Dissertations / Theses on the topic 'Reduced order models'

Le but de cette thèse est de construire des modèles réduits permettant de se substituer à des logiciels de simulation de systèmes physiques complexes. Ces modèles réduits devront être rapides à interroger et seront identifiés sur la base d’un nombre limité de calculs issus du logiciel de simulation. Ce travail s’inscrira ainsi dans le monde des méthodes d’apprentissage. Les modèles ainsi créés devront pouvoir être utilisés de manière autonome, sans avoir recours au logiciel de simulation. On s’intéressera principalement à deux types de physiques. Dans une première partie nous définirons une structure de modèle réduit adaptée à la mécanique linéaire. Dans une seconde partie nous travaillerons sur un modèle réduit dynamique développé pour la mécanique des fluides
The objective of this thesis is to build fast reduced order models able to replace a computationally intensive complex system simulation software. Those reduced order models will be identified using a reasonable amount of computations issued from the simulation software. This work enters therefore the field of learning methods. Once the models are built they should be usable in an autonomous way and should not rely on the simulation software. We will consider two kinds of physics. In a first chapter, we will address problems involving linear elasticity and develop an adequate reduced order model structure. In a second chapter, we will do the same work in the field of fluid dynamics

This dissertation uses spatial filtering to develop a large eddy simulation reduced order model (LES-ROM) framework for fluid flows. Proper orthogonal decomposition is utilized to extract the dominant spatial structures of the system. Within the general LES-ROM framework, two approaches are proposed to address the celebrated ROM closure problem. No phenomenological arguments (e.g., of eddy viscosity type) are used to develop these new ROM closure models. The first novel model is the approximate deconvolution ROM (AD-ROM), which uses methods from image processing and inverse problems to solve the ROM closure problem. The AD-ROM is investigated in the numerical simulation of a 3D flow past a circular cylinder at a Reynolds number $Re=1000$. The AD-ROM generates accurate results without any numerical dissipation mechanism. It also decreases the CPU time of the standard ROM by orders of magnitude. The second new model is the calibrated-filtered ROM (CF-ROM), which is a data-driven ROM. The available full order model results are used offline in an optimization problem to calibrate the ROM subfilter-scale stress tensor. The resulting CF-ROM is tested numerically in the simulation of the 1D Burgers equation with a small diffusion parameter. The numerical results show that the CF-ROM is more efficient than and as accurate as state-of-the-art ROM closure models.
Ph. D.

This thesis presents the a stabilized projection-based Reduced Order Model (ROM) formulation in low speed fluid flows using a Variational Multi-Scale (VMS) approach. To develop this formulation we use a Finite Element (FE) method for the Full Order Model (FOM) and a Proper Orthogonal Decomposition (POD) to construct the basis. Additional to the ROM formulation, we introduce two techniques that became possible using this approach: a mesh-based hyper-reduction that uses an Adaptive Mesh Refinement (AMR) approach, and a domain decomposition scheme for ROMs. To illustrate and test the proposed formulation we use five different models: a convection–diffusion–reaction, the incompressible Navier–Stokes, a Boussinesq approximation, a low Mach number model, and a three-field incompressible Navier–Stokes.
Esta tesis presenta un modelo de orden reducido estabilizado paran fluidos a baja velocidad utilizando un enfoque de multiescala variacional. Para desarrollar esta formulación utilizamos el método de elementos finitos para el modelo no reducido y una descomposición en autovalores del mismo para construir la base. Adicional a la formulación del modelo reducido, presentamos dos técnicas que podemos formular al utilizar este enfoque: una reducción adicional del dominio, basada en la reducción de la malla, donde usamos una técnica de refinamiento adaptativa y un esquema de descomposición de dominio para el modelo reducido. Para ilustrar y probar la formulación propuesta, utilizamos cuatro diferentes modelos fisicos: una ecuación de convección-difusión-reacción, la ecuación de Navier-Stokes para fluidos incompresibles, una aproximación de Boussinesq para la ecuación de Navier-Stokes, y una aproximación para números de Mach bajos de la ecuación de Navier-Stokes.

Reduced order models (ROMs) have become prevalent in many fields of physics as they offer the potential to simulate dynamical systems with substantially increased computation efficiency in comparison to standard techniques. Among the model reduction techniques, the proper orthogonal decomposition (POD) method has proven to be an efficient means of deriving a reduced basis for high-dimensional flow systems. The intrusive ROM (IROM) is normally derived by the POD and Galerkin projection methods. The IROM is appealing for non-linear and linear model reductions and has been successfully applied to numerous research fields. However, IROMs suffer from instability and non-linearity efficiency issues. In addition, they can be complex to code because they are intrusive. In most cases the source code describing the physical system has to be modified in order to generate the reduced order model. These modifications can be complex, especially in legacy codes, or may not be possible if the source code is not available (e.g. in some commercial software). To circumvent these shortcomings, non-intrusive approaches have been introduced into ROMs. The Non-Intrusive ROM (NIROM) is independent of the original physical system. The key contribution of this thesis are: Firstly, three novel NIROMs have been presented in this thesis: POD/Taylor series, POD-Smolyak and POD-RBF (radial basis function). Secondly, two NIROMs with varying material properties have been presented. Thirdly, these newly developed NIROMs were implemented and tested under the framework of an unstructured mesh finite element model (FLUIDITY) and a combined finite-discrete element method based solid model (Y2D). Fourthly, these NIROMs have been used to construct ROMs for multi-scale 3-D free surface flows, multi-phase porous media flows, fluid-structure interaction and blasting problems.

Thesis: Ph. D. in Computational Science for Energy Resources Engineering, Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, 2015.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 221-231).
Reservoir management and decision-making is often cast as an optimization problem where we seek to maximize the field's potential recovery while minimizing associated operational costs. Two reservoir management aspects are considered here, new well placement and production controls. Reservoir simulators are at the heart of this process as they aid in identifying best field development plans. The computational cost associated with managing realistic reservoirs is however substantial due to the significant number of unknowns evaluated by the simulator as well as the number of simulations required to achieve an optimal plan-it involves hundreds to thousands of reservoir simulation runs. Reduced-order models (ROM) are considered powerful techniques to address computational challenges associated with reservoir management decision-making. In this sense, they represent perfect alternatives that trade off accuracy for speed in a controllable manner. In this work, we focus on developing model-order reduction techniques that entail the use of proper orthogonal decomposition (POD), truncated balanced realization (TBR) and discrete empirical interpolation (DEIM) to accurately reproduce the full-order model (FOM) input/output behavior. POD allows for a concise representation of the FOM in terms of relatively few variables while TBR improves the overall stability and accuracy. DEIM improves the shortcomings of POD and TBR in the case of nonlinear PDEs, i.e., saturation equation, by retaining nonlinearities in lower dimensional space. Example cases demonstrate ROMs ability to reduce the computational costs by 0(100) while providing close overall agreement to FOM for instances with significant difference in boundary conditions (well placements and controls). ROMs are potentially perfect alternatives to FOMs in reservoir management intensive studies such as field development and optimization. However, ROMs presented in this thesis and the overall physics-based ROMs have the tendency to perform well within a restricted zone. This zone is generally dictated by the training simulations (with a specific set of boundary conditions) used to build the ROM. Therefore, special care is considered when implementing these training runs. To mitigate the heuristic process of implementing training runs (multiple boundary conditions training runs), we apply a trust-region approach that provides an adaptive framework to systemically retrain and update ROMs utilizing new solutions (flow) characteristics revealed during the course of the optimization run. The adaptive framework for determining the optimal well placements entails the development of a hybrid optimization algorithm, MCSMADS, that combines positive features of both local and global optimization methods. Typical FOM is used in conjunction with MCS to globally search the optimization surface while ROMs are used in conjunction with MADS to further improve the solution quality with minimum increase in computational costs. Well production controls are optimized sequentially via gradient-based trust-region approach. ROMs in this approach replace the FOM to find optimal solutions within a trust-region (subset of the optimization space). At the end of each trust-region optimization, the accuracy of the obtained solution is assessed and the ROM is updated. Both approaches are capable of handling nonlinear constraints. They are treated using a filter-based technique. The developed framework for adaptive ROMs is applied to two realistic field examples. The first example considers maximizing net present value (NPV) through sequentially optimizing well placements and controls while the second example considers maximizing recovery through minimizing Lorenz coefficient. Nonlinear constraints including well-to-well distance and field production limits are imposed in both examples. For all cases considered, the hybrid approach for well placement based on MCS-MADS was able to constantly provide better solution quality (up to 22% increase in NPV) when compared to standalone MCS with only 3% increase in computational costs. The incorporation of ROMs for well controls was shown to reduce computational cost by 96% with only 1% difference in solution quality when compared to FOM.
by Zeid M. Alghareeb.
Ph. D. in Computational Science for Energy Resources Engineering

The standard Fluid Structure Interaction coupling (velocity--pressure/displacement) is compared against two novel types of coupling, a Two Field coupling (velocity--pressure/displacement--pressure) and a Three Field coupling (velocity--pressure--stress/displacement--pressure--stress) in this way completing our set of, what we call, field to field equations, all stabilized by means of the VMS method using dynamic and orthogonal sub-scales. The solid Two field fluid structure interaction coupling formulation is benchmarked statically and dynamically. POD is applied to all three fluid structure interaction formulations to obtain reduced basis and asses their performance in a reduced space. Usual numerical benchmarks are shown comparing all three formulations. The three field fluid structure interaction coupling proves to provide very accurate results in both FOM and ROM spaces, making it a reliable formulation. Field to field pairing appears to be beneficial providing more accurate results in all cases shown. A reduced order model designed by means of a variational multi-scale method stabilized formulation has been applied successfully to fluid structure interaction problems in a strongly coupled partitioned solution scheme. Details of the formulation and the implementation both for the interaction problem and for the reduced models, for both the off-line and on-line phases, are shown. Results are obtained for cases in which both domains are reduced at the same time. Numerical results are presented for a semi-stationary and a fully transient case.
El acople estandar para casos de Interacción Fluido Estructura (Velocidad-Presión/Desplazamiento) se compara contra dos nuevas formas de acople, el primero de Dos Campos (Velocidad-Presión/Desplazamiento-Presión) y el segundo de Tres Campos (Velocidad-Presión-Esfuerzo/Desplazamiento-Presión-Esfuerzo) de esta forma completando lo que se ha llamado acoplamiento de Campo a Campo, todo estabilizado por medio del método VMS usando sub-escalas dínamicas y ortogonales. Se hacen comprobaciones estáticas y dínamicas para las dos nuevas formulaciones de sólidos (Dos y Tres campos). Se utiliza POD para obtener una base reducida y verificar el comportamiento de dichas formulaciones en el espacio reducido. La formulacion de Tres Campos resulta ser la mas precisa produciendo los resultados mas exactos tanto para los espacios FOM y ROM. La formulacion de Campo a Campo resulta ser beneficiosa al producir los resultados mas exactos en todas las pruebas realizadas. Un modelo estabilizado de orden reducido por medio del método de VMS ha sido aplicado satisfactoriamente a problemas de Interacción Fluido-Estructura en un modelos particionado de acople fuerte. Se muestran detalles de la formulación y su implementación tanto para casos de Interacción como para Problemas Reducidos para las fases de cálculo de base y ejecución del modelo. Se han obtenido resultados para problemas de Interacción en el cual se reducen ambos dominios al mismo tiempo. Se presentan resultados numéricos para ejemplos semi-transitorios y totalmente dinámicos.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2000.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 138-143).
Aeroelasticity is a critical consideration in the design of gas turbine engines, both for stability and forced response. Current aeroelastic models cannot provide high-fidelity aerodynamics in a form suitable for design or control applications. In this thesis low-order, high-fidelity aerodynamic models are developed using systematic model order reduction from computational fluid dynamic (CFD) methods. Reduction techniques are presented which use the proper orthogonal decomposition, and also a new approach for turbomachinery which is based on computing Arnoldi vectors. This method matches the input/output characteristic of the CFD model and includes the proper orthogonal decomposition as a special case. Here, reduction is applied to the linearized two-dimensional Euler equations, although the methodology applies to any linearized CFD model. Both methods make efficient use of linearity to compute the reduced-order basis on a single blade passage. The reduced-order models themselves are developed in the time domain for the full blade row and cast in state-space form. This makes the model appropriate for control applications and also facilitates coupling to other engine components. Moreover, because the full blade row is considered, the models can be applied to problems which lack cyclic symmetry. Although most aeroelastic analyses assume each blade to be identical, in practice variations in blade shape and structural properties exist due to manufacturing limitations and engine wear. These blade to blade variations, known as mistuning, have been shown to have a significant effect on compressor aeroelastic properties. A reduced-order aerodynamic model is developed for a twenty-blade transonic rotor operating in unsteady plunging motion, and coupled to a simple typical section structural model. Stability and forced response of the rotor to an inlet ow disturbance are computed and compared to results obtained using a constant coefficient model similar to those currently used in practice. Mistuning of this rotor and its effect on aeroelastic response is also considered. The simple models are found to inaccurately predict important aeroelastic results, while the relevant dynamics can be accurately captured by the reduced-order models with less than two hundred aerodynamic states. Models are also developed for a low-speed compressor stage in a stator/rotor configuration. The stator is shown to have a significant destabilizing effect on the aeroelastic system, and the results suggest that analysis of the rotor as an isolated blade row may provide inaccurate predictions.
by Karen Elizabeth Willcox.
Ph.D.

