Noise has become a main nuisance in urban areas to the point that according to the World Health Organization 40% of the European population is exposed to excessive noise levels, mainly due to ground transportation. There is therefore a need to find new ways to mitigate noise in urban areas. In this work, a possible device to achieve this goal is studied: a low-height noise barrier. It consists of a barrier typically less than one meter high placed close to a source, designed to decrease the noise level for nearby pedestrians and cyclists. This type of device is studied both numerically and experimentally. Tramway noise barriers are especially studied since the noise sources are in this case very close to the ground and can therefore be attenuated efficiently. The shape and the surface treatment of the barrier are optimized using a gradient-based method coupled to a 2D boundary element method (BEM). The optimization variables are the node coordinates of a control mesh and the parameters describing the surface impedance. Sensitivities are calculated efficiently using the adjoint state approach. Numerical results show that the shapes generated by the optimization algorithm tend to be quite irregular but provide a significant improvement of more than 5 dB(A) compared to simpler shapes. Utilizing an absorbing treatment on the source side of the barrier is shown to be efficient as well. This second point has been con firmed by scale model measurements. In addition, a full scale low height noise barrier prototype has been built and tested in situ close to a tramway track in Grenoble. Measurements show that the device provides more than 10 dB(A) of attenuation for a close receiver located at the typical height of human ears. These results therefore seem to con rm the applicability of such protections to efficiently decrease noise exposure in urban areas.

[Truncated abstract] The study reported in this thesis extends the scaled boundary finite-element method to firstorder and second-order wave diffraction problems. The scaled boundary finite-element method is a newly developed semi-analytical technique to solve systems of partial differential equations. It works by employing a special local coordinate system, called scaled boundary coordinate system, to define the computational field, and then weakening the partial differential equation in the circumferential direction with the standard finite elements whilst keeping the equation strong in the radial direction, finally analytically solving the resulting system of equations, termed the scaled boundary finite-element equation. This unique feature of the scaled boundary finite-element method enables it to combine many of advantages of the finite-element method and the boundaryelement method with the features of its own. ... In this thesis, both first-order and second-order solutions of wave diffraction problems are presented in the context of scaled boundary finite-element analysis. In the first-order wave diffraction analysis, the boundary-value problems governed by the Laplace equation or by the Helmholtz equation are considered. The solution methods for bounded domains and unbounded domains are described in detail. The solution process is implemented and validated by practical numerical examples. The numerical examples examined include well benchmarked problems such as wave reflection and transmission by a single horizontal structure and by two structures with a small gap, wave radiation induced by oscillating bodies in heave, sway and roll motions, wave diffraction by vertical structures with circular, elliptical, rectangular cross sections and harbour oscillation problems. The numerical results are compared with the available analytical solutions, numerical solutions with other conventional numerical methods and experimental results to demonstrate the accuracy and efficiency of the scaled boundary finite-element method. The computed results show that the scaled boundary finite-element method is able to accurately model the singularity of velocity field near sharp corners and to satisfy the radiation condition with ease. It is worth nothing that the scaled boundary finite-element method is completely free of irregular frequency problem that the Green's function methods often suffer from. For the second-order wave diffraction problem, this thesis develops solution schemes for both monochromatic wave and bichromatic wave cases, based on the analytical expression of first-order solution in the radial direction. It is found that the scaled boundary finiteelement method can produce accurate results of second-order wave loads, due to its high accuracy in calculating the first-order velocity field.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Em engenharia, a modelagem computacional desempenha um papel importante na concepção de produtos e no desenvolvimento de técnicas de atenuação de ruído. Nesse contexto, esta tese investiga a intensidade acústica gerada pela radiação sonora de superfícies vibrantes. De modo específico, a pesquisa enfoca a identificação das regiões de uma fonte sonora que contribuem efetivamente para potência sonora radiada para o campo afastado, quando a frequência de excitação ocorre abaixo da frequência crítica de coincidência. São descritas as fundamentações teóricas de duas diferentes abordagens. A primeira delas, denominada intensidade supersônica (analítica) é calculada via transformadas de Fourier para fontes sonoras com geometrias separáveis. A segunda, denominada intensidade útil (numérica) é calculada através do método dos elementos de contorno clássico para fontes com geometrias arbitrárias. Em ambas, a identificação das regiões é feita pela filtragem das ondas não propagantes (evanescentes). O trabalho está centrado em duas propostas, a saber. A primeira delas, é a apresentação implementação e análise de uma nova técnica numérica para o cálculo da grandeza intensidade útil. Essa técnica constitui uma variante do método dos