internal variables and their sensitivities in three- dimensional linear elasticity by the boundary...

Internal variables and their sensitivities in three- dimensionallinear elasticity by the boundary contour method

Subrata Mukherjee a,*, Xiaolan Shi b, Yu Xie Mukherjee b

a Department of Theoretical and Applied Mechanics, 220 Kimball Hall, Cornell University, Ithaca, NY 14853-1503, USAb DeHan Engineering Numerics, 95 Brown Road, Box 1016, Ithaca, NY 14850, USA

Abstract

A variant of the usual boundary element method (BEM), called the boundary contour method (BCM), has been presented in the

literature in recent years. In the BCM in three-dimensions, surface integrals on boundary elements of the usual BEM are transformed,

through an application of Stokes' theorem, into line integrals on the bounding contours of these elements. A new formulation for

design sensitivities in three-dimensional linear elasticity, based on the BCM, has been recently presented in Ref. [12]. This challenging

derivation is carried out by ®rst taking the material derivative of the regularized boundary integral equation (BIE) with respect to a

shape design variable, and then converting the resulting equation into its boundary contour version. The focus of [12] is the boundary

problem, i.e., evaluation of displacements, stresses and their sensitivities on the bounding surface of a body. The focus of the present

paper is the corresponding internal problem, i.e., analogous calculations at points inside a body. Numerical results for internal

variables and their sensitivities are presented here for selected examples. Ó 2000 Elsevier Science S.A. All rights reserved.

1. Introduction

An interesting variant of the boundary element method (BEM), called the boundary contour method(BCM), has been presented in the literature in recent years. The BCM achieves a further reduction in di-mension compared to the BEM, in that, for three-dimensional (3-D) linear elasticity problems, it onlyrequires numerical evaluations of line integrals over the closed bounding contours of the usual (surface)boundary elements. The central idea of the BCM is the exploitation of the divergence- free property of theusual BEM integrand and a very useful application of Stokes' theorem, to analytically convert surfaceintegrals on boundary elements to line integrals on closed contours that bound these elements. Lutz [1] ®rstproposed an application of this idea for the Laplace equation. Applications of the BCM in 2-D linearelasticity have been presented in Refs. [2±4]; while applications in 3-D linear elasticity appear in Refs. [5±7].

Design sensitivity coe�cients (DSCs), which are de®ned as rates of change of physical response quan-tities with respect to changes in design variables, are useful for various applications such as in judging therobustness of a given design, in reliability analysis and in solving inverse and design optimization problems.There are three methods for design sensitivity analysis (e.g., Haug et al. [8]), namely, the ®nite di�erenceapproach (FDA), the adjoint structure approach (ASA) and the direct di�erentiation approach (DDA).The DDA is of interest in this paper.

The BEM, which only requires surface meshing, is particularly e�cient for shape optimization problemswhere remeshing typically becomes necessary after each iterative step of an optimization process. There-fore, several researchers have used the BEM to develop e�cient techniques for computing DSCs. Besides

www.elsevier.com/locate/cmaComput. Methods Appl. Mech. Engrg. 187 (2000) 289±306

* Corresponding author. Tel.: +1-607-255-7143; fax: +1-607-255-2011.

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

0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 1 3 6 - X

having the same meshing advantage as in the conventional BEM, the BCM o�ers a further reduction indimension. Furthermore, stresses at boundary nodes can be recovered easily and accurately in the BCMfrom the global displacement shape functions. These advantages of the BCM make it very competitive withthe BEM in optimal shape design. Shape sensitivity analysis using the BCM has been presented for 2-Delasticity problems in Refs. [9,10]. Shape optimization for 2-D elasticity with the BCM is the subject ofRef. [11].

The goal of obtaining BCM sensitivity equations can be achieved in two equivalent ways. In the 2-Dwork by Phan et al. [9], design sensitivities are obtained by ®rst converting the discretized BIEs into theirboundary contour version, and then applying the DDA (using the concept of the material derivative) to thisBCM version. This approach, while relatively straightforward in principle, becomes extremely algebraicallyintensive for 3-D elasticity problems. Mukherjee et al. [12] o�ers a novel alternative derivation (for 3-Delasticity), using the opposite process, in which the DDA is ®rst applied to the regularized BIE and then theresulting equations are converted to their boundary contour version. This new derivation, together withnumerical results for boundary variables (displacements and stresses) and their sensitivities, is the primarycontribution of Ref. [12]. A similar derivation for 2-D elasticity appears in Ref. [10].

