## 2 The SF-iFEM Methodology  
### Calculation of Element Theoretical Strain Based on the Element Strain Function  
According to the form of the stiffened ship panel, the four-node shell element (iQS4) is used to divide it [18], as shown in Figure 1. The strain vector at any point in the element is mainly composed of six strain components, which are the tensile and compressive strain \(\varepsilon_{X}\) in the \(X\) direction, the tensile and compressive strain \(\varepsilon_{Y}\) in the \(Y\) direction, the shear strain \(\gamma_{XY}\) in the \(XY\) plane, the bending strain \(K_{X}\) in the \(X\) direction, the bending strain \(K_{Y}\) in the \(Y\) direction, and the generalized bending strain \(K_{XY}\) in the \(XY\) plane, as shown in Equation (1).  
\[\varepsilon=\begin{bmatrix}\varepsilon_{X}&\varepsilon_{Y}&\gamma_{XY}&K_{X}&K _{Y}&K_{XY}\end{bmatrix}^{T} \tag{1}\]  
where \(X\), \(Y\), and \(Z\) denote three mutually perpendicular coordinate axis directions in the global Cartesian coordinate system.  
According to the classical finite element theory, when the structure only undergoes elastic deformation, there is a functional relationship between the strain component of any point in the element and the strain components of the four element nodes in the element, which can be defined as the 'element strain function', as shown in Equation (2).  
\[\left\{\begin{array}{ll}\varepsilon_{X}=\sum\limits_{i=1}^{4}N_{i}\varepsilon _{Xi}&\varepsilon_{Y}=\sum\limits_{i=1}^{4}N_{i}\varepsilon_{Yi}&K_{XY}=\sum \limits_{i=1}^{4}N_{i}K_{XYi}\\ K_{X}=\sum\limits_{i=1}^{4}N_{i}K_{Xi}&K_{Y}=\sum\limits_{i=1}^{4}N_{i}K_{Yi}& \gamma_{XY}=\sum\limits_{i=1}^{4}N_{i}\gamma_{XYi}\end{array}\right.\left\{ \begin{array}{ll}N_{1}=\frac{(1-\xi)(1+\eta)}{4}&N_{2}=\frac{(1-\xi)(1-\eta) }{4}\\ N_{3}=\frac{(1+\xi)(1-\eta)}{4}&N_{4}=\frac{(1+\xi)(1+\eta)}{4}\end{array}\right. \tag{2}\]  
where \(N_{i}\) represents the interpolation function corresponding to the \(i\)th element node within the four-node shell element [19]. \(\xi\) and \(\eta\) are the coordinate values in the element's local isoparametric coordinate system.  
Using the four-node element strain deformation function matrix \(C\) and the node strain vector \(\varepsilon_{Node}\), the element theoretical strain \(\varepsilon^{e}\) is expressed as:  
\[\varepsilon^{e}=\varepsilon^{e}+z_{0}K^{e}=C_{Node}^{m}\varepsilon_{Node}+z_ {0}C_{Node}^{b}\varepsilon_{Node} \tag{3}\]  
where \(\varepsilon^{e}\) is the tensile and compressive strain of the element theory, \(K^{e}\) is the bending strain of the element theory, \(Z_{0}\) is the distance between the plate surface and the neutral layer of the structure, \(C^{m}{}_{Node}\) is the tensile and compressive strain deformation function matrix of  
Figure 1: Four-node shell element for ship stiffened panel division.  
the element node, \(C^{b}_{Node}\) is the bending strain deformation function matrix of the element node, \(\epsilon_{Node}\) is the node strain vector, as shown in Equation (4).  
\[\left\{\begin{array}{l}\varepsilon_{Node}=\left[\begin{array}{cccccccc} \varepsilon_{Node1}&\varepsilon_{Node2}&\varepsilon_{Node3}&\varepsilon_{Node4} \end{array}\right]^{T}\\ \varepsilon_{Node\ i}=\left[\begin{array}{cccccccc}\varepsilon_{Xi}& \varepsilon_{Yi}&\gamma_{XYi}&K_{Xi}&K_{Yi}&K_{XYi}\end{array}\right]^{T}\\ \end{array}\right.\,\ \left\{\begin{array}{l}C^{m}_{Node}=diag\Big{(}\begin{array}{cccccccc} M_{1}&0&M_{2}&0&M_{3}&0&M_{4}&0\\ C^{b}_{Node}=diag\Big{(}\begin{array}{cccccccc}0&M_{1}&0&M_{2}&0&M_{3}&0&M_{4} \end{array}\end{array}\Big{)}\\ M_{i}=diag\Big{(}\begin{array}{cccccccc}N_{i}&N_{i}&N_{i}\end{array}\Big{)} \end{array}\right. \tag{4}\]