Integral Methods in Science and Engineering - Springer Link

Integral Methods in Science and Engineering

Christian Constanda • Bardo E.J. Bodmann Haroldo F. de Campos Velho Editors

Integral Methods in Science and Engineering Progress in Numerical and Analytic Techniques

Editors Christian Constanda Department of Mathematics The University of Tulsa Tulsa, OK, USA

Bardo E.J. Bodmann Mechanical Engineering Federal University of Rio Grande do Sul Porto Alegre, RS, Brazil

Haroldo F. de Campos Velho Associate Laboratory for Computing and Applied Mathematics National Institute for Space Research São José dos Campos, SP, Brazil

ISBN 978-1-4614-7827-0 ISBN 978-1-4614-7828-7 (eBook) DOI 10.1007/978-1-4614-7828-7 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013943828 Mathematics Subject Classification (2010): 00B25, 35-06, 41-06, 44-06, 45-06, 65-06, 76-06, 86-06, 86A10 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)

To Tom Grasso for his professionalism and friendship, on his retirement from the world of publishing (CC)

Preface

The international conferences on Integral Methods in Science and Engineering (IMSE) are a forum where researchers in many theoretical and applied areas, whose working methodology includes integration, communicate their latest results and discoveries and create synergies based on their common interest in the use of a class of general—diverse but interconnected—mathematical procedures. The first 11 IMSE conferences took place in a variety of venues all over the world: 1985, 1990: University of Texas–Arlington, USA; 1993: Tohoku University, Sendai, Japan; 1996: University of Oulu, Finland; 1998: Michigan Technological University, Houghton, MI, USA; 2000: Banff, AB, Canada (organized by the University of Alberta, Edmonton); 2002: University of Saint-Étienne, France; 2004: University of Central Florida, Orlando, FL, USA; 2006: Niagara Falls, ON, Canada (organized by the University of Waterloo); 2008: University of Cantabria, Santander, Spain; 2010: University of Brighton, UK. The 2012 meeting, held in Bento Gonçalves, Rio Grande do Sul, Brazil, July 23–27, and attended by participants from 11 countries on 4 continents, enhanced even further the IMSE tradition as an important event on the international conference circuit, which makes it possible for scientists and engineers to talk about their research interests in a stimulating atmosphere of understanding and cooperation. As in the past, the organization of IMSE 2012 was of a very high standard; by way of acknowledgement, the participants wish to thank CNPq, CAPES, and FAPERGS for their financial support, and Dall’Onder Grande Hotel for special conditions and discounts, which ensured that the daily proceedings of the conference took place in pleasant surroundings. Special thanks are due to the members of the Local Organizing Committee: Bardo E.J. Bodmann (Federal University of Rio Grande do Sul), Chairman, Claudio Pellegrini (Federal University of São João Del Rey), Daniela Buske (Federal University of Pelotas), vii

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Fernando Carvalho (Federal University of Rio de Janeiro), Gervasio A. Degrazia (Federal University of Santa Maria), Haroldo F. de Campos Velho (National Institute for Space Research), Marco Túllio M.B. de Vilhena (Federal University of Rio Grande do Sul), Renato M. Cotta (Federal University of Rio de Janeiro), Ricardo C. Barros (Rio de Janeiro State University). A distinguishing feature of IMSE 2012 was the increased number of young researchers who attended and presented their work. It was both reassuring and gratifying to see that the new generation is ready to join in and help our particular field of scientific interest move forward. The next IMSE conference will be hosted by the Karlsruhe Institute of Technology, Germany, in July 2014. Further details will be posted in due course on the conference web site. The peer-reviewed chapters of this volume, arranged alphabetically by first author’s name, are an expansion of 26 papers from among those given in Bento Gonçalvez. The editors would like to thank the staff at Birkhäuser for their courteous and professional handling of the publication process. Tulsa, OK, USA Porto Alegre, RS, Brazil São José dos Campos, SP, Brazil

