The TiGL Geometry Library and its Current Mathematical Challenges

Transcription

1 The TiGL Geometry Library and its Current Mathematical Challenges Workshop DLR / Cologne University Dr. Martin Siggel Merlin Pelz Konstantin Rusch Dr. Jan Kleinert Simulation and Software Technology High Performance Computing DLR Cologne

2 DLR.de Chart 2 Outline What is the TiGL Library B-spline basics Surface Modelling: Gordon surfaces Alternative to Coons Patches to interpolate curve networks Solves problems with surface discontinuities Resulting geometries Fitting aircraft engines with Free Form Deformation (FFD) FFD Parametrization of aircraft engine cover The minimization problem + regularization Results Conclusion and Outlook

3 DLR.de Chart 3 TiGL What it does CPACS (Common Parametric Aircraft Configuration Schema) is an xml schema that describes the characteristics of an aircraft includes a parametric description for airplane or rotor-craft geometries TiGL generates geometries from CPACS files provides an interface for geometry manipulation and queries

4 DLR.de Chart 4 TiGL What it is TiGL is a C++ library with interfaces to C, Java, Python, Matlab and Fortran Uses Open CASCADE CAD kernel to represent airplane geometries as B-spline / NURBS surfaces Ships TiGL Viewer to visualize aircraft geometries or other CAD files. Is Open Source, Apache Joint development

5 DLR.de Chart 5 TiGL How it is used Aerodynamics Radar Signature CPACS Infrared Signature Structure and Aeroelastics

6 DLR.de Chart 6 TiGL How it is used VTK, BRep, IGES Mesh Generation for CFD Simulations Collada, STL Step, IGES Rendering, Visualization, 3D Printing Modeling with CAD Systems

7 DLR.de Chart 7 B-spline basics

8 DLR.de Chart 8 B-spline basics B-spline curves Definition: with: cc uu = PP ii cc NN ii dd (uu, tt) Control points PP ii cc B-spline basis functions NN ii dd (uu, tt) Knot vector tt, tt ii tt ii+1 nn ii=0 Linear in P!

9 DLR.de Chart 9 B-spline basics B-spline surfaces Definition: ss uu, vv = PP ss dd iiii NN uu dd ii uu, tt uu NN vv jj (vv, ttvv ) with: ss Control points PP iijj B-spline basis functions NN dd ii (uu, tt) Knot vectors tt uu, tt vv nn mm ii=0 jj=0 Linear in P!

10 DLR.de Chart 10 B-spline basics Algorithm: Knot Insertion Adds an entry to the knot vector t, without changing the curve shape Modifes control points and adds one control point! Knot insertion basis for many higher level algorithms

11 DLR.de Chart 11 B-spline basics Algorithm: B-Spline Curve Interpolation Given data points Pj, compute control points Ci, such that: nn CC ii NN ii dd uu jj, tt ii=0 NNNN PP = PP jj Solve a linear equation

12 DLR.de Chart 12 B-spline basics Algorithm: Surface Skinning Interpolates set of B-spline curves cc ii (uu) by B-spline surface ss(uu, vv) ss(uu, vv) Similar to curve interpolation Requires knot insertion to make input curves compatible

13 DLR.de Chart 13 Curve Network Interpolation with Gordon Surfaces

14 DLR.de Chart 14 Gordon Surfaces The curve network interpolation problem Definition of the problem: Given a network of profile and guide curves: Find surface that connects (interpolates) these curves Profiles ff ii (uu) Guides gg jj (vv) Curve network example

15 DLR.de Chart 15 Gordon Surfaces Motivation Before: Used OpenCASCADE implemention of Coons-patch based geometry generation Problems from aero simulations: Pressure oscillations on the wing Surface modelling probably the issue! Analysis: Generated surfaces have small bumps, waves and kinks The latter is caused by C1 or C2 discontinuities Conclusion: Must implement algorithm by our own: Gordon Surfaces

