Precomputing approach for a two-scale phase transition model
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| author | Eden, Michael Freudenberg, Tom Muntean, Adrian |
| author_facet | Eden, Michael Freudenberg, Tom Muntean, Adrian |
| contents | In this study, we employ analytical and numerical techniques to examine a phase transition model with moving boundaries. The model displays two relevant spatial scales pointing out to a macroscopic phase and a microscopic phase, interacting on disjoint inclusions. The shrinkage or the growth of the inclusions is governed by a modified Gibbs-Thomson law depending on the macroscopic temperature, but without accessing curvature information. We use the Hanzawa transformation to transform the problem onto a fixed reference domain. Then a fixed-point argument is employed to demonstrate the well-posedness of the system for a finite time interval. Due to the model's nonlinearities and the macroscopic parameters, which are given by differential equations that depend on the size of the inclusions, the problem is computationally expensive to solve numerically. We introduce a precomputing approach that solves multiple cell problems in an offline phase and uses an interpolation scheme afterward to determine the needed parameters. Additionally, we propose a semi-implicit time-stepping method to resolve the nonlinearity of the problem. We investigate the errors of both the precomputing and time-stepping procedures and verify the theoretical results via numerical simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Precomputing approach for a two-scale phase transition model Eden, Michael Freudenberg, Tom Muntean, Adrian Numerical Analysis Analysis of PDEs 35R37, 65M60, 80A22, 35K55 In this study, we employ analytical and numerical techniques to examine a phase transition model with moving boundaries. The model displays two relevant spatial scales pointing out to a macroscopic phase and a microscopic phase, interacting on disjoint inclusions. The shrinkage or the growth of the inclusions is governed by a modified Gibbs-Thomson law depending on the macroscopic temperature, but without accessing curvature information. We use the Hanzawa transformation to transform the problem onto a fixed reference domain. Then a fixed-point argument is employed to demonstrate the well-posedness of the system for a finite time interval. Due to the model's nonlinearities and the macroscopic parameters, which are given by differential equations that depend on the size of the inclusions, the problem is computationally expensive to solve numerically. We introduce a precomputing approach that solves multiple cell problems in an offline phase and uses an interpolation scheme afterward to determine the needed parameters. Additionally, we propose a semi-implicit time-stepping method to resolve the nonlinearity of the problem. We investigate the errors of both the precomputing and time-stepping procedures and verify the theoretical results via numerical simulations. |
| title | Precomputing approach for a two-scale phase transition model |
| topic | Numerical Analysis Analysis of PDEs 35R37, 65M60, 80A22, 35K55 |
| url | https://arxiv.org/abs/2407.21595 |