In both the BEM and the BCM, the process of obtaining the unspeci®ed boundary variables (dis-placements and tractions, then stresses) is carried out ®rst. This is then followed by calculation of variables(displacements and stresses) at speci®ed internal points where these quantities are desired. The same is truefor sensitivity analysis, namely that a boundary calculation (for unspeci®ed components of the sensitivitiesof tractions and displacements, then stresses) is followed by an internal one (for sensitivities of displace-ments and stresses). Ref. [12] addresses the boundary problem ± for both the appropriate variables and theirsensitivities. The current paper addresses the corresponding internal problem. Thus, this paper can beconsidered to be a companion paper to Ref. [12].

This paper is organized as follows. A brief review of the standard BCM for 3-D linear elasticity ispresented ®rst. The next section presents equations for the displacements, stresses, and curvatures at aninternal point. This is followed by a section that presents derivations for the sensitivities of displacements,and then stresses, at an internal point. Numerical results for internal variables and their sensitivities, forillustrative problems, follow. The last section of the paper presents some concluding remarks.

2. Boundary surface and boundary contour integral equations

A regularized form of the standard boundary integral equation (Rizzo [13]), for 3-D linear elasticity, canbe written as

0 �Z

oBUik�x; y�rij�y�� ÿ Rijk�x; y�fui�y� ÿ ui�x�g

�ej � dS�y� �

ZoB

Fk � dS�y�: �1�

Here, oB is the bounding surface of a body B (B is an open set) with in®nitesimal surface area dS � dSn,where n is the unit outward normal to oB at a point on it. The stress tensor is r, the displacement vector isu and ej �j � 1; 2; 3� are global Cartesian unit vectors. The BEM Kelvin kernels are written in terms of(boundary) source points x and ®eld points y. These are:

Rijk � ÿ 1

8p�1ÿ m�r2�1� ÿ 2m��r;idjk � r;jdik ÿ r;kdij� � 3r;ir;jr;k

�; �2�

Uik � 1

16pl�1ÿ m�r �3� ÿ 4m�dik � r;ir;k�; �3�

in terms of r, the Eucledian distance between the source and ®eld points x and y, and the shear modulus land Poisson's ratio m of the isotropic elastic solid. Also, d is the Kronecker delta and ;i � o=oyi: The range ofindices in these and all other equations in this paper is 1, 2, 3, unless speci®ed otherwise.

It has been shown in Refs. [2,5,6] that the integrand vector Fk in Eq. (1) is divergence free (except at thepoint of singularity x � y) and that the surface integral in it, over an open surface patch S 2 oB, can be

290 S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 289±306

converted to a contour integral around the bounding curve C of S, by applying Stokes' theorem. Therefore,vectors Vk exist such thatZ

SFk � dS �

IC

Vk � dr: �4�

Since the vectors Fk contain the unknown ®elds u and r, shape functions must be chosen for thesevariables, and potential functions derived for each linearly independent shape function, in order to de-termine the vectors Vk. Also, since the kernels in Eq. (1) are functions only of zk � yk ÿ xk (and not of thesource and ®eld coordinates separately), these shape functions must also be written in the coordinates zk inorder to determine the potential vectors Vk. Finally, these shape functions are global in nature and arechosen to satisfy, a priori, the Navier±Cauchy equations of equilibrium. The weights, in linear combina-tions of these shape functions, however, are de®ned piecewise on boundary elements.

Quadratic shape functions are used in this work. With

zk � yk ÿ xk �5�one has, on a boundary element:

ui �X27

a�1

bauai�y1; y2; y3� �X27

a�1

b̂a�x1; x2; x3�uai�z1; z2; z3�; �6�

rij �X27

a�1

baraij�y1; y2; y3� �X27

a�1

b̂a�x1; x2; x3�raij�z1; z2; z3�; �7�

where uai; raij (with i � 1; 2; 3 and a � 1; 2; . . . ; 27) are the shape functions and ba are the weights in thelinear combinations of the shape functions. Each boundary element has, associated with it, 27 constants ba

which are related to physical variables on that element. This set of b's di�er from one element to the next.The displacement shape functions for a � 1; 2; 3 are constants, those for a � 4; . . . ; 12 are of ®rst degree

and those for a � 13; . . . ; 27 are of second degree. There are a total of 27 linearly independent (vector)shape functions ua. The shape functions for the stresses are obtained from those for the displacementsthrough the use of Hooke's law. The shape functions uai and raij are given in Mukherjee et al. [6].

It is easy to show that the coordinate transformation (5) results in the constants b̂j being related to theba's as follows:

b̂i �X27

a�1

Sia�x1; x2; x3�ba; i � 1; 2; 3; �8�

b̂k �X27

a�1

Rna�x1; x2; x3�ba; k � 4; 5; . . . ; 12; n � k ÿ 3; �9�

b̂a � ba; a � 13; 14; . . . ; 27; �10�where

Sia � uai�x1; x2; x3�; i � 1; 2; 3; a � 1; 2; . . . ; 27;

Rka � oua`�y1; y2; y3�oyj

��x1;x2;x3�

; k � 1; 2; . . . ; 9; a � 1; 2; . . . ; 27

with j � 1� b�k ÿ 1�=3c and ` � k ÿ 3j� 3. Here, the symbol bnc, called the ¯oor of n, denotes the largestinteger less than or equal to n.

