A time–space multi-scale algorithm for transient structural nonlinear problems

Abstract The object of this paper is to propose a general time–space multi-scale method for the resolution of transient problems. The space multi-scale method is now of common use for static nonlinear analysis. A time multi-scale approach is also of some interest and has also been studied in the past years. For all transient problems, time and space scales are usually coupled, as we have to take care of wave propagation: hence, one should be able to consider time and space scales in a unique framework. A new approach is developed in this paper: the first one to give a general unique framework for time–space multi-scale method. Theoretical results for precision and convergence of the proposed multi-scale method are first presented. The method is then implemented with an algorithm for linear behaviour of the structures. An example consisting of different sub-domains where different numerical time integration schemes with different time steps are used is then presented. The paper shows afterwards how the method can be applied to nonlinear structural behaviour. The corresponding algorithms are then presented in the case of implicit or explicit schemes for time integration. The particular case where all the sub-domains are computed using explicit solvers is then presented because this particular case makes the proposed formulation particularly attractive. Applications to coupled implicit–explicit sub-domains are also presented.