Civil Engineering Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
x
Figure 1.6
Comparison of the exact and finite difference
solutions of Equation 1.4 with
f
0
A
1.
of the function between the calculated points is not known in the finite difference
method. One can, of course, linearly interpolate the values to produce an ap-
proximation to the curve of the exact solution but the manner of interpolation is
not an a priori determination in the finite difference method.
To contrast the finite difference method with the finite element method,
we note that, in the finite element method, the variation of the field variable in
the physical domain is an integral part of the procedure. That is, based on the
selected interpolation functions, the variation of the field variable throughout a
finite element is specified as an integral part of the problem formulation. In the
finite difference method, this is not the case: The field variable is computed at
specified points only. The major ramification of this contrast is that derivatives
(to a certain level) can be computed in the finite element approach, whereas the
finite difference method provides data only on the variable itself. In a structural
problem, for example, both methods provide displacement solutions, but the
finite element solution can be used to directly compute strain components (first
derivatives). To obtain strain data in the finite difference method requires addi-
tional considerations not inherent to the mathematical model.
There are also certain similarities between the two methods. The integration
points in the finite difference method are analogous to the nodes in a finite
element model. The variable of interest is explicitly evaluated at such points.
Also, as the integration step (step size) in the finite difference method is reduced,
the solution is expected to converge to the exact solution. This is similar to the
expected convergence of a finite element solution as the mesh of elements is
refined. In both cases, the refinement represents reduction of the mathematical
model from finite to infinitesimal. And in both cases, differential equations are
reduced to algebraic equations.
