Pergamon Mechamcs Research Communications, Vol. 28, No. 2, pp. 199-206, 2001

Copyright © 2001 Elsevier Science Ltd Pnnted m the USA. All rights reserved

0093-6413/01/S-see front matter

PII: S0093-6413(01)00163-X

AN E F F E C T I V E M E T H O D F O R C A L C U L A T I N G V A L U E S ON AND

N E A R B O U N D A R I E S IN T H E H Y B R I D D I S P L A C E M E N T B E M

H B CHEN" and P LU

Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, P R China

W P HOWSON I and F W WILLIAMS

Division of Structural Engineering, Cardiff School of Engineering, Cardiff University, Cardiff CF2 3TB, UK

(Received 17August 1999. accepted for print 24 January 2001)

Introduction

The hybrid displacement boundary element method (BEM) proposed by DeFigueiredo [1] is a kind of symmetric BEM formulation which has recently received much attention [2-5]. In this formulation, the fundamental solutions are used as the interpolation functions for the field variables, so as to transform the domain integral to the boundary. However, this kind of interpolation induces a higher singularity in the boundary integral equation (BIE) than for the normal BEM formulation Meanwhile, a strong boundary layer effect occurs when the physical values for points near the boundary are calculated and some boundary values, e.g the boundary tangent stress components in elastic problems, cannot be calculated directly Recently, Chen [6], Wang et al. [7] and Chen et al. [8] proposed an effective modified method for calculating physical values at points on and near the boundary in elastic BEM. In the present paper, this modified method is extended to overcome the difficulties discussed above, in the context of elasticity

In the hybrid displacement BEM, after the boundary unknowns are obtained, the displacements and stresses at points in the domain can be calculated by interpolation, w~thout further integration However, when the calculation point approaches the boundary, considerable errors occur due to the singular nature of the field interpolation functions, i e the fundamental solutions To overcome this difficulty, a set of uniform solutions is used to Indirectly determine the 'singular' interpolation function of the boundary node nearest to the calculatton point Moreover, this treatment can be used to calculate the boundary nodal values Numerical examples show that the treatment proposed m this paper is effective and easy to implement.

Currently vLstttng academic m the Dtvlslon of Structural Engmeermg, Cardtff School of Engmeermg, Cardiff Umversit). UK

corresponding author

199

200 H.B. CHEN, P. LU, W. P. HOWSON and F. W. WILLIAMS

F o n n u l a t m n o f the h y b n d d i s p l a c e m e n t B E M

Constder an elasticity problem with no body force and w~th a domain F2 bounded by a closed surface F The hybnd displacement BEM model is based on the modified variational principle [ 1 ]

n . . : fr ("' - ) a t - fF, i' 'ar , ( l )

where the displacements in the domain it,, the boundary displacements ~ and the

boundary tractions t, are independent of each other, i is the prescribed boundary traction

on F_,, the boundary displacements ~ satisfy the essential boundary condition, i.e. u : u

on l"t, where ~ is the prescribed boundary displacement, F = F~ + F 2 , and Coe ~ is the

fourth-order isotropic tensor of elastic constants Now the functions u,, ~" and 7 are approximated as the product of known functions

and unknown parameters In matrix form this gives , 7

u = U s in f2, (2)

= ~ r d on F, (3) ~- = ~ r p on F, (4)

where s(2N × 1), d(2N × 1) and p are vectors of unknown parameters, N is the number of

boundary nodes, T denotes transpose, U "r is a matrix the coefficients of which are displacements of the fundamental solution at a point in the domain due to a unit load acting at a boundary node, i e at an interpolation point, and • and W are matrices whose terms are interpolation functions generally used in finite or boundary elements.

Substituting equations (2)-(4) into equation (1) gives the functional I-IHB in terms of the

unknown vectors s, d and p. Omitting the intermediate steps, which may be found in [1],

this gives

I-IuB = l s r F s - p r G r s + p r L d - d r T , (5) 2

where

F = f r U ' T ' r d l " , G = frU*Wrdl -" , L = j'r~F~ rd r ' , T = frCI~idl-" (6)

is a matrix whose terms are the tractions obtained from the displacements U" Here T" corresponding to the fundamental solution, so that F is symmetric [1]. It should be noted that, in this procedure, the boundary interpolation nodes should be excluded from the considered domain, to enable the domain integral in equation (1) to be transformed to the boundary [ 1 ]

The stationary condition of YInB with respect to s, d and pgives

F s - Gp = 0, (7)

- G r s + Ld = 0, (8) Lrp - T = 0 (9)

From equation (8) s = G rLd = Rd (10)

(Note that in [1] G is always square, but this will not always be so [9,10], in which case it is necessary to proceed as described beneath equation (21) )

By using equations (7) and (10) to eliminate the vector p from equation (9), the final system equations for the solution of the problem can be wntten as

