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In the development of engineering products, it is necessary to predict dynamic properties of the modified structures. Achieving such predictions by using the structural properties of the original structure and information on the modifications is commonly referred to as structural modification analysis. In this thesis, &Ouml / zg&uuml / ven&rsquo / s Structural Modification Method and Sherman-Morrison Method are selected as exact methods for structural modifications to predict the dynamics of a locally modified structure. Also, a new structural modification method named as &ldquo / Extended Successive Matrix Inversion Method&rdquo / is developed in this study. These three methods are implemented in a software developed herein, called &ldquo / Structural Modification Toolbox&rdquo / . The software uses modal analysis results of MSC Nastran&copy / for the original structure and calculates the modified frequency response functions by any of the methods above. In order to validate the software, direct modal analysis results of MSC Nastran&copy / for the frequency response functions of the modified structure are used. The methods are compared in terms of computational time, and the effectivity of each method is studied as a function of modification size to determine which of these methods is more suitable. In order to investigate the application of the methods and compare their results with experimental ones, modal tests are conducted on a scaled aircraft structure. The solutions are compared with test results obtained from modified test structure. Additionally, the software is used for comparison of real aircraft test results and frequency response functions of the modified structure.

The increasing use of piezoelectric material in the last decade has led to the new discovery that piezoelectric devices may be used not only as sensors or actuators individually, but also as a sensor and actuator at the same time. This bi-directional phenomenon greatly expands the utility of piezoelectric devices, especially in the area of smart material systems and structures, such as structural dynamic measurement analysis, structural damage detection, active structural vibration and acoustic control, etc. Presented in this thesis is a new electromechanical approach for structural dynamic analysis. It uses PZT patches bonded on structures as active piezoelectric sensors, and acquires the structural information by measuring the electric admittance of the sensors. Because of the electromechanical coupling, the electric admittance is mechanically modulated by the structural dynamics through the piezoelectric effect. The structural dynamic characteristics can then be extracted using a mathematical model governing the interaction of PZT actuators and structures. In this thesis, a multi-input, multi-output mathematical model is derived based on the one-dimensional constitutive law of the PZT and the mechanical impedance approach. The model yields explicit expressions of structural mechanical mobility in terms of the measured electric admittance. Applications of the approach in smart structures are presented in the area of mechanical frequency response function extraction, structural modal analysis, and structural health monitoring. As a general issue related to the application of PZT sensor-actuators in smart structures, the energy conversion efficiency of PZT actuators has also been experimentally investigated. / Master of Science

Deux familles d'aldéhyde déshydrogénases (ALDH) phylogénétiquement et structuralement distinctes catalysent l'oxydation des aldéhydes : les ALDH phosphorylantes et les ALDH non phosphorylantes. Ces enzymes jouent un rôle essentiel au niveau cellulaire en intervenant au niveau du métabolisme et dans des processus de détoxication. En 2003, la résolution de la structure tridimensionnelle de l'enzyme bifonctionnelle 4-hydroxy-2-cétovalérate aldolase/acétaldéhyde déshydrogénase (DmpFG) de Pseudomonas sp. CF600 a permis l'identification d'une nouvelle famille d'ALDH : la sous-unité DmpF étant structuralement apparentée aux ALDH phosphorylantes alors qu'elle présente une activité de type non phosphorylante CoA-dépendante. Par la caractérisation enzymatique et structurale des orthologues MhpEF issus d'Escherichia coli et de Thermomonospora curvata, nos travaux montrent que les paramètres cinétiques de MhpF ne dépendent pas de son état oligomérique, ce qui est cas unique pour les ALDH. De plus, la résolution des structures cristallographiques de l'enzyme complexée avec du NAD+ ou du CoA, couplée à la structure en solution de la forme apoenzyme obtenue par SAXS montrent que le Rossmann fold s'accomode de la présence des cofacteurs par un vaste changement