A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional

Abstract

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the $$L^2$$ -distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed. The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem. The feasibility of the proposed method is illustrated through various numerical examples.