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Partial design space exploration strategies applied in preliminary design

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Abstract

During preliminary phases in product design, on the basis of strong physical hypotheses (e.g. isotherm, steady state), physical and functional requirements can be expressed as coarse-grained constraint-based models on a few degrees of freedom, possibly including several design criteria to optimize. Such models are usually handled by multi-objective optimization solvers in order to find design solutions giving the best trade-offs between design criteria. Another approach developed in this paper is to partially explore all the areas of the design space using an anytime interval branch-and-prune algorithm called IDFS such that the design criteria are converted into so-called \(\varepsilon \)-constraints. The expected result is a sample of solutions diversified in both the objective space and the design space. Several quality indicators are introduced in order to measure this diversity and compare IDFS with two state-of-the-art multi-objective optimization solvers NSGA-II and NSGA-III on three real-world case studies. The results show that IDFS is able to identify new close-to-optimal designs and permits a better understanding of the design space. This framework provides a promising alternative tool for decision making, in particular for integrating interaction in the preliminary design process.

Graphical Abstract

Partial exploration aims to compute a diversified subset of feasible solutions; We built an anytime branch and prune algorithm for partial design space exploration. We built a protocol to analyze diversity in both the design and the objective space. We compare partial exploration and optimization approaches on three design problems. Partial Exploration is a tool for decision makers to identify quasi-optimal designs.

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Notes

  1. https://pymoo.org/.

  2. http://www.ibex-lib.org/.

  3. https://pymoo.org/.

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Authors and Affiliations

  1. LS2N, UMR6004, Nantes Université, École Centrale Nantes, CNRS, 44000, Nantes, France

    Thomas Richard de Latour, Raphaël Chenouard & Laurent Granvilliers

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Appendix A Models

Appendix A Models

1.1 A1 Engine

$$\begin{aligned} CSP :\ {{\textbf {x}}} =&\, (b, c_r, d_I, d_E, w) \\ \left[b\right] =&\, [0,100] \\ \left[c_r\right] =&\, [0,100] \\ \left[d_I\right] =&\, [0,0.05] \\ \left[d_E\right] =&\, [0,1] \\ \left[w\right] =&\, [0,6.5] \\ subject \ to \\ c_1:&\ b-83.33 \le 0\\ c_2:&\ 76.9-b \le 0\\ c_3:&\ d_I+d_E-0.82b\le 0\\ c_4:&\ 0.83d_I-d_E\le 0\\ c_5:&\ d_E-0.89d_I \le 0\\ c_6:&\ 215.46-d_I^2 \le 0\\ c_7:&\ c_r-13.2+0.0045b \le 0\\ c_{8}:&\ 2b^3/1.18348e^6-1.3<0\\ c_{9}:&\ 2b^3/1.18348e^6-0.7>0\\ MOP :\\&\ min(ISFC,-BKW/V) \\ ISFC=&\,81.8964/(0.8595(1-c_r^{-0.33})\\&\quad +(0.83((8+4c_r)+1.5(c_r\\&\quad -1)(4\pi /1.859e^6)b^3)/(2+c_r)b))\\ BKW/V=&\,w[FMEP-3688\eta _t\eta _v]/120\\ where\\ FMEP=&\, 4.826(c_r-9.2)+(7.97+2.99e^5wb^{-2}\\&\quad +1.36e7bw^2)\\ \eta _t =&\, 0.8595(1-c_r^{-0.33})-S_v\sqrt{1.5/w}\\ \eta _v =&\, \eta _{vb}(1+5.96e^{-3}w^2)/((1+55.789/0.44)\\&\quad \times (w/(d_I)^2))^2\\ \eta _{vb} =&\,{\left\{ \begin{array}{ll} 0.067-0.038e^{w-5.25}, \ w \ge 5.25,\\ 0.637+0.13w-0.014w^2\\ +0.00066w^3, w < 5.25 \end{array}\right. }\\ S_v=&\,0.83(8+4c_r+1.0134e^{-5}(c_r-1)b^3)/((2+c_r)b)\\ Exploration: \\&BWK \ge 13400 \\&ISFC \le 223.2 \end{aligned}$$

