In present paper we consider crack problem in rectangle. Solution of the problem we define as -generalized one in the weighted Sobolev set. For calculation of approximate solution we construct the scheme of the weighted finite element method (FEM). We perform comparison of our method with the classic FEM on two model problems. Theoretical estimate of the convergence rate (ℎ) for constructed method in the norm of the Sobolev weight space is confirmed experimentally.
generalized solution. natural. It is not required to refine mesh or preliminary separate singularity as multiplicative term.
In the present paper for the crack problem a notion of -generalized solution in special weighted set is introduced. Such definition of the solution allowed us to construct the weighted FEM with a convergence rate (ℎ) in the norm of the weighted Sobolev space. A comparison of the constructed method with the classic FEM is carried out. Theoretical estimation of the convergence rate for the constructed method derived in [14] is confirmed in numerical experiment for two model problems. It is twice as much than for the classic FEM. At the same time, in the most of mesh nodes the absolute error for the approximate -generilized solution in a several decimal orders less than for the approximate 2. Weighted spaces and sets.
-generalized solution following boundary value problem of elasticity stated in displacements (crack problem): Ω Let Ω = (−0.7, 0.3) × [−1, 1] ⧵ [0, 0.3] × {0} be a two-dimensional domain with a crack Ω = [0, 0.3] × {0}, + and Ω
Assume that the domain Ω is a homogeneous isotropic body and strains are small. Consider the − are the crack sides, point (0, 0) is a crack tip. Denote Ω the boundary of Ω Ω ⊂ Ω , . − (2 ( ( )) +
( div )) = , ∈ Ω,
= , = 1, 2, ∈ Ω.
(
Denote by Ω′ closure of the -neighborhood of the point (0, 0) in the domain Ω ( 12 + 22) ≤ }. In Ω
′ we introduce a weight function ( ) as a distance to the point (0, 0) it to the rest of Ω with constant . Using ( ), we introduce weighted space 2, (Ω) consisting of : Ω′ = { ∈ Ω ∶ and extend Lebesque measurable functions with finite norm ‖ ‖ 2, (Ω) = , 2 ∈ , ≥ 0. Through 21, (Ω, ) we denote the set of functions that meet the following conditions:
− , ∈ Ω′, (a) ≤ 1 (b) || | | ≤ 1 | (c) |‖ ‖ 2, (Ω⧵Ω′) ≥ 2, 2 = const,
+1 − −1, ∈ Ω′, = 1, 2, 1 = const, with finite norm ‖ ‖ 21, (Ω) = ‖ ‖22, (Ω) + ‖‖ ( ‖ ‖ ‖ ‖ ‖ 2 1 ‖ 2, (Ω) + ‖‖ ‖ ‖ ‖ 2 ‖ 2, (Ω)) For the vector function = ( 1, 2) corresponding norm is calculated by formula ‖ ‖ 2 2 √ ∫ 2
The set consisting of traces of functions ∈ 21, (Ω, ) on Ω we denote by 21,/2( Ω, ): Norm in this set is defined by formula
2 ‖ ‖ 21, (Ω) = (‖ 1‖ 21, (Ω) + ‖ 2‖ 21, (Ω)) 21,/2( Ω, ) = { ∶ = | Ω, ∈ 21, (Ω, )}.
‖ ‖ 21,/2( Ω, ) = inf ‖ ‖ 21, (Ω).
| Ω=
equations (
The subset of set 21, (Ω, ) that contains functions with zero trace on Ω we denote by ̊ 1 2, (Ω, ).
The set of functions ∈ 2, (Ω) satisfying conditions (a) and (c) we denote by 2, (Ω, ). We assume that for some real number >
0 the following inequalities for the right hand sides of
∈ 2, (Ω, ), ∈ 21,/2( Ω, ), = 1, 2.
(
we designate ℎ and call mesh parameter.
=
for the problem (
We construct the scheme of the weighted FEM for calculation of approximate -generalized solution
We perform quasiuniform triangulation ℎ of the domain Ω coordinated with the crack Ω . For this, we decompose Ω into the set of rectangles by horizontal and vertical straight lines = , ∈ [−0.7, 0.3], ∈ [−1, 1]. Each rectangle we divide into two triangles by diagonal.
