4. Numerical Implementation with Finite Element Method

The mathematical description of most engineering problems mainly takes the form of integrals or partial differential equations (PDEs) defined on geometrically complicated domains of interest. Depending on the inherent physics of the problem and the corresponding mathematical formulation, numerical solutions to such models can be obtained by employing numerical methods. Finite difference method is the most common numerical technique for solving such mathematical problems. It replaces the PDEs by approximating them with difference equations using grid information [148]. This method requires high accuracy of the solution and it is difficult to implement when the geometry becomes more complex.

Considering the model equations proposed in Chapter 3 and the complicated interconnect structures described in Chapter 1, the finite element method (FEM) constitutes a powerful numerical analysis technique for obtaining approximate solutions to such problems. FEM is based on the discretization of the domain of interest into finite elements and uses variational methods to find an approximate solution within each element by minimizing an associated error function. It can be applied to a wide range of engineering problems and handles complex geometries with different types of boundary conditions.

In the next sections, after a brief introduction to the FEM, a rigorous mathematical analysis for the derivation of an approximate solution for a 1D boundary value problem is described. The mathematical background for the derivation of the FEM equations is provided by referring to several books [148,85,88,116,125,123,161]. Then, the description of the numerical implementation of the governing model equations for electromigration in a FEM-based simulation tool, such as COMSOL Multiphysics^®, is presented.