Page 345 - Process Modelling and Simulation With Finite Element Methods

332 Process Modelling and Simulation with Finite Element Methods

Boundary Mode and Boundary Settings
Mode wcu. Select domain 2,443, check
active in this subdomain, type ‘u’ into the
constraint variable entry box, and Apply
Mode wcv. Select domain 2,4-8, check
active in this subdomain, type ‘v’ into the
constraint variable entry box, and Apply
OK
Before solving the time evolution, we are going to try a new trick to improve
convergence. The biggest problem that was making the previous simulation
slow was the rapid change in the velocity and electric fields required in the first
few instances from the initial condition (no field, no flow) to practically
pseudosteady flow and field. The viscous and dielectric response times are
much faster than the diffusive and convective time scales, so this problem is
inherently stiff. In the computational modeling of the Navier-Stokes equations
for incompressible flow, this problem is encountered for the pressure. Since in
the incompressible approximation, sound speed is infinite, the pressure field
adjusts instantaneously to changes in velocity. Computationally, time stepping
over such widely different time scales leads to problems with stiffness and slow
convergence, requiring miniscule time steps. The SIMPLE algorithm (Patankar,
1980) circumvented this pitfall by staging the time stepping of the velocity with
rapid solution to the pressure field consistent with mass conservation and the
current velocity field by solving a separate elliptic equation for the pressure - the
Lighthill Poisson equation. The difference is that the corrections to the last
pressure field are not found - small changes on the order of the time step -- but
rather the pressure can be wholly different from that at the previous time step.
Instantaneous changes in the pressure that depend on the field everywhere in the
domain are thus catered for, and the Navier-Stokes simulation is no longer stiff.
The FEMLAB Navier-Stokes application mode has this fast Poisson solver for
pressure built in. But our electrostatic potential mode does not. So even though
@ should, in principle, change instantaneously to applied alterations in voltages,
which should change the slip velocity instantaneously, the ns mode will respond
on the incompressible time scale, but the electric field needs to be relaxed. So to
mimic the SIMPLE algorithm, we need to implement a fast elliptic solver for
flow and electric field while freezing the concentration profile. Once the
velocity field and electric field have been established, we can release the mass
transport. Our no electrokinetic relaxation time in the potential mode is
necessary for the model to advance steadily with only small changes to u,v,p and
phi at each time step. The fast elliptic step overcomes the rapid relaxation time
needed for abrupt changes in the electric field.