Efficient Shape and Topology Optimization for Random Exterior Bernoulli Free Boundary Problems Based on the Multimodes Monte Carlo Method

Shape and topology optimization with uncertainties typically requires to solve repeatedly the forward problems corresponding to many random samples during each optimization step, which occupies a large amount of computational cost. In this paper, we utilize the multimodes Monte Carlo method to develop efficient shape and topology optimization algorithms to solve exterior Bernoulli free boundary problems with random diffusion coefficients. The shape optimization with the shape functional of Kohn-Vogelius type and Neumann data-tracking type is considered to address the overdetermined problem. We construct the shape optimization algorithm to solve the Kohn-Vogelius problem by the shape gradient flow method. To tackle the Neumann data-tracking problem, the phase-field method of Allen-Cahn type is proposed for topology optimization to track the evolution of the interface. The optimization process can be accelerated to some extent by using the multimodes Monte Carlo method to solve the governing stochastic equation. Numerical examples in 2D and 3D are presented to show the effectiveness and efficiency of the proposed algorithms.