Copyright c 2001 Tech Science Press
CMES, vol.2, no.4, pp.423-433, 2001
A Pure Contour Formulation for the Meshless Local Boundary Integral Equation
Method in Thermoelasticity
J. Sladek 1 , V. Sladek1 and S.N. Atluri 2
1 Abstract
are analysed. In such a case, the temperature and
displacement fields are uncoupled [Balas et al.(1989)].
Then, in the first step, the temperature field is analysed.
If thermal fields (temperature and heat flux) are known,
the mechanical quantities (displacement and traction vector) are obtained from the solution of the local boundary
integral equations, which are reduced to the elastostatic
LBIE with known redefined body forces. The redefined
body force is proportional to the temperature gradients.
The domain integral with temperature gradients in stationary thermoelasticity can be transformed to boundary
integrals [Balas et al.(1989)]. The pure local boundary
integral equation formulation is computionally efficient
due to avoiding domain integration.
A new meshless method for solving stationary thermoelastic boundary value problems is proposed in the present
paper. The moving least square (MLS) method is used
for the approximation of physical quantities in the local
boundary integral equations (LBIE). In stationary thermoelasticity, the temperature and displacement fields are
uncoupled. In the first step, the temperature field, described by the Laplace equation, is analysed by the LBIE.
Then, the mechanical quantities are obtained from the solution of the LBIEs, which are reduced to elastostatic
ones with redefined body forces due to thermal loading. The domain integrals with temperature gradients
are transformed to boundary integrals. Numerical examples illustrate the implementation and performance of the Both displacement and traction vectors are unknown on
the local boundary. If a companion solution [Zhu et
present method.
al.(1998); Atluri et al.(2000)] to the Kelvin fundamental solution is introduced, in order to give a zero value of
2 Introduction
the final fundamental displacement on the local boundThe meshless discretization approach for continuum ary, the tractions are eliminated in the LBIEs allocated
mechanics problems has attracted much attention dur- to interior points. Displacements are approximated by
ing the past decade [Belytschko et al.(1996); Lin and the moving least-square (MLS) method [Belytschko et
Atluri(2000); Kim and Atluri(2000); Ching and Ba- al.(1994)]. The essential idea of MLS interpolants is that
tra(2001); and Gu and Liu(2001)]. By focusing only on it is only necessary to construct an array of nodes in the
the points, instead of the meshed elements as in the con- domain under consideration. The discussion devoted to
ventional FEM or BEM, the meshless approach has cer- the selection of two free parameters in the MLS approxtain advantages. Nodal points are randomly spread on the imation is presented via the numerical example repredomain of analysed body. Every node is surrounded by sented by a hollow cylinder under a thermal load.
a simple surface centered at the collocation point. Only
one nodal point is included into the subdomain. On the 3 Local boundary integral equations in stationary
thermoelasticity
surface of subdomains the local boundary integral equations are written. Then, the number of equations is equal
Consider a homogeneous, isotropic and perfectly elastic
to number of nodes.
body occupying the region Ω and bounded by the surface
In the present paper, stationary thermoelastic problems Γ . The governing equations in stationary thermoelastic1 Institute of Construction and Architecture, Slovak Academy of ity are given by [Balas et al.(1989)]
Sciences, 842 20 Bratislava, Slovakia
for Aerospace Research & Education, 7704 Boelter Hall,
University of California at Los Angeles, Los Angeles, CA 900951600, USA
2 Center
θ kk
Q
κ0
(1)
424
Copyright c 2001 Tech Science Press
CMES, vol.2, no.4, pp.423-433, 2001
∂θ
∂θ
(2)
λ µ u X γθ
θ y x
θ
x
y
dΓ
θ x
x y dΓ
∂n
∂n
where θ T T denotes the increase of temperature
with respect to the natural state T for which the stresses
θ x y κQ x dΩ
and deformations are equal to zero. Next, u and X deµui kk
k ki
i
i
Γ
0
Γ
0
i
i
note the components of the displacement vector and the
body force vector, respectively. The heat source is denoted by Q, while κ0 is the thermal diffusivity, and the
constant γ is expressed in terms of the Lame constants µ
and λ as
γ
2µ 3λ α
with α being the coefficient of linear thermal expansion.
Ω
The boundary integral equation (6) relates quantities (
θ - temperature and q ∂θ ∂n heat flux) on the global
boundary Γ . If, instead of the entire domain Ω of the
given problem, we consider a subdomain Ωs , which is
located entirely inside Ω , the following equation should
also hold over the subdomain Ωs
Ω
Q κ x θ x dΩ θ kk x
0
θ x ∂θ∂n x y dΓ θ x y κQ x dΩ
0
(3)
(6)
where n is the unit outward normal vector to the boundary Γ .
