4.3.1 Finite Element Method Procedure in COMSOL Multiphysics

The application of FEM for the modeling and simulation of different physical phenomena in interconnect structures by using COMSOL Multiphysics^® is discussed here. The finite element analysis in COMSOL Multiphysics^® is performed by following a procedural flow, as depicted in (4.5).

Figure 4.5: Basic steps to perform a finite element analysis in COMSOL Multiphysics^®.

The method can be split into three basic steps, namely pre-processing, solver, and post-processing [38]. The pre-processing step enables the user to build the model. Since it contains all the information regarding the FEM application for interconnect study, it is convenient to divide it into smaller sub-steps. The first sub-step involves the creation of the interconnect geometry (1D, 2D, or 3D), to represent the domain under study, and assign the material properties to the domain. Then, the physical environments of the problem under investigation are generated by assigning the underlying physics (or multiphysics), mathematical equations, and finite element formulation to the model. Typically, the latter is hidden among the core of the commercial software. After that, the application of appropriate loadings, boundary and initial conditions to the domain under study, as well as its discretization into finite elements, determines the matrix equation governing the model, in a similar form of the one presented in equation (4.27).

This step is followed by solving the set of algebraic equations, which provides the physics-related nodal solutions of the model. COMSOL Multiphysics^® mainly employs two methods to obtain the model solution, namely segregated step method and fully coupled method. The segregated step method generates the set of algebraic equations for every physical model under consideration and calculates the resulting solutions for each of those models by following a sequence specified by the user. The fully coupled method generates the single set of algebraic equations for all the involved physical models and implements them in a single iteration scheme which is repeated until convergence is reached.

Once the solutions of the problems are determined, the post-processing step enables the user to evaluate the results of the finite element analysis by means of plotting and data exporting tools.