Journal of Mathenmtical Sciences, Vol. 96, No. 1, 1999

N U M E R I C A L STUDY OF THE T W O - D I M E N S I O N A L P R O B L E M OF T H E T H E O R Y OF E L A S T I C I T Y USING T H E B O U N D A R Y - E L E M E N T M E T H O D

I. I. Diyak UDC 539.3.01:517.958

We propose a method of solving the two-dimensional problem of the theory of elasticity by a direct boundary- element method. We apply Galerkin~ method with linear and quadratic approximations of the forces and displacements. We give the numerical results of solving a model problem that shows the effectiveness of the proposed approach.

The Bubnov-Galerkin method is often applied to solve the boundary integral equations of elastostatic problems. Its use makes it possible to reduce the errors of computation near the edges and corners and also to predict them more precisely [3].

The relations of the direct boundary-element method, obtained from the duality theorem, are based on the boundary integral equation [1, 2]

Cij(~)uj(~) q- f Fij(x , ~)uj(x)dF(x) = ~ Gij(x , ~) pi(x)dF(x), i, j = I, 2. - (1)

F F

Here ~ = ~(x 1, x2) is the point of loading and x = x(x t, x2) is a point of the region lying on the boundary F of the region of the elastic body; uj(x) and pj(x) are respectively the displacement and stress on the boundary; F/j(x, ~)

and Go(x, 4) are fundamental solutions of the Kelvin equation. The matrix coefficients Cij(~ ) depend only on the

local geometry of the boundary F. If F is a smooth curve, then Cij(~ ) = 5ij/2. It is known that the kernels of the two-

dimensional problem have the following orders of singularity as x tends to ~:

Gq(x,~)=O(lnr) and Fq(x, ~):O(1/r), where r=lx- l The integral on the left-hand side of (1) is interpreted in the sense of the Cauchy principal value. Integrals of functions with a weak singularity always exist as improper integrals, but for integrals of functions with a strong singularity one must apply special methods of computation. At the same time the presence of singularities leads to diagonal dominance in the resolvent system of equations, which reduces its conditioning.

To obtain a numerical solution of the problem, we partition the boundary F of the region into Are boundary elements:

N, r = U r'n, I'nfqrm = O, m*n . (2)

n=l

On the element 1" n we define the components of the displacements and forces in the form of piecewise interpolating functions between the nodes of the elements:

N:-I N~'-I Ui(X)= Xtpkn(x)u:n , Pi(X ) kn kn = (x ) pi , x rn (3)

k=0 k---'0

In the cases of linear and quadratic approximation of the displacements and stresses we have N/n = 2, 3. Then the total number of degrees of freedom for the discrete model is

N, Nj N = E ~ , N ~ , Nd=2. (4)

nffil i=1

Translated from Matematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 60--63. Original article submitted June 15, 1994.

2847 t 072-3374/99/9601-2847522.00 �9 1999 Kluwer Academic/Plenum Publishers

Substituting the discrete expressions (3) into the boundary integral equations (1) and taking account of (2) and (4), we obtain

x x e--O j=l r ' n m=l e--0 j=l I'~n

,v, ,vT-~ ,%

-X X (5) m=l e=O j=l r=

where r/(~) is the remainder function connected with the error in interpolating the unknown values. Using the Bubnov-Galerkin method, we write

j" ~o ~" (~) ~ (~) dF(~) = 0, (6) r,

where i = 1, N d, k = O, N'~ - 1, n = 1, N e . Substituting (5) into (6), we obtain the equations

N~-t Na

e---O j=l r , r , r .

This system can be written as

:r N;'-I rr

re=l, e---'O j= l r n r . m;~-n

system of linear algebraic

~, uT-I nd

= X 2; m=l e'=0 j=l r n F m

(7)

F..knem am = (-~rknem em . i j uj --i] Pj (8)

Taking account of the boundary conditions, we represent Eq. (8) in standard form

Ay = b. (9)

Here A is a completely full nonsymn~tric matrix, y is the vector of unknown displacernents and forces at the boundary nodes, and b is the vector of the right-hand side. Solving the system (9), we obtain the values of the displacements and forces on the boundary. We t'md the displacements at an arbitrary point ~ inside the region using the formula

Uj(~) = f Gij(X , ~)pj(x)d~(x)-f F / j ( x , ~)uj(x)d~(x). (10) F F

From the Cauchy relations we have the following expressions for the strain tensor:

