High-order conforming finite elements on pyramids

Nigam, Nilima; Phillips, Joel

doi:10.1093/imanum/drr015

Cited by 30 publications

(58 citation statements)

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“…We shall follow [28, Page 1138] to choose a particular diagonal, which guarantees the non-degeneracy of the tetrahedra in the series of THP partitions. (1,12,1234,14,15,1562,0,1485), (14,1234,34,4, 1485, 0, 3487, 48), (12,2,23,1234,1562,26,2376,0), (1234, 23, 3, 34, 0, 2376, 37, 3487), where the new node at vertex ijkℓ is placed at the centroid of nodes with vertices i, j, k, and ℓ, and the new node at vertex 0 is placed at the centroid of the nodes for all eight vertices of the original hexahedron.…”

Section: Lowest-order Finite Elements Onmentioning

confidence: 99%

A lowest-order composite finite element exact sequence on pyramids

Ainsworth

2017

Computer Methods in Applied Mechanics and Engineering

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Abstract. Composite basis functions for pyramidal elements on the spaces H 1 (Ω), H(curl, Ω), H(div, Ω) and L 2 (Ω) are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart-Thomas-Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, ...) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements is successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions.

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Section: Lowest-order Finite Elements Onmentioning

confidence: 99%

A lowest-order composite finite element exact sequence on pyramids

Ainsworth

2017

Computer Methods in Applied Mechanics and Engineering

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“…In particular, we show that all the results in [20,21,2,15,25] on a tetrahedron, cube, and prism fit nicely within our construction. Moreover, a significant dimension reduction is obtained on the pyramidal commuting exact sequence in comparison with the exact sequence proposed in [22,23]. For a general polyhedron, our construction of of high-order commuting exact sequences is significantly more difficult than that of the cases already mentioned.…”

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confidence: 94%

“…Most of the previous work on the construction of commuting exact sequences (in three-space dimensions) focuses on the explicit construction of shape functions on one of four particular reference polyhedra, namely, the reference tetrahedron, hexahedron (cube), prism, and pyramid. See [20,21] for sequences on the reference tetrahedron and reference hexahedron, [2,15] on the reference hexahedron, [25] on the reference prism, and [22,23] on the reference pyramid. All of these spaces in these sequences are spanned by polynomial shape functions, except those in [22,23] which also contain rational shape functions.…”

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confidence: 99%

“…As a consequence, each of the four groups of sequences can be patched into in a hybrid polyhedral mesh Ω h = {K} where the elements K are suitably defined by affine mappings of the four reference polyhedral elements. This way of regrouping the sequences is motivated by the work proposed in [18], where one group of sequences that can be used on a hybrid mesh (with a more complicated pyramidal sequence from [22]) was recently carefully studied to construct "orientation embedded high-order shape functions".…”

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confidence: 99%

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A Systematic Construction of Finite Element Commuting Exact Sequences

Cockburn¹,

2017

SIAM J. Numer. Anal.

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We present a systematic construction of finite element exact sequences with a commuting diagram for the de Rham complex in one-, two-and three-space dimensions. We apply the construction in two-space dimensions to rediscover two families of exact sequences for triangles and three for squares, and to uncover one new family of exact sequence for squares and two new families of exact sequences for general polygonal elements. We apply the construction in three-space dimensions to rediscover two families of exact sequences for tetrahedra, three for cubes, and one for prisms; and to uncover four new families of exact sequences for pyramids, three for prisms, and one for cubes.

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“…In some circumstances, it may be desirable to use meshes composed of hexahedra and tetrahedra. If these meshes are to avoid hanging nodes then they will, in general, contain pyramids [2]. Regions that may be more critical to the analysis, such as boundary layers or regions of high stress may also be better served by hexahedra.…”

Section: Introductionmentioning

confidence: 99%

A 3D pyramid spline element

Chen

Wanji

2011

Acta Mech Sin

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In this paper, a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method, which achieves the second order completeness in Cartesian coordinates. Some appropriate examples were employed to evaluate the performance of the proposed element. The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted, and it is comparable to the Lagrange element Q27. It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.

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High-order conforming finite elements on pyramids

Cited by 30 publications

References 0 publications

A lowest-order composite finite element exact sequence on pyramids

A lowest-order composite finite element exact sequence on pyramids

A Systematic Construction of Finite Element Commuting Exact Sequences

A 3D pyramid spline element

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