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Über dieses Buch

This is the sixth volume in the series "Mathematics in Industrial Prob­ lems. " The motivation for these volumes is to foster interaction between Industry and Mathematics at the "grass roots level"; that is, at the level of specific problems. These problems come from Industry: they arise from models developed by the industrial scientists in ventures directed at the manufacture of new or improved products. At the same time, these prob­ lems have the potential for mathematical challenge and novelty. To identify such problems, I have visited industries and had discussions with their scientists. Some of the scientists have subsequently presented their problems in the IMA Seminar on Industrial Problems. The book is based on the seminar presentations and on questions raised in subse­ quent discussions. Each chapter is devoted to one of the talks and is self­ contained. The chapters usually provide references to the mathematical literature and a list of open problems which are of interest to the industrial scientists. For some problems a partial solution is indicated briefly. The last chapter of the book contains a short description of solutions to some of the problems raised in previous volumes, as well as references to papers in which such solutions have been published. The speakers in the seminar on Industrial Problems have given us at the IMA hours of delight and discovery. My thanks to Thomas Hoffend (3M), John Spence (Eastman Kodak Company), Marius Orlowski (Mo­ torola, Inc. ), Robert J.

Inhaltsverzeichnis

Frontmatter

1. Magnetization reversal in particulate recording media

Abstract
Magnetization is a material property traced to the spin motion of electrons. Magnetic particles are used in making magnetic tapes and disks for recording media; such particles are suspended in a binder and then painted onto the substrate (a film or a disk). Micromagnetics deals with magnetic phenomena at the “intermediate” scale, which includes such small particles as used in recording media. In micromagnetic phenomena at the scale of 0.01μm to 10μm, domains and domain-walls play a fundamental role in the model; these were described in [1]; the particles which will be considered in the present chapter are smaller, and do not support domain walls. On September 25, 1992 T. R. Hoffend, Jr. from 3M presented ongoing work [2] on micromagnetics based, vector hysteresis model for studying the collective effect of particle interaction and non uniformities on magnetization reversal processes in particulate recording media. The motivation for this work is to understand the link between single particle properties and ensemble properties and, in particular, to investigate the origins of measured differences in the response of various types of particulate recording media. He described the model, consisting of a coupled system of Landau—Lifshitz equations, and then presented some numerical results and posed open problems.
Avner Friedman

2. Variable forgetting factors in Kalman filtering

Abstract
Kodak Signature Color Proofing System is a machine that tests the color texture of a picture for separation of its components into magenta, cian, yellow and black. Actually the first three determine the color and the black component determines the brightness. We make an intuitive judgement on the color components at each pixel and feed the corresponding instructions to the machine. The machine then produces a print. If the print agrees with the original picture then the input information is stored away and is later transferred to the appropriate media where the picture is to appear in print (magazine, newspaper, book, etc.). If the print does not agree satisfactorily with the original picture then additional adjustments need to be made in determining the separation. The actual printing of the test prints is accomplished with toner fluid of the four colors mentioned above by application of electric field around the “printing station” within the machine. New toner is added typically after every 50 prints. The entire process must be assured of a high degree of consistency. It is inevitable however that there will be random variations in the moving parts of the machine, in the start-up conditions, and in the consistency of the toner. The goal of the manufacturer of the machine and of the machine operator is to minimize these variabilities.
Avner Friedman

3. Modeling of dopant diffusion networks

Abstract
Most of the materials used in semiconductor manufacturing have granular structure. The granularity can occur on various length scales ranging from amorphous (few nanometers) to columnar grain structures (tens or hundreds of micrometers). These scales are larger than the microscopic (atomic or molecular) scale but smaller than the macroscopic scale; one refers to these regimes as the mesoscopic regimes. Pantelides [1] points out that most industrial materials are polycrystalline, amorphous, or composite, and their properties are determined by the collective microstructure at the mesoscopic level. Yet the modeling of such materials have not been seriously investigated.
Avner Friedman

4. Statistical optics and effective medium theories of color

Abstract
Measurements of color are typically made by determining the spectral intensity of light diffusely reflected from images. In xerography color images are heterogeneous systems composed of light absorbing pigments of different colors and sizes suspended in a semitransparent binder. Classical textbook optics predicts the intensity of light specularly reflected from smooth homogeneous surfaces, but cannot provide the detailed information required for these complex optical systems. On October 30, 1992 Robert J. Meyer from Webster Research Center of Xerox has described how statistical optics and effective medium theories predict the intensity of light diffusely reflected from a rough surface of a body containing many small particles; such systems occur within the color image photocopier. He indicated some shortcomings of the dynamic effective medium theory and presented open problems.
Avner Friedman

