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 Rⁿ, ¿n ¿ {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 ¿/7 and ¿/10 discretizations. These analytical formulas are also very robust when applied to highly irregular triangles.