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.

When designing a control for a physical system described by a PDE, it is often necessary to reduce the size of the controller for the PDE system. This is done so that real time control can be achieved. One approach often taken by engineers is to reduce the approximating finite-dimensional system using a balanced reduction method known as balanced truncation and then design a control for the lower order system. The unsettling idea about this method is that it involves discarding information and then designing a control. What if valuable physical information were lost that would have allowed a more effective control to be designed? This paper will explore an alternate balanced reduction method called LQG balancing. This approach allows for the designing of a control on the full order approximating system and then reducing the control. Along the way, the basic ideas of feedback control design will be discussed, including system balancing and model reduction. Following, there will be mention of the linear Klein-Gordon equation and the development of the one-dimensional finite element approximation of the PDE. Finally, simulations and numerical experiments are used to discuss the differences between the two balanced reduction methods. / Master of Science

Steel moment frames are a popular seismic-force resisting system, but it is believed that they are susceptible to early fracture if there is a stress concentration in the plastic hinge region, also known as the protected zone. If a defect is present in this area, it may be repaired by grinding and/or welding, but little research has investigated how the repairs affect the performance of full-scale moment connections subjected to inelastic rotations. Thus, the goals of this research were to establish the performance of full-scale moment connections with repairs and defects, then develop a method for predicting fracture of the full-scale specimens using more economical cyclic bend tests. To do this, six full-scale reduced beam section (RBS) connections were tested having arrays of repairs or defects applied to the flanges. The repairs were 0.125 in. deep notches ground to a smooth taper and 0.25 in. deep notches ground to a smooth taper, welded, and ground smooth. The defects were sharp 0.25 in. and 0.375 in. notches. In addition, 54 bend tests were conducted on beam flange and bar stock coupons having the same repairs and defects, power actuated fasteners, puddle welds, and no artifacts. Finally, Coffin-Manson low-cycle fatigue relationships were calibrated using results from the cyclic bend tests with each artifact (repair, defect, or attachment method) and used in conjunction with estimates of full-scale plastic strain amplitudes to predict fracture of full-scale specimens. All four of the full-scale moment connections with repairs satisfied special moment frame qualification criteria (SMF). One full-scale specimen with sharp 0.25 in. notches satisfied SMF qualification criteria, but the flexural resistance dropped rapidly after the qualification cycle. On the other hand, the specimen with sharp 0.375 in. notches did not satisfy SMF qualification criteria due to ductile fractures propagating from the notches. The proposed method for predicting fracture of full-scale connections was validated using the six current and six previous full-scale RBS specimens. This method underpredicted fracture for eleven of the twelve specimens. The ratio of the actual to predicted cumulative story drift at fracture had a mean of 1.13 and a standard deviation of 0.19. / M.S. / Moment connections in steel structures resist earthquake loads by permanently deforming the material near the connection. This area is called the protected zone and is critical to the safety of the structure in an earthquake. Due to this importance, no defects are allowed near the connection, which can include gouges or notches. If a defect does occur, it must repaired by a grinding or welding. These are the required repair methods, but there have be no tests to determine how the repairs affect the strength and ductility of the connection. This research tested six full-scale moment connections with defects repaired by grinding and welding, as well as unrepaired defects. A correlation was also developed and validated between the full-scale tests and small-scale bend tests of steel bars with the same defects and repairs. This relationship is valuable because the small-scale tests are quicker and less expensive to conduct than the full-scale tests, meaning other defects or repairs could be easily tested in the future. All but one of the six full-scale specimens met the strength requirements and had adequate ductility. The one test specimen that failed had an unrepaired defect. The relationship between the full-scale and small-scale tests underpredicted fracture (a conservative estimate) for the five of the full-scale tests and overpredicted fracture (unconservative estimate) for one test.

