c 2003 Tech Science Press
Copyright CMES, vol.4, no.1, pp.5-20, 2003
On Simple Formulations of Weakly-Singular Traction & Displacement BIE, and
Their Solutions through Petrov-Galerkin Approaches
Z. D. Han1 and S. N. Atluri1
Abstract: Using the directly derived non-hyper singular integral equations for displacement gradients [as in
Okada, Rajiyah, and Atluri (1989a)], simple and straightforward derivations of weakly singular traction BIE’s for
solids undergoing small deformations are presented. A
large number of “intrinsic properties” of the fundamental solutions in elasticity are developed, and are used
in rendering the tBIE and dBIE to be only weaklysingular, in a very simple manner. The solutions of the
weakly singular tBIE and dBIE through either global
Petrov-Galerkin type “boundary element methods”, or,
alternatively, through the meshless local Petrov-Galerkin
(MLPG) methods, are discussed. As special cases, the
Galerkin type methods, which lead to symmetric systems
of equations, are also discussed.
1
dimensional solid), we denote the ( r ) type singularities
1
as being “weakly-singular”, the ( 2 ) type singularities
r
1
as being “strongly-singular”, the ( 3 ) type singularities
r
as being “hyper-singular”.
In the Kelvin solution, for a 3-D solid, it is well-known
that the displacement-vector is “weakly-singular”, and
the stress-tensor is “strongly-singular”. In the classical
formulation, the integral equation for the displacement
vector at any point [x ∈ Ω or ∂Ω] is “strongly singular”. If the displacement integral equation at any point
x ∈ Ω, is differentiated with respect to x ∈ Ω, one may
∂
∇x [∇
∇ = ei ] for any
obtain an integral equation for ∇
∂xi
x ∈ Ω. From this equation for the displacement gradients, one may derive a traction boundary integral equation [tBIE] at any x ∈ Ω or ∂Ω. It is seen that this tBIE
1 Introduction
is “hyper-singular”. Much has been written in the last
In the past 25 years, much has been written about the 10∼15 years on the “regularization” of the tBIE [i.e., renintegral equation formulations for the displacement and der the “hyper-singular” tBIE into a “weakly-singular”
traction vectors in a solid body. Much of this work has tBIE], through what appears to be laborious mathematibeen concentrated on linear elastic, homogenous, and cal exercises and “manipulations”. This literature is too
isotropic solids. The focus in these derivations is on large to discuss here, but excellent summaries may be
the “fundamental solution” in a linear elastic isotropic found in [Cruse and Richardson (1996); Bonnet, Maier,
solid, viz., the Kelvin solution for a unit point load ap- Polizzotto (1998); Li and Mear (1998)].
plied at an arbitrary location, in an arbitrary direction, In this paper, we revisit the Kelvin solution, and delinin an infinite linear elastic solid. The Kelvin solution is eate certain “fundamental properties” of this solution. By
well-understood, and is “singular”. For clarity, we de- using the global-weak-form [or the weighted-residualnote the various levels of singularity: If “r” is the dis- equation”] of the momentum balance laws of linear elastance between any arbitrary point (ξξ ) in the solid and ticity, corresponding to a point load, on which the Kelvin
(x) (the point at which the unit load is applied in a 3- solution is based, we derive an arbitrary number of these
“fundamental properties”, by simply using an arbitrary
1 Center for Aerospace Research & Education
number of different types of “test functions” in writing
University of California, Irvine
the weak-forms. These fundamental properties of the
5251 California Avenue, Suite 140
1
Kelvin solution, which otherwise has a ( 2 ) singularIrvine, CA 92612, USA
r
ity for tractions, are shown to be the key in-gradients in
Copyright: S. N. Atluri.
any “regularization” of the tBIE, which is derived by differentiating the strongly-singular displacement integral
This paper is dedicated to the memory of Dr. John W. Linequation.
coln of the U.S. Air Force Research Laboratory.
6
c 2003 Tech Science Press
Copyright On the other hand, as far back as 1989, Okada, Rajiyah,
and Atluri (1989a,b, 1991) have proposed a way to di∂
∇ u [∇
∇ = ei ], rather
rectly derive integral equations for ∇
∂xi
than first derive the dBIE, and then differentiate it with
respect to x as is most common is literature. Thus, one
may also, from Okada, Rajiyah and Atluri (1989a,b),
directly derive a tBIE. Thus, the directly derived tBIE
[Okada, Rajiyah and Atluri (1989a,b)] is only “stronglysingular”, as opposed to being “hyper-singular”. It is
shown in the present paper, that by using the “fundamental properties” of the Kelvin solution [which are
also derived in the present paper], one may “regularize”
the directly derived tBIE of Okada, Rajiyah and Atluri
(1989a,b) in a very straight-forward and simple manner.
In a like manner, it is shown here that the dBIE can also
be “regularized” in a very straight-forward and simple
manner.
It is also shown in this paper that the fundamental Kelvin
solution for the stress-tensor can be naturally split into 2
∗p
∗p
parts, which we denote here as φ i j and ψi j , respectively,
∗p
∗p
∗p
∗p
and write σi j = ψi j − φi j , where σi j is the Kelvin so∗p
lution for stresses, ψ i j is divergence free, and the diver∗p
gence of φi j is the Dirac function.
We also discuss the numerical solution, by discretization,
of the directly derived, tBIE of Okada, Rajiyah and Atluri
(1989a,b), as well as of the regularized, weakly-singular
dBIE. We write the general Petrov-Galerkin types of
weak-forms of these integral equations at ∂Ω. Thus, we
introduce an arbitrary test function w(x), x ∈ ∂Ω. If
w(x) us a Dirac function, and if the trial functions t(x)
and u(x) are interpolated in terms of their nodal values
over a contiguous (“non-overlapping”) set of elements
at ∂Ω [“boundary elements”], one obtains the popular
“Boundary Element Methods”. On the other hand, if
w(x) is chosen to be the same as the complementary [or
energy-conjugate] trial function [i.e., in the tBIE, we use
u(x) as the test function, and in the dBIE, we use t(x)
as the test function], we obtain the so-called “Symmetric Galerkin Approaches” for dBIE and tBIE [Bonnet,
Maier, Polizzotto (1998)]. On the other hand, one may
leave w(x) as arbitrary, and formulate a general PetrovGalerkin Approach. It is further shown that, in the “Symmetric Galerkin” approach to solving the directly derived
tBIE [Okada, Rajiyah and Atluri (1989a,b)], using the
natural split of σ ∗p
i j , and the use of the Stokes’ theorem
at ∂Ω when w(x) is a continuous function, the resulting
CMES, vol.4, no.1, pp.5-20, 2003
discrete formulation results in certain further algebraic
conveniences. These end results are found to be somewhat similar to the results in Li and Mear (1998), but the
present results are different from those in Li and Mear
(1998) in terms of the attendant kernel functions. However, the present formulations of the Patrov-Galerkin approaches to solving the directly derived tBIE [Okada,
Rajiyah and Atluri (1989a,b)] are simple and straightforward.
The structure of the paper is as follows. In Section 1,
we briefly discuss the well-known Galerkin vector potential for displacements is an elastic solid undergoing small
displacements. In Section 2, we derive the fundamental
∗p
solution σ i j for a point load in an infinite body, and point
∗p
∗p
out how it can be split into φ i j and ψi j . In Section 3, we
derive, following Okada, Rajiyah and Atluri (1989a,b),
the displacement equations (dBIE), and directly derive
the tBIE without differentiating the dBIE. In deriving
these, we use the notion of “unsymmetric weak-forms”
of differential equations, as first noted in Atluri (1985).
