Dispersion-minimized mass for isogeometric analysis

TL;DR

This paper introduces a dispersion-minimized mass for isogeometric analysis, significantly reducing eigenvalue errors and enhancing robustness in structural vibration simulations, especially for higher-order elements.

Contribution

The paper develops a novel dispersion-minimized mass formulation for isogeometric analysis, extending optimal quadrature techniques to arbitrary polynomial orders and demonstrating improved eigenvalue accuracy.

Findings

Eigenvalue error reduced to superconvergence order of 2p+2.

Dispersion-minimized mass improves robustness of isogeometric analysis.

Validation through numerical examples confirms theoretical error estimates.

Abstract

We introduce the dispersion-minimized mass for isogeometric analysis to approximate the structural vibration which we model as a second-order differential eigenvalue problem. The dispersion-minimized mass reduces the eigenvalue error significantly, from the optimum order of $2p$ to the superconvergence order of $2p+2$ for the $p$-th order isogeometric elements with maximum continuity, which in return leads to more robust of the isogeomectric analysis. We first establish the dispersion error for arbitrary polynomial order isogeometric elements. We derive the dispersion-minimized mass in one dimension by solving a $p$-dimensional local matrix problem for the $p$-th order approximation and then extend it to multiple dimensions on tensor-product grids. We show that the dispersion-minimized mass can also be obtained by approximating the mass matrix using optimally blended quadratures. We generalize the results of optimally blended quadratures from polynomial orders $p=1,\cdots, 7$ that were studied in \cite{calo2017dispersion} to arbitrary polynomial order isogeometric approximations. Various numerical examples validate the eigenvalue and eigenfunction error estimates we derive.