复合材料双曲度大网格高精度单元研究

基于四节点Timoshenko层合梁理论和八节点薄壳单元相关理论, 通过自编四节点板单元和八节点壳单元, 实现了加筋前后双曲度复合材料薄壳单元的大网格高精度计算。结果表明: 八节点单元精度稍高, 但四节点单元分析的时间更短, 因而效率更高, 更具有工程实用性; 筋条网格尺寸大小对计算精度影响小。研究成果可用于大规模复合材料双曲度结构设计和分析。     

Stability of collapsed 8-node 2-D and 20-node 3-D isoparametric finite elements

The collapse of ordinary finite elements to generate the desired strain singularity at the crack tip for fracture mechanics applications can lead to unwanted additional singularities in that area. Although neither the quarter-point nor the half-point 8-node two-dimensional (2-D) and 20-node three-dimensional (3-D) elements exhibit this behaviour, the present article proves that arbitrarily small deviations from the quarter-point element can be constructed which do have additional singularities. Since the general behaviour of the half-point element is not affected by small modifications, this element is better suited to match complex body geometries.     

Modified and Trefftz unsymmetric finite element models

The unsymmetric finite element method employs compatible test functions but incompatible trial functions. The pertinent 8-node quadrilateral and 20-node hexahedron unsymmetric elements possess exceptional immunity to mesh distortion. It was noted later that they are not invariant and the proposed remedy is to formulate the element stiffness matrix in a local frame and then transform the matrix back to the global frame. In this paper, a more efficient approach will be proposed to secure the invariance. To our best knowledge, unsymmetric 4-node quadrilateral and 8-node hexahedron do not exist. They will be devised by using the Trefftz functions as the trial function. Numerical examples show that the two elements also possess exceptional immunity to mesh distortion with respect to other advanced elements of the same nodal configurations.     

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.     

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。     

Solid elements with rotational degrees of freedom: Part 1—hexahedron elements

This is the first of a two part paper on three-dimensional finite elements with rotational degrees of freedom (DOF). Part I introduces an 8-node solid hexahedron element having three translational and three rotational DOF per node. The corner rotations are introduced by transformation of the midside translational DOF of a 20-node hexahedron element. The new element produces a much smaller effective band width of the global system equations than does the 20-node hexahedron element having midside nodes. A small penalty stiffness is introduced to augment the usual element stiffness so that no spurious zero energy modes are present. The new element passes the patch test and demonstrates greatly improved performance over elements of identical shape but having only translational DOF at the corner nodes.     

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.     

Plane isoparametric hybrid-stress elements: Invariance and optimal sampling

Formulation and applications of the hybrid-stress finite element model to plane elasticity problems are examined. Conditions for invariance of the element stiffness are established for two-dimensional problems, the results of which are easily extended to three-dimensional cases. Next, the hybrid-stress functional for a 3-D continuum is manipulated into a more convenient form in which the location of optimal stress/strain sampling points can be identified. To illustrate these concepts, 4- and 8-node plane isoparametric hybrid-stress elements which are invariant and of correct rank are developed and compared with existing hybrid-stress elements. For a 4-node element, lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation. Alternative 8-node elements are considered, and the best invariant element is shown to be one in which the stress compatibility equations are invoked. Results are also presented which demonstrate the validity of the optimal sampling points, the effects of reduced orders of numerical integration, and the behaviour of the elements for nearly incompressible materials.     

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.     

Finite element dispersion analysis for the three-dimensional second-order scalar wave equation

The dispersive properties of finite element semidiscretizations of the three-dimensional second-order scalar wave equation are examined for both plane and spherical waves. This analysis throws light on the performance and limitations of the finite element approximation over the entire spectrum of wavenumbers and provides guidance for optimal mesh discretization as well as mass representation. The 8-node trilinear element, 20-node serendipity element, 27-node triquadratic element and the linear and quadratic spherically symmetric elements are considered.     

Control of spurious mechanisms for 20-node and transition sub-integrated hexahedral elements

In this paper, control of spurious mechanisms for sub-integrated 20-node and transition hexahedral elements is devised by an assumed stress approach. The higher-order stress modes for stabilizing the sub-integrated elements are identified by examining the spurious mechanisms. With an admissible simplification of the flexibility matrix, row vectors of the leverage matrix can be used as stabilization vectors. Numerical examples for the stabilized 20-node element are presented. Accuracy of the derived element is far better than the fully integrated displacement elements. Meanwhile, the former also consumes marginally less CPU time than the latter.     

