WAVELET BASIS FEM FOR SOLUTION OF SOME TRANSIENT PROBLEMS IN STRUCTURAL DYNAMICS

Abstract

Thefinite element formulation for computingthe dynamic response of transient problems in structural dynamics has been studied. Conventionally, the solution of time dependent problems hasinvolved discretisation of thegoverning partial differential equations (PDE) independently in spaceandtimeby assuming a variable separable formof approximation, in which spatial discretisation of primary variables is usually carriedout first usingpoly nomial basis finite elements resulting in a set of ordinary differential equations (ODE) in time for the nodal values of primary variables. These ODEs are further approximated in time using an appropriate time integration scheme to compute the response of the nodal values. Generally direct time integration methods with or without mode superposition are commonlyemployed for temporal approximation of transient problems in structural dy namics applications. These direct time integration methodsare based on truncated Taylor series approximations of temporal evolution of piecewise continuous spatial approxima tion of primary variables. Since higher frequencies (or small time-scale variations) are not accurately resolved by a finite element mesh, some spurious oscillations creep into the computed solution of transient wave propagation problems whenever the time step of integration approaches the Courant-Friedrich-Lewy (CFL) limit. This is mainly due to inadequacy of time integration procedure to accurately resolve the motions of spatial finite element mesh at a particular time step. Further, these time integration algorithms suffer from the drawback of providing no control over the global error that gets accumu lated at the end of time duration of interest. In addition, the accuracy and stability of the computed response depends on choice of time step, which in turn depends on spectral characteristics of the structure. These time integration methods are computationally ex pensive for large scale problems and can lead to poor quality of computed solution due to the additional roundoff errors incurred by additional floating point computations at small time steps. Therefore to address these limitations of conventional approaches in providing reli able solutions to transient problems in structural dynamics an approach involving con ventional FEM for spatial approximation and an alternative formulation using wavelets for temporal approximation of semidiscrete system ofequations is proposed in this study. This approach provides acontinous approximation of temporal evolution of peice wise continous spatial approximation of primary variables thereby providing ameans to con trol global error. Further it provides a heirarchical form of representation ofsolution in time domain by exploiting the inherent multi-resolution capability ofwavelet basis func tions which facilitates to provide accurate representation of localised features ofthe prob lem. This formulation is examined for stability characteristics using energy method and validated by means ofnumerical testcases for various SDOF and MDOF systems arising in structural dynamics. It has been found that the response computed using the proposed formulation is energy conserving for undamped systems and L-stable for damped sys tems. Further it was found that the solutions computed compare favourably well with that ofNewmark (constant average acceleration method) and respective analytical solu tions and also provide a bound on global error. In addition this approach is found to provide better approximation than central difference methods for certain wave propaga tion problems.