A hybrid displacement variational formulation of BEM for elastostatics

Engineering Analysis with Boundary Elements 10 (1992) 219-224

A hybrid displacement variational formulation of BEM for elastostatics G. Davi' Dipartimento di Meccanica e Aeronautica, Facolta" di Ingegneria di Palermo, Viale delle Scienze -90128 Palermo - Italy

A variational formulation for symmetric positive definite BEM is derived by using a hybrid displacement functional. The functional is expressed in terms of domain and boundary variables, assumed as independent from one another. The resulting equations involve only boundary displacement variables and the approach gives symmetric and positive definite matrices which can be combined with finite elements. The accuracy and efficiency of the method is tested through comparison of some typical results. INTRODUCTION

variational formulation. The method is based on a modified functional using three independent variables, i.e. displacements and tractions on the boundary and displacements inside the domain. This approach uses classical fundamental solutions to interpolate the displacements in the domain. The fundamental solutions are due to a unit load acting at a source point on the boundary. The resulting system of equations is written in terms of boundary unknowns. The matrix of the solving system has the advantage of being symmetrical, but not positive definite. The formulation presented in this paper is analogous to the method used by DeFigueiredo & Brebbia. The displacements in the domain are expressed as a linear combination of 'regular fundamental solutions', which are obtained by setting the source point of every solution outside the domian. Making the modified functional stationary, we obtain an algebraic linear equation system, whose matrix is symmetric and positive definite. Moreover the coefficients are obtained by integrating functions which do not present any singularity on the boundary.

The solution of elastostatic equilibrium by using the direct approach of BEM leads to the numerical solution of a system of integral equations. These integral equations are obtained by using opportune solutions of the problem, so-called 'fundamental solutions'. 1-3 This procedure leads to sufficiently approximate results, with some computational advantage compared to other numerical methods. Nevertheless the matrix of the system is neither symmetric nor positive definite, and it thus shows certain disadvantages that the finite element method does not have. Fundamental properties of continuum, as the symmetry and definiteness of some tensor operators, are violated. As emphasized in Refs 10 and 11, basic principles holding for the continuum cannot be extended to BEM. Moreover direct BEM formulation does not consent the utilization of some numerical procedures, the properties of symmetry and definiteness being essential. Some authors have been able to overcome some of these difficulties.4-7 Recent papers on boundary element formulation have concentrated on the development of variational formulations 8 in which the field and boundary variables are taken to be independent of one another. 9-12 Polizzotto l°'ll has proposed a symmetric definite BEM formulation, using alternatively two energy principles, i.e. the Hellinger - Reissner principle and a boundary min-max principle ad-hoc formulated. At the same time, using a different approach, DeFigueiredo & Brebbia 12 have presented a new hybrid displacement

DIRECT BEM Let us consider a linear elastic body, subjected in the domain f~ to body forces f, however constrained on the boundary F1 and let it be loaded by tractions t on the free boundary F 2. Let u, e, ~r, be the displacement, strain and stress fields. The elastostatic problem can thus be represented by the following equations: Compatibility :

Engineering Analysis with Boundary Elements 0955-7997/92/$05.00

e= D u

© 1992 ElsevierSciencePublishers Ltd. 219

in f~

(1)

G. Davi'

220 Equilibrium : Dta = - f

in f~

t = Drur

on I"

(2) (3)

Constitutive : a=Ee=EDu

(4)

where -O/Ox: 0

0 0/0×2

0 0

0

0

O/Ox3

D=

O/Ox20/Ox:

0/03

0

010×~

0

0/0x3

O/0x2.

l

Dn =

(5)

0

0

a2

a3

0

O~2

0

a:

0

a3

0

a3

0

aI

a2

(6)

In eqn (6) al, oL2and •3 are the direction cosines of the outer normal at the boundary point. The direct formulation of the boundary element method is based on the reciprocity theorem. Let uj and tj be the displacements and the tractions due to a system of body forces fj. We then have:

Jr(.fi-tTu)dr= J (ffu-u:f)d~

(7)

Let the body forces fj. be so characterized that: f j - - A ( p - p o ) cj (Po)

(8)

where A is the Dirac function and the cj (P0) is the unit vector applied at the source point P0 in thejth direction. Substituting eqn (8) into eqn (7) we have: Jr

