Adaptive Analysis Of FEM For Forced Vibrations Of Bars Based On EEP Super-convergent Method

The forced vibration of bars is one of the basic problems in dynamic analysis of structures.It generally boils down to the initial-boundary value problems of partial differential equations(PDEs)in mathematics.The adaptive analysis of finite element method(FEM)has been one of the hot subjects on research of numerical analysis for years.The present paper extended the newly-developed adaptive strategies ofFEM which were based on the EEP super-convergent solutions into the forced vibration problems of bars,and proposed a method of adaptive finite element analysis for the three cases of forced vibrations of bars,i.e.the axial vibration of bar,the transverse vibration of Euler beam and of Timoshenko beam,which was proved to be efficient,reliable,versatile and general.The work of this paper is as follows:1.For the general form of initial value problem(IVP)of second-order ordinary differential equations(ODEs),the adaptive analysis of Galerkin FEMwas successfully implemented based on the EEP super-convergent solutions of simplified form.At first,the convergence orders of EEP solutions of simplified form was calculated and summarized,which proved that both the displacement and derivative solutions of EEP simplified form possess an accuracy of one-order higher convergence than the conventional finite element solutions on interior points of elements.After that,the self-adaptive analysis method was proposed and the corresponding calculation code was programmed,in which the EEP solution was used to estimate errors and direct mesh generations by replacing the unknown true solution.The results of typical numerical examples have showed that the method is efficient and reliable.A uniform and reasonable time mesh was automatically obtained with only one adaptive computation being applied on the entire time domain,and on this final mesh the errors of conventional finite element solution satisfy the specified error tolerance in maximum norm.This part is the basis of the present study.2.Taking the axial forced vibration of bars as the model problem,the basic idea,the implementation strategy and the computation method were proposed for the EEP adaptive analysis of such initial-boundary value problems of PDE.Firstly,the discretized procedure in space was given,which resulted in the second-order ODEs.Secondly,the EEP super-convergent solutions of simplified form was derived for elements in spatial dimension,and thirdly the adaptive strategy and algorithm were established and the code was programed.Representative examples were calculated and the validity and reliability of the proposed method were verified.The optimal time mesh and the optimal space mesh were obtained at the same time,on which errors of the FEM solution satisfy the error tolerance in maximum norm again.This is the core content of the present study.3.The adaptive approach proposed above was further successfully applied into the finite element adaptive solution of forced vibration in transverse direction for both Euler beams and Timoshenko beams(which are corresponding to the initial-boundary value problems of fourth-order PDE and second-order PDEs respectively).The discretized procedure of Galerkin FEM was introducted,the EEP formulae of simplified form were derived,the adaptive algorithm was constructed and the programing code was written.The approach was proved to be feasible and reliable by the representative examples.It is the first study to apply the EEP self-adaptive method into structural problems defined on two dimensions of both time and space.It is meaningful for both theoretical research and engineering application.