Dimension and basis construction for analysis-suitable G1 two-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.

Isogeometric analysis with C1 hierarchical functions on planar two-patch geometries

“…The second approach, on which we will focus, uses a particular class of regular C 0 multipatch parameterizations, called analysis-suitable G 1 multi-patch parameterization [11]. The class of analysis-suitable G 1 multi-patch geometries characterizes the regular C 0 multipatch parameterizations that allow the design of C 1 isogeometric spline spaces with optimal approximation properties, see [11,29], and includes for instance the subclass of bilinear multi-patch parameterizations [4,27,32]. An algorithm for the construction of analysissuitable G 1 parameterizations for complex multi-patch domains was presented in [29].…”

Refinable bi-quartics for design and analysis

“…However, nested refinement of the G k representation requires careful tracking of the original G k edges to ensure that a solution obtained at one level of refinement is not lost at the next finer one. Already when k = 1, the characterization of all possible degrees of freedom, even when joining just one pair of patches, is not easy [3]. For specific high-end surface constructions, where the degrees of freedom under refinement have been explicitly characterized, they are heterogeneously distributed [4].…”

Construction of analysis-suitableG1planar multi-patch parameterizations

“…This has motivated the use of a symbolic (exact arithmetic) solver. Our current and further research will focus on the analytic construction of a local and well conditioned basis for the spaces that fulfill the AS-G 1 constraints A G , extending the results of [7,16,17,20].…”

“…The second and third term in the objective function (17) are needed to obtain a non-singular linear system, and the influence of these terms in the minimization process are controlled by the positive weight λ β . The selection of the gluing data α ( ) , α ( ) , β ( ) and β ( ) guarantees that…”

“…[26,27,32]), the second strategy generates C 1 isogeometric spaces over multi-patch parameterizations, which are only C 0 at the patch interfaces (e.g. [3,4,7,16,17,20,24,25]). Similarly, [35] presents a construction of C k isogeometric spaces over patches having a polar layout, where they present details for k ≤ 2.…”