## 2 Literature Review of the Methods for Solving Plate Bending Problems
Division is defined by configuration of the area, solution typologies and mechanical properties of material. In each area approximate functions are taken as linear combinations of polynomial, trigonometric and exponential functions. For example, linear global or superelement has three degrees of freedom. Interpolation functions are taken in the form of second order polynomial. Boundary and continuity conditions at the common edges of the subareas are performed in manner of principle of virtual work. Fault of the method is difficulty of division of considered area into elements and selection of the functional for minimization. Global-local finite element method (GLFEM) [11] uses Ritz-Timoshenko method at the one part of the plate and finite element method at the another. Fault of this method is necessity of individual treatment of each problem and unstudied effectiveness. According to this method structure is first analyzed as a whole. Therefore, all the local details obtained from the global analysis are considered to be boundary conditions in local details analysis. The global-local analysis is especially useful in the stress concentration analysis, and so on. More information on global-local finite element method can be found in [44; 45; 46; 47; 48]. The extended finite element method (G/XFEM) [3; 49; 50; 51; 52; 53; 54; 55] is generalization of classical finite element method and it was especially developed for modeling structural problems with discontinuities. According to that generalized global approximation of the deflection of the plate is presented as [51]:  
\[\boldsymbol{\hat{u}}(\boldsymbol{x})=\underbrace{\sum_{j=1}^{N}\mathcal{N}_{j }(\boldsymbol{x})\boldsymbol{u}_{j}}_{\text{standard interpolation}}\quad+ \underbrace{\sum_{j=1}^{N}\mathcal{N}_{j}(\boldsymbol{x})\left(\sum_{i=2}^{ q}L_{ji}(\boldsymbol{x})\boldsymbol{b}_{ji}\right)}_{\text{enriched interpolation}} \tag{5}\]  
where \(\boldsymbol{u}_{j}\) is a nodal parameter associated with finite element shape functions \(\mathcal{N}_{j}(\boldsymbol{x})\), \(\boldsymbol{b}_{ji}\) are nodal parameters associated with G/XFEM shape functions--\(\mathcal{N}_{j}(\boldsymbol{x})\cdot L_{ji}(\boldsymbol{x})\). \(L_{ji}\) are so-called enrichments functions that hold the information about the problem solution. In the last ten years, there has been a growing interest in meshless methods [56; 57]. These methods do not demand of mesh as finite element ones. Area discretization is accomplished by a set of ordered or scattered nodes that are independent of the plate configuration. This approach does not cause difficulties in satisfying boundary conditions. The method is a simply, effective and attractive. Novel plate and shell analysis methods also include analysis of the Kirchhoff plate using rational Bezier triangles in isogeometric analysis coupled with a feature-preserving automatic meshing algorithm [58] and modeling using a NURBS-based isogeometric analysis (IGA) approach [59]. Some results obtained by analytical and numerical methods are given below. Rectangular uniformly loaded plate free supported at two opposite edges or rigidly clamped at the contour is considered in the paper [60]. To solution the problem [PERSON]'s method was used here. In each concrete case particular solution is taken separately. In similar manner the same plate free supported at the corners was considered in the paper [61]. The exact solution of orthotropic rectangular cantilever arbitrary loaded plate is obtained in [62]. Rectangular plate free supported at the contour was solved [63] using Bubnov-Galerkin variation method. Various kinds of outer load were considered. Using combination of functional series S. [PERSON] obtained original analytical solution of thin orthotropic plate rigidly clamped at the contour [8]. [PERSON] using spline approximation method obtained solution of parallelogram and trapeze plates [26]. Trefftz method boundary collocation was used to solve thin anisotropy plate of various geometry [29].