4.1. Comparative analysis of the generalized and Rν-generalized solutions

Results of numerical experiments presented in this subsection were obtained using the code "Proba-IV" [31] with regular meshes which were built by the following scheme:

Domain Ω was divided into squares by lines parallel to coordinate axis, with distance equal to 1/N between them, where N is a half of number of partitioning segments along the greater side;

Each square was subdivided into two triangles by the diagonal.

uh <sup>ν</sup>, <sup>1</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup> k¼1

Definition 2

is a function u<sup>h</sup>

holds, where u<sup>h</sup>

to the problem.

domain boundary [30].

regular part belongs to the W<sup>2</sup>

Problem A

Problem B

<sup>ν</sup> <sup>¼</sup> <sup>u</sup><sup>h</sup>

graded meshes of two kinds.

<sup>ν</sup>,1; uh ν, 2 � �

4. Results of numerical experiments

<sup>d</sup>2k�<sup>1</sup>Ψk, u<sup>h</sup>

300 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

and for arbitrary <sup>v</sup><sup>h</sup>ð Þ<sup>x</sup> <sup>∈</sup> <sup>V</sup><sup>h</sup> and <sup>ν</sup> <sup>&</sup>gt; <sup>β</sup> the integral identity

.

<sup>ν</sup>,<sup>2</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup> k¼1

a u<sup>h</sup>

<sup>d</sup>2kΨk, dj <sup>¼</sup> <sup>r</sup>�ν<sup>∗</sup>

<sup>ν</sup> ∈ V<sup>h</sup> such that it satisfies the boundary condition (3) in the nodes of the boundary ∂Ω

An approximate Rν-generalized solution to the problems (2)–(3) by the weighted finite-element method

<sup>ν</sup>; <sup>v</sup><sup>h</sup> � � <sup>¼</sup> <sup>l</sup> <sup>v</sup><sup>h</sup> � �,

In [18], it was shown that convergence rate of the approximate solution to the exact one does not depend on size of the re-entrant corner and is always equal to O hð Þ when weighted finiteelement method is used for finding an Rν-generalized solution to elasticity problem. The next section explains results of comparative numerical analysis for the model problems (2)–(3) of the weighted FEM using the classical finite-element method and the FEM with geometrically

In the domain, Ω is considered a Dirichlet problem for the Lamé system (2), (3) with constant coefficients λ ¼ 3 and μ ¼ 5. Two kinds of vector-function u ¼ ð Þ u1; u<sup>2</sup> were used as a solution

Components of the solution u of the model problem (2), (3) contain only a singular component

ð Þ <sup>x</sup><sup>1</sup> cos ð Þ <sup>x</sup><sup>2</sup> <sup>x</sup><sup>2</sup>

Singularity order of u1, u<sup>2</sup> corresponds to the size of the re-entrant corner γ ¼ 3π=2 on the

Solution u of the model problems (2, 3) contains both singular and regular components—

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2 � �<sup>0</sup>:<sup>3051</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2 � �<sup>0</sup>:<sup>3051</sup>

ð Þ <sup>x</sup><sup>2</sup> <sup>x</sup><sup>2</sup>

ð Þ <sup>x</sup><sup>1</sup> cos ð Þ <sup>x</sup><sup>2</sup> <sup>x</sup><sup>2</sup>

ð Þ <sup>x</sup><sup>2</sup> <sup>x</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2 � �<sup>0</sup>:<sup>3051</sup>,

<sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2 � �<sup>0</sup>:<sup>3051</sup>:

> <sup>þ</sup> <sup>x</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2 � �,

<sup>þ</sup> <sup>x</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2 � �:

<sup>u</sup><sup>1</sup> <sup>¼</sup> cos ð Þ <sup>x</sup><sup>1</sup> cos<sup>2</sup>

<sup>u</sup><sup>2</sup> <sup>¼</sup> cos<sup>2</sup>

<sup>2</sup>ð Þ Ω

<sup>u</sup><sup>1</sup> <sup>¼</sup> cos ð Þ <sup>x</sup><sup>1</sup> cos<sup>2</sup>

<sup>u</sup><sup>2</sup> <sup>¼</sup> cos2

P <sup>j</sup>þ<sup>1</sup> <sup>2</sup> ½ � � �

cj, j ¼ 1, …, 2n:

In this case, size of the mesh-step <sup>h</sup> could be computed by <sup>h</sup> <sup>¼</sup> ffiffiffi 2 <sup>p</sup> <sup>=</sup>N. Example of the regular mesh for N = 4 is presented in Figure 1.

