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Automatic Higher Order Mesh Generation and Movement Utilizing Spring-Field and Vector-Adding

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AUTOMATIC HIGHER ORDER MESH GENERATION AND MOVEMENT UTILIZING SPRING-FIELD AND VECTOR-ADDING By Tuo Liu C. Bruce Hilbert Kidambi Sreenivas Algorithms/Optimization Associate Professor of Branch Technology Mechanical Engineering (Chair) (Committee Member) James C. Newman III Steve L. Karman Jr. Professor of Computational Engineering Staff Specialist (Committee Member) Pointwise (Committee Member) AUTOMATIC HIGHER ORDER MESH GENERATION AND MOVEMENT UTILIZING SPRING-FIELD AND VECTOR-ADDING By Tuo Liu A Dissertation Submitted to the Faculty of the University of Tennessee at Chattanooga in Partial Fulfillment of the Requirements of the Degree of Doctor of Philosophy in Computational Engineering The University of Tennessee at Chattanooga Chattanooga, Tennessee December 2017 ii Copyright © 2017 By Tuo Liu All Rights Reserved iii ABSTRACT In this research, an automatic unstructured curved 2D mesh generation method is developed. It can handle complex geometries. Based on this 2D method, an extension to 3D geometries is developed. The methodology of this mesh generation is to provide near-body and off-body meshes by an advancing-layer method to avoid mesh quality problems in transition areas, while using spring models to complement the advancing-layers by smoothing the mesh, especially in medial axis regions. After generating the linear mesh, a higher-order finite-element ready, curved mesh is obtained by deforming the linear mesh through “pipes” using a simple Vector- Adding approach. Different from most curved viscous mesh generation approaches, this method eschews a linear or nonlinear elasticity analogy while still providing positive-Jacobian mesh elements. In addition, the transformation from linear to curved meshes can be achieved by only deforming the necessary edges of the elements near domain boundaries while retaining the remaining non-curved edges to the benefit of numerical solvers. The CFD results of 30P30N airfoil on the curved mesh are given and are in good agreement with the experimental data. Further, a new method for moving boundary problems with viscous mesh layers is also presented. This method is an extension of the mesh generation approach above. It can be used not only for high-order mesh moving but also for high-order mesh generation. Also, based on the curved mesh deformation strategy, the method can handle high-order mesh movement or regeneration with minimal extra computational time compared with linear mesh movement. Several 2D boundary motion cases are tested including boundary translations, boundary rotations, iv and boundary morphing. The CFD results on the curved mesh after rotation are given and are in good agreement with the experimental data. This technique is also tested through the optimization of an airfoil to achieve a maximized lift and drag ratio. The results demonstrate that this approach can handle large boundary movements while preserving a good mesh quality. v DEDICATION This dissertation is dedicated to my parents. Mom and Dad, I love you. vi ACKNOWLEDGMENTS I would like to express my special appreciation and thanks to my advisor Dr. C. Bruce Hilbert for devoting a great deal of time to answer my questions and offer suggestions, and his company Branch Technology for supporting him to do this. I also would like to express my gratitude to my previous advisors Dr. Li Wang and Dr. Steve Karman for their guidance and precious advice. Without their guidance and encouragement throughout my time as a graduate student, this work would not have been possible. I also thank my other committee members, Dr. Kidambi Sreenivas and Dr. James Newman for their guidance regarding this work. In addition, I would like to express my appreciation to Dr. Timothy Swafford for his support and for believing in me. My gratitude also goes to my wife Siman Wei. I cannot imagine sailing my vessel without her support and encouragement. Finally, I would like to thank all the teachers who impart knowledge to me, all my classmates with whom I study and play soccer and basketball, all the staff who offer me help and care, and all my friends, especially my friends in Chattanooga, who guided me towards more scenic lookout points when I hit the rocks. You light up my life. vii TABLE OF CONTENTS ABSTRACT ................................................................................................................................................. iv DEDICATION ............................................................................................................................................. vi ACKNOWLEDGMENTS .......................................................................................................................... vii LIST OF TABLES ....................................................................................................................................... xi LIST OF FIGURES .................................................................................................................................... xii LIST OF ABBREVIATIONS .................................................................................................................... xvi LIST OF SYMBOLS ................................................................................................................................ xvii CHAPTER I. INTRODUCTION ............................................................................................................................. 1 II. SPRING-FIELD AND VECTOR-ADDING ................................................................................... 11 2.1 Spring-Field ............................................................................................................................. 11 2.2 Vector-Adding ......................................................................................................................... 16 2.2.1 What is Vector-Adding? ................................................................................................ 16 2.2.2 Stop conditions ............................................................................................................. 19 III. 2D MESH GENERATION.............................................................................................................. 21 3.1 Boundary Mesh Generation Procedure .................................................................................... 22 3.1.1 Linear boundary mesh generation ................................................................................ 22 3.1.2 Curved boundary mesh generation .............................................................................. 25 3.1.3 Mesh results .................................................................................................................. 25 3.1.3.1 Bat Shape Geometry .......................................................................................... 26 3.1.3.2 30P30N airfoil ..................................................................................................... 27 3.2 Linear Domain Mesh Generation Procedure ........................................................................... 27 3.2.1 Procedure of the linear domain mesh generation ........................................................ 28 viii 3.2.2 Original AFSF method and mesh results ....................................................................... 30 3.2.2.1 Spring models ..................................................................................................... 30 3.2.2.2 Mesh results ....................................................................................................... 35 3.2.3 Improved AFSF method and mesh results .................................................................... 41 3.2.3.1 Two main adjustments ....................................................................................... 41 3.2.3.2 Mesh Results ...................................................................................................... 45 3.3 Curved Mesh Deformation Procedure ..................................................................................... 50 3.3.1 Positive Jacobian ........................................................................................................... 51 3.3.2 Mesh results .................................................................................................................. 55 3.3.3 CFD Results .................................................................................................................... 59 IV. 2D MESH MOVEMENT ................................................................................................................ 63 4.1 Further Adjustments in the Improved AFSF Method .............................................................. 63 4.2 Mesh Results ............................................................................................................................ 67 4.3 CFD Results ............................................................................................................................. 72 4.4 Optimization ............................................................................................................................ 76 V. 3D MESH GENERATION.............................................................................................................. 80 5.1 Boundary Surface Mesh Generation ........................................................................................ 82 5.2 Linear Domain Mesh Generation ............................................................................................
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