Divergent Integrals in Elastostatics: General Considerationsdownloads.hindawi.com/archive/2011/726402.pdfji x,y are called fundamental solutions for elastostatics. Applying to 3.9

International Scholarly Research NetworkISRN Applied MathematicsVolume 2011, Article ID 726402, 25 pagesdoi:10.5402/2011/726402

Research ArticleDivergent Integrals in Elastostatics:General Considerations

V. V. Zozulya

Materials Department, Centro de Investigacion Cientifica de Yucatan A.C., Calle 43,No. 130, Colonia Chuburna de Hidalgo, 97200 Merida, YUC, Mexico

Correspondence should be addressed to V. V. Zozulya, [email protected]

Received 4 April 2011; Accepted 31 May 2011

Academic Editors: S.-W. Chyuan and E. A. Navarro

Copyright q 2011 V. V. Zozulya. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This article considers weakly singular, singular, and hypersingular integrals, which arise whenthe boundary integral equation methods are used to solve problems in elastostatics. The mainequations related to formulation of the boundary integral equation and the boundary elementmethods in 2D and 3D elastostatics are discussed in details. For their regularization, an approachbased on the theory of distribution and the application of the Green theorem has been used. Theexpressions, which allow an easy calculation of the weakly singular, singular, and hypersingularintegrals, 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 inthe numerical solution of the BIE by the boundary element method (BEM) is a calculation ofthe divergent integrals. In mathematics divergent integrals have established theoretical basis.For example, the weakly singular integrals are considered as improper integrals; the singularintegrals are considered in the sense of Cauchy principal value (PV); the hypersingularintegrals had been considered by Hadamard as finite part integrals (FP). Different methodshave been developed for calculation of the divergent integrals. Usually different divergentintegrals need different methods for their calculation. Analysis of the most known methodsused for treatment of the different divergent integrals has been done in [3, 4, 6–8]. Thecorrect mathematical interpretation of the divergent integrals with different singularitieshas been done in terms of the theory of distributions (generalized functions). The theory ofdistributions provides a unified approach for the study of the divergent integrals and integraloperators with kernels containing different kind of singularities.

2 ISRN Applied Mathematics

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

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

2. Main Equations of Elastostatics

Lets consider a homogeneous, lineally elastic body, which in three-dimensional Euclideanspace R3 occupies volume V with smooth boundary ∂V . The region V is an open boundedsubset of the three-dimensional Euclidean spaceR3 with aC0,1 Lipschitzian regular boundary∂V . The boundary contains two parts ∂Vu and ∂Vp such that ∂Vu∩∂Vp = ∅ and ∂Vu∪∂Vp = ∂V .On the part ∂Vu are prescribed displacement ui(x)of the body points, and on the part ∂Vp areprescribed traction pi(x), respectively. The body may be affected by volume forces bi(x). Weassume that displacements of the body points and their gradients are small, so its stress-strain state is described by small strain deformation tensor εij(x). The strain tensor and thedisplacement vector are connected by Cauchy relations

εij =12(∂iuj + ∂jui

), (2.1)

where ∂/∂xi is a derivative with respect to space coordinates xi. The components of the straintensor must also satisfy the Saint-Venant’s relations

∂2klεij − ∂2ilεkj = ∂2kjεil − ∂2ijεkl. (2.2)

From the balance of impulse and the moment of impulse lows follow that the stresstensor is symmetric one and satisfy the equations of equilibrium

∂jσij + bi = 0i, ∀x ∈ V. (2.3)

Here and throughout the paper the summation convention applies to repeated indices.The tensor of deformation εij(x) and stress σij(x) are related by Hook’s law

σij = cijklεij . (2.4)

Here cijkl are elastic modules. In the case of homogeneous anisotropic medium they aresymmetric

cijkl = cjikl = cklij . (2.5)

ISRN Applied Mathematics 3

and satisfy condition of ellipticity

cijklεijεkl ≥ α1εijεij , ∀εij , ∀α1 > 0. (2.6)

In the case of homogeneous isotropic medium, the elastic modules have the form

cijkl = λδijδkl + μ(δikδjl + δilδjk

), (2.7)

where λ and μ are Lame constants, μ > 0 and λ > −μ, δij 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 whichmay be presented in the form

Aijuj + bi = 0, ∀x ∈ V. (2.8)

The differential operator Aij for homogeneous anisotropic medium has the form

Aij = cikjl∂k∂l, (2.9)

and for homogeneous isotropic medium has the form

Aij = μδij∂k∂k +(λ + μ

)∂i∂j . (2.10)

If the problem is defined in an infinite region, then solution of (2.8) must satisfyadditional conditions at the infinity in the form

uj(x) = O(r−1

), σij(x) = O

(r−2

)for r −→ ∞, (2.11)

where r =√x21 + x

22 + x

23 is the distance in the three-dimensional Euclidian space.

If the body occupied a finite region Vwith the boundary ∂V , it is necessary to establishboundary conditions. We consider the mixed boundary conditions in the form

ui(x) = ϕi(x), ∀x ∈ ∂Vu,pi(x) = σij(x)nj(x) = Pij

[uj(x)

]= ψi(x), ∀x ∈ ∂Vp.

(2.12)

The differential operator Pij : uj → pi is called stress operator. It transforms thedisplacements into the tractions. For homogeneous anisotropic and isotropic medium, theyhave the forms

Pij = cikjlnk∂l, Pij = λni∂k + μ(δij∂n + nk∂i

), (2.13)

respectively. Here ni are components of the outward normal vector, ∂n = ni∂i is a derivativein direction of the vector n(x) normal to the surface ∂Vp.


