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A new global sensitivity measure based on derivative-integral and variance decomposition and its application in structural crashworthiness

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Abstract

For the traditional variance-based global sensitivity analysis, the total effect of individual variable commonly involves the interactions with other variables. To further decompose the interactive effects, this paper proposes a new global sensitivity measure based on derivative-integral and variance decomposition. Firstly, the first-order sensitivity index only relating to individual variable and the high-order sensitivity indices involving with other interactive variables are analyzed through analysis of variance (ANOVA) representation. Then, the partial derivatives of high-order interaction terms of ANOVA representation with respect to individual variable and the integrals of the squares of partial derivative functions are calculated in the whole variable space. Consequently, to measure the contribution of individual variable to each high-order interaction term, the ratio of the square root of each integral to their sum is defined as the sensitivity weight factor. A high-order sensitivity index can be further decomposed into a series of sensitivity sub-indices by using the defined sensitivity weight factors. Accordingly, a new global sensitivity measure for individual variable is proposed by combining the first-order sensitivity index with the decomposed sensitivity sub-indices. Finally, three numerical examples and an engineering application are investigated to demonstrate the reasonability and superiority of the proposed sensitivity measure.

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Acknowledgments

We thank Prof. Libo Cao’s Group for their help in validating the simulation model of commercial vehicle model. We also thank Mr. Xingfu Wu for his assistants in establishing meta-models and editing graphs.

Funding

This work is supported by the National Key R&D Program of China (Grant No. 2017YFB1301300), the National Science Foundation of China (Grant Nos. 51621004, 11572115), independent research project of State Key Laboratory of Reliability and Intelligence Electrical Equipment, Hebei University of Technology (EERIZZ2018001), independent research project of State Key Laboratory of Advanced Design and Manufacturing for the vehicle body, Hunan University (51475003), the Applied Basic Research Project in Changzhou (CJ20179009).

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Authors and Affiliations

  1. State Key Laboratory of Reliability and Intelligence of Electrical Equipment, School of Electrical Engineering, Hebei University of Technology, Tianjin, 300401, People’s Republic of China

    Qiming Liu & Xu Han

  2. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, People’s Republic of China

    Jie Liu, Qiming Liu, Xu Han & Chao Jiang

  3. State Key Laboratory of Reliability and Intelligence of Electrical Equipment, School of Mechanical Engineering, Hebei University of Technology, Tianjin, 300401, People’s Republic of China

    Xu Han & Yourui Tao

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  1. Jie Liu
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  2. Qiming Liu
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Correspondence to Qiming Liu or Xu Han.

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Appendices

Appendix 1

1.1 Proof of the reasonability of high-order sensitivity index \( {\mathbf{S}}_{i_1L\kern0.14em {i}_s} \)

$$ {\displaystyle \begin{array}{l}{S}_{i_1L\kern0.24em {i}_s}^{i_1}+L+{S}_{i_1L\kern0.24em {i}_s}^{i_h}+L+{S}_{i_1L\kern0.24em {i}_s}^{i_s}\\ {}={S}_{i_1L\kern0.24em {i}_s}\cdot {k}_{i_1L\kern0.24em {i}_s}^{i_1}+L+{S}_{i_1L\kern0.24em {i}_s}\cdot {k}_{i_1L\kern0.24em {i}_s}^{i_h}+L+{S}_{i_1L\kern0.24em {i}_s}\cdot {k}_{i_1L\kern0.24em {i}_s}^{i_s}\\ {}={S}_{i_1L\kern0.24em {i}_s}\cdot \left({k}_{i_1L\kern0.24em {i}_s}^{i_1}+L+{k}_{i_1L\kern0.24em {i}_s}^{i_h}+L+{k}_{i_1L\kern0.24em {i}_s}^{i_s}\right)\\ {}={S}_{i_1L\kern0.24em {i}_s}\cdot \left(\frac{\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_1}}}{\sum \limits_{i_m={i}_1,L,{i}_s}\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_m}}}+L+\frac{\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_h}}}{\sum \limits_{i_m={i}_1,L,{i}_s}\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_m}}}+L+\frac{\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_s}}}{\sum \limits_{i_m={i}_1,L,{i}_s}\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_m}}}\right)\\ {}={S}_{i_1\cdots {i}_s}\cdot \frac{\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_1}}+L+\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_h}}+L+\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_s}}}{\sum \limits_{i_m={i}_1,L,{i}_s}\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_m}}}\\ {}={S}_{i_1L\kern0.24em {i}_s}\cdot \frac{\sum \limits_{i_m={i}_1,L,{i}_s}\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_m}}}{\sum \limits_{i_m={i}_1,L,{i}_s}\sqrt{M_{i_1L\kern0.24em {i}_s}^{i_m}}}\\ {}={S}_{i_1L\kern0.24em {i}_s}\end{array}} $$

1.2 Proof of the property that the sum of the reconstructed sensitivity indices is equal to 1

Proof of \( \sum \limits_{i_h=1}^nN{S}_{i_h}=1 \)

$$ {\displaystyle \begin{array}{l}\sum \limits_{i_h=1}^nN{S}_{i_h}\\ {}=\sum \limits_{i_h=1}^n\left(\sum \limits_{s=1}^n\sum \limits_{{}_1<L\kern0.36em <{i}_s}{S}_{i_1L\kern0.24em {i}_s}^{i_h}\right)=\sum \limits_{i_h=1}^n\left(\sum \limits_{s=1}^n\sum \limits_{i_1<L\kern0.36em <{i}_s}{S}_{i_1L\kern0.24em {i}_s}\cdot {k}_{i_1L\kern0.24em {i}_s}^{i_h}\right)\\ {}=\sum \limits_{s=1}^n\sum \limits_{i_1<L\kern0.36em <{i}_s}{S}_{i_1L\kern0.24em {i}_s}\cdot \left(\sum \limits_{i_1\le {i}_h\le {i}_s}{k}_{i_1L\kern0.24em {i}_s}^{i_h}\right)\\ {}=\sum \limits_{s=1}^n\sum \limits_{i_1<L\kern0.36em <{i}_s}{S}_{i_1L\kern0.24em {i}_s}\cdot 1\\ {}=1\end{array}} $$

Appendix 2.

Table 11 Thirty-five training samples generated using the Latin-hypercube sampling method
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Liu, J., Liu, Q., Han, X. et al. A new global sensitivity measure based on derivative-integral and variance decomposition and its application in structural crashworthiness. Struct Multidisc Optim 60, 2249–2264 (2019). https://doi.org/10.1007/s00158-019-02316-5

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