Decoupling uncertainty quantification from robust design optimization

Abstract

Robust design optimization (RDO) has been eminent in determining the optimal design of real-time complex systems under stochastic environment. Unlike conventional optimization, RDO involves uncertainty quantification which may render the procedure to be computationally intensive, if not prohibitive. In order to deal with such issues, an efficient approximation-based generalized RDO framework has been proposed. Since RDO formulation comprises of statistical terms of the performance functions, the proposed framework deals with approximation of those statistical quantities, rather than the performance functions. Consequently, the proposed framework allows transformation of the RDO problem to an equivalent deterministic one. As a result, unlike traditional surrogate-assisted RDO, the proposed framework yields desirable results in significantly less number of functional evaluations. For performing such response statistical approximation, two adaptive sparse refined Kriging-based computational models have been proposed. However, the generality of the proposed methodology allows any surrogate models to be employed within this framework, provided it is capable of capturing the functional non-linearity. Implementation of the proposed framework in three test examples and two finite element-based practical problems clearly illustrates its potential for further complicated applications.

Appendices

Appendix A. Spectral projected gradient algorithm (SPGL1)

The SPGL1 algorithm utilized in this work has been adopted from (Berg and Friedlander 2008). The algorithm is dependent on projection of the solution of each iterations on the feasible set {α| ‖α‖₁ ≤ τ}. This has been realized by the following operator as

$$ {\overline{P}}_{\tau}\left[{\mathbf{x}}^{\prime}\right]:\underset{\boldsymbol{\upalpha}}{\;\mathrm{argmin}\;}{\left\Vert {\mathbf{x}}^{\prime }-\boldsymbol{\upalpha} \right\Vert}_2\kern0.36em \mathrm{s}.\mathrm{t}.\kern0.48em {\left\Vert \boldsymbol{\upalpha} \right\Vert}_1\le \tau $$

(A.1)

which signify the projection of x^′ onto the ℓ₁-norm with radius τ. The SPGL1 technique has been described in a sequential manner in algorithm A.1. It should be noted that each iteration of algorithm A.1 search for the projection gradient ${\overline{P}}_{\tau}\left[{\boldsymbol{\upalpha}}_m-\beta {\mathbf{g}}_m^{\prime}\right]$, where ${g}_m^{\prime }$ is the current gradient for ${\left\Vert \boldsymbol{\uppsi} \boldsymbol{\upalpha} -\mathbf{d}\right\Vert}_2^2$. The criterion utilized for the line search ensures reduction in the objective function in at least L number of iterations. For implementation, MATLAB^® package SPGL1 (Berg and Friedlander 2007) has been employed here.

Appendix B. Description of the test problems investigated in section 5.1

1.1 B.1 Example 1: welded beam design

$$ \operatorname{Minimize}\kern1.44em f\left(\mathbf{x}\right)=1.10471{x}_1^2{x}_2+0.04811{x}_3{x}_4\left(14+{x}_2\right) $$

(B.1)

$$ {\displaystyle \begin{array}{l}\mathrm{s}.\mathrm{t}.\kern0.84em \\ {}\kern2.879999em {g}_1\left(\mathbf{x}\right)=t-{t}_{\mathrm{max}}\le 0\\ {}\kern2.879999em {g}_2\left(\mathbf{x}\right)=s-{s}_{\mathrm{max}}\le 0\\ {}\kern2.879999em {g}_3\left(\mathbf{x}\right)={x}_1-{x}_4\le 0\\ {}\kern2.879999em {g}_4\left(\mathbf{x}\right)=d-{d}_{\mathrm{max}}\le 0\\ {}\kern2.879999em {g}_5\left(\mathbf{x}\right)=P-{P}_c\le 0\end{array}} $$

(B.2)

$$ M=P\left(L+{x}_2/2\right) $$

(B.3)

$$ R=\sqrt{0.25\left({x}_2^2+{\left({x}_1+{x}_3\right)}^2\right)} $$

(B.4)

$$ J=\sqrt{2}{x}_1{x}_2\left({x}_2^2/12+0.25{\left({x}_1+{x}_3\right)}^2\right) $$

