Linear and Geometrically nonlinear analysis of plates and shells by a new refined non-conforming triangular plate/shell element

A refined non-conforming triangular plate/shell element for linear and geometrically nonlinear analysis of plates and shells is developed in this paper based on the refined non-conforming element method (RNEM). A conforming triangle membrane element with drilling degrees of freedom in Cartesian coordinates and the refined non-conforming triangular plate-bending element RT9, in which Kirchhoff kinematic assumption was adopted, are used to construct the present element. The displacement continuity condition along the interelement boundary is satisfied in an average sense for plate analysis, and the coupled displacement continuity requirement at the interelement is satisfied in an average sense, thereby improving the performance of the element for shell analysis. Selectively reduced integration with stabilization scheme is employed in this paper to avoid membrane locking. Numerical examples demonstrate that the present element behaves quite satisfactorily either for the linear analysis of plate bending problems and plane problems or for the geometrically nonlinear analysis of thin plates and shells with large displacement, moderate rotation but small strain.     

含两个分量的四边形单元面积坐标理论

为了便于构造抗畸变的四边形单元,建立了一套新的四边形单元面积坐标理论(QAC-2),并给出了相关的积分和微分公式。该坐标系作为自然坐标,具有明确的物理意义,且只含有两个相互独立的坐标分量,因此易于实现与直角坐标和等参坐标的沟通,便于理解和应用;两个坐标分量与直角坐标之间满足线性变换,在构造单元时易于选择完备的多项式序列,且多项式的完备次数不会随着网格的畸变而下降,因此可以保证单元的精度和抗畸变性能。     

Three-dimensional 8-node and 20-node refined hybrid isoparametric elements

In this paper a new hybrid variational principle with independent variables of strain, stress and displacement and with a weaker constraint condition of interelement continuity is proposed. Based on this functional, a general formulation of a refined hybrid isoparametric element method has been established by the orthogonal approach. The present formulation is a rational approach to be adopted for deriving high-performance three-dimensional hybrid isoparametric elements even up to the higher-order 20-node element. Several numerical examples are presented to show that the present elements RGH₈(8-node) and RGH₂₀(20-node) have high accuracy, excellent computational efficiency and less sensitivity to mesh distortion.     

Simple cylindrical and spherical finite elements

Finite element approximations are developed for three‐dimensional domains naturally represented in either cylindrical or spherical coordinates. Lines of constant radius, axial length, or angle are used to represent the domain and cast approximations that are natural for these geometries. As opposed to general isoparametric three‐dimensional elements generated in conventional parent space, these elements can be evaluated analytically and do not generate geometric discretization error. They also allow for anisotropic material coefficients that are frequently aligned in either cylindrical or spherical coordinates. Several examples are provided that show convergence properties and comparison with analytical solutions of the Poisson equation.     

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.     

一种新型的C_1阶协调任意四边形薄板弯曲单元

提出一种由单元协调边界位移直接插值单元位移的特殊插值法,并用该方法构造出一种新型的12节点参数C1阶协调任意四边形薄板弯曲单元。该特殊插值法分离了薄板单元完备性条件和C1阶连续条件的相互影响,从而才能直接构造出C1阶连续协调且完备薄板单元。理论证明该薄板单元具有完备性和C1阶连续性,数值分析表明其性能明显优于非协调单元。     

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

Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal's theorem: any 4‐node, 8‐DOF membrane element will either lock in in‐plane bending or fail to pass a C₀ patch test when the element's shape is an isosceles trapezoid. In this paper, a 4‐node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4‐node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates (x,y) and the second form of quadrilateral area coordinates (QACM‐II) (S,T) are applied together. The resulting element US‐ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first‐order patch test and the second‐order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM‐II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well‐known contradiction defined by MacNeal's theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element analysis of thermal stresses for laminated composites in terms of a new C0-type Reddy's theory

This article proposed an accurate C⁰-type Reddy's theory including effects of the transverse normal thermal strain to study the thermal behaviors of laminated composite plates. Although transverse normal thermal strain was taken into account, displacement variables are not increased as thermal loads can be included in the generalized force vector. Based on the proposed model, an eight-node quadrilateral isoparametric element is presented, in which the interelement C⁰ continuity conditions are satisfied. Numerical results show that the proposed model can produce accurate responses of laminated composite plates under temperature loads.     

基于整体-局部位移假设的高阶理论及其三角形板单元

本文推导一种基于整体-局部位移假设的高阶理论, 该理论满足层间位移、应力连续条件, 满足上、下自由表面条件。建立基于此高阶理论的三节点三角形层合板单元。数值计算结果表明此高阶理论能很好地描述剪切变形效应, 该位移单元不仅能很好地计算整体位移参数, 而且能很好地计算横向剪切应力。      

Variational principles for non‐conforming finite element methods

Variational principles with relaxed inter‐element continuity requirement for non‐conforming element methods in linear and non‐linear analyses are developed. Based on the principles, any non‐conforming element displacement can be used directly to derive the explicit expressions of non‐conforming displacement function, which can ensure the passage of the patch test C for the requirement of convergence Copyright © 2001 John Wiley & Sons, Ltd.     

