Convergence properties of relaxation versus the surface-Newton generalized-conjugate residual algorithm for self-consistent electromechanical analysis of 3-D micro-electro-mechanical structures

Electrostatic sensors and microactuators are typically controlled by applied voltages which create electrostatic forces that deform the structure. Therefore, accurately analyzing the performance of these sensors and actuators requires self-consistent electromechanical analysis. However, self-consistent electromechanical analysis is a difficult computational problem because the discretization grid used must track the electrostatically deformed boundaries of the structure. Self-consistent electromechanical analysis of complicated three-dimensional structures can be performed by combining a fast multipole-accelerated scheme for electrostatic analysis with a standard finite-element method for mechanical system analysis. There are two approaches for combining these analyses, one using a straight-forward relaxation scheme, and a second based on a surface-Newton method combined with a matrix-free generalized conjugate residual based solver (SNGCR). In this paper, the convergence properties of these two methods are examined. In particular, we show that relaxation will converge if the applied voltage is small enough, or if Young´s modulus is large enough, but will diverge otherwise. We also show by example that although the SNGCR algorithm is guaranteed to converge only given a sufficiently close initial guess, it converges much more frequently than relaxation