Divergent Integrals in Elastostatics: General Considerations

Abstract

This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed.

1. Introduction

A huge amount of publications is devoted to the boundary integral equation methods (BIM) and its application in science and engineering [1–5]. One of the main problems arising in the numerical solution of the BIE by the boundary element method (BEM) is a calculation of the divergent integrals. In mathematics divergent integrals have established theoretical basis. For example, the weakly singular integrals are considered as improper integrals; the singular integrals are considered in the sense of Cauchy principal value (PV); the hypersingular integrals had been considered by Hadamard as finite part integrals (FP). Different methods have been developed for calculation of the divergent integrals. Usually different divergent integrals need different methods for their calculation. Analysis of the most known methods used for treatment of the different divergent integrals has been done in [3, 4, 6–8]. The correct mathematical interpretation of the divergent integrals with different singularities has been done in terms of the theory of distributions (generalized functions). The theory of distributions provides a unified approach for the study of the divergent integrals and integral operators with kernels containing different kind of singularities.

In our previous publications [9–19], approach based on the theory of distributions has been developed for regularization of the divergent integrals with different singularities. The above mentioned approach for regularization of the hypersingular integrals has been used for the first time in [11]. Then it was further developed for static and dynamic problems of fracture mechanics in [15, 18, 19], respectively, and also in [9, 10, 13]. The equations presented in [12, 14] can be applied for a wide class of divergent integral regularizations for example for 2D and 3D elasticity [5, 14].

In the present paper, the approach for the divergent integral regularization based on the theory of distribution and Green’s theorem is further developed. The equations that enable easy calculation of the weakly singular, singular, and hypersingular integrals for different shape functions are presented here.

2. Main Equations of Elastostatics

Lets consider a homogeneous, lineally elastic body, which in three-dimensional Euclidean space ℜ³ occupies volume 𝑉 with smooth boundary 𝜕𝑉. The region 𝑉 is an open bounded subset of the three-dimensional Euclidean space ℜ³ with a 𝐶^0,1 Lipschitzian regular boundary 𝜕𝑉. The boundary contains two parts 𝜕𝑉_𝑢 and 𝜕𝑉_𝑝 such that 𝜕𝑉_𝑢∩𝜕𝑉_𝑝=∅ and 𝜕𝑉_𝑢∪𝜕𝑉_𝑝=𝜕𝑉. On the part 𝜕𝑉_𝑢 are prescribed displacement 𝑢_𝑖(𝐱)of the body points, and on the part 𝜕𝑉_𝑝 are prescribed traction 𝑝_𝑖(𝐱), respectively. The body may be affected by volume forces 𝑏_𝑖(𝐱). We assume that displacements of the body points and their gradients are small, so its stress-strain state is described by small strain deformation tensor 𝜀_𝑖𝑗(𝐱). The strain tensor and the displacement vector are connected by Cauchy relations𝜀_𝑖𝑗=12𝜕^𝑖𝑢_𝑗+𝜕_𝑗𝑢_𝑖,(2.1) where 𝜕/𝜕𝑥_𝑖 is a derivative with respect to space coordinates 𝑥_𝑖. The components of the strain tensor must also satisfy the Saint-Venant’s relations𝜕^2𝑘𝑙𝜀_𝑖𝑗−𝜕^2𝑖𝑙𝜀_𝑘𝑗=𝜕^2𝑘𝑗𝜀_𝑖𝑙−𝜕^2𝑖𝑗𝜀_𝑘𝑙.(2.2)

From the balance of impulse and the moment of impulse lows follow that the stress tensor is symmetric one and satisfy the equations of equilibrium𝜕_𝑗𝜎_𝑖𝑗+𝑏_𝑖=0_𝑖,∀𝐱∈𝑉.(2.3) Here and throughout the paper the summation convention applies to repeated indices.

The tensor of deformation 𝜀_𝑖𝑗(𝐱) and stress 𝜎_𝑖𝑗(𝐱) are related by Hook’s law𝜎_𝑖𝑗=𝑐_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗.(2.4) Here 𝑐_{𝑖𝑗𝑘𝑙} are elastic modules. In the case of homogeneous anisotropic medium they are symmetric 𝑐_{𝑖𝑗𝑘𝑙}=𝑐_{𝑗𝑖𝑘𝑙}=𝑐_{𝑘𝑙𝑖𝑗}.(2.5) and satisfy condition of ellipticity𝑐_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗𝜀_𝑘𝑙≥𝛼₁𝜀_𝑖𝑗𝜀_𝑖𝑗,∀𝜀_𝑖𝑗,∀𝛼₁>0.(2.6)

In the case of homogeneous isotropic medium, the elastic modules have the form 𝑐_{𝑖𝑗𝑘𝑙}=𝜆𝛿_𝑖𝑗𝛿_𝑘𝑙+𝜇𝛿_𝑖𝑘𝛿_𝑗𝑙+𝛿_𝑖𝑙𝛿_𝑗𝑘,(2.7) where 𝜆 and 𝜇 are Lame constants, 𝜇>0 and 𝜆>−𝜇, 𝛿_𝑖𝑗 is a Kronecker’s symbol.

Substituting stress tensor in (2.3) and using Hook’s law (2.4) and Cauchy relations (2.1), we obtain the differential equations of equilibrium in the form of displacements which may be presented in the form𝐴_𝑖𝑗𝑢_𝑗+𝑏_𝑖=0,∀𝐱∈𝑉.(2.8)

The differential operator 𝐴_𝑖𝑗 for homogeneous anisotropic medium has the form𝐴_𝑖𝑗=𝑐_{𝑖𝑘𝑗𝑙}𝜕_𝑘𝜕_𝑙,(2.9) and for homogeneous isotropic medium has the form𝐴_𝑖𝑗=𝜇𝛿_𝑖𝑗𝜕_𝑘𝜕_𝑘+(𝜆+𝜇)𝜕_𝑖𝜕_𝑗.(2.10)

If the problem is defined in an infinite region, then solution of (2.8) must satisfy additional conditions at the infinity in the form 𝑢_𝑗(𝐱)=𝑂𝑟⁻¹,𝜎_𝑖𝑗(𝐱)=𝑂𝑟⁻²for𝑟⟶∞,(2.11) where 𝑟=𝑥²¹+𝑥²²+𝑥²³ is the distance in the three-dimensional Euclidian space.

If the body occupied a finite region 𝑉with the boundary 𝜕𝑉, it is necessary to establish boundary conditions. We consider the mixed boundary conditions in the form 𝑝_𝑖𝑢_𝑖(𝐱)=𝜑_𝑖(𝐱),∀𝐱∈𝜕𝑉_𝑢,(𝐱)=𝜎_𝑖𝑗(𝐱)𝑛_𝑗(𝐱)=𝑃_𝑖𝑗𝑢_𝑗(𝐱)=𝜓_𝑖(𝐱),∀𝐱∈𝜕𝑉_𝑝.(2.12)

The differential operator 𝑃_𝑖𝑗∶𝑢_𝑗→𝑝_𝑖 is called stress operator. It transforms the displacements into the tractions. For homogeneous anisotropic and isotropic medium, they have the forms 𝑃_𝑖𝑗=𝑐_{𝑖𝑘𝑗𝑙}𝑛_𝑘𝜕_𝑙,𝑃_𝑖𝑗=𝜆𝑛_𝑖𝜕_𝑘+𝜇𝛿_𝑖𝑗𝜕_𝑛+𝑛_𝑘𝜕_𝑖,(2.13) respectively. Here 𝑛_𝑖 are components of the outward normal vector, 𝜕_𝑛=𝑛_𝑖𝜕_𝑖 is a derivative in direction of the vector 𝐧(𝐱) normal to the surface 𝜕𝑉_𝑝.

3. Integral Representations for Displacements and Traction

In order to establish integral representations for the displacements and tractions, let us consider bilinear form which depends on two fields of the strain tensor that correspond to two fields of the displacements 𝐮 and 𝐮^∗𝑎𝐮,𝐮^∗=𝑐_{𝑖𝑗𝑘𝑙}𝜀_𝑖𝑗(𝐮)𝜀_𝑘𝑙𝐮^∗.(3.1) Obviously that 𝑎𝐮,𝐮^∗=𝑎𝐮^∗,𝐮,𝑎(𝐮,𝐮)=𝜎_𝑖𝑗(𝐮)𝜀_𝑖𝑗(𝐮)≥𝛼₁𝜀_𝑖𝑗(𝐮).(3.2)

Integrating the equality (3.1) over the volume 𝑉 and applying the Gauss-Ostrogradskii formula, we will obtain_𝑉𝑎𝐮,𝐮^∗𝑑𝑉=_𝑉𝜎_𝑖𝑗(𝐮)𝜀_𝑘𝑙𝐮^∗𝑑𝑉=_𝜕𝑉𝜎_𝑖𝑗𝑛_𝑗𝑢^∗𝑖𝑑𝑆−_𝑉𝑢^∗𝑖𝜕_𝑗𝜎_𝑖𝑗𝑑𝑉.(3.3) Taking into account that 𝐴_𝑖𝑗𝑢_𝑗=𝜕_𝑗𝜎_𝑖𝑗 and 𝑝_𝑖=𝜎_𝑖𝑗𝑛_𝑗=𝑃_𝑖𝑗[𝑢_𝑗], we will find the first Betti’s theorem in the form_𝑉𝑢^∗𝑖𝐴_𝑖𝑗𝑢_𝑗𝑑𝑉=_𝑉𝑎𝐮,𝐮^∗𝑑𝑉−_𝜕𝑉𝑢^∗𝑖𝑃_𝑖𝑗𝑢_𝑗𝑑𝑆.(3.4) We will replace 𝑢_𝑖 and 𝑢^∗𝑖 in (3.4) and subtract resulting equation from (3.4). Because the form (3.1) is symmetrical one, we will obtain the second Betti’s theorem in the form_𝑉𝑢^∗𝑖𝐴_𝑖𝑗𝑢_𝑗−𝑢_𝑖𝐴_𝑖𝑗𝑢^∗𝑗𝑑𝑉=_𝜕𝑉𝑢_𝑖𝑃_𝑖𝑗𝑢^∗𝑗−𝑢^∗𝑖𝑃_𝑖𝑗𝑢_𝑗𝑑𝑆.(3.5) Taking into the account the definition of differential operator 𝐴_𝑖𝑗 given in (2.8) and the definition of differential operator 𝑃_𝑖𝑗 given in (2.12), we obtain relation _𝑉𝑏_𝑖𝑢^∗𝑖−𝑏^∗𝑖𝑢_𝑖𝑑𝑉=_𝜕𝑉𝑝^∗𝑖𝑢_𝑖−𝑝_𝑖𝑢^∗𝑖𝑑𝑆,(3.6) which is called the Betti’s reciprocal theorem.

