Parallel Tetrahedral Mesh Generation
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Review Article Survey Report on Space Filling Curves
International Journal of Modern Science and Technology Vol. 1, No. 8, November 2016. Page 264-268. http://www.ijmst.co/ ISSN: 2456-0235. Review Article Survey Report on Space Filling Curves R. Prethee, A. R. Rishivarman Department of Mathematics, Theivanai Ammal College for Women (Autonomous) Villupuram - 605 401. Tamilnadu, India. *Corresponding author’s e-mail: [email protected] Abstract Space-filling Curves have been extensively used as a mapping from the multi-dimensional space into the one-dimensional space. Space filling curve represent one of the oldest areas of fractal geometry. Mapping the multi-dimensional space into one-dimensional domain plays an important role in every application that involves multidimensional data. We describe the notion of space filling curves and describe some of the popularly used curves. There are numerous kinds of space filling curves. The difference between such curves is in their way of mapping to the one dimensional space. Selecting the appropriate curve for any application requires knowledge of the mapping scheme provided by each space filling curve. Space filling curves are the basis for scheduling has numerous advantages like scalability in terms of the number of scheduling parameters, ease of code development and maintenance. The present paper report on various space filling curves, classifications, and its applications. It elaborates the space filling curves and their applicability in scheduling, especially in transaction. Keywords: Space filling curve, Holder Continuity, Bi-Measure-Preserving Property, Transaction Scheduling. Introduction these other curves, sometimes space-filling In mathematical analysis, a space-filling curves are still referred to as Peano curves. curve is a curve whose range contains the entire Mathematical tools 2-dimensional unit square or more generally an The Euclidean Vector Norm n-dimensional unit hypercube. -
Arxiv:1911.09220V2 [Cs.MS] 13 Jul 2020
MFEM: A MODULAR FINITE ELEMENT METHODS LIBRARY ROBERT ANDERSON, JULIAN ANDREJ, ANDREW BARKER, JAMIE BRAMWELL, JEAN- SYLVAIN CAMIER, JAKUB CERVENY, VESELIN DOBREV, YOHANN DUDOUIT, AARON FISHER, TZANIO KOLEV, WILL PAZNER, MARK STOWELL, VLADIMIR TOMOV Lawrence Livermore National Laboratory, Livermore, USA IDO AKKERMAN Delft University of Technology, Netherlands JOHANN DAHM IBM Research { Almaden, Almaden, USA DAVID MEDINA Occalytics, LLC, Houston, USA STEFANO ZAMPINI King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Abstract. MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of dis- cretization approaches and emphasis on usability, portability, and high-performance computing efficiency. MFEM's goal is to provide application scientists with access to cutting-edge algorithms for high-order finite element mesh- ing, discretizations and linear solvers, while enabling researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel and GPU-accelerated settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library. arXiv:1911.09220v2 [cs.MS] 13 Jul 2020 1. Introduction The Finite Element Method (FEM) is a powerful discretization technique that uses general unstructured grids to approximate the solutions of many partial differential equations (PDEs). It has been exhaustively studied, both theoretically and in practice, in the past several decades [1, 2, 3, 4, 5, 6, 7, 8]. MFEM is an open-source, lightweight, modular and scalable software library for finite elements, featuring arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing (HPC) efficiency [9]. -
Meshing for the Finite Element Method
