Talk by Helmut Harbrecht

Shape optimization is indispensable for designing and constructing industrial components. Many problems that arise in application, particularly in structural mechanics and in the optimal control of distributed parameter systems, can be formulated as the minimization of functionals which are defined over a class of admissible domains.

The application of gradient based minimization algorithms involves the shape functionals' derivative with respect to the domain under consideration. Such derivatives can analytically be computed by means of shape calculus and enable the paradigm first optimize then discretize. Especially, by identifying the sought domain with a parametrization of its boundary, the solution of the shape optimization problem is equivalent to solving a nonlinear pseudodifferential equation for the unknown parametrization.

The present talk aims at surveying on analytical and numerical methods for shape optimization. In particular, besides several applications of shape optimization, the following items will be addressed:

• first and second order optimality conditions;
• discretization of shapes; 
• existence and convergence of approximate shapes;
• efficient numerical techniques to solve shape optimization problems.