The use of energy methods and variational principles is widespread in many fields of engineering of which structural mechanics and curve and surface design are two prominent examples. In principle many different types of function can be used as possible trial solutions to a given variational problem but where piecewise polynomial behaviour and user controlled cross segment continuity is either required or desirable, B-splines serve as a natural choice. Although there are many examples of the use of B-splines in such situations there is no common thread running through existing formulations that generalises from the one dimensional case through to two and three dimensions. We develop a unified approach to the representation of the minimisation equations for B-spline based functionals in tensor product form and apply these results to solving specific problems in geometric smoothing and finite element analysis using the Rayleigh-Ritz method. We focus on the development of algorithms for the exact computation of the minimisation matrices generated by finding stationary values of functionals involving integrals of squares and products of derivatives, and then use these to seek new variational based solutions to problems in the above fields. By using tensor notation we are able to generalise the methods and the algorithms from curves through to surfaces and volumes. The algorithms developed can be applied to other fields where a variational form of the problem exists and where such tensor product B-spline functions can be specified as potential solutions.

The use of energy methods and variational principles is widespread in many fields of engineering of which structural mechanics and curve and surface design are two prominent examples. In principle many different types of function can be used as possible trial solutions to a given variational problem but where piecewise polynomial behaviour and user controlled cross segment continuity is either required or desirable, B-splines serve as a natural choice. Although there are many examples of the use of B-splines in such situations there is no common thread running through existing formulations that generalises from the one dimensional case through to two and three dimensions. We develop a unified approach to the representation of the minimisation equations for B-spline based functionals in tensor product form and apply these results to solving specific problems in geometric smoothing and finite element analysis using the Rayleigh-Ritz method. We focus on the development of algorithms for the exact computation of the minimisation matrices generated by finding stationary values of functionals involving integrals of squares and products of derivatives, and then use these to seek new variational based solutions to problems in the above fields. By using tensor notation we are able to generalise the methods and the algorithms from curves through to surfaces and volumes. The algorithms developed can be applied to other fields where a variational form of the problem exists and where such tensor product B-spline functions can be specified as potential solutions.

This research discusses the mechanical characteristics of laminated piezoelectric actuators that are manufactured at an elevated temperature, to cure the adhesive bonding the layers together, or to cure the layers made of polymeric composite material, and then cooled to a service temperature. Mainly discussed are actuators that are composed of layers of passive materials and a layer of piezoelectric material. THUNDER (THin layer UNimorph ferroelectric DrivER and sensor) and LIPCA (LIghtweight Piezo-composite Curved Actuator) actuators, which consist of layers of metal, adhesive and piezoelectric material, and carbon-epoxy, glass-epoxy and piezoelectric material, respectively, are studied and investigated in detail to understand the thermal effects due to the elevated manufacturing temperature. Owing to the large out-of-plane deformations of the THUNDER actuators as a result of cooling to the service temperature, inclusion of geometric nonlinearities in the kinematic relations is taken into consideration for prediction of the thermally-induced deformations and residual stresses. The deformations and residual stresses are predicted by using a 23-term Rayleigh-Ritz approach and more rigorous, time-consuming, finite-element analyses performed with ABAQUS. The thermally-induced deformations of THUNDER actuators can result in multiple room-temperature manufactured shapes, whereas those of LIPCA actuators (LIPCA-C1 and LIPCA-C2) exhibit single room-temperature manufactured shape. Actuation responses of these actuators caused by a quasi-static electric field applied to the piezoelectric layer are also studied with the Rayleigh-Ritz approach. It is shown that geometrical nonlinearities play an important role in the actuation responses, and these nonlinearities can be controlled by the choice of actuator geometry and the materials in the passive layers. In addition, blocking forces representing load-carrying capability of THUNDER and LIPCA actuators are determined. Support conditions and again geometrical nonlinearities are vital factor in load-resisting performances. Amongst the actuators considered, the actuated deflection and blocking forces are compared. Finally, based on the outcome of this study, new criteria for designing a new type of laminated piezoelectric actuators with improvement of performance characteristics are proposed. / Ph. D.

