Page 126 - Process Modelling and Simulation With Finite Element Methods

Multiphysics 113

Max 8 09e-OC
Contour u2 (u2)
10’~
I I I I I I I I I I
0 6016
09 1 204
oa I a06
2 409
3 01 1
07 3 613
4 216
06 4 aia
5 421
05 6 023
6 625
04 7 228
7 a3
03 a 432
9 035
02 9 637
10 24
01 10 a4
11 44
0 12 05
03-02-01 0 01 0203040506070a09 I 11 1213 Min -0 00126
Figure 3.1 Steady state streamlines of hot walUcold wall buoyancy driven convection for Ra=l.
Buoyancy Driven Cavity Flow: Parametric Continuation
Our solution strategy for the hot wallkold wall problem to reach Ra=50 was to
build up elements of the solution piecemeal. Were we to try to start at Ra=50
directly, we would find that FEMLAB cannot find a solution. Why not? For a
nonlinear problem, the initial condition may not be in the “basin of attraction”
for the desired solution, so Newton’s Method could career far off. For it to work
well, Newton’s Method must start near a solution. For instance, the initial
solution for hydrostatic pressure and velocity noise for Ra=O was an essential
step. As a fully linear problem, it was readily solvable. It serves the important
purpose of introducing an asymmetric velocity profile (due to the numerical
noise of truncation error). This permits the solution for Ra=l, which is
qualitatively similar to the Ra=O in that it has a circulation, though a massive
change in scale. Even then, though qualitatively similar to the Ra=l solution,
Ra=50 was too far a leap from Ra=l to converge. The notion of traversing the
solution space to introduce various topological features consistent with the target
solution so as to be in its basin of attraction is similar to the established concept
of parametric continuation. In parametric continuation, one restarts the
simulation with a parametric value close to that of the saved, converged solution.
Because the solution at the new parameter values is not expected to be much
different than at the old parameter values, the Newton solver should converge
rapidly. This methodology only fails if the new parameter is close to a
bifurcation point - in which case multiple solutions are possible. The Jacobian
used by Newton’s Method is then very close to singular, so convergence may not
be achieved. Or if it is, which of the multiple solutions that is selected may not
be a priori predictable.