<p dir="ltr">Understanding the nonlinear rheological properties of food is essential for improving processes involving large-amplitude deformation such as pumping, extrusion, and consumer consumption. The development of mathematical analyses for analyzing these nonlinear responses has witnessed a notable upswing in the past decades. A novel mathematical analysis called "Sequence of Physical Processes" (SPP) was developed by Rogers et al. in 2011. Ever since, SPP has shown tremendous potential in characterizing and predicting the nonlinear rheological behavior of soft materials and polymers, yet more investigations are required to validate the efficacy of the SPP approach in the realm of food materials. Therefore, this thesis focuses on applying SPP method onto a range of food materials. Most importantly, we compared the analysis with the results obtained from the well-established Ewoldt-McKinley method of coupling “Fourier Transform with Chebyshev Decomposition” (FTC). As a result, it is found that SPP can provide a detailed picture of the material’s deformation history within an oscillation cycle. The time-dependent nature of SPP data allows a more accurate capture of important rheological transitions, which leads to a higher correlation with compositional and microstructural changes in comparison to the FTC method. Recognizing the potential of SPP analysis in studying food materials, this research emphasizes the necessity for further exploration across a diverse array of food types. The thesis contributes valuable insights to the evolving landscape of nonlinear rheological understanding, with the potential to improving methodologies in food processing.</p>

En general, encontrar una solución analítica de una ecuación diferencial parcial no es fácil, y más aún cuando ésta ecuación es no lineal. Debido a esto, surgieron varios métodos numéricos para encontrar una solución aproximada a la deseada. Los métodos numéricos más conocidos son: • Métodos de Diferencias Finitas que tuvo su gran auge en la década de 1950. • Métodos de Elementos Finitos que tuvo su gran auge en la década de 1960. • Métodos Espectrales que tuvo su gran auge en la década de 1970. Mientras que los métodos de diferencias finitas dan soluciones aproximadas en los puntos de la malla computacional elegida, los métodos de elementos finitos dan aproximaciones polinomiales continuas o continuas por partes en regiones poligonales (generalmente triangulares en dos dimensiones), mientras que los métodos espectrales brindan soluciones aproximadas en la forma de polinomios sobre todo su dominio. Los métodos espectrales son una clase de discretización espacial para ecuaciones diferenciales. Las componentes claves para su formulación son las funciones base (llamadas también funciones de aproximación o expansión) y las funciones de prueba. Las funciones base se usan para dar una representación aproximada de la solución. Las funciones de prueba se usan para asegurar que la ecuación diferencial y quizás algunas condiciones de frontera se cumplan tanto como sea posible por la serie truncada de expansión. Esto se consigue minimizando, con respecto a una norma adecuada, el residuo producido por el uso de la expansión truncada en lugar de la solución exacta. Los métodos espectrales tienen un amplio uso en diferentes áreas como: teoría cuántica ([31], [36]) basado en la ecuación Schrödinger que proporciona la descripción teórica de numerosos sistemas en química y física; teoría cinética basada en la ecuación de Boltzmann ([27], [32]) o en la ecuación de Fokker-Planck ([5], [45]); problemas en mecánica de fluidos ([4], [20], [42]). También hay importantes aplicaciones en el escape átomos de la atmósfera del planeta ([14], [51]) como la pérdida de carga de partículas de la tierra ([33], [43]) y del sol [11]. El presente trabajo pretende contribuir en sentar los fundamentos sobre métodos espectrales, para que sean aplicados en futuras investigaciones más elaboradas, así como brindar los códigos de implementación (en Matlab), los cuales raramente se encuentran en forma explícita en la literatura. Este trabajo está organizado de la siguiente manera: el Capítulo 1 abarca las propiedades más importantes de los polinomios ortogonales; en particular, los polinomios de Chebyshev, los cuales son adecuados para representar funciones de dominio finito y sus relaciones de recurrencia asociadas. Además, se presenta un breve repaso de las fórmulas de cuadratura gaussiana. En el Capítulo 2, se presenta en forma detallada los métodos espectrales polinomiales, útiles para problemas con condiciones de frontera no periódicas. Presentamos los métodos de Galerkin, Tau y de Colocación. En el Capítulo 3 se da ejemplos de la implementación numérica de la ecuación del calor usando los métodos de diferencias finitas y los métodos espectrales, usando los polinomios de Chebyshev. Además, se brindan los detalles necesarios para implementar la ecuación de Burger usando los métodos espectrales. / Tesis

