## 2 The Inverse Finite Element Method
In elements, where the experimental strain measures are unknown (due to the absence of experimental data), the coefficient vectors are assigned a relatively small value (\(\mathbf{w}_{e}=\mathbf{w}_{k}=\left\{10^{-4},\;10^{-4},\;10^{-4}\right\}\), \(\mathbf{w}_{g}=\left\{10^{-4},\;10^{-4}\right\}\)), and the corresponding squared norms reduce to  
\[ \begin{split}\left|\left|\mathbf{e}(\mathbf{u}^{\epsilon})- \mathbf{e}^{\epsilon}\right|\right|_{2}&=\frac{1}{A_{e}}\int_{A _{e}}\mathbf{e}(\mathbf{u}^{\epsilon})^{2}dA\;,\\ \left|\left|\mathbf{k}(\mathbf{u}^{\epsilon})-\mathbf{k}^{ \epsilon}\right|\right|&=\frac{(2t)^{2}}{A_{e}}\int_{A_{e}} \mathbf{k}(\mathbf{u}^{\epsilon})^{2}dA\;,\\ \left|\left|\mathbf{g}(\mathbf{u}^{\epsilon})-\mathbf{g}^{ \epsilon}\right|\right|&=\frac{1}{A_{e}}\int_{A_{e}}\mathbf{g}( \mathbf{u}^{\epsilon})^{2}dA.\end{split}  \tag{10}\]  
In cases where only specific components of membrane, bending, or transverse shear strains are experimentally measured at a point, two different strategies can be used for calculating the error functional of Equation (8). The first approach uses a suitable definition of the corresponding weighting coefficient vector. When using uniaxial strain sensors, where only specific components of the in-plane strains are measured at a point, the weighting coefficient vector for the membrane squared norms, \(\mathbf{w}_{e}\), is defined accordingly. For example, if the in-plane strain along the \(x\)-axis (\(\epsilon_{x0}^{\epsilon}\)) is the only strain component measured at a point. Then the corresponding weighting coefficient vector could be set as, \(\mathbf{w}_{e}=\left\{1,\;10^{-4},\;10^{-4}\right\}\). The use of a small value for the weighting coefficient reduces the contribution of those unknown strain components to the element error functional. The second approach employs a smoothing technique. The smoothing technique uses the existing \(\epsilon_{x0}^{\epsilon}\) strain measurements to obtain a smoothed value of \(\epsilon_{x0}^{\epsilon}\) at points with no strain data. Using this approach, all unknown components of the membrane, bending or transverse shear strain measures, not experimentally measured at a point, can be obtained by smoothing strain data from other measurement locations of the plate. The latter approach is used in the current work; a detailed explanation of the steps involved is discussed in Section 3. The transverse shear strain measure, \(\mathbf{g}^{\epsilon}\), cannot be obtained directly using experimental strain measurements. Hence, the transverse shear squared norm (defined in Equation (10)) is associated with a small value of the corresponding weighting coefficient vector (\(\mathbf{w}_{g}=\left\{10^{-4},\;10^{-4}\right\}\)) for all elements. The error functional, \(\Phi_{e}\), is solved by minimizing with respect to the nodal DOF, yielding a set of linear algebraic equations  
\[\frac{\partial\Phi_{e}(\mathbf{u}^{\epsilon})}{\partial\mathbf{u}^{\epsilon} }=\mathbf{k}^{\epsilon}\mathbf{u}^{\epsilon}-\mathbf{f}^{\epsilon}=0\;, \tag{11}\]  
where the matrix, \(\mathbf{k}^{\epsilon}\), and vector, \(\mathbf{f}^{\epsilon}\), are functions of the strain sensor positions and measured surface strain data, respectively. Both \(\mathbf{k}^{\epsilon}\) and \(\mathbf{f}^{\epsilon}\) can be expanded and given as  
\[ \begin{split}\mathbf{k}^{\epsilon}&=\frac{1}{A_{e}} \int_{A_{e}}\left[\mathbf{w}_{e}(\mathbf{B}^{m})^{T}\mathbf{B}^{m}+\mathbf{w} _{k}(2t)^{2}(\mathbf{B}^{b})^{T}\mathbf{B}^{b}+\mathbf{w}_{g}(\mathbf{B}^{s}) ^{T}\mathbf{B}^{s}\right]dA,\\ \mathbf{f}^{\epsilon}&=\frac{1}{A_{e}}\int_{A_{e}} \left[\mathbf{w}_{e}(\mathbf{B}^{m})^{T}\mathbf{e}^{\epsilon}+\mathbf{w}_{k} (2t)^{2}(\mathbf{B}^{b})^{T}\mathbf{k}^{\epsilon}+\mathbf{w}_{g}(\mathbf{B}^ {s})^{T}\mathbf{g}^{\epsilon}\right]dA.\end{split}  \tag{12}\]  
Since the above terms involve area integrals, a suitable numerical integration scheme was used.