16 DLR.de Chart 16 Gordon Surfaces Algorithm overview Create three different surfaces: 1) Create skinning surface interpolating the curves ff ii uu, ii: SS uu (uu, vv) 2) Create skinning surface interpolating the curves gg jj vv, jj: SS vv (uu, vv) 3) Create surface interpolating the intersection points of the curve network: TT(uu, vv)

17 DLR.de Chart 17 Gordon Surfaces Algorithm overview Gordon Surface is superposition of these three surfaces: GG(uu, vv) = SS uu (uu, vv) + SS vv (uu, vv) TT(uu, vv) SS uu (uu, vv) TT(uu, vv) GG(uu, vv) SS vv (uu, vv) + - = Convert to Gordon surface B-Spline / NURBS for the use in TiGL

18 DLR.de Chart 18 Gordon Surfaces Compatibility Conditions Condition for the algorithm: Curves must be compatible Compatibility condition All profile curves ff ii uu must intersect with a guide curve gg jj vv at exactly the same parameter value uu jj and vice versa: ff ii uu jj = g j v i, ii, jj Interpretation: The input curves ff ii uu, gg jj (vv) must be iso-parametric curves of the resulting surface

19 DLR.de Chart 19 Gordon Surfaces Compatibility Conditions, Example Compatible intersections of the guides with the profiles:

20 DLR.de Chart 20 Gordon Surfaces Reparameterization Compatibility condition very restrictive Compatibility practically never the case! All curves have to be reparametrized Reparameterization is the main issue!

21 DLR.de Chart 21 Gordon Surfaces Reparameterization Rough Idea: Actual intersections uu jj ii Find function rr ii such that rr ii uu jj ii = uu ii jj with: uu jj ii the parameter of the intersection of curve ff ii uu with curve gg jj (vv) uu ii jj the desired intersection parameter to achieve compatibility Desired intersections uu jj ii Reparametrized function is given by ff ii uu ff ii (rr ii uu ) Challenge: Find B-spline parameterization of ff (uu) ii

22 DLR.de Chart 22 Gordon Surfaces Quality Analysis Surface quality analysis with zebra stripe plot

23 DLR.de Chart 23 Gordon Surfaces Results, Nacelle (engine cover)

24 DLR.de Chart 24 Gordon Surfaces Results, Wing

25 DLR.de Chart 25 Gordon Surfaces Results, Arbitrary surface

26 DLR.de Chart 26 Gordon Surfaces Results, Coons vs. Gordon, Nacelle Coons Patches Gordon Surface (one half of the same nacelle)

27 DLR.de Chart 27 Gordon Surfaces Results, Coons vs. Gordon, Nacelle Coons Patches Gordon Surface

28 DLR.de Chart 28 Gordon Surfaces Results, Coons vs. Gordon, Nacelle Coons Patches Gordon Surface

29 DLR.de Chart 29 Gordon Surfaces Results, Coons vs. Gordon, Wing Coons Patches Gordon Surface

30 DLR.de Chart 30 Gordon Surfaces Results, Coons vs. Gordon, Wing Coons Patches

31 DLR.de Chart 31 Gordon Surfaces Results, Coons vs. Gordon, Wing Gordon Surface

32 DLR.de Chart 32 Gordon Surfaces Results, Coons vs. Gordon, Wing Coons Patches Gordon Surface

33 DLR.de Chart 33 Gordon Surfaces Results, Coons vs. Gordon, Arbitrary Surface Coons Patches Gordon Surface

34 DLR.de Chart 34 Gordon Surfaces Results, Coons vs. Gordon, Arbitrary Surface Coons Patches Gordon Surface

35 DLR.de Chart 35 Gordon Surfaces Results, Summary Gordon Surface method works! Better quality than previous Coons patch implementation Global interpolation: Smooth and natural interpolation of the curve network with a single surface But, Oscillation might occur!

