Understanding Gridshell Structures - Mannheim Multihalle Case Study
Tying connections between Graphics & Mechanics€¦ · elastic gridshell generated? Anatomy and...
Transcript of Tying connections between Graphics & Mechanics€¦ · elastic gridshell generated? Anatomy and...
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Elastic Rods: Tying connections between
Graphics & Mechanics
Bernhard Thomaszewski & Pedro Reis
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Mechanics of slender structures: ‘Buckliphobia’ & ‘Buckliphilia’:Our goal is to embrace mechanical instabilities of thin (soft) structures, guided primarily through precision model
experiments, towards understanding and exploiting novel functional mechanisms over a wide range of length scales.
With a focus on P.D.E.s (Precision Desktop Experiments)Plenty of opportunities
for discovery & innovation
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Research ThemesPhysical Surface Structured Materials Mechanism Design Robotics Visual Simulation
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Elastic Rods
Elastic Rods
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Filamentary Structures…
… in nature and engineering.
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Elastic Kirchhoff Rods
Gustav Kirchhoff Alfred Clebsch Augustus E. H. Love
“… the classical theory of thin rods as developed by Kirchhoff [1859, 1876] and Clebsch [1862], and presented by Love [1892, 1906].”
E.H. Dill. Kirchhoff’s Theory of Rods. Archive for History of Exact Sciences. Vol. 44, No. 1 (1992)
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Represent rod as an adapted framed curve Γ s = {𝜸 𝑠 ; 𝐹(𝒔)}
Elastic Kirchhoff Rods
11 mt =b22 mt =b
Bending:
21 mm =t
Twist:
• 𝜸 𝑠 :𝐑 → 𝐑3 and arc-length parameterized curve describing the rod’s centerline
• 𝐹 𝑠 = {𝒕 𝑠 ,𝒎1 𝑠 ,𝒎2 𝑠 } an orthonormal material frame adapted to centerline, i.e., 𝒕 = 𝜸′
dsstksbsbkE tb
l22
2
2
10
)(])()([)( ++=
Stored energy*:
* for an initially-straight, isotropic rod
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Rod Models from Graphics
[Gregoire & Schoemer ‘07]
[Spillmann & Teschner ‘07] [Spillmann & Teschner ‘08]
[Bertails et al. ‘06] [Bertails ‘09] [Casati & Bertails-Descoubes ‘13]
[Bergou et al. ‘08] [Bergou et al. ‘10][Pai ‘02]
[Umetani et al. ‘13]
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Thin Rods: Experiments
Simulation of ~100,000 hairs using DER
Thin Rods: Numerics
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Example 1: The elastic sewing machine & bacterial locomotion
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Pipe-
Lines
Under
The
Ocean
Deployment of pipelines/cables onto seabed
Deployment of subsea pipelines
https://www.pri.org/stories/2015-04-20/what-links-global-internet-wires-inside-tubes-no-bigger-garden-hose
Internet under-sea cable network
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Deployment of an elastic rod onto a moving substrate
Excellent agreement between experiments (PDEs)and simulations (DER) with no fitting parameters!
Translated coilingAlternating loopsMeandering
Exp
erim
ents
Sim
ula
tio
ns
` `
[PNAS 2014][Extreme Mech. Lett. 2015]
[J. App. Mech. 2015]
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Excellent agreement between experiments (PDEs)and simulations (DER) with no fitting parameters!
[PNAS 2014][Extreme Mech. Lett. 2015]
[J. App. Mech. 2015]
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Stepper Motor
LEDBacklighting
OrthogonalSLR Cameras
SoftFilament
GlycerinTank
Viscosity: Helical filament:
Length, LPitch, λradius Rn, λ
R. Sphaeroides
H. Berg (Harvard)
[J. Lighthill, "Flagellar hydrodynamics." SIAM Review (1976).]
► Couple DER code for rods with Lighthill's Slender Body Theory:
► Velocity u(s) at flagellum vs. force f(s) exerted by fluid.
local non-local
Cutoff
95% of bacteria in the ocean locomotive themselves by rotation of a single flexible
helical flagellum
Dynamics of rotating helical rods in viscous fluids Bacterial locomotion
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Dynamics of rotating helical rods in viscous fluids Bacterial locomotion
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Example 2:Structured Sheet Materials
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Natural Network Materials
Muscle tissue (actin filaments) Cancellous bone (trabeculae) Vein structure (chitin, resilin)
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Digital Network Materials
[Zhou et al. 2015]
[Schumacher et al. 2018]
[Carbon]
[Renishaw][Zhou et al. 2015]
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3D-Printed Fabric
Danit Peleg
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Elastic Isohedral Tilings
Schumacher, Marschner, Gross, Thomaszewski. Structured Sheet Materials. SIGGRAPH ‘18.