It is useful to note that the matrices S and R are functions of only the source point coordinates�x1; x2; x3�.

S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 187 (2000) 289±306 291

The procedure for designing boundary elements in the 3-D BCM is discussed in detail in Nagarajan et al.[5] and Mukherjee et al. [6]. A set of primary physical variables ak, whose number must match the number(here 27) of arti®cial variables bk on a boundary element, are chosen ®rst. The ®rst step in the BCM so-lution procedure is to determine the unspeci®ed primary physical variables in terms of those prescribedfrom the boundary conditions. Later, secondary physical variables as well as stresses, at boundary points,are obtained from a simple post-processing procedure. It is particularly easy to obtain surface variables,such as stresses and curvatures, in the BCM. This issue is discussed in Ref. [12].

A square invertible transformation matrix T relates the vectors a and b on element m according to theequation:

am � T

mbm: �11�

The CIM9 boundary element, shown in Fig. 1(c) in Nagarajan et al. [5], is used in the present work.Details of the shape functions and intrinsic coordinates, that are used to de®ne the geometry of theboundary elements, are available in Mukherjee et al. [6]. Also, the procedure for obtaining the vectorpotentials Vk, for nonsingular as well as singular integrands, are available in Refs. [5,6]. Finally, the reg-ularized BIE, Eq. (1), is converted into a regularized BCE that can be collocated (as in the usual BEM) atany point (including those on edges and corners). This equation is:

0 � 1

2

XM

m�1

X27

a�13

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�Tÿ1

m

am

� �a

�XM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�RT ÿ1

m

am

� �aÿ3

�XM

m62Sm�1

X3

a�1

ILm

Dajk dzj

� �S Tÿ1

m

am

��ÿ Tÿ1

P

aP��

a

�12�

with ILm

Dajk dzj � ÿZ

Sm

Rajkej � dS

� 1

8p�1ÿ m�I

Lm

�kijr;ar;i

rdzj � 1ÿ 2m

8p�1ÿ m�I

Lm

�akj1

rdzj � H

4pdak: �13�

Here Lm is the bounding contour of the surface element Sm. In the above, H is the solid angle (subtended bya surface element m at a collocation point x), which is de®ned as:

H �Z

Sm

r � dS

r3: �14�

Also, Tm

and am

are the transformation matrix and primary physical variable vectors on element m, TP

and aP

are the same quantities evaluated on any element that belongs to the set S of elements that contain thesource point x, and �ijk is the usual alternating symbol.

The procedure for obtaining an assembled discretized form of Eq. (12) is described in Refs. [6,7]. The®nal result is

Ka � 0; �15�which is written as

Ax � By; �16�where x contains the unknown and y the known (from the boundary conditions) values of the primaryphysical variables on the surface of the body. Once these equations are solved, the vector a is completely

known. Now, at a post processing step, ba

mcan be easily obtained on each boundary element from Eq. (11).


3. Variables at internal points

The starting point of this section is the standard displacement BIE collocated at an internal point x. ABCM version of this equation is obtained easily from Eq. (12). This is followed by a recapitulation of theBEM and BCM equations for displacement gradients at an internal point, the latter from Ref. [7]. Finally,one has analogous equations for the curvatures at an internal point. This derivation of curvatures is new.

3.1. Displacements

One starts with the usual boundary integral equation for linear elasticity at an internal point x 2 B. Thisequation is (see (1)):

uk�x; b� �Z

oBUik�x; y�rij�y; b�� ÿ Rijk�x; y�ui�y; b�

�nj�y� dS�y�: �17�

Here, b is a shape design variable and the spatial coordinates of the source and ®eld points depend on b, i.e.,x�b�; y�b�. Explicit designation of the shape parameter b is not necessary at this stage but is included forlater use in sensitivity analysis.

A comparison of the regularized BIE (1), its boundary contour version (12) and the internal BIE (17)immediately reveals the boundary contour version of (17). This equation is

uk�x� � 1

2

XM

m�1

X27

a�13

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�Tÿ1

m

am

� �a

�XM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�RT ÿ1

m

am

� �aÿ3

�XM

m�1

X3

a�1

ILm

Dajk dzj

� �STÿ1

m

am

� �a

: �18�

3.2. Displacement gradients

The starting point this time is the BIE (17) di�erentiated at a source point x. This well known equation is