In Section 4, we derive a large number of basic proper∗p
ties of σi j , by using the weak-forms [with different test
∗p
functions], of the balance laws for σ i j . We point out
how these methods can be used to derive the “basic properties” of the fundamental solutions for any problems of
mathematical physics [fluid mechanics, acoustics, electromagnetism, etc]. In Section 5, we discuss the “regularization” of tBIE, using these properties. We also discuss
the Petrov-Galerkin approach to discretize the “regularized” tBIE. In Section 6, we discuss the “regularization”
of dBIE, and the Petrov-Galerkin schemes to discretize
these regularized dBIE. In Section 7, we consider the important practical issue of the evaluation of displacements
and stresses near the surface. In Section 8, we briefly
mention the forthcoming papers of the authors, which
build upon the present results and to develop Meshless
Local Petrov Galerkin [MLPG] approaches [Atluri and
Zhu (1998), Atluri and Shen (2002a,b)].
2 Galerkin Vector Potential for Displacements in an
Elastic Solid Undergoing Small Deformations
Consider a linear elastic, homogeneous, isotropic body in
a domain Ω, with boundary ∂Ω. The Lame’ constants of
the linear elastic isotropic body are λ and µ; and the corresponding Young’s modulus and Poisson’s ratio are E
and υ, respectively. We use Cartesian coordinates ξ i , and
the attendant base vectors e i , to describe the geometry in
7
On Simple Formulations of Weakly-Singular tBIE & dBIE, and Petrov-Galerkin Approaches
Ω. The solid is assumed to undergo infinitesimal deformations. The displacement vector, strain-tensor, and the
stress-tensor in the elastic body are denoted as u, εε and
σ , respectively, with the corresponding dyadic representations, as follows:
u = ui ei
ε = εi j ei e j
(1)
where Φ is a scalar potential, and Ψ is a vector potential,
such that:
ϕ
Φ = ∇ ·ϕ
∇ ×ϕ
ϕ)
Ψ = −(∇
(12)
(13)
The displacements u Φ and uΨ have the properties,
Φ
∇ ×∇
∇Φ = 0
(2) ∇ × u = A∇
(14)
Ψ
σ = σi j ei e j
(3) ∇ · u = ∇ ·∇
∇ ×Ψ
Ψ=0
(15)
The equations of balance of linear and angular momen- Using (7c), (9c), (12)-(15) in (4)a, it is easily found (in
tum can be written as:
the absence of body force f) that:
∂
t
σ + f = 0; σ = σ ; ∇ = ei
(4)
∇ ·σ
σ = µ∇
∇2∇ 2ϕ = 0 or ∇ 2∇ 2ϕ = 0
(16)
∇ ·σ
∂ξi
(5)
σi j, j + fi = 0; σi j = σ ji
since
The strain-displacement relations are:
∇ · uΦ
∇ uΦ + uΦ∇ − 2 I∇
1
1
∇ ); εi j = (ui, j + u j,i )
(6)
ε = ∇ u + u∇
∇2 Φ +∇
∇ Φ∇
∇ − 2 I∇
∇ 2 Φ) = 0
= A(∇
(17)
2
2
The constitutive relations of an isotropic linear elastic homogeneous solid are:
∇ · u) + 2µ ε
σ = λ I (∇
∇ · u) + µ(∇
∇u + u∇
∇)
= λ I (∇
1
∇ · u) +∇
∇ u + u∇
∇ − 2 I (∇
∇ · u)]
= µ[ I (∇
A
(7a)
1 − 2υ
µ
=
λ + 2µ 2(1 − υ)
(18)
(7c)
∇∇ 2 Φ = 0
or ∇ 2 Φ = C
(19)
(8) 3 Fundamental Solutions in a Linear Elastic
Isotropic Homogeneous Infinite Medium
It is well known [Fung & Tong (2001)] that the displacement vector, which is a continuous function of ξξ , can be
derived, in general, from the Galerkin vector potential ϕϕ
such that:
1
∇ ·ϕ
ϕ)
∇ (∇
2(1 − υ)
∇Φ +∇
∇ ×Ψ
Ψ
= A∇
2
∇ ×∇
∇ ×ϕ
ϕ
∇ ϕ −∇
= A∇
2
2
∇ ϕ −∇
∇ (∇
∇ ·ϕ
ϕ)
∇ ϕ +∇
= A∇
u = ∇ 2ϕ −
Φ
∇Φ and uΨ = 0
u = uΦ = A∇
(7b) the balance equation (4) is simplified, as:
where
A=
For a curl-free solution, in which:
n(ȟ )
∂Ω
ȟ ∈ ∂Ω
ȟ ∈Ω
(9a)
(9b)
(9c)
Ω
x
ep
(9d)
Ψ
= u +u
(9e)
where, by definition,
∇Φ = A∇
∇2ϕ
uΦ = A∇
Ψ
(10)
Ψ = −∇
∇ ×∇
∇ ×ϕ
ϕ = ∇ ϕ −∇
∇ (∇
∇ ·ϕ
ϕ)
u = ∇ ×Ψ
2
Figure 1 : A solution domain with source point x and
(11) target point ξ
c 2003 Tech Science Press
Copyright 8
CMES, vol.4, no.1, pp.5-20, 2003
Consider a point unit load applied in an arbitrary direction e p at a generic location x in a linear elastic isotropic These stresses are seen to satisfy the balance laws:
homogeneous infinite medium as shown in Fig. 1. It is
well-known [Fung and Tong (2001)] that the displacement solution corresponding to this unit point load is σ∗p (x,ξξ) = µ(1 − υ)δ F ∗ = −µδ δ(x,ξξ);
p j ,kkii
pj
i j,i
given by the Galerkin vector displacement potential:
∗p
∗p
(27)
σi j = σ ji
(20)
ϕ ∗p = (1 − υ)F ∗ e p
∗p
We define two functions φ ∗p
i j and ψi j , as
where
F∗ =
r
8πµ(1 − υ)
for 3D problems
−r2 ln r
8πµ(1 − υ)
for 2D problems
(21a)
∗p
∗
φi j (x,ξξ) ≡ −µ(1 − υ)δ p j F,kki
and
F∗ =
(28a)
∗p
∗p
∗p
(21b) ψi j (x,ξξ) ≡ σi j (x,ξξ) + φi j (x,ξξ)
where r = ξξ − x
Thus, ϕ ∗p in (20) is the solution, in infinite space, to the
differential equation (in the coordinates ξξ),
∇2∇ 2ϕ ∗p + δ(x,ξξ)e p = 0
µ∇
σ (x,ξξ) + δ(x,ξξ)e p = 0
∇ ·σ
∂
∇ = ei
∂ξi
(28b)
Then, from (27), (28a) and (28b), it can be seen that:
(22) φ∗p (x,ξξ) = −σ∗p (x,ξξ)
i j,i
i j,i
∗
= µδ p j δ(x,ξξ)
= −µ(1 − υ)δ p j F,kkii
or
∇2∇ 2 F ∗ + δ(x,ξξ) = 0
µ(1 − υ)∇
∗
∗
∗
= µ[(1 − υ)δ pi F,kk
j + υδi j F,pkk − F,pi j ]
∗p
(23)
ψi j,i (x,ξξ) = 0 [ divergence of ψ ∗ (x,ξξ) = 0]
(29a)
(29b)
∗
The corresponding displacements are derived from the Hence, as a divergence free tensor, ψ (x, ξ) must be a
curl of another divergence free tensor. We choose to
