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Structural Safety

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / s t r u s a f e

System-reliability-based design and topology optimization of structuresunder constraints on rst-passage probabilityfi

Junho Chuna , Junho Songb,⁎, Glaucio H. Paulinoc

a School of Architecture, Syracuse University, Syracuse, NY, United Statesb Department of Civil and Environmental Engineering, Seoul National University, Seoul, Republic of Koreac School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, United States

A R T I C L E I N F O

Keywords:

Reliability-based design optimization

Reliability-based topology optimization

Stochastic excitation

Sequential compounding method

Parameter sensitivity

System reliability

First-passage probability

A B S T R A C T

For the purpose of reliability assessment of a structure subject to stochastic excitations, the probability of the

occurrence of at least one failure event over a time interval, i.e. the rst-passage probability, often needs to befi

evaluated. In this paper, a new method is proposed to incorporate constraints on the rst-passage probabilityfi

into reliability-based optimization of structural design or topology. For e cient evaluations of rst-passageffi fi

probability during the optimization, the failure event is described as a series system event consisting of in-

stantaneous failure events de ned at discrete time points. The probability of the series system event is thenfi

computed by use of a system reliability analysis method termed as the sequential compounding method. The

adjoint sensitivity formulation is derived for calculating the parameter sensitivity of the rst-passage probabilityfi

to facilitate the use of e cient gradient-based optimization algorithms. The proposed method is successfullyffi

demonstrated by numerical examples of a space truss and building structures subjected to stochastic earthquake

ground motions.

1. Introduction

Finding the optimal design of a structural system with regard tosafety, cost or performance is one of the most essential tasks in struc-

tural engineering practice. The optimal design should achieve majordesign objectives representing reliable operation and safety even under

stochastic excitations caused by natural hazards such as earthquakes

and wind loads. Due to inherent randomness in natural disasters,however, signi cant uncertainties may exist in the intensity and char-fi

acteristics of the excitations. Therefore, the performance of suchstructural systems needs to be assessed probabilistically during the

optimization process.To deal with uncertainties e ectively in structural design/topologyff

optimization, various optimization algorithms and frameworks were

developed recently. For instance, the so-called design/topologyrobust

optimization algorithms aim to reduce the sensitivity of the op-[ ] 1–3

timal performance of a structure with respect to the randomness of

interest. By contrast, design/topology optimizationReliability-based

[4 10] – aims to nd optimal solutions satisfying the probabilistic con-fi

straints on the structural performance indicators. So far, these studieshave been mainly focusing on accounting for uncertainties in static

loads representing typical load patterns of the structure. Recent studies

on structural optimization considering dynamic excitations employed a

small number of deterministic time histories representing possible fu-

ture realizations , or focused on partial descriptors of the dy-[ , ]11 12namic responses such as mode frequencies . These approaches have[13]

intrinsic limitations because (1) a single or small number of sampletime histories may not represent all possible realizations of stochastic

excitations, and (2) it is practically impossible to assess the probabilities

that the structural design does not satisfy the constraints on perfor-mances, i.e. failure probabilities using this approach. Therefore, the

probabilistic prediction of structural responses based on random vi-bration analysis is needed in the process for optimal design.

To overcome this technical challenge, the authors recently proposeda new method for topology optimization of structures under stochastic

excitations . In the proposed method, an e cient random vibration[14] ffi

analysis method based on the use of the discrete representation method[15] [16] and structural reliability theories (see for a review) were

integrated within a state-of-the-art topology optimization framework.

The authors also developed a system reliability-based topology opti-mization framework under stochastic excitations to cope with[17]

system failure events consisting of statistical dependent componentevents using the matrix-based system reliability method . The de-[18]

veloped method helps satisfy probabilistic constraints on a system

https://doi.org/10.1016/j.strusafe.2018.06.006

Received 5 September 2017; Accepted 17 June 2018

⁎ Corresponding author.

E-mail address: [email protected] (J. Song).

Structural Safety 76 (2019) 81–94

0167-4730/ © 2018 Elsevier Ltd. All rights reserved.

T

failure event, which consists of multiple limit-states de ned in terms offi

di erent locations, failure modes or time points as it optimizes aff

structural system.In these studies by the authors, the failure prob-instantaneous

abilities of the structure were evaluated at discrete time points.

However, to promote applications of design/topology optimization toengineering design practice, the rst-passage probability, i.e. thefi

probability of at least one occurrence of the failure over a time interval,needs to be estimated during the optimization process. Spence et al.

[19] proposed a framework for RBDO of linear systems constrained on

the rst-passage probability. This approach decouples the nested re-fi

liability analysis loop from the optimization loop by solving sub-opti-

mization problem formulated from simulation results. Bobby et al. [20]presented a simulation-based framework for topology optimization of

wind-excited building structures with the consideration of the rst-fi

passage probability.The rst-passage probability helps promote the use of the proposedfi

stochastic optimization framework for the design of the lateral load-resisting system or sizing structural elements under stochastic excita-

tions with a nite duration such as earthquake excitations. To this end,fi

this paper introduces a stochastic design and topology optimizationmethod that can handle probabilistic constraints on the rst-passagefi

probability, and demonstrates the method using numerical examples.

2. Random vibration analysis using discrete representation

method

In the aforementioned reliability-based design optimization frame-work under stochastic excitations , the authors proposed to[ , ]14 17

perform random vibration analysis by use of the discrete representationmethod in order to compute the instantaneous failure probability[15]

of the stochastic response at discrete time points. In the proposed ap-proach, for example, a zero-mean stationary Gaussian input excitation

process ( ) is discretized asf t

∑= ==

f t v s t ts v( ) ( ) ( )i

n

i i

1

T

(1)

where (s t) ( = [ s 1 ( ), , t … s n ( )]t T) is a vector of deterministic functions

that describe the spectral characteristics of the process, and v = [v1 ,

v2, …, vn ] T is a vector of uncorrelated standard normal random vari-ables. Among existing methods available to develop a discrete re-

presentation model in Eq. , a popular one for ground excitation(1)modeling is using a lter representing the characteristic of the soilfi

medium and a random pulse train. For example, if a ltered white noisefi

is used, the model in Eq. is constructed as(1)

