[Une stratégie d’évolution de maillage basée sur la méthode des lignes de niveaux pour l’optimisation de formes : survol et mise en pratique]
Cet article traite d’un cadre de travail récent dédié à la résolution numérique de problèmes d’optimisation de formes ; il s’illustre par une représentation exacte, maillée, de la forme à chaque itération du procédé, tout en laissant la place à une évolution arbitraire de celle-ci (y compris des changements de sa topologie). L’idée centrale de cette stratégie est de combiner deux représentations complémentaires de la forme : d’une part, celle-ci est maillée explicitement, de sorte qu’il est possible d’effectuer des calculs mécaniques précis par la méthode des éléments finis ; d’autre part, elle est décrite implicitement, par la méthode des lignes de niveaux, facilitant ainsi le suivi robuste de son évolution. Dans la première partie de ce travail, on résume les points saillants de cette stratégie numérique. Après avoir brièvement rappelé quelques notions de base – en lien, entre autres, avec l’optimisation de formes et le maillage – on décrit les schémas numériques mis en jeu, notamment pour la pratique de la méthode des lignes de niveaux, les algorithmes de remaillage, et l’algorithme d’optimisation numérique. Cette méthodologie est illustrée par des exemples numériques en deux et trois dimensions d’espace, dans différents contextes physiques. Dans la seconde partie de cet article, on propose une implémentation python open-source, simple mais efficace, de ce cadre de travail. Le code est détaillé de sorte que le lecteur puisse facilement intervenir dedans et le modifier pour traiter un problème de son choix.
This article revolves around a recent numerical framework for shape and topology optimization, which features an exact mesh of the shape at each iteration of the process, while still leaving the room for an arbitrary evolution of the latter (including changes in its topology). In a nutshell, two complementary representations of the shape are combined: on the one hand, it is meshed exactly, which allows for precise mechanical calculations based on the finite element method; on the other hand, it is described implicitly, using the level set method, which makes it possible to track its evolution in a robust way. In the first part of this work, we overview the main aspects of this numerical strategy. After a brief presentation of some necessary background material – related to shape optimization and meshing, among others – we describe the numerical schemes involved, notably when it comes to the practice of the level set method, the remeshing algorithms, and the considered optimization solver. This strategy is illustrated with 2d and 3d numerical examples in various physical contexts. In the second part of this article, we propose a simple albeit efficient python-based implementation of this framework. The code is described with a fair amount of details, and it is expected that the reader can easily elaborate upon the presented examples to tackle his own problems.
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Charles Dapogny 1 ; Florian Feppon 2
CC-BY 4.0
@article{CRMATH_2023__361_G8_1267_0,
author = {Charles Dapogny and Florian Feppon},
title = {Shape optimization using a level set based mesh evolution method: an overview and tutorial},
journal = {Comptes Rendus. Math\'ematique},
pages = {1267--1332},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
year = {2023},
doi = {10.5802/crmath.498},
language = {en},
}
TY - JOUR AU - Charles Dapogny AU - Florian Feppon TI - Shape optimization using a level set based mesh evolution method: an overview and tutorial JO - Comptes Rendus. Mathématique PY - 2023 SP - 1267 EP - 1332 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.498 LA - en ID - CRMATH_2023__361_G8_1267_0 ER -
Charles Dapogny; Florian Feppon. Shape optimization using a level set based mesh evolution method: an overview and tutorial. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1267-1332. doi : 10.5802/crmath.498. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.498/
[1] Numerical discretization of the first-order Hamilton-Jacobi equation on triangular meshes, Commun. Pure Appl. Math., Volume 49 (1996) no. 12, pp. 1339-1373 | DOI | MR | Zbl
[2] Sobolev spaces, Pure and Applied Mathematics, 140, Academic Press Inc., 2003
[3] Shape optimization by the homogenization method, 146, Springer, 2002 | DOI
