## 2 The Inverse Finite Element Method
\begin{array} []{l}\varepsilon_{xx}^{+}\\ \varepsilon_{yy}^{+}\\ \gamma_{xy}^{+}\end{array} \right\}_{j}+\left\{ \begin{array}{l}\varepsilon _{xx}^{-}\\ \varepsilon_{yy}^{-}\\ \gamma_{xy}^{-}\end{array} \right\}_{j}\right),\,j=1,\ldots,N, \tag{6}\]  
\[\mathbf{k}_{j}^{e}=\left\{ \begin{array}{l}\kappa_{x0}^{e}\\ \kappa_{y0}^{e}\\ \kappa_{xy0}^{e}\end{array} \right\}_{j}=\frac{1}{2i}\left(\left\{ \begin{array} []{l}\varepsilon_{xx}^{+}\\ \varepsilon_{yy}^{+}\\ \gamma_{xy}^{+}\end{array} \right\}_{j}-\left\{ \begin{array}{l}\varepsilon _{xx}^{-}\\ \varepsilon_{yy}^{-}\\ \gamma_{xy}^{-}\end{array} \right\}_{j}\right),\,j=1,\ldots,N, \tag{7}\]  
where \(\left\{\varepsilon_{xx}^{+},\ \varepsilon_{yy}^{+},\ \gamma_{xy}^{+}\right\}\) and \(\left\{\varepsilon_{xx}^{-},\ \varepsilon_{yy}^{-},\ \gamma_{xy}^{-}\right\}\), are the in-plane normal and shear strains measured on the top and bottom surface of the structure, respectively, and \(N\) refers to the total number of strain-sensor locations corresponding to the mid-plane coordinates (\(x_{j},\ y_{j}\)). The iFEM variational formulation is based on a weighted-least squares error functional that minimizes the least-square errors between the analytic and experimental strain measures. The structural domain is discretized using the customary finite element framework, and the individual inverse finite elements are formulated on the basis of the error functional, given as  
\[\Phi_{e}(\mathbf{u}^{e})=\mathbf{w}_{e}\big{|}\big{|}\mathbf{e}(\mathbf{u}^{e })-\mathbf{e}^{\epsilon}\big{|}\big{|}_{2}+\mathbf{w}_{k}\big{|}\big{|}\mathbf{ k}(\mathbf{u}^{e})-\mathbf{k}^{\epsilon}\big{|}\big{|}_{2}+\mathbf{w}_{g} \big{|}\big{|}\mathbf{g}(\mathbf{u}^{e})-\mathbf{g}^{\epsilon}\big{|}\big{|} _{2}, \tag{8}\]  
where \(\mathbf{w}_{e},\ \mathbf{w}_{k}\) and \(\mathbf{w}_{g}\) are vectors of weighting coefficients associated with the squared norms. The weighting coefficients are used to enforce a stronger or weaker correlation between the experimental strain measures and those described analytically. In elements where experimental strain measures are known, the coefficient vectors were assigned a value of unity (\(\mathbf{w}_{e}=\mathbf{w}_{k}=\{1,\ 1,\ 1\},\mathbf{w}_{g}=\{1,\ 1\}\)), and the squared norms are expressed as  
\[ \begin{split}&\Big{|}\big{|}\mathbf{e}(\mathbf{u}^{e})-\mathbf{e}^{ \epsilon}\big{|}\big{|}_{2}=\frac{1}{\ell_{\mathcal{A}}}\int_{\mathcal{A}_{e}} \big{[}\mathbf{e}(\mathbf{u}^{e})-\mathbf{e}^{\epsilon}\big{]}^{2}dA,\\ &\Big{|}\big{|}\mathbf{k}(\mathbf{u}^{e})-\mathbf{k}^{\epsilon} \Big{|}\Big{|}_{2}=\frac{(2\ell)^{2}}{A_{e}}\int_{A_{e}}\big{[}\mathbf{k}( \mathbf{u}^{e})-\mathbf{k}^{\epsilon}\big{]}^{2}dA,\\ &\Big{|}\big{|}\mathbf{g}(\mathbf{u}^{e})-\mathbf{g}^{\epsilon} \Big{|}\Big{|}_{2}=\frac{1}{\ell_{\mathcal{A}}}\int_{A_{e}}\big{[}\mathbf{g}( \mathbf{u}^{e})-\mathbf{g}^{\epsilon}\big{]}^{2}dA,\end{split}  \tag{9}\]  
where \(A_{e}\) is the element area. In specific situations of a pure membrane or bending behavior, Equations (6) and (7) can be simplified further. When the plate is only under in-plane loads, the in-plane normal and shear strains on the top and bottom surfaces of the plate are equal. Strain measurements on only one surface of the plate are required to calculate \(\mathbf{e}^{e}\) using Equation (6). In this case, the bending curvatures \(\mathbf{k}^{\epsilon}\) in Equation (7) vanish identically. Similarly, when the plate is under pure bending, the strains on the top and bottom surfaceshave the same magnitude with opposing signs (tension and compression). Measurements on only one surface of the plate are required to calculate \(\mathbf{k}^{\epsilon}\), whereas the membrane strain measures \(\mathbf{e}^{\epsilon}\) vanish identically.