Computer methods in applied

mechanios end engineering

ELSEVIER Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

Surface variables and their sensitivities in three-dimensional linear elasticity by the boundary contour method

Subrata Mukherjee a'*, Xiaolan Shi b, Yu Xie Mukherjee b'~ aDepartment of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, NY 14853, USA

~DeHan Engineering Numerics, 95 Brown Road, Box 1016, Ithaca, NY 14850, USA

Received 14 July 1998

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. The BCM employs global shape functions with the weights, in the linear combinations of these shape functions, being defined piecewise on boundary elements. A very useful consequence of this approach is that stresses and curvatures, at suitable points on the boundary of a body, can be easily obtained from a post-processing step of the standard BCM.

A new formulation for design sensitivities in three-dimensional linear elasticity, based on the BCM, is presented in this paper. This challenging derivation is carded out by first 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. Finally, numerical results for surface variables, as well as their sensitivities, are presented for selected illustrative examples. © 1999 Elsevier Science S.A. All rights reserved.

1. Introduction

The usual boundary element method (BEM), for three-dimensional (3-D) linear elasticity, requires numerical evaluations of surface integrals on boundary elements on the surface of a body (see, e.g. [1]). Nagarajan et al. [2,3] have recently proposed a novel approach, called the boundary contour method (BCM), that achieves a further reduction in dimension! The BCM, for 3-D linear elasticity problems, only requires numerical evaluation 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 the usual BEM integrand and a very useful application of Stokes ' theorem, to analytically convert surface integrals on boundary elements to line integrals on closed contours that bound these elements. Lutz [4] first proposed an application of this idea for the Laplace equation. Nagarajan et al. [2] generalized this idea to linear elasticity. Numerical results for two-dimensional (2-D) problems, with linear boundary elements, are presented in Nagarajan et al. [2], while results with quadratic boundary elements appear in [51. Three-dimensional elasticity problems, with quadratic boundary elements, is the subject of Nagarajan et al. [3], and Mukherjee et al. [6]. Sensitivity analysis for 2-D elasticity problems, based on the BCM, is presented in [7]. Hypersingular boundary contour formulations, for two-dimensional [8] and three-dimensional [9] linear elasticity, have been proposed recently.

Design sensitivity coefficients (DSCs), which are defined as rates of change of physical response quantities with respect to changes in design variables, are useful for various applications such as in judging the robustness

* Corresponding author. Professor. President.

0045-7825/99/$ - see front matter © 1999 Elsevier Science S.A. All rights reserved. PII: S0045-782'5 (98)00293-X

3 8 8 S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

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. [10]), namely, the finite difference approach (FDA), the adjoint structure approach (ASA) and the direct differentiation approach (DDA). The DDA is of interest in this paper.

Unlike the versatile finite element method (FEM) where the entire domain has to be meshed, the BEM only requires meshing on the boundary. This advantage makes the BEM efficient for shape optimization since meshing must be redone after each iterative step of an optimization process. Therefore, several researchers have used the BEM to develop efficient techniques for computing DSCs. A literature review of design sensitivity analysis using the BEM can be found in, among other places, [7].

Besides having the same meshing advantage as in the conventional BEM, the BCM offers a further reduction in dimension. Furthermore, unlike the BEM, stresses at boundary nodes can be recovered easily and accurately in the BCM from the global displacement shape functions. These advantages of the BCM are expected to make it very competitive with the BEM in optimal shape design. In fact, studies of design sensitivity analysis in 2-D elasticity problems by the BCM [7] have demonstrated the power and accuracy of this method for this class of problems.

The goal of obtaining BCM sensitivity equations can be achieved in two equivalent ways. In the 2-D work by Phan et al. [7], design sensitivities are obtained by first converting the discretized BIEs into their boundary contour version, and then applying the DDA (using the concept of the material derivative) to this BCM version. This approach, while relatively straightforward in principle, becomes extremely algebraically intensive for 3-D elasticity problems. The present study offers a novel alternative derivation, using the opposite process, in which the DDA is first applied to the regularized BIE and then the resulting equations are converted to their boundary contour version. It is important to point out that this process of converting the sensitivity BIE into a BCM form is quite challenging. This new derivation, together with numerical results, is the primary contribution of the present paper.

This paper is organized as follows. A brief review of the standard BCM for 3-D linear elasticity is presented first. This review mainly recapitulates the relevant equations that are needed for the subsequent sensitivity analysis. Section 3 presents simple formulae, for surface stresses and curvatures, that can be employed at a BCM post-processing step. Sections 4 and 5 are devoted to shape sensitivity analysis. The BCM sensitivity equations are derived carefully in Section 4 and sensitivities of surface stresses, as a simple post-processing step, is the subject of Section 5. Numerical results for surface variables and their sensitivities, for illustrative problems, are presented in Section 6. 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 [11], for 3-D linear elasticity, can be written as

0 = f~B [Uik(x' Y)°'iJ(Y) - ~'ijk(x' y ) {u i (Y) - U i ( X ) } ] e j ' d S ( y ) = - f~8 Fk "dS(y) (1)

