The momentindependent importance measure (IM) on the failure probability is important in system reliability engineering, and it is always influenced by the distribution parameters of inputs. For the purpose of identifying the influential distribution parameters, the parametric sensitivity of IM on the failure probability based on local and global sensitivity analysis technology is proposed. Then the definitions of the parametric sensitivities of IM on the failure probability are given, and their computational formulae are derived. The parametric sensitivity finds out how the IM can be changed by varying the distribution parameters, which provides an important reference to improve or modify the reliability properties. When the sensitivity indicator is larger, the basic distribution parameter becomes more important to the IM. Meanwhile, for the issue that the computational effort of the IM and its parametric sensitivity is usually too expensive, an active learning Kriging (ALK) solution is established in this study. Two numerical examples and two engineering examples are examined to demonstrate the significance of the proposed parametric sensitivity index, as well as the efficiency and precision of the calculation method.
Uncertainties existing in engineering analysis and design are inherently unavoidable in nature associated with the manufacturing error, material property, loads, and so forth. Fortunately, reliability analysis and sensitivity analysis are now available to deal with the uncertainty existing in design variables to improve the performance of a mechanical or structural system [
Sensitivity analysis (SA) is the study of how the output response of a model (numerical or otherwise) is affected by the input uncertainty, which can be classified into two groups: local SA and global SA [
Saltelli and Marivoet [
This momentindependent importance measure on failure probability is applied to quantify the average effect of the basic variables on the reliability of the model and obtain the importance ranking. The IM on the failure probability can be used in the priordesign stage for variables screening when a reliability design solution is yet identified and the postdesign stage for uncertainty reduction after an optimal design has been determined. Uncertain inputs inherent in most engineering problems are assumed as random variables obeying probabilistic distributions. Obviously, system reliability and reliability IM on failure probability are decided by distribution parameters. One can directly change the input’s IMs by controlling or modifying some input’s distribution parameters; namely, changing the input’s distribution parameters can also influence the failure probability, which would facilitate its use under various scenarios of design under uncertainty, for instance, in reliabilitybased design. It is necessary to further recognize effects of the distribution parameters within system reliability on the importance ranking. At present, Cui et al. [
Combined with the local SA technique of input parameters, the effects of the distribution parameters on the IM on failure probability can be introduced, by which IMs of the inputs can be controlled or modified by changing the distribution parameters. This can provide important guidance for robust design, reliabilitybased design, and reliabilitybased optimization in engineering. However, its solution still relies on the corresponding method for failure probability and the computation of the derivative operation on failure probability existing in the parametric sensitivity of IM.
The Monte Carlo simulation (MCS) procedure is easy to implement and is available for computing the parametric sensitivity of IM based purely on model evaluation [
The remainder of this work is organized as follows. Section
Consider a probabilistic reliability model
Based on the idea of the momentindependent importance analysis, the importance measure of basic variable
As the absolute value in (
In the reliability analysis, the failure domain of this structure system is defined as
Suppose the indicator function of this failure domain is given as
Then, the unconditional failure probability and conditional failure probability on
For the influential distribution parameter, it is significant to identify how it influences IM on failure probability. We suppose that each input only depends on one distribution parameter in order to simplify the notation in the following.
As stated above,
Influence of the distribution parameter on the
To analyze the effect of changing the
To analyze the effect of changing the
To compute the above formula, the derivatives of
Here, the effects of the distribution parameters on the IMs of input variables can be known, combining local and global sensitivity analysis technology. One can directly optimize the inputs’ IMs by controlling or changing some inputs’ distribution parameters. It can provide useful information for robust design, reliabilitybased design, and reliabilitybased optimization.
In computing the parametric sensitivity of IM, the derivative of the
Section
The present Kriging model expresses the unknown function
The mean of this component is zero, the variance is
Several types of correlation function
A set of
In (
The Kriging variance
However, at a given unknown point
The ALK method has been applied to different fields in engineering, like efficient global optimization (EGO) [
For the uncertainty of predicted value
First, if
As shown in Figure
Risk of the sign of
Covering the range (
If
And the ERF for the case
The ERF is employed to measure the potential possibility that the sign of the limit state function in a point
The ALKbased solution for the parametric sensitivity can be simply divided into five steps and provided as follows.
For a model
Construct an active learning Kriging model as follows.
The samples
Construct the Kriging model, compute the ERF of candidate points by (
Add the point with max value of ERF to the DOE and loop back to Step
Based on the active learning Kriging model, the corresponding
By fixing the
The IM
It can be seen that a large number of samples must be taken for providing precise estimates when calculating IM
In the next section, we introduce two numerical examples and two engineering examples for demonstrating the efficiency and precision of the calculation procedure and illustrating the engineering significance of the IM
The mathematical problem in Example
The estimates of the importance measure on the failure probability
Computational results of the importance measure
Method  Importance measure 
The numbers of function evaluations  


 
MC  0.00492  0.00537 

ALK  0.00476  0.00548  12 + 36 
Error  0.0033  0.0052 
The parametric sensitivities of the importance measure
Parametric sensitivity of IM  Method 






MC  0.00821  0.0179  0.00940  0.00610 
ALK  0.00858  0.0183  0.00947  0.00624  



MC  0.00741  −0.00301  4.594 
−3.576 
ALK  0.00752  −0.00293  4.648 
−3.470 
As revealed by Tables
Iteration history of the true value of added training points for Example
Sign of the response at each sample predicted for Example
From the results in Table
Ishigami function [
Computational results of the importance measure
Method  Importance measure  



