基于Woodbury非线性方法的迭代算法对比分析

在结构局部非线性求解过程中,刚度矩阵仅部分元素发生改变,此时切线刚度矩阵可写成初始刚度矩阵与其低秩修正矩阵和的形式,每个增量步的位移响应可用数学中快速求矩阵逆的Woodbury公式高效求解,但通常情况下迭代计算在结构非线性分析中是不可避免的,因此迭代算法的计算性能也对分析效率有重要影响。该文以基于Woodbury非线性方法为基础,分别采用Newton-Raphson （N-R）法、修正牛顿法、3阶两点法、4阶两点法及三点法求解其非线性平衡方程,并对比分析5种迭代算法的计算性能。利用算法时间复杂度理论,得到了5种迭代算法求解基于Woodbury非线性方法平衡方程的时间复杂度分析模型,定量对比了5种迭代算法的计算效率。通过2个数值算例,从收敛速度、时间复杂度和误差等方面对比了各迭代算法的计算性能,分析了各算法适用的非线性问题。最后,计算了5种算法求解基于Woodbury非线性方法平衡方程的综合性能指标。     

钢筋混凝土框架结构非线性静、动力分析的高效计算平台HSNAS(GPU)——Ⅰ程序开发

基于传统串行计算平台的有限元分析面临精度不足、耗时巨大的问题成为目前高层钢筋混凝土框架结构非线性动力时程分析面临的瓶颈,利用GPU强大的并行计算能力,开发了一种结构非线性有限元静力、动力分析的高精度和高效率分析平台HSNAS(GPU)。针对静力问题提出了适用于GPU计算,且能有效解决结构负刚度问题的位移增量迭代算法,开发了相应的GPU线性方程组并行求解器;针对动力问题,开发了GPU基于Newmark时间积分算法的动力分析软件平台,结合纤维模型单元技术,引入扭转、剪切变形以及材料非线性。算例模型表明,HSNAS(GPU)平台在满足精度条件下能有效地提高结构非线性静、动力分析的计算效率。     

样条函数线法分析壳体非线性问题

采用样条函数线法计算圆柱壳有具有封闭截面的一类壳体的几何线性问题，用样条函数插值交二维非线性偏微分问题化为一组用径向结线位移增量表示的非线性常微分方程，然后用常微分方程求解器迭代求解，文中导出了用于非线性分析的样条函数线增量方程，最后给出了算例。     

结构动力方程一种新的级数形式的解析解

将结构的位移及速度响应作为状态变量,采用Lyapunov(李雅普诺夫)人工小参数法求解状态方程,导出状态方程的一个新的级数形式的解析解,该解析解还可以推广到非线性动力方程的计算。将秦九韶算法引入级数解的计算,提高了计算的效率和稳定性,同时给出了算法的计算格式和步骤。该算法无需对转换矩阵H求逆,仅使用矩阵向量相乘,计算稳定,精度仅由收敛项数控制,很容易达到任意精度要求,而且适合并行计算及压缩存储。最后通过算例进一步证实了该算法的精度和效率。     

分段线性材料桁架的位移灵敏度

本文推导了分段线性材料桁架结构的位移对截面积的灵敏度，并给出三杆桁架实例。指出了在材料性质发生突变处位移对截面积的灵敏度不连续、位移等值面不光滑：讨论了，采用准则法求解优化问题时的灵敏度计算和迭代公式，旨在指出文献中的错误。     

充气膜结构零应力态求解

充气膜结构分析全过程包括9个状态和7个分析过程,零应力态是结构分析与设计的基础,将结构数值分析态和实际物理态有机联系起来。该文首次提出了一种从弹性平衡态到零应力态的逆解析数值分析方法,非线性协调矩阵广义逆法。首先,用膜线单元模拟膜面,将膜结构转化成为网格结构,由弹性平衡态预张力和材料参数,计算膜线无应力长度和伸长量;然后,基于杆系结构平衡矩阵理论和小变形假定,建立体系的协调方程,由协调矩阵M-P广义逆求解节点位移,逆向叠加求出新状态位形。根据新位形计算膜线张力向量和节点不平衡力向量,迭代求解零应力态。根据该算法,用MATLAB编制了计算程序。算例分析验证了该方法的正确性和高效性。该文对充气膜结构设计具有重要理论意义和实际指导价值。     

