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 2: boundary conditions

. The formulation of considering multipoint boundary problem includes three main components: a description of the domain occupied by the structure and the corresponding subdomains; description of the conditions inside the domain and inside subdomains; description of boundary conditions (for boundaries of domain and boundaries between subdomains). These boundary conditions (interface conditions) in under consideration in the distinctive paper


Analysis of options for boundary conditions
In practical applications the following variants of boundary conditions (interface conditions) between subdomains (2) from the first part of this paper are most often encountered (some of them are considered in this paper): interface "discrete-continual model -discrete model", "internal" boundary condition of the type "perfect contact"; interface "discrete model -discrete-continual model", "internal" boundary condition of the type "perfect contact"; interface "discrete-continual model -discrete-continual model", "internal" boundary condition of the type "perfect contact"; interface "discrete model -discrete model", "internal" boundary condition of the type "perfect contact"; interface "external boundary -discrete model", boundary condition of the type "hinged support"; interface "external boundary -discrete model", boundary condition of the type "free edge"; interface "external boundary -discrete-continual model", boundary condition of the type "hinged support"; interface "external boundary -discrete-continual model", boundary condition of the type "free edge"; interface "discrete model -external boundary", boundary condition of the type "hinged support"; interface "discrete model -external boundary", boundary condition of the type "free edge"; interface "discrete-continual model -external boundary", boundary condition of the type "hinged support"; interface "discrete-continual model -external boundary, boundary condition of the type "free edge". Of course, other variants of interface are possible, but similarly, somehow, in one way or another, as a rule, these variants are reduced to some combinations of the above-mentioned twelve [6][7][8][9][10][11][12][13][14][15][16].
Equations (1)-(4) can be rewritten in matrix form: where k B is matrix of boundary conditions of size The algorithm for computing of matrix k B under the conditions (5) is described below.
1. Elements of the matrix k B are determined by the formula where q p, G is Kronecker delta.
It should be noted that computing of the elements of the matrix k B by formula (6)  . For each fixed value of q , the actions listed below are performed.
2.1.1.1.2. As a vector of unknowns (see the formula (21) from the first part of this paper), we set It should be noted that computing of the elements of the matrix k B by formulas (9) corresponds to the boundary conditions (3)-(4).
3 Interface "discrete model -discrete-continual model", "internal" boundary condition of the type "perfect contact" Let's consider arbitrary boundary point   V (after corresponding averaging).

Equations (1)-(4) can be rewritten in matrix form
where k B is matrix of boundary conditions of size All other elements of matrix k B are equal to zero.
The algorithm for computing of matrix k B under the conditions (14) is described below.
1. Elements of the matrix k B are determined by the formula It should be noted that computing of the elements of the matrix k B by formula (15) corresponds to the boundary conditions (10)- (11 It should be noted that computing of the elements of the matrix k B by formulas (18) corresponds to the boundary conditions (12)-(13). (20) Equations (19) , which can be constructed in accordance with algorithms presented at Table 1 and Table 2 respectively. 5 Interface "discrete model -discrete model", "internal" boundary condition of the type "perfect contact" Let's consider arbitrary boundary point