Contribution to algorithmic strategies for solving coupled thermo-mechanical problems by an energy-consistent variational approach
Sources of thermo-mechanical coupling in visco-elastic materials are various: viscous dissipation, dependence of material characteristics on temperature, ... Numerical simulation of these kinds of coupling can be challenging especially when strong coupling effects are present. Various algorithmic approaches have been proposed in the literature for this type of problem, and fall within two alternative strategies: - Monolithic (or simultaneous) approaches that consists of resolving simultaneously mechanical and thermal balance equations, where time stepping algorithm is applied to the full problem of evolution. - Staggered approaches in which the coupled system is partitioned and each partition is treated by a different time stepping algorithm. The goal is to obtain a good compromise between the following aspects: precision, stability and com- putational cost. Monolithic schemes have the reputation of being unconditionally stable, but they may lead to impossibly large systems, and do not take advantage of the different time scales involved in the problem, and often lead to non-symmetric formulations. Staggered schemes were designed to overcome these drawbacks, but unfortunately, they are conditionally stable (limited in time step size). Recently, a new variational formulation of coupled thermo-mechanical boundary value problems has been proposed (Yang et al., 2006), allowing to write mechanical and thermal balance equations under the form of an optimisation problem of a scalar energy-like functional. This functional is analysed in the framework of a thermo-visco-elastic strongly coupled problem, considering at first a simplified problem, and then a more general 2D and 3D case. The variational approach has many advantages, in particular the fact that it leads to a symmetric numerical formulation has been exploited in deriving alternative optimization strategies. These various algorithmic schemes were tested and analysed aiming to find the one that exhibits the best computational costs.