A FORMULATION OF THE QS6 ELEMENT FOR LARGE ELASTIC DEFORMATIONS

The numerical simulation of processes undergoing finite deformations requires robust elements. For a broad range of applications these elements should have a good performance in bending dominated situations as well as in the case of incompressibility. The element should be insensitive against mesh distortions which frequently occurs during finite deformations. Furthermore, due to efficiency reasons a good coarse mesh accuracy in required in non-linear analysis. The QS6 element, developed in this paper, tries to fulfil the above-mentioned requirements. The performance is depicted by means of numerical examples.     

大转动平面梁有限元分析的共旋坐标法

虽然大转动平面梁单元已有很多,但其中许多太复杂,缺乏计算效率,值得改进。采用共旋坐标法准确的首次导出了平面梁单元发生大转动小应变时的非对称单元切线刚度矩阵,利用这一非对称的单元切线刚度矩阵由Newton-Raphson迭代法编制了一个FORTRAN程序NPFSAP,并获得了大转动梁、方形和圆形框架的高精度数值解,表明了这种非线性单元列式的正确性和非线性求解过程的收敛性,非对称单元切线刚度矩阵值得推介。     

塑性大变形有限元静水压力的改进算法

本文针对塑性大变形有限元中静水压力计算精度低的问题,研究了三维八节点等参元的偏应力导数佳点,提出了一种改进的静水压力间接积分算法。将该算法应用于金属塑性成型过程的有限元模拟,较好地解决了大步长情况下应力计算精度低的问题。     

A THREE-DIMENSIONAL FINITE ELEMENT FORMULATION FOR THERMOVISCOELASTIC ORTHOTROPIC MEDIA

This paper is concerned with the development of a numerical algorithm for the solution of the uncoupled, quasistatic initial/boundary value problem involving orthotropic linear viscoelastic media undergoing thermal and/or mechanical deformation. The constitutive equations, expressed in integral form involving the relaxation moduli, are transformed into an incremental algebraic form prior to development of the finite element formulation. This incrementalization is accomplished in closed form and results in a recursive relationship which leads to the need of solving a simple set of linear algebraic equations only for the extraction of the finite element solution. Use is made of a Dirichlet–Prony series representation of the relaxation moduli in order to derive the recursive relationship and thereby eliminate the storage problem that arises when dealing with materials possessing memory. Three illustrative example problems are included to demonstrate the method. © 1997 John Wiley & Sons, Ltd.     

A FINITE ELEMENT FORMULATION FOR 3-D CONTINUA USING THE CO-ROTATIONAL TECHNIQUE

The co-rotational technique is described for the three-dimensional analysis of continua. The technique exploits the proven technology of the best continua elements for linear analysis which are embedded into a formulation that applies an element-attached local co-ordinate frame that continuously moves and rotates with the element. The geometric non-linearity is then incorporated via the rotation of this local system. The method uses similar procedures to those recently described for 2-D continua elements but introduces concepts from a more conventional ‘continuum mechanics’ approach. The general framework for the co-rotational procedure is kept. However, a much neater formulation is derived, which readily allows the extension from two to three dimensions.     

A NEW FINITE ELEMENT FORMULATION FOR PLANAR ELASTIC DEFORMATION

For the stress analysis of planar deformable bodies, we usually refer to either plane stress or plane strain hypothesis. Three-dimensional analysis is required when neither hypothesis is applicable, e.g. bodies with finite thickness. In this paper, we derive an ‘exact’ solution for the plane stress problem based on a less restrictive hypothesis than σ_z=0. By requiring the out-plane stress σ_z to be a harmonic function, the three-dimensional solution is obtained. In addition, we present a two-dimensional finite element for planar analysis of problems where the thickness of the body 2h is comparable to other characteristic dimensions. This element is presented as a substitute for classical plane stress and plane strain finite elements. The typical plane stress and plane strain state are recovered in the case where h→0 and the case h→∞, respectively. As an example for the application of such formulation, the behaviour of a concrete gravity dam is investigated. It is shown that this structure, typically analysed by using plane strain hypothesis, has its out-plane stress underestimated. © 1997 John Wiley & Sons, Ltd.     

