4. Numerical Implementation

The mathematical description of physical phenomena very frequently consits of partial differential equations (PDE's) defined in a given domain of interest. Usually, these equations can be analytically solved only for very simple problems. Thus, for complex geometries and problems, involving variable material properties and general boundary conditions, numerical methods have to be applied.

Considering the model proposed in Chapter 3, the finite element method (FEM) has been chosen as numerical solving procedure. It presents a solid mathematical formulation for solving several types of PDE's and can handle complex geometries with different types of boundary conditions. Moreover, since it was originally devised for solving mechanical problems, it is rather convenient for the model implementation.

This chapter begins with a brief introduction to the finite element method, where the basic ideas are presented. A rigorous mathematical treatment is beyond the scope of this work and can be found elsewhere [151,152,153,154]. Then, the discretization of the model equations given in Chapter 3 is presented, followed by the description of the numerical implementation in a TCAD simulation tool.

- 4.1 The Finite Element Method

- 4.2 Discretization of the Model Equations
- 4.2.1 Discretization of Laplace's Equation
- 4.2.2 Discretization of the Thermal Equation
- 4.2.3 Discretization of the Vacancy Balance Equation
- 4.2.4 Discretization of the Mechanical Equations

- 4.3 Simulation in FEDOS

R. L. de Orio: Electromigration Modeling and Simulation