# Adaptive Deployable Thin Spherical Shell Reflectors
## 4 Thermal Response  
The unimorph design with the layer configuration shown in Figure 2a and the CTE properties of the last line of Table 2 is extremely sensitive to even small temperature changes, whether it be uniform or a gradient. In this section, we examine numerically how the thermal deformations can be reduced in such a way that they can be controlled actively with the piezoelectric layer with only a small share of the control budget (i.e., a set of voltages much smaller than the maximum allowed, \(V_{i}\ll V_{max}\)). Firstly, the design can be thermally balanced by adding a layer symmetrical to the PVDF-TrFE (Figure 2b) with equal value of  
\[\frac{Y}{1-v}\,.\,CTE\,.\,t \tag{8}\]  
where \(M=Y/(1-v)\) is the biaxial Young modulus and \(t\) is the layer thickness (note that the thermal balancing layer, being on the front side, would also have to satisfy the surface roughness requirements). Secondly, the thermal expansion of the substrate can be reduced, and even canceled, by using a zero-CTE polyimide such as the Novastrat-905 [19], available as a film sheet or liquid resin for spray and flow casting. In order to investigate how the foregoing design changes affect the control, we consider a spherical reflector of 1 m diameter with a radius of curvature of \(R_{c}=4\) m; the reflector is fixed along the local axes \(y\) and \(z\) of a central hole of 0.2 m (Figure 6a). The layer distribution is that of Figure 7a for the unimorph case and Figure 7b for the balanced design. The balanced design assumes a passive layer of PVDF-TrFE on the inner (optical) side of the reflector, although other options are possible, provided that Equation (8) is satisfied. The material properties of Table 2 are assumed in the numerical calculations [20], except for the case with a substrate of low CTE for which a near zero value of \(4\times 10^{-8}\) K\({}^{-1}\) is assumed. A linear temperature gradient going from \(-15^{\circ}\) K on one side of the reflector to \(+15^{\circ}\) K on the opposite side is assumed. A set of 120 control electrodes is assumed (Figure 6b); the voltage distribution which minimizes the RMS residual surface figure error of the reflector is computed by inverting the Jacobian between the 120 control voltages and the reflector surface displacements on a regular grid of 200 \(\times\) 200 points (with 31,000 points internal to the reflector shape); a simple Moore-Penrose pseudo-inverse is used in this case [7]. Columns 2 and 3 of Table 3 give, respectively, the peak to valley (PV) and the root mean squares (RMS) values of the surface figure error due to the thermal gradient of \(\Delta T=30^{\circ}\), respectively, for _(i)_ the unimorph design, _(ii)_ the balanced design with a substrate of large CTE, and _(iii)_ the balanced design with a substrate with a near zero CTE. Columns 4 to 5 give the PV and the RMS residual errors when the control has been applied, and column 6 gives the range of the voltage applied to the various electrodes, \(\Delta V=V_{max}-V_{min}\) (measure of the control budget). One sees that the addition of the thermal balancing layer and the near-zero CTE substrate reduce tremendously the residual error and the control budget. Figure 8 illustrates the case of the balanced layer design with near zero CTE substrate (last line in Table 3). Figure 8a shows the displacements along the \(z\) axis induced by the thermal gradient while Figure 8b shows the residual error when the control has been applied. Figure 8c shows the thermal displacements (in blue) and the residual error (in red) along the \(30^{\circ}\) cross section indicated on Figure 8a,b. Figure 8d shows the map of the correcting control voltages. The thermal displacement, residual error and voltage maps are similar to those of Figure 8 for both the unimorph and the thermally balanced high CTE design, except for the amplitudes, which are given in Table 3.  
_Actautos_**2022**, _11_, 198