In the past few years a new class of layered piezoceramic and piezoceramic-composite actuators, known as RAINBOW and GRAPHBOW, respectively, that are capable of achieving 100 times greater out-of-plane displacements than previously available has been developed. Prior to the development of RAINBOW and GRAPHBOW, large stacks of piezoelectric actuators, requiring complicated electronic drive circuits, were necessary to achieve the displacement now possible through the use of a single RAINBOW actuator. The major issues with both RAINBOW and GRAPHBOW are the prediction of their room-temperature shapes after processing, and their deformation response under application of electric field. In this research, a methodology for predicting the manufactured shapes of rectangular and disk-style RAINBOW and GRAPHBOW is developed. All of the predictive analyses developed are based on finding approximate displacement responses that minimize the total potential energy of the devices through the use of variational methods and the Rayleigh-Ritz technique. These analyses are based on classical layered plate theory and assumed the various layers exhibited linear elastic, temperature-independent behavior. Geometric nonlinearities are important and are included in the strain-displacement relations. Stability of the predicted shapes is determined by examining the second variation of the total potential energy. These models are easily modified to account for the deformations induced by actuation of the piezoceramic. The results indicate that for a given set of material properties, rectangular RAINBOW can have critical values of sidelength-to-thickness ratio (Lx/H or Ly/H) below which RAINBOW exhibits unique, or single-valued, spherical or domed shapes when cooled from the processing temperature to room temperature. For values of sidelength-to-thickness ratio greater than the critical value, RAINBOW exhibits multiple room-temperature shapes. Two of the shapes are stable and are, in general, near-cylindrical. The third shape is spherical and is unstable. Similarly, disk-style RAINBOW can have critical values of radius-to-thickness ratios (R/H) below which RAINBOW exhibits axisymmetric room-temperature shapes. For values of R/H greater than the critical value, disk-style RAINBOW exhibits two stable near-cylindrical shapes and one unstable axisymmetric shape. Moreover, it is found that for the set of material properties used in this study, the optimal reduced layer thickness would be at 55%, since the maximum change in curvature is achieved under the application of an electric field, while the relationship between the change in curvatures and the electric field is kept very close to being linear. In general, good agreement is found for comparisons between the predicted and manufactured shapes of RAINBOW. A multi-step thermoelastic analysis is developed to model the addition of the fiber-reinforced composite layer to RAINBOW to make GRAPHBOW. Results obtained for rectangular RAINBOW indicate that if the bifurcation temperature in the temperature-curvature relation is lower than the composite cure temperature, then a unique stable GRAPHBOW shape can be obtained. If the RAINBOW bifurcation temperature is above the composite cure temperature, multiple room-temperature GRAPHBOW shapes are obtained and saddle-node bifurcations can be encountered during the cooling to room temperature of [0°/RAINBOW], [RAINBOW/0o], and [0o2/RAINBOW]. Rectangular [RAINBOW/0o/90o] seems to be less likely to encounter saddle-node bifurcations. Furthermore, the unstable spherical RAINBOW configuration is converted to a stable near-cylindrical configuration. For the case considered of disk-style GRAPHBOW, three stable room-temperature shapes are obtained and the unstable axisymmetric RAINBOW configuration is also converted to a stable near-cylindrical configuration. For both rectangular and disk-style GRAPHBOW, the relationship between the major curvature and the electric field is shown to be very close to being linear. This characteristic would aid any dynamic analysis of RAINBOW or GRAPHBOW. / Ph. D.