Galerkin vector displacement potential, using (9a), as:
rewrite it in term of F ∗ from Eq. (28b), as:
1 ∗
∗p
∗
(24)
ui (x,ξξ) = (1 − υ)δ pi F,kk − F,pi
2
The gradients of the displacements in (24) are:
1 ∗
∗p
∗
ui, j (x,ξξ) = (1 − υ)δ pi F,kk
j − F,pi j
2
∗p
∗
∗
∗
ψi j (x,ξξ) = µ[(1 − υ)δ pi F,kk
j + υδi j F,pkk − F,pi j ]
(25)
The corresponding stresses in a linear elastic homogenous isotropic body are given by:
∗p
∗p
∗p
σi j (x,ξξ) ≡ Ei jkl εkl ≡ Ei jkl uk,l
∗
∗
∗
∗
= µ[(1 − υ)(δ pi F,kk
j − δi j F,pkk ) + δi j F,pkk − F,pi j ]
∗
∗
−δi j δ ps F,kks
)
= µ[(1 − υ)(δ pi δ js F,kks
∗
∗
− δi j δks F,pks
)]
−(δki δ js F,pks
∗
∗
∗
∗
= µ[(1 − υ)(δ piF,kk
j + δ p j F,kki ) + υδi j F,pkk − F,pi j ]
∗
∗
− etk j etis F,pks
]
= µ[(1 − υ)et p j etis F,kks
∗
∗
∗
= µ[(1 − υ)δ pi F,kk
j + υδi j F,pkk − F,pi j ]
∗
∗
− etk j F,pk
],s
= µetis [(1 − υ)et p j F,kk
∗
+ µ(1 − υ)δ p j F,kki
(26)
∗p
≡ eist Gt j,s
(30)
9
On Simple Formulations of Weakly-Singular tBIE & dBIE, and Petrov-Galerkin Approaches
From (32b), (32c), (32), the singularity in each of the
terms in Eq. (34) can be seen, for 3D problems, as:
where, by definition,
∗p
∗
∗
− eik j F,pk
]
Gi j (x,ξξ) = µ[(1 − υ)eip j F,kk
(31)
In a 3-dimensional linear elastic homogeneous body we
can easily derive the derivatives of F ∗ , from (21a), as:
r,p
=
8πµ(1 − υ)
1
∗
(δ pi − r,p r,i )
=
F,pi
8πµ(1 − υ)r
1
∗
=
F,kk
4πµ(1 − υ)r
1
∗
(δ pi r, j + δ p j r,i + δi j r,p
F,pi
j =−
8πµ(1 − υ)r 2
−3r,p r,i r, j )
1
∗
=−
r,i
F,kki
4πµ(1 − υ)r 2
1
∗
(δi j − 3r,i r, j )
F,kki
j=−
4πµ(1 − υ)r 3
F,p∗
(35a)
(35b)
(35c)
(32a)
and for 2D problems as:
(32b)
1
∗
∝ O( )
F,kki
r
(32c)
∗
∝ O(lnr)
F,kk
∗
F,pk
∝ O(lnr)
(36a)
(36b)
(36c)
(32d)
We write the kernel functions for 3D problems, from Eq.
(32e) (32), as:
(32f) u∗p
ξ) =
i (x,ξ
and for a 2-dimensional body,
1
(r + 2r lnr)r ,p
8πµ(1 − υ)
1
∗
[δ pi (1 + 2 lnr) + 2r ,p r,i ]
=−
F,pi
8πµ(1 − υ)
1
∗
(1 + ln r)
=−
F,kk
2πµ(1 − υ)
1
∗
(δ pi r, j + δ p j r,i + δi j r,p
F,pi
j =−
4πµ(1 − υ)r
−2r,p r,i r, j )
1
∗
r,i
=−
F,kki
2πµ(1 − υ)r
1
∗
(δi j − 2r,i r, j )
F,kki
j=−
2πµ(1 − υ)r 2
1
)
r2
1
∗
∝ O( )
F,kk
r
1
∗
∝ O( )
F,pk
r
∗
F,kki
∝ O(
ξ) =
G∗p
i j (x,ξ
F,p∗ = −
1
[(3 − 4υ)δip + r,i r,p ]
16πµ(1 − υ)r
(37)
1
[(1 − 2υ)eip j + eik j r,k r,p ]
8π(1 − υ)r
(38)
(33a)
(33b)
1
8π(1 − υ)r 2
[(1 − 2υ)(δi j r,p − δip r, j − δ jp r,i ) − 3r,i r, j r,p ]
ξ) =
σ∗p
i j (x,ξ
(39)
(33c)
and for 2D plane strain problems, from Eq. (33), as:
(33d)
(33e)
ξ) =
u∗p
i (x,ξ
1
[−(3 − 4υ) ln rδip + r,i r,p ]
8πµ(1 − υ)
ξ) =
G∗p
i j (x,ξ
1
[−(1 − 2υ) lnr eip j + eik j r,k r,p ]
4π(1 − υ)
(41)
(40)
(33f)
Thus, from (28a), (28b) and (30), one may see that:
∗p
∗p
∗p
σi j (x,ξξ) = ψi j (x,ξξ) − φi j (x,ξξ)
=
∗
∗
+ µeist [(1 − υ)et p j F,kk
− etk j F,pk
],s
1
4π(1 − υ)r
[(1 − 2υ)(δi j r,p − δip r, j − δ jp r,i ) − 2r,i r, j r,p ]
∗p
σi j (x,ξξ) =
∗
µ(1 − υ)δ p j F,kki
(34)
(42)
c 2003 Tech Science Press
Copyright 10
CMES, vol.4, no.1, pp.5-20, 2003
4 Displacement & Traction BIE: Derivations From
Unsymmetric Weak Forms of Balance Laws in
Elasticity
0=
The governing equations of momentum balance in a solid
undergoing small displacements are
σi j = σ ji
σi j,i + fi = 0;
(),i ≡
∂
∂ξi
=
=
(43)
Ω
∂Ω
∂Ω
+
For the present, we ignore the body forces f i ( but include
them later, when necessary). Thus, (43) is reduced to:
σi j,i = 0 in Ω
=
ni Ei jmn um,n u j dS −
ni Ei jmn um,n u j dS −
Ei jmn um u j,in dΩ
Ω
ni Ei jmn um,n u j dS −
Ei jmnum,n u j,i dΩ
Ω
nn Ei jmn um u j,i dS
∂Ω
∂Ω
+
(44)
Ei jmn um,niu j dΩ
nn Ei jmn um u j,i dS
∂Ω
um (Ei jmnu j,i ),n dΩ
(49)
Ω
For a homogeneous linear elastic isotropic homogeneous
solid, the constitutive equation is
Instead of the scalar weak form of Eq. (44), as in Eq.
(45) (48), we may also write a vector weak form of Eq. (44),
by using the tensor test functions u i,k [as originally proposed in Okada, Rajiyah, and Atluri (1988,1989)] as:
σi j = Ei jkl εkl = Ei jkl uk,l
where
1
εkl = (uk,l + ul,k )
2
(46)
Ω
σi j,i u j,k dΩ = 0 k = 1, 2, 3
(50)
and
Ei jkl = λδi j δkl + µ(δik δ jl + δil δ jk ),
By applying divergence theorem three times in Eq. (50),
(47) we may write:
with λ and µ being the Lame’s constants.
Let ui be the trial functions for displacements, to satisfy
Eq. (44), in terms of ui, when Eqs. (45)-(47) are used.
Let u j be the test functions to satisfy the momentum balance laws in terms of u i , in a weak form. The weak form
of the equilibrium Eq. (44) can then be written as,
Ω
σi j,i u j dΩ ≡
Ω
(Ei jmn um,n )i u j dΩ = 0
0=
=
=
=
Applying the divergence theorem two times 2 in Eq. (48),
we obtain:
we use the divergence theorem only once in Eq. (48), we obtain
the “symmetric” weak form:
Ω σi j u j,i dΩ − ∂Ω ni σi j u j dΩ =
0
Thus, in the symmetric weak form, both the trial functions u i as
well as the test functions u j are only required to be once differ
entiable. However, in the “unsymmetric weak form” of Eq. (49),
the test functions u j in Ω are required be twice-differentiable, while
there is no differentiability requirement on ui in Ω.
