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Abstracts and Talk Materials
Mathematical Modeling in Industry XVII
August 7 - 16, 2013

QR-codes are two-dimensional barcodes which gained popularity in recent years. These codes enable people to scan them using conventional cell-phone camera, and then get directed to appropriate web-site, or retrieve other information. Another place you often encounter these QR-codes is in boarding-tickets and similar applications, where a conventional scanner can retrieve the information from the code. (Figure 1)

These codes serve the function of connecting the user from a static medium (paper, Ad, picture) to a dynamic information source (web-site, email, reservation management system). HOWEVER, their big drawback is that these are not human readable forms. From looking at your boarding ticket barcode, you cannot tell to which flight, which airline, or any other info.

In this project we will investigate a novel method for incorporating visually-significant data in the barcode, without deteriorating their machine readability.

As an example, see (Figure 2) below: Both were done using automated algorithm, and all can easily be scanned using your standard QR-app (you probably have one on your cell or tablet).

In this project we will investigate both practical and theoretical aspects, and depending on the team spirit, will look even into market viability of such an idea.

Figure 1: Common usage of QR-codes.

Figure 2: Possible usage for boarding. Scan those: They all work!!!

Figure 3: Error Diffusion method. Scan those: They all work!!!

Some of the items we will consider:
1. Algorithm design and analysis:
• a. Information-theory and random-processes- Bit-Error-Rate for the resulting QR as a function of Image model, scanning model, and channel deterioration.
• b. HalfToning - Optimal location, using Error-diffusion, Matrix based, and other method. (Figure 3)
• c. Color blending.
2. Practical aspects:
• a. Image modifications and integration.
• b. Quality measure.
• a. Where and how to integrate it.
• b. Market for this application.
• c. Web service for that? Business model?
Pre-requisites:

None of the below are must-haves, and definitely no single person needs to possess all. However, various aspects of the project will involve the following: Image processing, e.g. Matlab; basic communication theory; random processes; basic coding and information theory.

Keywords:

Barcodes, Image processing, Random processes, Coding, Information.

About thirty years ago, the problem of the localization of rapid plastic deformation of a number of polycrystalline metals into adiabatic shear bands began to receive considerable attention, mainly due to its connection with military applications, such as armor penetration. However, the phenomenon had been observed much earlier in metal machining operations (see Figure 1). At faster cutting speeds and larger depths of cut, the strips of work material, removed by machining, called chips, were often observed to exhibit considerable structure, as a bifurcation took place in the material flow, from continuous, or fairly smooth chips, to serrated, or shear localized chips; see Figure 2(a).

The generally accepted explanation for the onset of this shear localization is that the tendency of a metal to work-harden with increasing deformation is overcome by a competing tendency for the material to soften, due to heat production caused by the rapid shearing. The term adiabatic refers to the fact that the time scale for heat conduction is much larger than the time scale for heat generation by plastic working. Recently, there has been a growing interest in the machining of amorphous metallic alloys, which are also called bulk metallic glasses (BMG). These materials differ from common polycrystalline metallic alloys, because their atoms do not assemble on a crystalline lattice, and as a result, they have unique physical, mechanical, and chemical properties. What is interesting is that a number of BMG’s have been found to produce shear-localized chips during machining operations; see Figure 2(b). Furthermore, a number of theoretical studies have argued that this strain localization is controlled not by rapid heating, but rather by a change in the concentration of free volume in the material.

In this project, we will investigate shear band formation in a BMG by studying a geometrically simple deformation of the material, a homogeneous shear flow. Our analysis will mimic some of the analysis that has been done to study adiabatic shear band formation in more common metallic alloys. We will perform a linear stability analysis of the homogeneous shear flow. The difficulty here is that, because the reference shear flow solution is time-dependent, so is the matrix of the linearization of the model equations about the reference solution. An approach that has been frequently used in this field is to apply the method of “frozen” coefficients, and this is where we will begin.

Figure 1. Schematic of a cutting process. The cutting tool moves across the workpart and a chip is formed in front of the tool.

