A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes

TL;DR

This paper proposes a boundary penalization method to eliminate spectral outliers in isogeometric analysis, enhancing the accuracy of eigenvalue approximations for Laplacian problems without significantly increasing computational cost.

Contribution

The paper introduces a boundary penalization technique that removes high-frequency spectral outliers in IGA, improving eigenvalue accuracy while maintaining the original function spaces.

Findings

Effective removal of spectral outliers in IGA eigenvalue problems.

Optimal convergence rates achieved for eigenvalues and eigenfunctions.

Analytical eigenpairs obtained for the penalized matrix problems.

Abstract

We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The technique removes the outliers appearing in the high-frequency region of the approximate spectrum when using the $C^{p-1}, p$-th ($p\ge3$) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multiple dimensional problems. To remove the outliers, we penalize the product of the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis to the Neumann eigenvalue problem (for $p\ge2$). The boundary penalization does not change the test and solution spaces. In the limiting case when the penalty goes to infinity, we perform the dispersion analysis of $C^2$ cubic elements for Dirichlet eigenvalue problem and $C^1$ quadratic elements for Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator.