International Applied Mechanics, Vol. 38, No. 9, 2002
To the Beginning of the Third Millennium
APPLICATION OF THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD
TO BOUNDARY-VALUE PROBLEMS
J. Sladek1, V. Sladek1, and S. N. Atluri2
UDC 539.3
A review of the meshless formulations based on local boundary integral equation (LBIE) methods is
presented. Physical quantities are approximated by the moving least-squares method. A summary of
recent developments in the application of the LBIE method to potential problems, elastostatics,
elastodynamics, thermoelasticity, and plate bending problems is given. The efficiency and generality of
the present formulation in a wide class of engineering problems are confirmed.
1. Introduction. The development of approximate methods for the numerical solution of boundary-value problems has
attracted the attention of engineers, physicists, and mathematicians for a long time. The finite-element method, used to model
complex problems in applied mechanics and related fields, is well established. It is a robust and thoroughly developed technique,
which is not without shortcomings, however. It is well known that the finite-element method relies on mesh discretization, which
complicates certain classes of problems. Loss of accuracy occurs when elements in the mesh become extremely skewed or
distorted. The boundary-element method has become an efficient and popular alternative to the finite-element method,
especially for stress-concentration problems, or for boundary-value problems wherein a part of the boundary extends to infinity.
In spite of the great success of the finite- and boundary-element methods as the most efficient numerical tools for the solution of
boundary-value problems in complex domains, there has been a growing interest in the so-called meshless methods over the past
decade. A number of methods have been proposed so far, including the smooth particle hydrodynamics (SPH) method [1, 2], the
diffuse-element (DE) method [3], the element-free Galerkin (EFG) method [4–8], the reproducing kernel particle (RKP) method
[9, 10], the moving least-squares reproducing kernel (MLSRK) method [11, 12], the partition-of-unity finite-element method
(PUFE) method [13–16], the hp-clouds method [19, 20], the finite-point method [21–24], the meshless local Petrov–Galerkin
(MLPG) method [25–32], the boundary-node (BN) method [33–37], the local boundary integral equation (LBIE) method
[38–47], and the method of finite spheres (which is a special case of the MLPG method where the local subdomains are chosen to
be spheres in 3-D) [48, 49]. The coupling of conventional and meshless computational methods to achieve solution efficiency
and accuracy is discussed in [7, 50–53]. Pardo [54] has given a quite different meshless formulation for linear elastostatics based
on a path integral, similar to the well-known Feynman path integrals in quantum mechanics. Belytschko et al [55], and later
Atluri and Shen [89], were the first to attempt to analyze the state of the art of this growing family of numerical procedures.
Meshless methods originated from the finite-difference (FD), finite-element (FE), and boundary-element (BE) methods, but,
contrastingly, they can treat an irregular distribution of points and require no costly mesh generation. In addition, since the
meshless methods use a functional basis and allow arbitrary placement of points, the solution and its derivatives may be found
directly where they are needed and more accurately compared with the FD, FE, and BE methods, which require differences and
interpolation. The term “meshless,” or “meshfree,” stems from the possibility of constructing an approximation or interpolation
scheme entirely from a set of nodes. The automatic generation of good-quality meshes required in conventional computational
methods presents significant difficulties in the analysis of engineering systems. These difficulties disappear when there is no
mesh. Therefore, meshless methods are very convenient for analyzing crack propagation [56–59] and large deformation
1Institute
of Construction and Architecture, Slovak Academy of Sciences, Bratislava, Slovakia. 2Department of
Mechanical and Aerospace Engineering, University of California at Irvine, USA. Translated from Prikladnaya Mekhanika,
Vol. 38, No. 9, pp. 3–26, September 2002. Original article submitted April 26, 2002.
1063-7095/02/3809-1025$27.00 ©2002 Plenum Publishing Corporation
1025
processes [10, 60–64], where remeshing at each step of the solution becomes prohibitively expensive in conventional
approaches.
The majority of meshless formulations are associated with the moving least-square (MLS) approximation of physical
quantities. One of the difficulties with the MLS interpolants is that they, generally, lack the delta function property of the usual
BE or FE method shape functions. This complicates the imposition of essential boundary conditions in the EFG method.
Improved EFG formulations are given in [65–68], which allow the direct imposition of essential boundary conditions.
Some of the above-mentioned meshless methods, namely the RKP, SPH, DE, and EFG methods, are not truly meshless.
All these “pseudo-meshless” methods use integration cells. Only the finite-point method, the MLPG method (and its special
variation, the so-called method of finite spheres), and the LBIE method are truly meshless. Further we will focus on these, truly
meshless methods. In the finite-point method, the weighted least-square interpolants are satisfied at collocation points. However,
methods based on point collocation are sensitive to the choice of proper collocation points. The MLPG methods are based on a
weak form computed over a local subdomain, which can be of any simple geometry such as a sphere in 3-D (and hence the name
the method of finite spheres) or a circle in 2-D for ease of integration. The trial and test function spaces can be either different or
the same. A method that employs a similar approach with boundary integral equations is the local boundary integral equation
method.
However, although considerable efforts have been made to develop meshless methods, the currently available
techniques are still computationally much less efficient than the well-established conventional discretization procedures. The
primary reason is that nonpolynomial shape functions are employed and the required numerical integration is very difficult to
perform accurately. Some progress has been done in [69, 70] in improving the accuracy of integration and performing a unified
stability analysis of meshless methods with Eulerian and Lagrangian kernels. The reliability of simulations is achieved by means
of self-adaptive techniques that refine cells, node density, and nodal influence regions [71, 72]. It is worth mentioning that the
cells are used just for integration and impose no restriction on shape or compatibility. This feature makes the meshless methods
especially suited for self-adaptive techniques.
In this review paper, we will briefly outline applications of the LBIE method to many engineering problems. The LBIE
method seems to be very promising for solving many boundary-value problems. It incorporates the well-known advantages of
the BE method and the generality in the use of the FE method. This can be demonstrated by an example of a continuously
nonhomogeneous elastic body. The fundamental solution to the governing equation of such a problem is not available in the
general case. If the Kelvin fundamental solution for a homogeneous material is used for the whole domain, a global
boundary-domain integral formulation can be derived. However, it brings some computational difficulties into the numerical
implementation. Alternatively, the local boundary integral equations apply to each subdomain related to nodes in the domain
being analyzed and used for the moving least-square approximation of displacements. Then, for each subdomain, we can use the
fundamental solution corresponding to the material constant for the subdomain in question. The LBIE method combined with the
MLS approximation has no restrictions due to missing fundamental solutions and, therefore, it can be applied to the general
boundary-value problem.
Local boundary integral equations supplemented with moving least-square interpolants lead to a reliable, effective
computational method with a wide application in many engineering fields. It results from the basic idea of the method. Nodal
points are randomly spread over the domain occupied by the body in question. Every node is surrounded by a simple surface (a
circle in 2-D or a sphere in 3-D) centered at the collocation point. Only one nodal point is included in the subdomain. Local
boundary equations are written for the surface of subdomains. In this case, the number of equations is equal to that of nodes. In
the MLS approximation, physical quantities are expressed in terms of certain shape functions and nonphysical unknowns
