Exploring Matrix Generation Strategies in Isogeometric Analysis

Abstract

Abstract. An important step in simulation via isogeometric analysis (IGA) is the assembly step, where the coefficients of the final linear system are generated. Typically, these coefficients are integrals of products of shape functions and their derivatives. Similarly to the finite element analysis (FEA), the standard choice for integral evaluation in IGA is Gaussian quadrature. Recent developments propose different quadrature rules, that reduce the number of quadrature points and weights used. We experiment with the existing methods for matrix generation. Furthermore we propose a new, quadrature-free approach, based on interpolation of the geometry factor and fast look-up operations for values of B-spline integrals. Our method builds upon the observation that exact integration is not required to achieve the optimal convergence rate of the solution. In particular, it suffices to generate the linear system within the order of accuracy matching the approximation order of the discretization space. We demonstrate that the best strategy is one that follows the above principle, resulting in expected accuracy and improved computational time.

Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

“…By Corollary 1, there are exactly two nodes inside (x k−1 , x k ). Due to Lemma 2.3, only the roots of the quadratic factor in (23) contribute to the computation of the nodes and hence solving Q k (α k ) = 0 with coefficients from (27) gives α k and β k . Combining these with (15) results in (29).…”

“…In the context of isogeometric analysis, quadrature rules for splines are important tools [1,21,22,24,28,34] because they are cheap and elegant alternatives to symbolic integration [20]. Recently, alternative methods of building mass and stiffness matrices have been proposed [27,28]. They exploit the observation that, under certain conditions, the optimal convergence rate of the linear system can be achieved despite the fact that the integration rule is not exact.…”

Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation

“…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]. While this approach provides a significant speedup when using larger polynomial degrees, it was found to provide major advantages with respect to using Gauss quadrature in the low degree case.…”

“…The additional difficulties associated with IGA are caused by the increased degree and the larger supports of the functions that occur in the integrals defining the matrix elements. Recently we introduced an interpolation-based approach that approximately transforms the integrands into piecewise polynomials and uses look-up tables to evaluate their integrals [15,16]. The present paper relies on this earlier work and proposes to use tensor methods to accelerate the assembly process further.…”

The Choice of Quadrature in Nurbs-Based Isogeometric Analysis

“…A recent paper [10] has highlighted that in order to calculate the integrals arising in Isogeometric analysis, one can try to substitute the integrand function with a proper combination of function with the property that the integral of these new functions are known. This general approach is similar to the one that uses modified moments in order to calculate the quadrature rules.…”