Introduction to the Finite Element Method in Electromagnetics

This series lecture is an introduction to the finite element method with applications in electromagnetics. The finite element method is a numerical method that is used to solve boundary-value problems characterized by a partial differential equation and a set of boundary conditions. The geometrical domain of a boundary-value problem is discretized using sub-domain elements, called the finite elements, and the differential equation is applied to a single element after it is brought to a “weak” integro-differential form. A set of shape functions is used to represent the primary unknown variable in the element domain. A set of linear equations is obtained for each element in the discretized domain. A global matrix system is formed after the assembly of all elements. This lecture is divided into two chapters. Chapter 1 describes one-dimensional boundary-value problems with applications to electrostatic problems described by the Poisson's equation. The accuracy of the finite element methodis evaluated for linear and higher order elements by computing the numerical error based on two different definitions. Chapter 2 describes two-dimensional boundary-value problems in the areas of electrostatics and electrodynamics (time-harmonic problems). For the second category, an absorbing boundary condition was imposed at the exterior boundary to simulate undisturbed wave propagation toward infinity. Computations of the numerical error were performed in order to evaluate the accuracy and effectiveness of the method in solving electromagnetic problems. Both chapters are accompanied by a number of Matlab codes which can be used by the reader to solve one- and two-dimensional boundary-value problems. These codes can be downloaded from the publisher's URL: www.morganclaypool.com/page/polycarpou This lecture is written primarily for the nonexpert engineer or the undergraduate or graduate student who wants to learn, for the first time, the finite element method with applications to electromagnetics. It is also targeted for research engineers who have knowledge of other numerical techniques and want to familiarize themselves with the finite element method. The lecture begins with the basics of the method, including formulating a boundary-value problem using a weighted-residual method and the Galerkin approach, and continues with imposing all three types of boundary conditions including absorbing boundary conditions. Another important topic of emphasis is the development of shape functions including those of higher order. In simple words, this series lecture provides the reader with all information necessary for someone to apply successfully the finite element method to one- and two-dimensional boundary-value problems in electromagnetics. It is suitable for newcomers in the field of finite elements in electromagnetics.

Anastasis C. Polycarpou received the B.S. (with summa cum laude), M.S., and Ph.D. degrees in Electrical Engineering from Arizona State University in 1992, 1994, and 1998, respectively. During his graduate studies, he was with the Telecommunications Research Center (TRC) of ASU where he worked on various research projects sponsored by government organizations and private companies such as the US Navy, US Army, Boeing, Sikorsky, and a few more. In the summer of 1998, he joined the Department of Electrical Engineering of ASU as an Associate Research Faculty where he performed research on a variety of subjects in the broad area of electromagnetics. While being at ASU, he worked on the development and enhancement of numerical methods, in particular the Finite Element Method (FEM) and the Method of Moments (MoM), for the analysis of complex electromagnetic problems such as microwave circuits, interconnects and electronic packaging, cavity-backed slot antennas in the presence of magnetized ferrites, and helicopter electromagnetics. He wrote a multipurpose three[1]dimensional finite element code using edge elements to solve geometrically complex scattering and radiation problems. The code utilizes advanced iterative techniques in linear algebra for the solution of extremely large indefinite matrix systems. Dr. Polycarpou has published more than 40 journals and conference proceedings and two chapters in books. He is currently an Associate Professor at Intercollege in Cyprus. His research areas of interest include numerical methods in electromagnetics and specifically the Finite Element Method and the Method of Moments, antenna analysis and design, electromagnetic theory, and ferrite materials.