Postdoctoral Associates
David Salac
I obtained my Bachelor's degree in Mechanical Engineering from Michigan Technological University in 2002. During 2001 I had an internship at Kaydon Corporation translating bearing analysis software from APL to Visual Basic. In the fall of 2002 I joined the Mechanical Engineering graduate degree program at the University of Michigan in Ann Arbor. After studying modeling techniques for self-assembled material systems, I completed my PhD in the spring of 2007. Starting in September 2007 I became a post-doctoral researcher in the RTG program in the Department of Engineering Sciences and Applied Mathematics at Northwestern University in Evanston. I am currently working on two main activities involving interfacial phenomena and multiple length scales; (i) motion of lipid bilayer vesicles and (ii) the freezing of foams. In addition, my initial investigations into these two topics has led to development of a new numerical method.
The first activity deals with the motion of lipid bilayer vesicles under the combined effects of external fluid flow and lipid diffusion. Here lipid molecules introduced into an aqueous environment congregate to form two mono-molecular layers held together by weak forces. To minimize the excess energy these layers close resulting in the formation of bag-like structures called vesicles. Lipid bilayer vesicles have become a prime model system, both experimentally and computationally, for the investigation of biological membrane systems.
My work on this topic involves the numerical modeling of these lipid systems. This is a numerically challenging system to model accurately. Unlike other physical systems which may be volume conserving (e.g. bubbles in a fluid), this lipid system is typically considered lipid conserving. This constraint is modeled as a constraint not on the total volume but the surface area of the vesicle. Additionally, the size of the vesicle is many orders of magnitude larger than the thickness of the lipid bilayer, yet it is necessary to model both the global motion of the vesicle due to the external fluid flow and the movement of the lipids within the interface itself. A numerical modeling technique is needed which has high resolution on the interface but can also handle large scale systems which may undergo large topological changes. Initial investigations have shown that level set methods are a prime candidate due to the ease with which level set methods handle interfacial motion. An exhaustive search of currently available level set schemes has not revealed any truly suitable methods. Thus it was necessary to develop such a method which I have done and which is described below.
The second activity is an investigation of the freezing of liquid foams. These foams have a very high gas content resulting in very little liquid between gaseous bubbles. A freezing front is then forced through the foam in the hopes of locking the bubbles in place. If the speed at which the front advances is not selected correctly the liquid may drain, resulting in the coalescence of the bubbles.
To better understand this freezing process I am currently developing a numerical model of the system. There is a large size discrepancy between the size of the bubbles and the thickness of the liquid between them. To accurately describe the freezing front we require high accuracy at the bubble interface where we will have a junction between the gas, liquid, and solid phases. As with the lipid bilayer system previously described, we require a numerical scheme which is both highly accurate and capable of handling very large scale systems. Initial work has shown that the same numerical scheme being developed for the lipid bilayer system can be applied for the case of freezing foams.
To simulate the two problems described above, I found it necessary to implement a novel numerical technique for moving interfaces. This method is a level set scheme, based on a combination of a modified Fast Marching Method scheme and adaptive non-graded Cartesian grids. Unlike standard fast marching level set schemes, which are not capable of advancing a front through time, this implementation can advance an interface in time schemes. The use of adaptive non-graded Cartesian grids reduces the memory footprint of the method, affording the opportunity to model large scale systems and those with many different length scales.
Michael Gratton
I received my BS in Mathematics from Harvey Mudd College in 2002, working with Andrew Bernoff on mixing in internal flow. I then attended Duke University, earning my Ph.D in Mathematics in 2008. My dissertation, supervised by Thomas Witelski, studied a coarsening problem for thin fluid droplets. This work involved breaking a nonlinear PDE (the lubrication equation) into a set of ordinary differential equations (ODEs) for the evolution of each drop. The aggregate behavior of these ODEs could then be used to make global predictions about the system.
I began my postdoctoral fellowship at the Department of Mathematical Sciences and Applied Mathematics at Northwestern University in Fall of 2008. Currently I study the dynamics of freezing foams.
