This paper presents the first application of the direct parametrisation method for invariant manifolds to a fully coupled multiphysics problem involving the nonlinear vibrations of deformable structures subjected to an electrostatic field. The formulation proposed is intended for model order reduction of electrostatically actuated resonating Micro-Electro-Mechanical Systems (MEMS). The continuous problem is first rewritten in a manner that can be directly handled by the parametrisation method, which relies upon automated asymptotic expansions. A new mixed fully Lagrangian formulation is thus proposed which contains only explicit polynomial nonlinearities, which is then discretised in the framework of finite element procedures. Validation is performed on the classical parallel plate configuration, where different formulations using either the general framework, or an approximation of the electrostatic field due to the geometric configuration selected, are compared. Reduced-order models along these formulations are also compared to full-order simulations operated with a time integration approach. Numerical results show a remarkable performance both in terms of accuracy and wealth of nonlinear effects that can be accounted for. In particular, the transition from hardening to softening behaviour of the primary resonance while increasing the constant voltage component of the electric actuation, is recovered. Secondary resonances leading to superharmonic and parametric resonances are also investigated with the reduced-order model.

翻译：本文首次将不变流形直接参数化方法应用于完全耦合的多物理场问题，该问题涉及受静电场作用可变形结构的非线性振动。所提出的方法旨在对静电驱动谐振微机电系统(MEMS)进行模型降阶。首先将连续问题重写为适合参数化方法直接处理的形式，该方法依赖于自动渐近展开。为此提出了一种新的混合全拉格朗日格式，其中仅包含显式多项式非线性项，随后在有限元框架内进行离散化。通过经典平行板构型进行验证，比较了采用通用框架或基于所选几何构型近似静电场时不同格式的表现。还将基于这些格式的降阶模型与采用时间积分方法进行全阶仿真的结果进行了对比。数值结果表明，该方法在精度和可描述的非线性效应丰富性方面均表现出色。特别地，成功复现了随着电致动恒定电压分量增大，主共振从硬化行为向软化行为的转变。此外，还利用降阶模型研究了导致超谐共振和参数共振的次级共振现象。