Christian Constanda Bardo E.J. Bodmann Haroldo F. de Campos Velho

The International Steering Committee of IMSE: C. Constanda (The University of Tulsa), Chairman M. Ahues (University of Saint-Étienne) B. Bodmann (Federal University of Rio Grande do Sul) H. de Campos Velho (INPE, Saõ José dos Campos) P. Harris (University of Brighton) A. Kirsch (Karlsruhe Institute of Technology) M. Lanza de Cristoforis (University of Padova) S. Mikhailov (Brunel University) D. Mitrea (University of Missouri-Columbia) A. Nastase (RWTH Aachen University) D. Natroshvili (Georgian Technical University) M. Pérez (University of Cantabria) K. Ruotsalainen (University of Oulu) O. Shoham (The University of Tulsa)

Contents

1

2

3

Multiphase Flow Splitting in Looped Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . L. Alvarez, R.S. Mohan, O. Shoham, L. Gomez, and C. Avila 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Experimental Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Model Development. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Green’s Function Decomposition Method for Transport Equation . . . F.S. Azevedo, E. Sauter, M. Thompson, and M.T. Vilhena 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Reformulation as an Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Anisotropic Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Calculation of the Coefficient of W l,k (and Wσ ). . . . . . 2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Neutron Transport and New Computational Methods: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Barbarino, S. Dulla, and P. Ravetto 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Integral Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The AN Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Boundary Element Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Spectral Element Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Comparison of Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 9 10 12 13 15 15 16 23 23 26 27 33 39 41 41 42 45 47 49 51

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3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

6

7

Scale Invariance and Some Limits in Transport Phenomenology: Existence of a Spontaneous Scale . . . . . . . . . . . . . . . . . . . . . B.E.J. Bodmann, M.T. Vilhena, J.R.S. Zabadal, L.P. Luna de Oliveira, and A. Schuck 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A Geometric Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Hyperspace Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 SO(4,2) Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Coherent Structures from a Diffusion-Type Model . . . . . . . . . . . . . . . . B.E.J. Bodmann, J.R.S. Zabadal, A. Schuck, M.T. Vilhena, and R. Quadros 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Motivation from “Arm-Waving Arguments” . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A Coherent Constituent–Mediator Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Concept of Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Modeling Coherent Fluid Constituents . . . . . . . . . . . . . . . . . . . . 5.3.3 Modeling a Coherent Interaction Mediator. . . . . . . . . . . . . . . . 5.4 A Simple Model with Coherence Content . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Simulation of the Dynamics of Molecular Markers Involved in Cell Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Calvez, N. Meunier, N. Muller, and R. Voituriez 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Two-Dimensional Case: The Model with Dynamical Exchange of Markers at the Boundary. . . . . . . . 6.1.3 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Two-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Study of Computational Radiative Fluxes in a Heterogeneous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Q. de Camargo, B.E.J. Bodmann, M.T. Vilhena, and C.F. Segatto 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Radiative-Conductive Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 55 57

57 58 60 61 62 63 65

65 66 67 67 68 68 69 72 73 75 75 77 79 80 81 81 84 88 88 89 91

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7.3 Solution by Decomposition Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.4 Problem Parameter and Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8

9

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11

A Novel Approach to the Hankel Transform Inversion of the Neutron Diffusion Problem Using the Parseval Identity . . . . . . . . . . . J.C.L. Fernandes, M.T. Vilhena, and B.E.J. Bodmann 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Multi-group Steady State Neutron Diffusion . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Hankel-Transformed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Fast Flux Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 The Thermal Flux Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Multi-regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is Convergence Acceleration Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . B.D. Ganapol 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Simulation of Abnormal Protein Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Biophysical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Numerical Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Nuclear Reactor Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Reactor Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Fractal Pattern Phenomenology of Geological Fracture Signatures from a Scaling Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Gioveli, A.J. Strieder, B.E.J. Bodmann, M.T. Vilhena, and A.S. Athayde 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Geological Setting of the Studied Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Fractal Dimension and Self-similarity Analysis . . . . . . . . . . . . . . . 10.4 Structural Fracture Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Fracture Lineament Map Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 105 106 107 108 109 112 114 114 115 115 116 116 118 128 129 130 135 135 137

