1. Introduction
In recent years, advanced computational technologies allow to develop detailed simulation models for virtual analysis and design optimization of structural systems. A key issue in this respect is to identify significant parameters while considering inherent uncertainties associated with the geometry, the material property, and the structural load variables. A recognized way to account for the input uncertainty is resorting to the probability theory. This includes the use of the probability theory to quantify input random variables and the effective algorithm for uncertainty quantification of the multivariate stochastic model [
1,
2]. The reliability-based sensitivity analysis that evaluates the significance input random variables with respect to the structural failure probability has received considerable attentions [
3,
4]. Due to computationally demanding cost for the reliability-based sensitivity analysis with a rather small failure probability, numerical evaluation of the sensitivity index becomes a challenging task. To this end, the paper presents an effective approach for the reliability-based sensitivity analysis based on the principle of maximum entropy (MaxEnt) and the fractional moment.
An accurate estimation for the structural failure probability is a precondition for the reliability-based sensitivity analysis. In engineering realities, the structural failure probability is usually defined based on a multivariate performance function
, i.e.,
. Herein, the input random vector
consists of all input random variables, whereas the failure domain is defined as
. Particulary, a numerical transformation is necessary to determine statistically independent random variables [
5]. Note that the reliability-based sensitivity index is mathematically defined as the partial derivative of
with respect to the mean and the standard deviation of input random variables, i.e.,
and
(as
). Therefore, a positively defined sensitivity index implies an increase of the distribution parameter will determine an increased structural failure probability, whereas a negative valued sensitivity index implies an inverse controlling effect of the distribution parameter on the structural failure probability. Note that the sensitivity index for the standard deviation is always negative, and an increase of variability of input random variables will generally increase the variation of a structural response, which further increases the exceeding probability of the model response quantity with respect to a predefined response threshold as shown in numerical examples.
Numerical evaluation of the reliability-based sensitivity index depends largely on an accurate estimation of the structural failure probability. In this regard, the first/second-order reliability method was developed in the literature for an effective estimation of the structural reliability index [
6,
7,
8]. In addition, Bucher and Bourgund [
9] proposed to approximate
with a regression model to deal with implicit performance functions. Similar techniques, e.g., the polynomial chaos expansion [
10], the Kriging approximation [
11], and the artificial neutral network, etc. were reported in the literature [
12,
13]. Once a surrogate model of the structural performance function is analytically or numerically available, the subsequent reliability and reliability-based sensitivity analysis can be alternatively realized by the brute-force Monte-Carlo simulation and the response surface model. However, if the structural reliability result is gradually varied during the design optimization process, one has to develop new surrogate models for the updated structural reliability result [
14]. This motives the entropy-based approach for the reliability-based sensitivity analysis in this paper.
The reliability-based sensitivity index has been widely used to rank the significance of input random variables. Specially, the variance-based global sensitivity method was investigated in many literatures [
15,
16]. Based on the variance decomposition of a generalized multivariate structural model, it is possible to express the total output response variance as a combination of variance components that are related to each group of input random variables and their combinations. Instead, the reliability-based sensitivity index pays major attention on the relation between distribution parameters and the structural failure probability. In this respect, the application of the variance-based sensitivity result will be rather limited, if the response distribution function of a structural model is highly skewed [
17].
To effectively realize the reliability-based sensitivity analysis, Guo and Du [
18] proposed to use the FORM-based approach that is based on a linear approximation of the performance function at the most probable failure point. Song and Lu [
19] investigated the subset simulation and the variance reduction technique for the probabilistic sensitivity analysis [
20]. Since the reliability-based sensitivity analysis is always limited to a predefined level of the structural failure probability, one way to determine the overall sensitivity result is to repeat the whole simulation procedure many times for a various realizations of the structural failure probability value, which is referred to as the distribution-based sensitivity analysis in the literature [
21]. To this end, the paper presents an effective approach for the reliability-based sensitivity analysis based on the MaxEnt approach. The structural response distribution is first estimated by using the entropy optimization. Contrary to integer moments that are used in previous investigations, the fractional moment that is approximated by using the multiplicative dimensional reduction method is employed to derive probability distribution of a multivariate structural model for the sensitivity analysis.
