# Coupling Various Problems

11.3.1 Finite Element Modelling: Monolithical Approach

Modelling the coupling between different phenomena should imply the need to model each of them and, simultaneously, all the interactions between them. A first approach consists in developing new finite element and constitutive laws especially dedicated to the physical coupled problem to be modelled. This approach allows taking accurately all the coupling terms into account. However there are some draw­backs that will be discussed in Section 11.3.4. Constitutive equations for coupled phenomena will be discussed in the following sections.

The number of basic unknowns and, consequently, the number of degrees of freedom – d. o.f. – per node are increased. This has a direct effect on the computer time used for solving the equation system (up to the third power of the total d. o.f. number). Coupled problems are highly time consuming.

Isoparametric finite elements will often be considered. However, some specific difficulties may be encountered for specific problems. Nodal forces or fluxes are computed in the same way as for decoupled problems. However, the stiffness matrix evaluation is much more complex, as interactions between the different phenomena are to be taken into account. Remember that the stiffness or iteration matrix, Eq. 11.25, is the derivative of internal nodal forces/fluxes with respect to the nodal unknowns (displacements/pressures/etc…). The complexity is illustrated by the following scheme of the stiffness matrix, restricted to the coupling between two problems.

The part of the stiffness matrix in cells 1-1 and 2-2 are similar, or simpler, than the ones involved in uncoupled problems. The two other cells, 1-2 and 2-1, are new and may be of a greater complexity. Remember also that the derivative considers internal nodal forces/fluxes as obtained numerically, i. e. taking into account all numerical integration/derivation procedures. On the other hand, the large difference of orders of magnitude between different terms may cause troubles in solving the problem and so needs to be checked.

Numerical convergence of the Newton-Raphson process has to be evaluated care­fully. It is generally based on some norms of the out-of-balance forces/fluxes. How­ever, coupling often implies the mixing of different kinds of d. o.f., which may not be compared without precaution. Convergence has to be obtained for each basic problem modelled, not only for one, which would then predominate in the computed indicator.