supported by the University of Ottawa and The Fields Institute

held at University of Ottawa

Abstracts:

Nasir Uddine Ahmed

Michel Delfour

Ian
Frigaard

Mohamed Masmoudi

Arian Novruzi

Michel Pierre

Alexandru
Tamasan

Brian Wetton

Jianying
Zhang

Nassirudine Ahmed,
University of Ottawa, Canada

Title:
Mathematical Modeling and Optimal Control of Artificial Heart

We discuss some fundamental problems arising in the study of optimal
control of artificial hearts. Though it is well known that blood plasma
is a non Newtonian fluid, its dynamics can be be approximated by the
Navier-Stokes equation for incompressible viscous fluid. Using Navier-Stokes
equation for the flow dynamics, we present a mathematical formulation
of the problem focusing particularly on hemolysis caused by excessive
shear stresses and turbulence in the flow field, and other flow related
problems such as blood clotting due to stagnation, and platelet activation
and thrombus formation due to presence of recirculation zones in the
cavity. Another closely related problem of significant interest is shape
optimization of currently available devices like LVAD ( left ventricular
assist device) with a view to minimizing hemolysis and thrombosis etc.

Michel Delfour, CRM, Montreal, Canada (http://www.crm.umontreal.ca/~delfour/)

This mini-course will provide an overview
of recent results on the handling of the geometry as a modeling, optimization,
or control variable with illustrative examples and applications. The
increased interest in theoretical studies in this general area are
motivated by numerous technological developments or phenomenological
studies and the fact that such problems are quite different from their
analogues involving only vectors of scalars or functions. Special
tools and constructions are definitely required. In the past few decades
the mathematical and computational communities have made considerable
contributions to this general area of activity by nicely intertwining
theoretical and numerical methods from optimal design, control theory,
optimization, geometry, partial differential equations, free and moving
boundary problems, and image processing. ...

Click here for a complete courses description.

Ian Frigaard,
UBC, Vancouver, Canada (http://www.mech.ubc.ca/~frigaard/)

Title:
The maximal layer of static mud on the walls of a cemented well

Coauthors

Sylvia Leimgruber (University of Innsbruck, Austria)|Otmar Scherzer
(University of Innsbruck, Austria)

In constructing an oil well a long cylindrical steel pipe is cemented
into the well, to provide structural integrity. The cement forms an
hydraulic seal with the surrounding rock formation, maintaining well
productivity. The process of placing the cement involves pumping down
the centre of the steel pipe, with the fluids returning up the outside,
in the eccentric annular duct between the pipe and rock formation. This
space is full of drilling mud, which typically has a yield stress. This
means that the drilling mud may be left on the sides of the annular
space, after displacement, in the form of static layers of residual
mud. Here we consider the problem of determining the maximal residual
mud layer after displacement. The talk will focus on formulation of
the problem as a shape optimisation problem and on our initial computed
results.

Mohamed Masmoudi, Universite Paul Sabatier, Toulouse, France (http://www.mip.ups-tlse.fr/)

We first give an overview on topological optimization. We will show that in topological optimization, unknown domains are defined imlicitely by the support of the positive part of a level set function:

- the level set function is the material density (up to an additive constant) for the topological optimization via the homogenization theory (N. Kikuchi, M. Bendsoe, G. Allaire, .) - the builtin level set function for the level set method (Osher, Santosa, Sethian, Allaire, .)

- the topological gradient provided by the "topological asymptotic expansion".

The latter method has, in addition, a fundamental property. At convergence, the positiveness of the topological gradient (the level set function) is a necessary and sufficient local optimality condition.

We will focus this lecture on the basic concepts of topological asymptotic expansion and the related algorithms.

More precisely, topological optimization is a 0-1 optimization problem. Determining an optimal domain is equivalent to finding its charachteristic function. At first sight, this is a non differential problem. But using variation calculus methods it becomes possible to derive the variation of a functional when we switch the characteristic function from zero to one or from one to zero in a small region of the domain. This is called the topological asymptotic expansion. Then it will be possible to build fast algorithms using this gradient type information. The iterative algorithm, that we will present, solves the topological optimality condition.