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3D-Printed Tilings
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Rod Network Mechanics
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Simulation
Discrete Elastic Rods ([Bergou '08,’10])
+ Extension to Networks ([Perez ‘15], [Zehnder ‘16])
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Mechanical Characterization
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Mechanical Characterization
……
𝝐1, 𝝈1
𝝐2, 𝝈2
𝝐𝑛, 𝝈𝑛
𝜿1,𝑊1
𝜿2,𝑊2
𝜿𝑛,𝑊𝑛
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Macromechanical Model
ℂ(𝝐1, 𝝈1, 𝝐2, 𝝈2, … , 𝝐𝑛, 𝝈𝑛)
𝔹(𝜿1,𝑊1, 𝜿2,𝑊2, … , 𝜿𝑛,𝑊𝑛)
Membrane
Bending
𝑊 = 𝜖 ∶ ℂ ∶ 𝜖 + 𝜅 ∶ 𝔹 ∶ 𝜅
Membrane Bending
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Macromechanical Representation
Young’s modulus Poisson’s ratio Bending stiffness
0.1
% s
tra
in1
0%
str
ain
0.1
m-1
cu
rva
ture
5m
-1cu
rva
ture
exact
fitted
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Exploration
http://www.structuredsheets.com
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Inverse Design
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Example 3:Elastic gridshells
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Shaping through buckling in elastic gridshells
Forum Café Gridshell for Solidays Festival, Paris (2011)
thinkshell.fr/elastic-gridshell/
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Mannheim Multihalle, Germany
Frei Otto, 1975
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Anatomy and Actuation of an Elastic Gridshell
Footprint (rest configuration)* Quadrilateral grid
Joints:* VPS* diameter* heightRods:
* Nitinol* *
Boundary points: * 3D printed ball joints* Pinned B.C.s
Original boundary of footprint
Actuated boundary
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Footprint (rest configuration)* Quadrilateral grid
Joints:* VPS* diameter* heightRods:
* Nitinol* *
Boundary points: * 3D printed ball joints* Pinned B.C.s
Given a target shape, what should be the footprint (inverse design)?
Given a footprint, what is the elastic gridshell generated?
Anatomy and Actuation of an Elastic Gridshell
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Fabrication of a
hemispherical
gridshell
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3D Digital Scanning (NextEngine)
Excellent quantitative agreement between exps. & DER.
Multiplicity of states for same input parameters.
Physical Experiments v. DER
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Probability, Chebyshev polynomials, number theory
Pafnuty Chebyshev (1821-94)
Probability, Chebyshev polynomials, number theory
How do textiles drape?
Weft
Warp
Chebyshev Net (1878):
Inextensible rods in directions:
(Shearing) angle changes of tangent vectors at a point deform the metric:
Maps 2D-to-3D:
Theory of Chebyshev Nets
For [Chebyshev, 1878]Gauss equation for Chebyshev nets:
3D 2D
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2DFootprint
Actuatedhemisphericalgridshell
3D Digital Scanning
Experiments
Chebyshev HemisphereDER simulations
Excellent quantitative agreement! Max. deviation ~2%
Spherical Elastic Gridshells
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Strainer
Cut
FlattenedStrainer
ChebyshevPrediction
Let’s cut a pasta strainer
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Quarter-sphere + Cylinder Quarter-sphere + Saddle
Building with blocks: more complex gridshells
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For Hemispherical shell:[Reissner, 1946]
Q: What is for hemispherical gridshell?
For Euler-Bernoulli beam:
Rigidity for a gridshell under point indentation?
Amazing collapse into 3 master curves!
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Example 4:Self-deploying Surfaces
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Self-Deploying Surfaces
Paik lab (RRL), EPFL
NASA
Kovac et al. (2009)
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Self-Deploying SurfacesPerez, Otaduy, Thomaszewski. Kirchhoff Plateau Surfaces. SIGGRAPH ‘17.
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Nonlinear Mechanics
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Constrained Design Space
Minimal surfaces have zero mean curvature throughout,
𝜅1 + 𝜅22
= 0
Minimal surfaces are either• locally flat
𝜅1 = 𝜅2 = 0• or saddle-shaped
𝜅1 = −𝜅2 ≠ 0
Frei Otto
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Kirchhoff Plateau Problem
www.exploratorium.edu
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Computational Model
Nonlinear Mechanics
Rod control points and cross sections
Parameters
Coupled membrane and rod elements (collocation)
Simulation Mesh
Discrete Elastic Rods
CST finite elements with orthotropic material
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Forward Design
Deployed StateDesign Space
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Inverse Design
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Exploring Design Variations
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Concluding thoughts…Computer graphics tools and physically-based simulation
offer unprecedented opportunities as quantitatively predictive
tools in experimental mechanics.
Timely engineering applications & physical scenarios and
novel experimental mechanics tools can challenge and
push physically-based simulation into new grounds.
Let’s
Keep
MixingBernhard Thomaszewski
Pedro Reis