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Block-Based Adaptive Mesh Refinement Finite-Volume Scheme for Hybrid Multi-Block Meshes

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Block-Based Adaptive Mesh Refinement Finite-Volume Scheme for Hybrid Multi-Block Meshes Block-Based Adaptive Mesh Refinement Finite-Volume Scheme for Hybrid Multi-Block Meshes by Jason Z.X. Zheng A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science Graduate Department of Aerospace Engineering University of Toronto Copyright © 2012 by Jason Z.X. Zheng Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-92937-7 Our file Notre référence ISBN: 978-0-494-92937-7 NOTICE: AVIS: The author has granted a non- L'auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque 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 télécommunication ou par l'Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distrbute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriété du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette thèse. Ni thesis. Neither the thesis nor la thèse ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent être imprimés ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformément à la loi canadienne sur la Privacy Act some supporting forms protection de la vie privée, quelques may have been removed from this formulaires secondaires ont été enlevés de thesis. cette thèse. 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. Abstract Block-Based Adaptive Mesh Refinement Finite-Volume Scheme for Hybrid Multi-Block Meshes Jason Z.X. Zheng Masters of Applied Science Graduate Department of Aerospace Engineering University of Toronto 2012 A block-based adaptive mesh refinement (AMR) finite-volume scheme is developed for solution of hyperbolic conservation laws on two-dimensional hybrid multi-block meshes. A Godunov- type upwind finite-volume spatial-discretization scheme, with piecewise limited linear recon- struction and Riemann-solver based flux functions, is applied to the quadrilateral cells of a hybrid multi-block mesh and these computational cells are embedded in either body-fitted structured or general unstructured grid partitions of the hybrid grid. A hierarchical quadtree data structure is used to allow local refinement of the individual subdomains based on heuris- tic physics-based refinement criteria. An efficient and scalable parallel implementation of the proposed algorithm is achieved via domain decomposition. The performance of the proposed scheme is demonstrated through application to solution of the compressible Euler equations for a number of flow configurations and regimes in two space dimensions. The efficiency of the AMR procedure and accuracy, robustness, and scalability of the hybrid mesh scheme are assessed. ii Acknowledgements I would like to express my deepest gratitude to a number of people who have guided and supported me throughout my M.A.Sc study. Without them, the completion of this thesis would not have been possible. First of all, I would like to thank my family for the unconditional support they have offered me throughout my career. They are the source of my movitation and keep me striving for the better. Without them, I wouldn't have been able to achieve so much. I would also like to thank Professor C. P. T. Groth for his insightful guidance and financial support throughout this work. For that, I am truly grateful. A special thanks also goes out to Professor J. J. Gottlieb for his inspirational advices, which have made all the differences in my life. An extended thank-you goes to Marc Charest for taking the time to help me out whenever I am stuck in my research. Your help is deeply appreciated. It should be acknowledged that computational resources for performing all of the calculations reported herein were provided by the SciNet High Performance Computing Consortium at the University of Toronto and Compute/Calcul Canada through funding from the Canada Foundation for Innovation (CFI) and the Province of Ontario, Canada. Finally, I would like to thank my friends, colleagues, and everyone from the Institute for the amazing experience in this graduate study! Toronto, January 2012 Jason Z.X. Zheng iii Contents Abstract ii Acknowledgments iii Contents iv List of Tables viii List of Figures ix 1 Introduction 1 1.1 Background and Motivation..............................2 1.1.1 Adaptive Mesh Refinement on Structured Mesh...............2 1.1.2 Adaptive Mesh Refinement on Unstructured Mesh.............5 1.1.3 Hybrid Mesh and Adaptive Mesh Refinement................6 1.2 Scope of Present Study.................................7 1.3 Overview of Thesis...................................8 2 Godunov-Type Finite-Volume Methods9 2.1 Conservation Equations................................9 2.1.1 Navier-Stokes Equations for a Cartesian Coordinate Frame........ 