[pt] O método híbrido de elementos finitos, proposto por Pian com base no potencial de Hellinger-Reissner, provou ser um avanço conceitual entre as formulações de discretização, tendo sido explorado extensivamente desde então por códigos acadêmicos e comerciais, também levando em conta uma série independente dos mais recentes desenvolvimentos chamados métodos de Trefftz. O método híbrido de elementos de contorno é uma generalização bem sucedida da formulação original de Pian, em que funções de Green são usadas como funções de interpolação no domínio, possibilitando assim a modelagem robusta e precisa de formas arbitrárias submetidas a vários tipos de ações.Mais recentemente, uma proposição de Przemieniecki - para a análise geral de vibração livre de elementos de treliça e viga - foi incorporada à formulação de elementos híbridos de contorno e estendida para a análise de problemas dependentes do tempo fazendo uso de um processo de superposição modal avançada que leva em conta condições iniciais gerais assim como ações de corpo gerais, além de efeitos inerciais. A presente contribuição pretende trazer para elementos finitos os melhoramentos conceituais obtidos no contexto do método híbrido de elementos de contorno. Uma grande família de macro elementos finitos híbridos é introduzida para o tratamento unificado em 2D e 3D, de problemas estáticos e transientes de elasticidade e potencial com base nas soluções fundamentais não-singulares. É também mostrado que materiais nãohomogêneos, como os novos materiais com gradação funcional, podem ser tratados consistentemente, pelo menos para problemas de potencial. Alguns exemplos numéricos simples são apresentados como ilustração dos desenvolvimentos teóricos. / [en] The hybrid finite element method, proposed by Pian on the basis of the Hellinger-Reissner potential, has proved itself a conceptual breakthrough among the discretization formulations, and has been extensively explored both academically and in commercial codes also taking into account an independent series of more recent developments called Trefftz methods. The hybrid boundary element method is a successful generalization of Pian´s original formulation, in which Green´s functions are taken as interpolation functions in the domain, thus enabling the robust and accurate modeling of arbitrarily shaped bodies submitted to several types of actions. More recently, a proposition by Przemieniecki - for the generalized free vibration analysis of truss and beam elements - was incorporated into the hybrid boundary element formulation and extended to the analysis of time-dependent problems by making use of an advanced mode superposition procedure that takes into account general initial conditions as well as general body actions, besides the inertial effect. The present contribution aims to bring to finite elements the conceptual improvements obtained in the frame of the hybrid boundary element method. A large family of hybrid, macro finite elements is introduced for the unified treatment of 2D and 3D, static and transient problems of elasticity and potential on the basis of nonsingular fundamental solutions. It is also shown that nonhomogeneous materials, as the novel functionally graded materials, may be dealt with consistently, at least for potential problems. Some simple numerical examples are shown to illustrate the theoretical developments.

No description available.

The selective laser melting (SLM) process allows for the control of unique part form and function characteristics not achievable with conventional manufacturing methods and has thus gained interest in several industries such as the aerospace and biomedical fields. The fabrication processing parameters selected to manufacture a given part influence the created material microstructure and the final mechanical performance of the part. Understanding the process-structure and structure-performance relationships is very important for the design and quality assurance of SLM parts. Image based analysis methods are commonly used to characterize material microstructures, but are very time consuming, traditionally requiring manual segmentation of imaged features. Two Python-based image analysis tools are developed here to automate the instance segmentation of manufacturing defects and subgranular cell features commonly found in SLM 316L stainless steel (SS) for quantitative analysis. A custom trained mask region-based convolution neural network (Mask R-CNN) model is used to segment cell features from scanning electron microscopy (SEM) images with an instance segmentation accuracy nearly identical to that of a human researcher, but about four orders of magnitude faster. The defect segmentation tool uses techniques from the OpenCV Python library to identify and segment defect instances from optical images. A melt pool structure generation tool is also developed to create custom melt-pool geometries based on a few user inputs with the ability to create functionally graded structures for use in a virtual testing framework. This tool allows for the study of complex melt-pool geometries and graded structures commonly seen in SLM parts and is applied to three finite element analyses to investigate the effects of different melt-pool geometries on part stress concentrations. / Master of Science / Recent advancements in additive manufacturing (AM) processes like the selective laser melting (SLM) process are revolutionizing the