On the analytic-numeric treatment of weakly singular integrals on arbitrary polygonal domains

An alternative analytical approach to calculate the weakly singular free-space static potential integral associated to uniform sources is presented. Arbitrary oriented flat polygons are considered as integration domains. The technique stands out by its mathematical simplicity and it is based on a novel integral transformation. The presented formula is equivalent to others existing in literature, being also concise and suitable within a singularity subtraction framework. Generalized Cartesian product rules built on the double exponential formula are utilized to integrate numerically the resulting analytical 2D potential integral. As a consequence, drawbacks associated to endpoint singularities in the derivative of the potential are tempered. Numerical examples within a surface integral equation-Method of Moments framework are finally provided.

[1]  On the locally continuous formulation of surface doublets , 1998 .

[2]  Xiaoye S. Li,et al.  A Comparison of Three High-Precision Quadrature Schemes , 2003, Exp. Math..

[3]  Juan R Mosig,et al.  Evaluation of Weakly Singular Integrals Via Generalized Cartesian Product Rules Based on the Double Exponential Formula , 2010, IEEE Transactions on Antennas and Propagation.

[4]  Thomas F. Eibert,et al.  On the calculation of potential integrals for linear source distributions on triangular domains , 1995 .

[5]  Branko M. Kolundzija,et al.  Comparison of different strategies for conversion of triangular mesh into quadrilateral mesh , 2010, Proceedings of the Fourth European Conference on Antennas and Propagation.

[7]  Line-Integral Approach to Computing Impedance Matrix Elements , 2007, IEEE Transactions on Antennas and Propagation.

[8]  A. Polimeridis,et al.  On the Direct Evaluation of Weakly Singular Integrals in Galerkin Mixed Potential Integral Equation Formulations , 2008, IEEE Transactions on Antennas and Propagation.

[9]  Luis Landesa,et al.  SUPERCOMPUTER AWARE APPROACH FOR THE SOLUTION OF CHALLENGING ELECTROMAGNETIC PROBLEMS , 2010 .

[10]  L. Gurel,et al.  SOLUTIONS OF LARGE-SCALE ELECTROMAGNETICS PROBLEMS USING AN ITERATIVE INNER-OUTER SCHEME WITH ORDINARY AND APPROXIMATE MULTILEVEL FAST MULTIPOLE ALGORITHMS , 2010 .

[11]  G. Lombardi,et al.  Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions , 2008, IEEE Transactions on Antennas and Propagation.

[12]  Shiwen Yang,et al.  FAST ANALYSIS OF MICROSTRIP ANTENNAS OVER A FREQUENCY BAND USING AN ACCURATE MOM MATRIX INTERPOLATION TECHNIQUE , 2010 .

[13]  D. Wilton,et al.  Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains , 1984 .

[14]  B. Kolundžija.  Electromagnetic modeling of composite metallic and dielectric structures , 1996, IEEE Antennas and Propagation Society International Symposium 1997. Digest.

[15]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

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

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

[18]  L. Perregrini,et al.  On the evaluation of the double surface integrals arising in the application of the boundary integral method to 3-D problems , 1997 .

[19]  E. Newman,et al.  A surface patch model for polygonal plates , 1982 .

[20]  Luis Landesa,et al.  MLFMA-FFT PARALLEL ALGORITHM FOR THE SO- LUTION OF LARGE-SCALE PROBLEMS IN ELECTRO- MAGNETICS (INVITED PAPER) , 2010 .

[21]  Roger F. Harrington,et al.  Field computation by moment methods , 1968 .

[22]  Yan Shi,et al.  MULTILEVEL GREEN'S FUNCTION INTERPOLATION METHOD SOLUTION OF VOLUME/SURFACE INTEGRAL EQUATION FOR MIXED CONDUCTING/BI-ISOTROPIC OBJECTS , 2010 .

[23]  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 .

[24]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[25]  P. J. Cullen,et al.  On the fully numerical evaluation of the linear-shape function times the 3D Green's function on a plane triangle , 1999 .

[26]  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.

[27]  A nonrectangular patch model for scattering from surfaces , 1979 .

[28]  B. Notaroš.  Higher Order Frequency-Domain Computational Electromagnetics , 2008, IEEE Transactions on Antennas and Propagation.

[29]  J. Mosig.  Arbitrarily shaped microstrip structures and their analysis with a mixed potential integral equation , 1988 .