Implicit coupling of heterogeneous and asynchronous time-schemes using a primal approach based on velocity continuity at the subdomain interface

Abstract

In the field of structural dynamics, it can be particularly interesting to consider a different time integrator and time scale in a different part of a problem (i.e. in the case of multi-physics problems, non smooth contact mechanics, seismic engineering with impacts, soil-structure interaction problems, or multiscale models using macro-element systems with a dynamic internal equilibrium). This paper presents a primal coupling algorithm based on a velocity gluing at the interface between two subdomains in order to be able to take into account both heterogeneous (different time schemes) and asynchronous (different time steps) time integrations (HATI). This algorithm allows for an implicit nonlinear resolution in providing the exact algorithmic tangent operator to maintain quadratic convergence for Newton-Raphson procedures. It is not intrusive in the finite element code as it only requires an interface element. The algorithm is presented in this paper for coupling different time schemes stemming from both Newmark families and Euler+ θ integration schemes (which can be very attractive when dealing with hard contact non smooth mechanics using complementarity methods). The proposed primal approach, which is based on imposing velocity continuity at the interface, is a viable alternative to the classical dual approaches since it is highly suitable for multiscale and sub-structuring models relying on subdomains with internal time integration schemes as well as for problems using macro-element families. The stability analysis exhibits a second order accuracy of the proposed approach. A selection of numerical examples under linear and nonlinear assumptions and for multiple degree-of-freedom system is provided; these examples show that no energy is being dissipated at the interface and overlap very closely with reference solutions. • A primal coupling algorithm based on a velocity gluing at the interface for implicit problems is introduced. • The tangent algorithm operator is derived even for multi-time stepping (i.e. asynchronous) and for various integration schemes using different integration parameters and order approximation (i.e. heterogeneous). • The stability analysis exhibits a second order accuracy of the proposed approach.