A pressurized arch-shell structural component made of flexible material is considered. The component is inflated with high internal pressure. The behavior of similar types of structures, such as a pair of leaning pressurized arches and pressurized arch-supported membrane shelters, has been investigated in the past. More recently, several types of pressurized structures have been incorporated as part of the framework for a variety of structural systems. Particularly, the U.S. Army has been investigating the use of large lightweight and transportable pressurized arch-shell structures to be used as maintenance shelters for vehicles, helicopters, and airplanes. The formulated equations using thin shell theory are applied to a pressurized arch-shell component. A numerical investigation based on the Rayleigh-Ritz method is utilized to determine the behavior of arch-shells under various types of loading. The types of loading include a uniformly distributed vertical load representing snow, a wind load, and a horizontal side load distributed along the arc length. Deflections, stress resultants, and moments at various locations are computed for two types of shapes: circular and non-circular arch-shells. / Master of Science

L’emploi de matériaux composites à matrice organique (CMO) pour la réalisation de structures aéronautiques « tièdes », peut exposer ces matériaux à l’action d’environnements agressifs, qui peuvent entrainer des phénomènes de vieillissement et de dégradation sévères associés à la diffusion d’espèces au sein du réseau macromoléculaire des matrices polymères. La complexité de la microstructure des CMO utilisés pour ces applications peut complexifier la compréhension de phénomènes de dégradation. Le vieillissement humide des CMO préoccupe particulièrement les industriels du secteur aéronautique ; la diffusion de l’eau dans la matrice polymère du composite peut entrainer des phénomènes de gonflement hygroscopique, des modifications des propriétés mécaniques. Des méthodes expérimentales existent pour la caractérisation de ces phénomènes et pour l’identification des paramètres associés : ces méthodes préconisent souvent l’emploi d’éprouvettes saturées en humidité,ce qui nécessite de longs temps de conditionnement et un nombre élevé d’échantillons. Cette thèse a pour but d’établir des protocoles d’identification rapide des propriétés diffuso-mécaniques de matériaux polymères et CMO pour applications aéronautiques. La démarche mise en place dans cette thèse s’organise autour de quatre chapitres. Le premier chapitre présente une étude bibliographique sur les outils de modélisation des couplages diffuso-mécaniques et sur les méthodes de caractérisation des propriétés diffuso-mécaniques. L’étude bibliographique permet de préciser le cadre de travail, qui prévoit l’emploi d’un modèle diffuso-mécanique faiblement couplé où la diffusion d’eau suit la loi de Fick et le comportement mécanique est hygroélastique linéaire, dépendant de la concentration en eau. Le second chapitre présente la mise en place et le développement d’une méthode d’identification rapide des propriétés de diffusion anisotrope, adaptée à des CMO à architecture complexe.La méthode s’appuie sur des mesures de prise de masse d’échantillons de CMO, l’anisotropie de diffusion de ces matériaux est obtenue par rotation des axes principaux d’orthotropie. La méthode proposée représente une extension de la « méthode de la pente » introduite par Shen et Springer pour l’identification des propriétés de diffusion de matériaux orthotropes, et basée sur l’exploitation des courbes gravimétriques aux temps courts. A travers cette méthode, les coefficients principaux et les axes principaux d’orthotropie peuvent être identifiés. Une discussion sur les conditions d’équivalence de la diffusion 3D à la diffusion 1D en fonction de l’épaisseur de l’échantillon est également présentée à la fin de ce chapitre. Le troisième chapitre explore à travers une étude numérique la possibilité d’identifier rapidement les propriétés mécaniques affectées par la concentration en eau de matériaux polymères à travers des essais mécaniques sur plaques minces avec des gradients de concentration en eau. Des essais de traction et de flexion sont considérés. Pour des matériaux isotropes, dans un cadre hygroélastique, il est montré que cette méthode permet l’identification du module d’Young et du coefficient de Poisson dépendant de la concentration en eau avec un gain remarquable de temps d’essai par rapport à des essais sur des échantillons saturés en humidité. Enfin, le dernier chapitre