![Page 1: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/1.jpg)
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
![Page 2: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/2.jpg)
Phononic Band Gaps Shape Mapping Results Resume
Motivation: Damping of Elastic Waves in Lattice Structures
?
Fabian Wein Band Gap Maximization via Shape Mapping
![Page 3: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/3.jpg)
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
![Page 4: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/4.jpg)
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
![Page 5: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/5.jpg)
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
![Page 6: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/6.jpg)
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
![Page 7: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/7.jpg)
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
![Page 8: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/8.jpg)
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
![Page 9: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/9.jpg)
Shape Mapping
![Page 10: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/10.jpg)
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
![Page 11: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/11.jpg)
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
![Page 12: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/12.jpg)
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
![Page 13: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/13.jpg)
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
![Page 14: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/14.jpg)
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
![Page 15: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/15.jpg)
Results
![Page 16: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/16.jpg)
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
![Page 17: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/17.jpg)
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
![Page 18: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/18.jpg)
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
![Page 19: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/19.jpg)
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
![Page 20: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/20.jpg)
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
![Page 21: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/21.jpg)
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
![Page 22: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/22.jpg)
Conclusions & Summary
![Page 23: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/23.jpg)
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
![Page 24: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/24.jpg)
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
![Page 25: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/25.jpg)
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
![Page 26: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/26.jpg)
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
![Page 27: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/27.jpg)
Phononic Band Gaps Shape Mapping Results Resume
Thank you for your attention
Fabian Wein Band Gap Maximization via Shape Mapping
![Page 28: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/28.jpg)
Phononic Band Gaps Shape Mapping Results Resume
Heat Tracking
Fabian Wein Band Gap Maximization via Shape Mapping
![Page 29: Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps](https://reader035.vdocuments.net/reader035/viewer/2022062523/5a6d734c7f8b9ade418b56c9/html5/thumbnails/29.jpg)
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