Shape Optimization for Consumer-Level 3D Printing

Size: px

Start display at page:

Download "Shape Optimization for Consumer-Level 3D Printing"

Penelope Webster
6 years ago
Views:

1 hape Optimization for Consumer-Level 3D Printing Przemyslaw Musialski TU Wien

2 Motivation 3D Modeling 3D Printing Przemyslaw Musialski 2

3 Motivation Przemyslaw Musialski 3

4 Przemyslaw Musialski 4

5 Example Przemyslaw Musialski 5

6 Goals 1. Optimize the shape to fulfill the desired goals Przemyslaw Musialski 6

7 Goals 1. Optimize the shape to fulfill the desired goals 2. Keep the input shape deformation minimal Przemyslaw Musialski 7

8 Related Work Optimization of Mass Properties [Prevost et al. 2013] [Baecher et al. 2014] tructural Optimization [tava et al. 2012] [Lu et al. 2014] Reduced Order Models [Pentland and Williams 1989] [von Tycowitch et al. 2013] Przemyslaw Musialski 8

9 hape Optimization

10 Input and Output Przemyslaw Musialski 10

11 Input and Output Input surface Przemyslaw Musialski 11

12 Input and Output input surface Przemyslaw Musialski 12

13 Input and Output input surface output: two surfaces outer offset surface inner offset surface solid body between and Przemyslaw Musialski 13

14 Offset urfaces surface deformation by offset: x i = x i + δ i v i Przemyslaw Musialski 14

15 Offset urfaces surface deformation by offset: x i = x i + δ i v i x i δ i v i for each vertex x i along v i x i add an individual offset δ i Przemyslaw Musialski 15

16 Offset urfaces surface deformation by offset: x i = x i + δ i v i x i δ i v i for each vertex x i along v i x i add an individual offset δ i Przemyslaw Musialski 16

17 Offset urfaces surface deformation by offset: x i = x i + δ i v i for each vertex x i along v i add an individual offset δ i Przemyslaw Musialski 17

18 Offset urfaces How far can we offset? Along which directions V? V Przemyslaw Musialski 18

19 Offset Bounds Przemyslaw Musialski 19

20 Offset Bounds global local Przemyslaw Musialski 20

21 Offset Bounds inside: skeleton Mean Curvature Flow [Tagliasacchi et al. 2012] Przemyslaw Musialski 21

22 Offset Vectors inside: skeleton Mean Curvature Flow [Tagliasacchi et al. 2012] offset along vectors v i V V Przemyslaw Musialski 22

23 Offset Vectors and Bounds inside: skeleton Mean Curvature Flow [Tagliasacchi et al. 2012] offset along vectors v i V outside a constant max. value V Przemyslaw Musialski 23

24 hape Optimization Problem minimize objective f as a function of δ : min δ f δ s. t. g(δ) subject to constraints g(δ) for example: f make shape float subject to g keep upright orientation Przemyslaw Musialski 24

25 hape Optimization Problem minimize objective f as a function of δ : min δ f (δ) n unknowns issues: problem is huge for large meshes scales very badly problem is underdetermined there exist many solutions (regularization needed) Przemyslaw Musialski 25

26 hape Optimization Problem minimize objective f as a function of δ : min δ f (δ) issues: problem is huge for large meshes scales very badly problem is underdetermined there exist many solutions (regularization needed) Przemyslaw Musialski 26

27 Order Reduction

28 Order Reduction order reduction: lower the dimensionality while preserving input-output behavior idea: project problem onto a lower dimensional space Manifold Harmonics Przemyslaw Musialski 28

29 Mesh Laplacian Input Mesh M Differential Operator Δ M Mesh Laplacian L M Przemyslaw Musialski 29

30 Manifold Harmonics diagonalization of the Laplacian matrix L pectral Theorem: L = ΓΛΓ T L Γ Λ Γ T Przemyslaw Musialski 30

