




Industrial Mathematics / Mathématiques industrielles (Huaxiong Huang, Organizer)
 DHAVIDE ARULIAH, Fields Institute, Toronto, Ontario
Fast solvers for electromagnetic problems in geophysical regimes

I will describe some work in computational electromagnetics that arises
within large scale inverse problems for geophysical prospecting.
Traditional formulations and discretizations of timeharmonic Maxwell's
equations in three dimensions used in the forwardmodelling leads to a
large, sparse system of linear algebraic equations that is difficult to
solve. That is, iterative methods applied to the linear system are slow
to converge, a major drawback in solving practical inverse problems.
Towards developing a multigrid preconditioner, I'll show a Fourier
analysis based on a finitevolume discretization of a vector potential
formulation of timeharmonic Maxwell's equations on a staggered grid in
three dimensions. Gridindependent bounds on the eigenvalue and
singular value ranges of the system obtained using a preconditioner
based on exact inversion of the dominant diagonal blocks of the
coefficient matrix can be proved. This result implies that a multigrid
preconditioner that uses single multigrid cycles to effect inversion of
the diagonal blocks also yields a preconditioned system with
l_{2}condition number bounded independent of the grid size.
Numerical experiments show that the somewhat restrictive assumptions of
the Fourier analysis do not prohibit it from describing the essential
local behavior of the preconditioned operator under consideration. A
very efficient, practical solver is obtained.
This is joint work with U. Ascher.
 DAN BEAMISH, Department of Mathematics and Statistics, York University,
Toronto, Ontario M3J 1P3
Analysis of delayed feedback arm movement

The Vector Integration to Endpoint (VITE) circuit describes a real time
neural network model which simulates behavioral and neurobiological
properties of planned arm movements by the interaction of two populations
of neurons. We generalize this model to include a delay between the
interacting populations and give conditions on how its presence affects
the accuracy of planned movements. We also show the existence of a
nonzero critical value for the delay where a transition between accurate
movement and target overshoot occurs. This critical value of the delay
depends on the movement speed and becomes arbitrarily large for
sufficiently slow movements. Thus neurobiological or artificial systems
modeled by the VITE circuit can tolerate an arbitrarily large delay if
the overall movement speed is sufficiently slow.
 LLOYD BRIDGE, University of British Columbia, Vancouver, British Columbia
A onedimensional study of condensation in a porous medium

The onedimensional steadystate heat and mass transfer of a twophase
zone in a watersaturated porous medium is studied. The system consists
of a sandwatervapour mixture in a tube that is heated from above and
cooled from below. Under certain conditions, a twophase zone of both
vapour and water exists in the middle of the tube. A model problem for
the temperature and the saturation profiles within this twophase zone
is formulated by allowing for an explicit temperature dependence for
the saturation vapour pressure together with an explicit saturation
dependence for the capillary pressure. A boundary layer analysis is
performed on this model in the asymptotic limit of a large vapour
pressure gradient. This asymptotic limit is similar to the large
activation energy limit commonly used in combustion problems. In this
limit, and away from any boundary layers, an uncoupled outer problem
for the saturation and the temperature is obtained. From this outer
problem it is shown that the temperature profile is slowly varying and
that the outer saturation profile agrees very well with that obtained
in the previous model of Udell [J. Heat Transfer, 105(1983)],
previous model of Udell [J. Heat Transfer, 105(1983), p. 485]
where strict isothermal conditions were assumed. The condensation and
evaporation occuring within the boundary layers near the edges of the
twophase zone is examined. Finally, an iterative method is described
that allows both the full and outer models of the twophase zone to be
coupled to the two singlephase zones consisting of either water or
vapour. Numerical computations are performed with realistic control
parameters for the entire system.
 YONGQIANG CAO, Department of Mathematics and Statistics, York University,
Toronto, Ontario M3J 1P3
Neural network for data mining

Most clustering algorithms do not work efficiently for data sets in
high dimensional spaces. Due to the inherent sparsity of data points,
it is not feasible to find interesting clusters in the original full
space of all dimensions, but pruning off dimensions in advance, as
most feature selection procedures do, may lead to significant loss of
information and thus render the clustering results unreliable.
In a recent project with Jianhong Wu, we propose a new neural network
architecture Projective Adaptive Resonance Theory (PART) in order to
provide a solution to this feasibilityreliability dilemma in clustering
data sets in high dimensional spaces. The architecture is based on the
well known ART developed by Carpenter and Grossberg, and a major
modification (selective output signaling) is provided in order to deal
with the inherent sparsity in the full space of the data points from
many datamining applications. Unlike PROCLUS (proposed by Aggarwal et. al
in 1999) and many other clustering algorithms, PART algorithm do not
require the number of clusters as input parameter, and in fact, PART
algorithm will find the number of clusters. Our simulations on high
dimensional synthetic data show that PART algorithm, with a wide range of
input parameters, enables us to find the correct number of clusters, the
correct centers of the clusters and the sufficiently large subsets of
dimensions where clusters are formed, so that we are able to fully
reproduce the original input clusters after a reassignment procedure.
We will also show that PART algorithm is based on rigorous mathematical
analysis of the dynamics for PART neural network model (a large scale
system of differential equations), and that in some ideal situations
which arise in many applications, PART does reproduce the original input
cluster structures.
 ROBERT ENEKEL, IBM
Fast pseudorandom number generation: mathematical innovation and
architectural exploitation

