The purpose of this article is to discuss several modern aspects of remeshing, which is the task of modifying an ill-shaped tetrahedral mesh with bad size elements so that it features an appropriate density of high-quality elements. After a brief sketch of classical stakes about meshes and local mesh operations, we notably expose (i) how the local size of the elements of a mesh can be adapted to a user-defined prescription (guided, e.g., by an error estimate attached to a numerical simulation), (ii) how a mesh can be deformed to efficiently track the motion of the underlying domain, (iii) how to construct a mesh of an implicitly-defined domain, and (iv) how remeshing procedures can be conducted in a parallel fashion when large-scale applications are targeted. These ideas are illustrated with several applications involving high-performance computing. In particular, we show how mesh adaptation and parallel remeshing strategies make it possible to achieve a high accuracy in large-scale simulations of complex flows, and how the aforementioned methods for meshing implicitly defined surfaces allow to represent faithfully intricate geophysical interfaces, and to account for the dramatic evolutions of shapes featured by shape optimization processes.
Mots clés : remeshing, implicit domain meshing, level-set discretization, topology optimization, mesh adaptation, h-adaptation, error estimator, metric, “lagrangian” mesh deformation, distributed memory parallel remeshing, hybrid RANS/LES, LES, geophysical inverse problem
@article{MSIA_2022__11_1_129_0,
author = {Balarac, G. and Basile, F. and B\'enard, P. and Bordeu, F. and Chapelier, J.-B. and Cirrottola, L. and Caumon, G. and Dapogny, C. and Frey, P. and Froehly, A. and Ghigliotti, G. and Laraufie, R. and Lartigue, G. and Legentil, C. and Mercier, R. and Moureau, V. and Nardoni, C. and Pertant, S. and Zakari, M.},
title = {Tetrahedral remeshing in the context of large-scale numerical simulation and high performance computing},
journal = {MathematicS In Action},
pages = {129--164},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {11},
number = {1},
year = {2022},
doi = {10.5802/msia.22},
language = {en},
url = {http://www.numdam.org/articles/10.5802/msia.22/}
}
TY - JOUR AU - Balarac, G. AU - Basile, F. AU - Bénard, P. AU - Bordeu, F. AU - Chapelier, J.-B. AU - Cirrottola, L. AU - Caumon, G. AU - Dapogny, C. AU - Frey, P. AU - Froehly, A. AU - Ghigliotti, G. AU - Laraufie, R. AU - Lartigue, G. AU - Legentil, C. AU - Mercier, R. AU - Moureau, V. AU - Nardoni, C. AU - Pertant, S. AU - Zakari, M. TI - Tetrahedral remeshing in the context of large-scale numerical simulation and high performance computing JO - MathematicS In Action PY - 2022 SP - 129 EP - 164 VL - 11 IS - 1 PB - Société de Mathématiques Appliquées et Industrielles UR - http://www.numdam.org/articles/10.5802/msia.22/ DO - 10.5802/msia.22 LA - en ID - MSIA_2022__11_1_129_0 ER -
%0 Journal Article %A Balarac, G. %A Basile, F. %A Bénard, P. %A Bordeu, F. %A Chapelier, J.-B. %A Cirrottola, L. %A Caumon, G. %A Dapogny, C. %A Frey, P. %A Froehly, A. %A Ghigliotti, G. %A Laraufie, R. %A Lartigue, G. %A Legentil, C. %A Mercier, R. %A Moureau, V. %A Nardoni, C. %A Pertant, S. %A Zakari, M. %T Tetrahedral remeshing in the context of large-scale numerical simulation and high performance computing %J MathematicS In Action %D 2022 %P 129-164 %V 11 %N 1 %I Société de Mathématiques Appliquées et Industrielles %U http://www.numdam.org/articles/10.5802/msia.22/ %R 10.5802/msia.22 %G en %F MSIA_2022__11_1_129_0