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

This dissertation investigates the learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the dimension of the problem. The first part answers some open questions for the binary classification problem in the framework of active learning. Given a random couple (X,Y) with unknown distribution P, the goal of binary classification is to predict a label Y based on the observation X. Prediction rule is constructed from a sequence of observations sampled from P. The concept of active learning can be informally characterized as follows: on every iteration, the algorithm is allowed to request a label Y for any instance X which it considers to be the most informative. The contribution of this work consists of two parts: first, we provide the minimax lower bounds for the performance of active learning methods. Second, we propose an active learning algorithm which attains nearly optimal rates over a broad class of underlying distributions and is adaptive with respect to the unknown parameters of the problem. The second part of this thesis is related to sparse recovery in the framework of dictionary learning. Let (X,Y) be a random couple with unknown distribution P. Given a collection of functions H, the goal of dictionary learning is to construct a prediction rule for Y given by a linear combination of the elements of H. The problem is sparse if there exists a good prediction rule that depends on a small number of functions from H. We propose an estimator of the unknown optimal prediction rule based on penalized empirical risk minimization algorithm. We show that the proposed estimator is able to take advantage of the possible sparse structure of the problem by providing probabilistic bounds for its performance.

Consider the minimally-supported D-optimal designs for dth degree polynomial regression with bounded and positive weight function on a compact interval. We show that the optimal design converges weakly to the arcsin distribution as d goes to infinity. Comparisons of the optimal design with the arcsin distribution and D-optimal arcsin support design by D-efficiencies are also given. We also show that if the design interval is [−1, 1], then the minimally-supported D-optimal design converges to the D-optimal arcsin support design with the specific weight function 1/¡Ô(£\-x^2), £\>1, as £\¡÷1+.