This thesis uses Proper Orthogonal Decomposition to model parametric and coupled systems. First, Proper Orthogonal Decomposition and its properties are introduced as well as how to numerically compute the decomposition. Next, a test case was used to show how well POD can be used to simulate and control a system. Finally, techniques for modeling a parametric system over a given range and a coupled system split into subdomains were explored, as well as numerical results. / Master of Science

<p dir="ltr">Thermo-acoustic interactions, characterized by the coupling between heat release and acoustic waves, are a phenomenon that can lead to combustion instability in high-speed propulsion devices. These interactions are highly undesirable as they can damage engine components and, in severe cases, cause catastrophic failure of the entire propulsion system. Mitigating these instabilities is crucial for ensuring reliable combustor operation. This work presents a computational investigation of combustion instability in Purdue's Continuously Variable Resonance Combustor (CVRC), focusing on the prediction of instability trend over the entire oxidizer-post length range. Computational fluid dynamics (CFD) studies in the past mainly focused on individual CVRC cases with specific oxidizer-post lengths. Those studies help understand the instability mechanism for individual CVRC cases but are limited in examining the applicability of model predictions over a wide range of instability conditions. No studies have been reported to assess the model predictivity over the entire oxidizer-post range in CVRC. </p><p dir="ltr">In this work, we first conduct a series of CFD simulations that cover the entire oxidizer-post length in CVRC to assess the models for a wide range of instability conditions. It is found that the CFD models generally fail to capture the instability trend over the entire oxidizer-post length although they can capture some individual cases. To understand the model failure, parametric studies are often deemed the first step of investigation. Such parameter studies, however, are expensive for CVRC since more than ten simulation cases to cover the entire oxidizer-post range are needed for each parametric study. Multiple parametric studies are typically needed to cover various uncertainties from numerics and physical models and those involved in the experimental conditions, making parametric studies for CVRC a computationally expensive task. Therefore, our focus next is on developing faster approaches.</p><p dir="ltr">The second part of this work is to develop a reduced-order model to quickly conduct the needed parametric studies. The developed reduced-order model leverages the instability mechanisms observed from the CFD simulations conducted in the first part. Monte Carlo approaches are employed to replace expensive CFD simulations by replicating the randomness in the combustor through statistical sampling. The developed reduced-order model is first validated by comparing its predictions with the CFD simulation results in a number of cases. The reduced-order model, despite its simplicity, reasonably reproduced the overall trend of instability from CFD simulations, making it an attractive alternative to the detailed model simulations for parametric studies. </p><p dir="ltr">The validated reduced-order model is then applied to parametric studies of CVRC to help identify the uncertainties of CFD predictions of CVRC. Four sets of parametric studies are conducted to provide a rapid examination of the effect of heat loss, the effect of oxidizer temperature, the effect of equivalence ratio, and the effect of turbulence on the instability predictions in CVRC. From the rapid reduced-order parametric studies, we found that the heat losses in upstream of the oxidizer inlet and the combustor wall are the two most contributing factors to the uncertainties of CFD model predictions. The turbulence level and the error involved in the equivalence ratio due to experimental uncertainties play an insignificant role in contributing to the CFD prediction uncertainties. </p><p dir="ltr">This work is a significant contribution to the combustion instability community by enabling an alternative rapid assessment of CFD model predictions. This capability facilitates the identification of major contributing factors of CFD modeling uncertainties with much less computational cost, thereby allowing for a more focused approach to CFD analysis and ultimately accelerating the improvement of CFD models for combustion instability studies. </p>

Yes / The coexistence of hermaphrodites and female-sterile individuals, or androdioecy, has been documented in only a handful of plants and animals. This study reports its existence in the plant species Cardamine amara (Brassicaceae), in which female-sterile individuals have shorter pistils than seed-producing hermaphrodites. Morphological analysis, in situ manual pollination, microsatellite genotyping and differential gene expression analysis using Arabidopsis microarrays were used to delimit variation between female-sterile individuals and hermaphrodites. Female sterility in C. amara appears to be caused by disrupted ovule development. It was associated with a 2.4- to 2.9-fold increase in clonal propagation. This made the pollen number of female-sterile genets more than double that of hermaphrodite genets, which fulfils a condition of co-existence predicted by simple androdioecy theories. When female-sterile individuals were observed in wild androdioecious populations, their ramet frequencies ranged from 5 to 54 %; however, their genet frequencies ranged from 11 to 29 %, which is consistent with the theoretically predicted upper limit of 50 %. The results suggest that a combination of sexual reproduction and increased asexual proliferation by female-sterile individuals probably explains the invasion and maintenance of female sterility in otherwise hermaphroditic populations. To our knowledge, this is the first report of the coexistence of female sterility and hermaphrodites in the Brassicaceae.