Thesis submitted in fulfilment of the requirements for the degree Master of Technology: Electrical Engineering in the Faculty of Engineering at the Cape Peninsula University of Technology 2013
The demand for electrical energy has kept on increasing, thus causing power systems to be more complex and bringing the challenging problems of electrical energy generation, transmission, stability, as well as storage to be examined more thoroughly. With the advent of high-speed computation and the desire to analyze increasingly complex behaviour in power systems, simulation techniques are gaining importance and prevalence. Nevertheless, while simulations of large, interconnected complex power systems are feasible, they remain time-consuming. Moreover, the models and parameters used in simulations are uncertain, due to measurement uncertainty, the need to represent a complex behaviour with low-order models, and the inherent changing nature of the power system. This research explores the use of a model reduction technique and the applications of a Real-Time Digital Simulator (RTDS) to reduce the uncertainty in large-scale complex power system models. The main goal of the research is to develop a reduced order model and to investigate the applications of the RTDS simulator in reduction of large, interconnected power systems models. The first stage of the study is to build and simulate the full model of the power system using the DigSILENT and RTDS simulators. The second phase is to apply model reduction technique to the full model and to determine the parameters in the reduced-order model as well as how the process of reduction increases this model uncertainty. In the third phase the results of the model reduction technique are compared based on the results of the original model - IEEE standard benchmark models has been used. The RTDS was used for comparative purposes. The thesis investigations use a particular model reduction technique as Coherency based Method. Though the method ideas are applicable more generally, a concrete demonstration of its principles is instructive and necessary. Further, while this particular technique is not relevant to every system, it does apply to a broad class of systems and illustrates the salient features of the proposed methodology. The results of the thesis can be used in the development of reduced models of complex power systems, simulation in real-time during power system operation, education at universities, and research. Keywords: IEEE benchmark models, reduced models, Coherency based Method, DigSILENT, RTDS, model uncertainty, power system stability

The present thesis work concerns the study of Monte Carlo (MC)-based data assimilation methods applied to the numerical simulation of complex hydrological models with stochastic parameters. The ensemble Kalman filter (EnKF) and the sequential importance resampling (SIR) are implemented in the CATHY model, a solver that couples the subsurface water flow in porous media with the surface water dynamics. A detailed comparison of the results given by the two filters in a synthetic test case highlights the main benefits and drawbacks associated to these techniques. A modification of the SIR update is suggested to improve the performance of the filter in case of small ensemble sizes and small variances of the measurement errors. With this modification, both filters are able to assimilate pressure head and streamflow measurements and correct model errors, such as biased initial and boundary conditions. SIR technique seems to be better suited for the simulations at hand as they do not make use of the Gaussian approximation inherent the EnKF method. Further research is needed, however, to assess the robustness of the particle filters methods in particular to ensure accuracy of the results even when relatively small ensemble sizes are employed. In the second part of the thesis the focus is shifted to reducing the computational burden associated with the construction of the MC realizations (which constitutes the core of the EnKF and SIR). With this goal, we analyze the computational saving associated to the use of reduced order models (RM) for the generation of the ensemble of solutions. The proper orthogonal decomposition (POD) is applied to the linear equations of the groundwater flow in saturated porous media with a randomly distributed recharge and random heterogeneous hydraulic conductivity. Several test cases are used to assess the errors on the ensemble statistics caused by the RM approximation. Particular attention is given to the efficient computation of the principal components that are needed to project the model equations in the reduced space. The greedy algorithm selects the snapshots in the set of the MC realizations in such a way that the final principal components are parameter independent. An innovative residual-based estimation of the error associated to the RM solution is used to assess the precision of the RM and to stop the iterations of the greedy algorithm. By way of numerical applications in synthetic and real scenarios, we demonstrate that this modified greedy algorithm determines the minimum number of principal components to use in the reduction and, thus, leads to important computational savings.
Questo lavoro di tesi riguarda lo studio di tecniche di assimilazione di dati basate sul metodo di Monte Carlo (MC) per la simulazione numerica di modelli idrologici in presenza di parametri stocastici. I metodi ensemble Kalman filter (EnKF) e sequential importance resampling (SIR) sono implementati nel modello CATHY, un modello idrologico che accoppia il flusso d'acqua sotterraneo in mezzi porosi con la dinamica del flusso d’acqua superficiale. Il confronto dettagliato dei risultati ottenuti con i due filtri in un caso test sintetico evidenzia i principali vantaggi e inconvenienti associati a queste tecniche. Per migliorare le prestazioni del metodo SIR, in questa tesi è proposta una modifica del passo di update che risulta fondamentale nei casi in cui si usi un ensemble di dimensioni ridotte e la varianza associata all'errore di misura sia piccola. Grazie a questa modifica, entrambi i filtri sono in grado di assimilare misure di carico piezometrico e portata, riducendo la propagazione temporale di errori di modellizzazione dovuti, ad esempio, all'utilizzo di condizioni iniziali o al contorno distorte. La tecnica SIR sembra essere più adeguata dell'EnKF per l’applicazione ai casi test presentati. Si dimostra infatti che l'ipotesi di Gaussianità, che contraddistingue il metodo EnKF, non è soddisfatta in questi casi test, rendendo preferibili metodi più generali come il SIR. Ulteriori approfondimenti sono comunque necessari per stabilire l'affidabilità dei metodi di tipo particle filter, in particolare per garantire l'accuratezza del filtro SIR anche quando viene usato un numero relativamente piccolo di realizzazioni. Siccome il passo di previsione dei metodi SIR ed EnKF è basato sul metodo di MC, la seconda parte della tesi riguarda il problema di ridurre gli onerosi tempi di calcolo associati alla costruzione delle realizzazioni di MC. Con questo obbiettivo, si analizza il risparmio in tempo computazione ottenuto dall'uso di modelli di ordine ridotto (RM) per la generazione dell'ensemble delle soluzioni. La tecnica proper orthogonal decomposition (POD) è applicata alle equazioni lineari del flusso d’acqua sotterraneo in mezzi porosi saturi con ricarica stocastica e distribuita spazialmente, oppure con conducibilità idraulica stocastica e descritta per zone. Gli errori di approssimazione introdotti dal modello ridotto sul calcolo delle singole realizzazioni di MC e sulle corrispondenti statistiche sono analizzati in diversi casi test al variare della distribuzione probabilistica dei parametri stocastici. Particolare attenzione è dedicata alla procedura di calcolo delle principal components che sono necessarie per la proiezione delle equazioni del modello nello spazio ridotto. Il greedy algorithm seleziona gli snapshots tra le realizzazioni di MC considerate, facendo in modo che le principal components finali siano indipendenti dalla particolare realizzazione dei parametri stocastici. Infine, viene introdotta una stima innovativa della norma dell'errore associato alla soluzione del modello ridotto. Tale stima, basata sul calcolo del residuo, è di fondamentale importanza per stimare la precisione del RM e, quindi, inferire sul numero di principal components da usare nella riduzione. Le applicazioni numeriche effettuate su casi test sintetici e reali dimostrano che il greedy algorithm così modificato determina un numero minore di principal components rispetto al metodo tradizionale, pur mantenendo la medesima accuratezza.

