Browsing by Subject "Mathematical models"
Now showing 1 - 18 of 18
Results Per Page
Sort Options
Item Adaptive multiscale modeling of polymeric materials using goal-oriented error estimation, Arlequin coupling, and goals algorithms(2008-05) Bauman, Paul Thomas, 1980-; Oden, J. Tinsley (John Tinsley), 1936-Scientific theories that explain how physical systems behave are described by mathematical models which provide the basis for computer simulations of events that occur in the physical universe. These models, being only mathematical characterizations of actual phenomena, are obviously subject to error because of the inherent limitations of all mathematical abstractions. In this work, new theory and methodologies are developed to quantify such modeling error in a special way that resolves a fundamental and standing issue: multiscale modeling, the development of models of events that transcend many spatial and temporal scales. Specifically, we devise the machinery for a posteriori estimates of relative modeling error between a model of fine scale and another of coarser scale, and we use this methodology as a general approach to multiscale problems. The target application is one of critical importance to nanomanufacturing: imprint lithography of semiconductor devices. The development of numerical methods for multiscale modeling has become one of the most important areas of computational science. Technological developments in the manufacturing of semiconductors hinge upon the ability to understand physical phenomena from the nanoscale to the microscale and beyond. Predictive simulation tools are critical to the advancement of nanomanufacturing semiconductor devices. In principle, they can displace expensive experiments and testing and optimize the design of the manufacturing process. The development of such tools rest on the edge of contemporary methods and high-performance computing capabilities and is a major open problem in computational science. In this dissertation, a molecular model is used to simulate the deformation of polymeric materials used in the fabrication of semiconductor devices. Algorithms are described which lead to a complex molecular model of polymer materials designed to produce an etch barrier, a critical component in imprint lithography approaches to semiconductor manufacturing. Each application of this so-called polymerization process leads to one realization of a lattice-type model of the polymer, a molecular statics model of enormous size and complexity. This is referred to as the base model for analyzing the deformation of the etch barrier, a critical feature of the manufacturing process. To reduce the size and complexity of this model, a sequence of coarser surrogate models is generated. These surrogates are the multiscale models critical to the successful computer simulation of the entire manufacturing process. The surrogate involves a combination of particle models, the molecular model of the polymer, and a coarse-scale model of the polymer as a nonlinear hyperelastic material. Coefficients for the nonlinear elastic continuum model are determined using numerical experiments on representative volume elements of the polymer model. Furthermore, a simple model of initial strain is incorporated in the continuum equations to model the inherit shrinking of the A coupled particle and continuum model is constructed using a special algorithm designed to provide constraints on a region of overlap between the continuum and particle models. This coupled model is based on the so-called Arlequin method that was introduced in the context of coupling two continuum models with differing levels of discretization. It is shown that the Arlequin problem for the particle-tocontinuum model is well posed in a one-dimensional setting involving linear harmonic springs coupled with a linearly elastic continuum. Several numerical examples are presented. Numerical experiments in three dimensions are also discussed in which the polymer model is coupled to a nonlinear elastic continuum. Error estimates in local quantities of interest are constructed in order to estimate the modeling error due to the approximation of the particle model by the coupled multiscale surrogate model. The estimates of the error are computed by solving an auxiliary adjoint, or dual, problem that incorporates as data the quantity of interest or its derivatives. The solution of the adjoint problem indicates how the error in the approximation of the polymer model inferences the error in the quantity of interest. The error in the quantity of interest represents the relative error between the value of the quantity evaluated for the base model, a quantity typically unavailable or intractable, and the value of the quantity of interest provided by the multiscale surrogate model. To estimate the error in the quantity of interest, a theorem is employed that establishes that the error coincides with the value of the residual functional acting on the adjoint solution plus a higher-order remainder. For each surrogate in a sequence of surrogates generated, the residual functional acting on various approximations of the adjoint is computed. These error estimates are used to construct an adaptive algorithm whereby the model is adapted by supplying additional fine-scale data in certain subdomains in order to reduce the error in the quantity of interest. The adaptation algorithm involves partitioning the domain and selecting which subdomains are to use the particle model, the continuum model, and where the two overlap. When the algorithm identifies that a region contributes a relatively large amount to the error in the quantity of interest, it is scheduled for refinement by switching the model for that region to the particle model. Numerical experiments