Singular perturbation analysis of a regularized MEMS model
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arXiv
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| Natura: | Preprint |
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2018
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| _version_ | 1866915924401979392 |
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| author | Iuorio, Annalisa Popovic, Nikola Szmolyan, Peter |
| author_facet | Iuorio, Annalisa Popovic, Nikola Szmolyan, Peter |
| contents | Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter $λ$. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called "touchdown" phenomenon, where the membrane establishes contact with the ground plate. Touchdown is known to imply the non-existence of steady state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter $\varepsilon$. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters $\varepsilon$ and $λ$ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_02069 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Singular perturbation analysis of a regularized MEMS model Iuorio, Annalisa Popovic, Nikola Szmolyan, Peter Dynamical Systems 34B16, 34C23, 34E05, 34E15, 34L30, 35K67, 74G10 Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter $λ$. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called "touchdown" phenomenon, where the membrane establishes contact with the ground plate. Touchdown is known to imply the non-existence of steady state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter $\varepsilon$. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters $\varepsilon$ and $λ$ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero. |
| title | Singular perturbation analysis of a regularized MEMS model |
| topic | Dynamical Systems 34B16, 34C23, 34E05, 34E15, 34L30, 35K67, 74G10 |
| url | https://arxiv.org/abs/1807.02069 |