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Mechanical behavior of composite bistable shell structure and surrogate-based optimal design

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Abstract

A challenge in designing a bistable structure is the need to have low energy input for state change while maximizing the load-carrying capability. Here, we present an optimization framework for a bistable structure such that the maximum load-bearing ability can be achieved based on the investigation on mechanical performance and surrogate models. Firstly, an analytical expression of radius for the second state of the bistable structure is derived and verified by numerical simulation using a two-point loading method. Then, the transforming process of a bistable structure is analyzed by the force-displacement curve, and the transformed load is identified as an indicator measuring the load-bearing capacity. Secondly, the influence of changing parameters, including length, ply angle, thickness, and radius of the bistable shell structure on the transformed load is carried out systematically to choose optimal design variables. Thirdly, the optimal model is established, targeting the transformed load with the constraint of coupling stress in the second stable state. Model selection is conducted to determine the surrogate model that maps design variables into objective and constraint functions. And then, the improved genetic algorithm is developed to solve the optimal model, and optimal results are analyzed and discussed. Ultimately, we achieve an optimal bistable structure with the maximum load-bearing capacity while satisfying constraint, which is validated by numerical simulation. These computational and optimal strategies can provide design ideas for new structural optimization design.

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Funding

This work was supported by the National Natural Science Foundation of China (11872080, 11172013), Beijing Natural Science Foundation (3192005), and Beijing Education Committee Development Project (SQKM201610005001).

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Authors and Affiliations

  1. Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing, 100124, China

    Yang Zhang, Hongling Ye, Bowen Li & Qingsheng Yang

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  1. Yang Zhang
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  2. Hongling Ye
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  3. Bowen Li
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  4. Qingsheng Yang
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Correspondence to Hongling Ye.

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Appendix

Appendix

Combined with the Gaussian function form in Table 3 with the training points in Table 2, the final form of the RBF-GS model can be obtained by solving (27)

$$ y(x)={\beta}_1{\mathrm{e}}^{\delta_1}+{\beta}_2{\mathrm{e}}^{\delta_2}+\dots +{\beta}_{27}{\mathrm{e}}^{\delta_{27}}+{b}_1{x}_1+{b}_2{x}_2+{b}_3{x}_3+{b}_0 $$
(A.1)

where

$$ {\displaystyle \begin{array}{c}{\delta}_1=-c\cdot \left[{\left({x}_1-{x}_{1,1}\right)}^2+{\left({x}_2-{x}_{1,2}\right)}^2+{\left({x}_3-{x}_{1,3}\right)}^2\right]\\ {}{\delta}_2=-c\cdot \left[{\left({x}_1-{x}_{2,1}\right)}^2+{\left({x}_2-{x}_{2,2}\right)}^2+{\left({x}_3-{x}_{2,3}\right)}^2\right]\\ {}\dots \\ {}{\delta}_{27}=-c\cdot \left[{\left({x}_1-{x}_{27,1}\right)}^2+{\left({x}_2-{x}_{27,2}\right)}^2+{\left({x}_3-{x}_{27,3}\right)}^2\right]\end{array}} $$
(A.2)

In this paper, c = 0.1. To make the paper clear, the coefficients in the above equation are attached in (A.3)–(A.6), respectively. Equations (A.3) and (A.4) represent the coefficients of the transformation load surrogate model, and (A.5) and (A.6) represent the coefficients of the maximum stress surrogate model.

Coefficients for the transformed load based on the RBF surrogate

$$ {\displaystyle \begin{array}{c}{\beta}_1=\Big[-0.08428,-\mathrm{15.3732,21.53591,11.74099},-13.2439,-\mathrm{9.72584,27.91396},-13.9384,\\ {}-25.3099,-6.51935,-\mathrm{5.30406,24.87283,6.155949},-\mathrm{8.63679,16.72813,21.83662},\\ {}-11.8259,-4.33821,-19.19,-\mathrm{18.4646,24.97922},-2.32732,-11.322,-2.27434,\\ {}13.97777,-\mathrm{8.89994,7.03655}\Big]\end{array}} $$
(A.3)
$$ {\mathbf{b}}_1=\left[0.689,\hbox{-} 6.7881,409.4371,\hbox{-} 103.49\right] $$
(A.4)

Coefficients for stress based on the RBF surrogate

$$ {\displaystyle \begin{array}{c}{\beta}_2=\Big[-33.57546,-0.93675,61.30411,-10.68648,-10.64946,-10.51601,40.56486,8.29826,\\ {}-31.30918,-26.10064,-3.37793,34.27713,-9.64921,-22.88281,-19.31415,36.17696,\\ {}\kern2em 4.58034,-18.69747,-21.40298,4.14278,45.98579,-2.6352,-13.50283,-20.79594,\\ {}\kern2em 41.86723,5.21025,-26.37521\Big]\end{array}} $$
(A.5)
$$ {\mathbf{b}}_2=\left[\hbox{-} 0.2457,\hbox{-} 27.7752,835.0198,637.3893\right] $$
(A.6)

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Zhang, Y., Ye, H., Li, B. et al. Mechanical behavior of composite bistable shell structure and surrogate-based optimal design. Struct Multidisc Optim 64, 303–320 (2021). https://doi.org/10.1007/s00158-021-02890-7

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