on several configurations representative of nano-features in semiconductor device fabrication demonstrate the effectiveness of the error estimate in controlling the modeling error as well as the ability of the adaptive algorithm to reduce the error in the quantity of interest. There are two major conclusions of this study: 1. an effective and well posed multiscale model that couples particle and continuum models can be constructed as a surrogate to molecular statics models of polymer networks and 2. an error estimate of the modeling error for such systems can be estimated with sufficient accuracy to provide the basis for very effective multiscale modeling procedures. The methodology developed in this study provides a general approach to multiscale modeling. The computational procedures, computer codes, and results could provide a powerful tool in understanding, designing, and optimizing an important class of semiconductormanufacturing processes. The study in this dissertation involves all three components of the CAM graduate program requirements: Area A, Applicable Mathematics; Area B, Numerical Analysis and Scientific Computation; and Area C, Mathematical Modeling and Applications. The multiscale modeling approach developed here is based on the construction of continuum surrogates and coupling them to molecular statics models of polymer as well as a posteriori estimates of error and their adaptive control. A detailed mathematical analysis is provided for the Arlequin method in the context of coupling particle and continuum models for a class of one-dimensional model problems. Algorithms are described and implemented that solve the adaptive, nonlinear problem proposed in the multiscale surrogate problem. Large scale, parallel computations for the base model are also shown. Finally, detailed studies of models relevant to applications to semiconductor manufacturing are presented.Item An algorithm for hyperbolic geometry(Texas Tech University, 1998-12) Tinney, Pheobe Alexis SamuelsNot availableItem Application of neural network control to distillation(Texas Tech University, 1997-05) Dutta, PriyabrataDistillation control is challenging due to its coupled, nonlinear, nonstationary, and slow dynamic behavior. Like distillation columns, most chemical processes are usually nonlinear and nonstationary. This nonlinearity greatly limits the effectiveness of linear controllers, especially when the process is operated away from the nominal operating region. Nonlinear controllers, based on phenomenological models, can be developed. However, it is still a very difficult task in real practice, in terms of computational power, to implement these controllers on-line, because the entire model needs to be solved within each control interval. Neural networks give us an alternative approach to model a nonlinear process, and a controller based on this model can overcome the issues of on-line computational problems. Besides nonlinearity, many practical control problems possess constraints on the input, state, and output variables. The ability to handle constraints is essential for any algorithm to be implemented on real processes. Thus strategies for constraint handling within model-based controllers have become one of the more popular research topics. In this dissertation, a constrained optimization technique for control which uses a neural network gain prediction approach has been developed and implemented on a laboratory distillation column as well as on a dynamic simulator. Here, the neural networks are trained based on a phenomenological model. Also, experimental results have been obtained to confirm the applicability of a neural network model-based controller using an inverse of a state-prediction approach that was developed and simulated earlier by Ramchandran and Rhinehart (1994). In addition, two separate single-input-singleoutput (SISO) controllers using the inverse of the state-prediction approach are implemented on the feed and reflux preheaters of the column.Item A Boundary Element Method for the strongly nonlinear analysis of ventilating water-entry and wave-body interaction problems(2009-08) Vinayan, Vimal; Kinnas, Spyros A.A two-dimensional Boundary Element Method (BEM) is developed to study the strongly nonlinear interaction between a surface-piercing body and the free-surface. The scheme is applied to problems with and without the possibility of ventilation resulting from the motion and geometric configuration of the surface-piercing body. The main emphasis of this research work is on the development of numerical methods to improve the performance prediction of surface-piercing propellers by including the whole range of free-surface nonlinearities. The scheme is applied to predict the ventilated cavity shapes resulting from the vertical and rotational motion of a blade-section with fully nonlinear free-surface boundary conditions. The current method is able to predict the ventilated cavity shapes for a wide range of angles of attack and Froude numbers, and is in good agreement with existing experimental results. Through a comparison with a linearized free-surface method, the current method highlights the shortcomings of the negative image approach used commonly in two-dimensional and three-dimensional numerical methods for surface-piercing hydrofoils or propellers. The current method with all its capabilities makes it a unique contribution to improving numerical tools for the performance prediction of surface-piercing propellers. The scheme is also applied to predict the roll and heave dynamics of two-dimensional Floating Production Storage and Offloading (FPSO) vessel hull sections within a potential flow framework. The development of the potential flow model is aimed at validating the free-surface dynamics of