Ω
∂Ω
∂Ω
∂Ω
+
Ω
ni Ei jmnum,n u j,k dS −
Ei jmn um,nk u j,i dΩ
∂Ω
∂Ω
nn Ei jmn um,k u j,i dS −
nk Ei jmnum,n u j,i dS
nk Ei jmnum,n u j,i dS
Ω
ni Ei jmnum,n u j,k dS −
∂Ω
Ei jmn um,n u j,ki dΩ
ni Ei jmnum,n u j,k dS −
∂Ω
(51)
ni Ei jmnum,n u j,k dS −
Ω
+
=
Ei jmn um,ni u j,k dΩ
∂Ω
+
(48)
2 If
∂Ω
nn Ei jmn um,k u j,i dS −
Ei jmnum,k u j,in dΩ
nk Ei jmnum,n u j,i dS
Ω
um,k (Ei jmnu j,i ),n dΩ
∗p
By taking the fundamental solution u i (x,ξξ) as the test
function u i (ξξ ), and with the consideration of its properties in Eq. (22), we re-write Eqs. (49), (51), respectively,
11
On Simple Formulations of Weakly-Singular tBIE & dBIE, and Petrov-Galerkin Approaches
as,
u p (x) =
−
≡
∂Ω
− Eabpk u p,k (x) = Eabpk
∗p
∂Ω
ξ) dS
ni (ξξ)Ei jmn um,n (ξξ)u∗p
j (x,ξ
∂Ω
nn (ξξ)Ei jmn um(ξξ)u j,i (x,ξξ) dS
∗p
t j (ξξ)u j (x,ξξ) dS −
∂Ω
+ Eabpk
um (ξξ)tm∗p (x,ξξ) dS
(52)
= Eabpk
− u p,k (x) =
−
+
∂Ω
∂Ω
∗p
∂Ω
ni (ξξ)Ei jmnum,n (ξξ)u j,k (x,ξξ) dS
∂Ω
∗p
∂Ω
t j (ξξ)u j,k (x,ξξ) dS
ξ) dS
[nn (ξξ)um,k (ξξ) − nk (ξξ)um,n (ξξ)]σ∗p
nm (x,ξ
t j (ξξ)u∗p,kj (x,ξξ) dS
∂Ω
ξ) dS
Dt um(ξξ)enkt σ∗p
nm (x,ξ
∂
∂ξs
Dt = nr erst
(53)
with respect to x k , we obtain:
u p,k (x) =
∂Ω
ξ) dS −
t j (ξξ)u∗p
j,k (x,ξ
(56)
Then Eq. (55) can be re-written as,
Eqs. (52) and (53) were originally given in [Okada,
Rajiyah, and Atluri (1989a,b)], and the notion of using
∗q
(x)
=
tq (ξξ)σab (x,ξξ) dS
−
σ
ab
unsymmetric weak-forms of the differential equations,
∂Ω
to obtain integral representations for displacements, was
ξ) dS
D p uq (ξξ)enl p Eabkl σ∗k
+
nq (x,ξ
presented in [Atluri (1985)]. It should be noted that the
∂Ω
integral equations for u p (x) and u p,k (x) as in Eqs. (52)
∗q
tq (ξξ)σab (x,ξξ) dS
≡
and (53) are derived independently of each other. On
∂Ω
the other hand, if we derive the integral equation for
D p uq (ξξ)Σ∗abpq (x,ξξ) dS
+
displacement-gradients, by directly differentiating u p (x)
∂Ω
in Eq. (52), i.e. by differentiating,
where by definition,
∗p
∗p
t j (ξξ)u j (x,ξξ) dS −
um (ξξ)tm (x,ξξ) dS
u p (x) =
ξ)
Σ∗i jpq (x,ξξ) = Ei jkl enl p σ∗k
∂Ω
∂Ω
nq (x,ξ
(55)
where the surface tangential operator D t is defined as,
ξ) dS
nkξ )Ei jmn um,n (ξξ)u∗p
j,i (x,ξ
ξ) dS
nn (ξξ)Ei jmn um,k (ξξ)u∗p
j,i (x,ξ
∂Ω
+ Eabpk
(57)
(58)
Contracting Eq. (57) with n a (x), we have
∂Ω
∗p
um(ξξ)tm,k
(x,ξξ) dS
(54)
∗p
t m,k (x,ξξ)
− tb (x) =
+
∂Ω
∂Ω
ξ) dS
tq (ξξ )na (x)σ∗q
ab(x,ξ
D p uq (ξξ)na (x)Σ∗abpq (x,ξξ) dS
is of
Thus, Eq. (54) is hypersingular, since
−3
O(r ) for 3D problems. On the other hand, the directly where the traction is defined as,
ξ) as in Eq. (53)
derived integral equations for u p,k (x,ξ
tb (x) = na (x)σab (x)
contain only singularities of O(r −2 ).
Eq. (52) is the original displacement BIE (dBIE) in
its strongly-singular form before any regularization. On
the other hand, Eq. (53) are the non-hypersingular
(“strongly-singular”) integral equations for displacement
gradients in a homogeneous linear elastic solid, as originally derived in Okada, Rajiyah, and Atluri (1989a,b). It
is but a simple extension to derive a non-hypersingular
integral equation for tractions in a linear elastic solid,
from Eq. (53),
(59)
(60)
5 Some Basic Properties of the Fundamental Solution
Consider a body of an infinite extent, subject to a point
force at a generic location x in the direction e p , as shown
in Fig. 1. The fundamental solution, in infinite space, of
the stress field, denoted by σσ∗p (x,ξξ), at any point ξ due
to this point load at x, is generated by the balance law,
from Eq. (22):
c 2003 Tech Science Press
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CMES, vol.4, no.1, pp.5-20, 2003
and thus, one obtains:
σ∗p (x,ξξ) + δ(x,ξξ )e p = 0
∇ ·σ
(61)
CPV
∂Ω
1
ξ)dS + δ p j = 0 x ∈ ∂Ω
t ∗p
j (x,ξ
2
(64b)
We write the weak form of Eq. (61) over the domain,
The second term on the right hand-side of Eq. (64a) reusing a constant c as a test function, as
sults from the principal value of the singular integral in
1
∗p
σ∗p (x,ξξ) c dΩ + e p c = 0
∇ ·σ
(62a) volving t j , which has a O( r2 ) singularity. Eq. (64b)
Ω
∗p
may also be physically explained as below. σ i j (and thus
or
∗p
t j ) are solutions due to a point load applied in an infi
σ∗p (x,ξξ)dS + e p = 0
n(ξξ ) ·σ
(62b) nite space. In reality, the point load can be assumed to be
∂Ω
distributed over a small-sphere, of radius ε, in an infinite
or
body. The tractions distributed over this sphere, that re
1
sult in a point load, are of O( 2 ); while the surface area
∗p
ε
ni (ξξ)σi j (x,ξξ)dS + δ p j = 0
(62c)
∂Ω
of the sphere is O(ε 2 ). As long as this sphere is inside
Ω, and while Ω is a part of the infinite space, the load
or
applied on Ω is still unity. Suppose x → x̂ at ∂Ω shown
∗p
t j (x,ξξ)dS + δ p j = 0 x ∈ Ω
(62d) in Fig. 2, then the sphere of radius ε is centered at the
∂Ω
boundary. As long as the boundary is smooth, only oneEq. (62d) is a “basic identity” of the fundamental solu- half of the sphere of radius ε is actually inside Ω, when
tion σ ∗p (x,ξξ). Eq. (62d) is simply an affirmation of the x → x̂ at ∂Ω. Thus while the load applied, in infinite
force balance law for Ω: if the point load is applied at a space, on a sphere of radius ε at x̂ ∈ ∂Ω, is still unity, the
point x ∈ Ω when Ω is entirely embedded in an infinite actual load applied on Ω is only 12 . Thus we obtain Eq.
