Optimization is a decision making process to solve problems in a number of fields including engineering mechanics. Bio-inspired optimization algorithms, including genetic algorithm (GA), have been studied for many years. There is a large literature on applying the GA to mechanics problems. However, disadvantages of the GA include the high computational cost and the inability to get the global optimal solution that can be found by using a honeybee-inspired optimization algorithm, called the New Nest-Site Selection (NeSS). We use the NeSS to find optimal parameters for three mechanics problems by following the three processes: screening, identifying relationships, and optimization. The screening process identifies significant parameters from a set of input parameters of interest. Then, relationships between the significant input parameters and responses are established. Finally, the optimization process searches for an optimal solution to achieve objectives of a problem. For the first two problems, we use the NeSS algorithm in conjunction with a third order shear and normal deformable plate theory (TSNDT), the finite element method (FEM), a one-step stress recovery scheme (SRS) and the Tsai-Wu failure criterion to find the stacking sequence of composite laminates and the topology and materials for doubly curved sandwich shells to maximize the first failure load. It is followed by the progressive failure analysis to determine the ultimate failure load. For the sandwich shell, we use the maximum transverse shear stress criterion for delineating failure of the core, and also study simultaneously maximizing the first failure load and minimizing the mass subject to certain constraints. For composite laminates, it is found that the first failure load for an optimally designed stacking sequence exceeds that for the typical [0°/90°]₅ laminate by about 36%. Moreover, the design for the optimal first failure load need not have the maximum ultimate load. For clamped laminates and sandwich shells, the ultimate load is about 50% higher than the first failure load. However, for simply supported edges the ultimate load is generally only about 10% higher than the first failure load. For the atmospheric spray process, we employ the NeSS algorithm to find optimal values of four process input parameters, namely the argon flow rate, the hydrogen flow rate, the powder feed rate and the current, that result in the desired mean particle temperature and the mean particle velocity when they reach the substrate. These optimal values give the desired mean particle temperature and the mean particle velocity within 5% of their target values. / Ph. D. / An optimization process iteratively searches for the best solution from all feasible solutions in the search space that satisfy prespecified criteria. Optimization problems consist of sets of parameters, constraints, and objective functions. Here we use a honeybee-inspired optimization algorithm, called the New Nest-Site Selection (NeSS), to find optimal parameters for three mechanics problems. In the first problem, we optimize the design of an assembly of layers of unidirectional fiber-reinforced materials called composite laminates. Because of their high specific strength and directional-dependent stiffness as compared to those of metals, the composite laminates are being increasingly used in aerospace and automotive industries. After having analyzed deformations of a composite laminate, a failure criterion is used to determine if any point in the structure has failed. The minimum load for which the failure criterion is satisfied at a point is called the first ply failure load. Here we determine the fiber orientation angle in each layer of a rectangular laminate deformed statically by transverse loads applied on the top surface that maximizes the first ply failure load. Subsequently, the load is incrementally increased for the optimally designed laminate and the strength of the failed elements is degraded till the structure cannot support any additional load. The maximum load a structure can support is called the ultimate load. It is found that for a laminate with all edges clamped, the ultimate load can be 40% more than the first ply failure load. We extend the above work to design an optimal geometry and an optimal combination of materials of the facesheets and the core that simultaneously maximizes the first failure load, minimizes the weight of a doubly curved sandwich shell, and satisfies pre-specified constraints. The doubly curved sandwich structure of interest here is comprised of two thin parallel unidirectional fiber-reinforced facesheets bonded to and enclosing a relatively thick mid-layer made of a material softer and lighter than that of the facesheets. The sandwich structures are widely used in aircraft, marine, automobile, and civilian infrastructures. It is found that optimal designs for doubly curved sandwich shells strongly depend upon how the shell edges are supported, and shells designed for the maximum first failure load need not have the maximum ultimate load. An atmospheric plasma spray process (APSP) has been successfully used to coat components for gas turbines, airframe, engines and drive trains, and silicon chips. In the APSP, coating powder is injected into the plasma, which is a mixture of ionized gases such as argon, hydrogen, and helium, through a powder port generally oriented perpendicular to the plasma jet axis. Through interactions with the plasma jet, the particles are accelerated, heated and partially melted before they strike the substrate and are deposited on it to form a coating. It is believed that the coating properties and its quality depend on the particles’ temperature and velocity when they hit the substrate. Here we determine optimum values of four input parameters, namely, the argon flow rate, the hydrogen flow rate, the current, and the powder feed rate to achieve the desirable mean particles’ temperature and the mean particles’ velocity. It is found that the four processes input parameters can be optimized to attain particles’ characteristics within 5% of their prespecified desired values.