Even though the power of supercomputers has increased extraordinarily, there is still an insatiable need for more advanced multi-disciplinary CFD simulations in the aircraft analysis and design fields. A particular current interest is in the realistic three-dimensional fully viscous turbulent flow simulation of the highly non-linear aspects of aero-icing. This highly complex simulation is still computationally too demanding in industry, especially when several runs, such as parametric studies, are needed. In order to make such compute-intensive simulations more affordable, this work presents a reduced order modeling approach, based on the "Proper Orthogonal Decomposition", (POD), method to predict a wider swath of flow fields and ice shapes based on a limited number of "snapshots" obtained from complete high-fidelity CFD computations. The procedure of the POD approach is to first decompose the fields into modes, using a limited number of full-calculations snapshots, and then to reconstruct the field and/or ice shapes using those decomposed modes for other conditions, leading to reduced order calculations. The use of the POD technique drastically reduces the computational cost and can provide a more complete map of the performance degradation of an iced aircraft over a wide range of flight and weather conditions.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 100-106).
Subsurface flow modeling is an indispensable task for reservoir management, but the associated computational cost is burdensome owing to model complexity and the fact that many simulation runs are required for its applications such as production optimization, uncertainty quantification, and history matching. To relieve the computational burden in reservoir flow modeling, a reduced-order modeling procedure based on hyper-reduction is presented. The procedure consists of three components: state reduction, constraint reduction, and nonlinearity treatment. State reduction based on proper orthogonal decomposition (POD) is considered, and the impact of state reduction, with different strategies for collecting snapshots, on accuracy and predictability is investigated. Petrov- Galerkin projection is used for constraint reduction, and a hyper-reduction that couples the Petrov-Galerkin projection and a 'gappy' reconstruction is applied for the nonlinearity treatment. The hyper-reduction method is a Gauss-Newton framework with approximated tensors (GNAT), and the main contribution of this study is the presentation of a procedure for applying the method to subsurface flow simulation. A fully implicit oil-water two-phase subsurface flow model in three-dimensional space is considered, and the application of the proposed hyper-reduced-order modeling procedure achieves a runtime speedup of more than 300 relative to the full-order method, which cannot be achieved when only constraint reduction is adopted. In addition, two types of sequential Bayesian filtering for history matching are considered to investigate the performance of the developed hyper-reduced-order model to relive the associated computational cost. First, an ensemble Kalman filter (EnKF) is considered for Gaussian system and a procedure embedding the hyper-reduced model (HRM) into the EnKF is presented. The use of the HRM for the EnKF significantly reduces the computational cost without much loss of accuracy, but the combination requires a few remedies such as clustering to find an optimum reduced-order model according to spatial similarity of geological condition, which causes an additional computation. For non-Gaussian system, an advanced particle filter, known as regularized particle filter (RPF), is considered because it does not take any distributional assumptions. Particle filtering has rarely been applied for reservoir history matching due to the fact that it is hard to locate the initial particles on highly probable regions of state spaces especially when large scale system is considered, which makes the required number of particles scale exponentially with the model dimension. To resolve the issues, reparameterization is adopted to reduce the order of the geological parameters. For the reparameterization, principal component analysis (PCA) is used to compute the reduced space of the model parameters, and by constraining the filtering analysis with the computed subspace the required number of initial particles can be reduced down to a manageable level. Consequently, a huge computational saving is achieved by embedding the HRM into the RPF. Furthermore, the additional cost of clustering required to identify the geospatially optimum reduced-order model is saved because the advanced particle filter allows to easily identify the groups of geospatially similar particles.
by Seonkyoo Yoon.
Ph. D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.
Includes bibliographical references (leaves 209-213).
With the advent of high-speed computation and the desire to analyze increasingly complex behavior in power systems, simulation techniques are gaining importance and prevalence. However, while simulations of large, interconnected power systems are feasible, they remain time-consuming. Additionally, the models and parameters used in simulations are uncertain, due to measurement uncertainty, the need to approximate complex behavior with low-order models and the inherent changing nature of the power system. This thesis explores the use of model reduction techniques to enable the study of uncertainty in large-scale power system models. The main goal of this thesis is to demonstrate that uncertainty analyses of transient simulations of large, interconnected power systems are possible. To achieve this, we demonstrate that a basic three stage approach to the problem yields useful results without significantly increasing the computational burden. The first stage is to reduce the order of the original power system model, which reduces simulation times and allows the system to be simulated multiple times in a reasonable time-frame. Second, the mechanics of the model reduction are closely studied; how uncertainties affect the reduction process and the parameters in the reduced-order model as well as how the process of reduction increases uncertainty are of particular interest. Third, the reduced-order model and its accompanying uncertainty description are used to study the uncertainty of the original model. Our demonstration uses a particular model reduction technique, synchronic modal equivalencing (SME), and a particular uncertainty analysis method, the probabilistic collocation method (PCM). Though our ideas are applicable more generally, a concrete demonstration of the principle is instructive and necessary. Further, while these particular techniques are not relevant to every system, they do apply to a broad class of systems and illustrate the salient features of our methodology. As mentioned above, a detailed analysis of the model reduction technique, in this case SME, is necessary. As an ancillary benefit of the thesis work, interesting theoretical results relevant to the SME algorithm, which is still under development, are derived.
by James R. Hockenberry.
Ph.D.

Le coût d’une simulation numérique aéroélastique peut devenir trop onéreuse lorsque une analyse paramétrique à haut fidélité est requise. Dans ce contexte, des Modèles d'Ordre Réduit (MOR) ont été développés en vue de réduire le coût de calcul des simulations numériques en préservant un haut niveau de précision. Ce travail de thèse porte sur la construction d'un MOR pour les équations de Navier-Stokes en tenant compte d'un maillage déformable dans le cas d'une application aéroélastique. Une base modale pour l'écoulement est obtenue via la Décomposition Orthogonale aux valeurs propres et une projection Galerkin est utilisée pour réduire le système d'équations de la mécanique des fluides. Pour pouvoir prendre en compte les non-linéarités des équation de Navier-Stokes une méthode de projection masquée est mise en œuvre et évaluée pour différent cas test avec maillage fixe. Le MOR est ensuite adapté pour prendre en compte des maillages déformables. Finalement, une méthode réduite spectrale en temps (ROTSM) a été formulée afin de répondre aux problèmes de stabilité qui concernent le MORs avec projection dans le domaine de la mécanique des fluides. Une évaluation du MOR obtenu est ensuite menée sur des études paramétriques pour des applications aéroélastiques
The numerical prediction of aeroelastic systems responses becomes unaffordable when parametric analyses with high-fidelity CFD are required. Reduced order modeling (ROM) methods have therefore been developed in view of reducing the costs of the numerical simulations while preserving a high level of accuracy. The present thesis focuses on the family of projection based methods for the compressible Navier-Stokes equations involving deforming meshes in the case of aeroelastic applications. A vector basis obtained by Proper Orthogonal Decomposition (POD) combined to a Galerkin projection of the system equations is used in order to build a ROM for fluid mechanics. Masked projection approaches are therefore implemented and assessed for different test cases with fixed boundaries in order to provide a fully nonlinear formulation for the projection-based ROMs. Then, the ROM is adapted in the case of deforming boundaries and aeroelastic applications in a parametric context. Finally, a Reduced Order Time Spectral Method (ROTSM) is formulated in order to address the stability issues which involve the projection-based ROMs for fluid mechanics applications

Lorsqu'elles vibrent avec de grandes amplitudes, les structures minces montrent un comportement non linéaire géométrique, provenant de la relation non linéaire entre les déformations et les déplacements. Les analyses des systèmes complets font appel à des calculs extrêmement couteux de telle sorte que l'établissement de modèles d'ordre réduit efficaces est un sujet d'intérêt majeur pour le calcul prédictif de vibrations de structures minces.Dans cette thèse, des méthodes non linéaires de réduction de modèle pour les structures discrétisées par la méthode des éléments finis et comportant une non-linéarité géométrique, sont étudiées. Trois méthodes non intrusives sont plus particulièrement examinées et systématiquement comparées: la méthode de condensation implicite, la méthode des dérivées modales, et la réduction sur variétés invariantes du système. Les analyses théoriques montrent que les deux premières méthodes ne peuvent donner de résultats fiables que sous hypothèse d'une séparation spectrale entre les fréquences propres des modes maitres et celles des modes esclaves. La méthode de réduction sur variétés invariantes permet quant à elle d'avoir une méthode directe, ne nécessitant pas de pré-calculs, ni d'hypothèses préalables sur les fréquences propres des modes esclaves, afin de fournir des résultats corrects.De nombreuses applications et de comparaisons numériques sont montrées sur diverses structures discrétisées avec la méthode des éléments finis. Pour appliquer la méthode des variétés invariantes, une méthode récemment développée, permet de proposer un calcul direct de la forme normale du problème, à partir de la base physique et donc des degrés de liberté du maillage éléments finis. Les exemples montrent clairement les avantage et inconvénients de chaque méthode, validant aussi les résultats théoriques montrés précédemment.Dans la dernière partie de la thèse, la dynamique non linéaire d'un système présentant une relation de résonance interne 1:2 est analysée, en tenant compte des termes cubiques. La forme normale réelle du problème est d'abord établie. Ensuite les branches de solution du problème sont analysées et comparées avec celles du modèle plus simple négligeant la non-linéarité cubique. Le comportement divergent observé lorsqu'on réduit le problème à un seul mode et que l'on cherche à prédire le comportement raidissant ou assouplissant, est ensuite étudié avec ce modèle plus complet
When vibrating with large amplitudes, thin structures experience geometric nonlinearity due to the nonlinear relationship between strains and displacements. Because full-order nonlinear analysis on geometrically nonlinear models are computationally very expensive, the derivation of efficient reduced-order models (ROMs) has always been a topic of interest.In this thesis, nonlinear reduction methods for building ROMs with geometric nonlinearity in the framework of the Finite Element (FE) procedure, are investigated. Three non-intrusive nonlinear reduction methods are specifically investigated and systematically compared. They are: implicit condensation and expansion (ICE), modal derivatives (MD), and the reduction to invariant manifold. Theoretical analysis shows that the first two methods can give reliable results only if a slow/fast assumption between slave and master coordinates holds. On the other hand, reduction to invariant manifolds allows proposing a simulation-free reduction method that can be applied without restricting assumptions on the frequencies of the slave modes.Numerical comparisons and numerous applications to continuous structures discretized with the FE procedure, are given subsequently. For application of the invariant manifold-based method, the computation is based on a direct application of the normal form to the physical space and hence to the nodes of the FE mesh, a method recently developed. The examples show the advantages and drawbacks of each reduction method when deriving ROM, and the results of the theoretical comparison are validated.Finally, the analysis of the dynamics of a system with 1:2 internal resonance and cubic nonlinearity is given in the last part of the thesis. The real normal form of the problem is first derived. Then the solution branches of the problem are investigated and compared to simpler solutions with the dynamics truncated at order two. The divergent behaviour of the hardening/softening characteristics for single-mode reduction is investigated with this more complete model

The lattice is a common frame applied in offshore structures benefiting from their economic construction and easily accessible, for example, the jacket platform of the offshore wind turbine. In order to accelerate the design process of a model, this thesis proposed a method to reduce the order of the offshore lattice structure models. The full order model is constructed based on the FEM. Applying the MDOF mass discrete concept lumps the structure mass on each story center. Meanwhile, apply the DSM to determine the displacement and the forces implied on the mass center for accomplishing the unitary displacement of each DOF of the mass center node. Finally, a reduced order model for the lattice structure will be obtained. That model composed of the stiffness matrix of the mass center node has the same fundamental flexural frequency as the original model. The model can be expressed by symbolic relationship as well as possible to package as a function for further quotation. All computation processes of this thesis were executed on the Python platform.

In aerodynamics, separated flows are often subject to strong instabilities which result in the shedding of large-scale vortices. Such low-frequency unsteadiness are commonly encountered in aeronautical applications and lead to detrimental effects such as severe structural vibrations or the generation of extensive noise pollution. This thesis aims at providing low-order models of such aerodynamic flows in order to design optimal control devices. The transitional backward-facing step flow is considered as a prototype of unstable separated flow. Firstly, the linear flow dynamics are examined using a global stability analysis. The flow is found to selectively amplify the upstream noise through the Kelvin-Helmholtz instability. Next, we use projection methods to construct low-order models of the linear two-dimensional dynamics of the flow. Three approaches are investigated: (i) the use of the least damped global modes, (ii) the Proper Orthogonal Decomposition (POD) and (iii) the balanced truncation. This thesis introduces a snapshot method in the frequency domain to compute the leading controllable, observable and balanced modes, as well as techniques to handle large fluid systems. Lastly, the question of the closed-loop control of the flow is addressed. An effective reduction of the perturbations is obtained by using a Linear Quadratic Gaussian compensator designed from a POD model
En aérodynamique, les écoulements décollés sont souvent sujets à de fortes instabilités qui provoquent l'apparition de grosses structures tourbillonnaires. Ces écoulements caractérisés par des instationnarités à basses fréquences sont couramment observés dans les applications aéronautiques et entraînent des effets néfastes tels que d'importantes vibrations des structures ou la génération de bruit. Cette thèse a pour objectif de fournir des modèles d'ordre réduit de tels écoulements aérodynamiques dans le but de concevoir des dispositifs de contrôle optimaux. Un écoulement transitionnel de marche descendante est considéré comme prototype d'écoulement décollé instable. Dans un premier temps, la dynamique linéaire de l'écoulement est étudiée à l'aide d'une analyse de stabilité globale. Nous montrons que l'écoulement amplifie de manière sélective le bruit amont par l'instabilité de Kelvin-Helmholtz. Ensuite, nous utilisons des méthodes de projection pour construire des modèles d'ordre réduit de la dynamique linéaire bidimensionnelle de l'écoulement. Trois approches sont étudiées : (i) l'utilisation des modes globaux les moins stables, (ii) la Décomposition Orthogonale Propre (POD) et (iii) la troncature équilibrée. Cette thèse introduit une méthode des clichés dans le domaine fréquentiel pour calculer les modes contrôlables, observables et équilibrés dominants, ainsi que des techniques pour traiter les systèmes fluides de grande taille. Finalement, nous traitons la question du contrôle en boucle fermée de l'écoulement. Une réduction conséquente des perturbations est obtenue en utilisant une commande Linéaire Quadratique Gaussienne conçue à partir d'un modèle POD

In addition to serving as a brief overview of aspects relevant to reduced-order modeling (in particular balanced-state and modal techniques) as applied to structural finite element models, this work produced tools for visualizing the relationship between the modes of a model and the states of its balanced representation. Specifically, error contour and mean error plots were developed that provide a designer with frequency response information absent from a typical analysis of a balanced model via its Hankel singular values. The plots were then used to analyze the controllability and observability aspects of finite element models of an illustrative system from a modal perspective -- this aided in the identification of computational artifacts in the models and helped predict points at which to halt the truncation of balanced states. Balanced reduced-order reference models of the illustrative system were implemented as part of a direct adaptive control algorithm to observe the effectiveness of the models. It was learned that the truncation point selected by observing the mean error plot produced the most satisfactory results overall -- the model closely approximated the dominant modes of the system and eliminated the computational artifacts. The problem of improving the performance of the system was also considered. The truncated balanced model was recast in modal form so that its damping could be increased, and the settling time decreased by about eighty percent.

