Advanced Building Simulation

98 Hensen
Computed pressure
c(0)
c(1)
c(2)
0 1 2 3
Iteration
c(3)
Exact solution
4 5
Figure 4.7 Example of successive computed values of the pressure and oscillating pressure
corrections at a single node.
where r is the ratio of C i for the current iteration to its value in the previous iteration.
The factor 1/(1 � r) is called a relaxation factor. The extrapolated value of node pressure
can be used in the next iteration. If it is used in the next iteration, then r is not
evaluated for that node in the following iteration but only in the one thereafter. In
this way, r is only evaluated with unrelaxed pressure correction values. This process
is similar to a Steffensen iteration (Conte and de Boor 1972), which is used with a
fixed-point iteration method for individual nonlinear equations. The iteration correction
method presented here gives a variable and node-dependent relaxation factor.
When the solution is close to convergence, Newton–Raphson iteration converges
quadratically. By limiting the application of relaxation factor to cases where r is less
than some value such as 0.5, it will not interfere with the rapid convergence.
However, there is some evidence that suggests that in a number of cases simple
under relaxation would provide even better convergence acceleration than the
Steffensen iteration (Walton 1990).
Some network simulation methods incorporate a feature to compute an initial pressure
vector from which the iterations will start. For instance (Walton 1989a,b) uses
linear pressure-flow relations for this. Reasons for refraining from this are as follows:
1 it is not possible to provide a linear pressure–flow relation for all envisaged flow
component types;
2 after the initial start, the previous time step results probably provide better iteration
starting values than those resulting from linear pressure–flow relations; and
3 this would impose an additional input burden upon the user.
According to Walton (1990) and Axley (1990), an initial pressure vector would also
be necessary for low flow velocities so that (a) flows are realistically modeled in the
laminar flow regimes, and (b) to avoid singular or nearly singular system Jacobians