径向基函数无网格配点法及其在岩石力学中的应用研究

发布时间：2019-01-27 12:21

【摘要】：由于岩石天然状态的复杂性，人们面对实际工程问题时很难得到解析解，，需要借助数值方法进行求解。无网格法作为一种新兴的数值方法，在处理复杂问题时，能避免传统有限元方法难以解决的许多难题。径向基函数法作为无网格法的一种，因其具有指数级收敛速度、形式简单、各向同性等优点备受瞩目。其形函数本身具备无穷阶可导且连续的性质，在求解偏微分方程时非常适合结合配点法等强形式算法进行计算，且不需要背景网格进行区域积分，能大大降低计算时间。然而，现有的关于径向基函数配点法的研究工作主要集中在算法本身的收敛性和求解边界值问题等方面，对于其求解动力问题的稳定性分析和非连续介质问题的应用研究较少。本论文基于von Neumann法提出了一种新的径向基函数配点法求解动力问题的稳定性评估算法，并将径向基函数配点法应用于非连续岩石结构承受静力和动力荷载问题。本文的主要研究工作如下： 1.推导了基于von Neumann法的径向基函数配点法求解动力问题的稳定性分析算法，定义了具体的稳定性参数来定量地对实际计算时如何选择合适的时间步长进行指导，并通过该参数对影响径向基函数配点法稳定性的各个因素进行了详细分析和讨论，对实际算例中出现的无条件不稳定情形的原因进行了分析，研究了径向基函数配点法求解动力问题稳定性的主要影响因素并给出了如何合理地选取径向基函数形函数形状参数及点距以提高计算稳定性的结论。 2.将径向基函数配点法应用于裂纹结构静力问题的求解，推导了径向基函数通过强形式配点法用于求解任意分布多裂纹结构承受复杂应力作用下裂纹结构的算法流程，建立了求解方程组，编写了FORTRAN静力计算程序,并对其计算结果进行了验证。 3.将径向基函数配点法应用于裂纹结构承受动力荷载问题的求解，推导了径向基函数配点法求解动力荷载作用下裂纹结构的算法流程，建立了求解方程组，编写了FORTRAN动力计算程序，并对其计算结果进行了验证。 4.通过应力外推法计算了径向基函数配点法数值解的应力强度因子，并以应力强度因子为指标，定量分析了静力问题中不同裂纹长度对应力强度因子的影响以及动力问题中不同外荷载频率对应力强度因子放大率的影响，其结论对实际工程施工时的结构安全评估具有一定的参考价值。
[Abstract]:Because of the complexity of the natural state of rock, it is difficult to obtain an analytical solution when people are faced with practical engineering problems, which need to be solved by numerical method. As a new numerical method, meshless method can avoid many difficult problems which are difficult to solve by traditional finite element method in dealing with complex problems. As a meshless method, radial basis function (RBF) method has attracted much attention for its advantages of exponential convergence rate, simple form and isotropy. The shape function itself has infinitely differentiable and continuous properties, so it is very suitable to solve partial differential equations with collocation method and other strong form algorithms, and it does not need background mesh for domain integration, which can greatly reduce the computation time. However, the existing researches on the radial basis function collocation method mainly focus on the convergence of the algorithm itself and the solution of boundary value problems, etc. There are few researches on the stability analysis of the radial basis function collocation method and the application of the discontinuous medium problem to the solution of the dynamic problem. In this paper, a new radial basis function collocation method is proposed to evaluate the stability of dynamic problems based on the von Neumann method, and the radial basis function collocation method is applied to the static and dynamic load problems of discontinuous rock structures. The main work of this paper is as follows: 1. The stability analysis algorithm of radial basis function collocation method based on von Neumann method for solving dynamic problems is derived, and specific stability parameters are defined to guide how to choose appropriate time step in practical calculation. The factors affecting the stability of radial basis function collocation method are analyzed and discussed in detail, and the causes of unconditional instability in practical examples are analyzed. The main factors affecting the stability of the radial basis function collocation method for solving dynamic problems are studied and the conclusion of how to select the radial basis function shape parameters and the distance between points to improve the stability is given. 2. The radial basis function collocation method is applied to solve the static problem of cracked structure. The algorithm flow of radial basis function (RBF) is derived by using the strong form collocation method to solve the crack structure with arbitrary distribution and multiple cracks subjected to complex stress. The equations are solved, the FORTRAN static calculation program is written, and the calculation results are verified. 3. The radial basis function collocation method is applied to solve the problem of crack structure subjected to dynamic load. The algorithm flow of radial basis function collocation method for solving crack structure under dynamic load is derived, and the solving equations are established. The FORTRAN dynamic calculation program is written and the calculation results are verified. 4. The stress intensity factor of the radial basis function collocation method is calculated by the stress extrapolation method, and the stress intensity factor is taken as the index. The influence of crack length on stress intensity factor in static problem and the effect of different frequency of external load on stress intensity factor in dynamic problem are analyzed quantitatively. The conclusion has certain reference value for the structural safety evaluation in actual engineering construction.
【学位授予单位】：上海交通大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：TU45

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