= , Obtained triangles we call finite elements and their vertices are nodes , = 1, … , . The set of internal is the set of boundary nodes. The longest side of all triangles For each node we introduce weighted basis function ( ) = ∗ ( ) ( ), = 1, … , , ∗ ∈ , where ( ) is a function linear on each finite element and ( ) is equal 1 in and equal 0 in all other nodes. Linear span of all built basis function we designate ℎ. Linear span of basis function associated with internal nodes we designate ̊ ℎ.
Vector-function ℎ = ( ℎ, 1, ℎ, 2) with components ,
ℎ
(
Components of approximate -generalized solution we write in the form ( ℎ, (0, )) = (0, ), = 1, … , .
Unknown coeficients , = 1, … , 2 can be found from the system of linear algebraic equations Remark. The diference of the weighted FEM is the weight function ( ) raise to some power ∗ in the finite element basis. This allowed us to approximate behavior of the solution near the singularity point better. Parameter ∗ as well as radius of the -neighborhood in definition of the weight function ( ) and its power in definition of the -generalized solution are governing parameters of the constructed weighted FEM. Varying these parameters, we are able to afect accuracy of the approximate -generalized solution. When we choose parameters close to the optimal ones, we get the best accuracy and maximum convergence rate of the constructed FEM that corresponds to the theoretical rate (ℎ). 4. Computer investigation of the model problems In present section we adduce computer investigation of two model problems using weighted FEM constructed in section 3. The results were obtained using the equipment of Shared Resource Center "Far Eastern Computing Resource" IACP FEB RAS (https://cc.dvo.ru) and of Shared Services Center “Data Center of FEB RAS” (Khabarovsk). Computation of approximate -generalized solution were realized by the program "Proba-IV", automatic startup of calculation series and sequent analysis of results were carried out by the software package [15].
The common algorithm of investigation is the following:
of sets of their fixed values.
hands of equation and boundary conditions (
ated set of values of governing parameters , , ∗. b) Computation of the approximate -generalized solution corresponding to the each cred) Analysis of results and detection of the optimal parameters set that minimize the computational error in the norm of the weighted space 21, (Ω). 3. Experimental evaluation of the convergence rate of approximate -generalized solution by the weighted FEM calculated with optimal values of governing parameters , , ∗ found in listbox 2d), comparison with the convergence rate of the approximate generalized solution derived by 4. Comparison of the absolute error in the mesh nodes of approximate -generalized and generthe classic FEM.
alized solutions. 4.1. Model problems Problem 1. For the problem 1 we choose the vector with only singular components: Dependence of relative error for approximate -generalized solution derived with indicated optimal parameters , , ∗, and of relative error for approximate generalized solution on the mesh parameter ℎ, model
Problem 2. For the problem 2 we choose the vector with both singular and regular components: √ √
2
Lamé parameters for both problems are = 578.923, = 384.615Pa, stress intensity factor = 4.2. Investigation of the convergence rate In the present subsection for two model problems we adduce results concerning convergence rate of norm of the weighted space 21, (Ω) when parameter = = 2.2, = = 0.062 by formula the approximate -generalized solution calculated by the weighted FEM with optimal parameters. For derived approximate -generalized solutions ℎ the error was calculated in the relative weighted = = ‖ − ℎ‖
21, (Ω) ‖ ‖ 21, (Ω) ‖ − ℎ‖ 21(Ω) ‖ ‖ 21(Ω) . . by formula
For approximate generalized solutions ℎ the error was calculated in the norm of the space 21(Ω) solution, and (ℎ) for approximate -generalized one.
On the Figure 1 we present graphs of and in the log scales.
