Machine Precision Evaluation of Singular and Nearly Singular Potential Integrals by Use of Gauss Quadrature Formulas for Rational Functions

A new technique for machine precision evaluation of singular and nearly singular potential integrals with 1/R singularities is presented. The numerical quadrature scheme is based on a new rational expression for the integrands, obtained by a cancellation procedure. In particular, by using library routines for Gauss quadrature of rational functions readily available in the literature, this new expression permits the exact numerical integration of singular static potentials associated with polynomial source distributions. The rules to achieve the desired numerical accuracy for singular and nearly singular static and dynamic potential integrals are presented and discussed, and several numerical examples are provided.

[1]  Thomas F. Eibert,et al.  Correction to “On the Calculation of Potential Integrals for Linear Source Distributions on Triangular Domains” , 2005 .

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

[3]  M. G. Duffy,et al.  Quadrature Over a Pyramid or Cube of Integrands with a Singularity at a Vertex , 1982 .

[4]  Seppo Järvenpää,et al.  Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra , 2003 .

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

[6]  David Elliott,et al.  A sinh transformation for evaluating nearly singular boundary element integrals , 2005 .

[7]  A General Gauss Theorem for Evaluating Singular Integrals over Polyhedral Domains , 1991 .

[8]  Roberto D. Graglia,et al.  Higher order interpolatory vector bases for computational electromagnetics," Special Issue on "Advanced Numerical Techniques in Electromagnetics , 1997 .

[9]  Wolfgang Hackbusch,et al.  On numerical cubatures of nearly singular surface integrals arising in BEM collocation , 1994, Computing.

[10]  Exact Quadrature of Singular and Nearly Singular Potential Integrals , 2007, 2007 International Conference on Electromagnetics in Advanced Applications.

[11]  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 .

[12]  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 .

[13]  Accurate evaluation of potential integrals with Gauss quadrature formulas for rational functions , 2007, 2007 IEEE Antennas and Propagation Society International Symposium.

[14]  D. E. Cormack,et al.  The Continuation Approach: A General Framework for the Analysis and Evaluation of Singular and Near-Singular Integrals , 1995, SIAM J. Appl. Math..

[15]  Roberto D. Graglia,et al.  Static and dynamic potential integrals for linearly varying source distributions in two- and three-dimensional problems , 1987 .

[16]  E. Becker,et al.  A conforming crack tip element with quadratic variation in the singular fields , 1978 .

[17]  Branislav M. Notaros,et al.  OPTIMIZED ENTIRE‐DOMAIN MOMENT‐METHOD ANALYSIS OF 3D DIELECTRIC SCATTERERS , 1997 .

[18]  Andrew F. Peterson,et al.  Higher order interpolatory vector bases on pyramidal elements , 1998 .

[19]  W. Hackbusch.  Integral Equations: Theory and Numerical Treatment , 1995 .

[20]  R. Graglia.  The use of parametric elements in the moment method solution of static and dynamic volume integral equations , 1988 .

[21]  D. A. Dunavant.  High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[22]  Walter Gautschi,et al.  Algorithm 793: GQRAT—Gauss quadrature for rational functions , 1999, TOMS.

[23]  D. Moreno,et al.  Theoretical and numerical treatment of surface integrals involving the free-space Green's function , 1993 .

[24]  A. Stroud,et al.  Gaussian quadrature formulas , 1966 .

[25]  Barbara M. Johnston,et al.  A sinh transformation for evaluating two‐dimensional nearly singular boundary element integrals , 2007 .

[26]  D. M. Tracey,et al.  Finite elements for determination of crack tip elastic stress intensity factors , 1971 .

[27]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[28]  R. Zich,et al.  Moment method with isoparametric elements for three-dimensional anisotropic scatterers , 1989, Proc. IEEE.

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

[30]  Jian-Ming Jin,et al.  The Finite Element Method in Electromagnetics , 1993 .

[31]  J. Thomas Beale,et al.  A Method for Computing Nearly Singular Integrals , 2000, SIAM J. Numer. Anal..

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