A mathematical programming algorithm for limit analysis

This paper deals with the limit analyses of perfect rigid-plastic continua. Based on the kinematic theorem of the limit analysis theory, a mathematical programming finite element formula for determining the upper bound load multiplier has been established, and an iteration algorithm proposed accordingly. In this algorithm the plastic and rigid zones are distinguished for every iteration step, and the goal function is modified gradually. The difficulties caused by the nonsmoothness of the goal function are overcome. Some examples solved by this algorithm are presented. The project supported by National Natural Science Foundation of China.     

基于自然单元法的极限上限分析

自然单元法是一种基于离散点集的Voronoi图和Delaunay三角化几何信息,以自然邻近插值为试函数的新型数值方法.相对于一般无网格法中常采用的移动最小二乘近似而言,自然邻近插值不涉及到复杂的矩阵求逆运算,更不需要任何人为的参数,可以提高计算效率.采用该方法构造的形函数满足Delta函数的性质,可以像有限元一样准确地施加边界条件,可以方便处理场函数及其导数的不连续性的问题.论文将自然单元法应用到极限上限分析中,编制了相应的计算程序,通过极限分析的几个经典算例进行了验证,同时采用类似于分片应力磨平的方式,编制相应的磨平程序,由计算点上的塑性耗散功外推得到了节点上的塑性耗散功的值,从而画出了极限状态下结构的塑性耗散功的分布云图.计算结果表明采用自然单元法求解极限上限分析具有稳定性好,精度高,收敛快等优点.     

Computational limit analysis of rigid-plastic bodies in plane strain

This paper deals with the computational methods for limit analysis of plane strain problems. The finite element mathematical programming formula (FE-MPF) for determining the upper bound load multiplier established by the authors earlier is adopted and modified for plane strain problems. The penalty method is used to impose the incompressibility constraint. The FE-MPF is solved by a direct iteration procedure without the need of a searching process. This algorithm is not sensitive to the volumetric locking effect. And it can be easily extended to the limit analysis of three dimensional problems. The results of numerical examples are satisfactory and show the stable convergency of the present algorithm.     

基于自然单元法的轴对称结构极限上限分析

自然单元法是一种以自然邻近插值为试函数的新兴无网格数值方法,其形函数的计算不涉及矩阵求逆,也不需要任何人为参数。为了充分发挥自然单元法的优势,本文基于极限分析上限定理建立了轴对称结构极限上限分析的整套求解算法。轴对称结构的位移场由自然邻近插值构造,并且采用罚函数法处理材料的不可压条件。为了消除目标函数非光滑所引起的数值困难,采用逐步识别刚性区和塑性区,并对两者用不同方法进行处理。数值算例结果表明,本文提出的轴对称结构极限上限分析方法是行之有效的。     

轴对称结构极限下限分析的自然单元法

作为一种基于自然邻近插值的新型无网格法,自然单元法克服了大多数无网格法难以施加本质边界条件的困难.将自然单元法与减缩基技术相结合,建立了一种轴对称结构极限下限分析的数值格式和求解算法.通过不断修正自平衡应力场,轴对称结构极限下限分析可转化为一系列的非线性数学规划子问题,并由复合形法求解.在每个非线性规划子问题中,自平衡应力场表示为一组带有待定系数的自平衡应力场基矢量的线性组合,并且这些自平衡应力场基矢量可由弹塑性增量分析的平衡迭代结果得到.算例结果表明,本文所提的轴对称结构极限下限分析方法行之有效.     

A general algorithm for plastic flow simulation by finite element limit analysis

Limit analysis has been rendered versatile in many structural and metal forming problems. In metal forming analysis, the slip-line method and the upper bound method have filled the role of limit analysis. As a breakthrough of the previous work, a computational approach to limit solutions is considered as the most challenging area.In the present work, a general algorithm for limit solutions of plastic flow is developed with the use of finite element limit analysis. The algorithm deals with a generalized Hölder inequality, a duality theorem, and combined smoothing and successive approximation in addition to a general procedure for finite element analysis. The algorithm is robust such that from any initial trial solution, the first iteration falls into a convex set which contains the exact solution (s) of the problem. The idea of the algorithm for limit solutions is extended from rigid⧹perfectly plastic materials to work-hardening materials by the nature of the limit formulation, which is also robust with numerically stable convergence and highly efficient computing time.     

弯管结构塑性极限分析的数值方法及应用

从塑性极限分析数值计算的角度,分析了多组载荷联合作用下弯管结构的塑性极限承载能力。为了克服塑性极限上限分析中目标函数非线性非光滑所导致的数值困难,提出了一种弯管结构塑性极限上限分析的无搜索优化迭代算法;采用一种改进的弯管单元并利用加载路径的径向射求解方案处理多组载荷系统。通过对典型弯管结构进行塑性极限分析得出了一些有价值的结论。     

理想塑性结构极限与安定分析的数值方法

极限分析和安全分析的近代发展方向是寻找通用性强，计算效率高的数值方法。本文介绍将有限单元法和数学规划法相结合的、同时适用于极限分析和安全分析的统一数值方法，包括下限格式和上限格式。     

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.     

