Finite Elements for Engineers with Ansys Applications

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.