elementos de contorno (MEC), tendo como base o fato de as aproximações para as variáveis acústicas pressão e velocidade normal serem tomadas como constantes em cada elemento. E também no modo peculiar de obter a velocidade constante através da média de um certo número de velocidades interiores a cada elemento. Por esse motivo, a técnica recebe o nome de método de elemento de contorno com velocidade média (AVBEMAverage Velocity Boundary Element Method). A segunda, é a obtenção da solução forma fechada do campo de velocidade normal para placas retangulares com oito diferentes combinações de condições contorno clássicas. Então, a intensidade supersônica é estimada e comparada à intensidade acústica. Nos ensaios numéricos, a comparação da intensidade útil obtida via MEC clássico e via AVBEM é mostrada para ilustrar a eficiência computacional da técnica aqui proposta, que traz como benefício adicional o fato de poder ser utilizada uma malha menos refinada para as simulações e, consequentemente, economia significativa de recursos computacionais. / In engineering computational modeling plays an important role in product design and development techniques for noise attenuation. In such context, this thesis investigates the acoustic intensity generated by sound radiation from vibrating surfaces. Specifically, the research focuses on the identification that effectively contribute to the radiated sound power into the far-field, especially when the driven frequency occurs below the critical coincidence frequency. The theoretical formulations of two different approaches are described. The first one, called supersonic intensity (analytical) is calculated via Fourier transforms for noise sources with separable geometries. The second, called useful intensity (numerical) is calculated by the method of boundary elements to classic fonts with arbitrary geometries. In both, the identification of regions is done by filtering the non propagating waves (evanescent). The work is focused on the presentation, implementation and analysis of a new numerical technique for calculating the magnitude useful intensity. This technique is a variant of the method-boundary element (BEM), based on the fact that the approximations for the acoustic pressure and normal speed variables are taken as constant in each element. Also in particular way to obtain constant speed by mean of a number of speeds interior of each element. For this reason, the technique is called the Average Velocity Boundary Element Method . The closed form solution of the normal velocity field for rectangular plates in eight cases with distinct combinations of classical boundary conditions is also obtained. Then, the supersonic intensity is estimated and compared to the acoustic intensity. Numerical experiments are performed comparing to the useful intensity via the conventional BEM with theAVBEM in order to illustrate the positive computational features of the method. The numerical results indicate that the proposed method is much more computationally effcient than its standard BEM counterpart, it enabling the use of a coarser mesh.

Les contacteurs et les relais sont utilisés dans de très nombreux dispositifs pour le contrôleet la commande à distance. La plupart sont équipés d’actionneurs électromagnétiques qu’il est nécessairede produire en quantité importante pour réaliser ces fonctions. Les exigences environnementalessur l’efficacité et la consommation énergétiques sont de plus en plus contraignantes. La conception deces actionneurs est donc un enjeu important. Cette thèse porte sur le développement d’une plateforme deconception dédiée au pré-dimensionnement d’actionneurs électromagnétiques. Cette plateforme reposesur le développement de méthodes de modélisation très rapides et suffisamment précises. La premièrepartie est consacrée à la résolution des équations de la magnétostatique en linéaire et non-linéaire par lecouplage de la méthode des intégrales de frontières avec la méthode de réseaux de réluctances. L’évaluationdes grandeurs globales telles que le flux magnétique traversant une bobine et la force est étudiée.Les résultats sont validés sur un relais bistable en rotation. La seconde partie concerne la simulationmulti-physique du composant dans son environnement système, pour prendre en compte les aspects mécaniqueet électrique. Une étude de faisabilité par optimisation est réalisée pour le composant seul, puispour l’ensemble composant-système. Enfin, une ouverture sur une autre méthode de modélisation plusgénérale est proposée avec la présentation d’un couplage original et performant entre les méthodes deséléments finis et des intégrales de frontière. / Contactors and relays are used in many devices for monitoring and remote controlling. Mostof them are equipped with electromagnetic actuators, which are produced in large quantities. The environmentalrequirements on efficiency and energy consumption have become more and more strict. Thedesign of these actuators is therefore an important issue. This thesis focuses on the development of adesign platform dedicated to the pre-sizing of electromagnetic actuators. This platform is based on theimplementation of very fast and sufficiently accurate modeling methods. The first part is devoted to theresolution of linear and nonlinear magnetostatic equations by the coupling between boundary integralmethod and reluctance network method. The evaluation of global quantities such as the magnetic fluxflowing through a coil and the force is investigated. The results are validated on a rotating bistable relay.The second part concerns the multi-physics simulation of the component in its environment by takinginto account the mechanical and electrical aspects. A feasibility study by optimization is performed bothfor the component alone and for the whole component-system. Finally, a general modeling method isproposed with the presentation of an original and efficient coupling between the finite element methodsand boundary integrals.

