# Browsing by Subject "Numerical analysis"

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Item A comparative analysis of flight crew vacation allocation models(2019-05) Liang, Zhaowei; Kutanoglu, ErhanShow more The airline industry has a long lasting history of using operations research for complex problems like crew scheduling, crew pairing, and aircraft tail assignment. However, the use of the optimization and operations research on crew vacation planning is not widespread. One of the most popular ways of assigning vacations currently is to let crew members bid for vacations using a heuristic preferential bidding system (PBS). This report will overview the existing problems in the crew vacation allocation domain. Then, it will introduce and compare an optimization based vacation allocation algorithm, an improved heuristic PBS model, and the original heuristic PBS model. Models will be compared using three performance measures as the number of unassigned vacation blocks, the number of crew members without any assigned vacation blocks, and the rank order of the preferences that are awarded to the crew members. This report also conducts a sensitivity analysis for the improved PBS model using the same performance measuresShow more Item A computational study of femtosecond laser sintering of copper nanoparticles(2021-12-07) Kim, Jaewoo, M.S. in Engineering; Wang, YaguoShow more Femtosecond Selective Laser Sintering (fs-SLS) has much advantage of finer resolution of 3D printing as the Heat Affected Zone (HAZ) of the irradiated samples is much narrower comparing to other laser sources. Recent studies report that fs- SLS technique suffers from ablation problem. The ultrafast nature of femtosecond laser causes different ablation mechanism that are not observed in nanosecond and continuous wave lasers. The theoretical understanding of the femtosecond laser ablation mechanisms of MNPs is the first step to develop a novel technique of fs-SLS. In this study, we develop a Two Temperature based ablation model considering two possible mechanisms; Thermal stress and Coulomb Explosion. We conduct simulation studies in different sizes of MNPs (10 nm, 100 nm and 1 μm) and investigate which ablation mechanism is dominant over the other in different particle sizes. Under high sintering fluences, 10 nm particles is both vulnerable to the thermal stress and Coulomb explosion while thermal stress is the dominant ablation mechanism in 100 nm and 1 μm. However, in lower fluence, the Coulomb Explosion is the dominant in 10 nm MNPs. We also found that there is a limit of ablation depth with Coulomb Explosion. After that, we conduct the double pulse sintering (DPS) simulation in 100 nm MNPs, and it shows that a significant reduction of thermal stress can be observed while there is no sign of suppression of Coulomb Explosion. We suggest that DPS could be the effective fs-SLS technique for the better sintering of MNPsShow more Item Analysis of a novel thermoelectric generator in the built environment(2011-08) Lozano, Adolfo; Webber, Michael E., 1971-; Schmidt, Philip S.Show more This study centered on a novel thermoelectric generator (TEG) integrated into the built environment. Designed by Watts Thermoelectric LLC, the TEG is essentially a novel assembly of thermoelectric modules whose required temperature differential is supplied by hot and cold streams of water flowing through the TEG. Per its recommended operating conditions, the TEG nominally generates 83 Watts of electrical power. In its default configuration in the built environment, solar-thermal energy serves as the TEG’s hot stream source and geothermal energy serves as its cold stream source. Two systems-level, thermodynamic analyses were performed, which were based on the TEG’s upcoming characterization testing, scheduled to occur later in 2011 in Detroit, Michigan. The first analysis considered the TEG coupled with a solar collector system. A numerical model of the coupled system was constructed in order to estimate the system’s annual energetic performance. It was determined numerically that over the course of a sample year, the solar collector system could deliver 39.73 megawatt-hours (MWh) of thermal energy to the TEG. The TEG converted that thermal energy into a net of 266.5 kilowatt-hours of electricity in that year. The second analysis focused on the TEG itself during operation with the purpose of providing a preliminary thermodynamic characterization of the TEG. Using experimental data, this analysis found the TEG’s operating efficiency to be 1.72%. Next, the annual emissions that would be avoided by implementing the zero-emission TEG were considered. The emission factor of Michigan’s electric grid, RFCM, was calculated to be 0.830 tons of carbon dioxide-equivalent (CO2e) per MWh, and with the TEG’s annual energy output, it was concluded that 0.221 tons CO2e would be avoided each year with the TEG. It is important to note that the TEG can be linearly scaled up by including additional modules. Thus, these benefits can be multiplied through the incorporation of more TEG units. Finally, the levelized cost of electricity (LCOE) of the TEG integrated into the built environment with the solar-thermal hot source and passive ground-based cold source was considered. The LCOE of the system was estimated to be approximately $8,404/MWh, which is substantially greater than