conformationnel. Enfin, l'étude du mécanisme catalytique et la résolution de la structure thioacylenzyme permettent d'identifierla MhpF comme étant un hybride des deux familles d'ALDH caractérisées jusqu'à présent / Two phylogenetically and structurally unrelated families of NAD(P)-dependent aldehyde dehydrogenases (ALDH) catalyze the oxidation of aldehydes into activated or non-activated acids. These enzymes are known to be involved in many biological functions such as cellular differentiation, central metabolism, or detoxification pathways. The crystal structure of the bifunctional enzyme, 4-hydroxy-2-ketovalerate aldolase (DmpG)/acetaldehyde dehydrogenase (DpmF) from Pseudomonas sp. CF600, leads to the identification of a new ALDH family. The DmpF subunit exhibits a non-phosphorylating CoA-dependent aldehyde dehydrogenase activity while its structure belongs to the phosphorylating ALDH superfamily. The kinetics of the MhpEF orthologs from Escherichia coli and Thermomonospora curvata show that the kinetic parameters of MhpF do not depend of its oligomeric state, which is unique for an ALDH. In addition, the crystal structures of the enzyme with NAD+ or CoA, as well as the solution structure of the apoenzyme using SAXS, reveal the dynamics of the overall Rossmann fold between apo or cofactors-bound conformers, which is necessary to carry on the catalytic cycle. Finally, the catalytic mechanism and the structure of the thioacylenzyme intermediates indicate that MhpF is a hybrid between both ALDH families characterized to date

This thesis contains three essays in microeconometrics and applied labor economics. In the first two essays, we estimate dynamic models of schooling choices and employment contract outcomes of the French population. The first essay focuses on the comparison between second-generation immigrants from Africa and their French-natives counterparts. We show that the gap in higher education attainments between those two sub-populations is mainly explained by parents' background and that schooling investment is the main determinant of the gap in permanent employment. The second essay investigates the role played by educational attainments on the employment contract transitions in the early career. We find that a first fixed term contract has a positive impact on the probability of employment in a permanent contract, except for a limited set of the population endowed with particular schooling attainments and unobserved characteristics. Globally, schooling attainments account for around one third of the variance in the probability of permanent employment. The third essay is devoted to the analysis of intra-firm promotions of American executives. We estimate a dynamic model of promotions, in which we disentangle the spurious and the causal impacts of the speed of past advancement. We find that the principal determinant of promotions is unobserved heterogeneity and that the speed of past advancement in the firm's hierarchy (fast tracks) does not have a causal impact on promotions. Functional area has a high explanatory power in promotion outcomes. / Cette thèse présente trois essais en microéconométrie et économie du travail appliquée. Dans les deux premiers essais, nous estimons des modèles dynamiques de choix d'éducation et de contrats de travail en France. Le premier essai s'intéresse à la comparaison entre immigrés de deuxième génération originaires d'Afrique et natifs de parents français. Nous montrons que l'écart dans l'accès aux diplômes d'éducation supérieure entre ces deux sous-populations est expliqué principalement par l'environnement parental et que l'investissement en scolarité est le principal déterminant de l'écart dans l'accès à l'emploi permanent. Le deuxième essai s'intéresse au rôle joué par la scolarité dans les transitions entre contrats de travail en début de carrière. Nous trouvons qu'un premier contrat à durée fixe a un impact positif sur la probabilité d'emploi dans un contrat permanent, excepté pour une partie limitée de la population, dotée de niveaux de scolarité et de caractéristiques inobservables particulières. Globalement, le niveau de scolarité atteint explique environ un tiers de la variance de la probabilité d'emploi permanent. Le troisième essai est dédié à l'analyse des promotions intra-firme de cadres américains. Nous estimons un modèle dynamique de promotion dans lequel nous séparons l'effet causal de l'effet artificiel de la vitesse des avancements passés. Nous trouvons que le principal déterminant des promotions est l'hétérogénéité individuelle inobservable et que la vitesse antérieure de progression dans la hiérarchie de la firme (fast tracks) n'a pas d'impact causal. La division d'appartenance dans l'entreprise a un fort pouvoir explicatif dans les promotions observées.

