A Solution to an Exterior Bernoulli Free Boundary problem using the Constitutive Error Cost Functional.

Abstract

A shape optimisation problem involves finding the minimum of the cost functional over an admissible domain. the cost functional is stated in form of an integral over a domain or its boundary, where the integrand depends smoothly on the solution of s boundary value problem. This work presents a numerical study of solving an exterior Bernoulli free boundary problem by minimising a constitutive error cost functional on a 2-dimensional domain. to obtain the shape gradient of the shape gradient of the cost functional, we introduce a change of geometry of domain by defining a transformation in a specific transformation in a shape direction V. Differentiability with respect to a small perturbation of the domain is obtained using shape Calculus, Domain and boundary transformation and perturbation of identity technique. The steepest descent method and the gradient information were used to formulate a boundary variation algorithm to solve the optimisation problem numerically using MATLAB and FreeFem ++. The analysis of the proposed reformulation is completed by addressing the existence of an optimal solution of the shape optimisation problem on star-shaped domains. The results shows that when we used a regularised direction vector V, we obtain a solution even if the domain considered disagrees with the conditions for existence of an optimal solution to the over-determined problem. Furthermore we found that making the valued of Lambda more negatives leads to decrease in radius of outer boundary when the boundaries are concentric circles.