Computational Design for Consumer-Level Fabrication Przemyslaw - - PowerPoint PPT Presentation

Computational Design for Consumer-Level Fabrication

Przemyslaw Musialski TU Wien

Motivation

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3D Printing… 3D Modeling…

Motivation

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Example

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Goals

• 1. Optimize the shape to

fulfill the desired goals

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Goals

• 1. Optimize the shape to

fulfill the desired goals

• 2. Keep the input shape

deformation minimal

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Shape Optimization

Input and Output

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Input and Output

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𝑻

• Input surface 𝑻

Input and Output

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𝑻

• input surface 𝑻

Input and Output

• input surface 𝑻
• output: two surfaces
• outer offset surface 𝑻
• inner offset surface 𝑻
• solid body between 𝑻 and 𝑻

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𝑻 𝑻 𝑻

𝑻

Offset Surfaces

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𝑻 𝑻

• surface deformation by offset:

𝒚𝒋 = 𝒚𝒋 + 𝜀𝑗𝒘𝒋

𝑻 𝑻 𝑻

Offset Surfaces

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• surface deformation by offset:
• for each vertex 𝒚𝒋
• along 𝒘𝒋
• add an individual offset 𝜀𝑗

𝒚𝒋 𝒚𝒋 𝒘𝒋 𝜀𝑗 𝑻 𝑻 𝑻

𝒚𝒋 = 𝒚𝒋 + 𝜀𝑗𝒘𝒋

Offset Surfaces

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• surface deformation by offset:
• for each vertex 𝒚𝒋
• along 𝒘𝒋
• add an individual offset 𝜀𝑗

𝒚𝒋 𝒚𝒋 𝒘𝒋 𝑻 𝑻 𝑻

𝒚𝒋 = 𝒚𝒋 + 𝜀𝑗𝒘𝒋

𝜀𝑗

Offset Surfaces

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𝑻 𝑻 𝑻

• surface deformation by offset:
• for each vertex 𝒚𝒋
• along 𝒘𝒋
• add an individual offset 𝜀𝑗

𝒚𝒋 = 𝒚𝒋 + 𝜀𝑗𝒘𝒋

Offset Surfaces

• How far can we offset?
• Along which directions 𝑾 ?

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𝑾 𝑻 𝑻 𝑻

Offset Bounds

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𝑻

Offset Bounds

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𝑻 local global

Offset Bounds

• inside: skeleton

𝑻

• Mean Curvature Flow

[Tagliasacchi et al. 2012]

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𝑻 𝑻

Offset Vectors

• inside: skeleton

𝑻

• Mean Curvature Flow

[Tagliasacchi et al. 2012]

• offset along vectors 𝒘𝒋 ∈ 𝑾

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𝑻 𝑾 𝑻

Offset Vectors and Bounds

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𝑾

• inside: skeleton

𝑻

• Mean Curvature Flow

[Tagliasacchi et al. 2012]

• offset along vectors 𝒘𝒋 ∈ 𝑾
• outside a constant max. value

𝑻 𝑻

Shape Optimization Problem

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• for example:
• 𝑔 ≔ make shape float

subject to

• 𝑕 ≔ keep upright orientation

𝑻 𝑻 𝑻

• minimize objective 𝑔 as a function of 𝜺 :
• subject to constraints 𝑕(𝜺)

min

𝜺 𝑔 𝜺

• s. t. 𝑕(𝜺)

Shape Optimization Problem

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𝑻 𝑻 𝑻

• minimize objective 𝑔 as a function of 𝜺 :
• issues:
• problem is huge for large meshes

 scales very badly

• problem is underdetermined

 there exist many solutions (regularization needed)

min

𝜺 𝑔 (𝜺)  𝑜 unknowns

Shape Optimization Problem

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𝑻 𝑻 𝑻

min

𝜺 𝑔 (𝜺)

• minimize objective 𝑔 as a function of 𝜺 :
• issues:
• problem is huge for large meshes

 scales very badly

• problem is underdetermined

 there exist many solutions (regularization needed)

Order Reduction

Order Reduction

• order reduction:
• lower the dimensionality while

preserving input-output behavior

• idea:
• project problem onto a lower

dimensional space

•  Manifold Harmonics

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Manifold Harmonics

• diagonalization of the Laplacian matrix L

 Spectral Theorem:

• generalization of the Fourier Transform

for scalar functions on surfaces

[VALLET, B. AND LÉVY, B. 2008. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27, 2, 251–260.]

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𝐌 = 𝚫𝚳𝚫𝐔

Manifold Harmonics

• diagonalization of the Laplacian matrix L

 Spectral Theorem:

• generalization of the Fourier Transform

for scalar functions on surfaces

[VALLET, B. AND LÉVY, B. 2008. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27, 2, 251–260.]

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𝐌 = 𝚫𝚳𝚫𝐔

Order Reduction

• project unknown offsets

𝜺 = 𝜀1, 𝜀2,…, 𝜀𝑜 𝑈onto 𝚫𝑙 :

• vector 𝜷 = 𝛽1, 𝛽2,…, 𝛽𝑙 𝑈

now contains the unknowns!

