Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes

Abstract

In the context of isogeometric analysis, globally C 1 isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [26]. There, the construction of a specific C 1 isogeometric spline space for the class of so-called analysis-suitable G 1 multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of C 1 spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable G 1 by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples.

Isogeometric Analysis on V-reps: First results

“…The only viable approach has been, until now, the use of spline basis on unstructured hexahedral (or tetrahedral) meshes, which has been object of several contributions [38,59,61,63]. In this context, the presence of extraordinary points and edges make the construction of regular B-spline functions preserving accuracy extremely challenging (see [62,33] and the references therein).…”

Dimension and basis construction for C2-smooth isogeometric spline spaces over bilinear-like GKapl1, Vitrih2 2018Journal of Computational and Applied MathematicsSelf Cite90470View full textAdd to dashboardCite

“…The proposed construction will work uniformly for all possible multi-patch configurations and is much simpler as for the entire C s -smooth space V s . Thereby, the design of the subspace W s will be based on the results of the two-patch case, and is motivated by the methods [27,28] and [32,33], where similar subspaces have been generated for a global smoothness of s = 1 and s = 2, respectively. There, it has been numerically shown that the corresponding subspaces possess as the entire C s -smooth space V s optimal approximation properties.…”

“…To generate the vertex subspaces W s Ξ (i) , we will distinguish between different types of vertices Ξ (i) , namely between inner and boundary vertices, and in the latter case also between boundary vertices of patch valency v i ≥ 3, v i = 2 and v i = 1. We will follow a similar approach as used in [27,28] and [32,33] for the construction of C 1 and C 2 -smooth isogeometric spline functions in the vicinity of the vertex Ξ (i) .…”

Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations

“…[11,12], and general quadrilateral meshes of arbitrary topology [9,10,48]. The recent survey article [29] provides more details about the single methods of the three approaches and also includes further possible constructions.…”