Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis

Abstract

We explore the use of various element-based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis. The rules studied encompass tensor-product Gauss and Gauss-Lobatto rules, and certain so-called monomial rules that do no possess a tensor-product structure. The objective of the study is to determine quadrature strategies, which enjoy the same accuracy and stability behavior as full Gauss quadrature, but with significantly fewer quadrature points. Several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard C 0 -continuous finite elements of the same order.

A parameter-free variational coupling approach for trimmed isogeometric thin shells

“…The accuracy of a discretized elastostatic boundary value problem predominantly depends on the accuracy of the lowest eigenvalues, which can be shown by a spectral representation of the solution coefficients [76]. Therefore, this observation supports the high fidelity results and excellent numerical properties of the non-symmetric Nitsche method that we have seen in the previous elastostatic benchmarks.…”

“…The plate is discretized by two B-spline patches, which consists of 8 × 8 and 16 × 16 elements. The two patches are coupled weakly with the non-symmetric Nitsche method, where we add the stabilization terms (76). Figure 32 plots the bending moment computed with the non-symmetric Nitsche method at different values C=C S =C N , which determines the level of stabilization via relations (77).…”

Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation

“…Special quadrature rules for spline functions [1,3,8,17] have been derived, but these rules are partially difficult to compute and provide only modest improvements. Reduced Bézier element quadrature rules for low degree discretizations have been investigated recently [19]. Another approach, which is based on spline projection and exact integration via look-up tables, has been presented in [15,16].…”

Dimensionally adaptive hp-finite element simulation and inversion of 2D magnetotelluric measurements

“…These methods have been extensively employed in the last years in applied mathematics to solve a variety of engineering problems. For example, the isogeometric analysis (IGA) [8,9] has recently experimented a huge explosion and it has been widely applied to the engineering industry, as well as the more recent Discontinuous Petrov-Galerkin (DPG) method initially proposed by Demkowicz and Gopalakrishnan [10,11], or the self-adaptive hp-Finite Element Method (FEM) [12,13] (where h stands for the element size and p for the polynomial order of approximation associated to each element). The latter one has been recently employed, for instance, to model the bone conduction of sound in the human head [14], or to simulate bend, step, and magic-T electromagnetic waveguide structures [15].…”