Mobiltelefoner har utvecklats i snabb takt över åren, från ett nytt koncept till en vardaglig självklarhet för många i bägge personlig och professional fattning. Efter smartphonens myntning i 1994 med IBM Simon Personal Communicator har mobiltelefonen evolveras i många olika riktningar. 2013 kom den först smartphonen med välvd skärm, Samsung Galaxy Round, vilket påbörjande en ny trend av böjda smartphones. Denna studie undersöker användaracceptans av den senaste typen av välvd skärm, sidoböjning, utifrån ett perspektiv av mänskliga nöjen. Studien föreslår att innovationen av sidoböjning inte avnjuts pragmatiskt utan idealistiskt, med fokus på adoptörens ideologiska och sociala begär. Innovationens framtid och utspridning utöver samhället slutdras som osäkert, med förslag på framtida forskning att vidareutveckla ämnets kunskapsbas. / Mobile phones have developed rapidly over the years, from a novel concept to an typical day certainty for many in both personal and professional use. Following the advent of the smartphone in 1994 with the IBM Simon Personal Comunicator, the mobile phone has evolved in a number of directions. In 2013, the Samsung Galaxy Round was launched, the first smartphone featuring a curved display, sparking a new trend of curved-display smartphones. This study investigates the user acceptance of the latest type of curved displays, curved sides, from a perspective of human pleasures. The studie suggests the innovation of curved sides is not enjoyed pragmatically, but idealistically, with a focus on the adopter’s ideological and social desires. The future of the innovation and its diffusion throughout society is concluded to be uncertain, with suggestions of future research to further develop the subject’s base of knowledge.

Today there is a great need for communication between people and even between things, using systems onboard e.g. aircraft, cars, ships and satellites. As a consequence, these communications needs require antennas mounted on or integrated in the surfaces of different vehicles or platforms, i.e. conformal antennas. In order to ensure proper operation of the communication systems it is important to be able to determine the characteristics of these antennas. This thesis is about the analysis and design of conformal antennas using high frequency asymptotic methods. Commonly used eigenfunction solutions or numerical methods such as FDTD, FEM or MoM are difficult to use for arbitrarily shaped electrically large surfaces. However, the high frequency approximation approach together with an accurate ray tracing procedure offers a convenient solution for these surfaces. The geodesics (ray paths) on the surfaces are key parameters in the analysis and they are discussed in detail. In the first part of the thesis singly and doubly curved perfectly electrical conducting (PEC) surfaces are studied, with respect to the mutual coupling among aperture type elements. A synthesis problem is also considered where the effect of the mutual coupling is taken into account. As expected, the mutual coupling must be included in the synthesis procedure to be able to realize the prescribed pattern, especially in the shaped main lobe. Furthermore, the polarization of the antenna elements is very important when considering antennas on generally shaped surfaces. For such antennas the polarization must most likely be controlled in some way for a proper function. For verification of the results two experimental antennas were built at Ericsson Microwave Systems AB, Mölndal, Sweden. The first antenna is a circular cylinder with an array of rectangular waveguide fed apertures and the second antenna is a doubly curved surface (paraboloid) with circular waveguide fed apertures. It is found that it is possible to obtain very accurate results with the asymptotic method when it is combined with the Method of Moments, i.e. a hybrid method is used. The agreement compared to measurements is good at levels as low as –80 dB in many cases. The second part of the thesis is about the development of a high frequency approximation for surface field calculations on a dielectric covered PEC circular cylinder. When using conformal antennas in practice they have to be covered with a radome for protection and with the method developed here this cover can be included in the analysis. The method is a combination of two different solutions, one valid in the non-paraxial region of the cylinder and the other is valid in the paraxial region. The method is verified against measurements and reference results obtained from a spectral domain moment method code. / QC 20100616

