · I. R. Efimov The George Washington University, 800 22nd, NW, Washington, DC, USA e-mail: [email protected] V. Steckmeister · N. Voigt Institute of Pharmacology and Toxicology, University Medical Center Göttingen, Robert-Koch-Str. 40, 37075 Göttingen, Germany e-mail: [email protected] N. Voigt e-mail: [email protected] V. Steckmeister · I. Kutschka · N. Voigt DZHK (German Center for Cardiovascular Research), Partner Site Göttingen, Robert-Koch-Str. 40, 37075 Göttingen, Germany e-mail: [email protected] I. Kutschka Department of Thoracic and Cardiovascular Surgery, University Medical Center, Robert-Koch-Str. 40, 37075 Göttingen, Germany N. Voigt Cluster of Excellence “Multiscale Bioimaging: From Molecular Machines to Networks of © 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 214, https://doi.org/10.1007/978-981-33-4709-0_19

223

224

A. V. Pikunov et al.

rescaling. In the present study, we have benchmarked two widely used electrophysiological models of the human atrium: the Maleckar and the Grandi models. In particular, patch-clamp AP recordings from human atrial myocytes were fitted by the genetic algorithm (GA) to test the models’ versatility. We have shown that the Maleckar model results in a more accurate fitting of heart rate dependence of action potential duration (APD) and resting potential (RP). On the other hand, both models demonstrate the poor fitting of the plateau phase and spike-and-dome morphologies. We propose that modifications to L-type calcium current–voltage relationships are required to improve atrial models’ fidelity. Keywords Atrial fibrillation · Genetic algorithms · Model personalization · Model benchmarking

19.1 Introduction Atrial fibrillation (AF) is the most common chronic arrhythmia encountered in clinical practice [1]. Today, the radio-frequency catheter ablation is considered to be one of the most effective treatment options, however, it is often followed by postoperational complications [2]. On the other hand, several groups reported that personalized mathematical simulation of the AP propagation in atria is capable to reproduce clinical measurements and consequently might be used to improve the outcome of the operation by choosing optimal ablation targets [3, 4]. One important challenge that remains in the field is inter-subject AP variability present in cardiac electrophysiological recordings [5]. Indeed, despite the variety of AP waveform morphologies recorded from atrial myocytes [6–12], both experimental and theoretical studies are mostly focusing on reporting the averaged characteristics of AP. To accurately describe the phenotypic variability, the baseline mathematical model should be able to reproduce these differences by the means of model parameter rescaling. While several groups benchmarked the existing human atrial models [5, 13], the question of whether these models are flexible enough to be personalized to a particular AP waveform remains to be answered. Previously, we have developed a genetic algorithm-based (GA-based) solution that was capable of precise personalization of the O’Hara-Rudy model [14] using optical AP recordings of the human left ventricular wedge preparations [15]. In the present preliminary research, we have utilized similar techniques to investigate the capability of the Maleckar [16] and the Grandi [17] atrial AP models to describe the variability of the AP morphology in healthy and diseased human atria. The chapter is organized as follows. Section 19.2 presents mathematical methods and data acquisition. The results are discussed in Sect. 19.3. We discuss modifications that may improve model fidelity in Sect. 19.4. Brief conclusion is given in Sect. 19.5. Excitable Cells” (MBExC), University of Goettingen, Robert-Koch-Str. 40, 37075 Göttingen, Germany

19 Personalization of Mathematical Models of Human Atrial …

225

19.2 Methods Sections 19.2.1 and 19.2.2 describe experimental conditions and details of mathematical modeling, respectively.

19.2.1 Experimental Data Acquisition Human tissue samples and myocyte isolation Right atrial appendages were obtained from patients undergoing cardiac surgery. Experimental protocols were approved by the ethics committee of the University Medical Center Göttingen (No. 4/11/18). Each patient gave a written informed consent. The patient samples were grouped in four blinded sets according to the diagnosis: sinus rhythm (SR), postoperative AF, paroxysmal AF, and chronic AF (referred to as groups 1–4 below). Excised right atrial appendages were subjected to a standard protocol for myocyte isolation [18]. Right atrial myocytes were suspended in EGTA-free storage solution for subsequent simultaneous measurement of membrane current/potential and intracellular Ca2+ concentration. Measurements of cellular electrophysiology Only rod-shaped myocytes with clear striations and defined margins were selected for measurements of [Ca2+ ]i and cellular electrophysiology. Cells were loaded with 10 μmol/L Fluo-3-AM (invitrogen; 10 min loading and 30 min de-esterification) for simultaneous measurements of AP and intracellular [Ca2+ ]i . APs were recorded using current-clamp configuration. To evoke APs, 1 ms current pulses of 1.5–2 threshold strength were applied. Sometimes, RP was too low to generate a proper AP since Nachannels were inactivated. In such cases, the small artificial hyperpolarizing current was introduced leading lower RP. The pacing cycle length (PCL) was decreased in a stepwise manner, starting at 2064 ms and subsequently 1032, 516, 344, 258, 206, 172, 148, 129 ms. During experiments, myocytes were perfused with a bath solution at 37 °C containing (in mmol/L): CaCl2 2, glucose 10, HEPES 10, KCl 4, MgCl2 1, NaCl 140, probenecid 2; pH = 7.35. The pipette solution contained (in mmol/L): EGTA 0.02, GTP-Tris 0.1, HEPES 10, K-aspartate 92, KCl 48, Mg-ATP 1, Na2 ATP 4, Fluo-3 pentapotassium salt 0.1; pH = 7.2. About 750 μl/50 ml KOH 1M was added to the pipette solution to adjust pH. For the bath solution pH adjustment, approximately, 3 ml of NaOH 1M was added 1 l tyrode solution. Pipette resistances in the range of 5–7 M were utilized. Intracellular Ca2+ measurements were not considered in the present study.

226

A. V. Pikunov et al.

19.2.2 Computer Simulations The custom C++ code replicating the Maleckar [16] and the Grandi [17] models was used to simulate AP of human atrial myocytes. The parameter values were fitted via GA as described in [15]. Briefly, a population of models with random parameter values was generated. Each model was paced for 9 beats at each experimental PCL before the fitness function—root mean square error (RMSE) evaluation. Next, the model population was modified by mutation and crossover genetic operators, followed by tournament selection. The cycle was repeated for each new generation. AP waveforms at PCLs of 2064, 1032, 516, 344, 258, 206, and 172 ms were used as the input traces for GA runs. Although intracellular Na, K, and sarcoplasmic Ca concentrations were modified by genetic operators, Na and K concentrations were limited to narrow ranges corresponding to experimental pipette solutions ([7, 9] and [130, 150] mM, respectively). Two additional parameters were adjusted by GA: the amplitude of the stimulation current (I st ) and the constant micropipette current distorting AP waveform (I bl ). The latter accounts for possible experimental artifacts and represents a combination of the hyperpolarizing current introduced to control the RP level and the leak current. The ranges for I st and I bl were set to [−80, −20] pA/pF and [−1, 1] pA/pF, respectively. Multipliers for model ionic current magnitudes were limited to [0.001, 10] range. The total number of organisms in a population was 320 with 24 elite organisms that were not modified by genetic operators. The number of generations in a GA run was set to 400. Simulations were performed on the cluster of The Moscow Institute of Physics and Technology (Intel Xeon Processor E5-2690, 2.90 GHz) and the George Washington University’s Pegasus cluster (Intel Xeon Gold 6148 Processor, 3.70 GHz). In the case of equal numbers of organisms and CPU cores, a single GA run took about 5 h for the Maleckar model and 20 h for the Grandi model.

19.3 Results Overview of the experimental recordings is given in Sect. 19.3.1. Section 19.3.2 contains results of the mathematical modeling.

19.3.1 Experimental AP Waveforms Experimental AP waveforms and dependencies of APA, APD80, and RP on PCL for all four groups of cells are shown in Fig. 19.1. We have observed a substantial difference in the AP morphologies not only across patients of the different groups but also among the patients of the same group. For example, Group 1 Cell 1 demonstrates a short ventricular-like plateau phase at +25 mV, while other cells from the same

19 Personalization of Mathematical Models of Human Atrial …

227

Fig. 19.1 AP waveforms, APD80, APA, and RP for cells of four groups (a–d) were recorded in the patch-clamp experiment. AP waveforms are drawn at PCL 2064 ms. PCLs 2064, 1032, 516, 344, 258, 206, and 172 ms were used for APD80, APA, and RP curves

228

A. V. Pikunov et al.

group have a more atrial-like triangulated AP waveform (Fig. 19.1a). Other groups of patients demonstrate both triangulated and spike-and-dome waveforms that were previously characterized as typical to AF and SR, respectively [7, 9]. It should be noted that in some cases RP dependence on PCL is nonmonotonic (e.g., Cells 1 and 2 from Group 2) resulting in corresponding distortions of APA and APD curves (Fig. 19.1b). Such RP behavior could be explained by the effect of the I bl described in the previous section.

19.3.2 Genetic Algorithm Runs Model comparison Output model RMSE scores for the Maleckar and the Grandi models were 3.5 (2.5– 3.8) mV and 4.3 (3.3–4.8) mV, respectively (errors will be given hereinafter as a median followed by an interquartile range in brackets). Although the overall difference in RMSE scores was not significant, it should be noted that in every single case the Maleckar model reproduced the experimental AP with lesser RMSE than the Grandi model. On the other hand, GA output concentrations for the Grandi model were closer to the steady-state solution (data is not shown). Representative fitting results for both models are shown in Fig. 19.2: triangular waveform (Fig. 19.2a) and spike-and-dome waveform (Fig. 19.2b, c). For the cases shown in Fig. 19.2a, b, the fitting is relatively accurate for both models, however, the Maleckar model replicated overall AP morphology better, which is especially evident for larger PCLs. Figure 19.2c exemplifies the case when the Maleckar model failed to capture spike-and-dome morphology at large PCLs still being able to fit the APs at smaller PCLs, while the Grandi model did not reproduce the experiment within the whole range of the PCLs. Figure 19.3 compares the fitting errors for common electrophysiological markers (APD, APA, and RP). The errors of APD80 and APD60 of the Grandi model have a similar trend, varying from more positive values at PCLs less than 1032 ms to more negative at PCLs 1032 and 2064 ms (Fig. 19.3a, b). The difference between the two models is more prominent for APD60: the Grandi model interquartile range for the APD60 error was about twice bigger than the same value for the Maleckar model within all the PCLs (excluding PCL 1032 ms). The overall APA error (Fig. 19.3c) for the Maleckar model equaled −8.6 (−15.1 to −2.3) mV and depended weakly on the PCL, while the Grandi model reproduced the APA with greater errors’ range at bigger PCLs (e.g., 0.9 (−17.4 to 3.8) mV for PCL 2064 ms). The RP error (Fig. 19.3d) for the Grandi model monotonically depended on PCL and raised from −2.8 (−6.7 to −0.2) mV at PCL 172 ms to 1.7 (0.4–2.3) mV at PCL 2064 ms. Generally, the RP of the Maleckar model slightly outreached the experimental level (0.90 (−0.63 to 2.25) mV).

19 Personalization of Mathematical Models of Human Atrial …

229

Fig. 19.2 Comparison of the model and experimental AP waveforms: a Group 3 Cell 1, b Group 4 Cell 4, c Group 3 Cell 2

Relying on the foregoing comparison, we considered utilizing the Maleckar model for further analysis, since this model was capable of reproducing APs more accurately as compared with the Grandi model.

230

A. V. Pikunov et al.

Fig. 19.3 Errors for the Maleckar and the Grandi models in reproducing: a APD80, b APD60, c APA, d RP. Outliers are not shown

Analysis of GA runs The GA output parameters of the Maleckar model are demonstrated in Fig. 19.4. In case of I Na , I Kur , and I K1 , relatively narrow conductivity variation was required to explain the diversity of waveforms that we have observed in the experiment (median (interquartile range): 0.71 (0.63–0.83), 0.67 (0.54–0.89), and 0.84 (0.72–1.17) for

19 Personalization of Mathematical Models of Human Atrial …

231

Fig. 19.4 GA output parameters for the Maleckar model: a I K1 , b I NaK , c I Ca,L , d I to , e I Kur , f I NCX , g I Na , h I Kr , i I Ks . Transparency of marker correlates with the fitting accuracy: bigger RMSE, more transparent the marker. Magnitudes lower 0.01 are indicated by arrows with a number of precedents

232

A. V. Pikunov et al.

I Na , I Kur , and I K1 , respectively) (Fig. 19.4a, e, g). Similar parameters’ ranges were reported previously by [5]. On the other hand, several ionic currents required extreme changes to model parameters to replicate the experimental AP waveform. In particular, I Ca,L conductivity was less than 10% of the original Maleckar model conductivity, in 12 cases out of 20 (Fig. 19.4c). On the contrary, I Ks and I Kr increase was above 500% of the original model value in 11 cases out of 20 for each of both currents (Fig. 19.4 h, i). As discussed below, we hypothesize that these discrepancies in model parameters might arise because of inaccuracies in ionic currents formulations.

19.4 Discussion Long-term persistent AF is associated with extensive atrial remodeling and, in particular, with changes in ionic channels conductivities [19]. These changes shorten the APD [7–10, 20], causing a decrease in effective reentrant wavelength, which affects the sustainability of arrhythmia [21]. Therefore, quantification of the gradual remodeling associated with a transition from healthy to diseased phenotype is important. Previously, using human ventricular AP recordings, we have demonstrated that GA accurately determines conductivities of high-amplitude ionic currents [15]. Applying GA to the atrial recordings from patients on different stages of AF development, we pursued twofold goals. First, to quantify the aforementioned remodeling of atrial ionic currents. Second, to personalize single-cell models for further simulations with real patient’s heart geometry, we focused on an investigation of the arrhythmogenesis during AF. The obvious limitation of personalization technique is that the current–voltage relationships and kinetics of ionic currents in the baseline model must correctly reproduce the experimental data. Otherwise, inaccurate formulations of an ionic current essentially reduce the identifiability of model parameters: the best waveform fitting results from an implausible combination of ionic currents conductivities which compensate for inaccuracies in current–voltage relationships. We compared the two most widely used atrial models in terms of the ability to describe a broad spectrum of healthy and diseased atrial AP morphologies. While we observed that the Maleckar model, in general, resulted in a better fitting in comparison with the Grandi model (Figs. 19.2 and 19.3), both of them failed to replicate all the variety of AP waveforms. This was especially evident in cases of a spikeand-dome morphology with a “deep notch” following an initial depolarization spike (Fig. 19.2c). On the other hand, we have observed very low GA output I Ca,L conductivity for the Maleckar model as shown in Fig. 19.4c. Taken together, denoted facts might indicate a substantial error in the formulation of I Ca,L . Altered expression of the auxiliary subunits of the L-type calcium channel (namely α1C\α2δ1 and α1C\α2δ2 encoded by CACNA2D1 and CACNA2D2 genes, respectively) was previously reported for patients with AF [22]. According to studies of the differential expression, the CACNA2D1 expression is higher in ventricles, while the CACNA2D2 is typical for atria [23]. As shown in Fig. 19.5c, current–voltage

19 Personalization of Mathematical Models of Human Atrial …

233

Fig. 19.5 Results of GA runs with the hybrid model: a model fitting for Group 1 Cell 1, b model fitting for Group 3 Cell 2, c current–voltage relationships of atrial and ventricular I Ca,L . Adapted from [24] and [25], d I Ca,L activation curves for the Maleckar and ToR-ORd models

234

A. V. Pikunov et al.

relationships of atrial [24] and ventricular [25] I Ca,L differ. Primarily, this difference is caused by the corresponding activation curves (Fig. 19.5d). We hypothesize that I Ca,L in atria is presented as a composition of several subpopulations of ionic channels formed by different subunits and, consequently, characterized by different current– voltage relationships. The balance between these subpopulations may change during AF development [22]. Given the different current–voltage relationships of corresponding ionic currents, the alterations in the expression profile cannot be replicated by a model with a single I Ca,L subpopulation. In line with the proposed hypothesis, we have introduced an additional ventricular subpopulation of I Ca,L to the Maleckar model. The I Ca,L formulation was adapted from [26]. The representative results of GA run with the hybrid model are presented in Fig. 19.5. In general, this modification indeed resulted in a more accurate fitting in comparison with the original model. For example, we have observed a 40% decrease in the fitting error (RMSE) for Group 1 Cell 1 (Fig. 19.5a). Model AP approximated experimental waveforms much better in the region from −25 to 25 mV. However, the introduction of the ventricular I Ca,L to the Maleckar model did not improve the fitting of the spike-and-dome morphology (Fig. 19.5b), although RMSE was 10% lower. The reason for such inefficacy can be explained via I Ca,L current–voltage relationships and activation curves (Fig. 19.5c, d). The notch following the spike repolarizes the cell as low as −25 mV, while both subpopulations of I Ca,L are essentially deactivated at this voltage level. We conclude that the absence of substantial depolarization current at −25 mV in the model prohibits the following “dome” observed in the experiment. It should be reiterated here that the presence of two subpopulations of I Ca,L most probably affects the voltage-clamp experiments underlying the atrial models. Given that α2δ1 subunit expression is not exclusive to ventricles, actual atrial-specific α2δ2 activation might be shifted even further to negative voltages than the one measured in experiments. We also suspect a change of the balance between these two isoforms to result in a proarrhythmic substrate in the case of the long-term persistent AF. As presented in dynamic patch-clamp experiments [27], the alterations in the atrial I Ca,L current–voltage relationships might result in early after depolarizations and trigger the arrhythmia.

