Volumetric Michell Trusses for Parametric Design & Fabrication
Rahul Arora, University of Torontowith
Alec Jacobson, Timothy R Langlois, Yijiang Huang, Caitlin Mueller, Wojciech Matusik, Ariel Shamir, Karan Singh, David IW Levin
Structural Optimization
The design of optimal load-carrying structures
2Engineer image by GraphicMama-team from Pixabay.
Structural optimization algorithm
Load
Fixed Boundary
Topology Optimization
Structural optimization methods that can introduce topological changes
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A user-centric approach to structural optimization.
By generating a parametrized output, our method generates structures that can be easily controlled and edited a posteriori.
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Background
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[A.G.M. Michell. 1904. The limits of economy of material in frame-structures.]
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[A.G.M. Michell. 1904. The limits of economy of material in frame-structures.]
Position material along the directions of principal stresses
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Stress magnitude visualization
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𝜎 ≡ 𝑑×𝑑 symmetric matrixStress 𝑑 = 2 in 2D, 𝑑 = 3 in 3D
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𝜎 = 𝑄)Λ𝑄+Stress (Symmetric matrix)
Orientation(rotation matrix)
Scaling (diagonal matrix)
Eigendecomposition of a stress matrix
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𝜎 = 𝑄)Λ𝑄+Stress (Symmetric matrix)
Orientation(rotation matrix)
Scaling (diagonal matrix)
Eigendecomposition of a stress matrix
Principal stress directions
𝒗𝟏𝒗𝟐
Computed eigenvectors
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𝒗𝟏𝒗𝟐
Computed eigenvectors
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Cross-field symmetry
𝒗𝟐
𝒗𝟏
Computed eigenvectors
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Cross-field symmetry
𝒗𝟏
𝒗𝟐
Computed eigenvectors
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Cross-field symmetry
𝒗𝟏
𝒗𝟐
Computed eigenvectors
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𝒗𝟐
𝒗𝟏
Isotropic stress tensor(tensor field singularity)
Method
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Algorithm: Overview
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Step 0: Problem Specification Step 1: Stress-field Step 2: Stress-aligned frame-field
Step 3: Texture parametrization Step 4: Truss layout Finally: Editing & fabrication
Later…
1. Stress Field Computation
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Problem description Stress field
2. Stress-Aligned Frame-Field Generation
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Stress field Smooth frame field
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𝑅 =| | |𝒓𝟏 𝒓𝟐 𝒓𝟑| | | 3×3Frame
(𝑑×𝑑 rotation matrix)
𝒓𝟏, 𝒓𝟐, 𝒓𝟑 are unit vectors
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𝜎 = 𝑄)Λ𝑄+Stress (Symmetric matrix)
𝜎5 = 𝑄Λ𝑄+Orientation
(rotation matrix)
Scaling (diagonal matrix)
Positive-definite matrix
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𝑅 =| | |𝒓𝟏 𝒓𝟐 𝒓𝟑| | | 3×3
Frame(𝑑×𝑑 rotation matrix)
𝒓𝟏, 𝒓𝟐, 𝒓𝟑 are unit vectors
𝐸789:; 𝑅 = (𝒓𝟏+𝜎5𝒓𝟏) ⁄? @ + (𝒓𝟐+𝜎5𝒓𝟐) ⁄? @ + (𝒓𝟑+𝜎5𝒓𝟑) ⁄? @
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𝜆? ≠ 𝜆@ ≠ 𝜆D
𝜆? = 𝜆@ ≠ 𝜆Dor
𝜆? ≠ 𝜆@ = 𝜆D 37
𝜆? = 𝜆@ ≠ 𝜆Dor
𝜆? ≠ 𝜆@ = 𝜆D 38
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𝜆? = 𝜆@ = 𝜆D
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𝐸 = E9F?
T
𝐸GHIJK9 + 𝛼𝐸MNOOPQ
Summed over all tets
Laplacian-based smoothness cost
𝛼
3. Texture Parametrization
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Frame field Frame-aligned parametrization
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𝜙:M → ℝ3ℝ3-valued
parametrization
∇𝜙 = 𝑅
4. Truss Layout Extraction
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Global parametrization Truss layout
Results
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— 𝑑 orthogonal families of smooth end-to-end curves—Curves in each family are identified with a pair of integers—Each curve itself is parametrized
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— 𝑑 orthogonal families of smooth end-to-end curves—Curves in each family are identified with 𝑑 − 1 integers—Each curve itself is parametrized
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— 𝑑 orthogonal families of smooth end-to-end curves—Curves in each family are identified with a pair of integers—Each curve itself is parametrized
55Image Source: William Dwight Whitney The Century
Dictionary: An Encyclopedic Lexicon of the English Language (New York, NY: The Century Co., 1911)
Entasis (Greek/Roman architecture)
Structural Tests
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Ours
Regular grid(unoptimized)
GRAND3[Zegard and Paulino 2015]
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Limitations and Future Work
—Manufacturing constraints not accounted for—Wire-bend each curve—Generate construction sequences for dowel assembly
—Sizing optimization
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Acknowledgements
We thank Lawson Fulton and Sarah Kushner for their immense help with renderingthe results, Peter Hamilton for narrating the video, and other members of the DGP labfor helping with the structural tests.
This research was funded in part by NSERC Discovery (RGPIN-2017-05524, RGPIN-2017-05235, RGPAS-2017-507938), NSERC Accelerator (RGPAS-2017-507909), NewFrontiers in Research Fund (NFRFE–201), UofT Connaught Fund 03114, CanadianFoundation for Innovations John Evans Leadership Fund, the Ontario Early ResearchAward program, the Canada Research Chairs program, the Fields Centre forQuantitative Analysis and Modelling, the Adobe Research Fellowship program, andgifts by Adobe Systems, Autodesk and MESH Inc.
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Open-source! (MIT License)
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https://github.com/rarora7777/VolumetricTruss
Project page https://www.dgp.toronto.edu/projects/michellCode https://github.com/rarora7777/VolumetricTrussContact [email protected]
Volumetric Michell Trusses for Parametric Design & FabricationRahul Arora, Alec Jacobson, Timothy R. Langlois, Yijiang Huang, Caitlin Mueller, Wojciech Matusik, Ariel Shamir, Karan Singh, David I.W. Levin
