Reduced integration at superconvergent points in isogeometric analysis

- We propose a new reduced quadrature rule for IGA based on variational collocation.
- Integration is performed on the weighted residual form and not on the variational form.
- The quadrature points are superconvergent points.
- Two quadrature points per parametric direction are used for any (odd) degree.
- Results are very close to those obtained with Galerkin and accurate quadrature.

We propose a new reduced integration rule for isogeometric analysis (IGA) based on the concept of variational collocation. It has been recently shown that, if a discrete space is constructed by smooth and pointwise non-negative basis functions, there exists a set of points - named Cauchy-Galerkin (CG) points - such that collocation performed at these points can reproduce the Galerkin solution of various boundary value problems exactly. Since CG points are not known a-priori, estimates are necessary in practice and can be found based on superconvergence theory. In this contribution, we explore the use of estimated CG points (i.e. superconvergent points) as numerical quadrature points to obtain an efficient and stable reduced quadrature rule in IGA. We use the weighted residual formulation as basis for our new quadrature rule, so that the proposed approach can be considered intermediate between the standard (accurately integrated) Galerkin variational formulation and the direct evaluation of the strong form in collocation approaches. The performance of the method is demonstrated by several examples. For odd degrees of discretization, we obtain spatial convergence rates and accuracy very close to those of accurately integrated standard Galerkin with a quadrature rule of two points per parametric direction independently of the degree.