vkr�x; b� � uk;r�x; b� � ÿZ

oBUik;r�x; y�rij�y; b�� ÿ Rijk;r�x; y�ui�y; b�

�nj�y� dS�y�: �19�

The BCM version of (19) (see Ref. [7]) is

uk;r�x� � ÿXM

m�1

X27

a�13

ILm

raijUik

ÿ�ÿ uaiRijk

��jrt dzt

�Tÿ1

m

am

� �a

�XM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�R;rTÿ1

m

am

� �aÿ3

ÿXM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jrt dzt

�RT ÿ1

m

am

� �aÿ3

�XM

m�1

X3

a�1

ILm

Dajk dzj

� �S;rTÿ1

m

am

� �a

�XM

m�1

X3

a�1

ILm

Rajk�jrt dzt

� �ST ÿ1

m

am

� �a

: �20�

3.3. Curvatures

The BIE for curvature at an internal point x can be obtained by di�erentiating (1) twice with respect to asource point x. The result is


uk;rp�x� �Z

oBUik;rp�x; y�rij�y�� ÿ Rijk;rp�x; y�ui�y�

�nj�y� dS�y�: �21�

Integration by parts, twice, is used to transfer the derivatives from the kernels to the primary variables.The result is

uk;rp�x� �Z

oBUik;r�x; y�rij�y�� ÿ Rijk;r�x; y�ui�y�

�;p

nj�y� dS�y�

ÿZ

oBUik�x; y�rij;p�y�� ÿ Rijk�x; y�ui;p�y�

�;rnj�y� dS�y�

�Z

oBUik�x; y�rij;rp�y�� ÿ Rijk�x; y�ui;rp�y�

�nj�y� dS�y�: �22�

Let the integrals on the right-hand side of the above equation be called I1; I2 and I3, respectively. Theseintegrals are now transformed to contour integrals by the use of Stokes' theorem.

The ®rst step is to convert I1 and I2 into contour integrals. De®ne

F �1�jkr � Uik;r�x; y�rij�y� ÿ Rijk;r�x; y�ui�y�; �23�

F �2�jkp � Uik�x; y�rij;p�y� ÿ Rijk�x; y�ui;p�y�: �24�

Let oB � [Sm and let Lm be the bounding contour of the boundary element Sm. Using Stokes' theorem inthe form:Z

Sm

�F;rnj ÿ F;jnr� dS�y� �I

Lm

�qjrF dyq �25�

with F � F �1�jkr and F �2�jkp , successively, and noting that F �1�jkr;j � 0 and F �2�jkp;j � 0, one gets

I1 �XM

m�1

ILm

Uik;r�x; y�rij�y�� ÿ Rijk;r�x; y�ui�y�

��jpt dyt; �26�

I2 � ÿXM

m�1

ILm

�Uik�x; y�rij;p�y� ÿ Rijk�x; y�ui;p�y��jrt dyt: �27�

With regard to the integral I3, one observes that, for quadratic displacement shape functions, ui;pr areconstants and rij;pr vanish on a given boundary element. From Eq. (13), one has

ÿZ

Sm

Rijk�x; y�nj�y� dS�y� �I

Lm

Dijk dyj: �28�

Further, on element m one show that

ui;rp � Cirp � S;rpbm

� �i

: �29�

Using Eqs. (26)±(29) in Eq. (22), one ®nally obtains a contour integral representation for the curvatures.That is


uk;rp�x� �XM

m�1

X27

a�13

ILm

raijUik;r

ÿ�ÿ uaiRijk;r

��jpt dzt

�Tÿ1

m

am

� �a

ÿXM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jrt dzt

�R;pTÿ1

m

am

� �aÿ3

�XM

m�1

X12

a�4

ILm

raijUik;r

ÿ�ÿ uaiRijk;r

��jpt dzt

�RTÿ1

m

am

� �aÿ3

�XM

m�1

X3

a�1

ILm

Rajk�jrt dzt

� �S;pTÿ1

m

am

� �a

ÿXM

m�1

X3

a�1

ILm

Rajk;r�jpt dzt

� �ST ÿ1

m

am

� �a

�XM

m�1

X3

a�1

ILm

Dajk dzj

� �S;rpTÿ1

m

am

� �a

; �30�

where, since the source point x is ®xed within each integral, dyt has been replaced by dzt in Eq. (30).

3.4. Evaluation of variables at a regular o�-contour boundary point

As mentioned earlier, this paper is primarily concerned with evaluation of variables of interest at pointsinside a body. Nevertheless, it is interesting to point out that it is possible to take the limits of Eqs. (18), (20)and (30) as x! x̂, where x̂ 2 oB is a regular, o�-contour boundary point (ROCBP), i.e., x̂ is a regular pointon oB that does not lie on a contour Lm. In this case, Eqs. (18), (20) and (30) must be modi®ed in thefollowing ways:

1. The left-hand sides of the above equations must be multiplied by 1=2, i.e., the left-hand side of (18) willbecome �1=2�uk�x̂� etc.