5. Amorphous and polysilicon devices

Abstract
Polysilicon is made up of silicon grains, each grain being a crystal in which the silicon atoms are arranged in a periodic structure. The distance between 0 two silicon atoms in the crystal is 2 – 3 Å, and the typical size of a grain in polysilicon is 1, 000 – 10, 000 Å. Thus each grain contains many millions of atoms. On the other hand, in amorphous silicon the atoms are typically grouped in 4 – 6 atoms, and there is no discerned periodic structure. In order to pacify the dangling bonds some hydrogen is added (approximately 20%); this is called hydrodegenation. Amorphous silicon is relatively easy to make, and it can be deposited on a large area. Polysilicon is harder to deposit on a large area; and pure silicon crystals are still harder to deposit (and more expensive), being more readily susceptible to faults. Amorphous silicon is a poor conductor, and is therefore not used in high-speed computer chips. Thin-film transistors (TFTs) fabricated from hydrogenated amorphous silicon (a — Si) and polycrystalline silicon (polysilicon, poly-Si) are now used in many commercial large-area electronic applications such as in flat panel display, printing and scanning (e.g., in fax machines).
Avner Friedman

6. Modeling the performance of a piezoelectric gyroscope

Abstract
Conventional gyroscopes are based on high speed rotor devices. Vibrating gyroscopes offer advantages over conventional ones: they are free of troublesome bearings, require low power with instant start-up time, and they can often be packaged in a small working volume. Furthermore, they are not sensitive to linear acceleration and thus are apt to better detect angular acceleration. Such devices are used in avionic navigation and have possible future use in steering control for automobiles.
Avner Friedman

7. Particle simulations for xerographic development

Abstract
The electrophotographic system of a copier was presented in some detail in [1; Chapter 17]. Figure 7.1 gives a schematic description of the photocopy process. The image is created when the document is exposed to light. The photocopy comes out on the belt after the latent (electric) image of the document is developed by means of toner, and then the developed visible image is transferred onto a rolling paper, on the belt; the toner is then diffused onto the paper.
Avner Friedman

8. A statistical dynamic theory of glassy polymers

Abstract
The physical properties of solid polymers vary slowly in time, so that the long time behavior and thermal history become a major concern. One would like to understand how aging affects the performance of viscoelastic material. On January 8, 1993 T.S. Chow from Xerox Webster Research Center described a unified approach toward understanding the structural and mechanical properties of glassy polymers. The glassy state relaxation is derived by analysis of local configurational rearrangements of molecular segments and “hole” motion; it is based on statistical mechanical theory of glasses. He then applied his general approach to study the effects of stress-strain relationships of solid polymers, under variable conditions of aging, temperature and filler concentration in composites. The calculated results are in good agreement with experiment. This chapter is based on his presentation, his recent review article [1], and other previous articles.
Avner Friedman

9. Simulation of magnetic recording and playback processes

Abstract
Recording information on magnetic media (disks, tapes) was first demonstrated almost 100 years ago. Since then many inventions and advancements have been made. Yet, a thorough understanding of the physical mechanisms involved is still lacking. The motivation for modeling the magnetic recording and performance is to gain better understanding of the physical processes so as to better predict the performance of magnetically recorded media. Once accurate models are developed one can turn to the important issues of design and optimization. On January 15, 1993 Robert L. Brott from 3M has described ongoing work on the modeling and simulation of magnetic recording. The model for the recording process is given in terms of complicated nonlinear hysteretic integral equation. The playback process is modeled relatively simply. He described his algorithm, and posed some open problems.
Avner Friedman

10. Sunglitter in oceanic remote detection

Abstract
One of the problems in oceanic remote sensing is how to reliably detect or distinguish a signal of interest from the random sources of noise arising in both the environment and the sensor. For example, under what conditions can a small capsized boat be detected in high-altitude optical imagery (e.g., by sight from an airplane) in the presence sunglitter and subsurface backscattering. In such an application, the design of the optimal signal-processing algorithm and the evaluation of its performance depend critically upon in the statistics of the noise in the sensing process. Diffuse optical noise can usually be adequately modeled as normally distributed. But sunglitter is known to be extremely non-normal and, therefore, its statistical characterization as a function of the viewing geometry, wind speed, and sensor resolution are critical to the design of an effective detector.
Avner Friedman