defined at some randomly spread nodes. Contrary to the classical computational (FE and BE) methods, the coupling on a global
scale is established by approximation. The local boundary integral equations provide local constraint equations in assembling the
global stiffness matrix. Using a modified test function, which vanishes on the circular part of the subdomain boundary, it is
possible to eliminate the derivatives of primary fields (traction vector, flux, etc.) completely if the LBIEs are formulated for the
interior nodes. They are eliminated partially if the LBIEs are formulated for the nodal points on the global boundary. However,
on the global boundary, half the boundary quantities are due to the boundary conditions in a well-posed boundary-value problem.
Therefore, the formulation based on the LBIEs and the MLS approximation is unique for the general boundary-value problem.
The present paper summarizes the recent developments in the LBIE application to potential problems, elastostatics for
homogeneous and nonhomogeneous bodies, elastodynamics, thermoelasticity, and plate bending problems. The local boundary
1026
uh (x )
ui
u$i
u
boundary node
x1
x2
xi
x
Fig. 1
integral equations employ the MLS approximation of physical quantities. In the next section, we will discuss important
equations of the MLS method.
2. MLS Approximation Scheme. In general, a meshless method uses local interpolation to represent the trial function
with values (or fictitious values) of an unknown variable at some randomly spread nodes. Among such schemes is the moving
least-square approximation, which will be described here. Consider a subdomain Ω x that is the neighborhood of a point x and the
domain of the MLS approximation for the trial function at x, which is located in the problem domain Ω. To approximate the
distribution of the function u in Ω over a number of randomly located nodes {x a }, a = 1, 2,... , n, the MLS approximant u h ( x ) of u,
∀ x ∈ Ω x , is defined by
u h ( x ) = pT ( x )a( x )
∀ x ∈Ω x ,
(1)
where pT ( x ) = [ p1 ( x ), p 2 ( x ),... , p m ( x )] is a complete monomial basis of order m, and a( x ) is a vector with coefficients
a j ( x ), j = 1, 2, ..., m, which are functions of the space coordinates x = [ x1 , x 2 , x 3 ]T . For example, linear basis m = 3 for
pT ( x ) = [1, x1 , x 2 ]
(2a)
pT ( x ) = [1, x1 , x 2 , ( x1 ) 2 , x1 x 2 , ( x 2 ) 2 ] .
(2b)
and quadratic basis m = 6 for
The vector a( x ) is determined by minimizing a weighted discrete L2 norm defined as
n
[
]
J ( x ) = ∑ w a ( x ) pT ( x a )a( x ) − u$ a ,
a =1
(3)
where w a ( x ) is a weight function associated with the node a, with w a ( x ) > 0 . Recall that n is the number of nodes in Ω x for
which the weight functions w a ( x ) > 0 and u$ a are fictitious nodal values and are not nodal values of the unknown trial function
u h ( x ) in general (the difference between u i and u$1 in the moving least-square approximation is shown in Fig. 1 for a simple
one-dimensional case). The stationarity of J in Eq. (3) with respect to a( x ) leads to the following linear relation between a( x )
$
and u:
A( x )a( x ) = B ( x )u$ ,
(4)
where
n
A( x ) = ∑ w a ( x ) p( x a ) pT ( x a ),
a =1
1027
local boundary ∂Ω s = Ω ′s = Ls
subdomain Ω s = Ω ′s
ri
Ωx
X
node zi
Ls
∂Ω s′ = Ls ∪ Γs
Γs
support of node zi
∂Ω s′
Ω s′
Fig. 2
[
]
B ( x ) = w1 ( x ) p( x1 ), w 2 ( x ) p( x 2 ),... , w n ( x ) p( x n ) .
(5)
The MLS approximation is well defined only when the matrix A in Eq. (4) is nonsingular. A necessary condition to be
satisfied is that at least m weight functions must be nonzero (i.e., n ≥ m) for each sample point x ∈Ω and that the nodes in Ω x must
not be arranged in a special pattern such as along a straight line.
Solving Eq. (4) for a( x ) and substituting it into Eq. (1), we obtain the relation
n
u h ( x ) = ΦT ( x )u$ = ∑ φa ( x )$u a ,
u$ a ≠ u( x a ) ∧ u$ a ≠ u h ( x a ),
(6)
a =1
where
ΦT ( x ) = pT ( x ) A −1 ( x )B ( x )
(7)
and φa ( x ) is usually called the shape function of the MLS approximation corresponding to the nodal point x a .From Eqs. (5) and
(7), it may be seen that φa ( x ) = 0 when w a ( x ) = 0 . In practical applications, w a ( x ) is generally chosen to be nonzero over the
support of the nodal points x a . The support of the nodal point xa is usually taken to be a circle of radius r a centered at x a (see
Fig. 2). This radius is an important parameter of the MLS approximation because it determines the range of interaction
(coupling) between the degrees of freedom defined at nodes.
A numerical analysis may employ both Gaussian and spline weight functions with compact supports [38–47]. The
Gaussian weight function can be written as
[
]
[
]
 exp −( d a / c a ) 2 − exp −( r a / c a ) 2 ,
wa ( x) = 
 0, d a ≥ r a ,
0 ≤ d a ≤ ra ,
(8)
where d a =| x − x a | , c a is a constant controlling the shape of the weight function w a , and r a is the size of the support. This size
should be large enough to have a sufficient number of nodes in the domain of definition to ensure that the matrix A is regular. The
partial derivatives of the MLS shape functions are as follows [4]:
m
[
]
φ,ak = ∑ p ,jk ( A −1 B ) ja + p j ( A −1 B ,k + A,−k1 B ) ja ,
j =1
where A,−k1 = ( A −1 ) ,k is the derivative of the inverse of A with respect to x k , given by
1028
(9)
A,−k1 = − A −1 A,k A −1 .
To avoid discontinuities in the shape functions due to the presence of cracks, the diffraction method [73,74] can be used
to modify d a in the weight function.
3. Local Boundary Integral Formulation for Potential Problems.
3.1. Poisson Equation. Although the present approach is fully general as applied to nonlinear boundary-value
problems, we will use only the linear Poisson equation to demonstrate the formulation. A nonlinear boundary-value problem was
analyzed in [75]. The Poisson equation can be written as
∇ 2 u( x ) = p( x ),
x ∈Ω ,
(10)
where p( x ) is a given source function, and the domain Q is enclosed by Γ = Γu ∪ Γq , with the boundary conditions
u=u
Γu ,
on
∂u
≡q=q
∂n
on
(11)
Γq ,
(12)
where u and q are prescribed potential and normal fluxes, respectively, on the essential boundary Γu and on the flux boundary
Γq , respectively, and n is the outward normal vector. A weak formulation of the problem may be written as
∫ u * (∇ 2 u − p) dΩ = 0,
(13)
Ω
where u * is the test function and u is the trial function. If the test function that satisfies the equation
∇u * ( x, y ) + δ( x, y ) = 0,
(14)
where δ( x, y ) is the Dirac delta function, is used, then the following integral representation can be obtained:
u( y ) = ∫ u * ( x, y )
Γ
∂u( x )
∂u * ( x, y )
dΓ − ∫ u( x )
dΓ − ∫ u * ( x, y ) p( x )dΩ.
∂n
∂n
Γ
(15)
Ω
If, instead of the entire domain Ω, we consider a subdomain Ω s located entirely inside Ω, then the following equation
must hold on the subdomain Ω s :
u( y ) =
∫ u * ( x, y )
∂Ω s
∂u( x )
dΓ −
∂n
∫ u( x )
∂Ω s
∂u * ( x, y )
dΓ −
∂n
∫ u * ( x, y ) p(x )dΩ ,
(16)
Ωs
where ∂Ω s is the boundary of the subdomain Ω s .
In the original boundary-value problem, either a potential or a flux may be specified at each point of the global boundary
Γ; therefore, the problem is well-posed. However, none of them is known a priori along the local boundary ∂Ω s .To eliminate the
flux variable from the integral representation (16), Atluri and his co-workers [38] introduced a companion solution into the
fundamental solution in such a way that the final modified fundamental solution became zero on the circular boundary ∂Ω s . The
modified fundamental solution to the Laplace equation can easily be derived and for 2-D problems is given by
r
1
u~ * ( x, y ) = ln 0 ,
2π
r
(17)
where r =| x − y| and r0 is the radius of the local subdomain Ω s . Considering that
u~ * ( r )
r = r0
= 0,
(18)
we can rewrite Eq. (16) as
1029
u( y ) = −
∫ u( x )
∂Ω s
∂u~ *
( x, y )dΓ −
∂n
∫ u * ( xy ) p( x )dΩ
(19)
Ωs
for the source point located inside Ω.
When the source point y is located on the global boundary Γ, the subdomain can still be taken as a part of the circular
domain whose boundary consists of a circular part Ls and a part of the boundary line Γs on which a nodal point is located
( ∂Ω s = Ls ∪ Γs ). It should be noted that the modified fundamental solution u~ * does not vanish along the line Γs . Then, the local
boundary integral equation takes the following form for the nodes ζ ∈ Γs ⊂ Γ:
u( ζ) + ∫ u( x )
Ls
∂u~ *
( x, ζ )dΓ + lim
y→ ζ
∂n
∫ u( x )
Γs
∂u~ *
( x, y )dΓ − ∫ u~ * ( x , ζ )q( x )dΓ = − ∫ u~ * ( x, ζ ) p( x )dΩ.
∂n
(20)
Ωs
Γs
Although LBIE (20) may be reduced to a nonsingular form, it is more appropriate to use this limit form because the
MLS approximation involves numerical integration [40]. Let Γs = Γsu ∪ Γst , where Γsu and Γst are the finite parts of Γs on
which a potential or a flux, respectively, is defined. When the MLS approximation is used, the discretized LBIEs (19) and (20)
collocated at y b ∈Ω and ζ b ∈ Γst become


