Finite Element Method for the Lamé System in Domain with a Crack Viktor A. Rukavishnikov¹, Andrew O. Mosolapov¹, Elena I. Rukavishnikova¹ ¹Computing Center of Far-Eastern Branch, Russian Academy of Sciences, Khabarovsk, Russia, vark0102@mail.ru 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. 1 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 O ( h1/ 2 ) , and in the case of Dirichlet conditions on one side and Newmann on the other one convergence rate is O ( h1/ 4 ) . In recent 30 years, on the base of classic works (see, for example, [1-4]) 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, [5-7]). Taking into account these results and on the base of weak statement of the problem, many special numerical methods with the convergence speed O ( h ) in the norm of Sobolev space W 2 (  ) were developed, such as: smoothed FEM [8] is based on the on the modification of 1 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 [12-16]. 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 [17-22]. On the basis of obtained theoretical results, various modifications of the weighted finite element method for different problems were designed and investigated [23-30]. 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 O ( h ) , that is twice greater than for the classical finite element method. 2 Problem Statement 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) Copyright © 2019 for the individual papers by the papers' authors. Copyright © 2019 for the volume as a collection by its editors. This volume and its papers are published under the Creative Commons License Attribution 4.0 International (CC BY 4.0). In: Sergey I. Smagin, Alexander A. Zatsarinnyy (eds.): V International Conference Information Technologies and High- Performance Computing (ITHPC-2019), Khabarovsk, Russia, 16-19 Sep, 2019, published at http://ceur-ws.org 133 Mathematical Modeling in Physics and Technology ______________________________________________________________________________________ ui = qi  i = 1 2 x   (2) Here u = (u1  u 2 ) 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 2, (   ) , W21/2 (   ) (see, for example, [19] 1 and [22]). For the corresponding spaces and sets of vector-functions we use notations W2 (  ) , L 2  (   ) , 1 W 2  (   ) . 1 Assume that the right hand sides of (1) and (2) meet the following conditions: 1 2 f  L 2  (  ) qi  W2  (  ) 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  W2 (  ) is an R -generalized solution to the problem (1),(2), if almost 1 everywhere on  it satisfies boundary conditions (2) and for any v  W 2  (   ) the integral identity 1 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. 2i 2j By means of strains x = − 1 + , y = −1 + , 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 2 2 nodes. Let  h = K T h K , h= is a longest length of the finite elements sides. We denote the set of all N triangulation nodes as P =  Pk  k =1 among which { Pk }kk ==1n is the set of internal nodes and { Pk }kk == nn +' 1 is the set of k =n ' 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. k =n We define the set V h = span{ k }k =1 , V = [V ] h h 2 . It Vh , we separate a subset V h =  = v  V h  vi ( Pk ) Pk  = 0 i = 1 2 .  We say that the function u  V h is an approximate R -generalized solution by the weighted finite element h  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 a ( u  v ) = l ( v ) h h h holds. Components of approximate solution u h will be found in the form n n u 1 =  d 2 k −1 k  u  2 =  d 2 k k  −  dj =  ( P j +1  ) c j  j = 1 … 2 n  h h 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 . 4 Numerical Experiment In the present section, the results of numerical experiment for a model problem are presented. We used the vector- 1 r        4  ( +  ) function u = (1 +  )  cos   ( − cos( )), sin   ( − cos( ))  as a solution,  = , E= , E 2    2   2   + 2   + 2  = 3 − 4 (see [10]). Lamé parameters  = 3.0 ,  = 5.0 . 134 Mathematical Modeling in Physics and Technology ______________________________________________________________________________________ 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 (  ) , W2, ( ) 1 h and in the triangulation nodes. To do this, for the derived approximate generalized u and R -generalized u h u − u h u − uh W2 (  ) 1 W 21, (  ) solutions a relative errors  = ,  = were computed, respectively. In the nodes u W 21 (  ) u W 1 (  ) 2 , Pi , i = 1, , n an absolute errors  ij = u j ( Pi ) − u hj ( Pi ) ,  ij = 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. Table 1: Dependence of relative errors  ,   for the approximate generalized and R -generalized (  = 0.015 ,  = 1.8 ,  * = 0.0 ) solutions, respectively, on the N . N 32 64 128 256 512 1024 −1 −2 −2 −2 −2 −2  1.130 10 1.41 8.006  10 1.41 5.666 10 1.41 4.009  10 1.41 2.835  10 1.41 2.005  10 −1 −2 −2 −2 −3 −3   1.024  10 1.22 8.347 10 2.10 4.003 10 2.18 1.836 10 2.04 9.002 10 1.99 4.523 10 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 ) . Figure 1: Dependence of relative errors  (left) for the approximate generalized and   (right) for the approximate R - generalized (  = 0.015 ,  = 1.8 ,  * = 0.0 ) solutions on the N . 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  ij 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 . Distribution of the absolute error for component u ,1 of −5 h the approximate R -generalized solution and for component u1h of the approximate generalized solution for 𝑁 = 256, N = 512 and N = 1024 are depicted on Fig. 2. Corresponding results for u , 2 , u 2h are depicted on Figure 3. h Table 2: The number of nodes Pi  { Pk }kn =1 in percentage of their total number, where the absolute errors  i1 ( n1 , approximate R -generalized solution) and  i1 ( n1 , approximate generalized solution), i = 1, , n , are less than the limit value  = 2  10 − 5 . N 32 64 128 256 512 1024  n 1 15.344% 14.148% 42.347% 86.736% 98.819% 99.740% n1 15.556% 29.286% 48.185% 74.523% 95.170% 99.449% 135 Mathematical Modeling in Physics and Technology ______________________________________________________________________________________ N = 256 N = 512 N = 1024 (a) N = 256 N = 512 N = 1024 (b) Figure 2: The errors  i1 and  i1 for approximate R -generalized  = 0.015 ,  = 1.8 ,  * = 0.0 (a) and generalized (b) solutions, respectively, on meshes with different partition number N . Table 3. The number of nodes Pi  { Pk }kn =1 in percentage of their total number, where the absolute errors  i2 ( n2 , approximate R -generalized solution) and  i 2 ( n 2 , approximate generalized solution), i = 1, , n , are less than the −5 limit value  = 2  10 . N 32 64 128 256 512 1024  n2 17.989% 13.665% 31.721 55.842% 94.834% 99.689% n2 17.143% 24.155% 34.790% 47.768% 63.128% 79.038% N = 256 N = 512 N = 1024 (a) N = 256 N = 512 N = 1024 (b) Figure 3: The errors  i2 and  i 2 for approximate R -generalized  = 0.015 ,  = 1.8 ,  * = 0.0 (a) and generalized (b) solutions, respectively, on meshes with different partition number N . 136 Mathematical Modeling in Physics and Technology ______________________________________________________________________________________ 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 W2, ( ) , 1 whereas approximate generalized one converges with the rate O ( h ) in the norm of the space W2 (  ) ; 1/ 2 1 • absolute error  ij 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 This research was supported in through computational resources provided by the Shared Facility Center "Data Center of FEB RAS". References 1. Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris, rma 22 edition (1992) 2. 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