DOCSLIB.ORG

A New Computational Framework for Particle and Spin Simulations Based

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A New Computational Framework for Particle and Spin Simulations Based Draft for PRE September 15, 2018 A new computational framework for particle and spin simulations based on the stochastic Galerkin method J. Slim,1 F. Rathmann,2, ∗ and D. Heberling1, 3 1Institut für Hochfrequenztechnik, RWTH Aachen University, 52056 Aachen, Germany 2Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany 3JARA–FAME (Forces and Matter Experiments), Forschungszentrum Jülich and RWTH Aachen University, Germany An implementation of the Polynomial Chaos Expansion is introduced here as a fast solver of the equations of beam and spin motion inside an RF Wien filter. The device shall be used to search for the deuteron electric dipole moment in the COSY storage ring. The new approach is based on the stochastic Galerkin method, and it is shown that this constitutes a new and very powerful alternative to the commonly used Monte-Carlo methods. I. INTRODUCTION AND MOTIVATION the electromagnetic performance of the above mentioned RF Wien filter under mechanical uncertainties [3]. This The JEDI1 collaboration aims at a measurement of the investigation made use of the so-called Polynomial Chaos electric dipole moment (EDM) of charged hadronic par- Expansion (PCE) [4], as an efficient and yet accurate al- ticles, such as deuterons and protons. In the near future, ternative to the MC method. a first direct EDM measurement of the deuteron will be In this paper, we present a computational framework carried out at the COoler SYnchrotron COSY, a conven- to further reduce the complexity of the problem in terms tional magnetic storage ring. To this end, an RF Wien of the required number of tracking simulations. The same filter [1] has recently been installed, which will be oper- methodology as in Ref. [3] is used here, but as we have ated at some harmonic of the spin precession frequency, access to the equations governing the system, here an whereby the sensitivity to the deuteron EDM is substan- intrusive version of the PCE is employed. tially enhanced. This paper is organized as follows. Section II briefly In order to eliminate the false EDM signals, it is of reviews the basic theoretical foundations of the PCE. crucial importance to understand the sources of mag- Section III introduces the stochastic Galerkin method netic imperfections in the accelerator ring [2]. Particle (SGM) and applies it to the beam and spin equations. and spin tracking simulation constitute powerful tools to Section IV describes the main steps required to perform do so. Conducting a realistic simulation for a machine the SGM. Section V presents the results and compares that includes mechanical, electrical and electromagnetic them quantitatively to the equivalent MC results. Sec- uncertainties of all of its elements is indeed a challenging tion VI summarizes and concludes the paper. task. The well-known Monte-Carlo (MC) simulation as a tool to conduct such a study is computationally very expensive. The dimension (number of random parame- II. POLYNOMIAL CHAOS EXPANSION ters) in such simulations is particularly large. To further clarify the complexity of the problem, just consider that The polynomial chaos expansion (PCE) is a stochas- the current in a dipole magnet is not perfectly uniform, tic spectral method that allows for stochastically varying and this will alter the magnetic fields, which will affect physical entities Y, as a response of some random input the spin motion and orbit motion of the particles. It ξ to be represented in terms of orthogonal polynomials. is therefore important to quantify such uncertainties be- PCE permits Y to be expanded into a series of orthogonal forehand for the intended EDM experiment. In addition, polynomials of degree p (the expansion order) as function many other factors like the mechanical fabrication tol- of the input variables ξ. Thus it follows that erance and the accuracy of the alignment contribute as arXiv:1707.09274v1 [physics.comp-ph] 28 Jul 2017 well. N The principle problem with the MC method is the ne- Y = aiΨi(ξ). (1) i cessity to repeat the simulation 10000 or even tens of X millions of times. For limited-budget projects such as Here the Ψi are the multivariate orthogonal polynomi- ours, this is therefore not an option. Recently, as a first als (the basis functions) of degree p, and the ai are the step toward a full systematic analysis of the future EDM expansion coefficients to be computed. The orthogonal experiments at COSY, we conducted a study to quantify polynomials can be the Hermite, Legendre, Laguerre, or any other set of orthogonal polynomials, depending on the probabilistic distribution of the random input vari- ∗ Corresponding author: [email protected] ables ξ. This paper deals with physical entities Y that 1 Jülich Electric Dipole moment Investigations, include electromagnetic fields with uncertainties, particle http://collaborations.fz-juelich.de/ikp/jedi/. positions, velocities and spin vectors. 