An unsymmetric 8‐node hexahedral element with high distortion tolerance

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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.

Section: Discussionmentioning

confidence: 99%

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

Cen

et al. 2018

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“…Note that, as the independently assumed rotation z  and the physical rotation derived from displacements have different orders of interpolation, severe locking behavior may be observed when full-integration scheme is used for the last penalty stiffness in Equation (47). To overcome this problem, the selective reduced integration procedure suggested in [39] is employed here: the last penalty term is calculated by using the one-point Gauss quadrature strategy, whilst other integrations are operated by using the full quadrature scheme.…”

Section: The Element Stiffness Matrixmentioning

confidence: 99%

“…Recently, Shang and Ouyang [1] proposed a simple and robust 4-node 12-DOF quadrilateral membrane element US-Q4 for the classical elastic problems based on the unsymmetric finite element method [42][43][44]. The unsymmetric FEM, which has been successfully applied to various applications in past years [45][46][47][48][49], employs different interpolations for the test and trial functions in the element formulation. Demonstrated by numerical tests [1], the element US-Q4 exhibits good numerical accuracy and resistance to mesh distortions, even when the element shape is severely distorted into concave quadrilateral or triangle.…”

Section: Introductionmentioning

confidence: 99%

A simple unsymmetric 4‐node 12‐DOF membrane element for the modified couple stress theory

Shang

Qian

Cen

et al. 2019

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SummaryIn this work, the recently proposed unsymmetric 4‐node 12‐DOF (degree‐of‐freedom) membrane element (Shang and Ouyang, Int J Numer Methods Eng 113(10): 1589‐1606, 2018), which has demonstrated excellent performance for the classical elastic problems, is further extended for the modified couple stress theory, to account for the size effect of materials. This is achieved via two formulation developments. Firstly, by using the penalty function method, the kinematic relations between the element's nodal drilling DOFs and the true physical rotations are enforced. Consequently, the continuity requirement for the modified couple stress theory is satisfied in weak sense, and the symmetric curvature test function can be easily derived from the gradients of the drilling DOFs. Secondly, the couple stress field that satisfies a priori the related equilibrium equations is adopted as the energy conjugate trial function to formulate the element for the modified couple stress theory. As demonstrated by a series of benchmark tests, the new element can efficiently capture the size‐dependent responses of materials and is robust to mesh distortions. Moreover, as the new element uses only three conventional DOFs per node, it can be readily incorporated into the standard finite element program framework and commonly available finite element programs.

“…The benchmark problem proposed in Reference is modified here to test the effects of warping on element performance. As shown in Figure , the 90° pre‐twisted beam is clamped at the root and subjected to an in‐plane or out‐plane tip force.…”

Section: Numerical Testsmentioning

confidence: 99%

“…In the recent work, a novel quadrilateral membrane element with four nodes and 12 DOFs has been developed in the context of the modified couple stress elasticity by using the unsymmetric FEM . Numerical tests show that this membrane element which satisfies the weak C 1 continuity condition possesses good accuracy and distortion tolerance when analyzing the size‐dependent plain strain problems.…”

Section: Introductionmentioning

confidence: 99%

8‐node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

Shang

Jia

2020

A new C 0 8-node 48-DOF hexahedral element is developed for analysis of size-dependent problems in the context of the modified couple stress theory by extending the methodology proposed in our recent work (Shang et al., Int J Numer Methods Eng 119(9): 807-825, 2019) to the three-dimensional (3D) cases.There are two major innovations in the present formulation. First, the independent nodal rotation degrees of freedom (DOFs) are employed to enhance the standard 3D isoparametric interpolation for obtaining the displacement and strain test functions, as well as to approximatively design the physical rotation field for deriving the curvature test function. Second, the equilibrium stress functions instead of the analytical functions are used to formulate the stress trial function whilst the couple stress trial function is directly obtained from the curvature test function by using the constitutive relationship. Besides, the penalty function is introduced into the virtual work principle for enforcing the C 1 continuity condition in weak sense. Several benchmark examples are examined and the numerical results demonstrate that the element can simulate the size-dependent mechanical behaviors well, exhibiting satisfactory accuracy and low susceptibility to mesh distortion. K E Y W O R D Shexahedral element, modified couple stress theory, rotation degree of freedom, size-dependent, unsymmetric FEM INTRODUCTIONMany experimental observations have shown that the micro/nano structures, which have been widely used in modern engineering applications, may experience size-dependent mechanical behaviors. Various non-classical higher-order continuum theories, such as the strain gradient theory (SGT) and the couple stress theory (CST), have been developed for describing such size-dependent phenomena. 1-7 However, as these higher-order theories are more complex than the classical one, the analytical or semi-analytical solutions are available only for restricted problems with simple geometries and certain boundary conditions. Therefore, the numerical approaches with high accuracy and efficiency are clearly required for the solution procedure, in which the well-accepted finite element method (FEM) is usually recognized as a very efficient choice. In recent year, the isogeometric method 8-11 and the meshless method, 12,13 which can formulate highly continuous basis functions, have also been applied to the size-dependent problems. However, the computation expenses Int J Numer Methods Eng. 2020;121:2683-2700. wileyonlinelibrary.com/journal/nme