A Time-integration Method Based On The Galerkin Weak Form For Nonlinear Dynamic Equations With Adaptive Analysis

For nonlinear dynamic problems in solid mechanics and structural engineering,after space discretization the governing equations to be solved often come down to a set of non-linear second-order ordinary differential equations(ODEs).It is still a hot and difficult research point in this field to give accurate,stable and reliable numerical solutions for them.In this paper,a time-integration method with high accuracy and excellent numerical stability is proposed,and the corresponding adaptive strategy in time domain is established with the element energy projection(EEP)method introduced.This adaptive strategy has been successfully applied to the non-linear vibration of discrete systems and that of large-deflection skeletal structures.Further more,an improved scheme of this time-integration algorithm is proposed,which can effectively filter out the high frequency modal components of the system.Its superior characteristics in numerical dissipation and numerical stability are demonstrated with representative examples.The main work of this paper is as follows:1.Taking the nonlinear second-order dynamic equations as the model problem,a reliable algorithm named GGW(Generalized Galerkin Weak Form)method is proposed.Firstly,based on Newton-Raphson iteration,weighted residual method and Galerkin finite element method(FEM),the basic theory and formulae of GGW method are expounded and derived.Then its numerical stability,period elongation(PE)and amplitude decay(AD)rate are analyzed.Finally,the accuracy and non-linear numerical stability are compared between GGW method and Newmark method for some typical problems.The results show that GGW method possesses high accuracy and stability,which can still give stable numerical solutions even when Newmark method appears to be unstable.2.Based on the basic theory of GGW method and the superconvergence formula of EEP,a set of adaptive solution strategy for the nonlinear dynamic equations are established and successfully applied to non-linear vibrations of both discrete systems and large-deflection skeletal structures.Firstly,the concept of approximate ideal linear problem is introduced and the superconvergence of EEP solution in time element is verified.Then a simple,convenient and reliable strategy for step-size adjustment in time is constructed.By using the correction technology of end-node displacements,the length of time domain to be effectively solved has been greatly prolonged.Finally,the effectiveness and reliability of the algorithm are verified by typical numerical examples.3.In order to solve the defect that GGW method has no numerical damping,a time integral method named GGW-? method possessing controllable numerical damping is proposed by suitable modification of GGW with linear elements.Firstly,the solution scheme of GGW-? method is derived,and then the numerical stability,PE and AD rate are analyzed using spectral radius theory,together with the numerical damping characteristics being studied.Finally,the superior numerical dissipation characteristics of the algorithm are verified through numerical examples including the large-deflection problems of Euler beam and skeletal structures.GGW-? method keeps to be stable and reliable even when Newmark method obviously appears to be unstable.