Many numerically intensive computations done in a scientific computing
environment require large quantities of uniformly distributed pseudorandom
numbers. Largescale computations on parallel processors pose additional
demands, such as independent generation of pseudorandom numbers on each
processor to avoid communication overhead, or coordination between the
independent generators to provide consistency during program development.
This talk shows how mathematical innovation, the fused multiplyadd
instruction, loop unrolling, and floatingpoint ``tricks'' can result in a
uniprocessor speed improvement of over 50 times over generic algorithms,
while retaining bitwise agreement with existing, proven, random number
generators. The result is a multiplicative congruential random number
generator with modulus 2^{k}, k £ 52, and period k2, that runs at a
rate of 40 million uniformly distributed random numbers in the interval
(0,1) per second on RS/6000 POWER2 Model 590 processors, or one number
every 3 machine cycles. In addition, the algorithms are ``embarrassingly
parallel'', so that a 250node IBM SP2 computer can generate 10 billion
uniform random numbers per second.
This is joint work with Ramesh Agarwal, Fred Gustavson, Alok Kothari and
Mohammad Zubair. Other pseudorandom number generators resulting from this
work cover the interval (1,1), have a full period of 2^{k}, or have
modulus 2^{k}1. Our algorithms are used in the IBM XL Fortran and XLHPF
(High Performance Fortran) RANDOM_NUMBER functions.
 DAVID FIELD, GM
Assessing the quality of triangulation

To solve real world problems such as distribution of heat under the
hood of an automobile, engineers often use the finite element method
for solving partial differential equations. Starting with a
geometrical description of an underhood compartment, the numerical
solution often relies on collections of triangles called meshes. This
talk emphasizes the generation good finite element meshes. An
investigation into the quality of these meshes provides opportunities
to apply mathematics in constructing meshes and in evaluating the
quality of constructed meshes.
 IAN FRIGAARD, Department of Mathematics, University of British Columbia,
Vancouver, British Columbia V6T 1Z2
NonNewtonian displacement flows in the cementing of oil wells

In cementing an oil well a series of nonNewtonian fluids are pumped
through a narrow eccentric annulus, in an effort to displace the
drilling mud and achieve zonal isolation of the well. These flows are
typically laminar and the fluids involved are shearthinning and
predominantly viscoplastic. A range of interesting displacement flows
result. The aim of this talk is to give an overview of the different
problems that result and outline what efforts are being undertaken to
resolve them.
 DONG LIANG, York University, York, Ontario
Numerical modelling of a 2D moving liquid drop/bubble on
a solid surface

In this talk we consider the motion of a twodimensional liquid drop or
bubble on a solid surface exposed to a shear flow. Peskin proposed an
elegant and efficient method for simulating blood flow in the heart,
which can be generalized to solve other problems with moving
interfaces. Unverdi and Tryggvason calculated the rising bubble problem
by using a fronttracking method based on Peskin's method. The main
challenge of numerical modelling the motion of a liquid drop on a solid
surface, in addition to capturing the moving interface between the
liquid and gas phases, is to incorporate conditions at the moving
contact lines. In our study of the problem, the free surface between
two liquid phases is handled by a fronttracking method; the moving
contact line is modelled by a slip velocity and contact angle
hysteresis is included; and the local forces are introduced at the
moving contact lines based on a relationship of slip coefficient,
moving contact angle and contact line speed. Several numerical
examples are also given. This is a joint work with Huaxiong Huang and
Brian Wetton.
 MINISYMPOSIUM

 KEITH PROMISLOW, Simon Fraser University, British Columbia
Front dynamics in PEM fuel cells

We consider models of water management in PEM fuel cells, which involve
phase change and twophase flow in porous media. The Teflonation of
fuel cell electrodes creates a nonwetting porous media, and renders
low water saturations immobile. The dynamics of wetting fronts in 1
and 2D are further complicated by condensation and evaporation layers.
We discuss the proper formulation of the problem, asymtotics of the
steady states, and some of the numerical difficulties in 1 and 2D
resolution of the front dynamics.
 DAN RYAN, University of British Columbia, Vancouver, British Columbia
Flow of a viscoplastic fluid in a wavy walled channel