High-fidelity computer simulations of childbirth are time consuming, making them impractical for guiding decision-making during obstetric emergencies. The complex geometry, micro-structure, and large finite deformations undergone by the vagina during childbirth result in material and geometric nonlinearities, complicated boundary conditions, and nonhomogeneities within finite element (FE) simulations. Such nonlinearities pose a significant challenge for numerical solvers, increasing the computational time. Simplifying assumptions can reduce the computational time significantly, but this usually comes at the expense of simulation accuracy. The work herein proposed the use of reduced order modeling (ROM) techniques to create surrogate models that capture experimentally-measured displacement fields of rat vaginal tissue during inflation testing in order to attain both the accuracy of higher-fidelity models and the speed of lower-fidelity simulations. The proper orthogonal decomposition (POD) method was used to extract the significant information from FE simulations generated by varying the luminal pressure and the parameters that introduce the anisotropy in the selected constitutive model. In our first study, a new data-driven (DD) variational multiscale (VMS) ROM framework was extended to obtain the displacement fields of rat vaginal tissue subjected to ramping luminal pressure. For comparison purposes, we also investigated the classical Galerkin ROM (G-ROM). In our numerical study, both the G-ROM and the DD-VMS-ROM decreased the FE computational cost by orders of magnitude without a significant decrease in numerical accuracy. Furthermore, the DD-VMS-ROM improved the G-ROM accuracy at a modest computational overhead. Our numerical investigation showed that ROM had the potential to provide efficient and accurate computational tools to describe vaginal deformations, with the ultimate goal of improving maternal health. Our second study compared two common computational strategies for surrogate modeling, physics-based G-ROM and data-driven machine learning (ML), for decreasing the cost of FE simulations of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. Since there are many methods associated with each modeling approach, to provide a fair and natural comparison, we selected a basic model from each category. From the ROM strategies, we considered a simplified G-ROM that is based on the linearization of the underlying nonlinear FE equations. From the ML strategies, we selected a feed-forward dense neural network (DNN) to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the POD coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical comparisons of G-ROM, ML, and POD-ML took place in the reconstructive regime. The numerical results showed that the G-ROM outperformed the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The POD-ML model improved on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model did not improve on the error performance of the ML. In our last study, we expanded our investigation of ML methods for surrogate modeling by comparing the performance of a DNN similar to what was used previously to that of a convolutional neural network (CNN) using 1-D convolution on the input parameters from FE simulations of active vaginal tearing. The new FE simulations utilized a custom continuum damage model that provided material damage and failure properties to an existing anisotropic hyperelastic constitutive model to replicate experimentally-observed tear propagation behaviors. We employed our DNN and CNN models to create mappings from constitutive model parameters, geometric properties of the propagating tear, and luminal pressure values to either the full FE displacement history or the POD coefficients of the displacement history. The root-mean-square error (RMSE) with respect to the FE displacement history achieved by full order output ML predictions was reproducible with POD-ML using a basis of only dimension l=10. Additionally, an order of magnitude reduction in offline time was observed using POD-ML over full-order ML with minimal difference between DNN and CNN architectures. Differences in online computational costs between ML and POD-ML were found to be negligible, but the DNNs produced predictions slightly faster than the CNNs, though both online times were on the same order of magnitude. While convolution did not significantly aid the regression task at hand, POD-ML was demonstrated to be an efficient and effective approach for surrogate modeling of the FE tear propagation model, approximating the displacement history with RMSE less than 0.1 mm and generating results 7 orders of magnitude faster than the FE model. This set of baseline numerical investigations serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, material properties, and tissue damage mechanics of the human vagina, as well as their changes during labor. / Doctor of Philosophy / Computer simulations of childbirth are extremely time-consuming, making them impractical for guiding decision-making by obstetricians when a patient is entering labor. The complex geometry, material microstructure, and large deformations undergone by the vagina during childbirth result in material and geometric properties that are challenging to mathematically model. Consequently, numerical solver methods (e.g., finite elements) require large amounts of time to simulate childbirth. Simplifying assumptions can reduce computational time, but this simplification usually comes at the expense of simulation accuracy. The work of this dissertation proposes the use of several techniques to reduce model complexity and create accurate approximations and predictions of results from full-order models (FOMs) with profound reductions in computational time. Our first study used reduced order models (ROMs) to extract the significant information from a FOM of the rat vagina subjected to inflation. We compared a basic ROM and an advanced, data-driven ROM. Our second study compared the basic ROM to a basic machine learning (ML) technique for approximating a FOM that simulated inflation of the rat vagina with a pre-imposed tear. A hybrid technique incorporating elements of both ROM and ML to approximate FOM results was also considered. Our final study made use of ML and hybrid techniques using a more advanced neural network (a convolutional neural network). These ML models were used to predict the results of a FOM simulation of vaginal tear propagation. These numerical investigations serve as a starting point for future development of computer simulations using state-of-the-art ROM and ML strategies as well as more realistic models for the mechanics of the human vagina during childbirth.

