2. Rv-generalized solution

Let <sup>Ω</sup> ¼ �ð Þ� � <sup>1</sup>; <sup>1</sup> ð Þ <sup>1</sup>; <sup>1</sup> <sup>∖</sup>½ �� � <sup>0</sup>; <sup>1</sup> ½ � <sup>1</sup>; <sup>0</sup> <sup>⊂</sup>R<sup>2</sup> be an L-shaped domain with boundary <sup>∂</sup><sup>Ω</sup> containing re-entrant corner of 3π/2 with the vertex located in the point O(0,0), Ω ¼ Ω∪∂Ω:

Denote by <sup>Ω</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>∈</sup> <sup>Ω</sup> : <sup>x</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>x</sup><sup>2</sup> 2 � �<sup>1</sup>=<sup>2</sup> ≤ δ < 1 n o a part of <sup>δ</sup>-neighbourhood of the point (0,0) laying in the Ω. A weight function r(x) can be introduced that coincides with the distance to the origin in Ω<sup>0</sup> , and equals δ for x∈ Ω Ω<sup>0</sup> .

Let W<sup>1</sup> <sup>2</sup>,αð Þ Ω; δ be the set of functions satisfying the following conditions:


with the norm

$$\|\|u\|\|\_{W^{1}\_{2,a}(\Omega)} = \left(\sum\_{|\lambda| \le 1} \int\_{\Omega} \rho^{2a} \left| D^{\lambda} u \right|^2 d\mathbf{x}\right)^{1/2},\tag{1}$$

where <sup>D</sup><sup>λ</sup> <sup>¼</sup> <sup>∂</sup><sup>∣</sup>λ<sup>∣</sup> =∂x<sup>λ</sup><sup>1</sup> <sup>1</sup> <sup>∂</sup>x<sup>λ</sup><sup>2</sup> <sup>2</sup> , λ = (λ1,λ2), and |λ|=λ1+λ2; λ1, λ<sup>2</sup> are nonnegative integers, and α is a nonnegative real number.

Let L2,αð Þ Ω; δ be the set of functions satisfying conditions (a) and (b) with the norm

$$\|\|u\|\|\_{L\_{2,a}(\Omega)} = \left(\int\_{\Omega} \rho^{2\alpha} u^2 \, d\mathbf{x}\right)^{1/2}$$

The set W � 1 <sup>2</sup>,αð Þ <sup>Ω</sup>; <sup>δ</sup> <sup>⊂</sup> <sup>W</sup><sup>1</sup> <sup>2</sup>,αð Þ Ω; δ is defined as the closure in norm (1) of the set C0ð Þ Ω; δ of infinitely differentiable and finite in Ω functions satisfying conditions (a) and (b).

One can say that φ ∈ W<sup>1</sup>=<sup>2</sup> <sup>2</sup>,αð Þ <sup>∂</sup>Ω; <sup>δ</sup> if there exists a function <sup>Φ</sup> from <sup>W</sup><sup>1</sup> <sup>2</sup>,αð Þ Ω; δ such that Φð Þj x <sup>∂</sup><sup>Ω</sup> ¼ φð Þx and

$$\|q\|\_{W^{1/2}\_{2,a}(\partial\Omega,\delta)} = \inf\_{\Phi|\_{\partial\Omega} = \phi} \|\Phi\|\_{W^{1}\_{2,a}(\Omega,\delta)}.$$

For the corresponding spaces and sets of vector-functions are used notationsW<sup>1</sup> <sup>2</sup>,αð Þ Ω; δ , L2,αð Þ Ω; δ , W � 1 <sup>2</sup>,αð Þ Ω; δ .

Let u = (u1,u2) be a vector-function of displacements. Assume that Ω is a homogeneous isotropic body and the strains are small. Consider a boundary value problem for the displacement field u for the Lamé system with constant coefficients λ and μ:

$$-\left(2\operatorname{div}\left(\mu\varepsilon(\mathbf{u})\right) + \nabla(\lambda\operatorname{div}\mathbf{u})\right) \succcurlyeq \mathbf{f}, \quad \mathbf{x} \in \Omega,\tag{2}$$

:

$$
\mu\_i = q\_{i\prime} \quad \text{ x} \in \partial \Omega,\tag{3}
$$

Here, <sup>ε</sup>(u) is a strain tensor with components <sup>ε</sup>ijð Þ¼ <sup>u</sup> <sup>1</sup> 2 ∂ui ∂xj <sup>þ</sup> <sup>∂</sup>uj ∂xi � �.

