Deconvolução de Euler: passado, presente e futuro – um tutorial

Neste tutorial apresentamos uma revisão da deconvolução de Euler que consiste de três partes. Na primeira parte, recordamos o papel da clássica formulação da deconvolução de Euler 2D e 3D como um método para localizar automaticamente fontes de campos potenciais anômalas e apontamos as dificuldades desta formulação: a presença de uma indesejável nuvem de soluções, o critério empírico usado para determinar o índice estrutural (um parâmetro relacionado com a natureza da fonte anômala), a exeqüibilidade da aplicação da deconvolução de Euler a levantamentos magnéticos terrestres, e a determinação do mergulho e do contraste de susceptibilidade magnética de contatos geológicos (ou o produto do contraste de susceptibilidade e a espessura quando aplicado a dique fino). Na segunda parte, apresentamos as recentes melhorias objetivando minimizar algumas dificuldades apresentadas na primeira parte deste tutorial. Entre estas melhorias incluem-se: i) a seleção das soluções essencialmente associadas com observações apresentando alta razão sinal-ruído; ii) o uso da correlação entre a estimativa do nível de base da anomalia e a própria anomalia observada ou a combinação da deconvolução de Euler com o sinal analítico para determinação do índice estrutural; iii) a combinação dos resultados de (i) e (ii), permitindo estimar o índice estrutural independentemente do número de soluções; desta forma, um menor número de observações (tal como em levantamentos terrestres) pode ser usado; iv) a introdução de equações adicionais independentes da equação de Euler que permitem estimar o mergulho e o contraste de susceptibilidade das fontes magnéticas 2D. Na terceira parte apresentaremos um prognóstico sobre futuros desenvolvimentos a curto e médio prazo envolvendo a deconvolução de Euler. As principais perspectivas são: i) novos ataques aos problemas selecionados na segunda parte deste tutorial; ii) desenvolvimento de métodos que permitam considerar interferências de fontes localizadas ao lado ou acima da fonte principal, e iii) uso das estimativas de localização da fonte anômala produzidas pela deconvolução de Euler como vínculos em métodos de inversão para obter a delineação das fontes em um ambiente computacional amigável.

Estruturação litosférica da Província Borborema ao longo da Transecta Angüera (BA) Macau (RN), com base em dados gravimétricos

Since 2005, geophysical surveys have been carried out in the Precambri-an Borborema Province, along two transects with 800 km long each one. A pool of Brazilian public universities and institutions has been acquired deep refrac-tion seismic, gravity and magnetotelluric, with the purpose to model the conti-nental lithosphere of the region. This paper present the gravity survey of the second transect, that crosses the Borborema Province from SW to NE, passing through the São Francisco Craton, Transversal and Meridional zones and Rio Grande do Norte Domain, in the Setentrional Zone. In this way, it cuts some important geologic structures, like the limit of the São Francis Craton and the Borborema Province, Paleozoic and Mesozoic sedimentary basins of Tucano, Jatobá and Potiguar and the extensive Pernambuco and Patos shear zones. Recognition techniques gravity sources in the subsurface, such as spectral analysis and Euler Deconvolution, were applied to the Bouguer anomalies, as well as their regional and residual components. These techniques provided in-formation on possible anomalous bodies, which correlated with pre-existing geological and geophysical data, subsidized a 2.5 D gravity modeling of the lithosphere beneath the Borborema Province and its southern limit with the São Francisco Craton.

Evidências da herança geotectônica pré-cambriana na geração da barcia potiguar: um estudo geofísico multdisciplinar

The 3D gravity modeling of the Potiguar rift basin consisted of a digital processing of gravity and aeromagnetic data, subsidized by the results of Euler deconvolution of gravity and magnetic data and the interpretation of seismic lines and wells descriptions. The gravity database is a compilation of independent geophysical surveys conducted by several universities, research institutions and governmental agencies. The aeromagnetic data are from the Bacia Potiguar and Plataforma Continental do Nordeste projects, obtained from the Brazilian Petroleum Agency (ANP). The solutions of the Euler Deconvolution allowed the analysis of the behavior of the rift main limits. While the integrated interpretation of seismic lines provided the delimitating horizons of the sedimentary formations and the basement top. The integration of these data allowed a 3D gravity modeling of basement topography, allowing the identification of a series of internal structures of the Potiguar rift, as well intra-basement structures without the gravity effect of the rift. The proposed inversion procedure of the gravity data allowed to identify the main structural features of the Potiguar rift, elongated in the NE-SW direction, and its southern and eastern faulted edges, where the sedimentary infill reachs thicknesses up to 5500 m. The southern boundary is marked by the Apodi and Baixa Grande faults. These faults seem to be a single NW-SE oriented fault with a strong bend to NE-SW direction. In addition, the eastern boundary of the rift is conditioned by the NE-SW trending Carnaubais fault system. It was also observed NW-SE oriented faults, which acted as transfer faults to the extensional efforts during the basin formation. In the central part of the residual anomaly map without the gravity effect of the rift stands out a NW-SE trending gravity high, corresponding to the Orós-Jaguaribe belt lithotypes. We also observe a gravity maximum parallel to the Carnaubais fault system. This anomaly is aligned to the eastern limit of the rift and reflects the contact of different crustal blocks, limited by the eastern ward counterpart of the Portalegre Shear Zone

Insights into the morphology of the Serra da Cangalha impact structure from geophysical modeling