11. Computer simulations of electrorheological fluids

Abstract
The term electrorheological (ER) fluid applies generally to any liquid that exhibits a marked change in rheological behavior in an external electric field. The ER fluids are typically colloidal suspensions of micron-sized polarizable particles in low conductivity liquids. A large electric field aligns the particles into chains and columns parallel to the field, thereby increasing the resistance to shear. The ER effect is currently being widely studied for variety of automotive products such as shock absorbers, clutches, and engine mounts. It has potential applications also in other vibration damping devices as they occur, for instance, in buildings and other structures which resist shock (e.g. of earthquake) and in designing quiet submarines.
Avner Friedman

12. Local flaws in permeation reducing barrier layers

Abstract
One of the concerns in designing a plastic fuel tank is to make the surface of the container such that it reduces the permeation of fuel. In order to improve the barrier properties of the surface, the inner surface is coated with appropriate thin barrier layer. This is often done by blow molding process. The process has local flaws: many small patches do not get covered by the thin barrier material. These “holes” have been observed by Mark Nulman and modelled by Giuseppe Rossi [1], both from Ford Motor Company. On March 5, 1993 Rossi described the model, and presented results in case of circular holes. He also posed the problem of studying the effect of general shaped “holes” and general distribution of “holes” in the inner surface of the container on the imperbeability of the surface of the container.
Avner Friedman

13. Iterative solution methods on the Cray YMP/C90

Abstract
Solution of a large sparse or dense linear systems often constitutes the most time consuming step when solving large multidimensional industrial problems on modern high performance supercomputers. In most practical cases these problems produce linear systems with ill-conditioned and off-diagonal dominant coefficient matrices. As the problem size grows, especially in the 3-dimensional case, direct methods for solving these systems (such as triangular decomposition) begin to lose their effectiveness due to the large arithmetic costs and memory requirements. These difficulties motivate resorting to iterative solution methods which are, at least theoretically, more efficient with respect to the arithmetic cost and memory requirements. On March 12, 1993 Qasim Sheikh from Cray Research presented numerical experiments with iterative solvers that are capable of solving large linear systems which arise in industrial applications. He then compared the effectiveness and some performance aspects of these iterative methods with other state of the art, highly optimized, iterative as well as direct solvers. The main results of his talk are based on a recent paper [1] written jointly with Kharchenko, Nikishin, Yeremin and Heroux.
Avner Friedman

14. Track reconstruction and data fusion

Abstract
A ship at sea finds its way using a combination of its own instruments and a collection of external navigation systems. If all the navigation data over the course of a voyage are recorded, how well can one reconstruct the track of the ship? On March 26, 1993 William J. Satzer from 3M described an approach to track reconstruction by way of optimal smoothing theory. The goal is (i) to provide an accurate and reliable “truth track” to serve as a basis for analysis of long range ocean sensors, and (ii) to develop means for using disparate data sources to form a coherent picture of an environment.
Avner Friedman

15. Approximation to Boltzmann transport in semiconductor devices

Abstract
There is hierarchy of models of electron transport in semiconductor device. A complete physical description is provided by the Schrödinger equations, which are numerically too intensive to implement. A simplified model is the drift-diffusion model, which has been successfully used to design devices larger than 0.25μ. The Boltzmann transport equation is a great simplification over the Schrödinger equations model, yet it contains much of the physics; see [1]. The solution of the Boltzmann equation is obtained by computationally expensive Monte Carlo techniques [2]. This motivated several simplified approaches, notably the hydrodynamic model and the drift-diffusion model (see [3] and the references given there). Another approximation to the Boltzmann equation has been developed by Hagan, Cox and Wagner [4] (see also [5; Chap. 11] and the last chapter in this volume).
Avner Friedman

16. Systematic data fusion using the theory of random conditional sets

Abstract
In data fusion one tries to locate and/or classify unknown objects of many different types on the basis of different kinds of evidence, collected on an ongoing basis by many sensors having varying capabilities. This is illustrated in Figure 16.1.
Avner Friedman

