Outlier removal for isogeometric spectral approximation with the optimally-blended quadratures

Abstract

It is well-known that outliers appear in the high-frequency region in the approximate spectrum of isogeometric analysis of the second-order elliptic operator. Recently, the outliers have been eliminated by a boundary penalty technique. The essential idea is to impose extra conditions arising from the differential equation at the domain boundary. In this paper, we extend the idea to remove outliers in the superconvergent approximate spectrum of isogeometric analysis with optimally-blended quadrature rules. We show numerically that the eigenvalue errors are of superconvergence rate h 2p+2 and the overall spectrum is outlier-free. The condition number and stiffness of the resulting algebraic system are reduced significantly. Various numerical examples demonstrate the performance of the proposed method.

A variational approach based on perturbed eigenvalue analysis for improving spectral properties of isogeometric multipatch discretizations

“…The penalty approach of [13], using large values of penalty parameters, serves purely as an indicator for the outlier eigenvalues / frequencies such that these are identified eas- ily, as verified in subsequent sections in this paper. Another approach is the optimally-blended quadratures [14] that can suppress the boundary outlier [15,16,17] as well as the interior outlier frequencies [5].…”

“…In the next step, we will address the systematic choice of the open parameters α and β in (17). To approach this aspect, we consider the perturbed eigenvalue problem (17) of a fixed bar (unit length, unit axial stiffness, unit mass), discretized by a univariate multipatch discretization with quadratic C 1 B-splines (p = 2) and sufficient regularity, i.e.…”