Browsing by Author "Alina Chertock, Committee Member"

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Dimension Reduction: Modeling and Numerical Analysis of Two Applied Problems.
(2010-09-17) Collins, James; Pierre Gremaud, Committee Chair; Alina Chertock, Committee Member; Michael Shearer, Committee Member; Ralph Smith, Committee Member; Juei Tu, Committee Member
Dynamic Microstructural Characterization of High Strength Aluminum Alloys
(2008-05-09) Lee, William Morgan; Alina Chertock, Committee Member; Kara Peters, Committee Member; Mohammed A. Zikry, Committee Chair
The use of aluminum alloys for commercial and military applications has increased substantially due to the alloys' low areal density, toughness, and processability. It has recently been shown that an aluminum alloy, Al 2139, with copper, magnesium, and silver can be significantly toughened and strengthened by combinations of θ' and Ω precipitates and dispersed manganese particles. What has not been quantified are how these precipitates and dispersed particles affect behavior and what the material mechanisms and microstructural characteristics are that control the behavior of Al 2139 for strain-rates that span the quasi-static to high rates of strain. Hence, in this investigation, detailed transmission electron microscopy (TEM), scanning transmission electron microscopy (STEM), scanning electron microscopy (SEM), orientation imaging microscopy (OIM), and optical microscopy (OM) were used to delineate the different physical scales that range from the nano for the precipitates and dislocations to the micron for the dispersed particles, grain orientations and texture, grain-sizes, slip-bands, and grain-boundary orientations. The deformed specimens were from an Al 2139 plate that was impacted by 4340 steel fragmentation stimulating projectiles (FSPs) at impact velocities ranging from 813 to 1043 m⁄s. The majority of the projectiles were defeated by the Al 2139 plate, which is another indication of the alloy's potential for damage mitigation and projectile defeat and resistance. Based on this detailed microstructural characterization, mechanisms for projectile defeat and full penetration are proposed. Deformation and damage modes include petalling on the impact face, shear cracking through the middle section of the plate due to projectile penetration, and discing due to bending stresses at a spall plane near the back of the plate. Shear cracking appears to be GB related, and the discing is dependent on the rolling direction. The extent of these modes for cross-sections where the target was penetrated was greater than that in regions where the projectile was defeated. For projectile defeat, large and elongated grains and precipitate deformation due to dislocation interaction can lead to highly ductile performance, which resists discing failure and plate penetration. Large grains significantly reduce the fraction of GBs, which then reduces the amount of GB cracking due to intense shear accumulation and spall. The elongation of the grains due to rolling also increased the dislocation densities, and subsequently the ductility of the grains, which reduced tensile failure due to the bending in the discing regions. High angle GB's can also limit heterogeneous θ' precipitation at the GB's, which would reduce intergranular fracture. Precipitation of Ω also increases the spall strength and decreases localized shear through its multiple cutting interactions with dislocations at the matrix interface. Dispersed particles also increase the strength of the alloy in high strain-rate applications by resisting localized shear. The results of this study are a first step in developing a tailored methodology that can be used to optimize microstructural characteristics and behavior of aluminum alloys for optimal strength and toughness.
Flow of Thin Liquid Films with Surfactant: Analysis, Numerics, and Experiment.