Forward modeling is commonly applied to gravity field data of impact structures to determine the main gravity anomaly sources. In this context, we have developed 2.5-D gravity models of the Serra da Cangalha impact structure for the purpose of investigating geological bodies/structures underneath the crater. Interpretation of the models was supported by ground magnetic data acquired along profiles, as well as by high resolution aeromagnetic data. Ground magnetic data reveal the presence of short-wavelength anomalies probably related to shallow magnetic sources that could have been emplaced during the cratering process. Aeromagnetic data show that the basement underneath the crater occurs at an average depth of about 1.9 km, whereas in the region beneath the central uplift it is raised to 0.51 km below the current surface. These depths are also supported by 2.5-D gravity models showing a gentle relief for the basement beneath the central uplift area. Geophysical data were used to provide further constraints for numeral modeling of crater formation that provided important information on the structural modification that affected the rocks underneath the crater, as well as on shock-induced modifications of target rocks. The results showed that the morphology is consistent with the current observations of the crater and that Serra da Cangalha was formed by a meteorite of approximately 1.4 km diameter striking at 12 km s-1.

Timoshenko versus Euler beam theory: Pitfalls of a deterministic approach

The selection criteria for Euler-Bernoulli or Timoshenko beam theories are generally given by means of some deterministic rule involving beam dimensions. The Euler-Bernoulli beam theory is used to model the behavior of flexure-dominated (or ""long"") beams. The Timoshenko theory applies for shear-dominated (or ""short"") beams. In the mid-length range, both theories should be equivalent, and some agreement between them would be expected. Indeed, it is shown in the paper that, for some mid-length beams, the deterministic displacement responses for the two theories agrees very well. However, the article points out that the behavior of the two beam models is radically different in terms of uncertainty propagation. In the paper, some beam parameters are modeled as parameterized stochastic processes. The two formulations are implemented and solved via a Monte Carlo-Galerkin scheme. It is shown that, for uncertain elasticity modulus, propagation of uncertainty to the displacement response is much larger for Timoshenko beams than for Euler-Bernoulli beams. On the other hand, propagation of the uncertainty for random beam height is much larger for Euler beam displacements. Hence, any reliability or risk analysis becomes completely dependent on the beam theory employed. The authors believe this is not widely acknowledged by the structural safety or stochastic mechanics communities. (C) 2010 Elsevier Ltd. All rights reserved.

The composite Euler method for stiff stochastic differential equations

In this paper we present the composite Euler method for the strong solution of stochastic differential equations driven by d-dimensional Wiener processes. This method is a combination of the semi-implicit Euler method and the implicit Euler method. At each step either the semi-implicit Euler method or the implicit Euler method is used in order to obtain better stability properties. We give criteria for selecting the semi-implicit Euler method or the implicit Euler method. For the linear test equation, the convergence properties of the composite Euler method depend on the criteria for selecting the methods. Numerical results suggest that the convergence properties of the composite Euler method applied to nonlinear SDEs is the same as those applied to linear equations. The stability properties of the composite Euler method are shown to be far superior to those of the Euler methods, and numerical results show that the composite Euler method is a very promising method. (C) 2001 Elsevier Science B.V. All rights reserved.

Elastically restrained Bernoulli-Euler beams applied to rotary machinery modelling

Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending, the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length. Based on Rayleigh's quotient, an iterative strategy is developed to find the approximated torsional stiffness coefficients, which allows the reconciliation between the theoretical model results and the experimental ones, obtained through impact tests. The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh's quotient but also on the mode shapes, considering the shape functions defined in branches. Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.

Modelling a Rotating Shaft as an Elastically Restrained Bernoulli-Euler Beam

Industrial rotating machines may be exposed to severe dynamic excitations due to resonant working regimes. Dealing with the bending vibration, problem of a machine rotor, the shaft - and attached discs - can be simply modelled using the Bernoulli-Euler beam theory, as a continuous beam subjected to a specific set of boundary conditions. In this study, the authors recall Rayleigh's method to propose an iterative strategy, which allows for the determination of natural frequencies and mode shapes of continuous beams taking into account the effect of attached concentrated masses and rotational inertias, including different stiffness coefficients at the right and the left end sides. The algorithm starts with the exact solutions from Bernoulli-Euler's beam theory, which are then updated through Rayleigh's quotient parameters. Several loading cases are examined in comparison with the experimental data and examples are presented to illustrate the validity of the model and the accuracy of the obtained values.

Euler angles, Bryant angles and Euler parameters

This chapter deals with the different approaches for describing the rotational coordinates in spatial multibody systems. In this process, Euler angles and Bryant angles are briefly characterized. Particular emphasis is given to Euler parameters, which are utilized to describe the rotational coordinates in the present work. In addition, for all the types of coordinates considered in this chapter, a characterization of the transformation matrix is fully described.

Analytical and numerical investigation of the ultra-relativistic Euler equations

Magdeburg, Univ., Fak. für Mathematik, Diss., 2013

As fórmulas de Euler em projeção estereográfica e seu significado físico no levantamento de indicatrizes óticas

As fórmulas de Euler são deduzidas a partir de triângulos esféricos obtidos da representação estereográfica das rotações φ, θ, ψ. Um exemplo de aplicação a uma secção arbitrária de gipsita é incluído.

The Euler characteristic of a category

The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Rota's Möbius inversion from posets to categories.

Finiteness obstructions and Euler characteristics of categories

"Vegeu el resum a l'inici del document del fitxer adjunt."

Symmetric planar central configurations of five bodies: Euler plus two

We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration, the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass at the midpoint of one of the parallel sides.