∂u~ *
 a b
b )φa ( x )dΓ  u$ a = 0,
φ
(
y
)
+
(
x
,
y

∑
∫ ∂n

a =1 
∂Ω s

n


n
n

∂u~ *

( x, ζ b )φa ( x )dΓ  u$ a + ∑
∂n
a =1
Ls

∑  φa ( ζ b ) + ∫

a =1 
n
−∑
lim
y→ ζ b
∫
Γst
∂u~ *
( x, y b )φa ( x )dΓu$ a
∂n
∫ u * ( x, ζb )[n1 φ,a1 ( x ) + n 2 φ,a2 ( x )] dΓu$ a = ∫ u * ( x, ζb )q ( x )dΓ − ∫
~
~
a =1 Γsu
(21)
Γst
Γsu
∂u~ *
( x, ζ b )u ( x )dΓ.
∂n
(22)
The set of algebraic equations (21) and (22) should be supplemented with the equations
n
∑ φa ( ζb )$u a = u ( ζb ),
a =1
ζ b ∈ Γsu ,
(23)
to form a complete set for computation of the fictitious unknowns u$ a at all nodal points.
Many numerical results are given in [38] to illustrate the implementation and convergence of the LBIE approach.
Numerical analyses revealed that the accuracy is best when the ratio r a / c a = 4 in the Gaussian weight function. The
well-known problem described by the Poisson equation is a stationary temperature distribution in a solid body. Let us consider
an example of how to compute the radial temperature distribution in a hollow cylinder subjected to a thermal gradient. Different,
yet constant temperatures are specified on the internal and external surfaces of the cylinder (Fig. 3). Owing to the symmetry of
the problem, it is sufficient to analyze only a quarter of the cylinder. Analytically, the temperature distribution is given by
θ ( r ) = θ1 +
∆θ
ln r * ,
ln R 2 / R1
where r * = r / R1 and ∆θ = θ 2 − θ1 .
The numerical analysis involved a cylinder with radii R1 = 8 and R2 = 10 and 40 boundary nodes and 56 additional
interior nodes. The radii of the subdomains are constant on both artificial cuts (due to symmetry), r a = rloc = 0.39. On the
remaining (circular) part of the boundary, the subdomains are of constant size too, with rloc = 0.49. Also c a = 1.5 and r a / c a = 4.
The numerical LBIE and analytical results for the temperature and temperature gradient are compared in Fig. 4. As is seen, they
1030
21
θ, θ,1
R2
u1 = 0
θ =1
t1 , t2 = 0
t2 = 0
analytical temperature
analytical θ,1
LBIE results for θ
LBIE for θ,1
0.8
q=0
0.6
26
0.4
R1
x2
θ=0
0.2
40
x1
0
1
u2 = 0, t1 = 0, q = 0
6
8.0
8.5
Fig. 3
9.0
9.5
10.0 r
Fig. 4
are in excellent agreement. Over the whole radius, the relative errors of the temperature and the temperature gradient are lower
than 0.001% and 0.05%, respectively.
3.2. Helmholtz Equation. The Helmholtz equation is frequently encountered in various fields of engineering and
physics. It is used for analyzing acoustics, wave diffraction, vibration of membranes, electromagnetic fields, etc. The Helmholtz
equation has the form
∆u( x ) + k 2 u( x ) = 0,
(24)
where u( x ) is the potential function and k is the wave number, k = ω/ c, with ωand c being the angular frequency and the velocity
of wave propagation, respectively.
Since the Helmholtz equation is self-adjoint, we can derive a boundary integral equation by the weighted residual
method. The weak formulation of Eq. (24) can be expressed as
∫ u * ( x )(∇ 2 + k 2 )u( x )dΩ = 0,
(25)
ω
where u * ( x ) is the weight field (test function). Applying the Gauss divergence theorem to the integral in Eq. (25), we obtain
∂u
∂u
∫ (u ,ii + k 2 u )u * ( x )dΩ = ∫ ∂n ( x )u * ( x )dΓ − ∫ u * ( x ) ∂n ( x )dΓ + ∫ (u ,*ii + k 2 u * )u( x )dΩ.
Ω
Γ
Γ
(26)
Ω
If the test function satisfies the governing equation
u ,*ii ( x, y ) + k 2 u * ( xy ) = −δ( x, y ),
(27)
then directly from Eq. (26) we can obtain the integral representation of the potential function u( y ) governed by the Helmholtz
equation
u( y ) = ∫
Γ
∂u
∂u *
(η )u * (η − y )dΓ − ∫ u(η )
(η, y )dΓ ,
∂n
∂n
(28)
Γ
where the fundamental solution for 2-D problems is given by
u * (η − y ) =
1
K 0 ( ikr ),
2π
r =|η − y|.
(29)
1031
In Eq. (29), K 0 ( ikr ) is a modified Bessel function of the second kind and zero order. Using the properties of Bessel
functions, we can write
∂u *
1
(η, y ) = − r, j n j ikK 1 ( ikr ).
∂n
2π
(30)
The global boundary integral equation (28) is identical to BIE (15) valid for problems described by the Poisson
equation. Therefore, LBIEs are the same for problems governed by the Poisson and/or Helmholtz equations. Only their
fundamental solutions are different. To avoid a flux problem, we have introduced a modified fundamental solution in the
previous section. A modified fundamental solution can easily be derived for the Poisson equation if the subdomain is a circle. For
the Helmholtz equation, however, we failed to find a modified fundamental solution. Thus, both potential and normal derivatives
(flux) occur on the boundary of subdomain ∂Ω s . In the MLS approximation, both the potential and the flux can be approximated
using a single fictitious parameter (approximant). The discretized equations collocated at y b ∈Ω and ζ b ∈ Γst are
n


∑  φa ( y b ) + ∫
a =1 

∂Ω s
∂u *
( x, y b )φa ( x )dΓ −
∂n

* ( x , y b )φa ( x )n dΓ  u$ a = 0,
u

k
∫
,k

∂Ω s
(31)


