Precision Analysis Of Fundamental Solution And Solver Optimization Of FM-BEM For 3-D Elastic Problems

The Boundary Element Method(BEM)is an important numerical method,which has the advantages of reducing and semi parsing.However,the coefficient matrix of the linear equation formed by the BEM is an asymmetric full rank array,which makes the calculation time of the algorithm too long.The Fast Multipole Method(FMM)injected vitality into the BEM,and formed the Fast Multipole Boundary Element Method(FM-BEM)combined with the BEM.It updated the computational structure of the BEM and improved the computation efficiency,making the BEM suitable for large-scale engineering problems.The main contents of the paper are summarized as follows:Firstly,on the basis of the truncation error of the 1? and its normal derivative,the truncation error estimation formula of the fundamental solution of displacement and stress is derived,and the relation between truncated number and relative distance is obtained.Combined with the results,the value of the truncated number and relative distance in practical engineering calculation is given.Then for the linear systems formed by FM-BEM,based on Variable Parameter Restarting Generalized Minimal Residual algorithm(VRP-GMRES(m)),this paper proposes a truncated solver named VRP-IGMRES(m)algorithm.The convergence of the algorithm is proved.The numerical examples show that the VRP-IGMRES(m)algorithm can improve the calculation efficiency under the premise of ensuring the calculation precision,and the optimal truncation ratio is obtained.By comparing two examples,it is found that the new algorithm is more suitable for solving large-scale complex problems.Finally,the dissertation uses VRP-IGMRES(m)algorithm to solve the three-dimensional elastic frictional contact problem.Based on the point contact constraint equation,a mathematical programming model is established.Then a optimizational VRP-IGMRES(m)algorithm is proposed by introducing the objective function of the model into VRP-IGMRES(m)algorithm.The numerical example verifies the validity of the optimized VRP-IGMRES(m)algorithm to solve the friction contact problem.