current generation technologies. Note that this calculation was based on one particular configuration with a particular and narrow set of assumptions, and is not intended to be a general conclusion about TEG systems overall. It was concluded that while solar-thermal energy systems can sustain the TEG, they are capital-intensive and therefore not economically suitable for the TEG given the assumptions of this analysis. In the end, because of the large costs associated with the solar-thermal system, waste heat recovery is proposed as a potentially more cost-effective provider of the TEG’s hot stream source.Show more Item Conservative spectral methods for Fokker-Planck-Landau type equations : simulations, long-time behaviour and error estimates(2020-05-08) Pennie, Clark Alexander; Martinez Gamba, Irene, 1957-; Chen, Thomas; Morrison, Philip; Ren, Kui; Tsai, Yen-HsiShow more The focus of this thesis is to investigate a conservative spectral method for solving Fokker-Planck-Landau type (F. P. L.) equations as a model for plasmas, when coupled to the Vlasov-Poisson equation in the mean-field limit, modelling particle interactions extending from Coulomb to hard sphere potentials. This study will range from numerical examples, that emphasise the strength and accuracy of the method, to a rigorous proof showing that approximations from the numerical scheme converge to analytical solutions. In particular, two sets of novel simulations are included. The first presents benchmark results of decay rates to statistical equilibrium in transport plasma models for Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The other studies the essentially unexplored phenomenon of the plasma sheath for Coulomb interactions, exhibiting the formation of a strong field due to charge separation. These topics will be arranged in three major projects: 1. Entropy decay rates for the conservative spectral scheme modelling Fokker-Planck-Landau type flows in the mean field limit. Benchmark simulations of decay rates to statistical equilibrium are created for F.P.L. equations associated to Coulomb particle interactions, as well as with Maxwell type and hard sphere interactions. The qualitative decay to the equilibrium Maxwell-Boltzmann distribution is studied in detail through relative entropy for all three types of particle interactions by means of a conservative hybrid spectral and discontinuous Galerkin scheme, adapted from Chenglong Zhang’s thesis in 2014. More precisely, the Coulomb case shows that there is a degenerate spectrum, with a decay rate close to the law of two thirds predicted by upper estimates in work by Strain and Guo in 2006, while the Maxwell type and hard sphere examples both exhibit a spectral gap as predicted by Desvillettes and Villani in 2000. Such decay rate behaviour indicates that the analytical estimates for the Coulomb case is sharp while, still to this date, there is no analytical proof of sharp degenerate spectral behaviour for the F.P.L. operator. Simulations are presented, both for the space-homogeneous case of just particle potential interactions and the space-inhomogeneous case with the mean field coupling through the Poisson equation for total charges in periodic domains. New explicit derivations of spectral collisional weights are presented in the case of Maxwell type and hard sphere interactions and the stability of all three scenarios, including Coulomb interactions, is investigated. 2. Convergence and error estimates for the conservative spectral method for Fokker-Planck-Landau equations. Error estimates are rigorously derived for a semi-discrete version of the conservative spectral method for approximating the space-homogeneous F.P.L. equation associated to hard potentials. The analysis included shows that the semi-discrete problem has a unique solution with bounded moments. In addition, the derivatives of such a solution up to any order also remain bounded in L² spaces globally time, under certain conditions. These estimates, combined with spectral projection control, are enough to obtain error estimates to the analytical solution and convergence to equilibrium states. It should be noted that this is the first time that an error estimate has been produced for any numerical method which approximates F.P.L. equations associated to any range of potentials. 3. Modelling charge separation with the Landau equation. A model for the plasma sheath is investigated using the space-inhomogeneous linear Landau equation (namely, the F.P.L. equation associated Coulomb interactions), modelling interactions between positive and negative particles. Some theory has been established for the plasma sheath, but this is the first time that an attempt has been made to simulate it with the Landau equation. The particular design of the kinetic model is described and an attempt made to capture physical phenomena associated to separation of charges. Several parameters within the model are varied to try and explain the creation of the sheaths.Show more Item Coupled flow systems, adjoint techniques and uncertainty quantification(2012-08) Garg, Vikram Vinod, 1985-; Carey, Graham F.; Prudhomme, Serge M.; Dawson, Clint N.; Gamba, Irene; Ghattas, Omar; Oden, J. Tinsley; Carey, VarisShow more Coupled systems are ubiquitous in modern engineering and science. Such systems can encompass fluid dynamics, structural mechanics, chemical species