Assume that the right-hand sides of (2), (3) satisfy the conditions

$$\mathbf{f} \in \mathbf{L}\_{2,\beta}(\Omega, \delta), \quad q\_i \in \mathcal{W}\_{2,\beta}^{1/2}(\partial \Omega, \delta), \quad i = 1, 2, \quad \beta > 0. \tag{4}$$

Denoted by

By using meshes refined toward the singularity point, it is possible to construct schemes of the finite-element method with the first order of the rate of convergence of the approximate

In [4, 5], for boundary value problems with strongly singular solutions for which a generalized

define the solution as a Rv-generalized one. The existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted Sobolev spaces and sets were proved [5–10], the weighted finite-element method was built, and its convergence rate was investigated [11–15]. In this chapter, for the Lamé system in domains containing re-entrant corners we will state construction and investigation of the weighted FEM for determination of the Rv-generalized solution [16, 17]. Convergence rate of this method did not depend on the corner size and was equal O(h) (see [18], Theorem 2.1). For the elasticity problems with solutions of two types with both singular and regular components and with singular component only—a comparative numerical analysis of the weighted finite-element method, the classic FEM, and the FEM with meshes geometrically refined toward the singularity point is performed. For the first two methods, the theoretical convergence rate estimations were confirmed. In addition, it was established that FEM with graded meshes failed on high dimensional meshes but weighted FEM stably found approximate solution with theoretical accuracy under the same computational conditions. The mentioned failure can be explained by a small size of steps of the graded mesh in a neighbourhood of the singular point. As a result, for the majority of nodes, the weighted finite-element method allows to find solution with absolute error which is by one or

Let <sup>Ω</sup> ¼ �ð Þ� � <sup>1</sup>; <sup>1</sup> ð Þ <sup>1</sup>; <sup>1</sup> <sup>∖</sup>½ �� � <sup>0</sup>; <sup>1</sup> ½ � <sup>1</sup>; <sup>0</sup> <sup>⊂</sup>R<sup>2</sup> be an L-shaped domain with boundary <sup>∂</sup><sup>Ω</sup> containing re-entrant corner of 3π/2 with the vertex located in the point O(0,0), Ω ¼ Ω∪∂Ω:

laying in the Ω. A weight function r(x) can be introduced that coincides with the distance to

a part of δ-neighbourhood of the point (0,0)

, where k = 0,1 and c<sup>1</sup> is a positive constant independent on

1 A

1=2

, (1)

≤ δ < 1

.

<sup>2</sup>,αð Þ Ω; δ be the set of functions satisfying the following conditions:

<sup>2</sup>,αð Þ <sup>Ω</sup> <sup>¼</sup> <sup>X</sup>

0 @

∣λ∣ ≤ 1

ð

r<sup>2</sup><sup>α</sup> D<sup>λ</sup>u � � � � 2 dx

Ω

, it was proposed to

solution could not be defined and it does not belong to the Sobolev space H<sup>1</sup>

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

two orders of magnitude less than that for the FEM with graded meshes.

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

, and equals δ for x∈ Ω Ω<sup>0</sup>

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

� <sup>≤</sup> <sup>c</sup>1ð Þ <sup>δ</sup>=rð Þ<sup>x</sup> <sup>α</sup>þ<sup>k</sup> for <sup>x</sup><sup>∈</sup> <sup>Ω</sup><sup>0</sup>

n o

solution to the exact one [1–3].

2. Rv-generalized solution

Denote by <sup>Ω</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup><sup>∈</sup> <sup>Ω</sup> : <sup>x</sup><sup>2</sup>

b. k k<sup>u</sup> <sup>L</sup>2,<sup>α</sup> <sup>Ω</sup>�Ω<sup>0</sup> ð Þ <sup>≥</sup> <sup>c</sup><sup>2</sup> <sup>&</sup>gt; <sup>0</sup>,