To summarize, the objective of this paper is to present an entropy-based approach for reliability-based sensitivity analysis for a structural model function represented by using multivariate random variables. The principle of maximum entropy with fractional moment (ME-FM) is used to determine an accurate estimation result for the structural response distribution. The moment-based and the distribution- based sensitivity measures are derived to rank the significance of input random variables. Several examples in the literature are presented to demonstrate potential applications of this moment and the reliability-based sensitivity method.
The rest of the manuscript is organized as follows.
Section 2 briefly summarizes probability measures that are used in this paper to rank the significance of an input random variable. With the multiplicative dimensional reduction method (M-DRM), sensitivity indicators based on the moment and the reliability results are derived in
Section 3. Three examples in the literature are presented in
Section 4 to examine the effectiveness of this approach against the brute-force Monte-Carlo simulation method, and conclusions are summarized in
Section 5.
4. Numerical Examples
Engineering applications of the proposed approach for the moment-based and the reliability-based sensitivity analysis are illustrated by three examples in this section. Numerical examples presented in
Section 4.1 and
Section 4.2 are explicitly defined with respect to the input random variables, whereas the natural frequency function of a vehicle frame structure in
Section 4.3 is defined as an implicit function of geometry and material random variables. Compared with benchmark results provided by the brute-force Monte-Carlo simulation method, the performance of this MaxEnt approach is examined as follows.
To rank the importance of each distribution parameter, the following sensitivity indices in the literature [
40] are used in this paper:
where,
and
are the mean and the standard deviation of an
ith input random variable, and
denotes an
th order moment of the structural response function
. Specially, results for the reliability-based sensitivity index are dimensionless by multiplying with the standard deviation of the input random variable, and only results for the mean-value based moment sensitivity indices (
), i.e.,
and
are presented in numerical examples for the sake of brevity.
4.1. Reliability-Based Sensitivity Analysis A Cantilever Tube Structure
This example considers the reliability-based sensitivity analysis of a cantilever tube structure depicted in
Figure 1. The performance function is defined as
where,
denotes the material yield strength, and
represents the structural maximum stress. Failure events of the cantilever tube structure are specified as
.
With external forces
,
,
P, and
T, the maximum von-Mises stress on the top surface of the tube is given as
Herein, the torsion stress is defined as
, whereas the normal stress
is given as
Specially, the parameters are given as
. Therefore, the bending moment
M is determined as
Note that
d and
t represent the outside diameter and the thickness of the tube, respectively. The probabilistic characteristic of input random variables are summarized in
Table 1.
Figure 2 depicts the moment-based sensitivity result for the performance function in Equation (
39). It is observed that distribution parameters
,
and
influence the performance function negatively, where other mean-value parameters, i.e.,
(
) have shown positive sensitivity results. Moreover,
is identified as the most significant uncertain factor among all input variables to manipulate the mean-value response of the performance function, which is also justified by the reliability-based sensitivity result as follows.
To derive probability distribution of structural performance function
, the procedure summarized in Equation (
19) was implemented. Results for the MaxEnt parameter are summarized in
Table 2, whereas
Figure 3 presents simulation results for the probability distribution. With three-order fractional moments of the structural performance function i.e.,
, an accurate result for the probability distribution of
is determined as shown in
Figure 3. With the determined result for the probability of exceedance, the structural failure probability is estimated as
, which is fairly close to the benchmark result
.
Figure 4 presents results for the reliability-based sensitivity analysis of the cantilever tube example. The simulation result is determined based on
rounds brute-force Monte-Carlo simulations with
samples in each. Estimation results for
and
(
) are agreed well with the benchmark result. The Gaussian quadrature scheme was employed to evaluate low-variate integrals for the fractional moment and the moment-based sensitivity analysis. The total number of functional evaluations is
in this example, which is rather small as compared to that of the brute-force Monte-Carlo simulation based on
samples.
