A method for selecting surrogate models in crashworthiness optimization

Abstract

Surrogate model or response surface based design optimization has been widely adopted as a common process in automotive industry, as large-scale, high fidelity models are often required. However, most surrogate models are built by using a limited number of design points without considering data uncertainty. In addition, the selection of surrogate model in the literature is often arbitrary. This paper presents a Bayesian metric to complement root mean square error for selecting the best surrogate model among several candidates in a library under data uncertainty. A strategy for automatically selecting the best surrogate model and determining a reasonable sample size was proposed for design optimization of large-scale complex problems. Lastly, a vehicle example with full-frontal and offset-frontal impacts was presented to demonstrate the proposed methodology.

Appendices

Appendix A: Review of SSR and RBF surrogate models

Radial basis function (RBF) was first introduced by Hardy to fit irregular topographic contours of geographical data in 1971 (Hardy 1971). Many researchers have compared RBF models to other surrogate models, e.g., KG, SVR, and so on. Similar to Kriging, RBF is a linear combination of a radically symmetric function based on Euclidean distance to obtain an approximate response function. A classical radial basis function model A(x, a) with fitting noise-free data (Forrester et al. 2008) is expressed as:

$$ A({{\bf x,a}})={\bf a}^T{{\boldsymbol \uppsi }}=\sum\limits_{i=1}^n {a_i } \psi \left(\left\| {{{\bf x}}-{{\bf x}}_i } \right\|\right) $$

(13)

where x _i denotes the ith of the n basis function centre and $\boldsymbol{\uppsi}$ is the n-vector containing the values of the basis functions ψ themselves, evaluated at the Euclidean distance between the prediction site x and the center x _i of the basis function. Generally, different types of basis function are discussion in Forrester et al. (2008). In this paper, the study is restricted to the following two basis functions:

$$ \mbox{Gaussian}:\psi (r)=\exp (-r^2/2\alpha^2) $$

(14)

$$ \mbox{Multiquadric}:\psi (r)=(r^2+\alpha^2)^{1/2} $$

(15)

where α is the model parameter and $r=\left\| {{{\bf x}}-{{\bf x}}_i } \right\|$ is the Euclidean distance. Details on algorithm of solving the (13) can be referred to Forrester et al. (2008). If the response is corrupted by noise data, Poggio and Girosi (1990) introduce a regularization parameter λ. This is added to the main diagonal matrix ψ. As a result, the approximation will no longer pass through the training points and a will be the least squares solution of (16).

$$ {\bf a}=({{\boldsymbol{\uppsi} }}+\lambda {{\bf E}})^{-1}{{\bf y}} $$

(16)

where E is an n ×n identity matrix, and y is the response data. Keane and Nair (2005) gave a detailed discussion on solving parameter λ.

Subset selection regression (SSR) is useful for two reasons: variance reduction and simplicity (Gu 2001). Various procedures have been used in an attempt to find the best subset of a series of predictor equations. Gu (2001) proposed a sequential replacement algorithm in conjunction with Residual Sum of Squares (RSS) checking criterion to get the best fitting regression model. The basic idea is that once two or more terms have been selected, it is determined that any of those terms can be replaced with another that gives a smaller RSS (Myers 1990). The procedure must converge as each replacement reduces the RSS that is bounded below. Sequential replacement algorithm is normally used in conjunction with stepwise selection. It is obtained by taking the stepwise selection and applying a replacement procedure after each new term is added. The theory and application of SSR method were summarized in Gu (2001).

