# A Refined Shape Sensing Method for Skin Antenna Structure Based on Inverse Finite Element Method
## 2 Inverse Finite Element Formulation  
Consider a plate with thickness of \(2t\) as the analytical model, and the structure can be described by the general Cartesian coordinate system, as shown in Figure 1.  
According to the Mindlin plate theory, the three-directional displacements relevant to arbitrary points can be expressed [13]. \[u_{x}(x,y,z) =u(x,y)+z\theta_{y}(x,y) \tag{1}\] \[u_{y}(x,y,z) =v(x,y)+z\theta_{x}(x,y)\] (2) \[u_{z}(x,y,z) =w(x,y) \tag{3}\]  
where, \(u_{x},u_{y}\) are the plane displacements and \(u_{z}\) is the transverse displacement across the thickness orientation; \(\theta_{x},\theta_{y}\) are rotations around the x- and y-axes, respectively; and z is the distance from the selected points to mid-plane of the structure. The linear strain components are expressed by appropriate derivatives of the displacements. \[ \begin{split} E&=\{e_{1},e_{2},e_{3}\}^{T}=\left\{u _{x,x},v_{y,y},u_{x,y}+v_{y,x}\right\}^{T}\\ B&=\{e_{4},e_{5},e_{6}\}^{T}=\left\{\theta_{x,y}, \theta_{y,x},\theta_{x,x}+\theta_{y,y}\right\}^{T}\\ G&=\{e_{7},e_{8}\}^{T}=\left\{w_{z,x}+\theta_{y}, w_{z,y}+\theta_{x}\right\}^{T}\end{split}  \tag{4}\]  
where, E, B, G indicates membrane, bending curvatures, and transverse-shear strain measures, respectively. According to the strain relation between any point and three main directions from material mechanics, the strain at any point of structure can be expressed. \[\varepsilon(x_{i},y_{i},\theta)=\varepsilon_{x}\cos^{2}\theta+\varepsilon_{y} \sin^{2}\theta-\gamma_{xy}\sin\theta\cos\theta \tag{5}\]  
Based on the functional theory, the Euclidean distance relationship between the measured strain and the theoretical strain can be expressed. \[ \begin{array}{rl}\phi(u)&=\sum\limits_{i=1}^{n}w_{i}(\left\|\varepsilon(x _{i},y_{i},\theta_{i})-\varepsilon^{\varepsilon}(x_{i},y_{i},\theta_{i}) \right\|^{2}+\left\|\varepsilon(x_{i},z_{i},\theta_{i})-\varepsilon^{ \varepsilon}(x_{i},z_{i},\theta_{i})\right\|^{2}\\ &+\left\|\varepsilon(y_{i},z_{i},\theta_{i})-\varepsilon^{\varepsilon}(y_{i },z_{i},\theta_{i})\right\|^{2})\end{array}  \tag{6}\]  
The \(\varepsilon^{\varepsilon}\) is the measured strain from sensor, \(\varepsilon\) is the theoretical strains associated with location, which is pasted angle along axis (x, y, z), respectively, and \(n\) denotes the numbers of sensors, \(w_{i}\) is the weighting constants controlling the completeness between the analytic strains and their experimentally measured values. Based on interpolation method, the deformation of points in the element can be expressed linearly by the shape function of element and the deformation of nodes. The result of any points in element can be expressed as follows:  
\[\{u,v,w\}=\sum\limits_{i=1}^{m}N_{i}u_{i}^{\varepsilon},(i=1,2,\ldots m) \tag{7}\]  
Figure 1: Plate model notion. where, \(N_{i}\) indicates the interpolation shape function of element, \(m\) is the numbers of element nodes, and \(u_{i}^{e}\) is the degree of freedom of element node expressed by nodal vector. \[u_{i}^{e}=\left[ \begin{array}{cccccc}u_{i}&v_{i}&w_{i}&\theta_{xi}&\theta_{yi}& \theta_{zi}\end{array} \right]^{T},i=(1,2,\ldots m) \tag{8}\]  
Introducing Equations (4) and (5) and into Equation (6), and taking the minimum value of it about node degrees of freedom, gives rise to the function of shape sensing simultaneously. \[k^{e}u^{e}=f^{e} \tag{9}\]  
where, \(k^{e}\) is the element matrix related to only the location layout of strain sensors and the \(f^{e}\) vector is a function of the number of strain sensors in the element as well as the measured angle values. These can be explicitly written, respectively, as  
\[k^{e}=w_{i}\sum_{i=1}^{n}\left(E^{e}+z\cdot B^{e}\right)^{T}m_{i}^{T}(E+z\cdot B ^{e})+r\iint\left(G^{e}\right)^{T}G^{e}dxdy \tag{10}\]  
\[f^{e}=w_{i}\sum_{i=1}^{n}\left(E^{e}+z\cdot B^{e}\right)^{T}m_{i}^{T}\cdot \epsilon_{i}^{e} \tag{11}\]  
where, \(E^{e},B^{e},G^{e}\) can be listed by Appendix A, \(m_{i}\) is the angle cosine vector, and \(r\) is usually assigned a minimum value [13].