牛顿迭代一致性算法及其在板弹塑性有限元分析中的应用

本文简略讨论了有限载荷增量弹塑性有限元分析中传统切线刚度法丧失精度和牛顿迭代平方收敛速度的原因,并提出保持牛顿迭代平方收敛速度、保证一阶精度和无条件稳定性的一致性算法.一致性算法具备以下两个特征:1)采用路径无关计算格式;2)采用一致弹塑性切线模量。根据一致性算法构造出以弯矩和曲率为基本变量的弹塑性板弯曲有限元NIDKQ元。数值结果表明NIDKQ元具有令人满意的精度,同时验证了有限载荷增量下牛顿迭代一致性算法的平方收敛率特性,而传统切线刚度法随着塑性区的扩展将大大降低收敛速度。     

一致切线刚度法在三维弹塑性有限元分析中的应用

本文提出了一致切线刚度法，并把它应用于三维弹塑性有限元分析问题。从而解决了增量迭代弹塑性有限元分析方法中长期存在的速度慢、精度低问题，一致切线刚度法满足加卸载互补准则，即没有应力漂移现象，具有一阶精度、二阶迭代收敛速度、计算量少和无条件稳定等优点，借助算例对一致切线刚度法和传统切线刚度法（包括路径相关和路径无关两种结构变量更新格式）从计算精度、迭代收敛速度和计算量等几方面进行了比较。     

弹塑性有限元增量分析的自动选择步长和用予测本步刚度阵求解

本文利用本步刚度参数(current stiffness parameter)概念,对改进弹塑性有限元增量分析的效率和精度提出了两个具体措施。 1.自动选择每个增量步的步长,可在保证精度的前提下大量缩减总的增量步数,并有效地解决了计算结构极限载荷的问题。 2.用予测获得的本步刚度阵代替现行的起点切线刚度阵求解本步载荷增量,可大量缩减每个步长的迭代次数,提高收敛速度。     

非线性弹性及全量弹塑性有限元分析中的一致性切线模量

本文导出了非线性弹性及全量弹塑性有限元分析中的一致性切线模量,从而可以保持牛顿迭代法固有的平方收敛速度.指出了某些文献中关于切线模量的不正确表述,并以数值算例验证了本文方法的正确有效.     

三维材料非线性问题的分区混合元法

本文构造了用于空间弹塑性分析的杂交混合元。导出了增量形式的分区混合变分原理,由此构造了用于空间弹塑性分析的分区混合元,并提出了一个加速求解非线性有限元方程组的修正的牛顿-拉夫森法迭代收敛的-维搜索格式。文中算例表明:杂交混合元用于空间弹塑性分析可以得到比位移元和杂交元更好的精度;分区混合元兼有较好的精度和算时较省的长处,适用于非线性问题的数值分析;并验证了该一维搜索法是一个有效地加速迭代收敛的方法。     

岩土弹塑性有限元分析的隐式积分弹性刚度法

本文提出了用于岩土弹塑性有限元分析的隐式积分弹性刚度算法。该算法既具有隐式积分法精度好、效率高、无条件稳定等优点，也具有弹性刚度法中刚度矩阵正定、对称的特点，更重要的是它避免了传统切线刚度法在处理岩土非相关联塑性流动和屈服面“角点”所遇到的非对称性和奇异性计算问题。通过算例分析了该算法的精度、效率     

岩土弹塑性有限元分析的隐式积分弹性刚度法

本文提出了用于岩土弹塑性有限元分析的隐式积分弹性刚度算法。该算法既具有隐式积分法精度好，效率高，无条件稳定等优点，也具有弹性刚度矩阵正定、对称的特点，更重要的是它避免了传统切线刚度法在处理岩土非相关联塑性流动和屈服面“角角”所遇到的非对称性和奇异性计算问题。通过算例分析了该算法的精度、效率。     

弹塑性体脆断相场模型的一致性无单元伽辽金法

采用无单元伽辽金法（EFG）对弹塑性体脆性断裂的相场模型进行了数值实现。利用无单元法便于构建高阶近似函数的优势,位移和相场均采用二阶移动最小二乘（MLS）近似。刚度阵的数值积分采用更为高效的二阶一致三点积分格式QC3（Quadratically Consistent 3-point integration scheme）。本构算法采用Newton-Raphson迭代和弹塑性一致性切线模量。数值结果表明了本文方法模拟弹塑性体脆性断裂的有效性。     

求解钢压杆极限承载力的有限条塑性系数增量初应力法

本文提出了有限条塑性系数增量初应力法，用于分析钢压杆的弹塑性稳定极限承载力，该法采用分级加载，用有限条法建立结构的增量平衡方程；在塑性范围，引入截面的塑性系数对弹性刚度进行折减得到结构的弹性刚度矩阵；用修正的Ｎｅｗｔｏ－Ｒａｐｈｓｏｎ方法迭代求解，数值结果表明，本法效率较高，与钢压杆试验结果吻合良好，可以考虑残余应力和载荷偏心的影响，可望实现大型超静定结构的极限载力分析。     

一种高效率弹塑性有限元分析新方法：增量杂交／混合修正弦线模量法

本文在文献[1]给出的放松应力增量平衡约束的修正余能广义变分原理基础上,提出一种高效率的弹塑性有限元分析的新方法——增量杂交/混合修正弦线模量法。该法保持了文[1]方法的全部优点,而在迭代过程中,依据材料的单向拉伸应力—应变关系,不断改变过渡区和塑性区单元柔度矩阵和塑性矩阵中的弹性模量;并在体积压缩模量不变假设下,相应地改变过渡区单元矩阵中的泊松系数。从而大大降低了迭代收敛次数和单刚计算量,提高了多类变量弹塑性有限元分析的计算效率和收敛精度。     

