LINEAR ELASTIC MESH DEFORMATION VIA LOCALIZED ORTHOTROPIC MATERIAL PROPERTIES OPTIMIZED BY THE ADJOINT METHOD By William Lawton Shoemake Dr. Bruce Hilbert Dr. Kidambi Sreenivas Algorithms and Optimization Engineer, Associate Professor of Mechanical Branch Technology, Inc. Engineering (Chair) (Committee Member) Dr. Matt Matthews Dr. James Charles Newman III Professor of Mathematics; Associate Dean, Professor; Department Head of Mechanical College of Arts and Sciences Engineering (Committee Member) (Committee Member) LINEAR ELASTIC MESH DEFORMATION VIA LOCALIZED ORTHOTROPIC MATERIAL PROPERTIES OPTIMIZED BY THE ADJOINT METHOD By William Lawton Shoemake 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 William Lawton Shoemake All Rights Reserved iii ABSTRACT The finite element method has been shown to be a powerful tool in computational engi- neering with recent application to electromagnetics and fluid dynamics. However, achieving the high orders of accuracy easily available to the finite element method has proven difficult due to conforming higher-order meshes to curved geometries. If higher-order nodes are not placed on the surface of the geometry error is introduced into the simulated solution. This barrier is largely a non-issue for inviscid meshes where a mid-edge node can be projected onto the nearest geometry surface with minimal detrimental side effects. Viscous meshes however have to deform most of the boundary layers in order to avoid inverting the surface elements and to maintain an acceptable mesh quality. This research focuses on extending the application of the linear elastic analogy to this mesh movement problem by attributing orthotropic material properties individually to each node or element. This technique allows each node or element to behave differently under the stress of conforming to the boundary. These localized material properties are determined using the adjoint optimization method. To better determine mesh quality, a new mesh metric called Metric3 is introduced. This new metric resembles the included angle metric and is based on an element’s isoparametric transformation matrix. iv DEDICATION This dissertation is dedicated to my amazing wife Kristen Karman-Shoemake. v ACKNOWLEDGEMENTS I would like to thank my adviser Dr. Bruce Hilbert for his time, effort, and for keeping me pointed in the right direction when there are too many leads to chase, also to my committee members Dr. James C. Newman III, Dr. John V. Matthews III, and Dr. Abdollah Arabshahi. The professors and students of the SimCenter both past and present deserve recognition for creating an environment so conducive to research and higher education. I would also like to thank Ethan Hereth but more importantly Kim Sapp. Branch Technology, Inc. also deserves recognition for allowing Dr. Hilbert to serve as a graduate adviser. I would like to thank my parents, Jerry and Sandra Shoemake, and brother Brian for their encouragement to follow my interests and support in getting an advanced degree in ’squares and triangles’. And finally I would like to thank my wife, Kristen, for her love and support as we faced graduate school together. vi TABLE OF CONTENTS ABSTRACT....................................................................................................................................iv DEDICATION.................................................................................................................................v ACKNOWLEDGEMENTS............................................................................................................vi LIST OF FIGURES........................................................................................................................ix CHAPTER 1. INTRODUCTION...............................................................................................................1 2. METHODOLOGY..............................................................................................................8 2.1 Governing Equations................................................................................................8 2.2 Discretization......................................................................................................... 12 2.3 Material Property Determination........................................................................... 16 2.3.1 Discrete Material Property Determination................................................ 16 2.3.2 Continuous Material Property Determination........................................... 23 2.4 Quality Metrics...................................................................................................... 25 2.4.1 Relative Change in Wall Distance.............................................................. 25 2.4.2 Scaled Jacobian.......................................................................................... 27 2.4.3 Metric3....................................................................................................... 29 2.5 Adjoint Optimization............................................................................................. 31 2.5.1 Implementation.......................................................................................... 36 3. RESULTS.......................................................................................................................... 38 3.1 Orthotropic Perturbation Study.............................................................................. 39 3.2 Globally Defined Material Properties.................................................................... 50 3.2.1 Viscous Circle............................................................................................ 50 3.2.2 Viscous Potato........................................................................................... 67 3.2.3 30P30N Front Slat..................................................................................... 85 vii 3.3 Discretely Defined Material Properties............................................................... 105 3.3.1 Viscous Potato......................................................................................... 107 3.3.2 30P30N Front Slat................................................................................... 114 3.4 Continuously Defined Material Properties.......................................................... 118 3.4.1 Viscous Potato......................................................................................... 119 3.4.2 30P30N Front Slat................................................................................... 127 3.5 Adjoint Optimized Material Properties............................................................... 131 3.5.1 Discretely Defined Material Properties................................................... 136 3.5.2 Continuously Defined Material Properties.............................................. 146 4. CONCLUSION................................................................................................................ 153 4.1 Recommendations for Future Work..................................................................... 154 REFERENCES............................................................................................................................ 156 APPENDIX A. STATICALLY TYPED SMALL MATRIX LIBRARY................................................... 159 VITA............................................................................................................................................ 164 viii LIST OF FIGURES 1.1 Radar cross section of a perfect electric conducting sphere................................................3 1.2 Circular geometry with viscous mesh..................................................................................5 1.3 Mesh inversion after projecting higher-order boundary points............................................6 2.1 Example of an SVD ellipse on a simple quadrilateral....................................................... 18 2.2 Least squares fit ellipse of a quadrilateral.......................................................................... 23 2.3 Example of change in relative wall distance...................................................................... 27 2.4 Example of scaled Jacobian contour.................................................................................. 29 2.5 Example of M3 contour...................................................................................................... 31 3.1 8 point circle mesh............................................................................................................. 40 3.2 8 point circle: isotropic elasticity quality metrics data and contours................................ 43 3.3 Perturbation Study: effects of E1 on quality metrics. Left column shows results for E1 = 0:5 and right column shows results for E1 = 1:5. First row shows ∆%dw , second row shows Js , third column shows M3 ........................................ 46 3.4 Perturbation Study: effects of E2 on quality metrics. Left column shows results for E2 = 0:5 and right column shows results for E2 = 1:5. First row shows ∆%dw , second row shows Js , third row shows M3 ............................................. 47 3.5 Perturbation Study: effects of ν21 on quality metrics. Left