space, the tractions exerted by the surrounding infinite (64b).
body on the finite-body Ω should be equilibrated with
the applied point force at x inside Ω.
n(ȟ )
∂Ω
Secondly, if we consider an arbitrary function u(x) in
Ω as the test function, we may write the corresponding
weak-form, from Eq.(61), as:
Ω
∇ ·σ
σ ∗p (x,ξξ) + δ(x,ξξ)e p ] · u(x)dΩ = 0
[∇
ȟ ∈Ω
(63a)
x̂
x
or
∂Ω
ȟ ∈ ∂Ω
ep
Ω
σ ∗p (x,ξξ)] · u(x)dS + e p · u(x) = 0
[n(ξξ) ·σ
(63b)
t∗p (x,ξξ) · u(x)dS + e p · u(x) = 0 x ∈ Ω
(63c)
∗p
t j (x,ξξ)u j (x)dS + u p (x) = 0 x ∈ Ω
(63d) We can write Eqs. (62d) and (63d) for x ∈ ∂Ω, with Eq.
(64), as:
or
∂Ω
Figure 2 : A loading point x approaching the boundary
or
∂Ω
CPV
1
Once the point x approaches a smooth boundary, i.e. x ∈
t ∗p
(x,ξξ)dS + δ p j = 0
x ∈ ∂Ω
j
∂Ω, the first term in Eq. (62d) can be written in a Cauchy ∂Ω
2
CPV
CPV
, as,
Principal value (CPV) integral, denoted by
1
∗p
t j (x,ξξ)u j (x)dS + δ p j u j (x) = 0
CPV
2
∂Ω
1
∗p
∗p
t j (x,ξξ)dS =
t j (x,ξξ)dS − δ p j (64a)
lim
2
x→∂Ω ∂Ω
∂Ω
(65)
x ∈ ∂Ω
(66)
13
On Simple Formulations of Weakly-Singular tBIE & dBIE, and Petrov-Galerkin Approaches
or
From Eq. (29a), we also see that
−
CPV
∂Ω
ξ)dS
ni (ξξ)σ∗p
i j (x,ξ
=
CPV
∂Ω
ξ)dS
ni (ξξ)φ∗p
i j (x,ξ
∂Ω
or
(67a) or
−
CPV
∂Ω
ξ)dS
t ∗p
j (x,ξ
=
CPV
∂Ω
∂Ω
ξ)dS
ni (ξξ)φ∗p
i j (x,ξ
ξ)um,k (x)dΩ + u p,k (x) = 0
nn (ξξ)Ei jmnu∗p
j,i (x,ξ
(73c)
tm∗p (x,ξξ)um,k (x)dΩ + u p,k (x) = 0 x ∈ Ω
(73d)
(67b) In addition, we may observe that the first two terms in
Eq. (53) have the following identity, as:
for both x ∈ Ω and x ∈ ∂Ω. Thus, we can write identities
∗p
for φi j which are similar to those in Eqs. (62d) and (63d)
∗p
ni (ξξ)Ei jmnum,n (x)u j,k(x,ξξ) dS
∗p
for σi j , as:
∂Ω
CPV
∂Ω
∗p
ni (ξξ )φi j (x,ξξ)dS −Cδ p j
=0
(68)
=
∂Ω
Ω
∗p
Ω
(69)
1
2
for x ∈ ∂Ω and C = 1 for x ∈ Ω.
The corresponding equations for Ψ ∗p
i j can also be written
as,
∂Ω
Ei jmnum,n (x)u j,ki(x,ξξ) dS
Ei jmnum,n (x)u j,ik (x,ξξ) dS = 0
(74)
By adding Eq. (74) into Eq. (73c), we obtain,
where C =
nk (ξξ)Ei jmn um,n (x)u j,i (x,ξξ) dS
∗p
−
∗p
ni (ξξ )φi j (x,ξξ)u j (x)dS −Cu p (x) = 0
∂Ω
and
CPV
∗p
−
∗p
ni (ξξ)Ψi j (x,ξξ)dS
=0
∗p
∂Ω
ni (ξξ)Ei jmnum,n (x)u j,k(x,ξξ) dS
−
+
(70)
∗p
∂Ω
∂Ω
nk (ξξ)Ei jmn um,n (x)u j,i (x,ξξ) dS
ξ)dΩ
nn (ξξ )Ei jmnum,k (x)u∗p
j,i (x,ξ
+ u p,k (x) = 0
(75a)
and
∂Ω
or
(71) ξ) · u j (x)dS = 0
ni (ξξ)Ψ∗p
i j (x,ξ
Third, we consider the weak form of Eq. (61), and consider the test functions to be the gradients of an arbitrary
function u(ξξ ) in Ω. This function u(ξξ ) is so chosen that
it has constant gradients, as:
Ω
∂Ω
∗p
ni (ξξ)σi j (x,ξξ)u j,k (x)dΩ + u p,k (x) = 0
∗p
∂Ω
enkt Dt um (x)Ei jmnu j,i (x,ξξ) dS
+ u p,k (x) = 0
(75b)
∗q
∂Ω
n p (ξξ)σ pq (x)σab (x,ξξ)dS
+
(73a)
∂Ω
D p uq (x)Σ∗abpq (x,ξξ) dS + σab (x) = 0
(76)
where Σ∗abpq is defined in Eq. (58).
Forth more, we use the displacement field u(ξξ ) in Ω as
the test function and write the weak form of Eq. (61) as,
Applying the divergence theorem, we obtain,
+
(72) Then, the weak form of Eq. (61) may be written as:
∗p
p
[σi j,i (x,ξξ) + δ(x,ξξ )e j ]u j,k (x)dΩ = 0
ni (ξξ)Ei jmnum,n (x)u j,k(x,ξξ) dS
Multiplying Eq. (75b) by E abpk , we obtain the following
identity for the corresponding stresses σσ(x), as:
u j,k (ξξ) = u j,k (x)
∗p
∂Ω
(73b)
∗p
Ω
[σi j,i(x,ξξ) + δ(x,ξξ)e j ]u j (ξξ)dΩ = 0
p
(77a)
c 2003 Tech Science Press
Copyright 14
CMES, vol.4, no.1, pp.5-20, 2003
(82)
or
0 =
∂Ω
−
=
Ω
∂Ω
−
=
∂Ω
which is the same as Eq. (62c) in the basic identity.
