Shape Optimisation and Applications Workshop
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/)
Title: Introduction to shape analysis and optmization.

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/)
Title: Numerical methods for shape optimisation problems. Topological derivatives.

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.

Jianying Zhang, UBC, Vancouver, Canada
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.



Shape Optimisation and Applications Workshop
supported by the University of Ottawa and The Fields Institute
held at University of Ottawa

Abstracts:

Nassirudine 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
Title: Introduction to shape analysis and optmization.

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
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
Title: Numerical methods for shape optimisation problems. Topological derivatives.

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
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
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
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
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.

Jianying Zhang, UBC, Vancouver, Canada
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.