K d - T - - 0 , (11) where

BOUNDARY VALUE CALCULATIONS WITH BEM 201

K = R' FR (12) Note that the system matrix K is symmetric and that equation (11) has exactly the same form as that obtained by the finite element method

After the system equation (11) has been solved, the nodal displacement vector d can be obtained and hence the displacements and stresses at internal points can be evaluated without further integration Therefore, the displacements at internal points can be interpolated directly by using equation (2) and the related stresses can be calculated from

. Y o = a s , (13)

where the coefficients of the matrix a" are the stresses obtained from the displacements U', corresponding to the fundamental solution

The modified calculation for displacements

The formulations in the above section and the modified method proposed in this paper are valid for both two dimensional (2D) and three dimensional (3D) problems However, for simplicity the modified formulation is presented for the 2D case only Thus equation (2) can be expanded as

N

u(k) -- U'rs = ~ U~q'r sq , (14) q=l

*7 ~ - - q where U,, Lu,-'~ ° .kq , s 2 u22 j sq - , k denotes the point where the displacements are

calculated, q denotes one of the N boundary interpolation nodes, u,~ ~ is the displacement

fundamental solution, which denotes the displacement at point k in direction j owing to a unit force acting at point q in direction t; and s'q is the interpolation parameter for the

fundamental solution at node q in direction t As the calculation point k approaches the boundary from within the domain, e g it approaches a boundary node p, the components of U~p vary sharply, owing to the singular nature of the fundamental solution. Hence the

displacements calculated will cease to have acceptable accuracy within a certain distance from the boundary To overcome this difficulty, two uniform solutions, i e. two rigid movement solutions of the domain along the two axes x t and x 2 , are introduced to

indirectly determine the 'singular' interpolation function U~p, i e

{ u ' (x) l={~} and Iu ' (x) l=~ '0 l (15) u 2(x)j [u s(x)j [ l J '

where x is any point within the domain ~ or on its boundary F. Substituting these two rigid movement solutions, i e equation (15), into equation (14)

and combining the related coefficients together gives N u:,, "'- -' Ukq Sq)S: , (16)

q - I

q : p

where I=[0 ~3, go = [sq (l) so (2)], and Sq(l)and sq (2 )a re the interpolation

parameter vectors of node q, owing to the rigid movement solutions of the domain considered along the axes x~ and x2, respectively; and Sq is obtained from

S : R I - - [ g , r "'" S t ] r, (17)

202 H.B. CHEN, P. LU, W. P. HOWSON and F. W. WILLIAMS

where =fI ,, By usingequatlon 16 rst

equation (14) can be used to calculate the displacements for points near the boundary It should be noted that using the redirect calculation of equation (16) to find the mam interpolation parameter matrix U~q tends to make equation (14) well conditioned

Furthermore, this method can be used to calculate displacements at the boundary nodes

The modified calculation for stresses

Similarly, a modified treatment can be obtained for calculating stresses at points on and near the boundary, as follows For the 2D case, equation (13) can be expanded as

N *T o ( k ) = o'Ts = Z o ~ s ~ , (18)

q = l

where ak0"r _- ~o~2 2.~q ° ; ~ I and cr,j I*kq is the stress fundamental solution, which denotes thef t *kq

stress component at point k owing to a unit force acting at node q in direction i When the calculation point k, at which the stresses are calculated, approaches a boundary node p from within the domain, the components of o~p vary sharply and the calculated results become

completely incorrect within a certain distance from the boundary To overcome the difficulty, the main submatrix o~p is determined indirectly by use of the three special uniform

solutions

c,_~(x)l= , J o 2 , ( x ) l = and la~2(x)~= , (19)

where x is any point m the domain f2 or on its boundary F Substituting these three uniform solutions, i e equation (19), into equation (18) and

combining the related coefficients together gives N

"T = ( I - .r ~,.qsq)~;' (20) I~ kp q = l

q~t p

where I= 1 ' gq =[sql ~ So,_: So,:], sq,~ is the vector sq for the first uniform

0

solut,on ofequat,on (19), , e {cL,(x) o2,(x) ot,_(x)} 7 = {1 0 0} r , and sLmilarly sq2:

and Sq~ z are for the other two umform solutions ofequatton (19) It is obvtous that Sp ts a

non-square matrix and so its inverse cannot be calculated directly. As Se is a row fully

posed matrix, its mverse exists in the general inverse sense as follows [10,11 ] I ~ T ~ ~ T I ,o : s~ (S~S~) (21)

Therefore by use of equation (20), equation (18) can be used to calculate stresses for points near the boundary Again, the above treatment can also be used to calculate stresses at the boundary nodes Note that G is not square for the following two examples, due to the traction discontinuity at the corner nodes, and therefore the G r of equation (10) must be

BOUNDARY VALUE CALCULATIONS WITH BEM 203

calculated analogously to equation (21)

Numer i ca l e x a m p l e s

Two examples are given to show the validity of the treatment proposed in this paper In both, the modulus of elasticity E = 200GPa and Poisson's ratio v = 025 Cubic spline interpolation is used for the boundary variables