Calculations were performed for different values of N. Optimal parameters δ, ν, and ν<sup>∗</sup> were obtained by the program complex [32]. Generalized solution was determined by the integral equality (5) for ν ¼ 0.

One calculated the errors <sup>e</sup> <sup>¼</sup> ð Þ¼ <sup>e</sup>1;e<sup>2</sup> <sup>u</sup><sup>1</sup> � uh <sup>1</sup>; <sup>u</sup><sup>2</sup> � <sup>u</sup><sup>h</sup> 2 � � and <sup>e</sup><sup>ν</sup> <sup>¼</sup> ð Þ¼ <sup>e</sup>ν, <sup>1</sup>;eν, <sup>2</sup> <sup>u</sup><sup>1</sup> � uh <sup>ν</sup>, <sup>1</sup>; � <sup>u</sup><sup>2</sup> � uh <sup>ν</sup>, <sup>2</sup><sup>Þ</sup> of numerical approximation to the generalized <sup>u</sup><sup>h</sup> <sup>¼</sup> uh <sup>1</sup>; uh 2 � � and Rν-generalized uh <sup>ν</sup> <sup>¼</sup> <sup>u</sup><sup>h</sup> <sup>1</sup>, <sup>ν</sup>; uh 2, ν � � solutions, respectively. Problems A and B in Tables 1 and <sup>4</sup>, respectively, present values of relative errors of the generalized solution in the norm of the Sobolev space

W<sup>1</sup> <sup>2</sup> <sup>η</sup> <sup>¼</sup> k k<sup>e</sup> <sup>W</sup><sup>1</sup> 2 k ku <sup>W</sup><sup>1</sup> 2 � � and the <sup>R</sup>ν-generalized one in the norm of the weighted Sobolev space <sup>W</sup><sup>1</sup> 2,ν k k e<sup>ν</sup> <sup>W</sup><sup>1</sup> !

η<sup>ν</sup> ¼ 2,ν k ku <sup>W</sup><sup>1</sup> 2,ν with different values of h. In addition, these tables contain ratios between error

norms, obtained on meshes with step reducing twice. Figures 2 and 3 show the convergence rates of the generalized and Rν-generalized solutions to the corresponding problems with the logarithmic scale. The dashed line in the figures corresponds to convergence with the rate O hð Þ. Tables 2 and 3 (Problem A) and Tables 5 and 6 (Problem B) give limit values: number of nodes where |e1|, |e2|, |ev,1|, and |ev,2| belong to the giving range, this number in percentage to the total number of nodes, and pictures of the absolute error distribution in the domain Ω.

Figure 1. Example of regular mesh (a), and graded meshes I (b) and II (c) (N = 4, κ ¼ 0:4).

#### 4.1.1. Problem A


4.1.2. Problem B

solution of the problem B on the mesh step.

2N 128 256 512 1024 2048 4096 h 1.105e-2 5.524e-3 2.762e-3 1.381e-3 6.905e-4 3.453e-4 η 2.849e-2 1.54 1.850e-2 1.53 1.205e-2 1.53 7.870e-3 1.53 5.146e-3 1.53 3.367e-3 η<sup>ν</sup> 2.868e-2 1.57 1.827e-2 1.65 1.107e-2 2.16 5.117e-3 2.21 2.319e-3 1.98 1.171e-3

Weighted Finite-Element Method for Elasticity Problems with Singularity

http://dx.doi.org/10.5772/intechopen.72733

303

Table 4. Dependence of relative errors of the generalized ð Þ<sup>η</sup> and <sup>R</sup>ν-generalized <sup>η</sup><sup>ν</sup> ð Þ <sup>ð</sup><sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:<sup>16</sup>

