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Tetrahedral remeshing in the context of large-scale numerical simulation and high performance computing. (English) Zbl 1498.76051

Summary: 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.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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[1] Acoustic Reference Nozzle with Mach 0.97, Unheated Jet Flow, , Accessed: 2021-05-04
[2] ANSA. The advanced CAE pre-processing software for complete model build up, , Accessed: 2021-05-04
[3] Mmg version 5.5.2
[4] Alauzet, Frédéric, A changing-topology moving mesh technique for large displacements, Engineering with Computers, 30, 2, 175-200 (2014) · doi:10.1007/s00366-013-0340-z
[5] Allaire, Grégoire; Dapogny, Charles; Frey, Pascal, Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Eng., 282, 22-53 (2014) · Zbl 1423.74739 · doi:10.1016/j.cma.2014.08.028
[6] Allaire, Grégoire; Dapogny, Charles; Jouve, François, Geometric partial differential equations. Part II, 22, Shape and topology optimization, 1-132 (2020), Elsevier/North Holland · Zbl 1475.49048
[7] Allaire, Grégoire; Jouve, François; Toader, Anca-Maria, A level-set method for shape optimization, C. R. Math. Acad. Sci. Paris, 334, 12, 1125-1130 (2002) · Zbl 1115.49306 · doi:10.1016/S1631-073X(02)02412-3
[8] Allaire, Grégoire; Jouve, François; Toader, Anca-Maria, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194, 1, 363-393 (2004) · Zbl 1136.74368 · doi:10.1016/j.jcp.2003.09.032
[9] Allaire, Grégoire; Schoenauer, Marc, Conception optimale de structures, 58 (2007), Springer · Zbl 1132.49033
[10] Anquez, P.; Zakari, M.; Caumon, G., Comparing Three DFN Simplification Strategies for Two-Phase Flow Applications, ECMOR XVII, 2020, 1-21 (2020), EAGE Publications · doi:10.3997/2214-4609.202035112
[11] Baker, Timothy J., Mesh movement and metamorphosis, Engineering with Computers, 18, 3, 188-198 (2002) · doi:10.1007/s003660200017
[12] Basile, Francesca; Chapelier, Jean-Baptiste; de la Llave Plata, Marta; Laraufie, Romain; Frey, Pascal, A high-order h-adaptive discontinuous Galerkin method for unstructured grids based on a posteriori error estimation., AIAA Scitech 2021 Forum, 1696 (2021) · doi:10.2514/6.2021-1696
[13] Bassi, F.; Botti, L.; Colombo, A.; Crivellini, A.; Franciolini, M.; Ghidoni, A.; Noventa, G., A p-adaptive matrix-free discontinuous Galerkin method for the implicit LES of incompressible transitional flows, Flow, Turbulence and Combustion, 105, 2, 437-470 (2020) · doi:10.1007/s10494-020-00178-2
[14] Benard, Pierre; Balarac, Guillaume; Moureau, Vincent; Dobrzynski, Cécile; Lartigue, Ghislain; D’Angelo, Yves, Mesh adaptation for large-eddy simulations in complex geometries, Int. J. Numer. Methods Fluids, 81, 12, 719-740 (2016) · doi:10.1002/fld.4204
[15] Benard, Pierre; Lartigue, Ghislain; Moureau, Vincent; Mercier, Renaud, Large-Eddy Simulation of the lean-premixed PRECCINSTA burner with wall heat loss, Symposium (International) on Combustion, 37, 4, 5233-5243 (2019) · doi:10.1016/j.proci.2018.07.026
[16] Bendsoe, Martin Philip; Sigmund, Ole, Topology optimization: theory, methods, and applications (2013), Springer
[17] Bernard, Paul-Emile; Chevaugeon, Nicolas; Legat, Vincent; Deleersnijder, Eric; Remacle, Jean-François, High-order h-adaptive discontinuous Galerkin methods for ocean modelling, Ocean Dynamics, 57, 2, 109-121 (2007) · doi:10.1007/s10236-006-0093-y