It is seen that the governing equations are uncoupled and
they can be solved separately. In the first step we solve
the Poisson equation (1). The weak formulation of the θ y
equation can be written as
0
∂θ
x θ x y dΓ
∂n
∂Ωs
0
Ωs
∂Ωs
(7)
where ∂Ωs is the boundary of the subdomain Ωs .
where θ x is the test function and θ x is the trial func- In the original boundary value problem, either the temtion for temperature.
perature θ x or the heat flux q ∂θ ∂n may be specApplying the Gauss divergence theorem to the first term ified at every point on the global boundary Γ , which
makes a well-posed problem. But none of them is known
in the in the integral given by eq. (3), we have
a priori along the local boundary ∂Ωs . To eliminate the
flux variable from the integral representation (7), Atluri
and his co-workers (1998) introduced a ‘companion soθ kk x θ x dΩ
lution’ to the fundamental solution in such a way that the
Ω
final modified fundamental solution is zero on the circu∂θ
∂θ
x θ x dΓ
θx
x dΓ
θ kk x θ x dΩ lar boundary ∂Ωs . The modified fundamental solution
∂n
∂n
can be easily derived for the Laplace equation and for
Γ
Γ
Ω
(4) 2-d problems is given by
If the test function (weighting field) satisfies the followθ̃ x y
ing equation
1 r0
ln
2π r
(8)
where r x y and r0 is the radius of the local subdomain
Ωs .
θ kk x y δ x y
0
(5)
Taking into account that
where δ x y is the Dirac delta function, the following
integral representation for temperature can be obtained
0
(9)
θ̃ r r r
from eqs. (3) – (5)
0
425
A pure contour formulation for the meshless LBIE method in thermoelasticity
one can rewrite eq. (7) into the form
∂Ωs
( %$ λ $ µ u ( x0/ u x dΩ
.
$ t x1$ γn x θ x+, u x dΓ t x u x dΓ
'
(13)
γθ( x X x+, u x dΩ
'
θ y θ̃ x y κQ x dΩ
∂θ̃
θx
x y dΓ
∂n
µui kk x
Ω
0
Ωs
i
(10)
i
k ki
i
i
Γ
i
i
for the source point located inside Ω .
i
i
Γ
i
Ω
When the source point y is located on the global boundary Γ , the subdomain can still be taken as a part of a cir- where
cular domain with the boundary composed of a circular
part Ls and the piece of boundary line Γs on which the
nodal point is lying ∂Ωs Ls Γs . It should be noted t σ n
µui k nk λuk k ni µuk ink γδi j n j θ
ij j
i
that along the line Γs the modified fundamental solution
θ̃ is not zero. Then, the local boundary integral equation
is the traction vector.
for nodes ζ Γs Γ becomes
If the weight field is selected as the elastostatical fundamental solutions of the elastostatical governing equation
"
! %$ θ x ∂∂nθ̃ x ζ dΓ
∂θ̃ θ x
$ lim
x y dΓ θ̃ x ζ q x dΓ ∂n
&
θ̃ x ζ κQ x dΩ
( ( 1$2 λ $ µ U ( x y
Γs
i.e.
3
Γs
0
ui x
3 In the second step the Lame-Navier governing equation
(2) is solved. The same procedure, as in the case of the
Poisson equation, is repeated. The weak formulation is
given as
$ 3
(14)
Ti j x y e j y
y t x dΓ Ui j x
Γ
Ti j x y u j x dΓ
j
Γ
y dΓ
γn j x θ x Ui j x
Γ
Ω
( )$* λ$ µ u ( x)$ X x+ γθ( x-, u x dΩ Ui j x y e j y and ti x
y
δi j δ x
k j ki
(11) with e j y being the unit orthogonal base vectors, one
obtains the integral representation of the displacement
field
Although the LBIE (11) can be rewritten into a nonsingular form, it is more appropriate to use this limit form
because of the numerical integration when the MLS- ui y
approximation is employed [Sladek et al.(2000)].