Figure 2. Chart of η for the generalized (squared line) and of η<sup>ν</sup> for Rν-generalized (circled line) (δ=0.0029, ν=1.2, ν\*=0.16)

Figure 3. Chart of η for the generalized (squared line) and of η<sup>ν</sup> for Rν-generalized (circled line) ðδ ¼ 0:0029, ν ¼ 1:2,

<sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16<sup>Þ</sup> solutions to the problem B in dependence on the number of subdivisions 2N.

solutions to the problem A in dependence on the number of subdivisions 2N.

Table 1. Dependence of relative errors of the generalized (η) and <sup>R</sup>ν-generalized (ην) (<sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16) solution to problem A on mesh step.


Table 2. Number, percentage equivalence, and distribution of nodes where absolute errors ∣ei∣ ð Þ i ¼ 1; 2 of finding components of the approximate generalized solution to problem A are not less than given limit values, 2N ¼ 4096.


Table 3. Number, percentage equivalence, and distribution of nodes where absolute errors ∣eν,i∣ ð Þ i ¼ 1; 2 of finding components of the approximate <sup>R</sup>ν-generalized solution to problem A <sup>ð</sup><sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16<sup>Þ</sup> are not less than given limit values, 2N ¼ 4096.


4.1.2. Problem B

4.1.1. Problem A

solution to problem A on mesh step.

given limit values, 2N ¼ 4096.

∣e1∣ ∣e2∣ Limit values ∣e1∣ ∣e2∣

Table 2. Number, percentage equivalence, and distribution of nodes where absolute errors ∣ei∣ ð Þ i ¼ 1; 2 of finding components of the approximate generalized solution to problem A are not less than given limit values, 2N ¼ 4096.

∣e1∣ ∣e2∣ Limit values ∣e1∣ ∣e2∣

2N 128 256 512 1024 2048 4096 h 1.105e-2 5.524e-3 2.762e-3 1.381e-3 6.905e-4 3.453e-4 η 6.963e-2 1.52 4.579e-2 1.52 3.007e-2 1.52 1.972e-2 1.53 1.293e-2 1.53 8.476e-3 η<sup>ν</sup> 7.011e-2 1.55 4.522e-2 1.64 2.756e-2 2.17 1.272e-2 2.21 5.745e-3 1.98 2.902e-3

302 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

Distribution Number % Number

Table 1. Dependence of relative errors of the generalized (η) and <sup>R</sup>ν-generalized (ην) (<sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16)

Distribution % Number % Number

Table 3. Number, percentage equivalence, and distribution of nodes where absolute errors ∣eν,i∣ ð Þ i ¼ 1; 2 of finding components of the approximate <sup>R</sup>ν-generalized solution to problem A <sup>ð</sup><sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16<sup>Þ</sup> are not less than

<sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>6</sup> 0.033 4102 0.033 4102 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>6</sup> 0.764 96075 0.764 96075 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>7</sup> 2.457 308985 2.457 308985 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>7</sup> 21.704 2729186 21.704 2729186 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>8</sup> 12.589 1582976 12.589 1582974 ≥ 0 62.454 7853397 62.454 7853399

<sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>6</sup> 48.077 6045579 48.077 6045579 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>6</sup> 29.387 3695290 29.387 3695290 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>7</sup> 6.724 845468 6.724 845468 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>7</sup> 9.624 1210192 9.624 1210192 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>8</sup> 2.564 322449 2.564 322449 ≥ 0 3.624 455743 3.624 455743

Table 4. Dependence of relative errors of the generalized ð Þ<sup>η</sup> and <sup>R</sup>ν-generalized <sup>η</sup><sup>ν</sup> ð Þ <sup>ð</sup><sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:<sup>16</sup> solution of the problem B on the mesh step.

