We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally ...C4-smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the L2-norm as well as to equivalents of the Hs-seminorms for 1≤s≤4.
In the studied case of spline degree p=9, the numerical results indicate in case of the Greville points a convergence of order O(hp−3) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order O(hp−2) for all (semi)norms except for the equivalent of the H4-seminorm, where the order O(hp−3) is anyway optimal.
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.
In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in Anitescu et al. (2015) and ...the variational collocation method presented in Gomez and De Lorenzis (2016). The focus is on smoothest B-splines/NURBS approximations, i.e, having global Cp−1 continuity for polynomial degree p. Within the framework of Gomez and De Lorenzis (2016), we select as collocation points a subset of those considered in Anitescu et al. (2015), which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behavior with respect to Gomez and De Lorenzis (2016), achieving optimal L2-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in Anitescu et al. (2015), where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.
•We propose an optimally convergent isogeometric collocation scheme (odd degrees only).•The proposed collocation points are a subset of the Galerkin superconvergent ones.•Robustness and comparison with other collocation schemes assessed numerically.•Convergence proof not yet available and numerical evidence not yet conclusive.
We present an isogeometric framework based on collocation to construct a C2-smooth approximation of the solution of the Poisson’s equation over planar bilinearly parameterized multi-patch domains. ...The construction of the used globally C2-smooth discretization space for the partial differential equation is simple and works uniformly for all possible multi-patch configurations. The basis of the C2-smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case 1, which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. 2–4) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.
•Isogeometric collocation method for solving Poisson’s equation over planar multi-patch domains.•As discretization space a globally C2-smooth isogeometric spline space is developed and used.•Two different choices of collocation points are numerically investigated.•Numerical experiments demonstrate the potential of the collocation method.
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