3. Integral Representations for Displacements and Traction

In order to establish integral representations for the displacements and tractions, let usconsider bilinear form which depends on two fields of the strain tensor that correspond totwo fields of the displacements u and u∗

a(u,u∗) = cijklεij(u)εkl(u∗). (3.1)

Obviously that

a(u,u∗) = a(u∗,u), a(u,u) = σij(u)εij(u) ≥ α1εij(u). (3.2)

Integrating the equality (3.1) over the volume V and applying the Gauss-Ostrogradskiiformula, we will obtain

∫

V

a(u,u∗)dV =∫

V

σij(u)εkl(u∗)dV =∫

∂V

σijnju∗i dS −

∫

V

u∗i ∂jσijdV. (3.3)

Taking into account that Aijuj = ∂jσij and pi = σijnj = Pij[uj], we will find the first Betti’stheorem in the form

∫

V

u∗i AijujdV =∫

V

a(u,u∗)dV −∫

∂V

u∗i Pij[uj]dS. (3.4)

Wewill replace ui 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∗i Aijuj − uiAiju

∗j

)dV =

∫

∂V

(uiPij

[u∗j]− u∗i Pij

[uj])dS. (3.5)

Taking into the account the definition of differential operator Aij given in (2.8) and thedefinition of differential operator Pij given in (2.12), we obtain relation

∫

V

(biu

∗i − b∗i ui

)dV =

∫

∂V

(p∗i ui − piu∗i

)dS, (3.6)

which is called the Betti’s reciprocal theorem.This theorem is usually used for obtaining integral representations for the displace-

ments and traction vectors. To do that, we consider solution of the elliptic partial differentialequation (2.8) in an infinite space for the body force b∗i (x) = δijδ(x − y)

AijUkj(x − y) + δki(x − y) = 0, ∀x,y ∈ R3. (3.7)

Now considering that

u∗i (x) = Uij(x − y), p∗i (x) = Pij[u∗j (x)

]=Wij(x,y), (3.8)


from (3.6)we obtain the integral representation for the displacements vector

ui(y) =∫

∂V

(pi(x)Uji(x − y) − uj(x)Wji(x,y)

)dS +

∫

V

bi(x)Uji(x − y)dV, (3.9)

which is called Somigliana’s identity. The kernels Uji(x − y) and Wji(x,y) are calledfundamental solutions for elastostatics.

Applying to (3.9) the differential operator Pij , we will find integral representation forthe traction in the form

pi(y) =∫

∂V

(pi(x)Kji(x,y) − uj(x)Fji(x,y)

)dS +

∫

V

bi(x)Kji(x,y)dV. (3.10)

The kernels Kji(x,y) and Fji(x,y) may be obtained applying the differential operator Pij tothe kernelsUji(x − y) andWji(x,y), respectively.

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

4. Fundamental Solutions

In order to find the fundamental solutions Uij(x − y) for the differential operator Aij , weconsider the differential equations of elastostatics in the form displacements (3.7). Solutionsof these equations are called the fundamental solutions.

In 3D elastostatics, they have the form

Uij(x − y) =1

16πμ(1 − υ)r((3 − 4υ)δij + ∂ir∂jr

). (4.1)

In 2D elastostatics they have the form

Uij(x − y) =1

8πμ(1 − υ)((3 − 4υ)δij ln

1r+ ∂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/∂xi = −∂r/∂yi = (xi − yi)/r.The kernelsWij(x,y) from (3.9)may be obtained by applying toUij(x − y) differential

operator

Pik[·, (x)] = λni(x)∂k[·] + μ⌊δiknj(x)∂j[·] + nk(x)∂i[·]

⌋, (4.3)

as it is shown here

Wij(x,y) = λni(x)∂kUkj(x,y) + μnk(x)⌊∂kUij(x,y) + ∂iUkj(x,y)

⌋. (4.4)


Then after some transformations and simplifications, the expression for the kernelsWij(x,y)will have the following form

Wij(x,y) =−1

4π(1 − υ)rα(nk(x)∂kr

((1 − 2υ)δij + β∂ir∂jr

)+ (1 − 2υ)

(ni(x)∂jr − nj(x)∂ir

)).

(4.5)

The kernelsKij(x,y) and Fij(x,y) from (3.10)may be obtained by applying differentialoperator

Pik[·, (y)] = λni(y)∂k + μ⌊δiknj(y)∂j + nk(y)∂i

⌋(4.6)

toUij(x − y) andWij(x,y), respectively

Kij(x,y) = λni(y)∂kUjk(x,y) + μnk(y)⌊∂kUji(x,y) + ∂iUjk(x,y)

⌋,

Fij(x,y) = λni(y)∂kWjk(x,y) + μnk(y)⌊∂kWji(x,y) + ∂iWjk(x,y)

⌋.

(4.7)

Then after some transformations and simplifications the expression for the kernels Kij(x,y)will have the form

Kij(x,y) =1

4π(1 − υ)rα(nk(y)∂kr

((1 − 2υ)δij + β∂ir∂jr

)+ (1 − 2υ)

(ni(y)∂jr − nj(y)∂ir

)),

(4.8)

and for the kernels, Fij(x,y) will have the form

Fij(x,y)

=μ

2απ(1 − υ)rβ(βnk(x)∂kr

((1 − 2υ)ni(y)∂jr + υ

(δijnk(y)∂kr + nj(y)∂ir

)

−γnk(y)∂kr∂ir∂jr)+ βν

(ni(x)nk(y)∂kr∂jr + nk(x)nk(y)∂ir∂jr

)

+(1 − 2υ)(βnj(x)nk(y)∂kr∂ir + nk(x)nk(y)δij + ni(x)nj(y)

) − (1 − 4υ)nj(x)ni(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 ofdivergent 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 toinfinity when r → 0. More detailed analysis of (4.1)–(4.9) gives us the following results.


