ShapeWright: Finite Element Based Free-Form Shape Design by George Celniker S.M., Mechanical Engineering Massachusetts Institute of Technology January, 1981 B.S., Mechanical Engineering Massachusetts Institute of Technology May, 1979 Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mechanical Engineering at the Massachusetts Institute of Technology September, 1990 © Massachusetts Institute of Technology, 1990, All rights reserved. Signature of Author . ... , , , Department of Mechariical Engineering August 3, 1990 Certified by Professor David C. Gossard Thesis Supervisor Accepted by Ain A. Sonin Chairman, Department Committee MASSACHUITTS INSTITOiI OF TECHIvn .:.. NOV 0 8 1990 LIBRARIEb ShapeWright: Finite Element Based Free-Form Shape Design by George Celniker Submitted to the Department of Mechanical Engineering on September, 1990 inpartial fulfillment of the requirements for the degree of Doctor of Philosophy inMechanical Engineering. Abstract The mathematical foundations and an implementation of a new free-form shape design paradigm are developed. The finite element method is applied to generate shapes that minimize a surface energy function subject to user-specified geometric constraints while responding to user-specified forces. Because of the energy minimization these surfaces opportunistically seek globally fair shapes. It is proposed that shape fairness is a global and not a local property. Very fair shapes with C1 continuity are designed. Two finite elements, a curve segment and a triangular surface element, are developed and used as the geometric primitives of the approach. The curve segments yield piecewise C1 Hermite cubic polynomials. The surface element edges have cubic variations in position and parabolic variations in surface normal and can be combined to yield piecewise C1 surfaces. These are called deformable curves and surfaces because they respond to user inputs much like elastic beams and membranes. The properties of these geometric primitives have been designed to enable a three phase interactive approach to defining very fair free-form shapes. The shape's character lines are created with deformable curve segments. These character lines are then "skinned" with a deformable surface. The final shape is sculpted interactively by applying loads to the surface to control the surface shape between character lines. Very fair free-form shapes have such applications as the design of automobiles, ships, ceramics, and air-supported buildings as well as the modeling of natural forms. Thesis Advisor: Prof. David Gossard Thesis Committee: Prof. Tony Patera Prof. Alex Pentland Dedication To Matthew Benjamin and to those who follow. Acknowledgements My motivations and drive to explore that have resulted in this effort, are in no small part due to my associations with my family and friends. Before all others, I thank my wife, Nancy Stevens, for encouraging me to make the changes to do the things that are important. I thank my parents and sisters for their unquestioning love and for having made the world a valuable place for me. I thank all my friends in the extended M.I.T. community that have made this such an interesting and enjoyable time in my life. The direction and quality of this work greatly benefited from the input of the members of my committee, Sandy Pentland and Tony Patera, and in particular from the advice and guidance of my thesis advisor, David Gossard. I also thank David-Gossard for the support he has given in furthering my career. The time and opportunity to pursue this study were provided by the people of the Schlumberger Technology Corporation. I thank them for having supported me throughout this endeavor and for showing their confidence and belief in my abilities. This research was made possible in part by grants from the Ford Motor Company, and the Eastman Kodak Company. Table of Contents A bstract ............................................................................................. 3 D edication ..................................................... ........... ......................... Acknowledgements ............................................................................... ........ Table of Contents .......................... ................... .. .................................. 5 List of Figures ............................................................. ...................... 7 N om enclature ..................................... ...................... ........ .............. 1 Introduction ....................................... .............................................. 13 1.1 Computer Aided Geometric Design .................................................. 13 1.2 The ShapeWright Design Paradigm ................................ 17 1.3 ShapeWright Design Issues .............................................................. 18 1.4 Scope of W ork ............................................................................ 20 2 Previous Work ... ........................ ..... ..................................................22 2.1 Precursors to Energy-Based Shape Design............................... ... 22 2.2 The Foundations of Computer Aided Geometric Design .......................... 25 2.3 Shape Building and the User Load Problem ............................................ 26 2.4 Parametric Shape Technical Issues .................. .... ................ ............... 28 2.5 Topology................................................................................34 3 Deformable Models....................................................39 3.1 Parametric Description of Shape ............................................ 39 3.2 Modeling of an Elastically Deformable Surface ....................................... 48 3.3 The Energy Functionals..... ............ ........ .... ............ 49 3.4 Properties of the Energy-Based Deformable Models .................. ........ 50 3.5 Minimize a Functional or Solve a Differential Equation...........................56 3.6 Time, Mass and Dynamics ...................................... 59 3.7 Forces ...................................................................................... 61 3.8 Soap Films, Plane Strain, Plates and Shells.................................. 64 3.9 Chapter 3 Summ ary ..................................................................... 69 4 Curve and Surface Geometric Primitives .................................................... 71 4.1 Finite Difference Solutions .................................. .......................... 72 4.2 The Finite Element Method..................................... ................. 77 4.3 Mass, Damping and Integrating Through Time ................ ...................... 84 4.4 A Deformable Surface Primitive ........................................................ 88 4.5 A Deformable Curve Primitive......................................................... 114 4.6 Chapter 4 Summ ary ...................................................... .............. 118 5 Survey of Constraint Methods ........................................................... 120 5.1 Reduced Transformation Equations ................................................... 121 5.2 Penalty Methods .............................................................................. 123 5.3 Lagrange M ultipliers ....................................................... 125 5.4 Stabilized Constraints .................................................................... 129 5.5 Stabilized Parametric Constraints ..................................................... 133 5.6 Chapter 5 Summary ................................................................... 137 6 Implementations .......................................................... 139 6.1 A Finite Difference Program for Deformable Surface Design, GGT.............139 6.2 A Finite Element Program for Deformable Surfaces, ECS .......................... 151 6.3 A Finite Element Program for Deformable Curves, CUBIC.......................162 6.4 Rendering ......................................................... 167 7 Conclusions and Future Work ............................................................. 171 7.1 Thesis Summary................................ ....................... 171 7.2 Conclusions ....................................................................... ... 173 7.3 Future Directions ......................................................................... 176 R eferences ...................................................................................... 179 List of Figures Figure 1.1) Current free-form design paradigms ............................................ 14 Figure 1.2) The ShapeWright design paradigm ............................................. 18 Figure 1.3) Geometric constraints needed for skinning ..................................19 Figure 2.1) Trimmed and constrained surface patches for complicated shapes..........38 Figure 3.1) The parametric tangents and normal to a regular surface ..................... 42 Figure 3.2) Curves in the parametric plane map to curves fixed in the surface .......... 43 Figure 3.3) The curvature and normal for a curve fixed in a surface......................45 Figure 3.4) Solutions for avoiding an obstacle . ................................. 61 Figure 3.5) External forces, a survey ................. ............... ............... 62 Figure 4.1) Discretizing the uv domain for a finite difference solution ..................