19.5 Conclusions In this short report, we have applied the previously developed GA [15] for the human atrial AP models’ personalization. We have compared the Maleckar model [16] and the Grandi model [17] capability to reproduce the variety of experimental AP morphologies. The former produced a tighter fit, however, in several cases with a spike-and-dome morphology both models failed to fit the experimental data. These discrepancies between model and experiment might be explained by imprecision in I Ca,L formulation. In particular, we have observed that including both ventricular and atrial subpopulations of I Ca,L improves models’ fidelity. We hypothesize that the change of the balance between two auxiliary I Ca,L subunits take place during AF

19 Personalization of Mathematical Models of Human Atrial …

235

development, however, further theoretical and experimental evidence is required to test these assumptions that we will address in the future studies. Acknowledgements The authors thank Ines Müller and Stefanie Kestel for excellent technical assistance and the cardiac surgeons of the Department of Thoracic and Cardiovascular Surgery, University Medical Center Göttingen for kindly providing human atrial tissue samples. This work was supported by grants from the German Center for Cardiovascular Research (DZHK), the Deutsche Forschungsgemeinschaft to Niels Voigt (DFG, German Research Foundation, VO 1568/3-1, IRTG1816 RP12, SFB1002 TPA13 and under Germany’s Excellence Strateg— EXC 2067/1-390729940), from the from the Else-Kröner-Fresenius Foundation to Niels Voigt (EKFS 2016_A20), from NIH/NHLBI (U01 HL141074) and Leducq Foundation (RHYTHM) to Igor R. Efimov, computer model optimization was supported by Russian Scientific Foundation grant 18-71-10058 to Roman A. Syunyaev.

References 1. Benjamin, E.J., Wolf, P.A., D’Agostino, R.B., Silbershatz, H., Kannel, W.B., Levy, D.: Impact of atrial fibrillation on the risk of death: the Framingham heart study. Circulation 98(10), 946–952 (1998) 2. Steffel, J., Verhamme, P., Potpara, T.S., Albaladejo, P., Antz, M., Desteghe, L., Haeusler, K.G., Oldgren, J., Reinecke, H., Roldan-Schilling, V., Rowell, N.: The 2018 European heart rhythm association practical guide on the use of non-vitamin K antagonist oral anticoagulants in patients with atrial fibrillation. Eur. Heart J. 39(16), 1330–1393 (2018) 3. Boyle, P.M., Zghaib, T., Zahid, S., Ali, R.L., Deng, D., Franceschi, W.H., Hakim, J.B., Murphy, M.J., Prakosa, A., Zimmerman, S.L., Ashikaga, H., Marine, J.E., Kolandaivelu, A., Nazarian, S., Spragg, D.D., Calkins, H., Trayanova, N.A.: Computationally guided personalized targeted ablation of persistent atrial fibrillation. Nat. Biomed. Eng. 3(11), 870–879 (2019) 4. Corrado, C., Williams, S., Karim, R., Plank, G., O’Neill, M., Niederer, S.: A work flow to build and validate patient specific left atrium electrophysiology models from catheter measurements. Med. Image Anal. 47, 153–163 (2018) 5. Sánchez, C., Bueno-Orovio, A., Wettwer, E., Loose, S., Simon, J., Ravens, U., Pueyo, E., Rodriguez, B.: Inter-subject variability in human atrial action potential in sinus rhythm versus chronic atrial fibrillation. PLoS ONE 9(8), e105897 (2014) 6. Fermini, B.E., Wang, Z.H., Duan, D.A., Nattel, S.T.: Differences in rate dependence of transient outward current in rabbit and human atrium. Am. J. Physiol. Heart Circulatory Physiol. 263(6), H1747–H1754 (1992) 7. Van Wagoner, D.R., Pond, A.L., Lamorgese, M., Rossie, S.S., McCarthy, P.M., Nerbonne, J.M.: Atrial L-type Ca2+ currents and human atrial fibrillation. Circ. Res. 85(5), 428–436 (1999) 8. Voigt, N., Li, N., Wang, Q., Wang, W., Trafford, A.W., Abu-Taha, I., Sun, Q., Wieland, T., Ravens, U., Nattel, S., Wehrens, X.H., Dobrev, D.: Enhanced sarcoplasmic reticulum Ca2+ leak and increased Na+-Ca2+ exchanger function underlie delayed afterdepolarizations in patients with chronic atrial fibrillation. Circulation 125(17), 2059–2070 (2012) 9. Wettwer, E., Hála, O., Christ, T., Heubach, J.F., Dobrev, D., Knaut, M., Varró, A., Ravens, U.: Role of I Kur in controlling action potential shape and contractility in the human atrium: influence of chronic atrial fibrillation. Circulation 110(16), 2299–2306 (2004) 10. Schmidt, C., Wiedmann, F., Voigt, N., Zhou, X.B., Heijman, J., Lang, S., Albert, V., Kallenberger, S., Ruhparwar. A., Szabo, G., Kallenbach, K., Karck, M., Borggrefe, M., Biliczki, P., Ehrlich, J.R., Baczko, I., Lugenbiel, P., Schweizer, P.A., Donner, B.C., Katus, H.A., Dobrev, D., Thomas, D.: Upregulation of K2P3. 1K+ current causes action potential shortening in patients with chronic atrial fibrillation. Circulation 132(2), 82–92 (2015)

236

A. V. Pikunov et al.

11. Schmidt, C., Wiedmann, F., Zhou, X.B., Heijman, J., Voigt, N., Ratte, A., Lang, S., Kallenberger, S.M., Campana, C., Weymann, A., De Simone, R., Szabo, G., Ruhparwar, A., Kallenbach, K., Karck, M., Ehrlich, J.R., Baczko, I., Borggrefe, M., Ravens, U., Dobrev, D., Katus, H.A., Thomas, D.: Inverse remodelling of K2P3.1 K+ channel expression and action potential duration in left ventricular dysfunction and atrial fibrillation: implications for patient-specific antiarrhythmic drug therapy. Eur. Heart J. 38, 1764–1774 (2017) 12. Voigt, N., Heijman, J., Wang, Q., Chiang, D.Y., Li, N., Karck, M., Wehrens, X.H., Nattel, S., Dobrev, D.: Cellular and molecular mechanisms of atrial arrhythmogenesis in patients with paroxysmal atrial fibrillation. Circulation 129(2), 145–156 (2014) 13. Wilhelms, M., Hettmann, H., Maleckar, M.M.C., Koivumäki, J.T., Dössel, O., Seemann, G.: Benchmarking electrophysiological models of human atrial myocytes. Front. Physiol. 3, 487 (2013) 14. O’Hara, T., Virág, L., Varró, A., Rudy, Y.: Simulation of the undiseased human cardiac ventricular action potential: model formulation and experimental validation. PLoS Comput. Biol. 7(5), e1002061 (2011) 15. Smirnov, D., Pikunov, A., Syunyaev, R., Deviatiiarov, R., Gusev, O., Aras, K., Gams, A., Koppel, A., Efimov, I.R.: Genetic algorithm-based personalized models of human cardiac action potential. PLoS ONE 15(5), e0231695 (2020) 16. Maleckar, M.M., Greenstein, J.L., Trayanova, N.A., Giles, W.R.: Mathematical simulations of ligand-gated and cell-type specific effects on the action potential of human atrium. Prog. Biophys. Mol. Biol. 98(2–3), 161–170 (2008) 17. Grandi, E., Pandit, S.V., Voigt, N., Workman, A.J., Dobrev, D., Jalife, J., Bers, D.M.: Human atrial action potential and Ca2+ model: sinus rhythm and chronic atrial fibrillation. Circ. Res. 109(9), 1055–1066 (2011) 18. Fakuade, F.E., Steckmeister, V., Seibertz, F., Gronwald, J., Kestel, S., Menzel, J., Pronto, J.R.D., Taha, K., Haghighi, F., Kensah, G., Pearman, C.M., Wiedmann, F., Teske, A.J., Schmidt, C., Dibb, K.M., El-Essawi, A., Danner, B.C., Baraki, H., Schwappach, B., Kutschka, I., Mason, F.E., Voigt, N.: Altered atrial cytosolic calcium handling contributes to the development of postoperative atrial fibrillation. Cardiovasc. Res. cvaa162 (2020) 19. Nattel, S., Heijman, J., Zhou, L., Dobrev, D.: Molecular basis of atrial fibrillation pathophysiology and therapy: a translational perspective. Circ. Res. 127(1), 51–72 (2020) 20. Bosch, R.F., Zeng, X., Grammer, J.B., Popovic, K., Mewis, C., Kühlkamp, V.: Ionic mechanisms of electrical remodeling in human atrial fibrillation. Cardiovasc. Res. 44(1), 121–131 (1999) 21. Aras, K.K., Faye, N.R., Cathey, B., Efimov, I.R.: Critical volume of human myocardium necessary to maintain ventricular fibrillation. Circ. Arrhythmia Electrophysiol. 11(11), e006692 (2018) 22. Gaborit, N., Steenman, M., Lamirault, G., Meur, N.L., Bouter, S.L., Lande, G., Léger, J., Charpentier, F., Christ, T., Dobrev, D., Escande, D., Nattel, S., Demolombe, S.: Human atrial ion channel and transporter subunit gene-expression remodeling associated with valvular heart disease and atrial fibrillation. Circulation 112, 471–481 (2005) 23. Gaborit, N., Le Bouter, S., Szuts, V., Varro, A., Escande, D., Nattel, S., Demolombe, S.: Regional and tissue specific transcript signatures of ion channel genes in the non-diseased human heart. J. Physiol. 582(2), 675–693 (2007) 24. Li, G.R., Nattel, S.: Properties of human atrial ICa at physiological temperatures and relevance to action potential. Am. J. Physiol. Heart Circulatory Physiol. 272(1), H227–H235 (1997) 25. Magyar, J., Iost, N., Körtvély, Á., Bányász, T., Virág, L., Szigligeti, P., Varró, A., Opincariu, M., Szécsi, J., Papp, J.G., Nánási, P.P.: Effects of endothelin-1 on calcium and potassium currents in undiseased human ventricular myocytes. Pflügers Arch. 441(1), 144–149 (2000) 26. Tomek, J., Bueno-Orovio, A., Passini, E., Zhou, X., Minchole, A., Britton, O., Bartolucci, C., Severi, S., Shrier, A., Virag, L., Varro, A., Rodriguez, B.: Development, calibration, and validation of a novel human ventricular myocyte model in health, disease, and drug block. Elife 8, e48890 (2019) 27. Kettlewell, S., Saxena, P., Dempster, J., Colman, M.A., Myles, R.C., Smith, G.L., Workman, A.J.: Dynamic clamping human and rabbit atrial calcium current: narrowing ICaL window abolishes early afterdepolarizations. J. Physiol. 597(14), 3619–3638 (2019)

Chapter 20

Computational Study of the Effect of Blood Viscosity to the Coronary Blood Flow by 1D Haemodynamics Approach Sergey S. Simakov

and Timur M. Gamilov

Abstract The integrated rheological properties of the blood are determined by various factors including hematocrit, red blood cells deformation, pressure, temperature, etc. These factors affect various approaches to simulate friction caused by blood viscosity. We demonstrate how different approaches affect coronary blood flow (CBF) in one-dimensional model. The difference in CBF for two most common approaches in 1D simulation (parabolic profile and flattened profile) can exceed 10%. Keywords Blood rheology · Blood viscosity · Anticoagulants · Coronary blood flow · Coronary circulation model

20.1 Introduction One of the main directions of the scientific school of Russian academician O. M. Belotserkovski was devoted to the mathematical modeling in life sciences based on the approaches of continuum mechanics. The results were successfully applied to many practical tasks in medicine and biology. In particular, grid-characteristic method, which was developed by Russian academician A. S. Kholodov, provided background for 1D network dynamical computational models of haemodynamics [1]. In this chapter, we discuss some aspects of these models, which concerns rheological properties of the blood.

S. S. Simakov (B) · T. M. Gamilov Moscow Institute of Physics and Technology, National Research University, 9, Institutsky per., Dolgoprudny, Moscow Region 141701, Russian Federation e-mail: [email protected] T. M. Gamilov e-mail: [email protected] Sechenov University, 8/2 Trubetskaya ul., Moscow 119991, 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 214, https://doi.org/10.1007/978-981-33-4709-0_20

237

238

S. S. Simakov and T. M. Gamilov

In continuum mechanics, all viscous fluids are classified as Newtonian and nonNewtonian. The viscosity of Newtonian fluids is independent of shear rate and shear stress. In non-Newtonian fluids, the apparent viscosity is not constant. It can be calculated as the ratio of shear rate to shear stress, which is nonlinear. The flow of non-Newtonian fluids is different from the Newtonian one due to the internal molecular structure and interaction [1–3]. The blood flow is the major determinant of sufficient tissue perfusion. Thus, it is important to understand the factors affecting the blood flow as a continuum. In different cases, blood is considered as a single-phase fluid, a two-phase fluid, suspension (fluid–solid mixture), or emulsion (fluid–fluid mixture) [4]. In any case, one part of the blood is plasma, and the other part consists of different cells. Plasma is generally considered as a Newtonian fluid. The normal range of plasma viscosity lies between 1.10 and 1.35 cP at 37 °C. Red blood cells (RBCs) are substantially bigger, than the other blood components. Normally, RBC is a biconcave disc with typical dimensions of 6–8 µm in diameter and 2 µm thick. White blood cells (WBCs) vary in size. Inactivated neutrophil is almost spherical with a diameter of 8 µm. Platelet is a disc with a diameter of 2 µm. These cells occupy less than 1% of the total blood volume. Hence, in microcirculation, the size of blood vessels is the same order or less than the size of RBCs. The other blood components are not significant for the blood rheology. The integrated rheological properties of the blood are primarily determined by the plasma viscosity, hematocrit, RBC rheology, RBC membrane elasticity, and RBCs interaction (e.g., aggregation, rouleaux formation).The properties of WBC and other cells play a less important role. Some extended reviews on the physical and mathematical description of blood viscosity can be found in [5–8]. The presence of cellular components in plasma disturbs the flow streamlines and, thereby, increases the total blood viscosity relative to the pure plasma. Thus, an increased number of cellular components in the blood cause the increase in the viscosity. For example, blood viscosity is exponentially proportional to the ratio of RBC volume to the blood volume (hematocrit). In many cases, RBCs behave in the blood like fluid drops. They align along the streamlines. Increased shear rate results in deformation and reorientation of RBCs relative to the streamlines, which also causes a viscosity increase. At moderate to high shear rates, RBCs tend to align and form layers in the flow, which results in the decrease of the viscosity. Low flow conditions and the presence of fibrinogen and globulins cause RBCs aggregation in a structure similar to coin stacks (rouleaux). Rouleaux and its branched 3D aggregates behave like huge single particles, thereby significantly increasing viscosity. Normally, WBCs produce a negligible effect on total blood viscosity. Nevertheless, biologically activated by inflammation or by extraneous molecules, WBCs and platelets may cause substantial changes to the flow due to their extensive biochemical, morphological, and mechanical alterations [8–10]. The near-zero flow or stasis conditions may result in the flow termination under nonzero shear stress (yield stress phenomenon). The chapter is organized as follows. Section 20.2 includes brief review of approaches to simulate blood viscosity in reduced order models. Section 20.3 contains main equations of 1D coronary circulation model including blood viscosity

20 Computational Study of the Effect of Blood Viscosity …

239

effects. Section 20.4 represents results of coronary blood flow simulations. We analyze the results and possible improvements of the model in Sect. 20.5.

20.2 Methodology of Modeling Blood Viscosity in Reduced Order Models One may conclude that in normal conditions in most parts of the arterial system, blood flow exhibits the properties of Newtonian fluid. Non-Newtonian effects become noticeable in high and low flow conditions: in small vessels, venous system, downstream stenoses, and saccular aneurysms. Non-Newtonian effects are especially important in the microcirculation. Particle methods are still computationally expensive for blood flow simulations in a microcirculatory network at the macroscopic scale. The low Reynolds numbers of the flow regimes allow to apply well-known Poiseuille pressure drop condition for the steady laminar flow of Newtonian fluid in a single vessel [6, 11–13]: P = R Q, R =

128Lμ , π D4

(20.1)

where P is pressure drop over the vessel’s terminal points, Q is the flow rate, R is hydraulic resistance, L and D are the length and diameter of the vessel, respectively, and μ is viscosity. The length of the vessel is obviously constant. Thus, the relationship between the pressure drop over the vessel and flow through it is characterized by the vessel diameter and viscosity. The diameter variations are associated with the growth, regulatory modulation, metabolic demands, and diseases [14]. The viscosity is, probably, the most uncertain parameter, which depends on various factors, including vessel’s diameter, blood temperature and composition, in particular, RBCs volume fraction H (hematocrit), flow rate, pathological change of the elastic properties of RBC membranes due to sickle cell anemia, diabetes, etc. All these and other factors are modeled within the considered approach as some effective (apparent) viscosity, which can be a nonlinear function of various parameters μ = μapp (Q, D, H, . . .). In different cases, this function can be derived from fundamental principles of continuum mechanics, from simulations by some particle method, or from experimental observations. Thus, the study of non-Newtonian rheological properties of blood in microvessels is the central problem of modeling blood flow in microcirculatory network. In vitro analysis of the flow of RBC, suspension in glass tubes allows to derive the following empirical relationship [15, 16]: μapp =1+ μp



 μ˜ (1 − HD )C − 1 , −1 μp (1 − H˜ D )C − 1

(20.2)

240

S. S. Simakov and T. M. Gamilov

where μp is the viscosity of plasma, HD is the ratio of the volume flux of RBCs to the total volume flux of blood (discharge hematocrit), μ˜ is the apparent viscosity of ˜ C(D) are the functions of the blood for a discharge hematocrit H˜ D = 0.45, μ(D), vessel diameter D. See [15] for more details on this model. In vivo analysis of living microvessels results in much higher values, which are determined by the following empirical relationship [13, 17]:  2  2    μapp D D μ˜ (1 − HD )C − 1 = 1+ −1 . μp μp D − D0 (1 − H˜ D )C − 1 D − D0

(20.3)

The reason for the difference D − D0 is the presence of the glycocalyx layer on the inner surface of endothelial layer of the microvessels (D0 = 1.1 µm), which slows down the plasma flow in the cell-free layer. One of the possible reasons of changes in blood rheology is drug therapy, e.g., anticoagulation therapy changes blood viscosity and blood flow. Therefore, it modifies associated oxygen and nutrients delivery to the tissues. The myocardium is one of the most important and sensitive tissues in the organism. Changes in blood viscosity can have a drastic effect on myocardium blood supply. In this work, we use a onedimensional model of coronary circulation to demonstrate these changes in coronary blood flow for various values of blood viscosity.

