Page 135 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

STEADY STATE HEAT CONDUCTION IN MULTI-DIMENSIONS
T 2 127
T
T 3 2 T 1
T 1
T 4 T
T T 1 2
T 1 2
Cross section T 5
T 6

(a) Two dimensional plane (b) Three dimensional (c) Axisymmetric
geometry domain configuration

Figure 5.1 Examples of heat conduction in two-dimensional, three-dimensional and
axisymmetric geometries

is referred to as being axisymmetric. For instance, the revolution of a rectangular sur-
◦
face through 360 , with respect to a vertical axis, produces a vertical cylinder as shown
in Figure 5.1(c). Therefore, the heat conduction equations need to be written in three-
dimensional cylindrical coordinates for such a system. However, if no significant variation
in temperature is expected in the circumferential direction (θ direction), which is often the
case, the problem can be reduced to two dimensions, and a solution based on the shaded
rectangular plane in Figure 5.1(c) is sufficient.
Unlike one-dimensional problems, two- and three-dimensional situations are usually
geometrically complex and expensive to solve. The complexity of the problem is increased
in multi-dimensions by the occurrence of irregular geometry shapes and the appropriate
implementation of boundary conditions on their boundaries. In the case of complicated
geometries, it is often necessary to use unstructured meshes (unstructured meshes are gen-
erated employing arbitrarily generated points in a domain, see Chapter 10) to divide the
domain into finite elements. Fortunately, owing to present-day computing capabilities, even
complex three-dimensional problems can be solved on a standard personal computer (PC).
In the following sections, we demonstrate the solution of multi-dimensional steady state
problems with relevant examples.

5.2 Two-dimensional Plane Problems

5.2.1 Triangular elements

The simplest finite element discretization that can be employed in two dimensions is by
using linear triangular elements. In Chapter 3, we discussed the use of triangular elements