Hybrid stress tetrahedral elements with Allman's rotational D.O.F.s

Sze, K. Y.; Pan, Yingdong

doi:10.1002/(sici)1097-0207(20000710)48:7<1055::aid-nme916>3.0.co;2-p

Cited by 26 publications

(43 citation statements)

References 29 publications

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“…skew-symmetric] stress at element level. In particular, the displacement shape functions N reported in [2] and the element stress shape functions P e and Q e proposed in [3] are adopted, whereas linear Lagrangian interpolation, given by N ω , is assumed for the rotational field. Distinctive feature of the present formulation is a Gauss-point-discontinuous strain interpolation.…”

Section: Mixed Variational Formulation and Finite Element Approximationmentioning

confidence: 99%

Mixed Tetrahedral Elements for the Analysis of Structures with Material and Geometric Nonlinearities

Nodargi

Bisegna

2015

Proc Appl Math and Mech

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A new mixed tetrahedral element, particularly suited for the analysis of structures exhibiting nonlinear material and geometric behavior, is here presented. Its derivation is based on a Hu-Washizu type formulation, including also rotation and skewsymmetric stress fields as independent variables, instrumental to equip the element with nodal rotational degrees of freedom. A Gauss-point-discontinuous interpolation is selected for the total strain field, in order to account for its possibly highly nonlinear spatial distribution due to inelastic strains. Accordingly, the resulting tetrahedron can properly describe inelastic effects occurring over a space scale smaller than the element size. An original and efficient iterative procedure is proposed to perform the element state determination. Mixed variational formulation and finite element approximationLet B ⊂ R 3 be the domain occupied by a three-dimensional continuum body whose kinematics is described by displacement u, infinitesimal strain ε and rotation ω. Denote by σ [resp. τ ] the symmetric [resp. skew-symmetric] part of the stress tensor field, and suppose the material constituting the body to be endowed with a convex (free or incremental) energy density ψ. As usual, the boundary of the domain is partitioned in the two disjoint sets Γ D and Γ N , where Dirichlet and Neumann conditions are imposed, respectively. Without loss of generality, homogeneous Dirichlet condition on Γ D are assumed. Once introduced the following functional spaces:the equilibrium problem is stated adopting the Hu-Washizu functional F : V × W × E × S × T → R such that (e.g., see [1]):where µ is a typical stiffness modulus, γ is an arbitrary positive non-dimensional parameter, D and L are the differential operators for deriving, respectively, strain and rotation from displacement field, and F ext is the load potential contribution. To proceed with finite element formulation on a given tetrahedral mesh B = e B e , interpolation spaces consistent with those ones listed in (1) have to be introduced. Switching to vector notation and denoting by χ e the characteristic function associated to finite element B e , interpolation spaces for displacements and stresses are selected to be:where a ∈ R n a collects the nodal DOFs, including three Allman's rotations per node, and β e ∈ R n β [resp. α e ∈ R n α ] are the interpolation parameters of symmetric [resp. skew-symmetric] stress at element level. In particular, the displacement shape functions N reported in [2] and the element stress shape functions P e and Q e proposed in [3] are adopted, whereas linear Lagrangian interpolation, given by N ω , is assumed for the rotational field. Distinctive feature of the present formulation is a Gauss-point-discontinuous strain interpolation. Denoting by e e g ∈ R 6 the strain at Gauss point g of the element e, say x e g , the following choice is made (e.g., see [4]):where δ x − x e g is a Dirac delta measure centered at x e g . The substitution of (3) and (4) into the functional (2) yields its finite element a...

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Section: Mixed Variational Formulation and Finite Element Approximationmentioning

confidence: 99%

Mixed Tetrahedral Elements for the Analysis of Structures with Material and Geometric Nonlinearities

Nodargi

Bisegna

2015

Proc Appl Math and Mech

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“…The element has vertex rotational dofs and an assumed strain field that is determined by a least square strain extraction. HT4R18: the hybrid stress tetrahedral element with Allman's rotational dofs developed by Sze and Pan [2000], employing 18 symmetric stress modes. HT4R14: the hybrid stress tetrahedral element with Allman's rotational dofs developed by Sze and Pan [2000], employing 14 symmetric stress modes.…”