137 140 142 149 150 151 153

Spectral Boundary Homogenization Problems in Perforated Domains with Robin Boundary Conditions and Large Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 D. Gómez, M.E. Pérez, and T.A. Shaposhnikova 11.1 Introduction and Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . 155 11.2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

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11.3 Convergence Results for α = 2 and κ > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Convergence Results for α ∈ [1, 2) and κ = 2(α − 1) . . . . . . . . . . . . . 11.5 Bounds for Other Values of α and κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

13

14

A Finite Element Formulation of the Total Variation Method for Denoising a Set of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P.J. Harris and K. Chen 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Formulation of the Nonlinear Differential Equation. . . . . . . . . . . . . . . . 12.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Convergence of the Multi-group Isotropic Neutron LT SN Nodal Solution in Cartesian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . E.B. Hauser, R.P. Pazos, and M.T. Vilhena 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Two-Group Discrete Ordinate (SN ) Approximation to the Transport Equation in X, Y Geometry . . . . . . 13.3 The Multigroup Nodal LTSN Formulation in a Rectangle . . . . . . . . . 13.4 Error Bounds for the Discrete Ordinates Nodal Method and Two Energy Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Integration with Singularity by Taylor Series . . . . . . . . . . . . . . H. Hirayama 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 The Arithmetic of Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Basic Functions of Taylor Series. . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Integration of Singular Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Integrals with Algebraic and Logarithmic Singularity . . . . 14.3.2 Cauchy Principal Value Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Hadamard Finite–Part Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Integration with Algebraic and Logarithmic Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Cauchy Principal Value Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Hadamard Finite Part Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 167 171 173 175 175 175 176 178 181 181 183 183 184 185 189 192 193 195 195 196 196 197 198 198 199 200 200 201 201 201 202 203 203

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16

17

Numerical Solutions of the 1D Convection–Diffusion– Reaction and the Burgers Equation Using Implicit Multi-stage and Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.A. Ladeia and N.M.L. Romeiro 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Statement of the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 1D Convection–Diffusion–Reaction Equation . . . . . . . . . . . . 15.2.2 Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.3 Finite Element Method via Least Squares . . . . . . . . . . . . . . . . . 15.3.4 Finite Element Method via Galerkin Procedure. . . . . . . . . . . 15.3.5 Finite Element Method via Streamline-Upwind Petrov–Galerkin Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.6 Linearization of the Convective Term . . . . . . . . . . . . . . . . . . . . . 15.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 1D Convection–Diffusion–Reaction Equation . . . . . . . . . . . . 15.4.2 The Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Reconstruction of Monoenergetic Neutron Angular Flux in Non-multiplying Slabs Using Diffusion Synthetic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.S. Mansur and R.C. Barros 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 The Spatial and the Angular Reconstruction Schemes of the SND Coarse-Mesh Numerical Solution . . . . . . . . . . . . . . . . . . . . . . 16.2.1 The Spatial Reconstruction Scheme . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 The Angular Reconstruction Scheme . . . . . . . . . . . . . . . . . . . . . . 16.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Fractional Neutron Point Kinetics Equations . . . . . . . . . . . . . . . . . . M. Schramm, C.Z. Petersen, M.T. Vilhena, B.E.J. Bodmann, and A.C.M. Alvim 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Derivation of the Fractional Neutron Point Kinetics Equations . . . . 17.3 The Solution of the FNPK Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Case A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Case B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.3 Case C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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205 205 206 206 206 207 207 208 208 208

209 209 210 210 212 213 215

217 217 218 218 223 224 226 227 229

229 231 235 238 238 239 239

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17.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 18