Here, aB is the bounding surface of a body B (B is an open set) with infinitesimal surface area dS = dSn, where n is the unit outward normal to 0B at a point on it. The stress tensor is tr, the displacement vector is u and ej ( j = 1, 2, 3) are global Cartesian unit vectors. The BEM Kelvin kernels are written in terms of (boundary) source points P and field points Q. These are

1 ~ i j k - - 8w(1 - - p ) r 2 [ ( 1 - - 2~')(r,i~. k q- rj6~k - r,k~ij ) -k- 3r irjr.k ] (2)

1 Uik - 16~r/~(1 -- ~,)r [(3 -- 4v)6ik + r.irk ] (3)

in terms of r, the Eucledian distance between the source and field points x and y, and the shear modulus/z and Poisson's ratio ~, of the isotropic elastic solid. Also, 8 is the Knonecker delta and .i = O/Oyi. The range of indices in these and all other equations in this paper is 1, 2, 3, unless specified otherwise.

It has been shown in [2,3,6] that the integrand vector F~ in Eq. (1) is divergence free (except at the point of

S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402 389

singularity P = Q) and that the surface integral in it, over an open surface patch S E OB, can be converted to a contour integral around the bounding curve C of S, by applying Stokes' theorem. Therefore, vectors V k exist such that

f s F k ' d S = ~ c V ~ ' d r (4)

Since the vectors F k contain the unknown fields u and tr, shape functions must be chosen for these variables, and potential functions derived for each linearly independent shape function, in order to determine the vectors V k. Also, since the kernels in Eq. (1) are functions only of Zk = Y k - X~ (and not of the source and field coordinates separately), these shape functions must also be written in the coordinates zk in order to determine the potential vectors V k. Finally, these shape functions are global in nature and are chosen to satisfy, a priori, the Navie r -Cauchy equations of equilibrium. The weights, in linear combinations of these shape functions, however, are defined piecewise on boundary elements.

Quadratic shape functions are used in this work. With

zk = Yk - xk (5)

one has, on a boundary element:

27 27

Ui = ~ fl~-Uai(Yl, Y2' Y3)----- Z ~a(Xi,X2, X3)Uai(Zl,Z2, Z3) (6) ~ = 1 o t = l

27 27

Orij = Z flaO'aij(Yl , YZ' Y 3 ) = Z ~ce(XI,X2, X3)O'aij(Zl,Z2, Z3) (7) ~=l ~=1

where u~, tr~ i (with i = 1, 2, 3 and a = 1, 2 . . . . . 27) are the shape functions and fl~ are the weights in the linear combinations of the shape functions. Each boundary element has, associated with it, 27 constants fl~ which will be related to physical variables on that element. This set of f l ' s differ from one element to the next.

The displacement shape functions for a = l, 2, 3 are constants, those for ce = 4 . . . . . 12 are of first degree and those for a --- 13 . . . . . 27 are of second degree. There are a total of 27 linearly independent (vector) shape functions ~ . The shape functions for the stresses are obtained from those for the displacements through the use of Hooke ' s law. The shape functions u,~ and ~'-0 are given in [6].

It is easy to show that the coordinate transformation (5) results in the constants/3j being related to the/3~'s as follows :

27

~i : Z Siot(Xl, X 2, X3) • , i -- 1, 2, 3 (8) a.=l

27

~k = ~ R,,(x~,x2, x3)fl,, k = 4 , 5 . . . . . 12, n = k - 3 (9) 4 = 1

~o : [ L , ~ = 13, 14 . . . . . 27 ( 1 0 )

where

Si~ ='u,i(x~,xz,x3), i = 1 , 2 , 3 , ce = 1,2 . . . . . 27

Rk~, O-U,~e(Yt, Y2, Y3) I k = 1, 2, 9 , a = 1, 2, 27 i

Oyj ~,.xz.x3)

with j = 1 + L(k - 1) /3J and g = k - 3j + 3. Here, the symbol LnJ, called the floor of n, denotes the largest integer 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 (x~,x2,x3). The procedure for designing boundary elements in the 3-D BCM is discussed in detail in [3,6]. A set of

primary physical variables ak, whose number must match the number (here 27) of artificial var iables/3 k on a boundary element, are chosen first. The first step in the BCM solution procedure is to determine the unspecified primary physical variables in terms of those prescribed from the boundary conditions. Later, secondary physical variables as well as stresses, at boundary points, are obtained from a simple post-processing procedure. Unlike

3 9 0 S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

in the standard BEM, it is particularly easy to obtain surface variables, such as stresses and curvatures, in the BCM. This issue is discussed in the next section.