Syfte Studien syftar till att klarlägga hur stavisättningen och stakfasen påverkas vid användandet av en böjd jämfört med en rak stavdesign vid stakning i hög hastighet på rullskidor på rullband. Följande frågeställningar har undersökts:  Leder en böjd stavdesign till att stavisättningen sker längre fram?  Leder en böjd stavdesign till att stakfasens absoluta längd förändras?  Leder en böjd stavdesign till att stakfasens relativa längd förändras? Metod Tio manliga längdskidåkare (ålder 25,3 ± 5,1 år, kroppslängd 182,1 ± 8,6 cm, kroppsvikt 78 ± 6,5 kg) deltog i studien. Forskningspersonerna genomförde ett Vmax-protokoll stakning på rullskidor på rullband, med rak såväl som med böjd stavdesign. Rörelsedata samlades in och analyserades för att fastställa eventuella skillnader i stavisättning, stakfas, återförandefas och total cykeltid. Resultat En signifikant skillnad kan påvisas gällande stavisättning, där en böjd stavdesign leder till en stavisättning längre framför bindningen jämfört med en rak stavdesign. Ingen signifikant skillnad kan påvisas rörande stakfasen, återförandefasen eller total cykeltid. Slutsatser Studiens visar att stavspetsens rörelsebana flyttas vid nyttjande av en böjd stavdesign i riktning framåt. Dock påverkas inte stakfasen, återförandefasen eller total cykeltid, vilket innebär att stakfasens längd, i absoluta såväl som relativa termer, är densamma vid stakning med rak och böjd stavdesign. / Purpose statement The purpose of the study is to examine how the pole plant and pole phase are affected by using a pole of curved, compared to straight, design for double poling at high speed on roller skis at a treadmill. The following research questions have been examined:  Does a curved pole design result in a pole plant further forward?  Does a curved pole design result in a change in the absolute length of the pole phase?  Does a curved pole design result in a change in the relative length of the pole phase? Methodology Ten male cross country skiers (age 25,3 ± 5,1 years, height 182,1 ± 8,6 cm, BW 78 ± 6,5 kg) took part in the study. The test subjects performed a Vmax-test double poling on roller skis on a treadmill, with straight and curved poles. Movement data was collected and analysed to examine potential differences regarding pole plant, pole phase, recovery phase and total cycle time. Results A significant difference was found regarding the pole plant, as a curved pole design resulted in a pole plant further forward compared to a straight design. No significant difference was found related to the pole phase, recovery phase or total cycle time. Conclusion The study shows that the path of the pole tip is moved forward when using poles of a curved design. There is no change in the pole phase, the recovery phase or the total cycle time, thus the length of the pole phase, absolute as well as relative, is unaffected by pole design when double poling.

Les tôles courbes sont de plus en plus fréquemment utilisées dans le domaine des ouvrages d'art à la fois pour raisons esthétiques et pour leurs performances mécaniques. Leur dimensionnement est cependant difficile du fait de l'absence de méthode dédiée dans les normes européennes, en particulier en ce qui concerne le voilement de l'Eurocode 3. Le recours à la modélisation par éléments finis est donc nécessaire mais bien souvent coûteux. La présente étude vise ainsi à développer une méthode simplifiée pour le dimensionnement des panneaux cylindriques soumis à une compression uniaxiale. La méthodologie utilisée est basée sur la démarche générale de l'Eurocode 3 pour tous les types de vérifications de stabilité. L'expression analytique obtenue est en bon accord avec les standards des recommandations de l'Eurocode. Cependant, cette démarche peut être difficile à reproduire dans la pratique quotidienne des ingénieurs face à un nouveau problème tel qu'une combinaison de chargements par exemple. Une méthode alternative basée sur la méthode des plans d'expérience numérique a donc été développée. Basée sur des outils statistiques, l'application de cette méthode permet facilement d'obtenir un résultat avec une bonne précision tout en nécessitant que peu simulations / The use of curved panels for the construction of steel bridges has become more and more popular due to structural performance, traffic and aesthetic requirements. Their design is however made difficult due to a lack of specifications, especially in European