International Journal of Aerospace Engineering

Volume 2019, Article ID 3768746, 11 pages

https://doi.org/10.1155/2019/3768746

## Sensitivity Study on Optimizing Thickness Distribution for a Membrane Reflector

^{1}School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, China^{2}College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China

Correspondence should be addressed to He Huang; nc.ude.upwn@3260gnauheh

Received 14 July 2019; Revised 22 September 2019; Accepted 17 October 2019; Published 23 November 2019

Academic Editor: Linda L. Vahala

Copyright © 2019 He Huang and Fu-ling Guan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Inflatable membrane reflectors are widely used in space and terrestrial deployable antennas. The mechanical properties of the inflated membrane reflector, however, are often a limiting factor in the level of the surface accuracy that can be achieved. In this study, membrane structural analysis is combined with sizing optimization to tailor the thickness of the membrane to improve surface accuracy of an axisymmetric parabolic inflated membrane reflector. Two reflector surface accuracy evaluation methods are employed in the optimizer and researched about their effect on the sensitivity analysis. Gradient-based optimization is employed together with a simplified sensitivity analysis, and the resulting algorithm is demonstrated on a membrane reflector case study. A linear thickness filter is used to avoid checkerboard patterns, and optimized solutions are computationally shown to achieve a feasible level of surface accuracy.

#### 1. Introduction

Inflatable reflector concepts have been considered and employed for the space and terrestrial deployable antennas, including QUASAT [1], IN-STEP [2], ARISE [3], NEXRAD [4], and other missions [5], due to their ultra-lightweight and high package ratio. These characteristics allow the scale of antennas to expand substantially while reducing the cost for launch, as compared with other types of deployable antennas. However, the mechanical properties of inflatable membrane structures restrict the range of the microwave bandwidth that can be achieved and, therefore, the range of application of these antennas [5].

The surface accuracy of the reflector is one of the most important factors influencing the gain and microwave bandwidth of the antenna, attracting the attention of previous research [6]. The so-called W-profile error, defined as the deviation from a parabolic profile due to the inflation, leads to a reduction in surface accuracy of the inflated membrane reflector. Jerkins and Marker [7] have studied the reflector figure error from plane sheet to curved surface, while Greschik et al. [8–11] have studied the structural properties of the flat and curved axisymmetric membranes for the inflated reflectors and the initially approximate and exact parabolic shape design with different parametric assembling. In particular, in Ref. [10], the sensitivity study of the membrane reflectors has been conducted and stated that nonuniform thickness perturbations have effects on reflector surface errors. Naboulsi [12] has investigated geometric imperfections associated with inflated structures using geometric nonlinear finite element (FE). Kitano and Ishida [13] have obtained the discrete distribution of optimum thickness for circular membrane reflector. In order to decrease the W-profile error, DeSmidt et al. and Hill et al. [14, 15] have developed active control of the reflector using optimized gore/seam cable-actuated shape control. Coleman et al. [16] have studied the effects of the elastic tendon boundary support and reflector dimensions on surface accuracy of the inflatable antenna by an ideal parabolic form and two different flat panel design patterns. Bouzidi and Lecieux [17] have optimized the shape parameter design of initial geometry for a space inflatable membrane reflector by a numerical downhill simplex method. Wang et al. [18] have employed a kind of support ribs stretched by the tension system to improve the surface accuracy of the inflatable deployable reflector. Huang and Guan [5] have proposed an iterative method working from the initial profile solution for reflector precision improvement.

There is no gainsaying that these methods have improved surface accuracy, but error magnitudes are still relatively large. Meanwhile, 3D printing technology is being processed in the manufacturing industry, but this application is not used in membrane reflector and relative sensitivity study is seldom carried on deeply. This has motivated the current study focusing on membrane reflector sensitivity and using optimization to tailor the membrane thickness to further enhance surface accuracy. There are many optimization methods for wrinkle-free design of membrane structures such as stress-based topology optimization [19], multimaterial topology optimization approach [20], and global shape optimization [21]. Although wrinkles are avoided in membrane reflector by engineering means such as changing border forms, the thoughts and methods of optimization and sensitivity analysis can be referred. In particular, we use the Method of Moving Asymptotes (MMA) by Svanberg [22], an optimizer that is widely used by the topology optimization community [23], together with the adjoint method for much more specific sensitivity analysis compared with Ref. [10]. The design variables are the thicknesses of membrane elements, and the objective function is the minimum surface accuracy of inflated reflector. The inflated reflector deformation is computed by using the finite element method.