Un algorithme efficace d'intégration plastique pour un matériau obéissant au critère anisotrope de HillAn efficient algorithm for plastic integration of material satisfying the Hill's anisotropic yield criterion.

This Note deals with an efficient algorithm to carry out the plastic integration and compute the stresses due to large strains for materials satisfying the Hill's anisotropic yield criterion. The classical algorithm of plastic integration such as ‘Return Mapping Method’ is largely used for nonlinear analyses of structures and numerical simulations of forming processes, but it requires an iterative schema and may have convergence problems. A new direct algorithm based on a scalar method is developed which allows us to directly obtain the plastic multiplier without an iteration procedure; thus the computation time is largely reduced and the numerical problems are avoided. To cite this article: I. Titeux et al., C. R. Mecanique 332 (2004).     

Kinematic limit analysis of frictional materials using nonlinear programming

In this paper, a nonlinear numerical technique is developed to calculate the plastic limit loads and failure modes of frictional materials by means of mathematical programming, limit analysis and the conventional displacement-based finite element method. The analysis is based on a general yield function which can take the form of the Mohr–Coulomb or Drucker–Prager criterion. By using an associated flow rule, a general nonlinear yield criterion can be directly introduced into the kinematic theorem of limit analysis without linearization. The plastic dissipation power can then be expressed in terms of kinematically admissible velocity fields and a nonlinear optimization formulation is obtained. The nonlinear formulation only has one constraint and requires considerably less computational effort than a linear programming formulation. The calculation is based entirely on kinematically admissible velocities without calculation of the stress field. The finite element formulation of kinematic limit analysis is developed and solved as a nonlinear mathematical programming problem subject to a single equality constraint. The objective function corresponds to the plastic dissipation power which is then minimized to give an upper bound to the true limit load. An effective, direct iterative algorithm for kinematic limit analysis is proposed in this paper to solve the resulting nonlinear mathematical programming problem. The effectiveness and efficiency of the proposed method have been illustrated through a number of numerical examples.     

确定复合材料宏观屈服准则的细观力学方法

运用细观力学中的均匀化方法，分析了含周期性微结构复合材料的宏观屈服准则，并对Hill-Tsai准则进行了修正。从基于复合材料细观结构的代表性胞元入手，运用塑性极限理论中的机动分析以及有限元方法，计算了细观结构的极限载荷域。通过宏细观尺度对应关系，得到复合材料的宏观屈服准则。     

Reduced augmented Lagrangian bi-conjugate gradient method for impact-contact problems

To avoid the numerical oscillation of the penalty method and non-compatibility with explicit operators of conventional Lagrange multiplier methods used in transient contact problems to enforce surface contact conditions, a new approach to enforcing surface contact constraints for the transient nonlinear finite element problems, referred to as “the reduced augmented Lagrangian bi-conjugate gradient method (ALCG)”, is developed in this paper. Based on the nonlinear constrained optimization theory and is compatible with the explicit time integration scheme, this approach can also be used in implicit scheme naturally. The new surface contact constraint method presented has significant advantages over the widely adopted penalty function methods and the conventional Lagrangian multiplier methods. The surface contact constraints are satisfied more accurately for each step by the algorithm, so the oscillation of numerical solution for the explicit scheme is depressed. Through the development of new iteration strategy for solving nonlinear equations, ALCG method improves the computational efficiency greatly. Project supported by State Education Commission Doctoral Foundation and Natural Science Foundation of Liaoning Province.     

自然单元法研究进展

自然单元法是一种基于Voronoi图和Delaunay三角化几何结构,以自然邻点插值为试函数的一种新型数值方法.其既具有无网格方法和经典有限元方法的优点,又克服了两者的一些缺陷,是一种发展前景广阔的求解微分方程的数值方法.自然单元法的形函数满足插值性质,可以像有限元法一样直接施加本质边界条件,不存在基于移动最小二乘拟合的无网格方法不能直接施加本质边界条件的难题.由于自然单元法是无网格方法,可以方便处理有限元方法较难处理的一些问题,例如移动边界和大变形等问题.自然单元法与其他数值方法的最根本区别于其插值格式的不同.将自然邻点插值用于Galerkin过程,就得到基于Voronoi结构的自然单元Galerkin法.自然邻点插值有自然邻点Sibson插值和Laplace插值(非Sibson插值)两种.Laplace插值比Sibson插值在计算上要简单的多,并且不论对凸的或非凸的区域都能精确施加本质边界条件.以Laplace插值为试函数的自然单元法在数值实施上比以Sibson插值为试函数的自然单元法简单.本文对基于Voronoi结构的自然邻点插值和自然单元法的基本思想作了介绍,综述了国内外关于自然单元法的研究成果,总结了自然单元法的优点和尚需解决的问题.      

