Dual distance transformations for evaluating domain integrals in the boundary integral equation of transient heat conduction

Accurate computation of domain integrals is indeed crucial for effective boundary element analysis of transient heat conduction problems. The progressive reduction of the time step can cause the time-dependent kernel within the integral to oscillate rapidly and exhibit near-singularity. Additionally, integration over the sub-triangular element with a large angle and large side length ratio will result in the circumferential near-singularity. The traditional distance transformation technique, while effective, is limited to the nearly singular integrals and addresses the near-singularity in only one direction. To enhance the accuracy of the domain integral computations, a dual distance transformations method is introduced. The (α, β) coordinate transformation is initially employed to separate the near-singularity. An appropriate distance transformation is then proposed to eliminate the near-singularity associated with the variable α. Furthermore, the circumferential near-singularity induced by the sub-triangle with poor shape is analyzed. Considering the integral form related to the variable β, another new distance transformation is finally established to mitigate the effect of the element shape. With the presented dual distance transformations method, the domain integrals can be accurately computed for different time steps and source point locations. The accuracy and efficiency of the proposed method are effectively proven through several numerical examples.