The remainder of this paper is organized as follows. The importance and foundation of two surface accuracy evaluation methods for the objective function are discussed in Section 2. The sizing optimization formulation is stated in Section 3, and the sensitivity analysis based on two methods is presented in Section 4. A membrane reflector case study is used to illustrate the method in Section 5, with concluding remarks following in Section 6.

#### 2. Surface Accuracy

One of the most important parameters for the antenna is the antenna gain, defined as the ratio of the power transmitted by the antenna to the power of an ideal isotropic radiator. With the mean phase plane chosen as the phase reference plane and the simplification during the prediction of the gain reduction and pattern degradation, the reflector gain of the Ruze equation [24] is given as where is the no-error gain axial value as , is a statistical phase error calculated from the mean phase plane, is the aperture efficiency, is the diameter of the reflector aperture, is the wavelength, and is the effective reflector tolerance as the structural parameter to approximately quantify the reflector surface efficiency.

When the effective gain is described in units of decibels (dB), the equation can be expanded as [6]

From this expression, it is clear that the effective gain can be kept large if is kept as low as possible. Therefore, the evaluation of this effective reflector tolerance is important and plays a significant role in the objective function and sensitivity analysis of reflector optimization.

##### 2.1. HPL-E Method

Generally, the interpretation of the parameter is presented as the Root Mean Square Half-Path-Length Error (HPL-E) Method of the microwave rays distributed over and reflected by the surface [6]. This is considered to be the most significant of all these separate efficiency terms and consequently motivates the use of root mean square (RMS), familiarly surface accuracy, as the objective function for the optimization formulation.

Based on the surface tolerance mentioned in Ref. [6], the HPL-E of an arbitrary point () on the reflector can be derived by the geometric path length analysis as where is the normal vector on the point as (assume axis is the focus axis), is the displacement vector of the point as , and is the value of normal vector in the axial direction. It means that the HPL-E is the product of the normal value of displacement vector and the value of normal vector in the axial direction.

There is a best-fit parabolic profile, namely, the mean phase plane, of each deformed surface as the phase reference plane during the tolerance theory deriving [6]. Depending on this phase reference plane, a new half-path-length error can be computed as where , , and are , , and , respectively, as aforementioned; the vector is the displacement difference between the best-fit parabolic profile and the original parabolic profile. Thereinto, the best-fit parabolic profile usually can be determined by six parameters, i.e., translations , , and in three axes, rotations and along the axis and , respectively, and focus-length change parameter .

Thus, there is a Root Mean Square HPL-E depending on the best-fit parabolic profile as where is the total number of the points on the reflector. This , the so-called RMS, is an important indicator for structural engineers to evaluate the reflector efficiency as aforementioned in this section.

##### 2.2. BFP-E Method

The HPL-E Method is a kind of traditional method to evaluate the reflector efficiency. It is apparently dependent on the displacement vector which is supposed to be tiny as shown in equation (4). When it comes to high-precision reflector, this HPL-E Method should be reconsidered to discuss its effectivity for the reflector surface accuracy evaluation.

The membrane reflector loaded by the inflation pressure has large displacements that cause the geometry nonlinear property. In other words, it might be improper to use the HPL-E Method with this large displacement vector to indicate the reflector efficiency of membrane structures. One more problem is the parameter, specifically the thickness here in the following sensitivity analysis, influencing the structural property and displacement, which means that it might be not comparable for each optimal result by using the HPL-E Method.

Therefore, another method, namely, the Best-Fit-Profile Error (BFP-E) Method, is stated here to evaluate the reflector surface accuracy. The fundamental theory of the BFP-E Method is to find a best-fit parabolic profile for a deformed reflector and get the RMS of the deviation error between the deformed reflector and the best-fit profile. It could indicate the surface roughness for the deformed surface without the displacement vector.

The specific steps of the BFP-E Method are explained as follows. Firstly, it is easy to get the best-fit profile computed from the deformed reflector by several algorithms, like the least square method. Secondly, a best-fit-profile error can be defined as the norm of deviation vector for each point and shown as where and are the coordinates of the deformed profile and the best-fit profile. Finally, a Root Mean Square BFP-E can be calculated by substituting into equation (5).

