In uncertainty analysis and reliability-based multidisciplinary design and optimization (RBMDO) of engineering structures, the saddlepoint approximation (SA) method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables involved in SA should be easy to handle. Additionally, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. In order to take advantage of the SA method and the moment matching method to enhance the efficiency of design and optimization, a fourth-moment saddlepoint approximation (FMSA) method is introduced into RBMDO. In FMSA, the approximate cumulative generating functions are constructed based on the first four moments of the limit state function. The probability density function and cumulative distribution function are estimated based on this approximate cumulative generating function. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of sequence optimization and reliability assessment, which is based on the performance measure approach strategy. Two engineering examples are introduced to verify the effectiveness of proposed method.
The progress of science and technology has put forward higher reliability requirements for engineering structural systems [
Uncertainty-based design optimization problems are mainly divided into two categories: robust design optimization and reliability design optimization. Robust design optimization focuses on guaranteed performance, looking for designs that are relatively insensitive to changes in uncertain variables. The purpose is to make the design solution robust when the design variables are degraded. The main observation is the tail of the PDF. Reliability design optimization focuses on the possibility of system failure, mainly to obtain a design that satisfies a given reliability. The main observation is the center of the PDF [
Among reliability calculation methods, the moment method approximates the distribution of random response by fitting the first few random moments based on a type of hypothetical distribution. It reduces the calculation difficulty of design under uncertainty [
Based on these problems, this study combines the moment method with the SA. It proposes an improved reliability-based multidisciplinary design and optimization (RBMDO) combined with fourth-moment saddlepoint approximation (FMSA) method (RBMDO-FMSA). In the FMSA method, the CGF is constructed based on the first four moments of limit state function (LSF) [
The structure of the study can be briefly summarized as: in
Compared with FORM and SORM, the moment method does not have the problem of finding the derivative and design point of the performance function. It directly uses the value of the performance function at some feature points to approximate the failure probability. The moment method can also be directly used for reliability analysis of single and multiple failure mode systems [
Here,
Normalize the random variable
Then in
When the first four-order center distance of the performance function is known, according to the high-order moment standardization technique (HOMST), the reliability index
The failure probability is expressed as
A new higher accuracy reliability index takes into account the parameters ignored by HOMST, which is expressed as
The use of the SA method for progressive analysis is efficient and practical. The SA has the following characteristics: (1) convenient calculation and strong operability; (2) good overall approximation effect to the function, especially the tail probability distribution; (3) SA method is useful when the PDF is known but the calculation of the CDF is difficult [
Firstly, given the performance function
Take the logarithm of MGF to get CGF in
By deriving CGF, the saddlepoint
According to the
Alternative equation for calculating the failure probability is as
This study combines the moment method and the saddlepoint approximation, then introduces an improved FMSA.
Assuming that Y is a random variable, its MGF and CGF are denoted by
Then, the first four moments of the standard variable
According to the saddlepoint approximation method, the CGF can be modeled as
Therefore, the first four derivatives of CGF can be derived as
Then combine with the
If
Then in
This is the CGF of a standard normal variable whose second derivative is
When
According to the
The saddlepoint
Bring the
According to the
Then according to the
Therefore, the CDF can be expressed as
When
This section first introduces the SORA and the PMA. Finally, the RBMDO-FMSA model is proposed.
The SORA method is a decoupling method that can efficiently solve reliability design optimization problems [
The SORA method uses a single-cycle strategy to perform continuous deterministic optimization cycles and reliability analysis. In each cycle, optimization and reliability analysis do not interfere with each other. Reliability analysis is used to verify the feasibility of probabilistic constraints after optimization. The key of this method is to continuously revise the constraints in the optimization with the results obtained through the reliability analysis. Keep it close to the expected probability constraints, realize the optimal design as quickly as possible, reduce the number of optimizations. Thereby reducing the number of reliability analyses.
In RBMDO, the PMA has the advantages of high calculation efficiency, good stability and wide application range [
The process of RBMDO-FMSA using SORA is as follows:
Step 1. Deterministic MDO.
The initial values Step 2. Reliability analysis. I. Linearization of the LSF by first-order Taylor expansion at the MLP point to minimize the accuracy of reliability analysis. When
Ⅱ. Solve the percent performance
Solving
In FMSA, the CGF
The MLP needs the corresponding percent performance
Step 3. Modified MDO.
Through the obtained MLP, the shift vector can be derived as
Then, the RBMDO problem is transformed into an MDO problem. Deterministic optimization can be performed in the next cycle of SORA. The optimization process is as
In the next cycle of SORA, after the MDO problem of the equation is solved, the reliability analysis is performed again. This process is repeated until the optimization converges.
