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Multi-phase structural optimization via a level set method∗∗∗

Published online by Cambridge University Press:  27 March 2014

G. Allaire ,
C. Dapogny ,
G. Delgado  and
G. Michailidis
G. Allaire
Affiliation:
CMAP, UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France
C. Dapogny
Affiliation:
UPMC Univ Paris 06, UMR 7598, Laboratoire J.-L. Lions, 75005 Paris, France. [email protected] Renault DREAM-DELT’A, Guyancourt, France
G. Delgado
Affiliation:
CMAP, UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France EADS Innovation Works, Suresnes, France
G. Michailidis
Affiliation:
CMAP, UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France Renault DREAM-DELT’A, Guyancourt, France
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Abstract

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 optimizationmulti-materialssigned distance function
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 576 - 611
Copyright
© EDP Sciences, SMAI, 2014

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References

G. Allaire, Shape optimization by the homogenization method. Springer Verlag, New York (2001).
G. Allaire, Conception optimale de structures, vol. 58 of Mathématiques et Applications. Springer, Heidelberg (2006).
Allaire, G. and Castro, C., A new approach for the optimal distribution of assemblies in a nuclear reactor. Numerische Mathematik 89 (2001) 1–29. Google Scholar
Allaire, G., Jouve, F. and Toader, A.M., Structural optimization using shape sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363–393. Google Scholar
Allaire, G., Jouve, F. and Van Goethem, N., Damage evolution in brittle materials by shape and topological sensitivity analysis. J. Comput. Phys. 230 (2011) 5010–5044. Google Scholar
Allaire, G. and Gutierrez, S., Optimal Design in Small Amplitude Homogenization. ESAIM: M2AN 41 (2007) 543–574. Google Scholar
L. Ambrosio, 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.
Ambrosio, L. and Buttazzo, G., An optimal design problem with perimeter penalization. Calc. Var. 1 (1993) 55–69. Google Scholar
M. Bendsoe and O. Sigmund, Topology Optimization. Theory, Methods, and Applications. Springer Verlag, New York (2003).
Bernardi, Ch. and Pironneau, O., Sensitivity of Darcy’s law to discontinuities. Chinese Ann. Math. Ser. B 24 (2003) 205–214. Google Scholar
H.-J. Butt, K. Graf and M. Kappl, Physics and Chemistry of Interfaces. Wiley (2003).
Cannarsa, P. and Cardaliaguet, P., Representation of equilibrium solutions to the table problem for growing sandpiles. J. Eur. Math. Soc. 6 (2004) 1–30. Google Scholar
Céa, J., Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. Math. Model. Numer. 20 (1986) 371–420. Google Scholar
Chambolle, A., A density result in two-dimensional linearized elasticity and applications. Arch. Ration. Mech. Anal. 167 (2003) 211–233. Google Scholar
I. Chavel, Riemannian Geometry, a modern introduction, 2nd Edn. Cambridge University Press (2006).
Chenais, D., On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189–289. Google Scholar
A. Cherkaev, Variational Methods for Structural Optimization. Springer, New York (2000).
C. Dapogny, Ph.D. thesis, Université Pierre et Marie Curie. In preparation.
de Gournay, F., Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343–367. Google Scholar
G. Delgado, Ph.D. thesis, Ecole Polytechnique. In preparation.
M.C. Delfour and J.-P. Zolesio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edition. SIAM, Philadelphia (2011).
Delfour, M.C. and Zolesio, J.-P., Shape identification via metrics constructed from the oriented distance function. Control and Cybernetics 34 (2005) 137–164. Google Scholar
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press (1992).
Federer, H., Curvature Measures. Trans. Amer. Math. Soc. 93 (1959) 418–491. Google Scholar
J. Gomes and O. Faugeras, Reconciling distance functions and level sets, Scale-Space Theories in Computer Vision. Springer (1999) 70–81.
Haslinger, J. and Dvorak, J., Optimum composite material design. RAIRO M2AN 29 (1995) 657–686. Google Scholar
A. Henrot and M. Pierre, Variation et optimisation de formes, une analyse géométrique, vol. 48 of Mathématiques et Applications. Springer, Heidelberg (2005).
Hettlich, F. and Rundell, W., The determination of a discontinuity in a conductivity from a single boundary measurement. Inverse Problems 14 (1998) 67–82. Google Scholar