Cette thèse de doctorat est consacrée à l'étude des problèmes directe et inverse en électrophysiologie cardiaque. Comme les équations qui décrivent l'activité électrique du coeur peuvent être très couteuses en temps de calcul, une attention particulière est apportée aux méthodes d'ordre réduit et à leur applications aux modèles de l'électrophysiologie.Dans un premier temps, nous introduisons les modèles mathématiques et numériques de l'électrophysiologie cardiaque. Ces modèles nous permettent de réaliser des simulations numériques que nous validons à l'aide de plusieurs critères qualitatifs et quantitatifs trouvés dans la littérature médicale. Comme notre modèle prend en compte les oreillettes et les ventricules, nous sommes capables de reproduire des cycles complets d'électrocardiogrammes (ECG) à la fois dans des conditions saines et dans des cas pathologiques.Ensuite, plusieurs méthodes d'ordre réduit sont étudiées pour la résolution des équations de l'électrophysiologie. La méthode Proper Orthogonal Decomposition (POD) est appliquée pour la discrétisation des équations de l'électrophysiologie dans plusieurs configurations, comme par exemple la simulation d'un infarctus du myocarde. De plus, cette méthode est utilisée pour résoudre quelques problèmes d'identification de paramètres comme localiser un infarctus à partir de mesures d'un électrocardiogramme ou simuler une courbe de restitution. Pour contourner les limitations de la POD, une nouvelle méthode basée sur des couples de Lax approchés (Approximated Lax Pairs, ALP) est utilisée. Cette méthode est appliquée aux problèmes directe et inverse. Pour finir, un nouvel algorithme, basé sur les méthodes ALP et l'interpolation empirique discrète, est proposé. Cette nouvelle approche améliore significativement l'efficacité de l'algorithme original ALP et nous permet de considérer des modèles plus complexes utilisés en électrophysiologie cardiaque
This PhD thesis is dedicated to the investigation of the forward and the inverse problem of cardiac electrophysiology. Since the equations that describe the electrical activity of the heart can be very demanding from a computational point of view, a particular attention is paid to the reduced order methods and to their application to the electrophysiology models. First, we introduce the mathematical and numerical models of electrophysiology and we implement them to provide for simulations that are validated against various qualitative and quantitative criteria found in the medical literature. Since our model takes into account atria and ventricles, we are able to reproduce full cycle Electrocardiograms (ECG) in healthy configurations and also in the case of several pathologies. Then, several reduced order methods are investigated for the resolution of the electrophysiology equations. The Proper orthogonal Decomposition (POD) method is applied for the discretization of the electrophysiology equations in several configurations, as for instance the simulation of a myocardial infarction. Also, the method is used in order to solve some parameters identification problems such as the identification of an infarcted zone using the Electrocardiogram measures and for the efficient simulation of restitution curves. To circumvent some limitations of the POD method, a new reduced order method based on the Approximated Lax Pairs (ALP) is investigated. This method is applied to the forward and inverse problems. Finally, a new reduced order algorithm is proposed, based on the ALP and the Discrete Empirical Interpolation methods. This new approach significantly improves the efficiency of the original ALP algorithm and allow us to consider more complex models used in electrophysiology

Numerical simulation tools are generally used in all modern engineering fields, especially those having difficulties in performing large number of practical experiments, such as biomechanics. Among the computational methods, Finite Element (FE) is an essential tool. Nowadays, the fast-growing computational techniques, from the upgrading hardware to the emerging of novel algorithm, have already enabled extensive applications in biomechanics. For applications that require fast response and/or multiple queries, Reduced Order Modelling (ROM) methods have been developed based on existing methods such as FE, and have eventually enabled real-time numerical simulation for a large variety of engineering problems. In this thesis, several novel computational techniques are developed to explore the capability of Proper Generalised Decomposition (PGD), which is an important approach of ROM. To assess the usability of the PGD-based ROM for biomechanical applications, a real human femur bone is chosen to study its mechanical behaviour as an example. Standard image-based modelling procedure in biomechanics is performed to create an FE model which is then validated with in vitro experimental results. As a basis of this work, the medical image processing has to be performed, in order to generate an available FE model. This model is validated according to data collected from a previously performed \textit{in vitro} experimental test. The full procedure of image-based model generation and the validation of generated model is described in Chapter 2. As a major objective of this thesis, a non-intrusive scheme for the PGD framework is developed in Chapter 3. It is implemented using in-house developed Matlab (Mathworks, USA) code to conduct the PGD work flow, and calling Abaqus as an external solver for devised fictitious mechanical problems. The transformation of data from computed tomography (CT) image set to FE model including inhomogeneous material properties is subjected to some physical constraints, and when applying the load, there are also geometric constraints limiting the locations where load could be applied. These constraints will lead to a constrained parameter space, which possibly has difficulty to be separated in a Cartesian fashion. Therefore, a novel strategy to separate the parameters in a collective manner is proposed in Chapter 4. Chapter 5 details a comprehensive application in biomechanics, the methodologies proposed in Chapter 3 and 4 are applied on the practical model generated in Chapter 2. As a typical application of the PGD vademecum, a material property identification problem is discussed. Further PGD vademecum is generated using the identified material properties with variable loading locations, and with this vademecum, real-time mechanical response of the femur is available. In addition, for the purpose of extending the methodologies to orthotropic materials, which is commonly used in biomechanics, in Chapter 6 another linear elastic model is investigated with the non-intrusive PGD scheme. Nowadays, isogeometric analysis (IGA) is a very popular tool in computational mechanics. It is appealing to take advantage of non-uniform rational B-splines (NURBS) to discretise the model. For PGD, using B-splines for the discretisation of the parameter space could improve the quality of vademecum, especially for problems involving sensitivities with respect to the parameters during the online computations. It is important and necessary to extend the PGD framework to nonlinear solid mechanics, because most biological soft tissues have been observed nonlinear mechanical behaviours. Consequently, in Chapter 7 we have developed a PGD framework for the St.Venant-Kirchhoff constitutive model using the Picard linearisation which is consistent with the fixed-point iteration algorithm commonly used in PGD. In Chapter 8, conclusive remarks are addressed as well as forecasts of possible future works.

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2004.
Includes bibliographical references (p. 75-79).
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Computational fluid dynamics (CFD) is now widely used throughout the fluid dynamics community and yields accurate models for problems of interest. However, due to its high computational cost, CFD is limited for some applications. Therefore, model reduction has been used to derive low-order models that replicate CFD behavior over a restricted range of inputs, and various frameworks have been developed. Unfortunately, the majority of those methods are limited to linear cases and do not properly handle reduction of nonlinear systems. In order to overcome restrictions of weak nonlinearity and the costly representation of the system's nonlinearity found in other nonlinear reduction approaches, a trajectory piecewise-linear (TPWL) scheme is developed for a CFD model of the two-dimensional Euler equations. The approach uses a weighted combination of linearized models to represent the nonlinear CFD system. Using a set of training trajectories obtained via a simulation of the nonlinear CFD model, algorithms are presented for linearization point selection and weighting of the models. Using the same training trajectories to provide a snapshot ensemble, the proper orthogonal decomposition (POD) is used to create a reduced-space basis, onto which the TPWL model is projected. This projection yields an efficient reduced-order model of the nonlinear system, which does not require the evaluation of any full-order system residuals, while capturing a large portion of the nonlinear space. The method is applied to the case of flow through an actively controlled supersonic diffuser. Convergence of the TPWL approach is presented for both full-order and reduced-order cases.
(cont.) The TPWL approach and the POD combine naturally to form an efficient reduction procedure and the methodology is found to yield accurate results, including cases with significant shock motion. Reduced-order PWL models are shown to be three orders of magnitude more efficient than the nonlinear CFD for simulation of a representative test case.
by David Gratton.
S.M.

Lean-premixed combustion has the advantage of low emissions for modern gas turbines, but it is susceptible to thermoacoustic instabilities, which can result in large amplitude pressure oscillations in the combustion chamber. The thermoacoustic limit cycle is generated by the unsteady heat release dynamics coupled to the combustor acoustics. In this dissertation, we focused on reduced-order modeling of the dynamics of a laminar premixed flame. From first principles of combustion dynamics, a physically-based, reduced-order, nonlinear model was developed based on the proper orthogonal decomposition technique and generalized Galerkin method. In addition, the describing function for the flame was measured experimentally and used to identify an empirical nonlinear flame model. Furthermore, a linear acoustic model was developed and identified for the Rijke tube experiment. Closed-loop thermoacoustic modeling using the first principles flame model coupled to the linear acoustics successfully reproduced the linear instability and predicted the thermoacoustic limit cycle amplitude. With the measured experimental flame data and the modeled linear acoustics, the describing function technique was applied for limit cycle analysis. The thermoacoustic limit cycle amplitude was predicted with reasonable accuracy, and the closed-loop model also predicted the performance for a phase shift controller. Some problems found in the predictions for high heat release cases were documented.
Ph. D.