Quadrupedal robots play a vital role in various applications, from search and rescue operations to exploration in challenging terrains. However, locomotion tasks involving unknown payload transportation on rough terrains pose significant challenges, requiring adaptive control strategies to ensure stability and performance. This dissertation contributes to the advancement of adaptive motion planning and control solutions that enable quadrupedal robots to traverse unknown rough environments while tasked with transporting unknown payloads. In the first project, a novel hierarchical planning and control framework for robust payload transportation by quadrupedal robots is developed. This framework integrates an adaptive model predictive control (AMPC) algorithm with a gradient-descent-based adaptive updating law applied to reduced-order locomotion (i.e., template) models. At the high level of the control hierarchy, an indirect adaptive law estimates unknown parameters of the reduced-order locomotion model under varying payloads, ensuring stability during trajectory planning. The optimal trajectories generated by the AMPC are then passed to a low-level and full-order nonlinear whole-body controller (WBC) for tracking. Extensive numerical investigations and hardware experiments on the A1 quadru[pedal robot validate the framework's capabilities, showcasing significant improvements in payload transportation on both flat and rough terrains compared to conventional MPC strategies. Specifically, the robot demonstrates proficiency in transporting unmodeled, unknown static payloads up to 109% of its own mass in experiments on flat terrains and 91% on rough experimental terrains. Moreover, the robot successfully manages dynamic payloads with 73% of its mass on rough terrains. Adaptive controllers must also address external disturbances inherent in real-world environments. Therefore, the second project introduces a hierarchical planning and control scheme with an adaptive L1 nonlinear model predictive control (ANMPC) at the high level, which integrates nonlinear MPC (NMPC) with an L1 adaptive controller. The prescribed optimal state and control input profiles generated by the ANMPC are then fed to the low-level nonlinear WBC. This approach aims to stabilize locomotion gaits in the presence of parametric uncertainties and external disturbances. The proposed controller is analyzed to accommodate uncertainties and external disturbances. Comprehensive numerical simulations and experimental validations on the A1 quadrupedal robot demonstrate its effectiveness on rough terrains. Numerical results suggest that ANMPC significantly improves the stability of the gaits in the presence of uncertainties and external disturbances compared to NMPC and AMPC. The robot can carry payloads up to 109% of its own mass on its trunk on flat and rough terrains. Simulation results show that the robot achieves a maximum payload capacity of 26.3 (kg), which is equivalent to 211% of its own mass on rough terrains with uncertainties and disturbances. / Doctor of Philosophy / In the rapidly advancing domain of robotics, there is a growing demand for intelligent robotic systems capable of adeptly addressing novel and unforeseen scenarios, such as uneven paths or external forces applied to the robots, like kicks and hits. This necessitates robots with the capability to handle diverse tasks with precision, particularly in the domains of object transportation and navigation through unknown terrains in applications such as search and rescue operations or cargo handling. This dissertation introduces innovative motion planning and control frameworks designed to imbue robots with adaptive capabilities, enabling them to adapt to real-world unanticipated scenarios and uncertainties during their movement, particularly when carrying unknown payloads. In the first project, a new framework is developed to enhance payload transportation by quadrupedal robots. This framework integrates an adaptive model predictive control (AMPC) algorithm with a gradient-descent-based adaptive updating law. Through extensive experiments and simulations, the framework shows remarkable improvements in payload transportation on both flat and rough terrains. The robot successfully transports payloads exceeding its own mass by up to 109% on flat terrains and 91% on rough terrains. Recognizing the need to address uncertainties in real-world environments, the second project introduces a hierarchical planning and control scheme with adaptive L1 nonlinear model predictive control (ANMPC). This approach stabilizes legged locomotion in the presence of uncertainties and disturbances. Results demonstrate that ANMPC significantly improves gait stability compared to existing methods. The robot achieves a payload capacity of up to 109% of its own mass on both experimental flat and rough terrains and reaches a maximum of 26.3 kg (around 212% of its own mass) on rough terrain simulations with uncertainties and disturbances.