Parallel Finite Volume Computation on General Meshes

197 Pages·2020·12.631 MB·English

by  Yuri Vassilevski, Kirill Terekhov, Kirill Nikitin, Ivan Kapyrin

Yuri Vassilevski
Kirill Terekhov
Kirill Nikitin
Ivan Kapyrin
Parallel Finite
Volume
Computation
on General
Meshes
Parallel Finite Volume Computation on General
Meshes
Yuri Vassilevski Kirill Terekhov
(cid:129) (cid:129)
Kirill Nikitin Ivan Kapyrin
(cid:129)
Parallel Finite Volume
Computation on General
Meshes
123
YuriVassilevski Kirill Terekhov
MarchukInstituteofNumericalMathematics MarchukInstituteofNumericalMathematics
of the Russian Academy of Sciencesand of the Russian Academy of Sciences
MoscowInstituteofPhysicsandTechnology Moscow,Russia
andSechenov University
Moscow,Russia IvanKapyrin
MarchukInstituteofNumericalMathematics
Kirill Nikitin andNuclear SafetyInstitute ofthe Russian
MarchukInstituteofNumericalMathematics Academy of Sciences
of the Russian Academy of Sciencesand Moscow,Russia
Moscow State University
Moscow,Russia
ISBN978-3-030-47231-3 ISBN978-3-030-47232-0 (eBook)
https://doi.org/10.1007/978-3-030-47232-0
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Acknowledgements
This book would not appear without dedications of many colleagues and collab-
orators.Thecoauthorsofourpaperscitedinthebibliographyareourcollaborators
to whom we are greatly indebted. In particular, we wish to thank Konstantin
Lipnikov,DaniilSvyatskiy,AlexanderDanilov,AlexeyChernyshenko,Konstantin
Novikov, Vasiliy Kramarenko, Ruslan Yanbarisov, Mikhail Shashkov, Bradley
Mallison,HamdiTchelepi,andMaximOlshanskiifortheircontributiontoourjoint
papersandcooperativeresearchonthenonlinearFVmethods.Wearealsograteful
to Mary Wheeler, Serguei Maliassov, Ilya Mishev, Roland Masson, and Denis
VoskovforilluminatingdiscussionsonapplicationsoftheFVmethodsinreservoir
simulation; Jerome Jaffre and Alexander Rastorguev for introduction to simulation
of radionuclides subsurface migration; Igor Linge and Sergey Utkin for practical
RW disposal safety assessment problem formulations and methodology; Vitaly
Volpert and Anass Bouchnita for joint development of the multi-physics model of
blood flow coagulation; Denis Anuprienko, Fedor Grigorev, Georgiy Neuvazhaev,
ViktorSuskin,andalltheGeRacodedevelopersforthecooperationingroundwater
flow and radionuclides transport models development and their verification. We
owe to our colleagues Igor Kaporin, Igor Konshin, Vadim Chugunov, Sergey
Goreinov, and Sergey Kharchenko for fruitful discussions on incomplete LU fac-
torizationmethodsandcontributiontoINMOSTsoftware.Weareinagreatdebtto
Yuri Kuznetsov who introduced us to the world of mathematical modeling years
ago.
A large part of the work presented here was supported by the ExxonMobil
Corporation within a long-term research project at the Marchuk Institute of
Numerical Mathematics of the Russian Academy of Sciences (INM RAS). We
acknowledge the financial support by the Russian Science Foundation projects
18-71-10111, 19-71-10094, and the Russian Foundation for Basic Research pro-
jects 18-31-20048 and 19-31-90110, the RAS Research program 26 “Basics of
algorithms and software for high performance computing” and the world-class
research center “Moscow Center for Fundamental and Applied Mathematics”.
Nuclear Safety Institute of the Russian Academy of Sciences (IBRAE) provided
v
vi Acknowledgements
resources for the GeRa code development. We thank INM RAS and IBRAE, for
administrative support of our research within the above projects.
Finally, we would like to thank all Springer team for making this publication
possible.And,mostimportantly,wearegratefultoourfamiliesforthepatienceand
irretrievably lost time, otherwise spent by the authors writing the code and text
of the book.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Structure and Overview of the Book . . . . . . . . . . . . . . . . . . . . . . 2
2 Monotone Finite Volume Method on General Meshes . . . . . . . . . . . 5
2.1 Cell-Centered Finite Volume Method on General Meshes. . . . . . . 5
2.2 Monotone Two-Point Flux Approximation Based on Finite
Differences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Monotone Two-Point Flux Approximation Based on Gradient
Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Monotone Multi-Point Flux Approximation . . . . . . . . . . . . . . . . . 15
2.5 Generalizations to Convection–Diffusion Equation . . . . . . . . . . . . 19
2.6 Generalization to Diffusion Problem in Mixed Formulation. . . . . . 21
2.7 Generalization to Navier–Stokes Equations . . . . . . . . . . . . . . . . . 24
2.8 Analysis of Monotone FV Methods. . . . . . . . . . . . . . . . . . . . . . . 30
2.8.1 Two-Point Flux Approximations. . . . . . . . . . . . . . . . . . . . 31
2.8.2 Multi-Point Flux Approximation. . . . . . . . . . . . . . . . . . . . 31
2.9 Numerical Features of Monotone FV Methods. . . . . . . . . . . . . . . 33
3 Application of MFV in Reservoir Simulation . . . . . . . . . . . . . . . . . . 39
3.1 Subsurface Flow Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 Single-Phase Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.3 Three-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.4 Well Model and Boundary Conditions . . . . . . . . . . . . . . . 43