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

m mm a = T3 (11)

The CIM9 boundary element, shown in Fig. l(c) in [3], is used in the present work. The displacement u is the primary physical variable at the three corner nodes C i, and the three midside nodes Mi, while tractions are primary variables at the internal nodes I i. Thus, there are a total of 27 primary variables. The BCM equations are collocated at the 6 peripheral nodes as well as at the centroid of the element. In a typical discretization procedure, some of the peripheral nodes may lie on corners or edges, while the internal nodes are always located at regular points where the boundary 0B is locally smooth. It is of obvious advantage to have to deal only with displacement components, that are always continuous, on edges and comers, while having traction components only at regular boundary points. It is important to restate that the BCM is versatile enough to handle any well-posed problem in linear elasticity--all the secondary variables can be easily determined by simple post-processing once the primary BCM equations are solved.

Details of the shape functions and intrinsic coordinates, that are used to define the geometry of the boundary elements, are available in [6]. Also, the procedure for obtaining the vector potentials V k, for nonsingular as well as singular integrands, are available in [3,6]. Finally, the regularized BIE, Eq. ( 1 ), is converted into a regularized BCE that can be collocated (as in the usual BEM) at any point (including those on edges and corners). This equation is

a = 1 3 m

"~ (actijUik ---Ucti~ijk)SntZn dz, [RT a ] , ~ _ 3 m = 1 o l = 4 m

+ D,~jk [S(T - la - l - T a ) ] ~ ( 1 2 ) r n = l a = l m r n ~ ' b v

with

~L~ D"jk dzj Js m Z~jkei • f

dS l

1 ~L r,~r,, 1- -2~ ~L 1 O 8~r(1 - ~) o E*iJ--7-dzJ + 8~(~---~,) ~ ~ , j ? -~z j + T ~ - ~ k (13)

Here, L,. is the bounding contour of the surface element S,.. In the above, O is the solid angle (subtended by a surface element m at a collocation point x), which is defined as

fs r ' d S O = 3 (14) m r

Also, T and a are the transformation matrix and primary physical variable vectors on element m, and ~ are the same quantities evaluated on any element that belongs to the set 6e of elements that contain the source point x, and %k is the usual alternating symbol.

The procedure for obtaining an assembled discretized form of Eq. (12) is described in [6,9]. The final result is

Ka = 0 (15)

which is written as

A x = B y (16)

where x contains the unknown and y the known (from the boundary conditions) values of the primary physical

S. Mukherjee et al. I Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402 391

variables on the surface of the body. Once these equations are solved, the vector a is completely known. Now, at m a post processing step, '84 can be easily obtained on each boundary element from Eq. (11).

3. Boundary stresses and curvatures

A very useful consequence of using global shape functions is that, once the standard BCM is solved, it is very easy to obtain the stresses and curvatures at a regular off-contour boundary point (ROCBP) on the bounding surface of a body. Here, a point at an edge or corner is called a corner point while at a regular point the boundary is locally smooth. Also, a boundary point can lie on or away from a boundary contour. The former is called a contour point, the latter an off-contour boundary point. A point inside a body is called an internal point. m ^m

First, one obtains '8~ from Eq. (11), then uses Eqs. (8) and (9) to get '8~,¢x = 1,2 . . . . . 12. The curvatures, which are piecewise constant on each boundary element, are obtained by direct differentiation of Eq. (6). Finally, one has the following results.

3.1. Surface values of displacements

[u,(x)] = /~3 x

3.2. Surface values of displacement gradients

L [ °u' (x)] ax, .& '8,2jx

3.3. Surface values of curvatures

O2Ul ] =[2('813 +'820)

Oxi Oxs ] [ Symmetric

Ox,. Ox s J LSymmetri c

o,, l__[ Ox~ Ox s I [_Symmetric

K2'815 k'KIB16 K1'817 +K2'818] 2'814 '825 /

2'819 J

(17)

(18)

(19)

K113,3 + K2'8, 4 '826 ] 2('816 + ,82l ) K2'823 + KI '824 / (20)

2'822 J

'827 Kz'819 + K,'Szo] 2'823 K,'82, +K2&2| (21)

2('8,7 + '824) J

with K l = -4 (1 - v) and K 2 = -2 (1 - v). Of course, Hooke's law can be used to obtain the stress components from the displacement gradients.

4. BCM shape sensitivity analysis

The starting point in this section is the standard BIE (1) collocated at an internal point x. The sensitivity (total or material derivative) of this equation is taken with respect to a design variable b. The resulting sensitivity BIE is split into three parts. The first part vanishes and the surface integrals in the second and third parts are systematically converted into contour integrals by using Stokes' theorem.

392 S. M u k h e r j e e et al. / Compu t . M e t h o d s App l . M e c h . Engrg . 173 ( 1 9 9 9 ) 3 8 7 - 4 0 2

4.1. Sensitivity of the BIE

One starts with the usual regularized boundary integral equation (1) for linear elasticity which is valid at both an internal point x E B as well as at a boundary point x E OB. An internal point x E B is considered first. Eq. ( l ) is rewritten as

0 = foe [Uik(x' Y)~J(Y' b) - ~jk(x, y)(ui(y, b) - ui(x, b))]nj(y) dS(y) (22)

As mentioned above, b is a shape design variable and the spatial coordinates of the source and field points depend on b, i.e. x(b), y(b).