This section gives two examples to verify the proposed method. The results obtained by the proposed method will be compared with existing methods. Other methods include mean first-order second-moment method (MVFOSM), first-order reliability method (FORM), second-order reliability method (SORM) and Monte Carlo simulation method (MCS) [
This is a one-leg wellhead platform composed of a deep well foundation, a tower body and two upper and lower decks [
The example has 7 basic random variables. The specific information is shown in
Variable | Name | Distribution | Mean | Coefficient |
---|---|---|---|---|
Equipment weight | Normality | 1.94 × 106 | 0.1 | |
Limit wave height | Extreme value type I | 15.4 | 0.07 | |
Flow rate | Extreme value type I | 1.10 | 0.13 | |
Limit wind speed | Lognormal | 67.5 | 0.1 | |
Drag coefficient | Normality | 1.83 | 0.1 | |
Mass force coefficient | Normality | 2.90 | 0.1 | |
Bow to extremes | Lognormal | 1.88 × 108 | 0.13 |
0.656 | 0.716 | −0.240 | 0 | 0 | |
−0.673 | 0.697 | 0.245 | 0 | 0 | |
0.343 | 0.001 | 0.940 | 0 | 0 | |
0 | 0 | 0 | 0.856 | 0.516 | |
0 | 0 | 0 | −0.516 | 0.856 |
Here only the yield failure of each failure point is considered. The load effect takes into account the axial force
The mathematical model of the optimal design is as
The results obtained by different methods are shown in
Variable | FMSA | MCS | MVFOSM | FORM | SORM |
---|---|---|---|---|---|
0.025 | 0.025 | 0.025 | 0.025 | 0.025 | |
0.032 | 0.030 | 0.027 | 0.025 | 0.028 | |
0.028 | 0.031 | 0.035 | 0.040 | 0.033 | |
0.028 | 0.031 | 0.034 | 0.040 | 0.032 | |
0.028 | 0.031 | 0.034 | 0.040 | 0.033 | |
2.26 | 2.23 | 2.08 | 2.00 | 2.19 | |
3.66 | 3.59 | 3.24 | 3.00 | 3.40 | |
4.20 | 4.25 | 4.96 | 5.00 | 4.86 | |
9.8 | 9.0 | 8.0 | 7.0 | 8.6 | |
17.0 | 14.0 | 10.8 | 9.0 | 13.7 | |
24.0 | 25.4 | 32.6 | 34.0 | 28.3 | |
38.4 | 38.6 | 37.5 | 37.0 | 37.7 |
This is an engineering evaluation example of NASA standard MDO test [
Variable | Description | Distribution | Mean | Standard deviation | Lower bound | Upper bound |
---|---|---|---|---|---|---|
Gear face width | – | – | – | 2.6 | 3.6 | |
Teeth module | – | – | – | 0.3 | 1.0 | |
Number of teeth of pinion | – | – | – | 17 | 28 | |
Distance between Bearings A | Gumbel | 0.001 |
7.3 | 8.3 | ||
Distance between Bearings B | Gumbel | 0.001 |
7.3 | 8.3 | ||
Diameter of Shaft A | Gumbel | 0.001 |
2.9 | 3.9 | ||
Diameter of Shaft B | Gumbel | 0.001 |
5 | 5.5 |
FMSA | MCS | FORM | SORM | MVFOSM | |
---|---|---|---|---|---|
3.426 | 3.427 | 3.424 | 3.425 | 3.423 | |
0.649 | 0.650 | 0.645 | 0.646 | 0.644 | |
18 | 18 | 18 | 18 | 18 | |
7.300 | 7.300 | 7.300 | 7.300 | 7.300 | |
7.688 | 7.688 | 7.686 | 7.687 | 7.685 | |
3.322 | 3.320 | 3.323 | 3.322 | 3.323 | |
5.264 | 5.264 | 5.263 | 5.263 | 5.263 | |
2882 | 2888 | 2866 | 2877 | 2859 |
It can be seen from
In uncertainty analysis and RBMDO of engineering structures, the SA method can be utilized to enhance the accuracy and efficiency of reliability evaluation. However, the random variables in SA should be easy to handle. Moreover, the corresponding saddlepoint equation should not be complicated. Both of them limit the application of SA for engineering problems. The moment method can construct an approximate cumulative distribution function of the performance function based on the first few statistical moments. However, the traditional moment matching method is not very accurate generally. To solve these problems, SA is combined with the moment method. An improved RBMDO-FMSA method is proposed to take the advantage of above methods. In FMSA, the approximate CGF is constructed based on the first four moments of the LSF. Then, the PDF and CDF are estimated based on this approximate CGF. Furthermore, the FMSA method is introduced and combined into RBMDO within the framework of SORA, which is based on the PMA strategy. The corresponding formulation RBMDO-FMSA effectively improves the efficiency and accuracy.