20.3 Blood Viscosity in 1D Model Hereinafter, we discuss 1D model of coronary circulation. We present the general equations and assumptions of the model in Sect. 20.3.1. Section 20.3.2 contains a description of friction force caused by blood viscosity.

20.3.1 1D Model of Haemodynamics with Application to Coronary Circulation Computational one-dimensional (1D) network model of blood flow in coronary vessels is used in [17]. Brief summary of this method is described in this section. Details can be found in [7, 8, 17, 18]. The approach is based on the model of viscous incompressible fluid flow through the network of elastic tubes. The flow in single linear vessel or vessel’s segment is described by mass and momentum balance equations: ∂ Sk /∂t + ∂(Sk u k )/∂ x = 0,

(20.4)

20 Computational Study of the Effect of Blood Viscosity …

241

  ∂u k /∂t + ∂ u 2k /2 + pk /ρ /∂ x = ψfr ,

(20.5)

where k is the index of the vessel, t is the time, x is the coordinate along the vessel, ρ is the blood density, Sk is the vessel cross section area, pk is the blood pressure, u k is the linear velocity averaged over the cross section, ψfr is the friction force. Elastic properties of the vessel wall material are described by function: pk (Sk ) = ρm ck2 f (Sk ),

(20.6)

where f (Sk ) is a monotone S-like function [18], ρm is the density of the vessel’s wall material, ck is the velocity of small disturbances propagation in the vessel wall. At the vessel’s junction composed by M vessels with indices k = k1 , k2 , .., k M , mass conservation condition is used together with the total pressure conservation in the form: M 

εki Q ki = 0,

(20.7)

i=1

u 2ki 2

+

u 2k j pk j pk i = + , i, j = 1 . . . M, i = j, ρ 2 ρ

(20.8)

where Q ki is the blood flow at the boundary of a vessel with index ki , εki = 1 for incoming vessels and εki = −1 for outgoing vessels. At the input to the arterial part of the network the blood flow was set as predefined function: Q in (t) = u(t, 0)S(t, 0),

(20.9)

where Q in (t) is the heart ejection profile. The shape of the profile and depends on the heart rate (HR) [18]. Stroke volume (SV) decreases with HR, as a result, cardiac output (CO) peaks around 120 bpm (Table 20.1). The outflow boundary conditions assume that a terminal artery with index k is connected to the venous pressure reservoir with the pressure pv = 8 mmHg by the hydraulic resistance Rk : pk − pv = R k Q k .

(20.10)

Table 20.1 Cardiac output and stroke volume for various heart rates HR, bpm

40

60

80

100

120

140

160

SV, ml

92

82

72

62

52

42

32

CO, ml/s

61.3

82

96

103.3

104

98

85.3

242

S. S. Simakov and T. M. Gamilov

Fig. 20.1 A 3D Anatomical model of the coronary arteries

Hydraulic resistance is increased during systolic phase to simulate myocardium compression [18]. The computational domain is the network of vessels including aortic root, aorta, left and right coronary arteries and their branches (Fig. 20.1). Parameters of the aorta, aortic root, and coronary arteries correspond to the physiological values of an adult human.

20.3.2 Blood Viscosity and Friction The viscosity of blood depends on rheological properties of plasma, cellular components (RBCs), and volume fraction of RBCs (hematocrit). An apparent blood dynamic viscosity (μapp ) is given by Eq. 20.11, where μ is the dynamic viscosity of plasma and μrel is the relative viscosity that depends on hematocrit. μapp = μrel μ

(20.11)

In [19, 20], the blood is assumed to be Newtonian fluid with linear rheological relationship. This assumption works well for modeling blood flows in large vessels at high and medium shear stress under nonpathological conditions. Non-Newtonian effects in blood flows become significant at shear rates smaller than 100 s–1 [21]. Such nonlinear rheological effects are more likely to occur in veins, small arteries, and capillaries. The following expression gives the general form of the viscous friction in ψfr =

2τ , ρ R˜

(20.12)

20 Computational Study of the Effect of Blood Viscosity …

243

where τ is the shear stress and R˜ is the vessel’s radius. Given the axial velocity component v(x, r, t), the viscous friction term reduces to 1 ∂v ψfr = 2γ ν, γ = , R˜ ∂r r = R˜

(20.13)

where ν is the kinematic viscosity of blood (ν = μ ρ). For derivation, we refer to, e.g., [22, 23].

For parabolic profile, ψfr = −8π μu S, a good agreement with experimental data was observed in [24] for ψfr = −22π μu S (ninth degree polynomial profile). We remind that u is the axial velocity averaged over the cross section. The viscous friction that is invariant to vessel contraction or dilation was proposed in [25, 26]: 4π νu fr(η), S

(20.14)

2 η ≥ 1, (η + η−1 ) η < 1.

(20.15)

ψfr = −

where η = S S0 ,  fr(η) =

Many non-Newtonian rheology models for blood flows consider the dynamic viscosity dependent on the shear stress τ. For instance, Carreau–Yasuda model states that

(n−1)/ p , μ = μ∞ + (μ0 − μ∞ ) 1 + (λτ ) p

(20.16)

where λ = 0.438 s, n = 0.191, p = 0.409, μ0 = 51.9 mPa s, and μ∞ = 4.76 mPa s [27]. We refer to [5, 8, 28] for a review of more than 30 models of shear-dependent viscosity models.

20.4 Results of Computational Simulations In these numerical experiments, we study the effect of applying various friction models in CBF model. We compute CBF as an average blood flow through the left and right coronary arteries over a heart cycle with the duration of 1 s. The friction coefficient is related to the assumptions on the shape of the blood flow profile in a vessel. Within the same set of parameters, we compare four cases corresponding to different assumptions on a friction force:

244

• • • •

S. S. Simakov and T. M. Gamilov

A: ψfr = 0, the absence of friction.

B: ψfr = −8π μu S, parabolic profile. C: ψfr = −22π μu

S, almost flat profile [24]. D: ψfr = −100π μu S, extreme friction. For the subsequent analysis, we generalize it as ψfr = −α

π μu . S

(20.17)

We study CBF in a range of heart rate from 40 to 160 bpm. The values below 60 bpm are classified as bradycardia, while the values above 100 bpm are commonly associated with tachicardia. The heart rhythms up to 160 bpm are typical to the conditions associated with high-intensive sport and similar loads. Figure 20.2 shows the results of simulations of CBF for various heart rates and α = 0, 8, 22, 100 (see Eq. 20.17). The notations of different cases correspond to the above notations. Curve A corresponds to the no friction conditions, and curve D corresponds to extremely high friction or viscosity. The value α = 100 is beyond friction models (Eq. 20.14) and (Eq. 20.16) and other possible physiological conditions. We observe the continuous change of CBF curves depending on the increase of α. The other intermediate curves are not shown in Fig. 20.2 for the visual clarity. Cases B (parabolic profile) and C (almost flat profile) are common in simulations with one-dimensional models [24, 29, 30]. Figure 20.2 shows that the difference in CBF for these two cases can exceed 10%. Reducing friction to zero produces

Fig. 20.2 Coronary blood flow (CBF—averaged over time total blood flow through both coronary arteries) over the heart rate for different values of α (see Eq. 20.17). Curve A corresponds to α = 0, curve B corresponds to α = 8, curve C corresponds to α = 22, and curve D corresponds to α = 100

20 Computational Study of the Effect of Blood Viscosity …

245

additional increase of CBF by 10–12%. Thus, the sensitivity to the profile and friction force assumptions is in the range of 20%. CBF curve corresponding to some other value of α should be between the curves A and D. Thus, in principle, we can theoretically estimate the maximum and minimum CBF regardless of assumption on friction force model for the given structure of the vessels and other parameters. From Fig. 20.2, we also observe interesting phenomena. All CBF curves have the maximum value at the heart rate of 100 bpm. We have observed similar behavior for the average blood flow in the terminal arteries of lower extremities under the conditions of periodic physical activity (walking or running) [31], which are similar to the conditions of periodic myocardium contractions. Figure 20.3 shows the other interpretation of the simulation results of CBF for various heart rates and α = 0, 8, 22, 100 (see Eq. 20.17). Figure 20.2 represents the maximum possible CBF in the range from 40 to 160 bpm for a specific value of α. From Fig. 20.3, we observe sharp increase of CBF for the values of α between 0 and 22. We see that increase of α above the value 22 corresponding to the flat blood flow profile produces insignificant increase of CBF for all values of heart rates. The curves corresponding to the heart rates between 40 and 100 bpm lay between the curves A and B in Fig. 20.3. The curves corresponding to the heart rates between 100 and 160 bpm lay between the curves B and C in Fig. 20.3. We observe the continuous change of these curves. The other intermediate curves are not shown in Fig. 20.3 for the visual clarity. Thus, the no friction conditions, parabolic profile, and flat profile are the distinctive cases with substantially different effect to CBF.

Fig. 20.3 CBF over α (see Eq. 20.17) for different values of the heart rate. Curve A corresponds to 40 bpm, curve B corresponds to 100 bpm, and curve C corresponds to 160 bpm

246

S. S. Simakov and T. M. Gamilov

20.5 Conclusions Blood viscosity is affected by many factors: temperature, pressure [32], oxygen concentration [33], hematocrit, erythrocyte deformability [34], drug treatment, etc. Different models of friction force may seriously affect the results of computational simulations. On the other hand, the modified viscosity may be a reason of drug administration and other pathological factors and processes, e.g., administration of oral anticoagulants and antiaggregants may decrease the blood viscosity, while other drugs may increase blood viscosity both as intended treatment (hemostatic agents) or as side effects. It causes effect to the blood flow and associated tissues and especially myocardium supply with oxygen and other nutrients. Coronary blood flow is also extremely sensitive in the presence of atherosclerotic plaques [9]. The viscosity factor may play important role during evaluation of stenosis haemodynamic significance [35–37]. All these challenges will be a matter of our future research. In this work, we observe 20% variations of CBF depending on the assumptions of the blood flow profile and friction force coefficient. It means that specific attention should be paid to analysis of the type of the blood flow in a local region of vascular network before the simulations. In addition, it should be mentioned that actual blood flow profile is not static. It changes dynamically within a heart period. In the case of rigid tubes, it is known as Womersley profile. In compliant tubes, the pulsatile profile is even more complicated. Under these assumptions, one should include to the model both friction between the blood and the vascular wall and internal friction losses as well. CBF dependence on friction force coefficient shows that patient-specific simulations of coronary hemodynamics should take into account patient’s blood viscosity. Viscosity can be evaluated from hematocrit measurements. In case when a patient is under any drug therapy, the type and dosage of a drug should also be taken into account. In this work, we perform the simulations for different values of the heart rate. We neglected such physiological features as dependence of the ratio of systole to the heart period and stroke volume of the heart from the heart rate. It may produce correction of the presented values and should be studied in the future work. The other, especially, nonlinear effects of the blood viscosity should be considered although their importance increases with the decrease of the diameter of the vessels. It may cause noticeable effect in such small but important vessels as the arteries of the Circle of Willis and in the cases of severe stenotic lesions. Acknowledgements The work was supported by the Russian Foundation for Basic Research, grants No. 18-31-20048, 18-00-01524, 18-00-01661.

20 Computational Study of the Effect of Blood Viscosity …

247

References 1. Petrov, I.B., Favorskaya, A.V., Favorskaya, M.N., Simakov, S.S., Jain, L.C.: Development and applications of computational methods. 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. 3–7, Springer, Cham (2019) 2. Anand, M., Rajagopal, K.R.: A short review of advances in the modelling of blood rheology and clot formation. Fluids 2(3), 35.1–35.9 (2017) 3. Clarion, M., Deegan, M., Helton, T., Hudgins, J., Monteferrante, N., Ousley, E., Armstrong, M.: Contemporary modeling and analysis of steady state and transient human blood rheology. Rheol. Acta 57, 141–168 (2018) 4. Baskurt, A., Meiselman, H.: Blood rheology and hemodynamics. Semin. Thromb. Hemost. 29(5), 435–450 (2003) 5. Yilmaz, F., Gundogdu, M.: A critical review on blood flow in large arteries; relevance to blood rheology, viscosity models, and physiologic conditions. Korea Aust. Rheol. J. 20(4), 197–211 (2008) 6. Popel, A.S., Johnson, P.C.: Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69 (2005) 7. Vassilevski, Y., Olshanskii, M., Simakov, S., Kolobov, A., Danilov, A.: Personalized Computational Hemodynamics: Models, Methods, and Applications for Vascular Surgery and Antitumor Therapy. Academic Press, Cambridge (2020) 8. Bessonov, N., Sequeira, A., Simakov, S., Vassilevskii, Yu., Volpert, V.: Methods of blood flow modeling. Math. Model. Nat. Phenom. 11(1), 1–25 (2016) 9. Khatib, N.El., Kafi, O., Sequeira, A., Simakov, S., Vassilevskii, Yu., Volpert, V.: Mathematical modelling of atherosclerosis. Math. Model. Nat. Phenom. 14(6), 603.1–603.25 (2019) 10. Andreeva, A.A., Anand, M., Lobanov, A.I., Nikolaev, A.V., Panteleev, M.A., Susree, M.: Mathematical modelling of platelet rich plasma clotting. Pointwise unified model. Russ. J. Numer. Anal. Math. Model. 33(5), 265–276 (2018) 11. Stamatelos, S.K., Kim, E., Pathak, A.P., Aleksander, S.P.: A bioimage informatics based reconstruction of breast tumor microvasculature with computational blood flow predictions. Microvasc. Res. 91, 8–21 (2014) 12. Pozrikidis, C.: Numerical simulation of blood flow through microvascular capillary networks. Bull. Math. Biol. 71, 1520–1541 (2009) 13. Pries, A.R., Secomb, T.W., Gaehtgens, P., Gross, J.F.: Blood flow in microvascular networks. Experiments and simulation. Circ. Res. 67, 826–834 (1990) 14. Pries, A.R., Secomb, T.W.: Making microvascular networks work: angiogenesis, remodeling, and pruning. Physiology 29, 446–455 (2014) 15. Pries, A.R., Neuhaus, D., Gaehtgens, P.: Blood viscosity in tube flow: dependence on diameter and hematocrit. Am. J. Physiol. 263, H1770–H1778 (1992) 16. Pries, A.R., Secomb, T.W.: Microvascular blood viscosity in vivo and the endothelial surface layer. Am. J. Physiol. Heart Circulatory Physiol. 289, H2657–H2664 (2005) 17. Secomb, T.W.: Blood flow in the microcirculation. Annu. Rev. Fluid Mech. 49, 443–461 (2017) 18. Gamilov, T.M., Liang, F.Y., Simakov, S.S.: Mathematical modeling of the coronary circulation during cardiac pacing and tachycardia. Lobachevskii J. Math. 40, 448–458 (2019) 19. Cohen, A.T., Hamilton, M., Mitchell, S.A., Phatak, H., Liu, X., Bird, A., Tushabe, D., Batson, S.: Comparison of the novel oral anticoagulants apixaban, dabigatran, edoxaban, and rivaroxaban in the initial and long-term treatment and prevention of venous thromboembolism: systematic review and network meta-analysis. PLOS ONE 10(12), e0144856.1–e0144856.14 (2015) 20. Bodnar, T., Sequeira, A., Prosi, M.: On the shear-thinning and viscoelastic effects of blood flow under various flow rates. Appl. Math. Comput. 217(11), 5055–5067 (2011) 21. Johnston, B.M., Johnston, P.R., Corney, S., Kilpatrick, D.: Non-Newtonian blood flow in human right coronary arteries: transient simulations. J. Biomech. 39(6), 1116–1128 (2006)

