Unconditional stability in large deformation dynamic analysis of elastic structures with arbitrary nonlinear strain measure and multi-body coupling

Abstract

Stability is a fundamental feature of a time integration method in long simulations of elastic bodies in large deformations. It is well known that energy conservation is the key to achieve it. The implicit one-step conserving method of Simo and Tarnow is simple and effective for quadratic strain measures. However, the issue of unconditional stability does not yet have a definitive solution for general nonlinear strain measures and multi-body couplings. This article shows how to reach this goal by a simple modification of the Simo and Tarnow method. In practice, the mean internal force vector of the time step is evaluated using the average value of the stress at the end-points and an integral mean of the strain–displacement tangent operator over the step computed by time integration points. Compared to other proposals, the approach conserves energy for any structural model regardless of its spatial finite element discretization and does not require Lagrange multipliers, discrete derivatives, approximations of the strain–displacement law or special iterative solutions. Long term stability is proved using dynamic analyses of Reissner beams and Kirchhoff–Love shells, discretized spatially with Lagrangian and isogeometric elements respectively. Moreover, the proposed energy-conserving scheme is extended to multi-body analyses with nonlinear couplings by means of a penalty formulation, with focus on the rotational continuity for generic multi-patch Kirchhoff–Love shells.