On the Ersatz material approximation in level-set methods
ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 618-634.

The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire et al., J. Comput. Phys. 194 (2004) 363-393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the ersatz material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.

DOI : 10.1051/cocv/2009023
Classification : 49Q10, 34A55, 49Q12
Mots-clés : shape optimization, stability, second order shape derivative, level-set method, Ersatz material approximation 
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Dambrine, Marc; Kateb, Djalil. On the Ersatz material approximation in level-set methods. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 3, pp. 618-634. doi : 10.1051/cocv/2009023. http://www.numdam.org/articles/10.1051/cocv/2009023/

[1] L. Afraites, M. Dambrine, K. Eppler and K. Kateb, Detecting perfectly insulated obstacles by shape optimization techniques of order two. Discret. Contin. Dyn. Syst. - série B 8 (2007) 389-416. | Zbl

[2] L. Afraites, M. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography. SIAM J. Control Optim. 47 (2008) 1556-1590. | Zbl

[3] G. Allaire and F. Jouve, A level-set method for vibration and multiple loads in structural optimization. Comput. Methods Appl. Mech. Engrg. 194 (2005) 3269-3290. | Zbl

[4] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363-393. | Zbl

[5] P. Bernardoni, Outils et méthode de conception de structures mécaniques à déformations et actionnements répartis. Ph.D. Thesis, Université Paris VI, France (2004).

[6] D. Bucur, Do optimal shapes exist? Milan J. Math. 75 (2007) 379-398.

[7] P. Cardaliaguet and O. Ley, Some flows in shape optimization. Arch. Ration. Mech. Anal. 183 (2007) 21-58. | Zbl

[8] P. Cardaliaguet and O. Ley, On the energy of a flow arising in shape optimization. Interfaces Free Bound. 10 (2008) 221-241. | Zbl

[9] M. Dambrine, About the variations of the shape Hessian and sufficient conditions of stability for critical shapes. Revista Real Academia Ciencias-RACSAM 96 (2002) 95-121. | Zbl

[10] M. Dambrine and M. Pierre, About stability of equilibrium shapes. ESAIM: M2AN 34 (2000) 811-834. | Numdam | Zbl

[11] F. De Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343-367. | Zbl

[12] M. Delfour and J.P. Zolesio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization. SIAM (2001).

[13] J. Descloux, Stability of the solutions of the bidimensional magnetic shaping problem in abscence of surface tension. Eur. J. Mech. B Fluid. 10 (1991) 513-526. | Zbl

[14] K. Eppler and H. Harbrecht, A regularized newton method in electrical impedance tomography using hessian information. Control Cybern. 34 (2005) 203-225. | Zbl

[15] K. Eppler, H. Harbrecht and R. Schneider, On convergence in elliptic shape optimization. SIAM J. Control Optim. 46 (2007) 61-83.

[16] A. Henrot and M. Pierre, Variation et optimisation de formes, Mathématiques et Applications 48. Springer (2005). | Zbl

[17] F. Hettlich and W. Rundell, A second degree method for nonlinear inverse problems. SIAM J. Numer. Anal. 37 (1999) 587-620. | Zbl

[18] V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences 127. Springer (2006). | Zbl

[19] A. Kisch, The domain derivative and two applications in inverse scattering theory. Inverse Problems 9 (1993) 81-96. | Zbl

[20] S. Osher and J.A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79 (1988) 12-49. | Zbl

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