Dispersion-optimized quadrature rules for isogeometric analysis: modified inner products, their dispersion properties, and optimally blended schemes

TL;DR

This paper develops and analyzes optimized quadrature rules for isogeometric analysis, enhancing accuracy and convergence rates, especially for wave propagation problems, by modifying inner products and blending schemes.

Contribution

It introduces new blended quadrature rules with improved dispersion properties and provides a generalized eigenvalue theorem to quantify approximation errors.

Findings

Blended quadrature rules improve convergence rates by two orders.

The methods enhance accuracy and robustness for wave propagation.

The approach is effective on uniform and non-uniform meshes across polynomial orders.

Abstract

This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non-uniform meshes as well as for different polynomial orders and continuity of the basis. The optimally-blended schemes improve the convergence rate of the method by two orders with respect to the fully-integrated Galerkin method. The proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems.