Finite Element Based Electrostatic-Structural Coupled Analysis with Automated Mesh Morphing V. I. Zhulin 1 , S.J. Owen 2 , D.F. Ostergaard 1 1 ANSYS, Inc., 275 Technology Drive, Southpointe, Canonsburg PA. 15317 2 Sandia National Labs, Parallel Computing Sciences, Albuquerque, NM 87185-0847 ABSTRACT A co-simulation tool based on finite element principles has been developed to solve coupled electrostatic- structural problems. An automated mesh morphing algorithm has been employed to update the field mesh after structural deformation. The co-simulation tool has been successfully applied to the hysteric behavior of a MEMS switch. 1 INTRODUCTION In a coupled electrostatic-structural problem, electric forces act on mechanical elements resulting in structural deformations, which in turn, change the electric field and forces. Traditionally, the finite elements (FE) technique is used to solve the structural problem and boundary elements (BE) for the electric domain [1]. BE is preferred because the FE solution requires meshing the electrostatic domain. However, FE simulation of the electrostatic domain has definite advantages over BE when the domain is large, complicated, and contains dielectrics. This is the case for many practical micro electromechanical structures where computer memory requirements and solver speed make BE impractical. To exploit the advantages of FE simulation a mesh morphing technique and its control algorithm, commonly known as a co-simulation tool, has been developed to automate the updating of the electrostatic mesh. The morphing algorithm has been implemented in the ANSYS/Multiphysics program for solving 2D and 3D electrostatic-structural simulations. The paper will focus on the mesh morphing algorithm which now enables the use of finite element methods in the electrostatic domain. 2 CO-SIMULATION TOOL The co-simulation tool solves the electrostatic field domain and the structural domain separately. A simple relaxation algorithm [1] is used to ensure a self-consistent electromechanical analysis. Finite element methods are used for both the electrostatic and structural domains. Coupling is achieved by forces computed using the Maxwell Stress Tensor technique. Forces are passed from the electrostatic mesh to the structural nodes. The algorithm requires node-to-node compatibility between the electrostatic mesh and the structural mesh. The electrostatic domain can consist of second order h- elements or p-adaptive elements. The structural domain may consist of first or second order elements. Monitoring the change in stored energy in the electrostatic domain as well as the change in the maximum structural displacement regulates convergence. All linear and non-linear structural finite element methods available in the ANSYS program may be employed in a co-simulation problem. 3 MESH MORPHING ALGORITHM After each iteration of the co-simulation tool, it is necessary to adjust the electrostatic field finite element mesh. This was accomplished by developing a mesh morphing tool that automatically updates the electrostatic mesh based on the current structural displacements. This is done by either (1) keeping the original element connectivity and adjusting the node locations within the electrostatic domain, or (2) forming a completely new mesh within the electrostatic region while maintaining the boundary between structural and electrostatic domains. In the first approach, a combination of smoothing techniques are used to adjust node locations. Laplacian smoothing [2] is a technique commonly used for improving finite element meshes by iteratively adjusting node locations to the centroid of their surrounding nodes. The surrounding nodes are typically defined by those connected by a finite element edge. For this application, the Laplacian smoothing proved to be not as effective, tending to bunch elements around the structural deformations. A modified area/volume based smoothing technique was employed which adjusts node locations based on the centroid of the adjacent elements. For example, For node P at location P x , its smoothed location is defined as: = = = n i i n i i i w w 1 1 C P x (1) where C i is the centroid of element i adjacent P, and w i is the area or volume of element i . The total number of elements adjacent to P is given by n. For nodes on the boundary, the above definition may be used, but a subsequent projection to the associated curve or surface is