Meshing for the Finite Element Method Summer Seminar ISC5939 .......... John Burkardt Department of Scientific Computing Florida State University http://people.sc.fsu.edu/∼jburkardt/presentations/. mesh 2012 fsu.pdf 10/12 July 2012 1 / 119 FEM Meshing Meshing Computer Representations The Delaunay Triangulation TRIANGLE DISTMESH MESH2D Files and Graphics 1 2 2 D Problems 3D Problems Conclusion 2 / 119 MESHING: The finite element method begins by looking at a complicated region, and thinking of it as a mesh of smaller, simpler subregions. The subregions are simple, (perhaps triangles) so we understand their geometry; they are small because when we approximate the differential equations, our errors will be related to the size of the subregions. More, smaller subregions usually mean less total error. After we compute our solution, it is described in terms of the mesh. The simplest description uses piecewise linear functions, which we might expect to be a crude approximation. However, excellent results can be obtained as long as the mesh is small enough in places where the solution changes rapidly. 3 / 119 MESHING: Thus, even though the hard part of the finite element method involves considering abstract approximation spaces, sequences of approximating functions, the issue of boundary conditions, weak forms and so on, ...it all starts with a very simple idea: Given a geometric shape, break it into smaller, simpler shapes; fit the boundary, and be small in some places. Since this is such a simple idea, you might think there's no reason to worry about it much! 4 / 119 MESHING: Indeed, if we start by thinking of a 1D problem, such as modeling the temperature along a thin strand of wire that extends from A to B, our meshing problem is trivial: Choose N, the number of subregions or elements; Insert N-1 equally spaced nodes between A and B; Create N elements, the intervals between successive nodes. -
Fractal-Based Magnetic Resonance Imaging Coils for 3T Xenon Imaging Fractal-Based Magnetic Resonance Imaging Coils for 3T Xenon Imaging
Fractal-based magnetic resonance imaging coils for 3T Xenon imaging Fractal-based magnetic resonance imaging coils for 3T Xenon imaging By Jimmy Nguyen A Thesis Submitted to the School of Graduate Studies in thePartial Fulfillment of the Requirements for the Degree Master of Applied Science McMaster University © Copyright by Jimmy Nguyen 10 July 2020 McMaster University Master of Applied Science (2020) Hamilton, Ontario (Department of Electrical & Computer Engineering) TITLE: Fractal-based magnetic resonance imaging coils for 3T Xenon imaging AUTHOR: Jimmy Nguyen, B.Eng., (McMaster University) SUPERVISOR: Dr. Michael D. Noseworthy NUMBER OF PAGES: ix, 77 ii Abstract Traditional 1H lung imaging using MRI faces numerous challenges and difficulties due to low proton density and air-tissue susceptibility artifacts. New imaging techniques using inhaled xenon gas can overcome these challenges at the cost of lower signal to noise ratio. The signal to noise ratio determines reconstructed image quality andis an essential parameter in ensuring reliable results in MR imaging. The traditional RF surface coils used in MR imaging exhibit an inhomogeneous field, leading to reduced image quality. For the last few decades, fractal-shaped antennas have been used to optimize the performance of antennas for radiofrequency systems. Although widely used in radiofrequency identification systems, mobile phones, and other applications, fractal designs have yet to be fully researched in the MRI application space. The use of fractal geometries for RF coils may prove to be fruitful and thus prompts an investiga- tion as the main goal of this thesis. Preliminary simulation results and experimental validation results show that RF coils created using the Gosper and pentaflake offer improved signal to noise ratio and exhibit a more homogeneous field than that ofa traditional circular surface coil. -
Compression and Streaming of Polygon Meshes