2 The first step in building a polynomial chaos series is (FEM) [11, 12]. In 1921 Boris Galerkin, a Russian math- to determine the probabilistic distribution of the input ematician, proposed a method to solve differential equa- random variables and their number (the dimensionality tions based on functional analysis. In contrast to other of the problem). Once known, the multivariate orthogo- methods, such as finite difference (FD) schemes, the nal polynomials can be constructed using the three-terms Galerkin method does not solve the differential equa- recurrence properties that can be found e.g., in [5]. tions directly, but transforms them into a variational Oftentimes, the number of random variables is larger form (a functional) that is then minimized. The func- than unity, which implies that multivariate polynomials tions minimizing this functional are the solutions to the must be used instead of univariate ones. Obviously, the required differential equations. The variational form is expansion order must be known and, as explained in [6, constructed via the Galerkin projection techniques [in- 7], can be changed adaptively during processing. sert proper ref]. The expansion coefficients can be calculated using in- In this section, the construction of the variational trusive and non-intrusive methods. Non-intrusive meth- form of the beam and spin dynamic equations using the ods consider the deterministic code as a black box, i.e., stochastic Galerkin projection is described in detail. Ne- they do not alter the code nor the equations. The ex- glecting forces other than the electromagnetic ones acting pansion coefficients are calculated using multiple calls to on the charged particles, the beam equations read the deterministic code either via projection or regression. Both require a number of N realization pairs (ξ , Ψ ) (see d q 1 i i ~v = E~ + ~v × B~ − ~v ~v · E~ , and Eq. (1)). Projection requires the evaluation of expecta- dt mγ c2 (5) tion values and relies on the orthogonality of polynomials d ~r = ~v. to compute the coefficients in the form of dt E {YΨ} a = , (2) The expansion of Eq. (5) in Cartesian coordinates E 2 {Ψ } yields a linear system of six coupled ordinary differen- where the computation of the expectation values (E{·}) tial equations, necessitates the evaluation of integrals. Quadrature methods are one way to do so, and are commonly used d q 1 1 1 1 vx = Ex + vyBz − vzBy − vx(~v · E~ ) , in PCE analyses. Depending on the type of input distri- dt m γ γ γ c2γ bution, the corresponding quadrature rule can be used. The Gauss-Laguerre quadrature for instance [8], is used (6a) in the case of uniformly distributed random variables. d q 1 1 1 1 This projection method is widely known as non-intrusive vy = Ey + vzBx − vxBz − vy(~v · E~ ) , dt m γ γ γ c2γ spectral projection (NISP) [9]. (6b) Regression, on the other hand, estimates the coeffi- cients that minimize the functional difference between d q 1 1 1 1 v = E + v B − v B − v (~v · E~ ) , and the estimated response Yˆ and the actual response Y, dt z m γ z γ x y γ y x c2γ z given by (6c) 2 d a = arg min E Yˆ − Y . (3) x = vx , (6d) dt n o d The solution of Eq. (3), obtained by linear regression, y = vy , (6e) dt yields d T z = vz . (6f) ai = Ψ · Ψ · Ψ · Y. (4) dt In the context of this work, the regression method [10] Here, ~v denotes the velocity vector of the particles, q the provides more accurate results than the projection particle charge, m the mass, γ the Lorentz factor, and ~r method, it is therefore used here to calculate the expan- the position vector. sion coefficients ai, which are subsequently fed into the When intra-beam scattering and other collective ef- stochastic Galerkin solver. fects are neglected, the simulation of a beam of parti- cles is equivalent to individual simulations with different initial conditions. It is furthermore assumed that the III. STOCHASTIC GALERKIN METHOD number of single particle simulations is sufficiently large APPLIED TO BEAM AND SPIN DYNAMICS to describe the beam. These assumptions permit us to use stochastic methods to solve the differential equations A. Beam dynamics with random coefficients, or with uncertain input vari- ables, or even with random boundary values. According One of the most common methods to solve differ- to a probabilistic distribution, each individual particle of ential equations is the Galerkin-finite element method the population has a different initial position ~r and veloc- 3 (a) C14jl (b) C26jl (c) C37jl (d) C45jl (e) C69jl (f) C77jl (g) C86jl (h) C93jl (i) C105jl (j) C116jl FIG. 1. With the m = 5 dimensional problem and an expansion order of p = 4, the number of basis functions is P = 126 (see Eq. (7) of [3]), which results in a (126 × 126 × 126) Cijl tensor.