( $
µUi j kk x y
Ls
Ωs
( $
#
θζ
y ζ
'
( 4 X x, U x y dΩ
γθ j x
j
ij
(15)
The local boundary integral equation is considered on a
subdomain Ωs . On the artificial boundary ∂Ωs , both the
Ω
displacement and traction vectors are unknown. In or(12)
der to get rid of the traction vector in the integral over
∂Ωs , the concept of a ‘companion solution’ can be utiwhere ui x is a weight function.
lized successfully [Atluri et al.(2000)]. The companion
Applying the Gauss divergence theorem to the domain solution is associated with the fundamental solution Ui j
integral in eq. (12), one can write
and is defined as the solution to the following equations:
'
µui kk x
k ki
i
i
i
0
5
426
Copyright c 2001 Tech Science Press
CMES, vol.2, no.4, pp.423-433, 2001
6 7 0 on Ω8
Ũ 7 U on ∂Ω8
(16)
where Ω8 is a circle of radius r , which coincides
with Ω for interior points. The modified test function
U 9 7 U : Ũ has to satisfy the governing equation (14).
Then, the integral equation (15) is valid also for modified
fundamental solution U 9 . On a circle ∂Ω8 , this fundaσ̃i j
j
In stationary thermoelasticity the following governing
equations have to be satisfied [Balas et al.(1989)]
s
ij
ij
s
0
s
s
ij
ij
ij
s
ij
9 6 D ; λ D µ< U 9 6 7F: δ ; r< δ
µV 8 6 D ; λ D µ < V 8 6 7 γT6
(20)
T6 7G: δ ; r <
where V 8 is the fundamental displacement correspond-
µUi j kk
ij
k j ki
i kk
i
k ki
ii
i
mental solution is zero due to the second condition (16). ing to a point heat source (see the third equation in (20)).
Differentiating the second equation in (20) by ∂i one obHence, we can write
tains
; < 7*:>= T 9 ; x ? y< u ; x< dΓ:=
ui y
j
ij
Ωs
∂Ωs
@
6 ; < : ; <-A 9 ; : <
; D 2µ< ∇ V 8 6 7 γT6
γθ j x X j x Ui j x y dΩ λ
2
ii
(21)
ii
(17) From eq. (21) it follows directly
for the source point y located inside Ω . The explicit
γ
expression for modified test function and modified fun- V
T A
(22)
ii
λ 2µ
damental traction Ti j one can find in [Atluri et al.(2000)].
For ζ Γs Γ (source point located on the global boundwhere A is a harmonic function, i.e. ∇2 A 0 .
ary) the LBIE can be written as
Substituting eq. (22) into the second equation of (20) one
can write
B C
7
; <%D = T 9 ; x ? ζ< u ; x< dΓ D lim
E = T 9 ; x ? y< u ; x< dΓ
:= t ; x<1D γn ; x< θ ; x<+A U 9 ; x : ζ< dΓ 7
@
: = γθ6 ; x< : X ; x<+A U 9 ; x : ζ< dΩ
@
ui ζ
j
ij
y ζ
Γs
Ls
j
j
j
ij
8 7 λ D γ 2µ T6 : λ D µ µ A6
∇2Vi
ij
Γs
j
j
D
86 7 D
9
ij
Ωs
i
(23)
i
Replacing the right hand side (Dirac delta function) in
(18) the first equation of (20) by the Laplacian of temperature
(see the third equation) and differentiating according ∂i ,
we obtain the following relation
Next we will assume the body forces Xi to be zero. The
domain integral on the right hand sides of eqs. (17) and
2
(24)
(18) can be transformed to boundary integrals in the same λ 2µ Ui j kki ∇ T i
way as presented in the book [Balas et al.(1989)]. Applying the Gauss divergence theorem to the domain integral Now, replacing the gradient of temperature in (24) by the
corresponding terms given by eq. (23) we get
in eq. (17) one can write
; D < 96 7
γ
; λ D 2µ< ∇ U 9 6 7
= θ6 ; x< U 9 ; x : y< dΩ 7
j
Ωs
2
ij
= ; < ; < 9 ; : y< dΓ : γ= θ ; x< U 9 6 ; x : y< dΩ (19)
γ θ x n j x Ui j x
∂Ωs
ij j
Ωs
ij i
6
8 λ D γ 2µ D λ D µ µ λ D γ 2µ A6 I
H
∇2 ∇2V j
j
(25)
Taking into account ∇2 A
6 7
j
0 we can write
427
A pure contour formulation for the meshless LBIE method in thermoelasticity
JK L
Ui j i
Bj
The LBIE (29) and (30) based on a pure boundary formulation are more convenient for numerical implementation
(26) than those with the domain integral given in eqs. (17) and
(18).
N
M
1 2
∇ Vj
γ
where B j are harmonic functions, i.e. ∇2 B j
M
0.
