有限元极限分析法发展及其在岩土工程中的应用

有限元极限分析法兼有数值分析法与经典极限分析法两者的优点，特别适用于岩土工程的分析与设计。20世纪初，国内岩土工程界应用国际上通用程序，大力发展有限元极限分析法并拓宽其在岩土工程中的应用。在基本理论研究、提高计算精度、拓宽应用范围及工程实际应用等方面取得了很大成绩。重点介绍作者及其合作者的一些研究成果。主要包括岩土工程安全系数定义、方法原理、整体失稳判据、强度准则的推导、选用及提高计算精度等方面的研究。应用范围从二维扩大到三维，从均质土坡、土基扩大到有节理的岩质边坡与岩基，从稳定渗流扩大到不稳定渗流、从边坡与地基工程扩大到隧道、还用于寻找边（滑）坡中的多个潜在滑面，进行岩土与结构共同作用的支挡结构设计，计算机仿真地基承载板载荷试验等应用项目，以逐渐达到革新岩土工程设计方法的目的。     

自由板干缩应力的差分算法

该文针对自由板干缩应力求解中表面积大、计算复杂和工作量大等问题,以混凝土湿度扩散方程为基础,推导出自由板湿度场有限差分格式,并在湿度场求解基础上,根据干缩引起的弹性应力推导出自由板干缩应力差分公式。通过实例对自由板湿度和干缩应力进行求解。结果表明：差分算法求解自由板湿度和干缩应力场与有限元求解结果一致且效率较高,是一种简单高效的干缩应力计算方法。该方法可以进行高效准确的干缩应力评估,可为施工人员和相关工程人员提供一种新的干缩应力计算方案。     

Lemaitre等向硬化弹塑性损伤耦合本构模型积分算法及程序实现

涉及复杂材料弹塑性损伤问题数值计算研究时,不仅需要选择恰当预测损伤和破坏的本构模型,还需要有效和稳健的本构积分算法。首先,阐述了在热力学和连续介质力学框架下建立弹塑性损伤本构模型的基本步骤;其次,基于Lemaitre等向硬化弹塑性损伤耦合本构模型、相应的本构积分算法-完全隐式返回映射算法(Fully Return Mapping Algorithm)和一致切线模量,采用C++语言在Visual 6.0 环境下编制有限元本构求解程序,在塑性损伤修正步中求解返回映射方程时,选取一种简单的形式,只需迭代求解一个标量非线性方程,计算效率较高。最后,通过缺口圆棒数值算例初步验证了程序的正确性,并编制接口程序对计算结果进行可视化。研究结果表明积分算法的有效性及程序的正确性,Lemaitre等向硬化弹塑性损伤耦合本构模型能够较好地模拟韧性材料的破坏发展过程,可以求解类似的有限元边界值问题,为考虑损伤特性的韧性材料结构研究和设计奠定基础。     

Lemaitre等向硬化弹塑性损伤耦合本构模型积分算法及程序实现EI北大核心CSCD

涉及复杂材料弹塑性损伤问题数值计算研究时,不仅需要选择恰当预测损伤和破坏的本构模型,还需要有效和稳健的本构积分算法。首先,阐述了在热力学和连续介质力学框架下建立弹塑性损伤本构模型的基本步骤;其次,基于Lemaitre等向硬化弹塑性损伤耦合本构模型、相应的本构积分算法-完全隐式返回映射算法(Fully Return Mapping Algorithm)和一致切线模量,采用C++语言在Visual 6.0环境下编制有限元本构求解程序,在塑性损伤修正步中求解返回映射方程时,选取一种简单的形式,只需迭代求解一个标量非线性方程,计算效率较高。最后,通过缺口圆棒数值算例初步验证了程序的正确性,并编制接口程序对计算结果进行可视化。研究结果表明积分算法的有效性及程序的正确性,Lemaitre等向硬化弹塑性损伤耦合本构模型能够较好地模拟韧性材料的破坏发展过程,可以求解类似的有限元边界值问题,为考虑损伤特性的韧性材料结构研究和设计奠定基础。     

