Page 259 - Process Modelling and Simulation With Finite Element Methods

246 Process Modelling and Simulation with Finite Element Methods

FEMLAB. Line integrals, integral equations, and integro-differential equations
are the target applications in applied mathematics. These are all treatable with
recourse to FEMLAB’s coupling variable capability. Perhaps that is reason to
have titled this chapter “Extended multiphysics 11.” The title selected, however,
is probably more descriptive. As ever, we target illustrations in chemical
engineering of the use of the FEMLAB features. The most important treated
here are using lidar to detect position and spread of dense gas contaminant
clouds, population balance equations which are exemplary of IDES, the inverse
problem in electrical capacitance tomography, and the computation of non-local
heat transfer in a fiber composite medium.

Extended Multiphysics Revisited
When I read through the new features introduced with FEMLAB 2.2, I must
admit to being skeptical of extended multiphysics as something that I was likely
to use. Eventually, the utility of scalar coupling variables dawned on me, and
provided the impetus for Chapter 4. It also spawned our interest in a new
adventure for our research team, modeling microfluidics networks. Yet
FEMLAB provides two other conceptual constructs for coupling variables -
extruded and projected coupling variables. The examples of their use in the
Model Library are nearly all about postprocessing, i.e. to express solutions in
cross domain functionals to analyze particular features.
Rarely, however, coupling variables (extruded and projection) have been
incorporated in the model and solved for simultaneously with the independent
field variables. Multi-domain, multiple scale, and multiple process models are
not common in engineering mathematics and mathematical physics. Typically,
models are local in character - conceived of as a set of (partial) differential
equations and boundary and initial conditions that are well posed. These are
termed continuum models. Historically, this development has been predicated
on the use of analysis techniques that have some scope for treating this class of
models in closed form. Computational models, even in situations that are
treatable by continuum methods, are approximated by discrete interaction rules
that need not be local. Smooth particle hydrodynamics [l] and discrete element
methods [2] are growing in popularity, but older methods like molecular
dynamics simulations 131, Monte Carlo methods [4], microhydrodynamics [5],
cellular automata [6] and exact numerical simulation in gadplasma dynamics [7]
bridge the continuuddiscrete system gap in modelling distributed systems.
Another set of techniques is based on optimization theory to satisfy pde
constraints - penalizing the degree to which constraints are not satisfied. Mixed
integer nonlinear programming [8], genetic algorithms [9] and genetic
programming [lo] are all suitable for treating models of mixed
discrete/continuum systems. FEMLAB was formulated with a strong bias
towards continuum systems with pde constraints. Yet, conceptually, extended