Equation (26) can be rewritten into a more convenient
form
4 Moving least squares approximation and discretization
The moving least squares (MLS) approximation is generally considered as one of schemes to interpolate data
1 2
Ui j i
(27) with a reasonable accuracy. Consider the approximation
∇ Vj
γ
of a function u x in a domain Ω with a number scattered
nodes xi , i 1 2 n , the MLS approximant uh x of
2
where V j V j Ṽ j and ∇ Ṽ j γB j .
u , x Ω , can be defined by
The analytic expression of V j can be found in Appendix.
JK M
J
J M NO
J
Q R
Z [ M S S]\^\^\
_ T
M
Substituting eq. (27) into the domain integral in eq. (19)
and applying the Gauss divergence theorem we get
Q RM
Q R Q R _ xT Ω
(31)
where p Q x R M p Q x R`S p Q x R`S]\^\^\aS p Q x R-X is a complete
monomial basis ofW order m; and a Q x R is a vector containing coefficients a Q x R , j M 1 S 2 S]\^\^\^S m which are functions
of the space co-ordinates x M x S y S zX . For example, for
a 2-d problem:
W
uh x
pT x a x
T
γ
P θ Q xR U J K Q x O yR dΩ M
ij j
P θ Q xR V J K Q x O yR n Q xR dΓO P θK Q xR n Q xR V J Q x O yR dΓ (28)
j
j
∂Ωs
j
i
∂Ωs
Then, the pure boundary LBIE for an interior point can
be written as
2
m
T
Q R M 1 S x S yXS linear basis md M 3
Q R McW b 1 S x S yS>Q xR S xyS>Q yR quadratic basis m M 6 \
The coefficient vector a Q x R is obtained by performing a
pT x
pT x
2
2
weighted least-squares fit for the local approximation
Q R M O P T J Q x S yR u Q xR dΓ
ui y
1
j
Ωs
i j
Q R
j
ij
∂Ωs
Q R M ∑ w Q xR b p Q x R a Q xR O û d M
L P θ Q xR Z J Q x S yR dΓ O P q Q xR V J Q x O yR dΓ
e
Pa Q x R O ûX W Pa Q x R O ûX
(32)
W
W
where Z J M V J K n (see Appendix).
where w Q x R M w Q x O x R is the weight function assoThe alternative pure boundary integral equation for ζ T
ciated with the node i. The weight function w Q x O x R
Γ U Γ is given as
i
(29) J x
i
∂Ωs
∂Ωs
n
T
i
i
i
2
i 1
T
i j j
i
i
i
i
s
is defined as a monotonically decreasing function as the
distance between the evaluation point and the node increases. Thus, the weight function can be parametrized
by the distance di
x xi . In this paper, the weight
function is the Gaussian weight function given by
Q R L P T J Q x S ζR u Q xR dΓ L lim
V P T J Q x S yR u Q xR dΓ
O P t Q xR U J Q x O ζR dΓ M
ui ζ
j
ij
y ζ
Γs
Ls
j
ij
Mgf O f
j
ij
Γs
P θ Q xR Z J Q x O ζR O q Q xR V J Q x O ζYR X dΩ
W
i
Ωs
i
(30)
ij
w Q x R MGh
i
exp
0
b O Q d k c R d O exp b O Q r k c R d
1 O exp O Q r k c R X
W
i
i
2
i
i
i
2
i
2
0
l
di
di
m
l
ri
ri
n
428
Copyright c 2001 Tech Science Press
CMES, vol.2, no.4, pp.423-433, 2001
(33) where matrix N correspond to normal vector n
where ri is the radius of the domain of influence of the
i node and ci is a constant controlling the shape of the
Nx
weight function wi .
q rp„ƒ
n1 0 n2
0 n2 n1
In eq. (32) ûi denotes the fictitious nodal values and not
the nodal values of the unknown trial function in general. the stress-strain matrix D is given by
To find the coefficient a , we perform the extremum of J
by
1 ν
ν
0
2µ
D
ν
1 ν
0
1 2ν
0
0
1 2ν 2
(34)
∂J ∂a A x a x B x û 0
p s
o p q r q 4r s q r p
where matrices A q x r and B q x r
q r p
B q x rp
PT WP
Ax
p
s
†‡
s
q s ]r o
with for plane strain problem; ν p ν o4q 1 € ν r
are defined:
∑ wi q xr p q xi r pT q xi r
n
i 1
1
1
2
2
~
n
φj 1 0
0
φj 2
φj 2 φj 1
j
n
q rp
q r| p
∑ φi q xr ûi
n
ΦT x û
q r3p
T
Φ x
q r } q xr B q xr
p xA
1
(36)
The partial derivatives of the MLS shape functions are
obtained as
~ p ∑t  p ~ q A } Br € p q A } B~ € A~} Br 
φi k
m
1
jk
1
j
ji
k
j 1
k
1
ji
q y y^w ‚ r
(37)
Substituting the MLS approximation (35) into the definition formula of traction vector, we get
q rp N q xr D ∑ B q xr û s
t
n
j
j 1
j
n
q r
γn ∑ φ j x θ̂ j
t
j 1
n
∑
su
t
‹
 ŒŽ
∑
t
j 1
‹
 q w r q r ‘’ p
“
q r €G
∂θ̃
x yi φ j x dΓ θ̂ j
∂n
φ j yi
j 1
n
It should be noted that the MLS approximation is well defined only when the matrix A in eq. (34) is non-singular.