~/g)=I Bok(x' ~)P,(x)dr(x)-I c,/~, ~)u,(x)ar(x), F F

(10

where

l ( ~ G i j ~Gik '~ I ( O F 0 BF/k'~

Substituting these values into Hooke's law, we obtain expressions for the stresses

oil(g) = j" r , /x . ~)p,(~)~(x)- I e,~(x, ~)~,(x)~(.~). F F

Here

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. . [ 2gv , aGi,. (3Gq aGik ~1] [ 2gv 5 OF/ . , ('~F/j aF/t ~1

There are no singular integrals in the relations (10)-(12), so that one can apply standard quadratic schemes in its numerical implementation.

The algorithm defined above was implemented as a software package in FORTRAN 77 on an IBM PC/AT. The proposed Bubnov--Galerkin procedure for the direct boundary-

element method was applied to solve a boundary-value problem of elasticity theory in the case of two-dimensional strain. Consider a body whose cross section is shown in the figure. The conditions of clamping and loading are also shown here. The numerical results were obtained for Young's modulus E/P = 2.1.104 and Poisson coefficient v = 0.3.

To approximate the unknown force and displacement functions and also the boundary of the region both linear and quadratic approximations were chosen. At the comer points the method of doubling the nodes was applied. The elements of the matrices in the system (7) were computed by applying the standard Gaussian quadrature formulas, even for the diagonal elements. The number of Gaussian points needed for exact

�89

6 5

7 4

8 3

Element 1 2 0 0 0

P

xl

integration depends on the ratio of the size R of the element being integrated to the minimal distance r from a fixed element. It was established experimentally that satisfactory resuks are obtained if the order of integration of the outer integral is 4 and for the inner integral it is chosen from the following condkion [4]:

t 8, r/R<l,

NG= 4, l<r/R<4, [2, r/R>4.

The table gives the values of the vertical displacements uy and the stresses 6yy at an internal node A (1.0,0.5)

and the boundary node B(2.0,0.5), obtained by applying the proposed Bubnov--Galerkin procedure for the boundary- element method (BEM) and the finite-element method (FEM). The total number of degrees of freedom (D.o.f.) for different partitions on the boundary and firlite elements is also shown. The order of the approximating functions for the displacements and stresses in the boundary-element method is the same, while in the method of finite elements for stresses the order is chosen to be one less. In the application of linear approximations for the unknowns the stresses obtained in the boundary-element method are convergent.

Table

Parti- D.o.f tion

2 x 1 24 BEM 4 x 2 48

8 x 4 96

2 x 1 26 FEM 4 x 2 74

8 x 4 242

2 x l 12 4 x 2 24

BEM 8 • 48 16x 8 96

B(2.0, 0.5) A(1.0, 0.5) Uy ~D, Uy ~yy

-1.279 (-3) -4.893 (-I) -5.390 (-4) -5.109 (-1) -1.306 (-3) -5.140 (-1) -5.524 (-4) -5.067 (-1) -1.322 (-3) -5.034 (-1) -5.624 (-4) -5.079 (-1)

-1.205 (-3) -4.904 (-1) -4.919 (-4) -4.571 (-1) -1.305 (-3) -5.007 (-1) -5.521 (-4) -5.216 (-1) -1.329 (-3) -4.998 (-1) -5.678 (-4) -5.098 (-1)

B

-1.252 (-3) - 1.297 (-3) -1.32o (-3)

m

-4.932 (-1) -5.051 (-1) -5.010 (-1)

-4.288 (-4) -5.338 (-4) -5.541 (-4) -5.648 (-4)

-5.455 (-1) -5.117 (-1) -5.077 (-1) -5.081 (-1)

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Literature Cited

1. P. Bannerjee and R. Butterfield, Boundary-Element Methods in the Applied Sciences [Russian translation], Mir, Moscow (1984).

2. C. Brebbia and S. Walker, The Boundary-Element Method for Engineers, Pentech, London (1984). 3. G. Beer, "Implementation of combined boundary element finite element analysis with applications in

geomechanics," Devel. Boundary Elem. Math., 4, 191-225 (1986). 4. P. Parreira and M. Guiggiani, "On the implementation of the Galerkin approach in the boundary element

method," Comput. and Struct., 33, No. 1, 69-279 (1989).

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