Balarac, G.; Basile, F.; Bénard, P.; Bordeu, F.; Chapelier, J.-B.; Cirrottola, L.; Caumon, G.; Dapogny, C.; Frey, P.; Froehly, A.; Ghigliotti, G.; Laraufie, R.; Lartigue, G.; Legentil, C.; Mercier, R.; Moureau, V.; Nardoni, C.; Pertant, S.; Zakari, M. Tetrahedral remeshing in the context of large-scale numerical simulation and high performance computing. MathematicS In Action, Tome 11 (2022) no. 1, pp. 129-164. doi : 10.5802/msia.22. http://www.numdam.org/articles/10.5802/msia.22/
[1] Acoustic Reference Nozzle with Mach 0.97, Unheated Jet Flow, https://www.grc.nasa.gov/www/wind/valid/arn/index.html, Accessed: 2021-05-04
[2] ANSA. The advanced CAE pre-processing software for complete model build up, https://www.beta-cae.com/ansa.htm, Accessed: 2021-05-04
[3] Mmg version 5.5.2, https://github.com/MmgTools/mmg (swhid: swh:1:rel:fe173a75f45f079d363d5a82204c9737550c5d79)
[4] A changing-topology moving mesh technique for large displacements, Engineering with Computers, Volume 30 (2014) no. 2, pp. 175-200 | DOI
[5] Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Eng., Volume 282 (2014), pp. 22-53 | DOI | MR
[6] Shape and topology optimization, Geometric partial differential equations. Part II (Handbook of Numerical Analysis), Volume 22, Elsevier/North Holland, 2020, pp. 1-132 | Zbl
[7] A level-set method for shape optimization, C. R. Math. Acad. Sci. Paris, Volume 334 (2002) no. 12, pp. 1125-1130 | DOI | MR | Zbl
[8] Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., Volume 194 (2004) no. 1, pp. 363-393 | DOI | MR | Zbl
[9] Conception optimale de structures, 58, Springer, 2007
[10] Comparing Three DFN Simplification Strategies for Two-Phase Flow Applications, ECMOR XVII, Volume 2020, EAGE Publications (2020), pp. 1-21 | DOI
[11] Mesh movement and metamorphosis, Engineering with Computers, Volume 18 (2002) no. 3, pp. 188-198 | DOI
[12] A high-order h-adaptive discontinuous Galerkin method for unstructured grids based on a posteriori error estimation., AIAA Scitech 2021 Forum (2021), p. 1696 | DOI
[13] A p-adaptive matrix-free discontinuous Galerkin method for the implicit LES of incompressible transitional flows, Flow, Turbulence and Combustion, Volume 105 (2020) no. 2, pp. 437-470 | DOI
[14] Mesh adaptation for large-eddy simulations in complex geometries, Int. J. Numer. Methods Fluids, Volume 81 (2016) no. 12, pp. 719-740 | DOI | MR
[15] Large-Eddy Simulation of the lean-premixed PRECCINSTA burner with wall heat loss, Symposium (International) on Combustion, Volume 37 (2019) no. 4, pp. 5233-5243 | DOI
[16] Topology optimization: theory, methods, and applications, Springer, 2013
[17] High-order h-adaptive discontinuous Galerkin methods for ocean modelling, Ocean Dynamics, Volume 57 (2007) no. 2, pp. 109-121 | DOI
[18] A self-parametrizing partition model approach to tomographic inverse problems, Inverse Probl., Volume 25 (2009) no. 5, p. 055009 (Accessed 2017-04-30) | DOI | MR | Zbl
[19] Impact of a Stochastic Sequential Initiation of Fractures on the Spatial Correlations and Connectivity of Discrete Fracture Networks, J. Geophys. Res. Solid Earth, Volume 121 (2016) no. 8, pp. 5641-5658 | DOI
[20] Meshing, Geometric Modeling and Numerical Simulation 1: Form Functions, Triangulations and Geometric Modeling, John Wiley & Sons, 2017 | DOI
[21] Polygon mesh processing, CRC Press, 2010 | DOI
[22] Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets, J. Fluid Mech., Volume 851 (2018), pp. 83-124 | DOI | MR | Zbl
[23] The Dynamic Adaptation of Parallel Mesh-Based Computation, PPSC (1997)
[24] Three-Dimensional Implicit Stratigraphic Model Building From Remote Sensing Data on Tetrahedral Meshes: Theory and Application to a Regional Model of La Popa Basin, NE Mexico, IEEE Trans. Geosci. Rem. Sens., Volume 51 (2013) no. 3, pp. 1613-1621 | DOI
[25] Parallel Unstructured Mesh Adaptation Method for Moving Body Applications, AIAA J., Volume 43 (2005) no. 9, pp. 1937-1945 | DOI
