Topology Optimization of Structures under Constraints on First Passage Probability

Junho Chun*University of Illinois at Urbana-Champaign, USA

July 13th, 2015

Junho SongSeoul National University, Korea

Glaucio H. PaulinoGeorgia Institute of Technology, USA

ICASP1212th International Conference on Applications of Statistics

and Probability in Civil Engineering

2

Structural Engineering under Natural Hazards and Risks

Random Excitations

Random processNon-deterministic excitationsMany possibilities of the process

San Francisco Earthquake, 1907

0 5 10 15 20 25 30 35 40 45 50-600

-400

-200

0

200

400

600

800HYOGOKEN NANBU EQ - KOBE-JMA3.EW 1/17/1995 DT=0.02 Amax=617.14gal

x - time / DT = 0.02

Accele

rati

on

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al)

1. http://www.documentingreality.com

2. Photograph: Kimimasa Mayama/Reuters

1 2

Kobe Earthquake, 1995

One of the most fundamental requirements on building structures is to withstand variousuncertain loads such as earthquake ground motions, wind loads and ocean waves.

The structural design needs to ensure safe and reliable operations over a prolonged period oftime despite random excitations caused by hazardous events.

3

Motivation – Reliable Structural Design under Stochastic Excitations

Structural systemCourtesy of Skidmore, Owing and Merrill, LLP

Structural elements optimization Structural performance optimizationA

ccel

era

tio

n

Time, sEl

evat

ion

, mStory Displacement, mm

Structural Design

Research aims to find the optimal structure and system under stochastic excitations

4

Reliability-Based Design(RBDO) / Topology Optimization (RBTO) Formulation

Numerical Applications / Discussion

Outline

Discrete Representation Method First Passage Probability / Structural Engineering Constrains

5Der Kiureghian, A. (2000). The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Engineering Mechanics, 15(1),: 81-90.

1

( ) ( ) ( ) ( ) ( )n

T

i i

i

f t t v s t t t

s v

Modeling Ground Excitations - Filtered Gaussian Process

0

1 1

T

0

1

( ) ( ) ( )

( ) ( )

2π / ( ) ( )

t

n n

i i i f i

i i

n

i f i

i

f t v s t d

v s t W h t t t

t v h t t t t

s v

Discrete Representation of Stochastic Excitation

The stochastic excitation is represented by a linear combination of basis functions, s(t), withstandard normal independent random variables, v:

Stochastic ground excitations can be modeled by using a filter representing the characteristic of soil mediumand Gaussian process.

Gaussian process Soil Medium (Filter)Filter parameter: ωg, ζg

Ground acceleration (Filtered Gaussian Process)

6

Discrete Representation of Responses of Linear Structures

The convolution integral for determining the responses of linear systems subjected to thestationary process can be developed with the impulse response function.

0

( ) (τ) ( τ) τ

t

su t f h t d Dynamic Responses

T

1 10

( ) ( ) ( τ) τ ( ) ( )

t n n

i i s i i

i i

u t v s h t d v a t t

a v

Deterministic, time-dependent - filter + structure

Random, time-independent

Instantaneous Failure Probability

Failure event of a linear system at a certain time ti

T

0 0 0: ( , ) 0 : : ( )f i f i f iE g t u E u t u E t u a v

Failure Probability

0 0: ( , ) 0 β ,f f i iP E g t u t u

00β ,i

i

ut u

t

a

7

First Passage Probability

In the reliability analysis of dynamic system subjected to stochastic excitations, a significantproblem is to determine the first passage probability that any one of output states of interestexceeds a certain threshold value within a given time duration T.

0 0 0

1

( ) ( max | ( ) |) ( )n

n

fp sys t t i

i

P E P u u t P u t u

First passage probability is defining the problem as a series system problem such as:

Ssiger International Plaza Courtesy of Skidmore, Owing and Merrill, LLP

Stress Displacement

Song, J., and A. Der Kiureghian (2006). Joint first-passage probability and reliability of systems under stochastic excitation. J. Engineering Mechanics,

ASCE, 132(1):65-77.

Fujimura, K. and A. Der Kiureghian (2007). Tail-Equivalent Linearization Method for Nonlinear Random Vibration. Probabilistic Engineering Mechanics,

22: 63-76

?

8

RBDO/RBTO under Constraints on First Passage Probability

Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation.

