The design of globally
C
s
-smooth (
s
≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current ...and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des.
52–53
:75–89,
2017
, Kapl et al. Comput. Aided Geom. Des.
69
:55–75,
2019
and Kapl and Vitrih J. Comput. Appl. Math.
335
:289–311,
2018
, Kapl and Vitrih J. Comput. Appl. Math.
358
:385–404,
2019
and Kapl and Vitrih Comput. Methods Appl. Mech. Engrg.
360
:112684,
2020
for the construction of
C
1
-smooth and
C
2
-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of
C
s
-smooth isogeometric multi-patch spline spaces of degree
p
, inner regularity
r
and of a smoothness
s
≥ 1, with
p
≥ 2
s
+ 1 and
s
≤
r
≤
p
−
s
− 1. More precisely, we study for
s
≥ 1 the space of
C
s
-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular
C
s
-smooth subspace of the entire
C
s
-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this
C
s
-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the
C
s
-smooth spline functions to perform
L
2
approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed
C
s
-smooth subspace.
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.
We study the space of C1 isogeometric spline functions defined on trilinearly parameterized multi-patch volumes. Amongst others, we present a general framework for the design of the C1 isogeometric ...spline space and of an associated basis, which is based on the two-patch construction 7, and which works uniformly for any possible multi-patch configuration. The presented method is demonstrated in more detail on the basis of a particular subclass of trilinear multi-patch volumes, namely for the class of trilinearly parameterized multi-patch volumes with exactly one inner edge. For this specific subclass of trivariate multi-patch parameterizations, we further numerically compute the dimension of the resulting C1 isogeometric spline space and use the constructed C1 isogeometric basis functions to numerically explore the approximation properties of the C1 spline space by performing L2 approximation.
We present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a ...globally C2-smooth isogeometric spline space which is used as discretization space. The generated C2-smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and numerical examples indicate that they are well-conditioned. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are obtained by computing the null space of a small system of linear equations. Several examples demonstrate the potential of our approach for solving the triharmonic equation.
A particular class of planar two-patch geometries, called bilinear-like G2 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 G2 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 C2-smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C2-smoothness of an isogeometric function and the G2-smoothness of its graph surface (cf. Groisser and Peters (2015) and Kapl et al. (2015). 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 G2 two-patch parameterizations. Numerical results obtained by performing L2-approximation and solving Poisson’s equation indicate that already the subspace possesses optimal approximation properties.
•Introduction of bilinear-like G2 two-patch parameterizations .•Investigation of the C2 smooth isogeometric space over this specific geometry.•Computation of dimension of the space.•Construction of a simple explicitly given and well-conditioned basis of the space.•Numerical experiments show optimal rates of convergence.
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 study the linear space of Cs-smooth isogeometric functions defined on a multi-patch domain Ω⊂R2. We show that the construction of these functions is closely related to the concept of geometric ...continuity of surfaces, which has originated in geometric design. More precisely, the Cs-smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order (Gs-smoothness) of their graph surfaces. This motivates us to call them Cs-smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of C1-smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing L2 approximation and for solving Poisson’s equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.
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