∂u *
 a b
b )φa ( x )dΓ − u * ( x, ζ b )φa ( x )n dΓ  u$ a
x
φ
(
ζ
)
+
(
,
ζ

∑
k
∫ ∂n
∫
,k

a =1 
Ls
Ls

n
n
+∑
lim
∫
b
a =1 y→ ζ γ st
n
∂u *
( x, y )φa ( x )dΓu$ a − ∑
∂n
a =1
=
∫ u * ( x, ζb )q ( x )dΓ −
Γst
∫ u * ( x, ζb )[n1 φ,a1 ( x ) + n 2 φ,a2 ( x )] dΓu$ a
Γsu
∫
Γst
∂u *
( x, ζ b )u ( x )dΓ.
∂n
(32)
The set of algebraic equations (31) and (32) should be supplemented with the approximation equations (23) at ζ b ∈ Γsu .
If the weight field is selected in the form of the Trefftz function, we obtain a homogeneous Helmholtz equation. Such an
approach was used in [76], where also illustrative examples are given for square and circular patch tests.
4. Elastostatics.
4.1. Homogeneous Body. Consider an isotropic, homogeneous, linearly elastic continuum. Displacements in that
continuum under a static load are described by the Navier equation. The solution of the boundary-value problem for the Navier
equation can be found in an integral form known as the Somigliana identity [77]
u k ( y ) = ∫ [t i (η )U ik (η − y ) − u i (η ) Tik (η, y )]dΓ + ∫ bi ( x )U ik ( x − y)dΩ ,
Γ
(33)
Ω
whereU ik ( x, y ) is the fundamental (Kelvin) solution, Tik ( x, y ) is the corresponding fundamental traction, and bi ( x ) is the body
force vector. The Somigliana identity gives displacements at any internal point in terms of the boundary displacements u i and the
traction vector t i . If, instead of the entire domain Ω of the given problem, we consider a subdomain Ω s located entirely inside it
and containing the point y, then Eq. (33) takes the form
uk ( y ) =
∫ [t i (η )U ik (η − y ) − u i (η )Tik (η, y )]dΓ + ∫ bi ( x )U ik ( x − y )dΩ ,
∂Ω s
(34)
Ωs
where ∂Ω s is the boundary of the subdomain Ω s .
~
To get rid of the unknown tractions on the boundary ∂Ω s , a companion solution U ik is introduced, similarly to linear
potential problems. The companion solution is associated with the fundamental solutionU ik ( x, y ) and is defined as the solution
to the equation
1032
~
σ
ij , j = 0 on Ω ′s ,
~
U ik = U ik
on
∂Ω ′s ,
(35)
where Ω ′s and ∂Ω ′s are the same as those in Fig. 2. As usual, Ω ′s is taken as a circle in the present implementation.
~
* = U −U
The modified test functionU ik
ik
ik must satisfy the governing equation for the fundamental solution. Then, the
* .This fundamental solution is zero on the
integral representation (34) will be valid also for the modified fundamental solutionU ik
circle ∂Ω s because of the second condition in (35). Hence, we can write
uk ( y ) = −
∫ u i (η )Tik* (η, y )dΓη + ∫ bi ( x )U ik* ( x − y )dΩ
∂Ω s
(36)
Ωs
for a source point y located inside Ω and
u k ( ζ ) + ∫ u i (η )Tik* (η, ζ )dΓη + lim
y→ ζ
Ls
* (η, ζ )dΓ =
− ∫ t i (η )U ik
η
Γs
∫ u i (η )Tik* (η, y )dΓη
Γs
∫ bi ( x )U ik* ( x − ζ)dΩ
(37)
Ωs
for a source point located on the global boundary ζ ∈ Γs ⊂ Γ. Note that ∂Ω s = Ls ∪ Γs with Γs = ∂Ω s ∩ Γ.
By introducing the companion solution, we mainly aim at simplifying the formulation and reducing the computational
cost. The unknown traction vector t i (η ) on Γs (Fig. 2) may be considered an independent variable, and a simple approximation
scheme can be used. Thus, both the displacements and tractions on the global boundary may appear in the final algebraic
equations as independent unknown variables. If the direct differentiation of the displacement approximation is used for the
traction vector given by Hooke’s law, then only one unknown (displacement) will appear in the final algebraic equations.
Substituting the MLS approximation (6) for displacements into the traction vector t i , we get
n
t i (η ) = ∑ sija (η )$u aj ,
a =1
2ν


sija (η ) = µ n k (η )φ,ak (η )δ ij + n j (η )φ,ai (η ) +
n i (η )φ,aj (η ) ,
1− 2ν


(38)
 ν, for plane strain

where µ = E / 2(1+ ν ) and ν =  ν
with E andν being Young’s modulus and Poisson’s ratio, respectively.
 1+ ν , for plane stress,
Now, the local boundary integral equations (36) and (37) considered at the nodal points y b ∈Ω reduce to a linear
system of algebraic equations for unknown fictitious nodal values:
n
n
a =1
a =1
∑ φa ( y b )$u ka + ∑ u$ ia ∫ Tik* (η, y b )φa (η )dΓη =
n
Ls
n
n
∑ φa ( ζb )$u ka + ∑ ∫ Tik* (η, ζb )φa (η )dΓηu$ ia + ∑
a =1
n
−∑
a =1 Ls
a =1
∫ bi ( x )U ik* ( x − y b )dΩ x ,
Ωs
lim
y→ ζ b
∫ Tik* (η, y )φa (η )dΓηu$ ia
Γst
∫ U ik* (η, ζb )sija (η )dΓηu$ aj = − ∫ Tik* (η, y )u i (η )dΓη + ∫ U ik* (η, ζb )t i (η )dΓη + ∫ bi ( x )U ik* ( x − ζb )dΩ ,
a =1 Γsu
Γsu
Γst
(39a)
(39b)
Ωs
1033
where Γst and Γsu are the traction and displacement boundary sections of Γs with Γs = Γst ∪ Γsu , the prescribed quantities are
labeled by a bar, and Ls is a part of the local boundary δΩ s , which is not located on the global boundary Γ. Furthermore, the
discretized LBIE (39b) is supplemented by the approximation formula
n
u j ( ζ b ) = ∑ φa ( ζ b )$u aj
for
a =1
ζ b ∈ Γsu .
(40)
The limit of the singular integral over Γst in (39b) can be evaluated numerically by using a regular quadrature, provided
that the optimal transformation of the integration variable is carried out before the integration [40].
The convergence of the LBIE formulations in elastostatics was analyzed in [39] by examples of a square patch, a
cantilever beam, and an infinite plate with a circular hole. Analytical solutions are known for all the examples considered. Also
the radius of the support domain and the parameter c in the Gaussian weight function were optimized in [39] to get the best
accuracy of numerical results.
4.2. Continuously Nonhomogeneous Body. Consider an isotropic, linearly elastic continuum whose Young’s modulus
depends on the Cartesian coordinates and Poisson’s ratio is a constant. Moreover, we will assume that Young’s modulus is given
by a differentiable function E ( x ). Under these assumptions, we can write the tensor of material coefficients as
0 ,
c ijkl ( x ) = µ ( x )c ijkl
µ(x) =
E( x )
,
2(1+ ν )
(41)
0 = 2ν / (1 − 2ν )δ δ + δ δ + δ δ .
where c ijkl
ij kl
ik jl
il jk
Recall that ν should be replaced everywhere (except for Eq. (41)) by ν / (1+ ν ) in the case of plane-stress problems. The
0 corresponds to a homogeneous, isotropic, linearly elastic continuum with the shear modulus µ = 1 and Poisson’s
tensor c ijkl
0
ratio ν. After the stress tensor is expressed in terms of displacement gradients as
σ ij ( x ) = c ijkl ( x )u k ,l ( x ),
(42)
( c ijkl u k ,l ) ,l = −bi .
(43)
µ,j
1
0 u ( x ).
( x ) c ijkl
bi −
k ,l
µ(x)
µ
(44)
the equilibrium equations become
Hence and from (41), we may write
0 u
c ijkl
k , lj = −
0 , we obtain an expression for a nonhomogeneous
Eventually, substituting for the tensor of material constants, c ijkl
isotropic medium:
µu i ,kk +
1
2ν
u k ,ki = −bi − µ ,i
u k , k − µ , j ( u i , j + u j , i ).
1− 2ν
1− 2ν
(45)
Apparently, it is impossible to find a closed fundamental solution in the general form for the operator

µ , j (x) ∂
∂
0  ∂
c ijkl
+
 ∂x l ∂x j
µ ( x ) ∂x l


.