∫∫

= − =

= ⎧⎨⎩

− < < = …⩽−

−

−

f t h t τ W τ dτ t

s tπ t h t τ dτ t t t i n

t t

s v( ) ( ) ( ) ( )

( )2 Φ /Δ · ( ) , 1, ,

0

tf

it

tf i i

i

0T

0 1

1

i

i

1

(2)

in which ( ) denotes the white noise process whose power spectralW τ

density function is ΦWW (ω) = Φ0 , hf (·) is the impulse response function

of the lter, = fi Δt ti − t i−1, and denotes the number of the time in-n

tervals introduced for the given time period (0, ). The details of thet

derivation of Eq. are available in Chun et al. .(2) [14]

2.1. Response of linear system under stochastic excitations

The responses of linear systems to stochastic excitation can be de-

termined by the convolution integral consisting of their impulse re-sponse function and the discretized input process in Eq. . That is, a(1)

response time history ( ) of the linear system subjected to the sto-u t

chastic excitation ( ) is derived asf t

∫ ∫ ∑ ∑= − = − =

== =

u t f τ h t τ dτ v s τ h t τ dτ v a t

ta v

( ) ( ) ( ) ( ) ( ) ( )

( )

t

s

t

i

n

i i si

n

i i0 0

1 1

T (3)

where hs (·) is the impulse response function of the linear structuralsystem, and ( ) denotes a vector of deterministic basis functionsa t

∫= − = …a t s τ h t τ dτ i n( ) ( ) ( ) , 1, ,i

t

i s0 (4)

Deriving the impulse response function in a nite element settingfi

can be computationally challenging or cumbersome. To facilitate the

process, the authors proposed novel numerical procedures in Chun et al.[14].

2.2. Instantaneous failure probability of linear system under stochastic

excitations

In structural reliability analysis, the probability that the outcome of

a random vector is located inside the failure domain X Ω f , i.e. the failureprobability, is computed by an integral

∫=P f dx x( )f XΩf (5)

where f X( ) is the joint probability density function (PDF) of thex

random vector . The failure domain is de ned by the area where theX fi

limit-state function ( ), e.g. capacity minus demand, takes the negativeg x

sign. In general, computing the multi-fold integral in Eq. is non-(5)

trivial or computationally challenging. Structural reliability methodssuch as FORM and SORM (see for a review) transform the space of[16]

the random variable into the uncorrelated standard normal space .x v

Then, the limit-state function is approximated by a linear (FORM) or

quadratic function (SORM) at the design point, often alternativelytermed as the most probable failure point (MPP). For example, in

FORM, the failure probability is approximated as

= −P Φ[ β]f (6)

where is the reliability index, i.e. the shortest distance from the originβ

of the standard normal space to the linearized failure surface, and [·]Φ

denotes the cumulative distribution function (CDF) of the standardnormal distribution. Using the discrete representation method de-

scribed above, limit-state functions de ned for displacement or otherfi

structural responses can be described in the space of standard normal

random variable . For example, the instantaneous failure event v E f

de ned for a linear structure subjected to the Gaussian input process infi

Eq. is given by(1)

= ⩽ = −= −

E t u g t u g t u u u t

u t

v v v

a v

( , , ) { ( , , ) 0}, where ( , , ) ( )

( )

f k k k k

k

0 0 0 0

0T (7)

where u 0 is the prescribed threshold on the displacement response. Inthis case, the reliability index is computed from the geometric in-β

terpretation of the limit-state surface as a closed form expression [15]

=t uu

taβ( , )

‖ ( )‖k

k0

0

(8)

It is noted that the limit-state function in Eq. is linear in this(7)

case, and thus the failure probability by Eq. , i.e. (6) Pf = [ (Φ −β t k ,u0 )]

does not introduce errors caused by function approximation or requirenonlinear optimization to nd the design point. If the structure behavesfi

nonlinearly or the input process is non-Gaussian, one needs to use re-liability methods such as FORM or SORM to compute the failure

probability approximately. Using this discrete representation method,

one can reduce the computational cost of the random vibration ana-lysis, which should be repetitively performed during the optimization

processes to compute the instantaneous failure probability at each up-dated set of design variables.

J. Chun et al. Structural Safety 76 (2019) 81–94

82

2.3. First-passage probability of linear system under stochastic excitations

The rst-passage probability is commonly utilized to nd thefi fi

probability of the failure event described within a time interval

[21 23]– . One of the available approaches for formulating the rst-fi

passage probability P fp de nes the problem as a series system problem,fi

i.e.

⎜ ⎟= < = ⎛⎝⋃ > ⎞

⎠< < =P P u u t P u t u( max ( )) { ( ) }fp

t t k

n

k00 1

0n (9)

Using the discrete representation, the rst-passage probability of afi

system with nc limit-state functions (de ned for di erent failure modesfi ff

or locations) is described as

⎜ ⎟ ⎜ ⎟= ⎛⎝⋃ ⎞

⎠= ⎛

⎝⋃ ⋃ ⎞

⎠= = =P P E P E t u v( , , )fp

i

n

sysi

i

n

k

n

f k1 1 1

0

c c

i(10)

where Esysi denotes the rst-passage failure event regarding the -thfi i

constraint, E (·) denotes the instantaneous failure event of the -th limit-i

state function at time tk, and is the total number of discretized timen

points. To compute the rst-passage probability in Eq. , it is re-fi (10)

quired to evaluate the failure probabilities of the component events at

each time point within an interval. Moreover, an e cient, reliable andffi

robust algorithm is required to evaluate the system failure probability

with a proper consideration of statistical dependency between the

component events. It is also desirable to compute the parameter sen-sitivity of the series system failure probabilities in Eqs. to(9) and (10)

enable the use of e cient gradient-based optimizers. To address theseffi

requirements, the sequential compounding method (SCM; ) and the[24]

Chun-Song-Paulino (CSP; ) method are adopted in this study.[25]

3. Optimization of structures subjected to stochastic excitation

under rst-passage probability constraints

3.1. Structural design optimization

Reliability Based Design Optimization (RBDO) of a structure aims toachieve the optimal design under probabilistic constraints on uncertain

performance, arising from uncertainties in material properties or loads.The RBDO problem of a structure under rst-passage probability con-fi

straints can be formulated as a System Reliability Based Design

Optimization (SRBDO) problem , i.e.[9]

⎜ ⎟

⎜ ⎟

= ⎛⎝⋃ ⎞

⎠

= ⎛⎝⋃ ⩽ ⎞

⎠⩽ = …

⩽ ⩽+ + =

=

=

f

s.t P E P E t

P g t P i n

t t t t

d

d

d

d d d

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

min ( )

( ) ( , )

{ ( , ) 0} , 1, ,

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

obj

sysi

k

n

f k

k

n

i k f c

d

1

1

target

lower upper

i

i

(11)

where f obj ( ) denotes the objective function of the design, d dlower and

dupper are the lower and upper bounds of the vector of design variablesd, respectively. gi ( ) represents the limit-state function whose negative

sign indicates the violation of a given constraint, n c is the number of theconstraints, (P gi(·) 0) is the probability of the failure event, and≤

Pftarget is the target failure probability.

