cauchy principal values and finite parts of boundary integrals—revisited

Cauchy principal values and finite parts of boundary integrals—revisited

Subrata Mukherjeea,*, Yu Xie Mukherjeeb, Wenjing Yec

aDepartment of Theoretical and Applied Mechanics Kimball Hall, Cornell University, Ithaca, NY 14853, USAbAvant Analysis Technology 39 Hickory Circle Ithaca, NY 14850, USA

cWoodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

Received 27 January 2005; accepted 22 April 2005

Available online 29 June 2005

Abstract

The relationship between Finite Parts (FPs) and Cauchy Principal Values (CPVs) (when they exist) of certain integrals has been previously

studied by Toh and Mukherjee [Toh K-C, Mukherjee S. Hypersingular and finite part integrals in the boundary element method. Int J Solids

Struct 1994;31:2299–2312] and Mukherjee [Mukherjee S. CPV and HFP integrals and their applications in the boundary element method. Int

J Solids Struct 2000;37:6623–6634, Mukherjee S. Finite parts of singular and hypersingular integrals with irregular boundary source points.

Engrg Anal Bound Elem 2000;24:767–776]. This paper continues this study and presents and proves an interesting new relationship between

the CPV and FP of certain boundary integrals (on closed boundaries) that occur in Boundary Integral Equation (BIE) formulations of some

common Boundary Value Problems (BVPs) in science and engineering.

q 2005 Elsevier Ltd. All rights reserved.

1. Introduction

This paper is concerned with Finite Parts (FPs) and

Cauchy Principal Values (CPVs) (when they exist) of

certain strongly singular boundary integrals that commonly

occur in Boundary Integral Equation (BIE) formulations of

Boundary Value Problems (BVPs). Such matters have been

considered before in Toh and Mukherjee [1] and Mukherjee

[2,3]. A certain relationship between the FP and CPV (when

it exists) of strongly singular integrals was stated and proved

in [2]. The present paper is a continuation of the previous

work on this subject. In particular, a new relationship

between the CPV and FP of certain integrals, of interest in

the BIE formulation of certain BVPs, is stated and proved

here. Examples of BVPs chosen in this work are three-

dimensional (3D) exterior problems in linear elasticity,

Stokes’ flow and potential theory.

The rest of this paper is organized as follows. Certain

previously published relationships between FPs, CPVs, etc.

are reviewed in Section 2. A new theorem is presented in

Section 3. Examples of applications of this new theorem, in

0955-7997/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enganabound.2005.04.008

* Corresponding author. Tel.: C1 607 255 7143; fax: C607 255 2011.

E-mail address: [email protected] (S. Mukherjee).

elasticity, Stokes flow and potential theory, appear in

Section 4. Finally, a proof of the new theorem appears in

Section 5.

2. Definitions and relationships

Start with a few pertinent definitions. Let S be an (open or

closed) surface in R3 and the function Kðx; yÞ/R have its

only singularity at yZx of the form Kðx; yÞZOð1=r2Þ with

rZyKx and rZjrj. Also, let f : S/R be a regular

function of the class C0,a at x2S. Now, let Se be an

exclusion or inclusion neighborhood of x2S. If SZvB is a

closed surface and the region B is exterior to vB, then SðEÞe is

an exclusion neighborhood if the boundary point x is

approached from outside B; and conversely for SðIÞe . (See

Fig. 1 for the case where SZvB is a closed surface and the

region B, exterior to vB, is the domain of the problem where

the relevant partial differential or integral equations apply.

Also, B0 is interior to vB and B and B0 are open sets. Finally,

the unit normal n(y) to vB at y points away from the region

B). Here x2S is taken to be a regular point (i.e. S is locally

smooth at x) and the neighborhood Se chosen in a symmetric

manner consistent with the usual definition of a CPV

integral given below. Therefore, Se is the surface of a

hemisphere of radius e. Finally, it is assumed that this CPV

integral exists.

Engineering Analysis with Boundary Elements 29 (2005) 844–849

www.elsevier.com/locate/enganabound

http://www.elsevier.com/locate/enganabound

n(y)

S=∂BS=∂B

B0 B0x

y

x

y

n(y)

BB

S(E)

S(I)(a)(b)

P

PQ

Q

S

S

^

^

ε

ε

Fig. 1. (a) Exterior approach and exclusion zone. (b) Interior approach and inclusion zone.

