Topology Optimization of Electrostatically Actuated Micromechanical Structures with Accurate Electrostatic Modeling of the Interpolated Material Model

In this paper, we present a novel formulation for performing topology optimization of electrostatically actuated constrained elastic structures. We propose a new electrostatic-elastic formulation that uses the leaky capacitor model and material interpolation to define the material state at every point of a given design domain continuously between conductor and void states. The new formulation accurately captures the physical behavior when the material in between a conductor and a void is present during the iterative process of topology optimization. The method then uses the optimality criteria method to solve the optimization problem by iteratively pushing the state of the domain towards that of a conductor or a void in the appropriate regions. We present examples to illustrate the ability of the method in creating the stiffest structure under electrostatic force for different boundary conditions. INTRODUCTION The use of electrostatic force for actuation in microsystems is desirable because of the large amplitudes that are achieved at the micron scale as well as the ease of manufacturing and integration along with electronic components. In microelectromechanical systems (MEMS), an electrostatic force that is attractive in nature deforms the mechanical structure. The potential difference between different conductors determines the magnitude of force, which in turn controls the equilibrium positions of the constrained elastic structures of the conductors. The fact that silicon can be used for generating both the electrostatic force as well as the mechanical restoring force makes it ideal for fabricating low cost devices using any of the standard micromachining methods. To meet the growing demand for electrostatic actuators in microsystems, it is important that synthesis methods are 1 developed in order to automate or aid the process of generating new designs. Synthesis techniques are also important from the point of view of generating complex designs that are not easily visualized through intuition. Among the many popular design methods, topology optimization is one such technique that is gaining popularity due to its ability to adapt to situations that involve many different physical phenomena [1]. Topology optimization refers to the synthesis of structures in a given domain so as to optimize an objective function subject to one or more constraints. For instance, a volume constraint is usually used to limit the amount of material available to the optimization algorithm while generating structures. Topology optimization is quite powerful because it requires only essential information from the user and is able to generate optimal designs that are often readily manufacturable. This is particularly true in the case of planar designs (such as those used in microsystems) which may be easily fabricated by choosing appropriate mask layouts without too much additional cost. There are a number of methods for solving problems of topology optimization. One popular approach among these that we shall be using here is the SIMP (Simple Isotropic Material with Penalization) approach [2]. In this method, we discretize the domain into a set of finite elements and define a material interpolation parameter for each of these elements. This parameter, which takes values between zero and one, is raised to some power and is multiplied with the material property values to interpolate the material properties throughout the domain in a continuous manner. This continuous interpolation between material and no-material calls for accurate modeling of the physics of the problem under intermediate, interpolated state of material(s). This paper deals with one such problem in topology optimization. Copyright © 2006 by ASME TP 1 PT The dielectric permittivity is exactly unity only in vaccum; however, the value in air and most metallic conductors is only slightly greater than unity. Topology optimization has been applied to design situations that involve many diverse physical phenomena like those present in mechanical structures, electrothermal actuators, piezoelectric actuators and even optical media like photonic crystals [3]. However, for electrostatically actuated structures, topology optimization has been implemented only recently [4], possibly due to the lack of a physical model to smoothly interpolate the material state from a conductor to a dielectric or a void. In this paper, we propose an accurate formulation for a material interpolation model that uses the so-called leaky capacitor model to provide a physical basis for this interpolation. In the following sections, we shall explain this model and discuss how it may be applied to topology optimization to yield optimal structures. In the next section, we begin with a brief description of the electrostatic analysis and force-computation when a material is in an intermediate state between a conductor and a void, which was discussed in our recent past work [5]. This analysis method is combined with the new material interpolation model of this paper to lead to topology optimization of electrostatically actuated structures. BRIEF OVERVIEW OF THE ANALYSIS METHOD Consider the domain to be a region that has some finite and spatially varying value of conductivity. In topology optimization, the changing material properties of the domain are defined in terms of a material interpolation scheme. Let γ be such a parameter used for material interpolation. In other words, the value of γ varies spatially in order to interpolate material properties between those of a conductor and a void in the case of electrostatic analysis. Upon the application of electrostatic boundary conditions, when there is “intermediate” material we observe a distribution of current flowing through the domain due to the finite value of its conductivity. The flow of current through this inhomogeneous domain under steady state is given by ( ) ( ) 0 V σ ⋅ = ⋅ = J ∇ ∇ ∇ (1) where J is the current density, V the electrostatic potential and ( ) , , x y z σ the spatially varying value of conductivity that ideally varies between infinity (for a conductor) and zero (for an insulating dielectric or void). In order to model the electrostatic force that is generated in actuators, we note that when current flows through an inhomogeneous domain, electric charge accumulates at the regions of discontinuity giving rise to an electrostatic force. This will be a body force at these regions in contrast to familiar surface force of electrostatics. In this case, the electrostatic force is localized to regions wherever there is a discontinuity in either conductivity or permittivity of the medium (see Eq. (2)). Using the generalized electrostatic stress tensor [6], we write the expression for this force per unit volume es F as follows. 2 2 2 m m 1 1 2 2 es e E E ε ρ ε ρ ρ ⎛ ⎞ ∂ = − + ⎜ ⎟ ∂ ⎝ ⎠ F E ∇ ∇ (2) Here e ρ is the free electric charge density, E the electric field, ε the permittivity and m ρ the mass-density of the material. The third term in expression for the body force of electrostatics is like a hydrostatic force that is the same in all directions inside a dielectric medium. Since we are considering only resultant forces on the domain, the third term in Eq. (2) may be neglected. Neglecting the third term, the expression becomes identical to the force predicted from Maxwell’s electrostatic stress tensor. For a detailed discussion of the above, please see [6]. The electrostatic body force is applied on the mechanical structure. The same material interpolation that is used to interpolate conductivity is also used to perform the same task on the mechanical moduli of the material (e.g., Young’s modulus and Poisson’s ratio in isotropic materials). The deformation in the mechanical structure is computed using the elastostatic governing equation: ( ) es ∇⋅ + = S F 0 and boundary conditions. (3) Here, S is the stress tensor, which is the product of the constitutive elastic modulus tensor and the strain tensor. When the material interpolation parameter takes values in between its two extreme limits in certain regions, we see that these parts partially conduct current and store electrostatic energy as well. In lumped modeling, this is known as a leaky capacitor model and is represented by a resistor and a capacitor in parallel. As the conductivity values in the domain are pushed towards the limits (i.e., for piece-wise homogeneous conductorvoid combinations), in the absence of a conducting path across the potential difference, we see that the structure resembles an ideal capacitive configuration. In this situation, the electrostatic force is localized to the interface between the conducting and void regions and becomes identical to the electrostatic surface force that is found in electrostatic actuators. Thus this model allows for the continuous interpolation of electrostatic material state between the limits of a conductor and a void. An example is shown in Fig. 1. More analysis results are in [5]. MATERIAL INTERPOLATION In this paper, we interpolate the material state only between two cases, i.e. a conductor and a void, while assuming permittivity to be unity everywhereTP PT. It must be noted here, that the physical model allows for the independent interpolation of dielectric permittivity too [5], though we do not make use of that in our optimization in this paper. The material interpolation is done in the following manner to ensure that regions of high Copyright © 2006 by ASME conductivity (i.e. conductors) also comprise the mechanical structure with higher elastic moduli when compared with void regions.