Data-driven modelling and optimization of stiffeners on undevelopable curved surfaces

Abstract

Undevelopable stiffened curved shells have been widely used in engineering fields. The shape of the undevelopable curved surface is generally characterized with the non-straight generatrix and variable cross sections, which makes it challenging to automatically model and optimize stiffeners on the undevelopable curved surface. Therefore, the data-driven modelling and optimization framework are proposed for undevelopable stiffened curved shells in this paper. Firstly, a novel mesh deformation method is developed for the data-driven modelling of undevelopable stiffened curved shells based on RBF neural network machine learning method. Its main idea is to firstly define a developable curved shell (background mesh domain) having similar topological characteristics with the undevelopable curved shell (target mesh domain), and then train the mapping relationship between the background mesh domain and the target mesh domain by RBF neural network, and finally the complicated modelling problem of the undevelopable stiffened curved shell can be transformed into a simple modelling problem of developable stiffened curved shell by means of the mapping relationship. Moreover, based on the efficient global optimization (EGO) surrogate method, a data-driven layout optimization method is established for minimizing the structural weight of undevelopable stiffened curved shells. Finally, three representative optimization examples are carried out, including modelling and optimization of stiffeners on hyperbolic parabolic curved surfaces, blade-shaped curved surfaces and S-shaped variable cross-sectional curved surfaces. Optimal results indicate that the structural weight of undevelopable stiffened curved shells decreases significantly after the optimization, indicating the effectiveness of the proposed modelling and optimization framework.

Appendices

Appendix 1. RBFNN algorithm

The structure of RBFNN is displayed in Fig. 30. When the input is X_i, the output of the jth node in the hidden layer can be expressed as

$$ G\left({X}_i,{C}_j,{\sigma}_j\right)=\exp \left(-\left\Vert {X}_i-{C}_j\right\Vert /2{\sigma}_j^2\right) $$

(11)

where C_j = (c_j1, c_j2,…, c_jn)^T and σ_j are the center and width of the Gaussian function of the jth node in the hidden layer, respectively.

For an input X_i, the expected output Y_i is

$$ {Y}_i=\sum \limits_{j=1}^MG\left({X}_1,{C}_j,{\sigma}_j\right){\omega}_j+{e}_i $$

(12)

where ω_j represents the weight between the jth neuron of the hidden layer and output neuron. M is the neuron number in the hidden layer. e_i represents the fitting error.

Fig. 30

Schematic diagrams of RBF neural network

Appendix 2. EGO algorithm

The flow chart of EGO method is displayed in Fig. 31. Firstly, sampling points are generated in the design space by Latin hypercube sampling (LHS) method. Secondly, a kriging surrogate model is constructed based on sampling results. Thirdly, the expectation of improvement EI of the kriging surrogate model at each sampling point x is evaluated,

$$ EI(x)=\left({y}_{\mathrm{min}}-\hat{y}\left(\left.x\right)\right)\Phi \left(\frac{y_{\mathrm{min}}-\hat{y}(x)}{s(x)}\right)+s(x)\phi \left(\frac{y_{\mathrm{min}}-\hat{y}(x)}{s(x)}\right)\right) $$

(13)

where y_min represents the optimal result of each update, $\hat{y}(x)$ represents the prediction mean value of kriging surrogate model, and s(x) represents the prediction standard deviation of kriging surrogate model. Φ(·) and ϕ(·) are the cumulative density function and probability density function of a normal distribution, respectively. In this presented work, the kriging model considered in the EGO algorithm is based on the Gaussian kernel.

Fourthly, MIGA is employed to search for the optimal result aiming at maximizing the EI. Finally, the stop criterion is estimated, which includes three conditions: (1) The maximum EI value is smaller than the tolerance (1e-4 in this paper); (2) the distance between the current iteration point and the former iteration point is smaller than the tolerance (1e-4 in this paper); (3) the iteration number reaches the maximum value. Once one of these three conditions are satisfied, the EGO is stopped. If the stop criterion is not satisfied, the new sampling point can be chosen by maximizing the EI and added into the original kriging surrogate model. This process repeats until the stop criterion is satisfied.

Fig. 31

Flow chart of EGO method