248

S. S. Simakov and T. M. Gamilov

22. Formaggia, L., Gerbeau, J.-F., Nobile, F., Quarteroni, A.: On the coupling of 3D and 1D NavierStokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191(6–7), 561–582 (2001) 23. Sazonov, I., Nithiarasu, P.: A novel, FFT-based one-dimensional blood flow solution method for arterial network. Biomech. Model. Mechanobiol. 18, 1311–1334 (2019) 24. Alastruey, J., Parker, K.H., Peiro, J., Sherwin, S.J.: Lumped parameter outflow models for 1D blood flow simulations: effect on pulse waves and parameter estimation. Commun. Comput. Phys. 4(2), 317–336 (2008) 25. Kholodov, A.S.: Some dynamical models of multi-dimensional problems of respiratory and circulatory systems including their interaction and matter transport. In: Computer Models and Medicine Progress, pp. 127–163. Nauka, Moskva (in Russian) (2001) 26. Simakov, S.S., Kholodov, A.S.: Computational study of oxygen concentration in human blood under low frequency disturbances. Math. Models Comput. Simul. 1(2), 283–295 (2009) 27. Vosse, F.N., Hart, J., Oijen, C.H.G.A., Bessems, D., Gunther, T.W.M., Segal, A., Wolters, B.J.B.M., Stijnen, J.M.A., Baaijens, F.P.T.: Finite-element-based computational methods for cardiovascular fluid-structure interaction. J. Eng. Math. 47, 335–368 (2003) 28. Fatahian, E., Kordani, N., Fatahian, H.: The application of computational fluid dynamics (CFD) method and several rheological models of blood flow: a review. Gazi Univ. J. Sci. 31(4), 1213–1227 (2018) 29. Borzov, A.G., Mukhin, S.I., Sosnin, N.V.: Conservative schemes of matter transport in a system of vessels closed by the heart. Differ. Equ. 48(7), 919–928 (2012) 30. Quarteroni, A., Tuveri, M., Veneziani, A.: Computational vascular fluid dynamics: problems, models and methods. Comput. Vis. Sci. 2(4), 163–197 (2000) 31. Gamilov, T., Simakov, S.: Blood flow under mechanical stimulations. Adv. Intell. Syst. Comput. 1028, 143–150 (2020) 32. Çinar, Y., Senyol, ¸ A.M., Duman, K.: Blood viscosity and blood pressure: role of temperature and hyperglycemia. Am. J. Hypertens. 14(5), 433–438 (2001) 33. Valant, A.Z., Ziberna, L., Papaharilaou, Y., Anayiotos, A., Georgiou, G.C.: The influence of oxygen concentration on the rheological properties and flow of whole human blood. Rheol. Acta 55, 921–933 (2016) 34. Szendro, G., Golcman, L., Cristal, N.: Study of the factors affecting blood viscosity in patients with thromboangiitis obliterans. A preliminary report. J. Vasc. Surg. 7(6), 759–762 (1988) 35. Ge, X., Liu, Y., Yin, Z., Tu, S., Fan, Y., Vassilevski, Y., Simakov, S., Liang, F.: Comparison of instantaneous wave-free ratio (iFR) and fractional flow reserve (FFR) with respect to their sensitivities to cardiovascular factors: a computational model-based study. J. Interv. Cardiol. 2020, 4094121 (2020) 36. Gognieva, D.G., Gamilov, T.M., Pryamonosov, R.A., Vasilevsky, Y.V., Simakov, S.S., Liang, F., Ternovoy, S.K., Serova, N.S., Tebenkova, E.S., Sinitsyn, E.A., Pershina, E.S., Abugov, S.A., Mardanyan, G.V., Zakryan, N.V., Kirakosyan, V.R., Betelin, V.B., Mitina, Y.O., Gubina, A.Y., Shchekochikhin, D.Y., Syrkin, A.L., Kopylov, F.Y.: Noninvasive assessment of the fractional reserve of coronary blood flow with a one-dimensional mathematical model. Preliminary results of the pilot study. Russ. J. Cardiol. 24(3), 60–68 (2019) 37. Ge, X., Liang, F., Vassilevski, Y., Simakov, S.: Sensitivity of coronary flow reserve to cardiovascular parameters: a computational model-based study. In: 2018 IEEE-EMBS Conference Biomedical Engineering and Sciences, pp. 32–35. Kuching, Sarawak, Malaysia (2019)

Chapter 21

Simulation of the Human Head Ultrasound Study by Grid-Characteristic Method on Analytically Generated Curved Meshes Alena V. Favorskaya Abstract The solution of inverse problems of ultrasonic study of a human head requires a development of numerical methods for solving direct problems. The task is complicated by the fact that the tissue layers around the human brain are very thin and have different thicknesses, which requires both a long construction of computational meshes and development of novel modifications of numerical methods to solve this problem. We propose to solve the direct problem of ultrasonic investigation of a human head using the grid-characteristic computational method based on the structured curved grids with a sharply changing coordinate step. Coupled boundary-value problem of elastic and acoustic wave equations has been solved, e.g., cerebrospinal fluid is acoustic medium, and other tissues of a human head are elastic media. An analytical approach to constructing structured curvilinear meshes has been also developed and used. Still analytical formulae describing the human head tissue system have been developed on the basis of clinical data from MRI scans. Keywords Structured meshes · Grid-characteristic method · Curved meshes · Computational grids · Human head shape · Ultrasonic study · Analytical approach to meshes constructing

21.1 Introduction The study of human body using ultrasound has been used for a long time [1]. However, there are a lot of problems, e.g., ultrasound study of transcranial vessels of the brain [2–6], ultrasonic investigation of bones [7], and chest [8–10], which require to develop more accurate computational methods for solution inverse and direct problems of ultrasound.

A. V. Favorskaya (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 214, https://doi.org/10.1007/978-981-33-4709-0_21

249

250

A. V. Favorskaya

In this chapter, to solve the direct problem of ultrasonic flaw detection of a human head, the grid-characteristic method based on the structured curved grids with a sharply changing coordinate step was used. The work [11] is devoted to the gridcharacteristic method based on the separate structured regular computational grids with a sharp varying coordinate step. This work is its continuation, when instead of structured regular grids the structured curvilinear ones have been used [12]. The grid-characteristic method is well established for solving the problems of seismic prospecting [13, 14], ultrasonic non-destructive testing [15], and destruction of materials including the determination of the seismic resistance of structures [16, 17] and improving the safety of offshore facilities in the northern seas [18]. This chapter is organized as follows. The solved boundary-value problem is discussed in Sect. 21.2. Analytical expressions using to describe a human head tissues shape are presented in Sect. 21.3. Section 21.4 deals with covering the integration domain by the system of conformal structured curved meshes with sharp varying coordinate step. Section 21.5 is devoted to generation of structured curved grids using analytical approach. Some wave patterns [13] of solved numerical examples are given in Sect. 21.6. Section 21.7 concludes the chapter.

21.2 The Solved Boundary-Value Problem of Elastic and Acoustic Wave Equations The elastic wave equation was solved to describe elastic waves in brain, pia and dura maters, skull, skin, and fat (Fig. 21.1, Table 21.1) [19–23] using Eqs. 21.1 and 21.2. ρ

∂v(t, r) = (∇ · σ(t, r))T ∂t

(21.1)

    ∂σ(t, r) = ρ cP2 − 2cS2 (∇ · v(t, r))I + ρcS2 ∇ ⊗ v(t, r) + (∇ ⊗ v(t, r))T (21.2) ∂t To describe the acoustic waves in a human head in cerebrospinal fluid, the acoustic wave equation was solved: ρ

∂v(t, r) = −∇ p(t, r), ∂t

∂ p(t, r) = −ρcP2 (∇ · v(t, r)) ∂t

(21.3)

In Eqs. 21.1–21.3 and further in the text, t is the time being positive, r is the radiusvector lied in corresponding tissue in accordance with Fig. 21.1, v(t, r), σ(t, r), and p(t, r) are the unknown functions depending on time and coordinates, i.e., velocity (derivative of displacement on time), symmetric Cauchy stress tensor of second rank, and pressure, ρ is the density given in Table 21.1, cP and cS are the pressure P- and shear S-wave speed given in Table 21.1, respectively, ∇ is the gradient-vector, I is the unit tensor of second rank, ⊗ means the tensor product of vectors, (a ⊗ b)i j = ai b j .

21 Simulation of the Human Head Ultrasound …

251

Fig. 21.1 Mathematical models of human head: a variant No 1, green rectangle marks a wave patterns region, b variant No 2, c medium scale of variant No 1, d the nearest scale of variant No 1, e colors description

Table 21.1 Elastic and acoustic parameters of tissues Tissue

Skin and fat (No 5)

Skull (No 4)

Dura mater (No 3)

Cerebrospinal fluid (No 2)

Pia mater (No 1)

Brain (No 0)

P-wave speed, m/s

1,446

1,816

2,244

1,578

1,639

1,551

S-wave speed, m/s

537

1,088

677

−

494

100

Density, kg/m3

916

1,904

1,130

1,004

1,130

1,060

4.0

0.8

0.11

0.15

−

Width (W ), 2.5 mm

The boundary condition of given pressure was considered at the outer boundary of skin D: σ(t, r) · n(r) = P(t, r)n(r), r ∈ D,

(21.4)

252

A. V. Favorskaya

 ⎧ N ⎪ ⎪ P sin(2π f · t), ∈ S) ∩ t ≤ (r ⎨ 0 f  . P(t, r) = ⎪ N ⎪ ⎩ 0, (r ∈ / S) ∪ t > f

(21.5)

In Eq. 21.5 and further in the text, n is the outer normal to D depending on considered boundary point r, S is a place of contact of ultrasound source and skin. This source had diameter being equal to 4 cm and was placed in the intersection of horizontal axis and skin, f is its frequency being equal to 1 MHz, N is a number of periods in a wavelet being equal to 5. At the interfaces between brain and pia mater, dura mater and skull, skull and fat, the following contact condition was used: vL (t, r) = vR (t, r), σL (t, r) · m(r) = σR (t, r) · m(r), r ∈ B.

(21.6)

The following contact condition at the interfaces of between pia mater and cerebrospinal fluid, dura mater and cerebrospinal fluid was used: vA (t, r) · m(r) = vE (t, r) · m(r), σ(t, r) · m(r) + p(t, r)m(r) = 0, r ∈ B (21.7) In Eqs. 21.6 and 21.7 and further in the text, B is the considered boundary between tissues. In Eq. 21.6, indices L and R correspond to the left and right tissues, respectively, m(r) is the outer normal to the left contacting tissue depending on considered point r. In Eq. 21.7, index A corresponds to acoustic cerebrospinal fluid, index E corresponds to elastic dura or pia mater, m(r) is the outer normal to cerebrospinal fluid. The zero initial conditions were used.

21.3 Analytical Expressions of Human Head Tissues Shape Analytical formulae describing the human head tissue system have been developed on the basis of clinical data from MRI scans [24–26] in order to test the designed method of covering human head by hybrid conformal computational structured regular and curved meshes. Firstly, the ellipse approximation is used in accordance with Eq. 21.8, where widths of layers W are given in Table 21.1, parameters of the ellipse (brain) are A = 8 cm, B = 6 cm:



 x(n) (A + W (n)) cos α = y(n) (B + W (n)) sin α

(21.8)

Then, this elliptic structure is distorted in accordance with Eqs. 21.9–21.16.

21 Simulation of the Human Head Ultrasound …

253

Table 21.2 Used coefficients k n

0

1

2

3

4

5

6

7

k0

−90º

90º

45º

−30º

36º

−45º

270º

60º

k1

2

3

1

2

4

2

5

4

k2

15 μm

11 μm

11 μm

80 μm

250 μm

250 μm

50 μm

0.01%





 x(n) (A + W (n) + L(α, n)) cos α = y(n) (B + W (n) + L(α, n)) sin α L(α, n) = (α, n) + (α, n)

(21.9) (21.10)

⎧ k (n) · sin(k1 (n) · α + k0 (n)), n > NALL ⎪ ⎪ ⎨ 2 0 −1 (α, n) = N ⎪ (α, NALL + i) + k2 (n) · sin(k1 (n) · α + k0 (n)), n ≤ NALL ⎪ ⎩ i=0

(21.11)

(α, n) =

8+N 1 −1

H (α, i) +

i=0

⎧ N 2 −1

⎪ ⎪ ⎪ (α, 8 + N1 + i), n = NALL ⎪ ⎪ ⎨ i=0

N 3 −1 ⎪

⎪ ⎪ ⎪ (α, 8 + N1 + N2 i), n < (NALL − 1) ⎪ ⎩ (−1) · i=0

(21.12) (α, n) =

+∞

i=−1

H (α, n) + 2πi =

K 5 −1

H (α, n) + 2πi

(21.13)

i=K 4

⎧ K (n)HP (|α − K 1 (n)|/K 2 (n)), (α ≥ α0 ) ∪ (n ∈ [4, 7]) ⎪ ⎪ 0 ⎪ ⎨ K (n)H (|α − K (n)|/K (n)), (α < α ) ∪ (n ∈ [4, 7]) 0 P 1 3 0 H (α, n) = ⎪ K 0 (n)HN (2 · |α − K 1 (n)|/K 2 (n)), (α ≥ α0 ) ∪ (n ∈ / [4, 7]) ⎪ ⎪ ⎩ K 0 (n)HN (2 · |α − K 1 (n)|/K 3 (n)), (α < α0 ) ∪ (n ∈ / [4, 7]) (21.14) ⎧ ⎪ ⎨ 0, a > 2 HN (a) = (a − 2)2 , a ∈ [1, 2] (21.15) ⎪ ⎩ 2 2−a , a 1 HP (a) = (21.16) (a − 1)2 , a ≤ 1

254

A. V. Favorskaya

Table 21.3 Used coefficients K, geometry variant No 1, indices 0–7 n

0

1

2

3

4

5

6

7

K 0 , μm

100

40

20

40

20

20

20

1000

K1

60º

140º

205º

−20º

60º

140º

205º

−20º

K2

50º

60º

20º

50º

10º

10º

10º

10º

K3

60º

40º

50º

50º

10º

10º

10º

10º

K4

0

0

0

0

0

0

0

0

K5

1

1

1

2

1

1

1

2

The coefficients in Eqs. 21.11, 21.13, and 21.14 are given in Tables 21.2, 21.3, 21.4, 21.5 and 21.6. The angles K1 with numbers 0–7 are of particular importance. Let us call them β1 , β2 , β3 , β4 : β1 ≡ K 1 (0) ≡ K 1 (4), β2 ≡ K 1 (1) ≡ K 1 (5),

(21.17)

β3 ≡ K 1 (2) ≡ K 1 (6), β4 ≡ K 1 (3) ≡ K 1 (7).

(21.18)

These angles correspond to the corners in Fig. 21.3. The use of the “pointed hat” (Eq. 21.16) function in these corners allows the construction of a non-degenerate system of computational grids. These angles are chosen in such a way that on the real Table 21.4 Used coefficients K, geometry variant No 1, indices 8–15 n

8

9

10

11

12

13

14

15

K 0 , μm

100

20

20

20

40

40

40

50,000

K1

190º

120º

−120º

10º

180º

130º

70º

−120º

K2

70º

40º

50º

5º

3º

5º

10º

10º

K3

60º

70º

20º

5º

3º

20º

10º

10º

K4

0

0

1

0

0

0

0

1

K5

1

1

2

1

1

1

1

2

Table 21.5 Used coefficients K, geometry variant No 2, indices 0–7 n

0

1

2

3

4

5

6

7

K 0 , μm

100

40

20

40

20

20

20

1,000

K1

40º

150º

215º

−30º

40º

150º

215º

−30º

K2

50º

60º

20º

50º

10º

10º

10º

10º

K3

60º

40º

50º

50º

10º

10º

10º

10º

K4

0

0

0

0

0

0

0

0

K5

1

1

1

2

1

1

1

2

21 Simulation of the Human Head Ultrasound …

255

Table 21.6 Used coefficients K, geometry variant No 2, indices 8–15 No

8

9

10

11

12

13

14

15

K 0 , μm

60

20

20

20

40

20

40

50,000

K1

170º

120º

−120º

5º

185º

110º

−70º

−125º

K2

60º

40º

50º

7º

3º

15º

5º

7º

K3

50º

70º

20º

5º

4º

5º

10º

10º

K4

0

0

1

0

0

0

1

1

K5

1

1

2

1

1

1

2

2

shape of the head in all layers there would be curved protrusions. Using both “pointed hat” and “hat” (Eq. 21.15) functions in these angles at the same time smoothes out the effect of the “pointed hat” function. In order to comply with the periodicity, k 1 must be integers, and the function Eq. 21.13 is a series. Strictly speaking, the summation in Eq. 21.13 is carried out from − 1 to infinity. However, in computational practice, natural numbers K 4 and K 5 are determined, and summation within the limits [K 4 , K 5 ) will be enough (see Fig. 21.2c, d). The number of k might be any number greater than the number of layers. The number of K is organized as follows. The first four indices are “hats” (Eq. 21.15, Fig. 21.2a) for angles K 1 . The second four indices are “pointed hats” (Eq. 21.16, Fig. 21.2b) for these angles K 1 . In accordance with Eq. 21.12, N 1 , N 2 , and N 3 are the number of “hats” for all layers, number of “hats” for skin only, and number of “hats” for inner boundaries of skull and other inner tissues, respectively. One can use another types of additional “hat” functions to describe the concrete human head shape analytically. In both two variants of a human head geometry (Fig. 21.1a, b), the coefficients k are given in Table 21.2, N 0 = 2 (Eq. 21.11), N 1 = 1, N 2 = 2, N 3 = 5, the number of layers designated N ALL = 5. The coefficients K are given in Tables 21.3 and 21.4 for the first variant (Fig. 21.1a), and in Tables 21.5 and 21.6 for the second variant (Fig. 21.1b).

21.4 Covering the Integration Domain by the System of Conformal Structured Curved Meshes The grid-characteristic method on structured curvilinear grids involves the conversion of curvilinear grids into a system of meshes with unit cells [12]. The human head is a layered structure. To cover it with a system of conformal curvilinear grids, the following algorithm of covering was used (Fig. 21.3). This method of coverage allows the use of smooth coordinate transformations in each separate computational mesh instead of a covering method shown in Fig. 21.4. And between the computational grids marked with “yellow and red”, “green and

256

A. V. Favorskaya

Fig. 21.2 Functions for distortion of elliptic model of human head: a “hat” function, b “pointed hat” function, c “pointed hat” function, K 4 = 0, K 5 = 1, d “pointed hat” function, K 4 = 0, K 5 = 2, e “pointed hat” function, K 4 = − 1, K 5 = 3

blue”, and “violet and gray” colors, the boundary condition of complete adhesion is used. Fragments of real computational grids covering a human head are shown in Fig. 21.5. Time step in accordance with stability conditions [12] was equal to 1.5 ns. Notice that the detailed description of grid-characteristic method on structured curve grids one can find in [12].

21 Simulation of the Human Head Ultrasound …

257

Fig. 21.3 Good scheme of covering the integration domain by the system of conformal structured curved meshes: a elliptic layered structure, b the same structure after conversion [12]

Fig. 21.4 Bad scheme of covering the integration domain by the system of conformal structured curved meshes a elliptic layered structure, b the same structure after conversion [12]

21.5 Generation of Structured Curved Mesh by Analytical Approach To construct each separate computational grid discussed in Sect. 21.4, the following analytical approach was used. Firstly, one can construct analytically trapezoid with straight boundaries (this approach was discussed in [27]) and obtain coordinates x  (i, j), y  (i, j). Then, one can use Eqs. 21.19–21.21 and obtain the final curved mesh.

