•Fine density/design variable
mesh
Computational Science and Engineering
2013 Annual Meeting
Tailoring Structural Dynamic Behavior through
Topology Optimization with Multiresolution Polygons Evgueni T. Filipov, Junho Chun, Glaucio H. Paulino, and Junho Song
University of Illinois at Urbana-Champaign
Email: [email protected] Website: www.ghpaulino.com
Motivation
Conclusions
• Topology optimization can be used for advanced structural
eigenfrequency or forced vibration problems
• Polygonal elements allow for optimization of complex
domains and improved solutions
• Multiresolution approach provides a higher resolution of the
topology for a lower computational cost
• National Science Foundation
Coarse
displacement
mesh
Acknowledgements
Polygonal multiresolution framework
𝑀𝑆
𝐴𝑖𝜌𝑖𝑁𝑛𝑖=1
=∞
𝑀𝑆
𝐴𝑖𝜌𝑖𝑁𝑛𝑖=1
=0
Maximize ω1
Suspended
mass MS
Cantilever with a suspended mass
Arch with band-gap maximization
ω3 - ω2 = 83.7Hz 319.2Hz
Incr
easi
ng
su
spen
ded
mas
s
𝑀𝑆
𝐴𝑖𝜌𝑖𝑁𝑛𝑖=1
=0.01
Fine design
variable mesh
Fine density
variable mesh
Comparison of forced vibration results 1. Conventional coarse mesh
3. Conventional fine mesh
2. Multiresolution mesh
Computational time
for the three meshes
• Topology optimization can be used
to tailor cost effective structures or
microstructures with specialized
dynamic characteristics
Eigenfrequency optimization
Forced vibrations
Minimize dynamic
compliance:
Maximize lowest
eigenfrequency:
maxρ λ𝑚𝑖𝑛 min
𝑗=1,…𝐽 ω𝑗
2
𝑉 ρ = ρ𝑑𝑉 ≤ 𝑉𝑠Ω
s.t.
𝑉 ρ = ρ𝑑𝑉 ≤ 𝑉𝑠Ω
s.t.
minρΦ(ρ) = 𝐅𝑇𝐔 𝑑ω
ω𝑒
ω𝑠
𝐊 + 𝑖𝜔𝐂 − 𝜔2𝐌 𝐔 = 𝐅
0 50 100 1500
200
400
600
Iteration
Eig
enfr
equ
ency
(
)
8
7
6
5
4
3
2
1
0 50 100 150 20050
100
150
200
250
300
350
Iteration
Eig
en
freq
uen
cy
(
)
4
3
2
1
ω1=205 Hz
ω1=68 Hz
Tim
e (
seco
nd
s)
1 2 30
200
400
600
800
1000
1200
1400
1600
FE Analysis
Optimization
InitializationT
ime (
seco
nd
s)
1 2 30
200
400
600
800
1000
1200
1400
1600
FE Analysis
Optimization
Initialization
Tim
e (
seco
nd
s)
1 2 30
200
400
600
800
1000
1200
1400
1600
FE Analysis
Optimization
Initialization
Buildings designed to minimize
seismic or wind vibrations, e.g.
tune mass damper of Taipei 101
(Figure on right)
Aircraft wings created to avoid
resonance from turbulence
Frequency passband and stopband
filters for use in audio equipment
Potential applications
Superposed
meshes
Polygonal elements can:
• Model irregular domains
• Avoid instabilities in optimization
• Provide mesh independent
solutions
Nonconforming
sub-discretization
ω2 = 119.7Hz ω3 = 203.4Hz