[18] Bodin, T.; Sambridge, M.; Gallagher, K., A self-parametrizing partition model approach to tomographic inverse problems, Inverse Probl., 25, 5, 055009 (20170430) · Zbl 1162.62014 · doi:10.1088/0266-5611/25/5/055009
[19] Bonneau, François; Caumon, Guillaume; Renard, Philippe, Impact of a Stochastic Sequential Initiation of Fractures on the Spatial Correlations and Connectivity of Discrete Fracture Networks, J. Geophys. Res. Solid Earth, 121, 8, 5641-5658 (2016) · doi:10.1002/2015JB012451
[20] Borouchaki, Houman; George, Paul-Louis, Meshing, Geometric Modeling and Numerical Simulation 1: Form Functions, Triangulations and Geometric Modeling (2017), John Wiley & Sons · doi:10.1002/9781119384335
[21] Botsch, Mario; Kobbelt, Leif; Pauly, Mark; Alliez, Pierre; Lévy, Bruno, Polygon mesh processing (2010), CRC Press · doi:10.1201/b10688
[22] Brès, Guillaume; Jordan, Peter; Jaunet, Vincent; Le Rallic, Maxime; Cavalieri, André; Towne, Aaron; Lele, Sanjiva; Colonius, Tim; Schmidt, Oliver, Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets, J. Fluid Mech., 851, 83-124 (2018) · Zbl 1415.76555 · doi:10.1017/jfm.2018.476
[23] Castaños, José G.; Savage, John E., The Dynamic Adaptation of Parallel Mesh-Based Computation, PPSC (1997)
[24] Caumon, G.; Gray, G.; Antoine, C.; Titeux, M.-O., 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., 51, 3, 1613-1621 (2013) · doi:10.1109/TGRS.2012.2207727
[25] Cavallo, Peter A.; Sinha, Neeraj; Feldman, Gregory M., Parallel Unstructured Mesh Adaptation Method for Moving Body Applications, AIAA J., 43, 9, 1937-1945 (2005) · doi:10.2514/1.7818
[26] Chantelot, Pierre, Rebonds spéciaux de liquides (2018)
[27] Cheng, Siu-Wing; Dey, Tamal Krishna; Shewchuk, Jonathan; Sahni, Sartaj, Delaunay mesh generation (2013), CRC Press · Zbl 1298.65187
[28] Cherpeau, Nicoles; Caumon, Guillaume; Caers, Jef; Lévy, Bruno, Method for Stochastic Inverse Modeling of Fault Geometry and Connectivity Using Flow Data, Mathematical Geosciences, 44, 2, 147-168 (2012) · doi:10.1007/s11004-012-9389-2
[29] Chiu, Sung Nok; Stoyan, Dietrich; Kendall, W. S.; Mecke, Joseph, Stochastic geometry and its applications (2013), John Wiley & Sons · Zbl 1291.60005 · doi:10.1002/9781118658222
[30] Chrisochoides, Nikos; Nave, Démian, Parallel Delaunay mesh generation kernel, Int. J. Numer. Meth. Engng., 58, 2, 161-176 (2003) · Zbl 1035.65018 · doi:10.1002/nme.765
[31] Ciarlet, Philippe G., The finite element method for elliptic problems, 40 (2002), Society for Industrial and Applied Mathematics · Zbl 0999.65129 · doi:10.1137/1.9780898719208
[32] Cignoni, Paolo; Callieri, Marco; Corsini, Massimiliano; Dellepiane, Matteo; Ganovelli, Fabio; Ranzuglia, Guido, Meshlab: an open-source mesh processing tool., Eurographics Italian chapter conference, 2008, 129-136 (2008)
[33] Clausolles, Nicolas; Collon, Pauline; Caumon, Guillaume, Generating variable shapes of salt geobodies from seismic images and prior geological knowledge, Interpretation, 7, 4, T829-T841 (20190907) · doi:10.1190/int-2019-0032.1
[34] Colombo, A.; Manzinali, G.; Ghidoni, A.; Noventa, G.; Franciolini, M.; Crivellini, A.; Bassi, F., 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] Compère, Gaëtan; Remacle, Jean-François; Jansson, Johan; Hoffman, Johan, A mesh adaptation framework for dealing with large deforming meshes, Int. J. Numer. Meth. Engng., 82, 7, 843-867 (2010) · Zbl 1188.74093 · doi:10.1002/nme.2788