 
MC ( 
0.0474  0.0263  0.0107 
ALK ( 
0.0475  0.0261  0.0110 
Error  0.0053  0.0076  0.028 
The parametric sensitivities of the importance measure
Parametric sensitivity of IM  Method 








MC  −7.826 
7.826 
1.513 
−1.513 
1.513 
−1.513 
ALK  −7.752 
7.752 
1.537 
−1.537 
1.537 
−1.537 




MC  8.422 
−8.422 
−4.402 
4.402 
8.422 
−8.422 
ALK  8.418 
−8.418 
−4.398 
4.398 
8.418 
−8.418 




MC  3.425 
−3.425 
3.425 
−3.425 
−1.896 
1.896 
ALK  3.479 
−3.479 
3.480 
−3.479 
−1.914 
−1.914 
As revealed by Tables
Iteration history of the true value of added training points for Ishigami function by ALK.
From Examples
Except for that, Table
In order to test the applicability of the proposed method for problems with more random variables and expressed in a more engineering way, the roof truss is selected as example. The truss is simply illustrated as in Figure
Statistical properties of random variables for roof truss.
Random variable 








Mean 
20000  12  9.82 × 10^{−4}  0.04  1 × 10^{11}  1.2 × 10^{10} 


Coefficient of variation 
0.07  0.01  0.06  0.12  0.06  0.06 
Schematic diagram of a roof truss.
Considering the safety of the truss, the perpendicular deflection
For this highly nonlinear example, the computational results of the importance measure on failure probability measures by ALK and MCS are listed in Tables
Computational results of the importance measure
Method  Global reliability sensitivity indices  






 
MC ( 
0.05714  0.00483  0.02431  0.01148  0.00664  0.01329 
ALK (69)  0.05684  0.00471  0.02425  0.01151  0.00665  0.01328 
SDP (1024)  0.05044  0.00436  0.02749  0.01334  0.00691  0.01231 
The parametric sensitivities of the importance measure
Parametric sensitivity of IM  Method 

 




 

MC  2.361 
−1.369 
2.241 
5.221 
ALK  2.384 
−1.378 
2.256 
5.240 




MC  −0.432  −0.285  −4.947  −4.053 
ALK  −0.428  −0.271  −4.956  −4.057  



MC  −28.546  −7.802  −3.235 
−1.324 
ALK  −27.611  −8.096  −3.312 
−1.387 
Iteration history of the true value of added training points for a roof truss by ALK.
Additionally, it can be seen from Table
Due to space limitation, Table
A planar 10bar structure shown in Figure
Computational results of the importance measure
Method  Global reliability sensitivity indices  






 
MC ( 
0.0449  0.0382  0.0401  0.0242  0.00282  2.249 
ALK (69)  0.0443  0.0371  0.0396  0.0245  0.00265  2.411 
The parametric sensitivities of the importance measure
Basic variable 



 

MC  ALK  MC  ALK  MC  ALK  


−236.763  −240.579  19.812  19.820  17.774  17.756 

1235.778  1232.543  −1.637  −1.625  −1.471  −1.457  




0.230  0.241  0.223  0.220  −1.759 
−1.788 

−0.408  −0.412  −0.384  −0.397  −1.956 
−2.012 





−1.843 
−2.316 
1.477 
1.546 
−2.304 
−2.277 

−4.231 
−6.186 
−3.714 
−4.827 
−4.034 
−5.259 





2.186 
2.492 
2.127 
2.111 
2.620 
2.581 

−3.262 
−3.641 
−3.208 
−3.496 
−3.207 
−3.834 





6.210 
7.434 
5.812 
5.792 
7.420 
6.834 

−3.174 
−3.412 
−3.986 
−3.796 
−3.528 
−3.419 





−1.641 
−1.624 
−1.574 
−1.657 
−1.871 
−1.862 

−2.631 
−2.942 
−1.876 
−1.748 
2.576 
2.124 
Planar 10bar structure.
The finite element model of the planar 10bar structure.
From Tables
Iteration histories of the true value of added training points for 10bar structure.
As revealed in Table
This paper investigates the influence of the distribution parameters on the IM on failure probability. It is noted that the IM of basic variable not only is influenced by its distribution parameters but also is influenced by other basic variables’ distribution parameters. By further developing the presented momentindependent IM on failure probability, the parametric sensitivity of IM is first presented according to the derivative theory; thus, how the influential distribution parameters influence the influential IM can be made clear. Meanwhile, we can decrease the variability of the IM on failure probability by collecting the information and improving the understanding of those most influential parameters. The parametric sensitivity of
The computation of the IM on failure probability and its parametric sensitivity is often feasible by the MCS, but the computational cost of MCS method is tremendously large with small failure probability (10^{−3}–10^{−4} or smaller). For dealing with this problem, the ALK method is employed to calculate the IM and its parametric sensitivity. It can be seen by the numerical and engineering examples that the ALK method is more efficient than MC method. To ensure the computational accuracy, the large number of training points used in the traditional Kriging method is essential. Thanks to the existence of active learning process, the points which may greatly affect the metamodel’s fitting accuracy can be precisely selected, which can make the Kriging metamodel more accurate and the additional computational cost is acceptable. It is noticed that a small quantity of points in the interesting region are added to construct the Kriging predictor model until the Kriging model satisfies necessary accuracy. The computational results of several examples demonstrate that the proposed method is validated to be rational and efficient.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The financial support from National Natural Science Foundation of China (Grant no. 51205312), The Chinese Aerospace Support Foundation (Grant no. NBXW0001), and The NPU Foundation for Fundamental Research (Grant no. JC20110255) is gratefully acknowledged by the authors.