Integrated structural-electromagnetic optimization of cable mesh reflectors considering pattern degradation for random structural errors

Abstract

To alleviate the effects of random structural errors on the radiation performance and directly guide the structural design of cable mesh reflectors, an integrated structural-electromagnetic optimization procedure is proposed considering the radiation pattern degradation for random structural errors. Based on analytical expressions and the structural sensitivity concept, the radiation pattern is directly expressed as two matrix-form functions with respect to the random structural errors. By applying the pattern calculation into the multidisciplinary design and selecting the average boresight directivity as the objective function, an optimum result with better radiation performance compared with the traditional structural design is obtained. To reveal the fundamentals of the optimum result, the concept of sensitivity analysis of the average boresight directivity with respect to the random structural errors is introduced. The effectiveness and benefits of this study are demonstrated via an offset cable mesh reflector.

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Acknowledgments

The authors would like to thank the reviewers and editor for their very beneficial comments and suggestion, which helped a lot in improving this paper.

Funding

This work was supported by the National Natural Science Foundation of China No. 51705388 and Young Talent fund of University Association for Science and Technology in Shaanxi, China.

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Correspondence to Shuxin Zhang.

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The authors declare that they have no conflict of interest.

Replication of results

The presented results in both the structural and electromagnetic disciplines are produced by subroutines using our in-house codes, which compute the structural sensitivity matrices and the average radiation patterns for cable mesh reflectors with random structural errors, respectively. The optimization iteration is implemented with the help of fmincon in the optimization toolbox of MATLAB software. The code and data for producing the presented results will be made available by request.

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Appendix

Appendix

Sensitivity matrix of the facet link point displacements with respect to the structural cable length imperfections

The sensitivity matrix of the facet link point displacements with respect to the structural cable length imperfections in (6) is obtained as follows (Du et al. 2013):

$$ {\boldsymbol{\varGamma}}_t=-{\left({\boldsymbol{K}}_c^{11}\right)}^{-1}{\boldsymbol{K}}_s^1 $$
(24)

where \( {K}_c^{11} \) and \( {K}_s^1 \) are the partitioned matrices according to the interior facet link points in the cable mesh reflector.

The partitioned matrices are expressed as

$$ {\boldsymbol{K}}_c^{11}=\sum {\boldsymbol{K}}_{ck} $$
(25)
$$ {\boldsymbol{K}}_s^1=\sum {\boldsymbol{K}}_{sk} $$
(26)

where Σ is the standard finite element assembly operator, Kck is the element axial stiffness matrix, and Ksk is the geometric stiffness matrix due to the cable.

The element axial stiffness matrix Kck and the geometric stiffness matrix Ksk are respectively deduced as

$$ {\boldsymbol{K}}_{ck}=\left[\begin{array}{cc}{\boldsymbol{k}}_{ck}& -{\boldsymbol{k}}_{ck}\\ {}-{\boldsymbol{k}}_{ck}& {\boldsymbol{k}}_{ck}\end{array}\right] $$
(27)
$$ {\boldsymbol{K}}_{sk}=\left[\begin{array}{c}{\boldsymbol{k}}_{sk}\\ {}-{k}_{sk}\end{array}\right] $$
(28)

and

$$ {\boldsymbol{k}}_{ck}=\frac{EA}{L^3}\frac{2{L}_0-L}{L}\left({\boldsymbol{r}}_p-{\boldsymbol{r}}_q\right){\left({\boldsymbol{r}}_p-{\boldsymbol{r}}_q\right)}^T+\frac{EA}{L}\frac{L-{L}_0}{L}{\boldsymbol{I}}_{3\times 3} $$
(29)
$$ {\boldsymbol{k}}_{sk}=-\frac{EA}{L^2}\left({\boldsymbol{r}}_p-{\boldsymbol{r}}_q\right) $$
(30)

where E is the elastic modulus, A is the cable cross-sectional area, L is the cable length in the stress state, L0 is the cable initial dimension, rp and rq are the position vectors of the facet link points in one cable as shown in Fig. 17, and I3 × 3 is a 3 × 3 identity matrix.

Fig. 17
figure17

A cable element with facet link points

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Zhang, S., Duan, B. Integrated structural-electromagnetic optimization of cable mesh reflectors considering pattern degradation for random structural errors. Struct Multidisc Optim 61, 1621–1635 (2020). https://doi.org/10.1007/s00158-019-02439-9

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Keywords

  • Cable mesh reflectors
  • Random structural errors
  • Integrated structural-electromagnetic design