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Estimation of response expectation bounds under parametric p-boxes by1
combining Bayesian global optimization with unscented transform2
Chen Ding1, Chao Dang2, Matteo Broggi3, and Michael Beer4
3
1Institute for Risk and Reliability, Leibniz University Hannover, Callinstr. 34, Hannover 30167,4
Germany. Email: chen.ding@irz.uni-hannover.de5
2Institute for Risk and Reliability, Leibniz University Hannover, Callinstr. 34, Hannover 30167,6
Germany (corresponding author). Email: chao.dang@irz.uni-hannover.de7
3Institute for Risk and Reliability, Leibniz University Hannover, Callinstr. 34, Hannover 30167,8
Germany. Email: broggi@irz.uni-hannover.de9
4Institute for Risk and Reliability, Leibniz University Hannover, Callinstr. 34, Hannover 30167,10
Germany; Institute of Risk and Uncertainty, University of Liverpool, Peach St., Liverpool L6911
7ZF, UK; International Joint Research Center for Resilient Infrastructure & International Joint12
Research Center for Engineering Reliability and Stochastic Mechanics, Tongji University,13
Shanghai 200092, China. Email: beer@irz.uni-hannover.de14
ABSTRACT15
In engineering analysis, propagating parametric probability boxes (p-boxes) remains a challenge16
since a computationally expensive nested solution scheme is involved. To tackle this challenge, this17
paper proposes a novel optimization-integration method to propagate parametric p-boxes, mainly18
focusing on estimating the lower and upper bounds of structural response expectation for linear19
and moderately nonlinear problems. A model-based optimization scheme, named Bayesian global20
optimization, is first introduced to explore the space of distribution parameters. Subsequently, an21
efficient numerical integration method, named unscented transform, is employed to estimate the22
response expectation with a given set of distribution parameters. Compared to existing optimization-23
integration methods, the proposed method has three advantages. First, the response expectation24
1 Ding, October 19, 2023
bounds are successively estimated, allowing for the reuse of samples generated from the lower25
bound estimation in the upper bound estimation. Second, the approximation error introduced26
by the numerical integration method is considered. Third, computational efficiency in both the27
optimization and integration processes is improved. Four applications are investigated to validate28
the effectiveness of the proposed method, showing its ability to balance computational efficiency29
and accuracy when evaluating response expectation bounds.30
KEYWORD31
Imprecise probability propagation, Parametric probability box, Response expectation bounds,32
Bayesian global optimization, Unscented transform, Gaussian process33
INTRODUCTION34
Practical engineering problems are often rife with aleatory uncertainties that are irreducible and35
stem from random nature, such as the uncertainties in material properties, external loads, operating36
environments and etc. In many cases, the information for describing such uncertainties can be37
insufficient, ambiguous, fragmentary, or indeterminate (Beer et al. 2013). In this regard, epistemic38
uncertainties that result from a lack of knowledge or information should also be considered. Such39
mixed uncertainty can be characterized by the imprecise probability model such as the parametric40
probability box (p-box) model (Ferson and Hajagos 2004; Faes et al. 2021). For a parametric41
p-box model, aleatory uncertainty is represented by a set of probability distributions with known42
distribution types, while epistemic uncertainty is reflected by the imprecise distribution parameters43
that can be described by intervals. In order to reflect the influence of input imprecise probabilities44
on structural responses, imprecise probability propagation is of great significance in engineering45
structural analysis.46
In general, the state-of-the-art methods for propagating parametric p-boxes can be classified into47
two categories: double-loop methods and single-loop methods. As a straightforward approach, the48
double-loop method treats the epistemic and aleatory uncertainties through a nested loop structure.49
Double-loop Monte Carlo simulation (DLMCS) (Bruns and Paredis 2006) samples different sets50
2 Ding, October 19, 2023
of distribution parameters in the outer loop, and for each distribution parameter set, Monte Carlo51
simulation (MCS) is performed to estimate the output of interest in the inner loop. DLMCS works52
regardless of nonlinear properties and dimensionalities of the problem at hand. However, it is quite53
computationally expensive, since both loops require a considerably large number of samples in54
order to ensure the estimation accuracy of the output of interest. Although some improved DLMCS55
methods, such as interval Monte Carlo method (Zhang et al. 2010; Zhang et al. 2012) and the56
vertex-based DLMCS (Vertex-MCS) (Xiao et al. 2016), are developed to reduce the number of57
samples in total, their scopes of application and efficiency are limited.58
To improve the computational efficiency, an outer-loop optimization can be adopted, where59
imprecise distribution parameters are treated as design variables to be optimized and the lower and60
upper bounds on the output of interest are regarded as two separate optimization objectives. In the61
inner loop, the output of interest at a certain design point can be estimated by aleatory uncertainty62
propagation methods. Take the example of capturing the bounds on a response expectation,63
numerical integration methods can be adopted in the inner loop to evaluate the response expectation64
under fixed distribution parameters. The integration of outer-loop optimization and inner-loop65
numerical integration methods can be collectively referred to as optimization-integration methods.66
Typical optimization-integration methods are the optimized parameter sampling (OPS) (Bruns and67
Paredis 2006; Bruns 2006), optimized univariate dimension-reduction method (OUDRM) (Liu et al.68
2018) and optimized sparse grid numerical integration method (OSGNI) (Liu et al. 2019). Such69
existing methods rely on gradient-based optimizers, which can easily converge to a local optimum.70
In this sense, the resulting response expectation bounds may be underestimated. Although some71
global optimization algorithms, like the genetic algorithm (Pedroni and Zio 2015), can help mitigate72
this issue, the optimization process requires a large number of objective function calls, which can be73
time-consuming when dealing with objective functions that are expensive to evaluate. In the inner74
loop, some efficient numerical integration methods, such as univariate dimension-reduction method75
(Liu et al. 2018) and sparse grid numerical integration (Liu et al. 2019), are employed. Nevertheless,76
the computational efficiency within the inner loop can be further enhanced, especially for linear77
3 Ding, October 19, 2023