[36] Dapogny, Charles, Shape optimization, level set methods on unstructured meshes and mesh evolution (2013)
[37] Dapogny, Charles; Dobrzynski, Cécile; Frey, Pascal, Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems, J. Comput. Phys., 262, 358-378 (2014) · Zbl 1349.76598 · doi:10.1016/j.jcp.2014.01.005
[38] Davy, Philippe; Le Goc, Romain; Darcel, Caroline, 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, 118, 4, 1393-1407 (20170509) · doi:10.1002/jgrb.50120
[39] De Cougny, H. L.; Shephard, Mark S., Parallel refinement and coarsening of tetrahedral meshes, Int. J. Numer. Meth. Engng., 46, 7, 1101-1125 (1999) · Zbl 0964.76073 · doi:10.1002/(SICI)1097-0207(19991110)46:7<1101::AID-NME741>3.0.CO;2-E
[40] Deck, Sébastien, Recent improvements in the zonal detached eddy simulation (ZDES) formulation, Theor. Comput. Fluid Dyn., 26, 6, 523-550 (2012) · doi:10.1007/s00162-011-0240-z
[41] Desjardins, Olivier; Moureau, Vincent; Pitsch, Heinz, An accurate conservative level set/ghost fluid method for simulating turbulent atomization, J. Comput. Phys., 227, 18, 8395-8416 (2008) · Zbl 1256.76051 · doi:10.1016/j.jcp.2008.05.027
[42] Digonnet, Hugues; Coupez, Thierry; Laure, Patrice; Silva, Luisa, Massively parallel anisotropic mesh adaptation, Int. J. High Perform. Comput. Appl., 33, 1, 3-24 (2017) · doi:10.1177/1094342017693906
[43] Dobrzynski, Cécile, Adaptation de maillage anisotrope 3d et application à l’aéro-thermique des bâtiments (2005)
[44] Dobrzynski, Cécile; Frey, Pascal, Proceedings of the 17th international Meshing Roundtable, Anisotropic Delaunay mesh adaptation for unsteady simulations, 177-194 (2008) · doi:10.1007/978-3-540-87921-3_11
[45] Doi, Akio; Koide, Akio, An efficient method of triangulating equi-valued surfaces by using tetrahedral cells, IEICE Trans. Inf. Syst., 74, 1, 214-224 (1991)
[46] Dolejší, Vít, hp-DGFEM for nonlinear convection-diffusion problems, Math. Comput. Simul., 87, 87-118 (2013) · Zbl 1490.65268 · doi:10.1016/j.matcom.2013.03.001
[47] Durey, Guillaume; Magdelaine, Quentin; Casiulis, Mathias; Kwon, Hoon; Mazet, Julien; Chantelot, Pierre; Gauthier, Anaïs; Clanet, Christophe; Quéré, David, Droplets impaling on a cone, Phys. Rev. Fluids, 5, 11 (2020) · doi:10.1103/PhysRevFluids.5.110507
[48] Duysinx, Pierre; Sigmund, Ole, New developments in handling stress constraints in optimal material distribution, 7th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization, 4906 (1998) · doi:10.2514/6.1998-4906
[49] Ern, Alexandre; Guermond, Jean-Luc, Theory and practice of finite elements, 159 (2013), Springer
[50] Feppon, Florian, Shape and topology optimization of multiphysics systems (2019)
[51] Feppon, Florian; Allaire, Grégoire; Dapogny, Charles; Jolivet, Pierre, Topology optimization of thermal fluid-structure systems using body-fitted meshes and parallel computing, J. Comput. Phys., 109574 (2020) · Zbl 1437.74021 · doi:10.1016/j.jcp.2020.109574
[52] Flaherty, Joseph E.; Loy, Raymond. M.; Özturan, Can; Shephard, Mark S.; Szymanski, Boleskaw K.; Teresco, James D.; Ziantz, L. H., Parallel structures and dynamic load balancing for adaptive finite element computation, Appl. Numer. Math., 26, 1, 241-263 (1998) · Zbl 0901.76032 · doi:10.1016/s0168-9274(97)00094-9
