A level set method for Laplacian eigenvalue optimization subject to geometric constraints

We consider to solve numerically the shape optimization problems of Dirichlet Laplace eigenvalues subject to volume and perimeter constraints. By combining a level set method with the relaxation approach, the algorithm can perform shape and topological changes on a fixed grid. We use the volume expr...

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Bibliographic Details
Published inComputational optimization and applications Vol. 82; no. 2; pp. 499 - 524
Main Authors Qian, Meizhi, Zhu, Shengfeng
Format Journal Article
LanguageEnglish
Published New York Springer US 01.06.2022
Springer Nature B.V
Subjects
Algorithms
Convex and Discrete Geometry
Dirichlet problem
Eigenvalues
Finite element method
Geometric constraints
Management Science
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Shape optimization
Statistics
Eigenvalue optimization
Eulerian derivative
Relaxed approach
Level set method
Finite element
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Summary:We consider to solve numerically the shape optimization problems of Dirichlet Laplace eigenvalues subject to volume and perimeter constraints. By combining a level set method with the relaxation approach, the algorithm can perform shape and topological changes on a fixed grid. We use the volume expressions of Eulerian derivatives in shape gradient descent algorithms. Finite element methods are used for discretizations. Two and three-dimensional numerical examples are presented to illustrate the effectiveness of the algorithms.
ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-022-00371-1