and moderately nonlinear problems. Additionally, existing optimization-integration methods do78
not account for the approximation error introduced by numerical integration methods, potentially79
resulting in inaccuracies in the derived response expectation bounds. Furthermore, these methods80
are unable to fully leverage the data generated by the lower or upper estimation that has already81
been performed.82
On the other hand, to alleviate the computational burden of the double-loop framework, many83
single-loop methods have been recently developed, such as the extended Monte Carlo simula-84
tion (EMCS) (Wei et al. 2014), non-intrusive imprecise stochastic simulation (NISS) (Wei et al.85
2019), non-intrusive imprecise probabilistic integration (NIPI) (Wei et al. 2021b), collaborative86
and adaptive Bayesian optimization (CABO) (Wei et al. 2021a), and parallel Bayesian quadrature87
optimization (PBQO) (Dang et al. 2022a). Note that existing single-loop methods typically rely88
on constructing an augmented uncertainty space consisting of both aleatory and epistemic uncer-89
tainties, which increases the dimensionality to be dealt with. Although some single-loop methods90
such as CABO and PBQO may require less response function calls compared with double-loop91
methods, it becomes difficult to estimate the output of interest for problems with high-dimensional92
augmented uncertainty space. Besides, EMCS and NISS are not capable for propagating parametric93
p-boxes with distribution parameters that are supported in a wide range.94
Therefore, there is still a need to develop a method for parametric p-box propagation with not95
only reasonable accuracy and efficiency, but also fine applicability. The main focus of this paper96
is on capturing the response expectation bounds that reflect the effect of epistemic uncertainty on97
the statistical characteristics of the response. A new optimization-integration method is presented98
to estimate the response expectation bounds, especially for linear and moderately nonlinear prob-99
lems. The proposed method combines two advanced and efficient strategies to greatly reduce the100
computational efforts. Specifically speaking, to facilitate the optimization process, a model-based101
optimization scheme, named Bayesian global optimization (BGO) (Jones et al. 1998), is employed.102
By using the BGO, the original expensive-to-evaluate objective function can be predicted by an103
cheap-to-evaluate Bayesian model. To consider the effects of approximation errors introduced by104
4 Ding, October 19, 2023
numerical integration in estimating the response expectation, noisy Gaussian process (GP) model105
is adopted in this study. Such GP model can be updated adaptively according to an effective106
improvement strategy, so as to obtain the globally effective optima, i.e., the optimal distribution107
parameters corresponding to the response expectation bounds, with fewer computational efforts.108
When estimating response expectation on the set of distribution parameters obtained from the109
optimization process, a highly efficient numerical integration method, called unscented transform110
(UT) (Julier and Uhlmann 1997b; Wan and Van Der Merwe 2000; Jia et al. 2013), is implemented.111
UT is able to provide estimated results up to third degree of algebraic accuracy, which should112
be acceptable for linear and moderately nonlinear problems. Besides, the number of simulations113
grows only linearly with the dimension of p-box variables. It is worth mentioning that compared114
with existing optimization-integration methods, the proposed method takes into account the ap-115
proximation errors brought by UT. Moreover, the proposed method estimates the lower and upper116
response expectation bounds in a sequential manner, where the samples generated for the lower117
bound evaluation can be further reused for the upper bound evaluation to reduce unnecessary waste118
of computational efforts.119
The remaining of the paper is organized as follows: Section "Problem statement" introduces the120
mathematical formulation of the response expectation bounds considering input variables described121
by parametric p-boxes. Section "Proposed method" presents the proposed optimization-integration122
approach that combines the BGO with the UT. Section "Test examples" investigates four test123
examples to illustrate the feasibility of the proposed method. Conclusions are given in section124
"Concluding remarks".125
To enhance readability, a list of acronyms used in this paper is provided in Table 1.126
PROBLEM STATEMENT127
Consider a response function that describes the input-output relationship of a structural system128
as 𝑌=𝑔(𝑿). Here, 𝑔(·)represents a deterministic, continuous and real-valued mapping function;129
𝑿=𝑋1, 𝑋2, ..., 𝑋𝑛𝑠is an 𝑛𝑠dimensional input vector of variables, where each variable is130
characterized by a parametric p-box model; 𝑌denotes a scalar output of interest, which is also a p-131
5 Ding, October 19, 2023
box variable. Let us denote 𝜽=𝜃1, 𝜃2, ..., 𝜃𝑛𝜃as 𝑛𝜃-dimensional imprecise distribution parameter132
vector. The epistemic uncertainty in 𝜽is represented by a hyperrectangle, i.e., 𝜽=𝜽,¯
𝜽, where133
𝜽=𝜃1, 𝜃2, ..., 𝜃𝑛𝜃denotes the lower bound and ¯
𝜽=¯
𝜃1,¯
𝜃2, ..., ¯
𝜃𝑛𝜃is the upper bound. Then,134
the joint probability density function (PDF) of 𝑿can be represented by 𝑓(𝒙|𝜽). For convenience,135
all variables in 𝑿and distribution parameters in 𝜽are assumed to be mutually independent.136
Under the above setting, the probability distribution and any statistical moments of 𝑌are also137
functions of 𝜽. Taking the expectation of 𝑌as an example, it can be written as:138
𝑚(𝜽)=∫R𝑛𝑠
𝑔(𝒙)𝑓(𝒙|𝜽)d𝒙,(1)139
where 𝑚(𝜽)represents the expectation of 𝑌, the value of which depends on the value of 𝜽. It140
is noted that for some practical engineering applications, the analysts may be more interested in141
obtaining the bounds on 𝑚(𝜽)than in obtaining expressions for the response expectation function142
over the entire domain of 𝜽. This is because the response expectation bounds enable to provide a143
possible range reflecting the effect of epistemic uncertainty on expectation of 𝑌(Wei et al. 2021a).144
In addition, evaluating the bounds on response expectation can be much more easier than capturing145
the overall behavior of 𝑚(𝜽)over the full domain of 𝜽. In this regard, this paper focuses on the146
estimation of the response expectation bounds.147
The lower and upper bounds of response expectation can be obtained by finding the minimal148
and maximal values of 𝑚(𝜽)within the hyperrectangle 𝜽,¯
𝜽, which can be expressed as:149
𝑚=min
𝜽∈[𝜽,¯
𝜽]𝑚(𝜽),(2)150
151
𝑚=max
𝜽∈[𝜽,¯
𝜽]𝑚(𝜽),(3)152
where 𝑚and 𝑚denote the lower and upper bounds of 𝑚(𝜽), respectively. In the following, the153
optimal distribution parameter corresponding to 𝑚is denoted as 𝜽min , and the optimal distribution154
parameter corresponding to 𝑚is represented as 𝜽max .155
6 Ding, October 19, 2023
Note that for most cases, the analytical expression of 𝑚(𝜽)is usually difficult and even impossi-156