My work is aimed at understanding the evolution of the bubble size in the foam prior to freezing. Specifically, I am studying an idealized two-dimensional foam with nearly polygonal bubbles. The goal is to reduce the system to a coupled network of ODEs that yield estimates about the average bubble size.
Building this network model involves several steps. Foams may be locally decomposed into thin lamella sheets that connect plateau borders at the corners of bubbles. Due to surface tension, the plateau border regions draw in fluid from the lamella to reduce the curvature at the corner of a bubble. This effect gives a thinning law for the lamella. A key prediction of this system is the rupture criterion for a lamella. Anthony Anderson (at NWU) is studying the stability of these lamella to produce an estimate of the point at which they break. Lucian Brush (at the University of Washington) is studying the immediate post-break dynamics.
Meanwhile, I am generalizing the lamella drainage equations to describe non-symmetric cases. Using timescale estimates and mass-balance equations, many of these cases can be dealt with asymptotically. However, certain limits (broad plateau borders that separate bubbles with different numbers of sides, e.g. hexagonal from square) require numerical simulation of the Stokes equations.
In addition to the local dynamics, I am investigating the changes to the lamella network shape after the breakage of a lamella. The breakage of one lamella in a square array of bubbles creates a larger hexagonal bubble, but also stresses the neighbors. I am developing rules for the evolution of the lamella network shape that can be fed back to the local model of lamella drainage.
Erin Lennon
I graduated in 2003 from both the University of Washington with a BS in Chemical Engineering and Scripps College with a BA as a part of their dual-degree liberal arts-engineering program. While at UW, I studied interfacial properties of self-assembled systems with Professor Shaoyi Jiang. My graduate work was at UC Santa Barbara, where in 2008 I received my Ph.D. in Chemical Engineering with an emphasis in Computational Science and Engineering. My dissertation focused on developing novel techniques to study properties of inhomogeneous polymers on the mesoscale. I was supervised by Professor Glenn Fredrickson, in Chemical Engineering, and Professors Hector Ceniceros and Carlos Garcia-Cervera, in Mathematics.
In the Fall of 2008, I joined Northwestern University as a postdoctoral fellow in the Department of Engineering Sciences and Applied Mathematics. My research here is focused on modeling both the self-propagation of exothermic reactions and the behavior of self-assembled polymer systems.
In the first project I am focused on investigating the properties and stability of sequential combustion reactions that are chemically uncoupled but where the heat generated from one reaction fuels the second reaction. Two ways that these reactions can be modeled are as being localized to a single point, that is an interface-type reaction, or over a small spatial region. In both cases, one must solve a set of differential equations subject to conditions specified both far from the reactions as well as at the reaction boundaries. In the case of the time-independent, moving-coordinate system, it is straightforward to find a system of nonlinear equations that can be solved numerically to determine the temperature and reactant/product profiles across the region. The second step in investigating this model is through a dispersion relation to study the stability of the stationary wave. This can be done for perturbations along the wave front (the y-direction) and in time. I am working on this problem with an undergraduate ESAM major, Jaime Swartz and in parallel to a project led by Matthew Tanzy, a graduate student in ESAM.
For the second project, I have been developing and implementing a method for determining the osmotic pressure of a heterogeneous polymer systems. In these calculations, I use a field-theoretic framework to model individual polymers as elastic filaments that resist stretching and have contact interaction penalties between dissimilar monomers. The interest here is in systems with multiple phases, each of which can have sharp internal interfaces between volumes with different majority monomer types. To capture the complex phase behavior of these materials, one must determine the osmotic pressure of each phase---which will be constant across phases at equilibrium. In past work, researchers have used a mean-field approximation to map phase diagrams of systems of interest. However, in many physically relevant systems, this approximation fails and the methods for determining the osmotic pressure of large multi-scale systems become prohibitive. I am investigating and optimizing other techniques for determining the osmotic pressure of these systems. For this project I have collaborated with members of Ken Shull's group in Materials Science.