[26] Rebonds spéciaux de liquides, Ph. D. Thesis, Université Paris-Saclay (2018) (https://pastel.archives-ouvertes.fr/tel-02011789)
[27] Delaunay mesh generation, CRC Press, 2013
[28] Method for Stochastic Inverse Modeling of Fault Geometry and Connectivity Using Flow Data, Mathematical Geosciences, Volume 44 (2012) no. 2, pp. 147-168 | DOI
[29] Stochastic geometry and its applications, Wiley Series in Probability and Statistics, John Wiley & Sons, 2013 | DOI
[30] Parallel Delaunay mesh generation kernel, Int. J. Numer. Meth. Engng., Volume 58 (2003) no. 2, pp. 161-176 | DOI | Zbl
[31] The finite element method for elliptic problems, 40, Society for Industrial and Applied Mathematics, 2002 | DOI
[32] et al. Meshlab: an open-source mesh processing tool., Eurographics Italian chapter conference, Volume 2008, Salerno, Italy (2008), pp. 129-136
[33] Generating variable shapes of salt geobodies from seismic images and prior geological knowledge, Interpretation, Volume 7 (2019) no. 4, p. T829-T841 (Accessed 2019-09-07) | DOI
[34] A p-adaptive implicit discontinuous Galerkin method for the under-resolved simulation of compressible turbulent flows, 7nd European Conference on Computational Fluid Dynamics (2018)
[35] A mesh adaptation framework for dealing with large deforming meshes, Int. J. Numer. Meth. Engng., Volume 82 (2010) no. 7, pp. 843-867 | DOI | Zbl
[36] Shape optimization, level set methods on unstructured meshes and mesh evolution, Ph. D. Thesis, Paris 6 (2013)
[37] Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems, J. Comput. Phys., Volume 262 (2014), pp. 358-378 | DOI | MR | Zbl
[38] A model of fracture nucleation, growth and arrest, and consequences for fracture density and scaling: a discrete fracture network model, J. Geophys. Res. Solid Earth, Volume 118 (2013) no. 4, pp. 1393-1407 (Accessed 2017-05-09) | DOI
[39] Parallel refinement and coarsening of tetrahedral meshes, Int. J. Numer. Meth. Engng., Volume 46 (1999) no. 7, pp. 1101-1125 | DOI | MR | Zbl
[40] Recent improvements in the zonal detached eddy simulation (ZDES) formulation, Theor. Comput. Fluid Dyn., Volume 26 (2012) no. 6, pp. 523-550 | DOI
[41] An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys., Volume 227 (2008) no. 18, pp. 8395-8416 | DOI | MR | Zbl
[42] Massively parallel anisotropic mesh adaptation, Int. J. High Perform. Comput. Appl., Volume 33 (2017) no. 1, pp. 3-24 | DOI
[43] Adaptation de maillage anisotrope 3d et application à l’aéro-thermique des bâtiments, Ph. D. Thesis, Université Pierre et Marie Curie - Paris VI (2005) (https://hal.archives-ouvertes.fr/tel-00120327)
[44] Anisotropic Delaunay mesh adaptation for unsteady simulations, Proceedings of the 17th international Meshing Roundtable, 2008, pp. 177-194 | DOI
[45] An efficient method of triangulating equi-valued surfaces by using tetrahedral cells, IEICE Trans. Inf. Syst., Volume 74 (1991) no. 1, pp. 214-224
[46] hp-DGFEM for nonlinear convection-diffusion problems, Math. Comput. Simul., Volume 87 (2013), pp. 87-118 | DOI | MR | Zbl
[47] Droplets impaling on a cone, Phys. Rev. Fluids, Volume 5 (2020) no. 11 | DOI
[48] New developments in handling stress constraints in optimal material distribution, 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization (1998), p. 4906 | DOI
[49] Theory and practice of finite elements, 159, Springer, 2013
[50] Shape and topology optimization of multiphysics systems, Ph. D. Thesis, Université Paris Saclay, préparée à l’École polytechnique (2019)
[51] Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing, J. Comput. Phys. (2020), p. 109574 | DOI | MR | Zbl
[52] Parallel structures and dynamic load balancing for adaptive finite element computation, Appl. Numer. Math., Volume 26 (1998) no. 1, pp. 241-263 | DOI | MR | Zbl
