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Parametric Co-Design of Modular Free-Form 2-Manifolds

This conference presentation discusses the benefits and challenges of parametric co-design in creating modular, free-form 2-manifold geometries. The speaker shares their experiences and lessons learned in using this design approach.

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Parametric Co-Design of Modular Free-Form 2-Manifolds

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  1. CAD Conference, Vancouver, 2016 Parametric Co-Design of Modular Free-Form 2-Manifolds Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. Parametric Design Time saver; amplifier of productivity. I have used it for several decades . . . • to create geometries that “belong to the same family,” • to fine-tune geometries using visual feedback: some shapes from “Sculpture Generator I”:

  3. Sculpture Generator 1, GUI

  4. Pitfalls in Parametric Design Changing parameters unscrupulously may lead to unexpected results! • Self-intersections in “Sculpture Generator I” • Another example:

  5. Parametric Co-Design Things get more challenging and more dangerous when several components are involved in the design: • Components may interact in many unforeseen ways… • as in these examples of my“Super-Bottle” sculptures:

  6. “LEGO Knots” (2014) • A project, a couple of years ago. • Modular components, based on sweeps of a fixed square cross section, which could be joined together to form models of mathematical knots & of free-form tubular sculptures. Extend this idea to non-orientable surfaces . . .

  7. Surface Characterization • Single-sided = non-orientable (σ=1) • Genus (g) measures connectivity Torus σ=2; g=1 Möbius band σ=1; g=1 Klein bottle σ=1; g=2

  8. Higher-Genus Surfaces • Grafting together Klein bottles . . . • does not always yield a higher genus surface! Just a Torus σ=2; g=1 (Cliff Stoll) Klein “Knottle” σ=1; g=2 Connected Sum of 2 KB σ=1; g=4

  9. Modular Sculptures • Yellow sculpture is made from two identical 2-sided pieces . . . • Their combination can be: “KBM” σ=2; g=2 σ=1; g=4 • Expand modularity concept to more complicated models of higher genus. The key is a branching module with a “KBM.” • “Klein-Bottle Mouth”

  10. Overall Modular Structure • Need an overall plan! • Follow the edge graph of a (semi-) regular polyhedron. • These 3-arm (valence-3) branching modulesrestrict us to stay with “cubic” graphs (all nodes are of valence 3). • Cube is straight-forward: Start with this . . .

  11. Overall Plan: Cube Frame • Modules occupy the corners. • Module arms are at right angles. joint

  12. Various “KBM” Junction Modules • Many possibilities! • Explored several of them,fitting a tubular cube-frame, that have a similar style and can be seen to belong into the same family. • 4 models: Different ways to introduce branching:

  13. Cube-Frame Assemblies • 2 Variants of a cube frame: Double-sided σ=2; g=5 Single-sided σ=1; g=10

  14. Cube-Frame Sculpture • An artistic result: • The main focus of my talk is to tell you how I got to this point, and what I learned in this process.

  15. Co-Design • The individual components have to be defined with an overall structure in mind. • But the overall structure cannot be judged and perfected until one has some prototype components that can be joined into a first tentative assembly.  Iterative process • Some modules originally designed on an individual basisjust did not want to fit into an overall sculpture. • It would be nice, if one could edit them in context and adjust any parameters as one sees the whole sculpture.

  16. Modeling Approach • My KBM modules are simple polyhedral meshes that roughly define the desired geometry. • Smoothed with 2-3 steps of Catmull Clark subdivision. Aim for mostly quad facets and vertices of valence 4;  yields good predictable results. • Some of the geometry is procedurally generated: • Torus body: R, r, for major and minor radii;parameters m, n, for polygonal “circles”; • Connecting arms: progressive sweeps (with octagonal cross sections). • When the shape is OK, provide thickness:  offset surfaces.

  17. Initial Mesh Geometries • Coarse polyhedral meshes,to be smoothed and refined by CC-subdivision. • Main body is based on an octagonal toroid,specified by: R(major), r(minor), m=n=8. Type_D Type_B Type_C Type_A Thick armsare branched Thin armsbranchoutside Thin armsbranchin torus Thin armsbranchin torus hull

  18. Adjusting the Overall Framework • Adjusting three global framework parameters(in a tetrahedral frame): A B C D E Armbulge Default Smallerframe Armdiameter Moduletilting * * Adjusting individual model parameters  

  19. Fine-Tuning a Module Geometry • Modifying various parameters of module Type_C Default Displaced toroid Reducinginner armdiameter Matchingarm radiusat joints Changingsize & tiltof toroid Outer ends of arm stubs stay in fixed positions:(progressive sweep is locked to joint locations) All parameters can be adjusted while looking at whole sculpture!

