Coupled electrostatic-elastic analysis for topology optimization using material interpolation

In this paper, we present a novel analytical formulation for the coupled partial differential equations governing electrostatically actuated constrained elastic structures of inhomogeneous material composition. We also present a computationally efficient numerical framework for solving the coupled equations over a reference domain with a fixed finiteelement mesh. This serves two purposes: (i) a series of problems with varying geometries and piece-wise homogeneous and/or inhomogeneous material distribution can be solved with a single pre-processing step, (ii) topology optimization methods can be easily implemented by interpolating the material at each point in the reference domain from a void to a dielectric or a conductor. This is attained by considering the steady-state electrical current conduction equation with a ‘leaky capacitor’ model instead of the usual electrostatic equation. This formulation is amenable for both static and transient problems in the elastic domain coupled with the quasi-electrostatic electric field. The procedure is numerically implemented on the COMSOL Multiphysics® platform using the weak variational form of the governing equations. Examples have been presented to show the accuracy and versatility of the scheme. The accuracy of the scheme is validated for the special case of piece-wise homogeneous material in the limit of the leaky-capacitor model approaching the ideal case. 1. Background and Motivation Electrostatic force is one of the most important means of providing actuation in microsystems owing its popularity primarily to its favorable scaling at the micron scale and adaptability to most micromachining techniques. Its widespread use has spawned a variety of analytical and numerical methods for its analysis (e.g., [1]) and shape optimization (e.g., [2]). The comparative speeds and accuracies of various analysis algorithms come under close scrutiny in design situations involving multiple, iterative analysis steps such as those encountered in topology optimization or in the manual design of complex microsystem devices. To implement topology optimization for electrostatically actuated structures, a framework is required to allow any portion of a given region to be a conductor, dielectric, or void because every point can potentially be occupied by a conductor or a dielectric or by no material at all. Even though topology optimization-based synthesis methods have been reported for a variety of actuations in microsystems [3], such an attempt for electrostatically actuated microstructures is reported only recently [4]. The inability to smoothly interpolate the state of the material from a conductor to a dielectric or a void was perhaps a reason for this. In this paper, we introduce a new method for this purpose by modeling electrostatic domains as limiting cases of regions with spatially varying Institute of Physics Publishing Journal of Physics: Conference Series 34 (2006) 264–270 doi:10.1088/1742-6596/34/1/044 International MEMS Conference 2006 264 © 2006 IOP Publishing Ltd conductivity. An additional advantage of the inhomogeneous conductivity model of the electrostatics problem is that it allows the flexibility to simulate arbitrary material composition within a given region of any shape and topology. Furthermore, the finite element mesh on the region can be fixed even when the internal geometries and interfaces change. This enables quick multiple runs with changed geometry, materials and other parameters, which expedites the re-design process without the trouble of repeated pre-processing. 2. New Analytical Formulation In traditional coupled electrostatic-elastic analysis, a given domain is partitioned into regions composed of conductors with dielectrics and/or voids among them. By applying the electrical boundary conditions, we solve the Laplace equation in the void regions. The resulting electrostatic potential field ( ) yields the distribution of surface charge density ( s ) on the conductors’ boundaries. This helps compute the electrostatic forces ( es f ) on the interfaces, which are then used to solve for the elastic displacements ( u ) in the structural domain. The displacements so computed are coupled back into the electrostatic domain (because the domains void and struc would have changed now) and the cycle is repeated until a self-consistent solution is found. This process is illustrated in figure 1, where 0 is the dielectric permittivity of free space, S the elastic stress tensor and Y the Young’s modulus tensor of the material that forms the structure. Figure 1. The traditional electrostatic-elastic analysis process The framework shown in figure 1 serves well for shape optimization and size (or parameter) optimization in which the regions of conductors and voids are known a priori although the boundary may be changing. However, in topology optimization the internal geometries and interfaces among conductors, dielectric and voids are not known; in fact, they get determined during the process of optimization. As is well known in this field, topology optimization is equivalent to optimal material distribution [3, 5]. When continuous optimization algorithms are used, the material distribution needs to be varied smoothly. This calls for “material interpolation” where continuous “intermediate” state is defined in between the material and void states [6]. In this paper, we define the properties of such a smoothened material state in terms of a spatially-varying material selection parameter. In the limit, this leads to distinct selection of conductors and voids when suitable upper and lower bounds are imposed on this parameter. Conductor and void regions are thus demarcated by spatially varying values of this parameter in an iterative optimization procedure. The physical basis for such an interpolation model is explained next. 2.1. The leaky capacitor model Consider a region of inhomogeneous conductivity with some arbitrary distribution of electric current flowing through it. Assuming the absence of sources and sinks of current, we consider the continuity equation for the volume density of free electric charge in the domain e , in terms of the electric current density vector J . 0 e d dt J (1) 2 0 on void