Define, as in Eq. (1),

Fik(x, y, b) = U~k(X, y)~j (y , b) - X~jk(x, y)[ui(y, b) - u~(x, b)] (23)

Now, the (total) sensitivity of a function f(x(b), y(b), b), in a materials derivative sense, is defined as

~f=_df O f , Of Of db - Ox k xk + ~Yk }k + ~ (24)

while partial sensitivities of displacements and stresses are defined as

A OIA i A 0 % U i ~ - ~ (25) Ob ' ~r~j Ob

It should be noted that A , , A , U i = U i - - U i , k y k , O'ij = ~ j - - O'# ,ky k (26)

where, k =- O/Oy k. Taking the sensitivity (total derivative) of Eq. (1) with respect to b, one gets

, ( OFjk(x, y, b) 0 = x r J~ nj(y) dS(y)

a OXr

~ OFjk(x'y 'b)* fo d + 8 aYr y,.nj(y) dS(y) + B Fjk(x' y' b) - ~ [nj(y) dS(y)]

f0 A A A + [U~k(x, y)o-sj(y, b) - X~jk(x, y)(u,(y, b) - u~(x, b))]nj(y) dS(y) (27) B

It should be noted that the last integrand above is (OFjk(x, y, b))/Ob. The first integral on the right hand-side of Eq. (27) is zero because the integral in Eq. (1) vanishes for all

values of x C B. Let the second and third integrals together be called I k and the last integral Jk. Thus:

I, + Jk = 0 (28)

Each of these surface integrals will be converted to line integrals in the next two sections.

4.2. The integrals I k

The surface integral I k is converted to a sum of contour integrals in this section. As mentioned before, a point x, inside the body, is considered first. From Bonnet and Xiao [12] (see also [13]):

n.j = -yr#nr + Yr,mnrnmnj

dS , , dS - Y r.r -- Y r,m n ,-nm

SO that

d , , db [nj dS] = [ Y r , r n j - - Yr.jnr] dS

Using Eq. (31) in (27), one has:

(29)

(30)

(31)

S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402 393

1 k = [Fjk,,yrn j + Fjky,,,n j -- Fjky~.yn,l dS (32) B

Since ~k,; = 0 (except at a point of singularity--here x is an internal point), the above expression can be written

a s

f0 * * = - (Fykyr)4nr] dS [(Fj~y~) rnj (33) lk B '

Let 0B = kJ S m and let L m be the bounding contour of the boundary element S m. Using Stokes' theorem in the form (see [ 14]):

f s m ( f r n j - F . j n r ) d S ( y ) = ~ L m e q j r F d y q (34)

with F-= ~ k ~ , one gets

I k = ffq)rFjkYr dyq (35) m = l m

It is useful to state here that formula (35) is a general one, in the sense that, a surface integral I (over a closed surface OB) of any divergence-free vector function F(y ) , of the form:

1 = foe Fj (y)n j (y) dS(y) (36)

has the sensitivity expression:

= m =t ~ , ,F j (y )y , dy, (37) m

Of course, from Gauss' theorem, I = 0. Therefore, 1 = 0. One can show that

Eqj~Fjk dy o = Fjk.rn j dS = 0 ~ x r ~qjrFjk dyq = 0 m = l m B m = l m

since Yr = Xr + Zr, one can replace Yr with z~ in Eq. (35). Also, dyq = dZq s i n c e d X q = 0 at a fixed source point. Substituting Eq. (23) into (35) (with y replaced by z), one gets

Ik(x) = [U;k(x, y)%(y , b) - X~jk(x, y)(u;(y, b) - u;(x, b))l~, ,z , dz, (38) t n = t m

Since z, is O(r) as r = Ily - xl[ 0, the above expression is completely regular. Therefore, it remains valid at a boundary point x ~ OB.

4.3. The integral J~

The surface integral Jk is converted to a sum of contour integrals in this section. This time, x is allowed to be a boundary point, as well as, of course, an internal point.

4.3.1. Shape funct ions for partial sensi t iv i t ies--a simple example Series expressions for the partial sensitivities of displacements and stresses, in terms of global BCM shape

functions, are derived next. It is useful to start with a very simple example. Let

f ( y ) = ao + a l y + a2y 2 (39)

With the change of variables

y = x + z (40)

394 S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

one can write

^ 2 f(x, Z) = do(X) + dtl(x)z + az(x)z (41)

where

ao = ao + alx + a2 x2 =f(x)

d al = a l + 2azx = ~ f(x)

a 2 ~ a 2

(42)

It is easy to show that taking sensitivities of the above equations results in

f (Y ) = no + *lY + ~2Y z + al~ + 2a2y~v (43) A *

f * f (Y ) = f (Y ) - vY

= n 0 + t]jy + ~2y 2 (44)

As expected, the partial sensitivity off (x) is its sensitivity at a fixed point in space. Now, with the change of variables (40), one has

f (x, z) = ao(x) + a t(x)z + a2(x)z 2 (45)

, where a k, k = 0, 1, 2, are related to a~ in the same manner as t~ k are related to a k in Eq. (42).

