Base force element method of complementary energy principle for large rotation problems

Using the concept of the base forces, a new finite element method （base force element method, BFEM） based on the complementary energy principle is presented for accurate modeling of structures with large displacements and large rotations. First, the complementary energy of an element is described by taking the base forces as state variables, and is then separated into deformation and rotation parts for the case of large deformation. Second, the control equations of the BFEM based on the complementary energy principle are derived using the Lagrange multiplier method. Nonlinear procedure of the BFEM is then developed. Finally, several examples are analyzed to illustrate the reliability and accuracy of the BFEM.     

基面力概念在几何非线性余能有限元中的应用

以基面力为基本未知量描述一个弹性系统的应力状态并表征单元的余能,将大变形的余能分解为变形余能部分和转动余能部分,采用Lagrange乘子法放松单元的平衡方程,利用已有的弹性大变形余能原理建立了一种几何非线性显式有限元模型,编制了相应的几何非线性余能原理有限元程序. 数值算例表明:该方法具有较好的收敛性和计算精度,可进行大载荷步的大位移、大转动计算.      

基于余能原理的有限变形问题有限元列式

利用基面力概念,推导了一种基于余能原理的有限变形问题显式有限元列式,可应用于结构的大位移、大转动问题。以基面力为状态变量来表达单元的余能,将有限变形情况下的单元余能分解为变形余能部分和转动余能部分,利用Lagrange乘子法推导出余能原理有限元的控制方程,编制了相应的非线性有限元程序。通过算例分析,说明该列式和程序的可靠性和精确性。     

基于基面力概念的一种新型余能原理有限元方法

在高玉臣(2003)提出的单元柔度矩阵表达式的基础上,基于基面力的概念,利用余能原理和Lagrange乘子法推导出以基面力为基本未知量的余能原理有限元支配方程张量表达式和节点位移表达式,编制出相应的有限元程序,通过对一些典型的弹性理论问题的计算分析,数值解与理论解相吻合。研究表明,这种方法简单而有效,是有限元方法的一种新思路,具有较好的应用前景。     

基于扩展有限元的应力强度因子的位移外推法

针对平面裂纹问题,阐述了扩展有限元法的单元位移模式、推导了扩展有限元法的控制方程、介绍了特殊单元的数值积分技术.基于最小二乘法,建立了应力强度因子位移外推法的计算公式.利用MATLAB编写计算程序,对平面裂纹问题用扩展有限元法进行了计算.基于扩展有限元法的计算结果,分别利用位移外推法和相互作用积分法,对平面裂纹的应力强度因子进行了计算.计算结果表明,位移外推法比相互作用积分法能更方便和准确地计算平面裂纹的应力强度因子.     

黏弹性界面断裂分析的增量“加料”有限元法

提出了一种适用于黏弹性界面裂纹问题的增量“加料” 有限元方法. 利用弹性界面裂纹尖端位移场的解答,通过对应原理和拉普拉斯逆变换近似方法,得到了黏弹性界面裂纹的尖端位移场. 用该位移场构造了黏弹性界面裂纹“加料” 单元和过渡单元位移模式,推导了增量“加料” 有限元方程,求解有限元方程可获得应力强度因子和应变能释放率等断裂参量. 建立了典型黏弹性界面裂纹平面问题“加料” 有限元模型,计算结果表明,对于弹性/黏弹性界面裂纹和黏弹性/黏弹性界面裂纹,该方法都能得到相当精确地断裂参量,并能很好地反映蠕变和松弛特性,可推广应用于黏弹性界面断裂问题的计算分析.      

基于欧拉描述的两节点索单元非线性有限元法

本文针对柔性悬索结构几何非线性分析的特点,提出了一种用欧拉描述来表示的两节点索单元非线性有限元模型,在索元变形后的位置上由虚功能建立了非线有限元基本方程及切线刚度矩阵。这样建立的非线性有限元分析方法可充分考虑拉索的几何非线性特性的影响并给悬索结构的初始平衡分析带来方便,算例结果表明,本文方法是精确有效的。     

板弯曲与平面弹性有限元的同一性

本文建立平面弹性与板弯曲的相似理论，并用于将平面弹性的单元移植到板弯曲元，从而其分片试验，收敛性等性质也同时移植到板弯曲元，使两者处于同一水平上，同时又将此基于力法的板弯曲元入位移法的轨道，找出共相应的位移系统，并证明其适定性，从而为将此类单元装入位移法通用程序系统创造了条件。     

Non-linear stability analysis of imperfect thin-walled composite beams

The static non-linear behavior of thin-walled composite beams is analyzed considering the effect of initial imperfections. A simple approach is used for determining the influence of imperfection on the buckling, prebuckling and postbuckling behavior of thin-walled composite beams. The fundamental and secondary equilibrium paths of perfect and imperfect systems corresponding to a major imperfection are analyzed for the case where the perfect system has a stable symmetric bifurcation point. A geometrically non-linear theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field. An initial displacement, either in vertical or horizontal plane, is considered in presence of initial geometric imperfection. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results are presented for a simply supported beam subjected to axial or lateral load. It is shown in the examples that a major imperfection reduces the load-carrying capacity of thin-walled beams. The influence of this effect is analyzed for different fiber orientation angle of a symmetric balanced lamination. In addition, the postbuckling response obtained with the present beam model is compared with the results obtained with a shell finite element model (Abaqus).     

