Research On Radial Basis Collocation Meshfree Method And Its Application To Rock Mechanics

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.