The use of ANSYS to calculate the behaviour of sandwich structures

In this article, we use different models to compute displacements and stresses of a simply supported sandwich beam subjected to a uniform pressure. 8-node quadrilateral elements (Plane 82), multi-layered 8-node quadrilateral shell elements (Shell 91) and multi-layered 20-node cubic elements (Solid 46) are used. The influence of mesh refinement and of the ratio of Young's moduli of the layers are studied. Finally, a local Reissner method is presented and assessed, which permits an improvement in the accuracy of interface stresses for a high ratio of Young's moduli of the layers with Plane 82 elements.     

Three-dimensional hybrid-stress isoparametric quadratic displacement elements

Two 20-node quadratic displacement three-dimensional isoparametric elements are developed based on the hybrid-stress model. The elements differ in the stress interpolation used. In one case, the stress polynomials are selected to correspond approximately to the strain polynomials obtained from the displacement field, and a 57β stress field results. In the other element, complete cubic polynomials are used which are forced to satisfy the equilibrium equations and stress compatibility equations, and a 69β stress field results. Both elements possess correct rank, but only the 69β element is invariant. Results obtained using these two elements, and the corresponding 20-node assumed-displacement element, are compared and the 69β element is shown to be the better element. The 2 × 2 × 2 Gauss stations are also verified to be the optimal sampling points for these elements.     

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

 Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using <ty>Mathematica<ty> and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. Received: 8 January 2002 / Accepted: 12 July 2002 The support of NSF under grant number DMI-9820880 is gratefully acknowledged.     

粗网格划分下的箱梁三维实体有限元分析方法

针对现有箱梁分析方法普遍存在的计算精度与计算效率之间矛盾的问题,提出了粗网格划分下的箱梁三维实体有限元分析方法。在充分考虑箱梁受力变形特点的基础上,以修正的Hellinger-Reissner变分原理为基础,通过合理引入非协调位移插值项,构造出直角坐标系下的六面体八结点杂交应力单元8N21β和柱坐标系下的六面体八结点杂交应力单元8N21βc,分别用于粗网格划分下的直箱梁和曲线箱梁的三维实体有限元分析。数值算例表明:8N21β单元和8N21βc单元在粗网格划分下具有较高的计算精度,能有效提高箱梁三维实体有限元分析的计算效率。     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.     

Element by element a posteriori error estimation and improvement of stress solutions for two-dimensional elastic problems

A quantitative method of a posteriori error estimation based on finite element solutions is developed. The proposed method can estimate the error of solutions obtained by the finite element method for two-dimensional elastic problems with 4-node elements. Since the error is estimated element by element, the proposed method does not require a large memory or long computing time. Not only rectangular elements but also arbitrarily shaped 4-node elements can be used in this method for estimating the error in the finite element computation with high accuracy. The finite element solutions are improved by adding the estimated errors onto the original solutions. The proposed method can be utilized for any type of linear problem if an isoparametric finite element method is used.     

Problems related to application of eigenstrains in a finite element analysis

Accorsi has recently proposed a general method for modelling microstructural material discontinuities in a finite element analysis. In this work, it is shown that his method can be significantly simplified. In particular, the transformation strain (eigenstrain) plays no role in his application. Numerical results based on the simplified equations with 8-node and 9-node isoparametric elements are presented for the same examples as in his papers. The accuracy of the results is examined in comparison with the solution obtained from a complex mesh model.     

实体退化板单元及其在板的振动分析中的应用

经典板壳单元是由板壳理论构造出来的,而经典的板壳理论是在空间弹性理论的基础上考虑板壳的基本假定得来的。在空间等参数单元的基础上,直接引入板壳的基本假定,修改空间等参数单元的弹性矩阵,从而构造出适合于厚薄板壳分析的20结点实体退化板单元,并将其应用于开口圆柱薄壳的静力分析和厚薄板的固有振动分析。数值算例表明,该单元收敛快,稳定性好,具有较高的精度。此外,该单元还可以用于曲边变厚度板、壳体及层合板的振动分析。     

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.