way many products are manufactured. The geometric form and material microstructure of SLM parts can be controlled by manufacturing settings, referred to as fabrication processing parameters, in ways not previously possible via conventional manufacturing techniques such as machining and casting. The improved geometric control of SLM parts has enabled more complex part geometries as well as significant manufacturing cost savings for some parts. With improved control over the material microstructure, the mechanical performance of SLM parts can be finely tailored and optimized for a particular application. Complex functionally graded materials (FGM) can also easily be created with the SLM process by varying the fabrication processing parameters spatially within the manufactured part to improve mechanical performance for a desired application. The added control offered by the SLM process has created a need for understanding how changes in the fabrication processing parameters affect the material structure, and in turn, how the produced structure affects the mechanical properties of the part. This study presents three different tools developed for the automated characterization of SLM 316L stainless steel (SS) material structures and the generation of realistic material structures for numerical simulation of mechanical performance. A defect content tool is presented to automatically identify and create binary segmentations of defects in SLM parts, consisting of small air pockets within the volume of the parts, from digital optical images. A machine learning based instance segmentation tool is also trained on a custom data set and used to measure the size of nanoscale cell features unique to 316L (SS) and some other metal alloys processed with SLM from scanning electron microscopy (SEM) images. Both these tools automate the laborious process of segmenting individual objects of interest from hundreds or thousands of images and are shown to have an accuracy very close to that of manually produced results from a human. The results are also used to analyze three different samples produced with different fabrication processing parameters which showed similar process-structure relationships with other studies. The SLM structure generation tool is developed to create melt pool structures similar to those seen in SLM parts from the successive melting and solidification of material from the laser scanning path. This structural feature is unique to AM processes such as SLM, and the example test cases investigated in this study shows that changes in the melt pool structure geometry have a measurable effect, slightly above 10% difference, on the stress and strain response of the material when a tensile load is applied. The melt pool structure generation tool can create complex geometries capable of varying spatially to create FGMs from a few user inputs, and when applied to existing simulation methods for SLM parts, offers improved estimates for the mechanical response of SLM parts.

Functionally graded materials (FGMs) are inhomogeneous materials in which the material properties vary with respect to space. Research has been done by scientific community in developing techniques like photothermal radiometry (PTR) to measure the thermal conductivity and volumetric heat capacity of FGMs. One of the problems involved in the technique is to solve the inverse problem, i.e., estimating the thermal properties after the frequency scan has been obtained. The present work involves finding the unknown thermal conductivity and volumetric heat capacity of the FGMs by using finite volume method. By taking the flux entering the sample as periodic and solving the discretized 1-D thermal wave field equation at a frequency domain, one can obtain the complex temperatures at the surface of the sample for each frequency. These complex temperatures when solved for a range of frequencies gives the phase vs frequency scan which can then be compared to original frequency scan obtained from the PTR experiment by using a residual function. Brute force and gradient descent optimization methods have been implemented to estimate the unknown thermal conductivity and volumetric specific heat of the FGMs through minimization of the residual function. In general, the spatial composition profile of the FGMs can be approximated by using a smooth curve. Three functional forms namely Arctangent curve, Hermite curve, and Bezier curve are used in approximating the thermal conductivity and volumetric heat capacity distributions in the FGMs. The use of Hermite and Bezier curves gives the flexibility to control the slope of the curve i.e. the thermal property distribution along the thickness of the sample. Two-layered samples with constant thermal properties and three layered samples in which one of the layer has varying thermal properties with respect to thickness are considered. The program is written in Fortran and several test runs are performed. Results obtained are close to the original thermal property values with some deviation based on the stopping criteria used in the gradient descent algorithm. Calculating the gradients at each iteration takes considerable amount of time and if these gradient values are already available, the problem can be solved at a faster rate. One of the methods is extending automatic differentiation to complex numbers and calculating the gradient values ahead; this is left for future work.