On an isoparametric finite-element for composite laminates with finite rotations

For composite laminates consisting of an arbitrary number of orthotropic laminae first a finite-rotation theory is presented as basis of isoparametric finite-element formulations. The derivation is achieved by a Reissner-Mindlin type kinematic assumption which allows a constant shear deformation distribution across the thickness. The constitutive equations are presented in a general form such that orthotropic material behaviour with material axes varying arbitrarily across the thickness may easily be considered in numerical implementation, also when using curvilinear coordinates. Special attention is taken to predict the force distribution in the deformed shell structure. This theory is then transformed into a four-node isoparametric assumed-strain finite element. Unlike in the degeneration approach, interpolation polynomials are introduced directly for rotation variables determining the deformed position of the unit normal vector. The capability of the finite element developed to deal with strongly nonlinear situations is demonstrated by many examples. Also numerical results are presented permitting a systematical comparison of classical and isoparametric approaches concerning the numerical efficiency.     

Twenty-four–DOF four-node quadrilateral elements for gradient elasticity

Computational analysis of gradient elasticity often requires the trial solution to be C¹, yet constructing simple C¹ finite elements is not trivial. In this paper, three four-node 24-DOF quadrilateral elements for gradient elasticity analysis are devised by generalizing some of the advanced element formulations for thin-plate analysis. These include the discrete Kirchhoff method, a relaxed hybrid-stress method, and the hybrid-stress method with equilibrating internal force modes. The first two methods start with the derivation of a C⁰ displacement, which is quadratic complete in the Cartesian coordinates. In the first method, at the midside points are derived and interpolated together with those at the nodes. Strain is derived from the displacement interpolation yet the second-order displacement derivatives are derived from the displacement-gradient interpolation. In the second method, only the assumed constant double-stress modes are employed to enforce the continuity of the normal derivative of the displacement. In the third method, the equilibrating internal force modes require the C¹ displacement to be defined only along the element boundary and the domain interpolation can be avoided. Patch test involving linear stress and constant double stress as well as other tests are presented to validate the proposed elements.     

Stress and velocity in 2D transient elastodynamic analysis by the boundary element method

This paper is mainly concerned with the development of integral equations to compute stress and velocity components in transient elastodynamic analysis by the boundary element method. All expressions required are presented explicitly. The boundary is discretized by linear isoparametric elements whereas linear and constant time interpolation are assumed, respectively, for the displacement and traction components. Time integration is carried out analytically and the resulting expressions are presented. An assessment of the accuracy of the results provided by the present formulation can be seen at the end of the article, where two examples are presented.     

一种有效的分析任意空间形状曲杆单元的位移函数

利用经典弹性理论和微分几何、矩阵方法等数学理论,基于空间自然坐标系和随体坐标系,通过求解应变与位移之间关系的微分方程,得到了一种能完全反映任意空间形状圆截面曲杆单元刚体位移和常应变等模式的位移函数。给出了两个算例,通过将采用导出的位移函数建立的有限元解与解析解进行了比较,验证了它的正确性,同时通过将基于位移函数导出的有限元解与应用软件得到的解进行了比较,其计算精度,特别是应力计算精度大为改善,验证了它的有效性。由于计算效率高,提出的曲杆单元可望在三维大曲率井的钻柱非线性分析等工程实际中发挥作用。     

Refined nine-parameter triangular thin plate bending element by using refined direct stiffness method

Based on a new generalized variational principle, a refined direct stiffness method (RDSM) which can be directly used to improve non-conforming elements is proposed. The formulation is similar to that of the direct stiffness method (DSM), but the constraint condition of interelement continuity is satisfied in an average sense and as a result convergence and high accuracy are insured. The well-known BCIZ nine-parameter triangular thin plate bending element is refined by the RDSM to yield a new nine-parameter thin plate bending element RT₉. Numerical examples are presented to show that the present model passes the patch test and possesses high accuracy.     

基于高阶剪切变形理论的复合材料层合板精化单元法

基于高阶剪切变形层合板理论(该理论满足层间位移、应力连续条件)建立了一种精化方法,由此建立了三角形精化板单元.该单元满足单元间C1类弱连续条件,其收敛性得到保证,且具有列式简单、计算效率和精度高的优点.      

Dual analysis of plane stress problems by commonly based finite elements

Finite element models for linear elastic plane stress problems which provide, alternatively, a completely compatible displacement field or a precisely equilibrated stress field are developed. The basis for both models is a biquintic interpolation polynomial representing Airy's stress function over a triangular region. The polynomial coefficients are modified and grouped to establish compatibility while retaining equilibrium within each element. Nodal kinematic parameters are selected and matched to the stress function for the compatible (modified stiffness) model, while nodal stress function parameters are chosen for the equilibrium (modified flexibility) model. Constraints on the global freedoms, enforced by Lagrange multipliers, are introduced to augment nodal connectivity in establishing interelement compatibility in the ‘stiffness’ model and uniform stress transmission in the ‘flexibility’ model. Appropriate boundary conditions are formed for each model. Numerical solutions are obtained and assessed.     

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.     

采用面积坐标的四边形板弯曲单元

本文采用四边形面积坐标，并应用广义协调法构造出一个具有１２个自由度的四边形板弯曲单元。单元的挠度场以面积坐标多项式表示，对应于直角坐标ｘ，ｙ的完全三次式和部分四次式，因而单元是完备的广义协调的板单元。应用的１２个协调条件为挠度的四个点协调条件和四个边协调条件，以及法向转角的四个边协调条件。由于面积坐标和直角坐标之间为线性变换关系，因此单元刚度矩阵的推导相当简单。数值算例表明：本文单元具有高精度、收敛性、可靠性和对网格畸变不敏感的优点