Page 61 - Process Modelling and Simulation With Finite Element Methods

48 Process Modelling and Simulation with Finite Element Methods

This example should illustrate the importance of selecting the right solver
for your equations. If there is any dependence off on the dependent variables,
then the stationary nonlinear solver should be used. The linear solver is faster,
but it also presumes that the coefficients of the PDE do not depend on the
dependent variable u (else the problem would be nonlinear). When in doubt, use
the nonlinear solver. After all, (1.19) with R(u)=k u, is a linear problem, but
FEMLAB only finds the correct steady state solution with the nonlinear solver!
The slow convergence rate is also the consequence of the form of the model -
general mode with the exact Jacobian solver option for the nonlinear solver
converges in two iterations to the correct profile.
We argued that (1.20) is the finite difference matrix equation for this
problem, yet later applied the argument that (1.22) should describe the FEMLAB
finite element problem. Because we used Lagrange linear elements, in this
special case the finite element and finite difference matrix operators coincide, up
to the boundary conditions. To see this, we will take a foray into the MATLAB
representation of FEMLAB problems.
Pull down the File Menu and select Export FEM structure as ‘fern’. This
puts the current solution as a MATLAB data structure in the MATLAB
workspace. We can then manipulate it using the built-in MATLAB functions
and commands, as well as the special function set of FEMLAB.
In your MATLAB workspace, try the commands
>>x=fem.xmesh.p{l};
>>u=fem.sol.u;
>>plot (x,u)
This should pop up a MATLAB Figure plotting the solution u versus the array of
mesh points. No doubt your plot looks scrambled. This is because FEMLAB
stores the mesh points and the associated solution variables so as to make the
specification of the matrix equations sparse and compact. We can make sense of
the solution by ordering the mesh points and the solution:
In your MATLAB workspace, try the commands
>> [xx, idx] =sort (x) ;
>>plot (xx, u (idx) )
This plot should resemble Figure 1.6, with the exception that it represents your
last FEMLAB solution. In fact, we can only make sense of the solution format
of the fern structure so readily because this is a single dependent variable, one-
dimensional problem. Otherwise, multiple variables and dimensions leave a
mesh and solution structure that only FEMLAB tools/functions can readily
decode.