[Paper Review] Reduced order modelling of fully coupled electro-mechanical systems through invariant manifolds with applications to microstructures

Attilio Frangi, Alessio Colombo+2 more|arXiv (Cornell University)|Dec 22, 2023

Advanced MEMS and NEMS Technologies4 citations

TL;DR

This paper introduces a novel reduced order modeling approach for fully coupled electro-mechanical MEMS using the direct parametrization method for invariant manifolds (DPIM). By developing a mixed fully Lagrangian formulation with explicit polynomial nonlinearities, the method enables accurate, high-order model order reduction of nonlinear vibrations in electrostatically actuated microstructures, successfully capturing complex dynamics such as hardening-to-softening transitions and secondary resonances with high fidelity.

ABSTRACT

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.

Motivation & Objective

• To address the challenge of model order reduction in fully coupled electro-mechanical systems with intrinsic nonlinearities.
• To overcome the nonpolynomial nature of electrostatic forces in traditional formulations for use in asymptotic expansion-based reduction methods.
• To develop a general, geometry-agnostic formulation compatible with finite element discretization and invariant manifold techniques.
• To validate the reduced-order model against full-order simulations and benchmark solutions for nonlinear resonant behavior in MEMS.
• To enable accurate prediction of complex nonlinear phenomena such as hardening-to-softening transitions and secondary resonances in resonating microstructures.

Proposed method

• A new mixed fully Lagrangian formulation is proposed that rewrites the electro-mechanical problem in the original configuration, introducing an auxiliary field to represent the electric field.
• The formulation ensures all nonlinearities are explicit and polynomial, enabling direct application of the DPIM for arbitrary-order asymptotic expansions.
• The governing equations are semi-discretized via the finite element method, resulting in a system of differential-algebraic equations suitable for non-autonomous DPIM.
• The method uses automated asymptotic expansions to derive reduced-order models (ROMs) that capture nonlinear normal modes as invariant manifolds in phase space.
• Continuation techniques are applied to the ROMs to trace frequency-response curves and analyze bifurcations.
• The approach is validated by comparing ROM predictions with full-order simulations using time integration and analytical benchmarks for parallel plate configurations.

Experimental results

Research questions

• RQ1 Can the direct parametrization method for invariant manifolds be successfully extended to fully coupled electro-mechanical systems with nonpolynomial nonlinearities?
• RQ2 How can electrostatic forces, which are inherently nonpolynomial, be reformulated to enable high-order asymptotic expansions in model order reduction?
• RQ3 To what extent can the proposed ROM capture complex nonlinear dynamics such as hardening-to-softening transitions and secondary resonances in MEMS?
• RQ4 How does the order of non-autonomous terms in the expansion affect the accuracy and convergence of the ROM?
• RQ5 What is the computational efficiency of the proposed ROM compared to full-order finite element simulations?

Key findings

• The proposed mixed fully Lagrangian formulation successfully transforms nonpolynomial electrostatic nonlinearities into explicit polynomial terms, enabling application of the DPIM.
• The ROM accurately captures the transition from hardening to softening behavior in primary resonance as the DC voltage component increases, matching full-order simulation results.
• Secondary resonances, including superharmonic and parametric resonances, are successfully reproduced by the reduced-order model.
• The convergence of the ROM is achieved at order O(9,3), with higher-order terms beyond O(9,5) showing minimal effect on the solution.
• Computational cost is drastically reduced: ROM construction takes less than 1 minute for O(7,1) and up to 25 minutes for O(9,5), while each continuation curve takes 10 seconds to 1 minute to compute.
• Stiffness-proportional damping is essential for convergence of bifurcated branches, while mass-proportional damping leads to divergent solutions, highlighting the ROM's accurate damping representation.