[4] Shape optimization by the homogenization method, Numer. Math., Volume 76 (1997) no. 1, pp. 27-68 | DOI | MR | Zbl
[5] Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 17-18, pp. 999-1003 | DOI | Numdam | MR | Zbl
[6] A mesh evolution algorithm based on the level set method for geometry and topology optimization, Struct. Multidiscip. Optim., Volume 48 (2013) no. 4, pp. 711-715 | DOI | MR
[7] Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Eng., Volume 282 (2014), pp. 22-53 | DOI | MR
[8] Shape and topology optimization, Geometric partial differential equations. Part II (Handbook of Numerical Analysis), Volume 22, Elsevier; North-Holland, 2021, pp. 1-132 | DOI | Zbl
[9] Structural optimization using topological and shape sensitivity via a level set method, Control Cybern., Volume 34 (2005) no. 1, p. 59 | MR | Zbl
[10] Thickness control in structural optimization via a level set method, Struct. Multidiscip. Optim., Volume 53 (2016) no. 6, pp. 1349-1382 | DOI | MR
[11] Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., Volume 194 (2004) no. 1, pp. 363-393 | DOI | MR | Zbl
[12] Structural optimization with FreeFem++, Struct. Multidiscip. Optim., Volume 32 (2006) no. 3, pp. 173-181 | DOI | MR | Zbl
[13] Conception optimale de structures, 58, Springer, 2007
[14] A new algorithm for topology optimization using a level-set method, J. Comput. Phys., Volume 216 (2006) no. 2, pp. 573-588 | DOI | MR | Zbl
[15] Shape Optimization Problems, Springer, 2020 | DOI
[16] Domain optimization analysis in linear elastic problems: approach using traction method, JSME Int. J. Ser. A, Mech. Mater. Eng., Volume 39 (1996) no. 2, pp. 272-278
[17] Dynamic adaptation for deforming tetrahedral meshes, 14th Computational Fluid Dynamics Conference (1999), p. 3253
[18] Tetrahedral remeshing in the context of large-scale numerical simulation and high performance computing, MathS In Action, Volume 11 (2022), pp. 129-164 | DOI | MR | Zbl
[19] An algorithm for constrained optimization with applications to the design of mechanical structures, International Conference on Engineering Optimization, Springer (2018), pp. 272-284
[20] Three-dimensional CFD simulations with large displacement of the geometries using a connectivity-change moving mesh approach, Eng. Comput., Volume 35 (2019) no. 2, pp. 397-422 | DOI
[21] Numerical schemes for the Hamilton–Jacobi and level set equations on triangulated domains, J. Comput. Phys., Volume 145 (1998) no. 1, pp. 1-40 | DOI | MR | Zbl
[22] Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., Volume 71 (1988) no. 2, pp. 197-224 | DOI | MR | Zbl
[23] Topology optimization: theory, methods, and applications, Springer, 2013
[24] Phase-field approaches to structural topology optimization, Constrained optimization and optimal control for partial differential equations, Springer, 2012, pp. 245-256 | DOI | Zbl
[25] Meshing, Geometric Modeling and Numerical Simulation 1: Form Functions, Triangulations and Geometric Modeling, John Wiley & Sons, 2017 | DOI
[26] Polygon mesh processing, CRC Press, 2010 | DOI
[27] Design-dependent loads in topology optimization, ESAIM, Control Optim. Calc. Var., Volume 9 (2003), pp. 19-48 | DOI | Numdam | MR | Zbl
[28] Shape and topology optimization for maximum probability domains in quantum chemistry, Numer. Math. (2022), pp. 1-48
[29] Variational methods in some shape optimization problems, Springer, 2002
[30] An accurate anisotropic adaptation method for solving the level set advection equation, Int. J. Numer. Methods Fluids, Volume 70 (2012) no. 7, pp. 899-922 | DOI | MR | Zbl
[31] A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces Free Bound., Volume 5 (2003) no. 3, pp. 301-329 | DOI | MR | Zbl
[32] Shape optimization using the cut finite element method, Comput. Methods Appl. Mech. Eng., Volume 328 (2018), pp. 242-261 | DOI | MR | Zbl
[33] Computing Minimal Surfaces via Level Set Curvature Flow, J. Comput. Phys., Volume 106 (1993) no. 1, pp. 77-91 | DOI | MR | Zbl
[34] Combined shape and topology optimization of 3D structures, Computers & Graphics, Volume 46 (2015), pp. 25-35 | DOI
[35] Topology optimization using an explicit interface representation, Struct. Multidiscip. Optim., Volume 49 (2014) no. 3, pp. 387-399 | DOI | MR
[36] The finite element method for elliptic problems, 40, Society for Industrial and Applied Mathematics, 2002 | DOI
[37] On the ersatz material approximation in level-set methods, ESAIM, Control Optim. Calc. Var., Volume 16 (2010) no. 3, pp. 618-634 | DOI | Numdam | MR | Zbl
