用无网格伽辽金法求解稳定地下水流问题

详细介绍了无网格伽辽金法（EFGM）基本原理,并将其应用于非均质多孔介质中的稳定地下水流问题,用具体算例将无网格伽辽金法计算结果与传统有限元法（LFEM）计算结果作比较,计算表明无网格伽辽金法具有较高的精度。     

一种快速求解程函方程的间断伽辽金方法

作为一种在勘探地震学里近期发展起来的有限元类数值求解程函方程的算法,间断伽辽金方法通过拓展基函数空间提高数值精度,但存在计算效率低的问题.为了解决该问题,提出一种并行快速扫描间断伽辽金方法.算法在快速扫描过程中对节点的计算顺序进行Cuthill-McKee变换,将能够并行计算的节点分为不同任务集合,进而实现不同任务集的...     

岩土工程数值模拟新方法

论述了岩土工程数值模拟的新进展和近年来新出现的数值模拟方法,重点介绍了无网络伽辽金法（ＥＦＧＭ）和数值流形法（ＭＭ）的原理及其在岩土工程中的应用,指出了今后研究的重点方向。     

模拟裂纹传播的新方法——无网格伽辽金法

无网格伽辽金法采用移动最小二乘法构造位移函数，由于它脱离了单元的概念，因此特别适合岩体介质裂纹的传播分析，计算了拉伸荷载作用下裂纹尖端的应力集中，对J积分围线作了进一步的讨论；运用无网格伽辽金法模拟岩体介质中的不连续面，计算了压剪复合型裂纹的应力强度因子，运用不同的断裂准则对裂纹的传播进行了分析模型，数值结果表明该方法与实验结果符合得很好。     

基于三次插值的大地电磁自适应有限元二维正演模拟

大地电磁(MT)数值模拟中通常使用有限单元法,通过伽辽金(Galerkin)法将微分方程转化为与其等价的泛函形式,对泛函求取极值并在单元上定义插值基函数,得到节点上电磁场值的线性方程组,最终形成大型复对称稀疏矩阵。要达到较高的有限元计算精度,一般采用密集的网格或高次插值的方法,这样做大大的减慢了正演的速度。结合两者的优点利用三次插值和h-型自适应相结合的有限元法来实现MT的正演算法。首先从一个粗网格出发并利用三次插值,通过后验误差估计方法局部加密网格,在计算量较小的情况获得较高的计算精度。这种方法可以针对目标区域和介质分界面发生突变处进行网格加密,不需要全局加密网格。最后通过对国际标准模型COMMEMI-2D1的模拟,分别比较二次插值与三次插值的自适应网格数量和数值模拟结果,证明了三次插值自适应有限元算法的可行性。     

接触摩擦问题的数值模拟

无网格伽辽金法(EFGM)可脱离单元的概念,特别适合岩体裂纹面的接触摩擦分析。基于EFGM,在裂纹面引入罚参数,通过迭代计算,得到裂纹面真实的应力状态,从而模拟闭合裂纹的粘接、滑移和张开行为,数值结果表明该方法是合理可行的。     

间断伽辽金法(DGM)求解弹性地基梁问题

间断伽辽金法使用节点位移一类未知数作为测试函数,削弱了内部单元边界上的一阶及n阶导数的连续性,大大降低了构造形函数的难度,特别适合控制方程为高阶微分方程问题的求解。基于间断伽辽金法的基本原理,推导了弹性地基梁四阶微分控制方程的积分“弱”形式,编制了计算程序,进行了数值计算和收敛性分析。计算结果表明：用间断伽辽金法求解弹性地基梁问题是十分有效率的。     

含单个孔洞岩盐路基稳定性的无网格分析

用无网格伽辽金法计算了孔洞边缘危险点的应力。对含单个孔洞岩盐路基的稳定性进行了分析,给出了针对不同孔洞埋深的临界载荷。数值结果表明,EFGM对于解决含有单个孔洞的应力集中问题,是有效且灵活的。     

