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Mesh Free Reverse Engineering of Automotive Components

Mesh Free Reverse Engineering of Automotive Components. Yohan FOUGEROLLE UTK advisors: Andrei Gribok, Mongi A. Abidi 05-19-2004. Outline. Introduction Background: Forward/Inverse problem of analytical geometry Superquadrics and Supershapes R-Functions Algorithm Results Applications

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Mesh Free Reverse Engineering of Automotive Components

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  1. Mesh Free Reverse Engineering of Automotive Components Yohan FOUGEROLLE UTK advisors: Andrei Gribok, Mongi A. Abidi 05-19-2004

  2. Outline • Introduction • Background: Forward/Inverse problem of analytical geometry • Superquadrics and Supershapes • R-Functions • Algorithm • Results • Applications • Conclusions and Future Work

  3. Forward problem ? Increasing complexity Inverse problem • What kind of primitives? • Is this possible to represent complex objects with only one equation? If yes, how? Forward/Inverse problem of analytical geometry

  4. Why Mesh Free Modeling? • Differential properties for direct computation of normal vectors and curvature • Simulation • Boundary representation F(x,y,z) = 0 • results heavily depend on triangulation • speed • Accuracy problems • Storage and transmission. MESH  EQUATION

  5. with Implicit function Implicit function Superquadrics and Supershapes Superquadrics 3D parametric formula Supershapes 3D parametric formula

  6. CSG Tree of primitives Complex initial object without formula C Output: Boolean predicate How to go from Boolean representation to equation? A B D F E G F(A,B,C,D,E,F,G) ≥ 0 A possible decomposition into primitives (with known equations) Boolean Operations and Constructive Solid Geometry (CSG) Trees

  7. R-Conjunction (AND) R-Disjunction (OR) R-Negation (NOT) Plot of the function z=x1 ٨ x2 From Boolean expression to single equation using R-Functions • R-functions introduced by Rvachev (1962) • Key observation : Some real valued functions of real variables have their sign completely defined by the sign of their arguments. • Example y=x1x2, Counter example y=x1x2+1 • Lets treat sign as a Boolean Variable +/1,-/0

  8. Outline • Introduction • Algorithm • Results • Applications • Conclusions and Future Work

  9. Algorithm • Input: constructive solid geometry tree containing: • Primitives parameters • Boolean operations • Global deformations • Output: • Mesh representing the resulting surface • One implicit equation

  10. Flagged Points Subdivision Criterion Vertex Flagging Subdivision Algorithm Splitting and Merging Original objects Face Splitting Frontiers Merging To Parent Node Intersection criterion Vertex and Face Transfer Child Node Child Node Intersection Evaluation

  11. Outline • Introduction • Algorithm • Results • Applications • Conclusions and Future Work

  12. U Original CAD File - U U Bolt CSG tree Results : Bolt

  13. U U U - - - U U U U U U S1 S1 S1 S2 S2 S2 S3 S3 S3 S4 S4 S4 S5 S5 S5 One Equation?

  14. Max y z x Min Scalar Field Generated Section y=0 • Note : • Surface = zero set of F • Non Unique equation

  15. - U U U - U U Results : Piston Head U Superimposed Parts Original CAD File

  16. Max Min Sections for planes x=const Results : Piston Head

  17. Outline • Introduction • Algorithm • Results • Applications • Conclusions and Future Work

  18. Applications • Direct application • Volume computation using Monte Carlo integration • Area of arbitrary object section also possible • Simulation (heat, acoustic or vibration transmission)

  19. Conclusions • CSG Tree  Equation • Multiple Boolean Operations • Ability to fuse • Extract, cut, sculpt objects, etc. • Generic if primitives have both parametric and implicit definition (SQs, SShapes, Dupin Cyclides…) • Implicit form used to get the resulting equation • Parametric form used to generate the solutions • Only one resulting equation, and no longer a Boolean expression • Surface defined by F = 0 • Differential properties guaranteed (gradient and curvature directly computed)

  20. Potential Applications • Reverse Engineering • Application to solve boundary problems of mathematical physics • Sections with Free Form Surface • Heat, acoustic, or vibration transmission on complex structures • Optimization of fatigue life • Object and shape recognition (with prior knowledge of automotive parts as CAD models) • Volume and arbitrary section area • Packaging • Compression

  21. Future works • Automate the process: • Supershape recovery • R-function recovery CSG Tree EQUATION 3D Mesh • Iterative approach (primitive insertion) • Function differentiable Gradient based optimization available • Error Measure Provided

  22. Questions?

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