Pi

uTtdF-Ir2tfudF-cfu(p0)=-lr2UT'dF

+J

dF + I uff df~

(9)

fl

For P0 on the boundary, and for j = 1, 2, 3 eqn (9) provides the boundary integral equation, which binds between them the unknown boundary data. Introducing the prescribed boundary conditions: u=~

onFl

(11)

t = i

on F 2

(12)

- Iv2 firi dF

(15)

in which both the field and the boundary functions are independent, is made stationary by the solutions to the elastostatic problem. In eqn (15) the boundary tractions i are assigned on F2, while the boundary conditions for displacements are as follows: fi = ~

on Fl

(16)

The variation with respect to the independent variables, taking into account eqn (16), gives:

uf'dF (13)

Jr(i- t)rOu dr - Ir2(i- i)ro~ dr

(17)

The stationarity conditions of eqn (17) for arbitrary variations Ou in fl, Oi on I" and Off on F2, provide the following field and boundary equations:

Dto:-f

in Q

(18)

fi = u

on F

(19)

i= i

the integral equations can be written:

rl

PRINCIPLE

I I : J Q (1T ~IE E { - urf) dQ - Jr(U - fi)ri d r +

= t

Jr2

VARIATIONAL

The hybrid boundary model proposed in this paper is based on a modified variational principle) 2 Let u be the displacement field in the domain f~, and fi, i the displacement and the traction on the boundary. The functions are assumed independent of one another. The modified variational principle states that the functional

-

(10)

r2

(14)

To obtain a solution by BE the boundary is discretized into a finite number of elements and the boundary unknowns are approximated through a suitable set of interpolation functions. By locating the source point P0 on each boundary node, a linear system of algebraic equations is obtained, involving the unknown nodal values.

c@°)=Jrf:dfl=-I DrEDujdf~=-Jtjdr~ r

j uftdF- I t f u d F = e T f i ( p o ) - [ +.[ tjTfidF-Ifuffdf~ onP,

onF 2

OH = - .[ (Dra + f)rou df~ - .[r(u - fi) Tot dP

The components of vector ¢i are given by:

rl

tjrfidF-Jnuffdf~

rl

MODIFIED

0

cfu(P0) = [ ( u f t - tfu)

I

on F

(20)

on F2

(21)

Consequently, assuming that eqns (1), (3) and (4) are verified, and that the displacement boundary condition (16) is identically satisfied, the solution of the elastostatic problem is now given in terms of the functions u, fi and t which make II stationary.

A hybriddisplacementvariationalformulationof BEMforelastostatics

the matrix b* is square and regular and possesses an inverse 4 = (~.)-1. Therefore:

BEM FORMULATION

Let the body be discretized by boundary elements in such a way that the boundary functions fi and i are defined in terms of their nodal values, generalized displacements 6 and generalized tractions p : fi = N ' = [N1,

N2][66'2]

(22)

i = @p

(23)

where N and • are suitable shape functions. The boundary conditions for displacements, expressed in terms of generalized displacements, are as follows: 61 = 61

on FI

(24)

while 62 are the unknown generalized displacements on U2. The displacements u inside the domian f~ are expressed as a linear combination of regular fundamental solutions u*, which are obtained by setting the source point P0 of every solution outside the domain 9t: u = u*s

(25)

Substituting eqns (22), (23) and (25) into eqn (15), after integrating by parts, we have: II:~SrIrU*rp*dFs-PrJrq~ru*dFs

(26) The stationarity condition of eqn (26), with respect to the independent variables s, 62 and p, gives:

IrU*rp*dFs-Iuu*r~pdFP-I~ *rfdf~=Or

(27)

lrN~

(28)

dF p - J

d

s-

F2

N2rt d F = 0

=0

/29/

(30)

The relationship between the unknown parameters s and the generalized displacements 6 can be established evaluating eqn (30) at the nodal points. As the matrix of shape functions N at these points gives an identity matrix, we have: ~*s=6

S:

(i*)-16 : 46 :

(31)

where the elements of matrix ~* are the boundary values of the functions u* at the nodal points. If the number of source points is chosen to be equal to the number of nodes and if the functions u* are linearly independent,

[41,

42] [¢~1] 62

(32)

Substituting eqn (32) into eqn (30), we have: N = u*(~*) -l = u*[41,

42] = [Ul,

U2]

(33)