ble to obtain due to underlying complexity of Eq. (1). Hence, directly finding analytic solutions for157
the minimum and maximum values according to Eqs. (2)-(3) is rather difficult. Alternatively, we158
can resort to the optimization-integration method such that 𝜽can be first searched by optimization159
and the response expectation at a certain 𝜽found during the optimization process is then estimated160
based on a numerical integration method. Since the response function involved in Eq. (1) is161
usually a black-box function that is expensive to evaluate, a computationally efficient optimization-162
integration method that calls the response function as few as possible is highly desired. At the same163
time, such method should also enable to provide estimated response expectation bounds with ac-164
ceptable accuracy. To achieve this aim, a novel optimization-integration method will be developed165
in the following.166
PROPOSED METHOD167
In this section, a new optimization-integration method is developed to estimate the lower168
and upper bounds of 𝑚(𝜽)with reasonable accuracy and efficiency, where 𝑚and 𝑚are separately169
estimated one after the other. Note that the proposed method is able to make full use of the available170
information such that data obtained from the lower bound estimation can be further reused in the171
upper bound estimation, and thus avoiding unnecessary computational effort. Specifically, a model-172
based optimization method, named Bayesian global optimization (Jones et al. 1998), is employed173
in order to explore the space of distribution parameters. At each 𝜽found during the optimization174
process, one highly efficient numerical integration method, named unscented transform (Julier and175
Uhlmann 1997b; Wan and Van Der Merwe 2000; Jia et al. 2013), is introduced to evaluate 𝑚(𝜽).176
Bayesian global optimization177
By making use of BGO, our basic idea is to assume a Bayesian model to 𝑚(𝜽), and then178
update the Bayesian model successively with additional observations according to an efficient infill179
sampling criterion (Jones et al. 1998; Dang et al. 2022b). Such infill sampling criterion enables180
to fully exploit the available observations, and strike a good tradeoff between exploitation and181
exploration for the selection of the new updating observations. In the following, the Bayesian182
7 Ding, October 19, 2023
model and infill sampling criterion adopted in the optimization process of proposed optimization-183
integration method are introduced in detail.184
Gaussian process model185
Following a Bayesian approach, the expensive-to-evaluate response expectation function can be186
treated with a Bayesian model. Commonly, Gaussian process (GP) model (Williams and Rasmussen187
2006) is adopted in the BGO. Note that for most realistic modeling situations, we cannot obtain188
the true value of 𝑚(𝜽), but only a noisy version of 𝑚(𝜽). In this regard, we assume that the noisy189
version of 𝑚(𝜽), denoted as ˆ𝑚(𝜽), is equal to the true response expectation function 𝑚(𝜽)plus an190
additional noise 𝜖such that ˆ𝑚(𝜽)=𝑚(𝜽)+𝜖, where 𝜖is assumed to follow a zero-mean Gaussian191
distribution with variance 𝜎2
𝜖. The true response expectation 𝑚(𝜽)is assigned a GP prior such that192
𝑚0(𝜽)∼GP (𝛽(𝜽), 𝜅 (𝜽,𝜽′)), where 𝛽(𝜽)and 𝜅(𝜽,𝜽′)are the prior mean and covariance (also193
called kernel) functions, respectively. There are many different forms of prior mean and covariance194
functions, which can be found in Ref. (Williams and Rasmussen 2006). In this work, a constant195
prior mean is adopted such that 𝛽(𝜽)=𝛽0and 𝛽0∈R. The squared exponential kernel function196
is employed here, which can be expressed as:197
𝜅(𝜽,𝜽′)=𝜎2
0exp −1
2(𝜽−𝜽′)𝜮−1(𝜽−𝜽′)T,(4)198
where 𝜎2
0denotes the overall variance; 𝜮=diag 𝑙2
1, 𝑙2
2, ..., 𝑙2
𝑛𝜽is a diagonal matrix and 𝑙𝑖, 𝑖 =199
1, ..., 𝑛𝜽is the length scale in the 𝑖-th dimension. Under these settings, a total of 𝑛𝜽+3free200
parameters are involved inside the GP model, which are referred to as the hyperparameters 𝝍=201 𝛽0, 𝜎0, 𝜎𝜖, 𝑙1, ..., 𝑙𝑛𝜽and can be inferred from a set of observations.202
Suppose that we have obtained Nnoisy observations. Denote such training dataset as D=203 n𝚯,ˆ
M(𝚯)o, where 𝚯=𝜽(1);𝜽(2);...;𝜽(N)is a sample matrix with size (N × 𝑛𝜃), and ˆ
M(𝚯)=204 nˆ𝑚𝜽(1), ..., ˆ𝑚𝜽(N)oT
is an (N × 1)response expectation vector whose components have been205
evaluated. Based on the training dataset D, the hyperparameters can be optimally determined by206
8 Ding, October 19, 2023
maximizing the log marginal likelihood function (Williams and Rasmussen 2006):207
𝝍⋆=arg max
𝝍log 𝑝ˆ
M(𝚯)|𝚯,𝝍,(5)208
where209
log 𝑝ˆ
M(𝚯)|𝚯,𝝍=−1
2ˆ
M(𝚯)−𝛽0T𝑲+𝜎2
𝜖𝑰−1ˆ
M(𝚯)−𝛽0−1
2log 𝑲+𝜎2
𝜖𝑰−N
2log (2𝜋),
(6)210
in which 𝑲is an (N × N )covariance matrix with (𝑖, 𝑗)-th entry as 𝜅𝜽(𝑖),𝜽(𝑗);𝑰is an (N × N)
211
identity matrix. For more details, the interested readership may refer to Ref. (Williams and212
Rasmussen 2006).213
Once the hyperparameters are determined, a posterior distribution of 𝑚(𝜽)can be obtained by214
conditioning on D. At a new observation 𝜽, the posterior of 𝑚(𝜽)follows a normal distribution215
such that 𝑚N(𝜽)∼N𝜇N(𝜽), 𝜎2
N(𝜽). Here, the posterior mean 𝜇N(𝜽)is employed as the216
predictor of response expectation in the optimization process, and the posterior variance 𝜎2
N(𝜽)is217
the measure of prediction uncertainty. 𝜇N(𝜽)and 𝜎2
N(𝜽)can be expressed in closed form:218
𝜇N(𝜽)=𝛽(𝜽)+𝜿(𝜽,𝚯)𝑲+𝜎2
𝜖𝑰−1ˆ
M(𝚯)−𝛽(𝚯),(7)219
220
𝜎2
N(𝜽)=𝜅(𝜽,𝜽)−𝜿(𝜽,𝚯)𝑲+𝜎2
𝜖𝑰−1
𝜿(𝜽,𝚯)T,(8)221
in which 𝜿(𝜽,𝚯)is a (1× N )covariance vector between 𝜽and 𝚯, and its 𝑖-th component is222
𝜅𝜽,𝜽(𝑖);𝛽(𝚯)is an (N × 1)expectation vector with 𝑖-th component as 𝛽𝜽(𝑖).223
Expected improvement criterion224
To infer the response expectation bounds 𝑚and 𝑚from fewer training samples, an efficient225
infill sample strategy combined with the GP model is desired. Note that the aim of such strategy226
is to find the promising points where to evaluate the objective function by extracting as much as227
possible knowledge from the current posterior GP. Along this line, the expected improvement (EI)228
9 Ding, October 19, 2023
criterion (Jones et al. 1998) could be a good choice. By maximizing the EI, new update points229
can be selected by exploiting the best existing solutions from the GP model and exploring the230
undeveloped design space that may contain potential optima. In this regard, here we adopt the EI231
criterion to find 𝜽min and 𝜽max that respectively corresponding to 𝑚and 𝑚, where the lower bound232
𝑚is estimated first and the upper bound 𝑚is evaluated subsequently.233
EI criterion for lower bound optimization Let 𝜽⋆
min =𝑎𝑟𝑔 min1⩽𝑗⩽Nnˆ𝑚𝜽(𝑗)obe the current234
best solution to lower expectation bound 𝑚obtained from the training dataset D. We are aiming235
to search for a new sample point 𝜽that enables to bring about an improvement beyond the current236
lower response expectation bound at point 𝜽⋆
min. The expectation of such improvement conditional237
on 𝚯, denoted as LEI
min (𝜽), can be expressed as (Jones et al. 1998):238
LEI
min (𝜽)=Ehmax 𝜇N𝜽⋆
min−𝜇N(𝜽),0i=
Eh𝜇N𝜽⋆
