Burman, E;
Elfverson, D;
Hansbo, P;
Larson, MG;
Larsson, K;
(2017)
A Cut Finite Element Method for the Bernoulli Free Boundary Value Problem.
Computer Methods in Applied Mechanics and Engineering
, 317
pp. 598-618.
10.1016/j.cma.2016.12.021.
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Abstract
We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms is added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the $H^1$ Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the the velocity field in the $H^1$ norm. Finally, we present illustrating numerical results.
| Type: | Article |
|---|---|
| Title: | A Cut Finite Element Method for the Bernoulli Free Boundary Value Problem |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.cma.2016.12.021 |
| Publisher version: | http://dx.doi.org/10.1016/j.cma.2016.12.021 |
| Language: | English |
| Additional information: | © 2016 Elsevier B.V. All rights reserved. This manuscript version is made available under a Creative Commons Attribution Non-commercial Non-derivative 4.0 International license (CC BY-NC-ND 4.0). This license allows you to share, copy, distribute and transmit the work for personal and non-commercial use providing author and publisher attribution is clearly stated. Further details about CC BY licenses are available at https://creativecommons.org/licenses/. Access may be initially restricted by the publisher. |
| Keywords: | Free boundary value problem; CutFEM; Shape optimization; Level set; Fictitious domain method |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/1518358 |
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