Shape Optimization for Consumer-Level 3D Printing Przemyslaw - PowerPoint PPT Presentation

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

Motivation 3D Modeling… 3D Printing… Przemyslaw Musialski 2

Motivation Przemyslaw Musialski 3

Przemyslaw Musialski 4

Example Przemyslaw Musialski 5

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

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

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

Shape Optimization

Input and Output Przemyslaw Musialski 10

Input and Output • Input surface 𝑻 𝑻 Przemyslaw Musialski 11

Input and Output • input surface 𝑻 𝑻 Przemyslaw Musialski 12

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

Offset Surfaces • surface deformation by offset: 𝑻 𝑻 𝒚 𝒋 = 𝒚 𝒋 + 𝜀 𝑗 𝒘 𝒋 𝑻 𝑻 𝑻 𝑻 Przemyslaw Musialski 14

Offset Surfaces • surface deformation by offset: 𝒚 𝒋 𝑻 𝒚 𝒋 = 𝒚 𝒋 + 𝜀 𝑗 𝒘 𝒋 𝒘 𝒋 𝜀 𝑗 𝒚 𝒋 • for each vertex 𝒚 𝒋 𝑻 • along 𝒘 𝒋 • add an individual offset 𝜀 𝑗 𝑻 Przemyslaw Musialski 15

Offset Surfaces • surface deformation by offset: 𝒚 𝒋 𝑻 𝒚 𝒋 = 𝒚 𝒋 + 𝜀 𝑗 𝒘 𝒋 𝒘 𝒋 𝜀 𝑗 𝒚 𝒋 • for each vertex 𝒚 𝒋 𝑻 • along 𝒘 𝒋 • add an individual offset 𝜀 𝑗 𝑻 Przemyslaw Musialski 16

Offset Surfaces • surface deformation by offset: 𝑻 𝒚 𝒋 = 𝒚 𝒋 + 𝜀 𝑗 𝒘 𝒋 • for each vertex 𝒚 𝒋 𝑻 𝑻 • along 𝒘 𝒋 • add an individual offset 𝜀 𝑗 Przemyslaw Musialski 17

Offset Surfaces • How far can we offset? • Along which directions 𝑾 ? 𝑻 𝑻 𝑻 𝑾 Przemyslaw Musialski 18

Offset Bounds 𝑻 Przemyslaw Musialski 19

Offset Bounds global 𝑻 local Przemyslaw Musialski 20

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

Offset Vectors • inside: skeleton 𝑻 𝑻 • Mean Curvature Flow [Tagliasacchi et al. 2012] • offset along vectors 𝒘 𝒋 ∈ 𝑾 𝑻 𝑾 Przemyslaw Musialski 22

Offset Vectors and Bounds • inside: skeleton 𝑻 𝑻 • Mean Curvature Flow [Tagliasacchi et al. 2012] • offset along vectors 𝒘 𝒋 ∈ 𝑾 𝑻 • outside a constant max. value 𝑾 Przemyslaw Musialski 23

Shape Optimization Problem • minimize objective 𝑔 as a function of 𝜺 : min 𝜺 𝑔 𝜺 s. t. 𝑕(𝜺) 𝑻 • subject to constraints 𝑕(𝜺) 𝑻 • for example: 𝑻 • 𝑔 ≔ make shape float subject to • 𝑕 ≔ keep upright orientation Przemyslaw Musialski 24

Shape Optimization Problem • minimize objective 𝑔 as a function of 𝜺 : 𝜺 𝑔 (𝜺)  𝑜 unknowns min 𝑻 • issues: 𝑻 • problem is huge for large meshes 𝑻  scales very badly • problem is underdetermined  there exist many solutions (regularization needed) Przemyslaw Musialski 25

Shape Optimization Problem • minimize objective 𝑔 as a function of 𝜺 : min 𝜺 𝑔 (𝜺) 𝑻 • issues: 𝑻 • problem is huge for large meshes 𝑻  scales very badly • problem is underdetermined  there exist many solutions (regularization needed) Przemyslaw Musialski 26

Order Reduction

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

Mesh Laplacian Input Mesh 𝑁 Differential Operator 𝚬 𝑁 Mesh Laplacian 𝐌 𝑁 Przemyslaw Musialski 29

Manifold Harmonics • diagonalization of the Laplacian matrix L  Spectral Theorem: 𝐌 = 𝚫𝚳𝚫 𝐔 𝚫 𝐔 𝐌 𝚫 𝚳 Przemyslaw Musialski 30

Manifold Harmonics • diagonalization of the Laplacian matrix L  Spectral Theorem: 𝐌 = 𝚫𝚳𝚫 𝐔 • generalization of the Fourier Transform for scalar functions on surfaces [V ALLET , B. AND L ÉVY , B. 2008. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27 , 2, 251 – 260.] Przemyslaw Musialski 31

Manifold Harmonics • diagonalization of the Laplacian matrix L  Spectral Theorem: 𝐌 = 𝚫𝚳𝚫 𝐔 • generalization of the Fourier Transform for scalar functions on surfaces [V ALLET , B. AND L ÉVY , B. 2008. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27 , 2, 251 – 260.] Przemyslaw Musialski 32

Manifold Harmonics • eigenfunctions • 𝚫 = [ 𝜹 1 𝜹 2 … 𝜹 𝑜 ] 𝜹 1 𝜹 2 𝜹 3 • shape can be transformed to 𝜹 4 𝜹 5 𝜹 6 • 𝒀 = 𝚫 T 𝒀 𝜹 7 𝜹 𝑜 Przemyslaw Musialski 33