In cementing an oil well it is necessary to displace mud from the
annular space between the casing and the outer rock formation. In an
extreme case, layers of drilling mud are left immobile, stuck to the
walls of the annular space. Recent studies (Allouche, Frigaard and
Sona, 2000) have shown that these layers are nonuniform, exhibiting
small amplitude longwavelength fluctuations in the direction of flow.
Cemented annuli are generally long and thin. Assuming little azimuthal
flow, a section along the annulus is approximated by a channel with
wavy walls. This provides the motivation for our study.
We consider a small longamplitude perturbation from plane channel and
the effects on a Poiseuille flow of a Bingham fluid. A naive
application of lubricationlike scalings results in a leading order
velocity profile with a plug velocity at the channel centre, which
varies with length along the annulus. Since the rate of strain should
be zero in the plug, this leads to a contradiction and it is clear that
the analysis has broken down. This problem is resolved using a more
refined analysis. It is shown that for a small perturbation, a truly
unyielded plug remains at the centre of the channel. This true plug
region is connected to the sheared outer layer via a transition layer.
We determine expressions for the thickness of this layer.
Finally, we develop a twodimensional numerical solution of the flow
using the augmented Lagrangian method. This method has the advantage of
accurately representing unyielded regions of the flow.
 DONALD SCHWENDEMAN, Rensselaer Polytechnic Institute, Troy, New York, USA
Multiphase flow in a roll press nip: modeling and computations

A threephase ensembleaveraged model will be discussed for the flow of
water and air through a deformable porous matrix. The model predicts a
separation of the flow into saturated and unsaturated regions. A
closure of the model is proposed based on an experimentallymotivated
heuristic elastic law which allows largestrain nonlinear behavior to
be treated in a relatively straightforward way. The equations are
applied to flow in the ``nip'' area of a roll press machine whose
function is to squeeze water out of wet paper as part of the
manufacturing process. By exploiting the thinlayer limit suggested by
the geometry of the nip, the problem is reduced to a nonlinear
convectiondiffusion equation with one free boundary. A numerical
method is proposed for determining the flow and sample simulations are
presented.
 JOHN STOCKIE, UNB
Modelling condensation in porous media

Condensation is a complex phenomenon, involving phase change and
transport of mass, momentum and energy. We first develop a
mathematical model for the flow of a multicomponent, dry gas in a
porous medium, consisting of a coupled system of nonlinear partial
differential equations. This model is then extended to include
condensation and liquid water by the introduction of a simple
regularised condensation term, and by adding one more equation
governing the liquid transport. Migration of water through the porous
medium occurs on a much longer time scale than either condensation or
the gas motion, which makes the underlying problem extremely stiff. We
will discuss the impact this has on the development of efficient
numerical methods for solving the system of PDEs.
Our interest in this problem arises from the study of condensation and
gas transport in hydrogen fuel cells. However, very similar models
arise in other applications as diverse as kiln drying of wood,
transport of contaminants in groundwater, and thermoregulation of
honeybee clusters.
 REX WESTBROOK, University of Calgary, Calgary, Alberta
Sag bending

Sag bending is not a chronic geriatric condition but is a method used
in the manufacture of windshields. A sheet of glass rests on a shaped
frame and is heated from below. The glass sags under the action of
gravity. The aim is to place the heaters in a manner that will cause
the glass to sag to a specified target shape. The problem which is
being considered is that of an elastic plate with variable elastic
constants under the action of gravity. The heating controls the
elastic constants in a known manner so that the problem becomes a
control problem with the Young's modulus as the control and a measure
of the difference between the actual shape and the target shape as the
objective function. There are also upper and lower bound constraints
on the value of the Young's modulus. The company has a code but it
apparently does not perform well. There are many aspects of this
problem both in the mathematical formulation and in the numerical
schemes proposed for its solution some of which will be discussed in
the talk.
 BRIAN WETTON, Department of Mathematics, University of British Columbia,
Vancouver, British Columbia V6T 1Z2
Dynamical contact angle of a drop in steady state motion

We discuss the behaviour of the dynamical angle at three phase
(liquidsolidgas) contact points for the twodimensional steady state
motion of a liquid drop. We present a result giving this angle as a
function of the droplet speed in the form of a simple algebraic
expression. It is well known that near these contact points, there are
singular stresses unless the problem is regularized. The originality
of the work is that it deals directly with the singularity, using only
an ansatz on the interpretation of the singular integrals. This
problem came to our attention from our work modelling ``water
management'' in fuel cells. Some general remarks on fuel cell design
and simpler related models will be given. This is joint work with
Arian Novruzi and Huaxiong Huang.

