Garcke, Harald and Huettl, Paul and Kahle, Christian and Knopf, Patrik (2024) Sharp-Interface Limit of a Multi-phase Spectral Shape Optimization Problem for Elastic Structures. APPLIED MATHEMATICS AND OPTIMIZATION, 89 (1): 24. ISSN 0095-4616, 1432-0606
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We consider an optimization problem for the eigenvalues of a multi-material elastic structure that was previously introduced by Garcke et al. (Adv. Nonlinear Anal. 11:159-197, 2022). There, the elastic structure is represented by a vector-valued phase-field variable, and a corresponding optimality system consisting of a state equation and a gradient inequality was derived. In the present paper, we pass to the sharp-interface limit in this optimality system by the technique of formally matched asymptotics. Therefore, we derive suitable Lagrange multipliers to formulate the gradient inequality as a pointwise equality. Afterwards, we introduce inner and outer expansions, relate them by suitable matching conditions and formally pass to the sharp-interface limit by comparing the leading order terms in the state equation and in the gradient equality. Furthermore, the relation between these formally derived first-order conditions and results of Allaire and Jouve (Comput. Methods Appl. Mech. Eng. 194:3269-3290, 2005) obtained in the framework of classical shape calculus is discussed. Eventually, we provide numerical simulations for a variety of examples. In particular, we illustrate the sharp-interface limit and also consider a joint optimization problem of simultaneous compliance and eigenvalue optimization.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | HILLIARD-DARCY MODEL; NAVIER-STOKES FLOW; LEVEL SET METHODS; PHASE-FIELD; TOPOLOGY OPTIMIZATION; TUMOR-GROWTH; INTEGRABILITY; EIGENVALUES; GRADIENT; STRESSES; Shape and topology optimization; Structural optimization; Eigenvalue problem; Sharp-interface limit; Formally matched asymptotics; Phase-field models; Linear elasticity |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics Mathematics > Prof. Dr. Harald Garcke |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 25 Jul 2025 08:30 |
| Last Modified: | 25 Jul 2025 08:30 |
| URI: | https://pred.uni-regensburg.de/id/eprint/63762 |
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