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 $p$-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.

28 pages

References in corpus (3)

• Dispersion-optimized quadrature rules for isogeometric analysis: modified inner products, their dispersion properties, and optimally blended schemes
• Dispersion-minimizing quadrature rules for $C^1$ quadratic isogeometric analysis
• Spectral approximation properties of isogeometric analysis with variable continuity

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• Optimal spectral approximation of $2n$-order differential operators by mixed isogeometric analysis
• SoftFEM: revisiting the spectral finite element approximation of second-order elliptic operators
• Isogeometric spectral approximation for elliptic differential operators