31 Manifold Harmonics diagonalization of the Laplacian matrix L pectral Theorem: L = ΓΛΓ T generalization of the Fourier Transform for scalar functions on surfaces [VALLET, B. AND LÉVY, B pectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27, 2, ] Przemyslaw Musialski 31

32 Manifold Harmonics diagonalization of the Laplacian matrix L pectral Theorem: L = ΓΛΓ T generalization of the Fourier Transform for scalar functions on surfaces [VALLET, B. AND LÉVY, B pectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27, 2, ] Przemyslaw Musialski 32

33 Manifold Harmonics eigenfunctions Γ = [ γ 1 γ 2 γ n ] shape can be transformed to X = Γ T X γ 1 γ 2 γ 3 γ 4 γ 5 γ 6 γ 7 γ n Przemyslaw Musialski 33

34 Manifold Harmonics eigenfunctions Γ = [ γ 1 γ 2 γ n ] shape can be transformed to X = Γ T X reconstruction X k = Γ k X k with Γ k = [ γ 1 γ 2 γ k ] Przemyslaw Musialski 34

35 Manifold Harmonics eigenfunctions Γ = [ γ 1 γ 2 γ n ] shape can be transformed to X = Γ T X reconstruction X k = Γ k X k with Γ k = [ γ 1 γ 2 γ k ] Przemyslaw Musialski 35

36 Order Reduction project unknown offsets δ = δ 1, δ 2,, δ n T onto Γ k : α 1 γ 1 α 2 γ 2 α 3 γ 3 δ = Γ k α = α 4 γ 4 α 5 γ 5 α 6 γ 6 vector α = α 1, α 2,, α k T now contains the unknowns! α 7 γ 7 α k γ k Przemyslaw Musialski 36

37 Order Reduction project unknown offsets δ = δ 1, δ 2,, δ n T onto Γ k : x i = x i + δ i v i k α 1 γ 1 α 2 γ 2 α 3 γ 3 α 4 γ 4 α 5 γ 5 α 6 γ 6 x i = x i + j=1 α j γ ij v i α 7 γ 7 α k γ k Przemyslaw Musialski 37

38 Reduced hape Optimization Problem minimize objective f as a function of α: min α f (α) α 1 γ 1 α 2 γ 2 α 3 γ 3 (subject to constraints) α 4 γ 4 α 5 γ 5 α 6 γ 6 α 7 γ 7 α k γ k Przemyslaw Musialski 38

39 Reduced hape Optimization Problem minimize objective f as a function of α: min α f (α) k unknowns, k n α 1 γ 1 α 2 γ 2 α 3 γ 3 We deform only the low-frequencies and leave high-frequency details untouched! α 4 γ 4 α 5 γ 5 α 6 γ 6 δ = Γ k α α 7 γ 7 α k γ k Przemyslaw Musialski 39

40 Reduced hape Optimization Problem minimize objective f as a function of α: min α f (α) k unknowns, k n independent of mesh resolution implicit regularization numerically stable easy to implement δ = Γ k α Przemyslaw Musialski 40

41 Applications I: Mass Properties

42 Example Przemyslaw Musialski 42

43 Equilibrium F g = F b Buoyancy Force: F b Center of Buoyancy Center of Mass Mass Properties Gravity: F g P() Przemyslaw Musialski 43

44 Applications Gauss Divergence Theorem allows us to compute mass properties as a function of the surface Center of Buoyancy P m = M P x,y,z () = CoM = c x c y c z T P x 2,xy,,z 2 () = I = I x 2 I xy I xz I xy I y 2 I yz I xz I yz I z 2 Center of Mass Przemyslaw Musialski 44

45 Applications optimization problem Center of Buoyancy min α f P δ(α) an analytical gradient f = f P P δ δ α Center of Mass Przemyslaw Musialski 45

46 Applications static stability monostatic stability rotational stability static stability under storage volume and buoyancy Przemyslaw Musialski 46