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2018-03-15T20:24:53Z No. of bitstreams: 2 Dissertação - Jéssica Gabriela de Almeida Cunha - 2018.pdf: 3483915 bytes, checksum: 12c37e736c4d6f53761fc0255e6bff6d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-16T11:10:21Z (GMT) No. of bitstreams: 2 Dissertação - Jéssica Gabriela de Almeida Cunha - 2018.pdf: 3483915 bytes, checksum: 12c37e736c4d6f53761fc0255e6bff6d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-16T11:10:21Z (GMT). No. of bitstreams: 2 Dissertação - Jéssica Gabriela de Almeida Cunha - 2018.pdf: 3483915 bytes, checksum: 12c37e736c4d6f53761fc0255e6bff6d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-02-22 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / This work brings the use of grid of points in the resolution of cutting and packing problems that consider rectangular shaped items. The grids can be considered for mathematical programming models and heuristics, and they are independent of the problem. The following grids that are defined by the literature are considered for this work: canonical dissections (also known as normal patterns), reduced raster points, useful numbers, corner points, regular normal patterns, extreme points, and meet-in-the-middle patterns. The objective is to assess the influence of each grid on the resolution of cutting and packing problems, before and after applying reduction procedures, as the one related to update the items size. Theoretical results are obtained from relations of set and size between the grids, showing that the grid of normal patterns and useful numbers are equivalent and, thus, proving formally that the grid of reduced raster points ensures an optimal solution (this result has been formally opened in the literature). In addition, we propose a new procedure to reduce the size of grids. In order to validate the proposed procedure and evaluate the grids, we perform experiments over instances from the literature, where it is possible to observe that the grids of reduced raster points and meet-in-the-middle patterns are the smallest. Experiments were also conducted in a two-dimensional packing problem that uses an integer linear programming model to pack the items in points of a grid. The results indicate that using the reduction procedures it is possible to obtain optimal solutions quicker. / Este trabalho traz o uso de malhas de pontos na resolução de problemas de corte e empacotamento para itens com formato retangular. As malhas podem ser consideradas em modelos de programação matemática e heurísticas, sendo independentes do problema tratado. As seguintes malhas definidas pela literatura, canonical dissections (também conhecida por normal patterns), reduced raster points, useful numbers, corner points, regular normal patterns, extreme points e meet-in-the-middle patterns, são consideradas neste trabalho. O objetivo é apresentar relações que existem entre as malhas e analisar a influência delas sobre o tempo gasto na resolução de problemas de corte e empacotamento, antes e após aplicar procedimentos de redução, como atualizar o tamanho dos itens. Resultados teóricos são obtidos envolvendo relações de conjunto e tamanho entre as malhas, mostrando que a malha de normal patterns e useful numbers são equivalentes e, assim, permitindo provar formalmente que a malha de reduced raster points garante uma solução ótima (resultado que estava em aberto na literatura). Além disso, propõe-se um novo procedimento visando reduzir o tamanho das malhas. Como forma de validar o procedimento proposto e avaliar a redução que ele proporciona nas malhas, executam-se experimentos sobre instâncias da literatura, sendo possível observar que as malhas de reduced raster points e meet-in-the-middle patterns são as menores. Experimentos também foram realizados sobre um problema de empacotamento bidimensional que utiliza um modelo de programação linear inteira para empacotar os itens em pontos da malha. Os resultados indicam que utilizando os procedimentos de redução é possível obter soluções ótimas mais rapidamente.

