Design optimization of variable stiffness composites by using multi-fidelity surrogate models

Abstract

Variable stiffness (VS) composites made by fiber steering have received intensive attention due to the tailorability of the stiffness and strength properties. However, a significantly larger number of elements are required to converge for VS composites, which results in much longer runtime and makes the design and optimization more complicated. In this paper, an efficient design optimization method assisted by multi-fidelity surrogate models is presented for the buckling design of VS composites. To reduce the computational burden, a multi-fidelity surrogate model called hierarchical Kriging is constructed through a few expensive high-fidelity samples and many cheap low-fidelity samples. Fine and coarse finite element (FE) analysis is performed respectively to calculate the structural responses for corresponding datasets. The efficient global optimization based on a modified expected improvement criterion is employed and used to adaptively add new samples of variable-fidelity. Two case studies, a composite plate subjected to uniform uniaxial compression and a composite cylinder under pure bending, are investigated. The effects of different number of design variables and coarse FE model mesh density on the optimum configuration are studied to demonstrate the effectiveness and robustness of the method. The results indicate that the present method can remarkably reduce the number of high-fidelity FE evaluations and improve the optimization efficiency when compared with available methods in the literature. Additionally, the investigation in the mechanism of loading carrying capacity improvement shows that the increase is mainly due to the load redistribution.

Appendix

The EGO (Jones et al. 1998) begins by fitting a Kriging model based on the initial sample set. Then, it adds new sample points by maximizing the expected improvement (EI). The prediction of Kriging at a point x obeys a normal distribution $Y\left(\mathbf{x}\right)\sim N\left(\hat{y}\left(\mathbf{x}\right),{s}^2\left(\mathbf{x}\right)\right)$ and the improvement at a point x with respect to the current best function value y_min is:

$$ I\left(\mathbf{x}\right)=\max \left({y}_{\mathrm{min}}-Y\left(\mathbf{x}\right),0\right) $$

(A1)

The expression is a random variable because Y(x) is random variable. EI is the expectation of I(x) and can be written as:

$$ EI\left(\mathbf{x}\right)=\left\{\begin{array}{c}\left({y}_{\mathrm{min}}-\hat{y}\left(\mathbf{x}\right)\right)\Phi \left(\frac{y_{\mathrm{min}}-\hat{y}\left(\mathbf{x}\right)}{s\left(\mathbf{x}\right)}\right)+s\left(\mathbf{x}\right)\phi \left(\frac{y_{\mathrm{min}}-\hat{y}\left(\mathbf{x}\right)}{s\left(\mathbf{x}\right)}\right), if\ s\left(\mathbf{x}\right)>0\ \\ {}\kern7.75em 0,\kern8.25em if\ s\left(\mathbf{x}\right)=0\ \end{array}\right. $$

(A2)

where Φ and ϕ are the cumulative distribution function and probability density function of a standard normal distribution, y_min is the current optimum in each circle, $\hat{y}\left(\mathbf{x}\right)$ is the Kriging prediction, and s(x) is the root mean squared error.