Load more

Recommended publications
  • Application of Variational Methods and Galerkin Method in Solving Engineering Problems Represented by Ordinary Differential Equations
    International Journal of Mechanical And Production Engineering, ISSN: 2320-2092, Volume- 4, Issue-4, Apr.-2016 APPLICATION OF VARIATIONAL METHODS AND GALERKIN METHOD IN SOLVING ENGINEERING PROBLEMS REPRESENTED BY ORDINARY DIFFERENTIAL EQUATIONS 1B.V. SIVA PRASAD REDDY, 2K. RAJESH BABU 1,2Department of Mechanical Engineering, Sri Venkateswara University college of Engineering, Tirupati, India E-mail: [email protected], [email protected]; Abstract – Nowadays the accuracy of problem solving is very important. In olden days the Variational methods were used to solve all engineering problems like structural, heat transfer and fluid mechanics problems. With the emergence of Finite Element Method (FEM) those methods are become less important, although FEM is also an approximate method of numerical technique. The concept of variational methods is inducted to solve majority of engineering problems, which gives more accurate results than any other type of approximate methods. The engineering problems like uniform bar, beams, heat transfer and fluid flow problems are used in our daily life and they play an important role in the development of our society. To achieve drastic development in the society, it is a must to focus on adopting approximation methods that improve the accuracy of engineering solution. Of all the methods, Galerkin method is emerging as an alternative and more accurate method than those of Ritz, Rayleigh – Ritz methods. Any physical problem in nature can be transformed into an equivalent mathematical model by idealization process and describing its behavior by a suitable governing equation with associated boundary conditions. Against this backdrop, the present work focuses on application of different variational methods in solving ordinary differential equations.
    [Show full text]
  • Biodata Dr. JS
    Biodata Dr. J. S. Rao CEO, Innovative Engineering Designs and Simulation Global Solutions President, The Vibration Institute of India Chief Editor, Journal Vibration Engineering and Technologies 1039, 2nd Cross, BEL Layout, Block II Bangalore 560097 +91 98453 46503 [email protected] Also Chief Science Officer (consulting) Altair Engineering India Pvt. Ltd., Bangalore 560103 Contents Title Page Number 1. Experience 2 2. Education 2 3. Memberships of Scientific Bodies 3 4. Contributions to Scientific Community 3 5. Research Areas 5 6. Doctoral Theses 5 7. Review Work 6 8. Industrial Consultancy and Sponsored Work 6 9. Books 9 10. Awards 10 11. Congresses and Schools 11 12. National and International Seminars 14 13. Five Decades of Research Work 18 14. Journal Papers 31 15. Conference Papers 38 16. Contributions as Science Counselor 53 1 Dr. J.S. Rao 1. EXPERIENCE President Kumaraguru College of Technology, Coimbatore 2012-2016 Protem Chancellor K L University, Vijayawada 2011-2012 Director GMR Energy Ltd., Bangalore 2000-2012 CEO, Dynaspede Integrated Systems, Bangalore 2004-2005 Chief Technology Officer, QuEST, Bangalore 2001-2004 Professor of Mechanical Engineering The University of New South Wales, Sydney, Australia 1996 NSC Research Chair Professor National Chung Cheng University, Chia-Yi, Taiwan 1994-96 Professor of Mechanical Engineering Inst. fur Mech., Gesamthochschule, Kassel, Germany 1988 Sr. Technical Consultant, Stress Technology Inc., Adjunct Professor Mechanical Engineering Rochester Institute of Technology, Rochester, NY, USA 1980-81 Professor of Mechanical Engineering Concordia University, Montreal, Canada 1980 Professor of Mechanical Engineering Inst. Nationale des Sciences Appliquees, Lyon, France 1980 Science Counselor Indian Embassy, Washington DC 1984-89 Indian Institute of Technology, Delhi Professor of Mechanical Engineering 1975-2000 Faculty 1960-70 Professor of Mechanical Engineering Indian Institute of Technology, Kharagpur 1970-75 Post-Doctoral Commonwealth Fellow University of Surrey, Guildford, England 1968-70 2 Dr.
