A Smoothly Varying Quadrature Approach for 3D IgA-BEM Discretizations: Application to Stokes Flow Simulations

TL;DR

The paper tackles accurate and efficient IgA-BEM discretizations for 3D Stokes problems by introducing a smoothly varying quadrature strategy that unifies the handling of regular, singular, and nearly singular boundary integrals. It combines a distance-calibrated desingularization (DCT) with Chebyshev-based interpolation and support-wise direct spline integration, plus a Duffy desingularization for the singular subregion, enabling automatic node adaptation and reduced memory access. The approach is validated on Stokes kernel identities and exterior flow benchmarks, achieving superior convergence and substantial efficiency gains over prior methods. This framework advances high-fidelity boundary integral simulations on multipatch IgA geometries and lays groundwork for local refinement and Navier–Stokes extensions.

Abstract

We introduce a novel quadrature strategy for Isogeometric Analysis (IgA) boundary element discretizations, specifically tailored to collocation methods. Thanks to the dimensionality reduction and the natural handling of unbounded domains, boundary integral formulations are particularly appealing in the IgA framework. However, they require the evaluation of boundary integrals whose kernels exhibit singular or nearly singular behavior. Even when the kernel is not singular, its numerical evaluation becomes challenging whenever the integration region lies close to a collocation point. These integrals of polar and nearly singular functions represent the main computational difficulty of IgA-BEM and motivate the development of efficient and accurate quadrature rules. Unlike traditional methods that classify integrals as singular, nearly singular, or regular, our approach employs a desingularizing change of variables that smoothly adapts to the physical distance from singularities in the boundary integral kernels. The transformation intensifies near the polar point and progressively weakens when integrating over portions of the domain that are farther from it, ultimately leaving the integrand unchanged in the limit of a diametrically opposed region. This automatic calibration enhances accuracy and robustness by eliminating the traditional classification step, to which the approximation quality is often highly sensitive. Moreover, integration is performed directly over B-spline supports rather than over individual elements, reducing computational cost, particularly for higher-degree splines. The proposed method is validated through boundary element benchmarks for the three dimensional Stokes problem, where we achieve excellent convergence rates.

Figures (12)

• Figure 1:
• Figure 2: Functions belonging to the class \ref{['eq:alphaclass']}, for $\gamma = 1, \ldots, 10$. On the $x$ axis the normalized distance $\delta/\text{diam}(\Gamma)$. The higher $\gamma$, the faster the function grows for small values of $\delta$ and, symmetrically, the slower it grows for large values of $\delta$.
• Figure 3: Possible topological configurations for support subdivision in weakly singular integrals. $R_D$ is the box formed by the knots of the B-spline $\hat{B}$ enclosing the polar point $\hat{\pmb{y}}$. Depending on where such polar point is located, $R_D$ could be internal (a), on an edge (b) or in a corner (c) of $\mathop{\mathrm{supp}}\nolimits(\hat{B})$. The remaining part of the support is split into rectangular regions running anti-clockwise around $R_D$ as reported in the figures.
• Figure 4: Possible topological configurations for subdivision in triangles of region $R_D$, according to the position of the point $\hat{\pmb{y}} \in \mathop{\mathrm{supp}}\nolimits(\hat{B})$. In all the figures $R_D$ is the largest rectangular box drawn with solid lines and with $(a_1, b_1)$, $(a_2, b_2)$ as lower left and upper right corners, respectively. In (a) $\hat{\pmb{y}}$ is inside an element of the mesh in $\mathop{\mathrm{supp}}\nolimits(\hat{B})$ and $R_D$ is split into 4 triangles. In (b) $\hat{\pmb{y}}$ is on the left edge of $\mathop{\mathrm{supp}}\nolimits(\hat{B})$ but it is not a vertex of the mesh. $R_D$ is then split into 3 triangles. In (c) $\hat{\pmb{y}}$ is again on the left edge but this time it is also a vertex of the mesh inside $\mathop{\mathrm{supp}}\nolimits(\hat{B})$. Then $R_D$ is partitioned in 4 triangles. In (d) $\hat{\pmb{y}}$ is on a corner of $\mathop{\mathrm{supp}}\nolimits(\hat{B})$ and $R_D$ can only be halved in 2 triangles. In (e) $\hat{\pmb{y}}$ is on a meshline inside $\mathop{\mathrm{supp}}\nolimits(\hat{B})$ and $R_D$ is decomposed 6 triangles. Finally in (f) $\hat{\pmb{y}}$ is a mesh vertex inside $\mathop{\mathrm{supp}}\nolimits(\hat{B})$ and $R_D$ is split in 8 triangles. Rotations of the splittings illustrated in figures (b)--(e) cover the remaining cases.
• Figure 5: A general representation of the triangle on the right of $\hat{\pmb{y}}$ belonging to the collection $\mathcal{T}$ obtained splitting $R_D$.

Theorems & Definitions (2)

• Remark 2.1
• Remark 3.1