In this chapter we discuss the mathematical foundation for obtaining the finite element equations for a general engineering problem or a physical system. Before studying the material in this chapter, it is essential to review the concepts of matrix algebra and indicial notation discussed in Appendix A. The chapter will provide a detailed discussion of how to formulate finite element equations using variational principles and weighted residual methods. The development will start with simple one-dimensional problems and will then proceed to full three-dimensional cases. The focus will be on deriving the FE equations in linear elasticity and heat transfer applications. The equations developed and presented here will be the basis for our discussion in the following chapters: the linear elasticity and heat transfer chapters.

In this chapter, we provide a simple introduction to the finite element method (FEM) and how it is related to other solution methods for engineering and physics problems. Throughout this chapter and the next, we avoid rigorous mathematical developments and equations and use simple examples that are easily understood by all students. We identify five basic steps or stages for any type of finite element analysis. These are: modeling and discretization; formulation and element equations; assembly; boundary conditions and solution; and finally postprocessing. In commercial FE programs these steps are lumped into three stages: modeling; solution; and postprocessing. One of the five steps, namely the assembly process, is quite simple and straightforward. A student may actually write a simple and general program in just one page that will do the assembly process. On the other hand, most of the research done and the textbooks written in the finite element area involve one or more of the other basic steps or stages. Throughout the introduction of the basic steps of the FE method, we introduce definitions of conceptual terminology that are common in the FE field, e.g., elements, nodes, boundary conditions, degrees of freedom (DOFs). To enable the students to start the modeling step we highlight the common elements used in most commercial programs and their geometry, nodes and DOFs.

This chapter, then, provides a brief account of the history of the development of the FE method. This is presented in two parts: the history of the development of the formulation and algorithms of the method; and the history of the development of computer hardware and software related to the application of the method. The final section of the chapter presents some typical applications of the FEM, mostly from the work of the author. These are meant to give students an overview of the capabilities and limitations of the method in various fields.

This chapter highlights the applications of FE in conduction heat transfer analysis. We will consider three-dimensional transient conduction analysis with convection and radiation boundary conditions. We start by highlighting the basic equations for conduction analysis and the pertinent boundary conditions, namely: flux, temperature, convection and radiation boundary conditions. Then, we discuss the development of FE equations in three-dimensional conduction problems using the weighted residual technique, which was discussed in Chapter 5. This is followed by a discussion of solution methods, accuracy and stability of the solution for the transient heat transfer problem. Finally, we develop sample elements for use in heat transfer problems and present a concluding example that is followed by an Ansys/Workbench project.