Combined Semianalytical and Numerical Static Plate Analysis. Part 1: Formulation of the Problem and Approximation Models

. The distinctive paper is devoted to solution of multipoint (particularly, two-point) boundary problem of plate analysis (Kirchhoff model) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). As is known the Kirchhoff-Love theory of plates is a two-dimensional mathematical model that is normally used to determine the stresses and deformations in thin plates subjected to forces and moments. The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. FEM is used for approximation of all other subdomains (it is convenient to solve plate bending problems in terms of displacements). Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains are under consideration.


Introduction
As is known the Kirchhoff-Love plate theory extends the Euler / Bernoulli beam assumptions to the two-dimensional case. Every significant measure of rotation, force, moment is evaluated with respect to the vertical deflection. The main kinematic assumption is that plane surfaces remain plane and perpendicular to the mid-surface of the plate [7,10,15,17,18,21,26,34,38]. Thus the Kirchhoff-Love theory predicts a zero distribution of shear stresses along the thickness. Therefor it can only be applied in problems where the variation of such stresses is expected to be small and their mean value does not deviate from 0. Such can be considered the case of thin plates [12,23,25,27,28,32,37,42]. The distinctive paper is devoted to combined semianalytical and numerical static plate analysis. Solution of multipoint (particularly, two-point) boundary problem of plate analysis, based on combined application of finite element method (FEM) [11,13,14,19,22,24,36,39,40,41,43] and discrete-continual finite element method (DCFEM) [1,[3][4][5]29,30,35,46] is under consideration. The given domain, occupied by considering structure, is embordered by extended one within so-called method of extended domain [2]. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension ("basic" dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual (corresponding correct analytical solution is constructed). FEM is used for approximation of all other subdomains [6,8,9,16,31,33,44,45]. Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains are constructed. Besides, coupled multilevel approximation model for extended domain and brief information about software systems [20] and verification samples are presented in the second part of this paper.

Formulation of the problem and approximation
Let's consider multipoint (particularly, two-point) boundary problem of plate analysis within Kirchhoff model [12,23,25,27,28,32,37,42]. Several elements of corresponding notation system for corresponding two-point boundary problem are presented at figure 1. Let's  be domain occupied by structure, In accordance with the method of extended domain, proposed by Professor Alexander B. Zolotov [2], the given domain is embordered by extended one of arbitrary shape, particularly elementary. Let (2) Without loss of generality we suppose piecewise constancy of physical and geometrical parameters of one group of subdomains from (2) along coordinate 2 x ("basic" dimension). It is necessary to note that physical and geometrical parameters of structure can be changed arbitrarily along 1 x . Thus, it is recommended to use DCFEM for approximation of these subdomains (discrete-continual design model is introduced).
Let physical and geometrical parameters within other group of subdomains from (2) are arbitrary varying. FEM [44,45] can be used for approximation here. Combined application of DCFEM and FEM is advisable.
Operational formulation of the problem of plate analysis within Kirchhoff model and method of extended domain is used.
We can introduce the following notations for considering two-point boundary problem ( Fig. 1 is the number of discrete-continual finite elements; x ) of nodes of finite elements, which are used for approximation of are numbers of finite elements along 1 x and 2 x . Two-index notation system is used for numbering of discrete-continual finite elements. Typical number of has the form ) , ( i k , where k is the number of subdomain, i is the number of element (along 1 x ). Three-index system is used for numbering of finite elements. Typical number of has the form ) , , ( j i k , where k is the number of subdomain, i and j are numbers of elements (along 1 x and 2 x ). Let's

Approximation models for subdomains and domain
Discrete-continual approximation model within DCFEM presupposes mesh approximation for non-basic dimensions of extended domain (along 1 x ) while in the basic dimension (along 2 x ) problem remains continual. Thus extended subdomain 1  is divided into discrete-continual finite elements (3) Flexural rigidity, Poisson's ratio and bedding value for discrete-continual finite element are defined by formulas:  is the characteristic function of element i , 1  ; k h is thickness of plate; k Ẽ is the modulus of elasticity of material of plate. Let's 1 w be deflection of plate at subdomain 1  . Basic nodal unknown functions are the following functions:   (10) Correct analytical solution of (6) is defined by formula where is the fundamental matrix-function of system (5), which is constructed in the special form convenient for problems of structural mechanics [1];  is convolution notation; Flexural rigidity, Poisson's ratio and bedding value for finite element are defined by formulas: https://doi.org/10.1051/matecconf/201819601010 XXVII R-S-P Seminar 2018 , Theoretical Foundation of Civil Engineering   2  ,  ,  2  ,  ,  2  2  ,  ,  2  ,  ,  2  2  ,  ,  2  ,  ,  2   ; ; (14) where j i, , 2  is the characteristic function of element we have the following unknown functions: In other words, we find it convenient to use polynomials as form functions, which are defined by 12 coefficients (the fourth-order polynomials with several zero coefficients can be used). It should be noted that formula (17) has certain advantages. In particular, deflection ) , ( x x w with respect to normal to any boundary is described by third-order polynomial along this boundary (for instance, function ) , ( ). Since we have only two given values of deflection angles at these lines, the third-order polynomial is ambiguously determined and deflection angle may be discontinuous (i.e. continuity of the first-order derivatives at boundaries between several finite elements is not provided). We have so-called nonconforming form function and nonconforming finite elements.
We should introduce additional nodal basic unknown, i.e. nodal value of function (mixed derivative) ) , ( (18) in order to obtain conforming finite elements. Corresponding formula instead of (17) has the following form 3 where 2 K is global stiffness matrix of order 2 1 4 N N ; 2 R is global right-side vector of order 2 1 4 N N (global load vector); 2 U is global vector of nodal unknowns (subscript corresponds to the number of subdomain 2  ),