  20. Fabricating Physical Parts • When I was satisfied that I had a good overall design,I fabricated 4 pairs of 3-arm KBM moduleson a Fused Deposition Modeling (FDM) machine: Design file sent to printer: 2 Type_C modules, 12 connector rings.

  21. Physical Parts • Two Type_C modules: Thanks to the UC Berkeley Invention Lab and the Jacobs Institute for Design Innovation!

  22. Coupling Parts • Elasticity from:bending prongs torsional twisting Main stress areas Difficult to get right ! Depends heavily on print material & machine.

  23. Results: Cube Frames • Many possible placements of the KBM modules at the cube corners:

  24. Other (Semi-) Regular Cubic Graphs 3 of the Platonic Solids Truncated Platonic solids n-sidedprisms 3-wedge hosohedron

  25. Curved Connector Pieces • Example: “Connector_65.5” How they come out of the FDM machine Support material (black) needs to be removed

  26. Results: (Semi-) Regular Graphs 3-Sided Prism (g=8) 6 KBM + 6 conn._30 Tetrahedron (g=6) 4 KBM + 6 conn._39

  27. Results: Less Regular Structures • “Loopy” connections . . . Hosohedron σ=1; g=4 2 KBM + 3 conn. 109.5 Double-torus σ=2; g=2 2 KBM + 2 loops 270 3-sided prism σ=1; g=8 4 KBM + various conn.

  28. Other Regular Structures ? What about the remaining Platonic solids? • Dodedcahedron: Needs 20 valence-3 modules • Octahedron: Needs 6 valence-4 modules • Icosahedron: Needs 12 valence-5 modules • I wanted to study expanded Co-Design where I also had to deal with branching modules with different valences (e.g., 4-arm KBMs). • Can we build something interesting with relatively few such new components?

  29. Assemblies Using Valence-4 Parts • Constructing a 4-way “anti-pyramid” 3 new valence-4 parts would allow additional interesting assemblies:

  30. New 4-Way Branching Modules • Reuse some design details of the 3-arm modules!

  31. Realization of the 4-sided Anti-Pyramid • Using 6 3-arm KBMs and 2 4-arm KBMs:

  32. 3 More 4-arm Parts That is how they come out of the FDM machine:

  33. Assemblies of Multiple 4-arm Parts “5-Ring” σ=1 g=12 5 KBM + 10 conn. 80.3 Octahedron σ=1 g=14 6 KBM + 12 conn. 41.1

  34. Other Possible Assemblies • Enabled by the 4-arm KBM modules …

  35. Parametric Co-Design • Individual components have to be defined with the overall structure in mind – and vice versa! • Parametric design is somewhat tricky in any case, but gets even more challenging,when there exist multiple components that can be put together in many different ways. • In the case presented, I used parameters to fine tune the topology and geometry of a few components that assemble into many different tubular sculptures.

  36. System Design • What is the best collection of components that together form a complete, versatile “LEGO-like” building block set?

  37. An Optimal Building Block Set ?? • To optimally design the whole system, one really needs to know its ultimate scope! • Do we want to build icosahedral structures? then we also need 5-arm KBM junctions! • At what angles do we bring out the legs? • designed for the icosahedron (60) ? • or aimed at the triacontahedron (63.25). . .(but then we also need new 3-way KBMs). HOWEVER: A successful building block system is never finished… it wants to be expanded: Look at the LEGO system!

  38. A Somewhat Simpler Question: • What would I do differently if I had to start again?

  39. Rhombic Dodecahedron (genus 22/σ). • Needs 24 connectors that bend thru 5

  40. Always need curved connectors . . . • Always different bend angles!  Annoying!! 12× 80-connectors 3× 71 + 6× 5-connectors

  41. Adjustable Curved Connectors ?? • Flexible Hoses:

  42. Ball-&-Socket Joints • Principle: • Composed into tubular connectors:

  43. Aesthetic Compatibility ?? • Connectors should match with KBM modules! ???

  44. Custom-Designed Flex-Connector • Prototype – Proof of Concept: flexible meridian plane

  45. Flexing a 90 Connector • From about 30 to about 150

  46. Conclusions Parametric Design is a very useful tool: • It permits to quickly generate a whole lot of shapesthat all “belong to the same family.” • It permits to fine-tune and optimize parts of a designin the context of a complex overall system. • Here I have used it to simultaneously optimizethe overall structure of a sculptural assembly,as well as the geometry of the individual components. Hopefully, the details how this worked in my sculptureswill be useful to you at some point in your own designs.

  47. Q U E S T I O N S ? • Combining all 5 different KBMs

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