Imposing displacements in implicit direct time integration & a patch test

Abstract

We consider the transient analysis of finite element models when an implicit direct time integration scheme is used and focus on the accuracy of the response solution when time varying nodal displacements are imposed. The solutions of such problems are abundantly sought in engineering practice. We analyze the performance of the widely used Trapezoidal Rule, a specific case of Newmark time integration, and the performance of the Bathe method. The theoretical results show that the Trapezoidal Rule is unstable and this instability becomes more pronounced as the time step size for solution is decreased, whereas the Bathe method is stable. We illustrate these theoretical findings in the transient analyses of some finite element models. These investigations lead us to define a “patch test” useful for the general evaluation of direct time integration schemes.