J. K. White and N. R. Aluru

Department of Electrical Engineering and Computer Science

Massachusetts Institute of Technology

Cambridge, MA 02139

aluru@rle-vlsi.mit.edu

Designers of VLSI integrated circuits use hierarchical or mixed level
simulation that allow them to focus on the details of one section of a
design, while still efficiently simulating the entire circuit. Such tools
allow a relatively free mixture of behavioral, register-transfer level,
logical, and circuit-level descriptions of a given design. For designers of
MEMS, the need for the equivalent of mixed-level simulation is even more
pressing than it was for VLSI designers, because simulating even a single
device's performance usually requires a multi-level approach. For example,
while using the MIT MEMCAD system to analyze the Analog Devices comb-drive
based accelerometer, we used an *ad-hoc* approach to mixed-level
simulation. The electrostatic force-displacement relation for each of the
comb fingers was macro-modeled by table, and then this macro-model was
combined with a 3-D mechanical model of the polysilicon proof-mass plug
spring system.

The development of mixed-level MEMS simulation tools can not directly follow the hierarchical simulation approach used for VLSI, because the VLSI mixed-level paradigm is too narrow to address the needs of MEMS designers. In VLSI, the mixed-level approach is based on a single low-level description -- circuits -- and a single hierarchy of macro-models. There is no single low-level description in MEMS: designs involve a mixture of forces due to electrostatic fields, fluids, mechanical elasticity, etc. This lack of a single low-level description has two important ramifications: there is no organized approach either to generating or to coupling together different levels of MEMS representation.

MEMS requires a mixed-level approach, where both different physical systems and different levels of models can be coupled together in an organized fashion. In order to take an important step towards solving the mixed-level simulation problem, it is necessary to develop a software simulation system which will not only allow coupling between fluids, electrostatics, and mechanics, but will also allow a mixture of different physical regimes for the different energy domains. Once such mixed-regime approaches are developed, coupling to existing commercial circuit simulation tools like SPECTRE(Cadence) or SABER(Analogy), can provide full mixed-level simulation capabilities. In the rest of the paper we summarize our present efforts in developing algorithms for coupled-domain and mixed-regime simulation.

One approach to coupled-domain simulation is to use very general finite-element analysis approach. In such approaches, the unknowns in the various physical domains are represented by a sum of basis functions whose coefficients are determined by a Galerkin condition applied to the appropriate physical equations. The main short-coming of the finite-element approach is that it does not allow for individual selection of the most efficient simulation algorithms in each of the physical domains. For example, consider coupled electromechanics. The exterior field problem is most efficiently solved with accelerated boundary-element methods, but mechanical elastostatics is most efficiently treated using standard finite-element methods. Using finite-elements for both computations would be extremely expensive.

Another common approach to coupled-domain simulation is the relaxation scheme where the domains are solved separately and the solution is advanced iteratively until a self-consistent solution is found. The advantage of this iterative technique is that it allows the most efficient simulation algorithms to be used in each of the physical domains, and in addition, simulators can be coupled without being rewritten. The problem with the relaxation algorithm is that for a variety of applications, such as high-field electromechanics, the relaxation fails to converge.

Another approach that has been studied for coupled problems is the surface- Newton method. The key idea in this approach is to reduce the dimensionality of the coupled problem from 3-D to 2-D, where only the surface variables are involved in the coupled equations, and to apply a Newton method. For example, in coupled electromechanics, once the displacement of the structure surface is known, both the surface electrostatic force and the structure's interior displacements can be determined by decoupled electrostatic and mechanical analysis. Surface-Newton approach not only preserves the easy extensibility of the relaxation scheme but also eliminates the convergence problems encountered with the relaxation scheme. The tangent in the surface-Newton method is evaluated through the use of matrix-free conjugate-direction algorithm. Unfortunately, most simulation packages are not designed to allow for efficient computation of matrix-vector products as required in surface-Newton methods.

Our main point then, is that perhaps individual simulation packages should be redesigned, so that matrix-vector product computation is efficient, as this will allow for a "plug-and-play" approach to coupled-domain simulation. Then, much more rapid progress could be made on coupled fluid-mechanics-electrostatics-magnetics-circuits problems.

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