# Penalization and Regularization Techniques in Shape Optimization in Arbitrary Dimension

Peter Philip^{*} and Dan Tiba

Two classes of shape optimization problems are studied, governed by elliptic partial differential state equations. In one problem type, the PDE is to be solved in an unknown region, which is to be optimized; in the second problem type, an optimal layout problem, the unknown regions determine the coefficients of the PDE. The problems are attacked using regularization together with a penalty method to extend the PDE to a larger fixed domain, providing results for the existence of optimal shapes in arbitrary dimension, as well as convergence optimal shapes if the regularization parameter tends to zero. Error estimates are shown for the layout problem. A numerical algorithm is formulated, and a series of corresponding numerical experiments illustrate that the employed fixed domain method allows and handles topological changes in a natural way. The numerical results indicate a possible connection with an oil industry application.

Mathematics Subject Classification: 35J25 49J20 49M25 49Q10 65M60

Keywords: shape optimization; optimal control; fixed domain method; elliptic partial differential equation; optimal layout problem; error estimate; numerical simulation

Minisymposion: Shape Optimization and Free Boundaries