Analytical computation of impedance integrals with power-law Green's functions

In this contribution, a method is presented for reducing the number of subsequent integrations that occur in impedance integrals with Green's functions of the form R., with R the distance between source and observation point. The method allows the number of integrations to be reduced to 1 in the two dimensional case and 2 in the three dimensional case, irrespective of the number of subsequent integrations that were originally present. These last integrations can be done analytically using well-known results if nu is an element of Z, resulting in a computation that is free of numerical integrations. The dynamic Green's function can be treated in a semi-analytical way, by expanding it into a Taylor series in the wavenumber. The method can be applied if both the basis and test functions are polynomial functions with polygonal support and if certain non-parallelity conditions are satisfied

Accurate computation and tabulation of the scalar Green function for bi-anisotropic media and its derivatives

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A broadband stable addition theorem for the two dimensional MLFMA

Integral equations arising from the time-harmonic Maxwell equations contain the Green function of the Helmholtz equation as the integration kernel. The structure of this Green function has allowed the development of so-called fast multipole methods (FMMs), i.e. methods for accelerating the matrix-vector products that are required for the iterative solution of integral equations. Arguably the most widely used FMM is the multilevel fast multipole algorithm (MLFMA). It allows the simulation of electrically large structures that are intractable with direct or iterative solvers without acceleration. The practical importance of the MLFMA is made all the more clear by its implementation in various commercial EM software packages such as FEKO and CST Microwave studio

High precision evaluation of the selfpatch integral for linear basis functions on flat triangles

The application of integral equations for the frequency domain analysis of scattering problems requires the accurate evaluation of interaction integrals. Generally speaking, the most critical integral is the selfpatch. However, due to the non-smoothness of the Green function, this integral is also the toughest to calculate numerically. In previous work, the source and test integrals have been determined analytically for the 1/R singularity, i.e., the static kernel. In this work we extend this result to the terms of the form R-n, for all n is an element of {0, 1, 2, 3, 4} that occur in the Taylor expansion of the Green function. Numerical testing shows that truncating the Taylor series beyond n = 4 yields a highly accurate result for lambda/7 and lambda/10 discretizations. These analytical formulas are also very robust when applied to highly irregular triangles

Iteration-free computation of Gauss-Legendre quadrature nodes and weights

Gauss-Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss-Legendre weights is derived. Together, these two expansions provide a practical and fast iteration-free method to compute individual Gauss-Legendre node-weight pairs in O(1) complexity and with double precision accuracy. An expansion for the barycentric interpolation weights for the Gauss-Legendre nodes is also derived. A C++ implementation is available online

Computing Green's functions for bianisotropic materials

A storage data structure and strategy is proposed for the storage of Gegenbauer polynomial expansions, used in the numerical computation and storage of the bianisotropic scalar Green's function and its partial derivatives. The data structure allows the error to be controlled and keeps in check the computational complexity of the evaluation procedure

Analytical computation of impedance integrals with power-law Green's functions

In this contribution, a method is presented for reducing the number of subsequent integrations that occur in impedance integrals with Green's functions of the form R., with R the distance between source and observation point. The method allows the number of integrations to be reduced to 1 in the two dimensional case and 2 in the three dimensional case, irrespective of the number of subsequent integrations that were originally present. These last integrations can be done analytically using well-known results if nu is an element of Z, resulting in a computation that is free of numerical integrations. The dynamic Green's function can be treated in a semi-analytical way, by expanding it into a Taylor series in the wavenumber. The method can be applied if both the basis and test functions are polynomial functions with polygonal support and if certain non-parallelity conditions are satisfied