In this work, it is proposed a mechanical behavior analysis of the soil-structure interaction from the development of a computational code using a coupling static formulation of Boundary Element Method (BEM) and Finite Element Method (FEM) for the displacements and stress calculation in structures in contact to the half space. Thus, it is intended to model the structure using the bending plate finite element DKT (discrete Kirchhoff triangle) and applying the concepts of the Boundary Element Method (BEM) formulation to model the soil, considered as a half- infinite and/or infinite space and using Kelvin s fundamental solution. The coupling between the media is done using the sub-regions technique. From the computational code development some practical examples of engineering are implemented, such as: soil-structure interaction analysis in superficial and buried plate foundations and others engineering structures, study on the behavior of a half- infinite space from the application of a distributed and concentrated load, analysis of bodies submitted to bend and traction, among others applications. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Neste trabalho, propõe-se a análise do comportamento mecânico da interação solo -estrutura a partir do desenvolvimento de um código computacional utilizando-se uma formulação estática conjunta do Método dos Elementos de Contorno (MEC) e do Método dos Elementos Finitos (MEF) para o cálculo de deslocamentos e tensões em estruturas em contato com o meio semiinfinito. Assim sendo, pretende-se modelar a estrutura a partir de elementos finitos de placa DKT (discrete Kirchhoff triangle) e utilizar o conceito da formulação do Método dos Elementos de Contorno (MEC) para modelar o solo, considerando-o como um espaço semi- infinito e/ou infinito e utilizando a solução fundamental de Kelvin. O acoplamento entre os meios é feito aplicando-se a técnica de sub-regiões. A partir do desenvolvimento de um código computacional são processados alguns exemplos de engenharia tais como: análise da interação solo -estrutura em fundações de placa superficiais e enterradas e outras estruturas de engenharia, estudo do comportamento de um espaço semi- infinito a partir da aplicação de um carregamento distribuído e carga concentrada, análise de corpos submetidos à flexão e à tração, entre outras aplicações.

Il existe une grande quantité de méthodes numériques adaptées d’une part à la modélisation, et d'autre part à la résolution des équations de Maxwell. En particulier, la méthode des éléments nis de frontière (BEM), ou méthode des Moments (MoM), semble appropriée pour la mise en équation des phénomènes de diffraction par des objets parfaitement conducteurs, en limitant le cadre de l'étude à la frontière entre l'objet diffractant et le milieu extérieur. Cette méthode mène systématiquement à la résolution d’un système linéaire dense, que nous parvenons à compresser en l'approchant numériquement par une matrice hiérarchique creuse, appelée H-matrice. Cette approximation peut être complétée d'une ré-agglomération permettant d'améliorer la sparsité de la H-matrice et ainsi d'optimiser davantage la résolution du système traité. La hiérarchisation du système s'effectue en considérant la matrice traitée par blocs, que l'on peut ou non compresser selon une condition d'admissibilité. L'Approximation en Croix Adaptative (ACA) ou l'Approximation en Croix Hybride (HCA) sont deux méthodes de compression que l'on peut alors appliquer aux blocs admissibles. Il existe une grande quantité de méthodes numériques adaptées d’une part à la modélisation, et d'autre part à la résolution des équations de Maxwell. En particulier, la méthode des éléments finis de frontière (BEM), ou méthode des Moments (MoM), semble appropriée pour la mise en équation des phénomènes de diffraction par des objets parfaitement conducteurs, en limitant le cadre de l'étude à la frontière entre l'objet diffractant et le milieu extérieur. Cette méthode mène systématiquement à la résolution d’un système linéaire dense, que nous parvenons à compresser en l'approchant numériquement par une matrice hiérarchique creuse, appelée H-matrice. Cette approximation peut être complétée d'une ré-agglomération permettant d'améliorer la sparsité de la H-matrice et ainsi d'optimiser davantage la résolution du système traité. La hiérarchisation du système s'effectue en considérant la matrice traitée par blocs, que l'on peut ou non compresser selon une condition d'admissibilité. L'Approximation en Croix Adaptative (ACA) ou l'Approximation en Croix Hybride (HCA) sont deux méthodes de compression que l'on peut alors appliquer aux blocs admissibles. Le travail de cette thèse consiste dans un premier temps à valider le format H-matrice en 2D et en 3D en utilisant l'ACA, puis d'y appliquer la méthode HCA, encore peu exploitée. Nous pouvons alors résoudre le système linéaire issu de la BEM en utilisant différents solveurs, directs ou non, adaptés au format hiérarchique. En particulier, nous pourrons constater l'efficacité du préconditionnement LU hiérarchique sur un solveur itératif. Nous pourrons alors appliquer ce formalisme au cas des surfaces rugueuses ou encore des fibres à cristaux photoniques (PCF). Il sera également possible de paralléliser certaines opérations sur architecture partagée afin de réduire de nouveau le coût temporel de la résolution. / A lot of numerical methods are available for the modelization as well as the solution of the Maxwell's equations. In particular the boundary element