# Interdisciplinary Applied Mathematics

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There are broadly two levels of simulation in the field of microsystems.

1. Physical Level: At this level, the full behavior of the continuum is captured by means of highly meshed 3D simulations. These simulations involve many degrees of freedom. This level is primarily divided into process, material, and structural modeling. Multiple-energy domains (thermal, chemical, electrical, mechanical, etc.), large amplitude motions, and inherent nonlinearities (e.g., forces on capacitor plates; squeeze film damping) make physical-level simulation very complicated. Physical-level models, although very accurate, are computationally very expensive, and it is impractical to use them in the iterative microsystem design process.

2. System Level: In practice, every microdevice is connected to a full system (see, e.g., Figure 1.30). The use of highly comprehensive physical models at this level makes the simulations slow and computationally expensive. System level modeling requires low-order behavioral models of the various components of the system, which are electronic, micromechanical, fluidic, optical, chemical, biological, etc. In the general case, these system-level models are represented as a network of lumped elements analogous to the electric circuit components. More generally, they are represented as a small set of coupled ordinary differential equations that can be easily integrated in time. The results of detailed numerical simulations (with their enormous number of degrees of freedom) have to be projected onto spaces spanned by a very small number of appropriately chosen dynamical variables used in the system-level simulation.

In order to develop a system-level simulation framework, that is sufficiently simple, accurate, and robust, all processes involved need to be simulated at a comparable degree of accuracy and integrated seamlessly. That is, circuits, semiconductors, springs and masses, beams and membranes, as well as the flow field need to be simulated in a consistent and compatible way and in reasonable computational time! There are two well-known approaches for system-level simulation of microsystems. In the first approach, reduced-order models or macromodels for microdevices are combined with circuit simulation tools. In the second approach, the full-physics-based simulation tools for microdevices are directly combined with the circuit simulation tools. Since circuit simulation tools play an important role in system-level modeling, a brief description of circuit simulation tools is provided before we discuss the two approaches in more detail.

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