(2010-06-28) Peterson, Ellen; Michael Shearer, Committee Chair; Karen Daniels, Committee Member; Mansoor Haider, Committee Member; Alina Chertock, Committee Member
Inverse Problems of Matrix Data Reconstruction
(2010-03-26) Lin, Matthew Min-Hsiung; Moody Chu, Committee Chair; Pierre Gremaud, Committee Member; Alina Chertock, Committee Member; Dmitry Zenkov, Committee Member
Mathematical modeling is an indispensable task in almost every discipline of sciences. If a model for a specific phenomenon can be correctly established, then it empowers the practitioners to analyze, predict, and delegate an onward decision which may have important applications and consequences. However, since most of the information gathering devices or methods, including our best intellectual endeavor for understanding, have only ï¬ nite bandwidth, we cannot avoid the fact that the models employed often are not exact. A constant update or modification of an existing model based on the newest information therefore is in demand. Such a task is generally referred to as an inverse problem. While in a forward problem the concern usually is to express the behavior of a certain physical system in terms of its system parameters, in the inverse problem the concern is to express the parameters in term of the behavior. This thesis addresses a small portion of the mass domain of inverse problems. The specific focus has been on matrix data reconstruction subject to some intrinsic or prescribed constraints. The purpose of this investigation is to develop theoretic understanding and numerical algorithms for model reconstruction so that the inexactness and uncertainty are reduced while certain specific conditions are satisfied. Explained and illustrated in this thesis are some most frequently used methodologies of matrix data reconstruction so that for a given dataset, these techniques can be employed to construct or update various (known) structural properties, or to classify or purify certain (unknown) embedded characteristics. Areas of applications include, for example, the applied mechanics where systems of bodies move in response to the values of their known endogenous parameters and the medical or social sciences where the causes (variables) of the observed incidences neither are known a priori nor can be precisely quantified. All of these could be considered as an inverse problem of matrix data reconstruction. This research revolves around two specific topics â€“ quadratic inverse eigenvalue problems and low rank approximations â€“ and some other related problems, both in theory and in computation. An immediate and the most straightforward application of the quadratic inverse eigenvalue problem would be the construction of a vibration system from its observed or desirable dynamical behavior. Its mathematical model is associated to the quadratic matrix polynomial Q(Î») = M Î»^2 + C Î» + K whose eigenvalues and eigenvectors govern the vibrational behavoir. Tremendous complexities and difficulties in recovering cofficient matrices M , C , K arise when the predetermined inner-connectivity among its elements and the mandatory nonnegativity of its parameters must be taken into account. Considerable efforts have been taken to derive theory and numerical methods for solving inverse eigenvalue problems, but techniques developed thus far can handle the inverse problems only on a case by case basis. The ï¬ rst contribution in this investigation is an efficient, reliable semi-definite programming technique for inverse eigenvalues problems subject to specified spectral and structural constraints. Of particular concern is the issue of inexactness of the prescribed or measured eigeninformation, which is almost inevitable in practice, since inaccurate data will affect the solvability of this inverse eigenvalue problem. To address this issue, a second contribution in this investigation is a methodical approach by using the notion of truncated QR decomposition to ï¬ rst determine whether a nearby inverse problem is solvable and, if it is so, the method computes the approximate coefficient matrices while providing an estimate of the residual error. Both methods enjoy the advantages of preserving inter-connectivity structures and other important properties embedded in the original problems. More importantly, both approaches allow more ï¬‚exibilities and robustness in handling highly structured problems than other special-purpose algorithms. Low rank approximations have become increasingly important and ubiquitous in this era of information. Generally, there is no uniï¬ ed approach because the technique often is data type dependent. This research studies and proposes new factorization techniques for three different type of data. The ï¬ rst algorithm aims to perform a nonnegative matrix factorization of a nonnegative data matrix by recasting the problem geometrically as the approximation of a given polytope on the probability simplex by a simpler polytope with fewer facets. This view leads to a convenient way of decomposing the data by computing the proximity map which, in contrast to most existing algorithm where only an approximate map is used, ï¬ nds the unique and global minimum per iteration. The second algorithm investigates the factorization of integer matrices which is more realistic and important in informatics. Searching through the literature, it appears that there does not exist a suitable algorithm which can handle this type of problem well owing to its discrete nature. Two effective approaches for computing integer matrix factorization are proposed in this investigation â€” one is based on hamming distance and the other on Euclidean distance. A lower rank approximation of a matrix A âˆˆ Z^{mÃ—n} â‰ˆ U V with factors U âˆˆ Z_2^{mÃ—k}, V âˆˆ Z^{kÃ—n} , where columns of U are mutually exclusive and integer k < min{m, n} is given. The third algorithm concerns expressing a nonnegative matrix as the shortest sum of nonnegative rank one matrices, the so called nonnegative rank factorization. Till now, only a few abstract results which are somewhat too conceptual for numerical implementation have been developed in the literature. Employing the Wedderburn rank reduction formula, a numerical procedure detecting whether a nonnegative rank factorization exists is presented. In the event that such a factorization does not exist, it is able to compute the maximum nonnegative rank splitting. This thesis includes a detailed analysis of inverse eigenvalue problems and low rank factorizations. Some of the theories are classical, but new insights are obtained and their implementation for numerical computation are developed. On the other hand, this investigation leads to quite a few innovative algorithms which are effective and robust in tackling the otherwise very difficult inverse problems. The research is ongoing and several interesting research problems are identified in this thesis.
