Computing singular and near-singular integrals over curved boundary elements: The strongly singular case

TL;DR

The accuracy and robustness of the algorithm for computing strongly singular and near-singular surface integrals over curved triangular patches, based on singularity subtraction, the continuation approach, and transplanted Gauss quadrature are demonstrated.

Abstract

We present algorithms for computing strongly singular and near-singular surface integrals over curved triangular patches, based on singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy and robustness of our method for quadratic basis functions and quadratic triangles by integrating it into a boundary element code and solving several scattering problems in 3D. We also give numerical evidence that the utilization of curved boundary elements enhances computational efficiency compared to conventional planar elements.

Figures (11)

• Figure 1: An arbitrary planar triangle $\mathcal{T}$ is obtained from the planar reference triangle $\widehat{T}$ via the linear map $F$ defined in \ref{['eq:linear-map']}. The $\boldsymbol{\hat{a}}_j$'s verify $\lambda_i(\boldsymbol{\hat{a}}_j)=\delta_{ij}$, while the $\boldsymbol{a}_j$'s verify $\boldsymbol{a}_j=F(\boldsymbol{\hat{a}}_j)$.
• Figure 1: The triangle $\widehat{T}-\boldsymbol{\hat{x}}_0$ above is the reference triangle $\widehat{T}$ of \ref{['fig:planar-tri']} shifted by $\boldsymbol{\hat{x}}_0=(\hat{x}_0,\hat{y}_0)$. The (signed) distances $\hat{s}_j$ to the edges are given by $\hat{s}_1=\hat{y}_0$, $\hat{s}_2=\sqrt{2}/2(1-\hat{x}_0-\hat{y}_0)$, and $\hat{s}_3=\hat{x}_0$.
• Figure 1: We demonstrate our algorithms on the quadratic triangle $\mathcal{T}$ displayed on the left, which is defined by \ref{['eq:quad-tri']}--\ref{['eq:quad-tri2']} with parameters $a=0.6$, $b=0.7$, and $c=0.5$. For 4D integrals, we integrate over the two quadratic triangles $\mathcal{T}$ and $\mathcal{T}'=\mathcal{T}+5\times10^{-2}(1,1,0)$ shown on the right.
• Figure 2: A quadratic triangle $\mathcal{T}$ is obtained from $\widehat{T}$ via the quadratic map $F$\ref{['eq:quad-map']}. Note that $\boldsymbol{\hat{a}}_4$, $\boldsymbol{\hat{a}}_5$, and $\boldsymbol{\hat{a}}_6$ are the midpoints. Again, the $\boldsymbol{\hat{a}}_j$'s verify $\varphi_i(\boldsymbol{\hat{a}}_j)=\delta_{ij}$, while the $\boldsymbol{a}_j$'s verify $\boldsymbol{a}_j=F(\boldsymbol{\hat{a}}_j)$.
• Figure 2: When the singularity is exactly on the triangle, cancellation with the normal leads to a weakly singular kernel. Regularization is not needed to obtain convergence at a rate $\mathcal{O}(N^{-0.5})$, while $T_{-1}$ regularization accelerates the convergence to linear (left). The nearly singular case is harder---the integrand is numerically strongly singular and regularization is needed to converge (right).

Theorems & Definitions (2)

• Theorem 3.1: Continuation for strong singularities
• Proof 1