min−𝜇N(𝜽)i,if 𝜇N(𝜽)< 𝜇N𝜽⋆
min
0,otherwise
,
(9)239
where 𝜇N𝜽⋆
min=min1⩽𝑗⩽Nnˆ𝑚𝜽(𝑗)o. The analytical expression of the above EI function can240
be derived as (Jones et al. 1998):241
LEI
min (𝜽)=𝜇N𝜽⋆
min−𝜇N(𝜽)Φ©«
𝜇N𝜽⋆
min−𝜇N(𝜽)
𝜎N(𝜽)ª®®¬+𝜎N(𝜽)𝜙©«
𝜇N𝜽⋆
min−𝜇N(𝜽)
𝜎N(𝜽)ª®®¬
,
(10)242
where Φ(·)and 𝜙(·)represent the cumulative distribution function (CDF) and PDF of the standard243
normal distribution, respectively. The new sample point, denoted as 𝜽+
min, is determined by244
maximizing LEI
min (𝜽), i.e.,245
𝜽+
min =arg max
𝜽∈h𝜽,𝜽iLEI
min (𝜽).(11)246
The first term in Eq. (10) prefers the point related to smaller 𝜇N(𝜽), while the second term in Eq.247
(10) prefers the sample point that has larger prediction uncertainty 𝜎N(𝜽). Hence, a good tradeoff248
between model exploitation and exploration can be achieved by the EI criterion. Note that since249
10 Ding, October 19, 2023
LEI
min (𝜽)is usually multi-modal, additional global optimization algorithms are desired to solve Eq.250
(11). Herein, one recently developed global optimization algorithm, called Equilibrium Optimizer251
(EO) algorithm (Faramarzi et al. 2020), is employed.252
A stopping criterion is required here to indicate when to stop the lower bound optimization253
scheme. One common stopping criterion is to check whether the value of maximum EI is relatively254
small or not, i.e., max𝜽∈h𝜽,𝜽iLEI
min (𝜽)<E𝑚, where E𝑚denotes the stopping tolerance that is255
usually prescribed by the users based on their requirement. Since the magnitude of LEI
min (𝜽)is256
usually unknown in advance, an improved stopping criterion that measures the relative error of the257
maximal EI (Huang et al. 2006) is adopted here, such as:258
max𝜽∈h𝜽,𝜽iLEI
min (𝜽)
max1⩽𝑗⩽Nˆ𝑚𝜽(𝑗)−min1⩽𝑗⩽Nˆ𝑚𝜽(𝑗)<E1,(12)259
where ˆ𝑚𝜽(𝑗),1⩽𝑗⩽Nis the estimated expectation in the current D; the stopping tolerance260
E1is suggested to take the magnitude of 0.1% −1%. If Eq. (12) is not satisfied, 𝜽+
min and261
the corresponding response expectation ˆ𝑚𝜽+
minevaluated by a numerical integration method262
described in Section 4 are added to D, and then a new round of lower bound optimization is263
implemented based on the enriched D. To avoid possible premature convergence to suboptimal264
solutions, it is preferable to use a delayed judgement, i.e., to stop only when Eq. (12) is successively265
satisfied several times (e.g., three times).266
EI criterion for upper bound optimization Once the lower bound optimization scheme ends,267
the upper bound optimization starts based on the training dataset Dobtained from lower bound268
optimization. In this manner, the number of update points needed for upper bound estimation can269
be reduced and the current available training data can be further reused. Similarly, let 𝜽⋆
max =270
𝑎𝑟𝑔 max1⩽𝑗⩽Nnˆ𝑚𝜽(𝑗)obe the current best solution to the upper expectation bound 𝑚observed271
so far. Then, the location of next evaluation 𝜽is determined by maximizing the EI over the current272
11 Ding, October 19, 2023
maximum posterior response expectation 𝜇N𝜽⋆
max=max1⩽𝑗⩽Nnˆ𝑚𝜽(𝑗)o, i.e.,273
𝜽+
max =arg max
𝜽∈h𝜽,𝜽iLEI
max (𝜽),(13)274
where 𝜽+
max denotes the new update point associated with the upper response expectation bound.275
The corresponding EI function, denoted as LEI
max (𝜽), is defined in closed form (Dang et al. 2022b):276
LEI
max (𝜽)=Ehmax 𝜇N(𝜽)−𝜇N𝜽⋆
max,0i
=𝜇N(𝜽)−𝜇N𝜽⋆
maxΦ 𝜇N(𝜽)−𝜇N𝜽⋆
max
𝜎N(𝜽)!+𝜎N(𝜽)𝜙 𝜇N(𝜽)−𝜇N𝜽⋆
max
𝜎N(𝜽)!.(14)277
Here, EO algorithm (Faramarzi et al. 2020) is also employed to find 𝜽+
max.278
Similarly, the normalized version of stopping criterion for upper bound optimization scheme is279
adopted (Huang et al. 2006):280
max𝜽∈h𝜽,𝜽iLEI
max (𝜽)
max1⩽𝑗⩽Nˆ𝑚𝜽(𝑗)−min1⩽𝑗⩽Nˆ𝑚𝜽(𝑗)<E2,(15)281
where the stopping tolerance E2can take the same value as E1for convenience. If Eq. (15) is not282
satisfied, 𝜽+
max and corresponding response expectation ˆ𝑚𝜽+
maxestimated according to Section 4283
are added to the training dataset D. Then, another round of upper bound optimization is performed284
based on the enriched D. The optimization scheme stops only when Eq. (15) is satisfied for three285
times consecutively.286
Remark. Note that it is also possible to perform the upper bound optimization first and then the287
lower bound optimization. In this case, the training dataset Dobtained from the upper bound288
optimization will be used as the initial training dataset for the lower bound optimization.289
Unscented transform290
As observed from Eq. (1), the evaluation of 𝑚(𝜽)at a fixed design point 𝜽becomes a determin-291
istic but still difficult-to-evaluate integration. In this regard, one may resort to use the numerical292
12 Ding, October 19, 2023
integration method to approximate such deterministic integration. Denote the approximated solu-293
tion of 𝑚(𝜽)at a fixed observation 𝜽as ˆ𝑚. Using the numerical integration method, ˆ𝑚can be294
expressed as:295
ˆ𝑚=
𝑁𝑞
𝑟=1
𝑤𝑟𝑔𝝌𝑟,(16)296
in which 𝑁𝑞is the number of integration points; 𝝌𝑟is the 𝑟-th integration point, and 𝑤𝑟is the297
corresponding 𝑟-th weight.298
Under this setting, there is a need for an efficient method to evaluate ˆ𝑚. The unscented transform299
(UT) (Julier and Uhlmann 1997b; Wan and Van Der Merwe 2000; Jia et al. 2013) is adopted in our300
work, since UT is computationally more efficient while maintaining acceptable accuracy, compared301
to MCS, univariate dimension reduction method, and sparse grid numerical integration utilized302
in existing optimization-integration methods (Bruns and Paredis 2006; Liu et al. 2018; Liu et al.303
2019). The UT was first introduced by Jeffrey Uhlmann (Julier and Uhlmann 1997b) in the field of304
nonlinear Kalman filter, which enables to calculate the expectation of a random vector propagated305
through a nonlinear transformation. The basic idea of UT is to first select a finite number of sample306
points, also known as sigma points, transform these sigma points by a nonlinear transformation,307
and finally perform a weighted summation of the transformed sigma points to obtain an estimate308
of the response expectation. The sigma points are obtained by sampling in the original Gaussian309
distribution according to certain rules, and the corresponding weights satisfy the results of the310
weighted sum of the sigma points with the same mean and variance of the Gaussian distribution.311
Accordingly, the sigma points and corresponding weights can be respectively given by (Julier and312
Uhlmann 1997b; Julier and Uhlmann 1997a):313
𝜸1=[0, ..., 0]T, 𝑤1=𝜅
𝑛𝑠+𝜅, 𝑟 =1
𝜸𝑟=√𝑛𝑠+𝜅𝒆𝑟−1, 𝑤𝑟=1
2(𝑛𝑠+𝜅), 𝑟 =2,· ·· , 𝑛𝑠+1
𝜸𝑟=−√𝑛𝑠+𝜅𝒆𝑟−𝑛𝑠−1, 𝑤𝑟=1
2(𝑛𝑠+𝜅), 𝑟 =𝑛𝑠+2,·· · ,2𝑛𝑠+1,
(17)314
where 𝒆𝑟−1is the 𝑛𝑠-dimensional unit vector with the (𝑟−1)-th element being 1; 𝜅is the scaling315
13 Ding, October 19, 2023
factor that enables to tune the accuracy of moment approximations. According to Ref. (Julier316
and Uhlmann 1997b), it is suggested to take 𝜅=3−𝑛𝑠for sigma points following the Gaussian317
distribution. In this manner, we have318
𝜸1=[0, ..., 0]T, 𝑤1=3−𝑛𝑠
3, 𝑟 =1
𝜸𝑟=√3𝒆𝑟−1, 𝑤𝑟=1
6, 𝑟 =2,·· · , 𝑛𝑠+1
𝜸𝑟=−√3𝒆𝑟−𝑛𝑠−1, 𝑤𝑟=1
6, 𝑟 =𝑛𝑠+2,·· · ,2𝑛𝑠+1.
