Finite Element Methods for the Stretching and Bending of Thin Structures with Folding
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917820716023808 |
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| author | Bonito, Andrea Guignard, Diane Morvant, Angelique |
| author_facet | Bonito, Andrea Guignard, Diane Morvant, Angelique |
| contents | In [Bonito et al., J. Comput. Phys. (2022)], a local discontinuous Galerkin method was proposed for approximating the large bending of prestrained plates, and in [Bonito et al., IMA J. Numer. Anal. (2023)] the numerical properties of this method were explored. These works considered deformations driven predominantly by bending. Thus, a bending energy with a metric constraint was considered. We extend these results to the case of an energy with both a bending component and a nonconvex stretching component, and we also consider folding across a crease. The proposed discretization of this energy features a continuous finite element space, as well as a discrete Hessian operator. We establish the $Γ$-convergence of the discrete to the continuous energy and also present an energy-decreasing gradient flow for finding critical points of the discrete energy. Finally, we provide numerical simulations illustrating the convergence of minimizers and the capabilities of the model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04810 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite Element Methods for the Stretching and Bending of Thin Structures with Folding Bonito, Andrea Guignard, Diane Morvant, Angelique Numerical Analysis 65N12, 65N30, 74K20 In [Bonito et al., J. Comput. Phys. (2022)], a local discontinuous Galerkin method was proposed for approximating the large bending of prestrained plates, and in [Bonito et al., IMA J. Numer. Anal. (2023)] the numerical properties of this method were explored. These works considered deformations driven predominantly by bending. Thus, a bending energy with a metric constraint was considered. We extend these results to the case of an energy with both a bending component and a nonconvex stretching component, and we also consider folding across a crease. The proposed discretization of this energy features a continuous finite element space, as well as a discrete Hessian operator. We establish the $Γ$-convergence of the discrete to the continuous energy and also present an energy-decreasing gradient flow for finding critical points of the discrete energy. Finally, we provide numerical simulations illustrating the convergence of minimizers and the capabilities of the model. |
| title | Finite Element Methods for the Stretching and Bending of Thin Structures with Folding |
| topic | Numerical Analysis 65N12, 65N30, 74K20 |
| url | https://arxiv.org/abs/2311.04810 |