An (α, β, γ) distance transformation for the numerical computation of 3D domain integrals directly in transient boundary element analysis

In this paper, an (α, β, γ) distance transformation is introduced for direct computation of 3D domain integrals, which is essential when employing the time-dependent boundary element method for the transient heat conduction problems. The gradual reduction of the time step to zero in the time-dependent integral kernel may result in near-singularity. In this situation, the direct application of Gaussian quadrature is ineffective for accurately calculating the domain integrals. To address this issue, a novel distance transformation incorporating the (α, β, γ) coordinate transformation is presented. The (α, β, γ) coordinate transformation is initially employed to enhance the smoothness of the integral kernels. Subsequently, a novel distance transformation is developed, in which the time step replaces the shortest distance in the traditional distance transformation, further smoothing the integral kernels. Consequently, the near-singularity in the integrand is eliminated by the Jacobian generated through the new transformation, thereby achieving higher calculation accuracy, even with very the small time step. Numerical examples under various situations are presented, illustrating the advantages of the new method in comparison to other existing methods.