[pt] O crescimento dos problemas de vibrações excessivas em edificações oriundos de atividades humanas rítmicas tem conduzido à necessidade de desenvolvimento de critérios específicos para projetos estruturais submetidos à ação dessas cargas dinâmicas. Esta foi uma das motivações para o desenvolvimento de uma metodologia de análise para investigação da resposta dinâmica de sistemas estruturais de pisos mistos (aço-concreto). Objetiva-se verificar a influência da interação aço-concreto (interação total e parcial) e, bem como, das ligações estruturais (ligações viga-coluna e viga-viga), sobre a resposta dinâmica não linear de pisos mistos (aço-concreto). Deste modo, são utilizados três modelos de carregamento dinâmico representativos das atividades humanas rítmicas referentes a saltos à vontade e ginástica aeróbica. As cargas dinâmicas são obtidas por meio de testes experimentais e, também, com base em normas e recomendações internacionais de projeto. Com referência a interação aço-concreto, vários tipos de conectores (tipo stud e perfobond) são considerados ao longo da investigação. Mesmo sabendo-se que a análise de vibrações de pisos induzidas por atividades humanas tem sido alvo de inúmeras investigações ao longo dos últimos dois séculos (1828-2012), os autores desconhecem na literatura técnica um trabalho de pesquisa tão abrangente, no que diz respeito à análise dinâmica não linear de pisos mistos (aço-concreto) submetidos à ação de atividades humanas rítmicas, com base na consideração dos efeitos dos conectores (interação aço-concreto) e das ligações (viga-coluna e viga-viga). Diante do exposto, o quadro global investigado neste trabalho de pesquisa demonstra, claramente, um indicativo de que os critérios de projeto devem levar em conta o caráter dinâmico da excitação e, especialmente, a influência da interação aço-concreto e das ligações estruturais, no que tange a verificação dos estados limites de utilização (conforto humano), associados ao comportamento estrutural de pisos mistos de edificações. / [en] The increasing incidence of building vibration problems due to human activities led to a specific design criterion to be addressed in structural design. This was the main motivation for the development of a design methodology centred on the steel-concrete composite floors non-linear dynamic response submitted to loads due to human rhythmic activities. This way, the main objective of this work is to investigate the influence of steel-concrete interaction degree (from total to various levels of partial interaction) and the beam-to-column and beam-to-beam connections effect over the non-linear dynamic behaviour of composite floors. Thus, three dynamic loading models were utilized, in order to simulate human rhythmic activities such as jumping and aerobic gymnastics. The dynamic loads were obtained through experimental tests and based on international design codes and recommendations. Regarding the steel-concrete interaction degree, the stud and perfobond connectors are considered in this investigation. Even though this topic has been studied in the technical literature for nearly two centuries, the steel-concrete composite floors non-linear dynamic analysis submitted to human rhythmic activities has not yet been addressed so comprehensively, as far as the authors are concerned in this investigation, based on the consideration of the connectors (steel-concrete interaction degree) and connections (beam-to-column and beam-to-beam) effects. Considering all aspects mentioned before, the results have demonstrated that the design criteria should include the original nature of the dynamic excitation and, specially, the steel-concrete interaction degree and structural connections effects when the steel-concrete composite floors structural behaviour, related to the serviceability limit states (human comfort) are investigated.

Materiais piezelétricos geram deslocamentos ao serem excitados com potencial elétrico, bem como potencial elétrico ao serem submetidos a força ou pressão. Eles são amplamente utilizados em aplicações relacionadas principalmente com a área de Mecânica de Precisão, Mecatrônica e aquisição de imagens por Ultra-Som. Por outro lado, os Materiais com Gradação Funcional (MGF) são materiais avançados compostos, os quais são projetados de forma que sua composição varie gradualmente numa direção espacial. Esses materiais combinam as vantagens de certas características de cada fase constitutiva; por exemplo, alta resistência à temperatura dos materiais cerâmicos com alta resistência mecânica dos metais. Vários trabalhos têm mostrado as vantagens de aplicar o conceito MGF ao projeto de transdutores piezelétricos. Entre essas vantagens podem-se mencionar: (i) atuadores flextensionais ou bilaminares sem interface entre materiais (ex: PZT e Alumínio); (ii) suavização da distribuição de tensões mecânicas; e (iii) aumento da largura de banda e redução das ondas refletidas em transdutores de ultra-som, principalmente. No entanto, na mesma literatura se observa uma carência de métodos computacionais para a sua modelagem e o seu projeto otimizado e sistemático. Baseado nessas idéias, esta tese propõe a formulação e desenvolvimento de modelos analíticos, algoritmos de elementos finitos, e algoritmos de otimização topológica para projetar Transdutores Piezelétricos com Gradação Funcional (TPGF) inovadores. Adicionalmente, amostras de TPGFs são fabricadas mediante Spark Plasma Sintering SPS, sendo estudado o seu comportamento dinâmico e as suas características