Figure 2. Chips from two materials being machined.

(a) 52100 Bearing Steel (Grain Size:10’s of μm) (b) Nickel=Phosphorus (Amorphous)

References:
1. R.J. Clifton, J. Duffy, K.A. Hartley, and T.G. Shawki, On critical conditions for shear band formation at high strain rates, Scripta Metallurgica 18 (1984) 443-448.
2. L.H. Dai, M. Yan, L.F. Liu, and Y.L. Bai, Adiabatic shear banding instability in bulk metallic glasses, Applied Physics Letters 87 14196 (2005).
3. R. Huang, Z. Suo, J.H. Prevost, and W.D. Nix, Inhomogeneous deformation in metallic glasses, Journal of the Mechanics and Physics of Solids 50 (2002) 1011-1027.
4. M.Q. Jiang, L.H. Dai, Formation mechanism of lamellar chips during machining of bulk metallic glass, Acta Materialia 57 (2009) 2730-2738.
5. T.W. Wright, The Physics and Mathematics of Adiabatic Shear Bands, Cambridge University Press, 2002.

Prerequisites:

Interest in computing (Matlab or C/C++). Background in ordinary differential equations; some linear algebra.

Keywords:

Materials modeling, shear band formation, machining.

Lithium is the lightest metal (6.94 g/mole, specific gravity=0.53 g/cm3) and is highly electropositive (-3.05 V versus standard hydrogen electrode)I. Hence lithium-ion cells have higher specific energy (Wh/kg), higher volumetric energy density (Wh/l), higher specific power (W/kg) and higher volumetric power density (W/l) than other battery chemistries such as lead acid and Ni-MH as seen in Figure 1 and Figure 2II. Therefore lithium-ion cells are being used in advance electric vehicles (EV)III in recent years. Vehicles with different levels of electrification are in the portfolio of most major automobile manufacturers to reduce tail pipe emissions, deliver higher fuel economy, and increase energy security by reducing dependence on foreign oil while delivering vehicles with good performance, durability, safety, customer satisfaction and acceptable range and United States Advanced Battery Consortium (USABC)IV lists the goals for these advanced EV batteries.

The schematic of a typical dual lithium ion insertion ‘unit’ cell is given in Figure 3V. Copper current collector, negative porous electrode, separator, positive porous electrode and aluminum current collector are seen along the cell sandwich direction. The porous composite insertion electrodes consist of active material particles (into which lithium ion intercalates), binder, conductive carbon (if needed) and the pores filled with liquid electrolyte (lithium salt in an organic solvent). Electrolyte fills in the porous separator, which is an electronically insulating material. When the cell discharges, lithium ions shuttle from the negative insertion electrode to the positive insertion electrode and in the opposite direction when the cell is charged and therefore, this device is colloquially called the ‘rocking-chair’ cell. Detailed physics will be discussed at the workshop.

Lithium-ion cells are usually available in cylindrical, prismatic rigid can or pouch configurations. Individual cells are assembled into a module and several modules comprise a battery pack that is used in the electric vehicles. Battery management system and thermal management by either liquid or air cooling are other integral components of advance electric vehicles.

Models varying in complexity, fidelity and computational time are pursued at various scales (with coupling between them as needed) as shown in Figure 4VI for designing advance electric vehicle lithium-ion batteries. In this project, students will set up and solve a mathematical model to simulate the responses of interest such as lithium concentration, voltage, salt concentration, etc. during different modes of operation and compare their results with available experimental data for both validation purposes as well as to gain understanding of the underlying phenomena in these electrochemical systems. The complete project will be compiled as a technical report and presented by the students at the workshop.

Figure 1. Comparison of the different battery technologies in terms of volumetric and gravimetric energy densityI.

Figure 2. Ragone plots of various electrochemical energy storage and energy conversion devices.