A necessary condition for a well-defined MLS approximation is that at least m weight functions are non-zero
i e n m for each sample point.
tx
~
(35) In view of the uncoupled theory and the use of the MLSapproximation, the discretized LBIE (10) and (11) collocated at yi Ω and ζi Γst become
t
i 1
where
T
~
ˆ‰
Γst , where Γsu and Γst are finite parts of
Γs over which these (temperature and displacements) or
(heat flux and tractions) are prescribed, respectively.
s
uh x
for plane
stress problem, and
t
pvu w q xr p q x rxw w q xr p q x r`wzy^y^y^y^w w q xr p q x r+{ B q xrp ‡†
~
Solving this for a q x r and substituting it into eq. (31), we
Let Γ p Γ Š
get
PT W
ˆ‰
∂Ωs
 ŒŽ
0
 q w r q r ‘’
“
∂θ̃ 
q x w yr φ q xr dΓθ̂ s

∑ lim
t • ∂n
∑  θ̃ q x w ζ r–u n φ ~ q xr € n φ ~ q xr+{ dΓθ̂
t
 θ̃ q x w ζ r q q xr dΓ s  ∂θ̃ q x w ζ r θ q xr dΓ
q r €”
φ j ζi
Ls
(39a)
∂θ̃
x ζi φ j x dΓ θ̂ j
∂n
n
€
j 1 y ζi
s
j
j
Γst
n
i
p
j 1
Γsu
1 j1
i
Γst
Γsu
2 j2
∂n
i
j
(39b)
(38) The set of algebraic equations (39a) and (39b) should be
supplemented by the equations
429
A pure contour formulation for the meshless LBIE method in thermoelasticity
Now, the fictitious unknowns û j can be calculated by
solving the set of algebraic equations (40). The limit of
ζi Γsu
(39c) the singular integral over Γst can be evaluated numeri∑ φ j ζi θ̂ j θ ζi
j 1
cally, by using a regular quadrature accurately, provided
in order to get a complete set for computation of the fic- that an optimal transformation of the integration variable
is employed before the integration [Sladek et al.(2000)].
titious unknowns θ̂ j at all the nodal points.
n
˜ ™ š ˜ ™ ›
—
œ
Having known the thermal fields, we can calculate the
5 Numerical examples
fictitious unknowns û j . The discretized LBIE (29) and
(30) collocated at yi Ω and ζi Γst become
In this section numerical results will be presented to test
the accuracy of the present computational method.
œ
ž Ÿ
œ
∑ ˜ ™1 ¡ ¢ ˜ › ™ ˜ ™ £¥ ¤ š
—
∑θ̂ ¡ Z¢ ˜ x › y ™ φ ˜ x™+§ V¢ ˜ x › y ™©¨ n φ ª ˜ x™+§
— ¦
n
φ j yi
j 1
∂Ωs
n
j
j 1
i
j
1 j1
i
n
j 1
j 1
—
T x ζi φ j x dΓ û j
Ls
∑— lim
® ¡ T¢ ˜ x › y™ φ ˜ x™ dΓ û
§ ∑— ¡ U¢ ˜ x › ζ ™ N ˜ x™ DB ˜ x™ dΓ û š
j 1 y ζi
Γst
n
j
i
j 1
Γsu
j
i
Γsu
¡ Z¢ ˜ x › ζ ™ θ ˜ x™ dΓ §¡ V¢ ˜ x › ζ ™ q ˜ x™ dΓ
∑— θ̂ ¡ ± Z¢ ˜ x › ζ ™ φ ˜ x™+§ V¢ ˜ x › ζ ™²˜ n φ ª ˜ x™+§
°
i
j
j 1
Ls Γst
um
1
i
Γsu
n
Γst
i
j
i
1 j1
∑ φ j ˜ ζi ™ û j š
—
j 1
˜ ™
u ζi
ζi
œ
Γsu
šG§ ν ˜ 1 ν™ Ep x
1
ª ˜ ™z™+³ dΓ u š˜ 1 § ν ™ Ep x
n2 φ j 2 x
m
2
2
§
¢
Finally, the set of equations (40a) and (40b) is supplemented by the equations
n
˜ ™>š
2
(40b) Then, the analytical expression for the stress σ22 in the
patch under a uniform thermal load with fixed displaceT V and are used in- ments in x2 and x3 directions can be written as
Ti j Ui and Zi , respec-
¢› ¢› ¢
¢› ¢› ¢
where the matrix notations U
stead of the tensor notations Ui j
tively.