Figure 2. Chart of η for the generalized (squared line) and of η<sup>ν</sup> for Rν-generalized (circled line) (δ=0.0029, ν=1.2, ν\*=0.16) solutions to the problem A in dependence on the number of subdivisions 2N.

Figure 3. Chart of η for the generalized (squared line) and of η<sup>ν</sup> for Rν-generalized (circled line) ðδ ¼ 0:0029, ν ¼ 1:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16<sup>Þ</sup> solutions to the problem B in dependence on the number of subdivisions 2N.


Mesh II. Constructing process for this mesh differs from the one described earlier in the level-

The FEM solution obtained on described graded meshes converges with the first rate on the mesh step when the value of the parameter κ is less than the order of singularity [2, 33].

Calculations were performed for different values of N and κ. For each node, one calculated the

on meshes I and II, respectively. The values of relative errors of the generalized solution to the

presented in Tables 8 and 11, respectively. In addition, these tables contain ratios between error norms and between mesh steps obtained with nodes number increasing four times. Figures 4 and 5 show the convergence rates of the generalized solutions to the corresponding problems for meshes I and II with the logarithmic scale. Dashed line in the figures corresponds to convergence with the rate O(h) as in paragraph 1. Besides, for the problems A and B, Tables 9 and 12, respectively, contain limit values for the following data: number of nodes where ∣e1,II∣, ∣e2,II∣ belong to the giving range, this number in percentage to the total number of

� � are presented in Tables 7 and <sup>10</sup>, respectively, and for mesh II <sup>η</sup>II <sup>¼</sup> k k <sup>e</sup>II <sup>W</sup><sup>1</sup>

2N 128 256 512 1024 2048 4096

η<sup>I</sup> 2.659e-2 2.00 1.332e-2 2.00 6.675e-3 1.91 3.501e-3 0.75 4.650e-3 0.27 1.741e-2 h 0.062263 1.979 0.031459 1.99 0.015812 1.995 0.007926 1.997 0.003968 1.999 0.001985

η<sup>I</sup> 2.111e-2 2.00 1.057e-2 1.99 5.302e-3 1.78 2.971e-3 0.53 5.559e-3 0.26 2.154e-2 h 0.044928 1.986 0.02262 1.993 0.011349 1.997 0.005684 1.998 0.002845 1.999 0.001423

η<sup>I</sup> 1.990e-2 1.99 1.001e-2 1.99 5.038e-3 1.71 2.940e-3 0.46 6.401e-3 0.25 2.513e-2 h 0.034611 1.99 0.017387 1.995 0.008714 1.998 0.004362 1.999 0.0021823 1.999 0.001092

η<sup>I</sup> 2.315e-2 1.92 1.204e-2 1.93 6.254e-3 1.70 3.678e-3 0.50 7.292e-3 0.26 2.818e-2 h 0.030169 1.993 0.015135 1.997 0.007580 1.998 0.003793 1.999 0.0018973 1.9996 0.0009489

Table 7. Dependence of relative errors of the generalized solution to problem A with mesh I on the mesh step for different κ.

∣N � ½ � ð Þ xi þ 1 N ∣. In this case, new coordinates are determined

Weighted Finite-Element Method for Elasticity Problems with Singularity

<sup>I</sup> , u<sup>h</sup>

<sup>2</sup> for different values of h and κ for mesh

http://dx.doi.org/10.5772/intechopen.72733

II obtained

305

2 k ku <sup>W</sup><sup>1</sup> 2 � � are

II of the approximate generalized solutions u<sup>h</sup>

calculating mode. Here, <sup>l</sup> <sup>¼</sup> <sup>P</sup>

only for nodes with l ≤ N.

errors <sup>e</sup><sup>I</sup> <sup>¼</sup> <sup>u</sup> � <sup>u</sup><sup>h</sup>

2 k ku <sup>W</sup><sup>1</sup> 2

<sup>I</sup> <sup>η</sup><sup>I</sup> <sup>¼</sup> k k <sup>e</sup><sup>I</sup> <sup>W</sup><sup>1</sup>

4.2.1. Problem A

κ ¼ 0:3

κ ¼ 0:4

κ ¼ 0:5

κ ¼ 0:6

2 i¼1

<sup>I</sup> and <sup>e</sup>II <sup>¼</sup> <sup>u</sup> � <sup>u</sup><sup>h</sup>

problems A and B in the norm of the Sobolev space W<sup>1</sup>

Examples of meshes I and II are shown in Figure 1(b) and (c), respectively.

nodes, and pictures of the absolute error distribution in the domain Ω.