4.3 . l BCM shape functions The displacement and stress shape functions for the BCM are considered next. One starts with Eq. (6) for the displacements. Following the procedure outlined above for the simple example,

it is easy to show that

27

ui = ~ B~-u~i(Y~, Y2, Y3) = B,(x l ,x2 , x3)u~i(Zl,Z2, Z3) (46) a = l a = l

O ,

where, [3 o, a = 1, 2 . . . . . 27, are related to/3,, in the same manner as/3~ are related to 134 (see Eqs. (81, (9) and (10)), i.e.:

~. 27 *

~i = ~ Sia(Xl,X2, X3)~ot, i = 1 , 2 , 3 o t = l

<> 27 *

¢~k= ~ R.o(x,,xvx3)B~, k=4,5 . . . . . 12,

© ,

( 4 7 )

n = k - 3 (48)

f l ~ = f l ~ , a = 1 3 , 1 4 . . . . . 27 (49)

Of course, the partial sensitivities for the stresses can now be expressed as

= ~ ~°'=o(Yl, Y2, Y3) = fl~(x,,x2,x3)°',a(z~,z2, z3) (50/ Or 0 c r = l ~ = l

4.3.3. The final form of Jk The conversion procedure is entirely analogous to the derivation of the primary BCM equation. Series

expansions (46) and (50) are first substituted into the last integral on the right-hand side of the sensitivity BIE (27). The potential functions are the same as before. Finally, the surface integral Jk is converted into a sum of contour integrals. The result is

S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402 3 9 5

J , (x ) --- ~ m=,

M

= 1

m = , m~S

o t = 1 3

[~L (O'aijUik ---~ai'a~ijk)EjntZn dZ'] ~m or=4 m

a~= Daj k dzj [t~ Cr -- fla] I m

(51 )

~D It is interesting to comment on the physicat meaning of the quantities fl~ for the case when the surface source A © point P is a ROCB~ Comparing Eqs. ($)-( l l )) with (47)-(49), it is ciear that #k = ~ , k = l, 2, 3 . . . . . 2?.

Now, the quantities fl~ can be easily interpretedin terms of the partial sensitivities of displacements and their

derivatives from Eqs. (17)-(21). For example, fl~" = ZUk(P), k = 1, 2, 3, etc.

4.4. The BCM sensitivity equation

An explicit form of the BCM sensitivity equation is now derived. On any boundary element:

g

fl = T - I~ + (T-1)*a (52)

in which it is convenient to evaluate the sensitivity of T-z from the formula

(T-r) * = - T - I ~ T -z (53)

o , Expressions (47)-(49) for flu are substituted into (51), and (52) is used to write/3 in explicit form. Next, an explicit expression for I~ is obtained by substituting the series expressions (6) for u~ and (7) for o;y into (38). Finally, the explicit expression for I k (obtained from (38)) and the explicit expression for Jk (obtained from (51)) are input into the BIE sensitivity equation (27) (see also 28) evaluated at a general boundary point x E 0B. The result is

-u~i~i jk)5, ,z , dz, [T a] , m = l ~ = 1 3 m

+ ~ (°'~0 u~k - . a]~_ 3 -u~,Xij~)Sntz~ dzt [RT -tm = 1 o ~ : 4 m

-m=, -T-'a] ~=, o Z o j ~ ° , z ° d z , J t ( T - a e P

t n ~ b ~

-~- 2 mE=l - - (O'oHjeik---Uai~ijk)EjntZn dz t -~- ( T - ' ) * a ] a a = ' 3 m

+ (o',~ijUi, ---U,~r~ijk)~jntZ,, dz, [R(T-'~t m + (~'-)*a)],~_ 3 m = l o , = 4 m

+ O.j k dzj [ S ( T - ' ~ m -- ~-I~P "JV (T- ' )*a - (T ')*a)]~ (54) m = ' 4 = ' m m ~ 6 ~

Comparison of the above sensitivity equation (54) with the standard BCE (12) reveals that the integrals in its last three terms are identical to those in the standard BCE. Therefore, its discretized form can be written with the same coefficient matrix A as for the standard BCM, i.e.

K* = h (55)

where the right-hand side vector h = - K a can be computed from Eq. (54) by using the boundary values of the primary variables a that are known at this stage.

396 s. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

Finally, the usual switching of columns leads to

A* = B} + h (56)

where } contains the unknown and } the known values of boundary sensitivities. In many applications, ~ = 0.

5. Sensitivities of surface stresses

The first step is to use Eq. (52) to find /3 on each element. There are at least four ways to find stress sensitivities on the surface of a body.