“依赖应变历史材料”结构动力松弛法静力分析中规避虚假应变历史的非线性弹性增量算法

针对SHDM结构（由依赖应变历史材料制成的结构）有限元非线性静力平衡方程组动力松弛法（DRM）迭代求解时的积分点应力更新步骤,提出非线性弹性增量算法,即在一个静力增量步内固定材料的加卸载路径,使之在该增量步内成为非线性弹性材料。该应力更新算法能使包括收敛解在内的迭代序列中不含虚假应变历史。此外,该算法还可避免静力解答精度依赖于静力增量步长的局限性。通过三个SHDM结构的数值试验对该算法进行了验证。该算法可望对SHDM结构非线性有限元静力问题DRM分析技术的发展起促进作用。     

弹塑性隐式阻尼迭代法

作者给出了联立求解平衡方程、屈服条件和塑性流动方程的弱解形式, 同时提出了两种位移增量迭代算法:一是以应变增量和塑性因子为独立基本未知量的联立求解迭代算法;二是以应变增量为独立基本未知量的数值求解方法。为了避免弹塑性刚度矩阵的奇异性, 改善迭代收敛速度, 针对上述两种迭代算法本文提出了隐式阻尼迭代法, 并编写了相应脚本文件和计算流程, 并基于有限元程序自动生成系统(FEPG)生成了求解弹塑性问题的有限元源程序。最后, 通过算例验证了所提出的迭代法和计算程序的可靠性和有效性。     

柔性框架结构动力非线性分析的刚体准则法

大跨、高层等柔性结构,其动力响应往往表现出大位移、大转动等非线性特征。动力非线性问题的分析关键在于运动方程的高效稳定求解,以及单元大转动产生的结点力增量的有效计算。动力时程分析通常采用直接积分法,但对于强非线性动力问题,直接积分法难以兼顾计算精度与稳定性。该文基于几何非线性分析的刚体准则,针对杆件结构大转动小应变的非线性问题,提出了一种新型空间杆系结构动力非线性分析的刚体准则法。该方法采用满足刚体准则的空间非线性梁单元,结合HHT-α法求解结构运动方程,并将刚体准则植入动力增量方程的迭代求解过程以计算结点力增量。通过典型柔性框架算例结果表明,该文方法可以有效分析柔性框架结构的强动力非线性行为。与高精度单元相比,该文采用的单元刚度矩阵构造简明,计算过程简洁;与商业软件所用方法相比,单元数和迭代步少,精度高,适于工程应用。     

基于均匀七次B样条插值求解动力响应的逐步积分法

为求解结构动力响应提出新的时间积分法。该方法采用均匀七次B样条插值近似对局部时间域节点位移、速度及加速度进行离散。给出动力平衡方程求解的逆推格式与算法流程。分析表明,通过选择合适参数可提高计算精度,且使算法绝对稳定。计算数值解精度较高,明显优于传统的Newmark法与Wilson-θ法。     

On Numerical Simulation of Cyclic Viscoplastic and Viscoelastic Constitutive Laws with the Large Time Increment Method

Computations on structures having a strongly non-linear time-dependentbehavior, even nowadays, require considerable computational times. Thereduction of the computational costs is crucial for the use ofsimulations in industrial areas. The large time increment method, whichbreaks completely with traditional methods, is developed for thispurpose. This method is based on a two-stage iterative procedure, whichtakes into account the whole load in one increment time, irrespective ofthe loading time.The aim of this paper is to show and understand how this algorithmworks, and to assess its performance, for several classes ofconstitutive laws. This algorithm is tested on one-dimensional periodicloading problems. The theory is developed for a simple viscoelasticmodel and for one viscoplastic model using material state variables, andconstructed in the framework of thermodynamics of irreversibleprocesses.The numerical experiments allowed us to confirm the theoreticalbasis of this algorithm. The results are remarkable since the computingtime is reduced by a factor between 3 and 15 according to the loading,in comparison to other classical methods. Furthermore, the algorithmcorrects rapidly perturbations due to a bad initialisation.On the one hand we show that the Large Time Increment Method can beadapted to a wide range of models. On the other hand the efficiency hasbeen measured in several loading cases and for different constitutivelaws. This opens new research perspectives such as the adaptation ofthis algorithm to a finite elements code, in order to achievethree-dimensional computations.     