Yulia Peet
I received my B.S. degree in Applied Mathematics and Physics and M.S. degree in Aerospace Engineering from Moscow Institute of Physics and Technology in Russia, and one year later I also obtained a second Master's degree in Management from the State University Higher School of Economics in Moscow. Having decided that my heart still lies in scientific disciplines, I moved to California to pursue my Ph.D. studies at the Department of Aeronautics and Astronautics in Stanford, where I was involved in developing numerical simulations for analyzing the mixing of hot and cold turbulent streams with application to turbine blade cooling. After graduating from Stanford I spent some time at the University of Pierre and Marie Curie in Paris as a Postdoctoral Researcher, where I investigated the possibility of reducing the drag on surfaces in contact with a turbulent flow by adding organized micro-structures to the surface, both theoretically and numerically. Finally, before joining the Engineering Sciences and Applied Mathematics Department of Northwestern University as an RTG Postdoctoral Fellow in September 2009, I worked for several months at the Mathematics and Computer Science Division at Argonne National Laboratory as an Assistant Computational Scientist, developing domain decomposition methods for a spectral element fluid dynamics solver.
My research within the RTG program at ESAM Department at Northwestern involves studying the problem of intracranial aneurysms -- a dangerous medical condition involving a sac forming on the wall of an artery within the cerebral circulation system, primarily at points where the artery splits in two (bifurcates). The complex interplay between the biological processes in the arterial wall and hemodynamic stimuli on the vessel's wall accounts for our poor understanding of the formation and evolution of aneurysms, and consequently a lack of confidence in medical decisions concerning aneurysm treatment. I plan to study the aneurysm phenomenon computationally, investigating the interaction of the blood flow with the elastic vessel wall by numerically solving the nonlinear equations for both the fluid and solid parts of the system. I plan to use the spectral element flow solver developed at Argonne and add the fluid-structure interaction (FSI) capability to the solver. Interaction between the fluid and solid domains will be modeled using the Arbitrary Lagrangian-Eulerian (ALE) approach, whereby the interface is tracked explicitly and both the fluid and solid meshes are dynamically updated to account for the interface displacement. The developed FSI solver will be used to investigate the effects which influence the formation and the growth rate of intracranial aneurysms and to suggest practical guidelines for appropriate medical intervention.
Graduate Students
Anthony Anderson (2007-2009)
As an undergraduate at the University of Minnesota, I received my BS degree in Chemical Engineering. During that time, I was involved in a three year research project focused on developing adaptive methods to numerically simulate three-dimensional free-boundary problems in fluids dynamics, materials science, and biology. During this project my interests shifted towards applied mathematics, particularly the mathematical modeling of interfacial phenomena in physical systems.
Now, as a graduate student in ESAM at Northwestern University, I am working to model the freezing of metallic foams. The solidification of metallic foams is an industrially important process. When it is carried out successfully, a light-weight porous solid is produced which exhibits a high ratio of rigidity to specific weigh as well as excellent energy absorption on impact. These are desirable properties in a number of applications, particularly those in the automotive and aerospace industries.
The principle goal of the project is to study the dynamics and time scales involved in solidifying metallic foams. Unlike in aqueous foams (e.g. soap froth), there are no surfactants available to stabilize metallic foams. Hence, the principle difficulty in solidifying them comes from rapid coarsening ahead of the freezing front, which is the result of film rupture followed by coalescence of adjacent gas bubbles. Thus, interfacial phenomena play the crucial role in determining the evolution of the foam.
To understand the dynamics at the level of an individual lamella (thin liquid bridges between adjacent gas bubbles), we have developed approximate mathematical models (partial differential equations of reduced order) using asymptotic methods. These models include several effects which are present during freezing: capillarity, viscosity, thermocapillarity, volume-change upon solidification, and van der Waals attractions. We are also studying the conditions which lead to lamella rupture using stability analysis. This latter portion of the research was initiated in a one month internship with Professor Lucien Brush (Materials Science & Engineering - University of Washington) and continues to be the focus of ongoing research and collaboration.