(18)319
Based on the above sigma points and corresponding weights, the response expectation can be320
estimated such that:321
ˆ𝑚=
𝑁𝑞
𝑟=1
𝑤𝑟𝑔Γ−1(𝜸𝑟|𝜽),(19)322
where 𝑁𝑞=2𝑛𝑠+1, which is highly efficient to evaluate the response expectation; Γ−1(·|𝜽)rep-323
resents the isoprobabilistic transformation that transform sigma points from the standard Gaussian324
space to the original input random space. In this regard, the integration points can be regarded as the325
transformed sigma points, where the transformation relationship is 𝝌𝑟= Γ−1(𝜸𝑟|𝜽), 𝑟 =1, ..., 𝑁𝑞.326
It is worth mentioning that responses corresponding to those 𝑁𝑞sigma points involved in Eq. (19),327
i.e., 𝑔Γ−1(𝜸𝑟|𝜽), 𝑟 =1, ..., 𝑁𝑞, can be evaluated in parallel.328
Note that the sigma points are generated by matching the moments of Gaussian random variables329
up to the second order. In addition, all odd-ordered moments of a Gaussian variable are zero.330
Therefore, the UT is able to estimate response expectation at a fixed 𝜽up to the third order, regardless331
of the dimension of input variables (Julier and Uhlmann 1997a; Wan et al. 2001). Nevertheless,332
the UT is somehow difficult to adapt to problems with strong nonlinearities (Julier 2002). In333
this regard, it is reasonable to expect that the use of UT to evaluate the response expectation can334
have acceptable accuracy and efficiency for linear and moderately nonlinear problems. However,335
the approximation error introduced by UT needs to be considered, as it may affect the accuracy336
of the outer-loop optimization within the distribution parameter space. To take into account the337
approximation error in the optimization process, the estimated response expectation at a fixed338
14 Ding, October 19, 2023
design point 𝜽, i.e., ˆ𝑚𝜽(𝑖), is treated as a noisy observation in the training dataset D.339
Step-by-step procedure340
Once both the stopping criteria associated with the lower and upper bounds are satisfied in the341
optimization process, the lower and upper response expectation bounds can be obtained from the342
final training dataset D=n𝚯,ˆ
M(𝚯)o, such as:343
𝑚=min
1⩽𝑗⩽Nˆ𝑚𝜽(𝑗),(20)344
345
𝑚=max
1⩽𝑗⩽Nˆ𝑚𝜽(𝑗),(21)346
where the current Nis the sample size of the final obtained D. Accordingly, the total number347
of response function calls required by response expectation bound estimation is 𝑁=N × 𝑁𝑞=348
N × (2𝑛𝑠+1). To distinguish the stopping criterion involved in lower bound estimation with that349
involved in upper bound estimation, the first criterion is named criterion 1, and the second is named350
criterion 2 in the following. A flowchart of the proposed method is shown in Fig. 1. To illustrate351
the procedure of proposed method, here we take the evaluation of the lower bound of response352
expectation as an example, and a brief procedure is summarized as follows:353
354
Step 1: Initialization. Set the initial sample size Nini and stopping tolerance E1. Create355
the initial training set D=n𝚯,ˆ
M(𝚯)oof size Nini by two steps. First, randomly sample Nini
356
distribution parameters 𝜽from the hyperrectangle 𝜽,¯
𝜽by adopting the Latin hypercube sampling357
(LHS) method, and form 𝚯=𝜽(1),𝜽(2), ..., 𝜽(Nini)T. Then, employ the UT to estimate Nini
358
response expectations ˆ𝑚(𝜽)at each component of 𝚯, and accumulate these resultant estimated359
expectations as ˆ
M(𝚯)=nˆ𝑚𝜽(1), ..., ˆ𝑚𝜽(Nini)oT
. Denote the current sample size as N, where360
N=Nini at present.361
Step 2: Optimization for finding 𝜽+
min. This step involves first training a noisy GP model362
of 𝑚(𝜽)based on the current training set Dsuch that 𝑚N(𝜽)∼GP 𝜇N(𝜽), 𝜎2
N(𝜽). The363
training of noisy GP model is realized by using the fitrgp function in the Matlab “Statistic364
15 Ding, October 19, 2023
and Machine Learning Toolbox", where the initial value for the noise standard deviation 𝜎𝜖is365
set to be 0.002. Then, the current best lower bound 𝜇N𝜽⋆
minis specified by 𝜇N𝜽⋆
min=366
min n𝜇N𝜽(1), 𝜇N𝜽(2), ..., 𝜇N𝜽(N)o. The new update point 𝜽+
min is selected from the hyper-367
rectangle 𝜽,¯
𝜽by maximizing EI over 𝜇N𝜽⋆
min, where the EO algorithm is employed.368
Step 3: Check the stopping criterion 1. To accommodate stochastic evaluations, criterion 1 in369
Eq. (12) is checked by three times successively. If the criterion 1 is satisfied, end the updating370
process and output the current Das the initial training set for upper bound optimization; otherwise,371
go to step 4.372
Step 4: Evaluation of the response expectation at 𝜽+
min. The response expectation at the373
new update point 𝜽+
min, i.e., ˆ𝑚𝜽+
min, is evaluated by the UT according to Eq. (19). A total of374
𝑁𝑞=2𝑛𝑠+1sigma points and corresponding weights involved in UT are generated by Eq. (18).375
Step 5: Enrichment of the training dataset. The new update point 𝜽+
min and corresponding376
expectation value ˆ𝑚𝜽+
minare added into the training set D. Then, set N=N + 1and go to Step377
2 to perform a new round of optimization.378
TEST EXAMPLES379
In this section, four test examples are investigated to verify the feasibility of the proposed380
method. In all cases, the size of initial training dataset takes Nini =min {2𝑛𝜃,10}, and the stopping381
tolerances for both lower bound and upper bound estimations take E1=E2=E=0.002. To382
illustrate the advantages of the proposed method, two existing optimization-integration methods,383
i.e., OSGNI (Liu et al. 2019) and OUDRM (Liu et al. 2018), are performed for comparison in384
all examples. Both of these two methods employ the fmincon algorithm with sequential quadratic385
programming (SQP) method in Matlab for searching the optimal distribution parameters in the386
optimization process, where the termination tolerances for first-order optimality and step size are387
set to be 10−6. At a certain set of distribution parameters, the OSGNI adopts the sparse grid388
numerical integration method (SGNI) (Heiss and Winschel 2008) using the nested quadrature389
rule with Gaussian weights to evaluate the response expectation, where the accuracy level 𝑘acc
390
representing the order of polynomial used for fitting is prescribed. The OUDRM employs the391
16 Ding, October 19, 2023
univariate dimension reduction method (UDRM) (Rahman and Xu 2004) for expectation estimation,392
in which the number of Gauss-Hermite points used, denoted as 𝑁G, is given in advance. For all393
examples, 𝑘acc =3and 𝑁G=6, unless otherwise specified in the example. Furthermore, in the first394
two examples, we also compare the results obtained by DLMCS (Bruns and Paredis 2006) method,395
Vertex-MCS (Dong and Shah 1987) method, and OPS (Bruns and Paredis 2006) method. Note396
that the outer loop of OPS is also performed by adopting MATLAB function fmincon with SQP397
algorithm, while the inner loop of OPS employs the MCS with 104runs. All the above methods398
are implemented in MATLAB on the same computer with Intel Core i7-11800H at 2.30 GHz and399
32GB of RAM.400
Example 1: a two-dimensional toy example401
A two-dimensional toy example is first investigated, whose response function is given by:402
𝑦=𝑔(𝑥1, 𝑥2)=1+(𝑥1−1)3
9+(𝑥2−1)3
16 ,(22)403
where 𝑥1and 𝑥2are both Gaussian random variables with non-deterministic distribution parameters,404
i.e., mean and standard deviation. The mean parameters of 𝑥1and 𝑥2, denoted as 𝜇1and 𝜇2, take405
the same interval value [−1,3]. And both the standard deviation parameters of 𝑥1and 𝑥2, i.e., 𝜎1
406
and 𝜎2, are set as [0.5,3].407
In this example, the lower and upper bounds of response expectation are estimated by the analyt-408
ical method, DLMCS, Vertex-MCS, OPS, OSGNI, OUDRM and the proposed method. Since this409
example is simple, the OUDRM employs the UDRM using 𝑁G=2Gauss-Hermite points in the410
inner loop. To examine the robustness, each method is repeatedly performed 10 times. The average411