[53] FraC: A new conforming mesh method for discrete fracture networks, J. Comput. Phys., Volume 376 (2019), pp. 713-732 (Accessed 2021-08-29) | DOI | MR | Zbl
[54] 3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data, Computers & Geosciences, Volume 33 (2007) no. 7, pp. 932-943 (Accessed 2017-10-18) | DOI
[55] Mesh generation: application to finite elements, ISTE, 2007
[56] Geophysical inversion for 3D contact surface geometry, Geophysics, Volume 85 (2020) no. 6, p. K27-K45 (Accessed 2021-08-26) | DOI
[57] On the generation of turbulent inflow for hybrid RANS/LES jet flow simulations, Comput. Fluids, Volume 216 (2021), p. 104816 | DOI | MR | Zbl
[58] Explicit Discontinuous Galerkin Schemes with Adaptation in Space and Time, 36th CFD/ADIGMA course on hp-adaptive and hp-multigrid methods, VKI LS, 2009
[59] Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Meth. Engng., Volume 79 (2009) no. 11, pp. 1309-1331 | DOI | MR | Zbl
[60] Generalization of level-set inversion to an arbitrary number of geologic units in a regularized least-squares framework, Geophysics, Volume 86 (2021) no. 4, p. R623-R637 (Accessed 2021-08-26) | DOI
[61] Multi-scenario interpretations from sparse fault evidence using graph theory and geological rules, J. Geophys. Res. Solid Earth, Volume 126 (2021) no. 2, p. e2020JB020022 | DOI
[62] Shape Variation and Optimization, EMS Tracts in Mathematics, 28, European Mathematical Society, 2018 | DOI
[63] Shape Optimization Methods Locating Layer Interfaces in Geothermal Reservoirs, Energy Procedia, Volume 76 (2015), pp. 321-330 (Accessed 2018-07-13) | DOI
[64] A general gridding, discretization, and coarsening methodology for modeling flow in porous formations with discrete geological features, Adv. Water Resources, Volume 96 (2016), pp. 354-372 (Accessed 2017-04-25) | DOI
[65] Numerical Simulation of Water Injection in 2D Fractured Media Using Discrete-Fracture Model, All Days, SPE (2001), p. SPE-71615–MS | DOI
[66] Error estimation and hp-adaptive mesh refinement for discontinuous Galerkin methods, Adaptive high-order methods in computational fluid dynamics, World Scientific, 2011, pp. 67-94 | DOI | Zbl
[67] DLR-Project Digital-X-Next Generation CFD Solver’Flucs’, CEAS Aeronautical Journal (2016)
[68] Finite element mesh generation, CRC Press, 2014
[69] Marching cubes: A high resolution 3D surface construction algorithm, ACM SIGGRAPH Comput. Graph., Volume 21 (1987) no. 4, pp. 163-169 | DOI
[70] Continuous mesh framework part I: well-posed continuous interpolation error, SIAM J. Numer. Anal., Volume 49 (2011) no. 1, pp. 38-60 | DOI | MR | Zbl
[71] A posteriori error estimators for adaptive spectral element techniques, Proceedings of the Eighth GAMM-Conference on Numerical Methods in Fluid Mechanics, Springer (1990), pp. 333-342 | DOI | MR
[72] Topology-adaptive interface tracking using the deformable simplicial complex, ACM Trans. Graph., Volume 31 (2012) no. 3, pp. 1-12 | DOI
[73] Control-volume method for numerical simulation of two-phase immiscible flow in two- and three-dimensional discrete-fractured media: SIMULATION OF FLOW IN FRACTURED MEDIA, Water Resources Research, Volume 40 (2004) no. 7 (Accessed 2021-08-29) | DOI
[74] Design of a massively parallel CFD code for complex geometries, C. R. Méc. Acad. Sci. Paris, Volume 339 (2011) no. 2-3, pp. 141-148 | DOI | Zbl
[75] From Large-Eddy Simulation to Direct Numerical Simulation of a lean premixed swirl flame: Filtered laminar flame-PDF modeling, Combustion and Flame, Volume 158 (2011) no. 7, pp. 1340-1357 | DOI
[76] Sur le contrôle par un domaine géométrique (1976) Pré-publication du Laboratoire d’Analyse Numérique, (76015)
