C1 finite elements on non-tensor-product 2d and 3d manifolds

Nguyen, Thien; Karčiauskas, Kȩstutis; Peters, Jörg

doi:10.1016/j.amc.2015.06.103

Cited by 49 publications

(62 citation statements)

References 23 publications

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“…The resulting smooth surfaces are then used to construct C 1 isogeometric splines spaces, but which are in general not nested, see e.g. [29,41]. The method [41] employs the surface construction [30], which is based on a biquadratic C 1 spline surface and on a bicubic or biquartic G 1 surface cap depending on the valency of the corresponding extraordinary vertex.…”

Section: The Design Of C 1 Isogeometric Spacesmentioning

confidence: 99%

“…[20]. The resulting global C 1 -smoothness of the spaces then enables the solution of fourth order PDEs just via its weak form using a standard Galerkin discretization, see for example [4,15,24,28,41,49] for the biharmonic equation, [3,6,32,33,34] for the Kirchhoff-Love shell formulation, [18,19,36] for the Cahn-Hilliard equation and [17,43] for plane problems of first strain gradient elasticity.…”

Section: Introductionmentioning

confidence: 99%

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Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes

Kapl¹,

Sangalli²,

Takacs³

2019

The SMAI Journal of Computational Mathematics

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In the context of isogeometric analysis, globally C 1 isogeometric spaces over unstructured quadrilateral meshes allow the direct solution of fourth order partial differential equations on complex geometries via their Galerkin discretization. The design of such smooth spaces has been intensively studied in the last five years, in particular for the case of planar domains, and is still task of current research. In this paper, we first give a short survey of the developed methods and especially focus on the approach [26]. There, the construction of a specific C 1 isogeometric spline space for the class of so-called analysis-suitable G 1 multi-patch parametrizations is presented. This particular class of parameterizations comprises exactly those multi-patch geometries, which ensure the design of C 1 spaces with optimal approximation properties, and allows the representation of complex planar multi-patch domains. We present known results in a coherent framework, and also extend the construction to parametrizations that are not analysis-suitable G 1 by allowing higher-degree splines in the neighborhood of the extraordinary vertices and edges. Finally, we present numerical tests that illustrate the behavior of the proposed method on representative examples.

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Section: The Design Of C 1 Isogeometric Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes

Kapl¹,

Sangalli²,

Takacs³

2019

The SMAI Journal of Computational Mathematics

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“…The application and further development of these techniques is currently a very active area of research in isogeometric analysis, see e.g. [4][5][6][7][8][9][10][11][12][13][14][15]. Unfortunately, none of the techniques from computer-aided design seems to give optimal finite element convergence rates without further modifications, especially when applied to Kirchhoff-Love thin shells with arbitrary geometry.…”

Section: Introductionmentioning

confidence: 99%

Manifold-based isogeometric analysis basis functions with prescribed sharp features

Zhang

Cirak

2020

Computer Methods in Applied Mechanics and Engineering

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We introduce manifold-based basis functions for isogeometric analysis of surfaces with arbitrary smoothness, prescribed C 0 continuous creases and boundaries. The utility of the manifold-based surface construction techniques in isogeometric analysis was demonstrated in Majeed and Cirak (CMAME, 2017). The respective basis functions are derived by combining differential-geometric manifold techniques with conformal parametrisations and the partition of unity method. The connectivity of a given unstructured quadrilateral control mesh in R 3 is used to define a set of overlapping charts. Each vertex with its attached elements is assigned a corresponding conformally parametrised planar chart domain in R 2 so that a quadrilateral element is present on four different charts. On the collection of unconnected chart domains, the partition of unity method is used for approximation. The transition functions required for navigating between the chart domains are composed out of conformal maps. The necessary smooth partition of unity, or blending, functions for the charts are assembled from tensor-product B-spline pieces and require in contrast to earlier constructions no normalisation. Creases are introduced across user tagged edges of the control mesh. Planar chart domains that include creased edges or are adjacent to the domain boundary require special local polynomial approximants in the partition of unity method. Three different types of chart domain geometries are necessary to consider boundaries and arbitrary number and arrangement of creases. The new chart domain geometries are chosen so that it becomes trivial to establish local polynomial approximants that are always C 0 continuous across the tagged edges. The derived non-rational manifold-based basis functions correspond to the vertices of the mesh and may have an arbitrary number of creases and prescribed smoothness. This makes them particularly well suited for isogeometric analysis of Kirchhoff-Love thin shells with kinks, which require C 1 continuous basis functions that are C 0 continuous across the kinks. We demonstrate the convergence and utility of the new basis functions with linear and nonlinear beam, plate and shell examples.

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“…The methods [43,48,49] use a singular parameterization with patches in the vicinity of an extraordinary vertex, which belong to a specific class of degenerate (Bézier) patches introduced in [45], and that allow, despite having singularities, the design of globally C 1 isogeometric spaces. The techniques [33,34,42] are based on G 1 multi-patch surface constructions, where the obtained surface in the neighborhood of an extraordinary vertex consists of patches of slightly higher degree [33,42] and is generated by means of a particular subdivision scheme [34]. As a special case of the first approach can be seen the constructions in [41,47], that employ a polar framework to generate C 1 spline spaces.…”

Section: Introductionmentioning

confidence: 99%

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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C1 finite elements on non-tensor-product 2d and 3d manifolds

Cited by 49 publications

References 23 publications

Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes

Isogeometric analysis with $C^1$ functions on planar, unstructured quadrilateral meshes

Manifold-based isogeometric analysis basis functions with prescribed sharp features

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

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