ABSTRACT
The finite element method (FEM) allows complex, continuum problems to be modeled. The FEM was originally developed by structural engineers as an extension of matrix structural analysis. It has since been used in just about every field where differential equations are used to define problem behavior. Here we will use the physical behavior of stress flow to discuss how finite elements can be derived and how the derivation assumptions affect the results. This process can be abstracted to apply to other domains such as fluid flow, temperature flow, population changes, stellar physics, and electronic circuits. In these cases, the quantities developed here (stiffness, stress, displacement) need to be mapped into the problem domain described by the differential equation being used. The process of the FEM is to create a stiffness matrix (coefficients) and a set of loads (right-hand side). After that, the solution process is identical to that covered in any stiffness-based structural analysis textbook. Many excellent books covering the FEM exist; this section is intended only as a basic introduction (Bathe, 1995; McGuire et al., 2002; Zienkiewicz et al., 2000).