Solutions to trajectory optimization problems are carried out by the direct and indirect methods. Under broad heading of these methods, numerous algorithms such as collocation, direct, indirect and multiple shooting methods have been developed and reported in the literature. Each of these algorithms has certain advantages and limitations. For example, direct shooting technique is not suitable when the number of nonlinear programming variables is large. Indirect shooting method requires analytical derivatives of the control and co-states function and a poorly guessed initial condition can result in numerical unstable values of the adjoint variable. Multiple shooting techniques can alleviate some of these difficulties by breaking down the trajectory into several segments that help in reducing the non-linearity effects of early control on later parts of the trajectory. However, multiple shooting methods then have to handle more number of variables and constraints to satisfy the defects at the segment joints. The sie of the nonlinear programming problem in the collocation method is also large and proper locations of grid points are necessary to satisfy all the path constraints. Stochastic methods such as Genetic algorithms, on the other hand, also require large number of function evaluations before convergence. To overcome some of the limitations of the conventional methods, improved solution techniques are developed. Three improved methods are proposed for the solution of trajectory optimization problems. They are • a genetic algorithm employing dominance and diploidy concept. • a collocation method using chebyshev polynomials , and • a hybrid method that combines collocation and direct shooting technique A conventional binary-coded genetic algorithm uses a haploid chromosome, where a single string contains all the variable information in the coded from. A diploid, as the name suggests, uses pair of chromosomes to store the same characteristic feature. The diploid genetic algorithm uses a dominant map for decoding genotype into a stable, consistent phenotype. In dominance, one allele takes precedence over another. Diploidy and dominance helps in retaining the previous best solution discovered and shields them from harmful selection in a changing environment. Hence, diploid and dominance affect a king of long-term memory in the genetic algorithm. They allow alternate solutions to co-exist. One solution is expressed and the other is held in abeyance. In the improved diploid genetic algorithm, dominant and recessive genes are defined based on the fitness evaluation of each string. The genotype of fittest string is declared as the dominant map. The dominant map is dynamic in nature as it is replaced with a better individual in future generations. The concept of diploidy and dominance in the improved method mimics closer to the principles used in human genetics as compared to any such algorithms reported in the literature. It is observed that the improved diploid genetic algorithm is able to locate the optima for a given trajectory optimization problem with 10% lower computational time as compared to the haploid genetic algorithm. A parameter optimization problem arising from an optimal control problem where states and control are approximated by piecewise Chebyshev polynomials is well known. These polynomials are more accurate than the interpolating segments involving equal spaced data. In the collocation method involving Chebyshev polynomials, derivatives of two neighboring polynomials are matched with the dynamics at the nodal points. This leads to a large number of equality constraints in the optimization problem. In the improved method, derivative of the polynomial is also matched with the dynamics at the center of segments. Though is appears the problem size is merely increased, the additional computations improve the accuracy of the polynomial for a larger segment. The implicit integration step size is enhanced and overall size of the problem is brought down to one-fourth of the problem size defined with a conventional collocation method using Chebyshev polynomials. Hybrid method uses both collocation and direct shooting techniques. Advantages of both the methods are combined to give more synergy. Collocation method is used in the starting phase of the hybrid method. The disadvantage of standalone collocation method is that tuning of grid points is required to satisfy the path constraints. Nevertheless, collocation method does give a good guess required for the terminal phase of the hybrid method, which uses a direct shooting approach. Results show nearly 30% reduction in computation time for the hybrid approach as compared to a method in which direct shooting alone is used, for the same initial guess of control. The solutions obtained from the three improved methods are compared with an indirect method. The indirect method requires derivations of the control and adjoint equations, which are difficult and problem specific. Due to sensitivity of the costate variables, it is often difficult to find a solution through the indirect method. Nevertheless, these methods do provide an accurate result, which defines a benchmark for comparing the solutions obtained through the improved methods. Trajectory design and optimization of a RLV(Reusable Launch Vehicle) Demonstrator mission is considered as a test problem for evaluating the performance of the improved methods. The optimization problem is difficult than a conventional launch vehicle trajectory optimization problem because of the following two reasons. • aerodynamic lift forces in the RLV add one more dimension to the already complex launch vehicle optimization problem. • as RLV performs a sub orbital flight, the ascent phase trajectory influences the re-entry trajectory. Both the ascent and re-entry optimization problem of the RLV mission is addressed. It is observed that the hybrid method gives accurate results with least computational effort, as compared with other improved techniques for the trajectory optimization problem of RLV during its ascent flight. Hybrid method is then successfully used during the re-entry phase and in designing the feasible optimal trajectories under the dispersion conditions. Analytical solutions obtained from literature are used to compare the optimized trajectory during the re-entry phase. Trajectory optimization studies are also carried out for the off-nominal performances. Being a thrusting phase, the ascent trajectory is subjected to significant deviations, mainly arising out of solid booster performance dispersions. The performance index during rhe ascent phase is modified in a novel way for handling dispersions. It minimizes the state errors in a least square sense, defined at the burnout conditions ensure possibilities of safe re-entry trajectories. The optimal trajectories under dispersion conditions serve as a benchmark for validating the closed-loop guidance algorithm that is developed for the ascent phase flight. Finally, an on-line trajectory command-reshaping algorithm is developed which meets the flight objectives under the dispersion conditions. The guidance algorithm uses a pre-computed trajectory database along with some real-time measured parameters in generating the optimal steering profiles. The flight objectives are met under the dispersion conditions and the guidance generated steering profiles matches closely with the optimal trajectories.

[Truncated abstract] This thesis presents three different methods to investigate flexural wave propagation and scattering, power flow and transmission efficiencies, and dynamic stress concentration and fatigue failures in structural dynamics. The first method is based on the acoustical wave propagator (AWP) technique, which is the main part described in this thesis. Through the numerical implementation of the AWP, the complete information of the vibrating structure can be obtained including displacement, velocity, acceleration, bending moments, strain and stresses. The AWP technique has been applied to systems consisting of a one-dimensional stepped beam, a two-dimensional thin plate, a thin plate with a sharp change of section, a heterogeneous plate with multiple cylindrical patches, and a Mindlin?s plate with a reinforced rib. For this Mindlin?s plate structure, through the comparison of the results obtained by Mindlin?s thick plate theory and Kirchhoff?s classical thin plate theory, the difference of theoretical predicted results is investigated. As part of these investigations, reflection and transmission coefficients, power flow and transmission efficiencies in a onedimensional stepped beam, and power flow in a two-dimensional circular plate structure, are studied. In particular, this technique has been successfully extended to investigate wave propagation and scattering, and dynamic stress concentration at discontinuities. Potential applications are fatigue failure prediction and damage detection in complex structures. The second method is based on experimental techniques to investigate the structural response under impact loads, which consist of the waveform measuring technique in the time domain by using the WAVEVIEW software, and steady-state measurements by using the Polytec Laser Scanning Vibrometer (PLSV) in the frequency domain. The waveform measuring technique is introduced to obtain the waveform at different locations in the time domain. These experimental results can be used to verify the validity of predicted results obtained by the AWP technique. Furthermore, distributions of dynamic strain and stress in both near-field (close to discontinuities) and far-field regions are investigated for the study of the effects of the discontinuities on reflection and transmission coefficients in a one-dimensional stepped beam structure. Experimental results in the time domain can be easily transferred into those in the frequency domain by the fast Fourier transformation, and compared with those obtained by other researchers. This PLSV technique provides an accurate and efficient tool to investigate mode shape and power flow in some coupled structures, such as a ribbed plate. Through the finite differencing technique, autospectral and spatial of dynamic strain can be obtained. The third method considered uses the travelling wave solution method to solve reflection and transmission coefficients in a one-dimensional stepped beam structure in the time domain. In particular, analytical exact solutions of reflection and transmission coefficients under the given initial-value problem are derived. These analytical solutions together with experimental results can be used to compare with those obtained by the AWP technique.

