PSI - Issue 6

Pavel A. Akimov et al. / Procedia Structural Integrity 6 (2017) 182–189 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

187

6

Basic nodal unknown functions are displacement components

( ) 2 k k u u and their derivatives ( ) 1 ,

( ) 2 k k v v with respect ( ) 1 ,

to 2 x (superscript “ ( ) k ” hereinafter corresponds to the number of considered subdomain i.e. dc k k k i k i v v . 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 4 N first-order ordinary differential equations:    ). Thus for node ( , ) k i we have the following unknown functions: ( , ) 2 k i ( , ) 1 , k i u u and ( , ) 2 ( , ) 1 ,

( ) ~ ( ) 2 2 U x A U x R x k k k k    , ( ) 2

(20)

where k U is global vector of nodal unknowns (subscript “ ( ) k ” corresponds to the number of subdomain dc k k    ),

(21)

;

k k U U x 

T T u v ( ) [ ( ) ( ) ] 2  k T k

(22)

;

;

u u x 

u

u

... u (

v v x

k T ( ) [ ( ) ( ( ,1) v v

v ... (

( ) [ ( 

) (

)

k N T T ) ] ( , ) 1

)

k N T T ) ] ( , ) 1

 

k T

k ( ,2)

T

k ( ,2)

T

( ,1)

k

k

n

n

n

k

k

n

n

n

2

2

(23)

;

k i u u x u u ( ) [ ( , ) ( , ) ( , )   k i k i

k i v v x v v ( ) [ ( , ) ( , ) ( , )   k i k i

T ] ;

]

k i ( , ) 2

k i ( , ) 2

T

n

n

n

n

2

1

2

1

~ ( ) 2 R x k is the right-side vector of order

1 4 N ;

1 4 N . Correct analytical

k A is global matrix of coefficients of order

solution of (21) is defined by formula

( ) 2 2 2 U x E x C S x k k k k   ; ( ) ( )

(24)

E x

(       ) ( b x x x x

( ) 2

)

(25)

S x k

( ) ~ ( ) x R x 

( ) 2

k  

;

;

b

k

k

k

k

2, 1 k 

2

2,

2

k

2

2

( ) 2 x k  is the fundamental matrix-function of system (20), which is constructed in the special form convenient for problems of structural mechanics [1];  is convolution notation; k C is the vector of constants of order 1 4 N . Generally 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 [2]. 3. Software system and verification samples All methods and algorithms considered in this paper have been realized in software system. The main purpose of Analysis system CSASA2Dm (DCFEM + FEM) is semianalytical structural analysis (static structural analysis of deep beam within two-dimensional 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) with Intel MKL Library. Software system is designed for Microsoft Windows 8.1/10. Verification samples of deep beam analysis are presented in [1]. ANSYS Mechanical 15.0 (FEM) was used for verification purposes. We should note that the results of analysis obtained by the ANSYS Mechanical and CSASA2D generally agree well with each other. Besides, it is necessary to note that DCFEM is more effective in the most critical, vital, potentially dangerous areas of structure in terms of fracture (areas of the so-called edge effects), where some components of solution are rapidly changing functions and their rate of change in many cases can ’ t be adequately taken into account by the standard FEM. Let’s consider verification sample dealing with static analysis of three -dimensional deep beam loaded with two uniform loads over 1 x ( 100  P kN) with hinged ends (cross-sections) along basic dimension (Fig. 3).