Hierarchical stochastic metamodels based on moving least squares and polynomial chaos expansion

Abstract

While surrogate-based optimization has encountered a growing success in engineering design, the development of stochastic metamodels, i.e. capable of representing the complete random responses with respect to random inputs, is still an open issue, although they could be fruitfully used in optimization under uncertainty, both with single and multiple objectives. Therefore, the contribution of the paper is twofold. First, hierarchical stochastic metamodels based on moving least squares and spectral decomposition (by polynomial chaos expansion) are proposed in order to obtain a complete description of the random responses with respect to the deterministic and random input parameters. Then, these metamodels are incorporated into a novel multiobjective reliability-based formulation leaning on the concept of probabilistic nondominance. The whole procedure is applied to an analytical test case as well as to the design optimization of space truss structures, demonstrating the ability of the proposed method to provide accurate solutions at an affordable computational time.

Appendix

To give the closed-form expression of the sensitivities of the limit surfaces b with respect to ξ _i (Section 3.2), the chain rule is applied:

$$ \frac{\partial b}{\partial \xi_i} = \sum\limits_{j} \frac{\partial b}{\partial \psi_j} \frac{\partial \psi_j}{\partial \xi_i}, $$

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Since ψ _j is expressed as a product of Hermite polynomials of each variable ξ _i :

$$ \psi_j(\boldsymbol\xi) = \prod\limits_{l=1}^M H_{\kappa_l(j)}(\xi_l), $$

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the derivative of ψ _j with respect to a variable ξ _i is given by:

$$ \begin{array}{rll} \frac{\partial \psi_j}{\partial \xi_i} & =& \frac{\partial }{\partial \xi_i} \left( \prod\limits_{l=1}^M H_{\kappa_l(j)}(\xi_l) \right) \\ & =& \left(\prod\limits_{l=1,l\neq i}^M H_{\kappa_l(j)}(\xi_l) \right) \frac{\partial H_{\kappa_i(j)}}{\partial \xi_i}. \end{array} $$

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If κ _i (j) = 0, $\frac{\partial H_0}{\partial \xi_i}=0$; otherwise, the definition of Hermite polynomials leads to:

$$ \frac{\partial \psi_j}{\partial \xi_i} = \left(\prod\limits_{l=1,l\neq i}^M H_{\kappa_l(j)}(\xi_l) \right) \left( \kappa_i(j) H_{\kappa_i(j)-1}(\xi_i) \right). $$

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Therefore, we obtain explicitly:

$$ \begin{array}{rll}&& \frac{\partial b_\textrm{nondominance}}{\partial \xi_k} \\ \;\; & =& \frac{\partial \,}{\partial \xi_k} \left\{ \sum\limits_{i=1}^m \exp \left( \rho \sum\limits_{j=1}^{P-1} \psi_j(\boldsymbol\xi) \gamma_j^{f_i}({\bf x}) \right.\right. \\ && \quad \quad -\! \left.\left. \rho \eta_i \sqrt{\sum\limits_{j=1}^{P-1} E\left[\psi_j^2\right] \gamma_j^{f_i}({\bf x})^2} \;\right) - 1 \right\} \\ \;\; & =& \sum\limits_{i=1}^m \left\{ \exp \left(\dots\right) \rho \sum\limits_{j=1}^{P-1} \gamma_j^{f_i}({\bf x}) \frac{\partial \psi_j}{\partial \xi_k} \right\} \\ \;\; & =& \sum\limits_{i=1}^m \left\{ \exp \left(\dots\right) \rho \sum\limits_{j=1}^{P-1} \gamma_j^{f_i}({\bf x}) \right.\\ && \quad \quad\times\!\left. \left[ \prod\limits_{l=1,l\neq k}^M H_{\kappa_l(j)}(\xi_l)\! \right]\! \kappa_k(j) H_{\kappa_k(j)-1}(\xi_k)\! \right\} \end{array} $$

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and:

$$ \begin{array}{rll} \frac{\partial b_\textrm{safety}}{\partial \xi_k} &=& \sum\limits_{i=1}^m \left\{ \exp \left(\dots\right) \rho \sum_{j=1}^{P-1} \gamma_j^{g_i}({\bf x}) \right.\\ && \quad \times\!\left. \left[ \prod\limits_{l=1,l\neq k}^M H_{\kappa_l(j)}(\xi_l)\! \right]\! \kappa_k(j) H_{\kappa_k(j)-1}(\xi_k)\! \right\}, \end{array} \label{eq:safety_deriv_PCE} $$

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where the expression $(\kappa_k(j) H_{\kappa_k(j)-1})$ is equal to 0 when κ _k (j) = 0.