    [Show full text]
  • Professor Ilya Yakovich Shtaerman (1891-1962)
    Professor Ilya Yakovich Shtaerman (1891-1962) I.Y. Shtaerman (1891-1962) - Ph.D., professor, an expert in the field of mechanics, corresponding member of the USSR Academy of Sciences (1939). The scientific activity of I.Y. Shtaerman was formed under the leadership of Professor of Mechanics University of Kiev: P.V. Voronets. Main areas of research I.Y. Shtaerman were devoted to the study of problems of the theory of elasticity, structural mechanics and mathematics. Ilya Shtaerman was born April 19, 1891 in the city of Mogilev-Podolsky; in 1910 - graduated from high school in Kamenetz-Podolsk and in 1915 - graduated from the Faculty of Physics and Mathematics of Kiev University. He published "Differential equations of the plate, rolling without slipping on a fixed surface," which has developed some of the provisions of the master's thesis of his teacher, P.V.Voronets. In 1918 he graduated from the Faculty of Engineering KPI; 1918 - 1941 he worked in the KPI and taught at the Kiev Institute of Public Education; 1920-1934 - member of Applied Mechanics committee, Academy of Sciences of the Ukrainian SSR; 1924-1941r. - Professor, Head of Department of Theoretical Mechanics KPI. In 1930 he defended his doctoral thesis "On the integration of the differential equations of equilibrium of elastic shells." 1934-1943 - Researcher, Institute of Mathematics, Ukrainian Academy of Sciences; 1943 - Professor of the Moscow Institute of Civil Engineering. I.Y. Shtaerman developed a number of methods for solving the complex problems of the theory of elasticity. It was the first major study of this issue, as set out in Russian.
    [Show full text]
  • Time Spectral Methods - Towards Plasma Turbulence Modelling
    kth royal institute of technology Doctoral Thesis in Electrical Engineering Time Spectral Methods - Towards Plasma Turbulence Modelling KRISTOFFER LINDVALL Stockholm, Sweden 2021 Time Spectral Methods - Towards Plasma Turbulence Modelling KRISTOFFER LINDVALL Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Philosophy on Thursday, February 18, 2021, at 3:00 p.m. in F3, Lindstedsvägen 26, Stockholm. Doctoral Thesis in Electrical Engineering KTH Royal Institute of Technology Stockholm, Sweden 2021 © Kristoffer Lindvall © Jan Scheffel ISBN: 978-91-7873-759-8 TRITA-EECS-AVL-2021:7 Printed by: Universitetsservice US-AB, Sweden 2021 i Abstract Energy comes in two forms; potential energy and kinetic energy. Energy is stored as potential energy and released in the form of kinetic energy. This process of storage and release is the basic strategy of all energy alternatives in use today. This applies to solar, wind, fossil fuels, and the list goes on. Most of these come in diluted and scarce forms allowing only a portion of the energy to be used, which has prompted the quest for the original source, the Sun. As early as 1905 in the work by Albert Einstein on the connection between mass and energy, it has been seen theoretically that energy can be extracted from the process of fusing lighter elements into heavier elements. Later, this process of fusion was discovered to be the very source powering the Sun. Almost a century later, the work continues to make thermonuclear fusion energy a reality. Looking closer at the Sun, we see that it consists of a hot burning gas subject to electromagnetic fields, i.e.
