Introduction

Cracks frequently appear during production and use of different materials, parts and building structures. Presence of cracks has a great influence on durability of the constructions and a catastrophic consequence could arise from appearance and growth of cracks.

In many cases, a Lamé system is used as a foundation for mathematical modeling of the linear elastostatics in the cracked domain with different types of boundary conditions. It is well known that in the case of Dirichlet or Neumann boundary conditions on both sides of the crack the classic finite element method converges with the rate 1/ 2 () Oh , and in the case of Dirichlet conditions on one side and Newmann on the other one convergence rate is 1/ 4 () Oh . In recent 30 years, on the base of classic works (see, for example, [1][2][3][4]) many researches of properties of solutions to the Lamé equations posed in the cracked or non-convex domains were performed, asymptotic expansion of the solutions near singularity points were obtained (see, for example, [5][6][7]). Taking into account these results and on the base of weak statement of the problem, many special numerical methods with the convergence speed () Oh in the norm of Sobolev space 1 2 () W  were developed, such as: smoothed FEM [8] is based on the on the modification of the strain field; DPG FEM [9] suppose simultaneous approximation of the displacement and stress fields; extended FEM (Xfem) [10][11] is based on the addition of special function into the finite element space.

For the boundary value problems with singularity caused by different sources, such as degeneration of the input data or geometrical features of domain boundary, it was suggested to define the solution as R  -generalized one [12][13][14][15][16]. This approach allowed us to define suitable weighted space or set in which R  -generalized solution exists and is unique. Properties of R  -generalized solutions and weighted spaces where they are defined were deeply studied [17][18][19][20][21][22]. On the basis of obtained theoretical results, various modifications of the weighted finite element method for different problems were designed and investigated [23][24][25][26][27][28][29][30].

In the present paper, a Dirichlet problem for the Lamé system posed in 2D cracked domain is considered. The solution to this problem is defined as R  -generalized one in the special weighted functional set. For the numerical treatment of the problem, the weighted finite element method is constructed. Numerical experiment for the model problem is performed and a comparison of the weighted FEM with the classic FEM is carried out. Results of the series of computations showed that the approximate R  -generalized solution converges to the exact one in the norm of the weighted Sobolev space with the rate () Oh , that is twice greater than for the classical finite element method.

Problem Statement

Let



,  is a boundary of the  . In addition to the Cartesian coordinate system we also introduce polar coordinates with pole in the origin (0, 0) and polar axis codirectional with the Ox axis.

Assume that the strains are small. In  consider a boundary value problem for the Lamé system (see [10]):

(2 ( ( )) ( div ))

x   − +  =    div u u f(1)

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12 ii u q i x =  =    (2)

Here 12 () uu = u is the displacement field, ()

 u is the deformation tensor. Denote 2 2 1 2 12 { ( ) 1} x x x     =    +   a  -neighborhood of the (0 0)  .

We introduce the weight function () x  which coincides with the distance to the origin in   and equals to  for x\     . Using weight function, we define the weighted spaces 2 ()

L    , 1 2 () W    and sets 1 2 () W     , 1 2, () W    , 1/ 2 2 () W    

(see, for example, [19] and [22]). For the corresponding spaces and sets of vector-functions we use notations

1 2 ()     W , 2 ()     L , 1 2 ()     W .

Assume that the right hand sides of ( 1) and ( 2) meet the following conditions:

12 22 ( ) ( ) 1 2 0 i q W i             =     fL

(3) Introduce bilinear and linear forms respectively:

22 ( ) ( ) : ( ) ( ) a dx l dx        =  =  u v u v v fv .

Here ()  u denotes the stress tensor. We say that the function ) ( )

al  = u v v

holds for any fixed value of  , satisfying the inequality  

Weighted Finite Element Method

For the problem ( 1),( 2) on the base of notion of R  -generalized solution the weighted finite element method is constructed. For that, a quasi-uniform triangulation h T of the  is performed and a special weighted basis functions are introduced. We define the set

By means of strains1 span{ } h k n kk V  = = = , 2 [] hh V = V . It h V , we separate a subset h = V   ( ) 0 1 2 k h i k P V v P i  =    =  =  v .

We say that the function hh   uV is an approximate R  -generalized solution by the weighted finite element method to the problem (1),(2), if it satisfies boundary conditions (2) in nodes

Numerical Experiment

In the present section, the results of numerical experiment for a model problem are presented. We used the vector- It Table 1 for the different N the values of  ,   and the ratios between them on the adjacent meshes are presented. In Table 2 and Table 3 we present a number of nodes

function 1 (1 ) cos ( cos( )), sin ( cos( )) 2 2 2 r E             = + − −           u as a solution, 2    = + ,4 ( ) 2 E

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Remark. During numerical experiment for the current problem, in contrast to the problems in works [23][24][25][26][27][28][29][30], the founded optimal value * 0.0  = . Convergence rate () Oh is achieved thanks to the introduction of R  -generalized solution.

Conclusion

From the presented numerical results one can make the following conclusions:

• an approximate R  -generalized solution to the problem (1),( 2) with the Dirichlet boundary conditions on both sides of the crack converges to the exact one with the rate () Oh in the norm of the space