Using Linearization and Penalty Approach to Solve Optimal Shape Design Problem with an Obstacle

To obtain the best domain of an elliptic boundary control problems, with and without abstacle, two approches are presented. The based measure method, apply a linearization technique and find the optimal domain and trajectory via a solution of finite linear method by using an optimization search technique. In the second one, by introducing the penalty function and then emplaying the finite element method the optimal domain for the same problem determind. The comparison between two methods is done via presenting some numerical simulations.