Browsing by Author "Iliescu, Traian"
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- Adaptive Numerical Methods for Large Scale Simulations and Data AssimilationConstantinescu, Emil Mihai (Virginia Tech, 2008-05-26)Numerical simulation is necessary to understand natural phenomena, make assessments and predictions in various research and engineering fields, develop new technologies, etc. New algorithms are needed to take advantage of the increasing computational resources and utilize the emerging hardware and software infrastructure with maximum efficiency. Adaptive numerical discretization methods can accommodate problems with various physical, scale, and dynamic features by adjusting the resolution, order, and the type of method used to solve them. In applications that simulate real systems, the numerical accuracy of the solution is typically just one of the challenges. Measurements can be included in the simulation to constrain the numerical solution through a process called data assimilation in order to anchor the simulation in reality. In this thesis we investigate adaptive discretization methods and data assimilation approaches for large-scale numerical simulations. We develop and investigate novel multirate and implicit-explicit methods that are appropriate for multiscale and multiphysics numerical discretizations. We construct and explore data assimilation approaches for, but not restricted to, atmospheric chemistry applications. A generic approach for describing the structure of the uncertainty in initial conditions that can be applied to the most popular data assimilation approaches is also presented. We show that adaptive numerical methods can effectively address the discretization of large-scale problems. Data assimilation complements the adaptive numerical methods by correcting the numerical solution with real measurements. Test problems and large-scale numerical experiments validate the theoretical findings. Synergistic approaches that use adaptive numerical methods within a data assimilation framework need to be investigated in the future.
- Advanced Spectral Methods for Turbulent FlowsNasr Azadani, Leila (Virginia Tech, 2014-04-24)Although spectral methods have been in use for decades, there is still room for innovation, refinement and improvement of the methods in terms of efficiency and accuracy, for generalized homogeneous turbulent flows, and especially for specialized applications like the computation of atmospheric flows and numerical weather prediction. In this thesis, two such innovations are presented. First, inspired by the adaptive mesh refinement (AMR) technique, which was developed for the computation of fluid flows in physical space, an algorithm is presented for accelerating direct numerical simulation (DNS) of isotropic homogeneous turbulence in spectral space. In the adaptive spectral resolution (ASR) technique developed here the spectral resolution in spectral space is dynamically refined based on refinement criteria suited to the special features of isotropic homogeneous turbulence in two, and three dimensions. Applying ASR to computations of two- and three-dimensional turbulence allows significant savings in the computational time with little to no compromise in the accuracy of the solutions. In the second part of this thesis the effect of explicit filtering on large eddy simulation (LES) of atmospheric flows in spectral space is studied. Apply an explicit filter in addition to the implicit filter due to the computational grid and discretization schemes in LES of turbulent flows allows for better control of the numerical error and improvement in the accuracy of the results. Explicit filtering has been extensively applied in LES of turbulent flows in physical space while few studies have been done on explicitly filtered LES of turbulent flows in spectral space because of perceived limitations of the approach, which are shown here to be incorrect. Here, explicit filtering in LES of the turbulent barotropic vorticity equation (BVE) as a first model of the Earth's atmosphere in spectral space is studied. It is shown that explicit filtering increases the accuracy of the results over implicit filtering, particularly where the location of coherent structures is concerned.
