Unsymmetric extensions of Wilson's incompatible four-node quadrilateral and eight-node hexahedral elements

The unsymmetric finite element is based on the virtual work principle with different sets of test and trial functions. In this article, the incompatible four-node quadrilateral element and eight-node hexahedral element originated by Wilson et al. are extended to their unsymmetric forms. The isoparametric shape functions together with Wilson's incompatible functions are chosen as the test functions, while internal nodes at the middle of element sides/edges are added to generate the trial functions with quadratic completeness in the Cartesian coordinate system. A local area/volume coordinate frame is established so that the trial shape functions can be explicitly obtained. The key idea which avoids the matrix inversion is that the trial nodal shape functions are constructed by standard quadratic triangular/tetrahedral elements and then transformed in consistent with the quadrilateral/hexahedral elements. Numerical examples show that the present elements keep the merits of both incompatible and unsymmetric elements, that is, high numerical accuracy, insensitivity to mesh distortion, free of trapezoidal and volumetric locking, and easy implementation.     

广义协调六结点平面曲边单元研究

主要运用广义协调原理,针对计算平面曲边单元的有限元算法进行了研究,并且利用点、周混合协调条件构造了三种高性能六结点曲边单元。第一、二种单元在平面直角坐标内分别采用解析试函数和完全三次多项式构造,第三种单元在六结点等参单元Q6的基础上附加广义协调泡状位移而成。这三种单元均能通过强式分片试验,并且显示了良好的计算精度和抗畸变能力。     

非协调元性能分析的两个定理

在构造非协调元的过程中,必须遵守一定的构造规律。本文从基本力学观点出发,提出并证明了两个定理。定理一、如果某种类型的有限单元共有n个独立参与整体刚度运算的自由度,则该单元最多只能精确模拟n种弹性力学基本解。该定理说明了单元的精度从根本上受自身自由度限制的,并指出了现有的四边形四结点单元发展空间不大,而四边形八结点Q8单元以及三维八结点H8单元仍然具有较大的发展余地。定理二则认为四边形四结点内参型非协调元如果能够通过小片试验,则不可能在任意畸变状态下精确表示纯弯场。该定理表明了畸变问题的尝试是有限制的。以上的结论虽然是针对非协调元的构造来提出的,但从论证过程看,应对其它类型的有限单元也适用。定理一和定理二对于今后新型有限元的发展可以起到一定的指导作用。     

带转角自由度的四边形六面体单元

本文提出了三个带转角自由度单元，其中一个平面四边形单元，两个空间六面体单元。对平面单元每个结点有两个线位移自由度、一个转角自由度；对空间单元，每个结点有三个线位移自由度、三个转角自由度。这些单元列式简单，其中两个无多余零能模式，数值计算表明，它们的计算精度高。     

An unsymmetric 8-node hexahedral solid-shell element with high distortion tolerance: Geometric nonlinear formulations

A recent distortion-tolerant unsymmetric 8-node hexahedral solid-shell element US-ATFHS8, which takes the analytical solutions of linear elasticity as the trial functions, is successfully extended to geometric nonlinear analysis. This extension is based on the corotational (CR) approach due to its simplicity and high efficiency, especially for geometric nonlinear analysis where the strain is still small. Based on the assumption that the analytical trial functions can properly work in each increment during the nonlinear analysis, the incremental corotational formulations of the nonlinear solid-shell element US-ATFHS8 are derived within the updated Lagrangian (UL) framework, in which an appropriate updated strategy for linear analytical trial functions is proposed. Numerical examples show that the present nonlinear element US-ATFHS8 possesses excellent performance for various rigorous tests no matter whether regular or distorted mesh is used. Especially, it even performs well in some situations that other conventional elements cannot work.     

Hyperelastic finite deformation analysis with the unsymmetric finite element method containing homogeneous solutions of linear elasticity

A recent unsymmetric 4-node, 8-DOF plane finite element US-ATFQ4 is generalized to hyperelastic finite deformation analysis. Since the trial functions of US-ATFQ4 contain the homogenous closed analytical solutions of governing equations for linear elasticity, the key of the proposed strategy is how to deal with these linear analytical trial functions (ATFs) during the hyperelastic finite deformation analysis. Assuming that the ATFs can properly work in each increment, an algorithm for updating the deformation gradient interpolated by ATFs is designed. Furthermore, the update of the corresponding ATFs referred to current configuration is discussed with regard to the hyperelastic material model, and a specified model, neo-Hookean model, is employed to verify the present formulation of US-ATFQ4 for hyperelastic finite deformation analysis. Various examples show that the present formulation not only remain the high accuracy and mesh distortion tolerance in the geometrically nonlinear problems, but also possess excellent performance in the compressible or quasi-incompressible hyperelastic finite deformation problems where the strain is large.     