Example 1 A long thick-walled circular cylinder, with inner radius 10 mm and outer radius 25 mm, is subjected to internal pressure p = 100MPa under plane strain

deformation By use of symmetry, only one quarter of the cylinder was considered in the analysis, as shown in Figure ! Figure 2 shows the displacements and stresses calculated along the line Ol of Figure I It can be seen that the results calculated by the modified method proposed in this paper agree well with the exact solutions [12] for points on and near the boundary, whereas when the unmodified formulas (i e equations (14) or (18)) were used directly the calculated results were incorrect for points within a certain distance from the boundary It can also be seen that the results from the unmodified stress calculations begin to depart from the exact solutions at much greater distances from the boundary than is the case for the displacements, as expected because of the stronger singularity of the stress interpolation functions, i e of the fundamental stress solutions.

Example 2 A square plate with side length 2 m and a hole of radius 10 cm, see Figure 3, is subjected to uniform tension a = 100MPa along each side. By using symmetry, only one quarter of the plate was considered in the analysis. Figure 4 shows the displacements and stresses calculated along the line OI of Figure 3 The numerical results are compared with the exact solution [12] for an infinite plate with a hole of radius 10 cm and subjected to a uniform tension of ~ -- 100MPa at infinity It can be seen again that the modified method proposed in this paper gave quite satisfactory results for points on and near the boundary, whereas those calculated directly from equation (14) or (18) were incorrect when the calculation point was within a certain distance of the boundary

C o n c l u s i o n s

The hybrid displacement BEM is notable because it is deduced from the variational principle and therefore has a symmetric system matrix Those characteristics give potential advantages in improving the accuracy of the BEM analysis and its coupling with FEM. Therefore such formulation has recently received much attention However, due to the fundamental solutions being used as the interpolation functions for the field variables, this formulation has a higher boundary singularity and greater boundary layer effect than the normal BEM formulation In this paper, an effective treatment is proposed for the calculation of values on and near the boundary In the treatment, a series of uniform solutions are introduced to calculate the main singular submatrices indirectly and to modify the total results Two numerical examples show that the treatment is effective

204 H.B. CHEN, P. LU, W. P. HOWSON and F. W. WILLIAMS

. / / /

121 : : : : : H L 10em-k 15era ~1

--xl

Figure 1 A thick-walled cylinder subjected to internal pressure

1 0 0 ~ Ur( lO-3mm)

9 0

k A

80" ~ ~ ~ ~ x

70-

R ( m m )

100 105 110 115 120

(a) radial displacement u r

6 0 0 -

200

200

-60 t)

1000

100

o11,o22 (MPa)

~ J , ~ # v v v v v v v v v v v ~,t,

A - -

.e,

x R(mm)

105 110 115 120

3 2 0 0 -

2400 -

1600-

8 0 0 -

00

100

-a12(MPa) /x

1,, A

R(mm)

105 110 115 120

(b) stresses or, and o_,2 (c) stress - o ~:

Figure 2

KEY A

X

Displacements and stresses calculated at points along OI

- - e x a c t solution [12] calculated directly (t e only equations (14) or (18) were used)

calculated by the proposed modified method

B O U N D A R Y VALUE CALCULATIONS WITH BEM 205

Figure 3

~' X Z o = I 0 0 M P a

/ ' 0 ~ Z t

L , . ~

r i 0 " ~ " m 9 X l 0 = 9 0 c m

Rectangular plate with a hole and subjected to uniform tension

6 a u l . u 2 ( 1 0 - 2 m m )

6 4

6 2 -

R(crn) 6 0 - T - r - r . . . . r - r 7 ~ T T ' ~ - - - 1 - - T ~ I " - - [

100 105 110 115 120

(a) displacements u I and u 2

3 0 - - a l l,o22(10;ZMPa)

'~ 0 iV"=g '~C'~ t . . )=C~"K~_ _'~¢ _ . ,~¢. . ~ " . . ) ¢ _

1 0 -

R(cm) 30 - ' - " ' -; ~ - ~ T - -T -'~-- ~ ~' -- • . . . .

100 105 110 I t 5 120

7 0 -

4 0 -

1 0

-o~2(102MPa)

L

R(o-n) - 2 0 - - r , - r - - - q - - - r - r r [ ; - r - - ~ , , -!

100 1 0 5 1 1 0 115 1 2 0

(b) stresses o , j and o 2_~ (c) stress - o ,_~

Figure 4 Displacements and stresses calculated along OI

KEY solution for an infinite plate with a hole when subjected to uniform tension at infinity [12]

calculated directly (i e only equations (14) or (18) were used) calculated by the proposed modified method

206 H.B. CHEN, P. LU, W. P. HOWSON and F. W. WILLIAMS

Acknowledgements

The authors are grateful for support provided by the National Natural Science Foundation of China and the Cardiff Advanced Chinese Engineering Centre of Cardiff University, UK

References

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