## 2 The Inverse Finite Element Method  
The iFEM formulation for plate/shell structures is based on the kinematic assumptions of Mindlin plate theory [21; 22]. For a plate of thickness, \(2t\), lying in a cartesian coordinate system, the components of the displacement vector are expressed as  
\[ \begin{array}{l}u_{x}(x,y,z)=u+z\theta_{y}\;,\\ u_{y}(x,y,z)=v-z\theta_{x}\;,\\ u_{z}(x,y,z)=w\;,\end{array}  \tag{1}\]  
where \(u,\ v,\ w,\ \theta_{x}\), and \(\theta_{y}\) are the kinematic variables associated with the mid-plane of the plate and are used to describe the displacement vector at any point of the structure. Variables \(u,\ v\), and \(w\) are average displacements along the \(x,\ y\), and \(z\) axis, respectively; \(\theta_{x},\ \theta_{y}\) are the bending rotations about the \(x\) and \(y\) axis, respectively (see Figure 1). The linear strain displacement relations are used to obtain the strain field from the displacement assumptions of Equation (1)  
\[\left\{ \begin{array}{c}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xy}\end{array} \right\}=\left\{ \begin{array}{c}\varepsilon_{x0}\\ \varepsilon_{y0}\\ \gamma_{xy0}\end{array} \right\}+z\left\{ \begin{array}{c}\kappa_{x0}\\ \kappa_{y0}\\ \kappa_{xy0}\end{array} \right\}\equiv\mathbf{e}(\mathbf{u})+z\mathbf{k}( \mathbf{u})\;. \tag{2}\]  
The strain field of Equation (2) is represented by six strain measures; three membrane strain measures, \(\mathbf{e}(\mathbf{u})\), representing the in-plane stretching of the mid-plane, and three bending strain measures, \(\mathbf{k}(\mathbf{u})\), representing the bending and twisting curvatures. Additionally, [PERSON] theory gives rise to two transverse shear strain measures, \(\mathbf{g}(\mathbf{u})\). The eight strain measures are given as  
\[ \begin{array}{l}\mathbf{e}(\mathbf{u})=\left\{\varepsilon_{x0},\ \varepsilon_{y0},\ \gamma_{xy0}\right\}^{T}=\left\{u_{x,}\ v_{y,}\ u_{y}+v_{x}\right\}^{T},\\ \mathbf{k}(\mathbf{u})=\left\{\kappa_{x0},\ \kappa_{y0},\ \kappa_{xy0}\right\}^{T}= \left\{\theta_{y,x},\ -\theta_{x,y},\ -\theta_{x,x}+\theta_{y,y}\right\}^{T},\\ \mathbf{g}(\mathbf{u})=\left\{\gamma_{xx0},\ \gamma_{yx0}\right\}^{T}= \left\{w_{x}+\theta_{y},\ w_{y,}-\theta_{x}\right\}^{T}.\end{array}  \tag{3}\]  
Throughout this study, a four-node plate/shell element, iQS4 [25], is used. The element is formulated using a set of anisoparametric C\({}^{0}\)-continuous shape functions that enabled improved kinematic interdependency between the bending and shear deforma  
Figure 1: Sign conventions used for the kinematic variables of a 4-node quadrilateral inverse element. tions. Using the element shape functions, the strain measures can be expressed in terms of the element nodal degrees-of-freedom (DOF)  
\[\left\{ \begin{array}{l}\varepsilon_{xx}\\ \varepsilon_{yy}\\ \gamma_{xy}\end{array} \right\}\equiv\mathbf{e}(\mathbf{u}^{c})+z\mathbf{k}( \mathbf{u}^{c})=\mathbf{B}^{m}\mathbf{u}^{c}+z\mathbf{B}^{b}\mathbf{u}^{c}, \left\{ \begin{array}{l}\gamma_{xz}\\ \gamma_{yz}\end{array} \right\}\equiv\mathbf{g}(\mathbf{u}^{c})=\mathbf{B}^{s} \mathbf{u}^{c}, \tag{4}\]  
where \(\mathbf{B}^{m},\ \mathbf{B}^{b}\) and \(\mathbf{B}^{s}\) are matrices of shape function derivatives corresponding to the membrane, bending, and transverse shear strains, respectively. The vector, \(\mathbf{u}^{c}\), representing the element DOF of each element, \(e\), can be expressed as  
\[\mathbf{u}^{e}=[\mathbf{u}_{1}^{e}\ \mathbf{u}_{2}^{e}\ldots\mathbf{u}_{n}^{e}]^ {T}, \tag{5}\]  
where \(\mathbf{u}_{i}^{e}\) denotes the vector of nodal DOF for each node \(i\), and \(n\) denotes the total number of nodes of the element. The membrane and bending strain measures, \(\mathbf{e}_{j}^{e}\) and \(\mathbf{k}_{j}^{e}\), corresponding to experimental surface strain measurements at any location (\(x_{j},\ y_{j}\)), are evaluated as  
\[\mathbf{e}_{j}^{e}=\left\{ \begin{array}{l}\varepsilon_{x0}^{e}\\ \varepsilon_{y0}^{e}\\ \gamma_{xy0}^{e}\end{array} \right\}_{j}=\frac{1}{2}\left(\left\{