A Non-Standard Finite Element Method Using Boundary Integral Operators
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Coupling Finite and Boundary Element Methods for Static and Dynamic
Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma Coupling finite and boundary element methods for static and dynamic elastic problems with non-conforming interfaces Thomas Rüberg a, Martin Schanz b,* a Graz University of Technology, Institute for Structural Analysis, Graz, Austria b Graz University of Technology, Institute of Applied Mechanics, Technikerstr. 4, 8010 Graz, Austria article info abstract Article history: A coupling algorithm is presented, which allows for the flexible use of finite and boundary element meth- Received 1 February 2008 ods as local discretization methods. On the subdomain level, Dirichlet-to-Neumann maps are realized by Received in revised form 4 August 2008 means of each discretization method. Such maps are common for the treatment of static problems and Accepted 26 August 2008 are here transferred to dynamic problems. This is realized based on the similarity of the structure of Available online 5 September 2008 the systems of equations obtained after discretization in space and time. The global set of equations is then established by incorporating the interface conditions in a weighted sense by means of Lagrange Keywords: multipliers. Therefore, the interface continuity condition is relaxed and the interface meshes can be FEM–BEM coupling non-conforming. The field of application are problems from elastostatics and elastodynamics. Linear elastodynamics Ó Non-conforming interfaces 2008 Elsevier B.V. All rights reserved. FETI/BETI 1. Introduction main, this approach is often carried out only once for every time step which gives a staggering scheme. -
On Multigrid Methods for Solving Electromagnetic Scattering Problems
On Multigrid Methods for Solving Electromagnetic Scattering Problems Dissertation zur Erlangung des akademischen Grades eines Doktor der Ingenieurwissenschaften (Dr.-Ing.) der Technischen Fakultat¨ der Christian-Albrechts-Universitat¨ zu Kiel vorgelegt von Simona Gheorghe 2005 1. Gutachter: Prof. Dr.-Ing. L. Klinkenbusch 2. Gutachter: Prof. Dr. U. van Rienen Datum der mundliche¨ Prufung:¨ 20. Jan. 2006 Contents 1 Introductory remarks 3 1.1 General introduction . 3 1.2 Maxwell’s equations . 6 1.3 Boundary conditions . 7 1.3.1 Sommerfeld’s radiation condition . 9 1.4 Scattering problem (Model Problem I) . 10 1.5 Discontinuity in a parallel-plate waveguide (Model Problem II) . 11 1.6 Absorbing-boundary conditions . 12 1.6.1 Global radiation conditions . 13 1.6.2 Local radiation conditions . 18 1.7 Summary . 19 2 Coupling of FEM-BEM 21 2.1 Introduction . 21 2.2 Finite element formulation . 21 2.2.1 Discretization . 26 2.3 Boundary-element formulation . 28 3 4 CONTENTS 2.4 Coupling . 32 3 Iterative solvers for sparse matrices 35 3.1 Introduction . 35 3.2 Classical iterative methods . 36 3.3 Krylov subspace methods . 37 3.3.1 General projection methods . 37 3.3.2 Krylov subspace methods . 39 3.4 Preconditioning . 40 3.4.1 Matrix-based preconditioners . 41 3.4.2 Operator-based preconditioners . 42 3.5 Multigrid . 43 3.5.1 Full Multigrid . 47 4 Numerical results 49 4.1 Coupling between FEM and local/global boundary conditions . 49 4.1.1 Model problem I . 50 4.1.2 Model problem II . 63 4.2 Multigrid . 64 4.2.1 Theoretical considerations regarding the classical multi- grid behavior in the case of an indefinite problem . -
Coupling of the Finite Volume Element Method and the Boundary Element Method – an a Priori Convergence Result
COUPLING OF THE FINITE VOLUME ELEMENT METHOD AND THE BOUNDARY ELEMENT METHOD – AN A PRIORI CONVERGENCE RESULT CHRISTOPH ERATH∗ Abstract. The coupling of the finite volume element method and the boundary element method is an interesting approach to simulate a coupled system of a diffusion convection reaction process in an interior domain and a diffusion process in the corresponding unbounded exterior domain. This discrete system maintains naturally local conservation and a possible weighted upwind scheme guarantees the stability of the discrete system also for convection dominated problems. We show existence and uniqueness of the continuous system with appropriate transmission conditions on the coupling boundary, provide a convergence and an a priori analysis in an energy (semi-) norm, and an existence and an uniqueness result for the discrete system. All results are also valid for the upwind version. Numerical experiments show that our coupling is an efficient method for the numerical treatment of transmission problems, which can be also convection dominated. Key words. finite volume element method, boundary element method, coupling, existence and uniqueness, convergence, a priori estimate 1. Introduction. The transport of a concentration in a fluid or the heat propa- gation in an interior domain often follows a diffusion convection reaction process. The influence of such a system to the corresponding unbounded exterior domain can be described by a homogeneous diffusion process. Such a coupled system is usually solved by a numerical scheme and therefore, a method which ensures local conservation and stability with respect to the convection term is preferable. In the literature one can find various discretization schemes to solve similar prob- lems. -
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. -
Uncertainties in the Solutions to Boundary Element
