Shape Differentiability Under Non-linear PDE Constraints

We review available methods to compute shape sensitivities and apply these methods to a semi-linear model problem. This will reveal the difficulties of each method and will help to decide which approach should be used for a concrete applications.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  W. Ziemer.  Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , 1989 .

[3]  R. Correa,et al.  Directional derivative of a minimax function , 1985 .

[4]  M. C. Delfour,et al.  Shapes and Geometries - Metrics, Analysis, Differential Calculus, and Optimization, Second Edition , 2011, Advances in design and control.

[5]  Michel C. Delfour,et al.  Shape sensitivity analysis via min max differentiability , 1988 .

[6]  J. Zolésio,et al.  Introduction to shape optimization : shape sensitivity analysis , 1992 .

[7]  E. Ng,et al.  Adjoint methods for electromagnetic shape optimization of the low-loss cavity for the International Linear Collider , 2005 .

[8]  L. Younes.  Shapes and Diffeomorphisms , 2010 .

[9]  F. Hettlich.  The domain derivative of time‐harmonic electromagnetic waves at interfaces , 2012 .

[10]  R. Temam,et al.  Analyse convexe et problèmes variationnels , 1974 .

[11]  W. Ziemer.  Weakly differentiable functions , 1989 .

[12]  Helmut Harbrecht,et al.  Analytical and numerical methods in shape optimization , 2008 .

[13]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[14]  F. Tröltzsch.  Optimale Steuerung partieller Differentialgleichungen , 2005 .

[15]  E. Casas.  Boundary control of semilinear elliptic equations with pointwise state constraints , 1993 .

[16]  Kazufumi Ito,et al.  Variational approach to shape derivatives , 2008 .

[17]  John Cagnol,et al.  Shape optimization for the Maxwell equations under weaker regularity of the data , 2010 .

[18]  M. Delfour,et al.  Shape sensitivity analysis via a penalization method , 1988 .

[19]  Olivier Pantz,et al.  Sensibilité de l'équation de la chaleur aux sauts de conductivité , 2005 .

[20]  Michel C. Delfour,et al.  Introduction to Optimization and Semidifferential Calculus , 2012, MOS-SIAM Series on Optimization.

[21]  J. Cea.  Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût , 1986 .