Construction of analysis-suitableG1planar multi-patch parameterizations

Abstract

Isogeometric analysis allows to define shape functions of global C 1 continuity (or of higher continuity) over multi-patch geometries. The construction of such C 1 -smooth isogeometric functions is a non-trivial task and requires particular multi-patch parameterizations, so-called analysis-suitable G 1 (in short, AS-G 1 ) parameterizations, to ensure that the resulting C 1 isogeometric spaces possess optimal approximation properties, cf. [7]. In this work, we show through examples that it is possible to construct AS-G 1 multi-patch parameterizations of planar domains, given their boundary. More precisely, given a generic multi-patch geometry, we generate an AS-G 1 multi-patch parameterization possessing the same boundary, the same vertices and the same first derivatives at the vertices, and which is as close as possible to this initial geometry. Our algorithm is based on a quadratic optimization problem with linear side constraints. Numerical tests also confirm that C 1 isogeometric spaces over AS-G 1 multi-patch parameterized domains converge optimally under mesh refinement, while for generic parameterizations the convergence order is severely reduced.

A new mixed approach to Kirchhoff–Love shells

“…We do not explicitly build in the coupling condition by constructing a basis of the space V h , but incorporate it implicitly, by replacing V h by Vh and adding (18) as additional constraint. Since Vh , Q h and Λ h are finite dimensional, the bilinear forms a, b, c and d can be represented as matrices A h , B h , C h and D h acting on vectors of real numbers x h , u h and λ h representing the elements in Vh , Q h and Λ h , respectively, with respect to the chosen basis.…”

“…Alternatively, dG (discontinuous Galerkin) techniques can be used patch-wise, see, e.g., [15], where a variationally consistent Nitsche formulation, which weakly enforces coupling and continuity constraints among patches, is derived. Another approach are analysis suitable C 1 multi-patch isogeometric spaces, see, e.g., [18].…”

Manifold-based isogeometric analysis basis functions with prescribed sharp features

“…The application and further development of these techniques is currently a very active area of research in isogeometric analysis, see e.g. [4][5][6][7][8][9][10][11][12][13][14][15]. Unfortunately, none of the techniques from computer-aided design seems to give optimal finite element convergence rates without further modifications, especially when applied to Kirchhoff-Love thin shells with arbitrary geometry.…”