A CO-ROTATIONAL FORMULATION FOR 2-D CONTINUA INCLUDING INCOMPATIBLE MODES

The co-rotational finite element method is well known for decomposing the motion of an element into a rigid body motion and a strain-producing deformation. A key feature is the definition of the rotating frame and, partially for this reason, the method has almost exclusively been applied to beams and shells. A novel feature of the current paper is that the method is applied to continua—in the current case, two-dimensional. The main motivation is to allow the analyst to quickly introduce the latest and best linear elements into a non-linear context. To this end, in the current work, a set of ‘incompatible modes’ or ‘enhanced strains’ is added to the conventional four-noded elements. While the main body of the paper considers small strains, as a further novel aspect, it later applies the co-rotational method to problems with large strains (here via hyperelasticity) and, to this end, establishes a link between the co-rotational technique and a Biot stress formulation.     

一个用于大位移大转动非线性动力计算的显式梁元

为了数值模拟建筑结构倒塌过程中的梁柱构件,建立了一个具有大位移大转动非线性动力计算能力的显式梁元。该梁元基于显式有限元单元理论,采用更新拉格朗日列式,考虑了转动的不可交换性,选用共旋(Co-Rotational)方法分离单元刚体位移和变形位移,通过应力更新算法来考虑材料的非线性。算例表明该梁元力学性能良好,具有一定的工程应用价值。     

AN UPPER BOUND FINITE ELEMENT PROCEDURE FOR SOLVING LARGE PLANE STRAIN DEFORMATION

A unique and robust upper bound finite element procedure is developed for the analysis of large plastic deformation problems under plane strain condition. It can consistently treat problems with isotropic strain varying materials. It can also effectively solve problems with any initial ‘guessed’ velocity field, even from an random number generator. To explore and demonstrate the capability of this new approach, strip tension and plane strain compression problems are solved. For validation, the computed results are compared with existing analytical or experimental solutions in good agreement. The phenomenon of shear band formation can be simulated and, as expected, is found to develop more distinctly in strain softening materials than in perfectly plastic and strain hardening materials. © 1997 by John Wiley & Sons, Ltd.     

压电复合梁热机电耦合有限元模型

压电材料应用于航天结构形状或振动控制时,可能会受到热场、力场和电场的共同作用。为分析处于热场、力场和电场共同作用下的压电复合结构,文中基于高阶剪切变形理论、高阶电势模型和线性温度分布假设,利用虚功原理建立了压电复合梁结构的热-机-电耦合有限元模型。该模型可应用于热机电耦合压电复合结构的形状与振动控制研究。利用本文模型对压电双晶片梁、压电复合悬臂梁进行了数值仿真,仿真结果与文献给出的理论结果和实验值吻合良好,表明本文模型是正确有效的。     

A NEW FORMULATION OF ISOPARAMETRIC FINITE ELEMENTS AND THE RELATIONSHIP BETWEEN HYBRID STRESS ELEMENT AND INCOMPATIBLE ELEMENT

A new method of formulating isoparametric finite element is developed, and the element strains are proposed to be resolved into two parts, constant part and higher-order one. The new method indicates two important properties of isoparametric finite element, and the equivalent relationship between hybrid stress elements and incompatible elements. © 1997 by John Wiley & Sons, Ltd.     

A SIMPLE AND EFFICIENT MIXED FINITE ELEMENT FOR AXISYMMETRIC SHELL ANALYSIS

An efficient, perhaps simplest, three-noded mixed finite element is proposed for axisymmetric shell analysis. The key feature in the present formulation is to start with a better variational principle in which the independent unknowns are only the quantities that can be prescribed at the shell edges. If the consistency for field approximations is satisfied, no other numerical consideration is necessary in the present element. Several examples confirm the satisfactory numerical behaviour of the present mixed element.     

