An isogeometric C1 subspace on unstructured multi-patch planar domains

Kapl, Mario; Sangalli, Giancarlo; Takacs, Thomas

doi:10.1016/j.cagd.2019.01.002

Cited by 58 publications

(100 citation statements)

References 35 publications

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“…(b) Error in H 2 semi-norm generalize the construction to the multi-patch domain setting of [31], but this will require a different strategy with respect to the approach presented in this work since the basis functions of a single level may be locally linearly dependent. The geometry in Fig.2(a) for the examples in Section 5 is generated by following the algorithm in [29].…”

Section: Discussionmentioning

confidence: 99%

“…We show that the basis functions of the considered C 1 space on analysis-suitable G 1 two-patch parameterizations, which is a subspace of the space [28] inspired by [31], satisfy the required properties given in [18], and in particular that the basis functions are locally linearly independent (see Section 3.1 for details). Note that in case of a multi-patch domain, the general framework for the construction of hierarchical splines [18] cannot be used anymore, since the appropriate C 1 basis functions [31] can be locally linearly dependent. Therefore, the development of another approach as [18] would be needed for the multi-patch case, which is beyond the scope of this paper.…”

Section: Introductionmentioning

confidence: 94%

“…In this work, we extend for the case of two-patch domains the second approach from above to the construction of hierarchical C 1 isogeometric spaces on analysis-suitable G 1 geometries, using the abstract framework for the definition of hierarchical splines detailed in [18]. We show that the basis functions of the considered C 1 space on analysis-suitable G 1 two-patch parameterizations, which is a subspace of the space [28] inspired by [31], satisfy the required properties given in [18], and in particular that the basis functions are locally linearly independent (see Section 3.1 for details). Note that in case of a multi-patch domain, the general framework for the construction of hierarchical splines [18] cannot be used anymore, since the appropriate C 1 basis functions [31] can be locally linearly dependent.…”

Section: Introductionmentioning

confidence: 99%

“…While the methods in [4,27,32] are mainly restricted to (mapped) bilinear multi-patch parameterizations, the techniques [5,28,30,31,40] can also deal with more general multipatch geometries. An alternative but related approach comprises the constructions [9,10] for general C 0 multi-patch parameterizations, which increase the degree of the constructed spline functions in the neighborhood of the common interfaces to obtain C 1 isogeometric spaces with good approximation properties.…”

Section: Introductionmentioning

confidence: 99%

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

Bracco

Giannelli

Kapl

et al. 2020

Computers & Mathematics with Applications

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Adaptive isogeometric methods for the solution of partial differential equations rely on the construction of locally refinable spline spaces. A simple and efficient way to obtain these spaces is to apply the multi-level construction of hierarchical splines, that can be used on single-patch domains or in multi-patch domains with C 0 continuity across the patch interfaces. Due to the benefits of higher continuity in isogeometric methods, recent works investigated the construction of spline spaces with global C 1 continuity on two or more patches. In this paper, we show how these approaches can be combined with the hierarchical construction to obtain global C 1 continuous hierarchical splines on two-patch domains. A selection of numerical examples is presented to highlight the features and effectivity of the construction.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Bracco

Giannelli

Kapl

et al. 2020

Computers & Mathematics with Applications

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“…This will be achieved by additionally enforcing that the functions of the generated space W v (i) have to be C 4 -smooth at the vertex v (i) . This strategy can be seen as an extension of approach [23], where C 1 -smooth functions have been generated in the vicinity of a vertex v (i) by additionally enforcing C 2 -smoothness of the functions at the vertex.…”

Section: The Vertex Subspace W V (I)mentioning

confidence: 99%

Isogeometric collocation on planar multi-patch domains

Kapl

Vitrih

2020

Computer Methods in Applied Mechanics and Engineering

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We present an isogeometric framework based on collocation to construct a C 2 -smooth approximation of the solution of the Poisson's equation over planar bilinearly parameterized multi-patch domains. The construction of the used globally C 2 -smooth discretization space for the partial differential equation is simple and works uniformly for all possible multipatch configurations. The basis of the C 2 -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 [2], which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [1,15,32]) to the multi-patch case. 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. [17,35], due to the fact that an isogeometric function is C s -smooth on a multi-patch domain if and only if its associated multi-patch graph surface is G s -smooth, cf. [16,28].So far, the problem of isogeometric collocation has been mostly explored on one-patch domains, see e.g. [1, 2, 5, 13-15, 30-32, 36, 38]. For analyzing the convergence behavior of a particular approach, the errors are generally computed with respect to L ∞ , W 1,∞ and W 2,∞ norm, or equivalently with respect to L 2 , H 1 and H 2 norm, respectively. The study of isogeometric collocation methods has started in [2] by using the Greville and Demko abscissae as collocation points. In comparison to the Galerkin approach, both choices show a suboptimal convergence behavior with respect to L 2 (L ∞ ) norm, namely of orders O(h p ) and O(h p−1 ) under h-refinement for even and odd spline degree p, respectively, and additionally a suboptimal convergence order O(h p−1 ) with respect to H 1 (W 1,∞ ) norm but just in case of odd spline degree p.In [1], an isogeometric collocation method has been presented, which is based on the computation and on the use of specific collocation points called superconvergent points. The proposed technique possesses only for even spline degree p in case of the L 2 ...

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An interior penalty coupling strategy for isogeometric non-conformal Kirchhoff–Love shell patches

Guarino,

Antolin,

Milazzo

et al. 2024

Engineering with Computers

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This work focuses on the coupling of trimmed shell patches using Isogeometric Analysis, based on higher continuity splines that seamlessly meet the $$C^1$$ C 1 requirement of Kirchhoff–Love-based discretizations. Weak enforcement of coupling conditions is achieved through the symmetric interior penalty method, where the fluxes are computed using their correct variationally consistent expression that was only recently proposed and is unprecedentedly adopted herein in the context of coupling conditions. The constitutive relationship accounts for generically laminated materials, although the proposed tests are conducted under the assumption of uniform thickness and lamination sequence. Numerical experiments assess the method for an isotropic and a laminated plate, as well as an isotropic hyperbolic paraboloid shell from the new shell obstacle course. The boundary conditions and domain force are chosen to reproduce manufactured analytical solutions, which are taken as reference to compute rigorous convergence curves in the $$L^2$$ L 2 , $$H^1$$ H 1 , and $$H^2$$ H 2 norms, that closely approach optimal ones predicted by theory. Additionally, we conduct a final test on a complex structure comprising five intersecting laminated cylindrical shells, whose geometry is directly imported from a STEP file. The results exhibit excellent agreement with those obtained through commercial software, showcasing the method’s potential for real-world industrial applications.

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An isogeometric C1 subspace on unstructured multi-patch planar domains

Cited by 58 publications

References 35 publications

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

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

Isogeometric collocation on planar multi-patch domains

An interior penalty coupling strategy for isogeometric non-conformal Kirchhoff–Love shell patches

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