Half-day course held before EuroSimE, Grand Hotel di Como, Sunday April, 23rd, 2006.

Lecture notes:

- Introduction to Dynamic Systems and Model Reduction;
- Implicit Moment Matching via Arnoldi Process: Theory;
- Implicit Moment Matching via Arnoldi Process: Practice;
- Advanced Topics in Model Reduction.

A good starting point is the third lecture where you will find examples of using model reduction for different domains: Electro-thermal MEMS, Structural mechanics, Piezoelectric actuators for control, Pre-stressed small-signal analysis for RF-MEMS, Thermomechanical models, Acoustics including fluid-structure interactions.

The course goal is to learn how to use model reduction in practice. The course is made for engineers that use high-dimensional finite-element models during device simulation. The course is an overview on how to produce compact models directly from the FEM models for system-level simulation based on modern mathematical approaches (model order reduction).

The design of MEMS-based devices often depends heavily on 3D time-dependent simulation of coupled physical partial differential equations in the thermal, mechanical, electrical, etc. domains. This in turn requires the solution of very large systems of ordinary differential equations (ODEs), resulting from the spatial discretization of the device that comes from a finite element program.

However, instead of the "brute force" approach to integrate a large system of ODEs, one can use modern mathematical methods to drastically reduce the size of the simulation, and thereby achieve dramatic speedup in the calculation. It is also now possible to solve large simulations that were previously simply too large (due to lack of time, computer memory or computer speed).

It happens that in many cases important for practice the order of ODEs can be reduced by several orders of magnitude without sacrificing precision. These include the simulation of accelerometers and gyroscopes, and many different electro-thermal devices.

The main emphasis of the course is on IDEAS rather than on MATHEMATICS. During the course you will learn:

• Some basics of control theory (a short review of the essential facts);

• How to reduce a linear system of ordinary differential equations by the truncated balanced approximation, by moment-matching methods, and by methods based on low-rank Gramian approximations;

• How to deal with nonlinear systems of ordinary differential equations;

• How to preserve design variable during model reduction;

• How to use the SLICOT library in order to solve practical problems;

• How to use mor4ansys (IMTEK software) in order to produce compact device models directly from ANSYS models;

• What other software is available for model reduction.