## 3 Materials and Methods
Then functions \(f_{kps}(x_{s})\) take the form [69]  
\[f_{kps}(x_{s})=R_{kps\
u}B_{kps\
u}(x_{s}), \tag{15}\]  
In the above  
\[B_{kps1}(x_{s})=\frac{\cosh\left(\kappa_{kp(3-s)}x_{s}\right)}{\exp\left(\kappa _{kp(3-s)}a_{s}\right)},\quad B_{kps2}(x_{s})=\frac{x_{s}\sinh\left(\kappa_{ kp(3-s)}x_{s}\right)}{a_{s}\exp\left(\kappa_{kp(3-s)}a_{s}\right)}, \tag{16}\]  
Functions \(B_{kps\
u}(x_{s})\) are called the basic functions of the solution. According to (10) general solution of Equation (2) takes the form  
\[w_{0}(x_{1},x_{2})=R_{kpsv}\cdot W_{kpsv}(x_{1},x_{2}). \tag{17}\]  
Functions  
\[W_{kpsv}(x_{1},x_{2})=B_{kpsv}(x_{s})\cdot T_{kp(3-s)}(x_{3-s}) \tag{18}\]  
are dependent on the solution of the problem and are called _shape functions_ of the plate deflection. General solution of Equation (1) we obtain substituting (17) into expression (6)  
\[w(x_{1},x_{2})=R_{kpsv}W_{kpsv}(x_{1},x_{2})+W_{*}(x_{1},x_{2}). \tag{19}\]  
Here \(W_{*}(x_{1},x_{2})=w_{*}(x_{1},x_{2})\) is particular solution of equation (1). Functions \(W_{*}(x_{1},x_{2})\) can be present in the form:  
\[W_{*}(x_{1},x_{2})=C_{mnpq}\cdot W_{mnpq}(x_{1},x_{2}). \tag{20}\]  
Functions \(W_{mnpq}(x_{1},x_{2})\) are dependent on outer load and called _force functions_ of the plate deflection. For each specific problem they can be obtain using special procedures. With the deflection of the plate \(w\) and using relations known from theory of thin isotropic plates, we obtain the expressions for tangent displacements [71, 72]  
\[ \begin{split} u_{1}(x_{1},x_{2})&=R_{kpsv}U_{kpsv}(x _{1},x_{2})+U_{*}(x_{1},x_{2}),\\ u_{2}(x_{1},x_{2})&=R_{kpsv}V_{kpsv}(x_{1},x_{2})+V_ {*}(x_{1},x_{2}),\end{split}  \tag{21}\]  
moments  
\[ \begin{split} M_{11}(x_{1},x_{2})&=R_{kpsv}X_{kpsv}(x _{1},x_{2})+X_{*}(x_{1},x_{2}),\\ M_{22}(x_{1},x_{2})&=R_{kpsv}\chi_{kpsv}(x_{1},x_{2} )+Y_{*}(x_{1},x_{2}),\\ M_{12}(x_{1},x_{2})&=R_{kpsv}Z_{kpsv}(x_{1},x_{2} )+Z_{*}(x_{1},x_{2}),\end{split}  \tag{22}\]  
shearing forces  
\[ \begin{split} Q_{1}(x_{1},x_{2})&=R_{kpsv}H_{kpsv}(x _{1},x_{2})+H_{*}(x_{1},x_{2}),\\ Q_{2}(x_{1},x_{2})&=R_{kpsv}\chi_{kpsv}(x_{1},x_{2} )+G_{*}(x_{1},x_{2}),\end{split}  \tag{23}\]  
and generalized shearing forces  
\[ \begin{split} V_{1}(x_{1},x_{2})&=R_{kpsv}K_{kpsv}(x _{1},x_{2})+K_{*}(x_{1},x_{2}),\\ V_{2}(x_{1},x_{2})&=R_{kpsv}L_{kpsv}(x_{1},x_{2} )+L_{*}(x_{1},x_{2}).\end{split}  \tag{24}\]  
In the above functions \(U_{kpsv}\), \(V_{kpsv}\), and so on are partial derivatives of shape functions \(W_{kpsv}\). Similarly functions \(U_{*}\), \(V_{*}\), etc. are partial derivatives of force functions \(W_{*}\). They have been calculated using automatic differentiation [19, 74]. Expressions (19)-(24) constitute mathematical model of considered plate. According to this model plate is in internal equilibrium with outer load \(q(x_{1},x_{2})\), but boundary conditions are not satisfied here. Unknown coefficients \(R_{kpsv}\) are degrees of freedom of the plate. They are determined satisfying boundary conditions at the separate points of the contour. Let us consider technique of their generation. Due to the symmetry of the plate, we can consider only its quarter. At first, we introduce two sets \(X_{1}\) and \(X_{2}\) of points \((x_{1m},x_{2n})\) with numbers \((m,n)\) placed at the positive semi axes of the plate and called them initial points. Projections of these points onto the contour \(C\) are called _current nodes_ and designated \(K_{rm}\) and \(L_{rn}\) correspondingly, where \(r\) is number of the edge (Figure 1b). It is seen that edges parallel to coordinateaxes contain current nodes generated by either set \(X_{1}\) or set \(X_{2}\) but nodes placed at the inclined edge are generated as set \(X_{1}\) so \(X_{2}\). If two initial points of set \(X_{1}\) and \(X_{2}\) fall into one node at the inclined edge we have double current node. Projections of these points into the vertex \(V_{i}\), with number \(i\) we call _corner nodes_ and designated \(\overline{K}_{ir}\), \(\overline{L}_{ir}\)[69].