Examining a variety of quite different problems, including conventional moving structures, inductors and even protein molecules, presenters came up with surprisingly similar techniques, and a surprisingly close link to existing work in inductance extraction for purely electronic circuits.

Gang Li reported on work at the University of Illinois (Urbana, Ill.) on the analysis of moving MEMS structures such as beams and a comb-drive structure. The fundamental problem, Li suggested, was that both a mechanical analysis of the forces on and deflection of the mechanical parts, and an electrostatic analysis of the forces on the structure must be done to determine the state of the structure. But since the mechanical and electrical forces interact, ideally the two analyses must be done simultaneously. Conventionally, this is done by first doing the mechanical analysis, computing the deformed shape of the structure, computing the electrostatic forces for the deformed structure and then iterating the entire process until it converges.

Li proposed using a Lagrangian approach to analyze the mechanical and electrostatic state of the system in its undeformed state, and from that to iterate more quickly to the deformed state. This approach quickly turns into a system of integral equations based on Green's function.

In a strikingly similar paper, Shihhsien Kuo reported work at the Massachusetts Institute of Technology on electrostatic analysis of proteins in conductive solutions. The work helps predict, for instance, the dissolution energy of a protein or the binding energy between a protein and a receptor. Both issues would be significant for biochip applications or drug design.

The saline solution was modeled as a uniform region in which a linearized Poisson-Boltzmann equation applies, and the protein molecule as a region containing isolated point charges in which the Poisson equation itself applies. This formulation creates, once again, a system of integral equations based on Green's function, and opens the possibility of using fast solvers. Kuo demonstrated use of a precorrected fast Fourier transform to solve the equations, leading to analysis of both a simple and a quite large molecule.

In a third paper, presented by Yehia Massoud of Synopsys (Mountain View, Calif.), the problem of inductance extraction was explored for systems containing materials with high permeability. Inductance extraction is being developed widely in the industry, of course, but most current solutions look at self-inductance of normal interconnect materials.

Massoud pointed out that MEMS structures that used magnetic fields — actuators, motors or high-value inductors — tended to have components fabricated from high-permeability materials as well, and this violated the assumptions of most existing extraction approaches.

Following the now-familiar approach, Massoud separated the system under analysis into regions of low and high permeability, and then applied integral equations based on Green's function. In this case an important step was the realization that the entire magnetic field within the permeable material did not have to be modeled: it could be replaced with a factitious field of magnetic monopoles covering the surface of the material. This leads to a substantial reduction in the size of the problem. Once again the equations were solved numerically, this time with a preconditioning matrix and the GMRES iterative solver.

While none of the papers are likely to result in an EDA product in the near future, taken together they indicate the direction of analytical work on MEMS structures.

The trend seems to be that analysis of those structures will require careful seeding of the structures with seed points to be used in the analysis — a process analogous to, but mathematically quite different from, constructing a grid for finite element analysis — and the iterative solution of huge systems of integral equations.

The trend also suggests that eventually, characterizing the behavior of MEMS in detail may require use of a total-energy analysis approach such as the Lagrangian one that can simultaneously develop analytical expressions for the position of, shape of and electrostatic/magnetic forces acting on a structure.

From an earlier session of the conference came a series of papers on bringing MEMS into a conventional IC design flow. Papers examined developing lumped-parameter models of suspended MEMS structures, extracting an electromechanical MEMS schematic from a layout and layout-vs.-source verification for MEMS, and an investigation of electronic/MEMS co-design.