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.

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, K_ck is the element axial stiffness matrix, and K_sk is the geometric stiffness matrix due to the cable.

The element axial stiffness matrix K_ck and the geometric stiffness matrix K_sk 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, r_p and r_q are the position vectors of the facet link points in one cable as shown in Fig. 17, and I_{3 × 3} is a 3 × 3 identity matrix.

Fig. 17

A cable element with facet link points