Applied Numerical Mathematics
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A class of optimal shape design problems is considered where a part of the boundary of the domain represents the free parameter. The variable domain is parametrized by a class of functions in such a way that the optimal design problem results in an optimal control problem on a fixed domain. The functions for the parametrization of the domain are used as controls, and the corresponding states are then given by the solution of an elliptic boundary value problem on a fixed domain.Discretizing this control problem normally leads to a large-scale optimization problem, where the corresponding solution methods are characterized by the requirement of solving many boundary value problems. In spite of this interesting numerical challenge, until now little work has been done to derive more efficient algorithms by taking advantage of the specific structure of this kind of problem.In this report, Newton's method in function space is derived, resulting in an efficient algorithm for the discretized optimization problems. By using the specific structure of these optimal shape design problems, an efficient implementation of the numerical algorithm is introduced. The properties of this algorithm are compared with those of the gradient method using illustrative numerical examples.