An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Development of Revolute-Joint Element with Rotation Limits, Locking, Resistive Moment and Damping

. The revolute-joint is an important component of a mechanical system. Thus, the motion simulation of multibodies connected with revolute-joints is of common interest. A simple revolute-joint can be easily represented in a numerical way, whereby the bending moment at the particular joint equals zero. Formulation becomes complicated when involving bending limits, the resistive moment and the damping effect. Furthermore, several bodies connected with revolute-joints form a complex system. This has been particularly useful for simulating a multibody system. A novel revolute-joint model is proposed in this paper. Several possible methods are suggested to handle bending limits. The incor-poration of the resistive moment and damping effects are also presented. The numerical stability of the revolute element is discussed, and a stability criterion suggested. The revolute-joint element is incorporated into an explicit finite difference code developed by the authors. Numerical examples are shown for different methods and conditions.     

FEM considering local bending effects for flat sliding isolators

Presented in this paper are an advanced analytical model and finite element formulations including the local bending moment effect for the flat sliding system. This study reveals that the local bending effect is a very important factor. The results presented in this paper show that the design of the isolation without considering this effect may endanger structures during earthquakes. The new formulations taking into consideration the local effect provide an effective tool for the more realistic simulation of the behavior of flat-sliding-isolated structures subjected to earthquake loadings. The proposed finite element formulations can be applied directly for both two- and three-dimensional analyses without any further assumption, and can generally be placed at any location in the structure. The implementation of the proposed techniques in existing finite element computer codes is a simple task. Examples of structures equipped with flat-sliding isolators is also given to demonstrate the paramount findings through the proposed formulations. Results obtained from the numerical analyses suggest that the local bending moment effects are extremely important, and should be taken into account to assure the safety of isolated-structures during earthquakes.     

A third Strang lemma and an Aubin–Nitsche trick for schemes in fully discrete formulation

In this work, we present an abstract error analysis framework for the approximation of linear partial differential equation problems in weak formulation. We consider approximation methods in fully discrete formulation, where the discrete and continuous spaces are possibly not embedded in a common space. A proper notion of consistency is designed, and, under a classical inf–sup condition, it is shown to bound the approximation error. This error estimate result is in the spirit of Strang’s first and second lemmas, but applicable in situations not covered by these lemmas (because of a fully discrete approximation space). An improved estimate is also established in a weaker norm, using the Aubin–Nitsche trick. We then apply these abstract estimates to an anisotropic heterogeneous diffusion model and two classical families of schemes for this model: virtual element and finite volume methods. For each of these methods, we show that the abstract results yield new error estimates with a precise and mild dependency on the local anisotropy ratio. A key intermediate step to derive such estimates for virtual element methods is proving optimal approximation properties of the oblique elliptic projector in weighted Sobolev seminorms. This is a result whose interest goes beyond the specific model and methods considered here. We also obtain, to our knowledge, the first clear notion of consistency for finite volume methods, which leads to a generic error estimate involving the fluxes and valid for a wide range of finite volume schemes. An important application is the first error estimate for multi-point flux approximation L and G methods.     

Lateral buckling analysis of beams by the fem

Two new finite element formulations for the calculation of the lateral buckling load for elastic straight prismatic thin-walled open beams under conservative static loads, are presented. The stability criterion used is based on the positive definiteness of the second variation of the total potential energy. One formulation is suitable for sections where the initial bending is about a dominant major axis. The other finite element formulation takes account of initial bending curvature and essentially takes the form of a quadratic eigenvalue problem. Both formulations are tested with problems that have classical solutions or experimentally determined results and are shown to be accurate.     

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist. This work was partially supported by the NSF grants DMS-0410266 and DMS-0710831.     

A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections

In this paper a finite element formulation of eccentric space curved beams with arbitrary cross-sections is derived. Based on a Timoshenko beam kinematic, the strain measures are derived by exploitation of the Green-Lagrangean strain tensor. Thus, the formulation is conformed with existing nonlinear shell theories. Finite rotations are described by orthogonal transformations of the basis systems from the initial to the current configuration. Since for arbitrary cross-sections the centroid and shear center do not coincide, torsion bending coupling occurs in the linear as well as in the finite deformation case. The linearization of the boundary value formulation leads to a symmetric bilinear form for conservative loads. The resulting finite element model is characterized by 6 degrees of freedom at the nodes and therefore is fully compatible with existing shell elements. Since the reference curve lies arbitrarily to the line of centroids, the element can be used to model eccentric stiffener of shells with arbitrary cross-sections.     

The three-dimensional beam theory: Finite element formulation based on curvature

The article introduces a new finite element formulation of the three-dimensional ‘geometrically exact finite-strain beam theory’. The formulation employs the generalized virtual work principle with the pseudo-curvature vector as the only unknown function. The solution of the governing equations is obtained by using a combined Galerkin-collocation algorithm. The collocation ensures that the equilibrium and the constitutive internal force and moment vectors are equal at a set of chosen discrete points. In Newton’s iteration special update procedures for the pseudo-curvature and rotational vectors have to be employed because of the non-linearity of the configuration space. The accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples.     