Karchevsky, A.L., Reconstruction of pressure velocities and boundaries of thin layers in thinly-stratified layers. J. Inverse Ill-Posed Probl. 18 (2010) 371–388. Google Scholar
Lipton, R., Design of functionally graded composite structures in the presence of stress constraints. Int. J. Solids Structures 39 (2002) 2575–2586. Google Scholar
Mantegazza, C. and Menucci, A.C., Hamilton–Jacobi Equations and Distance Functions on Riemannian Manifolds. Appl. Math. Optim. 47 (2002) 1–25. Google Scholar
W. Mc Lean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000).
Mei, Y. and Wang, X., A level set method for structural topology optimization with multi-constraints and multi-materials. Acta Mechanica Sinica 20 (2004). Google Scholar
Mei, Y. and Wang, X., A level set method for structural topology optimization and its applications. Adv. Eng. software 35 (2004) 415–441. Google Scholar
G. Milton, The theory of composites. Cambridge University Press (2001).
F. Murat and J. Simon, Sur le contrôle par un domaine géométrique. Technical Report RR-76015. Laboratoire d’Analyse Numérique (1976).
J. Nocedal and S.J. Wright, Numerical optimization. Springer Science+ Business Media (2006).
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.
Ortega, J.M. and Rheinboldt, W.C., On discretization and differentiation of operators with application to Newton’s method. SIAM J. Numer. Anal. 3 (1966) 143–156. Google Scholar
Osher, S.J. and Sethian, J.A., Front propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 78 (1988) 12–49. Google Scholar
Pantz, O., Sensibilité de l’équation de la chaleur aux sauts de conductivité. C. R. Acad. Sci. Paris, Ser. I 341 (2005) 333–337. Google Scholar
O. Pironneau, F. Hecht and A. Le Hyaric, FreeFem++ version 2.15-1. Available on http://www.freefem.org/ff++/.
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).
Sigmund, O. and Torquato, S., Design of materials with extreme thermal expansion using a three-phase topology optimization method. J. Mech. Phys. Solids 45 (1997) 1037–1067. Google Scholar
Sigmund, O., Design of multiphysics actuators using topology optimization-part ii: Two-material structures. Comput. Methods Appl. Mech. Eng. 190 (2001) 6605–6627. Google Scholar
Sukumar, N., Chopp, D.L., Moes, N. and Belytschko, T., Modeling holes and inclusions by level sets in the extended finite element method. Comput. Methods Appl. Mech. Eng. 190 (2001) 6183–6200. Google Scholar
S. Suresh, A. Mortensen, Fundamentals of functionally graded materials. London, Institute of Materials (1998).
Šverak, V., On optimal shape design. J. Math. Pures Appl. 72 (1993) 537–551. Google Scholar
Swan, C.C., Kosaka, I., Voigt-Reuss topology optimization for structures with linear elastic material behaviors. Int. J. Numer. Methods Eng. 40 (1997). Google Scholar
L. Tarta, The general theory of homogenization. A personalized introduction, vol. 7 of Lecture Notes of the Unione Matematica Italiana. Springer-Verlag, Berlin, UMI, Bologna (2009).
R. Tilley, Understanding Solids: The Science of Materials. Wiley (2004).
N. Vermaak, G. Michailidis, Y. Brechet, G. Allaire, G. Parry and R. Estevez, Material Interface Effects on the Topology Optimization of Multi-Phase Structures Using A Level Set Method. submitted.
Vese, L.A. and Chan, T.F., A multiphase level set framework for image segmentation using the Mumford and Shah model. Int. J. Comput. Vision 50 (2002) 271–293. Google Scholar
Wang, M., Chen, S., Wang, X. and Mei, Y., Design of Multimaterial Compliant Mechanisms Using Level-Set Methods. J. Mech. Des. 127 (2005) 941–956. Google Scholar
Wang, M. and Wang, X., Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput. Methods Appl. Mech. Eng. 193 (2004) 469–496. Google Scholar
Wang, M. and Wang, X., A level-set based variational method for design and optimization of heterogeneous objetcs. Compututer-Aided Design 37 (2005) 321–337. Google Scholar
Yin, L. and Ananthasuresh, G.K., Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct. Multidiscip. Optim. 23 (2001) 49–62. Google Scholar
Zhou, S. and Li, Q., Computational design of multi-phase microstructural materials for extremal conductivity. Comput. Mater. Sci. 43 (2008) 549–564. Google Scholar
Zhou, S. and Wang, M.Y., Multimaterial structural optimization with a generalized Cahn–Hilliard model of multiphase transition. Struct. Multidisc. Optim. 33 (2007) 89–111. Google Scholar