Table 5. Number, percentage equivalence, and distribution of nodes where absolute errors ∣ei∣ ð Þ i ¼ 1; 2 of finding components of the approximate generalized solution to problem B are not less than given limit values, 2N ¼ 4096.


Table 6. Number, percentage equivalence, and distribution of nodes where absolute errors ∣eν,i∣ ð Þ i ¼ 1; 2 ) of finding components of the approximate <sup>R</sup>ν-generalized solution to problem B (ð<sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16<sup>Þ</sup> are not less than given limit values, 2N ¼ 4096.

#### 4.2. FEM with graded mesh: comparative analysis

This subsection presents results of error analysis for finding generalized solution to the problems A and B by the FEM with graded meshes of two kinds (for detailed information about graded meshes, see [2, 33, 34]).

Mesh I. This partitioning was built by the following scheme


Mesh II. Constructing process for this mesh differs from the one described earlier in the levelcalculating mode. Here, <sup>l</sup> <sup>¼</sup> <sup>P</sup> 2 i¼1 ∣N � ½ � ð Þ xi þ 1 N ∣. In this case, new coordinates are determined only for nodes with l ≤ N.

Examples of meshes I and II are shown in Figure 1(b) and (c), respectively.

The FEM solution obtained on described graded meshes converges with the first rate on the mesh step when the value of the parameter κ is less than the order of singularity [2, 33].

Calculations were performed for different values of N and κ. For each node, one calculated the errors <sup>e</sup><sup>I</sup> <sup>¼</sup> <sup>u</sup> � <sup>u</sup><sup>h</sup> <sup>I</sup> and <sup>e</sup>II <sup>¼</sup> <sup>u</sup> � <sup>u</sup><sup>h</sup> II of the approximate generalized solutions u<sup>h</sup> <sup>I</sup> , u<sup>h</sup> II obtained on meshes I and II, respectively. The values of relative errors of the generalized solution to the problems A and B in the norm of the Sobolev space W<sup>1</sup> <sup>2</sup> for different values of h and κ for mesh <sup>I</sup> <sup>η</sup><sup>I</sup> <sup>¼</sup> k k <sup>e</sup><sup>I</sup> <sup>W</sup><sup>1</sup> 2 k ku <sup>W</sup><sup>1</sup> 2 � � are presented in Tables 7 and <sup>10</sup>, respectively, and for mesh II <sup>η</sup>II <sup>¼</sup> k k <sup>e</sup>II <sup>W</sup><sup>1</sup> 2 k ku <sup>W</sup><sup>1</sup> 2 � � are presented in Tables 8 and 11, respectively. In addition, these tables contain ratios between error norms and between mesh steps obtained with nodes number increasing four times. Figures 4 and 5 show the convergence rates of the generalized solutions to the corresponding problems for meshes I and II with the logarithmic scale. Dashed line in the figures corresponds to convergence with the rate O(h) as in paragraph 1. Besides, for the problems A and B, Tables 9 and 12, respectively, contain limit values for the following data: number of nodes where ∣e1,II∣, ∣e2,II∣ belong to the giving range, this number in percentage to the total number of nodes, and pictures of the absolute error distribution in the domain Ω.



4.2. FEM with graded mesh: comparative analysis

Mesh I. This partitioning was built by the following scheme

graded meshes, see [2, 33, 34]).

given limit values, 2N ¼ 4096.

ð Þ <sup>l</sup>=<sup>N</sup> <sup>1</sup>=<sup>κ</sup> (<sup>i</sup> <sup>¼</sup> <sup>1</sup>, 2).