David Hansen (2007-2008)
I am a fifth-year graduate student in the Engineering Sciences and Applied Mathematics Department at Northwestern University. As an undergraduate, I studied mathematics and physics at Albion College in Albion, Michigan. Since coming to Northwestern, I have been conducting research with Sascha Hilgenfeldt, using mathematics to model problems in fluid mechanics.
Micrometer-sized bubbles attached to plane walls, when caused to oscillate, can stimulate powerful steady streaming flow in the surrounding fluid. This fluid flow can be harnessed to achieve transport and mixing in microfluidic devices and can be controlled from a distance. A variety of mathematical methods including asymptotic and Green's function solutions to the relevant PDEs, are used to understand this fluid flow and make predictions for the engineering of efficient microfluidic devices.
These investigations have recently led to exploration of an intriguing axisymmetric,low-Reynolds number flow related to bubble-driven streaming. Such flows can be driven by benign boundary conditions on a plane wall, but appear to require logarithmically singular velocities along the axis of symmetry (while still obeying the Stokes equations). My research involves numerical and theoretical analysis to understand this problem (which has awaited a definitive solution for 50 years) as well as its implications for situations such as the microbubble streaming problem above.
As part of the RTG program, I have been working with the Hilgenfeldt lab to write computer software to analyze experimental data. This software enables the researcher to extract data such as streamlines and foam evolution from experimental videos. This analysis will lead directly to quantitative data for comparison to some of my theoretical predictions for the microbubble streaming problem.
Brian Merkey (2007-2008)
My background lies at the intersection between mathematics and the sciences. I studied mathematics and physics as an undergrad, and have continued the study of applied mathematics as a graduate student. I'm very interested in using mathematics as a tool to further our understanding of different disciplines.
My current research work, being carried out with Prof. David Chopp in the ESAM department and Prof. Bruce Rittmann of Arizona State University, is in the biological sciences, in particular modeling the growth of the bacterial communities known as biofilms. Biofilms are the growth of bacteria attached to a surface, and are on the order of tens to hundreds of microns thick. Because of their small size it is difficult to study biofilms experimentally and so mathematical modeling is often employed as a means to gain a deeper understand of these systems.
Due to the highly nonlinear forms of the reaction kinetics, modeling biofilm growth for most systems involves solving a set of partial differential equations numerically. A number of different numerical tools are employed to facilitate such studies, from methods as simple as the Newton-Raphson technique, to using the level set method to track an arbitrarily-shaped interface, and using the immersed interface method to capture the behavior of discontinuous diffusion coefficients. For this reason mathematical modeling of biofilm growth is an interesting problem because there are so many different issues that bust be addressed, each of which can require a different approach.
I am currently involved with several different projects using mathematical modeling as a tool to better understand microscopic biofilms, including: studying the structure and composition of nitrifying biofilms, studying the behavior of a denitrifying biofilm in a membrane-fed biofilm reactor, and studying the microbial diversity and interaction between species in the gut microbiota.
These last two projects form the majority of my RTG internship work: I have spent a total of four weeks at Arizona State University's Biodesign Institute working with experimentalists investigating denitrification in the membrane biofilm reactor and with scientists studying the diversity of the gut microbiota. In such collaborations we each contribute our expertise: I contribute my mathematical modeling skills, and they contribute a deep knowledge of microbial systems and the experiments needed to investigate these systems. Together we are able to make more and much quicker progress than any of us would be able to do alone. In addition, we are each able to challenge the other's assumptions, which leads to a more robust understanding of the problem.
In addition to participating in my internship, I have spent time mentoring an undergraduate applied math major, Paul Park, sponsored by the RTG project. I have helped carve out a piece of work suitable for investigation by the student and have helped him learn some of the mathematical, numerical, and scientific skills needed to investigate these complex biological systems.
Following the completion of my degree, I will be starting a postdoc position with Prof. Barth Smets at the Technical University of Denmark just outside of Copenhagen. I will be working in a group studying horizontal gene transfer in biofilms, and will take the lead in modeling the transfer of genes within biofilm communities. I will work in close collaboration with experimentalists, and know that I will be well-prepared for such work because of my involvement with the RTG program.