micro-estruturais. Assim, através de modelagem, análise, simulação, projeto otimizado, fabricação e caracterização explora-se a potencialidade do conceito de MGF em TPGFs; em particular, evidenciam-se as melhoras que os TPGF podem trazer em aplicações de ensaios não-destrutivos e aquisição de imagens médicas por ultra-som, e no aumento da vida útil de transdutores piezelétricos flextensionais. / Piezoelectric materials generate displacements when they are excited by electrical potential and electrical potential when they are excited by force or pressure. These materials are widely applied in Precision Mechanics, Mechatronics, and Ultrasonic imaging areas. On the other hand, Functionally Graded Materials (FGM) are advanced materials, whose properties change continuously in a specified direction. These materials combine desirable features of their constituent phases; for instance, high temperature resistance typical of ceramics with mechanical strength of metals. Several works have shown the advantages of applying FGM concept to piezoelectric transducer design. These advantages are; for example: (i) flextensional actuators without interfaces (e.g. PZT and Aluminum); (ii) smoothing mechanical stresses; and (iii) increasing bandwidth and reducing reflected waves in ultrasonic transducers. However, in the literature, a lack of computational methods for modeling and systematic designing of Functionally Graded Piezoelectric Transducers (FGPT) is observed. According to above ideas, this work proposes the formulation and development of analytical models, finite element algorithms, and topology optimization algorithms to design novel Functionally Graded Piezoelectric Transducers (FGPT). In addition, FGPT samples are manufactured by using Spark Plasma Sintering SPS, where it is studied their dynamic behavior and their microstructural characteristics. Hence, by performing analysis, optimal designing, manufacturing and characterization, the FGM concept potential is explored for FGPTs; particularly, FGPTs can bring advantages in ultrasonic non-destructive testing and ultrasonic medical imaging, and increasing life-time of flextensional piezoelectric transducers.

Le bruit de roulement contribue fortement au bruit perçu à l'intérieur de l'habitacle des automobiles. Ce bruit a pour origine le contact du pneumatique sur une chaussée rugueuse. En basses fréquences (0-400 Hz), il est transmis dans l'habitacle du véhicule essentiellement par la voie solidienne. La méthode actuelle de prévision de ce bruit chez PSA Peugeot Citroën repose sur une approche mixte calcul-mesure longue, coûteuse et pas suffisamment prédictive. Pour contourner ces limitations, une filière purement numérique est envisagée. Elle demande de modéliser le comportement vibro-acoustique du pneumatique en prenant en compte les effets liés à la rotation et de résoudre le problème de contact avec une chaussée rugueuse. Concernant la modélisation d'un pneumatique en rotation, des formulations des effets tournants d'un solide déformable sont établies en utilisant une approche Arbitrairement Lagrangienne Eulérienne (ALE). Ces formulations sont validées par une application sur un nouveau modèle simplifié du pneumatique. Il s'agit d'un modèle d'anneau circulaire incluant les effets de cisaillement soumis localement à une charge représentative de la masse du véhicule. Un modèle plus complexe d'ensemble monté pneu/roue/cavité intégrant l'ensemble des effets liés à la rotation est également validé par une comparaison avec des essais. Ensuite, le contact avec une chaussée réelle est formulé par différentes approches permettant de réduire le temps de calcul pour une utilisation industrielle. En particulier, le calcul du contact est décomposé en un calcul statique non linéaire suivi d'un calcul dynamique linéaire. La validation du modèle de contact est réalisée par une comparaison calcul/essai. Les résultats sont très satisfaisants / The rolling noise contributes significantly to the noise inside cars. This noise comes from the tire/road contact. In low frequencies (0-400 Hz), it is mainly transmitted into the cabin through structural vibration. The current method used at PSA Peugeot Citroen to predict this noise, is a mixed simulation/experimental approach which is long, expensive and not sufficiently predictive. In order to overcome these difficult, a full numerical approach is considered. It requires modeling the tire vibration by taking into account the rotating effects and the contact with the rough surface. Concerning the model of rotating tire, a formulation of a deformable solid is constructed by using an Arbitrary Lagrangian Eulerian approach (ALE). This formulation is validated by an application on a new simplified tire model which is a circular ring including the shear stresses and the non linear effects due to the vehicle weight. A more complex model composed of tire/wheel/cavity including all the rotating effects is also validated by comparison with experiments. Then the contact with a real road is calculated by different approaches to get the acceptable computing time for industrial uses. In particular, the calculation of the contact is divided into a non-linear static analysis followed by a linear dynamic calculation. The validation of this model is successfully achieved by comparison test results

