Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties, graded meshes & variational crimes

TL;DR

This paper addresses the Laplace eigenvalue problem on circular sectors, which exhibit corner singularities that challenge standard discretizations. It proposes a single-patch isogeometric analysis with graded h-refinement in a polar-like parameterization to capture singularities near the conical point, accepting a variational crime due to low regularity but achieving robustness. The results show optimal convergence rates for eigenvalues and eigenfunctions, with smooth splines offering better spectral approximation for the lower part of the spectrum, and graded meshes especially beneficial for singular modes; a hierarchical refinement strategy further mitigates anisotropy and reduces unnecessary degrees of freedom. The work extends known spectral approximation properties of smooth splines from rectangular domains to circular sectors and provides a practical framework for accurate eigenproblem simulations on singular geometries, with clear directions for future theoretical and methodological developments.

Abstract

The Laplace eigenvalue problem on circular sectors has eigenfunctions with corner singularities. Standard methods may produce suboptimal approximation results. To address this issue, a novel numerical algorithm that enhances standard isogeometric analysis is proposed in this paper by using a single-patch graded mesh refinement scheme. Numerical tests demonstrate optimal convergence rates for both the eigenvalues and eigenfunctions. Furthermore, the results show that smooth splines possess a superior approximation constant compared to their $C^0$-continuous counterparts for the lower part of the Laplace spectrum. This is an extension of previous findings about excellent spectral approximation properties of smooth splines on rectangular domains to circular sectors. In addition, graded meshes prove to be particularly advantageous for an accurate approximation of a limited number of eigenvalues. The novel algorithm applied here has a drawback in the singularity of the isogeometric parameterization. It results in some basis functions not belonging to the solution space of the corresponding weak problem, which is considered a variational crime. However, the approach proves to be robust. Finally, a hierarchical mesh structure is presented to avoid anisotropic elements, omit redundant degrees of freedom and keep the number of basis functions contributing to the variational crime constant, independent of the mesh size. Numerical results validate the effectiveness of hierarchical mesh grading for the simulation of eigenfunctions with and without corner singularities.

Figures (19)

• Figure 1: Circular sectors $\Omega$ with angle $\omega$ and the corresponding boundaries. (a) $\omega = \frac{3}{2} \pi$. (b) $\omega=2\pi$.
• Figure 2: Illustration of $12$ Laplace eigenfunctions $u_{\nu_k,m}$ of the unit disk with crack. Columns from left to right: $\nu_k =0, \nu_k=\frac{1}{2}, \nu_k =1, \nu_k=\frac{3}{2}$. Rows from top to bottom: $m=1,m=2,m=3$.
• Figure 3: Visualization of the Bessel functions $J_{\nu_k}$ to the orders $\nu_k \in \{0,0.5,1, \dots, 5.5,6\}$ in the interval $[0,10]$.
• Figure 4: Examples of cubic B-spline functions with distinct regularity properties, defined by different knot vectors $\Xi$. (a): $\Xi=\{0,0,0,0,0.25,0.5,0.75,1,1,1,1\}$. (b): $\Xi=\{0,0,0,0,0.25,0.25,0.25,0.5,0.5,0.75,1,1,1,1\}$.
• Figure 5: (a) Deforming a unit square to a unit disk with crack. (b) Illustration of the control points $\boldsymbol c_{\boldsymbol i}$ for $\boldsymbol i \in \boldsymbol I_0 = \{\boldsymbol i = (i_1,i_2): 1 \leq i_1 \leq 2, 1\leq i_2 \leq 9\}$.

Theorems & Definitions (14)

• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof
• Remark 2
• Remark 3
• Example 5.1: Eigenfunction of Type (A)
• Example 5.2: Eigenfunction of Type (B)
• Example 5.3