A Novel Space-time Coupling Generalized Finite Difference Method And Its Application

The loads on engineering structures are mostly related to time history,so it is often necessary to consider the influence of time-related inertial terms when analyzing the mechanical properties and evaluating the safety of such structures.In numerical processing,independent differential processing is usually used for the time derivative term of this kind of dynamic equation,and numerical discrete techniques such as finite element or boundary element are used for the spatial domain.This method is difficult to capture discontinuities or sharp gradients in the solution accurately,which leads to some difficult problems in numerical results.Taking the time domain as an additional dimension of the space domain,a direct space-time discrete technique is proposed,which becomes an effective means to solve these problems.The generalized finite difference method(GFDM)is a typical collocation meshless numerical method based on the Taylor series expansion and moving least squares method,which directly transforms the differential equation into the algebraic equation for solving.In this dissertation,a novel space-time coupling generalized finite difference method(ST-GFDM)is proposed,which combines the direct space-time discrete technique with GFDM for the first time.The following three typical dynamic and heat conduction problems are effectively analyzed by this method.(1)Transient heat conduction problem: considering the transient heat conduction problem of general heterogeneous and complex geometrical structures under thermal shock,the numerical simulation is carried out by using this method.Compared with the corresponding analytical results and other numerical results,the proposed method is proved to have high accuracy and computational efficiency.And through error analysis,the influence of the two important parameters of the number of points and the number of adjacent points in the algorithm on the numerical calculation results is studied.(2)Elastic dynamic problem: two mathematical methods of introducing unknown velocity and directly adding initial velocity condition are proposed for the inertia term of the second derivative of time in the governing equation of elastodynamics.The error analysis and verification are carried out by a numerical example,and the optimal method is selected.The numerical simulations of elastic dynamic problems in homogeneous and heterogeneous materials is carried out by using the proposed method.Compared with the analytical results,the high accuracy of the proposed method in dealing with the elastic dynamic problems is verified.The effect of the time step on the numerical results is also studied.(3)Thermo-mechanical coupled elastic dynamic problem: a space-time coupled generalized finite difference method for the analysis of thermoelastic dynamic problems is developed.This method is used to solve the quasi-static thermoelastic problems in homogeneous and heterogeneous materials.At the same time,the thermoelastic dynamics problems under thermal shock and mechanical shock are solved by sequential coupling and fully coupled respectively.Through the results,the importance of the coupled solution of the temperature field and the displacement field is analyzed,and the effectiveness and accuracy of the method are also verified by comparison.