STATICS AWARE VORONOI GRID-SHELLS

Transcript

STATICS AWARE
VORONOI GRID-SHELLS
research seminar
Bath, 18 December 2014
Ph.D. student: Davide Tonelli
Host: Dr. C. J. K. Williams
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Davide Tonelli – [email protected]
Università di Pisa, Dip.to DESTeC
Statics Aware Voronoi Grid-Shells
Free-form grid-shells, designed with a novel
polygonal pattern, that are at the same time:
- innovative;
- aesthetically pleasing;
- structurally sound.
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Grid-Shells: Before and Now
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What is a Grid-Shell
A colander is still a shell.
A sieve is a grid-shell.
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A well shaped SHELL (i.e. that minimizes some
domain integral based-function, such as the
strain energy U)
with enough boundary support (i.e. restrained in
such a way that only deformations involving
length changes over the surface are allowed)
is a very effective structure.
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A well shaped GRID-SHELL
with enough boundary support
might not be a very effective structure
because many other variables affect its behaviour.
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Among these, the joint rigidity and the
grid TOPOLOGY
are the most crucial.
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What is a Voronoi diagram
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What is a Voronoi-like
Topology
It is a hex-dominant grid TOPOLOGY with
constant vertex valence (3).
It is the only topology that leads to semi-regular
meshes (i.e. not with the same polygon repeated)
with all regular vertices (i.e. with the same
valence 3).
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Voronoi-like Topology in Nature
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Voronoi-like Topology reproduced
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Topology and Mechanics
M_tot = Const
does not
guarantee
equivalence in
terms of statics.
M_tot = Const +
L_tot = Const
does.
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Every n-gonal polygon is a mechanism, unless n-3
vertices are rigid.
Whichever polygonal cell is less stiff than a triangle.
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Statics Awareness - Intuition
The grid-shell remeshing is driven by the statics
of the underlying surface.
THE INTUITION:
Making smaller cells over high stress regions and
stretching them along the principal stress
direction, in such a way that the global strain
energy U is minimized, yields grid-shells with
optimal statics.
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Statics Awareness - Pipeline
1- Find out the state of stress of the surface
underlying the grid-shell. The surface is FIXED;
2- Define local density and anisotropy functions
for sizing and shaping the cells, respectively,
according to the local state of stress.
3- Remesh the surface, thus obtaining an optimal
grid-shell (i.e. that minimizes U).
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Statics Awareness
1- F.E.M. linear elastic analysis of the continuous
surface underlying the grid-shell. The stress
tensor is computed at each shell element.
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Statics Awareness
2 – Definition of local transfer criteria.
A transition from the continuum shell to the discrete
optimal Voronoi grid-shell:
The local stress tensor ψf(p) is related to a tailored
metric tensor ψe(p) by means of the strain energy.
ψe brings about a new non-Euclidean metric over
the surface, that maps a unit circle on the tangent
plane at p into an optimal ellipse of radiuses (a,b),
aligned along the principal directions at p.
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Statics Awareness
2 – Definition of
local transfer
criteria.
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Statics Awareness
DENSITY:
Ud (|u|,L ) ∝ |u|
u|2 L3
The density strain energy is
quadratic in the principal stress norm
and cubic in the edge length.
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Statics Awareness
ANISOTROPY:
|v| 𝑏
|v| 𝑏
min Ua (
, )⟹
∝
|u| 𝑎
|u| 𝑎
The anisotropy strain energy
iss minimized when this
relationship holds.
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Statics Awareness
NEW METRIC:
D = (|u||v|) 2/3
gψ = W-T W-1
|u|
A=
|v|
W=
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Statics Awareness
3 – Surface Remeshing.
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Statics Awareness
Parameters D, A can be manually tuned (i.e. they
can be limited and rescaled within a userdefined scale, usually ranging from 1 to 4).
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Statics Aware Voronoi Grid-Shells
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Mesh Quality Improvement
Regularization
algorithm.
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Regularity – comparison of results
Shape - Up
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Reg = ∑di2 /A
Stat. Aw. Voro.
Planarity – comparison of results
Shape - Up
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Plan = 2dav /P
Stat. Aw. Voro.
Performances - preliminary assessment
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Botanic dataset
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Botanic dataset – axial forces N distribution
Utot = 81329 J
λ = 0.83, δ = 335 mm
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Utot = 32240 J
λ = 1.48, δ = 172 mm
Lilium dataset
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Lilium dataset – axial forces N distribution
Utot = 2999 J
λ = 1.42, δ = 77.6 mm
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Utot = 1846 J
λ = 2.97, δ = 37.6 mm
Performances - further assessment
Neumunster dataset
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British Museum dataset
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Aquadom dataset
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Botanic dataset
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A physical mock-up
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A physical mock-up
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Conclusions
This dissertation provides:
1- A thorough insight in the mechanics of
polygonal, hex-dominant grid-shells;
2- A proposal for a novel and effective kind of
free-form hex-dominant grid-shells:
the ‘Statics Aware Voronoi Grid-Shells’.
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Thank you for participating
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STATICS AWARE VORONOI GRID-SHELLS

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