M C, K , and represent the global mass, damping and sti ness ma-ff

trices of the structure, respectively, and ,ü u, and are the accel-u, f

eration, velocity, displacement and force vectors at time , respectively.t

A proportional damping model known as Rayleigh damping is‘ ’ [26]

used throughout this paper. In this approach, the damping matrix is

determined as a linear combination of the sti ness and mass matrix,ff

that is = C κ0 M + κ1 K. The coe cients ffi κ 0 and κ1 in the Rayleigh

damping model are determined so as to have certain modal dampingfactors. For earthquake ground excitations, the force vector in Eq. (11)

is determined by a vector of e ective earthquake forces, i.e.ff

=− =−t u t f tf d M d l M d l( , ) ( ) ( ) ( ) ( )g (12)

where represents the directional distribution of mass with unity re-l

sulting from a unit ground displacement and ug is the ground accel-

eration time history.

3.2. Structural topology optimization

Topology optimization (see for a review) aims to nd the op-[27] fi

timal material distributions in a design domain subjected to tractionsand displacement boundary conditions while satisfying given design

constraints. Thus, every point of a design domain is expected to re-present either an existence of material or a void region. The Solid Iso-

tropic Material with Penalization (SIMP; ) model, which is adopted[28]

in this study, considers a continuous material density in a design vari-able using the power function representation, i.e.

=ψ x x( ) p (13)

where is the penalization factor and is a density associated withp x

element in the nite element method setting. The topology optimi-e fi

zation solutions using the SIMP, or related models, may su er fromff

“ ” checkerboard patterns and mesh-dependency . To overcome[29]these problems, various methods have been proposed (e.g. ). In[ ]30–33

this study, a projection method is implemented to obtain a ltered[30] fi

density based on element design variables within the neighborhood

such as:

∑

∑= = ⎧

⎨⎩

⩾∼ ∈

∈

−

ρ

w r d

w rw r

r r( )

( ), ( )

if

0 otherwisee

j

j j

j

j

j

r r

rjN

N

minj

e

e

min

min

(14)

where d j denotes design variable of element ,j Ne represents the set ofelements within the radius rmin of element , (e w r j) is the weighting

function, and rj is the distance between the centroids of element and .j e

Using the SIMP model, the sti ness and mass matrix of element ff e

and their derivatives with respect to an element density are obtained as

follows in the element-based computational framework :[27]

= =

= =

∼ ∼ ∼ ∼

∼ ∼∂∂

− ∂∂

−∼∼

∼∼

ρ ρ ρ ρ

pρ qρ

K K M M

K M

( ) , ( )

,

e e ep

e e e eq

e

ρ

ρ ep

eρ

ρ eq

eK M

0 0

( ) 1 0 ( ) 1 0e e

e

e e

e (15)

where is the penalization parameter, andq Ke0 and Me

0 are computed by

∫ ∫= = ρ dK B D B M N NΩ , Ωe e e m e0

Ω

T0

0

Ω

T

e e (16)

where denotes the strain displacement matrix of the shape functionB –

derivatives in the domain Ω e of element , e m represents the mass

density of the material and N is the shape function of element e. D0 isthe elasticity tensor of the solid material, where the density is 1.

Topology optimization of a structure under stochastic excitation

with constraints on the rst-passage probability can be formulated asfi

⎜ ⎟

⎜ ⎟

= ⎛⎝⋃ ⎞

⎠

= ⎛⎝⋃ ⩽ ⎞

⎠⩽ = …

< ⩽ ⩽ ∀ ∈+ + =

∼

=

=

f

s t P E P E t

P g t P i n

ε ρ i

t t t t

ρ

ρ

ρ

M ρ u ρ C ρ u ρ K ρ u ρ f ρ

min (~)

. ( ) ( , ~)

{ ( , ~) 0} , 1, ,

0 1 Ω

with ( ~) ¨ ( , ~) (~) ( , ~) (~) ( , ~) ( , ~)

obj

sysi

k

n

f k

k

n

i k f c

i

d

1

1

target

i

i

(17)

where denotes the vector of design variables, is a set of nited Ω fi

element indices and ρ~ is the vector of ltered densities de ned as:fi fi

=ρ~ Pd (18)

where represents the ltering matrix whose element is determined byP fi


83

∑= = ⎧

⎨⎩∈

∈

w

w rw w r kP( )

( ), ( ) if N

0 otherwiselk

lk

j

jlk

k l

Nl (19)

A owchart for topology optimization of a structure constrained byfl

first-passage probability is provided in .Appendix A

Various engineering constraints can be incorporated into the aboveformulations of reliability-based design optimization and topology op-

timization under rst-passage probability. To promote applications offi

the proposed method to truss and building structures, engineering

constraints on stress in the bar and inter-story drift ratio are derived

below.

