岩石冲击破坏的流形元与有限元模拟对比分析

流形元是新出现的一种被较为广泛地应用于材料破坏模拟的数值分析方法。为了验证该方法在材料破坏模拟中的有效性,分别利用流形元与有限元两种不同的数值方法对岩石冲击破坏过程进行了模拟分析。模拟结果表明,流形元法对材料破坏的模拟结果更符合实际情况,这主要是由于该方法采用了两种不同的覆盖系统——数学覆盖和物理覆盖,并引入能够客观反映材料裂纹的产生与扩展准则,以及相应的块体运动理论。流形元的出现有望对材料破坏模拟开创出一条新的途径。     

岩石冲击破坏的流形元与有限元模拟对比分析

流形元是新出现的一种被较为广泛地应用于材料破坏模拟的数值分析方法。为了验证该方法在材料破坏模拟中的有效性,分别利用流形元与有限元两种不同的数值方法对岩石冲击破坏过程进行了模拟分析。模拟结果表明,流形元法对材料破坏的模拟结果更符合实际情况,这主要是由于该方法采用了两种不同的覆盖系统——数学覆盖和物理覆盖,并引入能够客观反映材料裂纹的产生与扩展准则,以及相应的块体运动理论。流形元的出现有望对材料破坏模拟开创出一条新的途径。     

一种提高数值解精度的新方法

本文探讨在数值流形方法中使用一种改进数值精度的数值方法，即通过提高数值流形方法中各物理覆盖上的覆盖函数的阶次来实现提高数值解的精度之目的，在整个过程中，单元网格保持不变。通过算例对本文的进行检验，数值结构表明了算法的正确性。     

饱和多孔介质动力分析的数值流形单元

基于数值流形方法中覆盖函数的基本思想,构造了适用于饱和多孔介质动力耦合分析的三节点平面流形单元,该单元满足Babuska-Brezzi稳定性准则与Zienkiewicz-Taylor分片试验条件,对于位移和孔隙压力具有不等阶的插值函数,且所有节点上具有相同自由度。用标准Galerkin法和Newmark法将饱和多孔介质动力基本方程在空间和时间上离散,得到饱和多孔介质动力分析的流形元离散的算法公式。数值结果表明,与传统有限元相比在孔隙流体不可压缩且非渗流的条件下,数值流形单元对于压力场的计算具有良好的数值稳定性。     

模拟三维裂纹问题的自适应多尺度扩展有限元法

为了在大型结构分析中考虑小裂纹或以小的代价提高裂纹附近求解精度,该文建立了分析三维裂纹问题的自适应多尺度扩展有限元法。基于恢复法评估三维扩展有限元后验误差,大于给定误差值的单元进行细化。所有尺度单元采用八结点六面体单元,采用六面体任意结点单元连接不同尺度单元。采用互作用积分法计算三维应力强度因子。三维I 型裂纹和I-II 复合型裂纹算例分析表明了该方法的正确性和有效性。     

节理性质对岩体爆破破坏模式影响的数值分析

针对目前常用的有限元和离散元等数值方法难以客观反映岩体中存在的大量断续节理和在外力作用下岩体破碎及块体运动的不足,提出了采用数值流形方法以解决目前岩体爆破模拟中存在的上述问题.数值流形方法采用数学网格与物理网格以形成求解流形单元,因而很容易反映岩体中存在的众多初始节理,采用断裂力学准则以模拟节理、裂纹扩展,采用DDA中的块体运动学理论以模拟块体运动.最后通过算例对比分析了完整岩体和节理岩体爆破破坏模式的差异,说明了节理存在对岩体爆破破坏模式有着重要影响,且其影响程度与节理的几何分布及物理力学性质有着密切关系.     

高阶流形方法及其应用

流形方法是一种可进行连续与非连续变形问题分析的灵活而有效的数值计算方法。本文详细地推导了二阶流形方法的具体计算列式,分别开发了一阶流形方法与二阶流形方法的计算程序．通过实例计算表明：提高覆盖函数的阶次可有效地提高流形方法的计算精度。     