- Advancing Maternal Health through Projection-based and Machine Learning Strategies for Reduced Order ModelingSnyder, William David (Virginia Tech, 2024-06-12)High-fidelity computer simulations of childbirth are time consuming, making them impractical for guiding decision-making during obstetric emergencies. The complex geometry, micro-structure, and large finite deformations undergone by the vagina during childbirth result in material and geometric nonlinearities, complicated boundary conditions, and nonhomogeneities within finite element (FE) simulations. Such nonlinearities pose a significant challenge for numerical solvers, increasing the computational time. Simplifying assumptions can reduce the computational time significantly, but this usually comes at the expense of simulation accuracy. The work herein proposed the use of reduced order modeling (ROM) techniques to create surrogate models that capture experimentally-measured displacement fields of rat vaginal tissue during inflation testing in order to attain both the accuracy of higher-fidelity models and the speed of lower-fidelity simulations. The proper orthogonal decomposition (POD) method was used to extract the significant information from FE simulations generated by varying the luminal pressure and the parameters that introduce the anisotropy in the selected constitutive model. In our first study, a new data-driven (DD) variational multiscale (VMS) ROM framework was extended to obtain the displacement fields of rat vaginal tissue subjected to ramping luminal pressure. For comparison purposes, we also investigated the classical Galerkin ROM (G-ROM). In our numerical study, both the G-ROM and the DD-VMS-ROM decreased the FE computational cost by orders of magnitude without a significant decrease in numerical accuracy. Furthermore, the DD-VMS-ROM improved the G-ROM accuracy at a modest computational overhead. Our numerical investigation showed that ROM had the potential to provide efficient and accurate computational tools to describe vaginal deformations, with the ultimate goal of improving maternal health. Our second study compared two common computational strategies for surrogate modeling, physics-based G-ROM and data-driven machine learning (ML), for decreasing the cost of FE simulations of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. Since there are many methods associated with each modeling approach, to provide a fair and natural comparison, we selected a basic model from each category. From the ROM strategies, we considered a simplified G-ROM that is based on the linearization of the underlying nonlinear FE equations. From the ML strategies, we selected a feed-forward dense neural network (DNN) to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the POD coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical comparisons of G-ROM, ML, and POD-ML took place in the reconstructive regime. The numerical results showed that the G-ROM outperformed the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The POD-ML model improved on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model did not improve on the error performance of the ML. In our last study, we expanded our investigation of ML methods for surrogate modeling by comparing the performance of a DNN similar to what was used previously to that of a convolutional neural network (CNN) using 1-D convolution on the input parameters from FE simulations of active vaginal tearing. The new FE simulations utilized a custom continuum damage model that provided material damage and failure properties to an existing anisotropic hyperelastic constitutive model to replicate experimentally-observed tear propagation behaviors. We employed our DNN and CNN models to create mappings from constitutive model parameters, geometric properties of the propagating tear, and luminal pressure values to either the full FE displacement history or the POD coefficients of the displacement history. The root-mean-square error (RMSE) with respect to the FE displacement history achieved by full order output ML predictions was reproducible with POD-ML using a basis of only dimension l=10. Additionally, an order of magnitude reduction in offline time was observed using POD-ML over full-order ML with minimal difference between DNN and CNN architectures. Differences in online computational costs between ML and POD-ML were found to be negligible, but the DNNs produced predictions slightly faster than the CNNs, though both online times were on the same order of magnitude. While convolution did not significantly aid the regression task at hand, POD-ML was demonstrated to be an efficient and effective approach for surrogate modeling of the FE tear propagation model, approximating the displacement history with RMSE less than 0.1 mm and generating results 7 orders of magnitude faster than the FE model. This set of baseline numerical investigations serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, material properties, and tissue damage mechanics of the human vagina, as well as their changes during labor.
- Analysis and Approximation of Viscoelastic and Thermoelastic Joint-Beam SystemsFulton, Brian I. (Virginia Tech, 2006-07-21)Rigidizable/Inflatable space structures have been the focus of renewed interest in recent years due to efficient packaging for transport. In this work, we examine new mathematical systems used to model small-scale joint dynamics for inflatable space truss structures. We investigate the regularity and asymptotic behavior of systems resulting from various damping models, including Kelvin-Voigt, Boltzmann, and thermoelastic damping. Approximation schemes will also be introduced. Finally, we look at optimal control for the Kelvin-Voigt model using a linear feedback regulator.