Alternative reduced integration avoiding spurious modes for 8-node quadrilateral and 20-node hexahedron finite elements

Integrating the isoparametric 8-node quadrilateral and the 20-node hexahedron elements with Gauss integration based on the 3 point rule produces stiff elements. The excessive stiffness is mainly due to locking phenomenon. One remedy to partly remove locking consists in using reduced integration. Mostly, 2 × 2 or 2 × 2 × 2 integration, respectively, is employed. The lower order integration introduces spurious element modes, however. These modes may deteriorate solutions for finite element models. To overcome this drawback alternative reduced integration procedures are presented. A 5-point rule for the quadrilateral is described. 9-point and 21-point procedures are introduced for the hexahedron. The performance of these procedures is studied by some test problems.     

基于四边形面积坐标的广义协调轴对称单元

利用了点组合广义协调和周广义协调条件,基于四边形面积坐标方法构造了含内参的4结点四边形空间轴对称单元AQACQ6。通过进一步对内参应变矩阵进行合理修正,从而形成新单元AQACQ6M,该单元能够通过强式分片检验。两种单元的位移场都达到对整体坐标的二次完备。数值算例表明:上述轴对称单元不仅精度高,而且抗网格畸变和几乎不可压缩问题能力优于等参单元,显示了面积坐标和广义协调理论的优越性。     

Development of eight-node quadrilateral membrane elements using the area coordinates method

 Two eight-node quadrilateral elements, namely, AQ8-I and AQ8-II, have been developed using the quadrilateral area coordinate and generalized conforming methods. Some appropriate examples were employed to evaluate the performance of the proposed elements. The numerical results show that the proposed elements are superior to the standard eight-node isoparametric element, thereafter called Q8. This is because the former does not only possess the same accuracy as the latter when regular meshes are employed for analysis, but is also very insensitive to mesh distortion, for which the Q8 element can not handle. It has also been demonstrated that the area coordinate method is an efficient tool for developing simple, effective and reliable serendipity plane membrane elements. Received 11 August 1999     

A family of simple and robust finite elements for linear and geometrically nonlinear analysis of laminated composite plates

A family of simple, displacement-based and shear-flexible triangular and quadrilateral flat plate/shell elements for linear and geometrically nonlinear analysis of thin to moderately thick laminate composite plates are introduced and summarized in this paper.

The developed elements are based on the first-order shear deformation theory (FSDT) and von-Karman’s large deflection theory, and total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from Timoshenko’s laminated composite beam functions, thus convergence can be ensured theoretically for very thin laminates and shear-locking problem is avoided naturally.

The flat triangular plate/shell element is of 3-node, 18-degree-of-freedom, and the plane displacement interpolation functions of the Allman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. The flat quadrilateral plate/shell element is of 4-node, 24-degree-of-freedom, and the linear displacement interpolation functions of a quadrilateral plane element with drilling degrees of freedom are taken as the in-plane displacements.

The developed elements are simple in formulation, free from shear-locking, and include conventional engineering degrees of freedom. Numerical examples demonstrate that the elements are convergent, not sensitive to mesh distortion, accurate and efficient for linear and geometric nonlinear analysis of thin to moderately thick laminates.     

A ‘FE-Meshfree’ TRIA3 element based on partition of unity for linear and geometry nonlinear analyses

A new 3-node triangular element is developed on the basis of partition of unity (PU) concept. The formulation employs the parametric shape functions of classical triangular element (TRIA3) to construct the PU and the least square point interpolation method to construct the local displacement approximation. The proposed element synergizes the individual merits of finite element method and meshfree method. Moreover, the usual linear dependence problem associated with PU finite elements is eliminated in the present element. Application of the element to several linear and geometric nonlinear problems shows that the proposed element gives a performance better than that of classical linear triangular as well as linear quadrilateral elements, and comparable to that of quadratic quadrilateral element. The proposed element does not necessitate a new mesh or additional nodes in the mesh. It uses the same mesh as the classical TRIA3 element and is able to give more accurate solution than the TRIA3 element.     

Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM‐II)

A novel strategy for developing low‐order membrane elements with analytical element stiffness matrices is proposed. First, some complete low‐order basic analytical solutions for plane stress problems are given in terms of the new quadrilateral area coordinates method (QACM‐II). Then, these solutions are taken as the trial functions for developing new membrane elements. Thus, the interpolation formulae for displacement fields naturally possess second‐order completeness in physical space (Cartesian coordinates). Finally, by introducing nodal conforming conditions, new 4‐node and 5‐node membrane elements with analytical element stiffness matrices are successfully constructed. The resulting models, denoted as QAC‐ATF4 and QAC‐ATF5, have high computational efficiency since the element stiffness matrices are formulated explicitly and no internal parameter is added. These two elements exhibit excellent performance in various bending problems with mesh distortion. It is demonstrated that the proposed strategy possesses advantages of both the analytical and the discrete method, and the QACM‐II is a powerful tool for constructing high‐performance quadrilateral finite element models. Copyright © 2008 John Wiley & Sons, Ltd.     

