We consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing “grey” regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials. Several 2-d examples of compliance minimization are numerically tested which allow us to compare the shape derivatives obtained in the sharp or smoothed interface cases.
Keywords: shape and topology optimization, multi-materials, signed distance function
@article{COCV_2014__20_2_576_0, author = {Allaire, G. and Dapogny, C. and Delgado, G. and Michailidis, G.}, title = {Multi-phase structural optimization \protect\emph{via }a level set method}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {576--611}, publisher = {EDP-Sciences}, volume = {20}, number = {2}, year = {2014}, doi = {10.1051/cocv/2013076}, zbl = {1287.49045}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2013076/} }
TY - JOUR AU - Allaire, G. AU - Dapogny, C. AU - Delgado, G. AU - Michailidis, G. TI - Multi-phase structural optimization via a level set method JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 576 EP - 611 VL - 20 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013076/ DO - 10.1051/cocv/2013076 LA - en ID - COCV_2014__20_2_576_0 ER -
%0 Journal Article %A Allaire, G. %A Dapogny, C. %A Delgado, G. %A Michailidis, G. %T Multi-phase structural optimization via a level set method %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 576-611 %V 20 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013076/ %R 10.1051/cocv/2013076 %G en %F COCV_2014__20_2_576_0
Allaire, G.; Dapogny, C.; Delgado, G.; Michailidis, G. Multi-phase structural optimization via a level set method. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, pp. 576-611. doi : 10.1051/cocv/2013076. http://archive.numdam.org/articles/10.1051/cocv/2013076/
[1] Shape optimization by the homogenization method. Springer Verlag, New York (2001). | MR | Zbl
,[2] Conception optimale de structures, vol. 58 of Mathématiques et Applications. Springer, Heidelberg (2006). | MR | Zbl
,[3] A new approach for the optimal distribution of assemblies in a nuclear reactor. Numerische Mathematik 89 (2001) 1-29. | MR | Zbl
and ,[4] Structural optimization using shape sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363-393. | MR | Zbl
, and ,[5] Damage evolution in brittle materials by shape and topological sensitivity analysis. J. Comput. Phys. 230 (2011) 5010-5044. | MR
, and ,[6] Optimal Design in Small Amplitude Homogenization. ESAIM: M2AN 41 (2007) 543-574. | Numdam | MR | Zbl
and ,[7] Lecture notes on geometric evolution problems, distance function and viscosity solutions, in Calculus of Variations and Partial Differential Equations, edited by G. Buttazo, A. Marino and M.K.V. Murthy. Springer (1999) 5-93. | Zbl
,[8] An optimal design problem with perimeter penalization. Calc. Var. 1 (1993) 55-69. | MR | Zbl
and ,[9] Topology Optimization. Theory, Methods, and Applications. Springer Verlag, New York (2003). | MR | Zbl
and ,[10] Sensitivity of Darcy's law to discontinuities. Chinese Ann. Math. Ser. B 24 (2003) 205-214. | MR | Zbl
and ,[11] Physics and Chemistry of Interfaces. Wiley (2003).
, and ,[12] Representation of equilibrium solutions to the table problem for growing sandpiles. J. Eur. Math. Soc. 6 (2004) 1-30. | MR | Zbl
and ,[13] Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. Math. Model. Numer. 20 (1986) 371-420. | Numdam | MR | Zbl
,[14] A density result in two-dimensional linearized elasticity and applications. Arch. Ration. Mech. Anal. 167 (2003) 211-233. | MR | Zbl
,[15] Riemannian Geometry, a modern introduction, 2nd Edn. Cambridge University Press (2006). | MR | Zbl
,[16] On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189-289. | MR | Zbl
,[17] Variational Methods for Structural Optimization. Springer, New York (2000). | MR | Zbl
,[18] Ph.D. thesis, Université Pierre et Marie Curie. In preparation.
,[19] Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343-367. | MR | Zbl
,[20] Ph.D. thesis, Ecole Polytechnique. In preparation.