基于向量式有限元的三角形薄板单元

向量式有限元是一种基于点值描述和向量力学理论的新型分析方法。该文基于向量式有限元基本原理,推导了三角形DKT薄板单元的基本公式,详细阐述了通过逆向运动处理薄板单元的平面内、外刚体位移从而获得单元节点纯变形位移的过程,以及进一步通过变形坐标系获得单元节点内力的求解方法;同时对质点的质量矩阵与惯性矩阵、应力计算的数值积分及插值方法、时间步长及阻尼参数的取值等问题提出了合理可行的处理方式。在此基础上编制了薄板单元的计算分析程序,并进行了算例验证。算例分析表明所编制的向量式有限元薄板单元程序可以很好地完成平板结构的静、动力分析,验证了理论推导的正确性和分析程序的可靠性。该文成果为进一步建立向量式有限元薄壳单元理论打下了必要的基础。     

有限一维弹性波动问题的公理化求解及其在自由场中的应用

在保证级数的一致收敛的前提下,可以利用Laplace变换对有限一维空间弹性动力学边值问题给出公理化的严格求解过程,此过程能够作为Lavrentieff与Hilbert等著名学者的工作的补充。这一解答能够应用于岩土工程中的自由场计算问题,并且能够从原则上避免截断误差。在针对特定问题的计算中,该方法的效率明显高于有限元,并且在一定程度上弥补了现有的自由场专用计算软件不能计算纯弹性体的缺点。同时,作为解析解算法,该方法对于数值算法的精度分析有一定意义。     

基于主从式并行遗传算法的岩土力学参数反分析方法

考虑到遗传算法的天然并行性和集群计算的高速并行性,提出了基于主从式并行遗传算法的岩土力学参数反分析方法。采用实数编码方法,缩短了个体编码的长度,减少了搜索空间;采用动态任务分配方案,可以避免处理器效率的不均衡;采取"松耦合"的方法将主从式并行遗传算法与FLAC程序进行耦合。基于C+MPI语言编写了反分析程序,并用标准弹性问题对程序进行了测试。测试结果表明,主从式并行遗传算法不仅能够准确地对岩土力学参数进行反分析,而且随着问题规模的增大可以得到接近线性的加速比。因此,针对适应度评价计算量大的岩土工程反分析问题,采用基于主从式并行遗传算法的岩土力学参数反分析方法,既保证了反分析的求解精度,又提高了反分析速度,满足工程上对于反分析的及时性需求,具有较强的应用价值。     

三维有限元并行EBE方法

采用Jacobi预处理,推导了基于EBE方法的预处理共轭梯度算法,给出了有限元EBE方法在分布存储并行机上的计算过程,可以实现整个三维有限元计算过程的并行化。编制了三维有限元求解的PFEM(ParallelFiniteElementMethod)程序,并在网络机群系统上实现。采用矩形截面悬臂梁的算例,对PFEM程序进行了数值测试,对串行计算和并行计算的效率进行了分析,最后将PFEM程序应用于二滩拱坝-地基系统的三维有限元数值计算中。结果表明,三维有限元EBE算法在求解过程中不需要集成整体刚度矩阵,有效地减少了对内存的需求,具有很好的并行性,可以有效地进行三维复杂结构的大规模数值分析。     

有限元神经网络的计算方法与电路实现

对弹性力学有限元而言,它对应于等式约束下的二次优化问题,进一步可以转化为无约束的二次优化问题。本文应用改进的Hopfieid神经网络来求解弹性力学有限元问题,使网络能量函数与待求解有限元问题的优化目标函数相对应,网络能量函数的极小点,即系统的稳定平衡点为优化目标函数的解。另外在理论基础上进行了计算机仿真和模拟电路实验,两者都表明上述方法可靠、有效,误差满足预定的要求。     

