Eighty Years of the Finite Element Method: Birth, Evolution, and Future
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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. -
Meshfree Methods Chapter 1 — Part 1: Introduction and a Historical Overview
MATH 590: Meshfree Methods Chapter 1 — Part 1: Introduction and a Historical Overview Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2014 [email protected] MATH 590 – Chapter 1 1 Outline 1 Introduction 2 Some Historical Remarks [email protected] MATH 590 – Chapter 1 2 Introduction General Meshfree Methods Meshfree Methods have gained much attention in recent years interdisciplinary field many traditional numerical methods (finite differences, finite elements or finite volumes) have trouble with high-dimensional problems meshfree methods can often handle changes in the geometry of the domain of interest (e.g., free surfaces, moving particles and large deformations) better independence from a mesh is a great advantage since mesh generation is one of the most time consuming parts of any mesh-based numerical simulation new generation of numerical tools [email protected] MATH 590 – Chapter 1 4 Introduction General Meshfree Methods Applications Original applications were in geodesy, geophysics, mapping, or meteorology Later, many other application areas numerical solution of PDEs in many engineering applications, computer graphics, optics, artificial intelligence, machine learning or statistical learning (neural networks or SVMs), signal and image processing, sampling theory, statistics (kriging), response surface or surrogate modeling, finance, optimization. [email protected] MATH 590 – Chapter 1 5 Introduction General Meshfree Methods Complicated Domains Recent paradigm shift in numerical simulation of fluid -
A Meshless Approach to Solving Partial Differential Equations Using the Finite Cloud Method for the Purposes of Computer Aided Design
A Meshless Approach to Solving Partial Differential Equations Using the Finite Cloud Method for the Purposes of Computer Aided Design by Daniel Rutherford Burke, B.Eng A Thesis submitted to the Faculty of Graduate and Post Doctoral Affairs in partial fulfilment of the requirements for the degree of Doctor of Philosophy Ottawa Carleton Institute for Electrical and Computer Engineering Department of Electronics Carleton University Ottawa, Ontario, Canada January 2013 Library and Archives Bibliotheque et Canada Archives Canada Published Heritage Direction du 1+1 Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-94524-7 Our file Notre reference ISBN: 978-0-494-94524-7 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distrbute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. -
A Plane Wave Method Based on Approximate Wave Directions for Two
A PLANE WAVE METHOD BASED ON APPROXIMATE WAVE DIRECTIONS FOR TWO DIMENSIONAL HELMHOLTZ EQUATIONS WITH LARGE WAVE NUMBERS QIYA HU AND ZEZHONG WANG Abstract. In this paper we present and analyse a high accuracy method for computing wave directions defined in the geometrical optics ansatz of Helmholtz equation with variable wave number. Then we define an “adaptive” plane wave space with small dimensions, in which each plane wave basis function is determined by such an approximate wave direction. We establish a best L2 approximation of the plane wave space for the analytic solutions of homogeneous Helmholtz equa- tions with large wave numbers and report some numerical results to illustrate the efficiency of the proposed method. Key words. Helmholtz equations, variable wave numbers, geometrical optics ansatz, approximate wave direction, plane wave space, best approximation AMS subject classifications. 65N30, 65N55. 1. Introduction In this paper we consider the following Helmholtz equation with impedance boundary condition u = (∆ + κ2(r))u(ω, r)= f(ω, r), r = (x, y) Ω, L − ∈ (1.1) ((∂n + iκ(r))u(ω, r)= g(ω, r), r ∂Ω, ∈ where Ω R2 is a bounded Lipchitz domain, n is the out normal vector on ∂Ω, ⊂ f L2(Ω) is the source term and κ(r) = ω , g L2(∂Ω). In applications, ω ∈ c(r) ∈ denotes the frequency and may be large, c(r) > 0 denotes the light speed, which is arXiv:2107.09797v1 [math.NA] 20 Jul 2021 usually a variable positive function. The number κ(r) is called the wave number. Helmholtz equation is the basic model in sound propagation. -
National Academy of Sciences July 1, 1979 Officers
