Civil Engineering Reference
In-Depth Information
7
Steady State Flow
7.1
Introduction
The five programs presented in this chapter solve steady state problems governed by
Laplace's equation (2.122). Typical examples of this type of problem include steady seepage
through soils and steady heat flow through a conductor. Examples are presented of planar
(confined and unconfined), axisymmetric, and three-dimensional flow. Unlike the problems
solved in Chapters 5 and 6, which gave vector fields of displacements, the dependent vari-
able in these problems is a scalar, generically called the
potential
which may represent, for
example, the total head in a seepage problem or the temperature in a heat flow analysis.
Each node therefore has only one degree of freedom associated with it.
Systems that are governed by Laplace's equation require boundary conditions to be pre-
scribed at all points around a closed domain. These boundary conditions commonly take
the form of fixed values of the potential or its first derivative normal to the boundary. The
problem amounts to finding the values of the potential at points within the closed domain.
Being “elliptic” in character, the solution of Laplace's equation quite closely resembles
the solution of equilibrium equations (2.57) in solid elasticity. Both methods ultimately
require the solution of a set of linear simultaneous equations. The element conductivity
matrix (analogous to the “stiffness” matrix in elasticity) can be formed numerically, as
described by equations (3.61) to (3.63) or “analytically” as discussed in Section 3.2.2.
Either way, the element matrices can be assembled into a global conductivity matrix which,
like its global elastic counterpart, is symmetrical, banded, and usually stored as a skyline.
Alternatively, element-by-element iterative strategies can be used. Taking the analogy with
Chapter 5 one stage further, “displacements” now become total heads and “loads” become
net nodal inflow.
Program 7.1 describes the solution of Laplace's equation over a set of 1D elements, that
can each have different lengths, areas, and permeabilities. The elements can be attached
end to end, or in any desired “network” of connections. Program 7.2 describes the solution
of Laplace's equation over a plane or axisymmetric 2D domain. Program 7.3 describes the
non-linear problem of free-surface flow, in which the mesh is allowed to deform iteratively
