Simulação do processo de transferência de calor não linear condução-radiação por meio de um esquema linear em diferenças finitas.

Neste trabalho o processo não linear de transmissão de calor condução-radiação é abordado num contexto bidimensional plano e simulado com o uso de um esquema linear em diferenças finitas. O problema original é tratado como o limite de uma sequencia de problemas lineares, do tipo condução-convecção. Este limite, cuja existência é comprovada, é facilmente obtido a partir de procedimentos básicos, accessíveis a qualquer estudante de engenharia, permitindo assim o emprego de hipóteses mais realistas, já que não se tem o limitante matemático para a abordagem numérica de uma equação diferencial parcial elíptica. Neste trabalho foi resolvido o problema de condução de calor em regime permanente em uma placa com condições de contorno convectivas e radioativas utilizando-se o software MatLab, vale ressaltar, que a mesma metodologia é aplicável para geometrias mais complexas.

Multilevel Training of Binary Morphological Operators

The design of binary morphological operators that are translation-invariant and locally defined by a finite neighborhood window corresponds to the problem of designing Boolean functions. As in any supervised classification problem, morphological operators designed from a training sample also suffer from overfitting. Large neighborhood tends to lead to performance degradation of the designed operator. This work proposes a multilevel design approach to deal with the issue of designing large neighborhood-based operators. The main idea is inspired by stacked generalization (a multilevel classifier design approach) and consists of, at each training level, combining the outcomes of the previous level operators. The final operator is a multilevel operator that ultimately depends on a larger neighborhood than of the individual operators that have been combined. Experimental results show that two-level operators obtained by combining operators designed on subwindows of a large window consistently outperform the single-level operators designed on the full window. They also show that iterating two-level operators is an effective multilevel approach to obtain better results.

Avaliação de métodos eletromagnéticos aplicando campos polarizados e focalizados

Em investigações geofísicas rasas que empregam os métodos eletromagnéticos indutivos mais avançados, alvos com baixo número de indução (Low Induction Number – LIN) produzem anomalias eletromagnéticas muito baixas e de difícil interpretação. Para suprir esta deficiência, neste trabalho são estudados a aplicabilidade de campos eletromagnéticos polarizados e focalizados – POLFOCEM como fonte primária de indução. Os campos E.M. focalizados e polarizados, vertical e horizontalmente, são obtidos pelas combinações vetoriais de pares de dipolos transmissores e, ocorrem na região central entre eles. A focalização é observada nesta região na profundidade de 0,25 do espaçamento entre esses transmissores – L. Portanto, máximos acoplamentos podem ser obtidos através da seleção da polarização de acordo com a geometria do alvo, ocorrendo um aumento na densidade de fluxo magnético sobre ele e, máximas anomalias produzidas. É utilizada uma metodologia numérica para o cômputo dessas anomalias por meio da técnica dos elementos finitos para solução do problema 2,5-D. Em todos os experimentos numéricos são realizadas comparações qualitativas e quantitativas entre as respostas obtidas pelos sistemas POLFOCEM e convencional, o qual emprega um único dipolo como transmissor (dipolo-dipolo). As anomalias produzidas pelo sistema POLFOCEM, em que os dipolos transmissores são acionados simultaneamente, correspondem à soma das anomalias produzidas por cada um desses dipolos independentes, caracterizando, desta forma, a linearidade dos campos eletromagnéticos. Os experimentos numéricos são realizados para alvos prismáticos bidimensionais com três diferentes inclinações, inseridos num semi-espaço resistivo, e para as freqüências das fontes na faixa das ondas de rádio. As anomalias assimétricas no sistema convencional, que se tornam simétricas no sistema POLFOCEM, apresentam valores menores em amplitude. Contudo, aquelas anomalias tanto assimétricas quanto simétricas que se tornam anti-simétricas apresentam valores maiores. Em decorrência dessas diminuições e aumentos nas amplitudes ocorrem rotações nos diagramas de Argand, no sentido horário e anti-horário para alvos com baixos valores de condutividade, respectivamente. Em experimentos de identificação de presença de dois alvos próximos, o sistema convencional é capaz de identificá-los primeiramente, prevalecendo o seu uso.