[53] Fourno, André; Ngo, Tri-Dat; Noetinger, Benoit; La Borderie, Christian, FraC: A new conforming mesh method for discrete fracture networks, J. Comput. Phys., 376, 713-732 (20210829) · Zbl 1416.65494 · doi:10.1016/j.jcp.2018.10.005
[54] Frank, Tobias; Tertois, Anne-Laure; Mallet, Jean-Laurent, 3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data, Computers & Geosciences, 33, 7, 932-943 (20171018) · doi:10.1016/j.cageo.2006.11.014
[55] Frey, Pascal; George, Paul-Louis, Mesh generation: application to finite elements (2007), ISTE
[56] Galley, Christopher G.; Lelièvre, Peter G.; Farquharson, Colin G., Geophysical inversion for 3D contact surface geometry, Geophysics, 85, 6, K27-K45 (20210826) · doi:10.1190/geo2019-0614.1
[57] Gand, Fabien; Huet, Maxime, On the generation of turbulent inflow for hybrid RANS/LES jet flow simulations, Comput. Fluids, 216, 104816 (2021) · Zbl 1521.76211 · doi:10.1016/j.compfluid.2020.104816
[58] Gassner, G.; Altmann, C.; Hindenlang, F.; Staudenmeier, M.; Munz, C. D., 36th CFD/ADIGMA course on hp-adaptive and hp-multigrid methods, VKI LS, Explicit Discontinuous Galerkin Schemes with Adaptation in Space and Time (2009)
[59] Geuzaine, Christophe; Remacle, Jean-François, Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities, Int. J. Numer. Meth. Engng., 79, 11, 1309-1331 (2009) · Zbl 1176.74181 · doi:10.1002/nme.2579
[60] Giraud, Jérémie; Lindsay, Mark; Jessell, Mark, Generalization of level-set inversion to an arbitrary number of geologic units in a regularized least-squares framework, Geophysics, 86, 4, R623-R637 (20210826) · doi:10.1190/geo2020-0263.1
[61] Godefroy, Gabriel; Caumon, Guillaume; Laurent, Gautier; Bonneau, François, Multi-scenario interpretations from sparse fault evidence using graph theory and geological rules, J. Geophys. Res. Solid Earth, 126, 2, e2020JB020022 (2021) · doi:10.1029/2020JB020022
[62] Henrot, Antoine; Pierre, Michel, Shape Variation and Optimization, 28 (2018), European Mathematical Society · Zbl 1392.49001 · doi:10.4171/178
[63] Huang, Simin; Wellmann, Florian; Marquart, Gabriele; Herty, Michael; Clauser, Christoph, Shape Optimization Methods Locating Layer Interfaces in Geothermal Reservoirs, Energy Procedia, 76, 321-330 (20180713) · doi:10.1016/j.egypro.2015.07.869
[64] Karimi-Fard, M.; Durlofsky, L. J., A general gridding, discretization, and coarsening methodology for modeling flow in porous formations with discrete geological features, Adv. Water Resources, 96, 354-372 (20170425) · doi:10.1016/j.advwatres.2016.07.019
[65] Karimi-Fard, M.; Firoozabadi, A., Numerical Simulation of Water Injection in 2D Fractured Media Using Discrete-Fracture Model, All Days, SPE-71615-MS (2001), SPE · doi:10.2118/71615-MS
[66] Leicht, Tobias; Hartmann, Ralf, Adaptive high-order methods in computational fluid dynamics, Error estimation and hp-adaptive mesh refinement for discontinuous Galerkin methods, 67-94 (2011), World Scientific · Zbl 1358.76042 · doi:10.1142/9789814313193_0003
[67] Leicht, Tobias; Jägersküpper, Jens; Vollmer, Daniel; Schwöppe, Axel; Hartmann, Ralf; Fiedler, Jens; Schlauch, Tobias, DLR-Project Digital-X-Next Generation CFD Solver’Flucs’, CEAS Aeronautical Journal (2016)
[68] Lo, Daniel S. H., Finite element mesh generation (2014), CRC Press
[69] Lorensen, William E.; Cline, Harvey E., Marching cubes: A high resolution 3D surface construction algorithm, ACM SIGGRAPH Comput. Graph., 21, 4, 163-169 (1987) · doi:10.1145/37402.37422