L’objectif final étant de développer des modèles d’ordre réduit pour les applications de combustion, des techniques d’apprentissage automatique non supervisées et supervisées ont été testées et combinées dans les travaux de la présente thèse pour l’extraction de caractéristiques et la construction de modèles d’ordre réduit. Ainsi, l’application de techniques pilotées par les données pour la détection des caractéristiques d’ensembles de données de combustion turbulente (simulation numérique directe) a été étudiée sur deux flammes H2 / CO: une évolution spatiale (DNS1) et une jet à évolution temporelle (DNS2). Des méthodes telles que l’analyse en composantes principales (ACP), l’analyse en composantes principales locales (LPCA), la factorisation matricielle non négative (NMF) et les autoencodeurs ont été explorées à cette fin. Il a été démontré que divers facteurs pouvaient affecter les performances de ces méthodes, tels que les critères utilisés pour le centrage et la mise à l’échelle des données d’origine ou le choix du nombre de dimensions dans les approximations de rang inférieur. Un ensemble de lignes directrices a été présenté qui peut aider le processus d’identification de caractéristiques physiques significatives à partir de données de flux réactifs turbulents. Des méthodes de compression de données telles que l’analyse en composantes principales (ACP) et les variations ont été combinées à des méthodes d’interpolation telles que le krigeage, pour la construction de modèles ordonnées à prix réduits et calculables pour la prédiction de l’état d’un système de combustion dans des conditions de fonctionnement inconnues ou des combinaisons de modèles valeurs de paramètre d’entrée. La méthodologie a d’abord été testée pour la prévision des flammes 1D avec un nombre croissant de paramètres d’entrée (rapport d’équivalence, composition du carburant et température d’entrée), avec des variantes de l’approche PCA classique, à savoir PCA contrainte et PCA locale, appliquée aux cas de combustion la première fois en combinaison avec une technique d’interpolation. Les résultats positifs de l’étude ont conduit à l’application de la méthodologie proposée aux flammes 2D avec deux paramètres d’entrée, à savoir la composition du combustible et la vitesse d’entrée, qui ont donné des résultats satisfaisants. Des alternatives aux méthodes non supervisées et supervisées choisies ont également été testées sur les mêmes données 2D. L’utilisation de la factorisation matricielle non négative (FNM) pour l’approximation de bas rang a été étudiée en raison de la capacité de la méthode à représenter des données à valeur positive, ce qui permet de ne pas enfreindre des lois physiques importantes telles que la positivité des fractions de masse d’espèces chimiques et comparée à la PCA. Comme méthodes supervisées alternatives, la combinaison de l’expansion du chaos polynomial (PCE) et du Kriging et l’utilisation de réseaux de neurones artificiels (RNA) ont été testées. Les résultats des travaux susmentionnés ont ouvert la voie au développement d’un jumeau numérique d’un four à combustion à partir d’un ensemble de simulations 3D. La combinaison de PCA et de Kriging a également été utilisée dans le contexte de la quantification de l’incertitude (UQ), en particulier dans le cadre de collaboration de données lié (B2B-DC), qui a conduit à l’introduction de la procédure B2B-DC à commande réduite. Comme pour la première fois, le centre de distribution B2B a été développé en termes de variables latentes et non en termes de variables physiques originales
With the final objective being to developreduced-order models for combustion applications,unsupervised and supervised machine learningtechniques were tested and combined in the workof the present Thesis for feature extraction and theconstruction of reduced-order models. Thus, the applicationof data-driven techniques for the detection offeatures from turbulent combustion data sets (directnumerical simulation) was investigated on two H2/COflames: a spatially-evolving (DNS1) and a temporallyevolvingjet (DNS2). Methods such as Principal ComponentAnalysis (PCA), Local Principal ComponentAnalysis (LPCA), Non-negative Matrix Factorization(NMF) and Autoencoders were explored for this purpose.It was shown that various factors could affectthe performance of these methods, such as the criteriaemployed for the centering and the scaling of theoriginal data or the choice of the number of dimensionsin the low-rank approximations. A set of guidelineswas presented that can aid the process ofidentifying meaningful physical features from turbulentreactive flows data. Data compression methods suchas Principal Component Analysis (PCA) and variationswere combined with interpolation methods suchas Kriging, for the construction of computationally affordablereduced-order models for the prediction ofthe state of a combustion system for unseen operatingconditions or combinations of model input parametervalues. The methodology was first tested forthe prediction of 1D flames with an increasing numberof input parameters (equivalence ratio, fuel compositionand inlet temperature), with variations of the classicPCA approach, namely constrained PCA and localPCA, being applied to combustion cases for the firsttime in combination with an interpolation technique.The positive outcome of the study led to the applicationof the proposed methodology to 2D flames withtwo input parameters, namely fuel composition andinlet velocity, which produced satisfactory results. Alternativesto the chosen unsupervised and supervisedmethods were also tested on the same 2D data.The use of non-negative matrix factorization (NMF) forlow-rank approximation was investigated because ofthe ability of the method to represent positive-valueddata, which helps the non-violation of important physicallaws such as positivity of chemical species massfractions, and compared to PCA. As alternative supervisedmethods, the combination of polynomial chaosexpansion (PCE) and Kriging and the use of artificialneural networks (ANNs) were tested. Results from thementioned work paved the way for the developmentof a digital twin of a combustion furnace from a setof 3D simulations. The combination of PCA and Krigingwas also employed in the context of uncertaintyquantification (UQ), specifically in the bound-to-bounddata collaboration framework (B2B-DC), which led tothe introduction of the reduced-order B2B-DC procedureas for the first time the B2B-DC was developedin terms of latent variables and not in terms of originalphysical variables

Numerical simulation tools are generally used in all modern engineering fields, especially those having difficulties in performing large number of practical experiments, such as biomechanics. Among the computational methods, Finite Element (FE) is an essential tool. Nowadays, the fast-growing computational techniques, from the upgrading hardware to the emerging of novel algorithm, have already enabled extensive applications in biomechanics, including mechanical analysis from musculoskeletal or cardiovascular system in macro scale to cell structures or tissue behaviours in micro scale. For applications that require fast response and/or multiple queries, Reduced Order Modelling (ROM) methods have been developed based on existing methods such as FE, and have eventually enabled real-time numerical simulation for a large variety of engineering problems. In this thesis, several novel computational techniques are developed to explore the capability of Proper Generalised Decomposition (PGD), which is an important approach of ROM. To assess the usability of the PGD-based ROM for biomechanical applications, a real human femur bone is chosen to study its mechanical behaviour as an example. Standard image-based modelling procedure in biomechanics is performed to create an FE model which is then validated with in vitro experimental results. As a major contribution, a non-intrusive scheme of the PGD framework is developed and implemented using commonly-used industrial software such as Matlab and Abaqus. It uses Abaqus as an external FE solver, which is called by in-house Matlab codes implementing the PGD algorithms. An example code is available at https://github.com/xizou/NIPGD. This scheme takes advantages of the maturity, robustness and availability of existing FE solvers, and demonstrates a great potential for being applied to industrial projects. To solve parametrised partial differential equations with a parameter space subjected to physical or geometric constraints, a novel strategy is proposed. This strategy provides an approach that collects the most correlated parameters, and then separates them into 2D/3D spaces, instead of separating the parameter space into tensor products of 1D spaces in a Cartesian fashion as it is done in conventional PGD framework. Inspired by the fast-developing methods of isogeometric analysis, it is interesting to borrow the isogeometric idea to exploit the ways of discretising the parameter space inside the PGD framework. The high continuity of B-spline shape functions enables more accurate results for the computation of sensitivities with respect to the parameters. A classical mechanical problem is investigated with orthotropic materials in 2D, with the intention of further application in biomechanics. In addition, an exploration of the generalisation of PGD to nonlinear problems in solid mechanics is presented as another main contribution. Following the large strain theory, Picard linearisation is used to establish a consistent PGD framework within total Lagrange formulation. As a preliminary example, the St.Venant-Kirchhoff constitutive model is adopted. A practical example of the femur bone simulation is provided, the material parameters are obtained through an identification problem using the PGD vademecum, and in a further step, another PGD vademecum is generated for real-time simulation accounting for various loading locations.
Numerical simulation tools are generally used in all modern engineering fields, especially those having difficulties in performing large number of practical experiments, such as biomechanics. Among the computational methods, Finite Element (FE) is an essential tool. Nowadays, the fast-growing computational techniques, from the upgrading hardware to the emerging of novel algorithm, have already enabled extensive applications in biomechanics, including mechanical analysis from musculoskeletal or cardiovascular system in macro scale to cell structures or tissue behaviours in micro scale. For applications that require fast response and/or multiple queries, Reduced Order Modelling (ROM) methods have been developed based on existing methods such as FE, and have eventually enabled real-time numerical simulation for a large variety of engineering problems. In this thesis, several novel computational techniques are developed to explore the capability of Proper Generalised Decomposition (PGD), which is an important approach of ROM. To assess the usability of the PGD-based ROM for biomechanical applications, a real human femur bone is chosen to study its mechanical behaviour as an example. Standard image-based modelling procedure in biomechanics is performed to create an FE model which is then validated with in vitro experimental results. As a major contribution, a non-intrusive scheme of the PGD framework is developed and implemented using commonly-used industrial software such as Matlab and Abaqus. It uses Abaqus as an external FE solver, which is called by in-house Matlab codes implementing the PGD algorithms. An example code is available at https://github.com/xizou/NIPGD. This scheme takes advantages of the maturity, robustness and availability of existing FE solvers, and demonstrates a great potential for being applied to industrial projects. To solve parametrised partial differential equations with a parameter space subjected to physical or geometric constraints, a novel strategy is proposed. This strategy provides an approach that collects the most correlated parameters, and then separates them into 2D/3D spaces, instead of separating the parameter space into tensor products of 1D spaces in a Cartesian fashion as it is done in conventional PGD framework. Inspired by the fast-developing methods of isogeometric analysis, it is interesting to borrow the isogeometric idea to exploit the ways of discretising the parameter space inside the PGD framework. The high continuity of B-spline shape functions enables more accurate results for the computation of sensitivities with respect to the parameters. A classical mechanical problem is investigated with orthotropic materials in 2D, with the intention of further application in biomechanics. In addition, an exploration of the generalisation of PGD to nonlinear problems in solid mechanics is presented as another main contribution. Following the large strain theory, Picard linearisation is used to establish a consistent PGD framework within total Lagrange formulation. As a preliminary example, the St.Venant-Kirchhoff constitutive model is adopted. A practical example of the femur bone simulation is provided, the material parameters are obtained through an identification problem using the PGD vademecum, and in a further step, another PGD vademecum is generated for real-time simulation accounting for various loading locations.

Cette thèse présente un algorithme automatique pour la construction d’un modèle NURBS volumique à partir d’un modèle représenté par ses bords (maillages ou splines). Ce type de modèle est indispensable dans le cadre de l’analyse isogéométrique utilisant les NURBS comme fonctions de forme. Le point d’entrée de l’algorithme est une triangulation du bord du modèle. Après deux étapes de décomposition, le modèle est approché par un polycube. Ensuite un paramétrage surfacique entre le bord du modèle et celui du polycube est établi en calculant un paramétrage global aligné à un champ de direction interpolant les directions de courbure principales du modèle. Finalement, le paramétrage volumique est obtenu en se basant sur ce paramétrage surfacique. Dans le contexte des études paramétriques basées sur des paramètres de formes géométriques, cette méthode peut être appliquée aux techniques de réduction de modèles pour obtenir la même représentation pour des objets ayant différentes géométries mais la même topologie
This thesis presents an effective method to automatically construct trivariate tensor-product spline models of complicated geometry and arbitrary topology. Our method takes as input a solid model defined by its triangulated boundary. Using cuboid decomposition, an initial polycube approximating the input boundary mesh is built. This polycube serves as the parametric domain of the tensor-product spline representation required for isogeometric analysis. The polycube's nodes and arcs decompose the input model locally into quadrangular patches, and globally into hexahedral domains. Using aligned global parameterization, the nodes are re-positioned and the arcs are re-routed across the surface in a way to achieve low overall patch distortion, and alignment to principal curvature directions and sharp features. The optimization process is based on one of the main contributions of this thesis: a novel way to design cross fields with topological (i.e., imposed singularities) and geometrical (i.e., imposed directions) constraints by solving only sparse linear systems. Based on the optimized polycube and parameterization, compatible B-spline boundary surfaces are reconstructed. Finally, the interior volumetric parameterization is computed using Coon's interpolation and the B-spline surfaces as boundary conditions. This method can be applied to reduced order modeling for parametric studies based on geometrical parameters. For models with the same topology but different geometries, this method allows to have the same representation: i.e., meshes (or parameterizations) with the same topology

An optimal solution of vortex-induced vibrations of structures would be a time-domain numerical simulation that simultaneously solves the fluid flow and structural response. Yet, the requirements in terms of computing power remains a major obstacle for implementing such a simulation. On the other hand, lower- or reduced-order models provide an alternative for determining structural response to forcing by fluid flow. The objective of this thesis is to provide a consistent approach for the development of reduced-order models for the lift and drag on oscillating cylinders and the identification of their parameters. Amplitudes and phases of higher-order spectral moments of the lift and drag coefficients data are combined with approximate solutions of the representative models to determine their parameters. The results show that the amplitude and phase of the trispectrum could be used to model the lift on the oscillating cylinder under different excitation conditions. Moreover, the amplitude and phase of the cross-bispectrum could be used to establish the lift-drag relation for oscillating cylinders. A forced van der Pol equation is used to represent the lift on a transversely oscillating cylinder, and a parametrically excited van der Pol equation is used to model the lift coefficient on an inline oscillating cylinder. All cases of excitations lead to close values for the damping and nonlinear parameters in the van der Pol equation. Consequently, and as shown in this thesis, different excitation cases could be used to identify the parameters in the governing equations. Moreover, the results show that the drag coefficient could be derived from the lift coefficient through a square relation that takes into account the effects of the forced motions.
Ph. D.

