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A Meshless Approach to Solving Partial Differential Equations Using the Finite Cloud Method for the Purposes of Computer Aided Design

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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. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. Canada Copyright © 2013 - Daniel Rutherford Burke, B.Eng ii Abstract Modelling tools which are able to solve partial differential equations with increasing accuracy, complexity and ease of use are essential for engineers. Two main methods of solving these types of problems are the Finite Difference Method, and the Finite Element Method, both of which rely on a mesh to discretize the domain or solution space. These meshed methods are widely used and studied. However, they suffer from a variety of problems related to their construction and rigidity. A third type of solution, known as meshless or meshfree methods, are able to avoid meshing prob­ lems and are currently being heavily researched. In this thesis a promising type of meshless method, the Finite Cloud Method, is investigated and a ‘C’ program imple­ menting the method has been written. The method is applied to a range of problems including scalar and vectorial equations, coupled field and both time independent and time dependent solutions. In particular, the method is extended to include inhomo- geneous domains (multiple materials) and spatially dependent material parameters. Physical situations addressed include: Heat Flow, Schrodinger’s Equation, Maxwell’s Equations and optical mode solving. Results for the various equation types have been very promising and in high agreement with both analytically and numerically solved solutions. As well, several improvements to the method have been developed and are detailed. The method is shown to be versatile, robust and highly accurate. Acknowledgments I wish to thank, first and foremost, my supervisor prof.,Tom Smy, without whom this thesis would not have been possible. His patience and clarity when explaining concepts from the very simple to the overly complicated were invaluable to my re­ search and progress throughout this work. He provided encouragement and insights when I was stuck on theoretical roadblocks, coding bugs or general thesis malaise, many times giving me the drive to continue on. His confidence in me gave me the confidence to pursue this degree and tackle every new challenge knowing that it could be surmounted. I am indebted to the other professors and graduate students in the department with whom I spent far too many hours discussing our current stumbling blocks, solving the world’s problems, or generally providing entertaining and often thought provoking distractions from our research. I thank the office staff including Blazenka, Anna and Sylvie for always being impossibly helpful with any questions and humouring me when I was unable to fill out forms on my own. I would also like to thank Nagui and Scott for sharing their technical expertise as well as providing more powerful servers when my simulations became too large and onerous for a laptop. Lastly, it is with great pleasure that I thank my family, friends and roommates for all of their support throughout this process. Thank you for your understanding when I was too busy, stressed or frustrated to go out or see you. Thank you for your encouragement and confidence when I didn’t think this could be done. Thank you iv for listening and pretending to care when I was venting about computer problems, debugging errors or describing my current impasses. Thank you. Table of Contents Abstract iii Acknowledgments iv Table of Contents vi List of Tables xi List of Figures xiv List of Acronyms xxiv List of Symbols xxv 1 Introduction 1 1.1 Partial differential equations ...................................................................... 1 1.2 Solving PDEs with a m esh ......................................................................... 4 1.2.1 Finite difference m e th o d ............................................................... 4 1.2.2 Finite element m e th o d ................................................................... 6 1.2.3 Boundary element m e th o d ............................................................ 8 1.2.4 Problems with m e s h e s ................................................................... 9 1.3 A meshless alternative ................................................................................ 10 1.4 Objectives and contributions ...................................................................... 12 2 Meshless methods 15 2.1 Meshless advantages .................................................................................. 16 2.2 Solution formulations .................................................................................. 17 2.3 Weak and strong formulation procedures ............................................... 18 2.4 Shape function approximation sch em es .................................................. 19 2.4.1 Smoothed particle hydrodynamics ................................................. 21 2.4.2 Moving least squares and kernel m ethods .................................... 24 3 Finite cloud method 26 3.1 Formulation ................................................................................................... 26 3.1.1 Derivatives of the shape fu n c tio n ................................................. 31 3.1.2 Kernel function & kernel fixing .................................................... 32 3.2 Previous FCM w ork ...................................................................................... 33 3.3 Diffusion exam ple ......................................................................................... 34 4 Improvements and initial tests 38 4.1 Improvements in implementation ............................................................ 38 4.1.1 Modified FCM ................................................................................. 38 4.1.2 Node Scaling .................................................................................... 44 4.1.3 Node scaling ....................................................................................... 44 4.1.4 Cloud fixing ....................................................................................... 46 4.1.5 Solution interpolation ........................................................................ 48 4.1.6 Multi-threading ................................................................................. 49 4.2 Im plem entation ............................................................................................ 50 4.2.1 Basic geometry element creation .................................................... 51 4.2.2 Merging s h a p e s ................................................................................. 54 4.2.3 Using Atar models for point generation ....................................... 57 4.2.4 Atar build of large optical model ................................................. 60 vii 4.3 Initial tests .................................................................................................. 64 4.3.1 1-D Poisson eq u atio n ....................................................................... 64 4.3.2 2-D Poisson eq u atio n ....................................................................... 67 5 Thermal diffusion models 72 5.1 Materially inhomogeneous F C M ..............................................................
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