transport and electrostatic effects among other components, all of which can be coupled in many different ways. In addition, such models are usually multiscale, making their numerical simulation challenging, and necessitating the use of adaptive modeling techniques. The multiscale, multiphysics models of electrosomotic flow (EOF) constitute a particularly challenging coupled flow system. A special feature of such models is that the coupling between the electric physics and hydrodynamics is via the boundary. Numerical simulations of coupled systems are typically targeted towards specific Quantities of Interest (QoIs). Adjoint-based approaches offer the possibility of QoI targeted adaptive mesh refinement and efficient parameter sensitivity analysis. The formulation of appropriate adjoint problems for EOF models is particularly challenging, due to the coupling of physics via the boundary as opposed to the interior of the domain. The well-posedness of the adjoint problem for such models is also non-trivial. One contribution of this dissertation is the derivation of an appropriate adjoint problem for slip EOF models, and the development of penalty-based, adjoint-consistent variational formulations of these models. We demonstrate the use of these formulations in the simulation of EOF flows in straight and T-shaped microchannels, in conjunction with goal-oriented mesh refinement and adjoint sensitivity analysis. Complex computational models may exhibit uncertain behavior due to various reasons, ranging from uncertainty in experimentally measured model parameters to imperfections in device geometry. The last decade has seen a growing interest in the field of Uncertainty Quantification (UQ), which seeks to determine the effect of input uncertainties on the system QoIs. Monte Carlo methods remain a popular computational approach for UQ due to their ease of use and "embarassingly parallel" nature. However, a major drawback of such methods is their slow convergence rate. The second contribution of this work is the introduction of a new Monte Carlo method which utilizes local sensitivity information to build accurate surrogate models. This new method, called the Local Sensitivity Derivative Enhanced Monte Carlo (LSDEMC) method can converge at a faster rate than plain Monte Carlo, especially for problems with a low to moderate number of uncertain parameters. Adjoint-based sensitivity analysis methods enable the computation of sensitivity derivatives at virtually no extra cost after the forward solve. Thus, the LSDEMC method, in conjuction with adjoint sensitivity derivative techniques can offer a robust and efficient alternative for UQ of complex systems. The efficiency of Monte Carlo methods can be further enhanced by using stratified sampling schemes such as Latin Hypercube Sampling (LHS). However, the non-incremental nature of LHS has been identified as one of the main obstacles in its application to certain classes of complex physical systems. Current incremental LHS strategies restrict the user to at least doubling the size of an existing LHS set to retain the convergence properties of LHS. The third contribution of this research is the development of a new Hierachical LHS algorithm, that creates designs which can be used to perform LHS studies in a more flexibly incremental setting, taking a step towards adaptive LHS methods.Show more Item A direct numerical simulation of fully developed turbulent channel flow with spanwise wall oscillation(2005) Zhou, Dongmei; Ball, K. S.; Bogard, David G.Show more Low-Reynolds-number, fully developed turbulent channel flow with wall motion has been simulated by direct numerical simulation to examine the effectiveness and the near-wall mechanics using spanwise wall oscillation to reduce friction drag. The three-dimensional unsteady Navier-Stokes (and energy) equations are solved using Fourier-Chebyshev-Tau spectral methods combined with a second-order semi-implicit time-advancement scheme. The effects of spatial resolution and computational box size on the computed turbulence and the drag reduction percentage were investigated. Finer spanwise resolution has a greater effect on achieving a better solution and the turbulent flow is well resolved for a spanwise grid spacing of Δ 3 <10 + x . It was also confirmed that the dynamics of turbulence in a natural full channel could be reproduced by a minimal channel. Parameter studies have been performed to examine the variation of drag reduction value with wall oscillation frequency, velocity amplitude, peak-to-peak amplitude, and oscillation orientation, and drag reduction data were discovered to correlate better with peak-to-peak amplitude for frequencies 01 > 0. + f in contrast to the previous finding of its correlation with peak-wall-speed. At the optimal wall oscillation conditions, net power savings of about 5% are obtained after the power input to move the wall is accounted for, even though more than 40% friction drag reduction has been achieved in the turbulence flow. Significant drag reduction is accompanied by the suppression of the turbulent bursting process and production of turbulence, and by a reduction in the intensity of streaks and streamwise vortices. A thickened viscous sublayer is indicated through the observed outward shift of statistical quantities such as velocity fluctuations and Reynolds shear stress in the