Yiyun Fan
A Shape-Newton method for free-boundary problems subject to the Bernoulli boundary condition
Fan, Yiyun; Billingham, John; van der Zee, Kristoffer
Authors
John Billingham
Professor KRISTOFFER VAN DER ZEE KG.VANDERZEE@NOTTINGHAM.AC.UK
PROFESSOR OF NUMERICAL ANALYSIS &COMPUTATIONAL APPLIED MATHEMATICS
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 |
Files
Fan Billingham VanderZee SISC2024-AuthorAcceptedManuscript
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Licence
https://creativecommons.org/licenses/by/4.0/
Publisher Licence URL
https://creativecommons.org/licenses/by/4.0/
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