In the 3D case with x → y,

Uij(x − y) −→ r−1, Wij(x,y) −→ r−2, Kij(x,y) −→ r−2, Fij(x,y) −→ r−3. (5.1)

In the 2D case with x → y,

Uij(x − y) −→ ln(r−1

), Wij(x,y) −→ r−1, Kij(x,y) −→ r−1, Fij(x,y) −→ r−2.

(5.2)

In order to investigate these functions and integrals with singular kernels, definition andclassification of the integrals with various singularities will be presented here.

Definition 5.1. Let one considers two points with coordinated x,y ∈ Rn (where n = 3 or n = 2)and region V with smooth boundary ∂V of the class C0,1. The boundary integrals of the types

∫

∂V

G(x,y)rα

ϕ(x)dS, α > 0, (5.3)

where G(x,y) is a finite function in Rn × ∂V and ϕ(x) is a finite function in ∂V , are weaklysingular 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 ∈ Rn (where n = 3 or n = 2)and region V with smooth boundary ∂V of the class C0,1. The boundary integrals of the types

∫

∂V

G(x,y) ln(r)ϕ(y)dy, (5.4)

where G(x,y) is a finite function in Rn × ∂V and ϕ(x) is a finite function in ∂V , are weaklysingular.

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. Wewill apply this definition of the integrals from (3.9) and (3.10).

Definition 5.3. Integrals in (3.9) with kernels Uij(x − y) are weakly singular and must beconsidered as improper

W.S.

∫

∂V

pi(x)Uij(x − y)dS = limε→ 0

∫

∂V \∂Vεpi(x)Uij(x − y)dS. (5.5)

Here ∂Vε is a part of the boundary, projection of which on tangential plane is contained in thecircle Cε(x) of the radio ε with center in the point x.


Definition 5.4. Integrals in (3.9) and (3.10) with kernels Wij(x,y) and Kij(x,y) are singularand must be considered in sense of the Cauchy principal values as

P.V.

∫

∂V

ui(x)Wij(x,y)dS = limε→ 0

∫

∂V \∂V (r<ε)ui(x)Wij(x,y)dS,

P.V.

∫

∂V

pi(x)Kij(x,y)dS = limε→ 0

∫

∂V \∂V (r<ε)pi(x)Kij(x,y)dS.

(5.6)

Here ∂V (r < ε) is a part of the boundary, projection of which on tangential plane is the circleCε(x) of the radio ε with center in the point x.

Definition 5.5. Integrals in (3.10) with kernels Fij(x,y) are hypersingular and must beconsidered in sense of the Hadamard finite part as

F.P.

∫

∂V

ui(x)Fji(x − y)dS = limε→ 0

(∫

∂V \∂V (r<ε)ui(x)Wji(x − y)dS + 2uj(x)

fj(x)∂V (r < ε)

)

. (5.7)

Here functions fj(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 Uij(x − y), are weakly singular, and, therefore, they are continuouseverywhere in the Rn and, therefore, may be continuously extended on the boundary ∂V .The potentials with the kernels Wij(x,y) and Kij(x,y) contain singular kernels, and theyjump when crossing the boundary ∂V . The potential with the kernels Fij(x,y) containhypersingular 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

pi(x)Uji(x − y)dS)±

=(∫

∂V

pi(x)Uji(x − y)dS)0

,

(∫

∂V

ui(x)Wji(x,y)dS)±

= ∓12uj(y) +

(∫

∂V

ui(x)Wji(x,y)dS)0

,

(∫

∂V

pi(x)Kji(x,y)dS)±

= ±12pj(y) +

(∫

∂V

pi(x)Kji(x,y)dS)0

,

(∫

∂V

ui(x)Fji(x − y)dS)±

=(∫

∂V

ui(x)Fji(x − y)dS)0

.

(5.8)

The symbols “±” and “∓” denote that two equalities, one with the top sign and the otherwith the bottom sign, are considered. The up index “0” points out that the direct value of thecorresponding potentials on the surface ∂V should be taken.


Now using integral representations for displacements and traction and boundaryproperties of the potentials, we can get boundary integral equations for elastostatics. Tendingy in (3.9) and (3.10) to the boundary ∂V and taking into consideration boundary propertiesof the potentials (5.8), we obtain representation of the displacements and traction vectors onthe boundary surface ∂V . On the smooth parts of the boundary, they have the following form

±12ui(y) =

∫

∂V

(pj(x)Uij(x − y)uj(x) −Wij(x,y)

)dS +

∫

V

pj(x)Uij(x − y)dV,

∓12pi(y) =

∫

∂V

(pj(x)Kij(x,y)uj(x) − Fij(x,y)

)dS +

∫

V

pj(x)Kij(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 forthe problems of elastostatics. The BIE are usually solved numerical transforming them intodiscrete system of finite dimensional equations. Such method is called the boundary elementmethod (BEM).

6. Projection Method and the BEM Equations

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

We consider two Banach spaces X and Y and functional equation in those spaces

A · 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 theexact equation, and its solution is the exact solution. We denote L(X,Y) Banach space of thelinear operators mapping from X in Y.

Let in X and Y act sequences of projection operators Ph and P′hsuch that

P2h = Ph, PhX = Xh, Xh ⊂ X,

(P′h

)2 = P′h, P′

hY = Yh, Yh ⊂ Y,(6.2)

where Xh and Yh are finite dimension subspaces of the Banach spaces X and Y; h ∈ R1 is aparameter of discretization.

Now we consider operator Ah ∈ L(Xh,Yh) mapping in finite dimensional subspacesXh and Yh and an approximate equation

Ah · uh = fh, Ah = P′h ·A · Ph, uh = Ph · u, fh = P′

h · f. (6.3)


Solution uh of (6.3) is the approximate solutions of (6.1). The general scheme of the ap-proached equations construction (6.3) is illustrated by the following diagram.