[38] The topological ligament in shape optimization: an approach based on thin tubular inhomogeneities asymptotics, SMAI J. Comput. Math. (2021), pp. 185-266 | MR | Zbl
[39] 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
[40] Computation of the signed distance function to a discrete contour on adapted triangulation, Calcolo, Volume 49 (2012) no. 3, pp. 193-219 | DOI | MR | Zbl
[41] Geometrical shape optimization in fluid mechanics using FreeFem++, Struct. Multidiscip. Optim. (2017), pp. 1-28
[42] Optimization of the shape of regions supporting boundary conditions, Numer. Math., Volume 146 (2020) no. 1, pp. 51-104 | DOI | MR | Zbl
[43] Velocity extension for the level-set method and multiple eigenvalues in shape optimization, SIAM J. Control Optim., Volume 45 (2006) no. 1, pp. 343-367 | DOI | MR | Zbl
[44] Isogeometric analysis for topology optimization with a phase field model, Arch. Comput. Methods Eng., Volume 19 (2012) no. 3, pp. 427-465 | DOI | MR | Zbl
[45] Shapes and geometries: metrics, analysis, differential calculus, and optimization, Society for Industrial and Applied Mathematics, 2011 | DOI
[46] Topology optimization of structures undergoing brittle fracture, J. Comput. Phys., Volume 458 (2022), 111048, 35 pages | DOI | MR | Zbl
[47] Topology optimization in quasi-static plasticity with hardening using a level-set method, Struct. Multidiscip. Optim., Volume 64 (2021) no. 5, pp. 3163-3191 | DOI | MR
[48] An efficient method of triangulating equi-valued surfaces by using tetrahedral cells, IEICE Trans. Inf. Syst., Volume 74 (1991) no. 1, pp. 214-224
[49] Introducing the sequential linear programming level-set method for topology optimization, Struct. Multidiscip. Optim., Volume 51 (2015) no. 3, pp. 631-643 | DOI | MR
[50] Generalized shape optimization using X-FEM and level set methods, IUTAM symposium on topological design optimization of structures, machines and materials, Springer (2006), pp. 23-32 | DOI
[51] Comparative study of three-dimensional wing drag minimization by different optimization techniques, J. Aircraft, Volume 46 (2009) no. 2, pp. 526-541 | DOI
[52] Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory, SIAM J. Numer. Anal., Volume 44 (2006) no. 2, pp. 753-778 | MR
[53] Topology optimization of continuum structures: a review, Appl. Mech. Rev., Volume 54 (2001) no. 4, pp. 331-390 | DOI
[54] Measure theory and fine properties of functions, CRC Press, 2015 | DOI
[55] Shape and topology optimization of multiphysics systems, Ph. D. Thesis, Université Paris-Saclay (ComUE) (2019)
[56] Shape optimization of a coupled thermal fluid–structure problem in a level set mesh evolution framework, SeMA J. (2019), pp. 1-46
[57] Null space gradient flows for constrained optimization with applications to shape optimization, ESAIM, Control Optim. Calc. Var., Volume 26 (2020), 90, 45 pages | DOI | MR | Zbl
[58] Topology optimization of thermal fluid–structure systems using body-fitted meshes and parallel computing, J. Comput. Phys. (2020), 109574, 29 pages | DOI | MR | Zbl
[59] Body-fitted topology optimization of 2D and 3D fluid-to-fluid heat exchangers, Comput. Methods Appl. Mech. Eng., Volume 376 (2021), 113638, 36 pages | DOI | MR | Zbl
[60] Mesh generation: application to finite elements, ISTE, 2007
[61] The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim., Volume 39 (2001) no. 6, pp. 1756-1778 | DOI | MR | Zbl
[62] Surface evolution equations, Springer, 2006
[63] OpenMDAO: an open-source framework for multidisciplinary design, analysis, and optimization, Struct. Multidiscip. Optim., Volume 59 (2019) no. 4, pp. 1075-1104 | DOI | MR
[64] Mémoire sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées, Mém. Sav. étrang., 33, Imprimerie nationale, 1908 | Numdam
[65] A method for time accurate turbulent compressible fluid flow simulation with moving boundary components employing local remeshing, Int. J. Numer. Methods Fluids, Volume 53 (2007) no. 8, pp. 1243-1266 | DOI | MR | Zbl
[66] New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3-4, pp. 251-266 | MR | Zbl
[67] Shape Variation and Optimization. A geometrical analysis, EMS Tracts in Mathematics, 28, European Mathematical Society, 2018 | DOI | Zbl
[68] What is the optimal shape of a pipe?, Arch. Ration. Mech. Anal., Volume 196 (2010) no. 1, pp. 281-302 | DOI | MR | Zbl
[69] Comparison of approximate shape gradients, BIT Numer. Math., Volume 55 (2015) no. 2, pp. 459-485 | DOI | MR | Zbl