基于水平集算法的扩展有限元方法研究

扩展有限元是一种以单位分解思想为基础,在常规有限元位移中加入跳跃函数和渐近位移场函数,以处理不连续问题的数值方法。将水平集算法应用到裂纹界面的描述及加强单元类型的判别,并与扩展有限元相结合,用于分析材料断裂问题。相比传统有限元,有限元网格与裂纹面位置相互独立,不需满足裂纹为单元边、裂尖为单元节点和在裂纹附近进行高密度的...     

A modification of the rigid finite element method and its application to the J-lay problem

This article presents the Rigid Finite Element Method (RFEM), which allows us to take into account the flexibility of a system. Beam-like structures are analyzed, in which large deformations occur. The RFEM has been developed many years ago and successfully applied to practical engineering problems. The main difference between this method and the classical Finite Element Method (FEM) is the element deformation during analysis. In RFEM, the finite elements generated in a discretization process are treated as nondeformable bodies, whilst in FEM the elements are deformable; in RFEM, flexible, mass-less elements with properly chosen coefficients are introduced. A modification of the stiffness coefficients used in RFEM is proposed and explained in the article. It is shown how these new coefficients applied in RFEM lead to the same energy of deformation as in the case when the system is discretized by the classical FEM. This means that the energy of deformation is identical to that obtained in FEM, which leads to identical deformations of the elements. It is of particular importance that the RFEM is a much simpler method, faster in calculations and easier to learn and interpret. Furthermore, the generation of the inertia and stiffness matrices is much faster than in FEM. Another advantage is relatively easy implementation for multicore processor architecture. The calculation examples investigated cover some practical problems related to the offshore pipe laying process. The J-lay method is simulated by the use of the author??s own computer model based on a modified RFEM. The model takes into account wave and sea current loads, hydrodynamic forces and material nonlinearity (plastic strains can develop during large deformation). The simulation results are compared with those obtained from the commercial package ANSYS.     

基于共旋坐标和力插值纤维单元的RC框架结构连续倒塌构造方法

悬链机制会使钢筋混凝土框架结构产生有助于抵抗连续倒塌的附加承载能力,对结构抗连续倒塌能力至关重要。悬链机制处于几何大变形和材料非线性下降段的状态下,需要同时考虑材料非线性和几何非线性,因此对数值分析模型提出了更高的要求。为了解决基于力插值的纤维单元同时处理材料非线性和几何非线性的问题,该文采用基于共旋坐标法,提出了一种基于共旋坐标法的力插值纤维单元。该单元在形成中把变形体和刚体分开,局部坐标系的变形体内采用纤维划分考虑材料非线性,然后加上刚体位移,从局部坐标系到整体坐标系的转换中采用共旋坐标法以考虑几何非线性,给出了二维单元形成原理及非线性求解过程。实例分析结果表明基于共旋坐标法的力插值纤维单元能够较准确的模拟RC框架结构连续倒塌,梁机制阶段主要是材料非线性起控制作用,悬链线机制阶段主要是几何非线性起控制作用。     

A time-based arc-length like method to remove step size effects during fracture propagation

An arc-length like method is presented which alters the size of the time increment when simulating crack propagation problems. By allowing the time increment to change during the time step a constraint can be imposed, which is used to enforce the fracture to propagate a single element length per time step. This removes the effect of the (interface) element size on propagating fractures, and therefore allows smooth fracture propagation during the simulation. The benefits of the scheme are demonstrated for three cases: mode-I crack propagation in a double cantilever beam, a shear fracture including inertial and viscoplastic effects in the surrounding material, and a pressurized fracture inside a poroelastic material. These cases highlight the ability of this scheme to obtain more accurate and nonoscillatory results for the force–displacement relation, to remove numerically induced stepwise fracture propagation, and to allow for arbitrary propagation velocities. An added benefit is that plastic strains surrounding a fracture are no longer affected by the (interface) element size.     