Manifold Harmonics • eigenfunctions • 𝚫 = [ 𝜹 1 𝜹 2 … 𝜹 𝑜 ] • shape can be transformed to • 𝒀 = 𝚫 T 𝒀 • reconstruction • 𝒀 𝑙 = 𝚫 𝑙 𝒀 𝑙 • with 𝚫 𝑙 = [ 𝜹 1 𝜹 2 … 𝜹 𝑙 ] Przemyslaw Musialski 34

Manifold Harmonics • eigenfunctions • 𝚫 = [ 𝜹 1 𝜹 2 … 𝜹 𝑜 ] • shape can be transformed to • 𝒀 = 𝚫 T 𝒀 • reconstruction • 𝒀 𝑙 = 𝚫 𝑙 𝒀 𝑙 • with 𝚫 𝑙 = [ 𝜹 1 𝜹 2 … 𝜹 𝑙 ] Przemyslaw Musialski 35

Order Reduction • project unknown offsets 𝜺 = 𝜀 1 , 𝜀 2 ,…, 𝜀 𝑜 𝑈 onto 𝚫 𝑙 : 𝛽 1 𝜹 1 𝛽 2 𝜹 2 𝛽 3 𝜹 3 = 𝜺 = 𝚫 𝑙 𝜷 𝛽 4 𝜹 4 𝛽 5 𝜹 5 𝛽 6 𝜹 6 𝛽 7 𝜹 7 𝛽 𝑙 𝜹 𝑙 • vector 𝜷 = 𝛽 1 , 𝛽 2 ,…, 𝛽 𝑙 𝑈 now contains the unknowns! Przemyslaw Musialski 36

Order Reduction • project unknown offsets 𝜺 = 𝜀 1 , 𝜀 2 ,…, 𝜀 𝑜 𝑈 onto 𝚫 𝑙 : 𝛽 1 𝜹 1 𝛽 2 𝜹 2 𝛽 3 𝜹 3 𝒚 𝒋 = 𝒚 𝒋 + 𝜀 𝑗 𝒘 𝒋 𝛽 4 𝜹 4 𝛽 5 𝜹 5 𝛽 6 𝜹 6 𝑙 𝒚 𝒋 = 𝒚 𝒋 + 𝛽 𝑘 𝛿 𝑗𝑘 𝒘 𝒋 𝛽 7 𝜹 7 𝛽 𝑙 𝜹 𝑙 𝑘=1 Przemyslaw Musialski 37

Reduced Shape Optimization Problem • minimize objective 𝑔 as a function of 𝜷 : min 𝜷 𝑔 (𝜷) 𝛽 1 𝜹 1 𝛽 2 𝜹 2 𝛽 3 𝜹 3 𝛽 4 𝜹 4 𝛽 5 𝜹 5 𝛽 6 𝜹 6 • (subject to constraints) 𝛽 7 𝜹 7 𝛽 𝑙 𝜹 𝑙 Przemyslaw Musialski 38

Reduced Shape Optimization Problem • minimize objective 𝑔 as a function of 𝜷 : 𝜷 𝑔 (𝜷)  𝑙 unknowns , 𝑙 ≪ 𝑜 min 𝛽 1 𝜹 1 𝛽 2 𝜹 2 𝛽 3 𝜹 3 𝛽 4 𝜹 4 𝛽 5 𝜹 5 𝛽 6 𝜹 6 𝜺 = 𝚫 𝑙 𝜷 We deform only the low-frequencies and 𝛽 7 𝜹 7 𝛽 𝑙 𝜹 𝑙 leave high-frequency details untouched! Przemyslaw Musialski 39

Reduced Shape Optimization Problem • minimize objective 𝑔 as a function of 𝜷 : 𝜷 𝑔 (𝜷)  𝑙 unknowns , 𝑙 ≪ 𝑜 min 𝜺 = 𝚫 𝑙 𝜷  independent of mesh resolution  implicit regularization  numerically stable  easy to implement Przemyslaw Musialski 40

Applications I: Mass Properties

Example Przemyslaw Musialski 42

Buoyancy Force: 𝑮 𝑐 Equilibrium 𝑮 𝑕 = 𝑮 𝑐 Center of Buoyancy Center of Mass Gravity: 𝑮 𝑕 Mass Properties 𝑸(𝑻) Przemyslaw Musialski 43

Applications Center of Buoyancy • Gauss’ Divergence Theorem • allows us to compute mass properties as a function of the surface 𝑸 𝒏 𝑻 = 𝑁 𝑸 𝒚,𝒛,𝒜 (𝑻) = 𝑫𝒑𝑵 = 𝑑 𝑦 𝑑 𝑧 𝑑 𝑨 𝑼 𝐽 𝑦 2 𝐽 𝑦𝑧 𝐽 𝑦𝑨 𝐽 𝑦𝑧 𝐽 𝑧 2 𝐽 𝑧𝑨 𝑸 𝒚 𝟑 ,𝒚𝒛,…,𝒜 𝟑 (𝑻) = 𝑱 = Center of Mass 𝐽 𝑦𝑨 𝐽 𝑧𝑨 𝐽 𝑨 2 Przemyslaw Musialski 44

Applications Center of Buoyancy • optimization problem min 𝑔 𝑸 𝑻 𝜺(𝜷) 𝜷 • an analytical gradient 𝛼𝑔 = 𝜖𝑔 𝜖𝑸 𝜖𝑻 𝜖𝜺 𝜖𝑸 𝜖𝑻 𝜖𝜺 𝜖𝜷 Center of Mass Przemyslaw Musialski 45

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

Spinning Turtle Przemyslaw Musialski 47

Rabbit Rolly-Polly Przemyslaw Musialski 48

Balanced Bottles Przemyslaw Musialski 49

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

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

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

Applications II: Modal Synthesis

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