Numerical Evaluation via Singularity Cancellation Schemes of Near-Singular Integrals Involving the Gradient of Helmholtz-Type Potentials

In this paper, we present a purely numerical procedure to evaluate strongly near-singular integrals involving the gradient of Helmholtz-type potentials for observation points at finite, arbitrarily small distances from the source domain. In the proposed approach the source domain is subdivided into a disc plus truncated subtriangles, and proper variable transformations are applied in each integration domain to exactly cancel the kernel singularity. A novel feature of the proposed angular transform is that required discrete values of the inverse transform, which is transcendental, are determined via a root-finding procedure; the same idea can also be applied to other transforms that arise in singularity cancellation methods. The resulting integral may then evaluated via a low order Gauss-Legendre quadrature scheme.

[1]  Francesca Vipiana,et al.  Optimized Numerical Evaluation of Singular and Near-Singular Potential Integrals Involving Junction Basis Functions , 2011, IEEE Transactions on Antennas and Propagation.

[2]  Clive Parini,et al.  Spherical near-field antenna measurements , 2014, Theory and Practice of Modern Antenna Range Measurements, 2nd Expanded Edition, Volume 2.

[3]  M.A. Khayat,et al.  Simple and Efficient Numerical Evaluation of Near-Hypersingular Integrals , 2008, IEEE Antennas and Wireless Propagation Letters.

[4]  P. Yla-Oijala,et al.  Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles , 2006, IEEE Transactions on Antennas and Propagation.

[5]  Numerical evaluation of near strongly singular integrals via singularity cancellation techniques , 2011, Proceedings of the 5th European Conference on Antennas and Propagation (EUCAP).

[6]  D. Wilton,et al.  Issues and Methods Concerning the Evaluation of Hypersingular and Near-Hypersingular Integrals in BEM Formulations , 2005 .

[7]  P. Yla-Oijala,et al.  Calculation of CFIE impedance matrix elements with RWG and n/spl times/RWG functions , 2003 .

[8]  Weng Cho Chew,et al.  Super‐hyper singularity treatment for solving 3D electric field integral equations , 2007 .

[9]  Direct numerical evaluation of near strongly-singular integrals , 2011, 2011 IEEE International Symposium on Antennas and Propagation (APSURSI).

[10]  M.A. Khayat,et al.  Numerical evaluation of singular and near-singular potential Integrals , 2005, IEEE Transactions on Antennas and Propagation.

[11]  J. R. Mosig,et al.  On the Direct Evaluation of Surface Integral Equation Impedance Matrix Elements Involving Point Singularities , 2011, IEEE Antennas and Wireless Propagation Letters.

[12]  L. Gurel,et al.  Singularity of the magnetic-field Integral equation and its extraction , 2005, IEEE Antennas and Wireless Propagation Letters.

[13]  D. Wilton,et al.  Electromagnetic scattering by surfaces of arbitrary shape , 1980 .

[14]  M.A. Khayat,et al.  An improved transformation and optimized sampling scheme for the numerical evaluation of singular and near-singular potentials , 2008, 2007 IEEE Antennas and Propagation Society International Symposium.

[15]  T. Eibert,et al.  Adaptive Singularity Cancellation for Efficient Treatment of Near-Singular and Near-Hypersingular Integrals in Surface Integral Equation Formulations , 2008, IEEE Transactions on Antennas and Propagation.

[16]  Roberto D. Graglia,et al.  On the numerical integration of the linear shape functions times the 3-D Green's function or its gradient on a plane triangle , 1993 .

[17]  J. Mosig,et al.  Fast and Accurate Computation of Hypersingular Integrals in Galerkin Surface Integral Equation Formulations via the Direct Evaluation Method , 2011, IEEE Transactions on Antennas and Propagation.

[18]  J. Mosig,et al.  On the efficient evaluation of hyper-singular integrals in Galerkin surface integral equation formulations via the direct evaluation method , 2010, 2010 IEEE Antennas and Propagation Society International Symposium.

[19]  W. Chew,et al.  On the Near-Interaction Elements in Integral Equation Solvers for Electromagnetic Scattering by Three-Dimensional Thin Objects , 2009, IEEE Transactions on Antennas and Propagation.