3.3. First-passage probability constraints on stress in bar elements

Consider a bar in a truss structure with the local node numbers 1e

and 2 denoting the end points of the bar as shown in . A unitFig. 1vector ne pointing from node 1 to node 2 is de ned asfi

⎜ ⎟= ⎛⎝

⎞⎠

ncosθ

sinθe

e

e (20)

The global displacement vectors of the end nodes of bar aree

written as

⎜ ⎟= ⎛⎝

⎞⎠

= ⎛⎝

⎞⎠

= ⎛⎝

⎞⎠

uu

uuu

u

uu u, where ,e

eg

eg e

g e x

e yeg e x

e y

1

21

1,

1, 22,

2, (21)

The stress ( ) in a truss element under stochastic excitations cant e

be computed from the stress strain relationship based on Hooke s law– ’

as follows:

= − = = −σ tD

Lt t

D

L

D

Lu t u td n u d u d B u d d( , ) ·( ( , ) ( , )) ( ( , ) ( , ))e

e

ee e

geg e

ee e

e

eel

el

2 1 2 1

(22)

where D e denotes Young s modulus, ’ L e is the length of the element ,e

ule1 ( ) and t u

le2( ) are end displacements along the truss axis, andt

= −B n n[ ]e e eT T (23)

The elongation in Eq. can be described by using the discrete(22)

representation form in Eq. , i.e.(3)

= −σ tD

Lt td a d v a d v( , ) ( ( , ) ( , ) )e

e

ee e2T

1T

(24)

The instantaneous failure probability at time t k is expressed in terms

of stress in the truss element ase

= ⩽ = − ⩽ = −P E t P g t P σ σ t td d d d( ( , )) ( ( , ) 0) ( ( , ) 0) Φ[ β ( , )]fe k e k e e k σe k0

(25)

where 0e denotes the threshold value of stress. From the geometric

representation associated with the failure event of element , the re-e

liability index at time tk is computed as

=−

=tL σ

D t t

L σ

D td

a d a d b dβ ( , )

·

‖ ( , ) ( , ) ‖

·

‖ ( , )‖σe k

e e

e k e k e

e e

e e k

0

1T

2T

0

(26)

The rst-passage probability of the stress limit state function is thenfi

computed as

⎜ ⎟ ⎜ ⎟= ⎛⎝⋃ ⎞

⎠= ⎛

⎝⋃ − ⩽ ⎞

⎠= − … …= −

= =

−

P E P E t P σ σ t

t t t ρ ρ ρ

d d

d d d

β R

_ ( ) ( , ) { ( , ) 0}

1 Φ [β ( , ), β ( , ), ,β ( , ); , , , ]

1 Φ [ , ]

fp σ sysk

n

fσ kk

n

e e k

n σe σe σe n n n

n σe

1 10

1 2 1,2 1,3 1,

(27)

where Φn denotes the multivariate normal CDF, i,j represents the cor-

relation coe cient between the normal random variables representingffi

failure event and , and and are the vectors of the reliability in-i j R

dices and the correlation coe cient matrix, respectively. The correla-ffi

tion coe cient matrix is constructed asffi R

=⎡

⎣⎢⎢

⋯⋮ ⋱ ⋮

⋯

⎤

⎦⎥⎥

=ρ ρ

ρ ρρ t tR α α, ( ) · ( )

n

n n n

k l k l

1,1 1,

,1 ,

,T

(28)

where (t i ) = a(t i)/|| (a ti )|| denotes the negative normalized gradientvector of the limit-state function evaluated at the design point which is

obtained by u0 · (a ti)/|| (a t i)||2. The multivariate CDF in Eq. and those(27)

in the following Sections 3.4 and 3.5 are computed by SCM . The[24]

CSP method whose details are presented in Section is used to[25] 5

compute the sensitivity of the multivariate CDF.

3.4. First-passage probability constraint on inter-story drift ratio

The rst-passage probability can be computed in terms of the inter-fi

story drift ratio, which is one of the signi cant design criteria infi

structural engineering, de ned asfi

=⎧

⎨⎪

⎩⎪

=

− = …−

t

H

t

Hi

t tH

i n

d

a d v

a d a dv

Δ ( , )

( , )for 2

( ( , ) ( , ) ) for 3, 4, ,

i

i

i

i

i ii

s

T

T1

T

(29)

where Δi denotes the story drift at oor level , fl i H i represents the story

height below level , and i n s is the number of story levels. The in-

stantaneous failure probability in terms of the inter story-drift ratios is

⎜ ⎟= ⩽ = ⎛⎝

− ⩽ ⎞⎠

= − = …

P E t P g t P ut

H

t i n

d dd

d

( ( , )) ( ( , ) 0)Δ ( , )

0

Φ[ β ( , )], 1, ,

f k ki k

i

k s

Δ 0Δ

Δ

i i i

i

Δ

(30)

where u 0Δi denotes a threshold value of the inter-story drift ratio, andβΔi represents the reliability index which can be computed as

=

⎧

⎨⎪

⎩⎪

=

−= …

−

t

H u

ti

H u

t ti n

da d

a d a d

β ( , )‖ ( , ) ‖

for 2

‖ ( , ) ( , ) ‖for 3, 4, ,

i k

i i

k i

i i

k i k is

Δ

0ΔT

0ΔT

1T

(31)

Finally, the rst-passage probability isfi

⎜ ⎟ ⎜ ⎟= ⎛⎝⋃ ⎞

⎠= ⎛

⎝⋃ − ⩽ ⎞

⎠= − … …= −

= =

−

{ }P E P E t P u

t t t ρ ρ ρ

d

d d d

β R

_ ( ) ( , ) 0

1 Φ [β ( , ), β ( , ), ,β ( , ); , , , ]

1 Φ [ , ]

fp sysk

n

f kk

nt

H

n n n n

n

d

1 10Δ

Δ ( , )

Δ 1 Δ 2 Δ 1,2 1,3 1,

Δ

i i ii k

i

i i i

i

Δ Δ

(32)

4. Calculating sensitivity of rst-passage probability

To use e cient gradient-based optimization algorithms for RBDO, itffi

is essential to calculate the sensitivity of the failure probability withrespect to various design parameters. In this paper, a sensitivity for-

mulation employing the adjoint method is derived for the rst-[34] fi

passage probability of a linear structure based on the discrete

Fig. 1. Bar geometry.


84

representation method. It is noted that the sensitivity of the system

failure probability with respect to a parameter is obtained by a chainθ

rule, i.e.

∑∂∂

=∂∂

∂∂=

P E P E( )

θ

( )

β (θ)·

β (θ)

θ

f sys

i

nf sys

i

i

1 (33)

Recently, Chun et al. proposed the CSP method to compute the[25] derivatives of the system failure probability with respect to the relia-

bility index based on the use of the SCM. The CSP method computes

sensitivities of parallel and series systems, as well as general systemswith respect to reliability indices e ciently and accurately.ffi

4.1. Sensitivity of system reliability using SCM

The CSP method computes parametric sensitivity of the system re-liability based on the SCM method. The main idea of the CSP method is

carrying out sensitivity analysis after the system failure event is sim-pli ed using the SCM. This idea is brie y explained using a seriesfi fl

system example formulated as an -fold integral in the correlatedn

standard normal space, i.e.

