![INTRODUCTION TO
BOUNDARY METHODS
This chapter provides a brief introduction to various topics that are of interest
in this book.
Boundary Element Method
Boundary integral equations (BIE), and the boundary element method (BEM),
based on BIEs, are mature methods for numerical analysis of a large variety of
problems in science and engineering. The standard BEM for linear problems
has the well-known dimensionality advantage in that only the two-dimensional
(2-D) bounding surface of a three-dimensional (3-D) body needs to be meshed
when this method is used. Examples of books on the subject, published dur-
ing the last 15 years, are Banerjee [4], Becker [9], Bonnet [14], Brebbia and
Dominguez [16], Chandra and Mukherjee [22], Gaul et al. [47], Hartmann [62],
Kane [68] and Parı́s and Cañas [121]. BEM topics of interest in this book are
finite parts (FP) in Chapter 1, error estimation in Chapter 2 and thin features
(cracks and thin objects) in Chapter 3.
Hypersingular Boundary Integral Equations
Hypersingular boundary integral equations (HBIEs) are derived from a differ-
entiated version of the usual boundary integral equations (BIEs). HBIEs have
diverse important applications and are the subject of considerable current re-
search (see, for example, Krishnasamy et al. [76], Tanaka et al. [162], Paulino
[122] and Chen and Hong [30] for recent surveys of the field). HBIEs, for exam-
ple, have been employed for the evaluation of boundary stresses (e.g. Guiggiani
[60], Wilde and Aliabadi [173], Zhao and Lan [185], Chati and Mukherjee [24]),
in wave scattering (e.g. Krishnasamy et al. [75]), in fracture mechanics (e.g.
Cruse [38], Gray et al. [54], Lutz et al. [89], Paulino [122], Gray and Paulino
[58], Mukherjee et al. [110]), to obtain symmetric Galerkin boundary element
formulations (e.g. Bonnet [14], Gray et al. [55], Gray and Paulino ([56], [57]), to
xiii
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![xiv INTRODUCTION TO BOUNDARY METHODS
evaluate nearly singular integrals (Mukherjee et al. [104]), to obtain the hyper-
singular boundary contour method (Phan et al. [131], Mukherjee and Mukherjee
[99]), to obtain the hypersingular boundary node method (Chati et al. [27]), and
for error analysis (Paulino et al. [123], Menon [95], Menon et al. [96], Chati et
al. [27], Paulino and Gray [125]) and adaptivity [28].
An elegant approach of regularizing singular and hypersingular integrals, us-
ing simple solutions, was first proposed by Rudolphi [143]. Several researchers
have used this idea to regularize hypersingular integrals before collocating an
HBIE at a regular boundary point. Examples are Cruse and Richardson [39],
Lutz et al. [89], Poon et al. [138], Mukherjee et al. [110] and Mukherjee [106].
The relationship between finite parts of strongly singular and hypersingular in-
tegrals, and the HBIE, is discussed in [168], [101] and [102]. A lively debate (e.g.
[92], [39]), on smoothness requirements on boundary variables for collocating
an HBIE on the boundary of a body, has apparently been concluded recently
[93]. An alternative way of satisfying this smoothness requirement is the use of
the hypersingular boundary node method (HBNM).
Mesh-Free Methods
Mesh-free (also called meshless) methods [82], that only require points rather
than elements to be specified in the physical domain, have tremendous potential
advantages over methods such as the finite element method (FEM) that require
discretization of a body into elements.
The idea of moving least squares (MLS) interpolants, for curve and surface
fitting, is described in a book by Lancaster and Salkauskas [78]. Nayroles et
al. [117] proposed a coupling of MLS interpolants with Galerkin procedures in
order to solve boundary value problems. They called their method the diffuse
element method (DEM) and applied it to two-dimensional (2-D) problems in
potential theory and linear elasticity.
During the relatively short span of less than a decade, great progress has
been made in solid mechanics applications of mesh-free methods. Mesh-free
methods proposed to date include the element-free Galerkin (EFG) method
[10, 11, 12, 13, 67, 174, 175, 176, 108], the reproducing-kernel particle method
(RKPM) [83, 84], h − p clouds [42, 43, 120], the meshless local Petrov-Galerkin
(MLPG) approach [3], the local boundary integral equation (LBIE) method
[152, 188], the meshless regular local boundary integral equation (MRLBIE)
method [189], the natural element method (NEM) [158, 160], the general-
ized finite element method (GFEM) [157], the extended finite element method
(X-FEM) [97, 41, 159], the method of finite spheres (MFS) [40], the finite
cloud method (FCM) [2], the boundary cloud method (BCLM) [79, 80], the
boundary point interpolation method (BPIM) [82], the boundary-only radial
basis function method (BRBFM) [32] and the boundary node method (BNM)
[107, 72, 25, 26, 27, 28, 52].