This theorem is usually used for obtaining integral representations for the displacements and traction vectors. To do that, we consider solution of the elliptic partial differential equation (2.8) in an infinite space for the body force 𝑏^∗𝑖(𝐱)=𝛿_𝑖𝑗𝛿(𝐱−𝐲)𝐴_𝑖𝑗𝑈_𝑘𝑗(𝐱−𝐲)+𝛿_𝑘𝑖(𝐱−𝐲)=0,∀𝐱,𝐲∈ℜ³.(3.7) Now considering that 𝑢^∗𝑖(𝐱)=𝑈_𝑖𝑗(𝐱−𝐲),𝑝^∗𝑖(𝐱)=𝑃_𝑖𝑗𝑢^∗𝑗(𝐱)=𝑊_𝑖𝑗(𝐱,𝐲),(3.8)

from (3.6) we obtain the integral representation for the displacements vector𝑢_𝑖(𝐲)=_𝜕𝑉𝑝_𝑖(𝐱)𝑈_𝑗𝑖(𝐱−𝐲)−𝑢_𝑗(𝐱)𝑊_𝑗𝑖(𝐱,𝐲)𝑑𝑆+_𝑉𝑏_𝑖(𝐱)𝑈_𝑗𝑖(𝐱−𝐲)𝑑𝑉,(3.9) which is called Somigliana’s identity. The kernels 𝑈_𝑗𝑖(𝐱−𝐲) and 𝑊_𝑗𝑖(𝐱,𝐲) are called fundamental solutions for elastostatics.

Applying to (3.9) the differential operator 𝑃_𝑖𝑗, we will find integral representation for the traction in the form𝑝_𝑖(𝐲)=_𝜕𝑉𝑝_𝑖(𝐱)𝐾_𝑗𝑖(𝐱,𝐲)−𝑢_𝑗(𝐱)𝐹_𝑗𝑖(𝐱,𝐲)𝑑𝑆+_𝑉𝑏_𝑖(𝐱)𝐾_𝑗𝑖(𝐱,𝐲)𝑑𝑉.(3.10) The kernels 𝐾_𝑗𝑖(𝐱,𝐲) and 𝐹_𝑗𝑖(𝐱,𝐲) may be obtained applying the differential operator 𝑃_𝑖𝑗 to the kernels 𝑈_𝑗𝑖(𝐱−𝐲) and 𝑊_𝑗𝑖(𝐱,𝐲), respectively.

The integral representations (3.9) and (3.10) are usually used for direct formulation of the boundary integral equations in elastostatics.

4. Fundamental Solutions

In order to find the fundamental solutions 𝑈_𝑖𝑗(𝐱−𝐲) for the differential operator 𝐴_𝑖𝑗, we consider the differential equations of elastostatics in the form displacements (3.7). Solutions of these equations are called the fundamental solutions.

In 3D elastostatics, they have the form𝑈_𝑖𝑗(𝐱−𝐲)=116𝜋𝜇(1−𝜐)𝑟(3−4𝜐)𝛿^𝑖𝑗+𝜕_𝑖𝑟𝜕_𝑗𝑟.(4.1)

In 2D elastostatics they have the form𝑈_𝑖𝑗(𝐱−𝐲)=18𝜋𝜇(1−𝜐)(3−4𝜐)𝛿^𝑖𝑗ln1𝑟+𝜕^𝑖𝑟𝜕_𝑗𝑟.(4.2) Here 𝑟=(𝑥₁−𝑦₁)²+(𝑥₂−𝑦₂)²+(𝑥₃−𝑦₃)² and 𝑟=(𝑥₁−𝑦₁)²+(𝑥₂−𝑦₂)² for 3D and 2D case, respectively, 𝜕_𝑖𝑟=𝜕𝑟/𝜕𝑥_𝑖=−𝜕𝑟/𝜕𝑦_𝑖=(𝑥_𝑖−𝑦_𝑖)/𝑟.

The kernels 𝑊_𝑖𝑗(𝐱,𝐲) from (3.9) may be obtained by applying to 𝑈_𝑖𝑗(𝐱−𝐲) differential operator 𝑃_𝑖𝑘[⋅,(𝐱)]=𝜆𝑛_𝑖(𝐱)𝜕_𝑘[⋅]+𝜇𝛿_𝑖𝑘𝑛_𝑗(𝐱)𝜕_𝑗[⋅]+𝑛_𝑘(𝐱)𝜕_𝑖[⋅],(4.3) as it is shown here𝑊_𝑖𝑗(𝐱,𝐲)=𝜆𝑛_𝑖(𝐱)𝜕_𝑘𝑈_𝑘𝑗(𝐱,𝐲)+𝜇𝑛_𝑘(𝐱)𝜕_𝑘𝑈_𝑖𝑗(𝐱,𝐲)+𝜕_𝑖𝑈_𝑘𝑗(𝐱,𝐲).(4.4) Then after some transformations and simplifications, the expression for the kernels 𝑊_𝑖𝑗(𝐱,𝐲) will have the following form𝑊_𝑖𝑗(𝐱,𝐲)=−14𝜋(1−𝜐)𝑟^𝛼𝑛_𝑘(𝐱)𝜕_𝑘𝑟(1−2𝜐)𝛿_𝑖𝑗+𝛽𝜕_𝑖𝑟𝜕_𝑗𝑟+(1−2𝜐)𝑛_𝑖(𝐱)𝜕_𝑗𝑟−𝑛_𝑗(𝐱)𝜕_𝑖𝑟.(4.5)

The kernels 𝐾_𝑖𝑗(𝐱,𝐲) and 𝐹_𝑖𝑗(𝐱,𝐲) from (3.10) may be obtained by applying differential operator 𝑃_𝑖𝑘⋅,(𝐲)=𝜆𝑛_𝑖(𝐲)𝜕_𝑘+𝜇𝛿_𝑖𝑘𝑛_𝑗(𝐲)𝜕_𝑗+𝑛_𝑘(𝐲)𝜕_𝑖(4.6) to 𝑈_𝑖𝑗(𝐱−𝐲) and 𝑊_𝑖𝑗(𝐱,𝐲), respectively𝐹𝐾_𝑖𝑗(𝐱,𝐲)=𝜆𝑛_𝑖(𝐲)𝜕_𝑘𝑈_𝑗𝑘(𝐱,𝐲)+𝜇𝑛_𝑘(𝐲)𝜕_𝑘𝑈_𝑗𝑖(𝐱,𝐲)+𝜕_𝑖𝑈_𝑗𝑘(𝐱,𝐲)_𝑖𝑗,(𝐱,𝐲)=𝜆𝑛_𝑖(𝐲)𝜕_𝑘𝑊_𝑗𝑘(𝐱,𝐲)+𝜇𝑛_𝑘(𝐲)𝜕_𝑘𝑊_𝑗𝑖(𝐱,𝐲)+𝜕_𝑖𝑊_𝑗𝑘(𝐱,𝐲).(4.7) Then after some transformations and simplifications the expression for the kernels 𝐾_𝑖𝑗(𝐱,𝐲) will have the form𝐾_𝑖𝑗(𝐱,𝐲)=14𝜋(1−𝜐)𝑟^𝛼𝑛_𝑘(𝐲)𝜕_𝑘𝑟(1−2𝜐)𝛿_𝑖𝑗+𝛽𝜕_𝑖𝑟𝜕_𝑗𝑟+(1−2𝜐)𝑛_𝑖(𝐲)𝜕_𝑗𝑟−𝑛_𝑗(𝐲)𝜕_𝑖𝑟,(4.8) and for the kernels, 𝐹_𝑖𝑗(𝐱,𝐲) will have the form𝐹_𝑖𝑗(𝐱=,𝐲)𝜇2𝛼𝜋(1−𝜐)𝑟^𝛽−𝛾𝑛_𝑘𝛽𝑛_𝑘(𝐱)𝜕_𝑘𝑟(1−2𝜐)𝑛_𝑖(𝐲)𝜕_𝑗𝑟+𝜐𝛿_𝑖𝑗𝑛_𝑘(𝐲)𝜕_𝑘𝑟+𝑛_𝑗(𝐲)𝜕_𝑖𝑟+(1−(𝐲2)𝜐𝜕)_𝑘𝑟𝛽𝜕𝑛_𝑖𝑟_𝑗𝜕_𝑗𝑟+𝛽𝜈𝑛_𝑖(𝐱)𝑛_𝑘(𝐲)𝜕_𝑘𝑟𝜕_𝑗𝑟+𝑛_𝑘(𝐱)𝑛_𝑘(𝐲)𝜕_𝑖𝑟𝜕_𝑗𝑟(𝐱)𝑛_𝑘(𝐲)𝜕_𝑘𝑟𝜕_𝑖𝑟+𝑛_𝑘(𝐱)𝑛_𝑘(𝐲)𝛿_𝑖𝑗+𝑛_𝑖(𝐱)𝑛_𝑗(𝐲)−(1−4𝜐)𝑛_𝑗(𝐱)𝑛_𝑖(𝐲).(4.9) In (4.5)–(4.9), 𝛼=2,1, 𝛽=3,2 and 𝛾=5,4 in 3D and 2D cases, respectively.

