A study on the formulation of dynamic problems by time domain finite element method using displacments and velocities as unknowns and its numerical stability

Abstract

Time integral algorithm is one of the major factors for numerical stability and precision. Many methods have been proposed to calculate time integration of an equation of motion. The Newmark family of algorithm, such as Newmark's beta method, is popular in practical calculations. In this study, however, some characteristics of FETD (finite element time domain) method with Bubnov-Galerkin formulation are investigated. Period and phase of numerical solutions calcualted by a GSSSS (Generalized Single Step Single Solve) algorithm with a cubic interpolation shape function are calculated and summarized. According to the calulated results, numerical damping can be observed, if a time step is around the third of a natural period, eigen values of a trainsient matrix are complex values, i.e., numerical damping is observed. If a time step exceed a natural period, numerical solutions will be unstable. On the other hand, a time step is less than one third of a natural period, numerical results show good agreement to the analytical solutions.