Compression and Streaming of Polygon Meshes by Martin Isenburg A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Computer Science. Chapel Hill 2005 Approved by: Jack Snoeyink, Advisor Craig Gotsman, Reader Peter Lindstrom, Reader Dinesh Manocha, Committee Member Ming Lin, Committee Member ii iii ABSTRACT MARTIN ISENBURG: Compression and Streaming of Polygon Meshes (Under the direction of Jack Snoeyink) Polygon meshes provide a simple way to represent three-dimensional surfaces and are the de-facto standard for interactive visualization of geometric models. Storing large polygon meshes in standard indexed formats results in files of substantial size. Such formats allow listing vertices and polygons in any order so that not only the mesh is stored but also the particular ordering of its elements. Mesh compression rearranges vertices and polygons into an order that allows more compact coding of the incidence between vertices and predictive compression of their positions. Previous schemes were designed for triangle meshes and polygonal faces were triangulated prior to compression. I show that polygon models can be encoded more compactly by avoiding the initial triangulation step. I describe two compression schemes that achieve better compression by encoding meshes directly in their polygonal representation. I demonstrate that the same holds true for volume meshes by extending one scheme to hexahedral meshes. Nowadays scientists create polygonal meshes of incredible size. Ironically, com- pression schemes are not capable|at least not on common desktop PCs|to deal with giga-byte size meshes that need compression the most. -
Mesh Compression
Mesh Compression Dissertation der Fakult¨at f¨ur Informatik der Eberhard-Karls-Universit¨at zu T¨ubingen zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) vorgelegt von Dipl.-Inform. Stefan Gumhold aus Tubingen¨ Tubingen¨ 2000 Tag der m¨undlichen Qualifikation: 19.Juli 2000 Dekan: Prof. Dr. Klaus-J¨orn Lange 1. Berichterstatter: Prof. Dr.-Ing. Wolfgang Straßer 2. Berichterstatter: Prof. Jarek Rossignac iii Zusammenfassung Die Kompression von Netzen ist eine weitgef¨acherte Forschungsrichtung mit Anwen- dungen in den verschiedensten Bereichen, wie zum Beispiel im Bereich der Hand- habung extrem großer Modelle, beim Austausch von dreidimensionalem Inhaltuber ¨ das Internet, im elektronischen Handel, als anpassungsf¨ahige Repr¨asentation f¨ur Vo- lumendatens¨atze usw. In dieser Arbeit wird das Verfahren der Cut-Border Machine beschrieben. Die Cut-Border Machine kodiert Netze, indem ein Teilbereich durch das Netz w¨achst (region growing). Kodiert wird die Art und Weise, wie neue Netzele- mente dem wachsenden Teilbereich einverleibt werden. Das Verfahren der Cut-Border Machine kann sowohl auf Dreiecksnetze als auch auf Tetraedernetze angewendet wer- den. Trotz der einfachen Struktur des Verfahrens kann eine sehr hohe Kompression- srate erzielt werden. Im Falle von Tetraedernetzen erreicht die Cut-Border Machine die beste Kompressionsrate von allen bekannten Verfahren. Die einfache Struktur der Cut-Border Machine erm¨oglicht einerseits die Realisierung direkt in Hardware und ist auch als Implementierung in Software extrem schnell. Auf der anderen Seite erlaubt die Einfachheit eine theoretische Analyse des Algorithmus. Gezeigt werden konnte, dass f¨ur ebene Triangulierungen eine leicht modifizierte Version der Cut-Border Machine lineare Laufzeiten in der Zahl der Knoten erzielt und dass die komprimierte Darstellung nur linearen Speicherbedarf ben¨otigt, d.h. -
MFEM: a Modular Finite Element Methods Library
MFEM: A Modular Finite Element Methods Library Robert Anderson1, Andrew Barker1, Jamie Bramwell1, Jakub Cerveny2, Johann Dahm3, Veselin Dobrev1,YohannDudouit1, Aaron Fisher1,TzanioKolev1,MarkStowell1,and Vladimir Tomov1 1Lawrence Livermore National Laboratory 2University of West Bohemia 3IBM Research July 2, 2018 Abstract MFEM is a free, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and emphasis on usability, portability, and high-performance computing efficiency. Its mission is to provide application scientists with access to cutting-edge algorithms for high-order finite element meshing, discretizations and linear solvers. MFEM also enables researchers to quickly and easily develop and test new algorithms in very general, fully unstructured, high-order, parallel settings. In this paper we describe the underlying algorithms and finite element abstractions provided by MFEM, discuss the software implementation, and illustrate various applications of the library. Contents 1 Introduction 3 2 Overview of the Finite Element Method 4 3Meshes 9 3.1 Conforming Meshes . 10 3.2 Non-Conforming Meshes . 