On the other hand, the fundamental displacementsU km ( r ) for an elastic homogeneous continuum (the Kelvin solution
for µ = 1) satisfy the equation
0 ∂ ∂ U
c ijkl
j l km ( y − x ) = −δ im δ( x − y ),
1034
(46)
and the corresponding fundamental tractions are
0 n (η )U
Tim (η, y ) = c ijkl
j
km , l (η − y ).
Following the derivation of the boundary-domain formulation [78], the integral representation of displacements for a
nonhomogencous elastic medium can be written as
[
]
u k ( y ) = ∫ t i* (η )U ik (η − y ) − u i (η )Tik (η, y ) dΓη + ∫ g i ( x )U ik ( x − y )dΩ x + Wk ( y ),
Γ
(47)
Ω
where the modified traction vector is defined by
t i (η ) = µ (η )t i* (η ) or
t i* (η ) = n j (η )c ijkl u k ,l (η )
(48)
and
Wk ( y ) = ∫
Ω
gi (x) =
1
bi ( x )U ik ( x − y )dΩ x ,
µ(x)
1  2ν

µ ,i ( x )u j , j ( x ) + µ , j ( x ) [u i , j ( x ) + u j ,i ( x )] .

µ ( x )  1− 2ν

(49)
Due to the singular behavior of the kernel Tik , the accuracy of the computed displacements deteriorates near the
boundary. This singularity can be removed by using the integral identity [77]
∫ Tik (η, y )dΓη = −δik .
(50)
Γ
In view of (50), we can pass to the limit in Eq. (47) as y → ζ ∈ Γ and derive the nonsingular integral equation [78]
∫ [u i (η ) − u i ( ζ)]Tik (η, ζ)dΓη − ∫ t i* (η )U ik (η − ζ)dΓη = ∫ g i* ( x )U ik (x − ζ)dΩ x + Wk ( ζ).
Γ
Γ
(51)
Ω
The boundary integral equation (51) should be supplemented with the integral representation of displacement gradients
at interior points in order to derive a unique set of equations that describe a boundary-value problem for a finite body with
nonhomogeneous material properties. Although the problem of singularities has been resolved successfully in such a
formulation, both the boundary and interior domains should be discretized [78]. Consequently, two sets of coupled algebraic
equations, for boundary and interior unknowns, have to be solved.
Another approach is to use local boundary integral equations valid on the boundaries of simple circular domains around
each of the nodal points (randomly arranged) within the domain being analyzed. The resultant set of algebraic equations is
sparse. Similarly to the homogeneous case, we consider a subdomain Ω s located entirely inside Ω and containing the point y.The
local boundary integral equation is
uk ( y ) =
∫ [t i* (η )U ik (η − y ) − u i (η )Tik (η, y )] dΓη + ∫ g i* ( x )U ik (x − y )dΩ x + Wk ( y ),
∂Ω s
(52)
Ωs
where ∂Ω s is the boundary of the subdomain Ω s .
Since the same fundamental solutionU ik has been used to derive LBIE (52), as in the homogeneous case, the modified
fundamental solutions for both homogeneous and nonhomogeneous cases must be the same. Hence, we can write
uk ( y ) = −
∫ u i (η )Tik* (η, y )dΓη + ∫ g i (η )U ik* ( x − y )dΩ s + Wk ( y )
∂Ω s
(53)
Ωs
for the source point y located inside Ω and
1035
u k ( ζ ) + ∫ u i (η )Tik* (η, ζ )dΓη + lim
y→ ζ
Ls
* (η, ζ )dΓ =
− ∫ t i* (η )U ik
η
Γs
∫ u i (η )Tik* (η, y )dΓη
Γs
∫ g i (η )U ik* ( x − ζ)dΩ x + Wk ( ζ)
(54)
Ωs
for the source point located on the global boundary ζ ∈ Γs ⊂ Γ.
The formulation based on local boundary integral equations was tested numerically for a quadrilateral cross-section
(2a×2a) subjected to a uniform tension σ 22 = p in the x2-direction, assuming that the material is nonhomogeneous [78]:
µ = µ 0 (1+ α | x 2 |) 2 ,
µ0 =
E0
.
2(1+ ν )
Several equidistant node distributions with the total number of nodes N = 121, 49, 25 were considered in the numerical
analysis. The stress norm error, defined as
a
 1
r = ∫ σ 22 ( x1 ,0)dx1 − pa
100[%],
0
 pa


is 1.73% for the (LBIE/MLS) results representing the finest node distribution. A similar accuracy has been achieved for the LBIE
formulation with polynomial approximation [41].
5. Elastodynamics. The conventional boundary-element method for transient elastodynamic problems includes the
Laplace- or Fourier-domain formulations, time-domain formulation, and the mass-matrix approach with domain discretization.
In the time-domain formulation, spatial and time discretizations are required. The fundamental solution is very complicated in
this case. It requires more computation time to evaluate integrals. In the Laplace-transform-domain formulation, the
fundamental solution is also complicated and several quasi-static problems have to be solved for various values of the Laplace
transform parameter. In the mass-matrix formulation, the static fundamental solution is used. However, it leads to a
boundary-domain integral formulation because the static fundamental solution is not the solution of the governing
elastodynamic equation. The domain integral of the inertia terms can be transformed into boundary integrals by using the
dual-reciprocity method [79, 80]. To improve the spatial approximation, it is necessary to consider interior nodes in addition to
the boundary ones. If the interior nodes are spread over the domain, it is convenient to use the meshless approximation.
The meshless method for dynamic problems was first applied in [6], where the element-free Galerkin method was used.
The use of the static fundamental solution leads to local integral equations (LIEs) with a domain integral, which can easily be
evaluated if the subdomain is a circle. The spatial variation of the displacements is approximated by the moving least-square
method. In the time-domain formulation, the domain integral in the LIEs contains displacement accelerations. After evaluating
the space integrals, we obtain a system of ordinary differential equations for certain nodal unknowns. This system can be solved
numerically by the Houbolt finite-difference scheme [8].
In the Laplace-transform approach, the LIEs involve a domain integral of dynamic terms arising from the inertia term
and the initial values. There are no problems with the numerical evaluation of this integral when the MLS approximation is
employed for spatial variation. Several quasi-static problems have to be solved for various values of the Laplace-transform
parameter. To obtain time-dependent values, the Durbin inversion method [82] is applied.
5.1. Local Integral Equations in the Time-Domain. Let us consider a linear elastodynamic problem on a domain Ω
bounded by a boundary Γ. The displacements are governed by the equation [77]
µu i ,kk + ( λ + µ )u k ,ki + bi = ρu&& i ,
(55)
where u i and bi are the components of the time-dependent displacements and the body-force vector, respectively, and ρ is the
mass density of the material. A weak formulation of Eq. (55) can be expressed as
∫ [µu i ,kk ( x, τ ) + (λ + µ )u i ,kk ( x, τ ) + bi ( x, τ ) − ρu&& i ( x, τ )] u i* ( x )dΩ = 0,
Ω
1036
(56)
where u i* ( x ) is a weight field.
Applying the Gauss divergence theorem to the domain integral in Eq. (56), we obtain
∫ [µu i ,kk ( x, τ ) + (λ + µ )u i*,kk ( x )] u i ( x, τ )dΩ
Ω
[
]
+ ∫ t i ( x , τ )u i* ( x ) − t i* ( x )u i ( x , τ ) dΓ = ∫ [ρu&& i ( x , τ ) − bi ( x, τ )]u i* ( x )dΩ ,
(57)
t i = µu i ,k n k + λu k ,k n i + µu k ,i n k
(58)
Γ
Ω
where
is the traction vector with a unit outward normal vector n i to the boundary Γ. The weighted traction vector t i* is defined by
Eq. (58), replacing the trial field u i by the weight field u i* . The weight field can be selected as the fundamental solution of the
governing elastostatic equation [77]
µU ij ,kk ( x, y ) + ( λ + µ )U kj ,ki ( x, y ) = −δij δ( x − y ),
(59)
with u i* ( x ) = U ij ( x − y )e j ( y ) and t i* ( x ) = Tij ( x, y )e j ( y ), where e j is a unit orthonormal base vector.
In this case, we can rewrite Eq. (57) to obtain the integral representation of the displacement
u j ( y, τ ) = ∫ t i ( x, τ )U ij ( x − y )dΓ − ∫ Tij ( x, y )u i ( x, τ )dΓ + ∫ [bi ( x , τ ) − ρu&& i ( x , τ )]U ij ( x − y )dΩ.
Γ
Γ
(60)
Ω
The integral equation (60) can be considered on a small subdomain Ω s ⊂ Ω. Then, we can write
u j ( y, τ ) =
∫ t i ( x, τ )U ij ( x − y )dΓ − ∫ Tij ( x, y )u i ( x, τ )dΓ + ∫ [bi ( x, τ ) − ρu&& i ( x, τ )]U ij ( x − y )dΩ ,
∂Ω s
∂Ω s
(61)
Ωs
where ∂Ω s is the boundary of the subdomain Ω s .
Replacing the fundamental solution U ij by the modified fundamental solutionU ij* in Eq. (61), we get
u j ( y, τ ) =
∫ Tij* ( x, y )u i ( x, τ )dΓ + ∫ [bi ( x, τ ) − ρu&& i ( x, τ )]U ij* ( x − y )dΩ
∂Ω s
(62)
Ωs
for the source point y inside Ω. The explicit expression for the modified test function and the modified fundamental traction Tij*
can be found in [39].
For the source point on the global boundary ζ ∈ Γs ⊂ Γ (Fig. 2), the LBIE can be written as
u j ( ζ, τ ) + ∫ Tij* ( x, ζ )u i ( x, τ )dΓ + lim
y→ ζ
Ls
−∫ t i ( x , τ )U ij* ( x − ζ )dΓ =
Γ
∫ Tij* ( x, y )u j ( x, τ )dΓ
Ls
∫ [bi ( x, τ ) − ρu&& i ( x, τ )]U ij* ( x − ζ)dΩ.
(63)
Ωs
When the MLS approximation is applied to displacements (6) and traction vectors (38), LBIEs (62) and (63) yield the
following set of discretized LIEs:



