Dimension and basis construction for C2-smooth isogeometric spline spaces over bilinear-like G

Abstract

A particular class of planar two-patch geometries, called bilinear-like G 2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G 2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations.We investigate the space of C 2 -smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. [12,20]). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G 2 two-patch parameterizations. Numerical results obtained by performing L 2 -approximation indicate that already the subspace possesses optimal approximation properties.

Isogeometric collocation on planar multi-patch domains

“…A further possible topic could be the extension of our approach to more general multi-patch domains such as bilinear-like parameterizations, cf. [26], or to multi-patch structured shells and multi-patch volumes. j1,j2 , i = 1, 2, 3, of the bicubic three-patch spline geometry shown in Fig.…”

“…We assume that p ≥ 5, 2 ≤ r ≤ p−3, and that the number of inner knots satisfies k ≥ 9−p p−r−2 . These assumptions are necessary to ensure that the constructed C 2 -smooth spline spaces in Section 2.3 will be h-refineable and well-defined, see also [26,27]. Since the geometry mappings F (i) , i ∈ I Ω , are bilinearly parameterized, we trivially have that…”

“…The space V has been studied in different configurations in [24][25][26]. There, it has been observed that the space V possesses a complex structure and its dimension depends on the initial geometry.…”

“…, j 2 = 0, 1, 2, 3, and g , respectively, across the interface Γ (i ) , cf. [26,Section 7], the first set of interpolation conditions in (14) uniquely determine all coefficients a Γ (i ) j 1 ,j 2 . Moreover, as the functions g Γ (i ) i ensure C 2 -smoothness across the interfaces Γ (i ) , we also obtain…”

“…Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.While in case of a one-patch domain, the higher continuity of the functions can be easily guaranteed by using spline functions with the desired smoothness within the patch, the construction of C s -smooth (s ≥ 1) isogeometric spline spaces over multi-patch domains is challenging, and is the task of current research, see e.g. [6-9, 20-23, 28, 29, 33, 34, 40, 41] for s = 1 and [24][25][26][27]39] for s = 2. The design of C s -smooth multi-patch spline spaces is linked to the concept of geometric continuity of multi-patch surfaces, cf.…”