Finite Element Method for the Lamé System in Domain with a Crack

Abstract

In the present paper a Dirichlet problem for the Lamé system in the 2D cracked domain is considered. Solution to this problem is defined as R generalized one in the weighted functional set. For determining of approximate solution, the weighted finite element method is built. Numerical analysis of the model problem showed that convergence rate of the approximate R -generalized solution to the exact one in the norm of the weighted Sobolev space amount to O (h) , that is twice greater than for the classical finite element method.

Introduction

Let  = [−11]  [−11] be a homogeneous isotropic body with the straight crack [01]  {0} ,  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]): −(2div( (u)) + ( div u)) = f  x   ( 1 ) ui = qi  i = 1 2 x   ( 2 ) Here u = (u1 u2 ) is the displacement field,  (u) is the deformation tensor.

Denote  = {x    ( x12 + x22 )12    1} 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 L2 () , W21 () and sets W21 ( ) , W 12, ( ) , W21/2 ( ) (see, for example, [19] and [22]). For the corresponding spaces and sets of vector-functions we use notations W21 ( ) , L 2 ( ) , W 12 ( ) .

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

f  L2 ( ) qi  W212 ( ) i = 1 2   0 ( 3 ) Introduce bilinear and linear forms respectively: a(u v) =   (u) :  ( 2 v) dx l ( v) =   2 fv dx .

  Here  (u ) denotes the stress tensor.

We say that the function u  W21 ( ) is an R -generalized solution to the problem ( 1 ),( 2 ), if almost everywhere on  it satisfies boundary conditions ( 2 ) and for any v  W 12 ( ) the integral identity a (u  v ) = l ( v ) holds for any fixed value of  , satisfying the inequality     3

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 T h of the  is performed and a special weighted basis functions are introduced.

By means of strains x = −1 + 2i , y = −1 + 2 j , i, j = 0..N  is divided on squares and each of them by the

N N diagonal is divided into two triangles K called finite elements, with vertices Pk (k = 1… n ') are called triangulation nodes. Let  h = KT h K , h = 2N2 is a longest length of the finite elements sides. We denote the set of all triangulation nodes as P =  Pk  kk ==1n ' among which {Pk }kk ==1n is the set of internal nodes and {Pk }kk ==nn +'1 is the set of boundary nodes.

For each Pk  {Pk }kk ==1n a weighted basis function  k ( x) =    ( x) k ( x) k = 1… n is introduced, where  k ( x ) is a linear on each finite element and  k (Pj ) =  kj , k  j = 1… n ,  kj denotes a Kronecker delta,   is a real number.

We define the set

V h = span{ k }kk ==1n ,

V h = [V h ]2 . It

V h , we separate a subset

V h = = v  V h  vi (Pk ) Pk  = 0 i = 1 2 .

We say that the function uh  V h 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 Pk  {Pk }kk ==nn +'1 and for all v h ( x)  V h and    the integral identity holds.

Components of approximate solution uh will be found in the form

n n uh1 =  d 2k −1 k  uh2 =  d 2k k  d j =  −  ( P j+1 )c j  j = 1… 2n

k =1 k =1  2  The unknowns d j are defined from the system of linear equations a(uh  k ) = l( k ) k = 1 2n .

a(uh  v h ) = l ( v h ) 4

Numerical Experiment

In the present section, the results of numerical experiment for a model problem are presented. We used the vectorfunction u = 1 r (1 +  )  cos    ( − cos( )), sin    ( − cos( ))  as a solution,  =  , E = 4 ( +  ) ,

E 2   2   2    + 2  + 2  = 3 − 4 (see [10]). Lamé parameters  = 3.0 ,  = 5.0 .

Computations were carried out on meshes with different partition numbers N . The errors of approximate generalized ( = 0 ,  ( x)  1 ) and R -generalized solutions were compared in the norms of spaces W21 ( ) , W21, () and in the triangulation nodes. To do this, for the derived approximate generalized u h and R -generalized uh u − u h

W21 () ,  = u − uh W21, () u W21 () solutions a relative errors  =

u W21, () Pi , i = 1, , n an absolute errors  ij = u j (Pi ) − u hj (Pi ) ,  ij = u , j (Pi ) − uh, j (Pi ) were calculated. A number and coordinates of the nodes Pi , where the absolute errors are less than the limit value  = 10−5 were counted. Results of numerical experiments are presented in tables and figures. Optimal parameters  , , * were derived by the software complex [31].

It Table 1 for the different N the values of  ,  and the ratios between them on the adjacent meshes are presented. were computed, respectively. In the nodes

On Figure 1 a convergence rates of the approximate generalized (left) and R - generalized (right) solutions are presented in the logarithmic scales. Solid lines designate convergence rate O (h) .

In Table 2 and Table 3 we present a number of nodes Pi  {Pk }kn=1 in percentage of their total number, where the absolute errors  ij for approximate R -generalized solution and  ij for the approximate generalized solution, i = 1, , n , j = 1, 2 are less than the limit value  = 2 10−5 . Distribution of the absolute error for component uh,1 of the approximate R -generalized solution and for component u1h of the approximate generalized solution for = 256, N = 512 and N = 1024 are depicted on Fig. 2. Corresponding results for uh,2 , u2h are depicted on Figure 3. N = 512 approximate R -generalized solution) and  i2 ( n2 , approximate generalized solution), i = 1, , n , are less than the limit value  = 2 10−5 . (b) solutions, respectively, on meshes with different partition number N .

N = 256

N = 512

N = 1024 N = 256

N = 512

N = 1024

Remark. During numerical experiment for the current problem, in contrast to the problems in works [23-30], the founded optimal value  * = 0.0 . Convergence rate O (h) is achieved thanks to the introduction of R -generalized solution. 5

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 O (h) in the norm of the space W21, () , whereas approximate generalized one converges with the rate O (h1/2 ) in the norm of the space W21 ( ) ; absolute error  ij of the approximate R -generalized solution in overwhelming majority of nodes is by one or two decimal orders less than absolute error  ij of the approximate generalized solution.

Acknowledgements References

This research was supported in through computational resources provided by the Shared Facility Center "Data Center of FEB RAS".

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