1. Introduction

The singularity of the solution to a boundary value problem can be caused by the degeneration of the input data (of the coefficients and right-hand sides of the equation and the boundary conditions), by the geometry of the boundary, or by the internal properties of the solution. The classic numerical methods, such as finite-difference method, finite- and boundary-element methods, have insufficient convergence rate due to singularity which has an influence on the regularity of the solution. It results in significant increase of the computational power and time required for calculation of the solution with the given accuracy. For example, the classic finite-element method allows the finding of the solution for the elasticity problem posed in a two-dimensional domain containing a re-entrant corner of on the boundary with convergence rate O(h1/2). In this case to compute the solution with the accuracy of 10�<sup>3</sup> requires a computational power that is one million times greater than in the case of the weighted finite-element method used for the solution of the same problem.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

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 solution to the exact one [1–3].

where <sup>D</sup><sup>λ</sup> <sup>¼</sup> <sup>∂</sup><sup>∣</sup>λ<sup>∣</sup>

The set W � 1 =∂x<sup>λ</sup><sup>1</sup> <sup>1</sup> <sup>∂</sup>x<sup>λ</sup><sup>2</sup>

<sup>2</sup>,αð Þ <sup>Ω</sup>; <sup>δ</sup> <sup>⊂</sup> <sup>W</sup><sup>1</sup>

One can say that φ ∈ W<sup>1</sup>=<sup>2</sup>

Φð Þj x <sup>∂</sup><sup>Ω</sup> ¼ φð Þx and

� 1 <sup>2</sup>,αð Þ Ω; δ .

L2,αð Þ Ω; δ , W

Denoted by

a1ð Þ¼ u; v

a2ð Þ¼ u; v

l1ð Þ¼ v

ð

<sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> � � <sup>∂</sup>u<sup>1</sup>

∂x<sup>1</sup>

∂ r<sup>2</sup><sup>ν</sup>v<sup>2</sup> � � ∂x<sup>2</sup>

f <sup>1</sup>v1dx, l2ð Þ¼ v

Ω

ð

<sup>λ</sup> <sup>∂</sup>u<sup>1</sup> ∂x<sup>1</sup>

Ω

r<sup>2</sup><sup>ν</sup>

ð

Ω

a nonnegative real number.

<sup>2</sup> , λ = (λ1,λ2), and |λ|=λ1+λ2; λ1, λ<sup>2</sup> are nonnegative integers, and α is

Weighted Finite-Element Method for Elasticity Problems with Singularity

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

1 A

1=2 :

<sup>2</sup>,αð Þ Ω; δ is defined as the closure in norm (1) of the set C0ð Þ Ω; δ of

<sup>2</sup>,αð Þ <sup>Ω</sup>;<sup>δ</sup> :

� <sup>2</sup>div μεð Þ <sup>u</sup> � � <sup>þ</sup> <sup>∇</sup>ð Þ <sup>λ</sup>div<sup>u</sup> � �<sup>¼</sup> <sup>f</sup>, x<sup>∈</sup> <sup>Ω</sup>, (2)

, x∈ ∂Ω, (3)

<sup>2</sup>,<sup>β</sup> ð Þ ∂Ω; δ , i ¼ 1, 2, β > 0: (4)

þ μ ∂u<sup>2</sup> ∂x<sup>1</sup>

þ μ ∂u<sup>2</sup> ∂x<sup>1</sup> ∂ r<sup>2</sup><sup>ν</sup>v<sup>1</sup> � � ∂x<sup>2</sup>

∂ r<sup>2</sup><sup>ν</sup>v<sup>2</sup> � � ∂x<sup>1</sup>

�

�

∂ r<sup>2</sup><sup>ν</sup>v<sup>1</sup> � � ∂x<sup>1</sup>

∂ r<sup>2</sup><sup>ν</sup>v<sup>2</sup> � � ∂x<sup>2</sup>

<sup>2</sup>,αð Þ Ω; δ such that

<sup>2</sup>,αð Þ Ω; δ ,

297

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

ð

0 @

r<sup>2</sup><sup>α</sup>u<sup>2</sup> dx

<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>

k k Φ <sup>W</sup><sup>1</sup>

2 ∂ui ∂xj <sup>þ</sup> <sup>∂</sup>uj ∂xi � �.

<sup>þ</sup> <sup>λ</sup> <sup>∂</sup>u<sup>2</sup> ∂x<sup>2</sup>

∂x<sup>2</sup>

∂ r<sup>2</sup><sup>ν</sup>v<sup>1</sup> � � ∂x<sup>2</sup>

dx, �

<sup>þ</sup> <sup>λ</sup> <sup>þ</sup> <sup>2</sup><sup>μ</sup> � � <sup>∂</sup>u<sup>2</sup>

dx, �

Ω

k k<sup>u</sup> <sup>L</sup>2,αð Þ <sup>Ω</sup> <sup>¼</sup>

infinitely differentiable and finite in Ω functions satisfying conditions (a) and (b).

<sup>2</sup>,αð Þ <sup>∂</sup>Ω;<sup>δ</sup> <sup>¼</sup> inf Φj <sup>∂</sup>Ω¼ϕ

For the corresponding spaces and sets of vector-functions are used notationsW<sup>1</sup>

ui ¼ qi

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 displace-

k k φ <sup>W</sup>1=<sup>2</sup>

ment field u for the Lamé system with constant coefficients λ and μ:

Here, <sup>ε</sup>(u) is a strain tensor with components <sup>ε</sup>ijð Þ¼ <sup>u</sup> <sup>1</sup>

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

<sup>f</sup><sup>∈</sup> <sup>L</sup>2,βð Þ <sup>Ω</sup>; <sup>δ</sup> , qi <sup>∈</sup> <sup>W</sup><sup>1</sup>=<sup>2</sup>

∂ r<sup>2</sup><sup>ν</sup>v<sup>1</sup> � � ∂x<sup>1</sup>

> þ μ ∂u<sup>1</sup> ∂x<sup>2</sup>

> > ð

r<sup>2</sup><sup>ν</sup> f <sup>2</sup>v2dx

Ω

þ μ ∂u<sup>1</sup> ∂x<sup>2</sup>

∂ r<sup>2</sup><sup>ν</sup>v<sup>2</sup> � � ∂x<sup>1</sup>

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 .

In [4, 5], for boundary value problems with strongly singular solutions for which a generalized solution could not be defined and it does not belong to the Sobolev space H<sup>1</sup> , it was proposed to 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 two orders of magnitude less than that for the FEM with graded meshes.