moving-wall channel flow. Drag reduction by spanwise wall oscillation is mainly due to the suppression of ejection-sweep motions and the disruption in the cycle of the turbulence selfsustaining process, starting with the wall streaks that are distorted and reduced in number and extent. The intensity and the number of vortical structures are also reduced by the wall motion. The suppression of the regeneration of new streamwise vortices above the wall in turn further suppresses the ejection-sweep motions, thus leading to the reduced skin-friction levels at the wall.Show more Item Effects of site geometry and ground motion intensity measures on lateral spread displacements(2020-09-10) Little, Michael Vernon; Rathje, Ellen M.; Cox, Brady R; El Mohtar, Chadi S; Johnson, Joel P; Stokoe, Kenneth HShow more Liquefaction-induced lateral spreading represents an important geohazard during earthquakes due to the significant displacements that are induced. These movements have the potential to cause significant damage to both the overlying infrastructure (e.g., buildings, bridges), as well as embedded infrastructure (e.g., pipelines). The objectives of this research are to use field observations and numerical simulations of lateral spreading to elucidate the important factors influencing lateral spread displacements. High resolution displacement data from the 2011 Christchurch earthquake are used with surface and subsurface data to evaluate the effectiveness of state-of-practice predictive models for lateral spread displacements and to identify topographic and geotechnical factors that have a significant influence on lateral spreading displacements. The results show that limitations in the existing lateral spread displacement predictive models are in need of improvements and that special care should be taken when evaluating lateral spreading displacements at sites with a topographic terrace and multiple free-faces. A set of 132 numerical models of free-face lateral spread sites are analyzed to evaluate the influence of the height of the free-face, the thickness of liquefiable layers, sloping ground behind the free-face, and the presence of a topographic terrace. The results indicate that the main factor driving lateral spread displacements is the combined height of the free-face and liquefiable soil layer thickness. Sloping ground behind the free-face can increase the displacements by as much as 1 m in the zone 50-100 m from the free-face. The presence of a topographic terrace at a distance of up to 400 m from the free-face can result in more than 0.5 m of increased displacements between the free-face and terrace. The results of these analyses are also used to develop a framework for predicting the surface displacement profile behind the free-face of a lateral spread site. Another set of 456 finite element simulations are performed to investigate the effects of earthquake intensity measures (IM) on the triggering of liquefaction and on lateral spreading displacements. The IMs that are best able to distinguish between triggering and non-triggering of liquefaction are cumulative absolute velocity with a minimum acceleration threshold of 50 cm/s² (CAV₅₀), normalized dissipated energy [equation], and Arias intensity (I [subscript a]). The IM with the most potential for predicting lateral spreading displacements is cumulative absolute velocity (CAV, defined without an acceleration threshold). Considering only the part of the motion after which liquefaction is triggered (IM [subscript post]) as the predictor of lateral spread displacements results in only a marginal improvement in predictive capacity. The largest increase to predictive capacity after selecting the appropriate IM comes from including information about the geometry of the site.Show more Item Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method(2007-08) Cottrell, John Austin, 1980-; Hughes, Thomas J. R.Show more This work discusses isogeometric analysis as a promising alternative to standard finite element analysis. Isogeometric analysis has emerged from the idea that the act of modeling a geometry exactly at the coarsest levels of discretization greatly simplifies the refinement process by obviating the need for a link to an external representation of that geometry. The NURBS based implementation of the method is described in detail with particular emphasis given to the numerous refinement possibilities, including the use of functions of higher-continuity and a new technique for local refinement. Examples are shown that highlight each of the major features of the technology: geometric flexibility, functions of high continuity, and local refinement. New numerical approaches are introduced for modeling the fine scales within the variational multiscale method. First, a general framework is presented for seeking solutions to differential equations in a way that approximates optimality in certain norms. More importantly, it makes possible for the first time the approximation of the fine-scale Green's functions arising in the formulation, leading to a better understanding of machinery of the variational multiscale method and opening new avenues for research in the field. Second, a simplified version of the approach, dubbed the "parameter-free variational multiscale method," is proposed that constitutes an efficient stabilized method, grounded in the variational