The simplest conforming anisotropic rectangular and cubic mixed finite elements for elasticity

Chen, Shaochun; Sun, Yanping; Zhao, Jikun

doi:10.1016/j.amc.2015.04.117

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…For 1 ≤ k ≤ n − 1, either the symmetric tensor spaces are enriched by proper high order H(div) bubble functions of piecewise polynomials [17] or the stabilization technique is employed [18]. Corresponding mixed elements on both rectangular and cuboid grids are constructed in [19], see [20,21] for the lowest order mixed elements; mixed elements on triangular prism grids are constructed in [22].…”

Section: Introductionmentioning

confidence: 99%

Nonconforming mixed finite elements for linear elasticity on simplicial grids

2018

Numerical Methods Partial

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This paper introduces a new family of nonconforming mixed finite elements for solving the linear elasticity equations on simplicial grids. Besides, this paper describes the construction of the lowest order basis functions. The construction only involves simple computations due to the new explicit stress shape function spaces and the procedure applies for high order cases. Numerical experiments for four benchmark problems in mechanics indicate the robust locking-free behavior and show that the lowest order nonconforming mixed method leads to smaller stress errors than the first and second order standard Galerkin methods for the nearly incompressible case. KEYWORDSlinear elasticity, mixed finite element, nonconforming simplicial element

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Section: Introductionmentioning

confidence: 99%

Nonconforming mixed finite elements for linear elasticity on simplicial grids

2018

Numerical Methods Partial

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“…The DOFs are 2 plus 1 in 1D, 7 plus 2 in 2D, and 15 plus 3 in 3D, and the error estimates for all variables are optimal. In [20][21][22], some conforming rectangular and cubic elements were presented, of which the lowest order elements have 8 plus 2 DOFs in 2D, and 18 plus 3 DOFs in 3D. In [23][24][25], some conforming elements on simplicial meshes were developed, and the lowest order elements only involve 18 plus 3 DOFs in 2D and 48 plus 6 in 3D.…”

Section: Introductionmentioning

confidence: 99%

New Residual Based Stabilization Method for the Elasticity Problem

Li¹

2018

AAMM

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In this paper, we consider the mixed finite element method (MFEM) of the elasticity problem in two and three dimensions (2D and 3D). We develop a new residual based stabilization method to overcome the inf-sup difficulty, and use Langrange elements to approximate the stress and displacement. The new method is unconditionally stable, and its stability can be obtained directly from Céa's lemma. Optimal error estimates for the H 1-norm of the displacement and H(div)-norm of the stress can be obtained at the same time. Numerical results show the excellent stability and accuracy of the new method.

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“…We also refer to [21] for two types of interior penalty mixed finite element methods by using nonconforming symmetric stress spaces, where the stability is established by introducing the conforming H(div) bubble spaces from [24] and nonconforming face-bubble spaces. Corresponding mixed elements on both rectangular and cuboid meshes were constructed in [23], also see [18,25] for the lowest order mixed elements, while the simplest nonconforming mixed element on n-rectangular meshes can be found in [26].…”

Section: Introductionmentioning

confidence: 99%

Conforming mixed triangular prism and nonconforming mixed tetrahedral elements for the linear elasticity problem

Hu¹,

Ma²

2016

Preprint

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We propose two families of mixed finite elements for solving the classical Hellinger-Reissner mixed problem of the linear elasticity equations in three dimensions. First, a family of conforming mixed triangular prism elements is constructed by product of elements on triangular meshes and elements in one dimension. The well-posedness is established for all elements with k ≥ 1, which are of k + 1 order convergence for both the stress and displacement. Besides, a family of reduced stress spaces is proposed by dropping the degrees of polynomial functions associated with faces. As a result, the lowest order conforming mixed triangular prism element has 93 plus 33 degrees of freedom on each element. Second, we construct a new family of nonconforming mixed tetrahedral elements. The shape function spaces of our stress spaces are different from those of the elements in literature.

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The simplest conforming anisotropic rectangular and cubic mixed finite elements for elasticity

Cited by 4 publications

References 24 publications

Nonconforming mixed finite elements for linear elasticity on simplicial grids

Nonconforming mixed finite elements for linear elasticity on simplicial grids

New Residual Based Stabilization Method for the Elasticity Problem

Conforming mixed triangular prism and nonconforming mixed tetrahedral elements for the linear elasticity problem

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