In [1] a weighted finite element method (WFEM) is constructed to find an approximate solution to the crack problem. We have shown that the reentrant corner 2 at the boundary of the domain does not afect the accuracy of finding the solution by this method. The approximate solution by the WFEM converges to the exact one with the rate of (ℎ). Three control parameters afect the accuracy of finding the approximate solution by the WFEM. In this paper we define the body of optimal parameters (BOP) in the WFEM for the crack problem. The error of the found approximate solution deviates from the smallest error by no more than a predetermined value when we choose parameters from the BOP. corner singularity, weighted finite element method, body of optimal parameters Numerical methods for finding solutions to problems in the theory of elasticity with a singularity (tearing, sliding modes) play an essential role in fracture mechanics (see, for example, [2, 3]). For the system of Lamé equations on a nonconvex bounded polygonal domain with Dirichlet conditions, it is known [4, 5, 6, 7] that the solution to this problem can be written Far Eastern Workshop on Computational Technologies and Intelligent Systems, March 2-3, 2021, Khabarovsk, Russia
( ) = ∑ ( , )Ψ ( ) + ( ), ∈ 22(Ω). the internal angle and Ψ ( ) is the cutof function.
≤ ≤ 2 Here ( ) = ( 1, 2), ( , ) = ( 1, 2), ( ) = ( 1, 2), 1, 2 are suficiently smooth functions, are at the singularities
, ( , ) are the polar coordinates at the point
A weighted finite element method (FEM) for finding an approximate solution to the problem of a crack or a Lamé system in a domain with a boundary containing an angle of 2 This method is based on the introduction of an -generalized solution (see, for example, [8, 9, 10, 11]). The reentrant corner 2 at the boundary of the domain does not afect accuracy of finding of the approximate solution by the WFEM in compare to the classical FEM and the method with a refined was proposed in [1]. mesh. The rate of convergence of an approximate solution by the WFEM to the exact one is (ℎ) in the norm of the space for the high accuracy of the 2, + /2(Ω) and in the weighted energy norm [1]. The determining factor 1
WFEM is the correct choice of parameters: is exponent of the weight function in an -generalized solution, ∗ is the exponent of the weight function in the basis of the ifnite element method (see, for example, [12, 13, 14, 15, 16], and is the radius of the neighborhood in 2 0.3 x 1 which the weight function is specified as the distance to the point of singularity during calculations. In this article the body of optimal parameters (BOP) for the weighted finite element method for the crack problem is determined. We have determined the body of parameters at which the error of the found approximate solution by the WFEM in the norm of the weighted Sobolev space difers from the smallest error by no more than 5%, 10%, and 15%. We noticed that the BOP depends on the dimension of mesh or mesh step. 2.
-generalized solution In [1] for finding of a displacement field boundary value problem for the Lamé system with coeficients and :
u = ( 1, 2) in the crack problem we considered the first closure of Ω, i.e. Ω = Ω ∪ Γ.
Comment 1. The solution of the problem (
( div )) = , ∈ Ω, u = q, ∈ Γ.
Here without loss of generality, we will assume that Ω is the rectangle shown in Fig. 1. Let Γ be a boundary of domain Ω and Γ ⊂ Γ be a crack with sides Γ+ and Γ+ . We denote Ω the
1 u( ) = 02 ( 0, 0) 0( 0) + ( ), where 0 is a distance to (0, 0). Therefore u ∈ 1+ 12 − (Ω) ( > methods one obtains an order of at most (ℎ 12 − ), where ℎ is the mesh step. 0) and for regular finite elements
In [1] we proposed the weighted finite element method that allows to find an approximate solution
to problem (
Let be a disk of radius > 0 with its centre in the point (0, 0), i.e. = { ∶ ( 12 + 22) 21 ≤ ≪ 1}
(
We introduce the weighted spaces with norms:
Let ( ) be a weight function that is positive everywhere, except in (0, 0), and satisfies the follow2 ‖ ‖ 2, (Ω) = ∑ | |≤ Ω ∫ 2 | |2, ‖ ‖22, ( Ω) ∫ 2
2
,
Ω
‖ ‖ 2,0(Ω) = ‖ ‖ 2 (Ω),
(
Ω functions. where =
1 1| | 22 , = ( 1, 2) and | | = 1 + 2, is a nonnegative integer, and is a real number.