2. The solid angle H in the expression for Dajk (see (13)) in these equations must be evaluated on asingular element by following the procedure described in Appendix E of Ref. [7].

4. Sensitivities of variables at internal points

Boundary contour integral equations of the sensitivities of internal displacements and stresses are de-rived in this section. These derivations are new.

4.1. Sensitivities of displacements

The starting point is Eq. (17), the displacement BIE at an internal point x. De®ne

Gjk�x; y; b� � Uik�x; y�rij�y; b� ÿ Rijk�x; y�ui�y; b�: �31�As mentioned before, b is a shape design variable and the sensitivity of a function f �x; y; b� is de®ned as

f�� df

db� of

oxkx�

k � ofoyk

y�

k �ofob

�32�

while the partial sensitivity of f is de®ned as

fM� of

ob: �33�


Taking the sensitivity of Eq. (17), one gets

u�

k�x� � x�

r

ZoB

oGjk�x; y; b�oxr

nj�y� dS�y� �Z

oB

oGjk�x; y; b�oyr

y�

rnj�y� dS�y�

�Z

oBGjk�x; y; b� d

dbnj�y� dS�y�� Z

oBUik�x; y�rij

M �y; b�h

ÿ Rijk�x; y�uMi�y; b�inj�y� dS�y�:

�34�Let the ®rst term on the right-hand side of Eq. (34) be called J1k, the second and third integrals together

be called J2k and the last integral be J3k. Therefore, one has

u�

k�x� � J1k � J2k � J3k: �35�It is obvious that

J1k � uk;r�x�x�r: �36�Using exactly the same procedure described in Mukherjee et al. [12], one gets

J2k �XM

m�1

ILm

�jntGjkz�

n dzt: �37�

Finally,

J3k � uM

k: �38�Substituting Eqs. (36), (37), (38) and (31) into (35), one obtains the expression

u�

k�x� � uk;p�x�x�p �XM

m�1

ILm

�Uik�x; y�rij�y; b� ÿ Rijk�x; y�ui�y; b��jntz�

n dzt � uM

k: �39�

An explicit form of Eq. (39) is obtained by writing the displacements and stresses in terms of their shapefunctions from Eqs. (6) and (7). Please refer to Ref. [12] for details of the treatment of the partial sensitivityuM

k: Finally, the boundary contour integral form of the displacement sensitivity equation is

u�

k�x� � uk;r�x�x�r �XM

m�1

X27

a�13

ILm

raijUik

ÿ�ÿ uaiRijk

��jntz�

n dzt

�Tÿ1

m

am

� �a

�XM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jntz�

n dzt

�RTÿ1

m

am

� �aÿ3

ÿXM

m�1

X3

a�1

ILm

Rajk�jntz�

n dzt

� �S Tÿ1

m

am

� �a

� 1

2

XM

m�1

X27

a�13

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�Tÿ1

m

a�m

�� Tÿ1

m� ��am�

a

�XM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�R Tÿ1

m

a�m

�� Tÿ1

m� ��am��

aÿ3

�XM

m�1

X3

a�1

ILm

Dajk dzj

� �S Tÿ1

m

a�m

�� Tÿ1

m� ��am��

a

: �40�

4.2. Sensitivities of displacement gradients

This time, the starting point is the displacement gradient BIE (19).Now, one de®nes

Hjkr�x; y� � Uik;r�x; y�rij�y; b� ÿ Rijk;r�x; y�ui�y; b�: �41�


One has Hjkr;j � 0 at an internal point since

Hjkr�x; y� � ÿ oGjk�x; y�oxr

�42�

and Gjk;j � 0.The exact same reasoning as in the previous section leads to an equation for the sensitivities of dis-

placement gradients at an internal point. That is

v�

kr�x� � uk;rp�x�x�p ÿXM

m�1

ILm

Uik;r�x; y�rij�y; b�� ÿ Rijk;r�x; y�ui�y; b�

��jntz�

n dzt

ÿZ

oBUik;r�x; y�rMij�y; b�h

ÿ Rijk;r�x; y�uMi�y; b�inj�y� dS�y�: �43�

It is very interesting to verify (43) from another point of view. Di�erentiating Eq. (39) with respect to xr,one gets

�u� k�;r�x� � uk;rp�x�x�p � uk;p�x��x�p�;r ÿXM

m�1

ILm

Uik;r�x; y�rij�y; b�� ÿ Rijk;r�x; y�ui�y; b�

��jntz�

n dzt

ÿZ

oBUik;r�x; y�rMij�y; b�h

ÿ Rijk;r�x; y�uMi�y; b�inj�y� dS�y�: �44�

Normally, one would expect another term on the right-hand side of the above equation, namely,

XM

m�1

ILm

Uik�x; y�rij�y; b�� ÿ Rijk�x; y�ui�y; b�

��jnt z

�n

� �;r

dzt:

However, it is proved in Ref. [12] that z�

n can be replaced by y�

n in the second term on the right-hand sideof Eq. (39). Since y

�n is only a function of yk, its derivative with respect to xr vanishes.