Implicit scheme for integrating constitutive model of unsaturated soils with coupling hydraulic and mechanical behavior

A constitutive model of unsaturated soils with coupling capillary hystere- sis and skeleton deformation is developed and implemented in a fully coupled transient hydro-mechanical finite-element model （computer code U-DYSAC2）. The obtained re- sults are compared with experimental results, showing that the proposed constitutive model can simulate the main mechanical and hydraulic behavior of unsaturated soils in a unified framework. The non-lineaxity of the soil-water characteristic relation is treated in a similar way of elastoplasticity. Two constitutive relations axe integrated by an implicit return-mapping scheme similar to that developed for saturated soils. A consistent tan- gential modulus is derived to preserve the asymptotic rate of the quadratic convergence of Newton＇s iteration. Combined with the integration of the constitutive model, a complete finite-element formulation of coupling hydro-mechanical problems for unsaturated soils is presented. A number of practical problems with different given initial and boundary conditions are analyzed to illustrate the performance and capabilities of the finite-element model.     

无额外自由度广义有限元非线性分析

将无额外自由度的广义有限元法由线弹性分析扩展到弹塑性大变形分析.局部强化函数的构建依赖于已有节点,不引入额外自由度,避免了线性相关性问题.在更新拉格朗日框架下,通过控制方程弱形式的线性化推导得到了节点内力的率形式,并分为材料和几何两部分.考虑超弹性和亚弹-塑性两种材料模型,采用Newton-Raphson迭代求解,给出...     

Olver迭代与Newton迭代的比较

Olver迭代是一个立方收敛的求根公式,而Newton迭代仅是平方收敛,但前者却不如后者为人们所熟知,以至于近来有作者其推导了一个新的高阶迭代公式,而实际就是Olver迭代公式却浑然不如。那么,到底是什么原因导致Olver迭代没有被广大的计算方法教科书介绍呢？本文对Newton迭代与Olver迭代做了详尽的分析,给出了两者各自的精度表达式,并对两者进行了比较,结论是：从计算效率及精度方面综合考虑,Olver迭代公式不如Newton迭代公式实用。     

A generalized approach to formulate the consistent tangent stiffness in plasticity with application to the GLD porous material model

It has been shown that the use of the consistent tangent moduli is crucial for preserving the quadratic convergence rate of the global Newton iterations in the solution of the incremental problem. In this paper, we present a general method to formulate the consistent tangent stiffness for plasticity. The robustness and efficiency of the proposed approach are examined by applying it to the isotropic material with J₂ flow plasticity and comparing the performance and the analysis results with the original implementation in the commercial finite element program ABAQUS. The proposed approach is then applied to an anisotropic porous plasticity model, the Gologanu–Leblond–Devaux model. Performance comparison between the consistent tangent stiffness and the conventional continuum tangent stiffness demonstrates significant improvement in convergence characteristics of the overall Newton iterations caused by using the consistent tangent matrix.     

Analytical removal of singularity and direct evaluation of singular integrals in 3-D elastoplastic finite deformation analysis by BEM

As a further development of the present authors' research work [1,2], in this paper a method of the so-called quadratic pentahedron polar co-ordinate transformation and analytical removal of singularity of Cauchy principal value singular integrals is proposed to evaluate the strongly singular integrals in the sense of Cauchy principal values and the weakly singular integrals over quadratic internal cells in 3-D elastoplastic finite deformation analysis by BEM. First, a quadratic pentahedron polar co-ordinate transformation technique is used to reduce the order of singularity of the singular integrals. Then, a form of Gauss' theorem is introduced to remove the singularity in the Cauchy principal value singular integrals analytically. Therefore, the evaluation of all those strongly and weakly singular integrals can be carried out by standard Gaussian quadrature accurately and efficiently. Numerical examples of the 3-D elastoplastic problem and 3-D finite deformation problem are given to demonstrate that the method possesses good accuracy and numerical stability, and is convenient to implement. The method in this paper can be applied extensively to evaluating the singular integrals over cubic and higher order elements.     

A comparative study of GLS finite elements with velocity and pressure equally interpolated for solving incompressible viscous flows

A comparative study of the bi‐linear and bi‐quadratic quadrilateral elements and the quadratic triangular element for solving incompressible viscous flows is presented. These elements make use of the stabilized finite element formulation of the Galerkin/least‐squares method to simulate the flows, with the pressure and velocity fields interpolated with equal orders. The tangent matrices are explicitly derived and the Newton–Raphson algorithm is employed to solve the resulting nonlinear equations. The numerical solutions of the classical lid‐driven cavity flow problem are obtained for Reynolds numbers between 1000 and 20 000 and the accuracy and converging rate of the different elements are compared. The influence on the numerical solution of the least square of incompressible condition is also studied. The numerical example shows that the quadratic triangular element exhibits a better compromise between accuracy and converging rate than the other two elements. Copyright © 2008 John Wiley & Sons, Ltd.     

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

A Newton multigrid method is developed for one-dimensional (1D) and two-dimensional (2D) steady-state shallow water equations (SWEs) with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.