采用面积坐标和基于假设转角的薄板元

采用四边形面积坐标方法,从假设转角位移场入手构造了两个广义协调四边形4结点薄板单元AΨQ-I和AΨQ-II。通过采用边界协调条件一次项与二次项分别协调使转角场实现了三次完备。与DKQ等同类单元相比,单元的精度和抗网格畸变能力都有很大提高。     

A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field

An 8‐node quadrilateral plane finite element is developed based on a novel unsymmetric formulation which is characterized by the use of two sets of shape functions, viz., the compatibility enforcing shape functions and completeness enforcing shape functions. The former are chosen to satisfy exactly the minimum inter‐ as well as intra‐element displacement continuity requirements, while the latter are chosen to satisfy all the (linear and higher order) completeness requirements so as to reproduce exactly a quadratic displacement field. Numerical results from test problems reveal that the new element is indeed capable of reproducing exactly a complete quadratic displacement field under all types of admissible mesh distortions. In this respect, the proposed 8‐node unsymmetric element emerges to be better than the existing symmetric QUAD8, QUAD8/9, QUAD9, QUAD12 and QUAD16 elements, and matches the performance of the quartic element, QUAD25. For test problems involving a cubic or higher order displacement field, the proposed element yields a solution accuracy that is comparable to or better than that of QUAD8, QUAD8/9 and QUAD9 elements. Furthermore, the element maintains a good accuracy even with the reduced 2× 2 numerical integration. Copyright © 2003 John Wiley & Sons, Ltd.     

A shape‐free 8‐node plane element unsymmetric analytical trial function method

The unsymmetric FEM is one of the effective techniques for developing finite element models immune to various mesh distortions. However, because of the inherent limitation of the metric shape functions, the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions. In this paper, by introducing the analytical trial function method used in the hybrid stress‐function element method, an effort was made to naturally eliminate these defects and improve accuracy. The key point of the new strategy is that the monomial terms (the trial functions) in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems. Furthermore, some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourth‐order completeness in Cartesian coordinates. The resulting element model, denoted by US‐ATFQ8, can still work well when interpolation failure modes for original unsymmetric element occur, and provide the invariance for the coordinate rotation. Numerical results show that the exact solutions for constant strain/stress, pure bending and linear bending problems can be obtained by the new element US‐ATFQ8 using arbitrary severely distorted meshes, and produce more accurate results for other more complicated problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Composite delamination modelling using a multi-layered solid element

This paper focuses on the latest development of a solid hexahedron element for composite delamination analysis. The 8-node solid is derived from a 20-node hexahedron. It is transformed into two physical independent 4-node shell elements according to the propagation of delamination process within the element.     

An assumed-stress hybrid 4-node shell element with drilling degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or ‘drilling’ degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not ‘lock’, possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.     

Axisymmetric quadrilateral elements for large deformation hyperelastic analysis

In this paper, axisymmetric 8-node and 9-node quadrilateral elements for large deformation hyperelastic analysis are devised. To alleviate the volumetric locking which may be encountered in nearly incompressible materials, a volumetric enhanced assumed strain (EAS) mode is incorporated in the eight-node and nine-node uniformly reduced-integrated (URI) elements. To control the compatible spurious zero energy mode in the 9-node element, a stabilization matrix is attained by using a hybrid-strain formulation and, after some simplification, the matrix can be programmed in the element subroutine without resorting to numerical integration. Numerical examples show the relative efficacy of the proposed elements and other popular eight-node elements. In view of the constraint index count, the two elements are analogous to the Q8/3P and Q9/3P elements based on the u–p hybrid/mixed formulation. However, the former elements are more straight forward than the latter elements in both formulation and programming implementation.     

An incompatible and unsymmetric four-node quadrilateral plane element with high numerical performance

Recent studies show that the unsymmetric finite element method exhibits excellent performance when the discretized meshes are severely distorted. In this article, a new unsymmetric 4-noded quadrilateral plane element is presented using both incompatible test functions and trial functions. Five internal nodes, one at the elemental central and four at the middle sides, are added to ensure the quadratic completeness of the elemental displacement field. Thereafter, the total nine nodes are applied to form the shape functions of trial function, and the Lagrange interpolation functions are adopted as the incompatible test shape functions of the internal nodes. The incompatible test displacements are then revised to satisfy the patch test. Numerical tests show that the present element can provide very good numerical accuracy with badly distorted meshes. Unlike the existing unsymmetric four-node plane elements in which the analytical stress fields are employed, the present element can be extended to boundary value problems of any differential equations with no difficulties.     

A 20‐node hexahedron element with enhanced distortion tolerance

This paper presents a distortion resistant 20‐node hexahedron element that employs two different sets of shape functions for the trial and test functions. The formulation seeks to satisfy the continuity and completeness requirements by exploiting the intrinsic properties of these two sets of shape functions. Several test problems are used to assess the performance of the element under various mesh distortions. The ability of the proposed as well as the classical 20‐node element to maintain solution accuracy under severe mesh distortions has been studied. The proposed element exhibits a very high tolerance to mesh distortions. In particular, for problems involving linear and quadratic displacement fields, the element is capable of reproducing exact solution under all admissible geometrical distortions of the mesh. For test problems involving higher‐order displacement fields, the performance of the present element is in general better than that of the classical element. Copyright © 2004 John Wiley & Sons, Ltd.