Now, using the well-known formula (which is valid for any su�ciently smooth function / ± see, forexample, Haug et al. [8]),

�/;r�� /�;r� ÿ /;p�x

�p�;r �45�

with / � uk, it is easy to show that Eqs. (43) and (44) are consistent.Finally, one writes the displacements and stresses in terms of their shape functions in order to obtain an

explicit form of Eq. (43). It should be noted that the second term in the right-hand side of Eq. (43) isanalogous to the integral on the right-hand side of Eq. (39) (with G replaced by H), while the last term inEq. (43), v

Mkr, is analogous to an expression for the displacement gradient at an internal point. The dis-

placement gradient BCE (20) is very useful for obtaining an explicit expression for this integral. The ®nalresult is

v�

kr�x� � uk;rp�x�x�p ÿXM

m�1

X27

a�13

ILm

raijUik;r

ÿ�ÿ uaiRijk;r

��jntz�

n dzt

�Tÿ1

m

am

� �a

ÿXM

m�1

X12

a�4

ILm

raijUik;r

ÿ�ÿ uaiRijk;r

��jntz�

n dzt

�RT ÿ1

m

am

� �aÿ3

�XM

m�1

X3

a�1

ILm

Rajk;r�jntz�

n dzt

� �S Tÿ1

m

am

� �a

ÿXM

m�1

X27

a�13

ILm

raijUik

ÿ�ÿ uaiRijk

��jrt dzt

�Tÿ1

m

a�m

�� Tÿ1

m� ��am�

a


�XM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jntzn dzt

�R;r Tÿ1

m

a�m

�� Tÿ1

m� ��am��

aÿ3

ÿXM

m�1

X12

a�4

ILm

raijUik

ÿ�ÿ uaiRijk

��jrt dzt

�R Tÿ1

m

a�m

�� Tÿ1

m� ��am��

aÿ3

�XM

m�1

X3

a�1

ILm

Dajk dzj

� �S;r Tÿ1

m

a�m

�� Tÿ1

m� ��am��

a

�XM

m�1

X3

a�1

ILm

Rajk�jrt dzt

� �S Tÿ1

m

a�m

�� Tÿ1

m� ��am��

a

: �46�

The curvatures uk;rp in the above equation can be obtained from (30). Stress sensitivities can be easilyobtained from v

�kr by using Hooke's law.

4.3. Sensitivities at a regular o�-contour boundary point

As before, one can take a limit of Eqs. (40) and (46) as an internal point x! x̂, where x̂ 2 oB is aROCBP. This is because, as an internal source point x approaches a ROCBP, the right-hand sides, with thesole exception of the surface integral expression for the solid angle, remain regular (and continuous) whilethe left-hand sides, u

�k and v

�kr, of course, also remain continuous. This is the beauty of Stokes' regular-

ization. Some changes in the above equations, however, become necessary because of the ``undesiarble''behavior of the solid angle expression. These changes are as follows:

1. The solid angle H in the expression for Dajk (see (13)) in these equations must be evaluated on asingular element by following the procedure described in Appendix E of Ref. [7].

2. The free term fk; k � 1; 2; 3; must be added to the right-hand side of (40) while the free termgkr; k � 1; 2; 3; r � 1; 2; 3; must be added to the right-hand side of (46). These terms are

fk � 1

2b}

k�x� �1

2uM

k�x� � 1

2S Tÿ1

P

a�P

"� Tÿ1

P� ��aP

!#k

;

gkr � 1

2

o b}

k

� �oxr

� 1

2

o�uMk�oxr� 1

2S;r Tÿ1

P

a�P

"� Tÿ1

P� ��aP

!#k

:

In the above expressions, P denotes a source point x and (see Ref. [12]),

b}

k �X27

a�1

Ska�x1; x2; x3�b�

a; k � 1; 2; 3:

5. Numerical results

Numerical results for internal stresses and their sensitivities, obtained form the BCM, are presented inthis section. Two examples are considered here. The ®rst is a hollow sphere under internal pressure. Thesecond is a rectangular block with a circular cylindrical hole subjected to remote tensile loading. The BCMmodels for both examples are fully three-dimensional. Numerical results are compared with analyticalsolutions [14] for both examples ± with the La �me solution for the sphere example and with the Kirschsolution (for an in®nite slab with a circular cylindrical hole in plane strain) for the second example.


5.1. Hollow sphere subjected to internal pressure

5.1.1. Geometry and meshThe chosen sphere has inner and outer radii a and b equal to 1 and 2 units, respectively. The internal

pressure is 1 unit and the design variable is the inside radius a of the sphere. The elastic constants are l � 1and m � 0:3, respectively.