multiscale framework, that is free of the ad hoc stabilization parameter selection that has plagued classical stabilized methods. Examples demonstrate the efficacy of the method for both linear and nonlinear equations.Show more Item Numerical analysis of reproducing kernel collocation method for linear nonlocal models(2020-06-19) Leng, Yu, Ph. D.; Foster, John T., Ph. D.; Tian, Xiaochuan, Ph. D.; Demkowicz, Leszek F; Sepehrnoori, Kamy; Sharma, Mukul M; Wheeler, Mary FShow more Hydraulic fracturing has played a major role in north America's “shale revolution” over the past decades. Modeling of the hydraulic fracture propagation is challenging. Peridynamics, a nonlocal theory of continuum mechanics, has been used to model complex hydraulic fracturing processes in recent years. While the peridynamics-based hydraulic fracturing model has shown promising simulations results, its current numerical discretization lacks any mathematical analysis. This dissertation is motivated by the numerical solution of the peridynamics-based hydraulic fracturing model. The major objective is to develop a robust numerical method, under the change of the modeling parameters, for linear nonlocal diffusion models and peridynamic Navier equation, which are decoupled models of the peridynamics-based hydraulic fracturing model. Reproducing kernel (RK) collocation method is of our interest due to its mesh-free nature. By choosing special RK support sizes, we have developed a RK collocation method for nonlocal models and numerical solutions converge to the nonlocal solution and also to the corresponding local limit independent of the modeling parameters as the nonlocal interactions vanish. Accurate evaluation of the stiffness matrix for nonlocal models is computationally prohibitive even for collocation method. To save computational costs, the concept of RK approximation is generalized to approximate integrals and the quasi-discrete nonlocal operator, which uses a finite number of symmetric quadrature points to evaluate the integral, is proposed. We have shown RK collocation on the quasi-discrete nonlocal diffusion and peridynamic Navier equation converge to their classical counterparts. Finally, for the pure displacement form of the classical linear elasticity model, finite element solutions deteriorate when the material is nearly incompressible. A common remedy is to introduce an additional variable, pressure, and rewrite the equation in a mixed formulation, but the discrete functional spaces need to satisfy the famous inf-sup condition. For the the mixed form of the quasi-discrete peridynamic Navier equation, the discretization obtained using RK collocation with equal order interpolation for displacements and pressure passes the inf-sup test; the solution does not suffer from instability. Hence, with the use of penalty techniques or artificial compressibility, the proposed RK collocation method is promising in solving the peridynamics-based hydraulic fracturing model, which has an embedded saddle-point problemShow more Item Numerical multiscale methods for boundary layer problems in fluid dynamics(2020-05-06) Carney, Sean Patrick; Engquist, Björn, 1945-; Moser, Robert D; Tsai, Yen-Hsi; Gamba, Irene M; Arbogast, Todd JShow more Physical processes typically occur over a wide range of scales in space and time. In many instances it is computationally preferable to couple models with differing levels of physical description for different portions of the domain in space and/or time. Such multiscale, hybrid strategies allow for an accurate representation of important physical phenomena where necessary while ensuring a feasible overall computational cost. In general, every multiphysics problem is different, and for every coupling strategy there are nontrivial mathematical and algorithmic details that must be worked out. However, many problems share similar features, for instance, asymptotically thin boundary layers and nonlinear interactions across scales, and sometimes strategies developed for one problem can be successfully applied to a related, but different area. This thesis develops numerical strategies for the accurate and efficient simulation of multiscale, boundary layer problems arising in fluid dynamics. Considered are four different physical models, namely viscous laminar flow over a rough surface, high Reynolds number wall-bounded turbulent flow, electrokinetic flows over charge-conducting surfaces, and upscaling in porous media flow. The first three of these situations typically involve asymptotically small, physical boundary layers while the fourth can involve computational boundary layers as a result of modeling artifacts. The numerical strategies are presented in the context of the heterogeneous multiscale method (HMM), which generally involves coupling a coarse-scale, macroscopic solver for bulk dynamics to a fine-scale, microscopic solver for the small and/or fast scales in space and/or time. The first section contains a coupled method for rough-wall laminar flow. The second describes a reduced-order, microscale model for the near-wall eddies present in high Reynolds number wall bounded turbulence. The third section concerns the application of stochastic, multiscale partial differential equation model known as Fluctuating Hydrodynamics to the mesoscale dynamics of electrokinetic flows. Finally, the fourth contains a method for