258

A. V. Favorskaya

a

b Fig. 21.5 Fragments of real computational grids: a colored like in Fig. 21.3, b in the corner, c layers, large scale, d the same as in Fig. 21.5c layers, small scale

21 Simulation of the Human Head Ultrasound …

c

d Fig. 21.5 (continued)

259

260

A. V. Favorskaya

Fig. 21.6 Generation structured curved mesh by analytical approach: a series of vectors V, b resulted structured curvilinear mesh



x  (i, j) y  (i, j)





 =

 x(i, j) y(i, j)

 =

x  (i, j) y  (i, j) x  (i, j) y  (i, j)



i , V00 , V01 , i ∈ [1, NX − 1], j ∈ [1, NY − 1] +F NX − 1





 +F



(21.19)

j , V10 , V11 , i ∈ [1, NX − 1], j ∈ [1, NY − 1] NY − 1

(21.20) F(c, V A , V B ) = V A + c(V B − V A )

(21.21)

Here, V are vectors pictured in Fig. 21.6. Note that each boundary line of the initial curvilinear domain is divided into NX − 1 or NY − 1 parts in accordance with the equal arc lengths of each part.

21.6 Numerical Examples In this section, the wave patterns [13] of velocity module are given in Figs. 21.7, 21.8, 21.9, 21.10 and 21.11 at different time moments in order to show a quality of the calculation by the developed modification of the grid-characteristic method on analytically constructed structured curvilinear grids with a sharply varying coordinate step.

Fig. 21.7 Wave patterns of velocity module: a 1.2 μs, b 2.1 μs, c 4.2 μs, d 4.5 μs

21 Simulation of the Human Head Ultrasound …

261

Fig. 21.8 Wave patterns of velocity module: a 4.8 μs, b 5.1 μs, c μs, d 5.4 μs

Fig. 21.9 Wave patterns of velocity module: a 5.7 μs, b 6 μs, c 6.3 μs, d 6.6 μs

Fig. 21.10 Wave patterns of velocity module: a 6.9 μs, b 7.2 μs, c 7.5 μs, d 7.8 μs

Fig. 21.11 Wave patterns of velocity module: a 8.1 μs, b 10.8 μs, c 15 μs, d 21 μs

All Figs. 21.7, 21.8, 21.9, 21.10 and 21.11 are given in rectangle pictured in green in Fig. 21.1a. One can see reflections (Figs. 21.7c, 21.7d, 21.8, 21.9, 21.10, 21.10 and 21.11), diffraction (Figs. 21.9d, 21.10, 21.11), and attenuation (Fig. 21.11) of elastic and acoustic wave processes.

262

A. V. Favorskaya

21.7 Conclusions The calculations have shown that the developed modification of the gridcharacteristic method on analytically constructed structured curvilinear grids with a sharply varying coordinate step is well applicable to solving the direct problems of ultrasound examination of human head. Also, the developed analytical approaches have proven themselves: • Description of the layered structure around the human brain. • Coverage of this structure by a system of separate computational grids with a sharply changing step. • Construction of a curvilinear structured computational grid. Such approaches allow to save the computational resources both at the preprocessing stage and at the stage of direct integration of the joint boundary-value problem of elastic and acoustic wave equations. Acknowledgements This work has been performed with the financial support of the Russian Science Foundation (project No. 20-71-10028). This work has been carried out using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC “Kurchatov Institute”, https://ckp.nrcki.ru/.

References 1. Bushberg, J., Seibert, A., Leidholdt, E., Boone, J.: The essential physics of medical imaging. Lippincott Williams & Wilkins (2011) 2. Sarkar, S., Ghosh, S., Sandip, K., Ghosh, S.K., Collier, A.: Role of transcranial Doppler ultrasonography in stroke. Postgraduate Med. J. 83(985), 683–689 (2007) 3. Rajamani, K., Gorman, M.: Trancranial Doppler in stroke. Biomed. Pharmacother. 55(5), 247– 257 (2001) 4. Vassilevski, Yu.V., Beklemysheva, K.A., Grigoriev, G.K., Kazakov, A.O., Kulberg, N.S., Petrov, I.B., Salamatova, V.Yu., Vasyukov, A.V.: Transcranial ultrasound of cerebral vessels in silico: proof of concept. Russian J. Numer. Anal. Math. Model. 31(5), 317–328 (2016) 5. Connor, C.W., Hynynen, K.H.: Patterns of thermal deposition in the skull during transcranial focused ultrasound surgery. IEEE Trans. Biomed. Eng. 51(10), 1693–1706 (2004) 6. Kyriakou, A., Neufeld, E., Werner, B., Székely, G., Kuster, N.: Full-wave acoustic and thermal modeling of transcranial ultrasound propagation and investigation of skull-induced aberration correction techniques: a feasibility study. J. Therapeutic Ultrasound 3(1), 11.1–11.18 (2015) 7. Kaufman, J., Luo, G., Siffert, R.: Ultrasound simulation in bone. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55(6), 1205–1218 (2008) 8. Mast, D., Hinkelman, L.M., Metlay, L.A., Orr, M.J., Waag, R.C.: Simulation of ultrasonic pulse propagation, distortion, and attenuation in the human chest wall. J. Acoustical Society of America 106(6), 3665–3677 (1999) 9. Salahura, G., Tillett, J.C., Metlay, L.A., Waag, R.C.: Large-scale propagation of ultrasound in a 3-D breast model based on high-resolution MRI data. IEEE Trans. Biomed. Eng. 57(6), 1273–1284 (2010)

21 Simulation of the Human Head Ultrasound …

263

10. Narasimhan, C., Ward, R., Kruse, K., Guddati, M., Mahinthakumar, G.: A high resolution computer model for sound propagation in the human thorax based on the Visible Human data set. Comput. Biol. Med. 34(2), 177–192 (2004) 11. Favorskaya, A.V., Zhdanov, M.S., Khokhlov, N.I., Petrov, I.B.: Modelling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the gridcharacteristic method. Geophys. Prospect. 66(8), 1485–1502 (2018) 12. Favorskaya, A.V., Khokhlov, N.I., Petrov, I.B.: Grid-characteristic method on joint structured regular and curved grids for modeling coupled elastic and acoustic wave phenomena in objects of complex shape. Lobachevskii J. Math. 41(4), 512–525 (2020) 13. Favorskaya, A.V., Petrov, I.B.: The use of full-wave numerical simulation for the investigation of fractured zones. Math. Models Comput. Simul. 11(4), 518–530 (2019) 14. Stognii, P., Petrov, I., Favorskaya, A.: The influence of the ice field on the seismic exploration in the Arctic region. Procedia Comput. Sci. 159, 870–877 (2019) 15. Favorskaya, A.V.: Elastic wave scattering on a gas-filled fracture perpendicular to plane Pwave front. In: Jain, L.C., Favorskaya, M.N., Nikitin, I.S., Reviznikov, D.L. (eds) Advances in Theory and Practice of Computational Mechanics: Proceedings of the 21st International Conference on Computational Mechanics and Modern Applied Software Systems, SIST, vol. 173, pp. 213–224. Springer, Singapore (2020) 16. Favorskaya, A., Golubev, V.: Study the elastic waves propagation in multistory buildings, taking into account dynamic destruction. In: Czarnowski, I., Howlett, R., Jain, L.C. (eds) Intelligent Decision Technologies: Proceedings of the 12th KES International Conference on Intelligent Decision Technologies, SIST, vol. 193, pp. 189–199. Springer, Singapore (2020) 17. Favoskaya, A.V., Petrov, I.B.: Calculation the earthquake stability of various structures using the grid-characteristic method. Radioelektronika, Nanosistemy, Informacionnye Tehnologii 11(2), 345–350 (2019) 18. Favorskaya A., Khokhlov, N.: Icebergs explosions for prevention of offshore collision: computer simulation and analysis. In: Czarnowski, I., Howlett, R., Jain, L.C. (eds) Intelligent Decision Technologies: Proceedings of the 12th KES International Conference on Intelligent Decision Technologies, SIST, vol. 193, pp. 201–210. Springer, Singapore (2020) 19. Duck, F.A.: Physical Properties of Tissue. Academic Press, London (1990) 20. Hoskins, P.R.: Physical properties of tissues relevant to arterial ultrasound imaging and blood velocity measurement. Ultrasound Med. Biol. 33(10), 1527–1539 (2007) 21. Pichardo, S., Sin, V.W., Hynynen, K.: Multi-frequency characterization of the speed of sound and attenuation coefficient for longitudinal transmission of freshly excised human skulls. Phys. Med. Biol. 56(1), 219–250 (2011) 22. Goss, S.A., Johnston, R.L., Dunn, F.: Comprehensive compilation of empirical ultrasonic properties of mammalian tissues. J. Acoust. Soc. Am. 64(2), 423–457 (1978) 23. White, P.J., Clement, G.T., Hynynen, K.: Longitudinal and shear mode ultrasound propagation in human skull bone. Ultrasound Med. Biol. 32(7), 1085–1096 (2006) 24. Hiltner, J., Fathi, M., Reusch, B.: An approach to use linguistic and model-based fuzzy expert knowledge for the analysis of MRT images. Image Vis. Comput. 19(4), 195–206 (2001) 25. Fuchs, M., Wagner, M., Wischmann, H.A., Ottenberg, K., Dossel, O.: Possibilities of functional brain imaging using a combination of MEG and MRT. In: Pantev, C., Elbert, T., Lütkenhöner, B. (eds.) Oscillatory event-related brain dynamics, NATO ASI Series, vol. 271, pp. 435–457. Springer, Boston, MA (1994) 26. Anatomy of the brain (MRI)—cross-sectional atlas of human anatomy, https://www.imaios. com/en/e-Anatomy/Head-and-Neck/Brain-MRI-3D. Last accessed 15 July 2020 27. Favorskaya, A.: Computation the bridges earthquake resistance by the grid-characteristic method. In: Czarnowski, I., Howlett, R., Jain, L.C. (eds) Intelligent Decision Technologies: Proceedings of the 12th KES International Conference on Intelligent Decision Technologies, SIST, vol. 193, pp. 179–187. Springer, Singapore (2020)

Chapter 22

Reaction–Diffusion Model of Coexistence of Viruses in the Space of Genotypes Cristina Leon

Abstract This work is devoted to the study of persistence and evolution of two viruses in the host organism taking into account characteristic aspects of viral dynamic such as virus mutation, replication, and genotype-dependent mortality, either natural or determined by an antiviral treatment. The proposed model consists of a system of nonlocal reaction–diffusion equations that describe the virus density distribution u(x, t) for the first virus and v(y, t) for the second one as functions of genotypes x and y considered as continuous variables, and of time t. These equations contain two integral terms characterizing the nonlocal competition of viruses for host cells. Each virus strain is considered as density distribution concentrated around some average genotype value. The analysis of the model shows the conditions of the coexistence of virus strains in the host organism. Keywords Virus coexistence · Virus density distribution · Nonlocal interaction · Reaction–diffusion model · Virus dynamic

22.1 Introduction Viral diseases pose a constant challenge owing, in particular, to the ability of certain viruses to rapidly vary their genetic structure. These alterations in the genetic composition result from the interaction of virus replication, recombination, or mutation and contribute to the appearance of new virus strains. This rapid virus evolution can make the immune response less effective and it can lead to the emergence of the resistance to the antiviral therapy [1]. A better comprehension of viral infection would allow the improvement of viral treatments and their effectiveness. The disease progression begins with the contamination of host cells starting the virus replication. The viral load in the host organism C. Leon (B) Peoples’ Friendship University of Russia (RUDN University), 6, Miklukho-Maklaya ul., Moscow 117198, 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 214, https://doi.org/10.1007/978-981-33-4709-0_22

265

266

C. Leon

depends on virus elimination, either by the immune response, virus natural death, or some antiviral treatment. A model describing the evolution of virus density considered as a function of its genotype is introduced in [2], and the conditions for the coexistence of virus strains are determined. In this paper, the proposed mathematical model assumes the existence of two viruses u, v in the host. This model considers the aforementioned processes that affect the concentration of virus in the host organism without taking into account the immune response. We consider the system of equations: 

∂u ∂t dv dt

= D1 ∂∂ xu2 + α1 u(1 − β1 I (u) − γ1 I (v)) − σ1 (x)u, 2 = D2 ∂∂ yv2 + α2 v(1 − β2 I (u) − γ2 I (v)) − σ2 (y)v. 2

(22.1)

These equations describe the evolution of virus densities depending on the genotypes x and y, respectively, considered as continuous variables and on time t. Here, D1 , D2 are the diffusion coefficients determined by the mutation rate, and parameters α1,2 , β1,2 , γ1,2 are positive constants. The second terms of these equations characterize virus reproduction. Here, 1 is a dimensionless caring capacity corresponding to the total number of cells, β1 I (u) and γ1 I (v) are the number of infected cells by virus u and v, respectively, which are proportional to the total viruses quantity: ∞

∞ u(x, t)dx and I (v) =

I (u) = −∞

v(y, t)dy −∞

The last term in the right-hand side of Eq. 22.1 describes virus natural death or its elimination by some antiviral treatment. Let us note that the death rate can depend on virus genotype x or y. Stationary solutions of the proposed model are found in [3], in the particular case of equal coefficients, α1 = α2 , β1 = β2 , γ1 = γ2 . In this work, we study the case where these equalities are not imposed. The virus strain is considered as density distribution concentrated around some genotype value. It is a non-negative solution of Eq. 22.1 that decays at infinity. In Sect. 22.2, we determine the conditions of the existence of positive stationary solutions of this model corresponding to coexistence of virus u and v. The analytical construction of such solutions is based on the assumption that σ1 (x), σ2 (y) are piecewise constant functions. In Sect. 22.3, the numerical analysis of system of Eq. 22.1 is carried out, which corroborates the results obtained analytically. In Sect. 22.4, we consider the generalization of the virus mortality functions σ1 (x), σ2 (y). In this case, the proof of the existence of solutions of Eq. 22.1 is carried out by the Leray–Schauder method. Conclusions are presented in Sect. 22.5.

22 Reaction–Diffusion Model of Coexistence of Viruses …

267

22.2 Analytical Solution of Virus Coexistence Stationary solutions of Eq. 22.1 considered on the whole axis satisfy the following system of equations: 

D1 u  + α1 u(1 − β1 I (u) − γ1 I (v)) − σ1 (x)u = 0, D2 v  + α2 v(1 − β2 I (u) − γ2 I (v)) − σ2 (y)v = 0,

(22.2)

where x, y adopt all real values and ∞ I (u) =

∞ u(x)dx,

I (v) =

−∞

v(y)dy. −∞

In order to find an analytical solution of this system, consider piece-wise constant functions   σ1 , |x| ≥ x1 σ2 , |y| ≥ y1 , σ2 (y) = , σ1 (x) = 0, |x| < x1 0, |y| < y1 where σ1 > α1 and σ2 > α2 . We look for a non-negative bounded solution of system of Eq. 22.2 decaying at infinity. Clearly, such solution can exist only if β1 I (u) − γ1 I (v) < 1 and β2 I (u) − γ2 I (v) < 1. Set α1 (1 − β1 I (u) − γ1 I (v)) = k12 , α2 (1 − β2 I (u) − γ2 I (v)) = k22 Then, we can rewrite Eq. 22.2 as follows: 

D1 u  + k12 u − σ1 (x)u = 0, D2 v  + k22 v − σ1 (y)v = 0.

We look for its solutions in the form: ⎧ u(x) = c2 eλ1 x for x ≤ −x1 , u(x) = c1 cos(t1 x) for |x| < x1 , ⎪ ⎪ ⎨ u(x) = c2 e−λ1 x for x ≥ x1 , ⎪ v(y) = c4 eλ2 y for y ≤ −y1 , v(y) = c3 cos(t2 y) for |y| < y1 , ⎪ ⎩ v(y) = c4 e−λ2 y for y ≥ y1 where c1 , c2 , c3, and c4 are positive constants, and k1 k2 t1 = √ , t2 = √ , λ 1 = D1 D2

σ1 − k12 , λ2 = D1



σ2 − k22 . D2

(22.3)

268

C. Leon

From the continuity of the solution and its first derivative at x = ±x1 and y = ±y1 , we obtain the following equalities: c1 cos(t1 x1 ) = c2 e−λ1 x1 , c1 t1 sin(t1 x1 ) = c2 λ1 e−λ1 x1 , c3 cos(t2 y1 ) = c4 e−λ2 y1 , c3 t2 sin(t2 y1 ) = c4 λ2 e−λ2 y1 .

(22.4)

Dividing the equations at right by the equations at left, we get equations with respect to k1 and k2 :



 k12 = σ1 cos2 k1 x1∗ , k22 = σ2 cos2 k2 y1∗ ,

(22.5)

where x1∗ = √xD1 and y1∗ = √yD1 . 1 2 From Eq. 22.3, we can see that α1 = α1 (β1 I (u) + γ1 I (v)) + k12 , where √ I (u), I (v) > 0. Therefore, k12 < α1 , and we look for a solution k1 < α1 of √ this equation. Similarly, we obtain k2 < α2 . Such solutions of Eq. 22.5 exist if x1∗ is greater than the critical value ξ1 , and y1∗ is greater than ξ2 . These critical values can be determined by the equalities 1 ξi∗ = √ cos−1 αi



 αi , i = 1, 2. σi

These solutions do not exist if x1∗ ≤ ξ1 or y1∗ ≤ ξ2 . We can now determine the integrals I (u) and I (v) : ∞ I (u) =

u(x)dx =

2c1 2c2 −λ1 x1 sin(t1 x1 ) + e , t1 λ1

(22.6)

v(y)dy =

2c3 2c4 −λ2 y1 sin(t2 y1 ) + e . t2 λ2

(22.7)

−∞

and ∞ I (v) = −∞

From Eqs. 22.6 and 22.7, considering the first relations in Eqs. 22.4, we have: I (u) = 2c1 I (v) =



1  t1 2c3 t12

sin(t1 x1 ) + sin(t2 y1 ) +

1 λ1 1 λ2

 cos(t1 x1 ) ,  cos(t2 y1 ) .