(B.5)

$$ {P}_c=64746.022\left(1-0.0282346{x}_3\right){x}_3{x}_4^3 $$

(B.6)

$$ {t}_1=P/\left(\sqrt{2}{x}_1{x}_2\right) $$

(B.7)

$$ {t}_2= MR/J $$

(B.8)

$$ t=\sqrt{t_1^2+{t}_1{t}_2{x}_2/R+{t}_2^2} $$

(B.9)

$$ S=6 PL/\left({x}_4{x}_3^2\right) $$

(B.10)

$$ d=2.1952/\left({x}_4{x}_3^3\right) $$

(B.11)

$$ {\displaystyle \begin{array}{l}P=6000,L=14,E=30\times {10}^6,G=12\times {10}^6,\\ {}{t}_{\mathrm{max}}=13600,{s}_{\mathrm{max}}=30000,{x}_{\mathrm{max}}=10,{d}_{\mathrm{max}}=0.25\\ {}0.125\le {x}_1\le 10,\kern0.72em 0.1\le {x}_i\le 10,\kern0.96em for\kern0.24em i=2,3,4.\end{array}} $$

(B.12)

1.2 B.2 Example 2: speed reducer

$$ \operatorname{Minimize}\kern2.28em f\left(\mathbf{x}\right)=0.7854{x}_1{x}_2^2A-1.508{x}_1B+7.477C+0.7854D $$

(B.13)

$$ {\displaystyle \begin{array}{l}\mathrm{where},\kern2.999999em A=3.3333{x}_3^2+14.9334{x}_3-43.0934\\ {}\kern4.679997em B={x}_6^2+{x}_7^2\\ {}\kern4.679997em C={x}_6^3+{x}_7^3\\ {}\kern4.679997em D={x}_4{x}_6^2+{x}_5{x}_7^2\end{array}} $$

(B.12)

$$ {\displaystyle \begin{array}{l}\mathrm{s}.\mathrm{t}.,\kern3.239999em {g}_1\left(\mathbf{x}\right)=\left(27-{x}_1{x}_2^2{x}_3\right)/27\le 0\\ {}\kern4.079998em {g}_2\left(\mathbf{x}\right)=\left(397.5-{x}_1{x}_2^2{x}_3^2\right)/397.5\le 0\\ {}\kern4.199998em {g}_3\left(\mathbf{x}\right)=\left(1.93-\left({x}_2{x}_6^4{x}_3\right)/{x}_4^3\right)/1.93\le 0\\ {}\kern4.199998em {g}_4\left(\mathbf{x}\right)=\left(1.93-\left({x}_2{x}_7^4{x}_3\right)/{x}_5^3\right)/1.93\le 0\\ {}\kern4.199998em {g}_5\left(\mathbf{x}\right)=\left(\left({A}_1/{B}_1\right)-1100\right)/1100\le 0\\ {}\kern4.199998em {g}_6\left(\mathbf{x}\right)=\left(\left({A}_2/{B}_2\right)-850\right)/850\le 0\\ {}\kern4.199998em {g}_7\left(\mathbf{x}\right)=\left({x}_2{x}_3-40\right)/40\le 0\\ {}\kern4.199998em {g}_8\left(\mathbf{x}\right)=\left(5-\left({x}_1/{x}_2\right)\right)/5\le 0\\ {}\kern4.199998em {g}_9\left(\mathbf{x}\right)=\left(\left({x}_1/{x}_2\right)-12\right)/12\le 0\\ {}\kern4.199998em {g}_{10}\left(\mathbf{x}\right)=\left(1.9+1.5{x}_6-{x}_4\right)/1.9\le 0\\ {}\kern4.199998em {g}_{11}\left(\mathbf{x}\right)=\left(1.9+1.1{x}_7-{x}_5\right)/1.9\le 0\end{array}} $$

