A Shape-Newton method for free-boundary problems subject to the Bernoulli boundary condition

Fan, Yiyun; Billingham, John; van der Zee, Kristoffer

Abstract

We develop a shape-Newton method for solving generic free-boundary problems where one of the free-boundary conditions is governed by the nonlinear Bernoulli equation. The method is a Newton-like scheme that employs shape derivatives of the governing equations. In particular, we derive the shape derivative of the Bernoulli equation, which turns out to depend on the curvature in a nontrivial manner. The resulting shape-Newton method allows one to update the position of the free boundary by solving a special linear boundary-value problem at each iteration. We prove solvability of the linearised problem under certain conditions of the data. We verify the effectiveness of the shape-Newton approach applied to free-surface flow over a submerged triangular obstacle using a finite element method on a deforming mesh. We observe superlinear convergence behaviour for our shape-Newton method as opposed to the unfavourable linear rate of traditional methods.

Citation

Fan, Y., Billingham, J., & van der Zee, K. (2024). A Shape-Newton method for free-boundary problems subject to the Bernoulli boundary condition. SIAM Journal on Scientific Computing, 46(6), A3599-A3627. https://doi.org/10.1137/23M1590263

• Journal Article Type: Article
• Acceptance Date: Aug 30, 2024
• Publication Date: 2024
• Deposit Date: Sep 16, 2024
• Publicly Available Date: Dec 31, 2024
• Journal: SIAM Journal on Scientific Computing
• Print ISSN: 1064-8275
• Electronic ISSN: 1095-7197
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 46
• Issue: 6
• Pages: A3599-A3627
• DOI: https://doi.org/10.1137/23M1590263
• Public URL: https://nottingham-repository.worktribe.com/output/39715821
• Publisher URL: https://epubs.siam.org/doi/abs/10.1137/23M1590263?journalCode=sjoce3