System Reliability‐Based Topology Optimization of Structures under Stochastic Excitations
SKIDMORE, OWINGS & MERRILL LLP
Junho Chun, Junho Song, Glaucio H. Paulino
Department of Civil and Environmental Engineering, University of Illinois at Urbana‐Champaign, U.S.A
Research Objective
• Find optimal bracing systems under stochastic excitation, particularly induced by earthquake ground motions.
• Develop topology optimization framework Integrated with random vibration theory and structural reliability analysis.
(courtesy of SOM)
• Evaluate the system‐level failure probability accurately considering statistical dependence between failure modes, locations and time points.
Discrete Representation Method
Sequential Compounding Method (SCM)
The discrete representation method discretizes a continuous stochastic process with a finite number of standard normal random variables.
SCM can compute the probability of a large‐system reliability problem efficiently and accurately.
vj
Discretization of Random Process
f (t )   (t )   vi si (t )   (t )  s(t ) v
T
i 1
αˆ (t0 )
v* (u0 , t0 )
Stochastic Response
t
n
MPP
n
u (t )    vi si ( )hs (t  τ)dτ   vi ai (t )  a(t )T v
0 i 1
β(u0 ,t0 )
i 1
Instantaneous Failure Probability
u0  a(t0 )T v  0
P  E f   P  u0  a  t0  v  0   P  g (u )  0 
Geometric representation of instantaneous failure probability
T




| u (t ) |)  P   u (ti )  u0 
 i 1

n
10
=3
-2
EL : ‐ m
0
-3
100
200
300
400
‐2
1
2
3
4
5
6
Time (seconds)
7
8
9
10
800
3
Normalized computational time
10
B
1
5
10
0.9
3
0.8
4
0.7
0.6
5
3
4
0.5
5
A
0.9
B
0.7
0.6
5
A
B
0.5
5
0
10
400
600
800
1000 1200
Number of Elements
1400
1600
14
1
E C  E5
1800
C
0
‐0.2
10
0.9
C
0.8
0.7
5
‐0.2
5
0.6
10
‐0.4
8
‐0.4
8
‐0.6
6
‐0.8
2
4
6
8
Adjoint method
0.5
5
D
Sequential compounding procedure
‐1.2
10
ED
‐1
2
‐1.2
C
‐0.8
4
‐1
‐0.6
2
4
6
8
10
Finite difference method
h = 20
Convergence History
3
h = 15
10
8
6
4
2
0
20
40
60
80
100
120
140
20
40
60
80
No.iteration
100
120
140
0.2
Psys
0.1
0
0
h = 10
Dynamic Performance
Ground acceleration
1000
LEVEL 02
EL : h m
f(t) (m/s2)
0
‐1000
0
h m
0.5
1
Initial system
1.5
2
2.5
3
Optimized system
3.5
4
2
0
0
10
20
30
 (rad /s)
40
Kanai‐Tajimi PSD
50
60
GROUND
EL : 0 m
Design domain and loading configuration
0.02
1/L1
0.02
1/L1
Kanai-Tajimi PSD, ()
0.5
5
1
2
10
 g =0.5
6
4
0.8
5m
8
4
A
70
 g =0.4
3
1
A
3
d
10
2
A
5
min Volume(ρ (d ))
 g =0.3
1
A
1000
12
The purpose of topology optimization is to identify the optimal
distribution of materials in a given design domain subjected to tractions
and boundary conditions while satisfying given design constraints..
12
900
0
Lateral Load Resisting System Design


t
s.t P   u (ti )  u0   P ( E sys : β1 ,...,β 70 )  Psys
(=2.3%)
 i 1

0  ρ (d, x )  1 x  , ti  [0, 3.5]
u(t , ρ )  C(ρ )u (t , ρ )  K (ρ )u(t , ρ )   Mlf (t )
with M (ρ )
5
E A  E B  E5
EL : 0 m
Stochastic Topology Optimization Framework
700
FDM
AJM
14
4
4
10
10 200
EL : ‐ m
0.7
4
EA   E3  E4   E5
‐1
6
GROUND
600
5
2
‐4
0
500
10
LEVEL i
0.8
3
5
Exact
Proposed method
12
LEVEL j
0.9
2
0.6
LEVEL k
EL : ‐ m
1
2
3
Volume (m )
u0
=2
4
2
u(t)
-1
Sensitivity Analysis
The probability that a stochastic response exceeds a given threshold at
least once for a given duration. It is often used to describe the
reliability of a system subjected to stochastic excitations.
4
10
Performance of SCM
The First Passage Probability
P ( E sys )  P (u0  max 0t tn
1
# of components, N
g (u )  u0 / a  t0   a  t0  / a  t0  v  β  u0 , t0   αˆ (t0 )  v
T
E1  E2   E3  E4   E5 Correlation coefficient matrix
1
10
vi
u0  u(t0 )  0 Safe domain
0
=1
u0  u(t0 )  0 Failure domain
Probability of Cut Set System P(Esys)
n
10
0
‐0.02
Φ0=700
Φ0=30
Φ0=4
0
0
‐0.02
1
2
Time (seconds)
3
0
1
2
Time (seconds)
3
Conclusion
Acknowledgements
• The sequential compounding method enables for an efficient and accurate computation of the failure probability of a large‐size system reliability problem and its parameter sensitivities.
• The developed topology optimization framework under constraints on the first passage probability provides ways to find optimal bracing systems that can resist future realization of stochastic processes with a desired level of reliability.
The authors gratefully acknowledge funding
provided by the National Science Foundation
(NSF) through project CMMI 1234243. We also
acknowledge support from the Donald B. and
Elizabeth M. Willett endowment at the
University of Illinois at Urbana‐Champaign.
Reference
• J. Chun, J. Song and G. H. Paulino, Topology optimization of structures under stochastic excitations. Computer Methods in Applied Mechanics and Engineering, 2013. Submitted, under review. • W.‐H. Kang and J. Song, Evaluation of multivariate normal integrals for general systems by sequential compounding, Structural Safety, 32(1), 35‐41, 2010. • J. Song and A. Der Kiureghian, Joint first‐passage probability and reliability of systems under stochastic excitation, ASCE Journal of Engineering Mechanics, 132(1), 65‐77, 2006. • A. Der Kiureghian, The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Engineering Mechanics, 15(1), 81–90, 2000.