    [Show full text]
  • Solomon Grigoryevich Mikhlin (1908-1990)
    Solomon Grigoryevich Mikhlin (1908-1990) S.G. Mikhlin was born in Kholmech, a Belorussian village, into a Jewish family of modest means: his real name was Zalman Girshevich Mikhlin, and he was the youngest of five children. He graduated from a secondary school in Gomel (Belorussia) in 1923 and entered the Leningrad State Pedagogical Institute, named after Herzen, in 1925. In January 1927 he became a second year student in the Department of Mathematics and Mechanics (MatMekh) of Leningrad State University after passing all the first year examinations without attending any lectures. Sergey Lvovich Sobolev studied in the same class as Mikhlin. Among their university professors were Nikolai Maximovich Günther and Vladimir Ivanovich Smirnov. The latter became Mikhlin's master thesis supervisor: the topic of the thesis, defended in 1929, was the convergence of double power series. In 1930 Mikhlin started his teaching career, working for short periods in several Leningrad institutes. In 1932 he obtained a position at the Seismological Institute of the USSR Academy of Sciences, where he worked till 1941. He was awarded the degree of "Doktor nauk" in Mathematics and Physics in 1935 (equivalent to the Doctor of Science), without having to earn the "Kandidat nauk" degree (equivalent to a Ph.D.), and finally in 1937 he was promoted to the rank of professor. During World War II he was a professor at the State Alma Ata University. In 1944 Mikhlin returned to Leningrad State University as full professor. From 1964 to 1986 he headed the Laboratory of Numerical Methods at the Research Institute of Mathematics and Mechanics of the same university.
    [Show full text]
  • A Non-Standard Finite Element Method Using Boundary Integral Operators
    JOHANNES KEPLER UNIVERSITAT¨ LINZ JKU Technisch-Naturwissenschaftliche Fakult¨at A Non-standard Finite Element Method using Boundary Integral Operators DISSERTATION zur Erlangung des akademischen Grades Doktor im Doktoratsstudium der Technischen Wissenschaften Eingereicht von: DI Clemens Hofreither Angefertigt am: Doktoratskolleg Computational Mathematics Beurteilung: O.Univ.-Prof. Dipl.-Ing. Dr. Ulrich Langer (Betreuung) Prof. Dr. Sergej Rjasanow Linz, Oktober, 2012 Acknowledgments This thesis was created during my employment at the Doctoral Program “Computational Mathematics” (W1214) at the Johannes Kepler University in Linz. First of all I would like to thank my advisor, Prof. Ulrich Langer, for interesting me in the research topic treated in this thesis, for giving me the chance to work on it within the framework of the Doctoral Program, and for his guidance and advice during my work on the thesis. Even before, during my graduate studies and supervising my diploma thesis, he has taught me a lot and shaped my academic career in a significant way. Furthermore, I am very grateful to Clemens Pechstein for the countless hours he has invested in the past four years into discussions which have deepened my insight into many topics. Without him, many results of this thesis would not have been possible. I want to thank all the other past and present students at the Doctoral Program for creating a very pleasant environment in which to discuss and work, as well as for the friendships which have developed during my time at the Program, making this time not only a productive, but also a highly enjoyable one. Furthermore, the director of the Doctoral Program, Prof.
    [Show full text]
  • University of Texas at Arlington Dissertation Template
    BUCKLING ANALYSIS OF THIN PLATES WITH OR WITHOUT A HOLE UNDER ARBITRARY BOUNDARY CONDITIONS USING THE GALERKIN METHOD by SRIDER THIRUPACHOOR COMERICA Presented to the Faculty of the Graduate School of The University of Texas at Arlington in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING THE UNIVERSITY OF TEXAS AT ARLINGTON DECEMBER 2011 Copyright © by Srider Thirupachoor Comerica 2011 All Rights Reserved ACKNOWLEDGEMENTS I am deeply indebted to my supervising professor, Dr. Seiichi Nomura for his guidance, patience, and motivation. I would like to sincerely thank him for his help and guidance throughout my thesis work. I would also like to thank Dr. Dereje Agonafer and Dr. A. Haji-Sheikh for serving on my thesis committee. Finally, I would like to thank my parents, sisters, and all my friends for their immense support and encouragement throughout my Master’s program. November 16, 2011 iii ABSTRACT BUCKLING ANALYSIS OF THIN PLATES WITH OR WITHOUT A HOLE UNDER ARBITRARY BOUNDARY CONDITIONS USING THE GALERKIN METHOD Srider Thirupachoor Comerica, M.S. The University of Texas at Arlington, 2011 Supervising Professor: Dr. Seiichi Nomura This thesis demonstrates how to find the critical buckling load value of thin plates with or without a hole under different boundary conditions using the Galerkin method. The use of symbolic software is essential due to the lengthy computations involved because of the complexity of the problems. Firstly, the lateral deflection of the plate is expressed in a series of polynomials each of which satisfies the given boundary conditions. Then by using the Galerkin method, the coefficients of these polynomials are found and with the help of symbolic algebra system, the matrices for the corresponding eigenvalue problem are built from which the buckling loads (eigenvalues) are determined.