Section: Tet10mentioning

confidence: 99%

“…One is the TET4R(X) developed by Pawlak et al [1991], where X implies the use of an assumed strain field. The other is the hybrid stress tetrahedron HT4R proposed by Sze and Pan [2000]. Both elements improve the conventional 4-node tetrahedron by introducing to each node three extra Allman's rotational dofs.…”

Section: Introductionmentioning

confidence: 99%

A Pu-Based 4-Node Quadratic Tetrahedon and Linear Dependences Elimination in Three-Dimensions

Tian

2006

Int. J. Comput. Methods

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A 4-node, quadratic tetrahedral element is presented. The development of the element is based on a partition-of-unity (PU) approximation. The local function of the PU approximation is constructed using the basic deformation components of rigid body movements, node-centered rotation, and locally defined linear strain fields. The linear dependence issue of the three-dimensional PU approximation is investigated through numerical experiments. It is found that the global matrix associated with the PU approximation constructed has a constant number of zero eigenvalues independent of mesh subdivision. A method of suppressing the zero eigenvalues is proposed. The element developed does not contain unsuppressed spurious zero energy modes and the condition number of the stiffness matrix of the element is upper bounded. Good accuracy and robustness of the 4-node tetrahedron are demonstrated through numerical examples.

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“…Accordingly, its formulation can be systematically set within the variational framework of Hu–Washizu principle . Motivated by the lack of frame invariance and the presence of some unsuppressed zero‐energy deformation modes exhibited by TET4RX, Sze and Pan proposed an alternative derivation, relying on a modified Hellinger–Reissner variational principle. In particular, following the approach reported in , skew‐symmetric stress modes enter the functional as Lagrange multipliers to enforce the compatibility between the skew‐symmetric part of the discrete displacement gradient and the independently interpolated rotation field.…”

Section: Introductionmentioning

confidence: 99%

“…Compared with its standard version, the present functional is modified to include independently interpolated rotations and skew‐symmetric stresses, the latter being introduced to enforce the rotational compatibility condition (e.g., see ). Starting from a quadratic Lagrangian approximation, the displacement interpolation is obtained by migration of mid‐side node DOFs into Allman's corner rotations , whereas a first‐order Lagrangian interpolation is adopted for rotation field . Following the approach reported in , skew‐symmetric stresses are properly selected to suppress all spurious zero‐energy deformation modes, and symmetric stresses are approximated by means of linear shape functions, constructed to guarantee frame invariance and to generate an isostatic element.…”

Section: Introductionmentioning

confidence: 99%

A mixed tetrahedral element with nodal rotations for large-displacement analysis of inelastic structures

Nodargi

Caselli

Artioli

et al. 2016

Int. J. Numer. Meth. Engng

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SUMMARYA novel mixed four-node tetrahedral finite element, equipped with nodal rotational degrees of freedom, is presented. Its formulation is based on a Hu-Washizu-type functional, suitable to the treatment of material nonlinearities. Rotation and skew-symmetric stress fields are assumed as independent variables, the latter entering the functional to impose rotational compatibility and suppress spurious modes. Exploiting the choice of equal interpolation for strain and symmetric stress fields, a robust element state determination procedure, requiring no element-level iteration, is proposed. The mixed element stability is assessed by means of an original and effective numerical test. The extension of the present formulation to geometric nonlinear problems is achieved through a polar decomposition-based corotational framework. After validation in both material and geometric nonlinear context, the element performances are investigated in demanding simulations involving complex shape memory alloy structures. Supported by the comparison with available linear and quadratic tetrahedrons and hexahedrons, the numerical results prove accuracy, robustness, and effectiveness of the proposed formulation.

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Hybrid stress tetrahedral elements with Allman's rotational D.O.F.s

Cited by 26 publications

References 29 publications

Mixed Tetrahedral Elements for the Analysis of Structures with Material and Geometric Nonlinearities

Mixed Tetrahedral Elements for the Analysis of Structures with Material and Geometric Nonlinearities

A Pu-Based 4-Node Quadratic Tetrahedon and Linear Dependences Elimination in Three-Dimensions

A mixed tetrahedral element with nodal rotations for large-displacement analysis of inelastic structures

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