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The phenomena considered in this section are much more complex as they associate multiphase fluid flow and hydro-mechanical coupling (cf. relevant sub-sections of Section 11.3.2) as well as temperature effects. All the features described in the pre­ceding section are to be considered here, associated with some new points. Heat diffusion has to be modelled. Temperature […]

The coupled flow of two, different, fluids in a partly saturated rigid media is now considered. Unsaturated soils provide a common example, where the two fluids are water and air. Often, the air phase is considered to be at constant pressure, which is generally a relevant approximation as air pressure doesn’t highly affect the water […]

A large number of different phenomena may be coupled. It is impossible to discuss here all potential terms of coupling, and we will restrict ourselves to some basic cases often implied in environmental geomaterial mechanics. In the following para­graphs, some fundamental aspects of potential coupling are briefly described. 11.3.2.1 Hydro-Mechanical Coupling In the case of […]

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 […]

Let us first consider a purely advective process. In this case, the transport is gov­erned by the advection Eq. 11.11 and by the balance Eq. 11.6. Associating these two equations, one obtains: (VTC). ffluld + C = 0 (11.45) diff v ‘ which is a hyperbolic differential equation. It cannot be solved by the finite […]

The theoretical analysis of a time integration scheme accuracy and stability is generally based on a simplified problem (Zienkiewicz et al., 1988). Let us con­sider diffusion phenomena restricted to the linear case. Introducing the discre — tised field (Eq. 11.15) into the constitutive equations gives Darcy law (Eq. 11.7) (neglecting here the gravity term for […]

For solid mechanics problems, the constitutive law form (Eqs. 11.3 and 11.4) is an incremental one and differs from the ones for diffusion problems (Eq. 11.7). The knowledge of the stress tensor at any time implies that a time-integrated constitutive law is required. The stress tensor is a state variable that is stored and transmitted […]

The time dimension appears in the form of a first order time derivative in the consti­tutive mechanical model (Eqs. 11.3, 11.4) and in the diffusion problems though the storage term (Eq. 11.6). We will here discuss the time integration procedure and the accuracy and stability problems that are involved. 11.2.6.1 Time Integration — Diffusion Problems […]

From Eq. 11.24, it appears that the stiffness matrix is a derivative of the internal forces: n Fint n FLhK] = — F — = — *ljBLjdvj (11.25) 1 2 1 Derivative of problem 1 nodal forces with respect to problem 1 nodal unknowns Derivative of problem 1 nodal forces with respect to problem 2 […]

Let us now concentrate on the finite element method. The fundamental equation to be solved is the equilibrium Eq. 11.1 (or the balance Eq. 11.6 for diffusion phenomena). As the numerical methods give an approximate solution, the equilib — rium/balance equation has to be solved with the best compromise. This is obtained by a global […]