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Volume 41, Issue 5
On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems

Chunxiao Liu & Shengfeng Zhu

DOI: 10.4208/jcm.2208-m2020-0142

J. Comp. Math., 41 (2023), pp. 956-979.

Published online: 2023-05

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  • Abstract

Shape gradient flows are widely used in numerical shape optimization algorithms. We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems. We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative. Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.

  • Keywords

Shape optimization, Shape gradient, Eulerian derivative, Finite element, Error estimate.

  • AMS Subject Headings

65D15, 65N30, 49Q12

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xxliu198431@126.com (Chunxiao Liu)

sfzhu@math.ecnu.edu.cn (Shengfeng Zhu)

  • BibTex
  • RIS
  • TXT
@Article{JCM-41-956, author = {Liu , Chunxiao and Zhu , Shengfeng}, title = {On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {5}, pages = {956--979}, abstract = {

Shape gradient flows are widely used in numerical shape optimization algorithms. We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems. We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative. Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2208-m2020-0142}, url = {http://global-sci.org/intro/article_detail/jcm/21681.html} }
TY - JOUR T1 - On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems AU - Liu , Chunxiao AU - Zhu , Shengfeng JO - Journal of Computational Mathematics VL - 5 SP - 956 EP - 979 PY - 2023 DA - 2023/05 SN - 41 DO - http://doi.org/10.4208/jcm.2208-m2020-0142 UR - https://global-sci.org/intro/article_detail/jcm/21681.html KW - Shape optimization, Shape gradient, Eulerian derivative, Finite element, Error estimate. AB -

Shape gradient flows are widely used in numerical shape optimization algorithms. We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems. We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative. Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.

Chunxiao Liu & Shengfeng Zhu. (2023). On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems. Journal of Computational Mathematics. 41 (5). 956-979. doi:10.4208/jcm.2208-m2020-0142
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