X ⊃ D A

Ph

AR A ⊂ Y

Ph

Xh ⊃ D AhAh

R Ah ⊂ Yh

(6.4)

Now let us consider operatorAh ∈ L(Xh,Yh)mapping in finite-dimensional subspacesXh and Yh and an approximate equation.

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

We assume that projection operators Ph and P′h converge to identity operators in X and

Y, respectively. It means that

limh→ 0

‖Ph · u − u‖X = 0 ∀u ∈ X,

limh→ 0

∥∥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 uh. In this case if

limh→ 0

‖Ah · uh −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 initialproblem (6.1).

Definition 6.2. Let for some sequence of the operators {Ah} mapping from Xh into Yh there isa constant γ > 0 such that stating from some h = h0 > 0

‖Ah · uh −A · u‖Y ≥ γ‖uh‖X ∀uh ∈ Xh (6.7)

then for sequence of the operators {Ah}, the condition of stability of the approximate solutionis 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 followingtheorem [21, 22].


Theorem 6.3. Let the following conditions be satisfied:

(1) the projection operators Ph and P′hconverge to identity operators in the Banach spaces X

and Y, respectively, as it stated in (6.5);

(2) the sequence of approximate operators {Ah} converges to A on each exact solution;

(3) the condition of stability (6.7) is satisfied for the sequence of operators {Ah}.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 uh ∈ Xh of the approximate equation (6.3);

(3) the sequence of the approximate equation {uh} converges to the exact one and takes place inthe estimation

‖uh − Ph · u‖Xh ≤ γ−1∥∥P′

hA · u −AhPhu∥∥Yh. (6.8)

Thus, using a projective method instead of the exact solution of (6.1) in functionalspace X, we can find the solution of the approximate equation (6.3) in finite dimensionalspace Xh. The functional spaces {X,Xh} and {Y,Yh} are related by means of the projectionoperators Ph ∈ L(X,Xh) and P′

h∈ L(Y,Yh), respectively. It is also important to construct

inverse operator P−1h ∈ L(Xh,X) which maps the finite dimensional space Xh into the initial

functional space X. Such operator refers to as the operator of interpolation. Because of Xh ⊂ X,the interpolation operator is not unique; moreover, for any two functional spaces Xh and X, itis possible to construct infinite set of interpolation operators.

Let us apply the projection method to the BIE of elastostatics and construct corres-ponding 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 displacementsand traction on the boundary of the region in the following way:

Uij · pj =∫

∂V

(Uij(x,y)pj(x)

)dS : Y(∂V ) −→ X(∂V ),

Wij · uj =∫

∂V

(Wij(x,y)uj(x)

)dS : X(∂V ) −→ Y(∂V ),

Kij · pj =∫

∂V

(Kij(x,y)pj(x)

)dS : Y(∂V ) −→ X(∂V ),

Fij · uj =∫

∂V

(Fij(x,y)uj(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 dimensionalfunctional spaces, we shall apply approximation by finite functions and splitting ∂V intofinite elements

∂V =N⋃

n=1

∂Vn, ∂Vn ∩ ∂Vk = ∅, if n/= k. (6.10)


Because ∂V is the boundary of the region, these elements are called boundary ele-ments. On each boundary element, we shall chooseQ nodes of interpolation. Local projectionoperators act from functional X(∂Vn) and Y(∂Vn) to the finite directional ones Xq(∂Vn) andYq∂Vn)

Puq : X(∂Vn) −→ Xq(∂Vn) ∀x ∈ ∂Vn,

Ppq : Y(∂Vn) −→ Yq(∂Vn) ∀x ∈ ∂Vn.(6.11)

Global projection operators are defined as the sum of the local projection operators

Punq =N∑

n=1

Puq , Ppnq =N∑

n=1

Ppq. (6.12)

They map X(∂V ) and Y(∂V ) to finite dimensional interpolations spaces

Punq : X(∂V ) −→ Xq

(N⋃

n=1

∂Vn

)

∀x ∈ ∂V,

Ppnq : Y(∂V ) −→ Yq

(N⋃

n=1

∂Vn

)

∀x ∈ ∂V.(6.13)

The local projection operators Pun and Ppn establish correspondence between vectors ofdisplacements and traction and their value on the nodes of interpolation of the boundaryelements ∂Vn

Puq · ui(x) ={uni

(xq), q = 1, . . . , Q

} ∀x ∈ ∂Vn,

Ppq · pi(x) ={pni

(xq), q = 1, . . . , Q

} ∀x ∈ ∂Vn.(6.14)

Similarly for the global operators, we have

Punq · ui(x) ={uni

(xq), q = 1, . . . , Q; n = 1, . . . ,N

} ∀x ∈ ∂V,

Ppnq · pi(x) ={pni

(xq), q = 1, . . . , Q; n = 1, . . . ,N

} ∀x ∈ ∂V.(6.15)

Let us construct local interpolation operators (Puq)−1 and (Ppq)

−1. For this purpose, we will

introduce systems of shape functions ϕnq(x) and ψnq(x) in the finite dimensional functionalspaces Xq(∂Vn) and Yq(∂Vn). Then the vectors of displacements and traction on the boundaryelement ∂Vn will be represented approximately in the form

ui(x) ≈Q∑

q=1

uni(xq)ϕnq(x), x ∈ ∂Vn,

pi(x) ≈Q∑

q=1

pni(xq)ϕnq(x), x ∈ ∂Vn,

(6.16)


and on the whole crack surface ∂V in the form

ui(x) ≈N∑

n=1

Q∑

q=1

uni(xq)ϕnq(x), x ∈

N⋃

n=1

∂Vn,

pi(x) ≈N∑

n=1

Q∑

q=1

pni(xq)ϕnq(x), x ∈

N⋃

n=1

∂Vn.