Ce mémoire de thèse concerne l'interpolation sur les variétés de Grassmann et ses applications à la réduction de modèles en mécanique et plus généralement aux systèmes d'équations aux dérivées partielles d'évolution. Après une description de la méthode POD, nous introduisons les fondements théoriques en géométrie des variétés de Grassmann, qui seront utilisés dans le reste de la thèse. Ce chapitre donne à ce mémoire à la fois une rigueur mathématique au niveau des algorithmes mis au point, leur domaine de validité ainsi qu'une estimation de l'erreur en distance grassmannienne, mais également un caractère auto-contenu "self-contained" du manuscrit. Ensuite, on présente la méthode d'interpolation sur les variétés de Grassmann introduite par David Amsallem et Charbel Farhat. Cette méthode sera le point de départ des méthodes d'interpolation que nous développerons dans les chapitres suivants. La méthode de Amsallem-Farhat consiste à choisir un point d'interpolation de référence, envoyer l'ensemble des points d'interpolation sur l'espace tangent en ce point de référence via l'application logarithme géodésique, effectuer une interpolation classique sur cet espace tangent, puis revenir à la variété de Grassmann via l'application exponentielle géodésique. On met en évidence par des essais numériques l'influence du point de référence sur la qualité des résultats. Dans notre premier travail, nous présentons une version grassmannienne d'un algorithme connu dans la littérature sous le nom de Pondération par Distance Inverse (IDW). Dans cette méthode, l'interpolé en un point donné est considéré comme le barycentre des points d'interpolation où les coefficients de pondération utilisés sont inversement "proportionnels" à la distance entre le point considéré et les points d'interpolation. Dans notre méthode, notée IDW-G, la distance géodésique sur la variété de Grassmann remplace la distance euclidienne dans le cadre standard des espaces euclidiens. L'avantage de notre algorithme, dont on a montré la convergence sous certaines conditions assez générales, est qu'il ne requiert pas de point de référence contrairement à la méthode de Amsallem-Farhat. Pour remédier au caractère itératif (point fixe) de notre première méthode, nous proposons une version directe via la notion de barycentre généralisé. Notons enfin que notre algorithme IDW-G dépend nécessairement du choix des coefficients de pondération utilisés. Dans notre second travail, nous proposons une méthode qui permet un choix optimal des coefficients de pondération, tenant compte de l'auto-corrélation spatiale de l'ensemble des points d'interpolation. Ainsi, chaque coefficient de pondération dépend de tous les points d'interpolation et non pas seulement de la distance entre le point considéré et un point d'interpolation. Il s'agit d'une version grassmannienne de la méthode de Krigeage, très utilisée en géostatique. La méthode de Krigeage grassmannienne utilise également le point de référence. Dans notre dernier travail, nous proposons une version grassmannienne de l'algorithme de Neville qui permet de calculer le polynôme d'interpolation de Lagrange de manière récursive via l'interpolation linéaire entre deux points. La généralisation de cet algorithme sur une variété grassmannienne est basée sur l'extension de l'interpolation entre deux points (géodésique/droite) que l'on sait faire de manière explicite. Cet algorithme ne requiert pas le choix d'un point de référence, il est facile d'implémentation et très rapide. De plus, les résultats numériques obtenus sont remarquables et nettement meilleurs que tous les algorithmes décrits dans ce mémoire. / This dissertation deals with interpolation on Grassmann manifolds and its applications to reduced order methods in mechanics and more generally for systems of evolution partial differential systems. After a description of the POD method, we introduce the theoretical tools of grassmannian geometry which will be used in the rest of the thesis. This chapter gives this dissertation a mathematical rigor in the performed algorithms, their validity domain, the error estimate with respect to the grassmannian distance on one hand and also a self-contained character to the manuscript. The interpolation on Grassmann manifolds method introduced by David Amsallem and Charbel Farhat is afterward presented. This method is the starting point of the interpolation methods that we will develop in this thesis. The method of Amsallem-Farhat consists in chosing a reference interpolation point, mapping forward all interpolation points on the tangent space of this reference point via the geodesic logarithm, performing a classical interpolation on this tangent space and mapping backward the interpolated point to the Grassmann manifold by the geodesic exponential function. We carry out the influence of the reference point on the quality of the results through numerical simulations. In our first work, we present a grassmannian version of the well-known Inverse Distance Weighting (IDW) algorithm. In this method, the interpolation on a point can be considered as the barycenter of the interpolation points where the used weights are inversely proportional to the distance between the considered point and the given interpolation points. In our method, denoted by IDW-G, the geodesic distance on the Grassmann manifold replaces the euclidean distance in the standard framework of euclidean spaces. The advantage of our algorithm that we show the convergence undersome general assumptions, does not require a reference point unlike the method of Amsallem-Farhat. Moreover, to carry out this, we finally proposed a direct method, thanks to the notion of generalized barycenter instead of an earlier iterative method. However, our IDW-G algorithm depends on the choice of the used weighting coefficients. The second work deals with an optimal choice of the weighting coefficients, which take into account of the spatial autocorrelation of all interpolation points. Thus, each weighting coefficient depends of all interpolation points an not only on the distance between the considered point and the interpolation point. It is a grassmannian version of the Kriging method, widely used in Geographic Information System (GIS). Our grassmannian Kriging method require also the choice of a reference point. In our last work, we develop a grassmannian version of Neville's method which allow the computation of the Lagrange interpolation polynomial in a recursive way via the linear interpolation of two points. The generalization of this algorithm to grassmannian manifolds is based on the extension of interpolation of two points (geodesic/straightline) that we can do explicitly. This algorithm does not require the choice of a reference point, it is easy to implement and very quick. Furthermore, the obtained numerical results are notable and better than all the algorithms described in this dissertation.