Download NM-SESES Tutorial - Numerical Modelling GmbH

28
SESES Tutorial September 2012
As first step, we initialize again to zero the solution in order to destroy the previously computed one. Then we substitute the default printing statement displaying
the residual norm with a statement printing the incremental norm of the temperature.
This is done by defining a user convergence criterion and by embedding a call to the
write function. Since the problem is still linear, the criterion always returns 1 meaning unconditional convergence. As last, we solve as before three times and obtain the
following output.
Solution newly initialized
@@@ Solving for the first time:
@@@ Solving for the second time:
@@@ Solving for the third time:
AbsIncr.Temp=1.86e+00
AbsIncr.Temp=6.60e-16
AbsIncr.Temp=1.77e-16
The numerical behavior is the same as before, but now we can state that the temperature is computed up to a precision of 10−15 . This is the precision of the numerical solution when solving the discretized governing equations, which is by far much smaller
than the precision with respect to the exact analytical solution of the governing equation.
Now that we have acquainted some feeling on norm values for numerical solutions
of the linear thermal problem, we change its character and turn it into a non-linear
problem. Here, we define the thermal conductivity to be a function of the temperature
κ = κ(T ) = 1 + T and try to compute a solution.
MaterialSpec Silicon Parameter KappaIso 1+Temp W/(m*K)
Convergence (* restore default *)
Write "@@@ Solving the non-linear problem:\n"
Solve Stationary
After restoring the default convergence criterion, this time the output will be something similar to
@@@ Solving the non-linear problem:
AbsResid.Temp=8.73e-01
AbsResid.Temp=5.52e-02
.
.
.
AbsResid.Temp=1.26e-04
AbsResid.Temp=6.58e-05
AbsResid.Temp=3.41e-05
AbsResid.Temp=1.77e-05
@@@@ SESES SOFT ERROR: Coupled (Newton-Raphson) loop did not converge
SESES recognizes the new problem to be non-linear and so it automatically starts a
Newton’s iteration consisting of linear solution steps. The convergence of the residual norm is low and since after 15 steps the convergence criterion is not fulfilled, the
Newton’s algorithm gives up and returns. The coupled Newton’s algorithm, however, converges in the vicinity of a solution quadratically but only if the derivatives of
the residual equations with respect to the solution are available. For this problem, we
have specified the heat conductivity to be a function of the temperature, but SESES
is not designed to figure out the derivative. Actually with some programming effort
it would be possible to obtain the derivative either in analytical form or numerically
by difference. However, the former method is rather complex and may result in long
and complex analytical expressions to evaluate while the latter may be numerically
unstable. For these reasons, we do not try to obtain the derivative from the function