The kernels (4.1)–(4.9) contain different kind singularities; therefore, corresponding integrals are divergent. Here we will investigate there singularities and develop methods of divergent integrals calculation.

5. Singularities, Boundary Properties, and Boundary Integral Equations

Simple observation shows that kernels in the integral representations (3.9) and (3.10) tend to infinity when 𝑟→0. More detailed analysis of (4.1)–(4.9) gives us the following results.

In the 3D case with 𝐱→𝐲,𝑈_𝑖𝑗(𝐱−𝐲)⟶𝑟⁻¹,𝑊_𝑖𝑗(𝐱,𝐲)⟶𝑟⁻²,𝐾_𝑖𝑗(𝐱,𝐲)⟶𝑟⁻²,𝐹_𝑖𝑗(𝐱,𝐲)⟶𝑟⁻³.(5.1)

In the 2D case with 𝐱→𝐲,𝑈_𝑖𝑗(𝐱−𝐲)⟶ln𝑟⁻¹,𝑊_𝑖𝑗(𝐱,𝐲)⟶𝑟⁻¹,𝐾_𝑖𝑗(𝐱,𝐲)⟶𝑟⁻¹,𝐹_𝑖𝑗(𝐱,𝐲)⟶𝑟⁻².(5.2) In order to investigate these functions and integrals with singular kernels, definition and classification of the integrals with various singularities will be presented here.

Definition 5.1. Let one considers two points with coordinated 𝐱,𝐲∈ℜ^𝑛 (where 𝑛=3 or 𝑛=2) and region 𝑉 with smooth boundary 𝜕𝑉 of the class 𝐶^0,1. The boundary integrals of the types _𝜕𝑉𝐺(𝐱,𝐲)𝑟^𝛼𝜑(𝐱)𝑑𝑆,𝛼>0,(5.3) where 𝐺(𝐱,𝐲) is a finite function in ℜ^𝑛×𝜕𝑉 and 𝜑(𝐱) is a finite function in 𝜕𝑉, are weakly singular for 𝛼<𝑛−1, singular for 𝛼=𝑛−1, and strongly singular of hypersingular for 𝛼>𝑛−1.

Definition 5.2. Let we consider two points with coordinated 𝐱,𝐲∈ℜ^𝑛 (where 𝑛=3 or 𝑛=2) and region 𝑉 with smooth boundary 𝜕𝑉 of the class 𝐶^0,1. The boundary integrals of the types _𝜕𝑉𝐺(𝐱,𝐲)ln(𝑟)𝜑(𝐲)𝑑𝐲,(5.4) where 𝐺(𝐱,𝐲) is a finite function in ℜ^𝑛×𝜕𝑉 and 𝜑(𝐱) is a finite function in 𝜕𝑉, are weakly singular.
The integrals with singularities can not be considered in usual (Riemann or Lebegue) sense. In order to such integrals have sense, it is necessary special consideration of them. We will apply this definition of the integrals from (3.9) and (3.10).

Definition 5.3. Integrals in (3.9) with kernels 𝑈_𝑖𝑗(𝐱−𝐲) are weakly singular and must be considered as improper 𝑊.𝑆._𝜕𝑉𝑝_𝑖(𝐱)𝑈_𝑖𝑗(𝐱−𝐲)𝑑𝑆=lim_𝜀→0_{𝜕𝑉⧵𝜕𝑉𝜀}𝑝_𝑖(𝐱)𝑈_𝑖𝑗(𝐱−𝐲)𝑑𝑆.(5.5) Here 𝜕𝑉_𝜀 is a part of the boundary, projection of which on tangential plane is contained in the circle 𝐶_𝜀(𝐱) of the radio 𝜀 with center in the point 𝐱.

Definition 5.4. Integrals in (3.9) and (3.10) with kernels 𝑊_𝑖𝑗(𝐱,𝐲) and 𝐾_𝑖𝑗(𝐱,𝐲) are singular and must be considered in sense of the Cauchy principal values as 𝑃.𝑉._𝜕𝑉𝑢_𝑖(𝐱)𝑊_𝑖𝑗(𝐱,𝐲)𝑑𝑆=lim_𝜀→0_{𝜕𝑉⧵𝜕𝑉(𝑟<𝜀)}𝑃.𝑉._𝜕𝑉𝑢_𝑖(𝐱)𝑊_𝑖𝑗(𝐱,𝐲)𝑑𝑆,𝑝_𝑖(𝐱)𝐾_𝑖𝑗(𝐱,𝐲)𝑑𝑆=lim_𝜀→0_{𝜕𝑉⧵𝜕𝑉(𝑟<𝜀)}𝑝_𝑖(𝐱)𝐾_𝑖𝑗(𝐱,𝐲)𝑑𝑆.(5.6) Here 𝜕𝑉(𝑟<𝜀) is a part of the boundary, projection of which on tangential plane is the circle 𝐶_𝜀(𝐱) of the radio 𝜀 with center in the point 𝐱.

Definition 5.5. Integrals in (3.10) with kernels 𝐹_𝑖𝑗(𝐱,𝐲) are hypersingular and must be considered in sense of the Hadamard finite part as 𝐹.𝑃._𝜕𝑉𝑢_𝑖(𝐱)𝐹_𝑗𝑖(𝐱−𝐲)𝑑𝑆=lim_𝜀→0_{𝜕𝑉⧵𝜕𝑉(𝑟<𝜀)}𝑢_𝑖(𝐱)𝑊_𝑗𝑖(𝐱−𝐲)𝑑𝑆+2𝑢_𝑗(𝐱)𝑓^𝑗(𝐱)𝜕𝑉(𝑟<𝜀).(5.7) Here functions 𝑓_𝑗(𝐱) are chosen from the condition of the limit existence.
Singular character of the channels in (3.9) and (3.10) determine boundary properties of the corresponding potentials. Analysis of these formulae show that the boundary potentials, with the kernels 𝑈_𝑖𝑗(𝐱−𝐲), are weakly singular, and, therefore, they are continuous everywhere in the ℜ^𝑛 and, therefore, may be continuously extended on the boundary 𝜕𝑉. The potentials with the kernels 𝑊_𝑖𝑗(𝐱,𝐲) and 𝐾_𝑖𝑗(𝐱,𝐲) contain singular kernels, and they jump when crossing the boundary 𝜕𝑉. The potential with the kernels 𝐹_𝑖𝑗(𝐱,𝐲) contain hypersingular kernels. They continuously cross the boundary 𝜕𝑉.
Boundary properties of these potentials have been studied extensively in [1, 2, 5, 8, 20] and so forth. Summary of these studies may be expressed by the equations _𝜕𝑉_𝜕𝑉𝑝_𝑖(𝐱)𝑈_𝑗𝑖(𝐱−𝐲)𝑑𝑆^±=_𝜕𝑉𝑝_𝑖(𝐱)𝑈_𝑗𝑖(𝐱−𝐲)𝑑𝑆⁰,𝑢_𝑖(𝐱)𝑊_𝑗𝑖(𝐱,𝐲)𝑑𝑆^±=∓12𝑢^𝑗_𝜕𝑉(𝐲)+_𝜕𝑉𝑢_𝑖(𝐱)𝑊_𝑗𝑖(𝐱,𝐲)𝑑𝑆⁰,𝑝_𝑖(𝐱)𝐾_𝑗𝑖(𝐱,𝐲)𝑑𝑆^±=±12𝑝^𝑗_𝜕𝑉(𝐲)+_𝜕𝑉𝑝_𝑖(𝐱)𝐾_𝑗𝑖(𝐱,𝐲)𝑑𝑆⁰,𝑢_𝑖(𝐱)𝐹_𝑗𝑖(𝐱−𝐲)𝑑𝑆^±=_𝜕𝑉𝑢_𝑖(𝐱)𝐹_𝑗𝑖(𝐱−𝐲)𝑑𝑆⁰.(5.8) The symbols “±” and “∓” denote that two equalities, one with the top sign and the other with the bottom sign, are considered. The up index “0” points out that the direct value of the corresponding potentials on the surface  𝜕𝑉 should be taken.
Now using integral representations for displacements and traction and boundary properties of the potentials, we can get boundary integral equations for elastostatics. Tending 𝐲 in (3.9) and (3.10) to the boundary 𝜕𝑉 and taking into consideration boundary properties of the potentials (5.8), we obtain representation of the displacements and traction vectors on the boundary surface 𝜕𝑉. On the smooth parts of the boundary, they have the following form ±12𝑢^𝑖∓1(𝐲)=_𝜕𝑉𝑝_𝑗(𝐱)𝑈_𝑖𝑗(𝐱−𝐲)𝑢_𝑗(𝐱)−𝑊_𝑖𝑗(𝐱,𝐲)𝑑𝑆+_𝑉𝑝_𝑗(𝐱)𝑈_𝑖𝑗(𝐱−𝐲)𝑑𝑉,2𝑝^𝑖(𝐲)=_𝜕𝑉𝑝_𝑗(𝐱)𝐾_𝑖𝑗(𝐱,𝐲)𝑢_𝑗(𝐱)−𝐹_𝑖𝑗(𝐱,𝐲)𝑑𝑆+_𝑉𝑝_𝑗(𝐱)𝐾_𝑖𝑗(𝐱,𝐲)𝑑𝑉.(5.9) The plus and minus signs in these equations are used for the interior and exterior problems, respectively. Together with boundary conditions, they are used for compositing the BIE for the problems of elastostatics. The BIE are usually solved numerical transforming them into discrete system of finite dimensional equations. Such method is called the boundary element method (BEM).

6. Projection Method and the BEM Equations

The main idea of the BEM consists in approximation of the BIE and further solution of that approximated finite dimensional BE system of equations. The mathematical essence of this approach is the so-called projection method. Let us outline some results from mathematical theory of the projection methods related to the approximation of the BIE. For more information, one can refer to [4, 21].

