Solving Numerically Stochastic Bernoulli Free Boundary Problems Based On Shape Optimization Methods

This paper is concerned with the numerical method of solving a Bernoulli free boundary value problem with stochastic internal boundary.The random variables on the boundary are parameterized by truncated Karhunen-Lo(?)ve expansion.The shape gradient,which is a high-dimensional integral,is approximated by random collocation method.Based on the shape gradient,the optimal free boundary is represented by the level set method.The state equation and the adjoint equation is discretized by the finite element method.Finally,we carry out relevant numerical experiments and compare with the deterministic Bernoulli free boundary problem,the results show that random-ness has influences on the free boundary problem.