[77] Discretizing two-dimensional complex fractured fields for incompressible two-phase flow, Int. J. Numer. Methods Fluids, Volume 65 (2011) no. 7, pp. 764-780 (Accessed 2021-08-29) | DOI | Zbl
[78] A comparison of refinement indicators for p-adaptive simulations of steady and unsteady flows using discontinuous Galerkin methods, J. Comput. Phys., Volume 376 (2019), pp. 508-533 | DOI | MR | Zbl
[79] Jet noise prediction with Eddy relaxation source model, 21st AIAA/CEAS Aeroacoustics Conference (2015), p. 2370 | DOI
[80] Parallel tetrahedral mesh adaptation with dynamic load balancing, Parallel Comput., Volume 26 (2000) no. 12, pp. 1583-1608 | DOI | Zbl
[81] Level set methods and dynamic implicit surfaces, 153, Springer, 2006
[82] Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., Volume 79 (1988) no. 1, pp. 12-49 | DOI | MR | Zbl
[83] Identification of uncertainties in the shape of geophysical objects with level sets and the adjoint method, Comput. Geosci., Volume 15 (2011) no. 4, pp. 737-753 (Accessed 2018-07-13) | DOI | Zbl
[84] Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Towards CFD 2030, AIAA AVIATION Forum, American Institute of Aeronautics and Astronautics, 2016 (art. 3323)
[85] Sub-cell shock capturing for discontinuous Galerkin methods, 44th AIAA Aerospace Sciences Meeting and Exhibit (2006), p. 112 | DOI
[86] A finite-volume method for simulating contact lines on unstructured meshes in a conservative level-set framework, J. Comput. Phys., Volume 444 (2021), p. 110582 | DOI | MR | Zbl
[87] An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems, SIAM Rev., Volume 45 (2003) no. 1, pp. 53-72 | DOI | MR | Zbl
[88] Adaptive mesh generation and visualization, Encyclopedia of Aerospace Engineering, 2010 | DOI
[89] Multiscale and multiresolution approaches in turbulence-LES, DES and Hybrid RANS/LES Methods: Applications and Guidelines, World Scientific, 2013 | DOI
[90] Fast marching methods, SIAM Rev., Volume 41 (1999) no. 2, pp. 199-235 | DOI | MR | Zbl
[91] Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, 3, Cambridge University Press, 1999
[92] What Is a Good Linear Finite Element? - Interpolation, Conditioning, Anisotropy, and Quality Measures, Proceedings of the 11th International Meshing Roundtable, Sandia National Laboratories (2002), pp. 115-126
[93] TetGen, a Delaunay-based quality tetrahedral mesh generator, ACM Trans. Math. Softw., Volume 41 (2015) no. 2, pp. 1-36 | MR | Zbl
[94] Introduction to shape optimization, Springer, 1992 | DOI
[95] Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach, Proceedings of first AFOSR international conference on DNS/LES, Greyden Press (1997)
[96] A one-equation turbulence model for aerodynamic flows, 30th aerospace sciences meeting and exhibit (1992), p. 439 | DOI
[97] CGAL User and Reference Manual, CGAL Editorial Board, 2021 (https://doc.cgal.org/5.3/Manual/packages.html)
[98] Handbook of grid generation, CRC Press, 1998 | DOI
[99] Anisotropic control of mesh generation based upon a Voronoi type method, Numerical grid generation in computational fluid dynamics and related fields, North-Holland, 1991, pp. 93-103
[100] Adjoint-based h-p adaptive discontinuous Galerkin methods for the 2D compressible Euler equations, J. Comput. Phys., Volume 228 (2009) no. 20, pp. 7643-7661 | DOI | MR | Zbl
[101] 3-D Structural geological models: Concepts, methods, and uncertainties, Adv. Geophys., Volume 59 (2018), pp. 1-121 | DOI
[102] A new computer code for discrete fracture network modelling, Computers & Geosciences, Volume 36 (2010) no. 3, pp. 292-301 (Accessed 2015-06-30) | DOI
[103] Assessing and visualizing uncertainty of 3D geological surfaces using level sets with stochastic motion, Computers & Geosciences, Volume 122 (2019), pp. 54-67 (Accessed 2021-03-02) | DOI
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