(6.17)

Finite-dimensional analogies for the integral operators (6.9) are operators which mapthe finite-dimensional functional spaces Xq(

⋃Nn=1 ∂Vn) and Yq(

⋃Nn=1 ∂Vn), from one to another

Unq

ij = Ppnq ·Uij · Punq : Yq(∂Vn) −→ Xq(∂Vn),

Wnq

ij = Ppnq ·Wij · Punq : Xq(∂Vn) −→ Yq(∂Vn),

Knq

ij = Ppnq ·Kij · Punq : Yq(∂Vn) −→ Xq(∂Vn),

Fnqij = Ppnq · Fij · Punq : Xq(∂Vn) −→ Yq(∂Vn).

(6.18)

Note that in contrast to differential operators, the integral operators are global, andthey 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-dimensionalrepresentations for the vectors of displacements and traction on the boundary in the form

12umi (yr) =

N∑

n=1

Q∑

q=1

Unji

(yr , xq

)pnj(xq) −Wn

ji

(xr , xq

)unj

(xq)+Ui(f,y, Vn),

12pmi (yr) =

N∑

n=1

Q∑

q=1

Knji

(yr , xq

)pnj(yq) − Fnji

(yr , xq

)unj

(xq)+Ki(f,y, Vn),

(6.19)

where

Unji

(yr , xq

)=∫

∂Vn

Uji(yr , x)ψnq(x)dS, Wnji

(yr , xq

)=∫

∂Vn

Wji(yr , x)ϕnq(x)dS,

Knji

(yr , xq

)=∫

∂Vn

Kji(yr , x)ψnq(x)dS, Fnji(yr , xq

)=∫

∂Vn

Fji(yr , x)ϕnq(x)dS.

(6.20)

The volume potentialsUi(f,y, Vn) andKi(f,y, Vn) 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 includesapproximation 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 forapproximation of these functions by the shape functions determined on small subdomainscalled boundary elements. In this senses the BEM is closely related to a finite element methodwhere the functions also belong to corresponding functional spaces and are approximated byfinite 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 onone FE can be done independently from other FEs. It means that it is possible to approximatefunction on an FE by means of its values on the nodes independently of the place occupiedconsidered the FE in the FE model and how behave the function on other FEs. Hence, it ispossible to create the catalogue of various FE or BE with arbitrary node values interpolationfunction. Then from this catalogue can be chosen FEs which are necessary for approximationof the function and domain of its definition. The same FE can be used for discrete models ofvarious functions or physical fields by determination of the necessary position of nodes inthe 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 fieldsfor which this area can be a domain of definition.

Let us consider how to construct an FE model of an area V ⊂ Rn and a BE model of itsboundary ∂V ⊂ Rn−1 (n = 2, 3). We fix in the area V finite number of points xq(g = 1, . . . , Q);these points refer to as global node points V (q) = {xq ∈ V : g = 1, . . . , Q}. We shall divide thearea V into finite number of subareas Vn(n = 1, . . . ,N) which are FEs. They have to satisfythe following conditions:

Vn ∪ Vm = ∅, m/=n, m, n = 1, 2, . . . ,N, V =N⋃

n=1

Vn. (7.1)

On each FE we introduce a local coordinate system ξ. The nodal points xq ∈ Vn in the localsystem of coordinates we designate by ξq. They are coordinates of nodal points in the localcoordinate system. Local and global coordinate are related in the following way:

xq =N∑

n=1

Λnξqn. (7.2)

Functions Λn depend on position of the nodal points in the FE and BE. They join individualFE together in a FE model. Borders of the FEs and position of the nodal points should be suchthat, 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 thefunction f(x) that belong to some functional space. The FE model of the area V is the domainof function which should be approximated. We denote function f(x) on the FE Vn by fn(x).Then

f(x) =N∑

n=1

fn(x). (7.3)

On each FE, the local functions fn(x) may be represented in the form

fn(x) ≈Q∑

q=1

fn(xq)ϕnq(ξ), (7.4)


where ϕnq(ξ) are interpolation polynomials or nodal functions of the FE with number n. Inthe nodal point with coordinates xq, they are equal to 1, and in other nodal points are equal tozero. Taking into account (7.3) and (7.4), the global approximation of the function f(x) lookslike

f(x) ≈N∑

n=1

Q∑

q=1

fn(xq)ϕnq(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 surfaceis usually curvilinear. But it is more convenient to use standard FE, whose surfaces coincidewith coordinate planes of the local coordinates system. Mathematically it means that it isnecessary to establish relation between local coordinates ξi, in which the element has a simpleappearance, and global xi where the FE represents more complex figure. Local coordinates ξishould be functions of global (ξi(x1, x2, x3)) ones, and on the contrary, global coordinatesshould be functions of (xi(ξ1, ξ2, ξ3))ones. In order to these maps be one-to-one, it is necessaryand sufficient that the Jacobians of transformations be nonzero

J = det

∣∣∣∣∣∂xi∂ξj

∣∣∣∣∣/= 0, J−1 = det

∣∣∣∣∣∂ξi∂xj

∣∣∣∣∣/= 0. (7.6)

The differential elements along coordinate axes are related by

dxi =

(∂xi∂ξj

)

dξj , dx = J(ξ)dξ,

dξi =

(∂ξi∂xj

)

dxj , dξ = J−1(x)dx.

(7.7)

The volume element in the R3 is transformed under the formula

dV = dx1dx2dx3 = J(ξ)dξ1dξ2dξ3, (7.8)

and the area element in the R2 is transformed under the formula

dA = dx1dx2 = det

∣∣∣∣∣∂xα∂ξβ

∣∣∣∣∣dξ1dξ2, α, β = 1, 2. (7.9)

The differential of the surface located in the R3 is defined by the expression

dS =(n21 + n

22 + n

23

)1/2dξ1dξ2, (7.10)


where

n1 =∂x1∂ξ1

∂x3∂ξ2

− ∂x2∂ξ2

∂x3∂ξ1

,

n2 =∂x3∂ξ1

∂x1∂ξ2

− ∂x1∂ξ1

∂x3∂ξ2

,

n3 =∂x1∂ξ1

∂x2∂ξ2

− ∂x2∂ξ1

∂x1∂ξ2

.