NATIONAL ACADEMY OF SCIENCES JULY 1, 1979 OFFICERS Term expires President-PHILIP HANDLER June 30, 1981 Vice-President-SAUNDERS MAC LANE June 30, 1981 Home Secretary-BRYCE CRAWFORD,JR. June 30, 1983 Foreign Secretary-THOMAS F. MALONE June 30, 1982 Treasurer-E. R. PIORE June 30, 1980 Executive Officer Comptroller Robert M. White David Williams COUNCIL Abelson, Philip H. (1981) Markert,C. L. (1980) Berg, Paul (1982) Nierenberg,William A. (1982) Berliner, Robert W. (1981) Piore, E. R. (1980) Bing, R. H. (1980) Ranney, H. M. (1980) Crawford,Bryce, Jr. (1983) Simon, Herbert A. (1981) Friedman, Herbert (1982) Solow, R. M. (1980) Handler, Philip (1981) Thomas, Lewis (1982) Mac Lane, Saunders (1981) Townes, Charles H. (1981) Malone, Thomas F. (1982) Downloaded by guest on September 30, 2021 SECTIONS The Academyis divided into the followingSections, to which membersare assigned at their own choice: (11) Mathematics (31) Engineering (12) Astronomy (32) Applied Biology (13) Physics (33) Applied Physical and (14) Chemistry Mathematical Sciences (15) Geology (41) Medical Genetics Hema- (16) Geophysics tology, and Oncology (21) Biochemistry (42) Medical Physiology, En- (22) Cellularand Develop- docrinology,and Me- mental Biology tabolism (23) Physiological and Phar- (43) Medical Microbiology macologicalSciences and Immunology (24) Neurobiology (51) Anthropology (25) Botany (52) Psychology (26) Genetics (53) Social and Political Sci- (27) Population Biology, Evo- ences lution, and Ecology (54) Economic Sciences In the alphabetical list of members,the numbersin parentheses, followingyear of election, indicate the respective Class and Section of the member. CLASSES The members of Sections are grouped in the following Classes: I. Physical and Mathematical Sciences (Sections 11, 12, 13, 14, 15, 16). -
(PFEM) for Solid Mechanics
Development and preliminary evaluation of the main features of the Particle Finite Element Method (PFEM) for solid mechanics Vasilis Papakrivopoulos Development and preliminary evaluation of the main features of the Particle Finite Element Method (PFEM) for solid mechanics by Vasilis Papakrivopoulos to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Monday December 10, 2018 at 2:00 PM. Student number: 4632567 Project duration: February 15, 2018 – December 10, 2018 Thesis committee: Dr. P.J.Vardon, TU Delft (chairman) Prof. dr. M.A. Hicks, TU Delft Dr. F.Pisanò, TU Delft J.L. Gonzalez Acosta, MSc, TU Delft (daily supervisor) An electronic version of this thesis is available at http://repository.tudelft.nl/. PREFACE This work concludes my academic career in the Technical University of Delft and marks the end of my stay in this beautiful town. During the two-year period of my master stud- ies I managed to obtain experiences and strengthen the theoretical background in the civil engineering field that was founded through my studies in the National Technical University of Athens. I was given the opportunity to come across various interesting and challenging topics, with this current project being the highlight of this course, and I was also able to slowly but steadily integrate into the Dutch society. In retrospect, I feel that I have made the right choice both personally and career-wise when I decided to move here and study in TU Delft. At this point, I would like to thank the graduation committee of my master thesis. -
The Mortar Element Method and the FETI Method. Catherine Lacour, Yvon Maday
Two different approaches for matching nonconforming grids: the mortar element method and the FETI method. Catherine Lacour, Yvon Maday To cite this version: Catherine Lacour, Yvon Maday. Two different approaches for matching nonconforming grids: the mortar element method and the FETI method.. BIT Numerical Mathematics, Springer Verlag, 1997, 37 (3), pp.720 – 738. 10.1007/BF02510249. hal-00369517 HAL Id: hal-00369517 https://hal.archives-ouvertes.fr/hal-00369517 Submitted on 20 Mar 2009 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. BIT 37:3 (1997), 720-738. TWO DIFFERENT APPROACHES FOR MATCHING NONCONFORMING GRIDS: THE MORTAR ELEMENT METHOD AND THE FETI METHOD * C. LACOUR I and Y. MADAY 1'2 10NERA, DI, 29, Avenue de la Division Leclerc F-92322, Chatillon Cedex, France. email: [email protected] 2ASCI, Batiment 506, Universitd Paris Sud, F-91405 Orsay Cedex France. email: [email protected] Abstract. When using domain decomposition in a finite element framework for the approxi- mation of second order elliptic or parabolic type problems, it has become appealing to tune the mesh of each subdomain to the local behaviour of the solution. The resulting discretization being then nonconforming, different approaches have been advocated to match the admissible discrete functions. -