Applying this displacement field in Eq. (81) to Eq. (79),
we have
ξ)u j,i (ξξ)dΩ + u p (x)
σ∗p
i j (x,ξ
ξ)u p (ξξ)dS
ni (ξξ)σ∗p
i j (x,ξ
Ω
−
ξ)u p (ξξ)dS
ni (ξξ)σ∗p
i j (x,ξ
Eabpq
ξ)u j,i (ξξ)dΩ + u p (x)
Ei jmn u∗p
m,n (x,ξ
(77b)
By differentiating it with respect to x k , one may obtain
∂Ω
Ω
ξ)dΩ + δab ] = 0
nm (ξξ)σ∗m
ab (x,ξ
2
u(ξξ) = ω [v × (v × r(x,ξξ)) + r(x,ξξ)]
3
or
1
u(ξξ) = ω [v ∗ r(x,ξξ))v − r(x,ξξ)]
3
ξ)u j (ξξ)dS
t ∗p
j,k (x,ξ
(83)
Now we take another special case in which u(ξξ ) is from
a displacement field without volume strain, and contains
no free rotation, as:
ξ)u j,i (ξξ)dS
nn (ξξ)Ei jmn u∗p
m (x,ξ
+u p (x)
∂Ω
− (λ + 2µ)[
ξ)u p (ξξ)dS
ni (ξξ)σ∗p
i j (x,ξ
∂Ω
∗p
t j,q (x,ξξ)r j (x,ξξ)dS
(84a)
(84b)
∗p
nn (ξξ)Ei jmnum,k (x,ξξ)u j,i (ξξ)dS + u p,k (x) = 0 (78) where ω is a constant and v is a constant vector with a
unit length. A similar identity for such problems can be
Multiplying Eq. (78) by E abpk , we have an identity easily obtained.
∗p ∗p ∗p
∗p
∗p
ξ), in terms of the non- All the identities for σ i j , t j , φi j and Ψi j are obtained
for the hypersingular t i, j (x,ξ
from the weak form of the fundamental solution, with
∗p
ξ ), as:
hypersingular σ i j (x,ξ
various test functions. They can be readily used in the
regularization of the stongly-singular and hypersingular
integrals in BIE equations.
−
∂Ω
ξ)u j (ξξ)dS
t ∗p
j,q (x,ξ
It is clear that “properties” similar to the above, can be
derived for the fundamental solutions for any set of par∗q
(79) tial differential equations, such as those that arise in fluid
− n p (ξξ)σ pq (ξξ)σab (x,ξξ)dΩ + σab (x) = 0
Ω
mechanics, acoustics, electromagnetism, etc. These will
be presented in subsequent papers by the present authors.
Clearly, one may derive any number of “properties” of
the fundamental solution in elasticity. Some of additional
6 Regularization of tBIE
properties are derived as follows:
Contracting Eq. (76) with n a (x), and using the resulting
By taking u(ξξ ) as the “radial displacement” field, as:
equation in Eq. (59), we can obtain the fully regularized
form of Eq. (59), as
u(ξξ) = α r(x,ξξ)
(80)
Eabpq
∂Ω
where α is a constant. We obtain from Eq. (80):
u j,k (ξξ) = u j,k (x) = αδ jk
0=
(81)
+
∗q
∂Ω
∂Ω
[tq(ξξ) − n p (ξξ)σ pq (x)]na(x)σab(x,ξξ) dS
[D p uq (ξξ) − (D p uq )(x)]na(x)Σ∗abpq (x,ξξ) dS (85)
By substituting Eq. (81) into Eq. (73d), we have
∂Ω
tm∗p (x,ξξ)δmk dΩ + δ pk
=
∂Ω
∗p
tk (x,ξξ)dΩ + δ pk = 0
We know that t q(ξξ) −n p (ξξ) σ pq (x) and D p uq (ξξ) −
(D p uq )(x) become O(r) when ξ → x̂ at ∂Ω and hence
15
On Simple Formulations of Weakly-Singular tBIE & dBIE, and Petrov-Galerkin Approaches
0=
e 3 (xˆ ) = n(xˆ )
x̂
∗q
∂Ω
[tq (ξξ) − tq (x)]na(x)σab (x,ξξ) dS
+ σ pq (x)
e 2 (xˆ ) = t (xˆ )
e1 (xˆ ) = s(xˆ )
+
∂Ω
∂Ω
ξ) dS
[n p (x) − n p (ξξ)]na(x)σ∗q
ab (x,ξ
[D p uq (ξξ) − (D p uq )(x)]na(x)Σ∗abpq (x,ξξ) dS
(87b)
With Eq. (86), Eq. (87) can be evaluated numerically as
tBIE in its fully-regularized form.
Figure 3 : Local coordinates at a boundary point x̂
Eq. (85) is weakly singular. Thus Eq. (85) can be applied as x → x̂ ∈ ∂Ω.
Eq. (87) is the regularized traction BIE, derivable
from the non-hypersingular integral representation for
tractions, as reported in Okada, Rajiyah, and Atluri
(1988,1989).
Eq. (85), may be satisfied in a weak-form at ∂Ω, using a
We define the local coordinates at point x as e 1 (x) = s(x), Petrov-Galerkin scheme, as:
e2 (x) = t(x) and e3 (x) = n(x), as shown in Fig. 3. From
the strain-displacement relations in (6) and the stress
∗q
strain relations in Eq. (7), we have σσ(x) in terms of u(x) 0 = wb (x)dSx
[tq(ξξ) − n p (ξξ)σ pq (x)]na(x)σab(x,ξξ) dSξ
∂Ω
∂Ω
and t(x), as:
+ wb (x)dSx
[D p uq (ξξ) − (D p uq )(x)]na(x)Σ∗abpq (x,ξξ) dSξ
∂Ω
∂Ω
(88)
λ
2λµ
t3 (x) +
[u1,1(x) + u2,2 (x)]
σ11 (x) =
λ + 2µ
λ + 2µ
+ 2µu1,1 (x)
λ
2λµ
t3 (x) +
[u1,1(x) + u2,2 (x)]
σ22 (x) =
λ + 2µ
λ + 2µ
+ 2µu2,2 (x)
where wb (x) is a test function. If w b (x) is chosen as a
Dirac delta function, i.e. wb (x) = δ(x, xm ) at ∂Ω, we obtain the standard “collocation” traction boundary element
method.
