# Dispersion-minimizing quadrature rules for $C^1$ quadratic isogeometric analysis

## TL;DR

This paper introduces optimized two-point quadrature rules for $C^1$ quadratic isogeometric analysis that significantly reduce dispersion errors and computational costs in wave propagation and structural vibration simulations.

## Contribution

It presents novel dispersion-minimizing quadrature rules requiring only two points per element, simplifying integration and improving accuracy in isogeometric analysis.

## Key findings

- Two-point quadrature rules reduce dispersion error effectively.
- Proposed rules outperform standard quadrature in numerical tests.
- Cost savings due to fewer quadrature points.

## Abstract

We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation. The rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error [1], and they are equivalent to the optimized blending rules we recently described. Our approach further simplifies the numerical integration: instead of blending two three-point standard quadrature rules, we construct directly a single two-point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions. Also, we present a 2.5-point rule for both uniform and non-uniform meshes with arbitrary boundary conditions. Consequently, we reduce the computational cost by using the proposed quadrature rules. Various numerical examples demonstrate the performance of these quadrature rules.