This thesis investigates novel algorithms for enabling the use of first-principle electrochemical models for battery monitoring and control in advanced battery management systems (BMSs). Specifically, the fast solution and state estimation of a high-fidelity spatially resolved thermal-electrochemical lithium-ion battery model commonly referred to as the pseudo two-dimensional (P2D) model are investigated. The partial-differential algebraic equations (PDAEs) constituting the model are spatially discretised using Chebyshev orthogonal collocation enabling fast and accurate simulations up to high C-rates. This implementation of the P2D model is then used in combination with an extended Kalman filter (EKF) algorithm modified for differential-algebraic equations (DAEs) to estimate the states of the model, e.g. lithium concentrations, overpotential. The state estimation algorithm is able to rapidly recover the model states from current, voltage and temperature measurements. Results show that the error on the state estimate falls below 1% in less than 200s despite a 30% error on battery initial state-of-charge (SoC) and additive measurement noise with 10mV and 0.5&deg;C standard deviations. The parameter accuracy of such first-principle models is of utmost importance for the trustworthy estimation of internal battery electrochemical states. Therefore, the identifiability of the simpler single particle (SP) electrochemical model is investigated both in principle and in practice. Grouping parameters and partially non-dimensionalising the SP model equations in order to understand the maximum expected degrees of freedom in the problem reveals that there are only six unique parameters in the SP model. The structural identifiability is then examined by asking whether the transfer function of the linearised SP model is unique. It is found that the model is unique provided that the electrode open circuit voltage curves have a non-zero gradient, the parameters are ordered, and that the behaviour of the kinetics of each electrode is lumped together into a single parameter which is the charge transfer resistance. The practical estimation of the SP model parameters from frequency-domain experimental data obtained by electrochemical impedance spectroscopy (EIS) is then investigated and shows that estimation at a single SoC is insufficient to obtain satisfactory results and EIS data at multiple SoCs must be combined.