(7.11)

The element of length of a contour in the R2 is defined by the expression

dl =

[(dx1dξ1

)2

+(dx2dξ1

)2]1/2

dξ1. (7.12)

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

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

7.1. Piecewise Constant Approximation

The piecewise constant approximation is the simplest one. Interpolation functions in this casedo not depend on the FE form and the dimension of the domain. They have the form

ϕnq(x) =

⎧⎨

⎩

1 ∀x ∈ Vn,0 ∀x /∈ Vn.

(7.13)

This approximation is linear independent and compact in the functional spaces in which it isnot 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 anddimension of the domain. Therefore, interpolation functions are called shape functions.

In 1D case, the FEs are linear, and their shape functions are

ϕnq =12

(1 + ξq1ξ1

), ξ

q

i = ±1. (7.14)

In 2D case for the rectangular FE, shape functions are

ϕnq =14

(1 + ξq1ξ1

)(1 + ξq2ξ2

), ξ

q

i = ±1. (7.15)


For the triangular FE, it is convenient to introduce system of coordinates whichconnected with the Cartesian by relation

xi = ξqxq

i ,∑

ξq = 1 , i = 1, 2, q = 1, 2, 3. (7.16)

For the triangular FE, the interpolation functions are

ϕnq = ξq. (7.17)

The local coordinates ξq are also called area coordinates.

7.3. Piecewise Quadratic Approximation

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

In 1D case, the FEs are bilinear, and their shape functions are

ϕnq =12ξq

1ξ1(1 + ξq1ξ1

) (q = 1, 2, 3

)(7.18)

at the end nodes and

ϕnq = 1 − ξ21 (7.19)

at the midpoint node.In 2D case for the rectangular FE, shape functions are

ϕnq =14

(1 + ξq1ξ1

)(1 + ξq2ξ2

)(ξq

1ξ1 − ξq

2ξ2 − 1) (

q = 1, . . . , 8)

(7.20)

at the angular nodes and

ϕnq =12

(1 − ξ21

)(1 + ξq2ξ2

)ξq

1 = 0,

ϕnq =12

(1 − ξ22

)(1 + ξq1ξ1

)ξq

2 = 0

(7.21)

at the side nodes.For the triangular FE, the interpolation functions are

ϕnq =(2ξq − 1

)ξq

(q = 1, . . . , 6

)(7.22)

at the angular nodes and

ϕnq = 4ξ1ξ2 · · · . (7.23)

at the side nodes.


In order to construct system of algebraic equation for the BEM (6.18), we have torepresent global coordinates as function of local ones. The most convenient way to do thatis to use the interpolation function ϕnq defined in (7.13)–(7.23). In this case, the globalcoordinates have the form

xi = xq

i ϕnq(ξ). (7.24)

Here xqi 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 transformthem to the finite dimensional BE equations (6.18). In order to do that transform, we haveto calculate integrals (6.19). One of the main problems that occur in this situation is thepresence of the divergent integrals. They can not be calculated in traditional way, for example,numerically using quadrature formulas. For example, the integrals with the kernelsUij(x−y)are weakly singular (WS). They have to be considered as improper integrals. The integralswith the kernelsWij(x,y) andKij(x,y) are singular. They have to be considered in the sense ofCauchy as principal values (PVs). The integrals with the kernels Fij(x,y) are hypersingular.They have to be considered in the sense of Hadamard as finite parts (FPs). Traditionalapproach 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 thetheory of distribution. This approach consists in application of the second Green theorem andtransformation of divergent integrals into regular ones. In [12] we have developed formulasfor regularization of the divergent integrals r−m in the form

F.P.

∫a

−a

ϕ(x)rm

dx =k−1∑

i=0(−1)i+1 d

i

dxiPirm−k

dk−1−iϕ(x)dxk−1−i

∣∣∣∣∣

x=a

x=−a+ (−1)k

∫a

−a

Pkrm−k

dkϕ(x)dxk

,

F.P.∫V

ϕ(x)rm

dV =k−1∑

i=0(−1)i+1

∫

∂V

[Δk−i−1ϕ(x)∂n

Pirm−2i −

Pirm−2i ∂nΔ

k−i−1ϕ(x)]dS

+(−1)k ∫V1

rm−2kΔk+1ϕ(x)dV,

(8.1)

for 1D and 2D cases, respectively. We will demonstrate here how this approach works in theproblems of elastostatics.

8.1. 1D Divergent Integrals

In 2D elastostatics after introducing local system of coordinates and simplification, alldivergent integrals can be presented in the form

J0 =∫b

a

ϕ(x) ln1xdx, Jk =

∫b

a

ϕ(x)xk

dx, k = 1, 2 (8.2)

Here ϕ(x) is the smooth function that depend on the shape of the BE and the interpolationpolynomials.


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 inthe form

∫b

a

dg(x)dx

ϕ(x)dx = ϕ(x)g(x)∣∣ba −

∫b

a

dϕ(x)dx

g(x)dx. (8.3)

Let in this formula be g(x) = x ln 1/x, dg(x)/dx = ln 1/x, then we obtain

J0 =W.S.

∫b

a

ϕ(x) ln1xdx = ϕ(x)x ln

1x

∣∣∣∣

b

a

−∫b

a

dϕ(x)dx

x ln1xdx. (8.4)

Obviously the integral on the left is divergent and on the right is regular one. For the linearBE and piecewise constant approximation ϕ(x) = 1, and we get

J0 =W.S.

