A new transformation technique for evaluating nearly singular integrals

Accurate evaluation of nearly singular integrals plays an important role in the overall accuracy of the Boundary Element Method (BEM). A new approach for the evaluation of nearly singular integrals particularly those with severe near singularity is described in this paper. This method utilizes a degenerate mapping to first reduce near singularity and then employs a variable transformation to further smooth out the resultant integrand. The accuracy and efficiency of the method are demonstrated through several examples that are commonly encountered in the applications of the BEM. Comparison of this method with some of the existing methods is also presented.

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

[2]  Ken Hayami,et al.  Variable Transformations for Nearly Singular Integrals in the Boundary Element Method , 2005 .

[3]  A NEW VARIABLE TRANSFORMATION FOR NUMERICAL , 1993 .

[4]  T. Cruse,et al.  Some notes on singular integral techniques in boundary element analysis , 1993 .

[5]  Ken Hayami.  A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals , 1992 .

[6]  L. F. Martha,et al.  Hypersingular integrals in boundary element fracture analysis , 1990 .

[7]  Yijun Liu.  ANALYSIS OF SHELL-LIKE STRUCTURES BY THE BOUNDARY ELEMENT METHOD BASED ON 3-D ELASTICITY: FORMULATION AND VERIFICATION , 1998 .

[8]  David Elliott,et al.  Sigmoidal Transformations and the Trapezoidal Rule , 1998 .

[9]  Peter Rex Johnston,et al.  Application of sigmoidal transformations to weakly singular and near-singular boundary element integrals , 1999 .

[10]  Some improvements of accuracy and efficiency in three dimensional direct boundary element method , 1992 .

[11]  J. Telles.  A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals , 1987 .

[12]  T. Cruse,et al.  Non-singular boundary integral equation implementation , 1993 .

[13]  Subrata Mukherjee,et al.  A mapping method for numerical evaluation of two-dimensional integrals with 1/r singularity , 1993 .

[14]  Siegfried Prössdorf,et al.  An algorithm for the approximate solution of integral equations of Mellin type , 1995 .

[15]  David Elliott,et al.  Error analysis for a sinh transformation used in evaluating nearly singular boundary element integrals , 2007 .

[16]  G. Kuhn,et al.  Efficient evaluation of volume integrals in the boundary element method , 1993 .

[17]  Ken Hayami,et al.  A numerical quadrature for nearly singular boundary element integrals , 1994 .

[18]  Peter Rex Johnston.  Semi-sigmoidal transformations for evaluating weakly singular boundary element integrals , 2000 .

[19]  Herbert A. Mang,et al.  A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method , 1985 .

[20]  Tg Davies,et al.  Boundary Element Programming in Mechanics , 2002 .

[21]  J. Telles,et al.  Third degree polynomial transformation for boundary element integrals: Further improvements , 1994 .

[22]  Avram Sidi,et al.  A New Variable Transformation for Numerical Integration , 1993 .

[23]  A generalisation of Telles' method for evaluating weakly singular boundary element integrals , 2001 .

[24]  David Elliott,et al.  Transformations for evaluating singular boundary element integrals , 2002 .

[25]  L. Gray,et al.  Evaluation of supersingular integrals: second‐order boundary derivatives , 2007 .

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

[27]  J. Watson,et al.  Effective numerical treatment of boundary integral equations: A formulation for three‐dimensional elastostatics , 1976 .

[28]  S. Mukherjee,et al.  Evaluation of nearly singular integrals in boundary element contour and node methods for three-dimensional linear elasticity , 2000 .

[29]  B. Yun.  A generalized non‐linear transformation for evaluating singular integrals , 2006 .

[30]  David Elliott,et al.  Numerical Integration IV , 1993 .

[31]  Peter Rex Johnston,et al.  A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals , 2003 .

[32]  Massimo Guiggiani,et al.  The evaluation of cauchy principal value integrals in the boundary element method-a review , 1991 .