极限下限分析的正交基无单元Galerkin法

基于极限分析的下限定理,建立了用正交基无单元Galerkin法进行理想弹塑性结构极 限分析的整套求解算法．下限分析所需的虚拟弹性应力场可由正交基无单元Galerkin法直接 得到,所需的自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模 拟．这些自平衡应力场基矢量可由弹塑性增量分析中的平衡迭代得到．通过对自平衡应力场 子空间的不断修正,整个问题的求解将化为一系列非线性数学规划子问题,并通过复合形法 进行求解．算例表明该方法有效地克服了维数障碍问题,使计算效率得到了充分的提高,是 切实可行的．     

动力弹塑性分析的无网格自然单元法

基于无网格自然单元法,提出了结构动力弹塑性响应分析的一条新途径．自然单元法是一种新兴的无网格数值计算方法,其实质是基于自然邻近插值的伽辽金法．自然单元法在本质边界条件的施加上较采用移动最小二乘法的无网格法具有明显的优势．在空间域上采用自然单元法离散,并运用加权余量法推导了动力弹塑性分析的离散控制方程．然后,采用预校正形式的Newmark法在时间域上进行求解．最后给出了数值算例,并验证了所提方法的有效性和正确性．     

三维弹塑性自然单元法算法实现

自然单元法是一种新兴的无网格数值计算方法,其实质是基于自然相邻插值(C∞)的伽辽金法。该方法计算精度与四边形或六面体单元有限元法相当,自然相邻插值函数比其他无网格法插值函数的计算速度快。由于自然相邻插值在凸域的边界上的相邻点之间是严格线性的,所以自然单元法在边界面的处理也相当简单。本文研究了在自然单元法中采用Von.Mises,Mohr-Coulomb和Drucker-Prager屈服准则解决三维弹塑性问题,并编制了相应计算程序,最后通过算例验证算法的正确性。     

A local search algorithm for natural neighbours in the natural element method

A local basis algorithm for searching natural neighbours in Natural Element Method (NEM) is presented for solving the elasticity problems in this paper. Comparison with the global sweep algorithm used in natural element method or Natural Neighbour Method (NNM) for searching natural neighbours, the proposed algorithm is more expedient and convenient in the constructions and computation of natural neighbour interpolations. In the proposed NEM based on local search, the Laplace (non-sibson) interpolations are constructed with respect to the natural neighbour nodes of the given point which have been locally defined. The shape functions from the Laplace approximations have the delta function property and the Laplace interpolants are strictly linear between adjacent nodes, which facilitate imposition of essential boundary conditions and treatment of material discontinuity with ease as it is in the conventional finite element method. The Laplace interpolants derived from the local algorithm and the global algorithm in NEM are identical because of the uniqueness of the Voronoi diagram. Numerical results and convergence studies also show that the present NEM based on local search algorithm possesses the same accuracy and rate of convergence as they are in previous NEM.     

Spinning rods, elliptical disks and solid ellipsoidal bodies: Elastic and plastic stresses and limit spins

The analysis of the stresses in one-, two- and three-dimensional spinning bodies is discussed in a systematic and comprehensive way. First elastic solutions are derived for rods, for elliptical-shaped flat disks and for ellipsoidal solid bodies spinning about their sideways axes. Then the spins for first plastic yield are found in each case using each of the Tresca and the von Mises yield conditions. Then upper and lower bounds on the maximum allowable limit spins where the body would globally fail assuming perfectly plastic behavior are derived. The elastic solutions at first yield always give a lower bound to that limit spin, but global failure generally does not occur until the spin is increased. A way to calculate an improved lower bound is illustrated. Upper bounds are found in a simple and new way. The method uses the fact that the volume-averaged stresses can be calculated directly from the loadings without the need for any actual stress solutions, and then it is proved that the use of those average stresses in the yield functions always gives an upper bound to the limit loads. That use of the statically determinate average stresses to obtain meaningful plastic upper bounds to limit loads is though to be a new method, and can be applied to any shape. Finally, several finite element calculations are used to determine the quantitative relations between the lower and upper bounds and the actual limit spins for ellipsoidal bodies.The results are of interest in the spin of planetary bodies, where they explain the nature of an average-stress approximate method, and in the analysis of spinning bodies in general. In addition, the approach gives a very interesting example of the utility of the limit analysis approaches of plasticity theories.     

用刚体元方法求空间极限平衡问题的上限解

刚体元方法是研究块体系的变形和运动的一种数值方法,可以用它来研究弹塑性物体的极限平衡问题,文中用通用的二维刚体元程序计算了正方形柱体顶面中心区域作用方形基础载荷情况的上限解,结果表明,这种方法较好的精度和广泛的适用性。