Christine Sample (2007-2008)
I am currently a fifth-year graduate student in the Department of Engineering Sciences and Applied Mathematics (ESAM) at Northwestern University. The shared analytical and numerical aspect of my undergraduate background as a mathematics and computer science student at Boston College facilitated a smooth transition into the field of applied mathematics.
My research as a doctoral student in the ESAM Department has focused on the nonlinear dynamics of interacting interfaces in electrochemical and biological systems. Under the guidance of my advisor, Prof. Sasha Golovin, I began my research studying the formation of nanoporous structure in electrochemical systems. The spontaneous arrangement of nanoscale pores in metal oxides is of interest due to its use in the field of nanotechnology. Through my research, I identified the instability mechanism that yields the formation of nanoporous arrays and studied the nonlinear dependence of the instability conditions on the system parameters.
The second part of my graduate work resides in the field of mathematical biology. Some of the basic properties of biological membranes can be understood by studying the behavior of lipid bilayers: the represent the simplest model for complex biological membranes. As part of the Research Training Group (RTG) program, I have been studying a biological double membrane system consisting of two coupled lipid bilayers, typical of some intracellular organelles such as mitochondria. We approached the study of a double membrane system in two ways. First, based on experimental results concluding that phase separation within a multicomponent membrane can lead to morphological transitions, we investigated the conditions under which the curvature and composition dependence of a double bilayer membrane can lead to complex structures. In reality, biomembranes are constantly out of equilibrium as a result of various physiological processes. Therefore, the second project examines the effect non-equilibrium chemical fluxes across the membranes have on the membrane morphology. Both projects are studied analytically using techniques such as linear stability and weakly nonlinear analysis, and numerically using a pseudo-spectral method.
I have been awarded the opportunity to fulfill my RTG internship with Prof. Stas Shvartsman at Princeton University. The Shvartsman lab is an interdisciplinary team that focuses on the quantitative analysis of embryonic development by building models of developing tissues and testing them experimentally. As part of my internship, I will begin formulating a mathematical model of the mechanics of epithelial tissues. I will work closely with scientists and engineers to create a model that can relate the mechanical properties of epithelia to morphogenesis (the creation of form or structure during the development); our subsequent predictions can be tested in Drosophilia experiments. Following my internship, I will attend he Training Workshop on CompuCell3D, a numerical solver for multi-cellular developmental modeling, at Indiana University. During the workshop, I will implement a simulation of the model formulated during my RTG internship. Upon the completion of my Ph.D., I will join the Shvartsman lab as a postdoctoral fellow and continue working on the epithelial mechanics problem.
Liam Stanton (2007-2009)
My undergraduate degree is in physics and mathematics from Boston College.
I am currently developing mathematical models to study the formation of nanoscale porous arrays in anodized aluminum oxide (AAO). The highly ordered self-assembly of these pores makes AAO an excellent template for other nanoscale structures such as quantum dots, nanowires and nanotubes. While many experimental advancements have been made in the fabrication and refinement of porous AAO, the theoretical understanding of the physical and chemical mechanisms involved is still in its infancy. I have proposed a model which incorporates the effects of field induced ionic flows within the oxide coupled to the nonlinear reactions at the metal-oxide and oxide electrolyte effects due to the electric field on the deformations of the oxide film and anode substrate.
I plan to continue the modeling of ionic flows within oxide materials as they apply to recent advancements in battery technology. In particular, I will be modeling lithium transport in LiFePO_4, which has been proposed as a promising cathode base for Li-ion batteries. The current paradigm for describing batter charging (or discharging) is the shrinking core model, which assumes bulk-transport limited ion insertion. The diffusion within crystalline LiFePO_4 is highly anisotropic and involves disparate timescales, and this leads to surface-reaction limited insertions. Hence, this model is invalid, and a new approach is necessary.
