We continue the study initiated in Hughes et al. (2010) in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. These rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points. We extend the algorithm presented in Hughes et al. (2010) with an improved starting guess, which combined with arbitrary precision arithmetic, results in the practical computation of quadrature rules for univariate non-uniform splines up to any precision. Explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors. We study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two- and three-dimensional diffusion–reaction problems using tensor product and hierarchically refined splines, and prove a theorem rigorously establishing the stability and accuracy of the reduced rules.
|Original language||English (US)|
|Number of pages||39|
|Journal||Computer Methods in Applied Mechanics and Engineering|
|State||Published - Apr 1 2017|
Bibliographical noteFunding Information:
T.J.R Hughes was supported by the Army Research Office under contract number W911NF-13-1-0220. Francesco Calabr? was partially supported by INdAM, through GNCS research projects. D. Schillinger was supported by the National Science Foundation through grant ACI-1565997. This support is gratefully acknowledged.
- Generalized Gaussian quadrature rules
- Isogeometric analysis
- Optimal quadrature rules
- Reduced quadrature rules