Adaptive gradient-enhanced kriging model for variable-stiffness composite panels using Isogeometric analysis

Abstract

Variable-stiffness panel is very promising for the cutout reinforcement of composite structures. However, due to the increase of design variables, the optimization of variable-stiffness panels becomes very challenging, even if surrogate model is utilized, because the fidelity of surrogate model is difficult to guarantee for high-dimensional problems. In this study, isogeometric analysis method (IGA) is employed to predict the buckling load of variable-stiffness panels, which can produce accurate prediction with less computational cost compared to traditional FEA, moreover, it can provide analytical sensitivity for optimization. On this basis, an adaptive gradient-enhanced kriging (GEK) model assisted by a novel multiple points infilling criterion is constructed for the global optimization of variable-stiffness composite panels. The proposed method is compared with traditional surrogate model, and results show that the proposed method can find a better optimum design in a more efficient manner. It can be concluded that the proposed method is able to fully explore the advantages of IGA including exact modelling, analysis and analytical sensitivity, which is particularly suitable for the design of variable-stiffness panels and other complex structures.

Appendix

The predicted function of GEK can be written as

$$ \widehat{y}(x)=\sum \limits_{i=1}^n{\omega}_i{y}_i+\sum \limits_{i=0}^n\sum \limits_{j=0}^d{\nu}_j^i\frac{\partial y\left({\boldsymbol{x}}^i\right)}{\partial {x}_j^i} $$

(30)

where ω _i denotes the weight coefficient of the ith response value, and ${\nu}_j^i$ denotes the weight coefficient the partial derivative of the jth dimensional design variable to the ith response value. The regression function is defined as

$$ {\displaystyle \begin{array}{l} Cov\left(Z\left({\boldsymbol{x}}^i\right),Z\left({\boldsymbol{x}}^j\right)\right)={\sigma}^2R\left({\boldsymbol{x}}^i,{\boldsymbol{x}}^j\right)\\ {} Cov\left(Z\left({\boldsymbol{x}}^i\right),\frac{\partial Z\left({\boldsymbol{x}}^j\right)}{\partial {\boldsymbol{x}}_k}\right)={\sigma}^2\frac{\partial R\left({\boldsymbol{x}}^i,{\boldsymbol{x}}^j\right)}{\partial {x}_k^j}=- Cov\left(\frac{\partial Z\left({\boldsymbol{x}}^i\right)}{\partial {\boldsymbol{x}}_k},Z\left({\boldsymbol{x}}^j\right)\right)\\ {} Cov\left(\frac{\partial Z\left({\boldsymbol{x}}^i\right)}{\partial {\boldsymbol{x}}_k},\frac{\partial Z\left({\boldsymbol{x}}^j\right)}{\partial {\boldsymbol{x}}_l}\right)={\sigma}^2\frac{\partial^2R\left({\boldsymbol{x}}^i,{\boldsymbol{x}}^j\right)}{\partial {x}_k^i\partial {x}_l^j}\end{array}} $$

(31)

So far, the weight coefficients can be calculated by solving the minimization problem about MSE(x) with the Lagrange multiplier approach. It can be found as follows

$$ \left\{\begin{array}{c}\sum \limits_{j=1}^n{\omega}_j Cov\left(Z\left({\boldsymbol{x}}^i\right),Z\left({\boldsymbol{x}}^j\right)\right)+\sum \limits_{j=1}^n{\nu}_i^j Cov\left(Z\left({\boldsymbol{x}}^i\right),\frac{\partial Z\left({\boldsymbol{x}}^j\right)}{\partial {x}_k}\right)+\frac{\mu }{2}= Cov\left(Z\left({\boldsymbol{x}}^i\right),Z\left(\boldsymbol{x}\right)\right),i=1,2\cdots, n\\ {}\sum \limits_{j=1}^n{\omega}_j Cov\left(\frac{\partial Z\left({\boldsymbol{x}}^i\right)}{\partial {x}_k},Z\left({\boldsymbol{x}}^j\right)\right)+\sum \limits_{j=1}^n{\nu}_i^j Cov\left(\frac{\partial Z\left({\boldsymbol{x}}^i\right)}{\partial {x}_k},\frac{\partial Z\left({\boldsymbol{x}}^j\right)}{\partial {x}_l}\right)= Cov\left(\frac{\partial Z\left({\boldsymbol{x}}^i\right)}{\partial {x}_k},Z\left(\boldsymbol{x}\right)\right),i=1,2\cdots, n\\ {}\sum \limits_{i=0}^n{\omega}_i=1\end{array}\right. $$

(32)

And μ is the Lagrange multipliers.

