Dispersion-minimizing quadrature rules for C1 quadratic isogeometric analysis

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 [8], 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.

Dispersion optimized quadratures for isogeometric analysis

“…The cost of the assembly of the tensor product matrix does not increase much even when we evaluate two quadrature rules instead of one when using the optimally-blended rules. Moreover, nonstandard quadrature rules (with a minimal number of quadrature points), which are equivalent to the optimally-blended quadrature rules in the sense of producing the same mass and stiffness matrices, are developed in [41] and they reduce computational cost; see also the p = 1 p = 2 p = 3 N λ reduced rules in [42].…”

Spectral approximation properties of isogeometric analysis with variable continuity

“…In particular, reduced integration may minimize the numerical dispersion of the finite element and isogeometric analysis approximations. Higher accuracy can be obtained using optimal blending schemes (blending the Gauss and Lobatto quadratures [2,7,33]) or dispersion-minimizing quadrature rules, which can be constructed to produce equivalent results [4,20]. This dispersion-minimizing integration improves the convergence rate of the resulting eigenvalues by two orders when compared against the fully-integrated finite, spectral, or isogeometric elements, while preserving the optimal convergence of the eigenfunctions.…”

Higher order stable generalized finite element method for the elliptic eigenvalue and source problems with an interface in 1D

“…Traditionally, differential eigenvalue problems are solved by using standard FEMs (c.f., [8,11,16,17,[36][37][38]42]), isogeometric analysis (c.f., [19,29,30]), discontinuous Galerkin methods (c.f., [4,26,27]), etc. We are not going to expand the literature review here and only mention the most-recent quadrature rule blending techniques developed in [3] for FEMs and in [13,15,20,21,39] for isogeometric analysis. The optimally-blended rules developed in these papers lead to two extra orders of superconvergence for the eigenvalues while maintaining the optimal convergence rates for the eigenfunctions.…”