b )δ +
* φa ( x , y b )dΓ  u$ a ( τ ) = b ( x )U * ( x − y b )dΩ
φ
(
y
T

 i
∑
ik
ik
∫ ik
∫ i

a =1 
∂
Ω
Ωs
s

n
1037
n
−∑
∫ ρ ( x )U ik* ( x − y b )φa ( x )dΩu&& ia ( τ )
for
y b ∈Ω,
(64)
a =1Ω s
n




∑  φ( ζb )δik + ylim
∫ Tik* ( x, y )φa ( x )dΓ + ∫ Tik* ( x, ζ)φa ( x )dΓ − ∫ sija ( x )U ik* ( x, ζb ) u$ aj ( τ )
→ζ b

a =1 
Γst
=
~

Γsu
Ls
~
∫ U ik* ( x − ζb )t i ( x )dΓ − ∫ Tik* ( x, ζb )u i ( x )dΓ + ∫ bi ( x )U ik* ( x − ζb )dΩ
Γst
Γsu
n
−∑
Ωs
∫ ρ ( x )U ik* ( x − ζb )φa ( x )dΩu&& ia ( τ )
for ζ b ∈ Γst ,
a =1Ω s
(65)
~
where the prescribed boundary densities are denoted by u~i and t i .
Finally, the set of equations (64) and (65) is supplemented by the equation
n
∑ φa ( ζb )$u ia ( τ ) = u~i ( ζb , τ ),
ζ b ∈ Γsu .
a =1
(66)
Depending on the boundary conditions prescribed, the system of ordinary differential equations (64) and (65) can be
rearranged so that all the known quantities appear on the right-hand side. Thus, the system takes on the following matrix form:
&& + Kx = P.
Lx
(67)
To solve this system of ordinary differential equations, we should perform a great amount of time integration. Here we
use the Houbolt method. The Houbolt finite-difference scheme [81] expresses the acceleration (&&u = &&)
x as
&&x τ + ∆τ =
2x τ + ∆τ − 5x τ + 4 x τ − ∆τ − x τ − ∆τ
∆τ 2
(68)
,
where ∆τ is the time step.
Substituting Eq. (68) into Eq. (67), we get a system of algebraic equations for the unknowns x τ + ∆τ :
1

 2
 ∆τ 2 L + K  x τ + ∆τ = L ∆τ 2 {5x τ − 4 x τ − ∆τ + x τ − 2 ∆τ } + P.
(69)
The value of the time step should be selected with due regard for the material parameters (propagation velocities) and
the time-dependence of the boundary conditions.
5.2. Local Integral Equations in the Laplace-Transform Domain. Applying the Laplace transform to the governing
elastodynamic equation (55), we obtain [77]
µu i ,kk + ( λ + µ )u k ,ki + Fi − ρp 2 u i = 0,
(70)
where Fi = bi ( x, p ) + pu i ( x ) + u& i ( x ) is a redefined body force in the Laplace-transform domain with initial boundary
conditions for the displacements u i ( x ) and the velocities u& i ( x ).
The integral representation of the displacements in the Laplace-transform domain is given by
[
]
u j ( y, p ) = ∫ t i ( x, p )U ij ( x − y )dΓ − ∫ Tij ( x, y )u i ( x, p )dΓ + ∫ Fi ( x, p ) − ρp 2 u i ( x, p ) U ij ( x − y )dΩ.
Γ
Γ
Ω
(71)
If we consider a subdomain Ω s instead of the entire domain Ω, then the following local integral equation must hold on
the subdomain:
1038
u j ( y, p ) =
∫ t i ( x, p )U ij ( x − y )dΓ − ∫ Tij ( x, y )u i ( x, p )dΓ + ∫ [Fi ( x, p ) − ρp 2 u i ( x, p )]U ij (x − y )dΩ.
∂Ω s
∂Ω s
(72)
Ωs
The integral equations (72) are considered for small subdomains Ω s ⊂ Ω. Hence, none of the boundary densities is
prescribed on ∂Ω s as long as ∂Ω s lies entirely inside Ω. This deficit in the boundary conditions inside the domain Ω can be
overcome by domain-type approximation of field variables. With such an approximation, the boundary-domain formulation (72)
does not complicate the evaluation of domain integrals over a simple subdomain.
Similarly to the time-domain LIE formulation, the integral with the traction vector in Eq. (72) can again be eliminated
by introducing a companion solution if the subdomain Ω s is entirely inside Ω. For the modified test function U ij* , LIE (72)
transforms into
u j ( y, p ) =
∫ Tij ( x, y )u i ( x, p )dΓ + ∫ [Fi ( x, p ) − ρp 2 u i ( x, p )]U ij* ( x − y )dΩ
∂Ω s
(73)
Ωs
for the source point y located inside Ω.
For ζ ∈ Γs ⊂ Γ (the source point is on the global boundary), the LBIE can be written as
u j ( ζ, p ) = ∫ Tij ( x, ζ )u i ( x, p )dΓ + lim
y→ ζ
Ls
− ∫ t i ( x, p )U ij* ( x − ζ )dΓ =
Γs
∫ Tij* ( x, y )u j ( x, p )dΓ
Γs
∫ [Fi ( x, p ) − ρp 2 u i ( x, p )]U ij* ( x − ζ)dΩ.
(74)
Ωs
The discretized LIEs are derived by substituting the MLS approximations (6) and (38) for displacements and traction in
the Laplace-transform domain into Eqs. (73) and (74). Their form is similar to the discretized equations (64)–(66) valid in the
time-domain.
The time-dependent values of any of the transformed variables are obtained by applying the inverse transform. There
are many Laplace transform inversion methods. Here, we use the method proposed by Durbin [82]. The calculation formula to be
used is as follows:
f (τ ) = 2
e sτ
T
L
 1