Development of a Coupling Approach for Multi-Physics Analyses of Fusion Reactors
Development of a coupling approach for multi-physics analyses of fusion reactors Zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften (Dr.-Ing.) bei der Fakultat¨ fur¨ Maschinenbau des Karlsruher Instituts fur¨ Technologie (KIT) genehmigte DISSERTATION von Yuefeng Qiu Datum der mundlichen¨ Prufung:¨ 12. 05. 2016 Referent: Prof. Dr. Stieglitz Korreferent: Prof. Dr. Moslang¨ This document is licensed under the Creative Commons Attribution – Share Alike 3.0 DE License (CC BY-SA 3.0 DE): http://creativecommons.org/licenses/by-sa/3.0/de/ Abstract Fusion reactors are complex systems which are built of many complex components and sub-systems with irregular geometries. Their design involves many interdependent multi- physics problems which require coupled neutronic, thermal hydraulic (TH) and structural mechanical (SM) analyses. In this work, an integrated system has been developed to achieve coupled multi-physics analyses of complex fusion reactor systems. An advanced Monte Carlo (MC) modeling approach has been first developed for converting complex models to MC models with hybrid constructive solid and unstructured mesh geometries. A Tessellation-Tetrahedralization approach has been proposed for generating accurate and efficient unstructured meshes for describing MC models. For coupled multi-physics analyses, a high-fidelity coupling approach has been developed for the physical conservative data mapping from MC meshes to TH and SM meshes. Interfaces have been implemented for the MC codes MCNP5/6, TRIPOLI-4 and Geant4, the CFD codes CFX and Fluent, and the FE analysis platform ANSYS Workbench. Furthermore, these approaches have been implemented and integrated into the SALOME simulation platform. Therefore, a coupling system has been developed, which covers the entire analysis cycle of CAD design, neutronic, TH and SM analyses. -
Name Cit Degree(S) Position at Time of Award Year(S)
Distinguished Alumni Awards (Alphabetical Listing) YEAR(S) AWARD NAME CIT DEGREE(S) POSITION AT TIME OF AWARD RECEIVED Fred Champion Professor Emeritus of Civil Engineering, Dr. Mihran S. Agbabian MS 1948 CE 2000 University of Southern California Dr. Bruce N. Ames PhD 1953 BI Professor/Biochemistry, University of California, Berkeley 1977 Assistant Director, Science, Information and Natural BS 1955 PH Resources Dr. John P. Andelin, Jr. 1991 PhD 1967 PH Office of Technology Assessment, Congress of the United States Mr. Moshe Arens MS 1953 ME President, Cybernetics, Inc.Savyon, Israel 1980 Former Director, Observatories of the Carnegie Institute of Dr. Horace Babcock BS 1934 CE 1994 Washington Dr. William F. Ballhaus PhD 1947 AE President, Beckman Instruments, Inc. 1978 YEAR(S) AWARD NAME CIT DEGREE(S) POSITION AT TIME OF AWARD RECEIVED Dr. Mary Baker PhD 1972 AME President, ATA Engineering 2014 Dr. Arnold O. Beckman PhD 1928 CH Chairman, Beckman Instruments, Inc. 1984 Physicist, Group Leader, Janelia Research Campus, Howard Dr. Eric Betzig BS 1983 PH 2016 Hughes Medical Institute Mr. Frank Borman MS 1957 AE Colonel, United States Air Force 1966 Dr. James Boyd BS 1927 EEC President, Cooper Range Company 1966 MS 1963 EE Dr. Robert W. Bower Professor, University of California, Davis 2001 PhD 1973 APH Professor and Head, Inorganic Materials Research, University Dr. Leo Brewer BS 1940 CH 1974 of California, Berkeley YEAR(S) AWARD NAME CIT DEGREE(S) POSITION AT TIME OF AWARD RECEIVED IBM Fellow, IBM Almaden Research Center, San Jose, CA. Dr. Richard G. Brewer BS 1951 CH 1994 Consulting Professor of Applied Physics, Stanford University MS 1949 AE Pigott Professor of Engineering, Department of Aeronautics Dr. -
Overview of Meshless Methods
Technical Article Overview of Meshless Methods Abstract— This article presents an overview of the main develop- ments of the mesh-free idea. A review of the main publications on the application of the meshless methods in Computational Electromagnetics is also given. I. INTRODUCTION Several meshless methods have been proposed over the last decade. Although these methods usually all bear the generic label “meshless”, not all are truly meshless. Some, such as those based on the Collocation Point technique, have no associated mesh but others, such as those based on the Galerkin method, actually do Fig. 1. EFG background cell structure. require an auxiliary mesh or cell structure. At the time of writing, the authors are not aware of any