Mathematical models describing real world processes are becoming increasingly complex to better match the dynamics of the true system. While this is a positive step towards more complete knowledge of our world, numerical evaluations of these models become increasingly computationally inefficient, requiring increased resources or time to evaluate. This has led to the need for simplified surrogates to these complex mathematical models. A growing surrogate modeling solution is with the usage of neural networks. Neural networks (NN) are known to generalize an approximation across a diverse dataset and minimize the solution along complex nonlinear boundaries. Additionally, these surrogate models can be found using only incomplete knowledge of the true dynamics. However, NN surrogates often suffer from a lack of interpretability, where the decisions made in the training process are not fully understood, and the roles of individual neurons are not well defined. We present two solutions towards this lack of interpretability. The first focuses on mimicking polynomial chaos (PC) modeling techniques, modifying the structure of a NN to produce polynomial approximations of the underlying dynamics. This methodology allows for an extractable meaning from the network and results in improvement in accuracy over traditional PC methods. Secondly, we examine the construction of a reduced order modeling scheme using NN autoencoders, guiding the decisions of the training process to better match the real dynamics. This guiding process is performed via a physics-informed (PI) penalty, resulting in a speed-up in training convergence, but still results in poor performance compared to traditional schemes.
Master of Science
The world is an elaborate system of relationships between diverse processes. To accurately represent these relationships, increasingly complex models are defined to better match what is physically seen. These complex models can lead to issues when trying to use them to predict a realistic outcome, either requiring immensely powerful computers to run the simulations or long amounts of time to present a solution. To fix this, surrogates or approximations to these complex models are used. These surrogate models aim to reduce the resources needed to calculate a solution while remaining as accurate to the more complex model as possible. One way to make these surrogate models is through neural networks. Neural networks try to simulate a brain, making connections between some input and output given to the network. In the case of surrogate modeling, the input is some current state of the true process, and the output is what is seen later from the same system. But much like the human brain, the reasoning behind why choices are made when connecting the input and outputs is often largely unknown. Within this paper, we seek to add meaning to neural network surrogate models in two different ways. In the first, we change what each piece in a neural network represents to build large polynomials (e.g., $x^5 + 4x^2 + 2$) to approximate the larger complex system. We show that the building of these polynomials via neural networks performs much better than traditional ways to construct them. For the second, we guide the choices made by the neural network by enforcing restrictions in what connections it can make. We do this by using additional information from the larger system to ensure the connections made focus on the most important information first before trying to match the less important patterns. This guiding process leads to more information being captured when the surrogate model is compressed into only a few dimensions compared to traditional methods. Additionally, it allows for a faster learning time compared to similar surrogate models without the information.

In multibody dynamics simulation of robots and vehicles it is common to model the systems as being composed of mainly rigid bodies with articulation joints. With the trend to more lightweight robots, however, the structural flexibility of the robots link’s needs to be considered for realistic dynamic simulations. The link’s geometries are complex and finite element models (FEM) are required to compute the deformations. However, FEM includes too many degrees of freedom for time-efficient dynamics simulation. A popular method is to generate reduced order models from the FE models, but with much fewer degrees of freedom, for fast and precise simulations. In this thesis a method for introducing reduced order models in rigid multibody systems was developed. The method is to divide a rigid body into two rigid bodies. Their relative movement is described by a six degree of freedom restoration force, determined with a reduced order model from Guyan reduction (static condensation). The method was validated for quasistatic deformation of a homogenous beam, a robot link arm with a more complex geometry and in multibody dynamics simulations. Finally the method was tested in simulation of a complete ABB robot with joint actuators, and any significant differences in the motion of the robot tool centre point due to replacing a rigid link arm by a flexible one was demonstrated.The method show good results for computing deformations of the homogenous beam, of the link arm and in the multibody simulation. The differences observed in simulation of a complete robot was expected and demonstrated the method to be applicable in robotic simulations.

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; and (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.
Doctor of Philosophy
Reduced order models (ROMs) are popular in physical and engineering applications: for example, ROMs are widely used in aircraft designing as it can greatly reduce computational cost for the aircraft's aeroelastic predictions while retaining good accuracy. However, for high Reynolds number turbulent flows, such as blood flows in arteries, oil transport in pipelines, and ocean currents, the standard ROMs may yield inaccurate results. In this dissertation, to improve ROM's accuracy for turbulent flows, we investigate three different types of ROMs. In this dissertation, both numerical and theoretical results show that the proposed new ROMs yield more accurate results than the standard ROM and thus can be more useful.

Des réglementations de plus en plus strictes et un intérêt environnemental grandissant ont poussé les constructeurs de moteurs aéronautiques à développer la génération actuelle de chambres de combustion, affichant des consommations et émissions de polluants plus basses que jamais. Cependant, les phases de conception de chambres modernes ont clairement mis en évidence que celles-ci sont plus susceptibles de développer des instabilités de combustion, où le couplage entre l'acoustique de la chambre et la flamme suscite de larges oscillations de pression ainsi que des vibrations de la structure. Ces instabilités peuvent endommager le moteur, et potentiellement entraîner sa destruction. Dans le même temps, de considérables avancées ont eu lieu dans le domaine de la simulation numérique, et la Mécanique des Fluides Numérique (MFN) a démontré sa capacité à reproduire la dynamique de flammes instationnaires et les instabilités de combustion observées dans les moteurs. Pourtant, même avec le matériel informatique moderne, le temps de calcul reste la contrainte clé de ces simulations haute-fidélité, qui demeurent très coûteuses. Typiquement, couvrir la totalité du domaine de fonctionnement pour un moteur industriel est encore hors de portée. Des modèles dits bas-ordre existent également, et prédire efficacement les instabilités de combustion par leur intermédiaire est envisageable à la condition d'une modélisation appropriée de l'interaction entre l'acoustique et la flamme. La méthode de modélisation la plus commune de cet élément critique est la fonction de transfert de flamme (FTF) qui lie les fluctuations de taux de dégagement de chaleur aux fluctuations de vitesse en un point donné. Cette fonction de transfert peut être obtenue à partir de modèles analytiques, mais très peu existent pour des flammes swirlées turbulentes. Une autre approche consiste à réaliser des mesures expérimentales ou des simulations haute fidélité coûteuses, réduisant à néant la capacité de prédiction rapide recherchée avec les méthodes bas-ordre. Cette thèse vise donc à développer des outils bas ordre à la fois rapides et fiables pour la modélisation des instabilités de combustion, ainsi qu'à améliorer la compréhension des mécanismes inhérents à la réponse acoustique d'une flamme swirlée. A cet effet, une approche hybride nouvelle est proposée, où un nombre réduit de simulations haute fidélité peut être utilisé pour déterminer les paramètres d'entrée d'un modèle analytique représentatif de la fonction de transfert d'une flamme swirlée prémélangée. Le modèle analytique s'appuie sur des travaux antérieurs traitant la flamme comme une interface perturbée, et prend en compte la conversion acoustique-vorticité à travers un swirler. La validité du modèle est mise à l'épreuve en déterminant les divers paramètres nécessaires associés à partir de simulations numériques réactives stationnaires et pulsées d'une flamme prémélangée swirlée académique. Il est également démontré que le modèle peut prendre en compte diverses amplitudes de perturbation. Enfin, des simulations haute-fidélité 3D d'une turbine à gaz industrielle alimentée par un combustible liquide sont réalisées afin de déterminer s'il est possible de prédire numériquement un mode d'instabilité de combustion observé lors des essais. Pour cela, un ensemble de simulations forcées est mené à bien afin de souligner l'importance de l'acquisition de la réponse de la flamme diphasique, en comparant les positions de référence utilisées pour mesurer les vitesses fluctuantes ainsi que l'amplitude et l'origine de la perturbation acoustique. L'applicabilité du modèle analytique à ce cas complexe est aussi étudiée. Les résultats montrent que l'analyse acoustique proposée prédit bien la présence d'un mode instable, mais que le modèle bas ordre nécessite davantage de développements pour étendre son domaine de validité présumé
Increasingly stringent regulations as well as environmental concerns have lead gas turbine powered engine manufacturers to develop the current generation of combustors, which feature lower than ever fuel consumption and pollutant emissions. However, modern combustor designs have been shown to be prone to combustion instabilities, where the coupling between acoustics of the combustor and the flame results in large pressure oscillations and vibrations within the combustion chamber. These instabilities can cause structural damages to the engine or even lead to its destruction. At the same time, considerable developments have been achieved in the numerical simulation domain, and Computational Fluid Dynamics (CFD) has proven capable of capturing unsteady flame dynamics and combustion instabilities for aforementioned engines. Still, even with the current large and fast increasing computing capabilities, time remains the key constraint for these high fidelity yet computationally intensive calculations. Typically, covering the entire range of operating conditions for an industrial engine is still out of reach. In that respect, low order models exist and can be efficient at predicting the occurrence of combustion instabilities, provided an adequate modeling of the flame/acoustics interaction as appearing in the system is available. This essential piece of information is usually recast as the so called Flame Transfer Function (FTF) relating heat release rate fluctuations to velocity fluctuations at a given point. One way to obtain this transfer function is to rely on analytical models, but few exist for turbulent swirling flames. Another way consists in performing costly experiments or numerical simulations, negating the requested fast prediction capabilities. This thesis therefore aims at providing fast, yet reliable methods to allow for low order combustion instabilities modeling. In that context, understanding the underlying mechanisms of swirling flame acoustic response is also targeted. To address this issue, a novel hybrid approach is first proposed based on a reduced set of high fidelity simulations that can be used to determine input parameters of an analytical model used to express the FTF of premixed swirling flames. The analytical model builds on previous works starting with a level-set description of the flame front dynamics while also accounting for the acoustic-vorticity conversion through a swirler. For such a model, validation is obtained using reacting stationary and pulsed numerical simulations of a laboratory scale premixed swirl stabilized flame. The model is also shown to be able to handle various perturbation amplitudes. At last, 3D high fidelity simulations of an industrial gas turbine powered by a swirled spray flame are performed to determine whether a combustion instability observed in experiments can be predicted using numerical analysis. To do so, a series of forced simulations is carried out in en effort to highlight the importance of the two-phase flow flame response evaluation. In that case, sensitivity to reference velocity perturbation probing positions as well as the amplitude and location of the acoustic perturbation source are investigated. The analytical FTF model derived in the context of a laboratory premixed swirled burner is furthermore gauged in this complex case. Results show that the unstable mode is predicted by the acoustic analysis, but that the flame model proposed needs further improvements to extend its applicability range and thus provide data relevant to actual aero-engines