Premultiplying eqn (27) by 4~',using eqns (32), (33) and (28), we obtain a set of linear equations: 4TA4262 = Ir2 NEri dF + Jfl N~'f df~ - 4rA4,6, (34) where A = [ u*rp * dF (35) JF Since the matrix A is symmetric and positive definite, the stiffness matrix K = 4~'A42

(36)

is symmetric and positive definite. The final system of linear eqns (34) is therefore symmetric, positive definite and the unknowns are boundary displacements only. Moreover the coefficients of matrix A, from which the stiffness matrix K is obtained, are calculated by integrating functions which do not present any singularity and the integrations are performed merely along the boundary. DISPLACEMENT AND STRESS FIELD After solving the system of eqns (34) the displacement field u can be easily calculated, without further integrations, by substituting the value of s from eqn (32) into eqn (25): u = N 6

(37)

The stresses a and the boundary tractions t are obtained by: = tr*4 6

The eqns (29) are satisfied, for any ~P, if: u*s = N 6

221

t = Dn tr* 4 6

(38) (39)

where the elements of the matrix a* are the stresses due to the regular fundamental solutions u*. In the following applications the results for displacements are in all cases accurate for internal as well as for boundary points, while the internal stresses show considerable error for points close to the boundary when coarse meshes are used. This is to be expected as a result of the BEM discretization, since the boundary displacement conditions of the continuum problem have been transformed into the conditions on the generalized displacements 6.

G. Davi'

222

NUMERICAL EXAMPLES

i d - Node

Source Point

""1 de

Fig. 1. Location of source point.

I I

I

I

q

I

Y=O

T 1

V:O

I

I

t

1

q

L-2 H

Example 1

H:5

X

I

In this section two examples are studied to investigate the accuracy of the proposed formulation. The first example is the beam simply supported, subjected to a uniform transverse load, the second example is the hollow cylinder under uniform internal pressure. In order to test the convergence of the method three meshes are used. The nodal point is the mid-node of the elements, the source point corresponding to this mid-node is located at an arbitrary distance from the nodal point, along the outer normal (Fig. 1). For each mesh three different regular fundamental solutions are used, 13 derived from the several distances of the source point. The ratio between the distance of the source point and the length of the element is assumed to be 1, 1.5 and 2, respectively. The test results indicate that the difference between the three approximations used is negligible.

I

The example analyses a beam, assumed to be simply supported, subjected to a uniform load q (Fig. 2). Due to the symmetry of the problem, only one half of the beam needs to be considered. Each side was discretized into two, five, and ten elements. The results, obtained for u = 0.1, are compared to the theoretical value of the maximum deflection:

Fig. 2. Beam transversely loaded: Geometry and boundary conditions. f=

22

5 L3 1-2 L] 2 q L 1.736 q L ( l + v ) 6 4 H 3 + ' 8 - - H -] G G (40)

S 2.0 cJ

Figure 3 demonstrates the convergence of the deflection when the number of elements increases. The results for displacements on the boundary and at internal points are presented in Fig. 4. Figures 5 and 6 show the distribution of normal and shear stresses along the cross section at different abscissas.

-~ 1.8

~ 1.6

-~t.t c

Example 2 z

1.C

,

,

,

,

8

,

,

,

,

,

,

,

,

,

,

,

,

2O

,

,

,

,

,

,

,

,

40

l~xlaor of Elements

Fig. 3. Convergence of maximum deflection (fG/q L). In order to avoid this problem, after evaluating the nodal displacements, the unknown boundary tractions can be obtained using eqns (13). By rearranging them one obtains a system of equations in terms of boundary tractions only.

The second example considered is a hollow cylinder under uniform internal pressure q (Fig. 7). Due to the symmetry of the problem only one quarter of the cylinder is examined, using three different meshes; 12, 24 and 40 elements, respectively. The results for displacements and stresses along the section A-A are shown in Figs 8 and 9, and are compared to the exact solution, 14 for v = 0-1. The results obtained show the accuracy and convergence when the elements increase.

A hybrid displacement variational formulation of BEiU for elastostatics

Fig.

4. Deformation

of the half beam.

223

Fig. 7. Hollow cylinder under internal pressure: Geometry and boundary conditions.

4.5 4.0 3.5 3.0 . 2.5 2.0 -

0 x-

1.5 -

AX-.3L

.2L

0

12 el.

A

24 e1.

0

40 el.

I

exact

2.4

2.6

I.0 0.5 -

0.35 -

Ncdimnsioml

mrrml

1.2

stresses

Fig. 5. Normal stress (a/q) distributions along the cross section at different abscissas.

