Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps

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


Motivation: Damping of Elastic Waves in Lattice Structures

?



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)



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)



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



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


Shape Mapping


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.; . . .



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



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



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



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


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


relative gap 2γ / (α-γ)normalized gap 2γ / α



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




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




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




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



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

%


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


Poisson’s ratio

volume fraction 0.5 . . . 0.7


Conclusions & Summary


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



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



Further Applications: Pressure Drop with Perimeter Constraint

benefit from design restriction



Further Applications: Overhang Constraints

based on a±w “slope” constraints (inspired by Oded Amir)



Thank you for your attention



Heat Tracking



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


Parametric Shape Optimization of Lattice Structures for Phononic Band Gaps

Engineering

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