We consider two Banach spaces 𝐗 and 𝐘 and functional equation in those spaces𝐀⋅𝐮=𝐟,𝐮∈𝐷(𝐀)⊂𝐗,𝐟∈𝑅(𝐀)⊂𝐘.(6.1) Here 𝐀∶𝐗→𝐘 is the linear operator mapping from Banach space 𝐗 in Banach space 𝐘, 𝐷(𝐀) is a domain, and 𝑅(𝐀) is a range of the operator 𝐀. The equation (6.1) is named the exact equation, and its solution is the exact solution. We denote 𝐿(𝐗,𝐘) Banach space of the linear operators mapping from 𝐗 in 𝐘.

Let in 𝐗 and 𝐘 act sequences of projection operators 𝐏_ℎ and 𝐏^ℎ such that 𝐏𝐏^2ℎ=𝐏_ℎ,𝐏_ℎ𝐗=𝐗_ℎ,𝐗_ℎ⊂𝐗,_ℎ²=𝐏^ℎ,𝐏^ℎ𝐘=𝐘_ℎ,𝐘_ℎ⊂𝐘,(6.2) where 𝐗_ℎ and 𝐘_ℎ are finite dimension subspaces of the Banach spaces 𝐗 and 𝐘; ℎ∈𝑅¹ is a parameter of discretization.

Now we consider operator 𝐀_ℎ∈𝐿(𝐗_ℎ,𝐘_ℎ) mapping in finite dimensional subspaces 𝐗_ℎ and 𝐘_ℎ and an approximate equation𝐀_ℎ⋅𝐮_ℎ=𝐟_ℎ,𝐀_ℎ=𝐏^ℎ⋅𝐀⋅𝐏_ℎ,𝐮_ℎ=𝐏_ℎ⋅𝐮,𝐟_ℎ=𝐏^ℎ⋅𝐟.(6.3) Solution 𝐮_ℎ of (6.3) is the approximate solutions of (6.1). The general scheme of the approached equations construction (6.3) is illustrated by the following diagram.(6.4)

Now let us consider operator 𝐀_ℎ∈𝐿(𝐗_ℎ,𝐘_ℎ) mapping in finite-dimensional subspaces 𝐗_ℎ and 𝐘_ℎ and an approximate equation.

Existence of the exact solution, convergence of the approximate solution to the exact one, and stability of the approximations are the main problems which arrived in application of the projection methods. In order to solve these problems, we have to formulate them mathematically.

We assume that projection operators 𝐏_ℎ and 𝐏^ℎ converge to identity operators in 𝐗 and 𝐘, respectively. It means thatlilmim_{ℎ→0ℎ→0}‖‖𝐏_ℎ⋅𝐮−𝐮‖‖_𝐗=0∀𝐮∈𝐗,‖‖𝐏^ℎ⋅𝐟−𝐟‖‖_𝐘=0∀𝐟∈𝐘.(6.5)

Definition 6.1. Let conditions (6.5) be satisfied and stating from some ℎ=ℎ₀>0 for any 𝐟∈𝐘, (6.3) has unique solution 𝐮_ℎ. In this case if lim_ℎ→0‖‖𝐀_ℎ⋅𝐮_ℎ−𝐀⋅𝐮‖‖_𝐘=0,(6.6) then the solution of the approximate problem (6.3) converges to the exact solution (6.1). It means that the projective method presented on diagram (6.4) is applicable to the initial problem (6.1).

Definition 6.2. Let for some sequence of the operators {𝐀_ℎ} mapping from 𝐗_ℎ into 𝐘_ℎ there is a constant 𝛾>0 such that stating from some ℎ=ℎ₀>0‖‖𝐀_ℎ⋅𝐮_ℎ−𝐀⋅𝐮‖‖_𝐘≥𝛾‖‖𝐮_ℎ‖‖_𝐗∀𝐮_ℎ∈𝐗_ℎ(6.7) then for sequence of the operators {𝐀_ℎ}, the condition of stability of the approximate solution is satisfied.
Conditions (6.5)–(6.7) are very important for formulation conditions of existence, convergence and stability of the approximate solution. These conditions contain the following theorem [21, 22].

Theorem 6.3. Let the following conditions be satisfied: (1)the projection operators 𝐏_ℎ and 𝐏^ℎ converge to identity operators in the Banach spaces 𝐗 and 𝐘, respectively, as it stated in (6.5);(2)the sequence of approximate operators {𝐀_ℎ} converges to 𝐀 on each exact solution;(3)the condition of stability (6.7) is satisfied for the sequence of operators {𝐀_ℎ}.
Then the following consequences take place: (1)the exact solution exists and it is unique;(2)for all enough small h exists a unique solution 𝐮_ℎ∈𝐗_ℎ of the approximate equation (6.3);(3)the sequence of the approximate equation {𝐮_ℎ} converges to the exact one and takes place in the estimation ‖‖𝐮_ℎ−𝐏_ℎ⋅𝐮‖‖_𝐗ℎ≤𝛾⁻¹‖‖𝐏^ℎ𝐀⋅𝐮−𝐀_ℎ𝐏_ℎ𝐮‖‖_𝐘ℎ.(6.8)

Thus, using a projective method instead of the exact solution of (6.1) in functional space 𝐗, we can find the solution of the approximate equation (6.3) in finite dimensional space 𝐗_ℎ. The functional spaces {𝐗,𝐗_ℎ} and {𝐘,𝐘_ℎ} are related by means of the projection operators 𝐏_ℎ∈𝐿(𝐗,𝐗_ℎ) and 𝐏^ℎ∈𝐿(𝐘,𝐘_ℎ), respectively. It is also important to construct inverse operator 𝐏^ℎ−1∈𝐿(𝐗_ℎ,𝐗) which maps the finite dimensional space 𝐗_ℎ into the initial functional space 𝐗. Such operator refers to as the operator of interpolation. Because of 𝐗_ℎ⊂𝐗, the interpolation operator is not unique; moreover, for any two functional spaces 𝐗_ℎ and 𝐗, it is possible to construct infinite set of interpolation operators.

Let us apply the projection method to the BIE of elastostatics and construct corresponding finite dimensional BE equations. It is known [21, 23] that integral operators in (5.9) map between two functional spaces 𝐗(𝜕𝑉) and 𝐻^−1/2(𝜕𝑉) that are trace of displacements and traction on the boundary of the region in the following way: 𝐖𝐔_𝑖𝑗⋅𝐩_𝑗=_𝜕𝑉𝑈_𝑖𝑗(𝐱,𝐲)𝑝_𝑗(𝐱)𝑑𝑆∶𝐘(𝜕𝑉)⟶𝐗(𝜕𝑉),_𝑖𝑗𝐊⋅𝐮_𝑗=_𝜕𝑉𝑊_𝑖𝑗(𝐱,𝐲)𝑢_𝑗(𝐱)𝑑𝑆∶𝐗(𝜕𝑉)⟶𝐘(𝜕𝑉),_𝑖𝑗𝐅⋅𝐩_𝑗=_𝜕𝑉𝐾_𝑖𝑗(𝐱,𝐲)𝑝_𝑗(𝐱)𝑑𝑆∶𝐘(𝜕𝑉)⟶𝐗(𝜕𝑉),_𝑖𝑗⋅𝐮_𝑗=_𝜕𝑉𝐹_𝑖𝑗(𝐱,𝐲)𝑢_𝑗(𝐱)𝑑𝑆∶𝐗(𝜕𝑉)⟶𝐘(𝜕𝑉).(6.9)

We have to construct finite dimensional functional spaces that correspond to 𝐗(𝜕𝑉) and 𝐘(𝜕𝑉) and the corresponding projection operators. To construct finite dimensional functional spaces, we shall apply approximation by finite functions and splitting 𝜕𝑉 into finite elements 𝜕𝑉=^𝑁_𝑛=1𝜕𝑉_𝑛,𝜕𝑉_𝑛∩𝜕𝑉_𝑘=∅,if𝑛≠𝑘.(6.10)