基于计算接触力学的粗颗粒土体材料细观性质模拟

采用离散元等数值模拟方法研究粗颗粒土体材料的细观力学行为,是近年来岩土力学的热点研究课题。提出了一种基于计算接触力学的粗颗粒土体材料细观力学行为模拟分析新方法。该法将土体颗粒剖分成一定数量的单元,通过计算接触力学方法模拟颗粒间的多体接触行为,是一种基于有限元变分原理的隐式数值求解方法,具有严格的理论基础。和离散元法方法相比,该方法在描述颗粒本身的力学特性方面具有优势,可以计算得到颗粒内部的应力分布情况,从而为颗粒破碎等细观行为的模拟计算提供依据。利用所发展的粗颗粒土体多体接触有限元计算程序,进行了不同围压的常规三轴数值试验,模拟计算结果符合粗颗粒土体材料三轴试验的一般规律,初步验证了所提出的计算方法的适用性。     

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.     

Mixed Approximation Scheme of the Finite-Element Method for the Solution of Two-Dimensional Problems of the Elasticity Theory

For the solution of two-dimensional boundary-value problems of the elasticity theory, a triangular finite element, ensuring stability and convergence of mixed approximation, is proposed. The system of resolving equations of the mixed method is derived with account of strict satisfaction of static boundary conditions at the surface. To solve matrix equations of the mixed method, various algorithms of the conjugate-gradient method with the pre-conditional matrix have been considered. Numerical data on convergence and accuracy of the solution for a number of test problems of the elasticity theory and fracture mechanics are given. The results obtained by the conventional and mixed finite-element method approaches are compared.     

Finite element solution algorithm for viscous incompressible fluid dynamics

A numerical solution algorithm employing the finite element concept of solid mechanics is derived for the transient laminar two-dimensional flow of an incompressible viscous fluid. Through dependent variable transformation, the problem is uniformly recast into the solution of a quasi-linear elliptic boundary value-problem, for which the finite element solution theory is established. The algorithm is uniquely user-oriented in accepting the generalized elliptic boundary condition specification of any non-coordinate-surface solution do main closure segment and on employing an arbitrarily irregular computational latticc. Numerical results are presented for several problems in internal flow illustrating solution accuracy, convergence, versatility and the ability to predict imbedded regions of recirculating flow.     

多边形比例边界有限元非线性高效分析方法

比例边界有限元法作为一种高精度的半解析数值求解方法,特别适合于求解无限域与应力奇异性等问题,多边形比例边界单元在模拟裂纹扩展过程、处理局部网格重剖分等方面相较于有限单元法具有明显优势。目前,比例边界有限元法更多关注的是线弹性问题的求解,而非线性比例边界单元的研究则处于起步阶段。该文将高效的隔离非线性有限元法用于比例边界单元的非线性分析,提出了一种高效的隔离非线性比例边界有限元法。该方法认为每个边界线单元覆盖的区域为相互独立的扇形子单元,其形函数以及应变-位移矩阵可通过半解析的弹性解获得;每个扇形区的非线性应变场通过设置非线性应变插值点来表达,引入非线性本构关系即可实现多边形比例边界单元高效非线性分析。多边形比例边界单元的刚度通过集成每个扇形子单元的刚度获取,扇形子单元的刚度可采用高斯积分方案进行求解,其精度保持不变。由于引入了较多的非线性应变插值点,舒尔补矩阵维数较大,该文采用Woodbury近似法对隔离非线性比例边界单元的控制方程进行求解。该方法对大规模非线性问题的计算具有较高的计算效率,数值算例验证了算法的正确性以及高效性,将该方法进行推广,对实际工程分析具有重要意义。     

ROBUST PRECONDITIONERS FOR LINEAR ELASTICITY FEM ANALYSES

This paper deals with two forms of preconditioner which can be easily used with a Conjugate Gradient solver to replace a direct solution subroutine in a traditional engineering finite element package; they are tested in such a package (FINAL) over a range of 2-D and 3-D elasticity problems from geotechnical engineering. Quadratic basis functions are used. A number of modifications to the basic Incomplete Choleski [IC(0)] factorization preconditioner are considered. An algorithm to reduce positive off-diagonal entries is shown in numerical experiments to ensure stability, but at the expense of slow convergence. An alternative algorithm of Jennings and Malik is more successful, and a relaxation parameter o is introduced which can make a further significant improvement in performance while maintaining stability. A heuristic for determining a near-optimal value of o is proposed. A second form of preconditioning, symmetrically scaled element by element, due to Bartelt, is also shown to perform robustly over a range of problems; it does not require assembly of the global stiffness matrix, and has great potential for parallelization. © 1997 by John Wiley & Sons, Ltd.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part II: finite element formulation and applications

