Numerical solution to the exterior Bernoulli problem using the Dirichlet-Robin energy gap cost functional approach in two and three dimensions

• Published in: Numerical algorithms Vol. 94; no. 1; pp. 175 - 227
• Main Author: Rabago, Julius Fergy T.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2023; Springer Nature B.V
• Subjects: Algebra; Algorithms; Boundary value problems; Computer Science; Dirichlet problem; Energy gap; Finite element method; Free boundaries; Iterative methods; Mathematical analysis; Minimax technique; Numeric Computing; Numerical Analysis; Optimization; Original Paper; Shape optimization; Theory of Computation; Free boundary; Shape derivative; 49K20; Bernoulli problem; Shape optimization; Minimax formulation; 65K10; Domain perturbation; 49Q10; Lagrangian method
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-023-01497-x

The exterior Bernoulli problem — a prototype stationary free boundary problem — is rephrased into a shape optimization setting using an energy-gap type cost functional that is subject to two auxiliary problems: a pure Dirichlet problem and a mixed Dirichlet-Robin boundary value problem. It is demonstrated here that depending on what method is used, the shape gradient of the cost functional may appear in a different form. The dissimilarity in structure comes from the way the adjoint variable was utilized in the computation — then resulting to a different adjoint problem. The shape derivative is first obtained via Delfour-Zolésio’s minimax formulation, and then by using the weak form of the Eulerian derivative of the states coupled with the adjoint method. The latter approach is accomplished by first showing the existence of the derivatives. A fast iterative scheme based on finite element method is then formulated to numerically solve the proposed shape optimization formulation. The feasibility of the method — highlighting its efficiency and practicality — is illustrated through numerical examples in two and three dimensions.