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20

On a Closed Form Solution of the Point Kinetics Equations with a Modified Temperature Feedback. . . . . . . . . . . . . . . . . . . . . J.J.A. Silva, B.E.J. Bodmann, M.T. Vilhena, and A.C.M. Alvim 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 The Kinetic Model with Modified Temperature Feedback . . . . . . . . . 18.2.1 Expansions of Pj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.2 Expansion of A j and B j in Terms of Adomian Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.3 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eulerian Modeling of Radionuclides in Surficial Waters: The Case of Ilha Grande Bay (RJ, Brazil) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.F. Lamego Simões Filho, A.S. de Aguiar, A.D. Soares, C.M.F. Lapa, and M.A.V. Wasserman 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Methodology and Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Hydrodynamical Modeling Approach . . . . . . . . . . . . . . . . . . . . . 19.2.2 Transport Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Input Data and Boundary Conditions for Simulations. . . . . . . . . . . . . . 19.3.1 Bathymetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.2 Astronomical Tide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.3 Wind Speed and Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.4 River Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3.5 Hydrodynamic Model Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 Transport Model Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractional Calculus: Application in Modeling and Control . . . . . . . . . . . J. Tenreiro Machado 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Main Mathematical Aspects of the Theory of Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Approximations to Fractional-Order Derivatives . . . . . . . . . . . . . . . . . . . 20.4 Fractional Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Fractional Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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245 246 249 249 250 251 257 257 259

259 260 260 263 265 265 265 266 267 268 269 276 276 279 279 280 285 288 289 291 291

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Modified Integral Equation Method for Stationary Plate Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.R. Thomson and C. Constanda 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 A Modified Matrix of Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . 21.3 Uniquely Solvable Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Modification with a Finite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonstandard Integral Equations for the Harmonic Oscillations of Thin Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.R. Thomson, C. Constanda, and D.R. Doty 22.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Modified Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Modified Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Genuine Analytical Solution for the SN Multi-group Neutron Equation in Planar Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.K. Tomaschewski, C.F. Segatto, and M.T. Vilhena 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Time-Dependent Multi-group Transport Equation for Heterogeneous Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Phase Flow Instabilities: Effect of Pressure Waves in a Pump–Pipe–Plenum–Choke System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.A.M. Vieira and M.G. Prado 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Single-Phase Flow Instabilities Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.1 Static Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.2 Dynamic Instability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 Single-Phase Flow Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.1 Incompressible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3.2 Compressible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4 Application and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4.1 Example 1: Phase Portrait, Incompressible Model. . . . . . . . 24.4.2 Example 2: Phase Portrait, Incompressible Model with Check-Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4.3 Example 3: Incompressible Versus Compressible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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297 297 299 302 306 308 311 311 313 314 321 324 327 329 329 330 333 335 338 341 341 347 347 350 352 352 352 354 354 356 357

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24.4.4

Example 4: Incompressible Versus Compressible Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.6 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

26

Two-Phase Flow Instabilities in Oil Wells: ESP Oscillatory Behavior and Casing-Heading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R.A.M. Vieira and M.G. Prado 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Two-Phase Flow Modeling Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Application and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.1 Example 1. ESP: Tubing and Annular Space Included in the Solution Domain. Stability Example . . . . . 25.3.2 Example 2. ESP: Neither Casing nor Annular Space Included in the Solution Domain. Instability Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.3 Example 3. ESP: Tubing and Annular Space Included in the Solution Domain. Instability Example . . . 25.3.4 Example 4: Natural Flowing Well. Casing Heading . . . . . . 25.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validating a Closed Form Advection–Diffusion Solution by Experiments: Tritium Dispersion after Emission from the Brazilian Angra Dos Reis Nuclear Power Plant . . . . . . . . . . . . . . . . . . . . G.J. Weymar, D. Buske, M.T. Vilhena, and B.E.J. Bodmann 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 The Advection–Diffusion Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3 A Closed Form Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.1 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.2 A Specific Case for Application . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.4 Experimental Data and Turbulent Parametrization . . . . . . . . . . . . . . . . . 26.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

