Distributed shape derivative via averaged adjoint method and applications
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1241-1267.

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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2015075
Classification : 49Q10, 35Q93, 35R30, 35R05
Mots clés : Shape optimization, distributed shape derivative, electrical impedance tomography, Lagrangian method, level set method
Laurain, Antoine 1 ; Sturm, Kevin 2

1 Technische Universität Berlin, Institut für Mathematik, Str. des 17. Juni 136, 10623 Berlin, Germany.
2 Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Straße 9, 45127 Essen, Germany.
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Laurain, Antoine; Sturm, Kevin. Distributed shape derivative via averaged adjoint method and applications. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1241-1267. doi : 10.1051/m2an/2015075. http://www.numdam.org/articles/10.1051/m2an/2015075/

D. Adalsteinsson and J.A. Sethian, A fast level set method for propagating interfaces. J. Comput. Phys. 118 (1995) 269–277. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

Z. Belhachmi and H. Meftahi, Shape sensitivity analysis for an interface problem via minimax differentiability. Appl. Math. Comput. 219 (2013) 6828–6842. | MR | Zbl

L. Borcea, Electrical impedance tomography. Inverse Problems 18 (2002) R99–R136. | DOI | MR | Zbl

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. | MR

A. Canelas, A. Laurain and A.A. Novotny, A new reconstruction method for the inverse potential problem. J. Comput. Phys. 268 (2014) 417–431. | DOI | MR | Zbl

A. Canelas, A. Laurain and A.A. Novotny, A new reconstruction method for the inverse source problem from partial boundary measurements. Inverse Problems 31 (2015) 075009. | DOI | MR | Zbl

J. Céa, Conception optimale ou identification de formes: calcul rapide de la dérivée directionnelle de la fonction coût. RAIRO M2AN 20 (1986) 371–402. | DOI | Numdam | MR | Zbl

M. Cheney, D. Isaacson and J.C. Newell, Electrical impedance tomography. SIAM Rev. 41 (1999) 85–101 (electronic). | DOI | MR | Zbl

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. | DOI | MR | Zbl

M.C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability. SIAM J. Control Optim. 26 (1988) 834–862. | DOI | MR | Zbl

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). | MR | Zbl

J.D. Eshelby, The elastic energy-momentum tensor. Special issue dedicated to A.E. Green. J. Elasticity 5 (1975) 321–335. | DOI | MR | Zbl

P. Fulmanski, A. Laurain and J.-F. Scheid, Level set method for shape optimization of Signorini problem. In MMAR Proceedings (2004) 71–75.

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | MR | Zbl

J. Hadamard, Mémoire sur le probleme d’analyse relatif a l’équilibre des plaques élastiques. In Mémoire des savants étrangers, 33, 1907, Œuvres de Jacques Hadamard. Editions du C.N.R.S., Paris (1968) 515–641. | JFM

A. Henrot and M. Pierre, Variation et optimisation de formes. Une analyse géométrique. [A geometric analysis]. Vol. 48 of Math. Appl. Springer, Berlin (2005). | MR | Zbl

F. Hettlich, The domain derivative of time-harmonic electromagnetic waves at interfaces. Math. Methods Appl. Sci. 35 (2012) 1681–1689. | DOI | MR | Zbl

M. Hintermüller and A. Laurain, Electrical impedance tomography: from topology to shape. Control Cybernet. 37 (2008) 913–933. | MR | Zbl

M. Hintermüller and A. Laurain, Multiphase image segmentation and modulation recovery based on shape and topological sensitivity. J. Math. Imaging Vision 35 (2009) 1–22. | DOI | MR | Zbl

M. Hintermüller and A. Laurain, Optimal shape design subject to elliptic variational inequalities. SIAM J. Control Optim. 49 (2011) 1015–1047. | DOI | MR | Zbl

M. Hintermüller, A. Laurain and A.A. Novotny, Second-order topological expansion for electrical impedance tomography. Adv. Comput. Math. (2011) 1–31. | MR | Zbl

M. Hintermüller, A. Laurain and I. Yousept, Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model. Inverse Problems 31 (2015) 065006. | DOI | MR | Zbl

R. Hiptmair, A. Paganini and S. Sargheini, Comparison of approximate shape gradients. BIT 55 (2015) 459–485. | DOI | MR | Zbl

D. Hömberg and J. Sokołowski, Optimal shape design of inductor coils for surface hardening. SIAM J. Control Optim. 42 (2003) 1087–1117 (electronic). | DOI | MR | Zbl

R. Kress, Inverse problems and conformal mapping. Complex Var. Elliptic Equ. 57 (2012) 301–316. | DOI | MR | Zbl

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

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). | MR | Zbl

M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen. Proc. Phys. Math. Soc. Japan 24 (1942) 551–559. | MR | Zbl

A.A. Novotny and J. Sokołowski, Topological derivatives in shape optimization. Interaction of Mechanics and Mathematics. Springer, Heidelberg (2013). | MR | Zbl

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. | DOI | MR | Zbl

S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton–Jacobi equations. SIAM J. Numer. Anal. 28 (1991) 907–922. | DOI | MR | Zbl

S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces. Vol. 153 of Applied Mathematical Sciences. Springer–Verlag, New York (2003). | MR | Zbl

O. Pantz, Sensibilité de l’équation de la chaleur aux sauts de conductivité. C. R. Math. Acad. Sci. Paris 341 (2005) 333–337. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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). | MR | Zbl

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). | MR | Zbl

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). | MR | Zbl

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. | DOI | MR | Zbl

K. Sturm, D. Hömberg and M. Hintermüller, Shape optimization for a sharp interface model of distortion compensation. WIAS-preprint 4 (2013) 807–822.

J.-P. Zolésio, Identification de domaines par déformations. Thèse de doctorat d’état, Université de Nice, France (1979).

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