propose à travers une étude numérique une méthode d’identification rapide des propriétés diffuso-mécaniques de matériaux isotropes basée sur l’emploi de plaques sollicitées par un champ asymétrique de concentration en eau. L’identification est ainsi effectuée à partir du suivi des déflexions engendrées par les champs de concentration. Le coefficient de dilatation hygroscopique et le module d’Young dépendant de la concentration en eau peuvent être identifiés durant l’essai, à l’état transitoire du conditionnement, avec un gain remarquable du temps d’essai par rapport à des essais sur échantillons saturés en humidité. / The use of organic matrix composite materials (OMC) for the realization of "warm" aeronautical structures, may expose these materials to aggressive environments: wet or gaseous environments,high temperatures, which may promote severe aging and degradation phenomena related to species diffusion within the macromolecular network of the polymer matrices. The complexity of the OMC microstructure used for these applications can complicate the understanding of degradation phenomena : for example, species diffusion can be isotropic, orthotropic or anisotropic, depending on the texture of the fibrous reinforcement. Humid aging of OMC is of particular concern for the aeronautical industry ; the diffusion of water in the polymer matrix of the composite may promote hygroscopic swelling, changes in mechanical properties (stiffness, strength). Experimental methods exist for the characterization of these phenomena and for the identification of the associated parameters : these methods often recommend the use of moisture saturated specimens, which require long conditioning times, sometimes often a relevant number of samples (1 sample for each saturated state), high costs. The aim of this work is to establish protocols for fast identification of the diffusomechanical properties of polymers and polymer based OMC materials for aeronautical applications. The approach implemented in this thesis is organized in four chapters. The first chapter presents a bibliographic study on coupled diffuso-mechanics modeling tools and on methods of characterization/identification of diffuso-mechanical properties, more particularly for OMC for aeronautical applications. The bibliographic study allows specifying the framework of the present research, which foresees the employment of a weakly coupled diffuso-mechanical model, where water diffusion follows the Fick’s law and the mechanical behavior is linear hygroelastic, depending on water content. The second chap-ter presents the setting up and the development of a method for fast identification of anisotropic diffusion properties, suitable for OMC with complex architecture, such as for instance, 2D or 3Dwoven OMC. The method relies on mass-gain measures of OMC samples, the diffusion anisotropy ofthese materials is obtained by rotating the axes of orthotropy. The proposed method represents an extension of the "slope method" introduced by Shen and Springer in the 1970s for the identification of the diffusion properties of orthotropic materials (such as laminated composites), and is based onthe exploitation of gravimetric curves at short times. Through this method, the principal coefficients and the principal axes of orthotropy can be identified. A discussion about the transition from 3Dto 1D diffusion as a function of the sample geometry is also presented at the end of this chapter. The third chapter explores through a numerical study the possibility of identifying in a fast way the mechanical properties affected by moisture of polymeric materials by the use of mechanical tests on thin plates with water concentration gradients. Traction and bending tests are taken into account.For isotropic materials, in a hygroelastic setting, it is showed that this method allows identifyng the water concentration dependent Young’s modulus and the Poisson’s ratio with a remarkable time gain compared to tests on moisture saturated samples. Finally, the last chapter proposes through a numerical study a method for fast identification of the diffuso-mechanical properties of isotropic materials based on the use of plates loaded by an asymmetric water concentration field. The identification is thus carried out from the monitoring deflections generated by the concentration fields. The moisturedependent hygroscopic expansion coefficient and Young’s modulus can be identified during the test,by exploiting the transient state of conditioning, with a remarkable time gain compared with moisture saturated samples.

