<p>Numerical methods, LQR control, an abstract formulation andreduced basis techniques for a system consisting of a thin cylindrical shellwith surface-mounted piezoceramic actuators are investigated.Donnell-Mushtari equations,modified to include Kelvin-Voigt damping and passive patch contributions,are used to model the system dynamics. The voltage-induced piezoceramicpatch contributions, used as input inthe control regime, enter the equations as externalforces and moments. Existence and uniqueness of solutions are demonstratedthrough variational and semigroup formulations of the system equations.The semigroup formulation is also used to establish theoretical controlresults and illustrate convergence of the finite dimensional controlsand Riccati operators.The spatial components of the state arediscretized using a Galerkin expansion resulting in an ordinarydifferential equation that can be readily marched in time by existingordinary differential equationsolvers.Full order approximation methods which employ standard basiselements such as cubic or linear splines result in large matrixdimensions rendering the system computationally expensive for real-timesimulations. To lessen on-line computational requirements, reducedbasis methods employing snapshots of the full order model as basisfunctions are investigated.As a first step in validating the model, a shell with obtainable analyticnatural frequencies and modes was considered. The derived frequenciesand modeswere then compared with numerical approximations using full order basisfunctions. Further testing on the static and dynamic performance of the fullorder model was carried out through the following steps:(i) choose true state solutions, (ii) solve for the forces in theequations that would lead to these known solutions, and (iii) comparenumerical results excited by the derived forces with the true solutions.Reduced order methods employing the Lagrange and theKarhunen-Loève proper orthogonal decomposition (POD)basis functions are implemented on the model. Finally, a statefeedback method was developed and investigated computationallyfor both the full order and reduced ordermodels.<P>

<p>A finite, homogeneous, irreducible Markov chain $\mC$ with transitionprobability matrix possesses a unique stationary distribution vector. The questions one can pose in the area of computation of Markov chains include the following:<br>- How does one compute the stationary distributions? <br>- How accurate is the resulting answer? <br>In this thesis, we try to provide answers to these questions. <br><br>The thesis is divided in two parts. The first part deals with the perturbation theory of finite, homogeneous, irreducible Markov Chains, which is related to the first question above. The purpose of this part is to analyze the sensitivity of the stationarydistribution vector to perturbations in the transition probabilitymatrix. The second part gives answers to the question of computing the stationarydistributions of nearly uncoupled Markov chains (NUMC). <P>

<p>This work describes the development of a mathematical model of ahigh-pressure chemical vapor deposition (HPCVD) reactor and nonlinearfeedback methodologies for control of the growth of thin films in thisreactor. Precise control of the film thickness and composition is highlydesirable, making real-time control of the deposition process veryimportant. The source vapor species transport is modeled by the standardgas dynamics partial differential equations, with species decompositionreactions, reduced down to a small number of ordinary differential equationsthrough use of the proper orthogonal decomposition technique. This systemis coupled with a reduced order model of the reactions on the surfaceinvolved in the source vapor decomposition and film deposition on thesubstrate wafer. Also modeled is the real-time observation technique usedto obtain a partial measurement of the deposition process. The utilization of reduced order models greatly simplifies the mathematicalformulation of the physical process so it can be solved quickly enough to beused for real-time model-based feedback control. This control problem isfairly complicated, however, because the surface reactions render the modelnonlinear. Several control methodologies for nonlinear systems are studiedin this work to determine which performs best on test examples similar tothe HPCVD problem. One chosen method is extended to a tracking control toforce certain film growth properties to follow desired trajectories. Thenonlinear control method is used also in the development of a stateestimator which uses the nonlinear partial observation of the nonlinearsystem to create an estimate of the actual state, which the feedback controlformula then can use to guide the HPCVD system. The nonlinear trackingcontrol and estimator techniques are implemented on the HPCVD model and theresults analyzed as to the effectiveness of the reduced order model andnonlinear control.<P>