Modelagem direta bidimensional do método magnetotelúrico com o método dos elementos finitos de arestas

Fizemos a modelagem direta 2D do método magnetotelúrico (MT) com o método dos elementos finitos (MEF) de arestas em termos dos campos primários e secundários. Para usarmos modelos de maior complexidade e diminuirmos o custo computacional utilizamos malhas não estruturadas. Nas malhas utilizadas, introduzimos quatro nós em torno de cada estação MT, constituindo um quadrado alinhado nas direções dos eixos cartesianos

Dense disparity map estimation respecting image discontinuities

[EN] We present an energy based approach to estimate a dense disparity map from a set of two weakly calibrated stereoscopic images while preserving its discontinuities resulting from image boundaries. We first derive a simplified expression for the disparity that allows us to estimate it from a stereo pair of images using an energy minimization approach. We assume that the epipolar geometry is known, and we include this information in the energy model. Discontinuities are preserved by means of a regularization term based on the Nagel-Enkelmann operator. We investigate the associated Euler-Lagrange equation of the energy functional, and we approach the solution of the underlying partial differential equation (PDE) using a gradient descent method The resulting parabolic problem has a unique solution. In order to reduce the risk to be trapped within some irrelevant local minima during the iterations, we use a focusing strategy based on a linear scalespace. Experimental results on both synthetic and real images arere presented to illustrate the capabilities of this PDE and scale-space based method.

Safety factors in sheet pile wall design: considerations by using traditional methods and FEM analysis

The purpose of the work is: define and calculate a factor of collapse related to traditional method to design sheet pile walls. Furthermore, we tried to find the parameters that most influence a finite element model representative of this problem. The text is structured in this way: from chapter 1 to 5, we analyzed a series of arguments which are usefull to understanding the problem, while the considerations mainly related to the purpose of the text are reported in the chapters from 6 to 10. In the first part of the document the following arguments are shown: what is a sheet pile wall, what are the codes to be followed for the design of these structures and what they say, how can be formulated a mathematical model of the soil, some fundamentals of finite element analysis, and finally, what are the traditional methods that support the design of sheet pile walls. In the chapter 6 we performed a parametric analysis, giving an answer to the second part of the purpose of the work. Comparing the results from a laboratory test for a cantilever sheet pile wall in a sandy soil, with those provided by a finite element model of the same problem, we concluded that:in modelling a sandy soil we should pay attention to the value of cohesion that we insert in the model (some programs, like Abaqus, don’t accept a null value for this parameter), friction angle and elastic modulus of the soil, they influence significantly the behavior of the system (structure-soil), others parameters, like the dilatancy angle or the Poisson’s ratio, they don’t seem influence it. The logical path that we followed in the second part of the text is reported here. We analyzed two different structures, the first is able to support an excavation of 4 m, while the second an excavation of 7 m. Both structures are first designed by using the traditional method, then these structures are implemented in a finite element program (Abaqus), and they are pushed to collapse by decreasing the friction angle of the soil. The factor of collapse is the ratio between tangents of the initial friction angle and of the friction angle at collapse. At the end, we performed a more detailed analysis of the first structure, observing that, the value of the factor of collapse is influenced by a wide range of parameters including: the value of the coefficients assumed in the traditional method and by the relative stiffness of the structure-soil system. In the majority of cases, we found that the value of the factor of collapse is between and 1.25 and 2. With some considerations, reported in the text, we can compare the values so far found, with the value of the safety factor proposed by the code (linked to the friction angle of the soil).

Multilevel Domain Decomposition Algorithms for Monolithic Fluid-Structure Interaction Problems with Application to Haemodynamics

Finite element techniques for solving the problem of fluid-structure interaction of an elastic solid material in a laminar incompressible viscous flow are described. The mathematical problem consists of the Navier-Stokes equations in the Arbitrary Lagrangian-Eulerian formulation coupled with a non-linear structure model, considering the problem as one continuum. The coupling between the structure and the fluid is enforced inside a monolithic framework which computes simultaneously for the fluid and the structure unknowns within a unique solver. We used the well-known Crouzeix-Raviart finite element pair for discretization in space and the method of lines for discretization in time. A stability result using the Backward-Euler time-stepping scheme for both fluid and solid part and the finite element method for the space discretization has been proved. The resulting linear system has been solved by multilevel domain decomposition techniques. Our strategy is to solve several local subproblems over subdomain patches using the Schur-complement or GMRES smoother within a multigrid iterative solver. For validation and evaluation of the accuracy of the proposed methodology, we present corresponding results for a set of two FSI benchmark configurations which describe the self-induced elastic deformation of a beam attached to a cylinder in a laminar channel flow, allowing stationary as well as periodically oscillating deformations, and for a benchmark proposed by COMSOL multiphysics where a narrow vertical structure attached to the bottom wall of a channel bends under the force due to both viscous drag and pressure. Then, as an example of fluid-structure interaction in biomedical problems, we considered the academic numerical test which consists in simulating the pressure wave propagation through a straight compliant vessel. All the tests show the applicability and the numerical efficiency of our approach to both two-dimensional and three-dimensional problems.

