Page 228 - Process Modelling and Simulation With Finite Element Methods

Chapter 6

GEOMETRIC CONTINUATION

W.B.J. ZIMMERMAN and A. F. ROUTH
Department of Chemical and Process Engineering, University of Sheffield,
Newcastle Street, Sheffield Sl 3JD United Kingdom

E-mail: w.zimmerman @ she& ac. uk

Geometric continuation occurs if the mesh of the domain must change from one solution
to the next due to variation of the geometry model. In this chapter, we take two examples
as paradigmatic - the additional pressure loss in a channel due to various size orifice
plates is an example of steady state geometric continuation. Conceptually, this problem
is little different from the parametric continuation by Rayleigh number in the Benard
problem of Chapter 5. The second example is a drying film with latex particles
embedded in the fluid. Two variations on the theme are computed. The noncumulative
model vanes the front initial position and solves for the time evolution from a uniform
surfactant concentration profile initially, with the front frozen at several different
positions independently. The cumulative model takes the surfactant concentration
profile from the previous front position as its initial value, and alternates solving the
transport model with a point source and moving of the front position. This operator
splitting technique is shown to be asymptotically convergent as the time increment for
these two partial steps shrinks. On a minor note, the film drying model implements a
weak term for the point source in a I-D geometry model using the boundary conditions.
The example is unique in that the FEMLAB manuals give only 2-D and 3-D point
sources using point mode.

6.1 Introduction

We have already seen several examples of parametric continuation - the
traversing in small steps of a range in a parameter, using the previous solution of
a nearby parameter value as the initial guess for the solution at the new value of
the parameter. As long as the parameter does not pass through a bifurcation
point, the new solution should be smoothly connected to the old one if the step in
parameter is small enough. Even if there is a bifurcation, however, the old
branch may still be a solution, as we found with Benard convection in Chapter 5.
Geometric continuation is qualitatively different from parametric
continuation in one important respect. In geometric continuation, the
geometrical change of the domain leads to the requirement of re-meshing with
each geometric parameter value. We should be careful to class as geometric
continuation changes in a parameter that do not lead to a similar geometry. For
instance, in pipe flow, it is well known that the flow is characterized by a
Reynolds number:

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