A finite element analysis model for material and geometrical non-linearities due to large plastic deformations of ductile materials is presented using the continuum damage mechanics approach. To overcome limitations of the conventional plastic analysis, a fourth-order tensor damage, defined in Part I of this paper to represent the stiffness degradation in the finite strain regime, is incorporated. General forms of an updated Lagrangian (U.L.) finite element procedure are formulated to solve the governing equations of the coupled elastic–plastic-damage analysis, and a computer program is developed for two-dimensional plane stress/strain problems. A numerical algorithm to treat the anisotropic damage is proposed in addition to the non-linear incremental solution algorithm of the U.L. formulation. Selected examples, compared with published results, show the validity of the presented finite element approach. Finally, the necking phenomenon of a plate with a hole is studied to explore plastic damage in large strain deformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Direct analysis of elastic contact using super elements

Solutions to contact problems are important in mechanical as well as in civil engineering, and even for the most simple problems there is still a need for research results. In the present paper we suggest an alternative finite element procedure and by examples show the need for more knowledge related to the compliance of contact surfaces. The most simple solutions are named Hertz solutions from 1882, and we use some of these solutions for comparison with our finite element results. As a function of the total contact force we find the size of the contact area, the distribution of the contact pressure, and the contact compliance. In models of finite size the compliance depends on the flexibility of the total model, including the boundary condition of the model, and therefore disagreement with the locally based analytical models is expected and found. With computational contact mechanics we can solve more advanced contact problems and treat models that are closer to physical reality. The finite element method is widely used and solutions are obtained by incrementation and/or iteration for these non-linear problems with unknown boundary conditions. Still with these advanced tools the solution is difficult because of extreme sensitivity. Here we present a direct analysis of elastic contact without incrementation and iteration, and the procedure is based on a finite element super element technique. This means that the contacting bodies can be analyzed independently, and are only coupled through a direct analysis with low order super element stiffness matrices. The examples of the present paper are restricted to axisymmetric problems with isotropic, elastic materials and excluding friction. Direct extensions to cases of non-isotropy, including laminates, and to plane and general 3D models are possible.     

基于梯度塑性理论的动力软化问题分析

对于动力应变软化问题,采用梯度塑性模型进行分析,该模型能够有效地克服软化材料在有限元分析中的网格依赖性问题。对于动力非线性软化方程的求解则利用于基于参变量变分原理的参数二次规划方法。对于结构动力方程时域上的求解则采用传统的Newmark方法。本文的算法与传统方法相比在求解基于梯度塑性模型的非线性动力软化问题时保证了已有参数二次规划算法的优良特性,有实现简单与稳定性好等优点。给出的数值算例证实了本文的理论工作与所研制程序的正确性。     

复杂边界单连通域共形映射解析建模研究

在工程实际中任意非规则复杂边界单连通域问题的数学模型建立难度大，导致求解和优化参数非常困难，根据共形映射原理，采用复变三角插值理论，利用法线迭代收敛方法，将任意复杂边界单连通域问题转化为单位圆域问题，系统地建立了二者间的共形映射函数，并对具有对称轴特性的单连通域问题进行了算法简化。     

三维双向及三向映射无限元

不同于以往的单向映射无限元，本文提出了一种能分别从两个和三个方向延伸到无穷远的多向映射无限元，它们不仅能很好地与三维等参数有限单元耦合，保持公共边界的连续性，而且具有物理概念明确、节点数目少、精度高、无需改动单元形状、减少计算准备工作量、便于网格自动形成等诸多优点，能广泛应用于岩土工程领域及其他力学问题的数值分析工作中。