任意截面预应力混凝土细长柱的非线性分析

提出了轴力和双向弯曲作用下任意截面混凝土和预应力混凝土细长柱的非线性有限元计算模型。分析时既考虑了由单元变形和轴力二次矩引起的几何非线性效应,也考虑了由材料非线性应力应变关系和截面刚度矩阵引起的材料非线性效应。推导了非线性全过程分析的标准有限元公式,得到的单元刚度矩阵可分割成三个子矩阵,分别反映了材料非线性、材料非线性和单元大位移的耦合、轴力二次矩等三种不同的非线性作用效应。计算分析结果和试验结果吻合较好。     

Substructuring in the implicit simulation of single point incremental sheet forming

This paper presents a direct substructuring method to reduce the computing time of implicit simulations of single point incremental forming (SPIF). Substructuring is used to divide the finite element (FE) mesh into several non-overlapping parts. Based on the hypothesis that plastic deformation is localized, the substructures are categorized into two groups: the plastic—nonlinear—substructures and the elastic—pseudo-linear—substructures. The plastic substructures assemble a part of the FE mesh that is in contact with the forming tool; they are iteratively updated respecting all nonlinearities. The elastic substructures model the elastic deformation of the rest of the FE mesh. For these substructures, the geometrical and the material behaviour are assumed linear within the increment. The stiffness matrices and the internal force vectors are calculated at the beginning of each increment then they are statically condensed to eliminate the internal degrees of freedom (DOF). In the iteration process the condensed stiffness matrices for the elastic substructures are kept constant. The condensed internal force vectors are updated by the multiplication of the condensed stiffness matrices and the displacement increments. After convergence, any geometrical and material nonlinearity for the elastic substructures are nonlinearly updated. The categorization of substructures in plastic and elastic domains is adapted during the simulation to capture the tool motion. The resulting, plastic and condensed elastic, set of equations is solved on a single processor. In an example with 1600 shell elements, the presented substructuring of the SPIF implicit simulation is 2.4 times faster than the classical implicit simulation.     

Lagrange constraints for transient finite element surface contact

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.     

Fractional derivative and hereditary combined model for memory effects on flexible polyurethane foam

In a quasi-static regime with cyclic loading, the force–displacement curve of flexible polyurethane exhibits complicated behavior: nonlinearity, visco-elasticity, hysteresis, residual force, etc. Beside nonlinearity and visco-elasticity, this material displays high dependence on the displacement rate and past loading history. Its dependence on compression rate helps to appropriately identify the force–displacement curve. Based on the new curve identification, the overall foam response is assumed to be a composite of a nonlinear elastic component and a visco-elastic component. The elastic component is expressed as a polynomial function in displacement, while the visco-elastic one is formulated according to the hereditary approach to represent the foam visco-elastic damping force during the loading phase and according to the fractional derivative approach during unloading to represent the visco-elastic residual force in the material. The focus of this study was to develop mathematical formulations and identification parameters to faithfully characterize the visco-elastic behavior of flexible polyurethane foam under multi-cycle compressive tests. A parameter calibration methodology based on the separation of the measurement data of each component force was established. This optimization process helps to avoid the parameter values admixture problem during the phase of numeric calculations of the same component force. The validity of the model results is checked according to the simulation accuracy, the physical significance of results and their agreement with the obtained force–displacement curve identification.     

基于有限质点法的单层球面网壳强震作用下连续倒塌破坏研究

有限质点法是以向量式结构力学为基础的崭新的结构分析方法。该文以该方法为基础,综合考虑结构的几何非线性、材料非线性和构件断裂,研究单层球面网壳结构在地震作用下的连续倒塌破坏全过程。该文介绍了有限质点法的基本理论,提出了该方法分析结构非线性问题和断裂问题的具体思路。采用直接输入法输入地震波,利用自编程序实现了单层球面网壳结构倒塌全过程模拟。研究了矢跨比对球面网壳强震作用下倒塌性能的影响,讨论了结构的倒塌时间和倒塌模式,并基于结构的倒塌模式通过局部加强法对单层球面网壳结构进行了抗倒塌设计。