∫= ⋃ ⋃⋯⋃ = ⎡

⎣⎢⋃ ⩽ ⎤

⎦⎥ =

=P E P E E E P Z φ dz R z( ) ( ) (β ) ( ; )series n

j

n

j j n1 21

Ωf

(34)

Suppose the -th component is compounded at the last step, i.e.k

compounded with the super-component ESk , which denotes the union of

all the component events except the -th one. Utilizing the formula fork

bi-variate normal CDF , the sensitivity of the series system failure[35]probability with respect to βk is obtained as [25]

∂∂

= − − −∂ − −

∂P E

φρ( )

β( β )

Φ [ β , β ; ]

β

series

kk

k S k S

k

2 ,k k

(35)

where ρk S, kis the updated correlation coe cient between ffi E k and ESk

obtained during the sequential compounding , and[24]

= − = − ⋃−

∈

−P E P Eβ Φ[ ( )] Φ[ ( )]S Sp S

p1 1

k k

k (36)

where S k denotes the index set of the components in ESk. Similarly, the

parameter sensitivities of parallel and cut-set systems were derived by

Chun et al. .[25]

4.2. Sensitivity of rst-passage probability in RBDO

To facilitate the use of a gradient-based optimizer, the sensitivity of

the rst-passage probability in RBDO is computed using the chain rule,fi

i.e.

∑

∑ ⎜ ⎟

∂∂

= ∂ −∂

= ⎛

⎝⎜∂ −

∂∂∂

⎞

⎠⎟

= ⎛⎝

∂∂

⎞⎠

=

=

P E

d d d

cd

β R β R d

d

( ) (1 Φ [ , ]) ( Φ [ , ])

β·

β ( )

·β ( )

fp sys

i

n

i j

nn

j i

j

n

jj

i

1

j

1 (37)

where c j = ( [ , ])/−Φ R β j can be computed using Eq. . The(35)

partial derivative ∂βj/ d i is obtained by

∑

∑

⎜ ⎟

∂∂

= −

⎛

⎝⎜

⎛⎝

∂∂

⎞⎠⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

=

=

d

C a ta t

d

a t

dd

d

d

β ( )· ( , )·

( , )

( , )

j

i

cst

k

j

k jk j

i

k

j

k j

1

1

2

3/2

(38)

where Ccst is the coe cient determined depending on the constraintffi

used in optimization, e.g.

=⎧

⎨⎩

C

L σ D

Hu

H u

/ : stress in bar

: drift ratio

: inter-story drift ratiocst

e e e

i

0

0Δ

0Δi (39)

When a uniform time step size is used, i.e. − = = …−t t t i nΔ , 1, 2, ,i i 1

and =t tn 0, Eq. can be rewritten from Eq. as follows (see more(38) (4) details of the derivation in of Chun et al. ):Appendix [14]

Table 1

Topology optimization problem ( ): ltering parameters for ground ex-Fig. 2 fi

citations and a threshold value of the probabilistic constraint.

Φ0 ωf ζ f t t (s) Δ (s) u0Δ

100 5 0.4 5 0.1 1/400π

Fig. 2. Sensitivity comparison: (a) geometry, loading condition, and locations where sensitivity is reported ( ), (b) sensitivities from the adjoint method (AJM),Table 2

and (c) sensitivities from the nite di erence method (FDM).fi ff


85

∑

∑

⎜ ⎟

∂∂

= −

⎛

⎝⎜

⎛⎝

∂∂

⎞⎠⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

= − +

= − +

d

C a ta t

d

a t

dd

d

d

β ( )· ( , )·

( , )

( , )

j

i

cst

k n j

n

k nk n

i

k n j

n

k n

1

1

2

3/2

(40)

Furthermore, the uniform step size leads to the followings for theparameter sensitivity in Eq. :(37)

∑ ∑

∑

⎜ ⎟ ⎜ ⎟

⎜ ⎟

∂∂

= ⎛⎝

∂∂

⎞⎠= ⎛

⎝⎜ ⎛

⎝∂

∂⎞⎠⎞

⎠⎟

= ⎛⎝

∂∂

⎞⎠

= =− +

− +

=

P E

dc

dC χ a t

a t

d

κ ta t

d

dd

d

dd

( )·

β ( )· ( , )

( , )

( , )( , )

fp sys

i j

n

jj

icst

l

n

l n l nn l n

i

s

n

s ns n

i

1 1

11

1

(41)

where

∑ ∑= −⎛

⎝⎜

⎛

⎝⎜

⎞

⎠⎟

⎞

⎠⎟ =

= = − +− +χ c a t κ t C χ a td d d/ ( , ) , ( , ) · · ( , )l

i l

n

i

k n i

n

k n s n cst n s s n

1

2

3/2

1

(42)

4.3. Sensitivity of rst-passage probability by adjoint variable method

The sensitivity in Eq. includes the implicitly de ned derivative(41) fi

term of ∂as (t n , )/d ∂di , s = 1, …, . Those implicit derivatives can ben

computed using the direct di erentiation method (DDM), the niteff fi

di erence method (FDM) or the adjoint variable method (AJM) .ff [34]

Table 2

Sensitivity comparison of rst-passage probability on a displacement constraint in topology optimization.fi

Δd FDM AJM

∂Pf /∂dA ∂P f /∂dB ∂P f/∂d C ∂P f/∂d A ∂Pf /∂d B ∂Pf /∂dC

1 × 10 1− − − − − −0.000452 0.000293 0.000569 0.000486 0.000312 0.000599

1 × 10−2− − −0.000483 0.000310 0.000596

1 × 10−3− − −0.000486 0.000312 0.000599

1 × 10−4− − −0.000487 0.000312 0.000599

1 × 10−5− − −0.000487 0.000312 0.000599

1 × 10−6− − −0.000486 0.000312 0.000599

1 × 10−7− − −0.000485 0.000309 0.000598

1 × 10−8− − −0.000479 0.000301 0.000619

1 × 10−9− − −0.000444 0.000391 0.000632

1 × 10−10 0.000250 0.000289 0.000012−

1 × 10−11− − −0.000166 0.001532 0.004141

1 × 10−12 0.050625 0.025424 0.036526

Fig. 3. Computational time comparison for sensitivity analysis by the FDM and

the AJM.

Fig. 4. A space truss dome example: (a) perspective view of the dome, (b) plan view and directions of applied ground accelerations and (c) element numbering

choices.


86

Chun et al. derived an approach of sensitivity calculations asso-[14] ciated with ∂as (t n, )/d ∂d i using the adjoint variable method. The nu-

merical tests con rmed superior performance of AJM compared tofi

DDM and FDM. Based on the AJM derivation, the sensitivity of the rst-fi

passage probability in Eq. is rewritten as(41)

∑∂∂

= ⎡⎣⎢∂∂

−∂∂

− + −∂

∂− − +

∂∂

+∂∂

+∂∂

⎤⎦⎥

+ ⎡⎣⎢

∂∂

+ ∂∂

⎤⎦⎥

+ ⎡⎣⎢

∂∂

⎤⎦⎥

=− +

− −

− −

−

−

P E

d dt η t

t

d

γ η tt

dγ η t

t

d

dt

dt

d

t

d

d

λA d

u df d

f d f d

B du d

E du d

λ B du d

E du d

λ E du d

( ) ( )· ( , ) (Δ )

( , )

(0.5 2 )(Δ )( , )

(0.5 )(Δ )( , )

( )· ( , )

( )· ( , )

( )·(0, )

( )·( , )

( )·(0, )

fp sys

i j

n

n ji

jj

i

j

i

j

i

ij

ij

ni i

ni

11

T 2

2 1 2 2

1 2

T 1

1T

(43)

where − +λn j 1 denotes the adjoint variable vector. A d( ), B d( ), and E d( )

represent followings respectively:

= + += − + − + + −= + − + − +

γ t η t

γ t γ η t

γ t γ η t

A d M d C d K d

B d M d C d K d

E d M d C d K d

( ) ( ) Δ · ( ) (Δ ) ( )

( ) 2 ( ) (1 2 )Δ ( ) (0.5 2 )(Δ ) ( )

( ) ( ) ( 1)Δ ( ) (0.5 )(Δ ) ( )

2

2

2 (44)

4.4. Sensitivity analysis of rst-passage probability in RBTO

Sensitivity analysis of rst-passage probability in stochastic fi topology

optimization is similar to the derivation for RBDO described above. The

main di erence of sensitivity analysis in topology optimization comesff

from the projection method to obtain the ltered density as shownfi

below.

∑

∑ ∑

∑

∑

⎜ ⎟

⎜ ⎟

∂∂

= ∂ −∂

= ⎛

⎝⎜∂ −

∂∂∂

⎞

⎠⎟

= ⎛

⎝⎜∂ −

∂⎛⎝

∂∂

∂∂

⎞⎠

⎞

⎠⎟

= ⎛

⎝⎜∂ −

∂⎛⎝

∂∂

⎞⎠

⎞

⎠⎟

= ⎛

⎝⎜∂ −

∂∂∂

⎞

⎠⎟

∼∼

=

= =

=

=

P E

d d d

ρ

ρ

d

β R β R ρ~

β R ρ~

β RP

ρ~

ρ

Pβ R ρ~

ρ

( ) (1 Φ [ , ]) ( Φ [ , ])

β·

β ( )

( Φ [ , ])

β·

β ( )·

( Φ [ , ])

β· ( ) ·

β ( )~

( ) ·( Φ [ , ])

β·

β ( )~

fp sys

i

n

i j

nn

j

j

i

j

nn

l

nj

l

l

i

j

nn

ji

ij

nn

j

j

1

1 j 1

1row

T j

rowT

1

e

th

th

(45)

Fig. 5. Geometry of a space truss dome: (a) basic grid, (b) section view along grid line 1 5, and (c) section view along grid line 3 7.– –

Table 3

Parameters for lters of ground motion models and constraints in optimizationfi

(space truss dome optimization example).

Φ0_g1 Φ0_g2 ωf ζ f t t (s) Δ (s) Initial cross-section

areas (m 2 )

Threshold value

4.0 3.0 5 0.4 6.0 0.06 0.25 π u0 xΔ = 1/800

u 0 yΔ = 1/800

Fig. 6. Optimized space truss dome corresponding to di erent angles of ground accelerations: (a) ff θg1 = 0°, θ g2 = 30°, (b) θ g1 = 0°, θ g2 = 60°, (c) θ g1 = 0°, θ g2 = 90°

(Color legends: A i = A min in green, 0.02 m2 < Ai ≤ 0.2 m 2 in blue, 0.2 m2 < Ai ≤ 0.4 m2 in brown, 0.4 m 2 < A in red). (For interpretation of the references to color

in this gure legend, the reader is referred to the web version of this article.)fi


87

Fig. 7. Optimized cross-sectional areas of truss elements corresponding to the ground accelerations applied at di erent angles.ff

Fig. 8. Convergence history: (a) volume and (b) rst-passage probability. Comparison between dynamic responses by the initial structure and the optimizedfi

structures: (c) randomly generated ground accelerations (θg1 = 0°, θ g2 = 30°), (d) drift ratio in the -direction and (e) drift ratio in the -direction.x y


88

where PT denotes the transpose of the ltering matrix in Eq. andfi (19) P( )

i rowTth is the th row vector of Pi- T. Thus,

∑∂∂

= ⎛

⎝⎜∂ −

∂∂∂

⎞

⎠⎟

=

P E

dP

β R ρ~

ρ

( ) ( Φ [ , ])

β·

β ( )~

fp sys

j

nn

j

jT

1 (46)

where the partial derivative of βj (·) with respect to an element density

can be computed as explained in Sections .4.2 and 4.3

4.5. Veri cation of calculated sensitivity

The adjoint sensitivity method derived for the rst-passage prob-fi

ability constraints is tested for the topology optimization problem inFig. 2(a) through comparison with the nite di erence method to verifyfi ff

accuracy and e ciency. The stochastic seismic acceleration (ffi f t) i smodeled as a ltered white-noise process using the Kanai-Tajimi lterfi fi

model with the intensity Φ0 [26,23]. The unit-impulse response func-

tion of the lter and the e ective force vector caused by the earthquakefi ff

excitation are determined as follows, respectively:

= −⎡

⎣⎢⎢

−

−−

− −⎤

⎦

⎥⎥

h t ζ ω tζ ω

ζω ζ t

ζ ω ω ζ t

( ) exp( )(2 1)

1sin( 1 · )

2 cos( 1 · )

f f ff f

f

f f

f f f f

KT

2

2

2

2

(47)

∫= − = − −( )t f t h t τ W τ dτf d M d l M d l( , ) ( ) ( ) ( ) · ( ) ( )t

f0

KT

(48)

Table 1 summarizes the Kanai-Tajimi lter parameters of dominantfi

frequency ω f and bandwidth ζf, the time interval of interest, and thethreshold value of the drift ratio at each time point.