an independently developed Navier Stokes Solver for predicting the roll characteristics of two-dimensional hull sections with bilge keels.Item Deterministic and stochastic nonlinear age-structured models(Texas Tech University, 1998-12) Block, Garry L.The Leslie age-structured population model is reviewed, as well as its stochastic analogue. Nonlinear adjustments in the form of the Ricker and Beverton-Holt densitydependent factors are made to these models. The deterministic density-dependent nonlinear models are discussed and compared to their stochastic nonlinear analogs.Item Development of consistent nonlinear models of flexible body systems(Texas Tech University, 1998-12) Eskridge, Steven ENot availableItem Facility planning and value of information using a tank reservoir model : a case study in reserve uncertainty(2010-05) Singh, Ashutosh; Jablonowski, Christopher J.; Groat, Charles G.This thesis presents a methodology to incorporate reservoir uncertainties and estimate the loss in project value when facility planning decisions are based on erroneous estimates of input variables. We propose a tank model along with integrated asset development model to simulate the concept selection process. The model endogenizes drilling decisions and includes an option to expand. Key decision variables included in the model are number of pre-drill wells, initial facility capacity and number of well slots. Comparison is made between project value derived under erroneous estimates for reserve size and under an alternate hypothesis. The results suggest loss in project value of up to 40% when reservoir estimates are erroneous. Moreover, both optimistic and pessimistic reserve estimates results in a loss in project value. However, loss in project value is bigger when reserve size is underestimated than when it is overestimated.Item Feasibility of isotropic inversion in orthorhombic media : the Barrett unconventional model(2016-05) Yanke, Andrew James; Spikes, Kyle; Sen, Mrinal K; Fomel, Sergey BGeophysicists often relegate shale reservoirs as having higher symmetries (e.g., transversely isotropic (TI) or isotropic) than what reality demonstrates. Routine application of TI (or even isotropic) algorithms to orthorhombic media neglects the associated errors because we never know the true model in practice. This thesis evaluates the viability of isotropic post-stack and pre-stack seismic inversion to orthorhombic media using the SEAM Barrett Unconventional Model, the most realistic depositional model to date. The Barrett Model contains buried topography, simulated stratigraphy, and designated reservoir zones with orthorhombic anisotropy. I inverted the Barrett data volume for isotropic elastic property cubes, which I compared to the model volume in each symmetry-plane of an orthorhombic medium. If the stacked seismic data contained only the near offsets, post-stack inversion resolved acoustic impedances that closely matched the true model both within and outside of the reservoir zones at all well locations. Anisotropy most affected the far offsets, so muting them predictably enhanced the post-stack inversion. I maintained all offsets for pre-stack inversion, but a parabolic radon filter eliminated nonhyperbolic behavior (rather than nonhyperbolic moveout analysis) at far offsets. The pre-stack impedance attributes adequately described the vertical heterogeneity of the true model at a cross-validation well, but the inverted values increasingly relied on the initial model with depth. The inverted density estimates experienced notable oscillations relative to the initial model, particularly where steep contrasts in elastic properties occurred. Mismatch of the inverted elastic properties at the well locations can be attributed to noise, thin layering effects, band limitation, steep contrasts in elastic properties, AVO behavior stacked into the data, an inaccurate starting model, and the effects of anisotropy. The most significant sources of error include small-scale reflectivity and comprehensive filtering of nonhyperbolic phenomena. Away from the well locations, the isotropic inversion gave no visual indication of reservoir geobodies, but it sufficiently described the elastic property variations near reservoir mid-sections. Moreover, I showed that the inverted elastic properties differ from their orthorhombic models by no more than 35%. The greatest misfits occurred near reservoir contacts and geobody locations. The computed impedance models in each symmetry-plane have distinctive differences, but isotropic inversion dismisses these variations entirely. I conclude that isotropic inversion should not be a surrogate for orthorhombic methods in data preconditioning and quantitative reservoir characterization.Item Investigation of flow streamlines of partially hydrolyzed polyacrylamide solutions through cylindrical porous media(Texas Tech University, 1984-05) Baumgarten, Gary ANot availableItem Light driven microactuators : design, fabrication, and mathematical modeling(2009-08) Han, Li-Hsin; Chen, ShaochenThis dissertation is concerned with design, fabrication, and mathematical modeling of three different microactuators driven by light. Compared to electricity, electromagnetic wave is a wireless source of power. A distant light source can be delivered, absorbed, and converted to generate a driving force for a microactuator. The study of light-driven microsystems, still at its early stage, is already expanding the horizon for the research of microsystems. The microactuators of this dissertation include micro-cantilevers driven by pulsed laser, photo-deformable microshells coated with gold nanospheres, and a nano-particles coated micro-turbine driven by visible light. Experimental investigation and theoretical analysis of these