1.2 A2 Actuator

$$\begin{aligned} CSP : \\ {{\textbf {x}}} =&\, (e,J_{cu},la,\beta ,P)\\ \left[e\right] =&\, [0.00001,0.005] \\ \left[J_{cu}\right] =&\, [1e5,1e7] \\ \left[la\right] =&\, [0.001,0.05] \\ \left[\beta \right] =&\, [0.8,1] \\ \left[P\right] =&\,[1,10] \\ subject \ to.\\ c_1:&\ D-2*(la+e) \ge 0 \\ c_2:&\ 0.001 \le E \le 0.05 \\ c_3:&\ 0.001 \le D \le 0.05 \\ c_4:&\ 0.001 \le \lambda \le 0.05 \\ c_5:&\ 0.001 \le K_f \le 0.05 \\ where \\ E =&\, 10^{11}/(0.7*(J_{cu}^2))\\ D =&\,0.1P/\pi \\ \lambda =&\, 0.015\pi D^2(D+E)B_e\sqrt{7^{11}\beta E}\\ Kf =&\, 1.5P\beta (e+E)/D\\ B_e =&\,(1.8la/(D\log {(D+2E)/(D-2(la+e))})\\ C =&\, D\pi \beta B_e/(6P)\\ MOP:\\&min(Vu,Va,Pj)\\ Vu =&\, \pi (D/\lambda )(D+E-e-la)(2C+E+e+la)\\ Va =&\,\pi \beta la(D/\lambda )(D-2e-la)\\ Pj =&\, 0.018e^{-6}\pi (D/\lambda )(D+E)10^{11}\\ Exploration \\ Vu \le&6.5e^{-4}\\ Va \le&1.5e^{-4}\\ Pj \le&45 \end{aligned}$$

1.3 A3 Water

$$\begin{aligned} CSP: \\&{{\textbf {x}}} = (x_1, x_2, x_3) \\ \left[x_1\right] =&\,[0.01,0.45]\\ \left[x_2\right] =&\,[0.01,0.10]\\ \left[x_3\right] =&\,[0.01,0.10]\\ subject \ to. \\ c_1:&\ 0.00139/(x_1x_2)+4.94x_3 - 0.08 \le 1 \\ c_2:&\ 0.000306/(x_1x_2)+1.082x_3 - 0.0986 \le 1 \\ c_3:&\ 12.307/(x_1x_2) + 49408.24x_3 \\&\ \ + 4051.02 \le 50000 \\ c_4:&\ 2.098/(x_1x_2) + 8046.33x_3 - 696.71 \le 16000 \\ c_5:&\ 2.138/(x_1x_2) + 7883.39x_3 - 705.04 \le 10000 \\ c_6:&\ 0.417/(x_1x_2) + 1721.26x_3 - 136.54 \le 2000 \\ c_7:&\ 0.164/(x_1x_2) + 631.13x_3 - 54.48 \le 550 \\ MOP:\\&min(f_1,f_2,f_3,f_4,f_5)\\ f_1 =&\, 106780.37(x_2 + x_3) + 61704.67 \\ f_2 =&\, 3000x_1 \\ f_3 =&\, 30570 0.02289x_2/(0.06 2289.0)0.65 \\ f_4 =&\, 250.0 2289.0exp(-39.75x_2 +9.9x_3 +2.74) \\ f_5 =&\, 25.0((1.39/(x_1x_2)) + 4940.0x_3 - 80.0) \\ Exploration \\&f_1 \le 8e4\\&f_2 \le 1.35e3\\&f_3 \le 1\\&f_4 \le 8.1e6\\&f_5 \le 3e4 \end{aligned}$$

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Richard de Latour, T., Chenouard, R. & Granvilliers, L. Partial design space exploration strategies applied in preliminary design. Int J Interact Des Manuf (2023). https://doi.org/10.1007/s12008-023-01377-7

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