š µ
š µ ¶ · ·
š¸˜ 1 ν™ αθ
§
j
i
§
The thermal strain tensor εt22 has to be eliminated by
the mechanical strain εm
22 due to the fixed displacement
in x2 direction. For a uniform load σ22 x1
p of a
square patch the displacement vector under plane strain
conditions are:
¡ U¢ ˜ x › ζ ™ t ˜ x™ dΓ §¯¡ T¢ ˜ x › ζ ™ u ˜ x™ dΓ
Γst
§
š µ ¶
εt22
n
j
˜ ´ ™
§
¢˜ › ™ ˜ ™
∑ φ j ˜ ζi ™ û j ∑ ¡
Consider the material in a quadrilateral cross-section
2a 2a subjected to a uniform thermal load θ and
fixed
displacements in x2 and x3 directions (see Fig.
x dΓ
1). The following material constants are assumed: Young
modulus
E 1 0 105 MPa , Poisson ratio ν 0 25 and
(40a)
coefficient of thermal expansion α 1 0 10 5 deg 1 .
If the bar subjected to a uniform thermal load is fixed in
x3 direction (plane strain conditions are assumed) the
strain tensor εt22 is given as
ª ˜ ™z«­¬
n2φj 2
∂Ωs
n
—
5.1 Patch test
T x yi φj x dΓ û j
σ22
§
š p
š §¹˜ 1 ν™ 1αEθ
§ ν µ
2
(41)
Owing to the symmetry, it is sufficient to analyse only
a quarter of the cross section. In meshless LBIE analysis
we have selected 16 nodes with an equidistant dis(40c)
tribution on the boundary of the analysed domain. To
430
º
Copyright c 2001 Tech Science Press
CMES, vol.2, no.4, pp.423-433, 2001
1.0
analytical solution
LBIE for ci=1.5
LBIE for ci=1.
*
σ ϕϕ
0.5
0.0
-0.5
Figure 1 : Square patch under a uniform temperature
load
-1.0
8.0
8.5
9.0
9.5
10.0
radius r
boundary nodes, 16 additional interior nodes are consid- Figure 2 : Comparison of numerical errors for stresses
ered (Fig. 1). The radii of circular subdomains were σ22 at internal points lying on x2 0 5 line
constant rloc 0 6 for boundary nodes and rloc 0 4
for interior nodes. We have also considered ci 1 and
ri ci 4 in the Gaussian weight function.
½ »
» ¼
»
» ¼
» ¼
To compare the accuracy of the numerical results for
stresses σ22 obtained by the LBIE method with the other
numerical method, we have analysed the same boundary value problem by the conventional boundary element
method based on the pure boundary integral equation
formulation [Balas et al.(1989)]. For discretization of
boundary we have used 8 quadratic elements. It is known
that accuracy of the numerical results in the conventional
BIE method is decreasing at points, which are close to the
boundary provided that no special treatment of nearlysingular integrals is adopted in the numerical integration.
For comparison of numerical results we have made analyses at points, which are on the line.
r
»À¿ Á
¿ Ã
¿ ¿ Â
σ22 σexact
22
%
σexact
22
where σexact
is taken from eq. (41).
22
5.2 Hollow cylinder
To compare the accuracy of the proposed numerical
scheme with analytical results, we present an example
of the computation of the radial distribution of the temperature and stresses in a hollow cylinder subjected to
a thermal gradient. On internal and external surfaces of
the hollow cylinder different, but constant temperatures
are prescribed (Fig. 3). Because of the symmetry of the
The dependence of the traction norm error, defined as problem, it is sufficient to analyse only a quarter of the
x2 0 5 . The shortest distance such points from the cylinder.
boundary, s, is equal or lower than 0.5 . Then, the ra- Analytical distributions of temperature and hoop stresses
tio of s to the element length d is s d 0 25 . In Fig. are given by
2 one can see that the lowest accuracy for the BIE results is at points which have the shortest distance to the
∆θ
mid-node of boundary element. The accuracy is chang- θ r
ln r
θ1
ln R2 R1
ing from 0.2% to 2.1% . On the other hand, the LBIE
» ¼
½ ¾ ¼
ÄÅ»
method yields the accuracy 0.05% at all the points of observation.