It is obvious that this method is not exactly precise as the HPL-E Method since the BFP-E Method is based on the concept of geometry surface instead of phase plane. However, this method can indicate the deformation state for any reflectors with the uniform quantification standard. It is convenient to conduct the following sensitivity research by utilizing this evaluation of the reflector efficiency.

#### 3. Optimization Model

The optimization formulation uses surface accuracy, namely, the RMS based on two methods (HPL-E Method and BFP-E Method), as the objective function. The thickness of each membrane element is the (dependent) design variables, which are expressed as a function of the independent design variables as described below. The considered reflector is assumed to be an axisymmetric parabolic reflector made of isotropic membrane material and loaded by inflation pressure. The final deformed shape of the reflector under this pressure is influenced by the stiffness and therefore the thickness of the membrane. The optimization problem can be written as where is the number of independent design variables, and are the lower and upper bounds on the thickness of a membrane element, respectively, is the global stiffness matrix for the membrane structure, is the nodal displacement vector, and is the nodal load vector. Note that the stiffness and applied pressure loads are functions of the displacement field , and thus, we have a geometrically nonlinear problem that is solved iteratively herein using the Newton-Raphson Method [25].

Here in equation (7), will be separately discussed as HPL-E and BFP-E by the HPL-E Method and the BFP-E Method, respectively, to conduct two optimization models. The results of the best RMS and can be computed for each optimization model to evaluate the optimal surface accuracy.

It is well known in the structural topology optimization community that using low-order finite elements, such as linear three node triangles, may lead to artificial solutions known as checkerboards. The checkerboard pattern consists of alternating solid and void elements or equivalently alternating elements with high and low thickness. The stiffness of this pattern is overestimated with low-order finite elements [26]. Density filters in topology optimization [25, 27, 28] are a popular means for circumventing this effect, in addition to eliminating a related issue of solution mesh dependency. We employ this idea here to relate the independent design variables to the thickness of membrane elements and refer to it as thickness filtering, although the logic is exactly the same as the density filter.

Defining as the radius of the thickness filter (over which the smoothing occurs), we denote the neighborhood set of an element as and define it as [28] where is the location of design variable and is the location of the centroid of .

These new design variables are then filtered onto the elemental thickness as where is the linear distance-based weighting function

The Method of Moving Asymptotes [22] is used to solve this problem with sensitivities computed as described in the following section.

#### 4. Sensitivity Analysis

The adjoint method is used to estimate the sensitivities of objective function [23]. The equilibrium condition is added to the objective function as where is any arbitrary vector.

The derivatives of the adjoint function in equation (11) is given as

Grouping like terms gives where we note and define the geometric stiffness matrix at the last iteration of the structural analysis [25].

In order to eliminate the last term in equation (13), the arbitrary adjoint variables are chosen to solve

The derivative then simplifies to where the derivative of the stiffness matrix with respect to the membrane thickness is given as where is the element stiffness matrix for element having unit thickness. This provides all of the terms needed to compute the derivatives in equation (16).

The derivatives of equation (16) with respect to the independent optimization variables, which are needed by the optimizer, are then simply given by the chain rule [27] as where

When computing adjoint variables in equation (16), the left term of equation (15) should be considered. Hence, the RMS based on the HPL-E Method and the BFP-E Method has different expression shown as the following.

##### 4.1. HPL-E Derivative

The partial derivatives of RMS HPL-E with respect to the displacements are needed to solve , as indicated in equation (15). This can be computed as where the displacement vector is listed in sequence as where is the displacements listed by the order of the degree of freedom (DOF) and has the mapping relationship with .

Here in equation (4), it is also noted that the six parameters for each best-fit parabolic profile are unique so that the vector can be determined by the parameters and have no relationship with the displacement vector . So equation (20) can be rewritten more simply according to the DOF for each element as

This simplification shows that the displacement has no relationship with the original profile and the best-fit parabolic profile. It is linear in this derivative.

##### 4.2. BFP-E Derivative

The partial derivatives of RMS BFP-E with respect to the displacements can be computed as

Considering equation (21), equation (23) can be rewritten more simply according to the DOF for each element as

An assumption is introduced here as which means that the displacement has no relationship with the best-fit profile geometry and it is approximately linear in this derivative.

##### 4.3. Optimization Process

The flowchart of this optimization is shown in Figure 1. Here, two optimization models are separately built by using the HPL-E Method and the BFP-E Method to evaluate the inflated reflector surface accuracy as the objective function.