The application of adaptive finite element method to dynamic problems is investigated. Since the early development and application of the finite element method, attempts have been made to obtain the information about finite element discretization errors for better solutions. Accompanying the efforts to evaluate the discretization errors, the adaptive finite element method for more effective solution became one of the popular branches of the finite element method during the last decade. Especially for the finite element analysis of dynamic problems, it is not reasonable to proceed with a fixed mesh and fixed time step as the locations of steep regions or damping out of the energy is changing from time to time. In this study both the kinetic and strain energy errors induced by space and time discretization were estimated in a consistent manner and controllered by the simultaneous use of the adaptive mesh generation and the automatic time stepping. In such a way, the best performance attainable by the finite element method can be achived. Following three major parts are intensively studied :
First, for dynamic problems, a system of ordinary differential equations is usually solved by direct time integration. Once a direct integration scheme is chosen, the accuracy of the integration depends only on the time step size. In practice, it is unreasonable to use a fixed time step in the whole dynamic process. Thus, automatic time stepping schemes for dynamic problems have drawn attention of many researchers recently including Zienkiewicz, Xie, Zeng, etc. In this paper, the global and local error estimates and adaptive time stepping for the various direct time integration in dynamic analysis are presented. A successive quadratic function is used for the locally exact value of the acceleration and the corresponding parameters for the function are obtained from accelerations at three time stations at every time stage. The local error can be estimated simply by comparing the solutions obtained by direct time integration method (Newmark method and Wilson method) with the locally exact solutions. Based on this local error estimate an adaptive time stepping technique in global sense is proposed. By using the quadratic approximation of acceleration, the error of the strain energy can be estimated as well as the error of the kinetic energy.
Second, in representing a mathematical continuum by finite elements, there are spatial discrepancies between the discretized model and the mathematical model. Such discrepancies can induces subsequent numerical error. To reduce the spatial discretization error efficiently, error estimations and adaptive algorithms on problems of elliptic type has been studied. To apply these algorithms to problems of hyperbolic type, however, it is necessary to update the spatial mesh in time, so that solution vectors of displacements, velocities, and accelerations of the previous time step need to be transferred to the newly designed mesh.
Third, both the kinetic and strain energy errors induced by space and time discretization were estimated in a consistent manner and controlled by the simultaneous use of the adaptive mesh generation and the automatic time stepping. Also an optimal ratio of spatial discretization error to temporal discretization error was discreased. In this study it was found that the best performance can be obtained when the specified spatial and temporal discretization errors have the same value.
Numerical studies on SDOF examples and one-dimensional examples are presented to demonstrate the accuracy of the proposed temporal error estimate technique and to verify the performance of the procedure.