Uncertainties in both loading and structural characteristics can adversely affect the response and reliability of a structure. Parameter uncertainties in structural dynamics can arise due to several sources. These include variations due to intrinsic material property variability, measurement errors, manufacturing and assembly errors, differences in modeling and solution procedures. Problems of structural dynamics with randomly distributed spatial inhomogeneities in elastic, mass, and damping properties, have been receiving wide attention. Several mathematical and computational issues include discretization of random fields, characterization of random eigensolutions, inversion of random matrices, solutions of stochastic boundary-value problems, and description of random matrix products. Difficulties are encountered when one has to include interaction between nonlinear and stochastic system characteristics, or if one is interested in controlling the system response. The study of structural systems including the effects of system nonlinearity in the presence of parameter uncertainties presents serious challenges and difficulties to designers and reliability engineers. In the analysis of large structures, the situation for substructuring frequently arises due to the repetition of identical assemblages (substructures), within a structure, and the general need to reduce the size of the problem, particularly in the case of non-linear inelastic dynamic analysis. A small reduction in the model size can have a large effect on the storage and time requirement. A primary structural dynamic system may be coupled to subsystems such as piping systems in a nuclear reactor or in a chemical plant. Usually subsystem in itself is quite complex and its modeling with finite elements may result in a large number of degrees of freedom. The reduced subsystem model should be of low-order yet capturing the essential dynamics of the subsystem for useful integration with the primary structure. There are two major issues to be studied: one, techniques for analyzing a complex structure into component subsystems, analyzing the individual sub-system dynamics, and from thereon determining the dynamics of the structure after assembling the subsystems. The nonlinearity due to support gap effects such as supports for piping system in nuclear reactors further complicates the problem. The second is the issue of reviewing the methods for reducing the model-order of the component subsystems such that the order of the global dynamics, after assembly, is within some predefined limits. In the reliability analysis of complex engineering structures, a very large number of the system parameters have to be considered as random variables. The parameter uncertainties are modeled as random variables and are assumed to be time independent. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. The procedure involves the reduction of the size of the vector of random variables before the calculation of failure probability. The objectives of this thesis are: 1.To use the available model reduction techniques in order to effectively reduce the size of the finite element model, and hence, compare the dynamic responses from such models. 2.Study of propagation of uncertainties in the reduced order/coupled stochastic finite element dynamic models. 3.Addressing the localized nonlinearities due to support gap effects in the built up structures, and also in cases of sudden change in soil behaviour under the footings. The irregularity in soil behaviour due to lateral escape of soil due to failure of quay walls/retaining walls/excavation in neighbouring site, etc. 4.To evolve a procedure for the reduction of size of the vector containing the random variables before the calculation of failure probability. In the reliability analysis of complex engineering structures, a very large number of the system parameters are considered to be random variables. Here the problem would be to reduce the number of random variables without sacrificing the accuracy of the reliability analysis. 5.To analyze the reduced nonlinear stochastic dynamic system (with phase space reduction), and effectively using the network pruning technique for the solution, and also to use filter theory (wavelet theory) for reducing the input earthquake record to save computational time and cost. It is believed that the techniques described provide highly useful insights into the manner structural uncertainties propagate. The cross-sectional area, length, modulus of elasticity and mass density of the structural components are assumed as random variables. Since both the random and design variables are expressed in a discretized parameter space, the stochastic sensitivity function can be modeled in a parallel way. The response of the structures in frequency domain is considered. This thesis is organized into seven chapters. This thesis deals with the reduced order models of the stochastic structural systems under deterministic/random loads. The Chapter 1 consists of a brief introduction to the field of study. In Chapter 2, an extensive literature survey based on the previous works on model order reduction and the response variability of the structural dynamic systems is presented. The discussion on parameter uncertainties, stochastic finite element method, and reliability analysis of structures is covered. The importance of reducing mechanical models for dynamic response variability, the systems with high-dimensional variables and reduction in random variables space, nonlinearity issues are discussed. The next few chapters from Chapter 3 to Chapter 6 are the main contributions in this thesis, on model reduction under various situations for both linear and nonlinear systems. After forming a framework for