Because 𝜕𝑉 is the boundary of the region, these elements are called boundary elements. On each boundary element, we shall choose 𝑄 nodes of interpolation. Local projection operators act from functional 𝐗(𝜕𝑉_𝑛) and 𝐘(𝜕𝑉_𝑛) to the finite directional ones 𝐗_𝑞(𝜕𝑉_𝑛) and 𝐘_𝑞𝜕𝑉_𝑛)𝐏𝐏^𝑢𝑞_𝑝𝑞∶𝐗𝜕𝑉_𝑛⟶𝐗_𝑞𝜕𝑉_𝑛∀𝐱∈𝜕𝑉_𝑛,∶𝐘𝜕𝑉_𝑛⟶𝐘_𝑞𝜕𝑉_𝑛∀𝐱∈𝜕𝑉_𝑛.(6.11) Global projection operators are defined as the sum of the local projection operators 𝐏^𝑢𝑛𝑞=^𝑁_𝑛=1𝐏^𝑢𝑞,𝐏^𝑝𝑛𝑞=^𝑁_𝑛=1𝐏^𝑝𝑞.(6.12) They map 𝐗(𝜕𝑉) and 𝐘(𝜕𝑉) to finite dimensional interpolations spaces 𝐏^𝑢𝑛𝑞∶𝐗(𝜕𝑉)⟶𝐗_𝑞^𝑁_𝑛=1𝜕𝑉_𝑛𝐏∀𝐱∈𝜕𝑉,_𝑝𝑛𝑞∶𝐘(𝜕𝑉)⟶𝐘_𝑞^𝑁_𝑛=1𝜕𝑉_𝑛∀𝐱∈𝜕𝑉.(6.13) The local projection operators 𝐏^𝑢𝑛 and 𝐏^𝑝𝑛 establish correspondence between vectors of displacements and traction and their value on the nodes of interpolation of the boundary elements 𝜕𝑉_𝑛𝐏𝐏^𝑢𝑞_𝑝𝑞⋅𝐮_𝑖(𝐱)=𝐮^𝑛𝑖𝐱_𝑞,𝑞=1,…,𝑄∀𝐱∈𝜕𝑉_𝑛,⋅𝐩_𝑖(𝐱)=𝐩^𝑛𝑖𝐱_𝑞,𝑞=1,…,𝑄∀𝐱∈𝜕𝑉_𝑛.(6.14) Similarly for the global operators, we have 𝐏𝐏^𝑢𝑛𝑞⋅𝐮_𝑖(𝐱)=𝐮^𝑛𝑖𝐱_𝑞,𝑞=1,…,𝑄;𝑛=1,…,𝑁∀𝐱∈𝜕𝑉,_𝑝𝑛𝑞⋅𝐩_𝑖(𝐱)=𝐩^𝑛𝑖𝐱_𝑞,𝑞=1,…,𝑄;𝑛=1,…,𝑁∀𝐱∈𝜕𝑉.(6.15) Let us construct local interpolation operators (𝐏^𝑢𝑞)⁻¹ and (𝐏^𝑝𝑞)⁻¹. For this purpose, we will introduce systems of shape functions 𝜑_𝑛𝑞(𝐱) and 𝜓_𝑛𝑞(𝐱) in the finite dimensional functional spaces 𝐗_𝑞(𝜕𝑉_𝑛) and 𝐘_𝑞(𝜕𝑉_𝑛). Then the vectors of displacements and traction on the boundary element 𝜕𝑉_𝑛 will be represented approximately in the form 𝑢_𝑖(𝐱)≈^𝑄_𝑞=1𝑝_𝑖𝑢^𝑛𝑖𝐱_𝑞𝜑_𝑛𝑞(𝐱),𝐱∈𝜕𝑉_𝑛,(𝐱)≈^𝑄_𝑞=1𝑝^𝑛𝑖𝐱_𝑞𝜑_𝑛𝑞(𝐱),𝐱∈𝜕𝑉_𝑛,(6.16) and on the whole crack surface 𝜕𝑉 in the form 𝑢_𝑖(𝐱)≈^𝑁_𝑄𝑛=1_𝑞=1𝑢^𝑛𝑖𝐱_𝑞𝜑_𝑛𝑞(𝐱),𝐱∈^𝑁_𝑛=1𝑝_𝑖𝜕𝑉_𝑛,(𝐱)≈^𝑁_𝑄𝑛=1_𝑞=1𝑝^𝑛𝑖𝐱_𝑞𝜑_𝑛𝑞(𝐱),𝐱∈^𝑁_𝑛=1𝜕𝑉_𝑛.(6.17)

Finite-dimensional analogies for the integral operators (6.9) are operators which map the finite-dimensional functional spaces 𝐗_𝑞(⋃^𝑁𝑛=1𝜕𝑉_𝑛) and 𝐘_𝑞(⋃^𝑁𝑛=1𝜕𝑉_𝑛), from one to another 𝐖𝐔^{𝑛𝑞𝑖𝑗}_{𝑛𝑞𝑖𝑗}=𝐏^𝑝𝑛𝑞⋅𝐔_𝑖𝑗⋅𝐏^𝑢𝑛𝑞∶𝐘_𝑞𝜕𝑉_𝑛⟶𝐗_𝑞𝜕𝑉_𝑛,𝐊_{𝑛𝑞𝑖𝑗}=𝐏^𝑝𝑛𝑞⋅𝐖_𝑖𝑗⋅𝐏^𝑢𝑛𝑞∶𝐗_𝑞𝜕𝑉_𝑛⟶𝐘_𝑞𝜕𝑉_𝑛,𝐅_{𝑛𝑞𝑖𝑗}=𝐏^𝑝𝑛𝑞⋅𝐊_𝑖𝑗⋅𝐏^𝑢𝑛𝑞∶𝐘_𝑞𝜕𝑉_𝑛⟶𝐗_𝑞𝜕𝑉_𝑛,=𝐏^𝑝𝑛𝑞⋅𝐅_𝑖𝑗⋅𝐏^𝑢𝑛𝑞∶𝐗_𝑞𝜕𝑉_𝑛⟶𝐘_𝑞𝜕𝑉_𝑛.(6.18)

Note that in contrast to differential operators, the integral operators are global, and they are defined in the entire space, that is, at every boundary element.

Substitution of the expressions (6.17) in (2.8) gives us the finite-dimensional representations for the vectors of displacements and traction on the boundary in the form 12𝑢^𝑚𝑖𝐲_𝑟=^𝑁_𝑄𝑛=1_𝑞=11𝑈^𝑛𝑗𝑖𝐲_𝑟,𝐱_𝑞𝑝^𝑛𝑗𝐱_𝑞−𝑊^𝑛𝑗𝑖𝐱_𝑟,𝐱_𝑞𝑢^𝑛𝑗𝐱_𝑞+𝑈_𝑖𝐟,𝐲,𝑉_𝑛,2𝑝^𝑚𝑖𝐲_𝑟=^𝑁_𝑄𝑛=1_𝑞=1𝐾^𝑛𝑗𝑖𝐲_𝑟,𝐱_𝑞𝑝^𝑛𝑗𝐲_𝑞−𝐹^𝑛𝑗𝑖𝐲_𝑟,𝐱_𝑞𝑢^𝑛𝑗𝐱_𝑞+𝐾_𝑖𝐟,𝐲,𝑉_𝑛,(6.19) where 𝑈𝐾^𝑛𝑗𝑖𝐲_𝑟,𝐱_𝑞=_𝜕𝑉𝑛𝑈_𝑗𝑖𝐲_𝑟,𝐱𝜓_𝑛𝑞(𝐱)𝑑𝑆,𝑊^𝑛𝑗𝑖𝐲_𝑟,𝐱_𝑞=_𝜕𝑉𝑛𝑊_𝑗𝑖𝐲_𝑟,𝐱𝜑_𝑛𝑞(𝐱)𝑑𝑆,_𝑛𝑗𝑖𝐲_𝑟,𝐱_𝑞=_𝜕𝑉𝑛𝐾_𝑗𝑖𝐲_𝑟,𝐱𝜓_𝑛𝑞(𝐱)𝑑𝑆,𝐹^𝑛𝑗𝑖𝐲_𝑟,𝐱_𝑞=_𝜕𝑉𝑛𝐹_𝑗𝑖𝐲_𝑟,𝐱𝜑_𝑛𝑞(𝐱)𝑑𝑆.(6.20) The volume potentials 𝑈_𝑖(𝐟,𝐲,𝑉_𝑛) and 𝐾_𝑖(𝐟,𝐲,𝑉_𝑛) depend on discretization of the 𝑉 domain. More detailed information on transition from the BIE to the BEM equations can be found in [1, 2, 4, 5, 20].

7. Boundary Elements and Approximation

The BEM can be treated as the approximate method for the BIE solution, which includes approximation of the functions that belong to some functional space by discrete finite model. This model comprises finite number of values of the considered functions which are used for approximation of these functions by the shape functions determined on small subdomains called boundary elements. In this senses the BEM is closely related to a finite element method where the functions also belong to corresponding functional spaces and are approximated by finite model. Below we shall speak about finite element approximations and finite elements (FE), keeping in mind that boundary elements are their specific case.

It is important to mention that the local approximation of the considered function on one FE can be done independently from other FEs. It means that it is possible to approximate function on an FE by means of its values on the nodes independently of the place occupied considered the FE in the FE model and how behave the function on other FEs. Hence, it is possible to create the catalogue of various FE or BE with arbitrary node values interpolation function. Then from this catalogue can be chosen FEs which are necessary for approximation of the function and domain of its definition. The same FE can be used for discrete models of various functions or physical fields by determination of the necessary position of nodes in the model and further definition of the node values of the function or physical field. Thus, finite models of an area and its boundary do not not depend on functions and physical fields for which this area can be a domain of definition.

Let us consider how to construct an FE model of an area 𝑉⊂𝑅^𝑛 and a BE model of its boundary 𝜕𝑉⊂𝑅^𝑛−1 (𝑛=2,3). We fix in the area 𝑉 finite number of points 𝐱^𝑞(𝑔=1,…,𝑄); these points refer to as global node points 𝑉(𝑞)={𝐱^𝑞∈𝑉∶𝑔=1,…,𝑄}. We shall divide the area 𝑉 into finite number of subareas 𝑉_𝑛(𝑛=1,…,𝑁) which are FEs. They have to satisfy the following conditions: 𝑉_𝑛∪𝑉_𝑚=∅,𝑚≠𝑛,𝑚,𝑛=1,2,…,𝑁,𝑉=^𝑁_𝑛=1𝑉_𝑛.(7.1) On each FE we introduce a local coordinate system 𝜉. The nodal points 𝐱^𝑞∈𝑉_𝑛 in the local system of coordinates we designate by 𝜉^𝑞. They are coordinates of nodal points in the local coordinate system. Local and global coordinate are related in the following way:𝐱^𝑞=^𝑁_𝑛=1𝚲_𝑛𝜉^𝑞𝑛.(7.2) Functions Λ_𝑛 depend on position of the nodal points in the FE and BE. They join individual FE together in a FE model. Borders of the FEs and position of the nodal points should be such that, after joining together, separate elements form discrete model of the area 𝑉.