For the model problem 1 in Table 1 for meshes with diferent parameter ℎ we adduce values of , corresponding optimal parameters , , ∗, values of and their ratios for adjacent meshes. Ad duced results confirm theoretical estimations of convergence rate (ℎ0.5) for approximate generalized Results on the convergence rate for model problem 2 are presented in Table 2 and Figure 2. .3 .1 .3e–1 .1e–1 .3e–2 h
0.031 0.062 1.3 0.1 0.0155 0.046 1.2 0.1 0.0077 0.023 1.2 0.1
0.0038 0.0116 1.6 0.2 line represents convergence rate (ℎ). Dependence of relative error for approximate -generalized solution derived with indicated optimal parameters , , ∗, and of relative error for approximate generalized solution on the mesh parameter ℎ, model 0.0019 0.0058 1.6 0.2 0.062 0.062 2.0 0.1 line represents convergence rate (ℎ).
ℎ 11,, %%
ℎ 2 ,, %% 11,, %%
ℎ 2 ,, %% 2 Values of 1 , 1 (in percents of total number of nodes) on meshes with diferent parameter ℎ, model problem 1. Values of 2 , 2 (in percents of total number of nodes) on meshes with diferent parameter ℎ, model problem 1. Values of 1 , 1 (in percents of total number of nodes) on meshes with diferent parameter ℎ, model problem 2. Values of 2 , 2 (in percents of total number of nodes) on meshes with diferent parameter ℎ, model problem 2. 0.031 4.3. Investigation of absolute error in mesh nodes In nodes , = 1, … , on meshes with diferent parameter ℎ for components of approximate , , ∗ and for components of apgeneralized solution obtained with optimal values of parameters proximate generalized solutions we calculated the absolute diferences between them and components in Table 4.
| | of exact solution: = | ( ) − , ( )|
ℎ of nodes where the absolute errors |, | and | = || ( ) − ℎ( )||, = 1, 2. We also counted the numbers ,
, respectively, are less than the limit value Δ = 5 ⋅ 10−8.
For model problem 1, in Table 3 we adduce values of 1 , 1 on meshes with diferent parameter ℎ .
Results for second component of approximate -generalized and generalized solution are presented | On Figure 3 and Figure 4 for model problem 1 we depict distribution in the domain Ω of 1, 1 and Values of 1 , 1
and 2 , 2 for model problem 2 are presented in Table 5 and Table 6 respectively. Distribution in the domain Ω of 1, 1 and 2, 2 for model problem 2 is fully similar to the model ℎ = 0.0077 ℎ = 0.0077
Computer investigation of the crack problem realized by the weighted FEM allows us to draw following conclusions: • Theoretical convergence rate (ℎ) of the approximate -generalized solution by the weighted FEM with optimal governing parameters to the exact one in the norm of the weighted space 21, (Ω) were experimentally confirmed. This is twice as much than convergence rate of the approximate generalized solution. • In most mesh nodes, absolute error of approximate -generalized solution by the weighted
FEM in several decimal orders less than absolute error of approximate generalized solution.
The reported study was supported by RSF according to the research project No. 21-11-00039. [11] V. A. Rukavishnikov, A. O. Mosolapov, E. I. Rukavishnikova, Weighted finite element method for elasticity problem with a crack, Computers and Structures 243 (2021) 106400. doi:10.1016/ j.compstruc.2020.106400. [12] V. A. Rukavishnikov, A. V. Rukavishnikov, New numerical method for the rotation form of the
Oseen problem with corner singularity, Symmetry 11 (2019) 54. doi:10.3390/sym11010054. [13] V. A. Rukavishnikov, O. P. Tkachenko, Dynamics of a fluid-filled curvilinear pipeline, Applied
Mathematics and Mechanics 39 (2018) 905–922. doi:10.1007/s10483-018-2338-9. [14] V. A. Rukavishnikov, Weighted FEM for two-dimensional elasticity problem with corner singularity, Lecture Notes in Computational Science and Engineering 112 (2016) 411–419. doi:10.1007/978-3-319-39929-4_39. [15] V. A. Rukavishnikov, O. V. Maslov, A. O. Mosolapov, S. G. Nikolaev, Automated software complex for determination of the optimal parameters set for the weighted finite element method on computer clusters, Computational Nanotechnology (2015) 9–19.