Composite materials and hybrid materials such as fiber-metal laminates, and functionally graded materials are increasingly common in application in aerospace structures. However, adhesive bonding of dissimilar materials makes these materials susceptible to delamination. The use of integrated Sinc methods for predicting interlaminar failure in laminated composites and hybrid material systems was examined. Because the Sinc methods first approximate the highest-order derivative in the governing equation, the in-plane derivatives of in-plane strain needed to obtain interlaminar stresses by integration of the equilibrium equations of 3D elasticity are known without post-processing. Interlaminar stresses obtained with the Sinc method based on Interpolation of Highest derivative were compared for the first-order and third-order shear deformable theories, the refined zigzag beam theory and the higher-order shear and normal deformable beam theory. The results indicate that the interlaminar stresses by the zigzag theory compare well with those obtained by a 3D finite element analysis, while the traditional equivalent single layer theories perform well for some laminates. The philosophy of the Sinc method based on Interpolation of Highest Derivative was extended to create a novel weak form based approach called the Integrated Local Petrov-Galerkin Sinc Method. The Integrated Local Petrov-Galerkin Sinc Method is easily utilized for boundary-value problem on non-rectangular domains as demonstrated for analysis of elastic and elastic-plastic plane-stress panels with elliptical notches. The numerical results showed excellent accuracy compared to similar results obtained with the finite element method. The Integrated Local Petrov-Galerkin Sinc Method was used to analyze interlaminar debonding of composite and fiber-metal laminated beams. A double-cantilever beam and a fixed-ratio mixed mode beam were analyzed using the Integrated Local Petrov-Galerkin Sinc Method and the results were shown to correlate well with those by the finite element method. An adaptive Sinc point distribution technique was implemented for the delamination analysis which significantly improved the methods accuracy for the present problem. Delamination of a GLARE, plane-strain specimen was also analyzed using the Integrated Local Petrov-Galerkin Sinc Method. The results correlate well with 2D, plane-strain analysis by the finite element method, including interlaminar stresses obtained by through-the-thickness integration of the equilibrium equations of 3D elasticity. / Ph. D.

Submitted by Angela Maria de Oliveira (amolivei@uepg.br) on 2019-02-27T11:49:16Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Gustavo Antoniacomi de Carvalho.pdf: 6310308 bytes, checksum: 8e1efd9d86bc5a1adf80b45bba2a3985 (MD5) / Made available in DSpace on 2019-02-27T11:49:16Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Gustavo Antoniacomi de Carvalho.pdf: 6310308 bytes, checksum: 8e1efd9d86bc5a1adf80b45bba2a3985 (MD5) Previous issue date: 2018-01-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Cerâmicas porosas vêm ganhando visibilidade devido a algumas aplicações tecnológicas interessantes, tais como a utilização em eletrólitos sólidos, ânodos de células a combustível, filtros cerâmicos e reposição óssea e dental. Dentre elas, há especial atenção ao estudo de materiais porosos com porosidade graduada, nos quais a quantidade de poros e a morfologia dos mesmos se alteram pelo volume do material. Nesse trabalho foi realizado o processamento e caracterização de materiais cerâmicos de alumina com porosidade graduada a partir das técnicas de freeze casting e colagem de barbotina, utilizando hidróxido de alumínio e amido de arroz como fases de sacrifício. Após a conformação das amostras por esses métodos, a porosidade foi caracterizada por microscopia eletrônica de varredura, pelas medidas de porosidade aparente feita pelo Princípio de Arquimedes e pela distribuição de tamanho de poros feita pela técnica de porosimetria de mercúrio. Foi avaliada também a resistência mecânica das amostras a partir de ensaio de compressão. Foi confirmada a relação entre as amostras processados isoladamente e suas respectivas camadas nas amostras graduadas. Foi observada também uma boa interação interfacial entre cada uma das camadas das amostras graduadas. A porosidade das amostras com porosidade graduada se manteve próxima do esperado, o valor esperado foi determinado a partir da média das amostras processadas isoladamente em relação às camadas do material com porosidade graduada. O ensaio mecânico demonstrou que não houve influência das interfaces dos materiais graduados na sua resistência à compressão. / Porous ceramics have been gaining visibility due to some interesting technological applications, such as its use as solid electrolytes, fuel cell anodes, ceramic filters and bone and dental reposition. Among them, there is special care in studying graded porosity materials, where the quantity of pores and pore morphology changes through the material volume. In this work the processing and characterization of alumina ceramic materials with functionally graded porosity by freeze casting and slip casting techniques using aluminum hydroxide and rice starch as sacrificial template was performed. After conformation, the porosity was characterized through electron scanning microscopy, apparent porosity through Archimedes method and median pore size through mercury porosimetry. The mechanical resistance was also obtained by compression testing. The analysis allowed to confirm the relation between each of the isolated samples’ microstructure and its respective layer in each of the graded materials, also, the graded materials shown good interfacial interaction between each of the layers. The porosity in graded materials kept close to the expected value, which was determined by the medium value of the porosities of the isolated samples respective to the graded material. Mechanical testing shown that there was no influence of the graded material interfaces in its compressive strength.