11 3.3 NURBS Meshes . 12 3.4 Parallel Meshes . 12 3.5 Supported Input and Output Formats . 13 1 4 Finite Element Spaces 13 4.1 FiniteElements....................................... 14 4.2 DiscretedeRhamComplex ................................ 16 4.3 High-OrderSpaces ..................................... 17 4.4 Visualization . 18 5 Finite Element Operators 18 5.1 DiscretizationMethods................................... 18 5.2 FiniteElementLinearSystems . 19 5.3 Operator Decomposition . 23 5.4 High-Order Partial Assembly . 25 6 High-Performance Computing 27 6.1 Parallel Meshes, Spaces, and Operators . 27 6.2 Scalable Linear Solvers . -
FELICITY: a MATLAB/C++ TOOLBOX for DEVELOPING FINITE ELEMENT METHODS and SIMULATION MODELING\Ast
SIAM J. SCI.COMPUT. \bigcircc 2018 Society for Industrial and Applied Mathematics Vol. 40, No. 2, pp. C234{C257 FELICITY: A MATLAB/C++ TOOLBOX FOR DEVELOPING FINITE ELEMENT METHODS AND SIMULATION MODELING\ast SHAWN W. WALKERy Abstract. This paper describes a MATLAB/C++ finite element toolbox, called FELICITY, for simulating various types of systems of partial differential equations (e.g., coupled elliptic/parabolic problems) using the finite element method. It uses MATLAB in an object-oriented way for high-level manipulation of data structures in finite element codes, while utilizing a domain-specific language (DSL) and code generation to automate low-level tasks such as matrix assembly (via the MATLAB mex interface). We describe the fundamental functionality of the toolbox's MATLAB interface, such as using higher order Lagrange (simplicial) meshes, defining finite element spaces, allocating degrees- of-freedom, assembling discrete bilinear and linear forms, and interpolation over meshes. Moreover, we describe in-depth how automatic code generation is implemented in FELICITY. Two example problems and their implementation are provided to demonstrate the ability of FELICITY to solve coupled problems with interacting subdomains of different co-dimension. Future improvements are also discussed. Key words. finite elements, coupled systems, geometric flows, code generation, MATLAB, open source software AMS subject \bfc \bfl \bfa \bfs \bfifi\bfc\bfa\bft\bfo\bfn. 68N30, 65N30, 65M60, 68N19, 68N20 DOI. 10.1137/17M1128745 1. Introduction. The development of numerical methods to solve partial dif- ferential equations (PDEs) continues to advance to address new domain areas and problems of increasing complexity. With this, the number of software packages has increased to address the needs for researching new methods and simulating large prob- lems. -
Mathematics As “Gate-Keeper” (?): Three Theoretical Perspectives That Aim Toward Empowering All Children with a Key to the Gate DAVID W
____ THE _____ MATHEMATICS ___ ________ EDUCATOR _____ Volume 14 Number 1 Spring 2004 MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA Editorial Staff A Note from the Editor Editor Dear TME readers, Holly Garrett Anthony Along with the editorial team, I present the first of two issues to be produced during my brief tenure as editor of Volume 14 of The Mathematics Educator. This issue showcases the work of both Associate Editors veteran and budding scholars in mathematics education. The articles range in topic and thus invite all Ginger Rhodes those vested in mathematics education to read on. Margaret Sloan Both David Stinson and Amy Hackenberg direct our attention toward equity and social justice in Erik Tillema mathematics education. Stinson discusses the “gatekeeping” status of mathematics, offers theoretical perspectives he believes can change this, and motivates mathematics educators at all levels to rethink Publication their roles in empowering students. Hackenberg’s review of Burton’s edited book, Which Way Social Stephen Bismarck Justice in Mathematics Education? is both critical and engaging. She artfully draws connections across Laurel Bleich chapters and applauds the picture of social justice painted by the diversity of voices therein. Dennis Hembree Two invited pieces, one by Chandra Orrill and the other by Sybilla Beckmann, ask mathematics Advisors educators to step outside themselves and reexamine features of PhD programs and elementary Denise S. Mewborn textbooks. Orrill’s title question invites mathematics educators to consider what we value in classroom Nicholas Oppong teaching, how we engage in and write about research on or with teachers, and what features of a PhD program can inform teacher education. -
New Development in Freefem++