Direct and inverse transient eddy current problems

Die vorliegende Arbeit behandelt Vorwärts- sowie Rückwärtstheorie transienter Wirbelstromprobleme. Transiente Anregungsströme induzieren elektromagnetische Felder, welche sogenannte Wirbelströme in leitfähigen Objekten erzeugen. Im Falle von sich langsam ändernden Feldern kann diese Wechselwirkung durch die Wirbelstromgleichung, einer Approximation an die Maxwell-Gleichungen, beschrieben werden. Diese ist eine lineare partielle Differentialgleichung mit nicht-glatten Koeffizientenfunktionen von gemischt parabolisch-elliptischem Typ. Das Vorwärtsproblem besteht darin, zu gegebener Anregung sowie den umgebungsbeschreibenden Koeffizientenfunktionen das elektrische Feld als distributionelle Lösung der Gleichung zu bestimmen. Umgekehrt können die Felder mit Messspulen gemessen werden. Das Ziel des Rückwärtsproblems ist es, aus diesen Messungen Informationen über leitfähige Objekte, also über die Koeffizientenfunktion, die diese beschreibt, zu gewinnen. In dieser Arbeit wird eine variationelle Lösungstheorie vorgestellt und die Wohlgestelltheit der Gleichung diskutiert. Darauf aufbauend wird das Verhalten der Lösung für verschwindende Leitfähigkeit studiert und die Linearisierbarkeit der Gleichung ohne leitfähiges Objekt in Richtung des Auftauchens eines leitfähigen Objektes gezeigt. Zur Regularisierung der Gleichung werden Modifikationen vorgeschlagen, welche ein voll parabolisches bzw. elliptisches Problem liefern. Diese werden verifiziert, indem die Konvergenz der Lösungen gezeigt wird. Zuletzt wird gezeigt, dass unter der Annahme von sonst homogenen Umgebungsparametern leitfähige Objekte eindeutig durch die Messungen lokalisiert werden können. Hierzu werden die Linear Sampling Methode sowie die Faktorisierungsmethode angewendet.

Adaptivity and a posteriori error control for bifurcation problems II: Incompressible fluid flow in open systems with Z_2 symmetry

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z_2 symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual-Weighted-Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.

Wave propagation analysis in carbon nanotube embedded composite using wavelet based spectral finite elements

In this paper, elastic wave propagation is studied in a nanocomposite reinforced with multiwall carbon nanotubes (CNTs). Analysis is performed on a representative volume element of square cross section. The frequency content of the exciting signal is at the terahertz level. Here, the composite is modeled as a higher order shear deformable beam using layerwise theory, to account for partial shear stress transfer between the CNTs and the matrix. The walls of the multiwall CNTs are considered to be connected throughout their length by distributed springs, whose stiffness is governed by the van der Waals force acting between the walls of nanotubes. The analyses in both the frequency and time domains are done using the wavelet-based spectral finite element method (WSFEM). The method uses the Daubechies wavelet basis approximation in time to reduce the governing PDE to a set of ODEs. These transformed ODEs are solved using a finite element (FE) technique by deriving an exact interpolating function in the transformed domain to obtain the exact dynamic stiffness matrix. Numerical analyses are performed to study the spectrum and dispersion relations for different matrix materials and also for different beam models. The effects of partial shear stress transfer between CNTs and matrix on the frequency response function (FRF) and the time response due to broadband impulse loading are investigated for different matrix materials. The simultaneous existence of four coupled propagating modes in a double-walled CNT-composite is also captured using modulated sinusoidal excitation.