microactuators showed interesting results. These microactuators were functioned based on cross-linked, multiple physics phenomenon, such as photo-heating, thermal expansion, photo-chemistry effect, plasomonics enhancement, and thermal convection in rarefied gas. These multiple physics effects dominate the function of a mechanical system, when the system size becomes small. The modeling results of the microactuators suggest that, to simulate a microscale mechanical system accurately, one has to take account the minimum dimension of the system and to consider the validity of a theoretical model. Examples of the building of different microstructures were shown to demonstrate the capacity of a digital-micromirror-device (DMD) based apparatus for three-dimensional, heterogeneous fabrication of polymeric microstructures.Item Mathematical modeling and kinematics: a study of emerging themes and their implications for learning mathematics through an inquiry-based approach(2004) Carrejo, David John; Marshall, Jill Ann; Petrosino, Anthony J. (Anthony Joseph), 1961-Item Optimization of the TIEC/AMTEC cascade cell(Texas Tech University, 1998-12) Malka, Vivek RaoA mathematical modeling of a system consisting of a cascade of a thermionic energy conversion (TIEC) device and an alkali metal thermal to electric converter (AMTEC) device has been performed to optimize the efficiency of the cell. The TIEC is heated by electron bombardment, which converts heat partially into electricity and rejects the remaining. The AMTEC utilizes this reject heat of the TIEC. Cascading these two cells provides lots of advantages. A mathematical model of the cascade converter has been developed to analyze the effects of key parameters such as power level, heat fluxes, and temperatures. In this effort, a 12-node system of nonlinear simultaneous equations has been constmcted which is solved by MATHCAD and a locally optimized efficiency has been derived. Thus, efficiency of the cascaded cell is improved, so that it is greater than the highest efficiency among the TIEC and AMTEC and lower than the sum of their individual efficiencies. The results were compared with the previous program written for the same problem.Item Representation of control systems using Java(Texas Tech University, 1998-12) Conn, James NealNot availableItem Size and shape effects for the nano/micro particle dynamics in the microcirculation(2010-08) Lee, Sei Young; Moser, Robert deLancey; Ferrari, Mauro, 1959-; Decuzzi, Paolo; Chen, Shaochen; Hidrovo, CarlosThe nano/micro particles have been widely used as a carrier of therapeutic and contrast imaging agents. The nano/micro particles have many advantages, such as, specificity, controlled release, multifunctionality and engineerability. By tuning the chemical, physical and geometrical properties, the efficacy of delivery of nano/micro particle can be improved. In this study, by analyzing the effect of physical and geometrical properties of particle, such as, size, shape, material property and flow condition, the optimal condition for particle delivery will be explored. The objectives of this study are (1) to develop predictive mathematical models and (2) experimental models for particle margination and adhesion, and (3) to find optimal particle geometry in terms of size and shape to enhance the efficiency of its delivery. The effect of particle size expressed in terms of Stokes number and shape, namely, spherical, ellipsoidal, hemispherical, discoidal and cylindrical particle on the particle trajectory is investigated. For discoidal and cylindrical particles, the effect of aspect ratio is also considered. To calculate particle trajectory in the linear shear flow near the substrate, Newton's law of motion is decomposed into hydrodynamic drag and resistance induced by particle motion. The drag and resistance is estimated through finite volume formulation using Fluent v6.3. Particle behavior in the linear shear flow does strongly depend on Stokes number. Spherical particle is transported following the streamline in the absence of external body force. However, non-spherical particles could across the streamline and marginate to the substrate. For non-spherical particles, the optimal [Stokes number] in terms of particle margination is observed; [Stokes number almost equal to] 20 for ellipsoidal, hemispherical and discoidal particle; [Stokes number almost equal to] 10 for cylindrical particle. For discoidal particle with [gamma subscript d]=0.2 shows fastest margination to the substrate. The effect of gravitational force is also considered with respect to the fluid direction. When the gravitational force is applied, mostly, gravitational force plays a dominant role for particle margination. However, using small particle aspect ratio ([gamma subscript d]=0.2 and 0.33), spontaneous drift induced by particle-fluid-substrate interaction could overcome gravitational effect in some cases ([Stokes number]=10, G=0.1). In addition the adhesion characteristic of spherical particle has been studied using in vitro micro fluidic chamber system with different particle size and flow condition. The experimental results are compared to the mathematical model developed by Decuzzi and Ferrari (Decuzzi and Ferrari, 2006) and in vivo test (Decuzzi et al., 2010). The optimal particle size for S=75 and 90 is found to be 4-5 [micrometer] through the in vitro non-specific interaction of spherical particle on the biological substrate. The suggested mathematical model has proven to be valid for current experimental condition. At the end, the mathematical model, in vitro flow chamber results and in vivo test have been compared and the scaling law for particle adhesion on the vessel wall has been confirmed.Item The dynamics and control of a planar articulating segmental