method (BEM), also known as Method of Moments (MoM), seems appropriate to put in equation the scattering problems by perfectly conducting objects, by restricting the study to the frontier between the diffracting object and its surrounding. This method automatically leads to a dense linear system which we are able to compress, numerically approaching it by a hierarchical sparse matrix, called H-matrix. This approximation can be completed with a coarsening which enhance the sparsity of the H -matrix and thus optimizes again the solution of the concerned system. The hierarchization of the system is done considering the concerned matrix by its blocks, which can or cannot be compressed according to an admissibility condition. The Adaptive Cross Approximation (ACA) or the Hybrid Cross Approximation (HCA) are among the possible compression methods available to compress the admissible blocks. This PhD thesis first focuses on the validation of the H-matrix format both in 2D and 3D using the ACA. We then apply to this format the HCA method, which is still quite unmined. Thus we can solve the linear system coming from the BEM using different direct and iterative solution methods which are adapted to suit the hierarchical format. In particular, we will observe the efficiency of the hierarchical LU preconditionning used to enhance an iterative solver. Thus we will be able to apply this formalism on cases such as rough surfaces or photonic crystal fibers (PCF). It will also be possible to make some operations parallel in order to further reduce the time cost of the solution.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Em engenharia, a modelagem computacional desempenha um papel importante na concepção de produtos e no desenvolvimento de técnicas de atenuação de ruído. Nesse contexto, esta tese investiga a intensidade acústica gerada pela radiação sonora de superfícies vibrantes. De modo específico, a pesquisa enfoca a identificação das regiões de uma fonte sonora que contribuem efetivamente para potência sonora radiada para o campo afastado, quando a frequência de excitação ocorre abaixo da frequência crítica de coincidência. São descritas as fundamentações teóricas de duas diferentes abordagens. A primeira delas, denominada intensidade supersônica (analítica) é calculada via transformadas de Fourier para fontes sonoras com geometrias separáveis. A segunda, denominada intensidade útil (numérica) é calculada através do método dos elementos de contorno clássico para fontes com geometrias arbitrárias. Em ambas, a identificação das regiões é feita pela filtragem das ondas não propagantes (evanescentes). O trabalho está centrado em duas propostas, a saber. A primeira delas, é a apresentação implementação e análise de uma nova técnica numérica para o cálculo da grandeza intensidade útil. Essa técnica constitui uma variante do método dos elementos de contorno (MEC), tendo como base o fato de as aproximações para as variáveis acústicas pressão e velocidade normal serem tomadas como constantes em cada elemento. E também no modo peculiar de obter a velocidade constante através da média de um certo número de velocidades interiores a cada elemento. Por esse motivo, a técnica recebe o nome de método de elemento de contorno com velocidade média (AVBEMAverage Velocity Boundary Element Method). A segunda, é a obtenção da solução forma fechada do campo de velocidade normal para placas retangulares com oito diferentes combinações de condições contorno clássicas. Então, a intensidade supersônica é estimada e comparada à intensidade acústica. Nos ensaios numéricos, a comparação da intensidade útil obtida via MEC clássico e via AVBEM é mostrada para ilustrar a eficiência computacional da técnica aqui proposta, que traz como benefício adicional o fato de poder ser utilizada uma malha menos refinada para as simulações e, consequentemente, economia significativa de recursos computacionais. / In engineering computational modeling plays an important role in product design and development techniques for noise attenuation. In such context, this thesis investigates the acoustic intensity generated by sound radiation from vibrating surfaces. Specifically, the research focuses on the identification that effectively contribute to the radiated sound power into the far-field, especially when the driven frequency occurs below the critical coincidence frequency. The theoretical formulations of two different approaches are described. The first one, called supersonic intensity (analytical) is calculated via Fourier transforms for noise sources with separable geometries. The second, called useful intensity (numerical) is calculated by the method of boundary elements to classic fonts with arbitrary geometries. In both, the identification of regions is done by filtering the non propagating waves (evanescent). The work is focused on the presentation, implementation and analysis of a new numerical technique for calculating the magnitude useful intensity. This technique is a variant of the method-boundary element (BEM), based on the fact that the approximations for the acoustic pressure and normal speed variables are taken as constant in each element. Also in particular way to obtain constant speed by mean of a number of speeds interior of each element. For this reason, the technique is called the Average Velocity Boundary Element Method . The closed form solution of the normal velocity field for rectangular plates in eight cases with distinct combinations of classical boundary conditions is also obtained. Then, the supersonic intensity is estimated and compared to the acoustic intensity. Numerical experiments are performed comparing to the useful intensity via the conventional BEM with theAVBEM in order to illustrate the positive computational features of the method. The numerical results indicate that the proposed method is much more computationally effcient than its standard BEM counterpart, it enabling the use of a coarser mesh.

