The monolithic approach of coupled phenomena implies identical space and time meshes for each phenomenon. This is not always possible, for various reasons. The coupled problems may have different numerical convergence properties, generally associated with different physical scales or non-linearities. For example, a coupled hydro-mechanical problem may need large time steps for the fluid diffusion problem, in order to allow, in each step, fluid diffusion over a long distance (of the order of magnitude of the finite elements). At the same time, strong non-linearities may occur in solid mechanics behaviour (strong elasto-plasticity changes, interface behaviour, strain localisation…) and then the numerical convergence will need short time-loading steps, which should be adapted automatically to the rate of convergence. Then, it is quite impossible to obtain numerical convergence for identical time and space meshes.
Research teams of different physical and numerical culture have progressively developed different modelling problems. As an example, fluid flow has been largely
developed using the finite difference method for hydrogeology problems including pollutant transport, and for oil reservoir engineering (see Section 11.2.3) taking multiphase fluid flow (oil, gas, condensate, water,…) into account. Coupling such fluid flow with geomechanics in a monolithical approach would imply implementation of all the physical features already developed respectively in finite elements and finite differences codes. The global human effort would be very large!
Coupled problems generally present a higher non-linearity level then uncoupled ones. Thus, inaccuracy in parameters or in the problem idealisation may cause degradations of the convergence performance. How can we solve such problems and obtain a convincing solution? First of all, a good strategy would be to start with the uncoupled modelling of the leading process, and to try to obtain a reasonable first approximation. Then, one can add a first level of coupling and complexity, followed by a second one… until the full solution is obtained.
However such a “trick” is not always sufficient. Staggered approaches may then give an interesting solution. In a staggered scheme, the different problems to be coupled are solved separately, with (depending on the cases) different space or time mesh, or different numerical codes. However, the coupling is ensured thanks to transfer of information between the separated models at regular meeting points. This concept is summarised in Fig. 11.3. It allows, theoretically, coupling of any models.
When using different spatial meshes, or when coupling finite elements and finite differences codes, the transfer of information often needs an interpolation procedure, as the information to be exchanged is not defined at the same points in the different meshes.
The accuracy of the coupling scheme will mainly depend on the information exchange frequency (which is limited by the lowest time step that can be used) and by the type of information exchanged. The stability and accuracy of the process has been checked by different authors (Turska & Schrefler, 1993; Zienkiewicz et al., 1988). It has been shown that a good choice of the information exchange may highly improve the procedure efficiency.
Time