results obtained by each method and the corresponding average total number of response function412
calls (denoted as 𝑁) are presented in Table 2, along with the average number of simulations associ-413
ated with the lower and upper bounds (denoted as 𝑁𝐿and 𝑁𝑈). Additionally, the coefficients of vari-414
ation (COVs) for the estimated bounds are reported. The analytical solution of response expectation415
can be easily derived as 𝜇true =1+1
9(𝜇1−1)(𝜇1−1)2+3𝜎12+1
16 (𝜇2−1)(𝜇2−1)2+3𝜎22,416
17 Ding, October 19, 2023
which provides the analytical lower and upper bounds of response expectation as 𝑚=−9.7639 and417
𝑚=11.7639, respectively. Note that since the UT has third-order algebraic accuracy, the response418
expectation estimated using the UT should be accurate for any given set of distributed parameters419
in this example. Compared with the analytical results, the proposed method obtains both the lower420
and upper bounds of response expectation in a robust and accurate manner. On average, only a421
total of 𝑁=𝑁L+𝑁U=76 +27 =103 response function calls are required, where Nini is included422
in 𝑁L. The OSGNI and OUDRM enable to provide quite accurate bound results, however, these423
two existing methods require more response function calls compared with the proposed method.424
In this sense, more computational efforts are required by the OSGNI and OUDRM. In addition,425
as observed from Table 2, the Vertex-MCS and OPS are able to give relatively accurate bounds,426
but both require more than one million samples, which is considerably expensive. Unfortunately,427
the traditional and widely used DLMCS is unable to obtain accurate lower and upper bounds on428
response expectation. Besides, the COVs of DLMCS results are larger than those by other methods.429
Example 2: a 120-bar spatial truss structure430
Example 2 investigates a 120-bar spatial truss structure subjected to seven vertical nodal loads431
(Dang et al. 2021), shown in Figure 2. In this figure, the nodes that bear vertical loads are marked432
with red circles and numbers. The vertical displacement of the top node of this structure is of433
interest in this example, which is analyzed by a finite element software, OpenSees. Each member is434
modeled as a truss element. A total of 48 nodes and 120 elements are involved in the finite element435
model. The Young’s modulus 𝐸0, cross-sectional area of element 𝐴and seven vertical nodal loads436
(i.e., 𝑃0, 𝑃2, 𝑃4, 𝑃6, 𝑃8, 𝑃10 , 𝑃12) are considered as input variables. Among them, 𝐸0,𝐴and 𝑃0are437
p-box variables, and 𝑃2, 𝑃4, 𝑃6, 𝑃8, 𝑃10 and 𝑃12 are aleatory variables. The description of these438
nine input variables is provided in Table 3.439
In this example, the expectation bounds of the response of interest are estimated by the Vertex-440
MCS, DLMCS, OPS, OSGNI, OUDRM and the proposed method, where the corresponding results441
are given in Table 4. We take the result obtained by the Vertex-MCS as the reference. As observed,442
the lower and upper response expectation bounds by the proposed method accord fairly well with443
18 Ding, October 19, 2023
the reference bounds. In addition, only 𝑁=𝑁𝐿+𝑁𝑈=247 +57 =304 response function calls are444
required in the proposed method, which is within affordable computational limits. Unfortunately,445
other selected double-loop methods, i.e., the DLMCS, OPS, OSGNI and OUDRM, can only provide446
narrower bounds on the response expectation but require much more computational efforts.447
Example 3: a jet engine turbine blade448
The third example consists of a jet engine turbine blade under pressure loading, as illustrated in449
Fig. 3a (MATLAB 2022). The turbine blade is governed by two mechanical boundary conditions,450
namely the pressure loads 𝑃1and 𝑃2on the pressure and suction sides caused by the surrounding451
high-pressure gases, and the fixed Dirichlet boundary condition on the left side. The model is452
discretized by adopting the linear tetrahedral elements with maximum element size as 0.01 m, as453
shown in Fig. 3b. A total of 21252 nodes and 11794 elements are involved. This turbine blade is454
assumed to be made by the nickel-based alloy (NIMONIC 90) material, where the Young’s modulus,455
coefficient of thermal expansion and Poisson’s ratio are represented by 𝐸,𝛼and 𝜈, respectively.456
Here, the maximum von Mises stress of the turbine blade caused by high pressure from surrounding457
gases is the response of interest, which can be obtained by performing linear stress analysis using458
the Matlab Partial Differential Equation (PDE) Toolbox (MATLAB 2022). Five p-box variables,459
i.e., {𝐸, 𝛼, 𝜈, 𝑃1, 𝑃2}, are considered in this example, of which the mean and standard deviation460
parameters are all bounded by intervals. The detailed description of these p-box variables is listed461
in Table 5. Fig. 3c shows a resultant von Mises stress nephogram obtained by performing one462
structural analysis with all input variables to be fixed at the midpoint of their mean parameter463
intervals. As seen, the maximum von Mises stress happens at the tip of the turbine blade.464
The OSGNI, OUDRM, Vertex-MCS and proposed method are employed in this example to465
estimate the lower and upper bounds of response expectation, whose results are provided in Table466
6. It can be observed that the proposed method is able to obtain the response expectation bounds467
that are almost identical to those by the Vertex-MCS. However, much fewer response function468
calls (specifically 𝑁=143 +33 =176) are required by the proposed method, indicating that the469
proposed method has comparable accuracy but higher computational efficiency. In comparison,470
19 Ding, October 19, 2023
the OSGNI and OUDRM give narrower response expectation bounds but cost considerably larger471
computational efforts. Note that although this example involves only linear stress analysis, the large472
number of discrete elements results in a relatively long time for one evaluation of the response.473
We record the total computational times for all the methods implemented in this example, which474
are also given in Table 6. It is found that the proposed method takes much less time than OSGNI,475
OUDRM, and Vertex-MCS. Thus, it indicates that the proposed method is more computationally476
efficient for this example.477
Example 4: a crash box in the vehicle478
Last example investigates the frontal impact problem of a crash box impacted by a moving479
planar impactor. The crash box is an important energy absorbing component installed at the front480
of the vehicle, which determines the crashworthiness and ensures the safety of the vehicle. A481
quarter of a symmetric crash box (Reid 1998) shown in Fig. 4 is considered in this example,482
which is analyzed by the LS-DYNA software in symmetric multiprocessing (SMP) version. The483
crash box is built as a tube with an uncertain shell thickness 𝑡, and adopts a steel-like material484
modeled by a piecewise linear plastic model with Possion’s ratio as 0.3, yield strength as 207 MPa,485
mass density as 7830 kg/m3, strain rate model as Cowper-Symmonds with parameter 𝐶=40486
and 𝑝=5, and an uncertain Young’s modulus 𝐸. The lower end of the crash box is fixed. A487
planar impactor, modeled as a rigid wall with imprecisely known mass 𝑀wall and initial velocity488
𝑣wall, crushes the crash box from the top downwards. Three triggers are applied to the crash box489
in order to achieve desired energy absorption and deformation pattern. The LS-DYNA keyword490
∗𝐶𝑂 𝑁𝑇 𝐴𝐶𝑇 _𝐴𝑈𝑇 𝑂 𝑀 𝐴𝑇 𝐼𝐶 _𝑆𝐼 𝑁 𝐺 𝐿 𝐸 _𝑆𝑈𝑅𝐹 𝐴𝐶 𝐸 is applied to formulate the contact between491
the impactor and the crash box. The simulation is terminated when the impactor stops moving492
or the total time reaches 15.01 ms. This example involves a total of 4 p-box variables with493