[70] Loseille, Adrien; Alauzet, Frédéric, Continuous mesh framework part I: well-posed continuous interpolation error, SIAM J. Numer. Anal., 49, 1, 38-60 (2011) · Zbl 1230.65018 · doi:10.1137/090754078
[71] Mavriplis, Catherine, A posteriori error estimators for adaptive spectral element techniques, Proceedings of the Eighth GAMM-Conference on Numerical Methods in Fluid Mechanics, 333-342 (1990) · doi:10.1007/978-3-663-13975-1_34
[72] Misztal, Marek Krzysztof; Bærentzen, Jakob Andreas, Topology-adaptive interface tracking using the deformable simplicial complex, ACM Trans. Graph., 31, 3, 1-12 (2012) · doi:10.1145/2167076.2167082
[73] Monteagudo, J. E. P.; Firoozabadi, A., 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, 40, 7 (20210829) · doi:10.1029/2003WR002996
[74] Moureau, Vincent; Domingo, Pascale; Vervisch, Luc, Design of a massively parallel CFD code for complex geometries, C. R. Méc. Acad. Sci. Paris, 339, 2-3, 141-148 (2011) · Zbl 1217.76054 · doi:10.1016/j.crme.2010.12.001
[75] Moureau, Vincent; Domingo, Pascale; Vervisch, Luc, From Large-Eddy Simulation to Direct Numerical Simulation of a lean premixed swirl flame: Filtered laminar flame-PDF modeling, Combustion and Flame, 158, 7, 1340-1357 (2011) · doi:10.1016/j.combustflame.2010.12.004
[76] Murat, F.; Simon, J., Sur le contrôle par un domaine géométrique (1976)
[77] Mustapha, Hussein; Dimitrakopoulos, Roussos, Discretizing two-dimensional complex fractured fields for incompressible two-phase flow, Int. J. Numer. Methods Fluids, 65, 7, 764-780 (20210829) · Zbl 1444.76119 · doi:10.1002/fld.2197
[78] Naddei, Fabio; de la Llave Plata, Marta; Couaillier, Vincent; Coquel, Frédéric, A comparison of refinement indicators for p-adaptive simulations of steady and unsteady flows using discontinuous Galerkin methods, J. Comput. Phys., 376, 508-533 (2019) · Zbl 1416.76122 · doi:10.1016/j.jcp.2018.09.045
[79] Neifeld, Andrej; Boenke, Dirk; Dierke, Juergen; Ewert, Roland, Jet noise prediction with Eddy relaxation source model, 21st AIAA/CEAS Aeroacoustics Conference, 2370 (2015) · doi:10.2514/6.2015-2370
[80] Oliker, Leonid; Biswas, Rupak; Gabow, Harold N., Parallel tetrahedral mesh adaptation with dynamic load balancing, Parallel Comput., 26, 12, 1583-1608 (2000) · Zbl 0948.68075 · doi:10.1016/s0167-8191(00)00047-8
[81] Osher, Stanley; Fedkiw, Ronald, Level set methods and dynamic implicit surfaces, 153 (2006), Springer
[82] Osher, Stanley; Sethian, James A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 1, 12-49 (1988) · Zbl 0659.65132 · doi:10.1016/0021-9991(88)90002-2
[83] Papadopoulos, Dimitris; Herty, Michael; Rath, Volker; Behr, Marek, Identification of uncertainties in the shape of geophysical objects with level sets and the adjoint method, Comput. Geosci., 15, 4, 737-753 (20180713) · Zbl 1237.86006 · doi:10.1007/s10596-011-9242-6
[84] Park, Michael A.; Loseille, Adrien; Krakos, Joshua; Michal, Todd R.; Alonso, Juan J., AIAA AVIATION Forum, Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Towards CFD 2030 (2016), American Institute of Aeronautics and Astronautics
[85] Persson, Per-Olof; Peraire, Jaime, Sub-cell shock capturing for discontinuous Galerkin methods, 44th AIAA Aerospace Sciences Meeting and Exhibit, 112 (2006) · doi:10.2514/6.2006-112