- Approximate Deconvolution Reduced Order ModelingXie, Xuping (Virginia Tech, 2015-11-03)This thesis proposes a large eddy simulation reduced order model (LES-ROM) framework for the numerical simulation of realistic flows. In this LES-ROM framework, the proper orthogonal decomposition (POD) is used to define the ROM basis and a POD differential filter is used to define the large ROM structures. An approximate deconvolution (AD) approach is used to solve the ROM closure problem and develop a new AD-ROM. This AD-ROM is tested in the numerical simulation of the one-dimensional Burgers equation with a small diffusion coefficient ( ν= 10⁻³).
- Approximate Deconvolution Reduced Order ModelingXie, X.; Wells, D.; Wang, Z.; Iliescu, Traian (2016-10-12)This paper proposes a large eddy simulation reduced order model(LES-ROM) framework for the numerical simulation of realistic flows. In this LES-ROM framework, the proper orthogonal decomposition(POD) is used to define the ROM basis and a POD differential filter is used to define the large ROM structures. An approximate deconvolution(AD) approach is used to solve the ROM closure problem and develop a new AD-ROM. This AD-ROM is tested in the numerical simulation of the one-dimensional Burgers equation with a small diffusion coefficient(10^{-3})
- Approximation of Parametric Dynamical SystemsCarracedo Rodriguez, Andrea (Virginia Tech, 2020-09-02)Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples.
- Atmospheric Lagrangian transport structures and their applications to aerobiologyBozorg Magham, Amir Ebrahim (Virginia Tech, 2014-02-21)Exploring the concepts of long range aerial transport of microorganisms is the main motivation of this study. For this purpose we use theories and concepts of dynamical systems in the context of geophysical fluid systems. We apply powerful notions such as finite-time Lyapunov exponent (FTLE) and the associated Lagrangian coherent structures (LCS) and we attempt to provide mathematical explanations and frameworks for some applied questions which are based on realistic concerns of atmospheric transport phenomena. Accordingly, we quantify the accuracy of prediction of FTLE-LCS features and we determine the sensitivity of such predictions to forecasting parameters. In addition, we consider the spatiotemporal resolution of the operational data sets and we propose the concept of probabilistic source and destination regions which leads to the definition of stochastic FTLE fields. Moreover, we put forward the idea of using ensemble forecasting to quantify the uncertainty of the forecast results. Finally, we investigate the statistical properties of localized measurements of atmospheric microbial structure and their connections to the concept of local FTLE time-series. Results of this study would pave the way for more efficient models and management strategies for the spread of infectious diseases affecting plants, domestic animals, and humans.
- Behavioral EMI-Models of Switched Power ConvertersBishnoi, Hemant (Virginia Tech, 2013-11-05)Measurement-based behavioral electromagnetic interference (EMI) models have been shown earlier to accurately capture the EMI behavior of switched power converters. These models are compact, linear, and run in frequency domain, enabling faster and more stable simulations compared to the detailed lumped circuit models. So far, the behavioral EMI modeling techniques are developed and applied to the converter's input side only. The resulting models are therefore referred to as "terminated EMI models". Under the condition that the output side of the converter remains fixed, these models can predict the input side EMI for any change in the impedance of the input side network. However, any change at the output side would require re-extraction of the behavioral model. Thus the terminated EMI models are incapable of predicting the change in the input side EMI due to changes at the output side of the converter or vice versa. The above mentioned limitation has been overcome by an "un-terminated EMI model" proposed in this dissertation. Un-terminated EMI models are developed here to predict both the common-mode (CM) and the differential (DM) noise currents at the input and the output sides of a motor-drive system. The modeling procedure itself has been simplified and now requires fewer measurements and results in less noise in the identified model parameters. Both CM and DM models are then combined to predict the total noise in the motor drive system. All models are validated by experiments and their limitations identified. A significant portion of this dissertation is then devoted to the application of behavioral EMI models in the design of EMI filters. Comprehensive design procedures are developed for both DM and CM filters in a motor-drive system. The filters designed using the proposed methods are experimentally shown to satisfy the DO-160 conducted emissions standards. The dissertation ends with a summary of contributions, limitations, and some future research directions.