薄板弯曲分析的多边形流形单元

一般的数值流形方法均采用三角形、四边形单元进行计算。对于工程中的有些实际问题, 多边形单元能更好的适应复杂计算域形状。为此, 研究了采用多边形流形单元进行数值计算的方法。采用任意几何区域的Delaunay三角网格构造出新的凸多边形网格, 并以此单元作为计算的流形单元。采用改进的Wachspress插值函数作为多边形流形单元的权函数。为说明该方法的有效性, 将该流形方法应用于薄板弯曲计算, 推导出用于薄板弯曲分析的流形格式和单元矩阵。计算结果表明:较一般有限元法, 计算精度和收敛速度有很大提高。     

基于ARLEQUIN方法和XFEM的结构多尺度模拟

该文研究考察ARLEQUIN方法与扩展有限单元法(XFEM)相结合进行结构的多尺度模拟。首先简要介绍了扩展有限单元法和ARLEQUIN方法的基本理论,基于Matlab软件编制了ARLEQUIN方法和扩展有限元方法相结合的多尺度分析程序。对一带边裂纹的板进行了应力分析并计算裂纹尖端附近应力强度因子,同时对一中心带圆孔的方板在受水平均布拉力作用下的孔口应力集中问题进行了分析。计算结果表明:该文编制的程序是正确的,将ARLEQUIN方法和XFEM相结合进行结构多尺度数值模拟是可行的。     

岩体多裂纹扩展演化过程数值流形方法研究

岩体中含有大量节理、裂隙、断层等各类结构面,结构面在应力作用下的扩展与贯通是导致岩体破坏的重要原因。数值流形方法 (NMM)可以有效模拟连续和非连续问题,然而,其在多裂纹动态扩展的模拟方面仍处于探索阶段。该文以线弹性断裂力学原理为基础,提出了一种基于高阶数值流形方法的多裂纹扩展模拟算法。通过在基函数中增加关键项来考虑裂纹尖端位移场的奇异性;裂纹尖端的应力强度因子则采用了J积分来计算;Ⅰ型-Ⅱ型混合裂纹的开裂和扩展方向依据最大周向拉应力准则来判断;采用假设-修正的多裂纹扩展算法解决了多裂纹的扩展问题。根据强化后的基函数,对于不符合单纯形积分形式的被积函数,采用了泰勒级数展开式计算近似解。通过多个静态裂纹扩展的经典问题的数值模拟对计算方法的合理性和计算精度及进行了验证。     

A decomposition technique of generalized degrees of freedom for mixedmode crack problems

The numerical manifold method (NMM) builds up a unified framework that is used to describe continuous and discontinuous problems; it is an attractive method for simulating a cracking phenomenon. Taking into account the differences between the generalized degrees of freedom of the physical patch and nodal displacement of the element in the NMM, a decomposition technique of generalized degrees of freedom is deduced for mixed mode crack problems. An analytic expression of the energy release rate, which is caused by a virtual crack extension technique, is proposed. The necessity of using a symmetric mesh is demonstrated in detail by analysing an additional error that had previously been overlooked. Because of this necessity, the local mathematical cover refinement is further applied. Finally, four comparison tests are given to illustrate the validity and practicality of the proposed method. The aforementioned aspects are all implemented in the high‐order NMM, so this study can be regarded as the development of the virtual crack extension technique and can also be seen as a prelude to an h‐version high‐order NMM. Copyright © 2017 John Wiley & Sons, Ltd.     

Modeling complex crack problems using the numerical manifold method

In the numerical manifold method, there are two kinds of covers, namely mathematical cover and physical cover. Mathematical covers are independent of the physical domain of the problem, over which weight functions are defined. Physical covers are the intersection of the mathematical covers and the physical domain, over which cover functions with unknowns to be determined are defined. With these two kinds of covers, the method is quite suitable for modeling discontinuous problems. In this paper, complex crack problems such as multiple branched and intersecting cracks are studied to exhibit the advantageous features of the numerical manifold method. Complex displacement discontinuities across crack surfaces are modeled by different cover functions in a natural and straightforward manner. For the crack tip singularity, the asymptotic near tip field is incorporated to the cover function of the singular physical cover. By virtue of the domain form of the interaction integral, the mixed mode stress intensity factors are evaluated for three typical examples. The excellent results show that the numerical manifold method is prominent in modeling the complex crack problems.     

Modeling bimaterial interface cracks using the numerical manifold method

The numerical manifold method (NMM) is explored for simulations of bimaterial interface cracks. Two special types of physical covers with customized cover functions are introduced to describe the weak discontinuity across the material interface, the partially cracked elements as well as the interface crack tip singularity. Three typical bimaterial crack problems are simulated. The mixed-mode stress intensity factors are evaluated by the virtue of the domain form of the interaction integral and compared with the available reference solutions. Good agreements have demonstrated the validity and accuracy of the developed program.     