A generic surface mesh on a one-eighth sphere is shown in Fig. 1. Three levels of discretization ± coarse,medium and ®ne, are used in this work. The mesh statistics are shown in Table 1.

5.1.2. Internal stressesNumerical results from the BCM, for the stress components rRR; rhh and r// along the line x1 � x2 � x3,

are compared with the corresponding analytical solutions in Fig. 2. These results are obtained from Eq. (20),together with Hooke's law. Of course, the boundary value problem (12) is solved ®rst before Eq. (20) isused. It is noted that standard spherical coordinates R; h; / are used here with x1 � R sin h cos /;x2 � R sin h sin /; x3 � R cos h:

The numerical results presented in Fig. 2 are seen to be very accurate. Although similar results have beenpresented previously in Ref. [7], Fig. 2 is included in this paper because the numerical results shown in this®gure are more accurate than those published previously where a di�erent mesh was used.

5.1.3. Sensitivity of internal radial displacementsFig. 3 shows the sensitivity of radial displacement along the line x1 � x2 � x3. These results are obtained

from Eq. (40) which is used after ®rst solving the sensitivity boundary value problem (Eq. (54) in Ref. [12]).As in Ref. [12], a linear velocity pro®le

R� � bÿ R

bÿ a�47�

Fig. 1. A typical mesh on the surface of a one-eighth sphere.


is used here. It is seen from Fig. 3 that the agreement between the numerical and analytical solutions is verygood.

5.1.4. Sensitivities of internal stressesSensitivities of the stress components rhh and rRR, along the line x1 � x2 � x3, are presented in Fig. 4.

Again, very good agreement is observed between the numerical and analytical solutions.

5.2. Rectangular block with cylindrical hole

5.2.1. Geometry and meshThe second example is concerned with a rectangular block with a cylindrical hole of circular cross-

section, loaded in uniform remote tension. The BCM model is fully 3-D but the imposed boundary con-ditions are chosen such that a state of plane strain prevails in the block and the numerical results obtainedfrom the BCM can be compared with Kirsch's analytical solution for the corresponding 2-D plane strainproblem. Of course, the analytical solution is only available for an in®nitesimal hole in a slab and this isreferred to as the ``exact'' solution in this section of the paper.

Table 1

Mesh statistics on a one-eighth sphere

Mesh Number of elements on each

Flat plane Curved surface Total

Coarse 12 9 54

Medium 36 36 180

Fine 64 64 320

Fig. 2. Stresses along the line x1 � x2 � x3. Exact solutions Ð. Numerical solution from the ®ne mesh: rhh � r// � � � �; rRR � � � �.


Fig. 3. Sensitivity of radial displacement along the line x1 � x2 � x3. Exact solution Ð. Numerical solution from the ®ne mesh: � � ��.

Fig. 4. Stress sensitivities along the line x1 � x2 � x3. Exact solutions Ð. Numerical solutions from the ®ne mesh:

r�

hh � � � �; r�

RR � � � �.


The side of a square face of the full block is 20 units, the hole diameter is 2 units and the thickness (in thex3 direction) is 6 units. One-eighth of the block is modeled in order to take advantage of symmetry. Themesh on the front and back faces (x3 � 3 and x3 � 0, respectively) of the one-eighth block is shown inFig. 5. Four layers of triangles (in the thickness direction) constitute the mesh on the remaining surfaces ofthe one-eighth block. The complete block is loaded by uniform remote tensions in the x1 and x2 directionswhile u3 � 0 on the faces normal to the x3 axis in order to simulate plane strain conditions. Of course,boundary conditions on the symmetry planes are applied in the computer model of the one-eighth block inthe usual way.

The design variable in this example is the hole radius a. The chosen design velocity distribution in theslab is linear along any radial direction in any square section normal to the x3 axis and is independent of thex3 coordinate. In other words, referring to Fig. 6, one has

R� � L= cos /ÿ R

L= cos /ÿ afor / < p=4 �48�

and

R� � L= sin /ÿ R

L= sin /ÿ afor / � p=4: �49�

Fig. 5. Mesh on a quarter of the front and back faces of a rectangular block with a circular cylindrical hole.


5.2.2. Internal stressesComparisons of numerical and analytical solutions for internal stresses in the block are presented in

Figs. 7 and 8. Two cases are considered: stresses along the line x1 � x2; x3 � 1:5 for uniaxial remote loadingin the x1 direction, and along the line x1 �

��3p

x2; x3 � 1:5 for equal biaxial loading. The agreement betweenthe analytical and numerical results is again seen to be very good.

Fig. 6. Design velocities for the second example.

0 1 2 3 4 5 6 7 8 9 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

X1 = X2

Str

ess

Com

pon

ents

Fig. 7. Stresses along the line x1 � x2; x3 � 1:5 for uniaxial loading in the x1 direction. Exact solutions Ð. Numerical solutions:

r11 � � � �; r22 � � � �.