5.1. Me thod one

Eq. (7) is differentiated to give 27 27

, * * , • •

O'ij : E /3~Ovaij(Y l , Y2, Y3) + E /3c~O'czij(Yl, Y2' Y 3 ) a = ] a = l

- - * * :g Note that ~,j(Yt, Y2, Y3) involves sensitivities of the field point coordinates (y~, Y2, Y3)'

(57)

5.2. Method two

Sensitivities of displacement gradients v,j-----uij are computed by differentiating Eq. (18). The result is

L~6 ~ 9 ~12Jx (58)

where, by differentiating Eq. (9), one has , 27 2x~7

2 R°o(x,,x2,x3)* o + z * x 2,x3)/3,~, k = 4 , 5 . . . . . 12, n = k - 3 (59) g = l e t=l

Note that/<,,(x l, x z, ~3) involves sensitivities of tile source point coordinates (x l, x2, x 3). Finally, Hookes's law is used to determine the stress sensitivities from the sensitivities of the displacement

gradients.

Method three

One writes

. A . l) ij : Oij ~- l')i),kXk

Now

(60)

Dij,k : Ui,jk

Also, from Eq. (18): A A A

L/36 A L2]

(61)

(62)

with, from Eqs. (9) and (48):

s. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402 397

0 27 , f3~=[3k= ~ R,~(x~,x2, Xa)fl~, k = 4 , 5 . . . . . 12, n = k - 3 (63)

a = l

Tl~e curvature expressions needed in Eq: (60) are available in terms o f ~ , ,z -- 13, 14 . . . . . 27, from Eqs. (19)-(21).

5.4..~4e~had f ~ , r

The starting point is, again, Eq. (60). TheA term V~j,k on the right-hand side of Eq. (60) is treated in the same fashion as in Section 5.3. The other term, v 0, is treated differently. It is first observed that the operators, k and A

~p ccmmute and that ~(P~ = 19~ for i = 1, 2, 3. Therefore, one has

A A A OR vii = (ui,j) = (ui).j = (fl i ) # (64)

It follows from Eq. (47) that

~ p 27 ,

vii =( /3 i ) j = ~ Si~.j(xl,x2, x3)fl~, i , j = 1,2 ,3 (65) ot=l

6. Numerical results

Numerical results for surface values of components of displacements, stresses, and their sensitivities, obtained from the BCM, are presented in this section. Numerical results for cubes, subjected to uniaxial loading

exact solutions. These are not presented here. Instead, attention is focussed on comparison of numerical and exact solutions for a hollow sphere subjected to internal pressure.

6.1. Geometry and mesh for a one-eighth sphere

The 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 # = 1 and u = 0.3,

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

--i

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

398 S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

Table 1 Mesh statistics on a one-eighth sphere

Mesh Number of elements

On each fiat plane On each curved surface Total

Coarse 12 9 54

Medium 36 36 180 Fine 64 64 320

6.2. Displacements and displacement sensitivities on the sphere surface

Numerical results for the radial displacement u R (from Eq. (12)), as a function of radius along the x 3 axis, are shown in Fig. 2. Standard spherical coordinates R, 0, ~b are used here with x~ = R sin 0 cos ~b, x 2 = R sin 0 sin ~b, x 3 = R cos 0. Both the coarse and medium meshes give perfect results within plotting accuracy. The exact solution is the well known-one due to Larfie [15].

Displacement sensitivities (from Eq. (54)), along various lines on the sphere surface, for different discretizations, appear in Figs. 3 -5 . A linear design velocity profile:

* b - R R - b - a (66)

is used here. It is seen that the numerical results for the coarse mesh exhibit large errors. The reason for this is under investigation. However, they do appear to converge to the exact solution with increasing mesh density. Please see Chandra and Mukherjee [16] for a discussion of analytical solutions for design sensitivities for various examples.

6.3. Stresses on the sphere surface

Numerical results for surface stresses, computed from Eq. (18)), are presented in Figs. 6 -8 . Results on the curved surfaces of the sphere are shown in Figs. 6 and 7, respectively, while those on the plane x~ = 0 appear on Fig. 8. The nodes are chosen at the centroids of the boundary elements in all cases. It is seen that the match between the exact solutions, and numerical solutions from the medium mesh, are excellent. The ease of obtaining surface stresses is a particularly attractive feature of the BCM.

0.35 . . . . . . .

. 0.3

i 0.25

0.2

0.15

0.1

0.05

0 i i i i J i i L i

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

R

Fig. 2. Radial displacement along the x 3 axis. Exact solution • Numerical solutions: coarse mesh ~ , medium mesh

0.7

• ~ 0.6

0.5

. ~ 0 . 4

~ o.3

:~ O.2

0.1

o O

O

J

1.1 i i I i ~ i i

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

R

Fig. 3. Sensitivity of radial displacement along the x 3 axis. Exact solution - - . Numerical solutions: coarse mesh oooo, medium mesh ****.

S. Mukherjee et al. Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402 399

~ 5

o

~ -5

.~ -10 'N

.~ -15

i -20

~ -25

D I

O

0 I i i | L i i i

1.1 1.2 1.3 1.4 !.5 1.6 1.7 1.8 1.9

R

Fig. 4. Percent errors in numerical solutions for u R along the x 3 axis. Coarse mesh o00% medium mesh ****.