Also, (35) can be written as a matrix function

$$ \left[\begin{array}{cc}\boldsymbol{R}& \boldsymbol{F}\\ {}{\boldsymbol{F}}^T& 0\end{array}\right]\left[\begin{array}{c}\boldsymbol{\nu} \\ {}\tilde{\mu}\end{array}\right]=\left[\begin{array}{c}{\boldsymbol{r}}_{\left(\boldsymbol{x}\right)}\\ {}1\end{array}\right] $$

(33)

where

$$ {\displaystyle \begin{array}{l}\tilde{\mu}=-\frac{\mu }{2{\sigma}^2}={\left({\boldsymbol{F}}^T{\boldsymbol{R}}^{-1}\boldsymbol{F}\right)}^{-1}\left({\boldsymbol{F}}^T{\boldsymbol{R}}^{-1}{\boldsymbol{r}}_{\left(\boldsymbol{x}\right)}-1\right)\\ {}\boldsymbol{F}={\left[1,\cdots, 1,0,0,\cdots, 0\right]}^{\boldsymbol{T}}{\upepsilon \mathrm{\Re}}^{n+n\times \mathit{\dim}}\\ {}\boldsymbol{\nu} ={\left[{\omega}_1,\cdots, {\omega}_n,{\nu}_1^1,{\nu}_2^1,\cdots, {\nu}_{dim}^n\right]}^T={\boldsymbol{R}}^{-1}\left({\boldsymbol{r}}_{\left(\boldsymbol{x}\right)}-F\tilde{\mu}\right)\end{array}} $$

The correlation vector and the correlation matrix can be expressed as

$$ {\displaystyle \begin{array}{l}{\boldsymbol{r}}_{\left(\boldsymbol{x}\right)}={\left[\boldsymbol{R}\left(\boldsymbol{x},{\boldsymbol{x}}^1\right)\cdots \boldsymbol{R}\left(\boldsymbol{x},{\boldsymbol{x}}^n\right),\frac{\partial R\left(\boldsymbol{x},{\boldsymbol{x}}^1\right)}{\partial {x}_1^1},\frac{\partial R\left(\boldsymbol{x},{\boldsymbol{x}}^1\right)}{\partial {x}_2^1},\cdots, \frac{\partial R\left(\boldsymbol{x},{\boldsymbol{x}}^1\right)}{\partial {x}_{dim}^1},\cdots \frac{\partial R\left(\boldsymbol{x},{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n}\right]}^T\in {\mathrm{\Re}}^{n+n\mathit{\dim}}\\ {}\boldsymbol{R}=\left[\begin{array}{cc}\begin{array}{ccc}\boldsymbol{R}\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)& \cdots & \boldsymbol{R}\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^n\right)\\ {}\vdots & \ddots & \vdots \\ {}\boldsymbol{R}\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^1\right)& \cdots & \boldsymbol{R}\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^n\right)\end{array}& \begin{array}{ccc}\frac{\partial R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_1^1}& \frac{\partial R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_2^1}& \cdots \kern0.5em \frac{\partial R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n}\\ {}\vdots & \vdots & \begin{array}{cc}\ddots & \vdots \end{array}\\ {}\frac{\partial R\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^1\right)}{\partial {x}_1^1}& \frac{\partial R\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^1\right)}{\partial {x}_2^1}& \begin{array}{cc}\cdots & \frac{\partial R\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n}\end{array}\end{array}\\ {}\begin{array}{ccc}\frac{\partial R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_1^1}& \cdots & \frac{\partial R\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^1\right)}{\partial {x}_1^1}\\ {}\frac{\partial R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_2^1}& \cdots & \frac{\partial R\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^1\right)}{\partial {x}_2^1}\\ {}\begin{array}{c}\vdots \\ {}\frac{\partial R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n}\end{array}& \begin{array}{c}\ddots \\ {}\cdots \end{array}& \begin{array}{c}\vdots \\ {}\frac{\partial R\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n}\end{array}\end{array}& \begin{array}{cccc}\frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_1^1\partial {x}_1^1}& \frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_1^1\partial {x}_2^1}& \cdots & \frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^n\right)}{\partial {x}_1^1\partial {x}_{dim}^n}\\ {}\frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_2^1\partial {x}_1^1}& \frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^1\right)}{\partial {x}_2^1\partial {x}_2^1}& \cdots & \frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^n\right)}{\partial {x}_2^1\partial {x}_{dim}^n}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}\frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n\partial {x}_1^1}& \frac{\partial^2R\left({\boldsymbol{x}}^1,{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n\partial {x}_2^1}& \cdots & \frac{\partial^2R\left({\boldsymbol{x}}^n,{\boldsymbol{x}}^n\right)}{\partial {x}_{dim}^n\partial {x}_{dim}^n}\end{array}\end{array}\right]\in {\mathrm{\Re}}^{\left(n+n\mathit{\dim}\right)\times \left(n+n\mathit{\dim}\right)}\end{array}} $$