[pt] Algumas máquinas produzem solicitações dinâmicas que são transferidas às fundações por meio de movimentos vibratórios. É necessário analisar e estudar esses movimentos vibratórios durante o projeto da fundação para evitar danos aos equipamentos, à própria fundação e às estruturas vizinhas. Este tipo de analise também é muito importante para a segurança e saúde ocupacional de pessoas que venham a frequentar tais estruturas. Por esta razão as frequências naturais de vibração são importantes para o dimensionamento dinâmico de fundações, para se evitar efeitos de ressonância e amplificação de deslocamentos. Neste trabalho, estacas embutidas em fundações elásticas de Winkler foram estudadas. As vibrações longitudinais e transversais foram investigadas. As frequências naturais foram calculadas pelo método de Rayleigh-Ritz considerando estacas como elementos de barra embutidas em fundação elástica no caso da avaliação de vibrações longitudinais, e como vigas embutidas em base elástica no caso da avaliação das vibrações transversais. Na modelagem dos elementos de barra e viga utilizaram-se para aproximações dos deslocamentos funções convencionais enriquecidas com n funções adicionais. As constantes de mola de Winkler foram determinadas por provas de carga verticais e horizontais. Os resultados das vibrações longitudinais foram comparados com medições in situ com estacas reais. Os valores reais da vibração longitudinal das estacas foram obtidos pela aplicação da técnica Fast Fourier Transform nas acelerações resultantes de ensaios de carregamentos dinâmicos, que impõem uma vibração forçada no sistema estaca-solo. / [en] Some machines create dynamic efforts that are transferred to foundations by vibratory movements. Is necessary to analyze and study these movements during foundation design in order avoid damages in equipment, in foundation and in neighbor structures. This analysis is also important to safety and health of persons. For this reason natural vibration frequencies are important to dynamic design of foundations, to avoid resonance effects and displacements amplification. In this work, piles embedded in Winkler elastic foundations were studied. The longitudinal and transversal vibrations were investigated. The natural frequencies are calculated by Rayleigh-Ritz method modelling piles as bars embedded in elastic foundations in the case of longitudinal vibration evaluation, and beams embedded in elastic foundations in the case of transversal vibration evaluation. The bars and beams elements are modeled using conventional shape function enriched with n additional functions. The Winkler spring constants were evaluated by vertical and horizontal load tests. The longitudinal vibration results were compared with in situ measurements in real piles. The longitudinal vibration real values of piles were obtained by Fast Fourier Transform in acceleration data of dynamic load tests that impose a forced vibration in soil-pile system.

[pt] O presente trabalho mostra uma versão do método Rayleigh-Ritz com funções especializadas para a análise de placas circulares e anulares finas e espessas sujeitas a cargas fora do plano e em plano. As funções de aproximação para deslocamentos são polinómios em direção radial combinada com funções trigonométricas na direção circunferencial. Um recurso conveniente é o uso de funções nodais lineares, que permitem a fácil consideração de cargas nodais e condições de contorno (incluindo forças seguidoras), enriquecidos por polinômios de ordem superior, sem inclusão de nós adicionais. O modelo permite a variação da espessura e é aplicado em MAPLE18, possibilitando o cálculo de deslocamentos e tensões sob carregamento constante e de variação linear, as frequências de vibração, cargas de flambagem com alguns efeitos do nível de carga conservativa e não conservativa. Os exemplos mostram a eficácia desta abordagem na análise de tal estrutura e leva um novo enfoque a este problema clássico, que apresenta comparações interessantes e originais que descrevem o efeito de deformação de cisalhamento, no caso de vibrações o efeito das rotações inerciais e variação de espessura em placas circulares e anulares, incluindo deslocamentos, momentos e forças de cisalhamento, frequências de vibração, cargas de flambagem e uma análise de cargas seguidoras tangenciais não conservativas na estabilidade, utilizando o critério dinâmico é executada. / [en] The present work shows a version of the Rayleigh-Ritz method with specialized functions for the analysis of thin and thick circular and annular plates subjected to out-of-plane and in-plane loads. The approximation functions for displacements are polynomials in the radial direction combined with trigonometric functions in the circumferential direction. A convenient feature is the use of linear nodal functions, which allows for easy consideration of nodal loads and boundary conditions (including follower forces), enriched by higher order polynomials without inclusion of additional nodes. The model allows for thickness variation and was implemented in MAPLE18, enabling the calculation of displacements and stresses under constant and linearly varying load, frequencies of vibration, buckling loads with a few commands and the effect of the level of conservative and non-conservative on load the stability. The examples show the effectiveness of this approach in the analysis of such structures and bring new light to this classical problem, presenting interesting and novel comparisons illustrating the effect of shear deformation, in case of vibrations of the inertial rotations analysis and thickness variation in circular and annular plates, including displacements, moments and shear forces, vibration frequencies, buckling loads and a stability analysis of non-conservative tangential follower loads, using the dynamic criterion is performed.