The coefficients c1 and c3 can be determined from Eqs. 22.3:





 t1 λ1 α1 γ1 α2 − k22 − α2 γ2 α1 − k12 , c1 = 2α1 α2 (β2 γ1 − β1 γ2 )(λ1 sin(t1 x1 ) + t1 cos(t1 x1 ))

(22.8)

22 Reaction–Diffusion Model of Coexistence of Viruses …

269

Fig. 22.1 Graphical solution of Eq. 22.2 for the values of parameters: D1 = D2 = α1 = α2 = 1. For the function u(x) : β1 = 0.2, γ1 = 0.1, σ1 = 2, x1 = 2, k1 = 0.576, c1 = 0.676224, c2 = 3.64075. The function u(x) is described by the blue and red solid lines that smoothly continue each other at point x1 . The component v(y) of the solution is similar





 t2 λ2 α1 β1 α2 − k22 − α2 β2 α1 − k12 . c3 = 2α1 α2 (β1 γ2 − β2 γ1 )(λ2 sin(t2 y1 ) + t2 cos(t2 y1 )) Similarly, coefficients c2 and c4 can be determined from Eq. 22.4. An example of the analytical solution is shown in Fig. 22.1. The existence of such solutions depends of the relation between the values of the coefficients α1,2 , β1,2 , and γ1,2 . If α1 > α2 , then both components of the solutions are positive for β1 ≥ β2

α1 − k12 α2 − k22

 and γ2 >

β2 γ1 . β1

(22.9)

If one of these conditions is not satisfied, then the solution  has a variable sign. If α −k 2 α1 = α2 , then the strictly inequality is required β1 > β2 α1 −k12 . The case where 2 2 α1 < α2 can be considered similarly.

22.3 Numerical Simulations of Virus Coexistence In numerical simulations, Eq. 22.1 are considered in the interval [0; L] with Neumann boundary conditions. The functions σ1 (x) and σ2 (y) are piece-wise constant (see below). If Eq. 22.9 is satisfied, then both components of the stationary solution are positive (Sect. 22.2), and the numerical solution with a positive initial condition

270

C. Leon

converges to this stationary solution. If this condition is not satisfied, and one of the components of the stationary solution has variable sign, then numerical solution converges to a stationary solution with one positive component, while the other component identically equals zero. Such solutions can be obtained from system of Eq. 22.2 if we set v(y) = 0 and solve the first equation with respect to u(x). Figure 22.2 shows the behavior of the solutions of Eq. 22.1 in the interval [0; 12]. For the first component u(x, t), the parameter values are D1 = α1 = 1, β1 = 0.2, γ1 = 0.1, where piece-wise constant function σ1 (x) = 0 for 4 < x < 8 and = 2, otherwise, initial condition = 1 for 5 < x < 7; for the component v(y, t), the parameter values are D2 = α2 = 1, β2 = 0.1, γ2 = 0.4, where σ2 (y) = 0 for 2 < x < 10 and = 6, otherwise, initial condition = 2 for 5 < x < 7. The behavior of this solution is characterized by a quick convergence to a stationary solution for which u(x), v(y) > 0. Figure 22.3 shows the behavior of the solution in the case when Eq. 22.9 are not satisfied. For the values of parameters β1 = 0.1, γ1 = 0.1, β2 = 0.1, γ2 = 0.07 with the other parameters and initial conditions the same as above, the solution converges to a stationary solution for which u(x) > 0, v(y) = 0.

Fig. 22.2 Solution u(x, t) and v(y, t) of system of Eq. 22.1 in numerical simulations converges to stationary solution with both positive components. The values of parameters are as follows: α1 = α2 = 1, β1 = 0.2, γ1 = 0.1, β2 = 0.1, γ2 = 0.4

22 Reaction–Diffusion Model of Coexistence of Viruses …

271

Fig. 22.3 Solution u(x, t) and v(y, t) of system of Eq. 22.1 in numerical simulations converges to stationary with one positive component, while the other component identically equal zero. The values of parameters are as follows: β1 = 0.1, γ1 = 0.1, β2 = 0.1, γ2 = 0.07

22.4 Generalization of the Virus Mortality Functions The previous obtained results are based on the explicit construction of solutions for piece-wise constant functions σ1,2 . In this section, we study a more general case where the functions σ1 (x), σ2 (y) are not piece-wise constant, and the analytical solution cannot be constructed. We will use the Leray–Schauder method based on the topological degree and a priori estimates of solutions. We now consider the system of equations 

u  + α1 u(1 − β1 I (u) − γ1 I (v)) − σ1 (x)u = 0, v  + α2 v(1 − β2 I (u) − γ2 I (v)) − σ2 (y)v = 0,

(22.10)

on the whole axis, where we suppose that D1 = D2 = 1, α1,2 , β1,2 , γ1,2 are positive ∞ ∞ constants, I (u) = u(x)dx, I (v) = v(y)dy and σ1 (x), σ2 (y) are a bounded −∞

−∞

non-negative sufficiently smooth functions. We look for positive solutions of this system with zero limits at infinity. Theorem Suppose that σ1 (x), σ2 (y) are sufficiently smooth bounded functions such that   σ1 > α1 , |x| ≥ x1 σ2 > α2 , |y| ≥ y1 , σ2 (y) = , σ1 (x) = |x| ≤ x0 |y| ≤ y0 0, 0,

272

C. Leon

where x1 > x0 > π2 and y1 > y0 > solutions decaying at infinity.

π . 2

Then, the system of Eq. 22.10has positive

We begin with a priori estimates of solutions. Lemma 22.1 Let u(x), v(y) be positive solutionsof Eq. 22.10, u(±∞) = v(±∞) = 0. Then β I (u) + γ I (v) < 1. Proof The proof of the lemma follows directly form the maximum principle. Let a(x) = α(1 − β I (u) − γ I (v)) − σ (x). For the first equation of system, if β I (u) + γ I (v) ≥ 1, then u(x) is a solution of the equation u  +a(x)u = 0 with a(x) ≤ 0 and a(x) ≡ 0. Therefore, u(x) cannot have positive maximum or negative minimum. Hence, u(x) ≡ 0. Similarly for v(y). Lemma 22.2 Suppose that σ1 (x) = σ1 > α1 for |x| ≥ x1 and σ2 (y) > α2 for |y| ≥ √ √ y1 with some positives σ1 , x1 , σ2 , y1 . Then, u(x1 ) < σ1 /2 and v(y1 ) < σ2 /2. Proof For x ≥ x1, the first equation from Eq. 22.10 writes as: u  − au = 0, where a = σ1 (1 − β1 I (u) − γ1 I (v)) < σ1 , 0 < a. Then, u(x) = c1 e

√ ax

√ ax

+ c2 e−

.

Since we look for bounded solutions, then c1 = 0 and √ a(x−x1 )

u(x) = u(x1 )e−

√ a(x+x1 )

for x ≥ x1 , and u(x) = u(−x1 )e−

for x < −x1 .

Therefore, ∞ I (u) > 2 x1

u(x1 ) u(x1 ) u(x)dx = √ > √ . σ1 a

It follows from Lemma 22.1 that u(x1 ) 1 > β I (u) + γ I (v) > 2 √ . σ1 A similar estimate can be obtained for v(y1 ). The lemma is proved. Lemma 22.3 Supposethat σ1 (x) is a continuous function and supx σ1 (x) ≤ M.Then, supx |u(x)|admits an estimate which depends only on M. Proof Solution u(x) of Eq. 22.10 satisfies the boundary value problem w  + b(x)w = 0, w(±x1 ) = u(±x1 )

22 Reaction–Diffusion Model of Coexistence of Viruses …

273

on the interval −x1 ≤ x ≤ x1 . Here, b(x) = 1−β I (u)−γ I (v)−σ1 (x) is a bounded continuous function, |b(x)| ≤ M + 1 ≡ m. According to the previous lemma, the boundary values of the solution are bounded. Therefore, it is sufficient to estimate the maximum of the solution inside the interval. Suppose that the function v(x) has a global maximum at some point x0 ∈ [−x1 , x1 ]. Then,  x    x x        w (x) =  w  (y)dy  ≤ w  (y)dy = |−b(y)w(y)|dy     x0

x0

x0

x

x |−b(y)||w(y)|dy ≤

= x0

m|w(y)|dy ≤ m|w(x0 )(x − x0 )| x0

= mv(x0 )|x − x0 |. Hence, for x > x0 x w(x) = w(x0 ) +

x



mv(x0 )|x − x0 |dx

w (y)dx ≥ w(x0 ) − x0

x0

x = w(x0 ) − mw(x0 ) = w(x0 ) 1 −

(x − x0 )dx x0

m(x − x0 )2 2

 ⇒ w(x) ≥ w(x0 )h(x),

0) . Denote by the interval [−x1 ; x1 ], where this function where h(x) = 1 − m(x−x 2 is  positive. Then, there exists such positive constant k independent of solution that

h(x)dx > k > 0, where the constant depends only on M and possibly on x 1 . Hence, 1 > I (w) > kw(x0 ). This inequality proves the lemma. We will use the topological degree theory to prove the existence of solutions. Lemma 22.3 above provides a priori estimates of solutions in Banach spaces. Consider the operator 2

 Aθ (u, v) =

u  + α1 u(1 − β1 I (u) − γ1 I (v)) − σ1,θ (x)u, v  + α2 v(1 − β2 I (u) − γ2 I (v)) − σ2,θ (y)v,

acting from the weighted Hölder space Cμ2+α (R) to the space Cμα (R). Here, 0 < α < 1, θ ∈ [0; 1] is a parameter. The space Cμ2+α (R) is defined as the set of functions u(x) such that u(x)μ(x) ∈ C 2+α (R). Let μ(x) = 1 + x 2 , this weight function is increasing at infinity with a polynomial rate. Introduction of weighted spaces allows the definition of topological degree for elliptic operators in unbounded domains [4].

274

C. Leon

Besides, the integrals I (u), I (v) are well defined due to it does not affect to the essential spectrum. We will suppose for simplicity that σθ (x) is an infinitely differentiable function with respect to x and θ . Other conditions will be specified later. Denote by L θ the operator obtained by linearization of the operator Aθ about u, v = 0 :  L θ (w, z) =

w  + α1 w − σ1,θ (x)w, z  + α2 z − σ2,θ (y)z.

Lemma 22.4 Suppose that the principal eigenvalue the operator L θ is positive for θ0 ≤ θ ≤ θ1 and for some θ0, θ1 . Then, there exist ε > 0 such that u m = supx u(x) ≥ ε for any positive solution of equation Aθ (0) = 0, θ0 ≤ θ ≤ θ1 . Proof Suppose that the affirmation of the lemma does not hold, then there may be a sequence of solution of equations u k (x) for θ = θk such that u m k → 0. We can assume that θk → θ∗ for some θ∗ ∈ [θ0 ; θ1 ]. Then, 0 = Aθk (u k ) = Aθk (0) + L θk u k + o(||u k ||) = L θk u k + o(||u k ||). Set wk = ||uu kk || . Then, L θk wk = o(1). Since L θk is proper with respect to w and θ, then the sequence wk is compact and we can choose a convergent subsequence wk → w0 . Hence, L θ∗ w0 = 0. Since the functions u k (x) are positive, then w0 (x) > 0 for all x. Therefore, the operator L θk has a zero eigenvalue with a positive eigenfunction. However, the only positive eigenfunction corresponds to the principal eigenvalue [5]. We obtain a contradiction with the assumption that the principal eigenvalue of the operator L θ∗ is positive. The same corresponds to the second equation of the system. Lemma 22.5 Suppose that σ1 (x) = σ1 > α1 for |x| ≥ x1 and σ2 (y) = σ2 > α2 f or |y| ≥ y1 with some positives σ1 , x1 , σ2 , y1 , and the principal eigenvalue of the problem 

u  + u − σ1 (x)u = λu, v  + v − σ2 (y)v = λv,

is positive. Then, the system of Eq. 22.10 has a positive solution converging to 0 at infinity. Proof Set σθ (x) = (1 − θ )σ (x) + θ σ1,2 . Since σ1 > α1 (or σ2 > α2 ), then the operator L θ when θ = 1 with 1 ≥ σ1 ≥ α1 has no stable solutions and with σ1 > 1 has the spectrum in the half-plane. Let us not that the essential spectrum Se (L θ ) of the operator L θ does not depend on θ , and Re Se (L θ ) ≤ −δ < 0 for some positive δ. Denote the principal eigenvalue of this operator by λ0 (θ ). According to the assumption of the lemma λ0 (0) > 0. The function of θ ∈ [0, 1] is a monotonically decreasing function and there exists such θ0 ∈ [0, 1] that λ0 (θ0 ) = 0, λ0 (θ ) > 0 for 0 < θ < θ0 , λ0 (θ ) < 0 for θ0 < θ < θ1 ,

22 Reaction–Diffusion Model of Coexistence of Viruses …

275

where θ1 is some value in the interval (θ0 , 1]. Since the eigenvalue can approach the essential spectrum, we cannot guarantee its existence for all θ ∈ [0, 1]. Let us consider the equation Aθ (u) = 0 in a small vicinity of the bifurcation point θ = θ0 . For this value of parameter, the trivial solution u = 0 losses its stability leading to the appearance of another solution u θ (x). This solution is positive since the principal eigenfunction w0 (x) is positive in according to Lemma 22.4. Furthermore, the solution, equals 1. Indeed, from the homotropy invariance of the degree, it follows that ind(0) + ind(u θ ) + ind(u˜ θ ) = 1 for all θ > θ0 and sufficiently close to θ0 . Here, u˜ θ (x). Since ind(0) = −1 being equal (−1)v , where v = 1 is the number of positive eigenvalues of the linearized operator, then ind(u θ ) = ind(u˜ θ ) = 1. It follows from Lemma 22.3 that ||u||C 2+α (R) < M0 for some positive constant M0 and for any positive solution u of the equation Aθ (u) = 0. Next, from Lemma 22.4, we conclude that ||u||C 2+α (R) > δ(θ ) for some positive δ(θ ), θ < θ0 . Consider the following domain 

= u ∈ C 2+α (R) , u(x) > 0, x ∈ R, δ0 < ||u||C 2+α (R) < M0 } for some δ0 > 0 sufficiently small. Choose θ2 < θ0 such that δ(θ ) > δ0 for 0 ≤ θ ≤ θ2 . Since A0 (u) = 0 for u ∈ δ , 0 ≥ θ ≤ θ2 , then the value of the degree

γ (Aθ , ) does not depend on θ ∈ [0, θ2 ]. Hence, γ (A0 , ) = γ Aθ2 , = ind u θ2 = 1, and the equation A0 (u) = 0 has a solution in . This conclusion proves the theorem.

22.5 Conclusions The main goal of this work is to study the conditions of coexistence of two viruses in the host organism. The proposed model is represented by a system of nonlocal reaction–diffusion equations that take into account virus mutation, reproduction, and mortality. The nonlocal terms describe virus multiplication and competition for uninfected host cell. Virus mortality depends on the genotype and it can be due to antiviral treatment or to its natural death. We consider virus strain as a density distribution concentrated around some genotype value. This is a non-negative solution decaying at infinity. The analysis of the model shows the existence of such solutions under some conditions on parameters. Namely, the admissible interval where virus multiplication rate is larger than its mortality rate should be sufficiently long, and the mutation rate should be small enough.

276

C. Leon

We identify the conditions to the existence of positive solutions in the case when the virus mortality functions σ1 (x), σ2 (y) are defined as a piece-wise constant functions, this is the conditions to the coexistence of viruses in the host organism. For a more realistic representation of the virus mortality functions, σ1 (x), σ2 (y) are considered as a bounded non-negative sufficiently smooth function. The existence of positive solutions with zero limits at infinity for this generalized σ1,2 is studied in Sect. 22.4. The obtained results are biologically realistic; however, it is necessary to bear in mind that this model is relatively simple for studying the dynamics of viruses. The model presented in this work does not consider the immune response and other complex aspects of the interaction of the virus in the host. On the other hand, this simplification allows us to quantitatively analyze some conditions for the coexistence of viruses that could be unavailable in a more sophisticated model. In general, the considered model reveals interesting aspects of the viral dynamics that will be contemplated in the future works.