The space ̊
2, (Ω) ⊂ 2, (Ω) is defined as a closure in the norm (
(b) ‖ ‖ 2, (Ω⧵Ω′ ) ≥ 2, 2 = const,
+| |
with norm (
Let 21, (Ω, ) be the set of functions satisfying the following conditions: ‖ ‖ 21,/2( Ω, ) = inf ‖Φ‖ 21, (Ω).
Φ|Γ= ′ , ∈ Ω , | | = 0, 1, 1 > 0 is a constant; By analogy, one can introduce sets for other spaces.
Definition 1. [10] Let the right-hand sides of (
∈ 2, (Ω), ∈ 21,/2( Ω), ≥ 0.
the integral identities
A function = ( 1, 2) from the space 21, + /2(Ω) is called an R -generalized solution to the
problem (
2 1 ( 2 1) + 1 ( 2 1) + 2 ( 2 1) + 2 1 2 ( 2 1) 1 2 ] Ω Ω 2( , 2) = ∫ [ 1 1 ( 2 2) + 1 ( 2 2) + ( + 2 )
2 ( 2 2) + 2 2 1 2 2 2 ( 2 2) 1 1 ]
= ∫ 2
Ω = ∫ 2 Ω = = 1 1
= 1( 1); 2 2 = 2( 2); holds for any fixed value of satisfying the inequality ≥ .
Comment 2. We notice that an -generalized solution has a sheaf of solutions in the neighborhood of the singularity point if it is defined in the weighted space 12, + /2(Ω) and does not take into account the additional properties of this solution (see, for example, [17]). In [10] we proved the uniqueness of an -generalized solution if it is defined in the set 2, + /2(Ω, ). 1
An -generalized solution satisfies conditions (a), (b) of the set 21, + /2(Ω, ). This follows from
the asymptotic of the solution to problem (
Comment 3. We proved that an -generalized solution is the same for diferent (see [10]).
Comment 4. In contrast to the weak solution of problem (
We perform a quasi-uniform triangulation of the domain Ω (see Fig. 2). Let is the union of all the triangles , = 1, … , ; ℎ is the maximal length of the sides of the triangles and it called mesh step. The vertices , = 1, … , of the triangles are nodes of the triangulation, { } = { 1, … , } and the point ∈ . Let = { } = is the set of internal triangulation nodes.
To each node ∈ we assig n=1the weighted function
∗ ̂ = ( ) , = 1, … , , where is linear function on each triangle , equal to 1 at the node and zero at all the other nodes, ∗ is a real number.
We introduce weighted vector basis { ( )} =1
=2 , where ( ) { ( ̂ ( ), 0), = 2 − 1, (0, ̂ ( )), = 2, = 1, ..., .
We denote by Vℎ the linear span { ( )} =1
=2 . In Vℎ we denote the subset ̊ ℎ = { ∈ ℎ ∶ ( ) = 0, ∈ Γ}.
Definition 2. A function ℎ in the space ℎ is called an approximate R -generalized solution
of the problem (
An approximate solution will be found in the form ( ℎ, ℎ) = ( ℎ)
2 ℎ = ∑ ( ),
=1 h -0.7 0.3 x 1 here = − ∗ ( [( +1)/2]) , = { ℎ, 1( [( +1)/2]), = 2 − 1 , = 1, … , , [( + 1)/2] is an integer part ℎ, 2( [( +1)/2]), = 2 of number ( + 1)/2.
Comment 5. Note that we have introduced into the basis the weight function raised to some power. The weight basis and an -generalized solution made it possible to find an approximate solution without loss of accuracy on quasi-uniform grids.
We proved that the approximate -generalized solution by the weighted FEM converges to the exact one with the first rate with respect to the mesh step ℎ [19, Theorem 8].
In [1] a numerical analysis was carried out for one model problem on grids of large and small dimensions.
We have obtained experimentally confirmation of the convergence rate of the approximate solution
to the exact one (ℎ) in the norm of the space 21, (Ω) and in the energy norm. In addition, the
smallness of the absolute error (10−7) in the overwhelming number of grid nodes was established.
4. Body of optimal parameters
4.1. Algorithm for determining BOP on grids of various dimensions
For calculation of the approximate -generalized solution by the weighted finite element method
we need to set the parameters , ∗, . These parameters can be arbitrary if they satisfy conditions
of the theorem on the existence and uniqueness of the -generalized solution and correspond to
the asymptotic properties of the solution. But if you want to find an approximate solution to the
problem with the smallest error, then these parameters should be close to optimal. Currently, there is
no algorithm for theoretical determination of such parameters. For problem (
Consider two model problems in the domain Ω:
(A) Boundary value problem (
Lamé coeficients are = 576.923, = 384.615, and stress intensity factor = 1.611.