A Novel Hybrid Scheme for Large Eddy Simulation of Turbulent Combustion Based on the One-Dimensional Turbulence Model
(2006-08-31) Cao, Shufen; Tarek Echekki, Committee Chair; Alina Chertock, Committee Member; William Roberts, Committee Member; Jack Edwards, Committee Member
A hybrid numerical scheme based on large eddy simulation (LES) and the one-dimensional turbulence (ODT) model for turbulent combustion is developed and validated. The ODT model resolves, both temporally and spatially, subgrid scale processes such as mixing, molecular transport, and chemistry. This model addresses the limitations of traditional models in representing strong local and transient phenomena such as ignition or extinction and processes strongly dependent on cross-correlations of different scalars. The ODT model formulation and numerical implementation involves the treatment of different processes governing the transport and chemistry for scalars and momentum through a combination of stochastic and deterministic solutions, which are implemented in parallel on the ODT domains. These domains are embedded in the LES computational domain. The ODT-based and the LES solutions provide a coupled set of solutions for scalars and momentum with redundancy in the way these quantities can be computed. The key processes included in the proposed formulation are: molecular processes consisting of reaction and diffusion, turbulent stirring, and filtered convection. In the present study, turbulent stirring is represented by random, instantaneous rearrangements of the fields of transported variables along a one-dimensional line via 'triplet maps', which emulate the rotational folding effects of turbulent eddies. Molecular diffusion and chemistry are solved deterministically through finite-difference solutions of the unsteady reaction-diffusion transport equation along the 1D domain. A novel method to incorporate 3D convection in ODT, denoted as 'node convection' combined with 'intra-node relaxation', is implemented. The Smagorinsky model is used as a subgrid stress closure model for LES. The coupling of LES and ODT is accomplished spatially by interpolating velocity information from LES to ODT and temporally at each LES time step. The problem of non-homogeneous autoignition in isotropic turbulence is used to validate the proposed model. This problem offers a stringent test for the proposed model because it exhibits different modes of combustion (from ignition kernels to premixed and non-premixed flames) and a complex coupling between turbulent transport and molecular processes, diffusion and reaction, under highly transient conditions. The validation is carried out in comparison of the LES-ODT results with results from Direct Numerical Simulations (DNS). Both low and high turbulence conditions are considered, with three Lewis number cases carried out for the high turbulence condition. Both volume-averaged statistics and mixture fraction-conditioned statistics show that LES-ODT is able to accurately predict not only the flame ignition and extinction, kernel propagation, transition between different burning modes, but also the turbulence and Lewis number effects. LES-ODT simulation results are in excellent agreement with DNS results. This is achieved with a significantly reduced computational cost compared to DNS.
Numerical Study of Two Problems in Fluid Flow: Cavitation and Cerebral Circulation
(2008-03-11) DeVault, Kristen Jean; Ralph Smith, Committee Member; Alina Chertock, Committee Member; C. Tim Kelley, Committee Member; Pierre Gremaud, Committee Chair; Mette S. Olufsen, Committee Member
Two different computational models of fluid flow are considered. First, the possibility of cavitation is investigated numerically in two and three dimensions for the spherically symmetric, barotropic, Navier-Stokes equations. A splitting method is derived in order to allow the use of known solutions to the corresponding inviscid Euler equations. Results indicate cavitation is possible in the presence of high Mach numbers. This work is intended to be a stepping off point in the search for analytic solutions showing cavitation in multi-dimensional compressible flows. Second, a blood flow model for circulation in the Circle of Willis (CoW) is derived. It is calibrated using ensemble Kalman filtering and validated against clinical data. The resulting model is then used to predict the effects of common anatomical variations within the CoW on blood perfusion in the brain, both under normal circumstances and in the event of a stroke.
Particle Size Segregation In Granular Avalanches: A Study In Shocks.
(2010-08-03) Giffen, Nicholas; Michael Shearer, Committee Chair; Ralph Smith, Committee Member; Pierre Gremaud, Committee Member; Alina Chertock, Committee Member