model reduction, local nonlinearities like support gaps in structural elements are considered. Next, the effect of reduction in number of random variables is tackled. Finally influence of network pruning and decomposition of input signals into low and high frequency parts are investigated. The details are as under. In Chapter 3, the issue of finite element model reduction is looked into. The generalized finite element analysis of the full model of a randomly parametered structure is carried out under a harmonic input. Different well accepted finite element model reduction techniques are used for FE model reduction in the stochastic dynamic system. The structural parameters like, mass density and modulus of elasticity of the structural elements are considered to be non-Gaussian random variables. Since the variables considered here are strictly positive, the probabilistic distribution of the random variables is assumed to be lognormal. The sensitivities in the eigen solutions are compared. The response statistics based on response of models in frequency domain are compared. The dynamic responses of the full FE model, separated into real and imaginary parts, are statistically compared with those from reduced FE models. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 4, the problem of coupling of substructures in a large and complex structure, and FE model reduction, e.g., component mode synthesis (CMS) is studied in the stochastic environment. Here again, the statistics of the response from full model and reduced models are compared. The issues of non-proportional damping, support gap effects and/local nonlinearity are considered in the stochastic sense. Monte Carlo simulation is done to validate the analysis results from SFEM. In Chapter 5, the reduction in size of the vector of random variables in the reliability analysis is attempted. Here, the relative entropy/ K-L divergence/mutual information, between the random variables is considered as a measure for ranking of random variables to study the influence of each random variable on the response/reliability of the structure. The probabilistic distribution of the random variables is considered to be lognormal. The reliability analysis is carried out with the well known Bucher and Bourgund algorithm (1990), along with the probabilistic model reduction of the stochastic structural dynamic systems, within the framework of response surface method. The reduction in number of random variables reduces the computational effort required to construct an approximate closed form expression in response surface approach. In Chapter 6, issues regarding the nonlinearity effects in the reduced stochastic structural dynamic systems (with phase space reduction), along with network pruning are attempted. The network pruning is also adopted for reduction in computational effort. The earthquake accelerogram is decomposed using Fast Mallat Algorithm (Wavelet theory) into smaller number of points and the dynamic analysis of structures is carried out against these reduced points, effectively reducing the computational time and cost. Chapter 7 outlines the contributions made in this thesis, together with a few suggestions made for further research. All the finite element codes were developed using MATLAB5.3. Final pages of the thesis contain the references made in the preparation of this thesis.

[pt] Funções Wavelet de suporte compacto têm sido recentemente aplicadas na resolução numérica de equações diferenciais com resultados bastante promissores. A partir do sucesso do uso das wavelets de Daubechies em diversos métodos como o de Galerkin, surgiram novas famílias de wavelets para a resolução de problemas específicos. Nesse contexto, vale destacar uma família de wavelets com características de funções interpoladoras chamadas Interpolets. Este trabalho tem como uma de suas contribuições a formulação de elementos finitos baseados em funções wavelet de Daubechies e interpolets de Deslauriers-Dubuc para sua utilização em problemas dinâmicos como a propagação de ondas em estruturas, além de problemas não-lineares como o cálculo de cargas críticas de flambagem para colunas e pórticos. A partir dessa formulação, o Método de Wavelet-Galerkin foi adaptado para a solução direta das equações diferenciais através de uma implementação que não depende da discretização do sistema em graus de liberdade (formulação sem-malha ou meshless). Este tipo de abordagem permite também explorar ao máximo as propriedades de multirresolução das wavelets. Diversos exemplos com descontinuidades e não-linearidades foram estudados com êxito. / [en] The use of compactly supported wavelet functions has become increasingly popular in the development of numerical solutions for differential equations. Daubechies wavelets have been successfully used as base functions in several schemes like the Galerkin Method. Meanwhile, Deslauriers and Dubuc developed a new wavelet family with interpolating characteristics called Interpolets. One important contribution presented in this work is the formulation of Finite Elements based in Daubechies wavelets and Deslauriers-Dubuc interpolets. Dynamic problems like wave propagation and structural stability problems were used as examples for the validation of the Finite Elements. The Wavelet-Galerkin Method has then been adapted for the direct solution of differential equations in a meshless formulation. This approach enables the use of a multiresolution analysis. Several examples with discontinuities and nonlinearities were studied successfully.