360 363 363 365 367 367 372 376 376

377 379 381 382 382 384

385 385 386 387 388 389 391 393 395 396

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Contributors

André S. de Aguiar Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil Lourdes Alvarez The University of Tulsa, Tulsa, OK, USA Antônio C.M. Alvim Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil Alexandre S. Athayde Federal University of Pelotas, Pelotas, RS, Brazil Carlos Avila Chevron ETC, Houston, TX, USA Fabio S. Azevedo Institute for Mathematics, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Andrea Barbarino Energy Department, Politecnico di Torino, Torino, Italy Ricardo C. Barros University of the State of Rio de Janeiro, Programa de Pósgraduação em Ciências Computacionais, Rio de Janeiro, RJ, Brazil Bardo E.J. Bodmann Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Daniela Buske Federal University of Pelotas, Pelotas, Rio Grande do Sul, Brazil Vincent Calvez École Normale Supérieure de Lyon, Lyon Cedex, France Dayana Q. de Camargo Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Ke Chen University of Liverpool, Liverpool, UK Christian Constanda The University of Tulsa, Tulsa, OK, USA D.R. Doty The University of Tulsa, Tulsa, OK, USA Sandra Dulla Energy Department, Politecnico di Torino, Torino, Italy

xvii

xviii

Contributors

Julio C.L. Fernandes Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Barry D. Ganapol The University of Arizona, Tucson, AZ, USA Izabel Gioveli Federal University of Fronteira Sul, Santo Ângelo, RS, Brazil D. Gómez Dpto. Matemáticas, Estadísitica y Computación, Universidad de Cantabria, Santander, Spain Luis Gomez–Morillo The University of Tulsa, Tulsa, OK, USA Paul J. Harris University of Brighton, Brighton, UK Eliete B. Hauser Pontifical Catholic University of Rio Grande do Sul, Porto Alegre, RS, Brazil Hiroshi Hirayama Department of Vehicle System Engineering, Faculty of Creative Engineering, Kanagawa Institute of Technology, Kanagawa, Japan Cibele A. Ladeia State University of Londrina, Londrina, Paraná, Brazil Celso M.F. Lapa Institute of Nuclear Engineering, Rio de Janeiro, RJ, Brazil Ralph S. Mansur University of the State of Rio de Janeiro, Programa de Pósgraduação em Ciências Computacionais, Rio de Janeiro, RJ, Brazil Nicolas Meunier Université Paris Descartes, Paris, France Ram S. Mohan The University of Tulsa, Tulsa, OK, USA Nicolas Muller Université Paris Descartes, Paris, France Luis P.L. de Oliveira University of Vale do Rio dos Sinos, São Leopoldo, RS, Brazil Rubén P. Pazos University of Santa Cruz do Sul, Santa Cruz do Sul, RS, Brazil Claudio Z. Petersen Federal University of Pelotas, Pelotas, RS, Brazil M. Eugenia Pérez Dpto. Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Santander, Spain Mauricio G. Prado The University of Tulsa, Tulsa, OK, USA Regis Quadros Federal University of Pelotas, Pelotas, Rio Grande do Sul, Brazil Piero Ravetto Energy Department, Politecnico di Torino, Torino, Italy Neyva M.L. Romeiro State University of Londrina, Londrina, Paraná, Brazil Esequia Sauter Institute for Mathematics, Federal Institute of Rio Grande do Sul, Porto Alegre, RS, Brazil Marcelo Schramm Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil

Contributors

xix

Adalberto Schuck Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Cynthia F. Segatto Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Tatiana A. Shaposhnikova Department of Differential Equations, Moscow State University, Moscow, Russia Ovadia Shoham The University of Tulsa, Tulsa, OK, USA Jeronimo J.A. Silva Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil Francisco F. Lamego Simões Filho Institute of Nuclear Engineering, Rio de Janeiro, RJ, Brazil Abner D. Soares National Commission for Nuclear Energy, Rio de Janeiro, RJ, Brazil Adelir J. Strieder Federal University of Pelotas, Pelotas, RS, Brazil José A. Tenreiro Machado Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Porto, Portugal Gavin R. Thomson A.C.C.A., Glasgow, UK Mark Thompson Institute for Mathematics, Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Fernanda K. Tomaschewski Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Rinaldo A.M. Vieira Petrobras, Rio de Janeiro, Brazil Marco T.B.M. de Vilhena Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Raphaël Voituriez Université Pierre et Marie Curie, Paris Cedex, France Maria A.V. Wasserman Institute of Nuclear Engineering, Rio de Janeiro, RJ, Brazil Guilherme J. Weymar Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil Jorge R.S. Zabadal Federal University of Rio Grande do Sul, Porto Alegre, RS, Brazil