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Distributed shape derivative via averaged adjoint method and applications. (English) Zbl 1347.49070
Summary: The structure theorem of Hadamard-Zolésio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. Actually, the domain representation, also known as distributed shape derivative, is more general than the boundary expression as it is well-defined for shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithms. In this paper, we describe several advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We identify a tensor representation of the distributed shape derivative, study its properties and show how it allows to recover the boundary expression directly. We use a novel Lagrangian approach, which is applicable to a large class of shape optimization problems, to compute the distributed shape derivative. We also apply the technique to retrieve the distributed shape derivative for electrical impedance tomography. Finally, we explain how to adapt the level set method to the distributed shape derivative framework and present numerical results.

MSC:
49Q10 Optimization of shapes other than minimal surfaces
49M30 Other numerical methods in calculus of variations (MSC2010)
65K10 Numerical optimization and variational techniques
35Q93 PDEs in connection with control and optimization
35R30 Inverse problems for PDEs
35R05 PDEs with low regular coefficients and/or low regular data
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FEniCS
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References:
[1] D. Adalsteinsson and J.A. Sethian, A fast level set method for propagating interfaces. J. Comput. Phys.118 (1995) 269-277. · Zbl 0823.65137
[2] L. Afraites, M. Dambrine and D. Kateb, Shape methods for the transmission problem with a single measurement. Numer. Funct. Anal. Optim.28 (2007) 519-551. · Zbl 1114.49020
[3] 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 1136.74368
[4] Z. Belhachmi and H. Meftahi, Shape sensitivity analysis for an interface problem via minimax differentiability. Appl. Math. Comput.219 (2013) 6828-6842. · Zbl 1286.65148
[5] L. Borcea, Electrical impedance tomography. Inverse Problems18 (2002) R99-R136. · Zbl 1031.35147
[6] A.-P. Calderón, On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980). Soc. Brasil. Mat., Rio de Janeiro (1980) 65-73.
[7] A. Canelas, A. Laurain and A.A. Novotny, A new reconstruction method for the inverse potential problem. J. Comput. Phys.268 (2014) 417-431. · Zbl 1349.35425
[8] A. Canelas, A. Laurain and A.A. Novotny, A new reconstruction method for the inverse source problem from partial boundary measurements. Inverse Problems31 (2015) 075009. · Zbl 1319.35296
[9] J. Céa, Conception optimale ou identification de formes: calcul rapide de la dérivée directionnelle de la fonction coût. RAIRO M2AN20 (1986) 371-402. · Zbl 0604.49003
[10] M. Cheney, D. Isaacson and J.C. Newell, Electrical impedance tomography. SIAM Rev.41 (1999) 85-101 (electronic). · Zbl 0927.35130
[11] E.T. Chung, T.F. Chan and X.-C. Tai, Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys.205 (2005) 357-372. · Zbl 1072.65143
[12] M.C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability. SIAM J. Control Optim.26 (1988) 834-862. · Zbl 0654.49010
[13] M.C. Delfour and J.-P. Zolésio, Shapes and geometries. Metrics, analysis, differential calculus, and optimization. Vol. 22 of Advances in Design and Control, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011). · Zbl 1251.49001
[14] J.D. Eshelby, The elastic energy-momentum tensor. Special issue dedicated to A.E. Green. J. Elasticity5 (1975) 321-335. · Zbl 0323.73011
[15] P. Fulmanski, A. Laurain and J.-F. Scheid, Level set method for shape optimization of Signorini problem. In MMAR Proceedings (2004) 71-75.
[16] P. Fulmański, A. Laurain, J.-F. Scheid and J. Sokołowski, A level set method in shape and topology optimization for variational inequalities. Int. J. Appl. Math. Comput. Sci.17 (2007) 413-430. · Zbl 1237.49058
[17] P. Fulmański, A. Laurain, J.-F. Scheid and J. Sokołowski, Level set method with topological derivatives in shape optimization. Int. J. Comput. Math.85 (2008) 1491-1514. · Zbl 1155.65055
[18] P. Fulmanski, A. Laurain, J.-F. Scheid and J. Sokolowski, Une méthode levelset en optimisation de formes. In CANUM 2006 - Congrès National d’Analyse Numérique. Vol. 22 of ESAIM Proc. Survey EDP Sciences, Les Ulis (2008) 162-168. · Zbl 1155.74034
[19] J. Hadamard, Mémoire sur le probleme d’analyse relatif a l’équilibre des plaques élastiques. In Mémoire des savants étrangers, 33, 1907, OEuvres de Jacques Hadamard. Editions du C.N.R.S., Paris (1968) 515-641.