0.7

- 0.6

i 0.5

0.4

0.3

.~ 0.2

0.1

0 I ~ i i i i i i i

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

R

Fig. 5. Sensitivity of radial displacement along the line x~ = 0,

x 2 = x 3. Exact solution - - . Numerical solutions: coarse mesh oo0% medium mesh ****, fine mesh + + + +.

1.5

1

0.5

-0.5

-1

-1.5

0.5

0.4

0.3

0.2

~ . 0.1

i o

~ -0.1

-0.2

-0.3

-0.4

-0.5 i

0 5 - 2 i i i i i i i i I i L i i

0 5 10 15 20 25 30 35 40 10 15 20 25 30 35 40

Nodes on the inner surface R=a Nodes on the outer surface R=b

Fig. 6. Stresses on the inner surface R = a. Exact solutions Fig. 7. Stresses on the outer surface R = b. Exact solutions

- - . Numerical solutions from the medium mesh: G~e = o'** - - . Numerical solutions from the medium mesh: ~80 = tr** * * * * O-RR o o o o . * * * * ~rgR o e o o ,

6.4. Stress sensitivities on the sphere surface

Stress sensitivities on the surface o f the sphere are calculated from Eqs. (58)-(59). Numerical solutions from the medium mesh, for stress sensitivities on the outer surface R = b, agree well with the exact solution (Fig. 9).

The situation, however, is somewhat tricky on the inner surface R = a. Here, one has:

*oo A 0o'0o * = o'oo + - - ~ R (67)

g~ A (and similarly for the other components of stress). The exact solution for o'oo = o-~,~ is 120/49, which is positive, while the convected term in Eq. (67) is - 1 2 / 7 , which is negative. As can be seen from Figs. 10-13, this situation makes it quite difficult to calculate ~'00 and ~ '~ accurately, especially when accuracy is measured in terms of percentage errors, because one must calculate the difference between two numbers that are reasonably close.

Table 2 shows percentage root mean square errors in o-oo and o-,~, respectively. These are defined as

400 S. Mukherjee et al. I Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0,8

-1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Nodes on the plane Xl=0

Fig. 8. Stresses on the plane x] = 0. Exact solutions - -

Numerical solutions from the medium mesh: o'00 = o'** ****, O'RR oo00,

8

O~

1

0.8

0.6

0.4

0 . 2

0

-0.2

-0.4

-0.6

-0.8

! | l | ~ I l l I I | I O VIlli I

i i i

-1 5 i0 i5 20 25 30 35 40

Nodes on the outer surface R=b

Fig. 9. Stress sensitivities on the outer surface R = b. Exact

solutions - - . Numerical solutions from the medium mesh: trRR + + + +, tr0o 000% tr,~ ****.

2 c~

0

~ -1

0 o 0 o 0 " 0 o 0 0 0 0 0 0 o 0 0 °

0 o O O f l o n o G ~, ~, o 0 o 0 0 ~

. 0 0

• o e * e I • • • •

• • e ~ ° ° • • •

o

a+++++ + ++1 ++÷++++++++++ ++++++ ÷++÷+

+

i i i L i i J

0 5 10 15 20 25 30 35 40

Nodes on the inner surface R=a

Fig. 10. Stress sensitivities on the inner surface R = a . Exact

solutions: - - . Numerical solutions from the medium mesh:

O-oo 000% (Otroo/OR) ~ + + + +, o-00 ****.

3 - o o o o

~oo%ooo~oooOOoOooOO°oO¢O~o~ ~oOoo~o°°Oooo o o~ oo o

2 ° 0 ° o

"~ 0 ": ~o -1

-2 '~'~-*÷.,-~+'0" *'H'~C'*'v*'c*~÷+÷"'c'H'÷÷+÷÷÷'~'c'~'++-,-'X'+'~'+'~"% ~ + 4

-3 2 0 ' ' ' 0 30 40 50 70

Nodes on the inner surface R=a

Fig. 11. Stress sensitivities on the inner surface R = a. Exact

solutions: - - . Numerical solutions from the fine mesh: o'~0 o00% (Otroo/dR) ~ + + + + , tr~o ****.

Table 2

Percentage root mean square errors in sensitivities of stress components

Medium mesh Fine mesh

R = b ~0~ 1.99 1 , 6 2

tr** 1.77 2.01

R = a ~ e 25.45 15.15 o-~,~ 26.49 14.88

S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402 401

O

3 0 ° O O o o O n o v O n o 0 ° o O o O o ° ° ° o &o u n ° n ° ° v

o 2

o

'~ 1 a~oe, eee . • * • . ee o .