- Calibrated Filtered Reduced Order ModelingXie, X.; Mohebujjaman, M.; Rebholz, L. G.; Iliescu, Traian (2017-02-20)We propose a calibrated filtered reduced order model (CF-ROM) framework for the numerical simulation of general nonlinear PDEs that are amenable to reduced order modeling. The novel CF-ROM framework consists of two steps: (i) In the first step, we use explicit ROM spatial filtering of the nonlinear PDE to construct a filtered ROM. This filtered ROM is low-dimensional, but is not closed (because of the nonlinearity in the given PDE). (ii) In the second step, we use a calibration procedure to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved modes. To this end, we use a linear or quadratic ansatz to model this interaction and close the filtered ROM. To find the new coefficients in the closed filtered ROM, we solve an optimization problem that minimizes the difference between the full order model data and our ansatz. Although we use a fluid dynamics setting to illustrate how to construct and use the CF-ROM framework, we emphasize that it is built on general ideas of spatial filtering and optimization and is independent of (restrictive) phenomenological arguments. Thus, the CF-ROM framework can be applied to a wide variety of PDEs.
- Characterization of the Mechanism of Drag Reduction Using a Karhunen-Loève Analysis on a Direct Numerical Simulation of Turbulent Pipe FlowDuggleby, Andrew Thomas (Virginia Tech, 2006-08-02)The objective of this study is to characterize the mechanism of drag reduction by comparing the dynamical eigenfunctions of a turbulent pipe flow against those of two known cases of drag reduced flows. The first is forced drag reduction by spanwise wall oscillation, and the second is natural drag reduction found in relaminarizing flow. The dynamics are examined through a Karhunen-Lo`eve (KL) expansion of the direct numerical simulation flow field results. The direct numerical simulation (DNS) is performed using NEK5000, a spectral element Navier-Stokes solver, the first exponentially convergent investigation of DNS of turbulence in a pipe. The base flow is performed at a Reynolds number of Re = 150, resulting in a KL dimension of D_KL = 2130. As in turbulent channel flow, propagating modes are found, characterized with constant phase speed, and contribute of 80.58% of the total fluctuating energy. Based upon wavenumber characteristics and coherent vorticity visualization, four subclasses of propagating modes and two subclasses of non-propagating modes are discovered, qualitatively similar to the horseshoe (hairpin) vortex structure reported in literature. The drag reduced case is performed at the same Reynolds number with a spanwise velocity A+ = 20, a period of T+ = 50, and is driven by a constant pressure gradient. This results in a increase of flow rate by 27 %, and the KL dimension is reduced to D_KL = 102, a 96% reduction. The propagating modes, in particular the wall modes, are pushed away from the wall, resulting in a 34% increase in their advection speed, and a shift away from the wall of the root-mean-square and Reynolds stress peaks. The relaminarizing case observes the chugging motion of the mean flow rate when the Reynolds number is barely turbulent, at Re = 95. This chugging motion is the relaminarization of the flow, resulting in an increased flow rate, and then before complete relaminarization, the flow regains its turbulent state. This occurs because the lift modes, which are responsible for the majority of the energy in the inertial range of the energy spectra, decrease by two or three orders of magnitude. The chugging ends when the wall modes restart the turbulent cascade, and the lift modes are repopulated with energy. A model for the energy path is developed, with energy going from the pressure gradient to the shear modes, then to the roll modes, then to the wall modes, and then finally to the lift modes. It is concluded that drag reduction in a flow can be achieved by disrupting any leg of this model, thus disrupting the self-sustaining mechanism of turbulence. The spanwise wall oscillation shortened the life span of the wall modes, thus limiting their ability to pass energy to the lift modes. Likewise, the low Reynolds number did not provide enough energy to sustain the lift modes, and so relaminarization began. The contribution of this work is twofold. Firstly, the structure of turbulent pipe flow is examined and visualized for the first time using the Karhunen-Lo`eve method. The second, and perhaps greatest contribution of this work, is that the mechanism of drag reduction has been characterized as the link between the wall modes and the lift modes. This will allow future work on developing real methods of drag reduction, and eventually porting it to high Reynolds number flows, like that of an oil pipeline at Re= 40, 000. To achieve this, certain questions remain to be answered, such as what is the most efficient method of disrupting the wall-lift mechanism? Is there a single structure that can be identified and manipulated that gives a similar eect? Once answered, this will allow for a new generation of pipelines to be developed, and considering the implications in petroleum industry alone, will result in a significant contribution to the economy of the world.
- Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural NetworkXie, Xuping; Webster, Clayton G.; Iliescu, Traian (MDPI, 2020-03-23)Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear system and construct a spatially filtered ROM. This filtered ROM is low-dimensional, but is not closed. (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models.
- Combining Data-driven and Theory-guided Models in Ensemble Data AssimilationPopov, Andrey Anatoliyevich (Virginia Tech, 2022-08-23)There once was a dream that data-driven models would replace their theory-guided counterparts. We have awoken from this dream. We now know that data cannot replace theory. Data-driven models still have their advantages, mainly in computational efficiency but also providing us with some special sauce that is unreachable by our current theories. This dissertation aims to provide a way in which both the accuracy of theory-guided models, and the computational efficiency of data-driven models can be combined. This combination of theory-guided and data-driven allows us to combine ideas from a much broader set of disciplines, and can help pave the way for robust and fast methods.
- Commutation Error in Reduced Order ModelingKoc, Birgul (Virginia Tech, 2018-10-01)We investigate the effect of spatial filtering on the recently proposed data-driven correction reduced order model (DDC-ROM). We compare two filters: the ROM projection, which was originally used to develop the DDC-ROM, and the ROM differential filter, which uses a Helmholtz operator to attenuate the small scales in the input signal. We focus on the following questions: ``Do filtering and differentiation with respect to space variable commute, when filtering is applied to the diffusion term?'' or in other words ``Do we have commutation error (CE) in the diffusion term?" and ``If so, is the commutation error data-driven correction ROM (CE-DDC-ROM) more accurate than the original DDC-ROM?'' If the CE exists, the DDC-ROM has two different correction terms: one comes from the diffusion term and the other from the nonlinear convection term. We investigate the DDC-ROM and the CE-DDC-ROM equipped with the two ROM spatial filters in the numerical simulation of the Burgers equation with different diffusion coefficients and two different initial conditions (smooth and non-smooth).
- The Complex Spatiotemporal Dynamics of a Shallow Fluid LayerO'Connor, Nicholas L. (Virginia Tech, 2008-05-05)The nonlinear and chaotic dynamics of a shallow fluid layer are investigated numerically using large-scale parallel numerical simulations. Two particular situations are studied in detail. First, a fluid layer is placed between rigid boundaries and heated from below to yield the chaotic dynamics of thermal convection rolls (Rayleigh-Bénard convection). Second is a free-surface fluid layer placed on a shaker to yield nonlinear surface waves (Faraday waves). In both cases the full governing partial differential equations are solved using parallel spectral element methods. Rayleigh-Bénard convection is studied in a cylindrical dish with realistic boundaries. The complete flow field is obtained as well as the spectrum of Lyapunov exponents and Lyapunov vectors. The Lyapunov exponents and their corresponding perturbation fields are used to determine when and where events occur that contribute most to the chaotic dynamics. Roll pinch-off and roll mergers are found to be the largest contributors. Two dimensional and three dimensional Faraday waves are studied with periodic boundary conditions. The full Navier-Stokes equations are solved including the complex dynamics of the free surface waves to gain a better understanding of the interplay between the viscous boundary layers, the nonlinear streaming flow, and the bulk flow. The vortices in the bulk flow are weak compared to the flow in the viscous boundary layers. The surface waves are found to be non-sinusoidal and the time evolution of the waves are explored for both large and small amplitude waves.