reducing the nonphysical, boundary or "resonance" error term that arises in the numerical homogenization of multiscale elliptic operators as in, for example, the modeling of porous media flows or steady state heat conduction. All sections contain a description of and motivations for the problem, numerical examples of the model presented, and a discussion of future research and open problems to be addressed.Show more Item Optimal transport for seismic inverse problems(2018-07-27) Yang, Yunan; Engquist, Björn, 1945-; Fomel, Sergey; Ghattas, Omar; Ren, Kui; Tsai, Yen-Hsi RichardShow more Seismic data contains interpretable information about subsurface properties, which are important for exploration geophysics. Full waveform inversion (FWI) is a nonlinear inverse technique that inverts the model parameters by minimizing the difference between the synthetic data from numerical simulations and the observed data at the surface of the earth. The least-squares norm of this difference is the traditional objective function for FWI, but it is sensitive to the initial model, the data spectrum, the noise in the measurement, and other issues related to optimization. The least-squares norm is a point-by-point comparison. Other misfit functions with a global feature have been proposed in the literature to achieve better convexity, but none of them is technically a metric. Here we apply the quadratic Wasserstein metric of the optimal transport theory to FWI. Both the amplitude differences and the phase mismatches are considered in this new misfit function. Mathematically, we prove the convexity of the quadratic Wasserstein metric concerning shift, dilation, and partial amplitude changes of data as well as its insensitivity to noise. Despite these good properties of the quadratic Wasserstein metric, solving optimal transport problem in higher dimension is challenging. We first compute the misfit globally by regarding it as a 2D optimal transport problem. Since the Monge-Ampère equation is rigorously related to the quadratic Wasserstein metric, we solve the 2D optimal transport problem by solving a fully nonlinear Monge-Ampère equation based on a monotone finite difference solver which has been proved to converge to the viscosity solution. To increase the resolution of the inversion, we further develop another method to compute the quadratic Wasserstein metric: trace-by-trace comparison based on the 1D optimal transport. The 1D technique can be solved accurately and efficiently and thus is more robust to handle more complicated problems with less computational cost. We also explore the connections between optimal transport and other misfit functions and explain the intrinsic features of the transport-based idea. Since optimal transport problem concerns nonnegative measures, we will also investigate the critical data normalization step which transforms the sign-changing wavefields into probability densities. This is the most important topic to address in applying optimal transport to seismic inversion. With the least-squares norm being the misfit function, FWI using the reflection data often results in migration-like features in the model updates. We argue that it is the inherent nonconvexity that prevents it from updating the kinematics with high-frequency data. Through numerical examples and discussions, we demonstrate that the better convexity of the quadratic Wasserstein metric can tackle the local minima generated by the high-wavenumber update which appears in addition to the known cycle-skipping issues caused by phase mismatches.Show more Item Stability of dual discretization methods for partial differential equations(2011-05) Gillette, Andrew Kruse; Bajaj, Chandrajit; Demkowicz, Leszek; Gonzalez, Oscar; Luecke, John; Reid, Alan; Vick, JamesShow more This thesis studies the approximation of solutions to partial differential equations (PDEs) over domains discretized by the dual of a simplicial mesh. While `primal' methods associate degrees of freedom (DoFs) of the solution with specific geometrical entities of a simplicial mesh (simplex vertices, edges, faces, etc.), a `dual discretization method' associates DoFs with the geometric duals of these objects. In a tetrahedral mesh, for instance, a primal method might assign DoFs to edges of tetrahedra while a dual method for the same problem would assign DoFs to edges connecting circumcenters of adjacent tetrahedra. Dual discretization methods have been proposed for various specific PDE problems, especially in the context of electromagnetics, but have not been analyzed using the full toolkit of modern numerical analysis as is considered here. The recent and still-developing theories of finite element exterior calculus (FEEC) and discrete exterior calculus (DEC) are shown to be essential in understanding the feasibility of dual methods. These theories treat the solutions of continuous PDEs as differential forms which are then discretized as cochains (vectors of DoFs) over a mesh. While the language of DEC is ideal for describing dual methods in a straightforward fashion, the results of FEEC are required for proving convergence results. Our results about dual methods are focused on two types of stability associated with PDE solvers: discretization and numerical. Discretization stability analyzes the convergence of the approximate solution from the discrete method to the continuous