Two FETI-based heterogeneous time step coupling methods for Newmark and α-schemes derived from the energy method

The purpose of this paper is to describe a general methodology to obtain multi-time step coupling methods for different time-integration schemes (Newmark, HHT-α, WBZ-α, CH-α) on the basis of energetic considerations. The general framework of the FETI method is considered for building the heterogeneous time step coupling methods by ensuring the gluing of subdomains with Lagrange multipliers. The proposed methods for coupling subdomains with different time integration schemes and time steps (fine and large time steps), are based on the pseudo-energy measure as introduced by Hughes in the so-called energy method employed for proving the stability of implicit-explicit algorithms. Assuming a given variation of the Lagrange multipliers over the large time step, kinematic conditions are derived by ensuring the zero pseudo-energy at the interface. Two methods will be derived in this paper. The first new method, for which the interface problem involving the Lagrange multipliers is solved at the large time scale, can handle the popular dissipative α-schemes (HHT-α, WBZ-α, CH-α); in the particular case of the Newmark schemes, the new method matches with the PH method proposed by Prakash and Hjelmstad. The second new method is based at the micro time scale. It can be viewed as an extension of the method proposed by Gravouil and Combescure (GC method) for the coupling of the α-schemes. Nonetheless, like the GC method, the new micro-time-based method turns out to be dissipative when non-dissipative second-order accurate schemes are involved, and looses one order of accuracy.