For practical engineering design problems, random variables tend to follow multimodal probability distributions when working at multiple operating conditions. For example, the structural fatigue stress of a steel bridge carrying both highway and railway traffic obeys the bimodal distribution. The existing popular reliability-based design optimization (RBDO) methods are mainly used to treat random variables with only unimodal distributions, which, therefore, tends to result in relatively large computational errors when multimodal random variables are involved. In this paper, a sequential approximate RBDO method is firstly proposed for engineering design involving multimodal random variables. The probability density function (PDF) of the response function is firstly calculated to assess the reliability of each probabilistic constraint, due to the existence of multimodal random variables. Then, a deterministic optimization problem is established to calculate candidate design points, based on the approximation that the response PDF demonstrates only transitional deformation in the optimization process. Three numerical examples and one engineering application of a thermal laptop design are presented to demonstrate the effectiveness of the proposed method.
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This work is supported by the National Natural Science Foundation of China (Grant No. 51805157, Grant No. 51725502 and Grant No. 51490662) and Hunan Natural Science Foundation (Grant No. 2019JJ40015).
Conflict of interest
The authors declare that they have no conflict of interest.
Replication of results
In order to facilitate the replication of results presented in this paper, the MATLAB code files of the steel column design in Sect. 4.1 are provided as the supplementary material, and brief descriptions are given to the function of each file, as shown in Table 14. The results of the other examples can be reproduced conveniently, by modifying the characteristics of the problems such as objective function, probabilistic constraints, distribution type, dimensionality, etc
Thirteen Matlab code files are provided to perform the proposed sequential approximate method and double-loop method effectively. “MainProgram.m” is the main program of the proposed method, which consists of eight subprograms, namely, “GaussQuad.m”, “HermitePoly.m”, “UnivarQuad.m”, “GMFun.m”, “ReliabAsm.m”, “PdfFun.m”, “GNonlinear.m”, “ObjFun.m”. “DoubleLoop.m” is the main program of the double-loop method, which consists of one subprogram, namely “DLNonlinear.m”. “GMM_result.mat” is used in the programs of both methods to define the GMM characteristics and “MaxEnt_Newton” is the tool box to calculate the response PDF. The function of each subprogram is illustrated in Table 14.
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Zhang, Z., Deng, W. & Jiang, C. Sequential approximate reliability-based design optimization for structures with multimodal random variables. Struct Multidisc Optim 62, 511–528 (2020). https://doi.org/10.1007/s00158-020-02507-5
- Reliability-based design optimization
- Multimodal random variable
- Sequential approximate optimization