S. Mukherjee et al. / Engineering Analysis with Boundary Elements 29 (2005) 844–849 845

CPV integral. The CPV of the integral

IZÐ

S Kðx; yÞfðyÞdSðyÞ, denoted by the symbol G, has the

usual definition:

6S

Kðx; yÞfðyÞdSðyÞ Z limSe/0

ðSKSe

Kðx; yÞfðyÞdSðyÞ (1)

FP integral. Following [1,2], the FP of the integral

IZÐ

S Kðx; yÞfðyÞdSðyÞ, denoted by the symbol E=, can be

defined as

4¼ðE or IÞ

S

Kðx; yÞfðyÞdSðyÞ

Z

ðSKSðE or IÞ

e

Kðx; yÞfðyÞdSðyÞC

ðSðE or IÞ

e

Kðx; yÞ½fðyÞ

KfðxÞ�dSðyÞCfðxÞAðSðE or IÞe Þ (2)

where

AðSðE or IÞe Þ Z 4¼

SðE or IÞe

Kðx; yÞdSðyÞ (3)

Several ways of evaluating AðSeÞ are described in [3].

It is important to mention here that (2) and (3) is

a somewhat restricted version of the FP definition given in

[1–3]. Here, SðE or IÞe is the surface of a hemisphere around

the singular point x (see Fig. 1) while in [3] one has S which

is a piece of S that contains the singular point and can even

be equal to the entire surface S! Only the situation Se/0,

however, is of interest here since the CPV of an integral (see

(1)) is only defined for this case. The limit Se/0, where Se

is the surface of a hemisphere, is equivalent to the case

S/0 where S is the piece PQ of the surface S in Fig. 1 (see,

also, Section 3.2 of [3]).

The FP and the LTB. There is a simple relationship

between the FP, as defined above, and the Limit to the

Boundary (LTB) of the strongly singular integral under

consideration. With x;S/x2S, one has:

4¼S

Kðx; yÞfðyÞdSðyÞ Z limx/x

ðS

Kðx; yÞfðyÞdsðyÞ (4)

Of course, x can approach x from either side of x and the

resulting FP is, in general, dependent on the side of the

approach. Eq. (4) is proved in [3].

In anticipation of what is to follow, the relationship

expressed in (4) is written for the case when SZvB is a

closed surface. The FP of the integral, for an exterior and an

interior approach (see Fig. 1), is defined as:

4¼ðEÞ

vB

Kðx; yÞfðyÞdSðyÞ Z limx2B0/x2vB

ðS


4¼ðIÞ

vB

Kðx; yÞfðyÞdSðyÞ Z limx2B/x2vB

ðS


Please see [4] for applications of the above definitions for

an interior problem.

The FP and the CPV. The FP and the CPV of the integral

under consideration are related as IFPZ ICPV C lime/0Ie [2].

In explicit form:

4¼ðE or IÞ

S


Z6S

Kðx; yÞfðyÞdSðyÞC lime/0

ðSðE or IÞ

e


It is noted that the CPV integral in (7) is independent of

the (exterior or interior) approach while the last integral in

(7) (and, therefore, the FP integral on the left hand side of

(7)) does depend on the approach and is interpreted as

exterior or interior depending on whether Se is an exclusion

or an inclusion zone (see Fig. 1).

Relationship (7) is proved in [2].

S. Mukherjee et al. / Engineering Analysis with Boundary Elements 29 (2005) 844–849846

3. A new theorem

The following relationship is true for certain boundary

integrals that occur in BIE formulations for certain BVPs in

science and engineering:

26vB


Z 4¼ðEÞ

vB

Kðx; yÞfðyÞdSðyÞC4¼ðIÞ

vB


Eq. (8) is implied and proved for the specific case of

Stokes flow in [5]. The present paper extends this idea to

BIE formulations for linear elasticity and potential theory

(in addition to Stokes flow) problems in Section 4. A proof

of the new theorem is given in Section 5.

A simple illustration of (8) is the solid angle example in

Section 2.4 of [2]. The solid angle subtended by a surface S

at a point x is

UðS; xÞ Z

ðS

rðx; yÞ$nðyÞ

r3ðx; yÞdSðyÞ (9)

where r(x, y)ZyKx, n(y) is the unit normal to S at y and

rZjrj. Let SZvB the surface of a complete sphere (the case

bZ0 in Fig. 2 of [2]), and take the limit x;vB/x2vB.

Now, one gets the results (see [2])

UIN Z 4p; UOUT Z 0; UCPV Z 2p (10)

where UIN is the result for x inside and UOUT the result for x

outside the sphere. Please note that UIN and UOUT are FP

integrals when x/x (see (5), (6) and (9)) and UCPV is the

corresponding CPV integral.

4. Applications of the new theorem

Applications of the new theorem (8), in exterior

problems of 3D linear elasticity, Stokes flow and potential

theory, are presented in this Section (see Fig. 2). In all cases,

it is assumed that the primary variable /0 as x/N.

∂B

B0

B

x

y

n

ξ ξ

Fig. 2. Notation for BIEs.