In the present work reduced order models (ROM) that are independent from the full order finite element models (FOM) considering geometrical non linearities are developed and applied to the dynamic study of rotating structures. The structure is considered to present nonlinear vibrations around the pre-stressed equilibrium induced by rotation enhancing the classical linearised approach. The reduced nonlinear forces are represented by a polynomial expansion obtained by the Stiffness. Evaluation Procedure (STEP) and then corrected by means of an original procedure by means of a Proper Orthogonal Decomposition (POD) that filters the full order nonlinear forces before projection. The latter model is named STEP with Correction (StepC). Different types of reduced basis are presented and tested. Some of these bases are parametrised with respect to the rotating velocity reducing considerably the construction of the ROM. The results obtained with the StepC ROM are in good agreement with the solutions of the FOM and are capable of reproducing the coupled motion of the structure. Furthermore they are more accurate than the classsical Linearised ROM solutions and than the STEP ROM without correction. The proposed StepC ROM provides the best compromise between accuracy and time consumption of the ROM
Dans le présent travail, des modèles d’ordre réduits (ROM) indépendant des modèles ́eléments finis d’haute fidélité (FOM) ont ́eté d ́eveloppés pour l’etude de la dynamique non linéaire des structures en rotation. Les vibrations de la structure autour de l’équilibre précontraint induit par la rotation sont considérées comme non linéaires, améliorant l’approche linéarisée classique. Les forces généralisées non linéaires sont approximées par un polynôme d’ordre trois obtenu avec la procédure Stiffness Evaluation Procedure (STEP). Ici, une approche originale est proposée pour corriger les forces non linéaires à l’aide d’une base de forces non linéaires obtenue avec une décomposition orthogonale aux valeurs propres (POD). Ce modèle est nommé STEP avec Correction (StepC). Différents types de base réduite sont présentés et testés. Certaines de ces bases sont paramétrées en fonction de la vitesse de rotation, ce qui réduit considérablement le temps de construction du modèle réduit. Les résultats obtenus avec le modèle StepC ROM sont en bon accord avec le FOM et sont capables de reproduire le couplage en déplacement entre les dégrés de liberté de la structure. De plus, elles sont plus précises que les solutions ROM linéarisées classiques et que le modèle STEP ROM sans correction. Le modèle StepC ROM proposé offre le meilleur compromis entre précision et temps de construction du ROM

This work presents novel formulations for models describing acoustically forced combustion in three disjoint regimes; highly turbulent, laminar, and the moderately turbulent flamelet regime. Particular emphasis is placed on simplification of the models to facilitate analytical solutions while still reflecting real phenomenology. Each derivation is treated by beginning with general reacting flow equations, identifying a small subset of physics thought to be dominant in the corresponding regime, and making appropriate simplifications. Each model is non-dimensionalized and both naturally occurring and popular dimensionless parameters are investigated. The well-stirred reactor (WSR) is used to characterize the highly turbulent regime. It is confirmed that, consistent with the regime to which it is ascribed for static predictions, the WSR is most appropriate to predict the dynamics of chemical kinetics. Both convection time and chemical time dynamics are derived as explicit closed-form functions of dimensionless quantities such as the Damk\"ohler number and several newly defined parameters. The plug-flow reactor (PFR) is applied to a laminar, burner stabilized flame, using a number of established approaches, but with new attention to developing simple albeit accurate expressions governing the flame's frequency response. The system is studied experimentally using a ceramic honeycomb burner, combusting a methane-air mixture, numerically using a nonlinear FEA solver, and analytically by exact solution of the simplified governing equations. Accurately capturing non-unity Lewis-number effects are essential to capturing both the static and the dynamic response of the flame. It is shown that the flame dynamics can be expressed solely in terms of static quantities. Finally, a Reynolds-averaged flamelet model is applied to a hypothetical burner stabilized flame with homogeneous, isotropic turbulence. Exact solution with a simplified turbulent reaction model parallels that of the plug flow reactor closely, demonstrating a relation between static quantities and the flame frequency response. Comparison with published experiments using considerably more complex flame geometries yields unexpected similarities in frequency scale, and phase behavior. The observed differences are attributed to specific physical phenomena that were deliberately omitted to simplify the model's derivation.
Ph. D.

Two circuit model extractors for complex multilayer microelectronic packages based on the Partial Element Equivalent Circuit (PEEC) technique, namely University of Arizona Effective Package Inductance Calculator (UAEPIC) and University of Arizona Effective Package Inductance and Capacitance Calculator (UAEPIC²), have been developed. The first one, UAEPIC, is based on the magneto-quasistatic assumption where the displacement current effect on the derivation of the electromagnetic field integral equation is neglected and thus the dominant inductive effects are modeled in order to extract the RL equivalent model. The second one, UAEPIC², uses a more rigorous electromagnetic model that accounts for displacement (yet nonretarded) electromagnetic effects to extract the RLC equivalent model of the given microelectronic package. The development of electrical models of packages of high complexity requires the numerical solution of linear systems of several thousands of equations. This makes the development of a broadband equivalent circuit to include skin effect computationally expensive. To circumvent this difficulty, two model order reduction techniques have been utilized. The method of Asymptotic Waveform Evaluation (AWE) has been incorporated in UAEPIC, and the Passive Reduced-order Interconnect Macromodeling Algorithm (PRIMA) has been applied to UAEPIC². Applications of AWE and PRIMA provide orders of magnitude reduction in computation labor and lead to a direct multiport Y-matrix representation in terms of the poles and residues. In this form, and using a special algorithm, the multiport, frequency-dependent equivalent circuit of the package can be incorporated efficiently in a SPICE-like circuit simulator. This simulation capability facilitates rapid and accurate simulations for the analysis of noise generation and signal degradation such as delay, cross-talk, power and ground bounces, and Simultaneous Switching Noise (SSN) in the package.

Les réseaux de distribution d’eau consistent en de grandes infrastructures réparties dans l’espace qui assurent la distribution d’eau potable en quantité et en qualité suffisantes. Les modèles mathématiques de ces systèmes sont caractérisés par un grand nombre de variables d’état et de paramètres dont la plupart sont incertains. Les temps de calcul peuvent s’avérer conséquents pour les réseaux de taille importante et la propagation d’incertitude par des méthodes de Monte Carlo. Par conséquent, les deux principaux objectifs de cette thèse sont l’étude des techniques de modélisation à ordre réduit par projection ainsi que la propagation spectrale des incertitudes des paramètres. La thèse donne tout d’abord un aperçu des méthodes mathématiques utilisées. Ensuite, les équations permanentes des réseaux hydrauliques sont présentées et une nouvelle méthode de calcul des sensibilités est dérivée sur la base de la méthode adjointe. Les objectifs spécifiques du développement de modèles d’ordre réduit sont l’application de méthodes basées sur la projection, le développement de stratégies d’échantillonnage adaptatives plus efficaces et l’utilisation de méthodes d’hyper-réduction pour l’évaluation rapide des termes résiduels non linéaires. Pour la propagation des incertitudes, des méthodes spectrales sont introduites dans le modèle hydraulique et un modèle hydraulique intrusif est formulé. Dans le but d’une analyse plus efficace des incertitudes des paramètres, la propagation spectrale est ensuite évaluée sur la base du modèle réduit. Les résultats montrent que les modèles d’ordre réduit basés sur des projections offrent un avantage considérable par rapport à l’effort de calcul. Bien que l’utilisation de l’échantillonnage adaptatif permette une utilisation plus efficace des états système pré-calculés, l’utilisation de méthodes d’hyper-réduction n’a pas permis d’améliorer la charge de calcul. La propagation des incertitudes des paramètres sur la base des méthodes spectrales est comparable aux simulations de Monte Carlo en termes de précision, tout en réduisant considérablement l’effort de calcul
Water distribution systems are large, spatially distributed infrastructures that ensure the distribution of potable water of sufficient quantity and quality. Mathematical models of these systems are characterized by a large number of state variables and parameter. Two major challenges are given by the time constraints for the solution and the uncertain character of the model parameters. The main objectives of this thesis are thus the investigation of projection based reduced order modelling techniques for the time efficient solution of the hydraulic system as well as the spectral propagation of parameter uncertainties for the improved quantification of uncertainties. The thesis gives an overview of the mathematical methods that are being used. This is followed by the definition and discussion of the hydraulic network model, for which a new method for the derivation of the sensitivities is presented based on the adjoint method. The specific objectives for the development of reduced order models are the application of projection based methods, the development of more efficient adaptive sampling strategies and the use of hyper-reduction methods for the fast evaluation of non-linear residual terms. For the propagation of uncertainties spectral methods are introduced to the hydraulic model and an intrusive hydraulic model is formulated. With the objective of a more efficient analysis of the parameter uncertainties, the spectral propagation is then evaluated on the basis of the reduced model. The results show that projection based reduced order models give a considerable benefit with respect to the computational effort. While the use of adaptive sampling resulted in a more efficient use of pre-calculated system states, the use of hyper-reduction methods could not improve the computational burden and has to be explored further. The propagation of the parameter uncertainties on the basis of the spectral methods is shown to be comparable to Monte Carlo simulations in accuracy, while significantly reducing the computational effort

COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Um dos resultados fundamentais na mecânica clássica é que, para sistemas lineares com n graus de liberdade, existem n modos de vibração ortogonais e que as freqüências naturais são independentes da amplitude de vibração. Além disso, qualquer movimento da estrutura pode ser obtido como uma combinação linear desses modos. No caso de sistemas não-lineares, isto não mais se verifica e a relação entre freqüência, amplitude e os modos de vibração precisa ser determinada. A obtenção dessas informações para estruturas se dá em geral pelo uso de programas de análise não-linear baseados em uma formulação em elementos finitos. Contudo, isto é um procedimento custoso computacionalmente. Uma abordagem mais viável é o uso de modelos discretos compatíveis de baixa dimensão, por meio dos quais as freqüências e os modos não- lineares são obtidos. Neste trabalho é proposto um procedimento para a derivação de modelos de redução de dimensão para vigas e pórticos planos esbeltos. As equações diferenciais de movimento são obtidas a partir da aplicação das técnicas variacionais a um funcional não-linear de energia. A obtenção do modelo se dá através do emprego dos métodos de Ritz ou Galerkin para a redução espacial e do balanço harmônico para redução no tempo. Os modos lineares são utilizados como uma primeira aproximação para os modos não-lineares. As relações freqüência-amplitude são satisfatoriamente obtidas para vibrações livre e forçada (não-amortecida e amortecida). Entretanto, essas curvas apresentam, em geral, no regime não-linear, pontos limites, sendo obtidas, portanto, com uso do método do controle de comprimento de arco. Uma correção para o modo- linear é obtida com uso dos métodos dos elementos finitos e da perturbação. Um estudo paramétrico e das condições de contorno é apresentado para vigas. O comportamento não-linear de pórticos em L é também analisado. Para esses pórticos é estudada a influência de cargas axiais e da geometria. Os resultados são comparados com soluções analíticas encontradas na literatura.
One of the fundamental results in classical mechanics is that linear systems with n degrees of freedom have n orthogonal vibration modes and n natural frequencies which are independent of the vibration amplitude. Any motion of the system can be obtained as a linear combination of these modes. This does not hold for nonlinear systems in which case amplitude dependent vibrations modes and frequencies must be obtained. One way of obtaining these informations for arbitrary structures is to use a nonlinear finite element software. However, this is a cumbersome and time consuming procedure. A better approach is to derive a consistent low dimensional model from which the nonlinear frequencies and mode shapes can be derived. In this work a procedure for the derivation of low dimensional models for slender beams and portal frames is proposed. The differential equations of motion are derived from the application of variational techniques to a nonlinear energy functional. The linear vibration modes are used as a first approximation for the nonlinear modes. The Galerkin and Ritz methods are used in the model for the spatial reduction and the harmonic balance method for the reduction in time domain. This allows the analysis of the free and forced (damped or undamped) vibrations of the structure in non- linear regime. However nonlinear resonance curves usually presents limit points. To obtain these curves, a methodology for the solution of non-linear equations based on an arc-length procedure is derived. Based on the finite element methods and using the basic ideas of the perturbation theory, a correction for the nonlinear vibration modes is derived. The influence of boundary conditions, geometric, and force parameters on the beam response is analyzed. The behavior of L frames is studied. For this kind of frame, the influence of axial loading and geometric parameters on the response is studied. The results are compared with analytical solutions found in the literature.