My external collaborator Dr. Martin Bazant (Mechanical Engineering, Stanford University) has proposed a phase-field formulation that is more appropriate for modeling the ion insertion of this system, but as it is only a leading order approximation, there is much room for improvement. I plan to extend this model to include many of the important system physics such as wave-defect interactions and stresses resulting from the significant lattice mismatch at the lithiated-unlithiated interface. I will also carry out an internship in the fall of 2008 with Professor Bazant at Stanford University. I anticipate that this internship will guide the theoretical research and provide useful insights to the important mechanisms present during ion insertion.
Jonathan Schwalbe (2008)
I am currently a third year graduate student in the Department of Engineering Sciences and Applied Mathematics (ESAM) at Northwestern University. Before my work began at Northwestern I received a B.S. in mathematics from Rensselaer Polytechnic Institute where I focused mainly on applied mathematics, which made moving from one program to another relatively smooth.
Under the guidance of Professor Michael Miksis, my current field of work lies in the broad categories of fluid mechanics and mathematical biology. A vesicle is a naturally occurring object that is bounded by a membrane, which has the same basic components as that of a living cell, a phospholipid bilayer. Various models that describe this membrane have been developed and studied in the past. I however have focused on the bilayer architecture model, which accounts for not only the bending and elastic properties of the lipid bilayer, but also allows for diffusion of the individual lipid molecules on the two surfaces. Lastly, the two monolayer sheets, which form the bilayer by weak covalent bonds, are allowed to slide over one another. The fluid mechanics enters the problem due to the presence of a fluid inside and outside of the vesicle and any imposed external flow fields.
When put together, all of the above means that mathematically I am dealing with a complex free boundary problem. The spatial scales under investigation here make the problem well suited for a continuum theory approach, with appropriate modifications. The dynamics of the liquids inside the vesicle and surrounding it are governed by the Stokes equations. In addition to these a coupled system of partial differential equations governs the motions of the lipids in the interface. Several physical constraints, along with the actual geometry of the membrane appear as part of the jump in the stress across the interface as well as lateral stress conditions within the membrane.
The specifics of my project are as follows. A quasi-spherical vesicle with the properties just described will contain a fluid which may have a different viscosity than the fluid in the exterior. For a while my work concerted the fluctuations for the membrane around the spherical equilibrium shape and the stability of the configuration under a wide range of parameter values.
With the completion of the near equilibrium fluctuations I imposed an unbounded shear flow on the vesicle. Similar results were found to that of the minimal model of the lipid bilayer membrane, which accounts for bending moments with hard constraints on total area and volume. Unlike droplets whose shape dynamics are governed by surface tension, vesicles, under shear flow, can enter a tumbling motion under certain conditions rather than just being broken apart like droplets. New shape dynamics were also found when other parameters not present in other models of lipid bilayer membranes were varied, such as the friction between the two monolayers.
Looking to the future, other imposed flow fields will be considered, such as unbounded Poiseuille flow. Formation of pores is a key physical behavior of vesicles, which is also going to be investigated using techniques developed from previous models.
With the support of the RTG program, this April I plan to visit the laboratory of Petia Vlahovska at Dartmouth College where she does not only analytical work with vesicle dynamics but also laboratory studies of artificially created vesicles. I hope that this hands-on experience will not only give me more physical insight into the problem, but also inspire new ideas and ways in which this research can go.
Matthew Paul Miklius (2008)
My background in Physics and Mathematics as an undergraduate at Seton Hall University helped pique my interest in developing mathematical models for physical systems—an interest that naturally led me to pursue a graduate degree in Applied Mathematics. I am currently a third-year gradate student in Northwestern University’s department of Engineering Sciences and Applied Mathematics (ESAM), working with ESAM professor Vladimir Volpert and external advisor Sascha Hilgenfeldt (University of Illinois at Urbana Champaign).
My research focuses on quantifying and explaining the two-dimensional geometric order found in epithelial tissues. The characteristic order exhibited by these tissues is of paramount importance for biological function. For example, the functionality of the drosophila (fruit-fly) wing is contingent on cellular-level structure exhibited by the wing epithelia.
