Computation of Miura surfaces with gradient Dirichlet boundary conditions
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916657017913344 |
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| author | Marazzato, Frederic |
| author_facet | Marazzato, Frederic |
| contents | Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry, gave suboptimal conditions for existence of solutions and proposed an $H^2$-conformal finite element method to approximate them. In this paper, the existence of Miura surfaces is studied using a gradient formulation. It is also proved that, under some hypotheses, the constraints propagate from the boundary to the interior of the domain. Then, a numerical method based on a stabilized least-square formulation, conforming finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge and numerical tests are performed to demonstrate its robustness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_05567 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Computation of Miura surfaces with gradient Dirichlet boundary conditions Marazzato, Frederic Numerical Analysis Analysis of PDEs 35J66, 65N12, 65N15, 65N30 and 74A05 Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry, gave suboptimal conditions for existence of solutions and proposed an $H^2$-conformal finite element method to approximate them. In this paper, the existence of Miura surfaces is studied using a gradient formulation. It is also proved that, under some hypotheses, the constraints propagate from the boundary to the interior of the domain. Then, a numerical method based on a stabilized least-square formulation, conforming finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge and numerical tests are performed to demonstrate its robustness. |
| title | Computation of Miura surfaces with gradient Dirichlet boundary conditions |
| topic | Numerical Analysis Analysis of PDEs 35J66, 65N12, 65N15, 65N30 and 74A05 |
| url | https://arxiv.org/abs/2209.05567 |