## 2 Materials and Methods
### Finite Difference in Variational Form  
The finite difference method was used to solve certain differential equation systems. Although it is a less frequently used method, it can be successfully used for static calculations alongside the finite element method. In the literature, there are a number of fundamental works on the method of finite differences [42; 43; 44; 45; 46]. The method is used in the calculations of plate structures [19; 57; 58], tanks [59; 60; 61; 62], and surface girders. It has been used and tested many times and gives satisfactory results in the theory of coatings and thin plates. Figure 1: Plate of the Hoff model (w–out of plane displacement, u and v–in plane displacements). The discretization of solutions for sandwich panels using the finite difference method in terms of variation consists of determining the minimum function of the elastic deformation energy of a bent three-layer panel. This function takes the form of (4) [1]:  
\[ \begin{array}{rl}V=\frac{1}{2}\iint_{A}\{2D&\left[\left(\frac{\partial^{2}w}{ \partial x^{2}}\right)^{2}+2v\frac{\partial w}{\partial x^{2}}\quad\frac{ \partial^{2}w}{\partial y^{2}}+\left(\frac{\partial^{2}w}{\partial y^{2}} \right)^{2}+2(1-v)\left(\frac{\partial^{2}w}{\partial x\partial y}\right)^{2} \right]\\ &+\frac{2 Eg}{1-v^{2}}\left(\frac{\partial u}{\partial x}\right)^{2}+2v\frac{ \partial u}{\partial x}\cdot\frac{\partial w}{\partial y}+\frac{1-v}{2}\left( \frac{\partial u}{\partial y}\right)^{2}+(1-v)\frac{\partial u}{\partial y} \cdot\frac{\partial w}{\partial x}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1-v}{2}\left(\frac{ \partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^ {2}\right]\\ &+2 hG_{w}[\frac{h^{2}}{H^{2}}+2\frac{h}{h^{2}}\cdot\frac{2h+\delta}{2h}\cdot \frac{2w}{\partial x}+\frac{(2h+\delta)^{2}}{4h^{4}}\left(\frac{\partial w}{ \partial x}\right)^{2}+\frac{v^{2}}{H^{2}}+2\frac{v}{H}^{2}\\ &\qquad\qquad\qquad\qquad\cdot\frac{2h+\delta}{2h}\cdot\frac{\partial w}{ \partial y}+\frac{(2h+\delta)^{2}}{4h^{4}}\left(\frac{\partial w}{\partial y} \right)^{2}\right]\}dxdy\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\iint_{A}(q_{z}w+q_{x}u +q_{y}v)dxdy.\end{array}  \tag{4}\]  
The derivatives in Equation (4) have been replaced with the appropriate differential schemes given e.g., in [7]. Through numerical integration, Function V was obtained as a function of several variables (5). \[V=V(w_{ik},u_{ik},v_{ik}), \tag{5}\]  
where \(w_{ik},\ u_{ik},\ and\ v_{ik}\) are components of the displacement vector in individual nodes of the adopted partition grid. The necessary condition for the existence of an extreme of functions of several variables is (6):  
\[\frac{\partial V}{\partial w_{ik}}=0,\ \frac{\partial V}{\partial u_{ik}}=0,\ \frac{\partial V}{\partial v_{ik}}=0. \tag{6}\]  
The sought displacements in the nodes of the grid are shown in Equation (5) in a quadratic form. From the dependence of (6), a symmetrical system of algebraic linear equations is obtained in order to determine the displacements, in which the number of equations is equal to the number of unknowns. The boundary conditions known from the theory of thin plates, such as a free edge, freely supported, or fixed, are difficult to transfer to sandwich panels due to ambiguity. In work [8], using the calculus of variations, it was shown that there are sixteen homogeneous natural boundary conditions for the three-layer Hoff model discussed in this work. The advantage of the finite difference method in the variational approach is that in the assumed system of unknowns, only the geometric boundary conditions should be met, and the static conditions are met naturally by the function itself. To write the second derivative of the deflection function \(w(x,\ y)\) in the finite difference method for the boundary point, it is necessary to extend beyond the edge by one interval of the partition grid, while for the first derivatives of \(u(x,\ y)\) and \(v(x,\ y)\) it is not necessary.