36 DLR.de Chart 36 Free-Form Deformation of Aircraft Nacelles

37 DLR.de Chart 37 Free Form Deformation of Nacelles Motivation Different possibilities to parametrize engine nacelles, such as Interpolation of curve network Free-Form Deformation Proof of concept: Show, that we can rebuild existing geometries from real aircraft with the FFD parameterization approach

38 DLR.de Chart 38 Free Form Deformation of Nacelles Concept Free-Form-Deformation (FFD) start with simple, rotationally symmetric geometry deform the geometry using FFD (parametrization = grid point displacements) 1. Define a so-called FFD-grid around the geometry: 2. Deform your geometry by moving the FFD points from their original places u t s

39 DLR.de Chart 39 Free Form Deformation of Nacelles Definition Each point on the engine surface s has local grid coordinates (s,t,u) Deformation of the point (s,t,u) using FFD: FF ffffff ss, tt, uu = BB ii,ll (ss)bb jj,mm (tt)bb kk,nn (uu) with: FFD grid points xx iijjjj Bernstein basis functions BB ii,nn (uu) ll ii mm jj nn kk xx iiiiii Linear in x!

40 DLR.de Chart 40 Free Form Deformation of Nacelles Initial Surface Generation Rotation of iso-parametric curve of reference surface around central axis Identical parametrization of reference geometry and created geometry Initial surface with iso-parametric curve in axial direction and rotational curve

41 DLR.de Chart 41 Free Form Deformation of Nacelles Basic Principle Modify FFD grid points xx iiiiii, such that initial surface fits reference surface :

42 DLR.de Chart 42 Free Form Deformation of Nacelles The Minimization Problem The discretized surface zz depends linearly on the control points xx. zz = BB xx Given the discretized reference surface as rr, the fit is given by the solution of BB xx rr 2 2 xx mmmmmm

43 DLR.de Chart 43 Free Form Deformation of Nacelles Regularization Problem: Matrix B has bad condition, resulting FFD grid xx distorted Remedy: Use modified L2 regularization with: BB xx rr λλ nn xx xx Regularization parameter λλ RR + Number of FFD grid points nn xx mmmmmm

44 DLR.de Chart 44 Free Form Deformation of Nacelles Solving the Minimization Problem Re-arrange minimization problem: BB xx rr λλ nn xx xx 2 0 Δxx 2 = BB Δxx rr BB xx λλ nn Δxx 2 2 = BB Δxx rr λλ nn Δxx 2 2 Δxx mmmmmm Solve using Singular Value Decomposition (SVD): Let BB = UUΣVV TT its SVD, with diagonal Matrix Σ, containing singular values Σ iiii, then Δxx = VVΣ UU TT rr, with Σ iiii = Σ iiii Σ iiii 2 +λλ/nn

45 DLR.de Chart 45 Free Form Deformation of Nacelles Results, best Fit Reference surface and initial surface Reference surface and deformed surface using 10x10x10 FFD grid points

46 DLR.de Chart 46 Free Form Deformation of Nacelles Results, Error of the Fit Optimiertes FFD Gitter mit 4 / 2 / 2 Punkten und deformierte Gondel

47 DLR.de Chart 47 Free Form Deformation of Nacelles Results, Regularization Distorted Grid (unregularized) Same FFD grid, regularization, with λλ = 0.5

48 DLR.de Chart 48 Free Form Deformation of Nacelles Results, Regularization λλ = 1.0 λλ = 0.01 λλ = 0.1

49 DLR.de Chart 49 Free Form Deformation of Nacelles Summary and Outlook Approximation engine surfaces with FFD method Method converges with rising number of FFD grid points Regularization is required due to bad conditioning Regularization has small influence on fit error, but large influence on the resulting grid quality Outlook Use B-spline basis function in FFD function also local deformation possible (already implemented but not presented here) Hierarchical optimization could replace regularization

50 DLR.de Chart 50 Questions?