2kπ 
2kπτ  
2kπ 
2kπτ


−
Re
f
(
s
)
+
Re f  s + i
− Im f  s + i
 cos
 sin

,
∑

2
T 
T
T 
T  


k = 0


{ }
(75)
where f ( p k ) stands for the value in the Laplace domain at a sample point p k = s + 2kπi / T , k = 0, 1, ..., L.
Good results have been obtained for sT = 5 and T / τ 0 = 30, where τ 0 is unit time. The accuracy of Durbin’s method is
satisfactory even for large times, and its only disadvantage is that it deals with complex data.
The accuracy and efficiency of the present method were tested numerically for a long strip under a uniaxial tension and
for a frame structure [83]. Both methods in the time and Laplace domains give practically identical time variations of the
displacements and tractions.
6. Plate Bending Problems.
6.1. Simply Supported and Clamped Plates. When a plate has clamped edges and/or simply supported straight edges, it
is possible to decompose the biharmonic equation into a system of two Poisson equations [84, 85] for the deflection and its
Laplacian, respectively. For simply supported plates, the boundary-value problem can be formulated for each of these equations,
which can be solved separately on the basis of the BIE formulation. Recall that the boundary densities in the “moment integral
equation” (the bending moment and the shear force or Kirchhoff equivalent shear force) for clamped plates are unknown. Thus,
this BIE does not allow a unique formulation of a boundary-value problem for a plate with clamped edges by means of the
standard BE method. Nevertheless, the system of BIEs for the deflection and the moment yields a unique formulation. The BIE
for the deflection is associated with the Poisson equation. It can be replaced also by the standard BIE for the deflection
corresponding to the biharmonic operator. The meshless approximation based on the MLS method is chosen in the proposed
formulation for both the deflection and its Laplacian.
1039
Consider a thin elastic plate subjected to a transverse load of intensity q defined on a bounded plane region B. In the
classical (Kirchhoff) theory governing the bending of thin elastic plates, the deflection w obeys the partial differential equation
∇ 2 ∇ 2 w( x ) =
q( x )
,
D
x ∈ B,
(76)
where D = Eh 3 / 12(1− ν 2 ) is the plate stiffness, E is Young’s modulus, ν is Poisson’s ratio, and h is the plate thickness.
The biharmonic governing equation (76) is decomposed into the two Poisson equations
−D∇ 2 w( x ) = m( x ),
(77)
∇ 2 m( x ) = −q( x ).
(78)
Generally, ∇ 2 w( x ) has no physical meaning. At the clamped edge, however, m( x ) is equal to the bending moment
M ( x ). At the simply supported edge, this is the case if the edge is straight. In what follows, we restrict ourselves to plate
boundaries Γ such that m( x ) = M ( x ). Then, the derivative of m( x ) with respect to the normal to the plate boundary is identical to
the shear force N ( x ). Moreover, for the clamped boundary, N ( x ) is also equal to the equivalent shear force because
N ( x ) = H ( x ). Owing to the symmetry of the geometry and loading, it is sometimes sufficient to solve the boundary-value
problem only on a part of the plate domain B. Then, the boundary ∂Ω of the domain Ω ⊂ B consists of two parts
∂Ω = ∂Ω e ∪ ∂Ω i , where ∂Ω e = ∂Ω ∩ Γ and ∂Ω i = ∂Ω − ∂Ω e ⊂ B. Since ∂Ω i is the symmetry line, ∂w / ∂n = 0 on this line.
Thus, the relations m( x ) = M ( x ) and ∂w / ∂n = N ( x ) hold on ∂Ω e ⊂ Γ and on ∂Ω i . The solutions of the Poisson equations (77)
and (78) can be reduced to the integral form [86]
∂U
 ∂m

m( y ) = ∫ q( x )U ( x − y )dΩ + ∫  ( x )U ( x − y ) − m( x )
( x, y ) dΓ ,
∂
n
∂
n


Ω
Γ
w( y ) =
1
∂U

 ∂w
m( x )U ( x − y )dΩ + ∫  ( x )U ( x − y ) − w( x )
( x, y ) dΓ ,
∫
D
∂n

 ∂n
Ω
(79)
(80)
Γ
1
1
ln is the fundamental solution of the Laplace equation for an infinite plane.
2π r
The integral equations (79) and (80) can be considered on small subdomains Ω s ⊂ B (Fig. 2). We can write
where y ∈Ω and the kernel U =
m( y ) =
∫ q( x )U ( x − y )dΩ + ∫
Ωs
w( y ) =
1
D
∂Ω s
∂U
 ∂m

 ∂n ( x )U ( x − y ) − m( x ) ∂n ( x, y ) dΓ ,
∫ m( x )U ( x − y )dΩ + ∫
Ωs
∂Ω s
∂U

 ∂w
 ∂n ( x )U ( x − y ) − w( x ) ∂n ( x, y ) dΓ ,
(81)
(82)
where y ∈Ω s .
The concept of a companion solution, known from the previous section, is also applicable here. The modified
fundamental solution is given by Eq. (17),
U* =
r
1
ln 0 .
2π
r
This yields the following simplified LBIEs:
m( y ) =
∫ q( x )U * ( x − y )dΩ − ∫ m( x )
Ωs
1040
∂Ω s
∂U *
( x, y )dΓ ,
∂n
(83)
1
D
w( y ) =
∫ m( x )U * ( x − y )dΩ −
Ωs
∫ w( x )
∂Ω s
∂U *
( x, y )dΓ
∂n
(84)
as long as y ∈Ω. For y = ζ ∈ ∂Ω, the LBIEs take the form
m( ζ ) =
∫ q( x )U * ( x − y )dΩ − ∫ m( x )
Ωs
m( ζ ) =
1
D
Ls
∂m
∂U *
( x )U * ( x − y )dΓ − lim
( x, ζ )dΓ + ∫
y→ ζ
∂n
∂n
γs
∫ m( x )U * ( x − ζ)dΩ − ∫ w( x )
Ωs
Ls
∂U
∫ m( x ) ∂n ( x, y )dΓ ,
(85)
γs
∂w
∂U *
( x )U * ( x − ζ )dΓ − lim
( x, ζ )dΓ + ∫
y→ ζ
∂n
∂n
γs
∂U
∫ w( x ) ∂n ( x, y )dΓ.
(86)
γs
Recall that γ s = Γs ⊂ Γ if the domain being analyzed is the whole plate, Ω = B ( ∂Ω = Γ ). Then m( ζ ) = M ( ζ ),
∂m
m( x ) = M ( x ), and
( x ) = N ( x ) at x ∈ Γs ⊂ Γ. Based on the MLS approximation, the discretization of LBIEs (83) and (85)
∂n
yields


∂U *
 a b
b )φa ( x )dΓ  m
a
*
b
φ
(
y
)
+
(
x
,
y
 $ = ∫ q( x )U ( x − y )dΩ ,

∑
∫ ∂n

a =1 
∂Ω s
Ωs

n
n


∑  φa ( ζb ) + ylim
∫
→ζ b
a =1 

γs
∂U *
( x, y b )φa ( x )dΓ +
∂n
∫
Ls
y a ∈Ω ,
(87)
∂U *
( x, ζ b )φa ( x )dΓ
∂n

 $a
− ∫ n k φ,ak ( x )U * ( x − ζ b )dΓ  m
= ∫ q( x )U * ( x − ζ b )dΩ ,

Ωs
Ωs
ζ b ∈ ∂Ω.
(88)
Similarly, the discretization of LBIEs (84) and (86) results in the following set of equations:
n


∑  φa ( y b ) + ∫
a =1 

∂Ω s

1
∂U *

( x, y b )φa ( x )dΓ  w$ a = ∫ m( x )U * ( x − y b )dΩ ,
∂n
D

Ωs


∑  φa ( ζb ) + ylim
→ζ b
a =1 

n
∫
γs
∂U *
( x, y b )φa ( x )dΓ +
∂n
∫
Ls
y a ∈Ω ,
(89)
∂U *
( x, ζ b )φa ( x )dΓ
∂n