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![xv
Boundary Node Method
S. Mukherjee, together with his research collaborators, has recently pioneered
a new computational approach called the boundary node method (BNM) [26,
25, 27, 28, 72, 107]. Other examples of boundary-based meshless methods are
the boundary cloud method (BCLM) [79, 80], the boundary point interpolation
method (BPIM) [82], the boundary only radial basis function method (BRBFM)
[32] and the local BIE (LBIE) [188] approach. The LBIE, however, is not a
boundary method since it requires evaluation of integrals over certain surfaces
(called Ls in [188]) that can be regarded as “closure surfaces” of boundary
elements.
The BNM is a combination of the MLS interpolation scheme and the stan-
dard boundary integral equation (BIE) method. The method divorces the tra-
ditional coupling between spatial discretization (meshing) and interpolation (as
commonly practiced in the FEM or in the BEM). Instead, a “diffuse” interpo-
lation, based on MLS interpolants, is used to represent the unknown functions;
and surface cells, with a very flexible structure (e.g. any cell can be arbitrarily
subdivided without affecting its neighbors [27]) are used for integration. Thus,
the BNM retains the meshless attribute of the EFG method and the dimen-
sionality advantage of the BEM. As a consequence, the BNM only requires the
specification of points on the 2-D bounding surface of a 3-D body (including
crack faces in fracture mechanics problems), together with surface cells for in-
tegration, thereby practically eliminating the meshing problem (see Figures i
and ii). The required cell structure is analogous to (but not the same as) a
tiling [139]. The only requirements are that the intersection of any two surface
cells is the null set and that the union of all the cells is the bounding surface of
the body. In contrast, the FEM needs volume meshing, the BEM needs surface
meshing, and the EFG needs points throughout the domain of a body.
It is important to point out another important advantage of MLS inter-
polants. They can be easily designed to be sufficiently smooth to suit a given
purpose, e.g. they can be made C1
or higher [10] in order to collocate the HBNM
at a point on the boundary of a body.
The BNM is described in Chapters 8 and 9 of this book.
Figure i: BNM with nodes and cells
(from [28])
Figure ii: BEM with nodes and elements
(from [28])
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![xvi INTRODUCTION TO BOUNDARY METHODS
Boundary Contour Method
The Method
The usual boundary element method (BEM), for three-dimensional (3-D) lin-
ear elasticity, requires numerical evaluations of surface integrals on boundary
elements on the surface of a body (see, for example, [98]). [115] (for 2-D linear
elasticity) and [116] (for 3-D linear elasticity) have recently proposed a novel
approach, called the boundary contour method (BCM), that achieves a further
reduction in dimension! The BCM, for 3-D linear elasticity problems, only re-
quires numerical evaluation of line integrals over the closed bounding contours
of the usual (surface) boundary elements.
The central idea of the BCM is the exploitation of the divergence-free prop-
erty of the usual BEM integrand and a very useful application of Stokes’ the-
orem, to analytically convert surface integrals on boundary elements to line
integrals on closed contours that bound these elements. [88] first proposed an
application of this idea for the Laplace equation and Nagarajan et al. gen-
eralized this idea to linear elasticity. Numerical results for two-dimensional
(2-D) problems, with linear boundary elements, are presented in [115], while
results with quadratic boundary elements appear in [129]. Three-dimensional
elasticity problems, with quadratic boundary elements, are the subject of [116]
and [109]. Hypersingular boundary contour formulations, for two-dimensional
[131] and three-dimensional [99] linear elasticity, have been proposed recently.
A symmetric Galerkin BCM for 2-D linear elasticity appears in [119]. Recent
work on the BCM is available in [31, 134, 135, 136, 186].
The BCM is described in Chapter 4 of this book.
Shape Sensitivity Analysis with the BCM and the HBCM
Design sensitivity coefficients (DSCs), which are defined as rates of change of
physical response quantities with respect to changes in design variables, are
useful for various applications such as in judging the robustness of a given
design, in reliability analysis and in solving inverse and design optimization
problems. There are three methods for design sensitivity analysis (e.g. [63]),
namely, the finite difference approach (FDA), the adjoint structure approach
(ASA) and the direct differentiation approach (DDA). The DDA is of interest
in this work.