(B.14)

$$ {\displaystyle \begin{array}{l}\mathrm{where},\kern1.92em {A}_1={\left[{\left(745{x}_4/\left({x}_2{x}_3\right)\right)}^2+\left(16.91\times {10}^6\right)\right]}^{0.5}\\ {}\kern3.599998em {B}_1=0.1{x}_6^3\\ {}\kern3.479999em {A}_2={\left[{\left(745{x}_5/\left({x}_2{x}_3\right)\right)}^2+\left(157.5\times {10}^6\right)\right]}^{0.5}\\ {}\kern3.479999em {B}_2=0.1{x}_7^3\end{array}} $$

(B.16)

$$ {\displaystyle \begin{array}{l}2.6\le {x}_1\le 3.6\\ {}0.7\le {x}_2\le 0.8\\ {}17\le {x}_3\le 28\\ {}7.3\le {x}_4,{x}_5\le 8.3\\ {}2.9\le {x}_6\le 3.9\\ {}5\le {x}_7\le 5.5\end{array}} $$

(B.17)

1.3 B.3 Example 3: vibrating platform

$$ {\displaystyle \begin{array}{l}\operatorname{minimize}\kern3.839998em \mathrm{cost}=2 bL\left({c}_1{t}_1+{c}_2{t}_2+{c}_3{t}_3\right)\\ {}\kern3.839998em \mathrm{and}\\ {}\operatorname{maximize}\kern3.599998em \mathrm{frequency}=\left(\frac{\pi }{2{L}^2}\right){\left(\frac{EI}{\mu}\right)}^{1/2}\end{array}} $$

(B.18)

$$ EI=\left(\frac{2b}{3}\right)\left[{E}_1{t}_1^3+{E}_2\left\{{\left({t}_1+{t}_2\right)}^3-{t}_1^3\right\}+{E}_3\left\{{\left({t}_1+{t}_2+{t}_3\right)}^3-{\left({t}_1+{t}_2\right)}^3\right\}\right] $$

(B.19)

$$ \mu =2b\left({\rho}_1{t}_1+{\rho}_2{t}_2+{\rho}_3{t}_3\right) $$

(B.20)

$$ 0\le {t}_1\le 0.5,0\le {t}_2\le 0.15,\mathrm{and}\kern0.24em 0\le {t}_3\le 0.05;\kern0.36em b=0.4m;L=4\;\mathrm{m} $$

(B.21)

Appendix C. Comparative assessment of the computational effort of proposed and conventional surrogate model-assisted RDO framework

In this section, a case study has been undertaken to compare the computational effort entailed by the proposed and conventional RDO framework. In doing so, the building frame problem previously executed as example 4 has been carried out. As the problem involves FE modelling, it will provide a realistic indication of the computational effort and thus, more relevant for the applicability of surrogate models. For a fair comparison, the proposed surrogate models PM1 and PM2 have been implemented in a conventional surrogate model-assisted RDO framework (HF approach) for solving the problem. The number of actual response evaluations have been presented in Table 12. It can be observed from Table 12 that the computational effort in terms of actual response evaluations utilized by proposed RDO framework is (2.5 × 10⁵/4.95 × 10⁵) = 50.5% in comparison to conventional HF-RDO framework.

Table 12 Comparison of the number of actual response evaluations (NARE) in conventional PM1/PM2-RDO and proposed PM1/PM2-RDO

In order to further investigate the performance of the proposed RDO framework in terms of CPU time as compared to the conventional surrogate-assisted RDO approach, the time of a single actual response evaluation is varied and other parameters such as n_s1, n_s2, n_{f − count}, n_s, and ${n}_{\alpha_w}$ are kept as constant for the above building frame problem. From the following figure (Fig. 11), it can be observed that the proposed framework utilizes 47.5–50.5% computational effort in terms of CPU time as compared to the conventional HF approach varying the time of an actual response from 1 to 1200 s.

Fig. 11

Comparison of computational effort in terms of CPU time required by the proposed RDO framework to that of conventional approach varying the time of one actual response evaluation from 1 to 1200 s, keeping other parameters n_s1, n_s2, n_{f − count}, n_s, and ${n}_{\alpha_w}$constant for the building frame problem (example 4)