Appendix B: Derivation of matrix K for RBF

According to (8), the matrix K is composed of two items as follows:

$$ \begin{array}{rll} {{\bf K}}_1 &=&\sum\limits_{k=1}^n {\left[ {\frac{\partial A(x_k ,{\bf a})}{\partial a_i }\frac{\partial A(x_k ,{\bf a})}{\partial a_j }} \right]}\\ &=&\sum\limits_{k=1}^n {\left[ {{\begin{array}{*{20}c} {\frac{\partial A(x_k ,{\bf a})}{\partial a_1 }} \hfill \\ \vdots \hfill \\ {\frac{\partial A(x_k ,{\bf a})}{\partial a_n }} \hfill \\ {\frac{\partial A(x_k ,{\bf a})}{\partial \alpha }} \hfill \\ \end{array} }} \right]}\\ &&\times{\left[ {{\begin{array}{*{20}c} {\frac{\partial A(x_k ,{\bf a})}{\partial a_1 }} \hfill & \cdots \hfill & {\frac{\partial A(x_k ,{\bf a})}{\partial a_n }} \hfill &{\frac{\partial A(x_k ,{\bf a})}{\partial \alpha }} \hfill \end{array} }} \right]} \end{array} $$

(17)

$$ \begin{array}{rll} {{\bf K}}_2 &=&\sum\limits_{k=1}^n {\left[ {(y_k -A(x_k ,{\bf a}))\frac{\partial^2A(x_k ,{\bf a})}{\partial a_i \partial a_j }} \right]}\\ &=&\sum\limits_{k=1}^n {(y_k -A(x_k ,{\bf a}))\left[ {\frac{\partial^2A(x_k ,{\bf a})}{\partial a_i \partial a_j }} \right]} \\ &=&\sum\limits_{k=1}^n {(y_k -A(x_k ,{\bf a}))}\\ &&\times {\left[ {{\begin{array}{*{20}c} 0 \hfill & \cdots \hfill & 0 \hfill & {\frac{\partial^2A(x_k ,{\bf a})}{\partial a_1 \partial \alpha }} \hfill \\ \hfill & \ddots \hfill & \vdots \hfill & \vdots \hfill \\ \hfill & \hfill & 0 \hfill & {\frac{\partial^2A(x_k ,{\bf a})}{\partial a_N \partial \alpha }} \hfill \\ {\it Symmetry} \hfill & \hfill & \hfill & {\frac{\partial^2A(x_k ,{\bf a})}{\partial \alpha^2}} \hfill \end{array} }} \right]} \\ &=&{{\bf O}} \end{array} $$

(18)

Thus K = [K _ij ] = K ₁  − K ₂  = K ₁  − O = K ₁ , to compute the matrix K, $\frac{\partial A(x_k ,{\bf a})}{\partial a_i }$ and $\frac{\partial A(x_k ,{\bf a})}{\partial \sigma }$ in (17) for RBF Gaussian and multiquadric function are denoted as:

For Gaussian basis function:

$$ \left\{ {{\begin{array}{l} \dfrac{\partial A(x_k ,{\bf a})}{\partial a_i }=\exp \left( {-\left\| {x-x_i } \right\|^2/2\alpha^2} \right) \\\\ \dfrac{\partial A(x_k ,{\bf a})}{\partial \alpha }=\dfrac{1}{\alpha ^3}\sum\limits_{i=1}^n {a_i (x-x_i )^2} \\\\ \,\,\,\quad\qquad\qquad\times\,\exp \left( {-\left\| {x-x_i } \right\|^2/2\alpha^2} \right) \end{array} }} \right.\;i=1,2,\cdots n. $$

(19)

For multiquadric basis function:

$$ \left\{ {{\begin{array}{l} {\dfrac{\partial A(x_k ,{\bf a})}{\partial a_i }=\left( {\left\| {x-x_i } \right\|^2+\alpha^2} \right)^{1/2}} \\\\ {\dfrac{\partial A(x_k ,{\bf a})}{\partial \alpha }=\sum\limits_{i=1}^n {a_i \dfrac{\alpha }{\left( {\left\| {x-x_i } \right\|^2+\alpha^2} \right)^{1/2}}} } \end{array} }} \right.\;i=1,2,\cdots n. $$

(20)