Having constructed FE model of the area 𝑉, we shall consider approximation of the function 𝑓(𝐱) that belong to some functional space. The FE model of the area 𝑉 is the domain of function which should be approximated. We denote function 𝑓(𝐱) on the FE 𝑉_𝑛 by 𝑓^𝑛(𝐱). Then𝑓(𝐱)=^𝑁_𝑛=1𝑓^𝑛(𝐱).(7.3) On each FE, the local functions 𝑓^𝑛(𝐱) may be represented in the form 𝑓^𝑛(𝐱)≈^𝑄_𝑞=1𝑓^𝑛(𝐱^𝑞)𝜑_𝑛𝑞(𝜉),(7.4) where 𝜑_𝑛𝑞(𝜉) are interpolation polynomials or nodal functions of the FE with number 𝑛. In the nodal point with coordinates 𝐱^𝑞, they are equal to 1, and in other nodal points are equal to zero. Taking into account (7.3) and (7.4), the global approximation of the function 𝑓(𝐱) looks like𝑓(𝐱)≈^𝑁_𝑄𝑛=1_𝑞=1𝑓^𝑛𝐱_𝑞𝜑_𝑛𝑞(𝜉).(7.5) If the nodal point 𝑞 belongs to several Fes, it is considered in these sums only once.

The FE and BE elements can be of various form and sizes, their surfaces can be curvilinear. The curvilinear FEs are very important in BEM because the boundary surface is usually curvilinear. But it is more convenient to use standard FE, whose surfaces coincide with coordinate planes of the local coordinates system. Mathematically it means that it is necessary to establish relation between local coordinates 𝜉_𝑖, in which the element has a simple appearance, and global 𝑥_𝑖 where the FE represents more complex figure. Local coordinates 𝜉_𝑖 should be functions of global (𝜉_𝑖(𝑥₁,𝑥₂,𝑥₃)) ones, and on the contrary, global coordinates should be functions of (𝑥_𝑖(𝜉₁,𝜉₂,𝜉₃))ones. In order to these maps be one-to-one, it is necessary and sufficient that the Jacobians of transformations be nonzero𝐽=det||||𝜕𝑥^𝑖𝜕𝜉_𝑗||||≠0,𝐽₋₁=det||||𝜕𝜉^𝑖𝜕𝑥_𝑗||||≠0.(7.6) The differential elements along coordinate axes are related by 𝑑𝑥_𝑖=𝜕𝑥^𝑖𝜕𝜉_𝑗𝑑𝜉^𝑗𝑑𝜉_𝑖,𝑑𝐱=𝐽(𝜉)𝑑𝜉,=𝜕𝜉^𝑖𝜕𝑥_𝑗𝑑𝑥^𝑗,𝑑𝜉=𝐽⁻¹(𝐱)𝑑𝐱.(7.7) The volume element in the ℜ³ is transformed under the formula𝑑𝑉=𝑑𝑥₁𝑑𝑥₂𝑑𝑥₃=𝐽(𝜉)𝑑𝜉₁𝑑𝜉₂𝑑𝜉₃,(7.8) and the area element in the ℜ² is transformed under the formula𝑑𝐴=𝑑𝑥₁𝑑𝑥₂=det||||𝜕𝑥^𝛼𝜕𝜉_𝛽||||𝑑𝜉₁𝑑𝜉₂,𝛼,𝛽=1,2.(7.9) The differential of the surface located in the ℜ³ is defined by the expression𝑑𝑆=𝑛²¹+𝑛²²+𝑛²³^1/2𝑑𝜉₁𝑑𝜉₂,(7.10) where𝑛₁=𝜕𝑥¹𝜕𝜉₁𝜕𝑥₃𝜕𝜉₂−𝜕𝑥²𝜕𝜉₂𝜕𝑥₃𝜕𝜉₁𝑛₂,=𝜕𝑥³𝜕𝜉₁𝜕𝑥₁𝜕𝜉₂−𝜕𝑥¹𝜕𝜉₁𝜕𝑥₃𝜕𝜉₂𝑛₃,=𝜕𝑥¹𝜕𝜉₁𝜕𝑥₂𝜕𝜉₂−𝜕𝑥²𝜕𝜉₁𝜕𝑥₁𝜕𝜉₂.(7.11) The element of length of a contour in the ℜ² is defined by the expression𝑑𝑙=𝑑𝑥¹𝑑𝜉₁²+𝑑𝑥²𝑑𝜉₁²^1/2𝑑𝜉₁.(7.12)

It is important to point attention that it is quite enough to consider standard FE which can be transformed to the necessary form by suitable transformation of coordinates. The FE approximation has to be linear independent and compact in the corresponding functional space.

We consider here some examples of the FE approximation, which are frequently used in the BEM.

7.1. Piecewise Constant Approximation

The piecewise constant approximation is the simplest one. Interpolation functions in this case do not depend on the FE form and the dimension of the domain. They have the form𝜑_𝑛𝑞(𝐱)=1∀𝐱∈𝑉^𝑛0∀𝐱∉𝑉_𝑛,.(7.13) This approximation is linear independent and compact in the functional spaces in which it is not necessary to approximate the derivative of the function.

7.2. Piecewise Linear Approximation

Interpolation functions in this case change linearly inside the FE and depend on its form and dimension of the domain. Therefore, interpolation functions are called shape functions.

In 1D case, the FEs are linear, and their shape functions are 𝜑_𝑛𝑞=121+𝜉^𝑞1𝜉₁,𝜉^𝑞𝑖=±1.(7.14)

In 2D case for the rectangular FE, shape functions are𝜑_𝑛𝑞=141+𝜉^𝑞1𝜉₁1+𝜉^𝑞2𝜉₂,𝜉^𝑞𝑖=±1.(7.15)

For the triangular FE, it is convenient to introduce system of coordinates which connected with the Cartesian by relation 𝑥_𝑖=𝜉_𝑞𝑥^𝑞𝑖,𝜉_𝑞=1,𝑖=1,2,𝑞=1,2,3.(7.16) For the triangular FE, the interpolation functions are 𝜑_𝑛𝑞=𝜉_𝑞.(7.17) The local coordinates 𝜉_𝑞 are also called area coordinates.

7.3. Piecewise Quadratic Approximation

Interpolation functions in this case change quadratically inside the FE and depend on its form and dimension of the domain.

In 1D case, the FEs are bilinear, and their shape functions are 𝜑_𝑛𝑞=12𝜉^𝑞1𝜉₁1+𝜉^𝑞1𝜉₁(𝑞=1,2,3)(7.18) at the end nodes and 𝜑_𝑛𝑞=1−𝜉²¹(7.19) at the midpoint node.

In 2D case for the rectangular FE, shape functions are𝜑_𝑛𝑞=141+𝜉^𝑞1𝜉₁1+𝜉^𝑞2𝜉₂𝜉^𝑞1𝜉₁−𝜉^𝑞2𝜉₂−1(𝑞=1,…,8)(7.20) at the angular nodes and 𝜑_𝑛𝑞=121−𝜉²¹1+𝜉^𝑞2𝜑𝜉₂𝜉^𝑞1=0,_𝑛𝑞=121−𝜉²²1+𝜉^𝑞1𝜉₁𝜉^𝑞2=0(7.21) at the side nodes.

For the triangular FE, the interpolation functions are 𝜑_𝑛𝑞=2𝜉_𝑞−1𝜉_𝑞(𝑞=1,…,6)(7.22) at the angular nodes and 𝜑_𝑛𝑞=4𝜉₁𝜉₂⋯.(7.23) at the side nodes.

In order to construct system of algebraic equation for the BEM (6.18), we have to represent global coordinates as function of local ones. The most convenient way to do that is to use the interpolation function 𝜑_𝑛𝑞 defined in (7.13)–(7.23). In this case, the global coordinates have the form𝑥_𝑖=𝑥^𝑞𝑖𝜑_𝑛𝑞(𝜉).(7.24) Here 𝑥^𝑞𝑖 is the global coordinate of nodal points for the FE number 𝑛.

8. Regularization of the Divergent Integrals

As it was mentioned above in order to solve the BIE (5.9) numerically, we have to transform them to the finite dimensional BE equations (6.18). In order to do that transform, we have to calculate integrals (6.19). One of the main problems that occur in this situation is the presence of the divergent integrals. They can not be calculated in traditional way, for example, numerically using quadrature formulas. For example, the integrals with the kernels 𝑈_𝑖𝑗(𝐱−𝐲) are weakly singular (WS). They have to be considered as improper integrals. The integrals with the kernels 𝑊_𝑖𝑗(𝐱,𝐲) and 𝐾_𝑖𝑗(𝐱,𝐲) are singular. They have to be considered in the sense of Cauchy as principal values (PVs). The integrals with the kernels 𝐹_𝑖𝑗(𝐱,𝐲) are hypersingular. They have to be considered in the sense of Hadamard as finite parts (FPs). Traditional approach to the divergent integral calculation may be found in [1, 2, 6, 8]. Following [11–16], we will develop here and apply the method of the divergent integral calculation based on the theory of distribution. This approach consists in application of the second Green theorem and transformation of divergent integrals into regular ones. In [12] we have developed formulas for regularization of the divergent integrals 𝑟^−𝑚 in the form𝐹.𝑃.^𝑎−𝑎𝜑(𝑥)𝑟^𝑚𝑑𝑥=^𝑘−1_𝑖=0(−1)^𝑖+1𝑑^𝑖𝑑𝑥^𝑖𝑃_𝑖𝑟^𝑚−𝑘𝑑^{𝑘−1−𝑖}𝜑(𝑥)𝑑𝑥^{𝑘−1−𝑖}||||^{𝑥=𝑎𝑥=−𝑎}+(−1)^𝑘^𝑎−𝑎𝑃_𝑘𝑟^𝑚−𝑘𝑑^𝑘𝜑(𝑥)𝑑𝑥^𝑘𝐹.𝑃.∫_𝑉,𝜑(𝐱)𝑟^𝑚𝑑𝑉=∑^𝑘−1_𝑖=0(−1)^𝑖+1_𝜕𝑉Δ^{𝑘−𝑖−1}𝜑(𝐱)𝜕_𝑛𝑃_𝑖𝑟^𝑚−2𝑖−𝑃^𝑖𝑟^𝑚−2𝑖+(−1)_𝑘𝜕_𝑛Δ^{𝑘−𝑖−1}𝜑(𝐱)𝑑𝑆∫_𝑉1𝑟^𝑚−2𝑘Δ^𝑘+1𝜑(𝐱)𝑑𝑉,(8.1) for 1D and 2D cases, respectively. We will demonstrate here how this approach works in the problems of elastostatics.