8 imprecise distribution parameters, i.e., {𝑀wall, 𝑣wall, 𝐸, 𝑡}, whose detailed description is listed494
in Table 7. Fig. 4c shows the deformation of the crash box under rigid wall impact, where495
{𝑀wall, 𝑣wall, 𝐸 , 𝑡}={800 kg,8.94 m/s,200 GPa,2 mm}. As observed, the impacted crash box496
deforms in a folding mode without global bending, showing its good ability to absorb the impact497
20 Ding, October 19, 2023
energy. In addition, under the same input conditions, the force-displacement curve of the rigid wall498
in the negative Z direction is illustrated in Fig. 5, which indicates that the investigated crash box499
undergoes irreversible nonlinear buckling deformation, and has a relatively strong nonlinearity.500
The output response of interest is the average force of the complete force-displacement curve501
measured at the rigid wall. Note that this example also requires a long computational time502
to perform a simulation. In order to demonstrate the effectiveness of the proposed method, a503
comparison is made between the results obtained from the proposed method, the Vertex-MCS,504
OSGNI and UDRM, as summarized in Table 8, alongside the respective total computational times.505
As observed, the proposed method enables to provide lower and upper bounds on the expectation of506
the averaged rigid wall force that are quite close to those of Vertex-MCS, while the proposed method507
requires much fewer response function calls, specifically 𝑁=162 +63 =225. In comparison,508
OSGNI and OUDRM produce narrower response expectation bounds but require a larger number509
of simulations. Moreover, the computational time for the proposed method is 2134.01 s, while the510
Vertex-MCS, OSGNI and OUDRM require 13245.27 s and 6639.07 s, respectively. Hence, this511
example illustrates that the proposed method can be applied not only to linear and weakly nonlinear512
problems, but also to problems with relatively strong nonlinearity.513
CONCLUDING REMARKS514
In this paper, an efficient optimization-integration method is developed for estimating the515
lower and upper bounds of response expectation for linear and moderately nonlinear problems516
with inputs characterized by parametric p-boxes. The proposed method combines the Bayesian517
global optimization (BGO) with a highly efficient numerical integration method named unscented518
transform (UT), to sequentially evaluate lower and upper bounds on response expectations. An519
adaptively refined noisy Gaussian process (GP) model is adopted to explore the space of distribution520
parameters considering the approximation error introduced by UT. Besides, the sequential design521
strategy of BGO allows the proposed method to reuse the samples generated by the lower bound522
estimation in the upper bound estimation. In the process of response expectation at a given set of523
distribution parameters, the UT is quite efficient and can obtain the estimates of response expectation524
21 Ding, October 19, 2023
up to third accuracy. Four test examples are investigated to demonstrate the applicability to both525
linear and moderately nonlinear problems. For all of these examples, the results obtained by the526
proposed method use a reasonable number of response function calls. In addition, the resultant527
response expectation bounds are almost the same as the provided reference results, showing the528
effectiveness of the proposed method. Compared with some existing double-loop methods such529
as DLMCS, Vertex-MCS, OPS, OSGNI and OUDRM, the proposed method is able to acquire530
the results with acceptable accuracy and higher computational efficiency. It can also be observed531
from the four test examples that the accuracy of the proposed method is mainly affected by the532
complexity and nonlinearity of the problem at hand. For simpler problems, increasing the level533
of epistemic uncertainty does not affect the accuracy, while for more complex problems, a higher534
level of epistemic uncertainty tends to have a greater impact on the accuracy of the results.535
Admittedly, since the approximated expectation by the UT has only up to third order accuracy,536
the proposed method is not suitable for addressing strong nonlinear problems and evaluating537
higher-order response moments. To mitigate this, we are actively exploring alternative numerical538
integration methods, such as the mixed degree cubature scheme (He et al. 2022), to capture higher-539
order moments within our framework. Besides, the BGO in the optimization process still suffers540
from the so-called “curse of dimensionality" problem, i.e., it may perform poorly for problems with541
more than 20 dimensions. Future work will focus on a time-saving method for evaluating bounds542
on higher-order response moments that is applicable to higher dimensional and stronger nonlinear543
problems.544
DATA AVAILABILITY STATEMENT545
All data, models, or code that support the findings of this study are available from the corre-546
sponding author upon reasonable request.547
ACKNOWLEDGMENTS548
Chen Ding acknowledges the support of the European Union’s Horizon 2020 research and549
innovation programme under Marie Sklodowska-Curie project GREYDIENT – Grant Agreement550
22 Ding, October 19, 2023
n°955393. Chao Dang thanks the support from the China Scholarship Council (CSC). Michael Beer551
appreciates the support of National Natural Science Foundation of China under grant 72271025.552
23 Ding, October 19, 2023
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List of Tables637
1 Listofacronyms ................................... 29638
2 Comparison of results by different methods (Example 1) . . . . . . . . . . . . . . 30639
3 Description of input variables (Example 2) . . . . . . . . . . . . . . . . . . . . . . 31640
4 Comparison of results by different methods (Example 2) . . . . . . . . . . . . . . 32641
5 Description of input variables (Example 3) . . . . . . . . . . . . . . . . . . . . . . 33642
6 Comparison of results by different methods (Example 3) . . . . . . . . . . . . . . 34643
7 Description of input variables (Example 4) . . . . . . . . . . . . . . . . . . . . . . 35644
8 Comparison of results by different methods (Example 4) . . . . . . . . . . . . . . 36645
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TABLE 1. List of acronyms
Acronym Definition
BGO Bayesian global optimization
CABO Collaborative and adaptive Bayesian optimization
COV Coefficient of variation
DLMCS Double-loop Monte Carlo simulation
EI Expected improvement
EMCS Extended Monte Carlo simulation
EO Equilibrium optimizer
GP Gaussian process
MCS Monte Carlo simulation
NIPI Non-intrusive imprecise probabilistic integration
NISS Non-intrusive imprecise stochastic simulation
OPS Optimized parameter sampling
OSGNI Optimized sparse grid numerical integration method
OUDRM Optimized univariate dimension-reduction method
p-box Probability box
PBQO Parallel Bayesian quadrature optimization
PDE Partial differential equation
PDF Probability density function
SGNI Sparse grid numerical integration method
SMP Symmetric multiprocessing
SQP Sequential quadratic programming
UDRM Univariate dimension-reduction method
UT Unscented transform
Vertex-MCS Vertex-based Monte Carlo simulation
29 Ding, October 19, 2023
TABLE 2. Comparison of results by different methods (Example 1)
Method 𝑚COV of 𝑚 𝑚 COV of 𝑚 𝑁
Analytical −9.7639 - 11.7639 - -
Vertex-MCS −9.7318 0.79% 11.7839 0.47% 16 ×105
DLMCS −8.1585 4.75% 9.8238 5.97% 104×104
OPS −9.8778 0.37% 11.6531 0.29% (896 +1063)×104=1.959 ×107
OSGNI −9.7639 0.00% 11.7639 0.00% 270 +270 =540
OUDRM −9.7639 0.00% 11.7639 0.00% 140 +27 =167
Proposed −9.7639 0.00% 11.7639 0.00% 76 +27 =103
Note: COVs relates to the coefficients of variation; for DLMCS and vertex-MCS, 𝑁=𝑁L×𝑁U;
for OPS, OSGNI, OUDRM and proposed method, 𝑁=𝑁L+𝑁U.