Microdispositivos piezelétricos tem uma vasta aplicação em mecânica de precisão, como, por exemplo, manipulação de células, microcirurgias, equipamentos de nanotecnologia e principalmente em microeletromecanismos (MEMS). Os microdispositivos piezelétricos considerados nesta tese essencialmente consistem de uma estrutura multi-flexível atuada por duas ou mais piezocerâmicas, que geram deslocamentos e forças em direções e regiões pré-determinadas do domínio, ou seja, a estrutura multi-flexível atua como um transformador mecânico amplificando e alterando os deslocamentos gerados pelas piezocerâmicas nos movimentos de atuação. O desenvolvimento destes microdispositivos piezelétricos em sua grande maioria não utiliza ferramentas sistemáticas e genéricas. A complexidade dos movimentos de atuação torna o desenvolvimento dos microdispositivos piezelétricos complexo, principalmente devido ao surgimento de movimentos indesejados ou acoplados durante a sua atuação. Portanto, é necessário um método sistemático e eficiente como o método de otimização topológica (MOT), que incorpore na sua formulação as principais exigências de projeto dos microdispositivos, como apresentado nesse trabalho. O MOT implementado é baseado na abordagem CAMD (Distribuição Contínua da Distribuição de Material), onde as pseudo-densidades são interpoladas nos nós de cada elemento finito, resultando numa distribuição contínua de material no domínio. Um método adjunto foi implementado para o cálculo das sensibilidades. São consideradas três formulações. A primeira denominada de MAPs (Multi-Atuadores Piezelétricos) considera as regiões piezocerâmicas fixas, otimizando apenas a estrutura multi-flexível no domínio de projeto. Nesta formulação materiais não-piezelétricos (como, por exemplo, Alumínio) e vazio são distribuídos no domínio de projeto, mantendo as regiões piezocerâmicas fixas e homogêneas. Para validar os resultados obtidos com essa formulação foram fabricados protótipos de nanoposicionadores $XY$, que foram caracterizados experimentalmente utilizando técnicas de interferometria laser, considerando excitação quasi-estática. No entanto, essa primeira formulação impõe restrições no problema, limitando a optimalidade da solução obtida pela otimização topológica. Assim, surgiu a necessidade de desenvolver uma segunda formulação, que permite distribuir simultaneamente material não-piezelétrico, piezelétrico e vazio no domínio de projeto, denominada de LOMPs (Localização Ótima do Material Piezelétrico). A formulação dos LOMPs obtém simultaneamente a localização do material piezelétrico na estrutura flexível otimizada pela OT, e inclui também uma variável de projeto para determinar o ângulo ótimo entre as direções de polarização e do campo elétrico. Nesta formulação como as posições dos eletrodos não são conhecidas, ``a priori\'\', é utilizado como abordagem aplicar um campo elétrico constante para determinar a localização do material piezelétrico e conseqüentemente dos eletrodos. Finalmente, foi explorado o conceito de materiais com gradação funcional (MGFs) no projeto dos MAPs. Os MGFs apresentam uma distribuição contínua de materiais na sua microestrutura, não possuindo interface entre os materiais distribuídos, o que possibilita aumentar a vida útil do dispositivo piezelétrico. Assim, foi implementado uma terceira formulação denominada de MAPs MGFs, que permite obter a gradação ótima de materiais piezelétricos e não-piezelétricos no domínio piezocerâmico dos MAPs, conjuntamente com a topologia da estrutura multi-flexível. Essa formulação foi estendida para projetar atuadores bilaminares MGFs. Todas as formulações desenvolvidas utilizam uma função multi-objetivo, que permite controlar a rigidez e a flexibilidade minimizando o movimento acoplado, de cada movimento de atuação. Os exemplos numéricos são limitados a modelos bi-dimensionais, utilizando o estado plano de tensões e deformações mecânicas e elétricas, uma vez que a grande maioria das aplicações dos microdispositivos piezelétricos são bi-dimensionais. / Microtools offer significant promise in a wide range of applications such as cell manipulation, microsurgery, nanotechnology processes, and many other fields. The microtools considered in this doctoral thesis essentially consist of a multi-flexible structure actuated by two or more piezoceramic devices that when each piezoceramic is actuated, it generates an output displacement and force at a specified point of the domain and direction. The multi-flexible structure acts as a mechanical transformer by amplifying and changing the direction of the piezoceramic output displacements. Thus, the