UC Irvine

Free boundary problems can be reformulated as shape optimization problems with partial differential equation (PDE) constraints, making them solvable through numerical shape optimization algorithms. The Smooth Extension Embedding Method (SEEM), a novel approach for solving PDEs, leverages straightforward discretization of complex domains and achieves a high degree of convergence for problems with smooth solutions. This makes SEEM a viable alternative to the finite element methods used in traditional numerical shape optimization algorithms, effectively overcoming key challenges such as mesh degeneration and low-order convergence. To enhance the stability of SEEM while maintaining its high-order accuracy for boundary value problems on non-smooth numerical representations of inherently smooth boundaries, we employ a regularized level set boundary approximation technique. This enhancement broadens the applicability of SEEM to the class of free boundary problems with a shape optimization approach. Through theoretical and practical examples, experiments show that our improved SEEM-based algorithm maintains high-order accuracy in shape gradient approximation and circumvents computational pitfalls like mesh degeneration that can arise during shape evolution. This makes the direct use of a simple shape gradient formula and an explicit gradient descent flow possible. The resulting algorithm employs straightforward domain grids throughout the shape optimization process, resulting in significant computational savings and ease of implementation.