∫b

a

ln∣∣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 integrationby parts (8.3). Let in this formula g(x) = − ln 1/x, (dg(x))/dx = 1/x, then we obtain

J1 = P.V.∫b

a

ϕ(x)x

dx =(dϕ(x)dx

x ln1x− ϕ(x) ln 1

x

)∣∣∣∣

b

a

−∫b

a

d2ϕ(x)dx2

x ln1xdx. (8.6)

Here also the integral on the left is divergent and on the right is regular one. For the linear BEand piecewise constant approximation ϕ(x) = 1, and we get

J1 = P.V.∫b

a

dx

x − y = ln∣∣∣∣b − ya − y

∣∣∣∣,(a < y < b

). (8.7)

Finally for the hypersingular integral J2 regularization, we will use formula (8.1) andthe above-obtained result for singular J1 regularization. Finally we get

J2 = F.P.∫b

a

ϕ(x)x2

dx =

(d2ϕ(x)dx2

x ln1x− dϕ(x)

dxln

1x− ϕ(x)

x

)∣∣∣∣∣

b

a

−∫b

a

d3ϕ(x)dx3

x ln1xdx. (8.8)

Here also the integral on the left is divergent and on the right is regular one. For the linear BEand piecewise constant approximation ϕ(x) = 1, and we get

J2 = F.P.∫b

a

dx(x − y)2

= − 1(b − y) +

1(a − y)

(a < y < b

). (8.9)


From this equation can be found Hadamard’s example of a function that is positive everywere in the integration region, but its integral is a negative one

F.P.

∫a

−a

dy

y2= − 2

a, a > 0. (8.10)

8.2. 2D Divergent Integrals

In 3D elastostatics after introducing local system of coordinates and simplification, alldivergent integrals can be presented in the form

Jl,mk =∫

Sn

xl1xm2

rkϕ(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 interpolationpolynomials.

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

J0,01 =W.S.

∫

V

ϕ(x)r

dV =∫

∂V

[ϕ(x)

rn2r

− r∂nϕ(x)]dS +

∫

V

rΔϕ(x)dV. (8.12)

Here rn = (xα − yα)nα, 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, wecan calculate this integral analytically. Introducing polar, coordinates we will get

J0,01 =12

∫

∂Sn

rnrdl =

∫2π

0

r

rr dϕ = πr. (8.13)

In order to regularize singular integral, we will use, relation 1/r2 = (1/2)Δ(ln r)2. Then inthe same way we get

∫

V

ϕ(x)r2

dV =12

∫

∂V

(ϕ(x)

2rn ln rr2

− (ln r)2∂nϕ(x))dS +

12

∫

V

(ln r)2Δϕ(x)dV. (8.14)

The volume integral on the right is weakly singular. Taking into account the relation (ln r)2 =(r2/6)Δ(ln r)4, we obtain regularization for this weakly singular integral

∫

V

(ln r)2Δϕ(x)dV =16

∫

∂V

(2Δϕ(x)rn(ln r)

3 − r2(ln r)4∂nΔϕ(x))dS

+16

∫

V

r2(ln r)4Δ2ϕ(x)dV.

(8.15)


For the linear BE and piecewise constant approximation ϕ(x) = 1 and circular area, wecan calculate singular integral in (8.14) analytically. Introducing polar coordinates, we willget

J0,02 =∫

∂Sn

rn ln rr2

dl =∫2π

0

r ln rr2

rdϕ = 2π ln r. (8.16)

Finally from (8.1) for k = 3, we get regularization for hupersingular integral

∫

V

ϕ(x)r3

dV =∫

∂V

[Δϕ(x)

rn2r

− ϕ(x)rnr3

− 1r∂nϕ(x) − r∂nΔϕ(x)

]dS +

∫

V

rΔ2ϕ(x)dV . (8.17)

For the linear BE and piecewise constant approximation ϕ(x) = 1 and circular area, wecan calculate this integral analytically. Introducing polar coordinates, we will get

J0,03 = −∫

∂Sn

rnr3dl = −

∫2π

0

r

r3r dϕ = −2π

r. (8.18)

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

8.2.2. Integrals with Kernels of the Type x2α/r

k, k = 3, 4, 5

The weakly singular integral with kernel x2α/r

3 is calculated taking into account the equationx2α/r

3 = (1/3)((2/r) −Δ(x2α/r)). It is easy to show that

J2,03 =W.S.

∫

V

ϕ(x)x21

r3dV =

23W.S.

∫

V

ϕ(x)r

dV − 13W.S.

∫

V

ϕ(x)Δx21

rdV. (8.19)

The first integral here is already calculated in (8.12). The second one may be presented in theform

∫

V

ϕ(x)Δx21

rdV =

∫

∂V

[

ϕ(x)

(2n1x1r

− x12rnr3

)

− x21

r∂nϕ(x)

]

dS +∫

V

x21

rΔϕ(x)dV . (8.20)

Combining the last two equations, finally we will get

J2,03 =13

∫

∂V

[

ϕ(x)

(2n1x1r

− x12rnr3

+rn2r

)

−(x21

r+ r

)

∂nϕ(x)

]

dS

+13

∫

V

(x21

r+ r

)

Δϕ(x)dV.

(8.21)


For the linear BE and piecewise constant approximation ϕ(x) = 1 and circular area wecan calculate this integral analytically. Introducing polar coordinates, we will get

J2,03 =13

∫

∂Sn

(x21rn

r3− 2x1n1

r+2rnr

)

dl

=13

(∫2π

0

(r cosϕ

)2r

r3rdϕ −

∫2π

0

2r(cosϕ

)2

rrdϕ + 2

∫2π

0

r

rrdϕ

)

= πr.