3.2 Time-Stepping and Nonlinear Systems. . . . . . . . . . . . . . . . . . . . . 43
3.2.1 IMPES Scheme for Two-Phase Flow . . . . . . . . . . . . . . . . 44
3.2.2 Fully Implicit Scheme for Three-Phase Flow. . . . . . . . . . . 45
vii
viii Contents
3.3 Simulation of Waterflood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 Non-orthogonal Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Discontinuous Tensor with High Anisotropy. . . . . . . . . . . 52
3.3.3 Computational Complexity. . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.4 Discrete Maximum Principle . . . . . . . . . . . . . . . . . . . . . . 54
3.3.5 Parallel Simulation on the Norne Field . . . . . . . . . . . . . . . 57
3.4 Flow in Fractured Media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 Embedded Discrete Fracture Method. . . . . . . . . . . . . . . . . 59
3.4.2 Analysis of the Monotone EDFM. . . . . . . . . . . . . . . . . . . 62
3.4.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Near-Well Correction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.1 Numerical Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Application of FVM in Modeling of Subsurface Radionuclide
Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Domains, Physics, and Mathematical Models for Subsurface
Radionuclide Migration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Flow in Unconfined and Unsaturated Conditions, Transport
in Vadose Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Mathematical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.2 Numerical Solution Aspects . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3 Reactive Transport Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.1 Mathematical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.2 Numerical Solution Aspects . . . . . . . . . . . . . . . . . . . . . . . 93
4.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.4 Density-Driven Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4.2 Numerical Solution Aspects . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Application of MFV in Modeling of Coagulation of Blood Flow. . . . 109
5.1 Model of Blood Flow and Coagulation . . . . . . . . . . . . . . . . . . . . 109
5.2 FV Discretization of Blood Coagulation Model . . . . . . . . . . . . . . 112
5.3 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1 Lid-Driven Cavity Flow. . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3.2 Flow over Cylinder at Low Reynolds Number . . . . . . . . . 120
5.3.3 Coagulation of Blood Flow in Microfluidic Capillaries . . . 121
6 INMOST Platform Technologies for Numerical Model
Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1 Maintenance of General Meshes . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Generation and Modification of General Meshes . . . . . . . . . . . . . 130
Contents ix
6.3 Parallel Mesh Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.1 Parallel Local Mesh Modifications . . . . . . . . . . . . . . . . . . 149
6.3.2 Mesh Balancing and Redistribution. . . . . . . . . . . . . . . . . . 152
6.3.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.4 Linear System Assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.5 INMOST Linear Solvers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.5.1 Parallel Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.5.2 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.5.3 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.5.4 Multi-Level Factorization. . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5.5 INMOST Linear Solver Routines . . . . . . . . . . . . . . . . . . . 166
6.6 Automatic Differentiation for Jacobian and Hessian
Calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.6.1 Basic Structures and Realization Details . . . . . . . . . . . . . . 168
6.6.2 Interfaces for Automatic Differentiation. . . . . . . . . . . . . . . 171
6.7 Nonlinear Solvers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7.1 Newton Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7.2 Line-Search Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.7.3 Anderson Acceleration Method. . . . . . . . . . . . . . . . . . . . . 174
6.7.4 Halley Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.8 Multi-Physics Model Assembly. . . . . . . . . . . . . . . . . . . . . . . . . . 176
References.... .... .... .... ..... .... .... .... .... .... ..... .... 179
Acronyms
DDF Density-driven flow
DFN Discrete fracture network
DGR Deep geological repository
DMP Discrete maximum principle
DSA Direct substitution approach
EBS Engineered barriers system
EDFM Embedded discrete fracture method
FV Finite volume
FVM Finite volume method
HLW High-level radioactive waste
IMPES Implicit pressure–explicit saturation
LILW Low- and intermediate-level radioactive waste
mEDFM Monotone embedded discrete fracture method
MFV Monotone finite volume method
MPFA Multi-point flux approximation
NMPFA Nonlinear multi-point flux approximation
NTPFA Nonlinear two-point flux approximation
NWC Near-well correction
PDE Partial differential equation
RHS Right-hand side
RW Radioactive waste
SIA Sequential iterative approach
SNIA Sequential non-iterative approach
TPFA Two-point flux approximation
xi