Galerkin‐based multi‐scale time integration for nonlinear structural dynamics

This paper deals with a GALERKIN‐based multi‐scale time integration of a viscoelastic rope model. Using HAMILTON's dynamical formulation, NEWTON's equation of motion as a second‐order partial differential equation is transformed into two coupled first order partial differential equations in time. The considered finite viscoelastic deformations are described by means of a deformation‐like internal variable determined by a first order ordinary differential equation in time. The corresponding multi‐scale time‐integration is based on a PETROV‐GALERKIN approximation of all time evolution equations, leading to a new family of time stepping schemes with different accuracy orders in the state variables. The resulting nonlinear algebraic time evolution equations are solved by a multi‐level NEWTON‐RAPHSON method. Realizing this transient numerical simulation, we also demonstrates a parallelized solution of the viscous evolution equation in CUDA©. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)