[pt] Este estudo apresenta um método computacional para modelar campos eletromagnéticos em guias de onda retangulares curvados com seção transversal uniforme, usando o método variacional de Rayleigh-Ritz. Potenciais aplicações desta pesquisa em engenharia incluem o projeto de alimentadores para antenas, conversores de modais na faixa de micro-ondas, filtros, entre outros. Embora vários modelos tenham sido propostos para resolver este problema, as técnicas numéricas convencionais baseadas em elementos finitos, diferenças finitas e volumes finitos requerem altos custos computacionais. Para superar esses problemas, foi desenvolvida uma formulação variacional para resolver as equações de Maxwell em um sistema de coordenadas toroidal local, por meio de um novo funcional introduzido neste trabalho. O funcional foi adaptado para domínios uniformemente curvados com seção transversal arbitrária, e investigações analíticas foram conduzidas para confirmar suas características estacionárias. O formalismo Rayleigh-Ritz foi utilizado para converter o funcional em um problema equivalente de autovalores e autovetores, usando uma expansão em harmônicas retangulares de um guia de onda reto como funções de base para modelar um guia de onda retangular curvo. Um algoritmo numérico foi desenvolvido em Matlab para validar nosso modelo, e os resultados foram comparados com soluções perturbacionais e numéricas de referência, demonstrando alta precisão e menor custo computacional. / [en] This study presents a computational method for modeling electromagnetic fields in curved rectangular waveguides with uniform cross-section, using the variational Rayleigh-Ritz method. The potential applications of this research in engineering include the design of feeders for antennas, microwave mode converter devices, filters, among others. While various models have been proposed to solve this problem, conventional numerical techniques based on finite elements, finite differences, and finite volumes require high computational costs. To overcome these issues, a variational formulation for solving Maxwell s equations in a local toroidal coordinate system was developed via a novel functional introduced in this work. The functional was adapted to handle uniformly bend domains with arbitrary cross-section, and analytical investigations were conducted to confirm its stationary characteristics. The Rayleigh-Ritz formalism was employed to convert the functional into an equivalent problem of eigenvalues and eigenvectors using an expansion in terms of rectangular harmonics of a straight waveguide as basis functions for modeling a bend rectangular waveguide. A numerical algorithm was developed in Matlab to validate our model, and the results were compared against reference perturbational and numerical solutions, demonstrating high accuracy and lower computational costs.

We present a unified and self-contained treatment of rational Krylov methods for approximating the product of a function of a linear operator with a vector. With the help of general rational Krylov decompositions we reveal the connections between seemingly different approximation methods, such as the Rayleigh–Ritz or shift-and-invert method, and derive new methods, for example a restarted rational Krylov method and a related method based on rational interpolation in prescribed nodes. Various theorems known for polynomial Krylov spaces are generalized to the rational Krylov case. Computational issues, such as the computation of so-called matrix Rayleigh quotients or parallel variants of rational Arnoldi algorithms, are discussed. We also present novel estimates for the error arising from inexact linear system solves and the approximation error of the Rayleigh–Ritz method. Rational Krylov methods involve several parameters and we discuss their optimal choice by considering the underlying rational approximation problems. In particular, we present different classes of optimal parameters and collect formulas for the associated convergence rates. Often the parameters leading to best convergence rates are not optimal in terms of computation time required by the resulting rational Krylov method. We explain this observation and present new approaches for computing parameters that are preferable for computations. We give a heuristic explanation of superlinear convergence effects observed with the Rayleigh–Ritz method, utilizing a new theory of the convergence of rational Ritz values. All theoretical results are tested and illustrated by numerical examples. Numerous links to the historical and recent literature are included.