- A Computational Framework for Assessing and Optimizing the Performance of Observational Networks in 4D-Var Data AssimilationCioaca, Alexandru (Virginia Tech, 2013-09-04)A deep scientific understanding of complex physical systems, such as the atmosphere, can be achieved neither by direct measurements nor by numerical simulations alone. Data assimilation is a rigorous procedure to fuse information from a priori knowledge of the system state, the physical laws governing the evolution of the system, and real measurements, all with associated error statistics. Data assimilation produces best (a posteriori) estimates of model states and parameter values, and results in considerably improved computer simulations. The acquisition and use of observations in data assimilation raises several important scientific questions related to optimal sensor network design, quantification of data impact, pruning redundant data, and identifying the most beneficial additional observations. These questions originate in operational data assimilation practice, and have started to attract considerable interest in the recent past. This dissertation advances the state of knowledge in four dimensional variational (4D-Var) - data assimilation by developing, implementing, and validating a novel computational framework for estimating observation impact and for optimizing sensor networks. The framework builds on the powerful methodologies of second-order adjoint modeling and the 4D-Var sensitivity equations. Efficient computational approaches for quantifying the observation impact include matrix free linear algebra algorithms and low-rank approximations of the sensitivities to observations. The sensor network configuration problem is formulated as a meta-optimization problem. Best values for parameters such as sensor location are obtained by optimizing a performance criterion, subject to the constraint posed by the 4D-Var optimization. Tractable computational solutions to this "optimization-constrained" optimization problem are provided. The results of this work can be directly applied to the deployment of intelligent sensors and adaptive observations, as well as to reducing the operating costs of measuring networks, while preserving their ability to capture the essential features of the system under consideration.
- Cross-Validation of Data-Driven Correction Reduced Order ModelingMou, Changhong (Virginia Tech, 2018-10-03)In this thesis, we develop a data-driven correction reduced order model (DDC-ROM) for numerical simulation of fluid flows. The general DDC-ROM involves two stages: (1) we apply ROM filtering (such as ROM projection) to the full order model (FOM) and construct the filtered ROM (F-ROM). (2) We use data-driven modeling to model the nonlinear interactions between resolved and unresolved modes, which solves the F-ROM's closure problem. In the DDC-ROM, a linear or quadratic ansatz is used in the data-driven modeling step. In this thesis, we propose a new cubic ansatz. To get the unknown coefficients in our ansatz, we solve an optimization problem that minimizes the difference between the FOM data and the ansatz. We test the new DDC-ROM in the numerical simulation of the one-dimensional Burgers equation with a small diffusion coefficient. Furthermore, we perform a cross-validation of the DDC-ROM to investigate whether it can be successful in computational settings that are different from the training regime.
- Data-Driven Filtered Reduced Order Modeling Of Fluid FlowsXie, X.; Mohebujjaman, M.; Rebholz, L. G.; Iliescu, Traian (2017-09-11)We propose a data-driven filtered reduced order model (DDF-ROM) framework for the numerical simulation of fluid flows. The novel DDF-ROM framework consists of two steps: (i) In the first step, we use explicit ROM spatial filtering of the nonlinear PDE to construct a filtered ROM. This filtered ROM is low-dimensional, but is not closed (because of the nonlinearity in the given PDE). (ii) In the second step, we use data-driven modeling to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved modes. To this end, we use a quadratic ansatz to model this interaction and close the filtered ROM. To find the new coefficients in the closed filtered ROM, we solve an optimization problem that minimizes the difference between the full order model data and our ansatz. We emphasize that the new DDF-ROM is built on general ideas of spatial filtering and optimization and is independent of (restrictive) phenomenological arguments. We investigate the DDF-ROM in the numerical simulation of a 2D channel flow past a circular cylinder at Reynolds number $Re=100$. The DDF-ROM is significantly more accurate than the standard projection ROM. Furthermore, the computational costs of the DDF-ROM and the standard projection ROM are similar, both costs being orders of magnitude lower than the computational cost of the full order model. We also compare the new DDF-ROM with modern ROM closure models in the numerical simulation of the 1D Burgers equation. The DDF-ROM is more accurate and significantly more efficient than these ROM closure models.