,[21] Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edition. SIAM, Philadelphia (2011). | MR | Zbl
and ,[22] Shape identification via metrics constructed from the oriented distance function. Control and Cybernetics 34 (2005) 137-164. | MR | Zbl
and ,[23] Measure theory and fine properties of functions. CRC Press (1992). | MR | Zbl
and ,[24] Curvature Measures. Trans. Amer. Math. Soc. 93 (1959) 418-491. | MR | Zbl
,[25] Reconciling distance functions and level sets, Scale-Space Theories in Computer Vision. Springer (1999) 70-81.
and ,[26] Optimum composite material design. RAIRO M2AN 29 (1995) 657-686. | Numdam | MR | Zbl
and ,[27] Variation et optimisation de formes, une analyse géométrique, vol. 48 of Mathématiques et Applications. Springer, Heidelberg (2005). | MR | Zbl
and ,[28] The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 14 (1998) 67-82. | MR | Zbl
and ,[29] Reconstruction of pressure velocities and boundaries of thin layers in thinly-stratified layers. J. Inverse Ill-Posed Probl. 18 (2010) 371-388. | MR | Zbl
,[30] Design of functionally graded composite structures in the presence of stress constraints. Int. J. Solids Structures 39 (2002) 2575-2586. | MR | Zbl
,[31] Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds. Appl. Math. Optim. 47 (2002) 1-25. | MR | Zbl
and ,[32] Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000). | MR | Zbl
,[33] A level set method for structural topology optimization with multi-constraints and multi-materials. Acta Mechanica Sinica 20 (2004). | MR | Zbl
and ,[34] A level set method for structural topology optimization and its applications. Adv. Eng. software 35 (2004) 415-441. | Zbl
and ,[35] The theory of composites. Cambridge University Press (2001). | MR | Zbl
,[36] Sur le contrôle par un domaine géométrique. Technical Report RR-76015. Laboratoire d'Analyse Numérique (1976).
and ,[37] Numerical optimization. Springer Science+ Business Media (2006). | MR | Zbl
and ,[38] ZH.O. Oralbekova, K.T. Iskakov and A.L. Karchevsky, Existence of the residual functional derivative with respect to a coordinate of gap point of medium, to appear in Appl. Comput. Math. | MR
[39] On discretization and differentiation of operators with application to Newton's method. SIAM J. Numer. Anal. 3 (1966) 143-156. | MR | Zbl
and ,[40] Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 78 (1988) 12-49. | MR | Zbl
and ,[41] Sensibilité de l'équation de la chaleur aux sauts de conductivité. C. R. Acad. Sci. Paris, Ser. I 341 (2005) 333-337. | MR | Zbl
,[42] FreeFem++ version 2.15-1. Available on http://www.freefem.org/ff++/.
, and ,[43] J.A. Sethian, Level-Set Methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision and materials science. Cambridge University Press (1999). | MR | Zbl
[44] Design of materials with extreme thermal expansion using a three-phase topology optimization method. J. Mech. Phys. Solids 45 (1997) 1037-1067. | MR
and ,[45] Design of multiphysics actuators using topology optimization-part ii: Two-material structures. Comput. Methods Appl. Mech. Eng. 190 (2001) 6605-6627. | Zbl
,[46] Modeling holes and inclusions by level sets in the extended finite element method. Comput. Methods Appl. Mech. Eng. 190 (2001) 6183-6200. | MR | Zbl
, , and ,[47] Fundamentals of functionally graded materials. London, Institute of Materials (1998).
, ,[48] On optimal shape design. J. Math. Pures Appl. 72 (1993) 537-551. | MR | Zbl
,[49] Voigt-Reuss topology optimization for structures with linear elastic material behaviors. Int. J. Numer. Methods Eng. 40 (1997). | MR | Zbl
, ,[50] The general theory of homogenization. A personalized introduction, vol. 7 of Lecture Notes of the Unione Matematica Italiana. Springer-Verlag, Berlin, UMI, Bologna (2009). | MR | Zbl
,[51] Understanding Solids: The Science of Materials. Wiley (2004).
,[52] Material Interface Effects on the Topology Optimization of Multi-Phase Structures Using A Level Set Method. submitted.
, , , , and ,[53] A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vision 50 (2002) 271-293. | Zbl
and ,[54] Design of Multimaterial Compliant Mechanisms Using Level-Set Methods. J. Mech. Des. 127 (2005) 941-956.
, , and ,[55] Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput. Methods Appl. Mech. Eng. 193 (2004) 469-496. | MR | Zbl
and ,[56] A level-set based variational method for design and optimization of heterogeneous objetcs. Compututer-Aided Design 37 (2005) 321-337.
and ,[57] Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct. Multidiscip. Optim. 23 (2001) 49-62.
and ,[58] Computational design of multi-phase microstructural materials for extremal conductivity. Comput. Mater. Sci. 43 (2008) 549-564.
and ,[59] Multimaterial structural optimization with a generalized Cahn-Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33 (2007) 89-111. | MR | Zbl
and ,Cited by Sources: