Time-Variant Reliability-Based Design Optimization via T-PFR Method

Journal of Mechanical Science and Technology 37 (3) 2023 DOI 10.1007/s12206-023-0224-0
1375
Journal of Mechanical Science and Technology 37 (3) 2023
Original Article
DOI 10.1007/s12206-023-0224-0
Keywords:
· T-RBDO
· PFR
· Equivalent inverse most probable point
· T-PFR
· Effective
Correspondence to:
Zhenzhong Chen
zhenzh.c[email protected].cn
Citation:
Wu, Z., Chen, Z., Chen, G., Li, X., Jiang,
C., Gan, X., Qiu, H., Gao, L. (2023). An
efficient time-variant reliability-based
design optimization method based on
probabilistic feasible region. Journal of
Mechanical Science and Technology 37
(3) (2023) 1375~1387.
http://doi.org/10.1007/s12206-023-0224-0
Received September 20th, 2022
Revised November 8th, 2022
Accepted November 19th, 2022
†
Recommended by Editor
Hyun-Gyu Kim
An efficient time-variant reliability-based
design optimization method based on
probabilistic feasible region
Zihao Wu
1
, Zhenzhong Chen
1
, Ge Chen
1
, Xiaoke Li
2
, Chen Jiang
3
, Xuehui Gan
1
, Haobo Qiu
3
and Liang Gao
3
1
College of Mechanical Engineering, Donghua University, 2999 Renmin Road North, Songjiang, Shanghai
201620, People’s Republic of China,
2
Henan Key Laboratory of Mechanical Equipment Intelligent Manu-
facturing, School of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, No.5
Dongfeng Road, Zhengzhou 450002, People’s Republic of China,
3
The State Key Laboratory of Digital
Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong
University of Science and Technology, 1037 Luoyu Road, Wuhan 430074, People’s Republic of China
Abstract Time-variant reliability-based design optimization (T-RBDO) issues are probabil-
istic design problems that assess the dependability of the design across time. The engineering
situation is most closely resembled by it. The approaches to resolving T-RBDO problems, nev-
ertheless, are convoluted and time-consuming. In this paper, an effective strategy for solving
RBDO issues are extended to resolving TRBDO problems in order to increase efficiency. To
solve T-RBDO challenges, the time-variant probabilistic feasible region (T-PFR) approach,
which is based on the PFR method for resolving RBDO issues, was presented. The idea of the
equivalent inverse most probable point (EIMPP) is put out in this methodology. Based on the
EIMPP, the probabilistic feasible region is likewise developed for time-variant constraints. The
effective tactic of the proposed method is to identify the non-active time-variant constraints
using the probabilistic feasible region and to forego conducting time reliability analysis on them.
Three mathematics and two engineering issues demonstrate that the suggested solution is
viable and effective.
1. Introduction
In reliability-based design optimization (RBDO) issues, the optimal design points of structures
satisfy the minimal economic cost and failure probability requirements. Consideration of design
variable uncertainty and evaluation of constraint failure probability are RBDO concerns [1-6].
RBDO requires analysis and optimization in order to determine the ideal design points. In the
analysis loop, the failure probability of design variables subject to probabilistic constraints is
determined. In the optimization loop, the process of computing the optimal design variables is
carried out. Among these, the reliability analysis cycle is difficult and time-consuming. Some
methods employ novel strategies to determine reliability. The reliability index method (RIA) [7-
11] seeks the most probable point (MPP) to determine reliability, whereas the performance
measure approach (PMA) [12-15] seeks the inverse of the MPP.
These two strategies are commonly regarded as the most fundamental solutions for RBDO
difficulties. The reliability analysis loop and design optimization are coupled in the preceding
approach, which is known as a double-loop strategy. The approaches [16-18] based on a sin-
gle-loop strategy to solve the coupled RBDO problems are highly efficient. Data form the reli-
ability analysis loop would be passed to the design optimization loop using the decoupled strat-
egy [19-22]. These two loops in the decoupled strategy are connected; they are not nested.
The sequential optimization and reliability assessment (SORA) [23] method is effective and
computes the shifting vector in the reliability analysis loop. The probabilistic constraints will be
relocated to the feasible zone based on the shifting vector. The probabilistic feasible region
© The Korean Society of Mechanical
Engineers and Springer-Verlag GmbH
Germany, part of Springer Nature 2023

Journal of Mechanical Science and Technology 37 (3) 2023 DOI 10.1007/s12206-023-0224-0
1376
(PFR) [24] approach is likewise an efficient decoupling method
that enhances the efficiency of solving RBDO issues by ex-
panding on SORA's central idea.
The basic static RBDO model only accounts for the uncer-
tainty of design factors. Typically, neither material qualities nor
loading processes are addressed. In fact, the mechanism of
material property deterioration would proceed over time, and
the load on the structure would not be constant over its exis-
tence. When the material qualities and load vary, the struc-
ture's dependability will also change to some degree. At the
conclusion of the design lifetime, the reliability of some struc-
tures may not fulfill criteria. In other words, the failure probabil-
ity of a structure is not a constant throughout the design life-
span, but rather a variable that fluctuates with time. Based on
the basic static RBDO, time-variable reliability-based design
optimization (T-RBDO) takes time into account [25-29]. Time is
a variable in T-RBDO issues, resulting in a three-layer nested
structure consisting of a time loop, reliability analysis section,
and design optimization. Some methods based on T-RBDO
problems are proposed in recent years. The sequential ap-
proximate time-variant reliability analysis and optimization
(SATO) [30] method decoupled the original nested procedure
into a sequential process. The time-invariant equivalent
method (TIEM) [31] was an efficient way to resolve the T-
RBDO problems by constructing an equivalent time-invariant
RBDO problem and updating the design point. A new T-RBDO
method [32] performed the time-variant reliability analysis on
global kriging models to balance the efficiency and accuracy.
In other words, time-variable reliability analysis (TRA) com-
ponents with T-RBDO issues will take time factors into account.
The original analytical loop for time-invariant dependability was
a single loop, whereas the TRA considers the time variable to
be a double-loop structure. Single-loop reliability analysis has a
significant amount of calculation load when solving RBDO is-
sues; thus, double-loop TRA has an even bigger level of calcu-
lation burden when addressing T-RBDO problems. In recent
years, a variety of strategies for effectively resolving the TRA
loop have been developed [33-35]. Calculating the probability
of failure is the most vital aspect of TRA. Methods for resolving
TRA loops may be categorized into three types based on the
various ways for assessing failure probability: time integration
[36], time discrete [37], and outcrossing rate [38]. In T-RBDO
problem-solving, the outcrossing rate is a frequent variable.
The PHI-2 approach [39] is an effective way to calculate the
outcrossing rate and time-variable failure probability. The PHI-2
approach is only required to calculate the dependability of two
instantaneous moments in order to get the outcrossing rate.
The failure probability is then addressed using a parallel sys-
tem problem calculation. Based on the technique, the PHI-2
method for solving the TRA loop is very precise and efficient.
The I-PHI2 approach [40] is proposed as a technique for re-
solving issues with unknown intervals that vary over time.
It is challenging to apply RBDO problem-solving techniques
to T-RBDO situations. In general, the factor preventing the
extension of RBDO techniques to T-RBDO is the inability of the
TRA loop-solving method to compute MPP. The MPP is essen-
tial for RBDO problem-solving techniques, which may be im-
plemented in a design optimization loop to obtain the optimal
design point. It is recommended that the idea of equivalent
most probable point (EMPP) replace MPP in TRA. The time-
dependent SORA (t-SORA) [41] technique leverages the
EMPP and extends the SORA method to T-RBDO situations
by including the SORA method. A comparable PMA-based T-
RBDO issue, based on the notion of EMPP, is formulated [42].
Without a doubt, the EMPP is necessary for resolving the T-
RBDO issues. In EMPP, several efficient RBDO techniques
may be extended to T-RBDO.
From addressing time-invariant RBDO issues, it is evident
that the effect of non-active probabilistic constraints on the opti-
mal design point is negligible. In some RBDO approaches, the
reliability analysis components for non-active probabilistic con-
straints still present. This results in increased computation on
the applicable techniques. There are typically non-active prob-
abilistic restrictions in T-RBDO situations, particularly in engi-
neering problems. The PFR technique is a decoupled solution
for time-invariant RBDO problems that takes into account non-
active probabilistic constraints and has high efficiency and pre-
cision. On the basis of the PFR approach, the time-variant PFR
(T-PFR) method is presented to address T-RBDO problems
with non-active probabilistic constraints. The proposed T-PFR
approach might detect inactive probabilistic constraints and
minimize the number of TRA requests for those constraints.
Because TRA loops need a great deal of processing power, the
T-PFR technique can increase efficiency by restricting the num-
ber of TRA loops for presently unused probabilistic constraints.
In Sec. 2, several fundamental T-RBDO ideas that will be
utilized in the proposed technique are introduced. In Sec. 3, the
comprehensive theoretical principles and practical implementa-
tion procedures of the T-PFR approach are presented. Three
numerical cases and two engineering cases are used to verify
the feasibility of the proposed T-PFR method. In Sec. 5, sev-
eral significant findings are outlined.
2. Basis
2.1 Time-variant reliability-based design optimi-
zation
In engineering applications, the limit state functions are time-
dependent under the effect of corrosion, fatigue, and degrada-
tion processes. T-RBDO problems are RBDO problems with
time-variable limit state functions and time-variable design
variables. The T-RBDO problem may be described using the
following model:
:
:()
: Prob{ ( ) 0, [0, ]} ( )>∃∈ ≥Φtar
find
min f
s.t. g ,t t T
β
μ
μ
μ
X
(1)
where
μ
means the mean value vector of design variable

Journal of Mechanical Science and Technology 37 (3) 2023 DOI 10.1007/s12206-023-0224-0
1377
X
, t is the variable of time,
T
means the design whole
lifecycle,
tar
β
means the target reliability index. ()f
μ
is the
target function,
()
g
,t
X
is time-variant limit state function,
Prob{} is the probability of the event and
Prob{ ( )
i
g
,t >X
,[,]}tT∃∈00
means the time-dependent probability function,
which means the probability that the time-variant state function
will be greater than zero (not failure) during the design whole
lifecycle.
()Φtar β
is the standard normal cumulative distribu-
tion function.
When it comes to calculating reliability, the methodologies for
dealing with T-RBDO concerns and RBDO problems diverge
from each other. TRA is a temporal-consuming two-layer loop
that takes time variables into account. The primary objective of
the TRA component is to determine the time-dependent failure
probability of the structure. According to the varied methodolo-
gies treat for time variable, the approaches for solving the time-
variant reliability analysis loop may be categorized into three
types: the time integration method, the time discrete method,
and the outcrossing rate method. The approach that is most
frequently utilized to address the problems with time-variant
reliability is the outcrossing rate approach.
2.2 Outcrossing rate and time-variant failure
probability in PHI-2 method
The definition of outcrossing rate for the limit state function is
the average number of transitions from the safe state to the
failure state per unit of time. The approach suggested in this
study utilizes the outcrossing rate method. The PHI-2 method,
which is based on the outcrossing rate, is an efficient TRA
method that is necessary for computing the time-varying failure
probability. The PHI-2 method's formulation of the outcrossing
rate may be written as follows:
2((), ( ),(, ))
() Φ−+ + =
tttttt
vt t
ββ ρ
(2)
where
()vt stands for outcrossing rate,
2((), ( ),Φ−+ttt
ββ
(, ))+tt t
ρ
means binormal cumulative distribution function,
t and +tt are two different time instants, ()t
β
and
()+tt
β
are corresponding instantaneous reliability indexes
calculated by first-order reliability method (FORM),
(, )+tt t
ρ
is the correlation coefficient between limit state functions at
t ,
and
+tt . Time-variant failure probability is calculated based
on outcrossing rate. The average number of times the limit
state function switches from the safe state to the failed state in
one unit of time is known as the crossing rate. The outcrossing
rate can be used by some hypothesis models to determine the
cumulative time-variant failure probability.
In the Markov hypothesis model, the crossing events of limit
state functions usually occur in groups and each group of
events is independent of each other. Based on Markov hy-
pothesis, the time-variant failure probability can be expressed
as follows:
0
()
( ) 1 Prob{ 0}=1-exp[- ]
1(0)
=− = −
∫T
f
f
vt
PT N dt
P
(3)
where,
()
f
P0 means the instantaneous failure probability of
the structure at the initial time.
2.3 Probabilistic feasible region
Some approaches to RBDO issues have a one-to-one rela-
tionship between the reliability analysis and the current design
point. In other words, after deterministic optimization, reliability
analysis will be performed for each probabilistic constraint func-
tion at each design point. However, some probabilistic con-
straint functions have infinitely reliable at the all-design points.
This means that the probabilistic constraint function's depend-
ability won't change over the course of the iterations. Inactive
constraints are a common term for such functions. Therefore,
there is no need to perform the reliability analysis section of the
inactive probabilistic constraint function excessively. This issue
can be effectively resolved using the PFR approach.
Probabilistic feasible region is an essential concept proposed
in PFR method and the region can be defined as follows:
*
*
*
tar
ββ
⎧
⎫
−
⎪
⎪
Ψ= < −
⎨
⎬
⎪
⎪
⎩⎭
μμ
μσ
(4)
where
Ψ means the probabilistic feasible region,
*
μ
is the
mean value of design point that has been calculated,
*
σ is
the standard deviation of design variable.
*
β
means the cor-
responding real reliability index,
tar
β
means the target reliabil-
ity index.
The probabilistic feasible region is employed to evaluate the
connection between the design point and the probabilistic limit
state function. When the present design point achieved falls
inside the probabilistic feasible area, it is regarded to fulfill the
dependability criteria. In other words, the probabilistic con-
straint function for the current design point is inactive. The PFR
method will not analyze the limit state function's reliability if it is
inactive. In other words, the reliability analysis for the design
point within the probabilistic feasible region would not be per-
formed. Therefore, the PFR approach can increase the effi-
ciency of RBDO problem resolution by minimizing the number
of reliability analysis calls.
3. Time-variant probabilistic feasible region
method
The efficient resolution of T-RBDO problems is the goal of
the proposed T-PFR approach. The core concept and innova-
tion of the T-PFR approach will be introduced in this part.
3.1 Equivalent target reliability index
Reliability analysis, the establishment of a probabilistic feasi-

Journal of Mechanical Science and Technology 37 (3) 2023 DOI 10.1007/s12206-023-0224-0
1378
ble region, and deterministic optimization are the fundamental
PFR approach steps. Target reliability index plays a significant
part in the phases of establishing a probabilistic feasible region
and deterministic optimization. This means that in the decoup-
led PFR method's solution to RBDO problems, the target reli-
ability index is crucial.
However, the target reliability index is typically constant over
the course of the lifespan. This means that the design point's
distance from the limit state function is fixed as well. However,
the limit state functions will evolve with time in T-RBDO issues.
The position of the performance function's limit state at the
beginning of the design lifecycle is unquestionably different
from the position of the performance function's limit state at the
end of the design lifecycle. The reliability of the design point
would not satisfy the requirement at the end of the design if it
were determined with a fixed target reliability index by the de-
terministic optimization part at the initial time. The target reli-
ability index of T-RBDO issues should be time-variable such
that the best design points fulfill the reliability requirement
across the whole lifespan.
The concept of equivalent target reliability index is put out in
light of the time-varying property of constraint function and the
requirement of objective reliability for design points during the
entire design lifecycle. The equivalent target reliability index
means that the reliability index of the design point obtained
according to this target reliability index can always meet the
requirements of the target reliability index with the change of
time. At each iteration, an equivalent target reliability index is
examined in conjunction with the set target reliability index and
the incremental reliability index. The following equation deter-
mines the equivalent target reliability index:
=+
Etar tar
βββ
(5)
where
E
tar
β means the equivalent target reliability index,
tar
β
means the fixed target reliability index and
β
means the
increment of reliability index. The definition of the increment of
reliability index can be expressed as
0
=−t
ββ β
. In this
equation,
0
β
means the reliability index at start time and
t
β
means the time-variant reliability index in the whole lifecycle.
3.2 Equivalent inverse most probable point
Based on the probabilistic feasible region that is created us-
ing current reliability index, the PFR technique is effective at
solving RBDO problems with multiple probabilistic constraints.
Unfortunately, while computing the time-variant failure probabil-
ity, the TRA approach does not produce MPP, which is used to
calculated the current reliability index. And that leads to the fact
that the probabilistic feasible region cannot be established after
the TRA process in T-RBDO problems. As a result, the PFR
approach is inadequate for addressing T-RBDO issues.
The effective PFR method, however, can cut down on the
number of reliability analyses performed on inactive probabilis-
tic functions. The T-RBDO problems can also be resolved us-
ing this PFR technique plan to cut down on TRA calls. It is
important to resolve the MPP issue and create the probabilistic
feasible region for time-variant probabilistic constraints in order
to adapt the efficient strategy in PFR method to address T-
RBDO problems.
Only MPP at the certain moment can be determined using
the PHI-2 approach. It is impossible to obtain the MPP corre-
sponding to the time-variant failure probability. However, the
decoupling strategy of PFR method places emphasis on the
inverse most probable point (IMPP), which is a kind of MPP.
The production of shifting vectors involves IMPP in a significant
way, which passes the information between reliability analysis
and deterministic optimization. The idea of equivalent inverse
most probable point (EIMPP) is put out since time-varying reli-
ability analysis does not include IMPP points.
The essential notion of IMPP and the concept of equivalent
target reliability index are the foundations upon which the
EMPP is offered. The calculation formula can be displayed as
below:
00
*/
EIMPP Etar MPP MPP
β =UUU (6)
where
E
IMPP
U means the equivalent IMPP and
0
MPP
U means
the MPP at initial moment in PHI-2 method.
3.3 Equivalent probabilistic feasible region
In the PFR method, probabilistic feasible regions are deter-
mined using design points and accompanying reliability indices.
The probabilistic feasible region might be a circle, a sphere, or
a super sphere based on Eq. (4) in Sec. 2.3. The radius of the
initial probabilistic feasible region equals the difference be-
tween the current and target reliability indices. This is the tech-
nique to ensure that the reliability index of a design point within
the probabilistic feasible zone exceeds the intended reliability
index.
The target reliability index of the T-RBDO issue, however, is
only an immediate reliability index. In other words, the design
point that achieves the immediate target reliability index re-
quirement may not meet the lifespan reliability need. So, the
equivalent target reliability index is substituted for the target
reliability index in T-RBDO issue.
The equivalent probabilistic feasible areas are the domains
in which the time-variant reliability index of the present design
point satisfies its criteria throughout the duration of its lifespan.
Therefore, the equivalent probabilistic feasible area is reformu-
lated as follows:
()
k
mtEtar
iii
ββ
⎧
⎫
−
⎪
⎪
Ψ= < −
⎨
⎬
⎪
⎪
⎩⎭
μμ
μσ
(7)
where
Ψm
i
means the m-th probabilistic feasible region of i-th
probabilistic constraint,
μ
is the design point,
t
i
β
represents
the time-variant reliability index.

Journal of Mechanical Science and Technology 37 (3) 2023 DOI 10.1007/s12206-023-0224-0
1379
As the process of iterative optimization advances, numerous
equivalent probabilistic feasible regions are formed. These
equivalent probabilistic feasible regions can be piled on top of
one another in the normal space. Therefore, the union of these
two equivalent probabilistic feasible regions is described by the
following expression:
12
...
mm
iii i
Ω=Ψ∪Ψ∪ ∪Ψ (8)
where
Ωm
i
is m-th probabilistic feasible region of i-th probabil-
istic constraint.
3.4 The detailed process of T-PFR method
Based on the EIMPP and equivalent probabilistic feasible re-
gions, the T-PFR approach may be broken down into seven
distinct components. Fig. 1 depicts the T-PFR method's flow-
chart.
Step 1 : Initialize the random design variables' mean and
variance parameters.
Step 2 : If the design point falls inside the probabilistic feasi-
ble region, go to step 5. Otherwise, continue in order.
Step 3 : Perform the TRA design points part. In the TRA loop,
the PHI-2 approach is utilized to examine and evaluate time-
variant reliability indices.
Step 4 : Determine the equivalent target reliability index and
equivalent inverse most probable point.
Step 5 : Establish an equivalent probabilistic feasible region
with an equivalent target reliability index and a time-variant
reliability index.
Step 6 : Calculate the current best design point using the de-
terministic design optimization iteration. The vector used for
shifting is determined by the comparable most likely point.
Step 7 : If the optimization loop concludes, the present de-
sign point will be the ideal design point on a global scale. If not,
go to step 2.
4. Applications
Five applications are utilized to demonstrate the efficacy of
the proposed T-PFR approach in this section. The t-SORA
approach is also utilized to solve these five applications to
demonstrate the efficiency and precision more easily.
4.1 Case 1: A two-constraint numerical exam-
ple
The numerical example may be utilized to clearly see all the
data and its variations. This classic example of two restricted
functions has been transformed into a T-RBDO issue in order
to demonstrate the practicability of the suggested solution. This
T-RBDO issue is represented by the following model:
12
22
12
1
21 2
11
22
2
:[,]
:() ( 3.7) ( 3.7)
: Prob{ ( ) 0, [0, ]} ( )
()
( ) ( 0.01) ( 0.01) 3
( 0.01 )sin(4( 0.01 ))
1.1( 0.01 )sin(2( 0.01 ))
( ,0.128 ), 1, 2
t
ii
find
min f
s.t. g ,t t T
g,t ab
g,t X tX t
aX t X t
bX t X t
XN i
μμ
μμ
β
μ
=
=− +−
>∃∈ ≥Φ
=− −
=+ ++ −
=+ +
=+ +
=∼
μ
μ
μ
X
X
X
12
0
12
03.7,0 4
1.80, [3.00,3.00].
tar tar
μμ
ββ
≤≤ ≤≤
== =μ
(9)
The design life cycle
T is 5 units of time and the time in-
crement
t
applied to the TRA loop is 0.0001 units of time.
Particularly, time in this scenario is only a unit of time, which
makes no sense. Fig. 2 displays the time-dependent limit state
functions. As time passes, the limitations gradually advance,
and the workable zone shrinks. The viable design zone is be-
tween the two limit state functions.
On the MATLAB R2020a platform, the T-PFR technique re-
solves this T-RBDO issue. The outcomes of the iterative ap-
proach are given in Table 1. From the starting design point (3.00,
3.00) to the optimal design point (2.80, 3.11), the design optimi-
zation procedure requires 5 iterations. Notable is the fact that
the number of TRA for probabilistic constraint 2 is extremely low,
and only in the initial iteration. In other words, probabilistic con-
Fig. 1. The flowchart of T-PFR method.