References 1. Bocharov, G., Volpert, V., Ludewig, B., Meyerhans, A.: Mathematical Immunology of Virus Infections. Springer International Publishing (2018) 2. Bessonov, N., Bocharov, G., Meyyerhans, A., Popov, V., Volpert, V.: Nonlocal reaction-diffusion model of viral evolution: Emergence of virus strains. Mathematics 8(1), 117.1–117.20 (2020) 3. Leon, C., Popov, V., Volpert, V.: Viruses competition in the genotype space. ITM Web of Conferences 31, 02002.1–02002.7 (2020) 4. Volpert, V.: Elliptic Partial Differential Equations. Vol. 1. Fredholm Theory of Elliptic Problems in Unbounded Domains. Birkhäuser, Basel (2011) 5. Volpert, V.: Elliptic Partial Differential Equations. Vol. 2. Reaction-Diffusion Equations. Birkhäuser, Basel (2014)

Chapter 23

Numerical Simulation of the Denture Prosthesis Integrity Under Typical Chewing Loads Sergey D. Arutyunov , Dmitry I. Grachev , Grigoriy G. Bagdasaryan , and Alexander D. Nikitin Abstract The chapter considers the problems of stress analysis for the removable prosthesis bases of the upper and lower jaws under typical chewing loads. The prosthesis structural integrity is studied. In this modeling, a physically justified boundary conditions were developed and applied. The influence of various technological features of the prosthesis’s geometry on its structural integrity is studied. The critical load values and its locations are identified for different types of chewing loads. It was shown that taking into account the compliance and expression of the torus area are very important for correct stress distribution calculation for the prosthesis of the upper jaw. Also, it is shown that the presence of torus area leads to additional longitudinal displacements of the back part of the prosthesis basis that leads to higher value of the tensile stresses at the front part of the structure. For the case of the lower jaw prosthesis’s basis, it is of great importance to take into account the non-uniform distribution of stiffness of the mucosa along the alveolar ridge. It is shown that for both models (upper and lower jaw), the maximum stress occurs under the chewing loads applied at the blocks of incisors and canines. In this case, maximum stresses, as a rule, are localized in the vicinity of technological notches or teeth-basis connection. It is shown that for all the studied configurations, the maximum stresses do not exceed the ultimate tensile strength (UTS) of the corresponding material, however, they can exceed their fatigue limits. Thus, the most likely mechanism of the prosthesis basis failure under chewing loads is fatigue of S. D. Arutyunov · D. I. Grachev · G. G. Bagdasaryan A.I. Evdokimov Moscow State University of Medicine and Dentistry, 20/1, Delegatsksya, Moscow 127473, Russian Federation e-mail: [email protected] D. I. Grachev e-mail: [email protected] G. G. Bagdasaryan e-mail: [email protected] S. D. Arutyunov · D. I. Grachev · G. G. Bagdasaryan · A. D. Nikitin (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 214, https://doi.org/10.1007/978-981-33-4709-0_23

277

278

S. D. Arutyunov et al.

material. In this chapter, we introduce the models of the upper and lower jaws that takes into account the physico-mechanical features of soft tissues. Keywords Mathematical modeling · Biomechanics · Denture · Prosthesis · Fracture

23.1 Introduction For a long time, the prosthesis shape and its implantation were designed based on the individual dentist experience and personal comfort feelings of patients. Sometimes, these criteria are not safety enough to guarantee the sufficient durability of the structure. It happens that in-service prosthesis fracture can appear much early compare to the average service life of such structures. The premature failures are often related to the specific individual structure of the oral cavity and its morphology. The second reason is the natural structure of the dentition and the feeling of comfort for the patient associated with such structure. In the first case, the dentist should be guided by the general empirical knowledge and own experience. At the same time, there are no clear permissible limits for varying the prosthesis geometry parameters. In the second case, the sole criterion of the appropriate shape for the prosthesis is a personal feeling of comfort for the patient. It happens that the artificial structure that is comfortable for the patient has an extremely low in-service life. The mathematically based recommendations for these parameter values are needed. With the development of the modern computer technologies in computer-aided design (CAD), computer-aided engineering (CAE) (or engineering calculations), and computer-aided manufacturing (CAM) (or production methods), these technologies are increasingly being introduced into the medical practice and, in particular, in dentistry. In the last decade, the use of finite element methods and numerical modeling in applied problems of dentistry, prosthetics, etc., has become common practice [1–5]. The main goals and objectives of the computer modeling in dentistry are a high degree of personalization of the models used for calculation [6, 7], while insufficient attention is paid to the general issues of the behavior of the dental structure (prosthesis, implant, bridge) and biological tissues (bone tissue, mucosa) [8, 9]. Moreover, there are limited information in the literature on the correct choice of physically justified boundary conditions [10]. It is obvious that the results of a personalized calculation of the stress–strain state are largely depend on the used model and boundary conditions. Thus, during the developing of a clinical method or new dental structure, it is very important to have a uniformed, standardized instruments—a physically-based model, boundary conditions, and models of the biological tissue behavior. One of the most difficult questions in this case is the modeling of the mechanical response of biological tissues to mechanical loads. This is due to the fact that the physico-mechanical properties of bones, mucous membranes, etc., are largely depend on the individual characteristics of the patient, gender, and, in addition, can change with age. Taken into account these factors, it becomes obvious

23 Numerical Simulation of the Denture Prosthesis …

279

that the development of general ideas about the behavior of dental structures, as well as determining the boundaries of variation in the physico-mechanical properties of biological tissues and their effect on the stress–strain state of the prosthesis, is extremely relevant. In this chapter, we propose models of the upper and lower jaws of the human, taking into account the physico-mechanical features of their tissues. Models of laminar dentures are proposed for modeling the stress state under the typical chewing loads that occur in dentures both due to biting off and chewing food. The influence of the technological notches of the denture and prosthesis geometry parameters on the structural integrity of the product is investigated. The chapter is organized as follows. Section 23.2 provides mathematical modeling of laminar dentures of the upper and lower jaws. The boundary conditions and chewing loads are given in Sect. 23.3. Section 23.4 presents results and discussions, while Sect. 23.5 finalizes the chapter.

23.2 Mathematical Modeling of Laminar Dentures of the Upper and Lower Jaws The 3D models of the upper and lower jaw prosthesis are presented in Fig. 23.1a, b respectively. Each model consists of a prosthesis blade or basis and dentition. The size of the model is corresponding to the size of a typical adult jaw. The prosthetic basis was assumed to be a constant thickness shell of 1 mm. The problem was solved in a linear elastic approach. The two different materials were used for this model: Polymethyl methacrylate acryl (PMMA) with lower (in pink color) and higher (in white color) mechanical properties, as shown in Table 23.1.

a

b

Fig. 23.1 CAD models of lamellar bases in the case of complete absence of teeth: a the upper jaw, b the lower jaw

280

S. D. Arutyunov et al.

Table 23.1 Mechanical properties of the acrylic plastics used for denture prosthesis Material

Application

Mechanical properties

Acrylic plastic R

Prosthesis basis

Young’s modulus—1000 MPa Density—1000 kg/m3 Poisson ratio—0.3

Acrylic plastic W

Dents

Young’s modulus—2000 MPa Density—1000 kg/m3 Poisson ratio—0.3

Both materials were assumed homogeneous, isotropic materials with the same density but different Young’s modulus. The ultimate tensile strength of PMMA materials is vary from 47 to 79 MPa. In the framework of the present study, the middle line value was taken—60 MPa. The material of the dentition is typically harder and more durable. The connection between the teeth and the prosthesis basis is made by a chemical bond, which will allow us to modeling it by the condition of full adhesion.

23.3 Boundary Conditions and Chewing Loads During the formulation of the boundary and loading conditions for both models, the results of compliance studies [11] for the mucous membranes of the upper and lower jaw were used. It is empirically shown that for the upper jaw the minimal compliance is observed in the area of a bone closure. It is the area of the torus. In Fig. 23.2a, this region is indicated by an oval area located on the symmetry axis of the denture. In this area, the compliance is assumed to be lower. Except this torus area, the boundary condition of elastic base is applied to the rest part of the internal surface (in blue color, Fig. 23.2a). It is assumed that the reaction of the mucosa depends on the magnitude of the external effort that is due to the physical reaction of soft tissues. Thus, the selected boundary conditions are physically and clinically justified.

a

b

Fig. 23.2 Schematic representation of areas on the lower surface of the base of the removable denture of the lower jaw: a oral area and b area used to set the boundary conditions

23 Numerical Simulation of the Denture Prosthesis …

281

In the case of a lower prosthesis basis, the distribution of compliance is a little more complex. In Fig. 23.2b, a schematic representation of the areas used to set the boundary conditions for the basis of the lower basis is shown. It was shown that minimal mucosal compliance is observed along the apex of the alveolar ridge. Moreover, the amount of compliance is heterogeneous along this ridge. It can be outlined four different sectors along the alveolar ridge at each side (left and right branches), within which the compliance is considered constant. To select the compliance values in these sections, the criterion of minimal basis skewness under uniform load on all teeth is used. The minimum value of compliance on the alveolar ridge is taken to be 80% of the value in the torus area. During the chewing load simulating, a complete cycle of biting and chewing of a food is reproduced. For this purpose, four separate tooth blocks were identified as block I (incisors), block II (canine), block III (premolars), and block IV (molars) (see Fig. 23.3). It is assumed that the maximum load is determined by the amount of muscle effort, taken equal to 100 N. Further, the magnitude of muscle effort is converted into the value of pressure acting on the corresponding tooth blocks. The load can be as symmetrical in Fig. 23.3a, as well asymmetric in Fig. 23.3b. In this case, the pressure value, as before, is calculated keeping constant the muscle effort. The block separation was the same for the upper and lower jaw. The corresponding pressure values are given in Table 23.2.

Fig. 23.3 Blocks of teeth: block 1 (incisors), block 2 (canine), block 3 (premolars), and block 4 (molars) to which the load is applied

Table 23.2 Loading parameters for symmetrical and non-symmetrical loadings Loading type/location Symmetrical loading, pressure, MPa Non-symmetrical loading, pressure, MPa

Block I

Block II

Block III

Block IV

5.2

4.2

0.9

0.5

10.4

8.4

1.8

1.0

282

S. D. Arutyunov et al.

23.4 Results and Discussion The calculations were carried out for the case of loading on tooth blocks from 1 to 4. The results are presented in Figs. 23.4, 23.5, 23.6 and 23.7. In the case of an asymmetric load on the first block (incisors), it is shown that the prosthesis sagging is most significant in the front of the prosthesis, where the highest stresses are also observed. Moreover, the nature of the stress distribution is antisymmetric, as shown in Fig. 23.4c. This effect is more pronounced in the case of the basis of the upper jaw prosthesis. The areas of maximum stresses are localized in the vicinity of technological notch used for cords passing. For the case of loading on the second block (canine), it is shown, Fig. 23.5, that the area of subsidence of the prosthesis basis becomes larger compared to the case of the load on the incisors. Moreover, this effect is more pronounced for the case of the basis of the prosthesis of the upper jaw. The calculation results show that subsidence is observed up to molars in the case of an upper jaw prosthesis and affect only premolars in the case of a lower jaw prosthesis. Stress fields have maximums on the load side. The effects of asymmetry in the case of load on the incisors are not observed.

Fig. 23.4 Calculation of vertical displacements under load on the incisors: a for the prosthesis basis of the upper jaw, b for the prosthesis basis of the lower jaw and the corresponding stress intensity, c the upper prosthesis, and d the lower prosthesis

23 Numerical Simulation of the Denture Prosthesis …

283

Fig. 23.5 Calculation of vertical displacements under load on the canine: a for the prosthesis basis of the upper jaw, b for the prosthesis basis of the lower jaw and the corresponding stress intensity, c the upper prosthesis, and d the lower prosthesis

For the case of loading on the premolars, Fig. 23.6, it is shown that the region of maximum vertical displacements is still vast, but already less pronounced than in the previous cases. The base of the lower jaw prosthesis is thus less sensitive to stresses on premolars. The maximum vertical displacements for him are almost half lower in comparison with the basis of the upper jaw. The maximum stresses in this case are mainly localized in the vicinity of loaded itching. However, small features are observed in the area of technological openings under the cords in the case of the basis of the lower jaw prosthesis. Thus, when chewing, write on premolars the risks of destruction of the base become much lower in comparison with cases of biting and chopping food. Potential destruction can be observed to be realized either directly in the tooth material or in the vicinity of their base. In Fig. 23.7, the results of vertical displacements calculations and stress intensity for the case of a load on molars are shown. The areas of vertical movement in this case move to the back of the basis. The displacement value is much lower compared to previous cases and does not exceed 80% of the observed early, when the loading was on the first blocks of teeth. The tendency to lower sensitivity of the basis of the lower prosthesis is preserved. The areas of maximum stresses during loading on molars are localized at the base of the corresponding teeth. Nevertheless, the maximum stresses remain small compared to the ultimate tensile strength of the material, which allows us to conclude that loading on molars cannot lead to failure under either static or

284

S. D. Arutyunov et al.

Fig. 23.6 Calculation of vertical displacements under load on the premolars: a for the prosthesis basis of the upper jaw, b for the prosthesis basis of the lower jaw and the corresponding stress intensity, c the upper prosthesis, and d the lower prosthesis

dynamic loading. In the case of the basis of the prosthesis of the upper jaw, the areas of maximum stress encircle the base of the teeth, and areas of increased stress can be detected in the basis. The greater tension of the prosthesis basis for the upper jaw is due to the presence of the shell-shaped part connecting the parts of the prosthesis resting on the alveolar process. The maximum stresses under typical chewing loads decrease with the advancement of the food coma from the front to the back of the prosthesis (blocks 1–4). The absolute values of the maximum stresses are following: 40.2 MPa (block 1), 18.8 MPa (block 2), 5.7 MPa (block 3), and 2.3 MPa (block 4). For all cases, the maximum stresses do not exceed the tensile strength, which cannot lead to structural failure in a single loading cycle. However, for the case of load on block 1 and block 2, the levels of maximum stresses are sufficient for failure under dynamic (fatigue) loading. The summary of the results is given in Table 23.3.

23 Numerical Simulation of the Denture Prosthesis …

285

Fig. 23.7 Calculation of vertical displacements under load on the molars: a for the prosthesis basis of the upper jaw, b for the prosthesis basis of the lower jaw and the corresponding stress intensity, c the upper prosthesis, and d the lower prosthesis

Table 23.3 Maximum stress intensity for different loading conditions Loading type/location

Block I

Block II

Block III

Block IV

Symmetrical loading, stress max, MPa

18.8

14.4

3

1.1

Non-symmetrical loading, stress max, MPa

40.2

22.1

5.7

2.3

23.5 Conclusions Models of denture prosthesis for the lower and upper jaws are presented. Physically justified boundary conditions have been developed and applied with taking into account the morphology of the soft tissues structure within an oral cavity. The symmetric and asymmetric loads on various tooth blocks that simulate a complete cycle of chewing food were studied. It was shown that between all the considered cases, the maximum stresses are corresponding to the loads on the incisors and fangs. In the case of loading on premolars and molars, the maximum stresses are significantly lower. It is shown that in the case of loading on the front tooth blocks, the parameters of technological notches of the prosthesis play an important role. The shift of teeth installation line lead to increase of stress intensity compared to the normal installation line. The oral shift of the installation line is more critical between

286

S. D. Arutyunov et al.

all the studied configurations. The tilt angle effect is less critical compared to the shift effect. In all considered cases, no stresses were found that were close to or exceed the ultimate tensile strength of the material. Thus, a potential fracture mechanism is reloading leading to the formation of fatigue cracks. Acknowledgements The work is realized in the framework of the state contract of the Institute for Computer Aided Design of the RAS. The authors are grateful to A. Zhuravlev for the significant contribution to the present work.

References 1. Iskanderov, R.M., Gvetadze, R.Sh., Butova, V.G., Andreeva, S.N., Timofeev, D.E.: The overall development strategy of dental laboratories equipped with CAD/CAM-systems. Stomotologia 98(2), 8–12 (2019) 2. Chumachenko, E.N., Arutyunov, S.D., Lebedenko, IYu., Ilinich, A.N.: Analysis of load distribution and the likelihood of irreversible changes in the jaw bone tissue during orthopedic treatment using dental intraosseous implants. Clinical Dentistry 2, 1–44 (2002) 3. Weisgeim, L.D., Scherbakov, L.N., Goncharov, A.A.: Influence of separate clinical aspects on the tensely deformed condition of the biomechanical system «combined prosthesis – supporting tissues». Stomatologia 4(16), 18–20 (2009) 4. Al-Ali, M.A., Al-Ali, M.A., Takezawa, A., Kitamura, M.: Topology optimization and fatigue analysis of temporomandibular joint prosthesis. World Journal of Mechanics 7(12), 323–339 (2017) 5. Dubova, L.V., Tsarev, V.N., Zolkina, Y.S., Malik, M.V., Nikitin, I.S., Chuev, V.P.: Comparative assessment of milled materials for temporary unremovable dentures supported by the isoelastic implants according to the experimental study of their stress-strain states and microbial adhesion. Clinical Dentistry 3(87), 74–78 (2019) 6. Arutyunov, S.D., Chumachenko, E.N., Lebedenko, IYu., Arutyunov, A.S.: Comparative analysis of the mathematical modeling results on the stress-strain state of various designs for pin dentures. Dentistry 2, 1–41 (2001) 7. Shanidze, Z.L., Muslov, S.A., Arutyunov, A.S., Astashina, N.B., Arutyunov, S.D.: Biomechanical approach to dental orthopedic treatment of patients with postoperative maxillary defect. Russ. J. Biomech. 24(1), 28–38 (2020) 8. Perelmuter, M.N.: Analysis of stress-strain state of dental implants by boundary integral equations method. PNRPU Mech. Bull. 2, 83–95 (2018) 9. Wilke, H., Claes, L., Steinemann, S.: The influence of various titanium surfaces on the interface shear strength between implants and bone. In: Heimk, U., Lee, A. (eds.) Clinical Implants Material Advances in Biomaterials, vol. 9. Elsevier Science Publishers BV, Amsterdam (1990) 10. Aruyunov, S.D., Grachev, D.I., Nikitin, A.D.: Mathematical modelling on the fracture of a laminar prosthesis basis under natural chewing loads. IOP Conf. Ser. Mater. Sci. Eng. 747, 012065.1–012065.6 (2020) 11. Kulazenko, V.I., Berezovskiy, S.S.: Clasp prosthetics. K.: Zdorovie (in Russian) (1975)

Chapter 24

Numerical Modeling of Elastic Wave Propagation in a Human Craniocerebral Area with Discontinuous Galerkin Method Katerina A. Beklemysheva

and Igor B. Petrov

Abstract Numerical modeling of dynamic mechanical behavior of biological tissues on a large scale is a pressing problem, which is required for the development of diagnostic and operational ultrasound technologies. Unlike static modeling, it does not necessarily require the introduction of complex material models, but the modeling of elastic waves on grids based on real human anatomy that increases the calculation time and requirements for numerical method stability. Discontinuous Galerkin method meets these requirements and allows to model elastic waves in craniocerebral area with high precision. Numerical results for several three-dimensional model problem statements are given in this chapter. Keywords Numerical modeling · Discontinuous Galerkin method · Impact damage · Biomechanics · Medical ultrasound