(8.22)

The singular integral with kernel x2α/r

4 is calculated taking into account thatΔx2α/r

4 =(1/4)((2/r2) −Δx2

α/r2). In this case,

J2,04 = P.V.∫

V

ϕ(x)x21

r2dV =

14

∫

V

ϕ(x)

(2r2

−Δx21

r2

)

dV =12

∫

V

ϕ(x)r2

dV − 14

∫

V

ϕ(x)Δx21

r2dV.

(8.23)


∫

V

ϕ(x)Δx1

2

r2dV =

∫

∂V

[

ϕ(x)

(2n1x1r2

− 2x12rnr4

)

− x12

r2∂nϕ(x)

]

dS +∫

V

x12

r2Δϕ(x)dV.

(8.24)


J2,04 =12

∫

∂V

[

ϕ(x)

(x1

2rnr4

− n1x1r2

+rn ln rr2

)

−(

(ln r)2

2− x1

2

2r2

)

∂nϕ(x)

]

dS

+14

∫

V

(

(ln r)2 − x12

r2

)

Δϕ(x)dV.

(8.25)


J2,04 =12

∫

∂Sn

[x1

2rnr4

− x1n1r2

+rn ln rr2

]

dl =∫2π

0

(r cosϕ

)2r

r4rdϕ −

∫2π

0

r(cosϕ

)2

r2rdϕ

+∫2π

0

r ln rr2

rdϕ

= 2π ln r.

(8.26)


The hypersingular integral with kernel x2α/r

5 is calculated taking into account thatx2α/r

5 = (1/3)((2/r3) −Δ(x2α/r

3)). In this case, it is easy to show that

J2,05 = F.P.∫

V

ϕ(x)x21

r5dV =

13

∫

V

ϕ(x)

(2r3

−Δx21

r3

)

dV =23

∫

V

ϕ(x)r3

dV − 13

∫

V

ϕ(x)Δx21

r3dV.

(8.27)


∫

V

ϕ(x)Δx21

r3dV =

∫

∂V

[

ϕ(x)

(2n1x1r3

− 3x12rnr5

)

− x21

r3∂nϕ(x)

]

dS +∫

V

x21

r3Δϕ(x)dV. (8.28)


J2,05 =23

∫

∂V

(

ϕ(x)

(n1x1r3

− rnr3

− 3x12rn2r5

)

−(

1r+x21

r3

)

∂nϕ(x)

)

dS

+23W.S.

∫

V

(1r− x2

1

r3

)

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, wecan calculate this integral analytically. Introducing polar coordinates, we will get

J2,05 =∫

∂Sn

(2rn3r3

+2x12rn3r5

− x1n1r3

)

dl =23

∫2π

0

r

r3rdϕ +

23

∫2π

0

(r cosϕ

)2r

r5rdϕ

−∫2π

0

r(cosϕ

)2

r3rdϕ

= −πr.

(8.30)

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

The weakly singular integral with kernel (x1x2)/r3 is calculated using (8.1). Taking intoaccount that (x1x2)/r3 = −1/3Δ(x1x2)/r, it is easy to show that

J1,13 =W.S.

∫

V

ϕ(x)x1x2r3

dV =13

∫

∂V

[ϕ(x)

(x1x2rnr3

− r∗r

)+x1x2r

∂nϕ(x)]dS

− 13

∫

V

x1x2r

Δϕ(x)dV.

(8.31)

Here r∗ = x1n2 + x2n1.



J1,13 =13

∫

∂Sn

[x1x2rnr3

− r∗r

]dl =

∫2π

0

r3 cosϕ sinϕr3

r dϕ −∫2π

0

2r cosϕ sinϕr

r dϕ = 0. (8.32)

The singular integral with kernel x1x2/r4 is calculated using equation (8.1). Takinginto account that x1x2/r4 = −(1/4)Δx1x2/r2, it is easy to show that

J1,14 = P.V.∫

V

ϕ(x)x1x2r4

dV =14

∫

∂V

[ϕ(x)

(2x1x2rnr4

− r∗r2

)− x1x2

r2∂nϕ(x)

]dS

− 14

∫

V

x1x2r2

Δϕ(x)dV.

(8.33)


J1,14 =14

∫

∂Sn

[2x1x2rnr4

− r∗r2

]dl =

14

∫2π

0

2r3 cosϕ sinϕr4

rdϕ − 14

∫2π

0

2r cosϕ sinϕr2

rdϕ = 0.

(8.34)

The hypersingular integral with kernel x1x2/r5 is calculated using (8.1). Taking intoaccount that x1x2/r5 = −(1/3)Δx1x2/r3, it is easy to show that

J1,15 = F.P.∫

V

ϕ(x)x1x2r5

dV =∫

∂V

[ϕ(x)

(x1x2rnr5

− r∗3r3

)− x1x2

r3∂nϕ(x)

]dS

− 13

∫

V

x1x2r3

Δϕ(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

J1,15 =∫

∂Sn

[x1x2rnr5

− r∗3r3

]dl =

∫2π

0

r3 cosϕ sinϕr5

r dϕ −∫2π

0

2r cosϕ sinϕ3r3

r dϕ = 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 arepresented when the boundary integral equation methods are used to solve problems inelastostatics. 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 equationsrelated to the boundary integral equation and boundary elements methods formulation in2D and 3D elastostatics have been discussed in details.

References

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[10] A. N. Guz and V. V. Zozulya, “Elastodynamic unilateral contact problems with friction for bodieswith cracks,” International Applied Mechanics, vol. 38, no. 8, pp. 3–45, 2002.

[11] V. V. Zozulya, “Integrals of Hadamard type in dynamic problems of crack theory,” Doklady AkademiiNauk Ukrainskoj SSR Serija A, no. 2, pp. 19–22, 1991 (Russian).

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of Studies in Applied Mechanics, Elsevier Scientific, Amsterdam, The Netherlands, 1980.

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