A Penalization and Regularization Technique in Shape Optimization Problems

We consider shape optimization problems, where the state is governed by elliptic partial differential equations (PDE). Using a regularization technique, unknown shapes are encoded via shape functions, turning the shape optimization into optimal control problems for the unknown functions. The method is studied for elliptic PDE to be solved in an unknown region (to be optimized), where the regularization technique together with a penalty method extends the PDE to a larger fixed domain. Additionally, the method is studied for the optimal layout problem, where the unknown regions determine the coefficients of the state equation. In both cases and in arbitrary dimension, the existence of optimal shapes is established for the regularized and the original problem, with convergence of optimal shapes if the regularization parameter tends to zero. Error estimates are proved for the layout problem. In the context of finite element approximations, convergence and differentiability properties are shown. A series of numerical experiments demonstrate the method computationally for an industrially relevant elliptic PDE with two unknown shapes, one giving the region where the PDE is solved, and the other determining the PDE’s coefficients. ∗Department of Mathematics, Ludwig-Maximilians University (LMU) Munich, Theresienstrasse 39, 80333 Munich, Germany, [email protected] †Institute of Mathematics (Romanian Academy), P.O.Box 1-764, RO-014700 Bucharest and Academy of Romanian Scientists, Splaiul Independentei 54, 050094 Bucharest, [email protected] . ‡Basque Center for Applied Mathematics, Alameda Mazarredo 14, E-48009 Bilbao, Basque Country, Spain