Les méthodes d'ingénierie acoustique (e.g. ISO 9613-2 ou CNOSSOS-EU) approchent efficacement les niveaux de bruit générés par les routes, les voies ferrées et les sources industrielles en milieu urbain. Cependant, ces approches d'ingénierie sont limitées à des géométries de forme simple, le plus souvent de section rectangulaire. Ce mémoire développe donc, et valide, une approche hybride permettant l'extension des méthodes d'ingénierie à des formes plus complexes, en introduisant un terme d’atténuation supplémentaire qui représente l'effet d'un objet réel comparé à un objet simple.Le calcul de cette atténuation supplémentaire nécessite des calculs de référence, permettant de quantifier la différence entre objets simple et complexe. Dans la mesure où il est trop onéreux, numériquement, '’effectuer ce calcul pour tous les chemins de propagation, l'atténuation supplémentaire est obtenue par interpolation de données stockées dans un tableau et évaluées pour un large jeu de positions de sources, de récepteurs et de fréquences. Dans notre approche, le calcul de référence utilise la méthode BEM en 2.5D, et permet ainsi de produire les niveaux de référence pour les géométries simple et complexe, tout en tabulant leur écart. Sur le principe, d'autres approches de référence pourraient être utilisées.Ce travail valide cette approche hybride pour un écran en forme de T avec un sol rigide, un sol absorbant et un cas avec bâtiments. Ces trois cas démontrent que l'approche hybride est plus précise que l'approche d’ingénierie standard dans des cas complexes. / Noise engineering methods (e.g. ISO 9613-2 or CNOSSOS-EU) efficiently approximate sound levels from roads, railways, and industrial sources in cities. However, engineering methods are limited to only simple box-shaped geometries. This dissertation develops and validates a hybrid method to extend the engineering methods to more complicated geometries by introducing an extra attenuation term that represents the influence of a real object compared to a simplified object.Calculating the extra attenuation term requires reference calculations to quantify the difference between the complex and simplified objects. Since performing a reference computation for each path is too computationally expensive, the extra attenuation term is linearly interpolated from a data table containing the corrections for many source and receiver positions and frequencies. The 2.5D boundary element method produces the levels for the real complex geometry and a simplified geometry, and subtracting these levels yields the corrections in the table.This dissertation validates this hybrid method for a T-barrier with hard ground, soft ground, and buildings. All three cases demonstrate that the hybrid method is more accurate than standard engineering methods for complex cases.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-07-18T12:41:11Z No. of bitstreams: 1 arquivototal.pdf: 2094299 bytes, checksum: 783d6f9949086fb75e4a51fc3adbd48a (MD5) / Made available in DSpace on 2017-07-18T12:41:11Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2094299 bytes, checksum: 783d6f9949086fb75e4a51fc3adbd48a (MD5) Previous issue date: 2016-04-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, new solutions based on the Boundary Element Method (BEM) are established for the linear analysis of bending and stability problems of Reddy-Bickford beams. All mathematical steps to write the BEM representation are properly presented: transformation of governing differential equations into equivalent integral equations, deduction of fundamental solutions, formation and solution of algebraic representation.In addition, elastic foundations (winkler and pasternak’s types) attached to Reddy-Bickford beams are solved by BEM as well. It is also addressed a convenient strategy for discontinuities in the area such as abrupt change in geometry of the cross section (stepped beam), intermediate axial load, intermediate supports (continuous beam). Numerical examples incorporating various types of discontinuities and boundary conditions in the field are presented to validate the solutions proposed BEM. / Neste trabalho, novas soluções, baseadas no Método dos Elementos de Contorno (MEC), são estabelecidas para a análise linear de problemas de flexão e estabilidade de barras de Bickford-Reddy. Todos os passos matemáticos para estabelecer a representação do MEC são apresentados: transformações das equações diferenciais governantes em equações integrais equivalentes, dedução das soluções fundamentais, obtenção e solução do sistema alébrico. Além disso, fundações elásticas (Winkler e Pasternak) em barras de Bickford-Reddy também são analisados pelo MEC. É também abordada uma conveniente estratégia para de discontinuidades no domínio tais como: mudança abrupta de geometria da seção transversal (viga escalonada), carga axial intermediária, apoios rígidos no domínio (viga contínua). Exemplos numéricos incorporando vários tipos de condições de contorno e discontinuidades no domínio são apresentadas para validar as soluções do MEC propostas.

