Physical Expansion Functions for Electromagnetic Integral-Equation Modeling
Numerical methods of electromagnetic (EM) modeling usually represent spatial and temporal variations of the field with piecewise polynomials of low order. We study the accuracy of several piecewise representations—constant, linear, quadratic, and cubic—using an integral equation to compute EM scattering from a thin wire. The actual scattered field of a uniform conductor should be holomorphic (i.e., have derivatives of all orders). Although holomorphism is impossible to achieve with piecewise polynomial expansions, we do ensure that the scattered electric field is continuous along the uniform conductor and has derivatives continuous to the order allowed by the polynomial basis. The higher-order bases dramatically improve the accuracy to which boundary conditions are satisfied. Higher-order expansions also reduce the degree to which the accuracy depends on the method used to solve the numerical equations. Moreover, convergence can be achieved with fewer discretization cells. Larger cell sizes, in turn, allow more accurate numerical evaluations of the integrals over the (singular) Green’s functions. We believe that these higher-order bases offer substantial improvements in 3-D numerical modeling of EM fields.