,~ • - e • • • e - eoo " ~ e e ° e e e •

~ 0

÷

-2 ÷+÷ +÷+÷÷÷++÷÷++++÷÷÷+++÷++÷+÷÷++++ +

- 3 i n i J a L i

0 5 10 15 20 25 30 35 40

Nodes on the inner surface R=a

Fig. 12. Stress sensitivities on the inner surface R = a. Exact solutions: - - . Numerical, solutions from the medium mesh: , o-e, * ~,oo~, (Oo'e,JOR) R + + + + , o-~, ****.

:~. 1

0

~-1 -2

0

0 0 0 0 0 0 ° ^ ~ ° 0 0 0 0 0 000 % O 0 0 0 0 0 0 0 0 ~ 0 °

. : --. . .:. . . :.-..'...-" .'. : ' . -

t+++-~-÷+++++++'~'++++~.+,m.~ ~ - ~ + + + + + + . ~ . + * + + + + + + .+-~.~-~ + + +

- 3 i i i i i I

0 10 20 30 40 50 60 70

Nodes on the inner surface R=a

Fig. 13. Stress sensitivities on the inner surface R = a. Exact solutions: ~ - . Numerical solutions from the fine mesh: o-~,~ 000% ( Oo'e,e, l OR ) R + + + + , o-e,e, ****.

E = f e ~ a c t i = 1 ( f e . . . . - - f i . . . . . ) 2 ( 6 8 )

It is seen that while the errors on the outer surface are very low, those on the inner surface, even with the fine mesh, are quite high. Further work on this problem, including the development of a different algorithm for calculation of surface stress sensitivities, is currently in progress.

7. Concluding remarks

Elegant new formulations for calculation of surface variables and their sensitivities, in 3-D linear elasticity, are presented in this paper. The analysis is based on the BCM. It is seen here that the BCM is very well suited for calculation of these quantities that are of significant practical importance in simulation and in design of 3-D elastic components and structures. Numerical results for a hollow sphere, subjected to internal pressure, are discussed in detail. Surface displacements and stresses are seen to be very accurate with the medium mesh. The accuracy of stress sensitivities, while excellent on a surface that remains stationary during a design perturbation, is not good on a surface that moves, even when a fine surface discretization is used. This issue is a subject of continuing research.

Acknowledgments

This research has been supported by the NSF SBIR Phase II grant DMI-9629076 to Delian Engineering Numerics. Sincere thanks are expressed to Dr. Ketan Shah for his valuable suggestions during the course of this work.

R e f e r e n c e s

[1] s. Mukherjee, Boundary Element Methods in Creep and Fracture (Elsevier Applied Science, London, 1982). [2] A. Nagarajan, E.D. Lutz and 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.

402 S. Mukherjee et al. / Comput. Methods Appl. Mech. Engrg. 173 (1999) 387-402

[3] A. Nagarajan, E.D. Lutz and S. Mukherjee, The boundary contour method for three-dimensional linear elasticity, ASME J. Appl. Mech. 63 (1996) 278-286.

[4] E.D. Lutz, Numerical methods for hypersingular and near-singular boundary integrals in fracture mechanics, Ph.D. Dissertation, Cornell University, Ithaca, NY, 1991.

[5] A.V. Phan, S. Mukherjee and J.R.R. Mayer, The boundary contour method for two-dimensional linear elasticity with quadratic boundary elements, Comput. Mech. 20 (1997) 310-319.

[6] Y.X. Mukherjee, S. Mukherjee, X. Shi and A. Nagarajan, The boundary contour method for three-dimensional linear elasticity with a new quadratic boundary element, Engrg. Anal. Boundary Elem. 20 (1997) 35-44.

[7] A,E Phan, S. Mukherjee and J.R.R. Mayer, A boundary contour formulation for design sensitivity analysis in two-dimensional linear elasticity, Int. J. Solids Struct. 35 (1998) 1981-1999.

[8] A.V. Phan, S. Mukherjee and J.R.R. Mayer, The hypersingular boundary contour method for two-dimensional linear elasticity, Acta Mechanica 130 (1998) 209-225.

[9] S. Mukherjee and Y.X. Mukherjee, The hypersingular boundary contour method for three dimensional linear elasticity, ASME J. Appl. Mech. 65 (1998) 300-309.

[10] E.J. Haug, K.K. Choi and V. Komkov, Design Sensitivity Analysis of Structural Systems (Academic Press, New York, 1986). [11] F.J. Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Qtly. Appl. Math. 25 (1967) 83-95. [12] M. Bonnet and H. Xiao, Computation of energy release rate using material differentiation of elastic BIE for 3-D elastic fracture, Engrg.

Anal. Boundary Elem. 15 (1995) 137-149. [13] H. Petryk and Z. Mr6z, Time derivatives of integrals and functionals defined on varying volume and surface elements, Arch. Mech. 5-6

(1986) 697-724. [14] K.-C. Toh and S. Mukherjee, Hypersingular and finite part integrals in the boundary element method, Int. J. Solids Struct. 31 (1994)

2299-2312. [15] S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, 3rd. edition (McGraw Hill, New York, 1970). [16] A. Chandra and S. Mukherjee, Boundary Element Methods in Manufacturing (Oxford, New York, 1997).