Improved trial methods for a class of generalized Bernoulli problems

Abstract

The aim of this article is to develop improved trial methods for the solution of a generalized exterior Bernoulli free boundary problem. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. Appropriate update rules are obtained by expanding the state's Dirichlet data at the actual boundary via a Taylor expansion of first and second order. The resulting trial methods converge linearly for both cases, although the trial method based on the second order Taylor expansion is much more robust. Nevertheless, via results of shape sensitivity analysis, we are able to modify the update rules such that their convergence is improved. The feasibility of the proposed trial methods and their performance is demonstrated by numerical results.

A second order convergent trial method for a free boundary problem in three dimensions

“…Otherwise, it coincides with an inexact Newton method. The method has been introduced in [16] in case of starlike domains in two spatial dimensions and boundary updates in the radial direction. Here, we provided its realization for arbitrary domains in three spatial dimensions and boundary updates in the normal direction.…”

“…Instead, a second order Taylor expansion can also be used which would lead to a more stable trial method that, however, is also only first order convergent, cf. [16] for the details.…”

“…The basic idea of the trial method which we propose in this article has been introduced and analyzed in [16]. There, a trial method has been proposed whose convergence towards the optimal solution is locally quadratic.…”

Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

“…In general, this choice depends on the physical properties of the free boundary problem under consideration. Among the existing methods for solving free boundary problems, such as the level set method [1,2] or the shape optimization method [3][4][5][6], we investigate here the trial method [6][7][8][9][10].…”

A scalar prototype problem of deep abrasive drilling with an infinite free boundary: an asymptotic modeling study

“…on a free boundary Γ . A number of approaches for numerical solution of stationary free boundary value problems have been developed, in particular, using trial methods [12,13], shape optimization [14], cost functional minimization [15], Newton's method [16], and level set techniques [17].…”