4.1. Regular BIEs at x not in vB

These equations are of the form:

0 Z

ðvB

Gijðx; yÞtjðyÞdSðyÞ

C

ðvB

Tijkðx; yÞnkðyÞvjðyÞdSðyÞ x2B0 (11)

viðxÞ Z

ðvB


C

ðvB

Tijkðx; yÞnkðyÞvjðyÞdSðyÞ x2B (12)

In (11) and (12), one has the following:

Elasticity: v, displacement; t, traction.

Stokes flow: v, velocity; t, traction.

Potential theory: v, potential, tZ(vv/vn). In this case,

Eq. (11), for example, is of the form:

0 Z

ðvB

Gðx; yÞtðyÞdSðyÞ

C

ðvB

Tkðx; yÞnkðyÞvðyÞdSðyÞ x2B0 (13)

The kernels in Eqs. (11)–(13) are given in Appendix A.

4.2. Regular BIEs at x in vB

FP form for exterior approach. Let x2B0/x2vB in

(11). Using (5), one gets:

0 Z

ðvB


C4¼ðEÞ

vB

Tijkðx; yÞnkðyÞvjðyÞdSðyÞ x2vB (14)

FP form for interior approach. Let x2B/x2vB in


viðxÞ Z

ðvB


C4¼ðIÞ

vB


CPV form. Using the new theorem (8), the mean of (14)

and (15) gives the CPV form (at a regular point x2vB):


viðxÞ

2Z

ðvB


C6vB


4.3. Gradient BIEs at x not in vB

First note that:

ðvB

vTijk

vxm

ðx; yÞnkðyÞdSðyÞ Z 0 (17)

Differentiate (11) with respect to xm and use (17) to get:

0 Z

ðvB

vGij

vxm

ðx; yÞtjðyÞdSðyÞC

ðvB

vTijk

vxm

ðx; yÞnkðyÞ

!½vjðyÞKvjðxÞ�dSðyÞ x2B0 (18)

Next, Differentiate (12) with respect to xm and use (17) to

get:

vvi

vxm

ðxÞ Z

ðvB

vGij

vxm

ðx; yÞtjðyÞdSðyÞC

ðvB

vTijk

vxm

ðx; yÞ

!nkðyÞ½vjðyÞKvjðxÞ�dSðyÞ x2B ð19Þ

The kernels in Eqs. (18) and (19) are given in Appendix B.

x

y

y

y2

y1

y3

n(y)

n(y)

S(E)

S(I)

ε

ε

ε

Fig. 3. Local coordinates.

4.4. Gradient BIEs at x in vB

FP form for exterior approach. Let x2B0/x2vB in


0 Z 4¼ðEÞ

vB

vGij

vxm

ðx; yÞtjðyÞdSðyÞC4¼ðEÞ

vB

vTijk

vxm

ðx; yÞnkðyÞ

!½vjðyÞKvjðxÞ�dSðyÞ x2vB (20)

FP form for interior approach. Let x2B/x2vB in


vvi

vxm

ðxÞ Z4¼ðIÞ

vB

vGij

vxm

ðx; yÞtjðyÞdSðyÞC4¼ðIÞ

vB

vTijk

vxm

ðx; yÞ

!nkðyÞ½vjðyÞKvjðxÞ�dSðyÞ x2vB ð21Þ

CPV form. Using the new theorem (8), the mean of (20)

and (21) gives the CPV form (at a regular point x/vB):

1

2

vvi

vxm

ðxÞZ6vB

vGij

vxm

ðx;yÞtjðyÞdSðyÞ

C6vB

vTijk

vxm

ðx;yÞnkðyÞ½vjðyÞKvjðxÞ�dSðyÞ x2vB

(22)

5. Proof of the new theorem

5.1. Nature of Kernels on Se

The regions SðEÞe and SðIÞ

e from Fig. 1 are plotted together

in Fig. 3 so that SðEÞe gSðIÞ

e is the surface of a sphere of radius

e. Local coordinates are defined, with the origin at x, as

shown in Fig. 3. The upper hemispherical surface SðEÞe has

y3O0 while the lower hemispherical surface SðIÞe has y3!0

In this local coordinate system, the following are true:

jrj Z r Z e; r;k Z yk=e;

nk ZKyk=e for y3O0; nk Z yk=e for y3!0(23)

Also, from a Taylor series expansion

vjðyÞ Z vjðxÞCvvj

vxn

ðxÞðyn KxnÞCh:o:t: (24)

so that one has, in the local coordinates of Fig. 3

vjðyÞKvjðxÞ Z CjnðxÞyn Ch:o:t: (25)