[70] Aerodynamic design via control theory, Recent advances in computational fluid dynamics (Princeton, NJ, 1988) (Lecture Notes in Engineering), Volume 43, Springer, 1989, pp. 377-401 | DOI | MR | Zbl
[71] Computational algorithms for aerodynamic analysis and design, Appl. Numer. Math., Volume 13 (1993) no. 5, pp. 383-422 | DOI | MR | Zbl
[72] Computing geodesic paths on manifolds, Proc. Natl. Acad. Sci. USA, Volume 95 (1998) no. 15, pp. 8431-8435 | DOI | MR | Zbl
[73] On a cellular developmental method for layout optimization via the two-point topological derivative, Struct. Multidiscip. Optim., Volume 64 (2021) no. 4, pp. 2343-2360 | DOI | MR
[74] Optimal design and relaxation of variational problems, I, Commun. Pure Appl. Math., Volume 39 (1986) no. 1, pp. 113-137 | DOI | MR | Zbl
[75] Matrix-free aerostructural optimization of aircraft wings, Struct. Multidiscip. Optim., Volume 53 (2016) no. 3, pp. 589-603 | DOI | MR
[76] A level set-based structural optimization code using FEniCS, Struct. Multidiscip. Optim., Volume 58 (2018) no. 3, pp. 1311-1334 | DOI | MR
[77] Optimal control of systems governed by partial differential equations, Grundlehren der Mathematischen Wissenschaften, 170, Springer, 1971 | DOI
[78] Marching cubes: A high resolution 3D surface construction algorithm, ACM SIGGRAPH Comput. Graph., Volume 21 (1987) no. 4, pp. 163-169 | DOI
[79] A fast algorithm for computing the closest point and distance transform (2000) (https://www.researchgate.net/publication/2393786_A_Fast_Algorithm_for_Computing_the_Closest_Point_and_Distance_Transform)
[80] Applied shape optimization for fluids, Numerical Mathematics and Scientific Computation, Oxford University Press, 2010
[81] Sur le contrôle par un domaine géométrique (1976) pré-publication du Laboratoire d’Analyse Numérique,(76015)
[82] The topological derivative of the Dirichlet integral due to formation of a thin ligament, Sib. Math. J., Volume 45 (2004) no. 2, pp. 341-355 | DOI
[83] Numerical optimization 2nd, Springer, 2006
[84] Topological derivatives in shape optimization, Springer, 2012
[85] Level set methods and dynamic implicit surfaces, 153, Springer, 2006
[86] Level set methods for optimization problems involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum, J. Comput. Phys., Volume 171 (2001) no. 1, pp. 272-288 | DOI | MR | Zbl
[87] 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
[88] On optimum profiles in Stokes flow, J. Fluid Mech., Volume 59 (1973) no. 1, pp. 117-128 | DOI | MR | Zbl
[89] Optimal shape design for elliptic systems, Springer, 1982
[90] Finite element methods for fluids, Wiley Publishing, 1989
[91] A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophys. J. Int., Volume 167 (2006) no. 2, pp. 495-503 | DOI
[92] Fast marching methods, SIAM Rev., Volume 41 (1999) no. 2, pp. 199-235 | DOI | MR | Zbl
[93] Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, 3, Cambridge University Press, 1999
[94] Structural boundary design via level set and immersed interface methods, J. Comput. Phys., Volume 163 (2000) no. 2, pp. 489-528 | DOI | MR | Zbl
[95] A 99 line topology optimization code written in Matlab, Struct. Multidiscip. Optim., Volume 21 (2001) no. 2, pp. 120-127 | DOI
[96] Introduction to shape optimization, Springer, 1992 | DOI
[97] Semi-Lagrangian methods for level set equations, J. Comput. Phys., Volume 151 (1999) no. 2, pp. 498-533 | DOI | MR | Zbl
[98] The method of moving asymptotesÑa new method for structural optimization, Int. J. Numer. Methods Eng., Volume 24 (1987) no. 2, pp. 359-373 | DOI | MR | Zbl
[99] Shape and topology optimization based on the phase field method and sensitivity analysis, J. Comput. Phys., Volume 229 (2010) no. 7, pp. 2697-2718 | DOI | MR | Zbl
[100] Rapid and accurate computation of the distance function using grids, J. Comput. Phys., Volume 178 (2002) no. 1, pp. 175-195 | DOI | MR | Zbl
[101] A comprehensive review of educational articles on structural and multidisciplinary optimization, Struct. Multidiscip. Optim., Volume 64 (2021) no. 5, pp. 2827-2880 | DOI
[102] A level set method for structural topology optimization, Comput. Methods Appl. Mech. Eng., Volume 192 (2003) no. 1-2, pp. 227-246 | DOI | MR | Zbl
[103] Topology optimization of thermoelastic structures using level set method, Comput. Mech., Volume 42 (2008) no. 6, pp. 837-857 | DOI | Zbl
[104] A fast sweeping method for eikonal equations, Math. Comput., Volume 74 (2005) no. 250, pp. 603-627 | DOI | MR | Zbl
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