    [Show full text]
  • A Brief Introduction to Pseudo-Spectral Methods: Application to Diffusion Problems Denys Dutykh
    A brief introduction to pseudo-spectral methods: application to diffusion problems Denys Dutykh To cite this version: Denys Dutykh. A brief introduction to pseudo-spectral methods: application to diffusion problems. Doctoral. Curitiba, Brazil. 2016, pp.55. cel-01256472v3 HAL Id: cel-01256472 https://cel.archives-ouvertes.fr/cel-01256472v3 Submitted on 17 Jun 2016 (v3), last revised 13 Feb 2019 (v4) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License Denys Dutykh CNRS–LAMA, Université Savoie Mont Blanc, France A brief introduction to pseudo-spectral methods: application to diffusion problems arXiv.org / hal Last modified: June 17, 2016 A brief introduction to pseudo-spectral methods: application to diffusion problems Denys Dutykh∗ Abstract. The topic of these notes could be easily expanded into a full one-semester course. Nevertheless we shall try to give some flavour along with theoretical bases of spec- tral and pseudo-spectral methods. The main focus is made on Fourier-type discretiza- tions, even if some indications on how to handle non-periodic problems via Tchebyshev and Legendre approaches are made as well.
    [Show full text]
  • Symmetric Galerkin Boundary Element Method Alok Sutradhar · Glaucio H
    Symmetric Galerkin Boundary Element Method Alok Sutradhar · Glaucio H. Paulino · Leonard J. Gray Symmetric Galerkin Boundary Element Method 123 Alok Sutradhar Prof. Glaucio H. Paulino Ohio State University University of Illinois at Department of Surgery Urbana-Champaign 410 W 10th Ave, Columbus OH Newmark Laboratory USA 205 North Mathews Avenue [email protected] Urbana IL 61801-2352 USA [email protected] Leonard J. Gray Oak Ridge National Laboratory POB 2008, Bldg 6012 MS 6367 Oak Ridge TN 37831-6367 [email protected] ISBN: 978-3-540-68770-2 e-ISBN: 978-3-540-68772-6 Library of Congress Control Number: 2008928276 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Printed on acid-free paper 987654321 springer.com
    [Show full text]
  • The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering Lecture Notes in Applied Mathematics and Mechanics
    Lecture Notes in Applied Mathematics and Mechanics 1 Erwin Stein Editor The History of Theoretical, Material and Computational Mechanics - Mathematics Meets Mechanics and Engineering Lecture Notes in Applied Mathematics and Mechanics Volume 1 Series Editors Alexander Mielke, Humboldt-Universität zu Berlin, Berlin, Germany e-mail: [email protected] Bob Svendsen, RWTH Aachen University, Aachen, Germany e-mail: [email protected] For further volumes: http://www.springer.com/series/11915 About this Series The Lecture Notes in Applied Mathematics and Mechanics LAMM are intended for an interdisciplinary readership in the fields of applied mathematics and mechan- ics. This series is published under the auspices of the International Association of Applied Mathematics and Mechanics (IAAMM; German GAMM). Topics of interest include for example focus areas of the IAAMM such as: foun- dations of mechanics, thermodynamics, material theory and modeling, multibody dynamics, structural mechanics, solid mechanics, biomechanics, damage, fracture, multiscale modeling and homogenization, fluid mechanics, gas dynamics, laminar flows and transition, turbulence and reactive flows, interface flows, acoustics, waves, applied analysis, mathematical modeling, calculus of variations, variational princi- ples applied operator theory, evolutionary equations, applied stochastics, systems with uncertainty, dynamical systems, control theory, optimization, applied and nu- merical linear algebra, analysis and numerics of ordinary and partial differential equations. Each contribution to the series is intended to be accessible to researchers in math- ematics and mechanics and is written in English. The aim of the series is to provide introductory texts for modern developments in applied mathematics and mechan- ics contributing to cross-fertilization. The Lecture Notes are aimed at researchers as well as advanced masters and PhD students in both mechanics and mathematics.