8.1.  1D Divergent Integrals

In 2D elastostatics after introducing local system of coordinates and simplification, all divergent integrals can be presented in the form 𝐽₀=^𝑏𝑎𝜑(𝑥)ln1𝑥𝑑𝑥,𝐽^𝑘=^𝑏𝑎𝜑(𝑥)𝑥^𝑘𝑑𝑥,𝑘=1,2(8.2) Here 𝜑(𝑥) is the smooth function that depend on the shape of the BE and the interpolation polynomials.

We consider first the weakly singular integral 𝐽₀. Because of logarithmical singularity, we can not use formula (8.1). Therefore, we start from the formula for integration by parts in the form^𝑏𝑎𝑑𝑔(𝑥)𝑑𝑥𝜑(𝑥)𝑑𝑥=𝜑(𝑥)𝑔(𝑥)||^𝑏𝑎−^𝑏𝑎𝑑𝜑(𝑥)𝑑𝑥𝑔(𝑥)𝑑𝑥.(8.3) Let in this formula be 𝑔(𝑥)=𝑥ln1/𝑥, 𝑑𝑔(𝑥)/𝑑𝑥=ln1/𝑥, then we obtain 𝐽₀=𝑊.𝑆.^𝑏𝑎𝜑(𝑥)ln1𝑥𝑑𝑥=𝜑(𝑥)𝑥ln𝑥1|||^𝑏𝑎−^𝑏𝑎𝑑𝜑(𝑥)𝑑𝑥𝑥ln𝑥1𝑑𝑥.(8.4) Obviously the integral on the left is divergent and on the right is regular one. For the linear BE and piecewise constant approximation 𝜑(𝑥)=1, and we get 𝐽₀=𝑊.𝑆.^𝑏𝑎ln||𝑥−𝑦||𝑑𝑥=(𝑏−𝑦)ln||𝑏−𝑦||−(𝑎−𝑦)ln||𝑎−𝑦||(𝑎<𝑦<𝑏).(8.5)

For the singular integral 𝐽₁ regularization, we will also use the formula for integration by parts (8.3). Let in this formula 𝑔(𝑥)=−ln1/𝑥, (𝑑𝑔(𝑥))/𝑑𝑥=1/𝑥, then we obtain 𝐽₁=𝑃.𝑉.^𝑏𝑎𝜑(𝑥)𝑥𝑑𝑥=𝑑𝜑𝑑𝑥(𝑥)𝑥ln𝑥1−𝜑(𝑥)ln𝑥1||||^𝑏𝑎−^𝑏𝑎𝑑²𝜑(𝑥)𝑑𝑥²𝑥ln1𝑥𝑑𝑥.(8.6) Here also the integral on the left is divergent and on the right is regular one. For the linear BE and piecewise constant approximation 𝜑(𝑥)=1, and we get 𝐽₁=𝑃.𝑉.^𝑏𝑎𝑑𝑥𝑥−𝑦=ln||||𝑎𝑏−−𝑦𝑦||||,(𝑎<𝑦<𝑏).(8.7)

Finally for the hypersingular integral 𝐽₂ regularization, we will use formula (8.1) and the above-obtained result for singular 𝐽₁ regularization. Finally we get𝐽₂=𝐹.𝑃.^𝑏𝑎𝜑(𝑥)𝑥²𝑑𝑥=𝑑²𝜑(𝑥)𝑑𝑥²𝑥ln1𝑥−𝑑𝜑𝑑𝑥(𝑥)ln𝑥1−𝜑𝑥(𝑥)||||^𝑏𝑎−^𝑏𝑎𝑑³𝜑(𝑥)𝑑𝑥³𝑥ln1𝑥𝑑𝑥.(8.8) Here also the integral on the left is divergent and on the right is regular one. For the linear BE and piecewise constant approximation 𝜑(𝑥)=1, and we get 𝐽₂=𝐹.𝑃.^𝑏𝑎𝑑𝑥(𝑥−𝑦)²=−1(𝑏−𝑦)+(𝑎−1𝑦)(𝑎<𝑦<𝑏).(8.9) From this equation can be found Hadamard’s example of a function that is positive every were in the integration region, but its integral is a negative one 𝐹.𝑃.^𝑎−𝑎𝑑𝑦𝑦²=−2𝑎,𝑎>0.(8.10)

8.2.  2D Divergent Integrals

In 3D elastostatics after introducing local system of coordinates and simplification, all divergent integrals can be presented in the form 𝐽^{𝑘𝑙,𝑚}=_𝑆𝑛𝑥^𝑙1𝑥^𝑚2𝑟^𝑘𝜑(𝑥)𝑑𝑆,𝑙,𝑚=0,1,2,𝑘=3,4,5.(8.11) Here 𝜑(𝑥) is the smooth function that depends on shape of the BE and interpolation polynomials.

8.2.1. Integrals with Kernels of the Type 𝑟^−𝑘,𝑘=1,2,3

From (8.1) for 𝑘=1, we get regularization for weakly singular integral 𝐽^10,0=𝑊.𝑆._𝑉𝜑(𝐱)𝑟𝑑𝑉=_𝜕𝑉𝜑(𝐱)2𝑟^𝑛𝑟−𝑟𝜕^𝑛𝜑(𝐱)𝑑𝑆+_𝑉𝑟Δ𝜑(𝐱)𝑑𝑉.(8.12) Here 𝑟_𝑛=(𝑥_𝛼−𝑦_𝛼)𝑛_𝛼, and the summation convention applies to repeated indices 𝛼=1,2. The integral on the left is divergent and on the right are regular ones.

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate this integral analytically. Introducing polar, coordinates we will get 𝐽^10,0=12_𝜕𝑆𝑛𝑟_𝑛𝑟𝑑𝑙=^02𝜋𝑟𝑟𝑟𝑑𝜑=𝜋𝑟.(8.13) In order to regularize singular integral, we will use, relation 1/𝑟²=(1/2)Δ(ln𝑟)². Then in the same way we get_𝑉𝜑(𝐱)𝑟²𝑑𝑉=12_𝜕𝑉𝜑(𝐱)2𝑟^𝑛ln𝑟𝑟²−(ln𝑟)²𝜕_𝑛𝜑(𝐱)𝑑𝑆+12_𝑉(ln𝑟)²Δ𝜑(𝐱)𝑑𝑉.(8.14) The volume integral on the right is weakly singular. Taking into account the relation (ln𝑟)²=(𝑟²/6)Δ(ln𝑟)⁴, we obtain regularization for this weakly singular integral_𝑉(ln𝑟)²Δ𝜑(𝐱)𝑑𝑉=16_𝜕𝑉2Δ𝜑(𝐱)𝑟_𝑛+1(ln𝑟)³−𝑟²(ln𝑟)⁴𝜕_𝑛Δ𝜑(𝐱)𝑑𝑆6_𝑉𝑟²(ln𝑟)⁴Δ²𝜑(𝐱)𝑑𝑉.(8.15)

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate singular integral in (8.14) analytically. Introducing polar coordinates, we will get 𝐽^20,0=_𝜕𝑆𝑛𝑟_𝑛ln𝑟𝑟²𝑑𝑙=^02𝜋𝑟ln𝑟𝑟²𝑟𝑑𝜑=2𝜋ln𝑟.(8.16) Finally from (8.1) for 𝑘=3, we get regularization for hupersingular integral _𝑉𝜑(𝐱)𝑟³𝑑𝑉=_𝜕𝑉Δ𝜑(𝐱)𝑟^𝑛2𝑟−𝜑(𝐱)𝑟𝑟^𝑛3−1𝑟𝜕^𝑛𝜑(𝐱)−𝑟𝜕_𝑛Δ𝜑(𝐱)𝑑𝑆+_𝑉𝑟Δ²𝜑(𝐱)𝑑𝑉.(8.17)

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get 𝐽^30,0=−_𝜕𝑆𝑛𝑟_𝑛𝑟³𝑑𝑙=−^02𝜋𝑟𝑟³𝑟𝑑𝜑=−2𝜋𝑟.(8.18)

Now using (8.1), any divergent integral with the kernels of the type 1/𝑟^𝑘 for any positive integer 𝑘 can be calculated.

8.2.2. Integrals with Kernels of the Type 𝑥^2𝛼/𝑟^𝑘,𝑘=3,4,5

The weakly singular integral with kernel 𝑥^2𝛼/𝑟³ is calculated taking into account the equation 𝑥^2𝛼/𝑟³=(1/3)((2/𝑟)−Δ(𝑥^2𝛼/𝑟)). It is easy to show that𝐽^32,0=𝑊.𝑆._𝑉𝜑(𝐱)𝑥²¹𝑟³𝑑𝑉=23𝑊.𝑆._𝑉𝜑(𝐱)𝑟𝑑𝑉−13𝑊.𝑆._𝑉𝜑(𝐱)Δ𝑥²¹𝑟𝑑𝑉.(8.19) The first integral here is already calculated in (8.12). The second one may be presented in the form_𝑉𝜑(𝐱)Δ𝑥²¹𝑟𝑑𝑉=_𝜕𝑉𝜑(𝐱)2𝑛¹𝑥₁𝑟−𝑥¹²𝑟_𝑛𝑟³−𝑥𝑟²¹𝜕^𝑛𝜑(𝐱)𝑑𝑆+_𝑉𝑥²¹𝑟Δ𝜑(𝐱)𝑑𝑉.(8.20) Combining the last two equations, finally we will get 𝐽^32,0=13_𝜕𝑉𝜑(𝐱)2𝑛¹𝑥₁𝑟−𝑥¹²𝑟_𝑛𝑟³+𝑟^𝑛2𝑟−𝑥𝑟²¹+𝑟𝜕^𝑛+1𝜑(𝐱)𝑑𝑆3_𝑉𝑥𝑟²¹+𝑟Δ𝜑(𝐱)𝑑𝑉.(8.21)