(34)

The predicted MSE of $\widehat{y}(x)$ is

$$ {s}_{\left(\boldsymbol{x}\right)}^2={\sigma}^2\Big(1+{\left({\boldsymbol{F}}^T{\boldsymbol{R}}^{-1}\boldsymbol{F}\right)}^{-1}\left(1-{\boldsymbol{F}}^T{\boldsymbol{R}}^{-1}{\boldsymbol{r}}_{(x)}\Big){}^2-{\boldsymbol{r}}_{(x)}^T{\boldsymbol{R}}^{-1}{\boldsymbol{r}}_{(x)}\right) $$

(35)

As both $\widehat{y}(x)$ and MSE are the functions of σ² and θ, θ can be calculated by the maximum likelihood estimation approach. $\widehat{y}(x)$ follows the multidimensional normal distribution as

$$ {\displaystyle \begin{array}{l}P\left(Y\left(\boldsymbol{x}\right)|\boldsymbol{\theta}, {\sigma}^2,\widehat{y}\left(\boldsymbol{x}\right)\right)=L\left(\boldsymbol{\theta}, {\sigma}^2|\widehat{y}\left(\boldsymbol{x}\right)=Y\left(\boldsymbol{x}\right)\right)=\frac{1}{{\left(2{\pi \sigma}^2\right)}^{\frac{n\left(1+\mathit{\dim}\right)}{2}}{\left|\boldsymbol{R}\right|}^{\frac{1}{2}}}\exp \left[-\frac{{\left(\boldsymbol{Y}-\beta \boldsymbol{F}\right)}^{\boldsymbol{T}}{\boldsymbol{R}}^{-1}\left(\boldsymbol{Y}-\beta \boldsymbol{F}\right)}{2{\sigma}^2}\right]\\ {}\ln \left(L\left(\boldsymbol{\theta}, {\sigma}^2,\widehat{y}\left(\boldsymbol{x}\right)|Y\left(\boldsymbol{x}\right)\right)\right)=-\frac{n\left(1+\mathit{\dim}\right)}{2}\ln \left(2\pi \right)-\frac{n\left(1+\mathit{\dim}\right)}{2}\ln \left({\sigma}^2\right)-\frac{1}{2}\ln \left|\boldsymbol{R}\right|-\frac{{\left(\boldsymbol{Y}-\beta \boldsymbol{F}\right)}^{\boldsymbol{T}}{\boldsymbol{R}}^{-1}\left(\boldsymbol{Y}-\beta \boldsymbol{F}\right)}{2{\sigma}^2}\end{array}} $$

(36)

Take the partial derivative with respect to σ² of this equation and let it zero, σ² and β can be obtained as

$$ {\displaystyle \begin{array}{l}{\sigma}^2=\frac{1}{n\left(1+\mathit{\dim}\right)}{\left(\boldsymbol{Y}-\beta \boldsymbol{F}\right)}^{\boldsymbol{T}}{\boldsymbol{R}}^{-1}\left(\boldsymbol{Y}-\beta \boldsymbol{F}\right)\\ {}\beta ={\left({\boldsymbol{F}}^T{\boldsymbol{R}}^{-1}\boldsymbol{F}\right)}^{-1}{\boldsymbol{F}}^T{\boldsymbol{R}}^{-1}\boldsymbol{Y}\end{array}} $$

(37)