    [Show full text]
  • Computational Framework for Particle and Spin Simulations Based on the Stochastic Galerkin Method
    PHYSICAL REVIEW E 96, 063301 (2017) Computational framework for particle and spin simulations based on the stochastic Galerkin method J. Slim,1 F. Rathmann,2,* and D. Heberling1,3 1Institut für Hochfrequenztechnik, RWTH Aachen University, 52056 Aachen, Germany 2Institut für Kernphysik, Forschungszentrum Jülich, 52425 Jülich, Germany 3JARA–FAME (Forces and Matter Experiments), Forschungszentrum Jülich and RWTH Aachen University, Aachen, Germany (Received 3 September 2017; published 4 December 2017) An implementation of the polynomial chaos expansion is introduced as a fast solver of the equations of beam and spin motion of charged particles in electromagnetic fields. We show that, based on the stochastic Galerkin method, our computational framework substantially reduces the required number of tracking calculations compared to the widely used Monte Carlo method. DOI: 10.1103/PhysRevE.96.063301 I. INTRODUCTION AND MOTIVATION mechanical uncertainties [11]. This investigation made use of the so-called polynomial chaos expansion (PCE) [12]asan The aim of the JEDI1 collaboration is to measure the electric efficient and yet accurate alternative to the MC method. dipole moment (EDM) of charged hadronic particles, such as In this paper, we describe a computational framework deuterons and protons. In the near future, a first direct EDM based on the stochastic Galerkin method to determine the measurement of the deuteron [1,2] will be carried out at the solutions of the equations of beam and spin motion. This COoler SYnchrotron COSY [3,4]. Toward that end, an RF approach substantially reduces the required number of tracking Wien filter [5] has recently been installed, which is operated simulations, but we still arrive at the same result as that of a at some harmonic of the spin precession frequency, whereby standard MC simulation.
    [Show full text]
  • Numerical and Analytical Aspects of POD-Based Reduced-Order Modeling in Computational Fluid Dynamics
    Numerical and Analytical Aspects of POD-Based Reduced-Order Modeling in Computational Fluid Dynamics Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) am Fachbereich Mathematik und Informatik der Freien Universit¨atBerlin von Swetlana Giere Berlin Juli 2016 1. Gutachter: Prof. Dr. Volker John, Freie Universit¨atBerlin und Weierstraß-Institut f¨urAngewandte Analysis und Stochastik, Berlin 2. Gutachter: Dr. Gabriel Barrenechea, University of Strathclyde, Glasgow Datum der Disputation: 19. Oktober 2016 Abstract This thesis studies projection-based reduced-order models (ROMs) in the context of computational fluid dynamics. Proper Orthogonal Decomposition (POD) is employed to compute the reduced-order basis from snapshots, which are assumed to represent the finite element solution of a partial differential equation. All investigations involve either convection-diffusion-reaction equation or the incompressible Navier{Stokes equations. The main contribution of the thesis can be divided into three parts. Firstly, a Streamline-Upwind Petrov{Galerkin reduced-order model (SUPG-ROM) is investigated theoretically and numerically for convection-dominated convection-diffusion- reaction equations. Numerical analysis is utilized to propose the scaling of the stabi- lization parameter for the SUPG-ROM. Two approaches are used: One based on the underlying finite element discretization and the other one based on the POD truncation. The resulting SUPG-ROMs and the standard Galerkin ROM are studied numerically on several convection-dominated test problems aiming at answering several questions. One of the choices for the stabilization parameter is recommended. Secondly, an alternative approach for the computation of the ROM initial condition is derived for problems, for which the standard approach, that is usually used in the liter- ature, results in the ROM initial condition being polluted by spurious oscillations.
    [Show full text]