Standards. The use of finite element modeling is often necessary. This however requires very sophisticated and expensive models in terme of calculation times. The present study aims thus at developing a method for predicting the ultimate strength of cylindrical curved panels subjected to uniform axial compression. This method is based on the formal procedure recommended by Eurocode 3 for all types of stability verifications. The agreement with the numerical results is thus very good. However, this approach seems complex to reproduce in practice user when confronting a new problem such a combinaison of loading. An alternative method based on design of computer experiment is thus developed. Based on statistic notions, this turnkey methodology provides good results with the fewest simulations possible

Submitted by Maria Cristina (library@lncc.br) on 2015-04-06T17:02:05Z No. of bitstreams: 1 Tese-Gilmar Garbugio.pdf: 673919 bytes, checksum: a4c6ba4e7e0a9da9bbd2bc2e537fdf37 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2015-04-06T17:02:27Z (GMT) No. of bitstreams: 1 Tese-Gilmar Garbugio.pdf: 673919 bytes, checksum: a4c6ba4e7e0a9da9bbd2bc2e537fdf37 (MD5) / Made available in DSpace on 2015-04-06T17:02:41Z (GMT). No. of bitstreams: 1 Tese-Gilmar Garbugio.pdf: 673919 bytes, checksum: a4c6ba4e7e0a9da9bbd2bc2e537fdf37 (MD5) Previous issue date: 2014-12-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / In this work we study the use the Fourier law for the heat equation, which produces an evolution equation of parabolic type, which in turn produces the so-called paradox of infinite propagation velocity. This thesis proposes alternative models to avoid this problem. To achieve this, developing physical criteria about modeling of elastic systems is necessary, so that the paradox of infinite propagation velocity does not occur. We discussed various theories of heat propagation, such as the Maxwell-Cattaneo law. The thermodynamic theory, called thermoelasticity III, was adapted for modeling curved beams. Once the usage of the studied thermoelastic models is justified, the method used to validate such models is the semigroup theory. We will study the qualitative properties of the corresponding thermoelastic models, in particular the uniform stability of the solution. Finally, we prove results for exponential and polynomial stability of solutions for the Bresse beam models. / Neste trabalho estudamos a equação do calor com a lei de Fourier, resultando em uma equação de evolução do tipo parabólica, e isso nos leva ao chamado paradoxo da velocidade de propagação infinita. A tese propõe modelos alternativos para evitar este problema. Para isto é necessário desenvolver critérios físicos, sobre a modelagem de sistemas elásticos, de tal forma que o paradoxo da velocidade infinita de propagação não aconteça. Discutimos diversas teorias da propagação do calor, como a lei de Maxwell-Cattaneo. A teoria termodinâmica, denominada termoelasticidade III, foi adaptada para a modelagem de vigas curvas. Uma vez justificados os modelos termoelásticos estudados, o método usado para validar tais modelos é a teoria de semigrupos. Estudaremos as propriedades qualitativas dos correspondentes modelos termoelásticos, como a estabilidade uniforme das soluções. Finalmente, provamos resultados de estabilidade exponencial e polinomial de soluções para os modelos de viga de Bresse.

O problema de como descrever a dinâmica efetiva de partículas ou campos confinados a um espaço-tempo de baixa dimensão é de interesse em muitas áreas da física. Vários métodos foram propostos na literatura para atacar este problema. Recentemente, foram relatadas algumas evidências experimentais que são consistentes com a chamada abordagem de potencial confinante. À luz destes resultados, o objetivo desta tese é o de construir uma teoria quântica para partículas confinadas em uma subvariedade, imersa em um espaço-tempo plano, empregando o cenário conceitual da abordagem de potencial confinante. A tese está dividida em duas partes. A primeira parte é dedicada exclusivamente ao estudo da mecânica quântica em uma subvariedade. Para esta finalidade, deduzimos as equações efetivas de Schrödinger, Dirac e Klein-Gordon, em uma subvariedade curva, na presença de um campo electromagnético externo. Examinamos as características singulares estas equações e apresentamos algumas aplicações em física da matéria condensada. Na segunda parte, partimos da mecânica quântica na subvariedade e então formulamos a teoria quântica de