It is assumed that CjnðxÞzOð1Þ. Using Eqs. (23)–(25) and

the expressions for the kernels in Appendices, it can be

shown that the strongly singular integrals in Eqs. (14)–(16)

and (20)–(22) (within multiplicative functions of x) can be

categorized into one of two kinds

S. Mukherjee et al. / Engineering Analysis with Boundary Elements 29 (2005) 844–849848

I1 Z

ðSðEÞe gSðIÞe

yiyjyk /ym

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{n terms

enC2dS; i; j; k;.;m Z 1; 2; or 3

Z

ðSðEÞe gSðIÞe

yn1

1 yn2

2 yn3

3

enC2dS; n Z n1 Cn2 Cn3; n is odd

(26)

I2 Z

ðSðEÞe gSðIÞe

ayiyjyk /ym

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{n terms

enC2dS; i; j; k;.;m Z 1; 2; or 3

Z

ðSðEÞe gSðIÞe

ay

n1

1 yn2

2 yn3

3

enC2dS; n Z n1 Cn2 Cn3; n is even

(27)

where aZK1 for y3O0 and aZ1 for y3!0.

5.2. Proof of theorem

Lemma 1. I1Z0

Proof. Consider the two cases:

(a) In the integrand of I1, let n1 and/or n2 be odd. Then the

integral vanishes on each (upper and lower) hemi-

spherical surface and I1Z0.

(b) In the integrand in I1, let n1 and n2 be even. Then n3

must be odd. In this case, the integrand is an odd

function on the two (upper and lower) hemispherical

surfaces. Therefore, I1Z0.

,

Lemma 2. I2Z0

Proof. Consider the two cases:

(a) In the integrand of I2, let n1 and/or n2 be odd. Then the

integral vanishes on each (upper and lower) hemi-

spherical surface and I2Z0.

(b) In the integrand in I2, let n1 and n2 be even. Then n3

must be even. In this case, the integrand, without a, is

an even function on the two (upper and lower)

hemispherical surfaces. Therefore, I2Z0.

Proof of the new theorem (8) follows from Lemmas 1

and 2 and Eq. (7). From (7), one has:

4¼ðEÞ

S


Z6S


ðSðEÞe


4¼ðIÞ

S


Z6S


ðSðIÞe


Add (28) and (29) and use Lemmas 1 and 2 to get (8).

Appendix A. Kernels in the Regular BIEs (11)–(13)

Elasticity:

Gijðx; yÞ Z1

16pmð1 KnÞr½ð3 K4nÞdij Cr;ir;j� (A1)

Tijkðx; yÞ Z1

8pð1 KnÞr2½ð1 K2nÞ½Kr;idjk Cr;jdik

Cr;kdij�C3r;ir;jr;k� (A2)

In the above, m is the shear modulus, n is the Poisson’s

ratio, rZjyKxj, r,kZvr/vykZ(ykKxk)/r and d is the

Kronecker delta.

Stokes flow:

The equations are the same as (A1) and (A2) with m the

dynamic viscosity and nZ1/2.

Potential theory:

Gðx; yÞ Z1

4pr; Tkðx; yÞ Z

r;k

4pr2(A3)

Appendix B. Kernels in the Gradient BIEs (18) and (19)

Elasticity:

vGij

vxm

ðx; yÞ Z1

16pmð1 KnÞr2½ð3 K4nÞdijr;m Kr;idjm

Kr;jdim C3r;ir;jr;m� (B1)

vTijk

vxm

ðx; yÞ Zð1 K2nÞ

8pð1 KnÞr3½dimdjk Kdjmdik Kdkmdij

K3ðr;ir;mdjk Kr;jr;mdik Kr;kr;mdijÞ�

K3

8pð1 KnÞr3½r;jr;kdim Cr;ir;kdjm

Cr;ir;jdkm K5r;ir;jr;kr;m� (B2)

Stokes flow

The equations are the same as (B1) and (B2) with m the

dynamic viscosity and nZ1/2.

Potential theory:


vG

vxm

ðx; yÞ Zr;m

4pr2;

vTk

vxm

ðx; yÞ Z1

4pr3½Kdkm C3r;kr;m�

(B3)

References

[1] Toh K-C, Mukherjee S. Hypersingular and finite part integrals in the

boundary element method. Int J Solids Struct 1994;31:2299–312.

[2] Mukherjee S. CPV and HFP integrals and their applications in

the boundary element method. Int J Solids Struct 2000;37:

6623–34.

[3] Mukherjee S. Finite parts of singular and hypersingular integrals with

irregular boundary source points. Eng Anal Bound Elem 2000;24:

767–76.

[4] Mukherjee S, Kulkarni SS. Mean value theorems for integral

equations in 2-D potential theory. Eng Anal Bound Elem 2003;27:

183–91.

[5] Ding J, Ye W. A fast integral approach for drag force calculation

due to oscillatory slip Stokes flows. Int J Num Meth Eng 2004;60:

1535–67.

cauchy principal values and finite parts of boundary integrals—revisited

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