An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem

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SUMMARYAmong all 3D 8-node hexahedral solid elements in current finite element library, the 'best' one can produce good results for bending problems using coarse regular meshes. However, once the mesh is distorted, the accuracy will drop dramatically. And how to solve this problem is still a challenge that remains outstanding. This paper develops an 8-node, 24-DOF (three conventional DOFs per node) hexahedral element based on the virtual work principle, in which two different sets of displacement fields are employed simultaneously to formulate an unsymmetric element stiffness matrix. The first set simply utilizes the formulations of the traditional 8-node tri-linear isoparametric element, while the second set mainly employs the analytical trial functions in terms of 3D oblique coordinates (R, S, T). The resulting element, denoted by US-ATFH8, contains no adjustable factor, and can be used for both isotropic and anisotropic cases. Numerical examples show it can strictly pass both the first-order (constant stress/strain) patch test and the second-order patch test for pure bending, remove the volume locking, and provide the

Section: Discussionmentioning

confidence: 99%

An unsymmetric 8‐node hexahedral element with high distortion tolerance

Zhou

Cen

Huang

et al. 2016

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“…Nodes 1, 2, 3, and 4 are the corner nodes; ( t x I , t y I ) and ( t S I , t T I ) are respectively the Cartesian coordinates and quadrilateral area coordinates (QACM‐II) (see Appendix A) of the corner node I ( I = 1, 2, 3, 4) at time t ; and u αI is the α ‐component ( α = 1, 2) of the displacement increments at node I . For the unsymmetric element US‐ATFQ4, 2 different sets of interpolation functions for displacement fields are simultaneously used. The first set is for the virtual displacement fields { δ u } and employs the shape functions of the traditional 4‐node bilinear isoparametric element, ie,

({}, δ boldu) = ({}, \begin{array}{l} δ u_{x} \\ δ u_{y} \end{array}) = ([], \overset{true‾}{boldN}) ({}, δ normalΔ q^{e}),

where

({}, δ normalΔ q^{e}) = {[δ u_{x 1} δ u_{y 1} δ u_{x 2} δ u_{y 2} δ u_{x 3} δ u_{y 3} δ u_{x 4} δ u_{y 4}]}^{T},

([], \overset{true‾}{boldN}) = ([], \begin{array}{c} {\overset{N}{true}}_{1} & 0 & {\overset{true‾}{N}}_{2} & 0 & {\overset{true‾}{N}}_{3} & 0 & {\overset{true‾}{N}}_{4} & 0 \\ 0 & {\overset{true‾}{N}}_{1} & 0 & {\overset{true‾}{N}}_{2} & 0 & {\overset{true‾}{N}}_{3} & 0 & {\overset{true‾}{N}}_{4} \end{array}),

with

…”

Section: Extension Of the Unsymmetric 4‐node 8‐dof Plane Element Us‐mentioning

confidence: 99%

“…The second set is for the real incremental displacement fields { u } and adopts a composite coordinate interpolation scheme with analytical trial functions

({}, u) = ({}, \begin{array}{l} u_{x} \\ u_{y} \end{array}) = ([], P) ({}, α) = ([], \begin{array}{c} 1 & 0 & ^{t} x & 0 & ^{t} y & 0 & ^{t} U_{7} & ^{t} U_{8} \\ 0 & 1 & 0 & ^{t} x & 0 & ^{t} y & ^{t} V_{7} & ^{t} V_{8} \end{array}) ({}, \begin{array}{c} α_{1} \\ α_{2} \\ α_{3} \\ ⋮ \\ α_{8} \end{array}),

where α i ( i = 1 ∼ 8) are 8 undetermined coefficients; t U 7 , t V 7 , t U 8 , and t U 8 are the linear displacement solutions for plane pure bending in arbitrary direction and in terms of the second form of quadrilateral area coordinates (QACM‐II) ( S , T ) (see Appendix A) at time t . The detailed expressions of t U 7 , t V 7 , t U 8 , and t U 8 are derived in the work of Cen et al and given in Appendix B. Substitution of nodal coordinates (including Cartesian and QACM‐II) and nodal displacement increments into Equation yields

({}, u) = ({}, \begin{array}{l} u_{x} \\ u_{y} \end{array}) = ([], ^{t}, \overset{true^}{boldN}) ({}, normalΔ q^{e}),

where

({}, normalΔ q^{e}) = [\begin{matrix} center \\ u_{x 1} \end{matrix}]

…”

Section: Extension Of the Unsymmetric 4‐node 8‐dof Plane Element Us‐mentioning

confidence: 99%

“…It is very interesting that whether the unsymmetric element US‐ATFQ4, which possesses high distortion‐resistance for linear elastic problems, can be extended to nonlinear applications. However, some researchers negate this extension, they think that the approach, which employs the analytical solutions satisfying all governing equations for linear elasticity, restricts the element to linear elastic analysis .…”

Section: Introductionmentioning

confidence: 99%

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High‐performance geometric nonlinear analysis with the unsymmetric 4‐node, 8‐DOF plane element US‐ATFQ4

Cen

et al. 2018

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Summary A recent unsymmetric 4‐node, 8‐DOF plane element US‐ATFQ4, which exhibits excellent precision and distortion‐resistance for linear elastic problems, is extended to geometric nonlinear analysis. Since the original linear element US‐ATFQ4 contains the analytical solutions for plane pure bending, how to modify such formulae into incremental forms for nonlinear applications and design an appropriate updated algorithm become the key of the whole job. First, the analytical trial functions should be updated at each iterative step in the framework of updated Lagrangian formulation that takes the configuration at the beginning of an incremental step as the reference configuration during that step. Second, an appropriate stress update algorithm in which the Cauchy stresses are updated by the Hughes‐Winget method is adopted to estimate current stress fields. Numerical examples show that the new nonlinear element US‐ATFQ4 also possesses amazing performance for geometric nonlinear analysis, no matter whether regular or distorted meshes are used. It again demonstrates the advantages of the unsymmetric finite element method with analytical trial functions.

“…An important work related to the development of distortion insensitive plane elements was proposed by Chen et al By applying the natural quadrilateral area coordinates, several 4‐node, 8‐node nonconforming plane membranes were formulated, in which AGQ6‐I and AGQ6‐II were the most representative ones. It is reported that these elements can produce results with high numerical precision in distorted mesh and were free of trapezoidal locking.…”

Section: Introductionmentioning

confidence: 99%

4‐node unsymmetric quadrilateral membrane element with drilling DOFs insensitive to severe mesh‐distortion

Shang

Ouyang

2017

Summary The unsymmetric finite element method is a promising technique to produce distortion‐immune finite elements. In this work, a simple but robust 4‐node 12‐DOF unsymmetric quadrilateral membrane element is formulated. The test function of this new element is determined by a concise isoparametric‐based displacement field that is enriched by the Allman‐type drilling degrees of freedom. Meanwhile, a rational stress field, instead of the displacement one in the original unsymmetric formulation, is directly adopted to be the element's trial function. This stress field is obtained based on the analytical solutions of the plane stress/strain problem and the quasi‐conforming technique. Thus, it can a priori satisfy related governing equations. Numerical tests show that the presented new unsymmetric element, named as US‐Q4θ, exhibits excellent capabilities in predicting results of both displacement and stress, in most cases, superior to other existing 4‐node element models. In particular, it can still work very well in severely distorted meshes even when the element shape deteriorates into concave quadrangle or degenerated triangle.