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area we can calculate this integral analytically. Introducing polar coordinates, we will get 𝐽^32,0=13_𝜕𝑆𝑛𝑥²¹𝑟_𝑛𝑟³−2𝑥¹𝑛₁=1𝑟+2𝑟𝑟^𝑛𝑑𝑙3^02𝜋(𝑟cos𝜑)²𝑟𝑟³𝑟𝑑𝜑−^02𝜋2𝑟(cos𝜑)²𝑟𝑟𝑑𝜑+2^02𝜋𝑟𝑟𝑟𝑑𝜑=𝜋𝑟.(8.22)

The singular integral with kernel 𝑥^2𝛼/𝑟⁴ is calculated taking into account that Δ𝑥^2𝛼/𝑟⁴=(1/4)((2/𝑟²)−Δ𝑥^2𝛼/𝑟²). In this case,𝐽^42,0=𝑃.𝑉._𝑉𝜑(𝐱)𝑥²¹𝑟²𝑑𝑉=14_𝑉𝜑(𝐱)2𝑟²−Δ𝑥²¹𝑟²𝑑𝑉=12_𝑉𝜑(𝐱)𝑟²𝑑𝑉−14_𝑉𝜑(𝐱)Δ𝑥²¹𝑟²𝑑𝑉.(8.23) The first integral here is already calculated in (8.14). The second one may be presented in the form_𝑉𝜑(𝐱)Δ𝑥¹²𝑟²𝑑𝑉=_𝜕𝑉𝜑(𝐱)2𝑛¹𝑥₁𝑟²−2𝑥¹²𝑟_𝑛𝑟⁴−𝑥𝑟¹²²𝜕_𝑛𝜑(𝐱)𝑑𝑆+_𝑉𝑥₁₂𝑟²Δ𝜑(𝐱)𝑑𝑉.(8.24) Combining the last two equations, finally we will get 𝐽^42,0=12_𝜕𝑉𝜑(𝐱)𝑥¹²𝑟_𝑛𝑟⁴−𝑛¹𝑥₁𝑟²+𝑟^𝑛ln𝑟𝑟²−(ln2𝑟)²−𝑥2𝑟¹²²𝜕^𝑛+1𝜑(𝐱)𝑑𝑆4_𝑉(ln𝑟)²−𝑥¹²𝑟²Δ𝜑(𝐱)𝑑𝑉.(8.25)

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get 𝐽^42,0=12_𝜕𝑆𝑛𝑥¹²𝑟_𝑛𝑟⁴−𝑥¹𝑛₁𝑟²+𝑟^𝑛ln𝑟𝑟²𝑑𝑙=^02𝜋(𝑟cos𝜑)²𝑟𝑟⁴𝑟𝑑𝜑−^02𝜋𝑟(cos𝜑)²𝑟²+𝑟𝑑𝜑_02𝜋𝑟ln𝑟𝑟²=2𝜋ln𝑟.𝑟𝑑𝜑(8.26)

The hypersingular integral with kernel 𝑥^2𝛼/𝑟⁵ is calculated taking into account that 𝑥^2𝛼/𝑟⁵=(1/3)((2/𝑟³)−Δ(𝑥^2𝛼/𝑟³)). In this case, it is easy to show that𝐽^52,0=𝐹.𝑃._𝑉𝜑(𝐱)𝑥²¹𝑟⁵𝑑𝑉=13_𝑉𝜑(𝐱)2𝑟³−Δ𝑥²¹𝑟³𝑑𝑉=23_𝑉𝜑(𝐱)𝑟³𝑑𝑉−13_𝑉𝜑(𝐱)Δ𝑥²¹𝑟³𝑑𝑉.(8.27) The first integral here is already calculated in (8.17). The second one may be presented in the form_𝑉𝜑(𝐱)Δ𝑥²¹𝑟³𝑑𝑉=_𝜕𝑉𝜑(𝐱)2𝑛¹𝑥₁𝑟³−3𝑥¹²𝑟_𝑛𝑟⁵−𝑥𝑟²¹³𝜕_𝑛𝜑(𝐱)𝑑𝑆+_𝑉𝑥²¹𝑟³Δ𝜑(𝐱)𝑑𝑉.(8.28) Combining the last two equations, finally we will get 𝐽^52,0=23_𝜕𝑉𝜑(𝐱)𝑛¹𝑥₁𝑟³−𝑟^𝑛𝑟³−3𝑥¹²𝑟_𝑛2𝑟⁵−1𝑟+𝑥𝑟²¹³𝜕^𝑛+2𝜑(𝐱)𝑑𝑆3𝑊.𝑆._𝑉1𝑟−𝑥𝑟²¹³Δ𝜑(𝐱)𝑑𝑉.(8.29) The volume integral here is weakly singular. Its regularization may be done using (8.12) and (8.21). For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get 𝐽^52,0=_𝜕𝑆𝑛32𝑟𝑟^𝑛3+2𝑥¹²𝑟_𝑛3𝑟⁵−𝑥¹𝑛₁𝑟³𝑑𝑙=23^02𝜋𝑟𝑟³𝑟𝑑𝜑+23^02𝜋(𝑟cos𝜑)²𝑟𝑟⁵−𝑟𝑑𝜑_02𝜋𝑟(cos𝜑)²𝑟³=−𝜋𝑟𝑑𝜑𝑟.(8.30)

8.2.3. Integrals with Kernels of the Type (𝑥₁𝑥₂)/𝑟^𝑘,𝑘=3,4,5

The weakly singular integral with kernel (𝑥₁𝑥₂)/𝑟³ is calculated using (8.1). Taking into account that (𝑥₁𝑥₂)/𝑟³=−1/3Δ(𝑥₁𝑥₂)/𝑟, it is easy to show that𝐽^31,1=𝑊.𝑆._𝑉𝜑(𝐱)𝑥¹𝑥₂𝑟³𝑑𝑉=13_𝜕𝑉𝜑(𝐱)𝑥¹𝑥₂𝑟_𝑛𝑟³−𝑟^∗𝑟+𝑥¹𝑥₂𝑟𝜕^𝑛−1𝜑(𝐱)𝑑𝑆3_𝑉𝑥₁𝑥₂𝑟Δ𝜑(𝐱)𝑑𝑉.(8.31) Here 𝑟_∗=𝑥₁𝑛₂+𝑥₂𝑛₁.

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get 𝐽^31,1=13_𝜕𝑆𝑛𝑥¹𝑥₂𝑟_𝑛𝑟³−𝑟^∗𝑟𝑑𝑙=^02𝜋𝑟³cos𝜑sin𝜑𝑟³𝑟𝑑𝜑−^02𝜋2𝑟cos𝜑sin𝜑𝑟𝑟𝑑𝜑=0.(8.32)

The singular integral with kernel 𝑥₁𝑥₂/𝑟⁴ is calculated using equation (8.1). Taking into account that 𝑥₁𝑥₂/𝑟⁴=−(1/4)Δ𝑥₁𝑥₂/𝑟², it is easy to show that𝐽^41,1=𝑃.𝑉._𝑉𝜑(𝐱)𝑥¹𝑥₂𝑟⁴𝑑𝑉=14_𝜕𝑉𝜑(𝐱)2𝑥¹𝑥₂𝑟_𝑛𝑟⁴−𝑟^∗𝑟²−𝑥¹𝑥₂𝑟²−1𝜕_𝑛𝜑(𝐱)𝑑𝑆4_𝑉𝑥₁𝑥₂𝑟²Δ𝜑(𝐱)𝑑𝑉.(8.33)

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get 𝐽^41,1=14_𝜕𝑆𝑛2𝑥¹𝑥₂𝑟_𝑛𝑟⁴−𝑟^∗𝑟²𝑑𝑙=14^02𝜋2𝑟³cos𝜑sin𝜑𝑟⁴𝑟𝑑𝜑−14^02𝜋2𝑟cos𝜑sin𝜑𝑟²𝑟𝑑𝜑=0.(8.34)

The hypersingular integral with kernel 𝑥₁𝑥₂/𝑟⁵ is calculated using (8.1). Taking into account that 𝑥₁𝑥₂/𝑟⁵=−(1/3)Δ𝑥₁𝑥₂/𝑟³, it is easy to show that𝐽^51,1=𝐹.𝑃._𝑉𝜑(𝐱)𝑥¹𝑥₂𝑟⁵𝑑𝑉=_𝜕𝑉𝜑(𝐱)𝑥¹𝑥₂𝑟_𝑛𝑟⁵−𝑟^∗3𝑟³−𝑥¹𝑥₂𝑟³−1𝜕_𝑛𝜑(𝐱)𝑑𝑆3_𝑉𝑥₁𝑥₂𝑟³Δ𝜑(𝐱)𝑑𝑉.(8.35) The volume integral here is weakly singular. Its regularization may be done using (8.31).

For the linear BE and piecewise constant approximation 𝜑(𝑥)=1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get 𝐽^51,1=_𝜕𝑆𝑛𝑥¹𝑥₂𝑟_𝑛𝑟⁵−𝑟^∗3𝑟³𝑑𝑙=^02𝜋𝑟³cos𝜑sin𝜑𝑟⁵𝑟𝑑𝜑−^02𝜋2𝑟cos𝜑sin𝜑3𝑟³𝑟𝑑𝜑=0.(8.36)

9. Conclusions

The method of regularization of the weakly singular, singular and hypersingular integrals, based on the second Green theorem, is developed here. These divergent integrals are presented when the boundary integral equation methods are used to solve problems in elastostatics. The equations that permit easy calculation of the weakly singular, singular, and hypersingular integrals for different shape functions are presented here. The divergent integrals over the circular domain have been calculated analytically. The main equations related to the boundary integral equation and boundary elements methods formulation in 2D and 3D elastostatics have been discussed in details.