<p>This work investigates the characteristics of a steady state flow of granular material,under the influence of gravity, in two and three dimensional hoppers of simple geometry.Simulations of such flows are of particular interest to various industries, such as the foodand mining industries, where the handling of large quantities of granular materials in hop-persand silos is routine. While understanding and simulation of time-dependent phenomenaare the ultimate goals in this field, those phenomena are still poorly understood and thustheir study is beyond the scope of this research. It has been observed that steady flowscan provide reasonable approximations, and the corresponding steady state model has con-sequentlybeen the focus of a great deal of research. Historically, these steady state modelshave been approached using only smooth radial fields, and even today most practical hop-perdesign uses these fields as their basis. Our work represents the first time that qualitynumerical methods have been brought to bear on the model equations in their original form,without assuming smoothness of the resulting fields. Two different, yet related, models forstress/velocity consisting of systems of hyperbolic conservation laws and algebraic relationsare considered and discussed. The radial stress and velocity fields, and the stability of thosefields, are studied briefly with both analytical and numerical results presented. More im-portantly,a Runge-Kutta Discontinuous Galerkin method is implemented and applied tovarious boundary value problems involving perturbed stress and velocity fields arising fromdiscontinuous changes in parameters such as hopper wall angle or hopper wall friction.<P>

<p><p>MINIMAIR, MANFRED. Resultants of Composed Polynomials.(Under the direction of Hoon Hong.)</p><p>The objective of this research has been to develop methods forcomputing resultants of composedpolynomials, efficiently, by utilizing their composition structure.By the resultant of several polynomials in several variables (one fewer variables than polynomials) we mean anirreducible polynomial in the coefficients ofthe polynomials that vanishes if theyhave a common zero.By a composed polynomial we mean the polynomial obtained from a given polynomial by replacing each variable by a polynomial.</p><p>The main motivation for this researchcomes from the following observations:Resultants of polynomialsare frequently computedin many areas of science andin applicationsbecause they are fundamentally utilized in solving systemsof polynomial equations.Further, polynomials arising in science and applicationsare often composed because humans tend to structure knowledge modularly and hierarchically.Thus, it is important to have theories and software librariesfor efficientlycomputing resultants of composed polynomials.</p><p>However,most existing mathematical theories do not adequately support composed polynomials and most algorithms as well as software libraries ignore the composition structure, thus suffering from enormous blow up in space and time.Thus, it is important to develop theories and software librariesfor efficientlycomputing resultants of composed polynomials.</p><p>The main finding of this research is thatresultants of composed polynomials can benicely factorized, namely, they can be factorized into products of powers of the resultants of the component polynomialsand of some of their parts. These factorizationscan be utilized to compute resultants of composed polynomialswith dramatically improved efficiency.</p><P>

<p>Three physiologically based pharmacokinetic (PBPK) models for thesystemic transport of inhaled trichloroethylene (TCE) are presented.The major focus ofthese modeling efforts is the disposition of TCE in the adiposetissue, where TCE is known to accumulate. Adipose tissue is highly heterogeneous, with wide variations in fat cell size, lipid composition, blood flow rates and cellpermeability. Since TCE is highly lipophilic, the uneven distributionof lipids in the adipose tissue may lead to an uneven distribution of TCEwithin the fat. These physiological characteristics suggest that thedynamics of TCE in the adipose tissue may depend on spatial variations within the tissueitself.<br> <br>The first PBPK model for inhaled TCE presented here is a system ofordinary differential equations which includes the standardperfusion-limited compartmental model for each of the adipose, brain,kidney, liver, muscle and remaining tissue compartments.Model simulations predict relatively rapiddecreases in TCE fat concentrations following exposure, which may notreflect the accumulation and relative persistence of TCE inside the fattissue. The second PBPK model is identical to the first except forthe adipose tissue compartment, which is modeled as a diffusion-limited compartment.Although this model yields various concentration profiles for TCE inthe adipose tissue depending on the value of the permeabilitycoefficient, this model may not be physically appropriate for TCE,which is highly lipophilic and has a low molecular weight. Moreover,neither of these two PBPK models is able to capture spatialvariation of TCE concentrations in adipose tissue as suggested bythe physiology.<br><br>The third model we present is a hybrid PBPK model with adispersion-type model for the transport of TCE in the adipose tissue. Thedispersion model is designed to account for the heterogeneities within fattissue, as well as the corresponding spatial variation of TCE concentrationsthat may occur. This partial differential equation model is based onthe dispersion model of Roberts and Rowland for hepatic uptake andelimination, adapted here for the specific physiology of adiposetissue. <br><br>Theoretical results are given for the well-posedness of a generalclass of abstract nonlinear parabolic systems which includes the TCEPBPK-hybrid model as a special case. Moreover, theoretical issues related to associated general least squares estimation problems are addressed,including the standard type of deterministic problem and aprobability-based identification problem that incorporates variability inparameters across a population. We also establish thetheoretical convergence of the Galerkin approximations used in our numericalschemes.<br><br>The qualitative behavior of the TCE PBPK-hybrid model is studied usingmodel simulations and parameter estimation techniques. In general, theTCE PBPK-hybrid model can generate various predictions of the dynamicsof TCE in adipose tissue by varying the adipose model parameters.These predictions include simulations that are similar to the expectedbehavior of TCE in the adipose tissue, which involves a rapid increaseof TCE adipocyte concentrations during the exposure period, followed by aslow decay of TCE levels over several hours.<br><br>Results are presented for several types of parameter estimationproblems associatedwith the TCE PBPK-hybrid model. We test theseestimation strategies using two types of simulated data: observationsrepresenting TCE concentrations from a single individual, andobservations that simulateinter-individual variability. The latter type of data, which iscommonly found in experiments related to toxicokinetics, assumesvariability in the parameters across a population, and may includeobservations from multiple individuals. Using both deterministic andprobability-based estimation techniques, we demonstrate thatthe probability-based estimation strategiesthat incorporate variability in the parameters may be best suited forestimating adipose model parameters that vary across the population.<P>

