Introduction

Undoubtedly, the finite element method represents one of the most significant achievements in the field of computational methods in the last century. Historically, it has its roots in the analysis of weight-critical framed aerospace structures. These framed structures were treated as an assemblage of one-dimensional members, for which the exact solutions to the differential equations for each member were well known. These solutions were cast in the form of a matrix relationship between the forces and displacements at the ends of the member. Hence, the method was initially termed matrix analysis of structures. Later, it was extended to include the analysis of continuum structures. Since continuum structures have complex geometries, they had to be subdivided into simple components or “elements” interconnected at nodes. It was at this stage in the development of the method that the term “finite element” appeared. However, unlike framed structures, closed form solutions to the differential equations governing the behavior of continuum elements were not available. Energy principles such as the theorem of virtual work or the principle of minimum potential energy, which were well known, combined with a piece-wise polynomial interpolation of the unknown displacement, were used to establish the matrix relationship between the forces and the interpolated displacements at the nodes numerically. In the late 1960s, when the method was recognized as being equivalent to a minimization process, it was reformulated in the form of weighted residuals and variational calculus, and expanded to the simulation of nonstructural problems in fluids, thermomechanics, and electromagnetics. More recently, the method is extended to cover multiphysics applications where, for example, it is possible to study the effects of temperature on electromagnetic properties that might affect the performance of electric motors.