Isogeometric analysis with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e1752" altimg="si839.svg"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math> hierarchical functions on planar two-patch geometries

Bracco, Cesare; Giannelli, Carlotta; Kapl, Mario; Vázquez, Rafael

doi:10.1016/j.camwa.2020.03.018

Cited by 13 publications

(2 citation statements)

References 47 publications

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“…In the future we want to study several aspects of the method, such as convergence and stability properties and extensions that result in nested spaces. This would allow an adaptive construction with THB-splines, following the work as in [3,4]. Moreover, we want to extend the construction to C 0 -non-conforming (nonmatching) interfaces and to surface domains.…”

Section: Discussionmentioning

confidence: 99%

An approximate $C^1$ multi-patch space for isogeometric analysis with a comparison to Nitsche's method

Weinmüller¹,

Takacs²

2022

Preprint

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We present an approximately C 1 -smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). The construction extends the one presented in [42] for two-patch domains. A key property of IGA is that it is simple to achieve high order smoothness within a single patch. However, to represent more complex geometries one often uses a multi-patch construction. In this case, the global continuity for the basis functions is in general only C 0 . Therefore, to obtain C 1 -smooth isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff-Love plate/shell formulations, using an isogeometric Galerkin method.Isogeometric spaces that are globally C 1 over multi-patch domains can be constructed as in [9,[19][20][21][22]. The constructions require geometry parametrizations that satisfy certain constraints along the interfaces, so-called analysis-suitable G 1 parametrizations. To allow C 1 spaces over more general multi-patch parametrizations, one needs to increase the polynomial degree and/or to relax the C 1 conditions. Thus, we define function spaces that are not exactly C 1 but only approximately. We adopt the construction for two-patch domains, as developed in [42], and extend it to more general multi-patch domains.We employ the construction for a biharmonic model problem and compare the results with Nitsche's method. We compare both methods over complex multi-patch domains with non-trivial interfaces. The numerical tests indicate that the proposed construction converges optimally under h-refinement, comparable to the solution using Nitsche's method. In contrast to weakly imposing coupling conditions, the approximate C 1 construction is explicit and no additional terms need to be introduced to stabilize the method/penalize the jump of the derivative at the interface. Thus, the new proposed method can be used more easily as no parameters need to be estimated.

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Section: Discussionmentioning

confidence: 99%

An approximate $C^1$ multi-patch space for isogeometric analysis with a comparison to Nitsche's method

Weinmüller¹,

Takacs²

2022

Preprint

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“…For the interested reader, we mention [163,164,127,128,205,196,148]. The analysis of adaptive methods in multi-patch domains with high continuity is beyond the current state of the art, with preliminary steps in [34], and in particular beyond the scope of this paper. Fig.…”

Section: Remarkmentioning

confidence: 99%

Mathematical Foundations of Isogeometric Analysis

et al. 2020

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isogeometric analysis, with special focus on the mathematical theory. This includes an overview of available spline technologies for the local resolution of possible singularities as well as the state-of-the-art formulation of convergence and quasi-optimality of adaptive algorithms for both the finite element method (FEM) and the boundary element method (BEM) in the frame of isogeometric analysis (IGA).

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

Cited by 13 publications

References 47 publications

An approximate $C^1$ multi-patch space for isogeometric analysis with a comparison to Nitsche's method

An approximate $C^1$ multi-patch space for isogeometric analysis with a comparison to Nitsche's method

Mathematical Foundations of Isogeometric Analysis

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

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