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𝜺 = 𝚫

𝑙𝜷

𝛽1𝜹1 𝛽2𝜹2 𝛽3𝜹3 𝛽4𝜹4 𝛽5𝜹5 𝛽6𝜹6 𝛽7𝜹7 𝛽𝑙𝜹𝑙

=

Order Reduction

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𝒚𝒋 = 𝒚𝒋 + 𝜀𝑗𝒘𝒋 𝒚𝒋 = 𝒚𝒋 +

𝑘=1 𝑙

𝛽𝑘𝛿𝑗𝑘 𝒘𝒋

• project unknown offsets

𝜺 = 𝜀1, 𝜀2,…, 𝜀𝑜 𝑈onto 𝚫𝑙 :

𝛽1𝜹1 𝛽2𝜹2 𝛽3𝜹3 𝛽4𝜹4 𝛽5𝜹5 𝛽6𝜹6 𝛽7𝜹7 𝛽𝑙𝜹𝑙

Reduced Shape Optimization Problem

• minimize objective 𝑔 as a function of 𝜷:
• (subject to constraints)

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min

𝜷 𝑔 (𝜷)

𝛽1𝜹1 𝛽2𝜹2 𝛽3𝜹3 𝛽4𝜹4 𝛽5𝜹5 𝛽6𝜹6 𝛽7𝜹7 𝛽𝑙𝜹𝑙

Reduced Shape Optimization Problem

• minimize objective 𝑔 as a function of 𝜷:

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min

𝜷 𝑔 (𝜷)  𝑙 unknowns, 𝑙 ≪ 𝑜

We deform only the low-frequencies and leave high-frequency details untouched! 𝛽1𝜹1 𝛽2𝜹2 𝛽3𝜹3 𝛽4𝜹4 𝛽5𝜹5 𝛽6𝜹6 𝛽7𝜹7 𝛽𝑙𝜹𝑙

𝜺 = 𝚫

𝑙𝜷

Reduced Shape Optimization Problem

• minimize objective 𝑔 as a function of 𝜷:

 independent of mesh resolution  implicit regularization  numerically stable  easy to implement

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𝜺 = 𝚫

𝑙𝜷

min

𝜷 𝑔 (𝜷)  𝑙 unknowns, 𝑙 ≪ 𝑜

Applications I: Mass Properties

Example

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Center of Mass Center of Buoyancy Gravity: 𝑮𝑕 Buoyancy Force: 𝑮𝑐

Equilibrium 𝑮𝑕 = 𝑮𝑐 Mass Properties 𝑸(𝑻)

Applications

• Gauss’ Divergence Theorem
• allows us to compute mass properties

as a function of the surface

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𝑸𝒏 𝑻 = 𝑁 𝑸𝒚,𝒛,𝒜(𝑻) = 𝑫𝒑𝑵 = 𝑑𝑦 𝑑𝑧 𝑑𝑨 𝑼 𝑸𝒚𝟑,𝒚𝒛,…,𝒜𝟑(𝑻) = 𝑱 = 𝐽𝑦2 𝐽𝑦𝑧 𝐽𝑦𝑨 𝐽𝑦𝑧 𝐽𝑧2 𝐽𝑧𝑨 𝐽𝑦𝑨 𝐽𝑧𝑨 𝐽𝑨2

Center of Mass Center of Buoyancy

Applications

• optimization problem
• an analytical gradient

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min

𝜷

𝑔 𝑸 𝑻 𝜺(𝜷) 𝛼𝑔 = 𝜖𝑔 𝜖𝑸 𝜖𝑸 𝜖𝑻 𝜖𝑻 𝜖𝜺 𝜖𝜺 𝜖𝜷

Center of Buoyancy Center of Mass

Applications

• static stability
• monostatic stability
• rotational stability
• static stability under storage
• volume and buoyancy

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Spinning Turtle

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Rabbit Rolly-Polly

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Balanced Bottles

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Applications II: Modal Synthesis

Natural Frequencies

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Frequency: 440 Hz Concert pitch A

Natural Frequencies

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1st mode (pitch) 2nd mode (1st overtone) 3rd mode (2nd overtone) 440 Hz 1060 Hz 2790 Hz

Overtone spectrum ⟺ characteristic sound of object …

Natural Frequencies

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• material
• shape

Natural modes depend on

Goal: “Modal Synthesis”

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shape

• ptimization

fabrication

Vertex-Normal Parametrization

• Large offsets and high curvatures

⟹ self-intersections

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Shape Parametrization

• Use Reduced Basis with Manifold Harmonics
• Define offsets 𝜺 as linear combination of basis functions 𝚫𝒍

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𝜺 = 𝚫

𝑙𝜷

= 𝒀 +

𝒋=𝟐 𝒍

Shape Optimization

• Use non-linear optimization routine (Matlab)
• 𝑞0 … target pitch
• 𝑞 ... pitch of incument solution
• 𝜷 … coefficient vector

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min

𝜷 𝑔 𝜷 = 𝑞 − 𝑞0 2

Fabrication

• Material
• good acoustic properties
• cast into complex shape
• Tin
• melting point of 230°C
• Young’s modulus of 50 GPa

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Fabrication

• Oval bell
• molds from sand
• Rabbit bell
• molds from caoutchouc

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Bell

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Bell

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Rabbit

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Rabbit

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Results

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• 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

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

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Conclusions

• we proposed a novel framework

for shape optimization

• we provide an elegant and

efficient basis-reduction

• we demonstrate the method by
• ptimizing
• mass properties
• natural frequencies

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Publications

• Musialski, P., Auzinger, T., Birsak, M., Wimmer, M. & Kobbelt, L.

Reduced-Order Shape Optimization Using Offset Surfaces. ACM

• Trans. Graph. (Proc. ACM SIGGRAPH 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 SIGGRAPH 2015 Posters - SIGGRAPH ’15 1–1

(ACM Press, 2015).

• Hafner, C. Optimization of Natural Frequencies for Fabrication-

Aware Shape Modeling, Master Thesis, TU-Wien (2015)

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Thank you!

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