基于组合杂交法的四边形薄板弯曲有限元

在研究工程结构的问题中,为了建立简睁高效的四边形薄板弯曲有限元算法,一种基于组合杂交变分原理的新型四边形杂交有限元CH18P被设计出来。单元首先独立插值力矩和挠度形函数,然后在能量协调的概念下使用挠度对力矩进行约束从而得到优化的力矩模式。单元只要求挠度及其偏导数在单元顶点处连续,从而降低了挠度设计的难度。由于挠度弱协调,并引入组合系数和能量协调条件,和其它杂交元相比,计算花费小,无条件稳定,并且保证收敛性。此外,单元还有很高的计算精度,并且对网格形状不敏感．实验证明是一种实用的有限元算法。     

空间机械臂柔性连杆的弯曲形状

研究单连杆旋转柔性空间机械臂的弯曲形状,以确定空间柔性机械臂末端的位置.在考虑连杆变形对力矩影响和曲率表示式中不忽略形状函数一阶导数平方的情况下,利用Taylor展式表示连杆的变形,导出了连杆变形的数学模型.通过降阶方法导出形状函数一阶导数的解析式,并给出了由此解析式利用数值积分确定形状函数的方法.给出了利用二分法确定变形后的连杆在其固连坐标系中最大横坐标的方法.针对一组参数进行了仿真,证实了这种方法确定空间柔性机械臂末端的位置的可行性.     

Isoparametric axisymmetric shell elements with temperature gradients for heat conduction

This paper presents a finite element formulation for axisymmetric shell heat conduction where temperature gradients through the shell thickness are retained as primary nodal variables. The element geometry is constructed using the coordinates of the nodes lying on the middle surface of the shell and the middle surface nodal point normals. The element temperature field is approximated in terms of element approximation functions, the nodal temperature, and the nodal temperature gradients. The weak formulation of the two-dimensional Fourier heat conduction equation in cylindrical coordinate system is constructed. The finite element properties of the shell element are then derived using the weak formulation and the element temperature field approximation. The formulation permits linear temperature gradients through the shell thickness. Distributed heat flux as well as convective boundaries are permitted on all four faces of the element. Furthermore, the element can also have internal heat generation as well as orthotropic material properties. The superiority of the formulation in terms of efficiency and accuracy is demonstrated. Numerical examples are presented and a comparison is made with the theoretical results.     

Finite element dynamic analysis of geometrically exact planar beams

We present a new strain-based finite element formulation for the dynamic analysis of highly flexible elastic planar beams. The formulation employs the geometrically exact Reissner planar beam theory which accounts for finite displacements and rotations, and finite membrane, shear and bending strains. The system of semi-discrete dynamic equations of motion is derived from the modified Hamilton principle in which only the strain variables are interpolated. Such a choice of the interpolated variables is an advantage over approaches, in which the displacements and rotations are interpolated, since the field consistency problem and related locking phenomena do not arise. The performance and accuracy of the formulation are illustrated by several numerical examples.     

Elastic-plastic analysis of composite beams with incomplete interaction by finite element method

This paper presents a nonlinear finite element analysis of composite beams with incomplete interaction. A simplified nonlinear model is assumed in this approach. This is applied to the elastic-plastic analysis of reinforced concrete beams and composite beams with incomplete interaction. The numerical results are compared with the test results and existing values based on other numerical methods, and found to be in good agreement. The elastic-plastic behavior of partial composite beams without shear connectors in the negative bending moment region is discussed by the proposed method.     

Stabilized finite element formulation for elastic–plastic finite deformations

This paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic–plastic solids, such as metals. An updated Lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation. This formulation circumvents the restriction on the displacement and pressure fields due to the Babuška–Brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit, typical in metal forming applications. Moreover, it facilitates the use of low order simplex elements (i.e. P1/P1), reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary. Linearization of the weak form is derived for implementation into a finite element code. Numerical experiments with P1/P1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis.     

Discrete energy method for the analysis of cylindrical shells

This paper presents an investigation into the use of the closely associated finite difference technique for the analysis of shell structures as a feasible alternative to the finite element method. The method discretises the total energy of the structure into energy due to extension and bending and that due to shear and twisting, contributed by two separate sets of rectangular elements formed by a suitable finite difference network. The derivatives in the corresponding energy expressions are replaced by their finite difference forms and the nodal displacements then constitute the undetermined parameters in the variational formulation. The formulation is also extended to a cylindrical shell element of rectangular planform. The results obtained by DEM are compared with existing results and they show excellent agreement.     

A spline wavelet finite element formulation of thin plate bending

The wavelet scaling functions of spline wavelets are used to construct the displacement interpolation functions of triangular and rectangular thin plate elements. The displacement shape functions are then expressed by spline wavelet functions. A spline wavelet finite element formulation of thin plate bending is developed by using the virtual work principle. Two numerical examples have shown that the bending deflections and moments of thin plates agree well with those obtained by the differential equations and conventional elements. It is demonstrated that the current spline wavelet finite element method (FEM) can achieve a high numerical accuracy and converges fast. The proposed spline wavelet finite element formulation has a wide range of applicability since it is developed in the same way like conventional displacement-based FEM.     

Response surface methodology for design of sheet forming parameters to control springback effects

This paper deals with the optimization of tools geometry in sheet metal forming in order to reduce the springback effects after forming. A response surface method (RSM) based on diffuse approximation is used; this technique has been proved more efficient than classical gradient based methods since it requires fewer iterations and convergence is guaranteed especially for nonlinear problems. A new improved Inverse Approach for the stamping simulation based on DKTRF shell element is presented. In the new version, the strains and stresses due to bending and unbending effects are calculated analytically from the final workpiece, especially on the die entrance radii for curvature changes. The bending/unbending moments and the final shape are used to calculate springback using a second incremental Approach based on the Updated Lagrangian Formulation. The benchmark on the “U” bending problem of NUMISHEET’93 has been used to validate the method, good results on the elimination of springback have been obtained. The final results are validated using STAMPACK^® and ABAQUS^® commercial codes.     

Higher order stabilized finite element method for hyperelastic finite deformation

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.