Figure 3. Schematic of a dual lithium-ion insertion cell sandwich

Figure 4. Phenomena in lithium-ion battery systems at different length scales

References:

1. J.M. Tarascon, M. Armand, Building Better Batteries, Nature, Vol. 414, November 2001.
2. Venkat Srinivasan, Batteries for Vehicular Applications, http://bestar.lbl.gov/venkat/files/batteries-for-vehicles.pdf
3. Ford's Electric Vehicle Technology, http://ford.com/technology/electric/
4. USCAR Energy Storage System Goals http://www.uscar.org/guest/article_view.php?articles_id=85
5. R. Chandrasekaran, Ford Technical Report, SRR 2012-0069, June 2012.
6. A Pesaran et al, Computer-Aided Engineering of Batteries for Designing Better Li-Ion Batteries http://www.nrel.gov/docs/fy12osti/53777.pdf

Prerequisites:

Interests in modeling, background in ODE and PDE.

Keywords:

lithium-ion cell, electrochemical energy storage, batteries, electric vehicles.

Partial differential equations (PDEs) describe a wide range of phenomena, for example fluid flow, heat transfer, and sound propagation, among others. As such, models comprised of sets of PDEs, coupled with suitable boundary and initial conditions, are frequently used to perform computer simulations in an effort to understand the behavior of solutions, as well as being essential components in various types of problems, like those dealing with control and optimal design. Thus, the numerical solution of such models, which is key in performing the computer simulations, is of fundamental importance.

In this project, we will begin using a simple PDE model for simulating air ow and heat transfer. We will go through the process of solving this model numerically, using open source software to aid with the tasks of mesh generation, solution of the equations, and post-processing of the numerical results. If time permits, we will expand the model to allow for the simulation of more complicated behavior and study dierences in the solutions obtained for each of the models and by variations in the boundary conditions.

Figure 1. Visualization of a sample domain, mesh, and cross-sections from a 3D numerical simulation of air velocity and temperature distribution.

References:

1. V. Lopez and H. F. Hamann, "Heat Transfer Modeling in Data Centers", Int. J. Heat Mass Transfer, Vol 54, 2011

2. http://sourceforge.net/projects/netgen-mesher/

3. http://openfoam.com/

4. http://code.enthought.com/projects/mayavi/

Prerequisites:

Computer programming experience in a language like C or C++; Basic knowledge about PDEs and numerical solution methods for PDEs would be helpful.

Keywords:

PDE-based simulations, PDEs, numerical methods for PDES, visualization

Sweep MRI is a new technological development to address the difficulty of imaging tissues and organs, such as bone, tendon, and lung, that give weak signals. In Sweep MRI, the RF (radio frequency) excitation is pulsed, while the background magnetic field gradient can be varied in time (see here for a schematic diagram of an MRI machine).  The resulting mathematical problem that needs to be solved is an inverse problem involving two steps [1].  The first step is determining the planar averages of the material properties of the object over a set of planes determined by the gradient of the background magnetic field.  The second step is to recover the spatial distribution of the material properties from these 'projections'.  Once the data are unraveled, the problem of image reconstruction can be viewed as that of Fourier inversion of a 3-d function from samples of its Fourier transform.

The goal of this project is to first study the model of the relationship between the measured data and the desired unknown image, and the limitations of this in the case where the measured data is incomplete.  We will start with the case where the magnetic field gradient is assumed to be constant in time.  The time-varying case will also be considered.  Issues such as resolution limit and reconstruction methods will be explored, as well as signal recovery form incomplete data measurements.

Below is a picture of a fetal mouse using SWIFT MRI.

References:
1. M. Weiger, F. Hennel, and K. Pruessmann, Sweep MRI with Algebraic Reconstruction, Magnetic Resonance in Medicine, 64, 2010, 1685-1695.
2. Z.-P. Liang and P. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Approach, IEEE Press, 1999.
3. C. Epstein, Introduction to the Mathematics of Medical Imaging, Pearson/Prentice Hall, 2003.
Prerequisites:

Basic image processing, Fourier analysis, MATLAB programming.

Keywords:

Magnetic Resonance Imaging, Fourier transforms, inverse problem, tomography.

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