(B) Boundary value problem (
Let us find for problems (A) and (B) the parameters , ∗, , which allow us to calculate an approximate solution by the weighted finite element method with the best accuracy on quasiuniform meshes of various dimensions. In Ω we built meshes with a step ℎ = 0.062, 0.031, 0.015, 0.0077, 0.0038, 0.0019
The set of optimal parameters will be discrete, as we form it from the results of numerical experiments carried out for given fixed values , ∗, . there were no significant changes in the results.
We chose ∗ equal to 0, 0.1, 0.2, 0.3, 0.4, 0.49. The values of were selected from the interval [0.5, 5.5] with a step of 0.1. The radius of the -neighborhood Ω ′ was equated to ℎ, 2ℎ, 3ℎ, …. Calculations were stopped or later disregarded when the error between the exact solution and the found approximate solution became larger than specified limiting error. The relative error was determined for all grids and parameters of WFEM in the weighted Sobolev norm and weighted energy norm with fixed and predetermined parameters ̄ = 2.2, ̄ = 0.062. Note that when choosing other parameters ̄ and ̄ , For each problem (A), (B) and each mesh we determined three parameters , ∗ , for which the relative error in the weighted Sobolev norm and weighted energy norm was the smallest. In addition, we formed sets of parameters 1 from the best error by no more than 5%(6.5%), 10% and 15%.
Comment 6. The ratio of the smallest errors was exactly two in both weighted norms on adjacent meshes for problems (A) and (B). This corresponds to a theoretical estimate of the convergence rate.
For each mesh the body of optimal parameters (BOP) is = ( ) ∩ an error that difers from the best error by no more than 6.5% on all meshes simultaneously. have determined triples of parameters , ∗, , which allow us to find an approximate solution with ( ) at which the relative errors difered 4.2. Results of numerical experiments Figures 3, 4, 5, 6 show the parameters , ∗, at which the errors difer from the best error by no more than 5% (green), 10% (yellow), 15% (red) at ℎ = 0.031, 0.015 and ℎ = 0.0038, 0.0019 respectively. We present the results for tasks A and B in Figures 3a – 6a and 3b – 6b, respectively. Figures 3c – 6c depict the sets , = 1, 2, 3. In Tables 1 and 2 we indicated the intervals of the parameters , ∗ the BOP at which the relative error in the norm of the weight space deviates from the best error by , of no more than the indicated values for ℎ = 0.031, 0.015 and ℎ = 0.0038, 0.0019 respectively. (3a)
(3b) (3c) the mesh with step ℎ = 0.031. the mesh with step ℎ = 0.015. Optimal parameters with a given error for the meshes with steps ℎ = 0.031, 0.015.
(4c) Optimal parameters with a given error for the meshes with steps ℎ = 0.0038, 0.0019.
We present the values of the parameters at which the deviation of the relative error from the best error does not exceed 5%, 5.5% and 6% for problem A on the mesh with a step ℎ = 0.0038 in Fig. 7. 5. Discussion of the results. Conclusion to the following conclusions: In this paper we defined the body of optimal parameters in the weighted finite element method to ifnd an approximate solution to the crack problem with high accuracy. Finding the BOP is based on a series of numerical experiments. We used the knowledge about the asymptotic behavior of the solution in the neighborhood of the singularity point and the conditions on the input data , of the existence and uniqueness theorem for the -generalized solution. The results of the experiments led 1. The proposed approach allows us to determine the BOP for the weighted finite element method (Figure 3–6, Table 1,2). 2. BOP depends on the dimension of the mesh (mesh step). 3. With a small deviation in the choice of parameters from the best parameters in WFEM, the relative error in the norm of the Sobolev weight space grows slightly (see Fig. 7). This indicates the stability of the process, i.e., a small change in the control parameters corresponds to a small increase in the relative error of the approximate solution. 4. If we choose the parameters = 0.062, = 2.0, ∗ = 0 then the error does not exceed 6.75% of the best value the error on all meshes simultaneously. In our opinion this fact is not important. The optimal parameters for carrying out calculations should be chosen depending on the dimension of the mesh (mesh step).
5. The proposed method with the found BOP can be used for calculating engineering problems.
I would like to thank Andrei Mosolapov and Artem Zubkov for their help in organizing and conducting numerical experiments.
The reported study was supported by RSF according to the research project No. 21-11-00039. Computational resources for the numerical experiments were provided by the Shared Services Center "Data Center of FEB RAS".