Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915941031346176 |
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| author | Jin, Bangti Li, Jing Xu, Yifeng Zhu, Shengfeng |
| author_facet | Jin, Bangti Li, Jing Xu, Yifeng Zhu, Shengfeng |
| contents | In this work, we develop an adaptive nonconforming finite element algorithm for the numerical approximation of phase-field parameterized topology optimization governed by the Stokes system. We employ the conforming linear finite element space to approximate the phase field, and the nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants to approximate the velocity field and the pressure field, respectively. We establish the convergence of the adaptive method, i.e., the sequence of minimizers contains a subsequence that converges to a solution of the first-order optimality system, and the associated subsequence of discrete pressure fields also converges. The analysis relies crucially on a new discrete compactness result of nonconforming linear finite elements over a sequence of adaptively generated meshes. We present numerical results for several examples to illustrate the performance of the algorithm, including a comparison with the uniform refinement strategy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13776 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow Jin, Bangti Li, Jing Xu, Yifeng Zhu, Shengfeng Numerical Analysis Optimization and Control In this work, we develop an adaptive nonconforming finite element algorithm for the numerical approximation of phase-field parameterized topology optimization governed by the Stokes system. We employ the conforming linear finite element space to approximate the phase field, and the nonconforming linear finite elements (Crouzeix-Raviart elements) and piecewise constants to approximate the velocity field and the pressure field, respectively. We establish the convergence of the adaptive method, i.e., the sequence of minimizers contains a subsequence that converges to a solution of the first-order optimality system, and the associated subsequence of discrete pressure fields also converges. The analysis relies crucially on a new discrete compactness result of nonconforming linear finite elements over a sequence of adaptively generated meshes. We present numerical results for several examples to illustrate the performance of the algorithm, including a comparison with the uniform refinement strategy. |
| title | Convergence Analysis of an Adaptive Nonconforming FEM for Phase-Field Dependent Topology Optimization in Stokes Flow |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2505.13776 |