We will present some real life applications in :

- shape optimal design,

- inverse problems and imaging,

- image processing.

In most applications the first iteration provides a good idea on the optimal shape.

Arian Novruzi, University of
Ottawa, Canada (http://www.mathstat.uottawa.ca/~anovr479)

Title:
Shape derivatives of elliptic PDE solutions and of integral shape functionals

We consider the differentiation with respect to the domain of
a second order elliptic boundary problem with Dirichlet boundary data,
and of shape functionals given as domain or boundary integrals.

The main results we will present are

- properities of local derivatives

- differentiation of an equation and of a boundary condition

- differentiation of a cost functional defined as a domain or boundary
integral

References:

- J. Simon, Differentiation with respect to the domain in boundary value
problems, Numer. Funct.Anal. and Oprimiz. 2(7&8), 649-687 (1980)

- M.C. Delfour and J.-P. Zolesio, Shapes and Geometries: analysis, differential
calculus, and optimization, SIAM series on Advances in Design and Control,
SIAM, Philadelphia, PA, USA 2001 (http://www.siam.org/catalog/index.htm,
http://www.ec-securehost.com/SIAM/DC04.html).

---------------------------

Arian Novruzi,
University of Ottawa, Canada

Title:
Shape optimisation of porous domain/free air transmission coefficient

The heart of HFC is the electrolyte membrane (M). On either side
of the membrane is a thin catalytic layer (CL). Next to each CL (cathode
and anode side) is the gas diffusion layer (GDL). This assembly is sandwiched
between long, flat graphite plates into which flow fields (channels)
have been etched. On the cathode side, a pressure drop from inlet to
outlet in the flow fields induces oxygenated air to flow down the graphite
channels, similarly on the anode, or fuel side, hydrogen gas is induced
to flow. The oxygen and hydrogen diffuse into the cathode and anode
carbon fiber papers respectively, and upon reaching their respective
CL, disassociate into ions. The hydrogen ions cross the M and react
with the oxygen ions, producing both liquid water and water vapor. The
separation of the oxidation process into two steps, the disassociation
at the anode and the reaction at the cathode, produces a useful electrical
potential difference between the anode and cathode which induces a flow
of electrons and generates electric current. The governing equations
for HFC are quite distinct from one domain to another. They include
Navier-Stokes equations and Darcy law coupled with mass conservation
law for each component, and the energy equation. Understanding and controlling
the effects of liquid water saturation on the electro-chemical reaction
is key to HFC performance. The proper hydration of the membrane, without
the flooding of the catalyst layer, is the goal of water managemnt and
key to HFC performance. We will present some 3D numerical results for
the so called ``dry model'', restricted to channel and GDL cathode.
In HFC one has to optimize the current distribution on the membrane,
which leads to increase the nembrane longevity. Current distribution
depends strongly on the gas fluxes in GDL. One can control these fluxes
by

- the transmission coefficient in channel/GDL interface,

- the shape of the channels or

- the permeability.

We will present some numerical results that give
the shape of porous matrix that maximize the permeability under the
constraint that transmission coefficient and the measure of the pores
are given.

References

- Radu Bradean, Keith Promislow, Brian Wetton. Transport phenomena in
the porous cathode of a proton exchange membrane fuel cell. Heat and
Mass transfer in porous fuel cell electrodes. Proceedings of the International
Symposium on Advances in Computational Heat Transfer, Queensland, Australia
(2201)

- W. Jager, A. Mikelic, N. Neuss. Asymptotic analysis of the laminar
viscous flow over a porous bed}, SIAM, J. Sci. Comp., Vol. 22, No. 6,
pp. 2006-20028.

- P. G. Saffman. On the boundary condition at the surface of a porous
medium. Studies in Appl. Math., Vol. L, No. 2, June 1971.