J. Numer. Math., Vol. 20, No. 3-4, pp. 251–265 (2012) DOI 10.1515/jnum-2012-0013 c de Gruyter 2012 New development in freefem++ F. HECHT∗ Received July 2, 2012 Abstract — This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer. Keywords: finite element, mesh adaptation, Schwarz domain decomposition, parallel comput- ing, freefem++ 1. Introduction This paper intends to give a small presentation of the software freefem++. A partial differential equation is a relation between a function of several variables and its (partial) derivatives. Many problems in physics, engineering, mathematics and even banking are modeled by one or several partial differen- tial equations. Freefem++ is a software to solve these equations numerically in dimen- sions two or three. As its name implies, it is a free software based on the Finite Element Method; it is not a package, it is an integrated product with its own high level programming language; it runs on most UNIX, WINDOWS and MacOs computers. -
Mofem: an Open Source, Parallel Finite Element Library
MoFEM: An open source, parallel finite element library Kaczmarczyk, Ł., Ullah, Z., Lewandowski, K., Meng, X., Zhou, X-Y., Athanasiadis, I., Nguyen, H., Chalons- Mouriesse, C-A., Richardson, E. J., Miur, E., Shvarts, A. G., Wakeni, M., & Pearce, C. J. (2020). MoFEM: An open source, parallel finite element library. The Journal of Open Source Software, 5(45). https://doi.org/doi.org/10.21105/joss.01441 Published in: The Journal of Open Source Software Document Version: Publisher's PDF, also known as Version of record Queen's University Belfast - Research Portal: Link to publication record in Queen's University Belfast Research Portal Publisher rights © 2020 The Authors. This is an open access article published under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the author and source are cited. General rights Copyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made to ensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in the Research Portal that you believe breaches copyright or violates any law, please contact [email protected]. Download date:06. -
An XFEM Based Fixed-Grid Approach to Fluid-Structure Interaction
TECHNISCHE UNIVERSITAT¨ MUNCHEN¨ Lehrstuhl fur¨ Numerische Mechanik An XFEM based fixed-grid approach to fluid-structure interaction Axel Gerstenberger Vollstandiger¨ Abdruck der von der Fakultat¨ fur¨ Maschinenwesen der Techni- schen Universitat¨ Munchen¨ zur Erlangung des akademischen Grades eines Doktor-Ingenieurs (Dr.-Ing.) genehmigten Dissertation. Vorsitzender: Univ.-Prof. Dr.-Ing. habil. Nikolaus A. Adams Prufer¨ der Dissertation: 1. Univ.-Prof. Dr.-Ing. Wolfgang A. Wall 2. Prof. Peter Hansbo, Ph.D. University of Gothenburg / Sweden Die Dissertation wurde am 14. 4. 2010 bei der Technischen Universitat¨ Munchen¨ eingereicht und durch die Fakultat¨ fur¨ Maschinenwesen am 21. 6. 2010 ange- nommen. Zusammenfassung Die vorliegende Arbeit behandelt ein Finite Elemente (FE) basiertes Festgit- terverfahren zur Simulation von dreidimensionaler Fluid-Struktur-Interak- tion (FSI) unter Berucksichtigung¨ großer Strukturdeformationen. FSI ist ein oberflachengekoppeltes¨ Mehrfeldproblem, bei welchem Stuktur- und Fluid- gebiete eine gemeinsame Oberflache¨ teilen. In dem vorgeschlagenen Festgit- terverfahren wird die Stromung¨ durch eine Eulersche Betrachtungsweise be- schrieben, wahrend¨ die Struktur wie ublich¨ in Lagrangscher Betrachtungs- weise formuliert wird. Die Fluid-Struktur-Grenzflache¨ kann sich dabei un- abhangig¨ von dem ortsfesten Fluidnetz bewegen, so dass keine Fluidnetzver- formungen auftreten und beliebige Grenzflachenbewegungen¨ moglich¨ sind. Durch die unveranderte¨ Lagrangsche Strukturformulierung liegt der Schwer- punkt der Arbeit