UNCERTAINTIES IN THE SOLUTIONS TO BOUNDARY ELEMENT METHOD: AN INTERVAL APPROACH by BARTLOMIEJ FRANCISZEK ZALEWSKI Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Adviser: Dr. Robert L. Mullen Department of Civil Engineering CASE WESTERN RESERVE UNIVERSITY August, 2008 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of ______________Bartlomiej Franciszek Zalewski______________ candidate for the Doctor of Philosophy degree *. (signed)_________________Robert L. Mullen________________ (chair of the committee) ______________Arthur A. Huckelbridge_______________ ______________Xiangwu (David) Zeng_______________ _________________Daniela Calvetti__________________ _________________Shad M. Sargand_________________ (date) _______4-30-2008________ *We also certify that written approval has been obtained for any proprietary material contained therein. With love I dedicate my Ph.D. dissertation to my parents. TABLE OF CONTENTS LIST OF TABLES ………………………………………………………………………. 5 LIST OF FIGURES ……………………………………………………………………… 6 ACKNOWLEDGEMENTS ………………………………………………………..…… 10 LIST OF SYMBOLS ……………………………………………………….……………11 ABSTRACT …………………………………………………………………………...... 13 CHAPTER I. INTRODUCTION ……………………………………………………………… 15 1.1 Background ………………………………………………...………… 15 1.2 Overview ……………………………………………..…………….… 20 1.3 Historical Background ………………………………………….……. 21 1.3.1 Historical Background of Boundary Element Method ……. 21 1.3.2 Historical Background of Interval -
The Boundary Element Method in Acoustics
The Boundary Element Method in Acoustics by Stephen Kirkup 1998/2007 . The Boundary Element Method in Acoustics by S. M. Kirkup First edition published in 1998 by Integrated Sound Software and is published in 2007 in electronic format (corrections and minor amendments on the 1998 print). Information on this and related publications and software (including the second edi- tion of this work, when complete) can be obtained from the web site http://www.boundary-element-method.com The author’s publications can generally be found through the web page http://www.kirkup.info/papers The nine subroutines and the corresponding nine example test programs are available in Fortran 77 on CD ABEMFULL. The original core routines can be downloaded from the web site. The learning package of subroutines ABEM2D is can be downloaded from the web site. See the web site for a full price list and alternative methods of obtaining the software. ISBN 0 953 4031 06 The 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 subject to the permission of the author. c Stephen Kirkup 1998-2007. Preface to 2007 Print Having run out of hard copies of the book a number of years ago, the author is pleased to publish an on-line version of the book in PDF format. A number of minor corrections have been made to the original, following feedback from a number of readers. -
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. -
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. -
A Boundary-Spheropolygon Element Method for Modelling Sub-Particle Stress and Particle Breakage
A Boundary-Spheropolygon Element Method for Modelling Sub-particle Stress and Particle Breakage YUPENG JIANG1, HANS J. HERRMANN2, FERNANDO ALONSO-MARROQUIN*1, 1. School of Civil Engineering, The University of Sydney, Australia 2. Computational Physics for Engineering Materials, ETH Zurich, Switzerland Abstract: We present a boundary-spheropolygon element method that combines the boundary element method (BEM) and the spheropolygon-based discrete element method (SDEM). The interaction between particles is simulated via the SDEM, and the sub-particle stress is calculated by BEM. The framework of BSEM is presented. Then, the accuracy and efficiency of the method are analysed by comparison with both analytical solutions and a well-established finite element method. The results demonstrate that BSEM can efficiently provide instant sub-particle stress for particles with an optimized compromise between computational time and accuracy, thus overcoming most of the disadvantages of existing numerical methods. Keywords: Boundary-Spheropolygon element method; Particle stress state; Particle breakage; Numerical simulation 1. Introduction Particle breakage is ubiquitous for granular materials. Breakage is caused by many reasons, such as strong compressive loading, large shear displacement, extreme temperature, dehydration or chemical effects. Based on laboratory studies, it is concluded that breakage causes changes in particle size distribution [1-4] and particle shape [5,6], which in turn change the state variables, such as density and compressibility, that govern the mechanical behaviour of the bulk granular material. These studies have provided a framework for studying particle breakage. However, due to the limitations in experimental observations, laboratory studies are not able to offer more detailed information about the breakage process. -
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. -
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. -
Optimal Additive Schwarz Preconditioning for Adaptive 2D IGA
OPTIMAL ADDITIVE SCHWARZ PRECONDITIONING FOR ADAPTIVE 2D IGA BOUNDARY ELEMENT METHODS THOMAS FUHRER,¨ GREGOR GANTNER, DIRK PRAETORIUS, AND STEFAN SCHIMANKO Abstract. We define and analyze (local) multilevel diagonal preconditioners for isogeo- metric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the re- finement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results. 1. Introduction In the last decade, the isogeometric analysis (IGA) had a strong impact on the field of scientific computing and numerical analysis. We refer, e.g., to the pioneering work [HCB05] and to [CHB09, BdVBSV14] for an introduction to the field. The basic idea is to utilize the same ansatz functions for approximations as are used for the description of the geometry by some computer aided design (CAD) program. Here, we consider the case, where the geometry is represented by rational splines. For certain problems, where the fundamental solution is known, the boundary element method (BEM) is attractive since CAD programs usually only provide a parametrization of the boundary ∂Ω and not of the volume Ω itself. Isogeometric BEM (IGABEM) has first been considered for 2D BEM in [PGK+09] and for 3D BEM in [SSE+13]. We refer to [SBTR12, PTC13, SBLT13, NZW+17] for numerical experiments, to [HR10, TM12, MZBF15, DHP16, DHK+18, DKSW18] for fast IGABEM based on wavelets, fast multipole, -matrices resp.