La méthode d'identification développée dans cette thèse est inspirée des travaux de D. Rémond. On considérera les données d'entrée suivante : la réponse de la structure, qui sera mesurée de manière discrète, et qui dépendra des dimensions de la structure (temps, espace) le modèle de comportement, qui sera exprimé sous forme d'une équation différentielle ou d'une équation aux dérivées partielles, les conditions aux limites ainsi que la source d'excitation seront considérées comme non mesurées, ou inconnues. La procédure d'identification est composée de trois étapes : la projection sur une base polynomiale orthogonale (polynômes de Chebyshev) du signal mesuré, la différentiation du signal mesuré, l'estimation de paramètres, en transformant l'équation de comportement en une équation algébrique. La poutre de Bernoulli a permis d'établir un lien entre l'ordre de troncature de la base polynomiale et le nombre d'ondes contenu dans le signal projeté. Sur un signal bruité, nous avons pu établir une valeur de nombre d'onde et d'ordre de troncature minimum pour assurer une estimation précise du paramètre à identifier. Grâce à l'exemple de la poutre de Timoshenko, nous avons pu réadapter la procédure d'identification à l'estimation de plusieurs paramètres. Trois paramètres dont les valeurs ont des ordres radicalement différents ont été estimés. Cet exemple illustre également la stratégie de régularisation à adopter avec ce type de problèmes. L'estimation de l'amortissement sur une poutre a été réalisée avec succès, que ce soit à l'aide de sa réponse transitoire ou à l'aide du régime établi. Le cas bidimensionnel de la plaque a également été traité. Il a permis d'établir un lien similaire au cas de la poutre de Bernoulli entre le nombre d'onde et l'ordre de troncature. Deux cas d'applications expérimentales ont été traités au cours de cette thèse. Le premier se base sur le modèle de la poutre de Bernoulli, appliqué à la détection de défaut. En effet on applique un procédé d'identification ayant pour hypothèse initiale la continuité de la structure. Dans le cas où celle-ci ne le serait pas on s'attend à observer une valeur aberrante du paramètre reconstruit. Le procédé permet de localiser avec succès le lieu de la discontinuité. Le second cas applicatif vise à reconstruire l'amortissement d'une structure 2D : une plaque libre-libre. On compare les résultats obtenus à l'aide de notre procédé d'identification à ceux obtenus par Ablitzer à l'aide de la méthode RIFF. Les deux méthodes permettent d'obtenir des résultats sensiblement proches. / The purpose of this work is to adapt and improve the continuous time identification method proposed by D. Rémond for continuous structures. D. Rémond clearly separated this identification method into three steps: signal expansion, signal differentiation and parameter estimation. In this study, both expansion and differentiation steps are drastically improved. An original differentiation method is developed and adapted to partial differentiation. The existing identification process is firstly adapted to continuous structure. Then the expansion and differentiation principle are presented. For this identification purpose a novel differentiation model was proposed. The aim of this novel operator was to limit the sensitivity of the method to the tuning parameter (truncation number). The precision enhancement using this novel operator was highlighted through different examples. An interesting property of Chebyshev polynomials was also brought to the fore : the use of an exact discrete expansion with the polynomials Gauss points. The Gauss points permit an accurate identification using a restricted number of sensors, limiting de facto the signal acquisition duration. In order to reduce the noise sensitivity of the method, a regularization step was added. This regularization step, named the instrumental variable, was inspired from the automation domain. The instrumental variable works as a filter. The identified parameter is recursively filtered through the structure model. The final result is the optimal parameter estimation for a given model. Different numerical applications are depicted. A focus is made on different practical particularities, such as the use of the steady-state response, the identification of multiple parameters, etc. The first experimental application is a crack detection on a beam. The second experimental application is the identification of damping on a plate.

Magnetorheological (MR) dampers are adaptive devices whose properties can be adjusted through the application of a controlled voltage signal. A semi-active suspension system incorporating MR dampers combines the advantages of both active and passive suspensions. For this reason, there has been a continuous effort to develop control algorithms for MR-damped vehicle suspension systems to meet the requirements of the automotive industry. The overall aims of this thesis are twofold: (i) The investigation of non-parametric techniques for the identification of the nonlinear dynamics of an MR damper. (ii) The implementation of these techniques in the investigation of MR damper control of a vehicle suspension system that makes minimal use of sensors, thereby reducing the implementation cost and increasing system reliability. The novel contributions of this thesis can be listed as follows: 1- Nonparametric identification modelling of an MR damper using Chebyshev polynomials to identify the damping force from both simulated and experimental data. 2- The neural network identification of both the direct and inverse dynamics of an MR damper through an experimental procedure. 3- The experimental evaluation of a neural network MR damper controller relative to previously proposed controllers. 4- The application of the neural-based damper controller trained through experimental data to a semi-active vehicle suspension system. 5- The development and evaluation of an improved control strategy for a semi-active car seat suspension system using an MR damper. Simulated and experimental validation data tests show that Chebyshev polynomials can be used to identify the damper force as an approximate function of the displacement, velocity and input voltage. Feed-forward and recurrent neural networks are used to model both the direct and inverse dynamics of MR dampers. It is shown that these neural networks are superior to Chebyshev polynomials and can reliably represent both the direct and inverse dynamic behaviours of MR dampers. The neural network models are shown to be reasonably robust against significant temperature variation. Experimental tests show that an MR damper controller based a recurrent neural network (RNN) model of its inverse dynamics is superior to conventional controllers in achieving a desired damping force, apart from being more cost-effective. This is confirmed by introducing such a controller into a semi-active suspension, in conjunction with an overall system controller based on the sliding mode control algorithm. Control performance criteria are evaluated in the time and frequency domains in order to quantify the suspension effectiveness under bump and random road excitations. A study using the modified Bouc-Wen model for the MR damper, and another study using an actual damper fitted in a hardware-in-the-loop- simulation (HILS), both show that the inverse RNN damper controller potentially gives significantly superior ride comfort and vehicle stability. It is also shown that a similar control strategy is highly effective when used for a semi-active car seat suspension system incorporating an MR damper.