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Phononic Band Gaps Shape Mapping Results Resume
Parametric Shape Optimization of Lattice Structuresfor Phononic Band Gaps
Fabian Wein and Michael StinglFriedrich-Alexander-University Erlangen-Nurnbeg (FAU)
WCSMO-12 2017
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Motivation: Damping of Elastic Waves in Lattice Structures
?
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Floquet-Bloch Wave Theory
periodic structure
square symmetry
wave vector k = (kx ,ky )
Hermitian EV problem(K(k)−ω2M
)Φ = 0
ky
kx
G X
M
Γ X M Γ
eig
en
fre
qu
en
cy
wave vector (IBZ)
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Floquet-Bloch Wave Theory
periodic structure
square symmetry
wave vector k = (kx ,ky )
Hermitian EV problem(K(k)−ω2M
)Φ = 0
ky
kx
G X
M
Γ X M Γ
eige
nfre
quen
cy
wave vector (IBZ)
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Floquet-Bloch Wave Theory
periodic structure
square symmetry
wave vector k = (kx ,ky )
Hermitian EV problem(K(k)−ω2M
)Φ = 0
ky
kx
G X
M
Γ X M Γ
eige
nfre
quen
cy
wave vector (IBZ)
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Floquet-Bloch Wave Theory
periodic structure
square symmetry
wave vector k = (kx ,ky )
Hermitian EV problem(K(k)−ω2M
)Φ = 0
ky
kx
G X
M
Γ X M Γ
eige
nfre
quen
cy
wave vector (IBZ)
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Phononic Band Gaps
0
100
200
300
400
500
600
700
800
900
1000
Γ X M Γ
eig
en
freq
ue
ncy in
Hz
wave vector (IBZ)
contrast 1:10first optimization: Sigmund, Jensen; 2003
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Lattice Structures with Phononic Band Gaps (Selection)
Manual
Warmuth, Korner; 2015
Non-gradient based optimization
Bilal, Hussein; 2011 & 2012
Dong, Wang, Zhang; 2017
Gradient based optimization
Halkjær, Sigmund, Jensen; 2006
Andreassen, Jensen; 2014
Fabian Wein Band Gap Maximization via Shape Mapping
Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Geometry Projection Methods
map from geometries to pseudo density fieldsensitivity analysis: basically “SIMP” + chain rule
(xi,yi,ri)
Kumar, Saxena; 2015 (MMOS)
Norato et al.; 2015
also Dunning et al.; . . .
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Shape Mapping
technically similar togeometry mapping
parametric shape optimization
horizontal/ vertical “stripes”
radically reduced design space
close control on design
Nx + 1 positional parameters a
Nx + 1 profile parameters w
piecewise linear interpolation
45→ thickness 2w√2
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
a1 a2 a3
a4
a5
a6
w1
w1
w2
w2
w3
w3w4
w4
w5
w5 w6
w6
FEMcell
p
p
S
integration points
ρ1 ρ2 ρ3...
p1
p1
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Differentiability
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
pseu
do d
ensi
ty ρ
space in m
a
2 w
tβ (x ,a,w) =
1− 1
exp (β (x−a+w)) + 1ifx < a
1
exp (β (x−a−w)) + 1else
ρe = Te(a,w,β ) = ρmin + (1−ρmin)∫
Ωe
tβ (x,a(x),w(x)) dx
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Overlapping
(a) max (b) tanh sum
(a) max: ρ ′ =∫
Ω maxs tβ (x,a,w) dx
(b) tanh sum: ρ ′ =∫
Ω min∗(1,∑s tβ (x,a,w)) dx
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Problem Formulation: Normalized Band Gap Maximization
maxa,w,α,γ
2γ
α
s.t. ωjl ≤ α− γ, 1≤ j ≤ 6, 1≤ l ≤ 3
ωjl ≥ α + γ, 1≤ j ≤ 6, 4≤ l ≤ 12(K(kj ,ρ)−ω
2jlM(ρ)
)Φjl = 0, 1≤ j ≤ 6, 1≤ l ≤ 12
ρe = Te(a,w,β )
|ai −ai+1| ≤ 1.1/N
|ai−1−2ai +ai+1| ≤ c∗/N
|wi−1−2wi +wi+1| ≤ c∗/N
ai ∈ [0,0.5], 1≤ i ≤ N/2
wi ∈ [W ∗−,0.2], 1≤ i ≤ N/2
square symmetry: half strip → four stripsFabian Wein Band Gap Maximization via Shape Mapping
Results
Phononic Band Gaps Shape Mapping Results Resume
Dependency on Minimal Profile Width
0
500
1000
1500
2000
2500
0.04 0.08 0.12 0.16 0.20
eig
en
fre
qu
en
cy in
Hz
minimal profile width
min mode 4max mode 3
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
0.04 0.08 0.12 0.16 0.200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
rela
tive
gap
2γ/ (
α-γ)
norm
aliz
ed g
ap 2
γ/ α
minimal profile width
relative gap 2γ / (α-γ)normalized gap 2γ / α
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Results: Minimal Profile Width 0.04
0
500
1000
1500
2000
2500
3000
3500
O A B C
eig
enfr
equency in H
z
wave vector (IBZ)
rel=8.32, norm=1.61, W ∗−=0.04/2, β = 300, c∗=0.05
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Results: Minimal Profile Width 0.08
0
500
1000
1500
2000
2500
3000
3500
O A B C
eig
enfr
equency in H
z
wave vector (IBZ)
rel=4.30, norm=1.37, W ∗−=0.08/2, β = 250, c∗=0.11
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Results: Minimal Profile Width 0.12
0
500
1000
1500
2000
2500
3000
3500
O A B C
eig
enfr
equency in H
z
wave vector (IBZ)
rel=2.16, norm=1.04, W ∗−=0.12/2, β = 350, c∗=0.09
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Results: Minimal Profile Width 0.16
0
500
1000
1500
2000
2500
3000
3500
O A B C
eig
enfr
equency in H
z
wave vector (IBZ)
rel=1.40, norm=0.85, W ∗−=0.16/2, β = 250, c∗=0.1
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Observed Properties
0
1
2
3
4
5
6
7
8
9
0.04 0.08 0.12 0.16 0.20
Yo
un
g’s
mo
du
lus E
1/2
in
%
minimal profile width
Young’s modulus-0.025
-0.020
-0.015
-0.010
-0.005
0.000
0.005
0.04 0.08 0.12 0.16 0.20
Po
isso
n’s
ra
tio
minimal profile width
Poisson’s ratio
volume fraction 0.5 . . . 0.7
Fabian Wein Band Gap Maximization via Shape Mapping
Conclusions & Summary
Phononic Band Gaps Shape Mapping Results Resume
Summary
Obtained band gap design
there appears to be a unique design principle
. . . within the limited design space
Technical details
band gap problem difficult to solve (SNOPT)
independent on curvature bound c∗ and smoothing parameter β
→ “random shot”
Shape mapping
close control on design . . . yet versatile
clearly defined grayness at interface
allows topological changes
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Further Applications: Tracking of Interface Driven Heat Source
inte
rfac
e he
at s
ourc
e
benefit from strict interface for interface driven heat source
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Further Applications: Pressure Drop with Perimeter Constraint
benefit from design restriction
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Further Applications: Overhang Constraints
based on a±w “slope” constraints (inspired by Oded Amir)
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Thank you for your attention
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Heat Tracking
Fabian Wein Band Gap Maximization via Shape Mapping
Phononic Band Gaps Shape Mapping Results Resume
Heat Tracking cont.
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
Fabian Wein Band Gap Maximization via Shape Mapping