Optimization Formulation

target

,

1

min ( )

. ( , ) : ( , ) 0 , 1,...,

with ( ) ( , ) ( ) ( , ) ( ) ( , ) ( , )

t

i isys

obj

n

fp i f k i k f c

k

lower upper

i

f

s t P E t g t P i n

t t t t

E

dd

d d

d d d

M d u d C d u d K d u d f d ( , )= ( ) ( )= ( ) ( )gt u t f t f d M d l M d l

Probabilistic Constraints in Structural Engineering

Stress Maximum Displacement Inter-Story Drift Ratio

Hearst Tower (New York City)http://www.sefindia.org/

Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation.

Objective function

Probabilistic constraints

9

Evaluation of First Passage Probability and Sensitivity Analysis

22 β ,

22,,

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,,

( β )ρ( ) 1βφ( β ) exp exp

β 1 ρ22 2 1 ρ

φ( β ) ( β ) 1 ρβ ; β ρ ;

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k k Ssys kk

k k Sk S

k k k SS k k S

vP Edv

1 2 6( ) ( ) ( )sysP E P E E E d z

3 6( ) ( )sys AP E P E E E

4 5 6( ) ( )sys BP E P E E E E

6( ) ( )sys DP E P E E

Chun, J., Song, J., Paulino, G.H. (2015) Parameter sensitivity of system reliability using sequential compounding method. Structural Safety. 55: 26–36.

Kang, WH,, Song, J. (2010). Evaluation of multivariate normal integrals for general systems by sequential compounding. Structural Safety. 32(1): 35–41.

Sensitivity

R

R

The sequential compounding method (SCM) is a system reliability method that compoundscomponent events coupled by union or intersection sequentially until a single compoundevent represents the system event

10

Sensitivity Analysis of First Passage Probability

Chun, J., Song, J., Paulino, G.H. (2015) Parameter sensitivity of system reliability using sequential compounding method. Structural Safety. 55: 26–36.

Adjoint method Finite difference method

Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation.

( ) ,fp sys n

i i

P E

d d

β R

11Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation.

Space truss dome

xy

z

Node of interest

x x-dir

y y-dir

target

,x dir

1

target

,y dir

1

2

min

. ( , ) :

( , )

in x-dir 0.00

:

023

in y-dir 0.00023

0.015 m

n

fp f k f

k

n

fp f k

s

f

k

ys

sys

s t P E t P

P E tE P

E

dweight

drift ratio

drift ratio

d

d

23 m

with ( ) ( , ) ( ) ( , ) ( ) ( , ) ( , )t t t t

d

M d u d C d u d K d u d f d

Size optimization under Constraints on First Passage Probability

12

Size Optimization - Results

Initial structure Optimized structure

Drift ratio time historyOptimized area of truss elements

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Topology Optimization under Constraints on First Passage Probability

Topology optimization aims to identify optimal material layouts of problems through mathematical programming while fulfilling given design constraints

Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation.

Tip Drift ratio Inter-story drift ratio 0.0062fP 0.0013fP 0.0062fP 0.0013fP

V=2.97 m3 V=3.63 m3 V=3.24 m3 V=3.97 m3

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Topology Optimization - Results

Drift ratio time historyConvergence history

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Ground Structure Method under Constraints on First Passage Probability

The ground structure method removes unnecessary members from a highly interconnected truss while keeping the nodal locations fixed.

16

Concluding Remarks

Optimization framework is proposed to incorporate the first passage probability into structural optimization and topology optimization

Sensitivity calculation of the probabilistic constraint on the first passage probability is derived to use efficient optimization algorithms

Optimized system can withstand future realization of stochastic processes with a desired level of reliability

Thank you for your attention

Junho Chunjchun8@Illinois.edu

Acknowledgement

• National Science Foundation (NSF) - CMMI 1234243• Civil Engineering Risk and Reliability Association (CERRA)

BACK-UP SLIDES

19

Topology Optimization under Constraints on First Passage Probability

Topology optimization aims to identify optimal material layouts of problems through mathematical programming while fulfilling given design constraints

Chun, J., Song, J., Paulino, G.H. System reliability-based design/topology optimization of structures constrained by first passage probability. In preparation.

Tip Drift ratio Inter-story drift ratio0.0062fP 0.0013fP 0.0062fP 0.0013fP