Orientador: Luiz Fernando Bovolato / Banca: Sérgio Kurokawa / Banca: Afonso José do Prado / Resumo: Este trabalho analisa a aplicação em linhas de transmissão do método dos elementos de contorno para cálculo de potencial e campo elétrico, com um enfoque em eletrostática. O método dos elementos de contorno baseia-se numa formulação integral que elimina a discretização do domínio, restando apenas o contorno, permitindo o cálculo do potencial e do campo elétrico no contorno e na região estudada. O trabalho configura-se como uma revisão sobre eletrostática, ressaltando as equações de Laplace e Poisson, que serão utilizadas para encontrar as equações integrais do contorno. Há também vários tópicos relacionados ao campo elétrico de linhas de transmissão, bem como, ás normas brasileiras e recomendações internacionais que devem ser utilizadas no projeto de linhas de transmissão. O método dos elementos de contorno utiliza tais equações integrais para encontrar o potencial e o campo no contorno, e após conhecidos o potencial e o campo no contorno, pode-se aplicar o método em todo o domínio, obtendo o potencial e o campo. Para isso, apenas o contorno do domínio de interesse deve ser discretizado, o que trás uma enorme vantagem sobre os métodos que utilizam formulação diferencial. Neste trabalho, serão descritas as principais características do código computacional desenvolvido e suas sub-rotinas mais importantes. Para validar o programa, os resultados serão comparados com aqueles calculados por um procedimento analítico, sendo mostrada a eficiência da discretização do solo. São apresentados os resultados obtidos da análise do campo elétrico gerado por algumas silhuetas de linhas de transmissão. Os valores do campo elétrico gerado por estruturas compactas são comparados com estruturas convencionais e estruturas reduzidas (semi-compactas), também serão comparados os valores do gradiente de potencial na superfície dos condutores e suas capacitâncias equivalentes. / Abstract: This paper analyses the application in transmission lines of the Boundary Element Method (BEM) of the calculation of potential and electric field, with a focus on electrostatic. The Boundary Element Method is based on an integral formulation that eliminates the discretisation of the domain, remaining only the contour, allowing the calculation of the potential and the electric field in the contour and in the region studied. The work is configured as revision on electrostatic, underscoring the equations of Laplace and Poisson, which will be used to find the integral equations of the contour. There are also several topics related to the electric field of transmission lines, as well as to the standards Brazilian and international recommendations to be used in the design of transmission lines. The Boundary Element Method uses such integral equations for finding the potential and electric field in the contour, and after having known the potential and electric field in the contour, the BEM can be applied in the whole domain, and getting the potential and electric field. Therefore, only the contours of the domain of interest should just be discretized, which backward an enormous advantage on the methods that use formulation differential. This paper will describe the main characteristics of computer code developed and their sub-routines more important. To validate the program, the results will be compared with those calculated by an analytic procedure, being shown the efficiency of discretisation of the soil. The results obtained from analysis of the electric field generated by some silhouettes of transmission lines are presented. The values of the electric field generated by compact structures are compared with conventional structures and reduced structures, also will be compared the values of the gradient of potential on the surface of the conductors and their equivalents capacitances. / Mestre