− ∫ n k φ,ak ( x )U * ( x − ζ b )dΓ  w$ a = ∫ m( x )U * ( x − ζ b )dΩ ,

γs
Ωs
ζ b ∈ ∂Ω,
(90)
$ a are the fictitious parameters for the deflection and bending moment, respectively, used in the MLS
where w$ a and m
approximations
n
w h ( x ) = ∑ φa ( x )w$ a ,
a =1
n
∂w h
( x ) = n k ( x ) ∑ φ,ak ( x )w$ a ,
∂n
a =1
n
$a,
mh ( x ) = ∑ φa ( x )m
a =1
n
∂mh
$ a.
( x ) = n k ( x ) ∑ φ,ak ( x )m
∂n
a =1
(91)
1041
The maximum relative error produced by the LBIE approach for a simply supported square plate was less than 0.1% for
the bending moment and less than 0.8% for the deflection. The results were compared with analytical ones [46].
6.2. Simply Supported and Clamped Plates on an Elastic Foundation. Consider a thin elastic plate on a Pasternak-type
foundation. The boundary-element method was first applied in [87, 88] to plates resting on a two-parameter foundation. Since
the governing partial differential equation is of the fourth order, the pure boundary integral formulation for the boundary-value
solution requires two boundary integral equations. The complexity and high-order singularity of the kernel functions require
sophisticated algorithms for numerical integration within the framework of the conventional BE method. Similarly to the
previous section, the governing equation is decomposed into two differential equations, one of which is the Poisson equation and
the other is the Helmholtz equation. The local boundary integral equations are applied to both equations. The meshless
approximation based on the MLS method is chosen in the proposed formulation for both the deflection and its Laplacian.
Assuming the validity of the classical Kirchhoff hypothesis of thin plates and considering the plate resting on an elastic
Pasternak-type foundation, we obtain the differential equation for the plate deflection,
D∇ 2 ∇ 2 w( x ) = q( x ) − kw( x ) + k p ∇ 2 w( x ),
x ∈ B,
(92)
where q is the transverse load, D = Eh 3 / 12(1− ν 2 ) is the plate stiffness, E is Young’s modulus, ν is Poisson’s ratio, and h is the
plate thickness. The Pasternak foundation is characterized by two parameters k and k p . If the parameter k p is vanishing, the
Pasternak foundation is reduced to that of Winkler-type. The governing equation (92) is decomposed into two equations
containing partial derivatives of the second order:
−D∇ 2 w( x ) = m( x ),
(93)
∇ 2 m( x ) − k p m( x ) = −q( x ) + kw( x ).
(94)
The solutions of the coupled Poisson and Helmholtz equations (93) and (94) can be expressed in an integral form as
w( y ) =
1
∂U

 ∂w
m( x )U ( x − y )dΩ + ∫  ( x )U ( x − y ) − w( x )
( x, y ) dΓ ,
∫
D
∂n

 ∂n
Ω
Γ
∂U
 ∂m

m( y ) = ∫ q( x )U ( x − y )dΩ + ∫  ( x )U ( x − y ) − m( x )
( x, y ) dΓ −∫ [kw( x ) + k p m( x )]U ( x − y )dΩ ,
∂n
 ∂n

Ω
Γ
Γ
(95)
(96)
where y ∈Ω and the kernel U ( x, y ) is the fundamental solution of the Laplace equation for an infinite plane.
If the internal point y approaches the boundary point ζ ∈ Γ,then the boundary integral equations follow from the integral
representations (95) and (96) as
w( ζ ) =
1
∂w
m( x )U ( x − y )dΩ + ∫
( x )U ( x − ζ ) − lim
∫
y→ ζ
D
∂n
Ω
Ω
(97)
∫ M ( x ) ∂n ( x, y )dΓ −∫ [kw( x ) + k p m( x )]U ( x − ζ)dΩ.
(98)
Γ
M ( ζ ) = ∫ q( x )U ( x − ζ )dΩ + ∫ N ( x )U ( x − ζ )dΓ − lim
Γ
y→ ζ
∂U
∫ w( x ) ∂n ( x, y )dΓ ,
Γ
∂U
Γ
Ω
The domain being analyzed can be covered (even partially) by a set of simple and regular subdomains. The boundary
integral equations (97) and (98) can be applied directly to each subdomain. All the integrals can easily be evaluated owing to the
regular shapes of the subdomain Ω s ⊂ B (Fig. 2). The concept of a companion solution from the previous sections is used here.
Based on the MLS approximation for m, w, and their derivatives (see Sect. 6.1), the local boundary integral equations reduce to
the following set of equations:
n


∑  φa ( y b ) + ∫
a =1 

1042
∂Ω s
n

1
∂U *

$ a ∫ φa ( x )U * ( x − y b )dΩ ,
( x, y b )φa ( x )dΓ  w$ a = ∑ m
D a =1
∂n

Ωs


∂U *
 a b
b )φa ( x )dΓ  m
a
φ
(
y
)
+
(
x
,
y
 $
∑
∫ ∂n

a =1 
∂Ω s

n
=
n


q
(
x
)
−

∑ (kw$ a + k p m$ a )φa ( x ) U * ( x − y b )dΩ for
∫

j =1
Ωs 
y b ∈Ω
(99)
and
n


∑  φa ( ζb ) + ylim
∫
→ζ b
a =1 

γs
∂U *
( x, y b )φa ( x )dΓ +
∂n
∫
Ls
∂U *
( x, ζ b )φa ( x )dΓ
∂n
n

1

$ a ∫ φa ( x )U * ( x − ζ b )dΩ ,
− ∫ n k φ,ak ( x )U * ( x − ζ b )dΓ  w$ a = ∑ m
D

a =1
γs
Ωs
n


∑  φa ( ζb ) + ylim
∫
→ζ b
a =1 

γs
∂U *
( x, y b )φa ( x )dΓ +
∂n

∂U *
 a
$
( x, ζ b )φa ( x )dΓ − ∫ n k φ,ak ( x )U * ( x − ζ b )dΓ  m
∂n

Ls
γs

∫
n


$ a φa ( x ) U a ( x − ζ b )dΩ for
= ∫ q( x ) − ∑ kw$ a + k p m

a =1
Ω 
(
)
ζ b ∈ ∂Ω.
(100)
If the plate is simply supported along the boundary Γ, then w( ζ ) = 0 and M ( ζ ) = 0 at ζ ∈ Γ. Then, the nodal values w$ a
and ma ( a = 1,... , n ) can be computed by solving the system of discretized LBIEs (99) and the collocation equations derived from
the approximation formulas
n
∑ φa ( ζb )w$ a = 0,
a =1
n
∑ φa ( ζb )m$ a = 0,
ζb ∈ Γ ,
(101)
a =1
which are much simpler than LBIEs (100).
For a clamped plate, LBIEs (100) at ζ b ∈ Γ can again be replaced by the collocation equations
n
∑ φa ( ζb )w$ a = 0,
a =1
n
n k ( ζ b ) ∑ φa ( ζ b )w$ a = 0.
(102)
a =1
Then, the system of local boundary integral equations (99) and the collocation equations (102) uniquely describe a
boundary-value problem for a clamped plate on an elastic foundation.
7. Conclusions. The paper summarizes the recent developments in the application of the local boundary integral
equation method to boundary-value problems for many engineering problems. The combination of LBIE and moving
least-square approximation of physical quantities leads to a reliable, efficient computational method with wide application to
many boundary-value problems. The well-known advantages of the conventional boundary-element method are incorporated
into the LBIE approach. On the other hand, the restriction of the BE method to problems with a known fundamental solution is
removed in the LBIE method. A simpler fundamental solution in the LBIE approach leads to a boundary-domain formulation.
Evaluating the domain integrals over regular subdomains is not difficult. However, a simpler fundamental solution increases the
computational efficiency due to much simpler numerical implementation. The present LBIE approach can also be applied to
problems where the fundamental solution, leading to a pure boundary integral formulation, is not known. Therefore, the
applicability of the present method is much wider than that of the conventional BE method. Since the LBIE approach is
combined with the meshless approximation, all the advantages valid for other meshless approaches are preserved. They are
mainly time saving in creating a mesh and flexibility in adapting the density of nodal points at any place of the problem domain
1043
such that the resolution and fidelity of the solution can be improved easily. The present method possesses a tremendous potential
for solving nonlinear and nonhomogeneous problems.
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