30 Ding, October 19, 2023
TABLE 3. Description of input variables (Example 2)
Variable Unit Description Distribution Mean Standard deviation
𝐸0MPa Young’s modulus Truncated Normal [164800,247200] [2060,10300]
𝐴mm2Cross section area Truncated Normal [900,1100] [1,5]
𝑃0kN Vertical load Lognormal [180,220] [2,10]
𝑃2kN Vertical load Lognormal 200 2
𝑃4kN Vertical load Lognormal 200 2
𝑃6kN Vertical load Lognormal 200 2
𝑃8kN Vertical load Lognormal 200 2
𝑃10 kN Vertical load Lognormal 200 2
𝑃12 kN Vertical load Lognormal 200 2
Note: Trancated Normal means the values are all positive.
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TABLE 4. Comparison of results by different methods (Example 2)
Method 𝑚(MPa) 𝑚(MPa) 𝑁
Vertex-MCS 25.2309 51.1475 64 ×104
DLMCS 25.9681 49.2774 103×104
OPS 27.8320 46.2661 (57 +75)×104=132 ×104
OSGNI 27.8153 46.2762 1304 +1304 =2608
OUDRM 27.8153 46.2762 432 +368 =800
Proposed 25.2393 51.1061 247 +57 =304
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TABLE 5. Description of input variables (Example 3)
Variable Unit Description Distribution Mean Standard deviation
𝐸Pa Young’s modulus Truncated Normal [204.3,249.7]×109[227,1135]×109
𝛼1/K Coefficient of thermal expansion Truncated Normal [1.143,1.397]×10−5[1.270,6.350]×10−7
𝜈- Poisson’s ratio Truncated Normal [0.243,0.297] [0.270,1.350]×10−2
𝑃1Pa Pressure load Truncated Normal [45,55]×104[5,25]×103
𝑃2Pa Pressure load Truncated Normal [405,495]×103[45,225]×102
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TABLE 6. Comparison of results by different methods (Example 3)
Method 𝑚(MPa) 𝑚(MPa) 𝑁CPU time
Vertex-MCS 93.8542 118.8419 1024 ×103804115.49 s
OSGNI 97.0640 115.0664 1122 +1122 =2244 7464.80 s
OUDRM 97.0640 115.0664 660 +572 =1232 5402.62 s
Proposed 94.1447 118.6347 143 +33 =176 821.28 s
Note: CPU time represents the total computational time.
34 Ding, October 19, 2023
TABLE 7. Description of input variables (Example 4)
Variable Unit Description Distribution Mean Standard deviation
𝑀wall kg Mass of the rigid wall Truncated Normal [760,840] [8,40]
𝑣wall m/s Velocity of the rigid wall Truncated Normal [8.10,9.90] [0.09,0.45]
𝐸GPa Young’s modulus Truncated Normal [195,205] [2,10]
𝑡mm shell thickness Truncated Normal [1.90,2.10] [0.02,0.10]
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TABLE 8. Comparison of results by different methods (Example 4)
Method 𝑚(kN) 𝑚(kN) 𝑁CPU time
Vertex-MCS 7.7243 9.4776 256 ×102166403.64 s
OSGNI 8.0823 9.2635 1353 +297 =1650 13245.27 s
OUDRM 8.0893 9.2379 672 +216 =888 6639.07 s
Proposed 7.8018 9.4252 162 +63 =225 2134.01 s
36 Ding, October 19, 2023
List of Figures646
1 Flowchart of the proposed method . . . . . . . . . . . . . . . . . . . . . . . . . . 38647
2 Diagram of 120-bar spatial frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 39648
3 Geometry, mesh diagram and von Mises stress nephogram of the jet engine turbine649
bladeunderpressureloads .............................. 40650
3a Geometry of the turbine blade (dimensions in m) . . . . . . . . . . . . . . 40651
3b Meshed model of the turbine blade . . . . . . . . . . . . . . . . . . . . . . 40652
3c Von Mises stress nephogram of structural analysis . . . . . . . . . . . . . 40653
4 Front and right view of the meshed model of a quarter of the crash box, and the654
deformation of the crash box under planar rigid wall impact . . . . . . . . . . . . . 41655
4a Front view of the meshed model . . . . . . . . . . . . . . . . . . . . . . . 41656
4b Right view of the meshed model . . . . . . . . . . . . . . . . . . . . . . . 41657
4c Deformation of the crash box under planar rigid wall impact . . . . . . . . 41658
5 Force-displacement curve of the planar impactor in the negative Z direction . . . . 42659
37 Ding, October 19, 2023
Lower bound optimization Upper bound optimization
Start
Set Nini and E1=E2=E, and let N=Nini
Generate initial training set D=nΘ,ˆ
M(Θ)o
by LHS and UT
Train a GP model
mN(θ)∼GP µN(θ), σ2
N(θ)with D
Learn θ+
min by maximizing EI,
i.e., θ+
min = maxθ∈[θ,θ]LEI
min (θ)
Criterion 1 satisfied?
Compute ˆmθ+
minby UT,
and add θ+
min and ˆmθ+
mininto D
N=N+ 1
Set the resultant training set Das the initial
training set for upper bound estimation
Train a GP model
mN(θ)∼GP µN(θ), σ2
N(θ)with D
Learn θ+
max by maximizing EI,
i.e., θ+
max = maxθ∈[θ,θ]LEI
max (θ)
Criterion 2 satisfied?
Compute ˆmθ+
maxby UT,
and add θ+
max and ˆmθ+
maxinto D
N=N+ 1
Obtain the bounds by Eqs. (20) and (21)
Stop
No
Yes
No
Yes
Figure 1: Flowchart of the proposed method
Fig. 1. Flowchart of the proposed method
38 Ding, October 19, 2023
Fig. 2. Diagram of 120-bar spatial frame
39 Ding, October 19, 2023
(a) Geometry of the turbine blade (dimen-
sions in m) (b) Meshed model of the turbine blade
(c) Von Mises stress nephogram of struc-
tural analysis
Fig. 3. Geometry, mesh diagram and von Mises stress nephogram of the jet engine turbine blade
under pressure loads
40 Ding, October 19, 2023
(a) Front view of the meshed model (b) Right view of the meshed model
(c) Deformation of the crash box under planar rigid
wall impact
Fig. 4. Front and right view of the meshed model of a quarter of the crash box, and the deformation
of the crash box under planar rigid wall impact
41 Ding, October 19, 2023
0 20 40 60 80 100 120 140
0
5
10
15
20
25
30
35
Fig. 5. Force-displacement curve of the planar impactor in the negative Z direction
42 Ding, October 19, 2023