## 3 Results  
The discrete nodal root mean square (RMS) error is used to evaluate the surface accuracy of a reflector. As shown in Figure 2, a geometric schematic of a reflector is given. The origin of the coordinate system is located at the vertex of the paraboloid. The \(y\)-axis passes the focal point of the paraboloid. A discrete root mean square (RMS) error of the reflector is defined as  
\[f_{\text{RMS}}=\sqrt{\frac{\sum_{i=1}^{k}\left(\delta_{i}{}^{Desired}-\delta_{ i}^{Deformed}\right)^{2}}{k}}, \tag{29}\]  
where \(\delta_{i}^{Deformed}\) and \(\delta_{i}{}^{Desired}\) are the deformed and desired \(z\)-coordinate values of the \(i\)-th node; and \(k\) is the number of nodes on the surface of the reflector. The deformed \(z\)-coordinate of the \(i\)-th node can be expressed as  
\[\delta_{i}^{Deformed}=\delta_{i}^{Initial}+\
abla\delta_{i}, \tag{30}\]  
where \(\
abla\delta_{i}\) is the displacement of the \(i\)-th node along the \(z\)-direction. Let \(\delta^{D}\) stand for a nodal position vector of the desired reflector surface, \(\delta^{I}\) stands for a nodal position vector of the initial reflector surface, and \(\
abla\delta\) stands for a nodal displacement vector. Assuming the total number of reflector nodal displacements is \(p\), the number of nodal displacements on the reflector surface is \(q\), and the number of voltage freedom degrees is \(r\), an equivalent objective function can be defined as  
Figure 2: Initial, desired and deformed surface positions of the reflector. \[f=\Big{[}\
abla\delta-\Big{(}\delta^{D}-\delta^{I}\Big{)}\Big{]}^{T}\Big{[}\
abla \delta-\Big{(}\delta^{D}-\delta^{I}\Big{)}\Big{]}=\
abla\delta^{T}\
abla\delta-2 \
abla\delta^{T}\Big{(}\delta^{D}-\delta^{I}\Big{)}+\Big{(}\delta^{D}-\delta^{ I}\Big{)}^{T}\Big{(}\delta^{D}-\delta^{I}\Big{)} \tag{31}\]  
where the reflector nodal displacements vector \(\
abla\delta\) is a \(q\)-dimensional vector and is determined only by the voltage vector \(\mathbf{U}\). According to Equation (28), the total nodal displacement vector can be denoted as  
\[\mathbf{d}=\mathbf{K_{m}}^{-1}(\mathbf{F}-\mathbf{K_{me}}\mathbf{U}), \tag{32}\]  
where \(\mathbf{K_{m}}\) is a \(p\)-by-\(p\) nonsingular matrix, \(\mathbf{F}\) is a \(p\)-dimensional forces vector, and \(\mathbf{U}\) is an \(r\)-dimensional voltages vector; \(\mathbf{K_{me}}\) is a \(p\)-by-\(r\) mechanical-electric matrix. The nodal displacements vector of the reflector surface \(\
abla\delta\) can be written as  
\[\
abla\delta=\mathbf{K_{s}}(\mathbf{F}-\mathbf{K_{me}}\mathbf{U}), \tag{33}\]  
where \(\mathbf{K_{s}}\) is a \(q\)-by-\(r\) submatrix of \(\mathbf{K_{m}}^{-1}\) that consists of the rows corresponding to the nodal displacements on the reflector surface in matrix \(\mathbf{K_{m}}^{-1}\). After substituting Equation (33) into Equation (31), the objective function \(f\) can be expressed as a function in terms of the voltage vector \(\mathbf{U}\):  
\[f(\mathbf{U})=\Big{(}\mathbf{F}^{T}-\mathbf{U}^{T}\mathbf{K_{me}}^{T}\Big{)}\Big{(}\mathbf{K_{s}} ^{T}\mathbf{K_{s}}\Big{)}(\mathbf{F}-\mathbf{K_{me}}\mathbf{U})=\mathbf{K_{s}}(\mathbf{F}-\mathbf{K_{me}} \mathbf{U})\Big{(}\delta^{D}-\delta^{I}\Big{)}+\Big{(}\delta^{D}-\delta^{I}\Big{)} ^{T}\Big{(}\delta^{D}-\delta^{I}\Big{)}. \tag{34}\]  
Equation (34) can be further simplified as  
\[(\mathbf{U})=\mathbf{U}^{T}\mathbf{A}\mathbf{U}-\mathbf{b}^{T}\mathbf{U}+c, \tag{35}\]  
where the parameter matrices are  
\[\mathbf{A}=\mathbf{K_{me}}^{T}\Big{(}\mathbf{K_{s}}^{T}\mathbf{K_{s}}\Big{)}\mathbf{K_{me}}, \tag{36}\]  
\[\mathbf{b}^{T}=2\Big{(}\delta^{D}-\delta^{I}\Big{)}^{T}\mathbf{K_{s}}\mathbf{K_{me}}+2\bm {F}^{T}\Big{(}\mathbf{K_{s}}^{T}\mathbf{K_{s}}\Big{)}\mathbf{K_{me}}, \tag{37}\]  
\[\delta_{i}^{Deformed}=\delta_{i}^{Initial}+\
abla\delta_{i}, \tag{30}\]0  
\(\mathbf{A}\) is an \(r\)-by-\(r\) positive definite matrix; \(\mathbf{b}\) is an r-dimensional vector; and c is a constant.