development of microtools requires the design of actuated flexible structures that can perform complex movements. The development of these microtools is still in the beginning and it can be strongly enhanced by using design tools. In addition, when multiple piezoceramic devices are involved, coupling effects in their movements become critical, especially the appearance of undesired movements, which makes the design task very complex. One way to avoid such undesirable effects is the use of a systematic design method, such as topology optimization, with appropriate formulation of the optimization problem. The topology optimization method implemented is based on the CAMD (Continuous Approximation of Material Distribution) approach where fictitious densities are interpolated at each finite element, providing a continuum material distribution in the domain. The corresponding sensitivity analysis is presented using the adjoint method. Three formulations are considered. The first formulation, called Piezoelectric Multi-Actuators (PMAs), keeps fixed piezoceramic positions in the design domain and only the flexible structure is designed by distributing some non-piezoelectric material (Aluminum, for example). $XY$ Piezoelectric Nanopositioner are manufactured and experimentally analyzed to validate the results of the topology optimization obtained using this formulation. Experimental analyses are conducted using laser interferometry to measure displacement, while considering a quasi-static excitation. However, this first formulation imposes a constraint to the position of piezoelectric material in the optimization problem limiting the optimality of the solution. Thus, the second formulation presented, called LOMPs, allows the simultaneous distribution of non-piezoelectric and piezoelectric material in the design domain, to achieve certain specified actuation movements. The optimization problem is posed as the simultaneous search for an optimal topology of a flexible structure as well as the optimal position of piezoceramics in the design domain and optimal rotation angle of piezoceramic material axes that maximize output displacements or output forces at a specified point of the domain and direction. When the distribution of a non-piezoelectric conductor material and a piezoceramic material is considered in the design domain, the electrode positions are not known ``a priori\'\'. To circumvent this problem, an electric field is applied as electrical excitation. Finally, the concept of functionally graded materials (FGM) is applied to PMAs design. FGMs are special materials that possess continuously graded properties without interfaces which can increase lifetime of piezoelectric devices. Thus, a third formulation is implemented to find the optimum gradation and polarization sign variation of piezoceramic FGMs, while simultaneously optimizing the multi-flexible structural configuration. This formulation is extended to design bimorph type FGM actuators. For all developed formulations, a multi-objective function is defined that controls the stiffness and flexibility, minimizing the coupling movement of each actuated movement. The present examples are limited to two-dimensional models because most part of the applications for such micro-tools are planar devices.

Nonlinear finite element models of functionally graded beams with power-law variation of material, accounting for the von-Karman geometric nonlinearity and temperature dependent material properties as well as microstructure dependent length scale have been developed using the Euler-Bernoulli as well as the first-order and third- order beam theories. To capture the size effect, a modified couple stress theory with one length scale parameter is used. Such theories play crucial role in predicting accurate deflections of micro- and nano-beam structures. A general third order beam theory for microstructure dependent beam has been developed for functionally graded beams for the first time using a modified couple stress theory with the von Karman nonlinear strain. Finite element models of the three beam theories have been developed. The thermo-mechanical coupling as well as the bending-stretching coupling play significant role in the deflection response. Numerical results are presented to show the effect of nonlinearity, power-law index, microstructural length scale, and boundary conditions on the bending response of beams under thermo-mechanical loads. In general, the effect of microstructural parameter is to stiffen the beam, while shear deformation has the effect of modeling more realistically as a flexible beam.