Semi-Lagrange time integration for PDE models of asian options

Semi-Lagrange time integration is used with the finite difference method to provide accurate stable prices for Asian options, with or without early exercise. These are combined with coordinate transformations for computational efficiency and compared with published results

Zur Konvergenz der Randpunktmethode

Das von Maz'ya eingeführte Approximationsverfahren, die Methode der näherungsweisen Näherungen (Approximate Approximations), kann auch zur numerischen Lösung von Randintegralgleichungen verwendet werden (Randpunktmethode). In diesem Fall hängen die Komponenten der Matrix des resultierenden Gleichungssystems zur Berechnung der Näherung für die Dichte nur von der Position der Randpunkte und der Richtung der äußeren Einheitsnormalen in diesen Punkten ab. Dieses numerisches Verfahren wird am Beispiel des Dirichlet Problems für die Laplace Gleichung und die Stokes Gleichungen in einem beschränkten zweidimensionalem Gebiet untersucht. Die Randpunktmethode umfasst drei Schritte: Im ersten Schritt wird die unbekannte Dichte durch eine Linearkombination von radialen, exponentiell abklingenden Basisfunktionen approximiert. Im zweiten Schritt wird die Integration über den Rand durch die Integration über die Tangenten in Randpunkten ersetzt. Für die auftretende Näherungspotentiale können sogar analytische Ausdrücke gewonnen werden. Im dritten Schritt wird das lineare Gleichungssystem gelöst, und eine Näherung für die unbekannte Dichte und damit auch für die Lösung der Randwertaufgabe konstruiert. Die Konvergenz dieses Verfahrens wird für glatte konvexe Gebiete nachgewiesen.

Klassifikationsgestützte on-line Adaption eines robusten beobachterbasierten Fehlerdiagnoseansatzes für nichtlineare Systeme

Diese Arbeit behandelt die Problemstellung der modellbasierten Fehlerdiagnose für Lipschitz-stetige nichtlineare Systeme mit Unsicherheiten. Es wird eine neue adaptive Fehlerdiagnosemethode vorgestellt. Erkenntnisse und Verfahren aus dem Bereich der Takagi-Sugeno (TS) Fuzzy-Modellbildung und des Beobachterentwurfs sowie der Sliding-Mode (SM) Theorie werden genutzt, um einen neuartigen robusten und nichtlinearen TS-SM-Beobachter zu entwickeln. Durch diese Zusammenführung lassen sich die jeweiligen Vorteile beider Ansätze miteinander kombinieren. Bedingungen zur Konvergenz des Beobachters werden als lineare Matrizenungleichungen (LMIs) abgeleitet. Diese Bedingungen garantieren zum einen die Stabilität und liefern zum anderen ein direktes Entwurfsverfahren für den Beobachter. Der Beobachterentwurf wird für die Fälle messbarer und nicht messbarer Prämissenvariablen angegeben. Durch die TS-Erweiterung des in dieser Arbeit verwendeten SM-Beobachters ist es möglich, den diskontinuierlichen Rückführterm mithilfe einer geeigneten kontinuierlichen Funktion zu approximieren und dieses Signal daraufhin zur Fehlerdiagnose auszuwerten. Dies liefert eine Methodik zur Aktor- und Sensorfehlerdiagnose nichtlinearer unsicherer Systeme. Gegenüber anderen Ansätzen erlaubt das Vorgehen eine quantitative Bestimmung und teilweise sogar exakte Rekonstruktion des Fehlersignalverlaufs. Darüber hinaus ermöglicht der Ansatz die Berechnung konstanter Fehlerschwellen direkt aus dem physikalischen Vorwissen über das betrachtete System. Durch eine Erweiterung um eine Betriebsphasenerkennung wird es möglich, die Schwellenwerte des Fehlerdiagnoseansatzes online an die aktuelle Betriebsphase anzupassen. Hierdurch ergibt sich in Betriebsphasen mit geringen Modellunsicherheiten eine deutlich erhöhte Fehlersensitivität. Zudem werden in Betriebsphasen mit großen Modellunsicherheiten Falschalarme vermieden. Die Kernidee besteht darin, die aktuelle Betriebsphase mittels eines Bayes-Klassikators in Echtzeit zu ermitteln und darüber die Fehlerschwellen an die a-priori de nierten Unsicherheiten der unterschiedlichen Betriebsphasen anzupassen. Die E ffektivität und Übertragbarkeit der vorgeschlagenen Ansätze werden einerseits am akademischen Beispiel des Pendelwagens und anderseits am Beispiel der Sensorfehlerdiagnose hydrostatisch angetriebener Radlader als praxisnahe Anwendung demonstriert.