<p>GABLONSKY, JÖRG MAXIMILIAN XAVER. Modifications of the DIRECTAlgorithm. (Under the direction of Carl Timothy Kelley.)This work presents improvements of a global optimization method for boundconstraint problems along with theoretical results. These improvements arestrongly biased towards local search. The global optimization method known asDIRECT was modified specifically for small-dimensional problems with few globalminima. The motivation for our work comes from our theoretical results regarding thebehavior of DIRECT. Specifically, we explain how DIRECT clusters its search neara global minimizer. An additional influence is our explanation of DIRECT'sbehavior for both constant and linear functions. We further improved the effectiveness of both DIRECT, and our modification, bycombining them with another global optimization method known as ImplicitFiltering. In addition to these improvements the methods were also extended tohandle problems where the objective function is defined solely on an unknownsubset of the bounding box. We demonstrate the increased effectiveness androbustness of our modifications using optimization problems from the natural gastransmission industry, as well as commonly used test problems from theliterature. <P>

<p>Benzene (C_6 H_6) is a highly flammable, colorless liquid. Ubiquitous exposures result from its presence in gasoline vapors, cigarette smoke, and industrial processes. Benzene increases the incidence of leukemia in humans when they are exposed to high doses for extended periods; however, leukemia risks in humans subjected to low exposures are uncertain. The exposure-dose-response relationship of benzene in humans is expected to be nonlinear because benzene undergoes a series of metabolic transformations, detoxifying and activating, resulting in multiple metabolites that exert toxic effects on the bone marrow. <br><br>We developed a physiologically based pharmacokinetic model for the uptake and elimination of benzene in mice to relate the concentration of inhaled and orally administered benzene to the tissue doses of benzene and its key metabolites, benzene oxide, phenol, and hydroquinone. Analysis was done to examine the existence and uniqueness of solutions of the system. We then formulated an inverse problem to obtain estimates for the unknown parameters; data from multiple independent laboratories and experiments were used.<br><br>The model was then revised to take into account the zonal distribution of enzymes and metabolisms in the liver, rather than assuming the liver is one homogeneous compartment, and extrahepatic metabolisms were also considered. Despite the sources of variability, the multicompartment metabolism model simulations matched the data reasonably well in most cases and improved results from the original model, showing that data for benzene metabolism and dosimetry.<br><br>Since benzene is a known human leukemogen, the toxicity of benzene in the bone marrow is of most importance. And because blood cells are produced in the bone marrow, we investigated the effects of benzene on hematopoiesis (blood cell production and development). An age-structured model was used to examine the process of erythropoiesis, the development of red blood cells. Again, the existence and uniqueness of the solution of the system of coupled partial and ordinary differential equations was proven. The system was then written in weak form and reduced from an infinite dimensional system to a system of ordinary differential equations by employing a finite element formulation. The control of this process is governed by the hormone erythropoietin. The form of the feedback of this hormone is unknown. Although a Hill function has previously been used to represent this feedback, it has no physiological basis, so an optimal control problem was formulated in order to find the optimal form of the feedback function and to track the total number of mature cells. Numerical results for both forms of the feedback function are presented.<P>