4.2. Reliability-Based Sensitivity Analysis of a Cracked Membrane
A thermal introduced crack is observed in a membrane due to variations of temperature in a heating environment. The membrane is heated with a permanent uniform temperature field
, whereas the temperature is reduced to the ambient temperature
T during a maintenance procedure. The heat drop implies the tension and the open of a mode I crack. In this regard, the stress intensity factor
of the crack produced by the heat variation can be evaluated as [
41]
Note that the probability distribution of the stress intensity factor
is estimated based on the MaxEnt procedure in Equation (
19). Therefore, with the material toughness parameter
, the performance function for reliability analysis of the membrane can be defined as
and the probability of exceedance of the cracked membrane is determined as
Herein, the probabilistic characteristic of input random variables are listed in
Table 3.
Figure 5 depicts numerical results for the mean-value based sensitivity index of the cracked membrane structure. Compared to benchmark results for
and
provided by the brute-force Monte-Carlo simulation, it is observed the high numerical accuracy of this proposed approach.
With the MaxEnt optimization and the fractional moment approach, the probability distribution for the stress intensity factor
of the membrane structure is depicted as shown in
Figure 6, whereas results for the MaxEnt parameters, i.e.,
and
are listed in
Table 4. Compared to the benchmark result provided by the brute-force MCS with
samples, it has highlighted numerical accuracy of this fractional moment based entropy approach in estimating the probability distribution for the multivariate intensity factor function.
Figure 7 further depicts results for the reliability-based sensitivity analysis of the cracked membrane provided by the proposed entropy approach. The mean-values of input random variables
,
and
contribute positively to the structural failure failure, whereas a minimization of the temperature difference, i.e., (
) is able to reduce the structural failure probability. Similar observations for the moment-based sensitivity result are presented in
Figure 5. Besides, the utility of the multiplicative dimensional reduction method needs 25 functional evaluations in total. This has demonstrated the high numerical efficiency of this approach as well.
4.3. Probabilistic Sensitivity Analysis for Fundamental Natural Frequency of a Vehicle Frame
The example considers the probabilistic sensitivity analysis of a vehicle frame that is depicted by a finite element model. The total length (
L) of the vehicle frame is equally spaced as six segments, i.e.,
. The shell element with the thickness
t is used to develop the finite element model for natural frequency analysis of the vehicle structure. The simulation model contains 2820 quadratic elements with
degrees of freedom. The probabilistic characteristic of input random variables are summarized in
Table 5, whereas the mean-value of
determines the fundamental natural frequency of the vehicle frame as
rad/s.
Figure 8 first presents the sensitivity results for the fundamental natural frequency of the vehicle frame. It has observed that an increase of mean-values
and
is able to increase the mean value response for the fundamental natural frequency of this vehicle frame. However, an increment of parameters
and
will reduce the mean-value natural frequency result. In this regard, distribution parameters, i.e.,
and
of the input random variable can be used to manipulate the response moment results for fundamental natural frequency of the vehicle frame structure.
To determine the response distribution of the structural natural frequency, the multiplicative dimensional reduction method is first used to determine fractional moments, whereas the MaxEnt optimization procedure is followed to determine an estimation for the probability distribution function for the natural frequency result, and the distribution parameters are summarized in
Table 6. Compared to the benchmark result provided by the brute-force Monte-Carlo simulation method, results depicted in
Figure 9 have confirmed the high accuracy of this entropy approach in estimating the response distribution of the structural natural frequency.
Figure 10 presents the reliability-based sensitivity result of the vehicle frame represented by various allowable threshold value of the fundamental natural frequency results. A close agreement between the estimation results of the probability distribution and the sensitivity curves has demonstrate the high accuracy of the proposed approach for probabilistic sensitivity analysis of the vehicle frame. Specially, the mean values of random variables
and
increase the natural frequency result positively, whereas
,
,
and
change the natural frequency result negatively. Specially,
and
almost contribute fairly small for the uncertain natural frequency result, and they can be further treated as deterministic parameters to reduce the dimension of input random variables. Therefore, based on the probability-based sensitivity result, it is possible to locate controlling variables to increase/decrease the structural fundamental natural frequency result to avoid potential failures (e.g., the resonance) of the vehicle frame structure in engineering realities.