Abstract
The design of globally Cs-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 C1-smooth and C2-smooth isogeometric spline spaces over particular planar multi-patch geometries to the case of Cs-smooth isogeometric multi-patch spline spaces of degree p, inner regularity r and of a smoothness s ≥ 1, with p ≥ 2s + 1 and s ≤ r ≤ p − s − 1. More precisely, we study for s ≥ 1 the space of Cs-smooth isogeometric spline functions defined on planar, bilinearly parameterized multi-patch domains, and generate a particular Cs-smooth subspace of the entire Cs-smooth isogeometric multi-patch spline space. We further present the construction of a basis for this Cs-smooth subspace, which consists of simple and locally supported functions. Moreover, we use the Cs-smooth spline functions to perform L2 approximation on bilinearly parameterized multi-patch domains, where the obtained numerical results indicate an optimal approximation power of the constructed Cs-smooth subspace.
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Acknowledgements
The authors wish to thank the anonymous reviewers for their comments that helped to improve the paper.
Funding
Open access funding provided by Carinthia University of Applied Sciences (CUAS). M. Kapl has been partially supported by the Austrian Science Fund (FWF) through the project P 33023-N. V. Vitrih has been partially supported by the research program P1-0404 and research projects J1-9186, J1-1715 from ARRS, Republic of Slovenia. This support is gratefully acknowledged.
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Appendix
Appendix
We present concrete examples of functions which have been introduced in Section 3.
Example 1
We give for the cases ℓ ∈{1, 2, 3} explicit expressions for the functions Ξℓ, ηℓ and 𝜃ℓ, which have been firstly considered in (12) and (15). Based on (12), (19) and Lemma 1, we get for ℓ = 1
for ℓ = 2
and for ℓ = 3
Example 2
We consider particular examples of the functions Aσ;ℓ and of the sets \( \mathcal {I}_{\boldsymbol {\sigma };\ell }\) introduced in (21) and (22), respectively. Let |σ|≤ 3. Then, the sets \( \mathcal {I}_{\boldsymbol {\sigma };3}\) as well as the functions Aσ;3 are equal to
and
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Kapl, M., Vitrih, V. Cs-smooth isogeometric spline spaces over planar bilinear multi-patch parameterizations. Adv Comput Math 47, 47 (2021). https://doi.org/10.1007/s10444-021-09868-5
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DOI: https://doi.org/10.1007/s10444-021-09868-5