[20] A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique. [A geometric analysis]. Vol. 48 of Math. Appl. Springer, Berlin (2005). · Zbl 1098.49001
[21] F. Hettlich, The domain derivative of time-harmonic electromagnetic waves at interfaces. Math. Methods Appl. Sci.35 (2012) 1681-1689. · Zbl 1247.35167
[22] M. Hintermüller and A. Laurain, Electrical impedance tomography: from topology to shape. Control Cybernet.37 (2008) 913-933. · Zbl 1194.49062
[23] M. Hintermüller and A. Laurain, Multiphase image segmentation and modulation recovery based on shape and topological sensitivity. J. Math. Imaging Vision35 (2009) 1-22. · Zbl 05788090
[24] M. Hintermüller and A. Laurain, Optimal shape design subject to elliptic variational inequalities. SIAM J. Control Optim.49 (2011) 1015-1047. · Zbl 1228.49045
[25] M. Hintermüller, A. Laurain and A.A. Novotny, Second-order topological expansion for electrical impedance tomography. Adv. Comput. Math. (2011) 1-31. · Zbl 1243.49049
[26] M. Hintermüller, A. Laurain and I. Yousept, Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model. Inverse Problems31 (2015) 065006. · Zbl 1335.35297
[27] R. Hiptmair, A. Paganini and S. Sargheini, Comparison of approximate shape gradients. BIT55 (2015) 459-485. · Zbl 1320.65099
[28] D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening. SIAM J. Control Optim.42 (2003) 1087-1117 (electronic). · Zbl 1042.49037
[29] R. Kress, Inverse problems and conformal mapping. Complex Var. Elliptic Equ.57 (2012) 301-316. · Zbl 1246.30016
[30] A. Logg, K.-A. Mardal and G.N. Wells, editors, Automated Solution of Differential Equations by the Finite Element Method. Vol. 84 of Lecture Notes in Computational Science and Engineering. Springer (2012). · Zbl 1247.65105
[31] J.L. Mueller and S. Siltanen, Linear and nonlinear inverse problems with practical applications. Vol. 10 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2012). · Zbl 1262.65124
[32] M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen. Proc. Phys. Math. Soc. Japan24 (1942) 551-559. · Zbl 0061.17204
[33] A.A. Novotny and J. Sokołowski, Topological derivatives in shape optimization. Interaction of Mechanics and Mathematics. Springer, Heidelberg (2013). · Zbl 1276.35002
[34] 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 0659.65132
[35] S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations. SIAM J. Numer. Anal.28 (1991) 907-922. · Zbl 0736.65066
[36] S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces. Vol. 153 of Applied Mathematical Sciences. Springer-Verlag, New York (2003). · Zbl 1026.76001
[37] O. Pantz, Sensibilité de l’équation de la chaleur aux sauts de conductivité. C. R. Math. Acad. Sci. Paris341 (2005) 333-337. · Zbl 1115.35053
[38] D. Peng, B. Merriman, S. Osher, H. Zhao and M. Kang, A PDE-based fast local level set method. J. Comput. Phys.155 (1999) 410-438. · Zbl 0964.76069
[39] M. Renardy and R.C. Rogers. An introduction to partial differential equations. Vol. 13 of Texts in Applied Mathematics, 2nd edn. Springer-Verlag, New York (2004). · Zbl 1072.35001
[40] J.A. Sethian, Level set methods and fast marching methods. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Vol. 3 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, second edition (1999). · Zbl 0973.76003
[41] J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization. Shape sensitivity analysis. Vol. 16 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1992).
[42] K. Sturm, Minimax Lagrangian approach to the differentiability of non-linear PDE constrained shape functions without saddle point assumption. SIAM J. Control Optim.53 (2015) 2017-2039. · Zbl 1327.49073
[43] K. Sturm, D. Hömberg and M. Hintermüller, Shape optimization for a sharp interface model of distortion compensation. WIAS-preprint4 (2013) 807-822.
[44] J.-P. Zolésio, Identification de domaines par déformations. Thèse de doctorat d’état, Université de Nice, France (1979).
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