Dispersion-minimized mass for isogeometric analysis

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 pth order isogeometric elements with maximum continuity, which in return leads to a more accurate method. We first establish the dispersion error, where the leading error term is explicitly written in terms of the stiffness and mass entries, 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 pth 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 lower order quadrature-blending results to arbitrary polynomial order isogeometric approximations as well as to arbitrary quadrature rules. Various numerical examples validate the eigenvalue and eigenfunction error estimates.