## 2 The SF-iFEM Methodology
### Strain Field Reconstruction Based on Node Strain Vectors  
Based on the previously outlined data, a least squares error function is formulated to establish the relationship between the theoretical and measured strains of the elements, as shown in Equation (8).  
\[\Phi(\varepsilon_{Node})=\left\|e^{\varepsilon}-e^{\varepsilon}\right\|^{2}+ \left\|K^{\varepsilon}-K^{\varepsilon}\right\|^{2} \tag{8}\]  
The error function \(\Phi(\varepsilon_{Node})\) is used to derive the partial derivative of the element node strain vector \(\varepsilon_{Node}\), so that the function value of the error function is minimized, and the structural strain pseudo-stiffness equation can be obtained, as shown in Equation (9).  
\[\frac{\partial\Phi(\varepsilon_{Node})}{\partial\varepsilon_{Node}}=K^{ \varepsilon}_{\varepsilon}\varepsilon_{Node}-F^{\varepsilon}_{\varepsilon}=0 \tag{9}\]  
where \(K^{\varepsilon}_{\varepsilon}\) is the structural strain pseudo-stiffness matrix, and \(F^{\varepsilon}_{\varepsilon}\) is the structural strain pseudo-load matrix, as shown in Equation (10).  
\[\left\{\begin{array}{l}K^{\varepsilon}_{e}=\int\limits_{A_{e}}\left[\left( C^{m}_{Node}\right)^{T}\!C^{m}_{Node}+4Z^{2}_{0}\!\left(C^{b}_{Node}\right)^{T}\!C^{b}_{ Node}\right]dA_{e}\\ F^{\varepsilon}_{\varepsilon}=\int\limits_{A_{e}}\left[\left(C^{m}_{Node} \right)^{T}\!e^{\varepsilon}+4Z^{2}_{0}\!\left(C^{b}_{Node}\right)^{T}\!K^{ \varepsilon}\right]dA_{e}\end{array}\right. \tag{10}\]  
where \(A_{e}\) is the area of the element.  
Finally, according to the initial element meshing scheme, the strain pseudo-stiffness equations for each element are assembled into the general equation of strain pseudo-stiffness [24], and the element node strain vector \(\varepsilon_{Node}\) of each element is directly solved. Combined with Equation (3), the strain component of any point in the structure is obtained; that is, the strain field information of the stiffened panel structure is reconstructed, as shown in Figure 3. This method addresses the challenge of low accuracy in reconstructing the strain field, stemming from errors in calculating the node displacement vector employed in the iFEM.  
Figure 3: Strain field reconstruction process for stiffened ship panels based on SF-iFEM.  
To assess the effectiveness of the proposed method in reconstructing the strain field, the simulated or experimentally measured strain serves as the reference value. Subsequently, calculations are conducted based on Equation (11) to determine the absolute error, relative error, root mean square error, and average relative error of the reconstructed strain. These metrics offer a comprehensive evaluation of the accuracy and performance of the strain field reconstruction method.  
\[\left\{\begin{aligned} AE(i)=\left|\varepsilon^{SF-iFEM}(i)- \varepsilon^{Ref}(i)\right|,&\quad RE(i)=100\%\times\left| \frac{\varepsilon^{Ref}(i)-\varepsilon^{SF-iFEM}(i)}{\varepsilon^{Ref}(i)} \right|\\ RMSE=\sqrt[2]{\frac{1}{n}\sum\limits_{i=1}^{n}\left(\varepsilon^{ SF-iFEM}(i)-\varepsilon^{Ref}(i)\right)^{2}},&\quad MRE=\frac{1}{n}\sum \limits_{i=1}^{n}RE(i)\end{aligned}\right. \tag{11}\]  
where \(AE(i)\) is the absolute error of strain reconstruction of the \(i\)th verification point, \(RMSE\) is the root mean square error of strain reconstruction results of n verification points, \(RE(i)\) is the relative error of strain reconstruction of the \(i\)th verification point, \(MRE\) is the average relative error of strain reconstruction results of n verification points, \(\varepsilon^{SF-iFEM}(i)\) is the reconstructed strain of the \(i\)th verification point, and \(\varepsilon^{Ref}(i)\) is the strain reference value of the \(i\)th verification point.