24.1 Introduction The development of diagnostic and operational ultrasound is hindered by the lack of stable, effective and reliable numerical models for dynamic behavior of tissues. We can highlight several issues that are responsible for that. The first one is the rheology of biological tissues. The complex cellular structure of soft tissues leads to a complex nonlinear behavior of a material during loading [1, 2]. This effect is more prominent during a static loading due to large deformations of a sample [3]. A very short dynamic loading during an ultrasound procedure does not cause large deformations, and a lot of research is limited to acoustic material model [4–6], which is linear and does not take into account shear waves, surface waves and K. A. Beklemysheva (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] © 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 214, https://doi.org/10.1007/978-981-33-4709-0_24

287

288

K. A. Beklemysheva and I. B. Petrov

other purely elastic effects. These waves dissipate very fast in soft tissues, and we are not able to consider them. This model allows to apply simple numerical methods like ray tracing [7]. It is very effective in terms of calculation time, since it does not model the full three-dimensional volume but only material surfaces [8]. In addition, graphics cards [9] can effectively parallelize it. This method used on complex biological grids can lead to numerical artifacts caused by ray discretization and the scarcity of them in areas that require the highest precision. Unlike ray tracing, grid methods like discontinuous Galerkin [10] or grid-characteristic method [11] allow to increase the amount of calculation nodes in these areas and effectively distinguish, for example, small inclusions. Ray tracing can be modified to dynamically adjust the density of rays, but it limits possible parallelization options [12]. Also, in existing modifications, ray tracing does not allow to use the elastic material model, which leads to the second big issue. The second issue is that we have to model elastic effects. We can limit ourselves to linear models, which is acceptable for dynamic modeling [2, 3], and an acoustic approach is very effective for modeling of the ultrasound in soft tissues [4–6]. However, the large-scale modeling that includes a whole region of a human body requires including bone tissues [13]. Some of bone tissues are highly porous, and we can effectively model them either by cavities or by non-reflective border condition depending on a tissue. Other bone tissues are very dense and rigid. Shear and surface waves appear in bone tissues even if they are surrounded by soft tissues due to refraction on material contact surface [14]. These waves do not dissipate fast enough in the dense bone tissue, so we have to consider them. The third, and very important, issue is that it is hard to determine the material parameters. Unlike construction materials like steel or concrete, biological tissues have a wide scattering of values of material parameters. Individual differences cause the scattering, and material parameters not only vary from person to person, but can even depend on a diet or a time of a day [1, 15]. Also, traditional methods of measuring the elastic material parameters require a lot of large-sized specimens which is impossible for living tissues. New methods of determining material parameters are necessary, and numerical modeling can be a very useful tool for that. If we introduce modern achievements in inverse problem solving and machine learning for the problem of material parameters [16], it will be of great help, but, primarily, a stable and effective numerical method for direct problems is required. In this chapter, we use the discontinuous Galerkin method for the modeling of a full human craniocerebral area. This numerical method was developed and used for modeling of construction materials [10, 17, 18] and ice [19]. It is very effective for a high-precision modeling of elastic wave effects and stable on complex biological grids even during long-term calculations. It has a long calculation time compared to grid-characteristic method [11] or ray tracing [7], but it can be parallelized and further improved. A calculation grid used in this chapter was developed in [20]. Several model problem statements are considered. To demonstrate method capabilities, we modeled impacts on different sides of the head—both temple, vertex, nape and forehead. We calculated three-dimensional patterns of stress tensor and velocity

24 Numerical Modeling of Elastic Wave Propagation …

289

vector distributions during the propagation of elastic waves inside the craniocerebral area. The chapter is organized as follows. Section 24.2 describes a numerical method, while Sect. 24.3 presents the numerical results. Section 24.4 concludes the chapter.

24.2 Numerical Method The isotropic linear elastic material used in this research is described by the system Eqs. 24.1–24.2 [10], where ρ is the material density, v is the velocity vector, ∇ is the del operator, σ is the stress tensor, λ and μ are the Lame material parameters, δ is the Kronecker delta. ρvi = ∇ j · σi j

(24.1)

   σi j = λδi j δkl + μ δik δ jl + δil δ jk εkl

(24.2)

The discontinuous Galerkin method on irregular grids is described in [10, 17, 19]. The main idea of this method is to approximate the variable vector value in a grid element by a polynome, introduce numerical fluxes through the grid element edges and solve a Riemann discontinuity problem for each edge [17]. As a system of basis polynomials, the orthogonal Dubiner polynomials of fifth order are used, while the fifth-order Dormand–Prince method with an adaptive step is used as an integrator. The time derivation is based on a high-order Runge–Kutta method, which is local and well suited for parallelization. The calculations were performed on two computers with six-core processors (Intel Core i7-5820 K CPU 3.30 GHz) and 64 GB RAM.

24.3 Numerical Results In this section, the problem statement is formulated in Sect. 24.3.1. Modeling of various types of strikes such as temple strike, nape strike, vertex strike and forehead strike is considered in Sects. 24.3.2–24.3.5, respectively.

24.3.1 Problem Statement A general view of the calculation area and a slice with marked materials are presented in Fig. 24.1. The short impact was modeled as an initial velocity in a small area at the impact point. All borders were considered as free.

290

K. A. Beklemysheva and I. B. Petrov

Fig. 24.1 Problem statement: a slice in YZ plane with marked materials, b general view of the calculation area

Table 24.1 Material parameters for model problems Name

Longitudinal waves speed, km/s

Shear waves speed, km/s

Density, g/cm3

Fat

1.4

0.7

0.9

Muscle

1.6

0.8

1.1

Brain

1.5

0.75

1.0

Bone

4.0

2.0

1.8

Vessels

1.6

0.8

1.06

Different impact points such as both temples, vertex, forehead and nape were considered. Temples are perspective points in terms of ultrasound [12] because the skull bone is thinner and does not contain porous tissues or large cavities. Nape, vertex and forehead strikes are common traumatic events, and their modeling is interesting not only for an ultrasound research, but also for sports and casual traumas. The material parameters are given in Table 24.1.

24.3.2 Temple Strike Figures 24.2 and 24.3 show the elastic wave pattern on a clip of the calculation area after a strike at the left temple. The strongest wave is propagating along the surface with a low speed. The fastest wave travels along the skull bones and reaches the other side of the calculation area approximately at the same time as the wave that travels through the brain tissue in the center.

24 Numerical Modeling of Elastic Wave Propagation …

291

Fig. 24.2 Dynamics of pressure distribution after a strike at the left temple on a clip of the calculation area. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

Fig. 24.3 Dynamics of velocity modulus distribution after a strike at the left temple on a clip of the calculation area. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

292

K. A. Beklemysheva and I. B. Petrov

Waves reflect from cavities inside the craniocerebral area creating a complex pattern of elastic waves. The dynamics of the wave pattern is clearer in Figs. 24.4 and 24.5, where we present the velocity modulus distribution on ZY and YZ planes through the impact point. The velocity distribution after a strike at the right temple is slightly different (see Figs. 24.6 and 24.7) because human anatomy is not exactly symmetrical.

Fig. 24.4 Dynamics of velocity modulus distribution after a strike at the left temple on a XZ plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

Fig. 24.5 Dynamics of velocity modulus distribution after a strike at the left temple on a XY plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

24 Numerical Modeling of Elastic Wave Propagation …

293

Fig. 24.6 Dynamics of velocity modulus distribution after a strike at the left temple on the surface of the calculation area. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

Fig. 24.7 Dynamics of velocity modulus distribution after a strike at the right temple on the surface of the calculation area. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

294

K. A. Beklemysheva and I. B. Petrov

24.3.3 Nape Strike The dynamics of the velocity modulus pattern after a strike at the nape is shown in Figs. 24.8 and 24.9. The clip visualization gives a general idea of waves’ pattern, and

Fig. 24.8 Dynamics of velocity modulus distribution after a strike at the nape on a clip of the calculation area. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

Fig. 24.9 Dynamics of velocity modulus distribution after a strike at the nape on a YZ plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500. The scale was adjusted to the velocity modulus in the bone tissue

24 Numerical Modeling of Elastic Wave Propagation …

295

the slice visualization gives a clear quantitative picture. As we can see, the waves in the bone tissue travel faster than in the brain. To observe this effect, we can look at the pressure distribution patterns in Figs. 24.10 and 24.11. Waves in the bone tissue

Fig. 24.10 Dynamics of pressure distribution after a strike at the nape on a YZ plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500. The scale was adjusted to the pressure in the bone tissue

Fig. 24.11 Dynamics of pressure distribution after a strike at the nape on a XY plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500. The scale was adjusted to the pressure in the bone tissue

296

K. A. Beklemysheva and I. B. Petrov

Fig. 24.12 Dynamics of velocity modulus distribution after a strike at the nape on a clip of the calculation area. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

travel from nape to the forehead very fast and leave trace waves in the brain area that interfere with the wave that is propagating from the impact point.

24.3.4 Vertex Strike The dynamics of the velocity modulus pattern after a strike at the vertex is shown in Fig. 24.12. The clip visualization gives a general idea of waves’ pattern.

24.3.5 Forehead Strike Analyzing the wave pattern in case of a forehead strike is more complex due to large cavities in the front part of the craniocerebral area. Figures 24.13 and 24.14 demonstrate the velocity modulus on different YZ planes near the impact point. The first one falls on the larger part of a big cavity, while the second one is on the smaller. This calculation demonstrates the necessity of full three-dimensional modeling. Cavities and material borders that do not appear on a slice still influence the calculation on corresponding nodes. The wave pattern for different slices is also significantly different.

24 Numerical Modeling of Elastic Wave Propagation …

297

Fig. 24.13 Dynamics of velocity modulus distribution after a strike at the forehead on the first YZ plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35500, i 40500, j 45500

Fig. 24.14 Dynamics of velocity modulus distribution after a strike at the forehead on the second YZ plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

Figures 24.15, 24.16 and 24.17 demonstrate the velocity modulus on different XY planes near the impact point. The first one has two cavities divided by a thin wall, the second one has one large cavity, and the third one does not have any cavities. The presence of thin walls complicates the wave pattern because of multiple reflections. For example, in case of ultrasound testing, these reflections will create a repeating noise and block the actual reflected signal.

298

K. A. Beklemysheva and I. B. Petrov

Fig. 24.15 Dynamics of velocity modulus distribution after a strike at the forehead on the first XY plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

Fig. 24.16 Dynamics of velocity modulus distribution after a strike at the forehead on the second XY plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500, i 40,500, j 45,500

24.4 Conclusions The discontinuous Galerkin method that was developed and used for construction materials is applied to modeling a full human craniocerebral area. It is stable during

24 Numerical Modeling of Elastic Wave Propagation …

299

Fig. 24.17 Dynamics of velocity modulus distribution after a strike at the forehead on the third XY plane through the impact point. Consecutive time steps are displayed: a 500, b 5500, c 10,500, d 15,500, e 20,500, f 25,500, g 30,500, h 35,500,i 40,500, j 45,500

long-term calculations on complex biological calculation grids and can be used for further research in this area. Several model problem statements were considered. To demonstrate capabilities of the used method, we modeled impacts on different sides of the head—both temples, vertex, forehead and nape. We calculated three-dimensional patterns of stress tensor and velocity vector distributions during the propagation of elastic waves inside the craniocerebral area. In this chapter, we presented velocity modulus and pressure distributions on the surface of the calculation area and on cuts and slices through the impact point. The qualitative pattern was analyzed. Acknowledgements This work was carried out with the financial support of Russian Foundation for Basic Research, project no. 18-29-02127.

References 1. Beklemysheva, K.A., Danilov, A.A., Petrov, I.B., Salamatova, V.Yu., Vassilevski, Yu.V., Vasyukov, A.V: Virtual blunt injury of human thorax: age-dependent response of vascular system. Russ. J. Numer. Anal. Math. Modelling 30(5), 259–268 (2015) 2. Pichardo, S., Sin, V.W., Hynynen, K.: Multi-frequency characterization of the speed of sound and attenuation coefficient for longitudinal transmission of freshly excised human skulls. Phys. Med. Biol. 56(1), 219–250 (2011) 3. Black, J., Hastings, G.: Handbook of Biomaterial Properties. Springer Science & Business Media (2013) 4. Hill, C. R.: Generation and structure of acoustic field. In: Hill, C. R., Bamber, J. C., Haar, G. R. (eds) Physical Principles of Medical Ultrasonics, pp. 41–68. Wiley, Chichester, UK (2004)

300

K. A. Beklemysheva and I. B. Petrov

5. Kyriakou, A., Neufeld, E., Werner, B., Szekely, G., Kuster, N.: Full-wave acoustic and thermal modeling of transcranial ultrasound propagation and investigation of skull-induced aberration correction techniques: a feasibility study. J. Ther. Ultrasound 3(11), arct. no. 11 (2015) 6. Narasimhan, C., Ward, R., Kruse, K.L., Guddati, M., Mahinthakumar, G.: A high resolution computer model for sound propagation in the human thorax based on the Visible Human data set. Comput. Biol. Med. 34(2), 177–192 (2004) 7. Spencer, G.H., Murty, M.V.R.K.: General ray tracing procedure. J. Opt. Soc. Amer. 52(6), 672–678 (1962) 8. Vassilevski, Yu., Beklemysheva, K., Grigoriev, G., Kulberg, N., Petrov, I., Vasyukov, A.: Numerical modelling of medical ultrasound: phantom-based verification. Russ. J. Numer. Anal. Math. Modelling 32(5), 339–346 (2017) 9. Leidenfrost, A., Ettrich, N., Gajewski, D., Kosloff, D.: Comparison of six different methods for calculating traveltimes. Geophys. Prosp. 47, 269–297 (1999) 10. Biryukov, V.A., Miryaha, V.A., Petrov, I.B., Khokhlov, N.I.: Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods. Comput. Math. Math. Phys. 56(6), 1086–1095 (2016) 11. Ivanov, A.M.; Khokhlov, N.I.: Efficient inter-process communication in parallel implementation of grid-characteristic method. 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. 91–102. Springer, Cham (2019) 12. Vassilevski, Yu., Beklemysheva, K., Grigoriev, G., Kulberg, N., Petrov, I., Vasyukov, A.: Numerical simulation of aberrated medical ultrasound signals. Russ. J. Numer. Anal. Math. Modelling 33(5), 277–288 (2018) 13. Madsen, E.L., Sathoff, H.J., Zagzebski, J.A.: Ultrasonic shear wave properties of soft tissues and tissue-like materials. J. Acoust. Soc. Am. 74(5), 1346–1355 (1983) 14. Aki, K., Richards, P.: Quantitative Seismology Theory and methods. W.H. Freeman and Company, San Francisco (1980) 15. Vassilevski, Yu., Beklemysheva, K., Petrov, I., Vasyukov, A., Danilov, A., Salamatova, V.: Transcranial ultrasound of cerebral vessels in silico: proof of concept. Russ. J. Numer. Anal. Math. Modelling 31(5), 259–268 (2016) 16. Favorskaya, A.V.: Investigation the material properties of a plate by laser ultrasound using the analysis of multiple waves. Comput. Res. Modeling 11(4), 653–673 (2019) 17. 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) 18. Pelties, C., de la Puente, J., Ampuero, J.-P., Brietzke, G., Käser M.: Three dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. Solid Earth 117, artc. no. B2 (2012) 19. Miryaha, V.A., Petrov, I.B.: Discontinuous Galerkin method for simulation of ice flow impact on vertical cylinder offshore structure. Matem. Mod. 30(9), 111–134 (2018) 20. Danilov, A., Ivanov, Y., Pryamonosov, R., Vassilevski, Y.: Methods of graph network reconstruction in personalized medicine. Int. J. Numer. Method Biomed. Eng. 32(8), artc. no. e02754 (2015)

Author Index

A Aksenov, Alexey G., 61 Aksenov, Vitalii V., 189 Aliev, Yusuf, 213 Andrushchenko, Viktor A., 199 Arutyunov, Sergey D., 277

B Babakov, Alexander V., 35 Bagdasaryan, Grigoriy G., 277 Beklemysheva, Katerina A., 149, 189, 287

C Churyakov, Mikhail, 137

Golubev, Vasily I., 87, 137, 175 Goreynov, Sergei, 213 Grachev, Dmitry I., 277 Guzev, Mikhail A., 75

J Jain, Lakhmi C., 1

K Kabanova, Anastasia S., 165 Khare, Rakesh Kumar, 175 Khokhlov, Nikolay I., 101, 125 Kosukhin, Oleg, 213 Kozhemyachenko, Anton A., 165 Kutschka, Ingo, 223

D Derbysheva, Tatiana N., 115 L Leon, Cristina, 265 E Efimov, Igor R., 223

F Favorskaya, Alena V., 1, 101, 165, 249 Favorskaya, Margarita N., 1 Fortova, Svetlana V., 15

M Miryakha, Vladislav A., 87 Moiseeva, Daria S., 47 Motorin, Andrey A., 47 Muratov, Maksim V., 115

G Gamilov, Timur M., 237 Golubeva, Yulia A., 87

N Nikitin, Alexander D., 277 Nikitin, Ilia S., 87

© The Editor(s) (if applicable) and 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 214, https://doi.org/10.1007/978-981-33-4709-0

301

302

Author Index

O Oparina, Elena I., 15

Syunyaev, Roman A., 223 Syzranova, Nina G., 199

P Petrov, Igor B., 1, 7, 101, 165, 287 Pikunov, Andrey V., 223

V Vassilevski, Yuri, 213 Vasyukov, Alexey V., 137 Voigt, Niels, 223

S Salamatova, Victoria, 213 Sharma, Amit, 175 Shipilov, Anatoly, 213 Simakov, Sergey S., 237 Steckmeister, Vanessa, 223 Stognii, Polina V., 101, 125 Stupitsky, Evgenii L., 47

W Wang, Lin, 213

Y Yurova, Alexandra, 213