the origin in Ω<sup>0</sup>

Let W<sup>1</sup>

a. D<sup>k</sup> u xð Þ � � �

k,

with the norm

$$\begin{split} a\_{1}(\mathbf{u},\mathbf{v}) &= \int\_{\Omega} \left[ (\lambda + 2\mu) \frac{\partial u\_{1}}{\partial \mathbf{x}\_{1}} \frac{\partial (\rho^{2v} v\_{1})}{\partial \mathbf{x}\_{1}} + \mu \frac{\partial u\_{1}}{\partial \mathbf{x}\_{2}} \frac{\partial (\rho^{2v} v\_{1})}{\partial \mathbf{x}\_{2}} + \lambda \frac{\partial u\_{2}}{\partial \mathbf{x}\_{2}} \frac{\partial (\rho^{2v} v\_{1})}{\partial \mathbf{x}\_{1}} + \mu \frac{\partial u\_{2}}{\partial \mathbf{x}\_{1}} \frac{\partial (\rho^{2v} v\_{2})}{\partial \mathbf{x}\_{2}} \right] d\mathbf{x}\_{1} \\ a\_{2}(\mathbf{u},\mathbf{v}) &= \int\_{\Omega} \left[ \lambda \frac{\partial u\_{1}}{\partial \mathbf{x}\_{1}} \frac{\partial (\rho^{2v} v\_{2})}{\partial \mathbf{x}\_{2}} + \mu \frac{\partial u\_{1}}{\partial \mathbf{x}\_{2}} \frac{\partial (\rho^{2v} v\_{2})}{\partial \mathbf{x}\_{1}} + (\lambda + 2\mu) \frac{\partial u\_{2}}{\partial \mathbf{x}\_{2}} \frac{\partial (\rho^{2v} v\_{2})}{\partial \mathbf{x}\_{2}} + \mu \frac{\partial u\_{2}}{\partial \mathbf{x}\_{1}} \frac{\partial (\rho^{2v} v\_{2})}{\partial \mathbf{x}\_{1}} \right] d\mathbf{x}\_{1} \\ l\_{1}(\mathbf{v}) &= \int\_{\Omega} \rho^{2v} f\_{1} v\_{1} d\mathbf{x}, \quad l\_{2}(\mathbf{v}) = \int\_{\Omega} \rho^{2v} f\_{2} v\_{2} d\mathbf{x} \end{split}$$

the bilinear and linear forms and að Þ¼ u; v ð Þ a1ð Þ u; v ; a2ð Þ u; v , lð Þ¼ v ð Þ l1ð Þ v ; l2ð Þ v .

#### Definition 1

A function u<sup>v</sup> from the set W<sup>1</sup> <sup>2</sup>,νð Þ Ω; δ is called an Rv-generalized solution to the problem (2), (3) if it satisfies boundary condition (3) almost everywhere on ∂Ω and for every v from W<sup>1</sup> <sup>2</sup>, <sup>ν</sup>ð Þ Ω; δ the integral identity

$$a(\mathbf{u}\_{\mathbf{v}}, \mathbf{v}) = l(\mathbf{v}) \tag{5}$$

holds for any fixed value of ν satisfying the inequality

$$
\nu \ge \beta. \tag{6}
$$

integrals in both parts of the integral equality. Taking into account the local character of the singularity, we define weight function as the distance to each singularity point inside the disk of radius δ centered in that points, and outside these disks the weight function equals δ. An exponent of the weight function in the definition of the Rν-generalized solution as well as weighted space containing this solution depend on the spaces to which problem initial data belongs, on geometrical features of the boundary (re-entrant corners), and on changing of the

In [13, 14], for the transformed system of Maxwell equations in the domain with re-entrant

element method was developed on the basis of introducing the Rν-generalized solution. Convergence rate of this method is O(h), and it does not depend on the size of singularity as

The proposed approach of introducing Rν-generalized solution allows to effectively find solutions not only to the boundary value problems with divergent Dirichlet integral but also to

A finite-element scheme for problems (2)–(3) is constructed relying on the definition of an Rνgeneralized solution. For this purpose, a quasi-uniform triangulation T<sup>h</sup> of Ω and introduction

The domain Ω is divided into a finite number of triangles K (called finite elements) with vertices Pk (<sup>k</sup> = 1,…,N), which are triangulation nodes. Denoted by <sup>Ω</sup><sup>h</sup> <sup>¼</sup> <sup>∪</sup><sup>K</sup> <sup>∈</sup>T<sup>h</sup> <sup>K</sup>—the union of all elements; here, h is the longest of their side lengths. It is required that the partition satisfies the conventional constraints imposed on triangulations [10]. Denote by P ¼ f g Pk

k¼N

ð Þx ϕkð Þx , k ¼ 1, …, n,

2, the weighted edge-based finite-

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

299

Weighted Finite-Element Method for Elasticity Problems with Singularity

<sup>k</sup>¼nþ1, the set of nodes belonging to the <sup>∂</sup>Ω.

<sup>¼</sup> <sup>δ</sup>kj, k, j <sup>¼</sup> <sup>1</sup>, …, n <sup>δ</sup>kj is the Kronecker

, one singled out the subset <sup>V</sup>�<sup>h</sup> <sup>¼</sup> <sup>v</sup><sup>∈</sup> <sup>V</sup><sup>h</sup>

k¼n

<sup>k</sup>¼<sup>1</sup> . Denote the

; vi

<sup>2</sup> and does not belong to

k¼n k¼1,

corner in which the solution does not depend on the space W<sup>1</sup>

problems with weak singularity when the solution belongs to the W<sup>1</sup>

boundary condition type.

the space W<sup>2</sup>

2.

opposed to other methods [28, 29].