�NÞl �1

This subsection presents results of error analysis for finding generalized solution to the problems A and B by the FEM with graded meshes of two kinds (for detailed information about

Table 6. Number, percentage equivalence, and distribution of nodes where absolute errors ∣eν,i∣ ð Þ i ¼ 1; 2 ) of finding components of the approximate <sup>R</sup>ν-generalized solution to problem B (ð<sup>δ</sup> <sup>¼</sup> <sup>0</sup>:0029, <sup>ν</sup> <sup>¼</sup> <sup>1</sup>:2, <sup>ν</sup><sup>∗</sup> <sup>¼</sup> <sup>0</sup>:16<sup>Þ</sup> are not less than

∣e1∣ ∣e2∣ Limit values ∣e1∣ ∣e2∣

304 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

Distribution Number % Number

Table 5. Number, percentage equivalence, and distribution of nodes where absolute errors ∣ei∣ ð Þ i ¼ 1; 2 of finding components of the approximate generalized solution to problem B are not less than given limit values, 2N ¼ 4096.

∣e1∣ ∣e2∣ Limit values ∣e1∣ ∣e2∣

Distribution % Number % Number

<sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>6</sup> 48.078 6045622 48.078 6045622 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>6</sup> 29.387 3695278 29.387 3695278 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>7</sup> 6.724 845466 6.724 845466 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>7</sup> 9.624 1210158 9.624 1210159 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>8</sup> 2.564 322439 2.564 322438 ≥ 0 3.624 455758 3.624 455758

<sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>6</sup> 0.033 4108 0.033 4108 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>6</sup> 0.771 96899 0.771 96899 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>7</sup> 2.481 311996 2.481 311996 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>7</sup> 21.789 2739862 21.789 2739863 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>8</sup> 12.588 1582876 12.588 1582876 ≥ 0 62.339 7838980 62.339 7838979

1. In the domain Ω, for a given N, regular mesh was constructed as described in section 4.1.

2. Level l ¼ max<sup>i</sup>¼<sup>1</sup>, <sup>2</sup> ð Þ jN � ½ �j ð Þ xi þ 1 N was determined for each node. Here, xi (i ¼ 1, 2) are

3. New coordinates of nodes of the graded mesh are calculated by the formula ð½ � ð Þ xi þ 1 N

initial node coordinates on the regular mesh, ½ �� means integer part.

Table 7. Dependence of relative errors of the generalized solution to problem A with mesh I on the mesh step for different κ.


2N 128 256 512 1024 2048 4096

η<sup>I</sup> 7.625e-3 1.99 3.839e-3 1.92 1.995e-3 0.87 2.301e-3 0.27 8.676e-3 0.25 3.454e-2 h 0.034611 1.99 0.017387 1.995 0.008714 1.998 0.004362 1.999 0.002182 1.999 0.001091

Weighted Finite-Element Method for Elasticity Problems with Singularity

http://dx.doi.org/10.5772/intechopen.72733

307

η<sup>I</sup> 9.330e-3 1.92 4.849e-3 1.88 2.584e-3 0.91 2.847e-3 0.28 1.016e-2 0.25 4.001e-2 h 0.034611 1.99 0.017387 1.995 0.008714 1.998 0.004362 1.999 0.002182 1.999 0.001091

Table 10. Dependence of relative errors of the generalized solution to problem B with mesh I on the mesh step for

ηII 5.963e-3 2.00 2.982e-3 2.00 1.492e-3 1.91 7.819e-4 0.77 1.013e-3 0.27 3.757e-3 h 0.05114 1.982 0.025805 1.99 0.012962 1.995 0.006496 1.998 0.003252 1.999 0.0016267

ηII 6.349e-3 2.00 3.178e-3 2.00 1.591e-3 1.87 8.490e-4 0.67 1.263e-3 0.26 4.805e-3 h 0.038606 1.988 0.019417 1.994 0.009737 1.997 0.004876 1.999 0.00244 1.999 0.0012203

