Reduced-Order Models for MEMS Applications

Abstract

We review the development of reduced-order models for MEMS devices. Based on their implementation procedures, we classify these reduced-order models into two broad categories: node and domain methods. Node methods use lower-order approximations of the system matrices found by evaluating the system equations at each node in the discretization mesh. Domainbased methods rely on modal analysis and the Galerkin method to rewrite the system equations in terms of domain-wide modes (eigenfunctions). We summarize the major contributions in the field and discuss the advantages and disadvantages of each implementation. We then present reduced-order models for microbeams and rectangular and circular microplates. Finally, we present reduced-order approaches to model squeeze-film and thermoelastic damping in MEMS and present analytical expressions for the damping coefficients. We validate these models by comparing their results with available theoretical and experimental results. State-of-the-ArtThe dynamics of MEMS are represented by partial-differential equations (PDEs) and associated boundary conditions. The most widely used method to treat these distributed-parameter problems is to reduce them to ordinary-differential equations (ODEs) in time and then solve the reduced equations either numerically or analytically. Three approaches are used in the reduction.• Idealization of the device flexible structural elements as rigid bodies.• Discretization using finite-element methods (FEM), boundary-element methods (BEM), or finitedifference methods (FDM). • Construction of reduced-order models (ROM).The first and second approaches, while lying at opposite extremes of complexity, are currently the most widely used. The pressure for better designs, less trial-and-error in the design process, and better device performance demands better models than idealized rigid bodies. Numerous researchers compared the pull-in voltage of electrostatically actuated cantilever [1] and clamped-clamped [2] microbeams obtained by solving the distributed-parameter system to those obtained using a spring-mass model and found that the spring-mass model underpredicts the pull-in voltage.Although FEM/BEM and FDM simulations are adequate for the analysis of the static deflections (equilibrium positions) of MEMS devices, they are inadequate for dynamic simulations because they require the time integration of thousands of second-order ODEs (one for each degree of freedom in the model). This is a very expensive process, making system-level simulation, device optimization, interactive design, and evolutionary design almost impossible. As a result, reduced-order modeling of MEMS is gaining attention as a way to balance the need for enough fidelity in the model against the numerical efficiency necessary to make the model of practical use in MEMS design.

Theoretical and Experimental Investigation of the Nonlinear Behavior of an Electrostatically Actuated In-Plane MEMS Arch

“…where, (5) is multiplied by the denominator of the electrostatic force term (1-w o -w) 2 in order to reduce the computational cost [31,33]. Then, substituting equation (8) into the resulting equation, multiplying by the mode shape φ j (x), and then integrating the outcome over the normalized domain…”

The Influence of Couple Stress Components and Electrostatic Actuation on Free Vibration Characteristics of Thin Micro-Plates

“…In this way, the governing differential equations of equilibrium are reduced to a set of algebraic equations which can be solved more easily than the governing equations themselves. It is to be mentioned here that utilizing the Galerkin weighted residual method for eliminating the spatial dependence is so usual in the creation of ROMs [3]. In this way, it is proved that employing linear mode-shapes of the system accelerates the convergence of the procedure and significantly reduces the computational costs [2,8,9].…”

“…One of the most important phenomena associated with electrically actuated micro-plates is pull-in instability which occurs when the input voltage reaches a critical value called pull-in voltage. In this manner, the elastic restoring force of the micro-plate cannot resist against the Coulomb attraction and it suddenly collapses toward the substrate underneath it [3].…”

“…To date, variety of mathematical procedures has been employed to uncover the static and dynamic behaviors of such systems [5][6][7]. Amongst all of these mathematical techniques, reduced order modelling does not suffer from long run-time and has a unique advantage in providing the ability of presenting analytical solutions for MEMS problems [3]. In general, creating reduced order models (ROMs) eliminates the spatial dependence in the governing equations and performs a transformation from the physical coordinates of the device to a set of generalized coordinates.…”

Investigation of the nonlinear static and dynamic behaviour of rectangular microplates under electrostatic actuation

“…FEM based software tools are accurate but computationally expensive, especially when it comes to study the nonlinear dynamic behavior. On the other hand Reduced Order Models (ROM) based on the Galerkin approach have got popularity during the last decades because of their accuracy and low computational cost [6][7][8][9][10]. They have the capability to reveal the effect of different design parameters very conveniently.…”