The relative error (accuracy) is defined as
σϕϕ
»
Æ
∆θ
Ç
½
Æ Ä 1½ rÇ Å Æ 1 Æ ln rÇ
Ä Á ÉÅ È Á Ä R ½ R Å lnR ½ R Ê
Eα
21 ν
1
1
2
1
2
2
1
2
(42)
431
A pure contour formulation for the meshless LBIE method in thermoelasticity
21
1.0
R2
u1=0
analytical temperature
analytical θ,1
LBIE results for θ
LBIE for θ,1
q=1.
t2=0
0.8
t1,t2=0
θ and θ,1
q=0
26
R1
0.6
0.4
q=0.
x2
0.2
40
x1
6
1
u2=0, t1=0 , q=0
0.0
Figure 3 : Nodes for the hollow cylinder subject to thermal load
8.0
8.5
9.0
9.5
10.0
radius r
ËÍÌ r Î R
1
and ∆θ
Ì
θ2
Ï
θ1 .
In the numerical analysis we have considered the following geometrical and material constants: cylinder radii
R1 8 , R2 10 ; Young modulus E 2 0 105 MPa;
Poisson ratio ν 0 25 and coefficient of thermal expansion α 1 67 10 5 deg 1 . We have used 40 nodes on
the boundary and 56 additional interior nodes. On both
artificial cuts (due to symmetry) the radii of subdomains
are constant rloc 0 39 . On the rest part of the boundary
(curvilinear surfaces) the size of subdomains is constant
too, rloc 0 49. Next, ci 1 5 and ri ci 4 are considered.
Ì
Ì
Ì Ð
Ì Ð Ñ Ò
Ì Ð
Ì Ð
Ì Ð Ñ
Ò
Ì Ð
The hoop stresses are normalized to obtain a nondimensional value according the formula
Ó Ï Ô
σË Ì
ϕϕ
were considered in the numerical analysis. The relative
errors for the hoop stresses on internal and external surfaces of the hollow cylinder are defined as
rI
Ì„Õ Ö × Ò Ö × Ö × Õ %Ù
Õ
Õ Ø
Î Ì
Numerical results for temperature and temperature gradient obtained by the LBIE method are compared with
analytical ones in Fig. 4. One can see an excellent agreement. The relative error for temperature in the whole
radius interval is less than 0.001% and for temperature
gradient is less than 0.05% .
21 ν
σϕϕ
∆θEα
Ï
σϕϕ RI
σexact
RI
ϕϕ
σexact
ϕϕ
RI
for I
Ì 1Ú 2 .
2.5
LBIE results
conventional BIE
2.0
1.5
r [%]
where r
Figure 4 : Temperature and temperature gradient along
x1 coordinate
1.0
0.5
0.0
Ë
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
The distribution of the non-dimensional σϕϕ along the
x1
radius of the hollow cylinder is shown in Fig. 5. One can
Figure 5 : Hoop stress distribution along the radius of
observe a quite good agreement of analytical results with
the hollow cylinder
LBIE ones. Various values of the weight function parameter ci at the constant size of the support domain ri 6
Ì
432
Û
Copyright c 2001 Tech Science Press
CMES, vol.2, no.4, pp.423-433, 2001
A variety of numerical experiments have been carried
out, to optimalize the size of support domain ri and the
weight function parameter ci . It follows from the analyses that ratio ri ci 4 is the most optimal. The wellknown boundary layer effect (lost accuracy of results at
points close to the boundary) for conventional BIE does
not arise in the LBIE with the MLS approximation. In
such a case, the accuracy does not dependent on the position of an evaluation point. It is a great advantage of
suggested method.
5.0
4.5
relative errors r1 and r2 [%]
Ü Ý
error for σϕϕ(R1)
error for σϕϕ(R2)
4.0
3.5
3.0
2.5
2.0
1.5
Appendix
1.0
0.5
0.0
1.0
1.2
1.4
1.6
1.8
2.0
weight function parameter ci
Figure 6 : Dependence of relative errors for hoop
stresses on the weight function parameter ci
In this Appendix we derive the explicit expression for
modified fundamental displacement corresponding to a
point heat source, U j . The fundamental displacement
has to satisfy eq. (27). The modified test function Ui j is
given by [Atluri et al.(2000)]
Þ
ÞÝ
Ui j
ß
Ui j
Þ
Ũi j
(43)
where
Figure 6 shows the influence of the weight function parameter ci on the accuracy of hoop stresses for both
1
cylinder surfaces. Accuracy on the whole ci interval is
Ui j
higher than 5% . Even for optimal of ci the accuracy is
8πµ 1
higher than 1% .
