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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.09303 |
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| _version_ | 1866913930569318400 |
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| author | Li, Qinfeng Yang, Hang |
| author_facet | Li, Qinfeng Yang, Hang |
| contents | This paper investigates shape optimization problems in the context of heat transfer, with a focus on the stability and non-optimality of round domains under Robin boundary conditions. Using the flow approach and Steklov eigenvalue estimates, we derive the necessary and sufficient stability conditions for a ball to maximize the averaged heat when the heat source is radially decreasing. Our results show that, counterintuitively, a ball may not be optimal for maximizing the averaged heat under heat convection, even with radially decreasing heat sources located on the center of the ball. Moreover, we identify stability-breaking phenomena by giving precise values of thresholds, which depend on the Robin coefficient, dimension, and volume constraints. Additionally, we demonstrate that a ball can maximize the averaged temperature under certain conditions and we also explore optimal shapes in thin insulation problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09303 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Heat Transfer Shape Optimization: Stability and Non-Optimality of the Ball Li, Qinfeng Yang, Hang Analysis of PDEs This paper investigates shape optimization problems in the context of heat transfer, with a focus on the stability and non-optimality of round domains under Robin boundary conditions. Using the flow approach and Steklov eigenvalue estimates, we derive the necessary and sufficient stability conditions for a ball to maximize the averaged heat when the heat source is radially decreasing. Our results show that, counterintuitively, a ball may not be optimal for maximizing the averaged heat under heat convection, even with radially decreasing heat sources located on the center of the ball. Moreover, we identify stability-breaking phenomena by giving precise values of thresholds, which depend on the Robin coefficient, dimension, and volume constraints. Additionally, we demonstrate that a ball can maximize the averaged temperature under certain conditions and we also explore optimal shapes in thin insulation problems. |
| title | Heat Transfer Shape Optimization: Stability and Non-Optimality of the Ball |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2307.09303 |