campos (TQC) na subvariedade. Mostramos que a TQC \"livre\" em uma subvariedade pode ser representada esquematicamente como uma teoria de campos livres, em um fundo curvo, mais um potencial escalar e um campo externo SO(n - m) de Yang-Mills. Calculamos para essa teoria a ação efetiva em ordem de um laço para bósons e férmions a temperatura e potencial químico não nulos, em todas as ordens, usando a expansão de Seely-DeWitt. Para teorias com interações, demonstramos que a teoria conhecida como eletrodinâmica quântica reduzida (RQEDd?,de) pode ser recuperada a partir da abordagem de potencial confinante. Para uma teoria bidimensional, propomos uma grande classe de extensões do modelo de Schwinger, em que a interação entre férmions vai além do potencial linear. Demonstramos que, notavelmente, essas extensões são exatamente solúveis para férmions sem massa e que não há geração de massa dinâmica para os férmions. Além disso, construímos uma nova família de teorias que podem ser exatamente bosonizadas. Também mostramos que RQED4,2 tem as características necessárias para ser identificada como uma teoria de campo efetivo para fios de grafeno. Finalmente, estudamos o efeito da interação de campos magnéticos e pseudomagnéticos no grafeno. Para este sistema, calculamos o condensado fermiônico, a densidade de carga induzida de vácuo, a ação efetiva em ordem de um laço e a magnetização. Demonstramos que a presença de um campo pseudo-magnético diferente de zero torna possível observar experimentalmente uma densidade de carga induzida de vácuo. Além disso, vamos mostrar que é possível ter controle sobre a magnetização, bem como a massa dinâmica para cada vale no grafeno, quer seja por deformações ou variando o campo magnético aplicado. / The problem of how to describe the effective dynamics of particles or fields confined to a lower dimensional curved space-time is of interest in many areas of physics. In the literature, several methods have been proposed to attack this problem. Recently it has been reported some experimental evidences that are consistent with the so-called confining potential approach. In light of these results, the purpose of this thesis is to construct a quantum theory for particles confined in a submanifold of a flat space-time employing the theoretical framework of the confining potential approach. The thesis is divided into two parts. The first one is dedicated exclusively to the study of quantum mechanics on a submanifold. For this purpose, we derive the effective Schrödinger, Dirac and Klein-Gordon equations on a curved submanifold, in the presence of an external electromagnetic field. We examine the singular features of these equations and present some applications to condensed matter. In the second part of this thesis, we start from the quantum mechanics on the submanifold and then we formulate the quantum field theory (QFT) on the submanifold. We will show that the \"free\" QFT on a submanifold can be schematically represented as a quantum theory of free fields on a curved background plus a scalar potential and an external SO(n-m) Yang-Mills field. For this theory, we compute the one-loop effective action for scalars and fermions at finite temperature and chemical potential to all orders using the Seely-DeWitt expansion. For interacting theories, we will prove that the theory known as reduced quantum electrodynamics (RQEDd?,de) can be recovered from the confining potential approach. For a two-dimensional theory, we propose a large class of extensions of the Schwinger model, in which the interaction between fermions goes beyond the linear potential. We demonstrate that, remarkably, these extensions are exactly solvable for massless fermions and that there is no dynamical mass generation for the fermions. Furthermore, we construct a new family of exactly bosonized theories. We also show that RQED4,2 has the necessary features to be identified as an effective field theory for graphene wires. Finally, we study the effect of an interplay of real and pseudomagnetic fields in graphene. We compute the fermion condensate, the induced vacuum charge density, the one-loop effective action and the magnetization, for this system. We will show that the presence of a non-zero pseudomagnetic field makes it possible, experimentally, to observe an induced vacuum charge density. Moreover, we will show that it is possible to have control over the magnetization as well as the dynamical mass for each valley in graphene, by straining or varying the applied magnetic field.

