Computational Design for Consumer-Level Fabrication Przemyslaw Musialski TU Wien
Motivation 3D Modeling… 3D Printing… Przemyslaw Musialski 2
Motivation Przemyslaw Musialski 3
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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
Shape Optimization
Input and Output Przemyslaw Musialski 9
Input and Output • Input surface 𝑻 𝑻 Przemyslaw Musialski 10
Input and Output • input surface 𝑻 𝑻 Przemyslaw Musialski 11
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 12
Offset Surfaces • surface deformation by offset: 𝑻 𝑻 𝒚 𝒋 = 𝒚 𝒋 + 𝜀 𝑗 𝒘 𝒋 𝑻 𝑻 𝑻 𝑻 Przemyslaw Musialski 13
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 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: 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 16
Offset Surfaces • How far can we offset? • Along which directions 𝑾 ? 𝑻 𝑻 𝑻 𝑾 Przemyslaw Musialski 17
Offset Bounds 𝑻 Przemyslaw Musialski 18
Offset Bounds global 𝑻 local Przemyslaw Musialski 19
Offset Bounds • inside: skeleton 𝑻 𝑻 • Mean Curvature Flow [Tagliasacchi et al. 2012] 𝑻 Przemyslaw Musialski 20
Offset Vectors • inside: skeleton 𝑻 𝑻 • Mean Curvature Flow [Tagliasacchi et al. 2012] • offset along vectors 𝒘 𝒋 ∈ 𝑾 𝑻 𝑾 Przemyslaw Musialski 21
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 22
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 23
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 24
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 25
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 27
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 28
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 29
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 30
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 31
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 32
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 33
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 34
Applications I: Mass Properties
Example Przemyslaw Musialski 36
Buoyancy Force: 𝑮 𝑐 Equilibrium 𝑮 = 𝑮 𝑐 Center of Buoyancy Center of Mass Gravity: 𝑮 Mass Properties 𝑸(𝑻) Przemyslaw Musialski 37
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 38
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 39
Applications • static stability • monostatic stability • rotational stability • static stability under storage • volume and buoyancy Przemyslaw Musialski 40
Spinning Turtle Przemyslaw Musialski 41
Rabbit Rolly-Polly Przemyslaw Musialski 42
Balanced Bottles Przemyslaw Musialski 43
Applications II: Modal Synthesis
Natural Frequencies Frequency: 440 Hz Concert pitch A Przemyslaw Musialski 45
Natural Frequencies 1 st mode 440 Hz (pitch) 2 nd mode 1060 Hz (1 st overtone) 3 rd mode 2790 Hz (2 nd overtone) … Overtone spectrum ⟺ characteristic sound of object Przemyslaw Musialski 46
Natural Frequencies Natural modes depend on • material • shape Przemyslaw Musialski 47
Goal: “Modal Synthesis” shape optimization fabrication Przemyslaw Musialski 48
Vertex-Normal Parametrization • Large offsets and high curvatures ⟹ self-intersections Przemyslaw Musialski 49
Shape Parametrization • Use Reduced Basis with Manifold Harmonics • Define offsets 𝜺 as linear combination of basis functions 𝚫 𝒍 𝜺 = 𝚫 𝑙 𝜷 𝒍 = 𝒀 + 𝒋=𝟐 Przemyslaw Musialski 50
Shape Optimization • Use non-linear optimization routine (Matlab) 𝜷 𝑔 𝜷 = 𝑞 − 𝑞 0 2 min • 𝑞 0 … target pitch • 𝑞 ... pitch of incument solution • 𝜷 … coefficient vector Przemyslaw Musialski 51
Fabrication • Material • good acoustic properties • cast into complex shape • Tin • melting point of 230°C • Young’s modulus of 50 GPa Przemyslaw Musialski 52
Fabrication • Oval bell • molds from sand • Rabbit bell • molds from caoutchouc Przemyslaw Musialski 53
Bell Przemyslaw Musialski 54
Bell Przemyslaw Musialski 55
Rabbit Przemyslaw Musialski 56
Rabbit Przemyslaw Musialski 57
Results • Aluminium plates • Median error of 1.7% • 0.7% with parameter estimation • Bell • Error of 2.8% • Rabbit Bell • Error of 11% • 6% with parameter estimation Przemyslaw Musialski 58
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