径向基函数无网格配点法及其在岩石力学中的应用研究
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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。
Due to the complexity of the rock natural situation, analytical solutioncan hardly be achieved in practical engineering and numerical methodsare usually hired as effective solution tools. As a kind of newly developednumerical methods, meshfree method can overcome many difficultieswhich trouble the traditional finite element method. Radial basis function,as a kind of meshfree method, received many researchers’ interest inrecent decades due to its characteristics of exponential convergence rate,simple expression and isotrope. Since the shape function of radial basisfunction and its arbitrary order differential functions are infinitelydifferentiable and continuous, it is very convenient for radial basisfunction to work together with collocation method by using strong formalgorithm when solving partial differential equations. Moreover, itrequires no background grid for domain integration, which can greatlyreduced the computational time.
However, the existing studies about radial basis collocation methodare mostly focused on the algorithm itself or its applications to boundaryvalue problems. There are only very few investigations on using radialbasis function to solve dynamic problems and stability analysis, as well asits application to discontinuous medium problem. This paper proposed anew stability evaluation algorithm for radial basis function solvingdynamic problems by introducing von Neumann method, and radial basiscollocation method has been applied to solve discontinuous rock massstructure under static and dynamic loading.
This paper is mainly devoted to the following contents:
1. A stability analysis algorithm based on von Neumann method forradial basis collocation method solving dynamic problems had beenproposed. A stability parameter for quantitative evaluation of choosingproper time step was defined. The effects on stability of every factor inradial basis function method were discussed in detail by the proposedstability parameter, and the reason causing the unconditionally unstablecases was also investigated. The major influence factor was concludedand some guidance on how to choose shape parameter of radial basisfunction and nodal distance were stated.
2. Radial basis collocation method had been applied to the crackstructure problem under static loading. The algorithm and procedure forradial basis collocation method solving arbitrarily distributedmultiple-crack structure under complex stress loading was advanced, thesolving equations were established, and a static FORTRAN program waswritten for implementation. The algorithm was demonstrated to possessgood accuracy by comparing the numerical results with the numericalsolutions by displacement discontinuity method (DDM).
3. The application of radial basis collocation method to crackstructure problem under dynamic loading was researched. The algorithmand procedure for radial basis collocation method solving crack structureunder dynamic loading was derived, the solving equations were founded,and a dynamic FORTRAN program was coded. A numerical case wasprovided for validation.
4. Stress intensity factor was calculated by stress extrapolationmethod for numerical results by radial basis collocation method. Theinfluence on stress intensity factor of different crack length for staticcases was investigated. By quantitatively analyzing the amplificationratio of stress intensity factor by different loading frequencies fordynamic cases, some useful conclusions were provided for safetyestimation in practical engineering constructions.
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。
Due to the complexity of the rock natural situation, analytical solutioncan hardly be achieved in practical engineering and numerical methodsare usually hired as effective solution tools. As a kind of newly developednumerical methods, meshfree method can overcome many difficultieswhich trouble the traditional finite element method. Radial basis function,as a kind of meshfree method, received many researchers’ interest inrecent decades due to its characteristics of exponential convergence rate,simple expression and isotrope. Since the shape function of radial basisfunction and its arbitrary order differential functions are infinitelydifferentiable and continuous, it is very convenient for radial basisfunction to work together with collocation method by using strong formalgorithm when solving partial differential equations. Moreover, itrequires no background grid for domain integration, which can greatlyreduced the computational time.
However, the existing studies about radial basis collocation methodare mostly focused on the algorithm itself or its applications to boundaryvalue problems. There are only very few investigations on using radialbasis function to solve dynamic problems and stability analysis, as well asits application to discontinuous medium problem. This paper proposed anew stability evaluation algorithm for radial basis function solvingdynamic problems by introducing von Neumann method, and radial basiscollocation method has been applied to solve discontinuous rock massstructure under static and dynamic loading.
This paper is mainly devoted to the following contents:
1. A stability analysis algorithm based on von Neumann method forradial basis collocation method solving dynamic problems had beenproposed. A stability parameter for quantitative evaluation of choosingproper time step was defined. The effects on stability of every factor inradial basis function method were discussed in detail by the proposedstability parameter, and the reason causing the unconditionally unstablecases was also investigated. The major influence factor was concludedand some guidance on how to choose shape parameter of radial basisfunction and nodal distance were stated.
2. Radial basis collocation method had been applied to the crackstructure problem under static loading. The algorithm and procedure forradial basis collocation method solving arbitrarily distributedmultiple-crack structure under complex stress loading was advanced, thesolving equations were established, and a static FORTRAN program waswritten for implementation. The algorithm was demonstrated to possessgood accuracy by comparing the numerical results with the numericalsolutions by displacement discontinuity method (DDM).
3. The application of radial basis collocation method to crackstructure problem under dynamic loading was researched. The algorithmand procedure for radial basis collocation method solving crack structureunder dynamic loading was derived, the solving equations were founded,and a dynamic FORTRAN program was coded. A numerical case wasprovided for validation.
4. Stress intensity factor was calculated by stress extrapolationmethod for numerical results by radial basis collocation method. Theinfluence on stress intensity factor of different crack length for staticcases was investigated. By quantitatively analyzing the amplificationratio of stress intensity factor by different loading frequencies fordynamic cases, some useful conclusions were provided for safetyestimation in practical engineering constructions.
引文
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