UC Berkeley

The main focus of this work is on the development of a high-order Eulerian-Lagrangian finite element method for the simulation of electro-mechanical systems. The coupled problem is solved by a staggered scheme, where the mechanical motion is discretized by standard Lagrangian finite elements, and the electrical field is solved on a fixed Eulerian grid with embedded boundary conditions. Traditional Lagrangian-Lagrangian or arbitrary Lagrangian-Eulerian (ALE) methods encounter deficiencies, for example, when dealing with mesh distortion due to large deformations, or topology changes due to contacting bodies. The presented Eulerian-Lagrangian approach addresses these issues in a natural way. Within this context we develop a high-order immersed boundary discontinuous-Galerkin (IB-DG) method, which is shown to be necessary for (i) the accurate representation of the electrical gradient along nonlinear boundary features such as singular corners, and (ii) to achieve full convergence during the iterative global solution. We develop an implicit scheme based on the mid-point rule, as well as an explicit scheme based on the centered-difference method, with the incorporation of energy conserving, frictionless contact algorithms for an elastic-to-rigid-surface contact. The performance of the proposed method is assessed for several benchmark tests: the electro-static force vector around a singular corner, the quasi-static pull-in of an electro-mechanically actuated switch, the excitation of a carbon nanotube at resonance, and the cyclic impact simulation of a micro-electro-mechanical resonant-switch. We report improved accuracy for the high-order method as compared to low-order methods, and linear convergence in the iterative solution of the staggered scheme. Additionally, we investigate a Newton-Krylov shooting scheme in order to directly find cyclic steady states of electro-mechanical devices excited at resonance-- as opposed to a naive time-stepping from zero initial conditions. For the examples discussed, we observe power law computational speed-ups of the form S=0.7xi^(-0.8), where xi is the damping ratio of the corresponding resonance frequency.