3. The weighted finite-element method

the set of triangulation internal nodes; by P ¼ f g Pk

where ϕkð Þx is linear on each finite element, ϕ<sup>k</sup> Pj

delta, and ν<sup>∗</sup> is a real number.

ð Þj Pk Pk <sup>∈</sup> <sup>∂</sup><sup>Ω</sup> ¼ 0, i ¼ 1, 2g.

corresponding vector set by <sup>V</sup><sup>h</sup> <sup>¼</sup> <sup>V</sup><sup>h</sup> <sup>2</sup>

placement vector components has the form

Each node Pk ∈P is associated with a function Ψ<sup>k</sup> of the form

<sup>Ψ</sup>kð Þ¼ <sup>x</sup> <sup>r</sup><sup>ν</sup><sup>∗</sup>

The set <sup>V</sup><sup>h</sup> is defined as the linear span of the system of basis functions f g <sup>Ψ</sup><sup>k</sup>

. In set V<sup>h</sup>

Associated with the constructed triangulation, the finite-element approximation of the dis-

of special basis functions are constructed.

In [17], for the boundary value problem (2)–(3) with homogeneous boundary conditions, existence and uniqueness of its Rv-generalized solution were established.

#### Theorem 1

Let condition (4) be satisfied. Then for any ν > β there always exists parameter δ such that the problem (2)–(3) with homogeneous boundary conditions has a unique Rv-generalized solution u<sup>v</sup> in the set W�<sup>1</sup> <sup>2</sup>,αð Þ Ω; δ . In this case

$$\|\|\mathbf{u}\_{\nu}\|\|\_{\mathbf{W}^{1}\_{2,r}(\Omega)} \leq c\_{3} \|\mathbf{f}\|\|\_{\mathbf{L}\_{2,\beta}(\Omega)^{\prime}}\tag{7}$$

where c3 is a positive constant independent of f.

Then for any ν > β, there always exists parameter δ such that the problem (2)–(3) with homogeneous boundary conditions has a unique Rv-generalized solution u<sup>v</sup> in the set W�<sup>1</sup> <sup>2</sup>,αð Þ Ω; δ .

#### Comment 1

At present, there exists a complete theory of classical solutions to boundary value problems with smooth initial data (equation coefficients, right hands of solution and boundary conditions) and with smooth enough domain boundary [19–22].

On the basis of the generalized solution-wide investigations of boundary value problems with discontinuous initial data and not smooth domain boundary were performed in Sobolev and different weighted spaces [23–26]. On the basis of the Galerkin method, theories of difference schemes, finite volumes, and finite-element method were developed to find approximate generalized solution [27].

Let us call boundary value problem a problem with strong singularity if its generalized solution could not be defined. This solution does not belong to the Sobolev space W<sup>1</sup> <sup>2</sup> (H<sup>1</sup> ), or, in other words, the Dirichlet integral of the solution diverges. In [4, 5], we suggested to define a solution to the boundary value problems with strong singularity as an Rv-generalized one in the weighted Sobolev space. The essence of this approach is in introducing weight function into the integral equality. The weight function coincides with the distance to the singular points in their neighbourhoods. The role (sense, mission) of this function is in suppressing of the solution singularity caused by the problem features and is in assuring convergence of integrals in both parts of the integral equality. Taking into account the local character of the singularity, we define weight function as the distance to each singularity point inside the disk of radius δ centered in that points, and outside these disks the weight function equals δ. An exponent of the weight function in the definition of the Rν-generalized solution as well as weighted space containing this solution depend on the spaces to which problem initial data belongs, on geometrical features of the boundary (re-entrant corners), and on changing of the boundary condition type.

In [13, 14], for the transformed system of Maxwell equations in the domain with re-entrant corner in which the solution does not depend on the space W<sup>1</sup> 2, the weighted edge-based finiteelement method was developed on the basis of introducing the Rν-generalized solution. Convergence rate of this method is O(h), and it does not depend on the size of singularity as opposed to other methods [28, 29].

The proposed approach of introducing Rν-generalized solution allows to effectively find solutions not only to the boundary value problems with divergent Dirichlet integral but also to problems with weak singularity when the solution belongs to the W<sup>1</sup> <sup>2</sup> and does not belong to the space W<sup>2</sup> 2.