High-flying unmanned reconnaissance and surveillance systems are now being used extensively in the United States military. Current development programs are producing demonstrations of next-generation unmanned flight systems that are designed to perform combat missions. Their use in first-strike combat operations will dictate operations in densely cluttered environments that include unknown obstacles and threats, and will require the use of terrain for masking. The demand for autonomy of operations in such environments dictates the need for advanced trajectory optimization capabilities. In addition, the ability to coordinate the movements of more than one aircraft in the same area is an emerging challenge. This thesis examines using an analytical reduced order formulation for trajectory generation for minimum time and terrain masking cases. First, pseudo-3D constant velocity equations of motion are used for path planning for a single vehicle. In addition, the inclusion of winds, moving targets and moving threats is considered. Then, this formulation is increased to using 3D equations of motion, both with a constant velocity and with a simplified varying velocity model. Next, the constant velocity equations of motion are expanded to include the simultaneous path planning of an unspecified number of vehicles, for both aircraft avoidance situations and formation flight cases.

Thesis (PhD)--Stellenbosch University, 2015.
ENGLISH ABSTRACT: Numerical simulations are widely used for predicting and optimising the exploitation of aquifers. They are also used to determine certain physical parameters, for example soil conductivity, by inverse calculations, where the model parameters are changed until the model results correspond optimally to measurements taken on site. The Richards’ equation describes the movement of an unsaturated fluid through porous media, and is characterised as a non-linear partial differential equation. The equation is subject to a number of parameters and is typically computationally expensive to solve. To determine the parameters in the Richards’ equation, inverse modelling studies often need to be undertaken. In these studies, the parameters of a numerical model are varied until the numerical response matches a measured response. Inverse modelling studies typically require 100’s of simulations, which implies that parameter optimisation in unsaturated case studies is common only in small or 1D problems in the literature. As a solution to overcome the computational expense incurred in inverse modelling, the use of Proper Orthogonal Decomposition (POD) as a Reduced Order Modelling (ROM) method is proposed in this thesis to speed-up individual simulations. An explanation of the Finite Element Method (FEM) is given using the Galerkin method, followed by a detailed explanation of the Galerkin POD approach. In the development of the Galerkin POD approach, the method of reducing matrices and vectors is shown, and the treatment of Neumann and Dirichlet boundary values is explained. The Galerkin POD method is applied to two case studies. The first case study is the Kogelberg site in the Table Mountain Group near Cape Town in South Africa. The response of the site is modelled at one well over the period of 2 years, and is assumed to be governed by saturated flow, making it a linear problem. The site is modelled as a 3D transient, homogeneous site, using 15 layers and ≈ 20000 nodes, using the FEM implemented on the open-source software FreeFem++. The model takes the evapotranspiration of the fynbos vegetation at the site into consideration, allowing the calculation of annual recharge into the aquifer. The ROM is created from high-fidelity responses taken over time at different parameter points, and speed-up times of ≈ 500 are achieved, corresponding to speed-up times found in the literature for linear problems. The purpose of the saturated groundwater model is to demonstrate that a POD-based ROM can approximate the full model response over the entire parameter domain, highlighting the excellent interpolation qualities and speed-up times of the Galerkin POD approach, when applied to linear problems. A second case study is undertaken on a synthetic unsaturated case study, using the Richards’ equation to describe the water movement. The model is a 2D transient model consisting of ≈ 5000 nodes, and is also created using FreeFem++. The Galerkin POD method is applied to the case study in order to replicate the high-fidelity response. This did not yield in any speed-up times, since the full matrices of non-linear problems need to be recreated at each time step in the transient simulation. Subsequently, a method is proposed in this thesis that adapts the Galerkin POD method by linearising the non-linear terms in the Richards’ equation, in a method named the Linearised Galerkin POD (LGP) method. This method is applied to the same 2D synthetic problem, and results in speed-up times in the range of 10 to 100. The adaptation, notably, does not use any interpolation techniques, favouring a code intrusive, but physics-based, approach. While the use of an intrusively linearised POD approach adds to the complexity of the ROM, it avoids the problem of finding kernel parameters typically present in interpolative POD approaches. Furthermore, the interpolation and possible extrapolation properties inherent to intrusive POD-based ROM’s are explored. The good extrapolation properties, within predetermined bounds, of intrusive POD’s allows for the development of an optimisation approach requiring a very small Design of Experiments (DOE) sets (e.g. with improved Latin Hypercube sampling). The optimisation method creates locally accurate models within the parameter space using Support Vector Classification (SVC). The region inside of the parameter space in which the optimiser is allowed to move is called the confidence region. This confidence region is chosen as the parameter region in which the ROM meets certain accuracy conditions. With the proposed optimisation technique, advantage is taken of the good extrapolation characteristics of the intrusive POD-based ROM’s. A further advantage of this optimisation approach is that the ROM is built on a set of high-fidelity responses obtained prior to the inverse modelling study, avoiding the need for full simulations during the inverse modelling study. In the methodologies and case studies presented in this thesis, initially infeasible inverse modelling problems are made possible by the use of the POD-based ROM’s. The speed up times and extrapolation properties of POD-based ROM’s are also shown to be favourable. In this research, the use of POD as a groundwater management tool for saturated and unsaturated sites is evident, and allows for the quick evaluation of different scenarios that would otherwise not be possible. It is proposed that a form of POD be implemented in conventional groundwater software to significantly reduce the time required for inverse modelling studies, thereby allowing for more effective groundwater management.
AFRIKAANSE OPSOMMING: Die Richards vergelyking beskryf die beweging van ’n vloeistof deur ’n onversadigde poreuse media, en word gekenmerk as ’n nie-lineêre parsiële differensiaalvergelyking. Die vergelyking is onderhewig aan ’n aantal parameters en is tipies berekeningsintensief om op te los. Om die parameters in die Richards vergelyking te bepaal, moet parameter optimering studies dikwels onderneem word. In hierdie studies, word die parameters van ’n numeriese model verander totdat die numeriese resultate die gemete resultate pas. Parameter optimering studies vereis in die orde van honderde simulasies, wat beteken dat studies wat gebruik maak van die Richards vergelyking net algemeen is in 1D probleme in die literatuur. As ’n oplossing vir die berekingskoste wat vereis word in parameter optimering studies, is die gebruik van Eie Ortogonale Ontbinding (POD) as ’n Verminderde Orde Model (ROM) in hierdie tesis voorgestel om individuele simulasies te versnel in die optimering konteks. Die Galerkin POD benadering is aanvanklik ondersoek en toegepas op die Richards vergelyking, en daarna is die tegniek getoets op verskeie gevallestudies. Die Galerkin POD metode word gedemonstreer op ’n hipotetiese gevallestudie waarin water beweging deur die Richards-vergelyking beskryf word. As gevolg van die nie-lineêre aard van die Richards vergelyking, het die Galerkin POD metode nie gelei tot beduidende vermindering in die berekeningskoste per simulasie nie. ’n Verdere gevallestudie word gedoen op ’n ware grootskaalse terrein in die Tafelberg Groep naby Kaapstad, Suid-Afrika, waar die grondwater beweging as versadig beskou word. Weens die lineêre aard van die vergelyking wat die beweging van versadigde water beskryf, is merkwaardige versnellings van > 500 in die ROM waargeneem in hierdie gevallestudie. Daarna was die die Galerkin POD metode aangepas deur die nie-lineêre terme in die Richards vergelyking te lineariseer. Die tegniek word die geLineariserde Galerkin POD (LGP) tegniek genoem. Die aanpassing het goeie resultate getoon, met versnellings groter as 50 keer wanneer die ROM met die oorspronklike simulasie vergelyk word. Al maak die tegniek gebruik van verder lineariseering, is die metode nogsteeds ’n fisika-gebaseerde benadering, en maak nie gebruik van interpolasie tegnieke nie. Die gebruik van ’n fisika-gebaseerde POD benaderings dra by tot die kompleksiteit van ’n volledige numeriese model, maar die kompleksiteit is geregverdig deur die merkwaardige versnellings in parameter optimerings studies. Verder word die interpolasie eienskappe, en moontlike ekstrapolasie eienskappe, inherent aan fisika-gebaseerde POD ROM tegnieke ondersoek in die navorsing. In die navorsing word ’n tegniek voorgestel waarin hierdie inherente eienskappe gebruik word om plaaslik akkurate modelle binne die parameter ruimte te skep. Die voorgestelde tegniek maak gebruik van ondersteunende vektor klassifikasie. Die grense van die plaaslik akkurate model word ’n vertrouens gebeid genoem. Hierdie vertrouens gebied is gekies as die parameter ruimte waarin die ROM voldoen aan vooraf uitgekiesde akkuraatheidsvereistes. Die optimeeringsbenadering vermy ook die uitvoer van volledige simulasies tydens die parameter optimering, deur gebruik te maak van ’n ROM wat gebaseer is op die resultate van ’n stel volledige simulasies, voordat die parameter optimering studie gedoen word. Die volledige simulasies word tipies uitgevoer op parameter punte wat gekies word deur ’n proses wat genoem word die ontwerp van eksperimente. Verdere hipotetiese grondwater gevallestudies is onderneem om die LGP en die plaaslik akkurate tegnieke te toets. In hierdie gevallestudies is die grondwater beweging weereens beskryf deur die Richards vergelyking. In die gevalle studie word komplekse en tyd-rowende modellerings probleme vervang deur ’n POD gebaseerde ROM, waarin individuele simulasies merkwaardig vinniger is. Die spoed en interpolasie/ekstrapolasie eienskappe blyk baie gunstig te wees. In hierdie navorsing is die gebruik van verminderde orde modelle as ’n grondwaterbestuursinstrument duidelik getoon, waarin voorsiening geskep word vir die vinnige evaluering van verskillende modellering situasies, wat andersins nie moontlik is nie. Daar word voorgestel dat ’n vorm van POD in konvensionele grondwater sagteware geïmplementeer word om aansienlike versnellings in parameter studies moontlik te maak, wat na meer effektiewe bestuur van grondwater sal lei.