*!

x--.,

1.4

I.6

I.8

2.0

2.2

I

2.8

3.0

Fig. 8. Radial displacement (u, G/q) distributions along the section A-A. Comparisons between meshes and exact solution.

0 I2 el.

L

A

0.8 -

0.7 .

‘!I %

24 el.

0 40 el.

0.6 . 0.5

y-=\<,

0.4 0.3 . Nortdinmsionol shear stresses Fig. 6. Shear stress (T/q) distributions along the cross section at different abscissas.

o*v .O

1’1-&ps&‘6 1.2’

I.4

1.6

I.8

2.0

2.2

2.4

2.6

2.8

:

Fig. 9. Normal stress (a/q) distributions along the section AA. Comparisons between meshes and exact solution.

224

G. Davi'

CONCLUSIONS This paper presents a symmetric and positive definite boundary element formulation. It is based on a hybrid displacement variational formulation, and involves domain and boundary variables, displacements and tractions, considered independent from one another. The final system of equations is written in terms of boundary displacements only and the matrix of the system is the symmetric and positive definite stiffness matrix. Moreover no integrations of singular functions are required. Numerical examples have shown that the proposed method is rapidly convergent and gives very accurate results. The displacement and stress fields are obtained without calculating any integrals. The results for stresses are more sensitive in the proximity of the interior points to the boundary, whereas the displacements are in all cases accurate for internal as well as for boundary points. REFERENCES 1. Rizzo, F.J. An integral equation approach to boundary value problems of classical elastostatic, Quart. Appl. Math., 1967, 25, 83-95. 2. Cruse, T.A. An improved boundary-integral equation method for three dimensional elastic stress analysis, Comp. and Struct., 1974, 4, 741-754. 3. Brebbia, C.A. & Walker, S. Boundary Element Technique in Engineering, Newnes-Butterowrths, 1980. 4. Hartmann, F. The derivation of stiffness matrices from integral equations, Proc. 3rd Int. Sem. Boundary Element Method, Irvine, California, ed. C.A. Brebbia, SpringerVerlag, Berlin, 1981.

5. Volait, F. Three dimensional super-element by the boundary integral equations method for elastostatic, Proc. 3rd Int. Sem. Boundary Element Method, Irvine, California, ed. C.A. Brebbia, Springer-Verlag, Berlin, 1981. 6. Beer, G. & Meek, J.L. The coupling of boundary and finite element method for infinite domain problems in elastoplasticity, Proc. 3rd Int. Sem. Boundary Element Method, Irvine, California, ed. C.A. Brebbia, Springer-Verlag, Berlin, 1981. 7. Tullberg, O. & Bolteus, L. A critical study of different boundary element stiffness matrices, Proc. 4th Int. Sere. Boundary Element Method in Engineering, ed. C.A. Brebbia, Springer-Verlag, Berlin, 1982. 8. Washizu, K. Variational Methods in Elasticity and Plasticity, 3rd Ed., Pergamon Press, Oxford, 1982. 9. Dumont, N.A. The hydbrid boundary element method, Proc. 9th Int. Conf. on BEM, Southampton, Computational Mechanics Publications, Southampton and Springer-Verlag, Berlin, 1987. 10. Polizzotto, C. A consistent formulation of the BEM within elastoplasticity, Advanced Boundary Element Methods, ed. T.A. Cruse, Springer-Verlag, Berlin and New York, 1988. 11. Polizzotto, C. Variational principles for boundary element formulations in structural mechanics, Proc. lOth Int. Conf. on BEM, Southampton, Computational Mechanics Publications, Southampton and Springer-Verlag, Berlin, 1988. 12. DeFigueiredo, T.G.B. & Brebbia, C.A. A new hybrid displacement variational formulation of BEM for elastostatics, Proc. 11th Int. Conf. on BEM, Cambridge, USA, Computational Mechanics Publications, Southampton and Springer-Verlag, Berlin, 1989. 13. Patterson, C. & Sheikh, M.A. Regular boundary integral equations for stress analysis, Proc. 3rd Int. Sem. Boundary Element Method, Irvine, California, ed. C.A. Brebbia, Springer-Verlag, Berlin, 1981. 14. Timoshenko, S.P. & Goodier, J.N. Theory of Elasticity, 3rd Ed., McGraw-Hill, 1982.