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Sensitivity-based adaptive sequential sampling for metamodel uncertainty reduction in multilevel systems

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Abstract

Decomposition-based technique is often used in the analysis and design of complex engineering systems for reducing the computational complexity by studying the subsystems decomposed from multilevel systems. Metamodels, as a replacement of original simulation models, can further alleviate the computational burden. However, discrepancy between the simulation models and metamodels, which is defined as metamodel uncertainty, may be introduced in the analysis process of multilevel systems owing to the lack of data. The metamodel uncertainties of sub-models will be further amplified because of the hierarchical uncertainty propagation and interaction between uncertainties, which will have a great impact on the system results. An adaptive sequential sampling strategy based on sensitivity is proposed in this paper so as to improve the prediction accuracy of system response. In this strategy, polynomial-chaos expansion is used to realize the forward propagation of metamodel uncertainty quantified by the Kriging model. The forward propagation is combined with optimization based on maximum variance criterion for searching the input locations that results in the largest variance of system response. Then, the indices of subsystems are obtained to make decisions about which subsystem needs extra samples by combining Karhunen-Loeve expansion and sensitivity analysis. The effectiveness of the proposed sequential sampling strategy method is verified by two mathematical examples and a multiscale composite material.

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Funding

This research is supported by the Key National Natural Science Foundation of China (Grant No. U1864211), National Natural Science Foundation of China (Grant No.11772191), and National Science Foundation for Young Scientists of China (Grant No.51705312).

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Authors and Affiliations

  1. The State Key Laboratory of Mechanical System and Vibration, Shanghai Key Laboratory of Digital Manufacture for Thin-walled Structures, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Can Xu, Ping Zhu & Mushi Li

  2. School of Design, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Zhao Liu

Authors
  1. Can Xu
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  2. Zhao Liu
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  3. Ping Zhu
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  4. Mushi Li
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Correspondence to Zhao Liu or Ping Zhu.

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Replication of results

The pseudocodes for the algorithms intended to facilitate the replication of results are provided in this section. The pseudocode of the M-MCS is presented below.

The pseudocode of PCE-quantification for multilevel metamodel uncertainty is presented below.

The pseudocode of the proposed adaptive sequential sampling strategy is presented below.

The toolbox SURROGATES is free software, which can be learned about through the website (http://sites.google.com/site/felipeacviana/surrogatestoolbox). The toolbox is employed to build the Kriging model using the following parameters: the correlation model is “corrgauss,” the regression model is “regpoly0,” the initial guess on “theta” is 0.5. The toolbox UQLab is completely free for academic users (https://www.uqlab.com). And the toolbox is used to construct the PCE models by the following parameters: the method is “LARS,” the hyperbolic norm is 1. The optimization is executed by MATLAB genetic algorithm tool with the following parameters: the “Generations” is 100, the “PopulationSize” is 100. The code for random filed representation is downloaded from the website (https://www.mathworks.com/matlabcentral/fileexchange/52372-random-field-representation-methods). Other related parameters for replicating this paper have been provided in Section 5.

In order to further understand the proposed strategy and replicate the results presented in this paper, the MATLAB codes for examples 1~2 are provided as the supplementary material. Table 3 listed the basic information of the code files. The results of numerical examples can be obtained by running the main programs, and the following subprograms need to be called while running the main programs. It should be note that the results of each run will vary depending on the initial samples generated based on the LHS method. Therefore, the attachment also provides the data set that obtains the figures in this article.

Table 3 Basic information of the code files in the supplementary material
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Appendix. Derivations of (9) and (10)

Appendix. Derivations of (9) and (10)

The metamodel uncertainty of the top-level sub-model in (7) is firstly considered; then, we have:

$$ {\displaystyle \begin{array}{l}{u}_{\mathrm{L}+\mathrm{U}}=E\left\{E\left[Z\left({u}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)+{\delta}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)\right)\left|{\delta}_Z\right.\right]\right\}\\ {}\kern3.5em =E\left\{{u}_Z\left({u}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)+{\delta}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)\right)\right\}\end{array}} $$
(34)

Considering the relationship between mean and variance which is shown:

$$ \mathrm{Var}(Z)=E\left({Z}^2\right)-E{(Z)}^2 $$
(35)

Equation (8) can be rewritten as:

$$ {\displaystyle \begin{array}{l}{\sigma}_{\mathrm{L}+\mathrm{U}}^2=\mathrm{Var}\left[Z\left({u}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)\right)\left|{\delta}_{\boldsymbol{Y}},{\delta}_Z\right.\right]\\ {}\kern4em =E\left[{Z}^2\left({u}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)\right)\left|{\delta}_{\boldsymbol{Y}},{\delta}_Z\right.\right]-E{\left[Z\left({u}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)\right)\left|{\delta}_{\boldsymbol{Y}},{\delta}_Z\right.\right]}^2\end{array}} $$
(36)

Similarly, the metamodel uncertainty of the top-level sub-model in (36) is firstly considered; then, we have:

$$ {\sigma}_{\mathrm{L}+\mathrm{U}}^2=E\left\{E\left[{Z}^2\left({u}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)+{\delta}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)\right)\left|{\delta}_Z\right.\right]\right\}-E{\left\{E\left[Z\left({u}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)+{\delta}_{\boldsymbol{Y}}\left(\boldsymbol{x}\right)\right)\left|{\delta}_Z\right.\right]\right\}}^2 $$
(37)

Use the relationship in (35) again. Equation (37) can be rewritten as:

$$ {\displaystyle \begin{array}{c}{\sigma}_{\mathrm{L}+\mathrm{U}}^2=E\left\{\mathrm{Var}\left[Z\left({u}_Y(x)+{\delta}_Y(x)\right)\left|{\delta}_Z\right.\right]+E{\left[Z\left({u}_Y(x)+{\delta}_Y(x)\right)\left|{\delta}_Z\right.\right]}^2\right\}\\ {}\kern3.63em -E{\left\{E\left[Z\left({u}_Y(x)+{\delta}_Y(x)\right)\left|{\delta}_Z\right.\right]\right\}}^2\\ {}\kern2.64em =E\left\{\mathrm{Var}\left[Z\left({u}_Y(x)+{\delta}_Y(x)\right)\left|{\delta}_Z\right.\right]\right\}+E\left\{E{\left[Z\left({u}_Y(x)+{\delta}_Y(x)\right)\left|{\delta}_Z\right.\right]}^2\right\}\\ {}\kern3.63em -E{\left\{E\left[Z\left({u}_Y(x)+{\delta}_Y(x)\right)\left|{\delta}_Z\right.\right]\right\}}^2\\ {}\kern2.64em =E\left\{{\sigma}_Z^2\left({u}_Y(x)+{\delta}_Y(x)\right)\right\}+E\left\{{u}_Z^2\left({u}_Y(x)+{\delta}_Y(x)\right)\right\}\\ {}\kern3.63em -E{\left\{{u}_Z\left({u}_Y(x)+{\delta}_Y(x)\right)\right\}}^2\\ {}\kern2.64em =E\left\{{\sigma}_Z^2\left({u}_Y(x)+{\delta}_Y(x)\right)\right\}+ Var\left\{{u}_Z\left({u}_Y(x)+{\delta}_Y(x)\right)\right\}\end{array}} $$
(38)

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Xu, C., Liu, Z., Zhu, P. et al. Sensitivity-based adaptive sequential sampling for metamodel uncertainty reduction in multilevel systems. Struct Multidisc Optim 62, 1473–1496 (2020). https://doi.org/10.1007/s00158-020-02673-6

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