σ33 (x) = t3 (x)
From Eqs. (7) and (26), Σ∗i jpq can be written in terms of
F ∗ as:
σ12 (x) = σ21 (x) = µ[u1,2 (x) + u1,2 (x)]
σ13 (x) = σ31 (x) = t1 (x)
σ23 (x) = σ32 (x) = t2 (x)
(86)
Thus, we can re-write Eq. (85) as:
0=
∂Ω
= µ2 [(−enip F, jnq − en jp F,inq + δiq e jpn F,bbn + δ jq eipn F,bbn )
{t j (ξξ) − t j (x)
+ υ(eqip Fbb j + eq jp F,bbi + 2δi j eqpn F,bbn
∗j
+ [ni (x) − ni (ξξ)]σi j (x)}na(x)σab (x,ξξ) dS
+
or
∂Ω
ξ)
Σ∗i jpq (x,ξξ) = Ei jkl enl p σ∗k
nq (x,ξ
2υ
δi j δkl + δik δ jl + δil δ jk ) ·
= µ2 (
1 − 2υ
∗
∗
∗
∗
+ δkq F,bbn
) + υδnq F,bbk
− F,knq
]
enl p [(1 − υ)(δknF,bbq
[D p uq (ξξ) − (D p uq )(x)]na(x)Σ∗abpq(x,ξξ)
− δiq e jpn F,bbn − δ jq eipn F,bbn )]
dS (87a)
= µ2 [(einp F, jqn − einp δ jq F,bbn + eint etqk e jpm F,kmn )
+ υ(einq δ jp F,bbn + e jnq δip F,bbn )]
(89)
c 2003 Tech Science Press
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CMES, vol.4, no.1, pp.5-20, 2003
We also have the divergence of Σ ∗i jpq as:
We have Hi∗jpq , by definition, as
Σ∗i jpq,i (x,ξξ)
Hi∗jpq (x,ξξ) = µ2 [(δip F, jq − δip δ jq F,bb + eiqk e jpm F,km)
= µ2 [(einp F, jqni − einp δ jq F,bbni + eint etqk e jpm F,kmni )
+ υ(einq δ jp F,bbni + e jnq δip F,bbni )]
+ υ(δiq δ jp F,bb + eipm e jmq F,bb )]
= µ2 υe jnq F,bbnp
= µ2 [−δi j F,pq + 2δip F, jq + 2δ jq F,ip − δ pq F,i j − 2δip δ jq F,bb
≡ Λ∗i jpq,i (x,ξξ)
(90)
where, by definition,
(91) e e F = (−δ δ + δ δ )F
ipm jmq ,bb
i j pq
iq p j ,bb
− δip F, jq + δi j F,pq + eiqk e jpm F,km
We observe that:
∂Ω
D p uq (ξξ)Λ∗i jpq (x,ξξ)dS
=−
=−
∂Ω
∂Ω
(95)
in which, the following results are used:
Λ∗i jpq (x,ξξ) = µ2 υe jiq F,bbp
+ 2υδiq δ jp F,bb + (1 − υ)δi j δ pq F,bb )]
(96a)
= +δip F, jq + δ jq F,ip − δi j F,pq − δ pq F,i j
+ δi j δ pq F,bb − δip δ jq F,bb
uq (ξξ)D p Λ∗i jpq (x,ξξ)dS
uq (ξξ)D p [µ2 υe jiq F,bb ],p dS = 0
(96b)
(92) With Eqs. (21a) and (32), we can write H i∗jpq for 3D problems as:
Then we have
∂Ω
D p uq (ξξ)Σ∗i jpq (x,ξξ) dS
=
≡
∂Ω
∂Ω
Hi∗jpq (x,ξξ) =
D p uq (ξξ)[Σ∗i jpq (x,ξξ) − Λ∗i jpq (x,ξξ)] dS
D p uq (ξξ)K∗i jpq (x,ξξ)
dS
µ
8π(1 − υ)r
[4υδiq δ jp − δip δ jq − 2υδi j δ pq
(93)
+ δi j r,p r,q + δ pq r,i r, j − 2δip r, j r,q − δ jq r,i r,p ]
where, by definition
and with Eq. (33) for 2D plain strain problems as:
K∗i jpq(x,ξξ) = Σ∗i jpq (x,ξξ) − Λ∗i jpq (x,ξξ)
µ
4π(1 − υ)
[−4υ ln rδiq δ jp + ln rδip δ jq + 2υ ln rδi j δ pq
= µ2 [(einp F, jqn − einp δ jq F,bbn + eint etqk e jpm F,kmn )
+ υ(einq δ jp F,bbn + e jnq δip F,bbn − e jiq F,bbp )]
= µ2 eint [(δt p F, jq − δt p δ jq F,bb + etqk e jpm F,km )
+ υ(δtq δ jp F,bb + et pm e jmq F,bb )],n
≡ eint Ht∗jpq,n (x,ξξ)
(97)
Hi∗jpq (x,ξξ) =
+ δi j r,p r,q + δ pq r,i r, j − 2δip r, j r,q − δ jq r,i r,p ]
(98)
From Eqs. (90) and (94), some properties of the kernel
functions can be found as following:
Λ∗ (x,ξξ)
Σ∗ (x,ξξ) = ∇ ·Λ
∇ ·Σ
∇ · K∗ (x,ξξ) = 0
∗
∗
(94) K (x,ξξ) = ∇ × H (x,ξξ)
(99a)
(99b)
(99c)
17
On Simple Formulations of Weakly-Singular tBIE & dBIE, and Petrov-Galerkin Approaches
We may take wb (x) to be any continuous function in
Eq. (88), and derive a Petrov-Galerkin boundary element
method. If wb (x) is continuous, one may use Stokes’
theorem, and write:
−
∂Ω
=
+
If the test function w b (x) is chosen to be identical to
a function that is energy-conjugate to t b , namely, the
trial function û b (x), we generate the symmetric Galerkin
BEM as [Han and Atluri (2002); Nikishkov, Park,
Atluri(2001)]:
wb (x)tb(x)dSx
∂Ω
∂Ω
wb (x)dSx
wb (x)dSx
∗q
∂Ω
∂Ω
−
∗q
tq (ξξ)na (x)[Ψab(x,ξξ) − φab (x,ξξ)]dSξ
1
2
∂Ω
=
D p uq (ξξ)na (x)K∗abpq (x,ξξ) dSξ (100)
tb (x)ûb(x)dSx
∂Ω
−
in which, Eqs. (76) and (93) are applied.
+
With the fact that
Da ûb (x)dSx
∂Ω
∂Ω
tq (ξξ) dSξ
∗q
∂Ω
tq (ξξ)Gab (x,ξξ) dSξ
CPV
∂Ω
Da ûb (x)dSx
∗q
na (x)ûb (ξξ)φab(x,ξξ)dSx
∂Ω
∗
D p uq (ξξ)Habpq
(x,ξξ) dSξ
(104)
∂
∂
=− ,
∂xi
∂ξi
(101)
The results in Eq. (103) are similar to those reported in
[Li and Mear (1998)] but are different from those in [Li
and with Eqs. (69) and (94), one may rewrite Eq. (100) and Mear (1998)] in the kernel functions appearing in Eq.
(104). However, here, we obtain these results in a very
as:
straightforward and simple manner.
1
−
2
∂Ω
=−
−
−
tb (x)wb (x)dSx
∂Ω
∂Ω
∂Ω
wb (x)dSx
tq (ξξ ) dSξ
wb (x)dSx
7 Regularization of dBIE
∂Ω
CPV
∂Ω
∂Ω
ξ) dSξ
tq (ξξ)Da G∗q
ab (x,ξ
∗q
na (x)wb (x)φab(x,ξξ)dSx
∗
D p uq (ξξ)Da Habpq
(x,ξξ) dSξ
∗q
(102)
We subtract Eq. (63d) from Eq. (52), and obtain,
∗q
where Gab is defined in Eq.(31); φ ab is defined in Eq.
∗
is defined in Eq. (95).
(28a), and Habpq
As wb (x) is continuous, one may use Stokes’ theorem,
and re-write Eq. (102) as:
−
1
tb (x)wb (x)dSx
2 ∂Ω
=
Da wb (x)dSx
−
+
∂Ω
∂Ω
∂Ω
tq (ξξ ) dSξ
In this section, we consider the regularization of the displacement BIE (52), in order to render it tractable for numerical implementation. We also consider the possibility
of satisfying the dBIE, at ∂Ω, in a weak form, through a
general Petrov-Galerkin scheme.
∗q
∂Ω
CPV
∂Ω
Da wb (x)dSx
tq (ξξ)Gab(x,ξξ) dSξ
∗q
na (x)wb (x)φab(x,ξξ)dSx
∂Ω
∗
D p uq (ξξ)Habpq
(x,ξξ) dSξ
0=
−
∂Ω
ξ) dS
t j (ξξ)u∗p
j (x,ξ
∂Ω
ξ) dS
ni (ξξ)[u j (ξξ) − u j (x)]σ∗p
i j (x,ξ
(105)
We know that u j (ξξ) − u j (x) becomes O(r) when ξ → x
and Eq. (105) becomes weakly singular. Then it can be
evaluated numerically, and applicable to point x on the
boundary ∂Ω. Eq. (105) is the well-known regularized
dBIE equation.