The sensitivities of the rst-passage probability with respect to thefi

design variables located at the three points A, B and C in (a) areFig. 2

computed by the proposed adjoint method and the FDM, respectively.

The structural columns represented by two vertical lines in (a) areFig. 2modeled by frame elements. Young s modulus = 21,000 MPa and’ E

mass density ρ m = 2,400 kg/m 3 are used as material properties for both

the quadrilateral and frame elements. The instantaneous failure eventat a discretized time point is considered in terms of an averaged drift

ratio evaluated at two nodes of interest. Thus, the rst-passage event isfi

de ned asfi

= ⋃ = ⋃ ⎛

⎝⎜ −

+⩽ ⎞

⎠⎟

= =E E t u

t t

Hρ~

a ρ~ a ρ~ v( , )

( ( , ) ( , ) )

20sys

k

n

f kk

nk Left k Right

1 10Δ

T T

iΔ

(49)

The sensitivities of the rst-passage probability constraint in Eq.fi

(49) Fig. 2are shown in (b) and (c), which show good agreements. Thesensitivities by the FDM employing a range of perturbations (from 10−1

to 10−12 ) are tabulated in for comparison with the results byTable 2 the AJM, and the in uence of the perturbation size on the results by thefl

Fig. 9. (a) Design domain and loading condition, (b) node of interest for a tip drift ratio constraint, and (c) nodes of interest for inter-story drift ratios.

Table 4

Parameters for ground motion lter model and constraints of topology opti-fi

mization (topology optimization example).

Φ0 ω f ζf t t (s) Δ (s) Init. density Colum size Thres. value

7.5 5 0.4 6.0 0.06 0.7 0.6 m × 0.6 m π u 0Δ = 1/50


89

Fig. 10. Topology optimization results from the four-story building example constrained by the rst-passage probability in term of the tip drift ratio constraint: (a)fi

βtarget = 1.5, P ftarget = 6.68%, (b) β target = 2.5, Pf

target = 0.62%, and (c) βtarget = 3.0, Pftarget = 0.13%.

Fig. 11. Topology optimization results from the four-story building example constrained by the rst-passage probabilities in terms of inter-story drift ratio: (a)fi

βtarget = 1.5, P ftarget = 6.68%, (b) β target = 2.5, Pf

target = 0.62%, and (c) βtarget = 3.0, Pftarget = 0.13%.


90

FDM. The computational costs of the two methods are compared inFig. 3 while varying the number of elements in the problem. The

computational costs are normalized by that of the AJM for the 100-element case. It is noted that the proposed AJM requires dramatically

less computational time than the FDM. It should also be noted that

unlike FDM, AJM does not require determining the perturbation size,for which an optimal choice is generally not known a priori.

5. Numerical applications

5.1. Space truss dome optimization

In this example, the weight of an asymmetric space truss dome

composed of 104 elements ( ) is minimized under constraints onFig. 4the rst-passage probability of the drift ratio evaluated at the node offi

interest, i.e. the highest elevation. The basic grid of the structure, plan

view, and section views are provided in . (c) shows theFig. 5 Fig. 4element numbering choices of the space truss dome. At each node of the

structure, additional masses (10,000 kg) representing non-structuralmasses such as claddings are equally applied. Young s modulus’

E = 210 GPa and mass density m = 7,850 kg are used as material

properties for each truss element. The ground acceleration is generatedby using the Kanai-Tajimi lter. The lter and optimization parametersfi fi

are presented in . The probabilistic constraint is de ned in termsTable 3 fi

of the tip drift ratio evaluated at the top ( = 15 m). For a loadingz

scenario, two direction components of earthquake ground excitations at

angles (θg1 , θ g2) shown in (b) are considered simultaneously andFig. 4applied to the structure. The target failure probability and a lower

bound of design variables are set to Pftarget = 0.0013

(β ftarget = [−Φ P f

target ]−1 = 3.0) and 0.02 m2. The optimization for-mulation considering multiple ground accelerations with constraints on

drift ratios in both - and -directions is developed asx y

⎜ ⎟

⎜ ⎟

⎛⎝⋃ ⩽ ⎞

⎠⩽

⎛⎝⋃ ⩽ ⎞

⎠⩽

⩽ ⩽+ + = −

−

−=

−

−=

−

f

s t P E t g t P

P E t g t P

t t t θ f t

θ f t

d

d d

d d

d

M d u d C d u d K d u d M d l

M d l

min ( )

. ( ( , ): ( , ) 0)

( ( , ): ( , ) 0)

0.02m 1.5m

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

( ) ( ) ( )

obj

fp x dirk

n

f k k f

fp y dirk

n

f k k f

d

,1

x dirtarget

,1

y dirtarget

2 2

g1 g1

g2 g2

x x

y y

(50)

Fig. 6 shows that the space truss domes optimized with xed fi θ g1

while varying θ g2 to three di erent angles. Optimal results from theff

case of the applied ground acceleration with θg1 = 0°, θg2 = 90° show

Fig. 12. Convergence history of the four-story building. First-passage probabilities of inter-story drift ratio constraints: (a) volume and (b) rst-passage probability offi

each inter-story drift ratio.