A finite element finite-strain formulation for modeling colliding blocks of Gent materials

The present paper is devoted to the analysis of the contact/impact problems with Coulomb friction and large deformation between two hyperelastic bodies of Gent model. The total Lagrangian formulation is adopted to describe the geometrically non-linear behavior. For the finite element implementation, the explicit expression of the incremental law of Gent model is derived. A first order algorithm is applied for the numerical integration of the time-discretized equation of motion. Efficiency and accuracy of the resulting method is illustrated on a two-dimensional static contact problem and a three-dimensional dynamic contact problem as compared with ANSYS simulations.     

The fundamental equations in finite element method of coupled thermo-elastic plane problem

The fundamental equations in finite element method for unsteady temperature field elastic plane problem are derived on the bases of variational principle of coupled thermoelastic problems. In these derivations, elastic plane is divided into three nodes triangular elements, and time interval is divided into linear time elements, in which all the variables, including displacements and temperatures at various nodal points, are varied linearly with time. Two coupled sets of linear algebraic equations of all the unknown displacements and temperatures at every nodal point in every instant (i.e. the terminal values of time elements) are obtained. They are the fundamental equations of the said problem.The total energy in elastic body not only contains the potential energy and heat energy but also contains the kinetic energy, if the rate of change of temperature field with respect to the time in thermoelastic problem is large enough. And the change of displacement is included in the equations of heat conduction. For this reason the variational principle of coupled thermoelastic problems is employed. [1] In this paper, expressions of this principle for plane problems are given. The discretization is carried on then, and Hamilton's action and the potential action of heat flow of elements are derived. Finally they are assembled, so as to get the polar value of the action. And thus the groups of linear algebraic equations in matrix form are obtained.     

On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables

A complementary-dual force-based finite element formulation is proposed for the geometrically exact quasi-static analysis of one-dimensional hyperelastic perfectly flexible cables lying in the two-dimensional space. This formulation employs as approximate functions the exact statically admissible force fields, i.e., those that satisfy the equilibrium differential equations in strong form, as well as the equilibrium boundary conditions. The formulation relies on a principle of total complementary energy only expressed in terms of force fields, being therefore called a pure principle. Under the assumption of stress-unilateral behavior, this principle can be regarded as being dual to the principle of minimum total potential energy, corresponding therefore to a maximum principle. Some numerical applications, including cables suspended from two and three points at the same level or at different levels, with both Hookean and Neo-Hookean material behaviors, are presented. As it will be shown, in contrast to the standard two-node displacement-based formulation derived from the principle of minimum total potential energy, the proposed dual force-based formulation is capable of providing the exact solution of a given problem only using a single finite element per cable. Both the proposed principle of pure complementary energy and its corresponding force-based finite element formulation can be easily extended to the case of cables lying in the three-dimensional space.     

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions.     

Time discontinuous Galerkin finite element method for dynamic analyses in saturated poro-elasto-plastic medium

A time-discontinuous Galerkin finite element method for dynamic analyses in saturated poro-elasto-plastic medium is proposed. As compared with the existing discontinuous Galerkin finite element methods, the distinct feature of the proposed method is that the continuity of the displacement vector at each discrete time instant is automatically ensured, whereas the discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously reduced, particularly, for material non-linear problems. Both the implicit and explicit algorithms to solve the derived formulations for material non-linear problems are developed. Numerical results show a good performance of the present method in eliminating spurious numerical oscillations and providing with much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. The project supported by the National Natural Science Foundation of China (19832010, 50278012, 10272027) and the National Key Basic Research and Development Program (973 Program, 2002CB412709)     

加强板的弯矩函数列式

本文首先谇薄板弯曲问题矩函数的物理意义,据此,将弯矩函数列式推广到具有加强条的薄板弯曲问题,给出了与平面弹性问题完全对应的余能原理。     

Complementary energy theorem in geometrically non-linear structural problems

The principle of stationary complementary energy for geometrically non-linear problems of beams and trusses is considered. The functional of complementary energy is formulated in terms of force quantities. The stationary conditions describe the conditions of compatibility of the structure. The applications include the problem of elastica, a truss, and a bending problem by use of beam elements.     

Finite deformations of Ogden's materials under impact loading

The present paper is devoted to the modeling of finite deformations of a hyperelastic body described by Ogden's model under contact/impact conditions. Frictional contact problems are solved by means of the bi-potential method. The first order algorithm is applied to integrate the equation of motion. The total Lagrangian formulation is adopted to describe the geometrically non-linear behavior. For the finite element implementation, the explicit expression of the tangent operator is derived including the case of repeated eigenvalues. A numerical example is given to illustrate efficiency and accuracy of the method.     

用于几何非线性分析的内参型非协调元法

本文把内参型非协调元的理论和方法推广应用于几何非线性分析。对方法的合理性、可行性、单元收敛条件及实用简化方案等诸方面作了初步探讨。文章以大位移平面问题为例进行了详细讨论,给出了有限元非线性列式的全过程和数值结果。     

弹塑性体中波传播问题的间断Galerkin有限元法

对结构动力学和波传播问题提出了一个时域间断的Galerkin有限元法．其主要特点是对问题的半离散场方程的节点基本未知向量及其时间导数向量在时间域中分别采用三次多项式和线性(P3-P1)插值，节点基本未知(位移)向量在离散的时间段之间将自动保证连续，而仅仅是它的时间导数(速度)向量存在间断．在非线性条件下，与现有的间断Galerkin有限元法相比。明显地节省了计算工作量．对所提出的间断Galerkin有限元法发展了弹塑性非线性问题的隐式和显式算法．数值计算结果表明了所提出方法的有效性，以及相对基于连续Galerkin有限元法的Newmark算法的计算结果的优越性．