Civil Engineering Reference
In-Depth Information
9
Coupled Problems
9.1
Introduction
In the previous Chapter, flow problems were treated in terms of a single dependent variable,
for example the “potential” or “total head”, and solutions involved only 1 degree of freedom
per node in the finite element mesh. While this simplification may be adequate in some
cases, it may be necessary to solve problems in which several degrees of freedom exist at the
nodes of the mesh and the several dependent variables, for example velocities and pressures,
or displacements and pressures, are “coupled” in the differential equations. Strictly speaking,
the equations of two- and three-dimensional elasticity involve coupling between the various
components of displacement, but the term “coupled problems” is really reserved for those
in which variables of entirely different types are interdependent.
Both steady state and transient problems are considered in this Chapter. As usual,
the former involves the solution of sets of simultaneous equations, as in Chapters 4 to
7. Program 9.1 describes a steady state solution of the Navier-Stokes equations (see
Sections 2.16, 3.11), in which the simultaneous equations are non-linear. An iterative pro-
cess is therefore necessary during which the equations are solved repeatedly until the
velocities and pressures have converged. As discussed in Section 3.11, these equations will
have unsymmetrical coefficient matrices.
Program 9.2 solves the same problem without any global matrix assembly using a
BiCGStab(l) iterative solver. In this case nested iterative processes are employed, with an
internal one for BiCGStab iterations and an external one until convergence of velocities
and pressures is obtained. The BiCGStab process is described in Section 3.5.3.
The remaining three programs describe coupled transient problems governed by the
“Biot” equations (see Sections 2.18, 3.12). These coupled equations are cast as (linear)
first order differential equations in the time variable, and solved by the implicit integration
techniques introduced in Chapter 8.
Program 9.3 describes analysis of poro-elastic materials subjected to incremental time-
dependent external loading (see Section 3.12.2). Program 9.4 enables investigations to be
made of poro-elastic-plastic materials and transient collapse problems, by extending the
