Inconsistency errors of constant velocity multi-time step integration algorithms

Abstract

Previously known multi-time step integration methods for finite element computations in structural dynamics have been shown to be unstable due to interpolation error propagation. New algorithms of multitime step integration based on constant velocity during sub cycling are investigated. The assumption of constant velocity gives linear variation of displacements so the errors connected to interpolation at the interface between different time step partitions are eliminated. As a consequence, the new constant velocity algorithms give bounded solutions and have been shown to be conditionally stable by their authors. However, numerical investigation demonstrates that if time steps close to the stability limit are used, the errors for higher natural modes are so huge that the obtained solutions can only be considered as incorrect. The main reason for this behaviour is that the constant velocity time integration algorithms are inconsistent. Displacements can be calculated either by direct integration or from the equation of motion leading to different solutions. Based on the numerical results it is concluded that use of time steps below stability limit is insufficient to assure proper solutions. Therefore, significant time step reductions are often required to assure acceptable error levels. As a consequence, the new sub cycling algorithms can be more expensive than ordinary time integration. Because they also lead to larger errors the constant velocity sub cycling algorithms are useless from practical point of view. Since sub cycling is available as an option in LS-DYNA a serious warning is issued to potential users.