---------------------

Arian Novruzi,
University of Ottawa, Canada

Title:
Structure of shape derivatives

We describe the precise structure of second "shape derivatives",
that is derivatives of functions whose argument is a variable subset
of R^N. One originality lies in the way the structure property is established:
the starting point is a new way of stating the well-known fact that
small regular perturbations of a given regular domain may be described
"uniquely" through normal deformations of the boundary of the domain.
The approach involves the implicit function theorem in a space of mappings
from R^N into itself and the corresponding first and second derivative
information. A consequence of this "normal representation" property
is that any shape functional may be described through a functional depending
on functions defined only on the boundary of the given domain. Differentiating
twice this representation leads to the structure theorem. We recover
the fact that, at critical shapes, the second derivative around the
given domain depends only on the normal component of the deformation
vector-field at its boundary.

Reference

A. Novruzi, M. Pierre, Structure of shape derivatives, Journal of Evolution
Equations", (2), 2002, no. 3, 365-382.

Michel Pierre, ENS Cachan, France (http://www.bretagne.ens-cachan.fr)

Title: About Regularity of Optimal Shapes

The goal of our
lectures in this workshop is to describe some results, ideas and
techniques concerning the study of the regularity of optimal shapes.
Most of the time, existence of these optimal shape is derived via the
use of functional analytic tools which generally provide optimal shape
with very poor regularity: they may be only open sets, sometimes even
only measurable sets, while we expect them to be regular or even very
regular like having an analytic boundary ...

Click here for a complete courses description and a list of references.

Alexandru
Tamasan, University of Toronto,
Canada (http://www.math.toronto.edu/tamasan/)

Title:
Reconstruction of the convection terms from boundary measurements

One tries to image the interior of a domain by making impedance measurements
on its boundary. The conductivity properties of the medium is encoded
in the covection coefficients. I will describe a two dimensional reconstruction
procedure based on generalized analytic functions which translates in
solving singular integrals on the boundary and weakly singular integrals
in the domain.

Brian Wetton,
UBC, Vancouver, Canada (http://www.math.ubc.ca/~wetton/)

Title:
Generalized Stefan velocities

Co-authors

Roger Donaldson, Applied and Computational Mathematics, Caltech

We consider elliptic problems in which the domain is separated into
two regions by a steady free boundary, on which mixed Dirichlet-Neumann
conditions are specified. Led by the classical Stefan condition applied
to change of phase models, we consider numerical methods which evolve
interfaces to the desired steady shape by using the residual in one
of the boundary conditions as a normal velocity. Using linear perturbation
analysis of simple cases, we show exactly which interfacial conditions
lead to well-posed problems and which choices of velocities lead to
convergent methods. Moreover, some velocities lead to methods having
superior numerical properties, an idea related to early work of Garabedian.
Analysis of a semi-discrete scheme in which the free boundary is approximated
by a cubic spline fit is presented, followed by an example computation.

Title: Level Set Method for Shape Optimization of Plate Piezoelectric Patch

We consider a closed-loop displacement feedback control system: a thin rectangular plate reinforced with two laminated piezoelectric patches, a sensor and an actuator. The sensor senses the vibration of the plate and generates a certain signal which is amplified and sent to the actuator. The actuator can then generate a corresponding signal which causes the plate to bend in the opposite direction and therefore balances its original vibration. The shape optimization task is to find the optimal shapes of the patches (under some constraint) in order to minimize the minimum vibration frequency. In the absence of mechanical excitations, the equation of motion of the plate with externally applied control moments is given by a fourth order hyperbolic PDE with simply supported boundary conditions. The singular behavior of the solution on the free boundaries (patch boundaries) leads to the main difficulty in handling this problem. We will present the numerical approach to this shape optimization problem using the level set method. And the numerical results will also be provided in the end.

supported by the University of Ottawa and The Fields Institute held at University of Ottawa |