4.1.1 Introduction to Finite Element Method

The development of scientific computing has increased the possibilities to efficiently solve specific mathematical and engineering problems through numerical methods implemented in computers by using TCAD [85]. Computer-implemented mathematical models allows one to simulate and analyze complicated systems in order to significantly improve the design and operation of a device or process. The simulation provides access to physical quantities that cannot be measured and strongly supports insight into the physical phenomenon [88]. The purpose of a simulation software is to reduce the number of tests that have to be run during the design and optimization of a device or process.

TCAD mathematical models need numerical methods. Normally, analytical solutions of the model equations can easily be found by making some assumptions in very simple cases. In general, numerical techniques have to be applied for problems with complicated geometries, loadings, and material properties in order to find approximate solutions for PDEs where exact analytical solutions can not be obtained. Numerical methods of analysis transform the PDE governing a physical problem to a set of equations of a discrete model of the problem that has to be solved [125]. Variational methods have been developed for this purpose. They provide simple means of finding approximate solutions to physical problems. The approximate solutions are continuous functions over particular domains and are obtained by a linear combination of basis functions and unknown coefficients. In the solution of a PDE by means of the variational method, the governing equation is transformed into a weighted-integral statement in order to determine the unknown coefficients by minimizing the error introduced in the approximate solution of the PDE.

FEM is a computational technique that makes use of variational methods. Using FEM as a numerical solving procedure is convenient for complex mathematical models in complicated geometrical domains. FEM is a standard numerical technique for solving a wide spectrum of problems which are described by PDEs, defined in a domain of interest, and subjected to specified boundary conditions. The domain is represented as an assembly of finite elements [116]. FEM can handle irregular geometries with different boundary conditions and material behaviors. It is extensively used for solving problems in many areas of science and engineering, such as diffusion processes and solid mechanics, and it is often integrated in CAD tools. Furthermore, its diversity and flexibility as an analysis tool permits its application to multidisciplinary problems, such as the electromigration phenomenon in microelectronic structures which involves electro-thermal-mechanical analysis [148]. FEM can therefore be employed for the implementation of the electromigration model for more complete investigation on different interconnect structures.