91

that the cross-sectional areas of bracings and vertical elements, espe-cially at a lower level, are increased to reduce displacements in both -x

and -directions and failure probabilities. Also, as the angle y θ g2 be-comes closer to θ g1, the optimal volume increases. By changing θ g2 from

90° to 60° (or 30°), the increase in the failure probability in the x-di-

rection is much higher than the decrease in the failure probability in they-direction. The optimized area of each element is plotted in Fig. 7. A s

expected, truss members, which are closely aligned to the appliedground accelerations are enlarged especially for lower levels. Thus,

truss members are sized more symmetrically in both x- and y- directions

for ground accelerations with θg1 = 0°, θ g2 = 90° compared toθg2 = 30° or θ g2 = 60°. (a) and (b) show convergence histories ofFig. 8

the volume and the rst-passage probability. The proposed methodfi

enables achieving the target failure probability with reduced volumes.

The comparison of dynamic responses of the initial structure and the

optimized structure is shown in (c) through (e) under randomlyFig. 8generated samples of ground excitations with the lter parameters re-fi

ported in . Overall reductions in the drift ratio in the optimizedTable 3structure are observed, which naturally reduces the likelihood of ex-

ceedance of the threshold value during the excitation.

5.2. Optimization of a bracing system using topology optimization

The previous numerical application of the bracing system is con-

sidered as optimization for a given structural layout. By contrast,size

topology optimization can identify the optimal bracing layout of a

structure. To demonstrate this optimization under rst-passage prob-fi

ability constraints, the proposed method is applied to the design do-main under earthquake excitations as shown in (a). During theFig. 9

optimization for minimizing volume, the rst-passage failures are de-fi

fined in terms of inter-story drift ratios at each level, and a tip drift ratioat the building height (see (b) and (c)). The structural columnsFig. 9

represented by two vertical lines shown in (a) are modeled byFig. 9frame elements whose densities remain unchanged throughout the

optimization process. Young s modulus = 21,000 MPa and mass’ E

density m = 2,400 kg/m 3 are used as material properties for both thequadrilateral and frame elements. The additional mass of 4,000 kg is

considered at each oor level as shown in (a). The damping ma-fl Fig. 9trix is constructed using a Rayleigh damping model with a 2% damping

ratio.Table 4 summarizes the Kanai-Tajimi lter parameters of dominantfi

frequency ωf and bandwidth ζf , column size, the time interval of in-

terest, and the threshold value u0 of the average drift ratio at each timepoint. The ltering radius is 0.25 m, and a prescribed density 0.7 isfi r

applied uniformly throughout the mesh. Topology optimization resultsare shown from di erent target failure probabilities of the inter-storyff

drift ratios, and the tip-drift ratio constraints are shown in Figs. 10 and

11. For the tip drift ratio constraint, the increase in the thicknesses inlower levels and additional branches of material distributions are ob-

served as the target failure probabilities decrease, whereas bracingpoints and topologies remain relatively the same for all three cases

under the tip drift ratio constraint in . On the other hand, Fig. 10 Fig. 11

shows that connections of topologies to each oor level can be checkedfl

for inter-story drift ratio constraints except the lowest level. As the

target failure probability is reduced, the second lowest intersectionpoint of bracing and column also decreases in elevation, such that in the

case of (c), the intersection point is on the second level. Inter-Fig. 11

section points of bracings for upper two levels in both constrainedoptimization problems are at the midpoint of two oors so that X shapesfl

of bracings with 90° are observed. At lower levels, the bracing inter-section points become higher. In addition, there is a signi cant increasefi

in material distribution in lower levels of the tip displacement con-straint, whereas overall thicknesses of bracings throughout the building

height are increased for the inter-story drift ratio constraint. Thus,optimization results show that reinforcing lower regions will be an ef-

ficient approach to control the tip displacement whereas adjusting each

bracing module will lead to successful designs of structures ful llingfi

inter-story drift ratio criteria (see ).Fig. 12

For constructability and aesthetic aspect of architecture, a patternrepetition constraint can be implemented in the proposed fra-[ , ] 33 36

mework. Therefore, di erent choices regarding the number of patternsff

or size of the primary region in optimization will result in varioustopologies, which can provide diverse options of solutions for the ar-

chitectural and engineering schematic design process.

6. Summary and concluding remarks

In this paper, an optimization framework is proposed to incorporate

the rst-passage probability into size optimization and topology opti-fi

mization of structures. Using the discrete representation method and

theories of structural system reliability, the rst-passage probability isfi

computed e ciently during the optimization process with a properffi

consideration of the statistical dependence between component failure

events.Parameter sensitivity formulation of the probabilistic constraint on

the rst-passage probability is also derived based on the adjoint methodfi

and sequential compounding method to facilitate the usage of e cientffi

optimization algorithms. The developed method is successfully applied

to the lateral bracing system of structures subjected to stochasticground motions to identify optimal member sizes under engineering

constraints associated with structural design criteria such as the stress,the displacement as well as the inter-story drift ratio. In the numerical

application of the space truss dome subject to simultaneous multiple

earthquake ground motions, the proposed optimization frameworkprovides reliable structural solutions for various loading scenarios.

Furthermore, the proposed method can be further extended to

consider the rst-passage probability constraint constructed by com-fi

bining di erent types of failure events such as di erent time points andff ff

locations as well as multiple design criteria. The optimized system canwithstand future realization of stochastic processes with a desired level

of reliability. In addition, numerical examples show that the proposedtopology optimization framework can provide an efficient way for

structural engineers to obtain optimal design solutions that satisfy

probabilistic constraints on the stochastic response in the conceptual(and schematic) design process.

The proposed method is based on the assumption of a stationaryprocess for the earthquake ground motions. The stochastic excitation

generated by natural hazards (e.g. earthquakes, hurricanes) can be non-

stationary and/or non-Gaussian. Thus, future research could focus ondevelopments of optimization frameworks under non-stationary sto-

chastic processes in the time domain as well as in frequency domain.

Acknowledgements

The authors gratefully acknowledge funding provided by the

National Science Foundation (NSF) through projects 1234243 and1663244. The second author acknowledges the support from the

Institute of Engineering Research at Seoul National University. The

third author acknowledges support from the Raymond Allen JonesChair at the Georgia Institute of Technology. The information provided

in this paper is the sole opinion of the authors and does not necessarilyre ect the views of the sponsoring agencies.fl


92

Appendix A

Flowchart of implementation for RBTO of structures constrained by rst-passage probability.fi

Fig. A1

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