à ß ν–á â à 4ν ß 3á lnrδ ã rä rä å
Ý
6 Conclusion
The local boundary integral equation (LBIE) method
with a meshless approximation has been applied successfully, to solve 2-d boundary value problems in stationary thermoelasticity. The moving least square method
is used for approximation of physical quantities. Pure
contour boundary integral equations, with circular subdomains, have been utilized as LBIE. Introducing the
modified fundamental solution for displacement, which
gives zero displacement values on the contour of the subdomain, we eliminate the traction quantity from the local
boundary integral equations. The modified fundamental
solution requires one to derive new corresponding fundamental solutions related to a point heat source. They are
used to develop a pure contour local boundary integral
formulation in stationary thermoelasticity. The suggested
method is computionally efficient, due to the missing domain integration.
Ũi j
Ý
ij
i
(44)
j
æ ç èéëê à 4ν ß 3á lnr ã æ ç ç è©ì 1 ß í3î δ ã
1
8πµ 1 ν
0
5 4ν
2 3 4ν
r2
r02
ij
ri r j
r02
(45)
and r0 is the radius of the circular subdomain Ωs .
Considering eq. (43) we can rewrite eq. (27) into the
form
äß
Ui j i
äÝ
Ũi j i
à ð ß Ṽ á
1 2
∇ Vj
γ
(46)
j
From (45) it follows directly
ä Ý 13 ßß
Ũi j i
2ν
1
4ν 2πµ 1
ä
rr j
ν r02
à ß á
(47)
Thus, the additional fundamental displacement Ṽ j has to
satisfy the following equation
ï
A pure contour formulation for the meshless LBIE method in thermoelasticity
∇2Ṽ j
ñ 13 òò
2ν
γ
4ν 2πµ 1
Ching, H.K., and Batra, R.C.(2001) Determination od
Crack Tip Fields in Linear Elastostatics by Meshless Lo(48) cal Petrov-Galerkin (MLPG) Method. CMES: Computer
Modeling in Engineering & Sciences Vol. 2, No. 2, pp
273-290.
rj
ν r02
ó ò ô
It can be easy shown that function
Ṽ j
ñ òò
1
3
2ν
γ
4ν 16πµ 1
r
r0
ó ò νëô õ ö
433
Gu, Y.T. and Liu, G.R.(2001) A Meshless Local PetrovGalerkin (MLPG) Formulation for Static and Free Vibration Analyses of Thin Plates. CMES: Computer Model(49) ing in Engineering & Sciences Vol. 2, No. 4, pp 463-476.
2
rj
Kim, H.G. and Atluri, S.N.(2000) Arbitrary Placement
of
Secondary Nodes and Error Control in the Meshless
satisfies eq. (48).
Local Petrov-Galerkin (MLPG) Method. CMES: ComThe analytic expression of V j is given in (Balas et al.,
puter Modeling in Engineering & Sciences Vol. 1, No. 3,
1989). Finally, we can write
11-32.
÷
øñ ò
Vj
ó ù
m
rj 1
8π
2 ln r
ô4ò
Lin, H. and Atluri, S.N.(2000) Meshless Local PetrovGalerkin (MLPG) Method for Convection-Diffusion
(50) Problems. CMES: Computer Modeling in Engineering
& Sciences Vol. 1, No. 2, 45-60
r2 r j
m
8π 3 4ν r02
ó ò ô
where
m
ñ λ ù γ 2µ ñ
ò
Sladek V., Sladek, J., Atluri S.N. and Van Kerr, R.
(2000) Numerical integration of singularities in a meshless implementation of local boundary integral equations.
Comput. Mech. 25: 394-403.
γ 1 2ν
2µ 1 ν
ò
Zhu T., Zhang J.D. and Atluri S.N. (1998) A local
boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput. Mech. 21: 223-235.
From (50) one can obtain
ø ñ Vøú n ñ
ò 8πm û ó 1 ù 2 lnrô n ù 2rú n rú ü ò
Zj
ji i
j
j i i
ó
ù
ô
m 2rini r j r2 n j
(51)
8π r02 3 4ν
ó ò ô
References
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The local boundary integral equation (LBIE) and its
meshless implementation for linear elasticity. Comput.
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Belytschko T., Krongauz Y., Organ D., Fleming M.
and Krysl P. (1996) Meshless methods; An overview
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