Spectral approximation properties of isogeometric analysis with variable continuity

TL;DR

This paper investigates how reducing local continuity in isogeometric analysis affects spectral approximation, revealing trade-offs between computational savings and approximation quality, and proposes optimized quadrature rules for improved accuracy.

Contribution

It introduces a detailed analysis of spectral properties with local continuity reduction and demonstrates how non-standard quadrature rules enhance eigenvalue accuracy in refined isogeometric analysis.

Findings

Continuity reduction leads to artefacts like stopping bands and outliers in spectra.

Optimal quadrature rules significantly reduce eigenvalue errors.

Refined isogeometric analysis maintains convergence rates comparable to maximum continuity basis.

Abstract

We study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis. Such continuity reduction results in a reduction in the interconnection between the degrees of freedom of the mesh, which allows for large savings in computational requirements during the solution of the resulting linear system. The continuity reduction results in extra degrees of freedom that modify the approximation properties of the method. The convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis. We show how the breaks in continuity and inhomogeneity of the basis lead to artefacts in the frequency spectra, such as stopping bands and outliers, and present a unified description of these effects in finite element method, isogeometric analysis, and refined isogeometric analysis. Accuracy of the refined isogeometric analysis approximations can be improved by using non-standard quadrature rules. In particular, optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence similar to classical isogeometric analysis.