- Data-Driven Variational Multiscale Reduced Order Modeling of Turbulent FlowsMou, Changhong (Virginia Tech, 2021-06-16)In this dissertation, we consider two different strategies for improving the projection-based reduced order model (ROM) accuracy: (I) adding closure terms to the standard ROM; (II) using Lagrangian data to improve the ROM basis. Following strategy (I), we propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $nu = 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs. Furthermore, we propose a new hybrid ROM framework for the numerical simulation of fluid flows. This hybrid framework incorporates two closure modeling strategies: (i) A structural closure modeling component that involves the recently proposed data-driven variational multiscale ROM approach, and (ii) A functional closure modeling component that introduces an artificial viscosity term. We also utilize physical constraints for the structural ROM operators in order to add robustness to the hybrid ROM. We perform a numerical investigation of the hybrid ROM for the three-dimensional turbulent channel flow at a Reynolds number $Re = 13,750$. In addition, we focus on the mathematical foundations of ROM closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. For strategy (II), we propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis.
- Dynamics of vortices in complex wakes: modeling, analysis, and experimentsBasu, Saikat (Virginia Tech, 2014-05-01)The thesis develops singly-periodic mathematical models for complex laminar wakes which are formed behind vortex-shedding bluff bodies. These wake structures exhibit a variety of patterns as the bodies oscillate or are in close proximity of one another. The most well-known formation comprises two counter-rotating vortices in each shedding cycle and is popularly known as the vk vortex street. Of the more complex configurations, as a specific example, this thesis investigates one of the most commonly occurring wake arrangements, which consists of two pairs of vortices in each shedding period. The paired vortices are, in general, counter-rotating and belong to a more general definition of the 2P mode, which involves periodic release of four vortices into the flow. The 2P arrangement can, primarily, be sub-classed into two types: one with a symmetric orientation of the two vortex pairs about the streamwise direction in a periodic domain and the other in which the two vortex pairs per period are placed in a staggered geometry about the wake centerline. The thesis explores the governing dynamics of such wakes and characterizes the corresponding relative vortex motion. In general, for both the symmetric as well as the staggered four vortex periodic arrangements, the thesis develops two-dimensional potential flow models (consisting of an integrable Hamiltonian system of point vortices) that consider spatially periodic arrays of four vortices with their strengths being +/-1 and +/-2. Vortex formations observed in the experiments inspire the assumed spatial symmetry. The models demonstrate a number of dynamic modes that are classified using a bifurcation analysis of the phase space topology, consisting of level curves of the Hamiltonian. Despite the vortex strengths in each pair being unequal in magnitude, some initial conditions lead to relative equilibrium when the vortex configuration moves with invariant size and shape. The scaled comparisons of the model results with experiments conducted in a flowing soap film with an airfoil, which was imparted with forced oscillations, are satisfactory and validate the reduced order modeling framework. The experiments have been performed by a collaborator group at the Department of Physics and Fluid Dynamics at the Technical University of Denmark (DTU), led by Dr. Anders Andersen. Similar experiments have also been run at Virginia Tech as part of this dissertation and the preliminary results are included in this treatise. The thesis also employs the same dynamical systems techniques, which have been applied to study the 2P regime dynamics, to develop a mathematical model for the P+S mode vortex wakes, with three vortices present in each shedding cycle. The model results have also been compared favorably with an experiment and the predictions regarding the vortex circulation data match well with the previous results from literature. Finally, the thesis introduces a novel concept of clean and renewable energy extraction from vortex-induced vibrations of bluff bodies. The slow-moving currents in the off-shore marine environments and riverine flows are beyond the operational capabilities of the more established hydrokinetic energy converters and the discussed technology promises to be a significant tool to generate useful power from these copiously available but previously untapped sources.
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