We can also use a Petrov-Galerkin scheme to write a
(103) weak-form for Eq. (105) as:
c 2003 Tech Science Press
Copyright 18
0=
∂Ω
w p (x)dSx
∂Ω
∗p
t j (ξξ)u j (x,ξξ) dS
CMES, vol.4, no.1, pp.5-20, 2003
8 Evaluation of displacements and stresses near the
surface
(106)
After obtaining the displacements and the stresses on the
boundary, we sometimes also need to evaluate the disw p (x)dSx
−
∂Ω
∂Ω
placements and stresses inside the domain. It is well
where w p (x) is a test function. If w p (x) is chosen as known that we need to evaluate the strongly singular ina Dirac delta function, i.e. w p (x) = δ(x, xm) at ∂Ω, we tegrals if Eqs. (52) and (53) are applied directly, if the
obtain the standard “collocation” displacement boundary point approaches the boundary.
Consider a generic domain point x ∈ Ω which is close to
element method.
∗p
∗p
∗p
∗p a boundary point x̂. By replacing the displacements in
Using (28b), i.e., σi j = ψi j (x,ξξ) − φi j (x,ξξ) [with Ψi j
Eq. (63d) with u p (x̂), Eq. (52) is re-written as,
∗p
is defined in Eq.(30). and φ i j is defined in Eq. (28a)], we
ξ) dS
ni (ξξ)[u j (ξξ) − u j (x)]σ∗p
i j (x,ξ
re-write Eq. (106), with Eq. (69), as:
u p (x) =
1
w p (x)u j (x)dSx =
2 ∂Ω
∗p
w p (x)dSx
t j (ξξ)u j (x,ξξ) dSξ
∂Ω
−
+
∂Ω
∂Ω
w p (x)dSx
w p (x)dSx
∂Ω
∗p
ni (ξξ)u j (ξξ)ψi j (x,ξξ) dSξ
CPV
∂Ω
∗p
ni (ξξ)u j (ξξ)φi j (x,ξξ) dSξ
Applying Stokes’ theorem to Eq. (107), we have
∂Ω
∂Ω
∗p
t j (ξξ)u j (x,ξξ) dS
ξ) dS + u p (x̂)
[um(ξξ) − um (x̂)]σ∗p
nm (x,ξ
(110)
We know that u m (ξξ) − um(x̂) becomes O(r) when ξ → x̂
and Eq. (110) becomes weakly singular. Then it can be
evaluated numerically.
∂Ω
−
(107) Eq. (53) is used for stress calculation. As a part of the solution, we have the known displacements and tractions at
point x̂ as u j (x̂) and t j (x̂), respectively. We first calculate
the stress σab (x̂) at point x̂ from Eq. (86).
From Eq. (76), it is ready to re-write Eq. (57) as,
1
w p (x)u p (x)dSx =
2 ∂Ω
∗p
w p (x)dSx
t j (ξξ)u j (x,ξξ) dSξ
∂Ω
+
+
∂Ω
− σab (x) =
∂Ω
∂Ω
w p (x)dSx
w p (ξξ)dSx
∗p
Di (ξξ)u j (ξξ)Gi j (x,ξξ) dSξ
∂Ω
CPV
∂Ω
∗p
ni (ξξ)u j (ξξ)φi j (x,ξξ) dSξ
+
∂Ω
∂Ω
ξ) dS
[tq (ξξ) − n p (ξξ)σ pq (x̂)]σ∗q
ab (x,ξ
[D p uq (ξξ) − (D p uq )(x̂)]Σ∗abpq (x,ξξ) dS − σab (x̂)
(111)
(108)
Again, we know that the singularities of terms
∗p
where Gi j is defined in Eq.(31)
tq(ξξ) −n p (ξξ)σ pq (x̂) and D p uq (ξξ) − (D p uq)(x̂) become
If w p (x) is chosen to be identical to a function which is O(r) when ξ → x̂. Then Eq. (111) becomes weakly sinenergy-conjugate to u p , viz., the trial function tˆp (x), we gular and can be evaluated numerically.
obtain the symmetric Galerkin dBEM, as
9 Closure
1
2
∗p
t j (ξξ)u j (x,ξξ) dSξ 1. We have presented simple and straight-forward for∂Ω
∂Ω
∂Ω
mulations for weakly-singular traction as well as dis
∗p
placement integral equations in a linear elastic solid un+
Di (ξξ)u j (ξξ)Gi j (x,ξξ) dSξ
tˆp (x)dSx
∂Ω
∂Ω
dergoing small displacements. Clearly, these formula
CPV
∗p
+
ni (ξξ)u j (ξξ)φi j (x,ξξ) dSξ
(109) tions can be extended to finite elasticity, large-strains,
tˆp (x)dSx
∂Ω
∂Ω
and rate-formulations of elastic-plastic solids undergoing
tˆp (x)u p (x)dSx =
tˆp (x)dSx
On Simple Formulations of Weakly-Singular tBIE & dBIE, and Petrov-Galerkin Approaches
large deformations, using the methodologies presented in
Okada, Rajiyah, and Atluri (1989b).
19
ing & Sciences, vol. 3 no. 1,pp. 11-52
Atluri, S. N.; Zhu, T. (1998): A new meshless local
2. The traditional traction boundary element, or dis- Petrov-Galerkin (MLPG) approach in computational meplacement boundary element methods can be derived chanics. Comput. Mech., vol. 22, pp.117-127.
from Eqs. (102) and (108), respectively. In these Bonnet, M.; Maier, G.; Polizzotto, C. (1998): Symmethods, one uses a “mesh” at ∂Ω, which consists of metric Galerkin boundary element methods, Appl. Mech.
a set of contiguous (non-overlapping) “elements”. In Rev., vol. 51, pp. 669-704.
general, the trial functions u b , tb , and the test functions
Cruse, T. A.; Richardson, J. D. (1996): Non-singular
wb at ∂Ω are interpolated in terms of their respective
Somigliana stress identities in elasticity, Int. J. Numer.
values at the nodes of the boundary elements. On
Meth. Engng., vol. 39, pp. 3273-3304.
the other hand, using the concepts of the meshless
local Petrov-Galerkin methods (MLPG) developed in Frangi, A.; Novati, G. (1996): Symmetric BE method in
Atluri et al [1998, 2002a,b], one may develop “mesh- two dimensional elasticity: evaluation of double integrals
less local Petrov-Galerkin boundary integral equation for curved elements, Comp. Mech. Vol. 19, pp. 58-68.
approaches”. Also, Eqs. (102) and (108) in general Fung, Y. C; Tong, P.(2001): Classical and Computainvolve double integrals over ∂Ω on their right-hand tional Solid Mechanics, World Scientific, 930 pages.
sides, where ∂Ω is the entire global boundary. However, Han. Z. D.; Atluri, S. N. (2002): SGBEM (for
in using MLPG, the second integral over ∂Ω in Eqs. Cracked Local Subdomain) – FEM (for uncracked global
(102) and (108) may be replaced by a local integral over Structure) Alternating Method for Analyzing 3D Surface
a sub-region of ∂Ω only. In evaluating the first integral Cracks and Their Fatigue-Growth. CMES: Computer
over the global ∂Ω, one may use “shadow-elements”, Modeling in Engineering & Sciences,vol. 3, no. 6, pp
or alternatively, one may also develop a truly meshless 699-716.
MLPG method for integral equations. They are the Han. Z. D.; Atluri, S. N. (2003): Truly Meshless Losubjects of our forthcoming papers [Atluri, Han, and cal Patrov-Galerkin (MLPG) solutions of traction & disShen (2003); Han and Atluri (2003)].
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Acknowledgements: This paper is based on research Li, S.; Mear, M.E (1998): Singularity-reduced inteperformed under the support of ARO, ONR, and NASA. gral equations for displacement discontinuities in threeThe authors gratefully acknowledge this support.
dimensional linear elastic media, Int. J. Fracture, vol.
93, pp. 87-114.
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CMES, vol.4, no.1, pp.5-20, 2003