<p>In the field of nondestructive evaluation, new and improved techniques are constantly being sought to facilitate the detection of hidden corrosion and flaws in structures such as airplanes and pipelines. In this dissertation, we explore the feasibility of detecting such damages by application of an eddy current based technique and reduced order modeling. <br><br>We begin by developing a model for a specific eddy current method in which we make some simplifying assumptions reducing the three-dimensional problem to a two-dimensional problem. (We do this for proof-of-concept.) Theoretical results are then presented which establish the existence and uniqueness of solutions as well as continuous dependence of the solution on the parameters which represent the damage. We further discuss theoretical issues concerning the least squares parameter estimation problem used in identifying the geometry of the damage. <br><br>To solve the identification problem, an optimization algorithm is employed which requires solving the forward problem numerous times. To implement these methods in a practical setting, the forward algorithm must be solved with extremely fast and accurate solution methods. Therefore in constructing these computational methods, we employ reduced order Proper Orthogonal Decomposition (POD) techniques which allows one to create a set of basis elements spanning a data set consisting of either numerical simulations or experimental data. <br><br>We investigate two different approaches in forming the POD approximation, a POD/Galerkin technique and a POD/Interpolation technique. We examine the error in the approximation using one approach versus the other as well as present results of the parameter estimation problem for both techniques. <br><br>Finally, results of the parameter estimation problem are given using both simulated data with relative noise added as well as experimental data obtained using a giant magnetoresistive (GMR) sensor. The experimental results are based on successfully using actual experimental data to form the POD basis elements (instead of numerical simulations) thus illustrating the effectiveness of this method on a wide range of applications. In both instances the methods are found to be efficient and robust. Furthermore, the methods were fast; our findings suggest a significant reduction in computational time. <P>

<p>SEGAL, REBECCA ANNE. Patterns of air flow and particle depositionin the diseased human lung. (Under the direction of Michael Shearer.)In this work, we investigate particle deposition and air flow in thehuman lung. In particular we are interested in how the motion ofparticulate matter and air is affected by the presence of lungdisease. Patients with compromised lung function are more sensitiveto air pollution; understanding the extent of that sensitivity canlead to more effective air quality standards. Also, understanding ofair flow andparticle trajectories could lead to the development of better aerosoldrugs to treat the lung diseases.We focus our efforts on twodiseases: chronic obstructive pulmonary disease (COPD) and bronchialtumors. Because COPD affects the majority of airways in a patientwith the disease, we are able to take a more global approach toanalyzing the effects of the disease. Using a FORTRAN codewhich computes total deposition in the lung over the course of onebreath, we modified the pre-existing code to account forthe difference between healthy subjects and patients with COPD. Usingthe model, itwas possible to isolate the different disease components of COPD andsimulate their effects separately. It was determined thatthe chronic bronchitis component of COPD was responsible for the increaseddeposition seen in COPD patients.While COPD affects the whole lung, tumors tend to belocalized to one or several airways. This led us to investigate theeffects of bronchial tumors in detail within these individualairways. Using a computational fluid dynamics package, FIDAP, wedefined a Weibel type branching network of airways.In particular, we modeled theairways of a four-year-old child.In the work with the tumors, we ran numerous simulations with variousinitial velocities and tumor locations. It was determined that tumorslocated on the carinal ridge had the dominant effect on the flow. Athigher initial velocities, areas of circulation developed downstreamfrom the tumors. Extensive simulations were run with a 2-D model. Theresults from the 2-D model were then compared with some initial 3-Dsimulations.In the development of the FIDAP model, we avoided thecomplications of flow past the larynx, by limiting the model togenerations 2-5 of the Weibel lung. We developed a realistic inletvelocity profile to be used as the input into the model. The skewednature ofthis inlet profile led to thequestion of boundary layer development and the determination of theentrance length needed to achieve fully developed parabolicflow. Simple scale analysis of the Navier-Stokes equations did notcapture the results we were seeing with the CFD simulations.We turned to a more quantitative, energy correctionanalysis to determine the theoretical entrance length.In conclusion, the presence of disease in the lunghas a large effect both on global deposition patterns and on localizedairflow patterns. This indicates the need for different protocolsregarding susceptibility of people to airborne pollutants that take intoaccount lung disease. It also suggests that treatment should accountfor changes in airflow in the diseased lung.<P>