La plupart des problèmes d'ingénierie des deux derniers siècles ont été résolus grâce à des modèles structuraux pour poutres, plaques et coques. Les théories classiques, tels que Euler-Bernoulli, Navier et de Saint-Venant pour les poutres, et Kirchhoff-Love et Mindlin-Reissner pour plaques et coques, ont permis de réduire le problème générique 3-D, dans le problème unidimensionnel pour les poutres et deux dimensionnelle pour les coques et les plaques. Théories raffinés d'ordre supérieur ont été proposées au cours du temps, comme les modèles classiques ne consentez pas à d'obtenir une complète domaine des contraintes et des déformations. La Carrera Unified Formulation (UF) a été proposé au cours de la dernière décennie, et permet de développer un grand nombre de théories structurelles avec un nombre variable d'inconnues principales au moyen d'une notation compacte et se référant à des nuclei fondamentales. Cette formulation unifiée permet de dériver carrément des modèles structurels d'ordre supérieur, pour les poutres, plaques et coques. Dans ce cadre, cette thèse vise à étendre la formulation pour l'analyse des structures fonctionnellement gradués (FGM), en introduisant aussi le problème thermo-mécanique, dans le cas des poutres fonctionnellement gradués. Suite à la formulation unifiée, les variables génériques déplacements sont écrits en termes de fonctions de base, qui multiplie les inconnues. Dans la deuxième partie de la thèse, de nouvelles fonctions de bases pour la modélisation des coques, qui représentent une approximation trigonométrique des variables déplacements, sont pris en compte / Most of the engineering problems of the last two centuries have been solved thanks to structural models for both beams, and for plates and shells. Classical theories, such as Euler-Bernoulli, Navier and De Saint-Venant for beams, and Kirchhoff-Love and Mindlin- Reissner for plates and shells, permitted to reduce the generic 3-D problem, in onedimensional one for beams and two-dimensional for shells and plates. Refined higher order theories have been proposed in the course of time, as the classical models do not consent to obtain a complete stress/strain field. Carrera Unified Formulation (UF) has been proposed during the last decade, and allows to develop a large number of structural theories with a variable number of main unknowns by means of a compact notation and referring to few fundamental nuclei. This Unified Formulation allows to derive straightforwardly higher-order structural models, for beams, plates and shells. In this framework, this thesis aims to extend the formulation for the analysis of Functionally Graded structures, introducing also the thermo-mechanical problem, in the case of functionally graded beams. Following the Unified Formulation, the generic displacements variables are written in terms of a base functions, which multiplies the unknowns. In the second part of the thesis, new bases functions for shells modelling, accounting for trigonometric approximation of the displacements variables, are considered.

Microdispositivos piezelétricos tem uma vasta aplicação em mecânica de precisão, como, por exemplo, manipulação de células, microcirurgias, equipamentos de nanotecnologia e principalmente em microeletromecanismos (MEMS). Os microdispositivos piezelétricos considerados nesta tese essencialmente consistem de uma estrutura multi-flexível atuada por duas ou mais piezocerâmicas, que geram deslocamentos e forças em direções e regiões pré-determinadas do domínio, ou seja, a estrutura multi-flexível atua como um transformador mecânico amplificando e alterando os deslocamentos gerados pelas piezocerâmicas nos movimentos de atuação. O desenvolvimento destes microdispositivos piezelétricos em sua grande maioria não utiliza ferramentas sistemáticas e genéricas. A complexidade dos movimentos de atuação torna o desenvolvimento dos microdispositivos piezelétricos complexo, principalmente devido ao surgimento de movimentos indesejados ou acoplados durante a sua atuação. Portanto, é necessário um método sistemático e eficiente como o método de otimização topológica (MOT), que incorpore na sua formulação as principais exigências de projeto dos microdispositivos, como apresentado nesse trabalho. O MOT implementado é baseado na abordagem CAMD (Distribuição Contínua da Distribuição de Material), onde as pseudo-densidades são interpoladas nos nós de cada elemento finito, resultando numa distribuição contínua de material no domínio. Um método adjunto foi implementado para o cálculo das sensibilidades. São consideradas três formulações. A primeira denominada de MAPs (Multi-Atuadores Piezelétricos) considera as regiões piezocerâmicas fixas, otimizando apenas a estrutura multi-flexível no domínio de projeto. Nesta formulação materiais não-piezelétricos (como, por exemplo, Alumínio) e vazio são distribuídos no domínio de projeto, mantendo as regiões piezocerâmicas fixas e homogêneas. Para validar os resultados obtidos com essa formulação foram fabricados protótipos de nanoposicionadores $XY$, que foram caracterizados experimentalmente utilizando técnicas de interferometria laser, considerando excitação quasi-estática. No entanto, essa primeira formulação impõe restrições no problema, limitando a optimalidade da solução obtida pela otimização topológica. Assim, surgiu a necessidade de desenvolver uma segunda formulação, que permite distribuir simultaneamente material não-piezelétrico, piezelétrico e vazio no domínio de