砂土液化变形的有限元-无网格耦合方法

通过构造Biot固结理论u-p方程的无网格伽辽金-有限元耦合方法,对砂土液化变形问题进行了数值模拟。对于饱和砂土,采用Oka等提出的弹塑性本构模型,同时采用更新的Lagrange计算格式推导了控制方程。耦合方法能够发挥有限元和无网格各自的优点,既避免了由于单元变形扭曲而引起的计算中断,也可节约计算时间,算例验证了该方法在地震液化问题中的有效性。     

无单元伽辽金法及其在瞬态温度场中的应用研究

无单元伽辽金法(EFGM)采用移动的最小二乘法构造形函数,和有限元相比,它只需结点信息而不需要单元信息.简述了无单元法的基础理论,推导出瞬态温度场的无单元法计算公式,采用罚函数法引入了第一类边界条件,编制了相应的计算程序.通过应用于经典的瞬态温度场例子,和有限元结果作比较,说明了无单元法具有精度高、前后处理简单等优越性,是一种具有较大发展潜力的新数值计算方法.     

Back analysis procedures for the interpretation of field measurements in geomechanics

A survey is presented of some recent developments of the numerical techniques for back analysis in the field of geomechanics, with particular reference to tunnelling problems. In the spirit of Terzaghi's observational design method, these techniques are seen as practical tools for interpreting the available field measurements, in order to reduce the uncertainties that in many instances affect the parameters governing the solution of complex geomechanics problems. Both deterministic and probabilistic viewpoints are considered and some significant applications to practical problems are illustrated.     

六面体覆盖的高阶数值流形方法的探讨

采用高阶近似位移覆盖函数,基于六面体数学覆盖网格建立了三维数值流形方法分析格式,给出了相应的子矩阵。利用MATLAB编制了与之对应的计算程序,对简单的地下洞室模型进行了计算,并将计算结果与其他数值分析方法结果进行了比较,证明了分析格式及相应程序的正确性和有效性。结果表明, 当数值流形方法的覆盖函数推广到高阶情况时,其求解精度会有相应的提高。最后,对该方法在隧道及地下工程的应用前景作了展望。     

杂交边界点方法求解动态断裂力学问题

结合杂交边界点法和双互易法则,推导出求解动力问题的纯边界类型无网格方法--双互易杂交边界点方法,并将该方法用于求解含中心裂纹的方板受瞬态载荷作用的问题。该方法将问题的解分为通解和特解两部分,通解使用杂交边界点法求解,特解则利用局部径向基函数近似,域内布点仅仅为了径向基插值,因此仍然是一种纯边界类型的无网格方法。同时,将移动最小二乘近似中的基函数扩充,使该方法能更好地模拟裂纹尖端应力场的奇异性,具有后处理简单、精度高的优点。数值算例表明了该方法的稳定性和有效性。     

Analytical solution for the 1D consolidation of unsaturated multi-layered soil

The 1D consolidation of unsaturated multi-layered soil is studied based on the theory proposed by Fredlund and Hasan, and an analytical solution for a typical boundary condition is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the eigenfunction and eigenvalue for the multi-layered problem are first derived through the transfer matrix method. Then, by using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution is obtained. The present method is applicable to various types of boundary conditions. Finally, numerical examples are provided to investigate the consolidation behavior of unsaturated multi-layered soil.     

