Search for efficient time integration methods in structural dynamics for finite element meshes with large variations of properties

Abstract: Direct time integration methods are the most commonly used solution algorithms for analysis of non-linear transient phenomena in finite element computations. One major disadvantage with these methods is that stability and accuracy of the computation are governed by the smallest element in the mesh assuming all other properties constant. Even a few small elements within a finite element mesh limits the time-step size that can be used in the computations. This, in turn, leads to many time-steps and large computational cost to cover even a short time range. The present thesis is devoted to a study of efficient time integration methods in structural dynamics for finite element meshes with large variations of properties. Both multi-time step techniques based on the Newmark family of methods and the space-time finite element method are examined. In the first part of the thesis multi-time step integration based on the Newmark family of methods are thoroughly described and a theoretical study of error propagation in the computations is given. The theoretical study indicates that errors introduced into the system by interpolation of nodal values at the interface between small an larger time-steps lead to high-frequency blow-up. This high-frequency blow-up makes the method unconditionally unstable and is termed numerical resonance. The instability of multi-time step methods is also illustrated in several numerical computations of longitudinal and transversal vibrations of a cantilever beam. It is also shown that numerical damping can eventually control the error propagation but will also alter the solution. These results were very disappointing since this study was intended to be the base for improvements of the method. Instead it was concluded that multi-time step methods were not to be used in any further work. The second part of this thesis presents an initial study of the space-time finite element method based on continuous Galerkin formulation. This method has in the literature been suggested as an alternative method for multi-time step computations. Basic formulations and a computational scheme are thoroughly described. Two triangular space-time bar elements are also derived. Numerical examples are presented to illustrate that stability problems can occur if these elements are used in meshes, unstructured in space. However, for structured meshes this technique gives reliable results, but the method is only conditionally stable. The critical time-step size can be computed using the CFL-condition. Simulations of multi-time step computations carried out with the space-time finite element method are shown to illustrate that this strategy can decrease the computational effort without reducing the accuracy of the results.