projeto, denominada de LOMPs (Localização Ótima do Material Piezelétrico). A formulação dos LOMPs obtém simultaneamente a localização do material piezelétrico na estrutura flexível otimizada pela OT, e inclui também uma variável de projeto para determinar o ângulo ótimo entre as direções de polarização e do campo elétrico. Nesta formulação como as posições dos eletrodos não são conhecidas, ``a priori\'\', é utilizado como abordagem aplicar um campo elétrico constante para determinar a localização do material piezelétrico e conseqüentemente dos eletrodos. Finalmente, foi explorado o conceito de materiais com gradação funcional (MGFs) no projeto dos MAPs. Os MGFs apresentam uma distribuição contínua de materiais na sua microestrutura, não possuindo interface entre os materiais distribuídos, o que possibilita aumentar a vida útil do dispositivo piezelétrico. Assim, foi implementado uma terceira formulação denominada de MAPs MGFs, que permite obter a gradação ótima de materiais piezelétricos e não-piezelétricos no domínio piezocerâmico dos MAPs, conjuntamente com a topologia da estrutura multi-flexível. Essa formulação foi estendida para projetar atuadores bilaminares MGFs. Todas as formulações desenvolvidas utilizam uma função multi-objetivo, que permite controlar a rigidez e a flexibilidade minimizando o movimento acoplado, de cada movimento de atuação. Os exemplos numéricos são limitados a modelos bi-dimensionais, utilizando o estado plano de tensões e deformações mecânicas e elétricas, uma vez que a grande maioria das aplicações dos microdispositivos piezelétricos são bi-dimensionais. / Microtools offer significant promise in a wide range of applications such as cell manipulation, microsurgery, nanotechnology processes, and many other fields. The microtools considered in this doctoral thesis essentially consist of a multi-flexible structure actuated by two or more piezoceramic devices that when each piezoceramic is actuated, it generates an output displacement and force at a specified point of the domain and direction. The multi-flexible structure acts as a mechanical transformer by amplifying and changing the direction of the piezoceramic output displacements. Thus, the development of microtools requires the design of actuated flexible structures that can perform complex movements. The development of these microtools is still in the beginning and it can be strongly enhanced by using design tools. In addition, when multiple piezoceramic devices are involved, coupling effects in their movements become critical, especially the appearance of undesired movements, which makes the design task very complex. One way to avoid such undesirable effects is the use of a systematic design method, such as topology optimization, with appropriate formulation of the optimization problem. The topology optimization method implemented is based on the CAMD (Continuous Approximation of Material Distribution) approach where fictitious densities are interpolated at each finite element, providing a continuum material distribution in the domain. The corresponding sensitivity analysis is presented using the adjoint method. Three formulations are considered. The first formulation, called Piezoelectric Multi-Actuators (PMAs), keeps fixed piezoceramic positions in the design domain and only the flexible structure is designed by distributing some non-piezoelectric material (Aluminum, for example). $XY$ Piezoelectric Nanopositioner are manufactured and experimentally analyzed to validate the results of the topology optimization obtained using this formulation. Experimental analyses are conducted using laser interferometry to measure displacement, while considering a quasi-static excitation. However, this first formulation imposes a constraint to the position of piezoelectric material in the optimization problem limiting the optimality of the solution. Thus, the second formulation presented, called LOMPs, allows the simultaneous distribution of non-piezoelectric and piezoelectric material in the design domain, to achieve certain specified actuation movements. The optimization problem is posed as the simultaneous search for an optimal topology of a flexible structure as well as the optimal position of piezoceramics in the design domain and optimal rotation angle of piezoceramic material axes that maximize output displacements or output forces at a specified point of the domain and direction. When the distribution of a non-piezoelectric conductor material and a piezoceramic material is considered in the design domain, the electrode positions are not known ``a priori\'\'. To circumvent this problem, an electric field is applied as electrical excitation. Finally, the concept of functionally graded materials (FGM) is applied to PMAs design. FGMs are special materials that possess continuously graded properties without interfaces which can increase lifetime of piezoelectric devices. Thus, a third formulation is implemented to find the optimum gradation and polarization sign variation of piezoceramic FGMs, while simultaneously optimizing the multi-flexible structural configuration. This formulation is extended to design bimorph type FGM actuators. For all developed formulations, a multi-objective function is defined that controls the stiffness and flexibility, minimizing the coupling movement of each actuated movement. The present examples are limited to two-dimensional models because most part of the applications for such micro-tools are planar devices.