5.2.3. Sensitivities of internal stressesThis last example is a di�cult one. It is concerned with stress sensitivities for the situation depicted in

Fig. 7.The numerical and analytical results for this problem are shown in Fig. 9. The internal point numerical

results, which are obtained from independent calculations, show acceptable agreement with the analyticalsolutions for the stress sensitivities, even though the analytical solutions vary rapidly near the hole.

6. Concluding remarks

A novel formulation for shape sensitivity analysis for 3-D linear elasticity problems, using the BCM, hasbeen presented in a recent paper Ref. [12] and in the present one. This derivation is carried out by ®rsttaking the material derivative of a relevant BIE with respect to a shape design variable, and then convertingthe resulting equation into its boundary contour version. The focus of Ref. [12] is the calculation of sen-sitivities on the surface of a body while sensitivities of internal variables is the focus of the present paper.

Numerical results are presented in this paper for two selected examples. The computational domain inboth these cases is fully 3-D, but the problems are chosen such that it is possible to compare the BCMnumerical results with analytical solutions for appropriate problems in lower dimensions. The numericalresults for the internal stresses are uniformly excellent in all cases while the stress sensitivity results areacceptably accurate.

It is well known that the BEM (as well as the BCM) has problems with convergence for ``nearly singular''cases such as those that occur when an internal point is very close to the surface of a body. These di�cultiesare typically managable for displacement calculations and become worse for stress calculations when thekernels become ``nearly hypersingular''. This issue has been addressed in the BEM context by many re-searchers (e.g., Refs. [1,15,16]) and various remedies have been suggested. The nearly singular problem,however, has not yet been addressed for the BCM. It is expected that use of the algorithm outlined, for

0 1 2 3 4 5 6 7 8 9 100.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

X1= 3 X2

Str

ess

Com

pon

ents

Fig. 8. Stresses along the line x1 ��3p

x2; x3 � 1:5 for equal biaxial loading. Exact solutions Ð. Numerical solutions:

r11 � � � �; r22 � � � �.


example, in [16] will deliver accurate stress values from the BCM, at internal points that are close to theboundary. This is a topic for future research.

Acknowledgements

This research has been supported by the NSF SBIR Phase II grant DMI-9629076 to DeHan EngineeringNumerics.

References

[1] E.D. Lutz, Numerical Methods for Hypersingular and Near-singular Boundary Integrals in Fracture Mechanics, Ph.D.

Dissertation, Cornell University, Ithaca, NY, 1991.

[2] A. Nagarajan, E.D. Lutz, S. Mukherjee, A novel boundary element method for linear elasticity with no numerical integration for

two-dimensional and line integrals for three-dimensional problems, ASME J. Appl. Mech. 61 (1994) 264±269.

[3] A.-V. Phan, S. Mukherjee, J.R.R. Mayer, The boundary contour method for two-dimensional linear elasticity with quadratic

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[4] A.-V. Phan, S. Mukherjee, J.R.R. Mayer, The hypersingular boundary contour method for two-dimensional linear elasticity, Acta

Mechanica 130 (1998) 209±225.

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[6] Y.X. Mukherjee, S. Mukherjee, X. Shi, A. Nagarajan, The boundary contour method for three-dimensional linear elasticity with a

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[8] E.J. Haug, K.K. Choi, V. Komkov, Design Sensitivity Analysis of Structural Systems, Academic Press, New York, 1986.

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linear elasticity, Int. J. Solids Struct. 35 (1998) 1981±1999.

Fig. 9. Stress sensitivities along the line x1 � x2; x3 � 1:5 for uniaxial loading in the x1 direction. Exact solutions: r�

11 Ð; r�

22 ± ± ±.

Numerical solutions: r�

11 � � � �; r�

22 � � � �.


[10] A.-V. Phan, S. Mukherjee, On design sensitivity analysis in linear elasticity by the boundary contour method, Engng. Anal. with

Boundary Elem. 23 (1999) 195±199.

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by the boundary contour method, Int. J. Num. Meth. Engng. 42 (1998) 1391±1407.

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boundary contour method, Comp. Meth. Appl. Mech. Engng. 173 (1999) 387±402.

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83±95.

[14] S.P Timoshenko, J.N. Goodier, Theory of Elasticity, 3rd. ed., McGraw-Hill, New York, 1970.

[15] G. Krishnasamy, F.J. Rizzo, Y. Liu, Boundary integral equations for thin bodies, Int. J. Num. Meth. Engng. 37 (1994) 107±121.

[16] Y. Liu, Analysis of shell-like structures by the boundary element method based on 3-D elasticity: Formulation and veri®cation,

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internal variables and their sensitivities in three- dimensional linear elasticity by the boundary...

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