A Fast Convergence Scheme for Coupled Energy Domains Simulation of Mems

Due to the massive demands from industry, the black-box based coupled energy domains simulation plays a more and more important role in the design of MEMS devices, and many numerical techniques have been developed so far for it. This paper presents a fast convergence scheme for coupled energy domains simulation of MEMS. This scheme is based on the relaxation approach but employs the Steffensen’s acceleration technique to speed up the convergence procedure. In this paper, the details of this scheme are described as well as the relaxation and the multilevel Newton methods. The numerical examples show that the proposed scheme has equivalent convergence performance, sometimes even more efficient, as the multilevel Newton method, and much better than the relaxation method, especially for strong coupled or nonlinear situations, while keeping the advantage of easy programming. INTRODUCTION The rapid development of Micro-Electro-Mechanical Systems (MEMS) has growing demands for Computer-Aided Design (CAD) tools. Due to the fact that all MEMS devices involve multiple coupled energy domains, there are two special requirements for MEMS CAD tools: firstly, thesee should have the ability to accomplish coupled energy domains simulation based on detailed models which are constructed using Finite Element Method (FEM) or other methods and; secondly, they should be ∗Address all correspondence to this author. able to utilize existing, commercially available and black-box based simulation tools for each individual energy domain [1]. Several numerical techniques have been developed for coupled energy domains simulation of MEMS devices. The widely used relaxation approach is very convenient and easy to program, furthermore, it is a black-box based method and therefore can be easily extended to include more coupled domains without the modification on commercial domain-specific tools [2]. However, many numerical examples indicated that the relaxation approach is extremely slow and sometimes fails to converge for strong coupling or nonlinear problems [3]. The Newton method can be very quick and efficient compared to the relaxation method, but it has no black-box capability and, hence, is not easy to be connected with existing single-domain analysis tools [4]. The multilevel Newton method permits the robust convergence properties of the Newton method to be realized in black-box architecture and, has been proven to be very accurate, efficient and convergent even when there is strong coupling between energy domains, but it has considerable computing loads for each iteration [5]. The objective of this paper is to introduce a fast convergence scheme for black-box based coupled energy domains simulation of MEMS. This scheme is based on the relaxation approach but employs the Steffensen’s acceleration technique to speed up the convergence procedure giving good convergence performance while keeping the advantages of the relaxation method. The rest of this paper is organized as follows. In the next section the general problem of coupled domain simulation will be introduced. The relaxation and the multilevel Newton meth1 Copyright c © 2006 by ASME ods will be reviewed followed by a detailed description of the fast convergence scheme. Then, these three approaches will be used to combine FEM-based mechanical analysis with Boundary Element Method (BEM) based electrostatic analysis on three numerical examples: a MEMS parallel plate capacitor, a cantilever micro-beam, and cross bars. The last section will give a conclusion. COUPLED ENERGY DOMAINS MEMS devices are able to realize their functions due to the interactions of multiple coupled energy domains within them. For example, the electromechanical coupling in MEMS comb drives and, the interaction between electrostatic, mechanical and fluidic domains in microresonator-based gyroscopes [6]. Therefore, simulation of these coupled energy domains has become an essential part of the design of MEMS devices. Electromechanical Problem Electromechanical system is a classical kind of MEMS devices. It usually involves a mechanical structure which undergoes deformation when subjected to electrostatics force and conversely, the electrostatics charge distribution is also changed due to the structural deformation. The equilibration is obtained until the mechanical and electrostatic force balance each other. A well known example of the electromechanical system is the cantilever micro-beam over a ground plane [5]. A schematic setup for this system is shown in Fig. 1. As the interactions between mechanical and electrostatic domains are the most typical coupling phenomena in MEMS, the discussions below will only focus on electromechanical systems.