A SPACE-TIME COUPLED p-VERSION LEAST SQUARES FINITE ELEMENT FORMULATION FOR UNSTEADY TWO-DIMENSIONAL NAVIER–STOKES EQUATIONS

This paper presents a p-version least squares finite element formulation for two-dimensional unsteady fluid flow described by Navier–Stokes equations where the effects of space and time are coupled. The dimensionless form of the Navier–Stokes equations are first cast into a set of first-order differential equations by introducing auxiliary variables. This permits the use of C⁰ element approximation. The element properties are derived by utilizing the p-version approximation functions in both space and time and then minimizing the error functional given by the space–time integral of the sum of squares of the errors resulting from the set of first-order differential equations. This results in a true space–time coupled least squares minimization procedure. The application of least squares minimization to the set of coupled first-order partial differential equations results in finding a solution vector {δ} which makes gradient of error functional with respect to {δ} a null vector. This is accomplished by using Newton's method with a line search. A time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step. Equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance. The space–time coupled p-version approximation functions provide the ability to control truncation error which, in turn, permits very large time steps. What literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space–time coupled approach. The generality, success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity. The results are compared with the analytical solutions and those reported in the literature. The formulation presented here is ideally suited for space–time adaptive procedures. The element error functional values provide a mechanism for adaptive h, p or hp refinements.     

SPACE–TIME SPECTRAL ELEMENT METHOD FOR OPTIMAL SLEWING OF A FLEXIBLE BEAM

A new computational approach to modelling and control of a flexible beam is proposed. The structural modelling and the control design problems are formulated in a unified mathematical framework that allows simultaneous structural and control design iterations that result in an optimal overall system performance. The method employs the space–time spectral elements for simultaneous space and time discretizations of a Timoshenko beam model. Dimensionless equations of motion are derived using Hamilton's principle of variable action and an integral formulation in the framework of space–time spectral elements is introduced. An optimal control problem formulated for the continuum model is transformed by the space–time spectral element formulation into an optimization problem in a finite-dimensional parameter space. Dynamic programming is then used to obtain both open and closed loop control laws. A simulation study shows good performance of the control law applied to the nominal model. It is also demonstrated that proper discretization yields performance robustness of the system with respect to modal truncation.     

DIRECT FORMULATION FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL GROUNDWATER POLLUTION MODELLING WITH A COMPARISON WITH CONVENTIONAL FINITE ELEMENT METHOD

Real world ground water pollution modelling deals with solute transport through anisotropic, heterogeneous media. The applicability of analytical solutions for such a real world system is extremely limited. As an effective tool, numerical models, such as finite difference and finite element methods, are usually employed to model field scenarios. Nevertheless, ground water pollution modelling is a hallenging task and frequently ends up with misleading results. Most of the time insufficient data are blamed for such erratic results. A recent investigation shows that the shortcomings of numerical formulations may be the major cause for many disputes and confusions in numerical analyses. In reality, a point injection of water in a static, homogeneous and isotropic groundwater system shows a radial dissipation of water forming a sphere; and a full-depth line injection shows a radial dissipation forming a cylinder. The finite difference method completely ignores this fundamental flow principles and allows water only to flow along orthogonal directions. To overcome this limitation, the finite element method was developed as a flexible approach in order to connect a node with the neighbouring nodes in various directions where water is assumed to flow in any directions along node connections. In a recent investigation, it has been found that the conventional finite element method does not keep the commitments; and its formulation techniques lead to a global matrix where a solution domain is not connected with all the neighbouring nodes and does not comply with the control-volume mass balance concept. A consistent finite element formulation approach which does not need imaginary mathematical formulation and overcomes the limitations of both the conventional finite difference and finite element methods has been developed. This method allows fluid flow and solute transport in a porous medium in radial directions. The global matrices for flow and transport obtained from this technique are field representative, diagonally dominant and easily convergent. The new method is robust, needs less mathematical computation and has many advantages over the conventional finite difference and finite element methods.     

考虑剪切效应三维纤维梁单元的几何非线性增量有限元分析

该文以三维连续介质力学和虚功原理为基础,推导了增量U.L.有限元列式,该列式保留了大位移刚度矩阵项,并对该刚度矩阵进行修正使其成为对称矩阵。根据增量U.L.列式,推导了三维纤维梁单元的刚度矩阵。该单元采用平截面假定,以轴向节点位移表示截面上任意一点的位移,并结合Timoshenko梁理论来考虑剪切效应。以上原理编制分析程序,通过几个算例分析,证明了该方法的精确性、通用性。