47 pinning Turtle Przemyslaw Musialski 47

48 Rabbit Rolly-Polly Przemyslaw Musialski 48

49 Balanced Bottles Przemyslaw Musialski 49

50 objective f Evaluation infeasible number of basis functions k Przemyslaw Musialski 50

51 objective f Evaluation and Performance infeasible time in seconds Przemyslaw Musialski 51

52 objective f Evaluation and Performance infeasible t=87s t=0.4s Przemyslaw Musialski 52 t=4.4s

53 Applications II: Modal ynthesis

54 Natural Frequencies I Frequency: 440 Hz Concert pitch A Przemyslaw Musialski 54

55 Natural Frequencies II 1 st mode (pitch) 440 Hz 2 nd mode (1 st overtone) 1060 Hz 3 rd mode (2 nd overtone) 2790 Hz Overtone spectrum characteristic sound of object Przemyslaw Musialski 55

56 Natural Frequencies III Natural modes depend on material shape Przemyslaw Musialski 56

57 Modal Analysis material properties finite element analysis experimental analysis 610 Hz 1461 Hz 1645 Hz 2701 Hz 3201 Hz Przemyslaw Musialski 57

58 Goal: Modal ynthesis shape optimization fabrication Przemyslaw Musialski 58

59 Vertex-Normal Parametrization Use offset surfaces Constant wall thickness outer surface = user input single optimization parameter δ controls offset magnitude Przemyslaw Musialski 60 inner surface is offset along vertex normals

60 Vertex-Normal Parametrization Large offsets and high curvatures self-intersections Przemyslaw Musialski 61

61 hape Parametrization Use Reduced Basis with Manifold Harmonics Przemyslaw Musialski 62

62 hape Parametrization Use Reduced Basis with Manifold Harmonics Define offsets δ as linear combination of basis functions Γ k δ = Γ k α = X + k i=1 Przemyslaw Musialski 63

63 hape Optimization Use non-linear optimization routine (Matlab) p 0 target pitch p... pitch of incument solution α coefficient vector min α f α = p p 0 2 Przemyslaw Musialski 64

64 Fabrication Material good acoustic properties cast into complex shape Tin melting point of 230 C Young s modulus of 50 GPa Przemyslaw Musialski 65

65 Fabrication Oval bell molds from sand Rabbit bell molds from caoutchouc Przemyslaw Musialski 66

66 Bell Przemyslaw Musialski 67

67 Bell Przemyslaw Musialski 68

68 Rabbit Przemyslaw Musialski 69

69 Rabbit Przemyslaw Musialski 70

70 Results Aluminium plates Bell Median error of 1.7% 0.7% with parameter estimation Error of 2.8% Rabbit Bell Error of 11% 6% with parameter estimation Przemyslaw Musialski 71

71 Discussion & Limitations skeleton dependence our method relies on the skeleton we use iterative mesh contraction (Mean Curvature Flow) design space limitation we can only offset a surface up to the skeleton Przemyslaw Musialski 72

72 Conclusions we proposed a novel framework for shape optimization we provide an elegant and efficient basis-reduction we demonstrate the method by optimizing mass properties natural frequencies Przemyslaw Musialski 73

73 Publications Musialski, P., Auzinger, T., Birsak, M., Wimmer, M. & Kobbelt, L. Reduced-Order hape Optimization Using Offset urfaces. ACM Trans. Graph. (Proc. ACM IGGRAPH 2015) 34, 102:1 102:9 (2015). Hafner, C., Musialski, P., Auzinger, T., Wimmer, M. & Kobbelt, L. Optimization of natural frequencies for fabrication-aware shape modeling. in ACM IGGRAPH 2015 Posters - IGGRAPH (ACM Press, 2015). Hafner, C. Optimization of Natural Frequencies for Fabrication- Aware hape Modeling, Master Thesis, TU-Wien (2015) Przemyslaw Musialski 74

74 Thank you! Przemyslaw Musialski 75

Shape Modeling and Geometry Processing

252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry