Extension of Three-dimensional Electromagnetic Finite Element and Method of Moments Analysis To Include Singular Fields

The Finite Element Method (FEM) and Method of Moments (MoM) are popular numerical methods for solving complex problems encountered in almost any field of engineering. The structure to be studied is divided into small cells (2D or 3D) and the relevant equations can be numerically solved. The modes of the structure are typically obtained by solving a generalized eigenvalue equation (FEM) or performing a matrix inversion (MoM); scattering problems formulated in term of integral equations are typically solved with MoM. For problems with smooth surfaces or other regular features, high order finite-element techniques based on the use of (hierarchical) curl-conforming for FEM or divergence-conforming vector bases for MoM successfully improve accuracy and efficiency (greatly reducing the dimension of the matrices and minimizing CPU time). High degree polynomial expansion functions often do not improve the solution accuracy, or do not provide as rapid convergence as anticipated, when dealing with geometries containing sharp edges or corners. The slow convergence observed in these cases is a consequence of the non-analytic nature of the solution in the vicinity of the singular point. To improve the accuracy of these problems, special basis functions are being developed that incorporate the singular field behavior.Conversely, by using a $5^{th}$ or $6^{th} $ order base with only four triangles one obtains far better results than with 2748 triangles and order 0 (“classic FEM” implementation). In the following the reader will be presented with the development of high order polynomial basis for 2D structures (typically waveguide cross sections) in terms of scalar, vector and full-wave analysis; then with especially developed functions which greatly enhance the accuracy of modes that are affected by corner singularities. Chapters 2 and 3 will show the development of high order polynomial bases for 3D FEM (used to study cavities) founded on tetrahedral or triangular prism based cells. Chapter 4 will provide the results from the development of divergence-conforming high order hierarchical singular bases for quadrangular cells.