model(Texas Tech University, 2000-08) Meador, C. DougA planar model of the human forearm complex is presented. The upper arm (humerus) is fixed in a position parallel to the body and the forearm (radius and ulna) is free to move in the sagittal plane. Two Hill-type actuators drive the motion of the forearm. These actuators represent flexors and extensors of the forearm. A stability analysis about a nominal trajectory is performed. A feedback scheme for stabilization is developed by using the linear quadratic regulator method. Finally, questions are answered concerning the role of muscle feedback during skeletal movements by analyzing the feedback gains.Item The economics of precision farming in the Texas High Plains(Texas Tech University, 2002-12) Watson, Susan ElizabethNot available.Item The effects of sedimentary basins on the dynamics of the East Antarctic Ice Sheet from enhanced groundwater and geothermal heat flow(2016-05) Gooch, Bradley Tyler; Blankenship, Donald D.; Dalziel, Ian W; Ghattas, Omar; Hesse, Marc A; Young, Duncan AIt is well known that ice sheets heavily influence groundwater systems, however, the impact of groundwater on ice sheet dynamics is not. This poorly understood aspect of ice-sheet hydrology is relevant to the subglacial hydrology of ice sheets lacking surface or englacial meltwater such as the East Antarctic Ice Sheet (EAIS). How groundwater systems redistribute geothermal heat at the base of an ice sheet is also largely unknown. Geothermal heat and subglacial hydrology are important basal processes controlling ice flow. Large sedimentary basins underlie the EAIS, which likely play host to many groundwater systems. I hypothesized that groundwater systems in these sedimentary basins may be the main water transport mechanism over water sheets (or films) at large scales in the interior of the ice sheet where basal melt rates are very low. I also hypothesized that these groundwater systems are likely important to the basal processes (specifically heat flux) and dynamics of the EAIS (particularly in rheological and sliding behavior). To test these, I created various one- and two-dimensional numerical models incorporating relevant datasets and conservative assumptions about the subsurface. The models ranged from simple groundwater and thermal simulations to a complex subsurface fluid and thermal model coupled to a fully dynamic ice sheet simulator. The models suggest that groundwater most likely has measurable effects on the dynamics of ice sheets like the EAIS. I have shown that probable groundwater systems underneath the interior of the EAIS can likely transport most of the meltwater produced and that groundwater can strongly affect the heat flux (positively, as well as, negatively) at the ice base under kilometers of relatively slow-moving ice. I have also not only shown that groundwater systems under the EAIS are strongly controlled by the ice sheet’s dynamics but that groundwater systems have a feedback to the ice dynamics, mostly through enhanced basal sliding and changes to the ice rheology. These results provide the justification to include groundwater in future simulations of the EAIS as well as a call to collect more data to better delineate its subsurface sedimentary basins – a critical input for groundwater and heat transport modeling.Item Two-person games for stochastic network interdiction : models, methods, and complexities(2009-12) Nehme, Michael Victor; Morton, David P.We describe a stochastic network interdiction problem in which an interdictor, subject to limited resources, installs radiation detectors at border checkpoints in a transportation network in order to minimize the probability that a smuggler of nuclear material can traverse the residual network undetected. The problems are stochastic because the smuggler's origin-destination pair, the mass and type of material being smuggled, and the level of shielding are known only through a probability distribution when the detectors are installed. We consider three variants of the problem. The first is a Stackelberg game which assumes that the smuggler chooses a maximum-reliability path through the network with full knowledge of detector locations. The second is a Cournot game in which the interdictor and the smuggler act simultaneously. The third is a "hybrid" game in which only a subset of detector locations is revealed to the smuggler. In the Stackelberg setting, the problem is NP-complete even if the interdictor can only install detectors at border checkpoints of a single country. However, we can compute wait-and-see bounds in polynomial time if the interdictor can only install detectors at border checkpoints of the origin and destination countries. We describe mixed-integer programming formulations and customized branch-and-bound algorithms which exploit this fact, and provide computational results which show that these specialized approaches are substantially faster than more straightforward integer-programming implementations. We also present some special properties of the single-country case and a complexity landscape for this family of problems. The Cournot variant of the problem is potentially challenging as the interdictor must place a probability distribution over an exponentially-sized set of feasible detector deployments. We use the equivalence of optimization and separation to show that the problem is polynomially solvable in the single-country case if the detectors have unit installation costs. We present a row-generation algorithm and a version of the weighted majority algorithm to solve such instances. We use an exact-penalty result to formulate a model in which some detectors are visible to the smuggler and others are not. This may be appropriate to model "decoy" detectors and detector upgrades.