An efficient lightweight design strategy for body-in-white based on implicit parameterization technique

Abstract

In the early design phase of vehicles, performing lightweight design of body-in-white (BIW) using shape, size and topology optimization is a challenge. The large amount of design parameters including size, shape of cross-sections and positions of various parts are the main contributors to the challenge, which will lead to the huge computational cost for running a large number of finite element (FE) simulations and function evaluations. To handle this problem, an efficient lightweight design strategy which integrates implicit parameterization technique, global sensitivity analysis (GSA) and Pareto set pursuing (PSP) algorithm is proposed in this paper to explore the lightweight design of BIW. Firstly, a full parameterized model of BIW is established with implicit parametrization technique via SFE-CONCEPT. Secondly, the GSA technique is used to reduce the dimensions of design space. Finally, the parameterized model of BIW is optimized by PSP method directly in the reduced design space. It is clearly shown that the optimized BIW structure signifies the noticeable improvement from the baseline model. The results demonstrate that the proposed method is capable in generating a well-distributed Pareto optimal frontier, and can largely reduce the design complexity of BIW lightweight.

Appendix A. Review of Sobol’s method

Let I ⁿ = [0,1]ⁿ represents n-dimensional unit hypercube and x = [x ₁,…,x _n ] is an n-dimensional design vector defined in the unit hypercube I ⁿ. According to Ref. (Sobol’ 2003), f(x) can be decomposed as

$$ f\left(\boldsymbol{x}\right)={f}_0+{\displaystyle \sum_i{f}_i\left({x}_i\right)}+{\displaystyle \sum_{1\le i<j\le n}{f}_{ij}\left({x}_i,{x}_j\right)}+\cdots +{f}_{12\cdots n}\left({x}_1,\cdots, {x}_n\right) $$

(A. 1)

(A. 1) is called ANOVA-representation of f(x), and

$$ {\displaystyle {\int}_0^1{f}_{i_1\cdots {i}_s}\left({x}_{i_1},\cdots, {x}_{i_s}\right)\mathrm{d}{x}_k}=0\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em k={i}_1,\cdots, {i}_s $$

(A. 2)

According to (A. 2), the members in (A. 1) are orthogonal which can be expressed as integrals of f(x). Indeed,

$$ \begin{array}{l}{\displaystyle \int f\left(\boldsymbol{x}\right)d\boldsymbol{x}}={f}_0\\ {}{\displaystyle \int f\left(\boldsymbol{x}\right){\displaystyle \prod_{k\ne i}\mathrm{d}{x}_k}}={f}_i\left({x}_i\right)+{f}_0\\ {}{\displaystyle \int f\left(\boldsymbol{x}\right){\displaystyle \prod_{k\ne i,j}\mathrm{d}{x}_k}}={f}_{ij}\left({x}_i,{x}_j\right)+{f}_i\left({x}_i\right)+{f}_j\left({x}_j\right)+{f}_0\end{array} $$

(A. 3)

and so on.

According to (A. 1) and integral operations, we have

$$ {\displaystyle \int {f}^2\left(\boldsymbol{x}\right)d\boldsymbol{x}}-{f}_0^2={\displaystyle \sum_{s=1}^n{\displaystyle \sum_{i_1<\cdots <{i}_s}^n{f^2}_{i_1\cdots {i}_s}\left({x}_{i_1},\cdots, {x}_{i_s}\right)}}d{x}_{i_1}\cdots {x}_{i_s} $$

(A. 4)

The total and partial variances can be respectively expressed as

$$ \begin{array}{l}D={\displaystyle \int {f}^2\left(\boldsymbol{x}\right)d\boldsymbol{x}}-{f}_0^2\\ {}{D}_{i_1\cdots {i}_s}={\displaystyle \int {f}_{i_1\cdots {i}_s}^2\left({x}_{i_1},\cdots, {x}_{i_s}\right)}\mathrm{d}{x}_{i_1}\cdots \mathrm{d}{x}_{i_s},\kern0.5em 1\le {i}_1<\cdots {i}_s\le n,\ s=1,\cdots, n\end{array} $$

(A. 5)

According to (A. 4), the total variance D can thus be decomposed into partial variances ${D}_{i_1\cdots {i}_s}$ associated with x ₁, x _2,…, x _n as

$$ D={\displaystyle \sum_{s=1}^n{\displaystyle \sum_{i_1<\cdots <{i}_s}^n{D}_{i_1\cdots {i}_s}}}={\displaystyle \sum_i{D}_i}+{\displaystyle \sum_{1\le i<j\le n}{D}_{ij}}+\cdots +{D}_{12\cdots n} $$

(A. 6)

It should be noted that f(x) and ${f}_{i_1\cdots {i}_s}\left({x}_{i_1},\cdots, {x}_{i_s}\right)$ are random variables with variances D and ${D}_{i_1\cdots {i}_s}$, respectively.