solution of the PDE as the maximum size of a mesh element goes to zero. Numerical stability analyzes the potential roundoff errors accrued when computing an approximate solution. We show that dual methods can attain the same approximation power with regard to discretization stability as primal methods and may, in some circumstances, offer improved numerical stability properties. A lengthier exposition of the approach and a detailed description of our results is given in the first chapter of the thesis.Show more Item Toward seamless multiscale computations(2013-05) Lee, Yoonsang, active 2013; Engquist, Björn, 1945-Show more Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields.Show more Item Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods(2018-06-25) Fuentes, Federico, Ph. D.; Demkowicz, Leszek; Babuska, Ivo M.; Caffarelli, Luis A.; Hughes, Thomas J. R.; Oden, J. Tinsley; Wilder, AletaShow more Discontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and are crafted to be numerically stable by implicitly computing a near-optimal discrete test space as a function of a discrete trial space. Moreover, they are completely general in the sense that they can be applied to a variety of variational formulations, including non-conventional ones that involve non-symmetric functional settings, such as ultraweak variational formulations. In most cases, these properties have been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like in convection-diffusion problems and other singularly perturbed problems. In this work, other features of DPG methods are systematically exploited and applied to different problems. More specifically, the versatility of DPG methods is elucidated by utilizing the underlying methodology to discretize four distinct variational formulations of the equations of linear elasticity. By taking advantage of interface variables inherent to DPG discretizations, an approach to coupling different variational formulations within the same domain is described and used to solve interesting problems. Moreover, the convenient algebraic structure in DPG methods is harnessed to develop a new family of numerical methods called discrete least-squares (DLS) finite element methods. These involve solving, with improved conditioning properties, a discrete least-squares problem associated with an overdetermined rectangular system of equations, instead of directly solving the usual square systems. Their utility is demonstrated with illustrative examples. Additionally, high-order polygonal DPG (PolyDPG) methods are devised by using the intrinsic discontinuities present in ultraweak formulations. The resulting methods can handle heavily distorted non-convex polygonal elements and discontinuous material properties. A polygonal adaptive strategy was also proposed and compared with standard techniques. Lastly, the natural high-order residual-based a posteriori error estimator ingrained within DPG methods was further applied to problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery.Show more Item Weakly non-local arbitrarily-shaped absorbing boundary conditions for acoustics and elastodynamics theory and numerical experiments(2004) Lee, Sanghoon; Kallivokas, Loukas F.Show more In this dissertation we discuss the performance of a family of local and weakly non-local in space and time absorbing boundary conditions, prescribed on trun cation boundaries of elliptical and ellipsoidal shape for the solution of two- and three-dimensional scalar wave equations, respectively, in both the time- and frequency-domains. The elliptical and ellipsoidal artiﬁcial boundaries are de rived as particular cases of general arbitrarily-shaped convex boundaries for which the absorbing conditions are developed. From the mathematical per spective, the development of the conditions is based on earlier work by Kalli vokas et al [72–77]; herein an incremental modiﬁcation is made to allow for the spatial variability of the conditions’ absorption characteristics. From the appli cations perspective, the obtained numerical results appear herein for the ﬁrst time. It is further shown that the conditions, via an operator-splitting scheme, lend themselves to easy incorporation in a variational form that, in turn, leads to a standard Galerkin ﬁnite element approach. The resulting wave absorbing ﬁnite elements are shown to preserve the sparsity and symmetry of standard ﬁnite element schemes in both the time- and frequency-domains. Herein, we also extend the applicability of elliptically-shaped truncation boundaries to semi-inﬁnite acoustic media. Numerical experiments for transient and time harmonic cases attest to the computational savings realized when elongated scatterers are surrounded by elliptically- or ellipsoidally-shaped boundaries, as opposed to the more commonly used circular or spherical truncation geome tries in either the full- or half-space cases (near-surface scatterers). Lastly, we treat the two-dimensional elastodynamics case based on a Helmholtz decomposition of the displacement vector ﬁeld. The decomposi tion allows for scalar wave equations to be written for the scalar and vector potential components. Thus, absorbing conditions similar to the ones writ ten for acoustics can be used for the elastodynamics case. The stability of the elastodynamics conditions for time-domain applications remains an open question.Show more