Adaptive Method for Shape Optimization

1. Abstract The aim of this work is to introduce an adaptive strategy to monitor the rate of convergence of a Newtonlike method in shape optimization. Numerical solution of this problem involves numerical representation of the domain, optimization algorithms and numerical solution of the state equation, a partial differential equation. Newtons like algorithms are often computationally intensive and the rate of convergence depends on how accurate the numerical solution of the state equation is. The adaptive procedure consists in a monitoring step and in a mesh refinement step. A contraction factor has been introduced and we give a criterion to decide when we need to refine the mesh. Thanks to a local a posteriori estimation of the error we known where mesh refinements are needed and then an adaptive algorithm is derived in order to have super linear rate of convergence for the shape optimization Newton-like method. The model problem concerns a cost function depending on the perimeter of unknown domain and the solution of an exterior Dirichlet problem. The equilibrium shape is shown to be the stationary state of the total energy under the constraint that the surface (the volume in 3-d) is prescribed. Numerical results of the adaptive technique applied to the model problem are analyzed. 2.