proposed formal classification of these techniques. This paper is therefore not concerned with any C. The Element-Free Galerkin (EFG) classification of these methods, instead its objective is to present In 1994 Belytschko and colleagues introduced the Element-Free an overview of the main developments of the mesh-free idea, Galerkin Method (EFG) [8], an extended version of Nayroles’s followed by a review of the main publications on the application method. The Element-Free Galerkin introduced a series of im- of meshless methods to Computational Electromagnetics. provements over the Diffuse Element Method formulation, such as II. MESHLESS METHODS -THE HISTORY • Proper determination of the approximation derivatives: A. The Smoothed Particle Hydrodynamics In DEM the derivatives of the approximation function The advent of the mesh free idea dates back from 1977, U h are obtained by considering the coefficients b of the with Monaghan and Gingold [1] and Lucy [2] developing a polynomial basis p as constants, such that Lagrangian method based on the Kernel Estimates method to h T model astrophysics problems. -
A Bddc Algorithm for the Stokes Problem with Weak 2 Galerkin Discretizations
1 A BDDC ALGORITHM FOR THE STOKES PROBLEM WITH WEAK 2 GALERKIN DISCRETIZATIONS 3 XUEMIN TU∗ AND BIN WANGy 4 Abstract. The BDDC (balancing domain decomposition by constraints) methods have been 5 applied successfully to solve the large sparse linear algebraic systems arising from conforming finite 6 element discretizations of second order elliptic and Stokes problems. In this paper, the Stokes 7 equations are discretized using the weak Galerkin method, a newly developed nonconforming finite 8 element method. A BDDC algorithm is designed to solve the linear system such obtained. Edge/face 9 velocity interface average and mean subdomain pressure are selected for the coarse problem. The 10 condition number bounds of the BDDC preconditioned operator are analyzed, and the same rate 11 of convergence is obtained as for conforming finite element methods. Numerical experiments are 12 conducted to verify the theoretical results. 13 Key words. Discontinuous Galerkin, HDG, weak Galerkin, domain decomposition, BDDC, 14 Stokes, Saddle point problems, benign subspace 15 AMS subject classifications. 65F10, 65N30, 65N55 16 1. Introduction. Numerical solution of saddle point problems using non over- 17 lapping domain decomposition methods have long been an active area of research; see, 18 e.g., [28, 15, 11, 10, 18, 29, 30, 16, 33, 17, 34, 27]. The Balancing Domain Decomposi- 19 tion by Constraints (BDDC) algorithm is an advanced variant of the non-overlapping 20 domain decomposition technique. It was first introduced by Dohrmann [5], and the 21 theoretical analysis was later given by Mandel and Dohrmann [20]. In this theoretical 22 development, optimal condition number bound was obtained for the BBDC opera- 23 tors proposed for symmetric positive definite systems. -
Computational Fluid and Solid Mechanics
COMPUTATIONAL FLUID AND SOLID MECHANICS Proceedings First MIT Conference on Computational Fluid and Solid Mechanics June 12-15,2001 Editor: K.J. Bathe Massachusetts Institute of Technology, Cambridge, MA, USA VOLUME 2 UNIVERSITATSBIBLIOTHEK HANNOVER TECHKUSCHE INFORMATIONSBIBLIOTHEK 2001 ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo Contents Volume 2 Preface v Session Organizers vi Fellowship Awardees vii Sponsors ix Fluids Achdou, Y., Pironneau, O., Valentin, E, Comparison of wall laws for unsteady incompressible Navier-Stokes equations over rough interfaces 762 Allik, H., Dees, R.N., Oppe, T.C., Duffy, D., Dual-level parallelization of structural acoustics computations 764 Altai, W., Chu, V, K-e Model simulation by Lagrangian block method 767 Alves, M.A., Oliveira, P.J., Pinho, F.T., Numerical simulations of viscoelastic flow around sharp corners 772 Badeau,A., CelikJ., A droplet formation model for stratified liquid-liquid shear flows 776 Balage, S., Saghir, M.Z., Buoyancy and Marangoni convections of Te-doped GaSb 779 Bauer, A.C., Patra, A.K., Preconditioners for parallel adaptive hp FEM for incompressible flows 782 Berger, S.A., Stroud, J.S., Flow in sclerotic carotid arteries 786 Bouhairie, S., Chu, V.H., Gehr, R., Heat transfer calculations of high-Reynolds-number flows around a circular cylinder 791 Cabral, E.L.L., Sabundjian, G, Hierarchical expansion method in the solution of the Navier-Stokes equations for incompressible fluids in laminar two-dimensional flow 795 Chaidron, G., Chinesta, E, On the