Multi-objective layout optimization for an orbital propellant depot

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The overall layout optimization design of an orbital propellant depot involves the optimization of shape, size, and positions of propellant tanks in functional module and the optimization of positions of equipment in service module, with the aim of making the carrying capacity of propellant, dry/wet ratio, and mass properties meet the allowable values. To alleviate the difficulty in dealing with the overall optimization problems involving two modules of the orbital propellant depot, a step-by-step modeling and solving strategy is presented. Two multi-objective optimization mathematical models for the tanks in functional module (model I) and the equipment in service module (model II) are constructed separately, which are solved one after another. In the solution process of the two models, model I is solved firstly and the obtained optimization solution is transmitted to model II as a known condition. We mainly focus on the layout optimization of equipment in the service module and give a batch component assignment and layout integration optimization method. In the proposed method, all the components are grouped firstly according to the functional subsystem, and then the obtained component groups are sorted in descending order of their feature values. Finally, the sorted component groups are added into the service module one by one for both assignment optimization and layout optimization. The computational results of the case study show that the obtained Pareto solutions meet the given allowable values of carrying capacity of propellant, dry/wet ratio, and mass properties of the orbital propellant depot.

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Appendix 1. Mass properties calculation.

  1. 1.

    Moment of inertia

$$ {\displaystyle \begin{array}{l}{I}_{x^{\prime }{x}^{\prime }}=\sum \limits_{i=0}^N\left({I}_{x^{{\prime\prime} },i}{\cos}^2{\upalpha}_i+{I}_{y^{{\prime\prime} },i}{\sin}^2{\upalpha}_i\right)\\ {}\kern1.6em +\sum \limits_{i=0}^N{m}_i\left({y}_i^2+{z}_i^2\right)-\left({y}_m^2+{z}_m^2\right)\sum \limits_{i=0}^N{m}_i\\ {}{I}_{y^{\prime }{y}^{\prime }}=\sum \limits_{i=0}^N\left({I}_{y^{{\prime\prime} },i}{\cos}^2{\upalpha}_i+{I}_{x^{{\prime\prime} },i}{\sin}^2{\upalpha}_i\right)\kern0.1em \\ {}\kern1.7em +\sum \limits_{i=0}^N{m}_i\left({x}_i^2+{z}_i^2\right)-\left({x}_m^2+{z}_m^2\right)\sum \limits_{i=0}^N{m}_i\\ {}{I}_{z^{\prime }{z}^{\prime }}=\sum \limits_{i=0}^N{I}_{z^{{\prime\prime} },i}+\sum \limits_{i=0}^N{m}_i\left({x}_i^2+{y}_i^2\right)\\ {}\kern1.6em -\left({x}_m^2+{y}_m^2\right)\sum \limits_{i=0}^N{m}_i\end{array}} $$

For cylinder component:

$$ {\displaystyle \begin{array}{l}{I}_{x^{{\prime\prime} },i}={I}_{y^{{\prime\prime} },i}={m}_i\left(3{r}_i^2+{h}_i^2\right)/12\\ {}{I}_{z^{{\prime\prime} },i}={m}_i{r}_i^2/2\end{array}} $$

For cuboid component:

$$ {\displaystyle \begin{array}{l}{I}_{x^{{\prime\prime} },i}={m}_i\left({b}_i^2+{h}_i^2\right)/12\\ {}{I}_{y^{{\prime\prime} },i}={m}_i\left({a}_i^2+{h}_i^2\right)/12\\ {}{I}_{z^{{\prime\prime} },i}={m}_i\left({b}_i^2+{a}_i^2\right)/12\end{array}} $$

For hemisphere component:

$$ {\displaystyle \begin{array}{l}{I}_{x^{{\prime\prime} },i}={I}_{y^{{\prime\prime} },i}=83{m}_i{r}_i^2/320\\ {}{I}_{z^{{\prime\prime} },i}=2{m}_i{r}_i^2/5\end{array}} $$

For component i, (xi, yi, zi) is the center of mass; αi is the arrangement angle; ai, bi, and hi denote the length, width, and height, respectively. (xm, ym, zm) is the calculated center of mass for the whole orbital propellant depot. ri denotes the radius of a cylinder or sphere component.

  1. 2.

    Center of mass

$$ {\displaystyle \begin{array}{l}{x}_m={\sum}_{i=0}^N{m}_i{x}_i/{\sum}_{i=0}^N{m}_i\\ {}{y}_m={\sum}_{i=0}^N{m}_i{y}_i/{\sum}_{i=0}^N{m}_i\\ {}{z}_m={\sum}_{i=0}^N{m}_i{z}_i/{\sum}_{i=0}^N{m}_i\end{array}} $$
  1. 3.

    Angle of inertia

$$ {\displaystyle \begin{array}{l}{\theta}_{x^{\prime }}=\arctan \left(2{I}_{x^{\prime }{y}^{\prime }}/\left({I}_{x^{\prime }{x}^{\prime }}-{I}_{y^{\prime }{y}^{\prime }}\right)\right)/2\\ {}{\theta}_{y^{\prime }}=\arctan \left(2{I}_{x^{\prime }{z}^{\prime }}/\left({I}_{z^{\prime }{z}^{\prime }}-{I}_{x^{\prime }{x}^{\prime }}\right)\right)/2\\ {}{\theta}_{z^{\prime }}=\arctan \left(2{I}_{y^{\prime }{z}^{\prime }}/\left({I}_{z^{\prime }{z}^{\prime }}-{I}_{y^{\prime }{y}^{\prime }}\right)\right)/2\end{array}} $$
  1. 4.

    Product of inertia

$$ {\displaystyle \begin{array}{l}{I}_{x^{\prime }{y}^{\prime }}(X)=\sum \left({m}_i{x}_i{y}_i+\right(\Big({J}_{x^{{\prime\prime} }i}+{m}_i\left({y}_i^2+{z}_i^2\right)\\ {}\kern3.799999em -{J}_{y^{{\prime\prime} }i}-{m}_i\left({x}_i^2+{z}_i^2\right)\left)/2\right)\sin 2{\alpha}_i\Big)\\ {}\kern3.799999em -{x}_m{y}_m\sum \limits_{i=0}^N{m}_i\\ {}{I}_{x^{\prime }{z}^{\prime }}(X)=\sum \limits_{i=0}^N{m}_i{x}_i{z}_i-{x}_m{z}_m\sum \limits_{i=0}^N{m}_i\\ {}{I}_{y^{\prime }{z}^{\prime }}(X)=\sum \limits_{i=0}^N{m}_i{y}_i{z}_i-{y}_m{z}_m\sum \limits_{i=0}^N{m}_i\end{array}} $$

Appendix 2.

Table 8 Component grouping table

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Xu, Z., Jiang, F., Zhong, C. et al. Multi-objective layout optimization for an orbital propellant depot. Struct Multidisc Optim 61, 207–223 (2020).

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  • Component assignment
  • Component layout
  • Multi-objective optimization
  • Orbital propellant depot