The first-order, k-order sensitivity indices and the total sensitivity index for the ith design variable are respectively given by

$$ {S}_i=\frac{D_i}{D},\kern0.5em {S}_{i_1\cdots {i}_k}=\frac{D_{i_1\cdots {i}_k}}{D},\kern0.5em {S}_i^{total}=1-\frac{D_{-i}}{D} $$

(A. 7)

where D _− i is the sum of all ${D}_{i_1\cdots {i}_k}$ terms without the i-th variable. The first-order sensitivity index represents the main effect of a design variable, and higher-order sensitivity indices can capture the effects of interactions among design variables.

Let x _i be the i-th variable of x, x _− i  = {x ₁, ⋯ x _{(i − 1)}, x _(i + 1), ⋯, x _n } is the set of n-1 complementary variables and x = [x _i , x _-i ]. According to Ref. (Sobol’ 1993), D _i can be calculated as

$$ \begin{array}{l}{D}_i={\displaystyle \int f\left(\boldsymbol{x}\right)f\left({x}_i,\ x{\prime}_{-i}\right)d\boldsymbol{x}}dx{\prime}_{-i}-{f}_0^2\\ {} where\kern0.5em dx{\prime}_{-i}={\displaystyle \prod_{k\ne i}\mathrm{d}{x}_k^{\prime }},\kern0.75em k=1,\cdots, n\end{array} $$

(A. 8)

Similarly, D _-i can be calculated as

$$ {D}_{-i}={\displaystyle \int f\left(\boldsymbol{x}\right)f\left({x}_i^{\prime },\ {x}_{\mathit{\hbox{-}}i}\right)d\boldsymbol{x}}d{x}_i^{\prime }-{f}_0^2 $$

(A. 9)

For computational expensive engineering problems, multi-dimensional integrals are necessary to obtain the Sobol’ indices when an exact form of the response is not available. Thus, Monte Carlo Sampling (MCS) is used to approximate the integrals. From (A. 3), (A. 5), (A. 7) and (A. 9), the mean value, total and partial variances can be numerically estimated by using MCS efficiently. Consider two independent random points π ₁ and π ₂ that uniformly distribute in I ⁿ, and let π ₁ = (x ⁽¹⁾ _i , x ⁽¹⁾_− i ), π ₂ = (x ⁽²⁾ _i , x ⁽²⁾_− i ), the model f(x) can be evaluated by two points: f(x ⁽¹⁾ _i , x ⁽¹⁾_− i ) and f(x ⁽¹⁾ _i , x ⁽²⁾_− i ).

Crude Monte Carlo estimation is obtained accurately if N → ∞:

$$ \begin{array}{l}{\widehat{f}}_0=\frac{1}{N}{\displaystyle \sum_{m=1}^Nf\left({x}_i^{(1m)},{x}_{-i}^{(1m)}\right)}\\ {}\widehat{D}=\frac{1}{N}{\displaystyle \sum_{m=1}^N{f}^2\left({x}_i^{(1m)},{x}_{-i}^{(1m)}\right)}-{\widehat{f}}_0^2\\ {}{\widehat{D}}_i=\frac{1}{N}{\displaystyle \sum_{m=1}^Nf\left({x}_i^{(1m)},{x}_{-i}^{(1m)}\right)}f\left({x}_i^{(1m)},{x}_{-i}^{(2m)}\right)-{\widehat{f}}_0^2\\ {}{\widehat{D}}_{-i}=\frac{1}{N}{\displaystyle \sum_{m=1}^Nf\left({x}_i^{(1m)},{x}_{-i}^{(1m)}\right)}f\left({x}_i^{(2m)},{x}_{-i}^{(1m)}\right)-{\widehat{f}}_0^2\end{array} $$

(A. 10)

where N is the number of sample points generated by Monte Carlo method (N = 100000 is selected in this paper), x ^(1m)_− i  = {x ^(1m)₁ , ⋯, x ^(1m)_{(i − 1)} , x ^(1m)_(i + 1) , ⋯, x ^(1m) _n } represents the set of n-1 complementary variables for the m-th sample point and x ^(1m) _i is the i-th variable for the m-th sample point.

According to (A. 7), the main effect and the total sensitivity index (total effect) for the i-th design variable can be respectively given by

$$ \begin{array}{l}\widehat{S_i}=\frac{{\widehat{D}}_i}{\widehat{D}}\\ {}{\widehat{S}}_i^{total}=1-\frac{{\widehat{D}}_{-i}}{\widehat{D}}\end{array} $$

(A. 11)