Numerical analysis of 2-D crack propagation problems using the numerical manifold method

The numerical manifold method is a cover-based method using mathematical covers that are independent of the physical domain. As the unknowns are defined on individual physical covers, the numerical manifold method is very suitable for modeling discontinuities. This paper focuses on modeling complex crack propagation problems containing multiple or branched cracks. The displacement discontinuity across crack surface is modeled by independent cover functions over different physical covers, while additional functions, extracted from the asymptotic near tip field, are incorporated into cover functions of singular physical covers to reflect the stress singularity around the crack tips. In evaluating the element matrices, Gaussian quadrature is used over the sub-triangles of the element, replacing the simplex integration over the whole element. First, the method is validated by evaluating the fracture parameters in two examples involving stationary cracks. The results show good agreement with the reference solutions available. Next, three crack propagation problems involving multiple and branched cracks are simulated. It is found that when the crack growth increment is taken to be 0.5h≤da≤0.75h, the crack growth paths converge consistently and are satisfactory.     

Numerical manifold space of Hermitian form and application to Kirchhoff's thin plate problems

For second‐order problems, where the behavior is described by second‐order partial differential equations, the numerical manifold method (NMM) has gained great success. Because of difficulties in the construction of the H ²‐regular Lagrangian partition of unity subordinate to the finite element cover; however, few applications of the NMM have been found to fourth‐order problems such as Kirchhoff's thin plate problems. Parallel to the finite element methods, this study constructs the numerical manifold space of the Hermitian form to solve fourth‐order problems. From the minimum potential principle, meanwhile, the mixed primal formulation and the penalized formulation fitted to the NMM for Kirchhoff's thin plate problems are derived. The typical examples indicate that by the proposed procedures, even those earliest developed elements in the finite element history, such as Zienkiewicz's plate element, regain their vigor. Copyright © 2013 John Wiley & Sons, Ltd.     

数值流形方法的数学推导及其应用

以往的数值流形方法都是以最小势能原理或变分原理为基础来建立求解方程的。但在实际工程中科技人员所遇到的有些实际问题,其控制方程所对应的泛函往往是难以找到的,在这些情况下就无法应用变分方法来建立数值流形方法的求解方程,而必须寻找较为一般的方法来推导数值流形方法的求解方程。因此,研究了如何从加权残数法出发建立数值流形方法的求解方程。在此过程中,通过建立弹性力学方程的数值流形方法,可以看出,通过选取适当的权函数,该方法最终的求解方程将转化为以最小势能原理或以变分原理为基础的离散形式。为了说明方法的有效性,求解了岩石试件中含单裂隙双边受拉的问题,并给出了裂隙尖端的应力强度因子和应力场的变化关系。     

Mixed mode fracture propagation by manifold method

The numerical manifold method combined with the virtual crack extension method is proposed to study the mixed mode fracture propagation. The manifold method is a new numerical method, and it provides a unified framework for solving problems dealing with both continuums and jointed materials. This new method can be considered as a generalized finite element method and discontinuous deformation analysis. One of the most innovative features of the method is that it employs both physical mesh and mathematical mesh to formulate the physical problem. These two meshes are separated and independent. They are inter-related through the application of weighting functions. A local mesh refinement and auto-remeshing schemes previously proposed by the authors are adopted in this study. The proposed model is first verified by comparing the numerical stress intensity factors with the benchmark solutions, and the results show satisfactory accuracy. The maximum tangential stress criterion is adopted and the mixed mode fracture propagation problems are then fully investigated. The numerical solutions by the present method agree well with the experimental results.     

无网格自动加密方法及其在土体裂缝分析中的应用

将无网格法和弥散裂缝模型结合用于裂缝分析时,需要较细密的节点分布。为了准确描述土体裂缝开展,提出了一种无网格自动加密方法。通过在等参单元上布点后变换到实际的背景积分单元中,实现了裂缝开展区域的局部加密。在背景积分单元中使用有限元插值来计算新增节点和高斯积分点的状态变量。将提出的方法与弥散裂缝模型相结合,可用于土体裂缝的分析。对土梁弯曲试验进行模拟计算,验证了方法的有效性。