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Industrial Mathematics / Mathématiques industrielles
(Huaxiong Huang, Organizer)

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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 time-harmonic Maxwell's equations in three dimensions used in the forward-modelling 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 finite-volume discretization of a vector potential formulation of time-harmonic Maxwell's equations on a staggered grid in three dimensions. Grid-independent 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 l₂-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 one-dimensional study of condensation in a porous medium

The one-dimensional steady-state heat and mass transfer of a two-phase zone in a water-saturated porous medium is studied. The system consists of a sand-water-vapour mixture in a tube that is heated from above and cooled from below. Under certain conditions, a two-phase 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 two-phase 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 two-phase zone is examined. Finally, an iterative method is described that allows both the full and outer models of the two-phase zone to be coupled to the two single-phase 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 feasibility-reliability 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 data-mining 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. Large-scale 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 multiply-add instruction, loop unrolling, and floating-point ``tricks'' can result in a uniprocessor speed improvement of over 50 times over generic algorithms, while retaining bit-wise agreement with existing, proven, random number generators. The result is a multiplicative congruential random number generator with modulus 2<sup>k</sup>, k £ 52, and period k-2, 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 250-node 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
Non-Newtonian displacement flows in the cementing of oil wells

In cementing an oil well a series of non-Newtonian 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 shear-thinning and predominantly visco-plastic. 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 two-dimensional 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 front-tracking 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 front-tracking 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.

MINI-SYMPOSIUM

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 two-phase flow in porous media. The Teflonation of fuel cell electrodes creates a non-wetting 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 visco-plastic 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 non-uniform, exhibiting small amplitude long-wavelength 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 long-amplitude perturbation from plane channel and the effects on a Poiseuille flow of a Bingham fluid. A naive application of lubrication-like 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 two-dimensional 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 three-phase ensemble-averaged 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 experimentally-motivated heuristic elastic law which allows large-strain 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 thin-layer limit suggested by the geometry of the nip, the problem is reduced to a nonlinear convection-diffusion 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 (liquid-solid-gas) contact points for the two-dimensional 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.