17. Micromagnetic simulation

Abstract
A brief introduction to micromagnetism was given in Chapter 17 of [1]; it included the description of domains and walls, and the derivation of the Landau—Lifshitz equation. Chapters 1 and 9 of this Volume deal with micromagnetic material which is used in magnetic tapes and disks. This material consists of magnetic particles suspended in a binder which is then painted onto a substrate such as polyester film (for tapes) or aluminum disk.
Avner Friedman

18. Dissolution kinetics with feedback

Abstract
Consider a solid particle composed of chemical A. the particle is in a solution of chemical B. As the particle dissolves, the A that enters the solution reacts with B to form chemical C. Then C diffuses back to the particle and adsorbs to the particle’s surface. The presence of C on the particle’s surface inhibits the dissolution, and ultimately shuts it down entirely. This process was recently modeled by Kam-Chuen Ng and David S. Ross from Eastman Kodak. The mathematical model is expressed as a Stefan problem for reaction-diffusion system. On May 7, 1993 David Ross presented the model, described some asymptotic analysis, both for large reaction rates and for small time. He concluded with several open problems regarding this system.
Avner Friedman

19. Wetting and adsorption at chemically heterogeneous surfaces

Abstract
Many industrial processes require control of wettability of a solid surface by various liquids. For example, the liquid may be a paint, a lubricant or a dye, and the solid may have smooth or rough surface. In this chapter the surface is fabricated by chemisorbing mixed alkanethiol monolayer on gold substrate from solution. Such surfaces can be used for controlling wetting, spreading, adhesion and friction of industrially engineered smooth surfaces. Alkanes are (CH 2) n groups, thiols refer to SH, and alkanethiol chains are X- (CH 2) n -SH molecules. In this chapter X is either OH or CH 3; OH is hydrophilic and CH 3 is hydrophobic. We shall deal with alkanethiol chains chemiosorbed from solution onto gold (Au), as shown in Figure 19.1.
Avner Friedman

20. Fluid and kinetic modeling for micromechanical structures

Abstract
Micromechanical structures are structures which are at least partially mechanical in nature and function, which are very small (usually less than 1mm), and are produced using semiconductor fabrication technology. Micromechanical technology is a growing industry. The devices can be replicated cheaply and fast, and their small size saves space and offers flexibility (by integrating many devices). Micromachines are used both as sensors and actuators. Present applications include, for instance, accelerometers for air-bag deployment in automobiles and pressure sensors in medical and industrial applications. There are, in general, three aspects to simulating micromechanical structures;
(i)
The mechanical aspect, relating to bending and stress of the elements of the structure;
 
(ii)
The electromagnetic aspect when voltages are applied to either stimulate or sense the motion of the structure (piezoelectric, piezoresistive and capacitive effects), and
 
(iii)
The fluid or kinetic aspect, depending on the relative dimensions, which is present when liquid or gas are not evacuated from the region of mechanical motion.
 
Avner Friedman

21. Modeling exhaust-gas oxygen sensors

Abstract
Catalytic converters cannot effectively burn off exhaust gases if the engine air-to-fuel ratio is either too rich or too lean. It is therefore necessary to control automobile engines to run at the stoichiometric point; that is, the air-to-fuel ratio at which all combustible gases can be burnt, leaving no oxygen at the end of the process.
Avner Friedman

22. 3D modeling of a smart power device

Abstract
Solid state devices are increasingly being used in the automobile industry, both as sensors and actuators. An accelerometer for crash detection in automobile, for example, was described in Chapter 20 of this Volume. In this chapter we shall consider a smart power device used for fuel injection. This is a 3D device. The power devices reside within the top few microns of the silicon substrate of the chip, are arranged in interdigitated fingers, as shown in Figure 22.1, and number in the thousands per chip. Heat generated by the devices poses a significant problem and must be conducted away quickly enough to prevent thermal runaway and burnout failure. On June 3, 1993 Leonard Borucki from Motorola described the device and developed a mathematical model. The problem can be formulated as a nonlinear time dependent heat equation with a source term in the fingers. Significant computational resources and time are required to calculate the location and magnitude of the temperature maximum, where failure is likely to occur first. A problem posed by Borucki is whether analytic methods can provide, more quickly, estimates of the maximum temperature.
Avner Friedman

23. Solutions to problems from previous parts

Without Abstract
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Backmatter

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