The rapid development of MEMS-based devices requires 3D time-dependent simulations for coupled physical domains (thermal, mechanical, electrical, etc.). This in turn requires the solution of high-dimensional ordinary differential equations (ODEs) that result from space discretization of the device. However, instead of a "brute force" approach to integrate a large system of ODEs, one can use modern mathematical methods to reduce the system's dimension. The goal of the present paper is to review them from an engineering perspective. It is shown that in many cases important for practice the order of ODEs can be reduced by several orders of magnitude almost without sacrificing precision.

An overview of electro-thermal modeling of microsystems is presented. We consider the most important coupling between thermal and electrical phenomena, and then focus on the industry's central concern, that of Joule heating. A description of different solution approaches for the heat transfer partial differential equation, which constitutes the central part of electro-thermal simulation, is given. We briefly review the analytical solutions and consider further the numerical approaches, which are based on spatial discretization of the thermal domain. Lastly, we describe the final level of approximation, the dynamic compact thermal modeling. We emphasize the formal model order reduction methods, because they directly follow the spatial discretization, and thus preserve the investment into the finite element modeling.

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Summary One of the main obstacles to including transient dynamic effects into the performance functions of a structural optimization is the high computational cost of each time-dependent simulation. The focus of this paper is on the application of model order reduction techniques to reduce the transient analysis time for the attainable optimization process. The software mor4ansys developed at IMTEK performs model reduction via the Arnoldi algorithm directly to ANSYS finite element models. We adopt a micro accelerometer as an example to demonstrate the advantages of this approach. The harmonic and transient results of a reduced model of the accelerometer yield very good agreement with those from the original high dimensional ANSYS model. The use of model reduction within the optimization iterations produces almost the same results as without order reduction and speeds up the total computation by about an order of magnitude.

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In this paper, we present three heuristic error indicators for the iterative model order reduction of electro-thermal MEMS models via the Arnoldi algorithm. Such error indicators help a designer to choose an optimal order of the reduced model, required to achieve a desired accuracy, and hence allow a completely automatic extraction of heattransfer macromodels for MEMS. We first suggest a convergence criterion between two successive reduced models of order and. We further propose to use a solution of the Lyapunov equations for reduced-order systems in each iteration, and alternatively to employ sequential model order reduction, which is based on consecutively applying Arnoldi and control-theory methods.

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