Page 41 - Process Modelling and Simulation With Finite Element Methods

28 Process Modelling and Simulation with Finite Element Methods

au ar
d -+-=F
a at ax
where T(u, ux) is in principle the same functionality as the coefficient form (1.3),
but is treated differently by the Solver routines. In Coefficient Mode, the
coefficients are treated as independent of u unless the numerical Jacobian is
used, which brings out some of the nonlinear dependency - iteration does the
rest. The exact Jacobian in General Mode differentiates both r and F with
respect to u symbolically in assembling the stiffness matrix. Typically, General
Mode requires fewer iterations for convergence than Coefficient Mode with the
numerical Jacobian. The use of the exact Jacobian below does not require any
special treatment to avoid a singular stiffness matrix in the treatment of the linear
terms as the coefficient mode did. In general, General Mode is more robust at
solving nonlinear problems than Coefficient Mode. It is my opinion that
Coefficient Mode is a ‘‘legacy’’ feature of FEMLAB - the PDE Toolbox of
MATLAB, in many ways a precursor to FEMLAB, uses coefficient
representations extensively. Further, the coefficient formulation with numerical
Jacobian is a long standing FEM methodology, so for benchmarking against
other codes, it is a useful formulation.
Here’s the recipe for General Mode - a minor modification of what we just
did.

Model Navigator 1-D geom., PDE modes, general form (nonlin stat)
Options Set Axes/Grid to [0,1]
Draw Name: Interval; Start = 0; Stop =1
Boundary Model Set both endpoints (domains) to Neumann BCs
Boundary Settings
Subdomain Model Set r = 0; da = 0; F = uA3+uA2-4*u+2
Subdomain Settings
Mesh mode Set Max element size, general = 1; Remesh
Solve Use default settings (nonlinear solver, exact Jacobian)
Post Process After five iterations, the solution is found. Click on the
graph to read out u=0.732051. Play with the initial
conditions to find the other two roots.

Although setting up this template (rtfindgen.m for root finiding of simple)
function of one variable was rather involved, and in face MATLAB haas a
simple procedure for root finding using the built-in function fzero and inline
declarations of functions, the FEMLAB GUI still provides many options and
flexibility to root finding that may not be available in other standard packages.
The next subsection applies our newly constructed nonlinear root finding scheme
to a common chemical engineering application, flash distillation, which clarifies
A FEW MORE FEATURES OF THE femlab gui.