Efficient Numerical Evaluation of Singular Integrals in Volume Integral Equations

We present a method for the numerical evaluation of 6D and 5D singular integrals appearing in Volume Integral Equations. It is an extension of the Sauter-Schwab/Taylor-Duffy strategy for singular triangle-triangle interaction integrals to singular tetrahedron-tetrahedron and triangle-tetrahedron interaction integrals. The general advantages of these kind of quadrature strategy is that they allow the use of different kinds of kernel and basis functions. They also work on curvilinear domains. They are all based on relative coordinates tranformation and splitting the integration domain into subdomains for which quadrature rules can be constructed. We show how to build these tensor-product quadrature rules in 6D and 5D and further show how to improve their efficiency by using quadrature rules defined over 2D, 3D and 4D simplices. Compared to the existing approach, which computes the integral over the subdomains as a sequence of 1D integrations, significant speedup can be achieved. The accuracy and convergence properties of the method are demonstrated by numerical experiments for 5D and 6D singular integrals. Additionally, we applied the new quadrature approach to the triangle-triangle interaction integrals appearing in Surface Integral Equations.