ηII 7.441e-3 1.98 3.756e-3 1.98 1.894e-3 1.83 1.037e-3 0.60 1.717e-3 0.26 6.606e-3 h 0.031006 1.99 0.015564 1.996 0.007797 1.998 0.003902 1.999 0.001952 1.9995 0.0009763

ηII 9.574e-3 1.91 5.000e-3 1.92 2.602e-3 1.85 1.409e-3 0.78 1.804e-3 0.27 6.660e-3 h 0.025906 1.995 0.012987 1.997 0.006502 1.999 0.003253 1.999 0.001627 1.9997 0.00081366

Table 11. Dependence of relative errors of the generalized solution to problem B with mesh II on the mesh step for

Figure 4. Chart of η<sup>I</sup> for mesh I (squared line) and of ηII for mesh II (circled line) for problem A depending on the number

κ ¼ 0:3

different κ.

κ ¼ 0:5

κ ¼ 0:6

κ ¼ 0:4

κ ¼ 0:5

κ ¼ 0:6

different κ.

of subdivisions 2N; κ ¼ 0:3.

Table 8. Dependence of relative errors of the generalized solution to problem A with mesh II on the mesh step for different κ.


Table 9. Number, percentage equivalence, and distribution of nodes where absolute errors ∣ei,II∣ ð Þ i ¼ 1; 2 of finding components of the approximate generalized solution to problem A obtained with mesh II ð Þ κ ¼ 0:5 are not less than given limit values, 2N ¼ 1024.

#### 4.2.2. Problem B



Table 10. Dependence of relative errors of the generalized solution to problem B with mesh I on the mesh step for different κ.


Table 11. Dependence of relative errors of the generalized solution to problem B with mesh II on the mesh step for different κ.

4.2.2. Problem B

κ ¼ 0:3

κ ¼ 0:4

given limit values, 2N ¼ 1024.

κ ¼ 0:3

κ ¼ 0:4

κ ¼ 0:5

κ ¼ 0:6

different κ.

2N 128 256 512 1024 2048 4096

306 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

ηII 2.392e-2 2.00 1.196e-2 2.00 5.982e-3 1.99 3.012e-3 1.46 2.059e-3 0.36 5.687e-3 h 0.05114 1.982 0.025805 1.99 0.012962 1.995 0.006496 1.998 0.003252 1.999 0.001627

ηII 1.974e-2 2.00 9.879e-3 2.00 4.942e-3 1.97 2.511e-3 1.16 2.167e-3 0.30 7.154e-3 h 0.038606 1.988 0.019417 1.994 0.009737 1.997 0.004876 1.999 0.00244 1.999 0.001220

ηII 1.954e-2 1.98 9.857e-3 1.99 4.963e-3 1.93 2.565e-3 0.94 2.726e-3 0.28 9.725e-3 h 0.031006 1.99 0.015564 1.996 0.007797 1.998 0.003902 1.999 0.001952 1.9995 0.000976

ηII 2.339e-2 1.91 1.225e-2 1.92 6.386e-3 1.90 3.368e-3 1.14 2.966e-3 0.31 9.712e-3 h 0.025906 1.995 0.012987 1.997 0.006502 1.999 0.003253 1.999 0.001627 1.9997 0.000814

Table 8. Dependence of relative errors of the generalized solution to problem A with mesh II on the mesh step for

Distribution % Number % Number

Table 9. Number, percentage equivalence, and distribution of nodes where absolute errors ∣ei,II∣ ð Þ i ¼ 1; 2 of finding components of the approximate generalized solution to problem A obtained with mesh II ð Þ κ ¼ 0:5 are not less than

2N 128 256 512 1024 2048 4096

η<sup>I</sup> 9.851e-3 1.99 4.955e-3 1.97 2.510e-3 1.36 1.845e-3 0.33 5.639e-3 0.25 2.247e-2 h 0.062263 1.979 0.031459 1.99 0.015812 1.995 0.007926 1.997 0.003968 1.999 0.001985

η<sup>I</sup> 7.712e-3 1.99 3.870e-3 1.95 1.988e-3 0.98 2.034e-3 0.28 7.218e-3 0.25 2.866e-2 h 0.044928 1.986 0.02262 1.993 0.011349 1.997 0.005684 1.998 0.002845 1.999 0.001423

<sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>6</sup> 0.001 6 0.001 6 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>6</sup> 35.524 278645 35.479 278292 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>7</sup> 13.631 106920 13.770 108011 <sup>≥</sup> <sup>1</sup><sup>e</sup> � <sup>7</sup> 33.363 261697 33.377 261808 <sup>≥</sup> <sup>5</sup><sup>e</sup> � <sup>8</sup> 7.020 55066 6.984 54782 ≥ 0 10.461 82051 10.389 81486

∣e1∣ ∣e2∣ Limit values ∣e1∣ ∣e2∣

Figure 4. Chart of η<sup>I</sup> for mesh I (squared line) and of ηII for mesh II (circled line) for problem A depending on the number of subdivisions 2N; κ ¼ 0:3.

For the approximate Rν-generalized solution obtained by the weighted finite-element method, an absolute error value is by one or two orders of magnitude less than the approximate generalized one obtained by the FEM or by the FEM with graded meshes; this holds for the

Weighted Finite-Element Method for Elasticity Problems with Singularity

http://dx.doi.org/10.5772/intechopen.72733

309

Viktor Anatolievich Rukavishnikov1,2\* and Elena Ivanovna Rukavishnikova1,2

2 Far Eastern State Transport University, Khabarovsk, Russian Federation

matical Methods in the Applied Sciences. 1996;19(1):63-85

Applied Mechanics and Engineering. 2013;253:252-273

550-560. DOI: 10.1134/S0012266107040131

boundary value problem belongs to the space W<sup>k</sup>þ<sup>2</sup>

2009;45(6):913-917. DOI: 10.1134/S0012266109060147

gence. Doklady Akademii Nauk SSSR. 1986;288(5):1058-1062

problem. Doklady Akademii Nauk SSSR. 1989;309(6):1318-1320

1 Far Eastern Branch, Russian Academy of Science, Khabarovsk, Russian Federation

[1] Szabó B, Babuška I. Finite Element Analysis. New York: Wiley; 1991. 368 p

[2] Apel T, Sändig A-M, Whiteman JR. Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Mathe-

[3] Nguyen-Xuan H, Liu GR, Bordas S, Natarajan S, Rabczuk T. An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order. Computer Methods in

[4] Rukavishnikov VA. The weight estimation of the speed of difference scheme conver-

[5] Rukavishnikov VA. On the differential properties of Rν-generalized solution of Dirichlet

[6] Rukavishnikov VA. On the uniqueness of the Rν-generalized solution of boundary value problems with noncoordinated degeneration of the initial data. Doklady Mathematics.

[7] Rukavishnikov VA, Ereklintsev AG. On the coercivity of the Rν-generalized solution of the first boundary value problem with coordinated degeneration of the input data.

[8] Rukavishnikov VA, Kuznetsova EV. A coercive estimate for a boundary value problem with noncoordinated degeneration of the input data. Differential Equations. 2007;43(4):

[9] Rukavishnikov VA, Kuznetsova EV. The Rν-generalized solution with a singularity of a

<sup>2</sup>,νþβ=2þkþ<sup>1</sup>ð Þ <sup>Ω</sup>; <sup>δ</sup> . Differential Equations.

Differential Equations. 2005;41(12):1757-1767. DOI: 10.1007/s10625-006-0012-5

overwhelming majority of nodes.

\*Address all correspondence to: vark0102@mail.ru

Author details

References

2001;63(1):68-70

Figure 5. Chart of η<sup>I</sup> for mesh I (squared line) and of ηII for mesh II (circled line) for problem B depending on the number of subdivisions 2N; κ ¼ 0:3.


Table 12. Number, percentage equivalence, and distribution of nodes where absolute errors ∣ei,II∣ ð Þ i ¼ 1; 2 of finding components of the approximate generalized solution to problem B obtained with mesh II ð Þ κ ¼ 0:5 are not less than given limit values, 2N ¼ 1024.
