About solution of multipoint boundary problem of static analysis of deep beam with the use of combined application of finite element method and discrete-continual finite element method. part 1: formulation of the problem and general principles of approximation

. This paper is devoted to formulation and general principles of approximation of multipoint boundary problem of static analysis of deep beam with the use of combined application of finite element method (FEM) discrete-continual finite element method (DCFEM). The field of application of DCFEM comprises structures with regular physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element approximation for non-basic dimension while in the basic dimension problem remains continual. DCFEM is based on analytical solutions of resulting multipoint boundary problems for systems of ordinary differential equations with piecewise-constant coefficients

So-called approximation parameter k U can be introduced in accordance with the following rule : 1 k U for approximation with the use of FEM ; 2 k U for approximation with the use of DCFEM .

About numbering of subdomains
Various approaches can be used for numbering of subdomains. The first approach provides separate numbering of subdomains with different types of approximation: is the corresponding number of subdomain with approximation with the use of DCFEM.
Various approaches can be used for numbering of subdomains. We can certainly construct inverse relationships by tabulating the results of calculations using formulas (3).
Thus we can rewrite (2) in the following form: The second approach, on the contrary, is based on a linked numbering of subdomains with different types of approximation. Formula (2) can be used, wherē while formulas (3) and (4) are not required. The second approach is used in this paper.

About numbering of finite elements and discrete-continual finite elements
Let's consider arbitrary subdomain fe k Z . We can introduce notation Z . Twoindex notation system is used for numbering of discrete-continual finite elements, which are used for approximation of dc k Z . Typical number of has the form ) , ( i k , where k is the number of subdomain, i is the number of element (along 1 x ).
We can introduce notation It should be noted that in the simples cases (such case in considered in the distinctive paper) discretization of structure is constant along 1 x throughout the domain (otherwise the mathematical constructions given below are substantially more complicated). We have 1 , ... 2, , 1 , In should be noted that notation from [3][4][5]30] is also used in this paper.

Discrete (finite element) approximation model for subdomain
Let's consider arbitrary subdomain fe k Z . Discrete (finite element) approximation model for the considering two-dimensional problems presupposes finite element approximation along 1 x and 2 x . Thus extended subdomain fe k Z is divided into finite elements, Lame constants for finite element are defined by formulas: . Bilinear approximation of unknowns is used within finite element (conventional plane rectangular 4-node finite element of two-dimensional problem of elasticity theory (Fig. 2)).

Fig. 2. Finite element and its local coordinate system.
Computing of partial derivatives of displacements, deformations and stresses within the finite element, nodal stresses and nodal deformations with allowance for averaging is described in [3][4][5]30].
As known, FEM is reduced to the solution of systems of where k U is global vector of nodal unknowns (subscript " ) (k " corresponds to the number of subdomain . Linear approximation is used for unknown functions within discrete-continual finite element.
DCFEM is reduced at some stage to the solution of systems of 1 4N first-order ordinary differential equations: where k U is global vector of nodal unknowns (subscript " ) (k " corresponds to the number of subdomain (23) , ) ( 2 x k H is the fundamental matrix-function of system (20), which is constructed in the special form convenient for problems of structural mechanics [1][2][3][4][5]35]; is convolution notation; k C is the vector of constants of order 1 4N .

Software and verification samples
We should stress that all methods and algorithms considered in this paper have been realized in software. The main purpose of Analysis system CSASA2Dm (DCFEM + FEM) is semianalytical structural analysis (static structural analysis of deep beam within twodimensional theory of elasticity), based on combined application of FEM and DCFEM. Programming environment is Microsoft Visual Studio 2013 Community and Intel Parallel Studio 2017XE (Fortran programming language [16]) with Intel MKL Library [8]. Software is designed for Microsoft Windows 8.1/10.