The aim of this work is present the phenomenon denoted entanglement harvesting. We begin by introducing entanglement historically. Following, we go beyond the one particle theory in flat spacetime and introduce Quantum Field Theory in Curved Spacetime, showing two famous consequences: the Unruh effect and the Hawking radiation. Finally, we analyze entanglement harvesting for two Unruh-deWitt detectors. In the fisrt example, we see that there is a \"sudden death\" point of entanglement harvesting when the detectors are near the BTZ black hole event horizon, due to redshift effect and Hawking radiation. Then, we compare the phenomenon for different scenarios, and find out that it is sensitive to the structure of spacetime. Finally, we see how detectors\' parameters affect it and find out that the smoothness of the switching of the detectors\' coupling to the field is extremely relevant. We also see how the parameters can be used to optimize entanglement harvested. / O objetivo desse trabalho é apresentar o fenômeno denotado colheita de emaranhamento. Primeiramente fazemos uma introdução histórica de emaranhamento de estados quânticos. Em seguida, introduzimos a Teoria Quântica de Campos no Espaço-tempo Curvo, como um passo além da teoria quântica de uma partícula no espaço-tempo plano, e demonstramos dois resultados famosos da teoria: o efeito Unruh e a radiação Hawking. Por fim, fazemos uma analise do fenômeno de colheita de emaranhamento para dois detectores Unruh-deWitt. Nosso primeiro exemplo mostra que há um ponto de \"morte súbita\" do fenômeno quando os detectores se aproximam do horizonte de eventos de um buraco negro de BTZ, que é uma consequência do efeito de redshift e da radiação Hawking. Em seguida, comparamos o fenômeno em cenários diferentes, e observamos que a colheita de emaranhamento é sensível à estrutura do espaço-tempo. Por último, analisamos como os parâmetros dos detectores afetam a colheita de emaranhamento, e vemos que a suavidade em que o acoplamento dos detectores com o campo é \"ligado\" e \"desligado\" é extremamente relevante. Também analisamos como podemos usar os parâmetros dos detectores para otimizar a quantidade de emaranhamento colhida.

Submitted by RODRIGO DE CASTRO LIMA null (castro.lima.rodrigo@gmail.com) on 2017-03-21T14:21:46Z No. of bitstreams: 1 Dissertação Rodrigo.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-22T14:43:07Z (GMT) No. of bitstreams: 1 lima_rc_me_guara.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) / Made available in DSpace on 2017-03-22T14:43:07Z (GMT). No. of bitstreams: 1 lima_rc_me_guara.pdf: 1498013 bytes, checksum: 43cddf583f4f6c01f01e4ada8c8e38bd (MD5) Previous issue date: 2017-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O campo espinorial ELKO associado às partículas de matéria de spin 1/2 e dimensão de massa 1, construídos em um conjunto completo de autoespinores de helicidade dual do operador conjugação de carga, é um canditado a descrever a matéria escura. Devido a sua natureza, possui diversas aplicações cosmológicas. Neste trabalho, o estudo de tais espinores é realizado sob universo de Friedmann-Robertson-Walker plano, homogênio e isotrópico. À luz da Teoria Quântica de Campos em Espaços-Tempo Curvos, com as transformações de Bogoliubov, obteve-se a densidade de partículas criadas em uma métrica modelo de regime assintóticamente plano no passado e no futuro, permitindo-se comparar com soluções exatas presentes na literatura para produção de partículas escalares e férmions de Dirac. Discute- se, também, a compatibilidade do fenômeno de criação de partículas provenientes da Teoria Quântica de Campos em Espaço Curvos com os estudos de Prigogine para descrição deste fenômeno à escala cosmológica, considerando-se um universo termodinamicamente aberto. / The ELKO spinor field associated with spin 1/2 particles and mass dimension 1, con- structed on a complete set of dual helicity eigenspinors of the charge conjugation operator, is a candidate for describing dark matter. Due to its nature, it has several cosmological applications. In this work, the study of such spinors is performed under a homogenous and isotropic Friedmann-Robertson-Walker universe. In the light of the Quantum Field Theory in Curved Space-Time, with the Bogoliubov transformations, we obtained the density of particles created in a model of asymptotically plane metric in the past and in the future, allowing to compare with exact solutions present in the literature for the production of scalar particles and Dirac fermions. It is also discussed the compatibility of the phenomenon of particle creation from the Quantum Field Theory in Curved Space with the studies of Prigogine to describe this phenomenon at the cosmological scale, considering a thermodynamically open universe.

Utilizando a representação funcional de schrodinger e um ansatz gaussiano simples, obtemos um conjunto de equações finitas para a matéria e gravitação em espaços homogêneos e inomogêneos. / Using the formalism of functional Schrödinger representation and a simple Ansatz, we obtain a set of finite equations to the matter and gravitation in homogeneous and inhomogeneous spaces.