[86] Pertant, Savinien; Bernard, Manuel; Ghigliotti, Giovanni; Balarac, Guillaume, A finite-volume method for simulating contact lines on unstructured meshes in a conservative level-set framework, J. Comput. Phys., 444, 110582 (2021) · Zbl 07515470 · doi:10.1016/j.jcp.2021.110582
[87] Remacle, Jean-François; Flaherty, Joseph E.; Shephard, Mark S., An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems, SIAM Rev., 45, 1, 53-72 (2003) · Zbl 1127.65323 · doi:10.1137/S00361445023830
[88] Remacle, Jean-François; Geuzaine, Christophe; Compère, Gaëtan; Helenbrook, B. T., Encyclopedia of Aerospace Engineering, Adaptive mesh generation and visualization (2010) · doi:10.1002/9780470686652.eae165
[89] Sagaut, Pierre; Terracol, Marc; Deck, Sébastien, Multiscale and multiresolution approaches in turbulence-LES, DES and Hybrid RANS/LES Methods: Applications and Guidelines (2013), World Scientific · Zbl 1275.76004 · doi:10.1142/p878
[90] Sethian, James A., Fast marching methods, SIAM Rev., 41, 2, 199-235 (1999) · Zbl 0926.65106 · doi:10.1137/S0036144598347059
[91] Sethian, James A., Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, 3 (1999), Cambridge University Press · Zbl 0973.76003
[92] Shewchuk, Jonathan Richard, What Is a Good Linear Finite Element? - Interpolation, Conditioning, Anisotropy, and Quality Measures, Proceedings of the 11th International Meshing Roundtable, 115-126 (2002), Sandia National Laboratories
[93] Si, Hang, TetGen, a Delaunay-based quality tetrahedral mesh generator, ACM Trans. Math. Softw., 41, 2, 1-36 (2015) · Zbl 1369.65157
[94] Sokolowski, Jan; Zolésio, Jean-Paul, Introduction to shape optimization (1992), Springer · Zbl 0761.73003 · doi:10.1007/978-3-642-58106-9
[95] Spalart, Philippe, Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach, Proceedings of first AFOSR international conference on DNS/LES (1997)
[96] Spalart, Philippe; Allmaras, Steven, A one-equation turbulence model for aerodynamic flows, 30th aerospace sciences meeting and exhibit, 439 (1992) · doi:10.2514/6.1992-439
[97] The CGAL Project, CGAL User and Reference Manual (2021), CGAL Editorial Board
[98] Thompson, Joe F.; Soni, Bharat K.; Weatherill, Nigel P., Handbook of grid generation (1998), CRC Press · doi:10.1201/9781420050349
[99] Vallet, M. G.; Hecht, F.; Mantel, B., Numerical grid generation in computational fluid dynamics and related fields, Anisotropic control of mesh generation based upon a Voronoi type method, 93-103 (1991), North-Holland · Zbl 0748.76008
[100] Wang, Li; Mavriplis, Dimitri J., Adjoint-based h-p adaptive discontinuous Galerkin methods for the 2D compressible Euler equations, J. Comput. Phys., 228, 20, 7643-7661 (2009) · Zbl 1391.76367 · doi:10.1016/j.jcp.2009.07.012
[101] Wellmann, Florian; Caumon, Guillaume, 3-D Structural geological models: Concepts, methods, and uncertainties, Adv. Geophys., 59, 1-121 (2018) · doi:10.1016/bs.agph.2018.09.001
[102] Xu, Chaoshui; Dowd, Peter, A new computer code for discrete fracture network modelling, Computers & Geosciences, 36, 3, 292-301 (20150630) · doi:10.1016/j.cageo.2009.05.012
[103] Yang, Liang; Hyde, David; Grujic, Ognjen; Scheidt, Celine; Caers, Jef, Assessing and visualizing uncertainty of 3D geological surfaces using level sets with stochastic motion, Computers & Geosciences, 122, 54-67 (20210302) · doi:10.1016/j.cageo.2018.10.006
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