10 2.1.2 Euler Equations for a Cartesian Coordinate Frame............. 11 iv 2.1.3 Euler Equations in Two Space Dimensions.................. 11 2.1.4 Thermodynamic Relations........................... 12 2.2 Euler Equations in One Space Dimension...................... 13 2.3 Godunov-Type Finite-Volume Schemes........................ 13 2.3.1 Solution Update: Godunov's First-Order Scheme.............. 14 2.4 Flux Evaluation..................................... 14 2.4.1 The Riemann Problem............................. 15 2.4.2 Riemann Solvers................................ 15 2.4.3 Comparison of Riemann Solvers....................... 18 2.5 Second-Order Godunov-Type Finite-Volume Schemes................ 19 2.5.1 Godunov's Theorem.............................. 21 2.5.2 Piecewise Linear Reconstruction....................... 21 2.5.3 Enforcing Monotonicity: Slope Limiters................... 22 2.6 Time Marching Schemes................................ 25 2.6.1 Semi-Discrete Form.............................. 25 2.6.2 Runge-Kutta Methods............................. 26 2.6.3 Courant Number................................ 27 2.7 Godunov-Type Finite-Volume Schemes in Multi-Dimensions.................................... 27 2.7.1 Two-Dimensional Example Solution..................... 29 3 Hybrid Mesh 31 3.1 Hybrid Mesh Generation................................ 34 3.1.1 Multi-Block Structured Mesh......................... 34 3.1.2 Unstructured Mesh............................... 34 3.1.3 Gmsh...................................... 35 v 3.2 Domain Decomposition and Multi-Block Partitioning................ 36 3.2.1 Metis...................................... 38 3.3 Godunov-Type Finite-Volume Scheme on Hybrid Mesh............... 41 3.3.1 Data Structure and Cell Connectivity.................... 41 3.3.2 Example Hybrid Mesh: Supersonic Flow Past a Circular Cylinder.... 42 4 Parallel AMR for Multi-Block Hybrid Meshes 44 4.1 Overview of Parallel AMR Scheme.......................... 44 4.2 Refinement and Coarsening of Solution Blocks................... 45 4.2.1 Refined Grid Generation for Unstructured Mesh Blocks.......... 46 4.2.2 Coarse Grid Generation for Unstructured Mesh Blocks........... 47 4.2.3 Refinement Criteria.............................. 47 4.2.4 Example: Refinement of an Unstructured Mesh............... 49 4.3 Solution Block Connectivity.............................. 49 4.3.1 Hierarchical Tree Data Structure....................... 49 4.3.2 Computation of Hybrid Root-Block Connectivity.............. 52 4.4 Exchange of Solution Information Between Blocks................. 52 4.4.1 Ghost-Cell Structure.............................. 52 4.4.2 Conservative Flux Corrections........................ 53 4.5 Domain Decomposition and Parallel Implementation................ 54 4.5.1 Domain Decomposition............................ 54 4.5.2 Load Balancing................................. 54 5 Numerical Results 56 5.1 Evaluation of AMR Efficiency............................. 57 5.2 Supersonic Channel Flow Past a Bump........................ 57 vi 5.2.1 Unstructured Mesh Results.......................... 57 5.2.2 Hybrid Mesh Results.............................. 59 5.3 Supersonic Flow Over a Circular Cylinder Without AMR............. 59 5.4 Supersonic Flow Past Two Circular Cylinders in a Channel With AMR..... 66 5.4.1 Symmetrical Case............................... 66 5.4.2 Asymmetrical Case............................... 68 6 Conclusions and Future Research 74 6.1 Conclusions....................................... 74 6.2 Future Research.................................... 75 References 86 vii List of Tables 3.1 Example of Edge Type Data Structure........................ 42 5.1 Combinations of structured and unstructured blocks used.............. 61 viii List of Figures 1.1 Illustration of (b) patch-based, (c) cell-based, and (d) block-based AMR tech- niques applied to a base Cartesian mesh (a) with cells flagged
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