By building a model that focuses on probabilistic local changes during the mitotic process, we have modeled tissue proliferation as a Markov process. This localized model accurately quantifies the global geometric order found in developmental-stage wing epithelia, and offers promise as a suitable model for modeling human epithelia, as well.
While the order in certain tissues like the drosophila wing can be understood from a purely stochastic point of view, the structure of other tissues like the retina of drosophila is deterministic—governed entirely by a mechanical energy functional. Our continued work on this project focuses on exploring the biological and biochemical differences (such as differential cell adhesion) in various tissues, and using these to foster a better understanding of the continuum of tissue states between the purely stochastic and purely deterministic.
Genevieve Brown (2008)
My B.S. degree is in applied mathematics from the University of California, Irvine. As an undergraduate, my first taste of research was in the area of mathematical biology, and I am still interested in using the techniques of applied mathematics to study physical systems.
Under the supervision of Prof. Mary Silber (ESAM) and external advisors Dr. Claire Postlethwaite (Math Department, University of Auckland) and Prof. Randy Freeman (EECS), I am currently focused on the feedback control of nonlinear dynamical systems. The feedback scheme we are studying (the so-called Pyragas-type control) involves the addition of time-delayed terms that exploit the symmetry of the system so that control can be achieved in a non-invasive manner. Pyragas-type control has been used successfully in chemical, laser, and mechanical experiments, but fewer theoretical results are available.
In particular, we are considering instabilities that arise due to a subcritical Hopf bifurcation. Using analytical methods from bifurcation theory, as well as numerical tools, I have investigated the optimal level of feedback control for two specific examples: the normal form of a subcritical Hopf bifurcation, and the Lorenz system with delay terms. Interestingly, the optimal choice of feedback for the Lorenz equations turns out to be identical to that of the simple normal form case.
The next steps in the research will involve trying to understand this feature more generally. A multiple-scales approach plays a crucial role in the analysis, because it allows for the reduction of an infinite-dimensional system with delay terms into a finite-dimensional system without delay terms. The goal is to obtain a more unified knowledge of how and when stabilization can be achieved, which may ultimately be used to inform experimental work.
Lisa Melanson (2008)
I am currently a third-year graduate student, working with Professor David Chopp, in the Department of Engineering Sciences and Applied Mathematics (ESAM) at Northwestern University. As an undergraduate, I studied mathematics and physics at the College of the Holy Cross in Worcester, MA.
My present research concerns the modeling of intracranial aneurysms, a project which lies in the area of computational fluid dynamics and mathematical biology. Aneurysms are bulbous dilatations of the vessel wall that can rupture and hemorrhage, causing a stroke. My work focuses on the investigation of the underlying mechanism of instability which results in the formation and subsequent rupture of aneurysms. Present treatment decisions are based largely on limited data, namely the size and location of the aneurysm, and there is a large size range in which the decision to operate is uncertain. Understanding which physiological features lead to instability has the potential to greatly aid in determining a neurosurgeon's course of action.
Because of the complexity of the highly nonlinear equations that govern the coupled blood/artery system, a number of numerical techniques are required to implement this model. The hemodynamics, governed by the Navier-Stokes equations, can be solved with a finite difference projection method, while the level set method allows us to track the moving nonlinear elastic walls as well as handle the irregular arterial geometries. Idealized arterial geometries have been considered, and we are working to include data extracted from the imaging of real aneurysms. In my current project, a rigid wall version of this model is investigated to determine the correlation between wall shear stress and remodeling of the arterial wall by biological reconstitution, two factors believed to promote rupture. This work will then be extended to include the fully coupled blood/vessel interaction with a nonlinear elastic wall model.
Through the RTG program, I have been able to collaborate with a group of professors, postdoctoral associates, and neurosurgeons, each of whom has contributed their expertise to provide a deeper understanding of this multifaceted problem. For instance, the addition of the neurosurgeons' knowledge of the physiology and biological processes involved in the growth of aneurysms has allowed us to continually reevaluate our model and ensure that it is physically relevant. In the future, this collaboration will provide us with the clinical data to determine the validity of our model and connect our findings to real aneurysms.
