Page 258 - Process Modelling and Simulation With Finite Element Methods

Chapter 7

COUPLING VARIABLES REVISITED: INVERSE PROBLEMS,
LINE INTEGRALS, INTEGRAL EQUATIONS, AND
INTEGRO-DIFFERENTIAL EQUATIONS

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

E-mail: w.zimmerman @ she5 ac. uk

In this chapter, coupling variables are explored in great depth with regard to their role in
solving inverse equations and integral equations of various types. Four important
applications are taken as example studies - using lidar to detect position and spread of
dense gas contaminant clouds, the inverse problem in electrical capacitance tomography,
the computation of non-local heat transfer in a fiber composite medium, and the
population balance equations in particle processing. En route, we encounter several
features of FEMLAB not previously explored - coupling to optimization tools through
MATLAB, extended meshes, using the time-dependent solver as an iterative tool for
stationary nonlinear models, and the ability to selectively activate/deactivate
multiphysics modes in coupled models. The latter is particularly useful if there is only
one-way coupling (as in the hydrodynamics around the catalyst supported on the pellet in
Chapter 3). In the case of the integral equations treated here, a fictitious dependent
variable on an auxiliary domain is set up. The domain is used by coupling variables for
various operations, but the dependent variable is never needed itself. So deactivating it
results in better conditioning the FEM approximation to the integral equation.

7.1 Introduction

We are already familiar with boundary and subdomain integration - options
available on FEMLAB’s post processing menu. Boundary integration is useful
for computing all manner of surface quantities: the charge on a body in
electrostatics and the drag on a body in hydrodynamics, for example.
Subdomain integration is typically used for averages and higher moments of
combinations of the degrees of freedom defined in the domain. These features
are reliable, and given the nature of the finite element method expressed through
an integral property, the Galerkin method (see Chapter 2), FEMLAB naturally
incorporates efficient and accurate integration schemes. Yet if the reader is
interested in numerical integration of arbitrary integrands or ODES, the built-in
MATLAB schemes are generally sufficient (see Chapter 1) and do not warrant
further discussion here. In this chapter, more complicated applications of
integral equations and theory are explored with an eye to computation within

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