Numerical simulation of undrained insertion problems in geotechnical engineering with the Particle Finite Element Method (PFEM)

The paper presents total-stress numerical analyses of large-displacement soil-structure interaction problems in geomechanics using the Particle Finite Element Method (PFEM). This method is characterized by frequent remeshing and the use of low order finite elements to evaluate the solution. Several important features of the method are: (i) a mixed formulation (displacement-mean pressure) stabilized numerically to alleviate the volumetric locking effects that are characteristic of low order elements when the medium is incompressible, (ii) a penalty method to prescribe the contact constraints between a rigid body and a deformable media combined with an implicit scheme to solve the tangential contact constraint, (iii) an explicit algorithm with adaptive substepping and correction of the yield surface drift to integrate the finite-strain multiplicative elasto-plastic constitutive relationship, and (iv) the mapping schemes to transfer information between successive discretizations. The performance of the method is demonstrated by several numerical examples, of increasing complexity, ranging from the insertion of a rigid strip footing to a rough cone penetration test. It is shown that the proposed method requires fewer computational resources than other numerical approaches addressing the same type of problems.     

Godunov格式下高精度二维水流-输运耦合模型

针对复杂流体运动中物质输运方程的数值求解面临地形复杂、数值阻尼过大以及数值振荡等难题,建立了Godunov格式下求解二维水流-输运方程的高精度耦合数学模型,提出了集成输运对流项的HLLC (Harten-Lax-van Leer-Contact)型近似黎曼算子,可同时计算水流通量及输运通量,不仅有效模拟了复杂地形上水流运动,而且解决了输运方程中对流项产生的数值阻尼过大和不稳定振荡等难题。采用水深-水位加权重构技术和Minmod限制器,提高了模型处理复杂混合流态的能力,同时结合Hancock预测-校正方法,使模型具有时空二阶精度。算例结果表明,模型精度高、稳定性好,能有效抑制数值阻尼,适合模拟实际复杂流体运动中物质的输运过程,具有较好的推广应用价值。     

Coupled meshfree and fractal finite element method for unbounded problems

This paper presents a coupling technique for integrating the element-free Galerkin method (EFGM) with the fractal finite element method (FFEM) to analyze unbounded problems in the half-space. FFEM is adopted to model the far field of an unbounded domain and EFGM is used in the near field. In the transition region interface elements are employed. The shape functions of interface elements which comprise both the element-free Galerkin and the finite element shape functions, satisfy the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary. The numerical results show that the proposed method performs extremely well converging rapidly to the analytical solution. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the weight function, the scaling parameter and the number of transformation terms, on the quality of the numerical solutions.     

光滑粒子流体动力学二阶算法精度研究

光滑粒子流体动力学(SPH)由于无需网格生成和拉格朗日特性,对求解带有自由表面和大变形的力学问题有优势。但是该方法存在计算精度不高,计算效率较低等缺点。为此重点对SPH方法的精度提高进行研究。介绍了传统算法的基本公式,根据误差分析指出该算法精度不高的原因,提出了SPH二阶精度算法。通过精度验证分析,证明了该方法的精度的确能够达到二阶。通过二维计算实例,给出传统方法和二阶方法在粒子均匀分布和非均匀分布时函数值以及函数的一、二阶导数的误差分布,证明二阶算法能够克服传统算法的一些缺点,且计算精度有较大提高。     

Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass-conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that cannot be localized, in general. This is built on our previous work on the generalized multiscale finite element method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain, taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast dependence in the convergence. The method’s convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.     

Accelerating strategies to the numerical simulation of large‐scale models for sequential excavation

In this paper, a novel combination of well‐established numerical procedures is explored in order to accelerate the simulation of sequential excavation. Usually, large‐scale models are used to represent these problems. Due to the high number of equations involved, the solver algorithm represents the critical aspect which makes the simulation very time consuming. The mutable nature of the excavation models makes this problem even more pronounced. To accomplish the representation of geometrical and mechanical aspects in an efficient and simple manner, the proposed solution employs the boundary element method with a multiple‐region strategy. Together with this representational system, a segmented storage scheme and a time‐ordered tracking of the changes form an adequate basis for the usage of fast updating methods instead of frontal solvers. The present development employs the Sherman–Morrison–Woodbury method to speed up the calculation due to sequential changes. The efficiency of the proposed framework is illustrated through the simulation of test examples of 2D and 3D models. Copyright © 2006 John Wiley & Sons, Ltd.