The goal of obtaining BCM sensitivity equations can be achieved in two
equivalent ways. In the 2-D work by [130], design sensitivities are obtained
by first converting the discretized BIEs into their boundary contour version,
and then applying the DDA (using the concept of the material derivative) to
this BCM version. This approach, while relatively straightforward in principle,
becomes extremely algebraically intensive for 3-D elasticity problems. [100]
offers a novel alternative derivation, using the opposite process, in which the
DDA is first applied to the regularized BIE and then the resulting equations
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![xvii
are converted to their boundary contour version. It is important to point out
that this process of converting the sensitivity BIE into a BCM form is quite
challenging. This new derivation, for sensitivities of surface variables [100], as
well as for internal variables [103], for 3-D elasticity problems, is presented in
Chapter 5 of this book. The reader is referred to [133] for a corresponding
derivation for 2-D elasticity
Shape Optimization with the BCM
Shape optimization refers to the optimal design of the shape of structural com-
ponents and is of great importance in mechanical engineering design. A typical
gradient-based shape optimization procedure is an iterative process in which
iterative improvements are carried out over successive designs until an optimal
design is accepted. A domain-based method such as the finite element method
(FEM) typically requires discretization of the entire domain of a body many
times during this iterative process. The BEM, however, only requires surface
discretization, so that mesh generation and remeshing procedures can be carried
out much more easily for the BEM than for the FEM. Also, surface stresses are
typically obtained very accurately in the BEM. As a result, the BEM has been
a popular method for shape optimization in linear mechanics. Some examples
are references [33], [145], [178], [144], [169], [177], [161] and the book [184].
In addition to having the same meshing advantages as the usual BEM, the
BCM, as explained above, offers a further reduction in dimension. Also, surface
stresses can be obtained very easily and accurately by the BCM without the
need for additional shape function differentiation as is commonly required with
the BEM. These properties make the BCM very attractive as the computational
engine for stress analysis for use in shape optimization. Shape optimization in
2-D linear elasticity, with the BCM, has been presented by [132]. The corre-
sponding 3-D problem is presented in [150] and is discussed in Chapter 6.
Error Estimation and Adaptivity
A particular strength of the finite element method (FEM) is the well-developed
theory of error estimation, and its use in adaptive methods (see, for example,
Ciarlet [34], Eriksson et al. [44]). In contrast, error estimation in the boundary
element method (BEM) is a subject that has attracted attention mainly over
the past decade, and much work remains to be done. For recent surveys on
error estimation and adaptivity in the BEM, see Sloan [155], Kita and Kamiya
[70], Liapis [81] and Paulino et al. [124].
Many error estimators in the BEM are essentially heuristic and, unlike for
the FEM, theoretical work in this field has been quite limited. Rank [140]
proposed error indicators and an adaptive algorithm for the BEM using tech-
niques similar to those used in the FEM. Most notable is the work of Yu and
Wendland [171, 172, 181, 182], who have presented local error estimates based
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![xviii INTRODUCTION TO BOUNDARY METHODS
on a linear error-residual relation that is very effective in the FEM. More re-
cently, Carstensen et al. [18, 21, 19, 20] have presented error estimates for
the BEM analogous to the approach of Eriksson [44] for the FEM. There are
numerous stumbling blocks in the development of a satisfactory theoretical
analysis of a generic boundary value problem (BVP). First, theoretical analy-
ses are easiest for Galerkin schemes, but most engineering codes, to date, use
collocation-based methods (see, for example, Banerjee [4]). Though one can
view collocation schemes as variants of Petrov-Galerkin methods, and, in fact,
numerous theoretical analyses exist for collocation methods (see, for example,
references in [155]), the mathematical analysis for this class of problems is
difficult. Theoretical analyses for mixed boundary conditions are limited and
involved (Wendland et al. [170]) and the presence of corners and cracks has
been a source of challenging problems for many years (Sloan [155], Costabel
and Stephan [35], Costabel et al. [36]). Of course, problems with corners and
mixed boundary conditions are the ones of most practical interest, and for such
situations one has to rely mostly on numerical experiments.
During the past few years, there has been a marked interest, among mathe-
maticians in the field, in extending analyses for the BEM with singular integrals
to hypersingular integrals ([21, 19, 156, 45]. For instance, Feistauer et al. [45]
have studied the solution of the exterior Neumann problem for the Helmholtz
equation formulated as an HBIE. Their paper contains a rigorous analysis of
hypersingular integral equations and addresses the problem of noncompatibil-
ity of the residual norm, where additional hypotheses are needed to design a
practical error estimate. These authors use residuals to estimate the error,
but they do not use the BIE and the HBIE simultaneously. Finally, Goldberg
and Bowman [51] have used superconvergence of the Sloan iterate [153, 154] to
show the asymptotic equivalence of the error and the residual. They have used
Galerkin methods, an iteration scheme that uses the same integral equation for
the approximation and for the iterates, and usual residuals in their work.
Paulino [122] and Paulino et al. [123] first proposed the idea of obtaining
a hypersingular residual by substituting the BEM solution of a problem into
the hypersingular BEM (HBEM) for the same problem; and then using this
residual as an element error estimator in the BEM. It has been proved that
([95], [96], [127]), under certain conditions, this residual is related to a measure
of the local error on a boundary element, and has been used to postulate local
error estimates on that element. This idea has been applied to the collocation
BEM ([123], [96], [127]) and to the symmetric Galerkin BEM ([125]). Recently,
residuals have been obtained in the context of the BNM [28] and used to obtain
local error estimates (at the element level) and then to drive an h-adaptive
mesh refinement process. An analogous approach for error estimation and h-
adaptivity, in the context of the BCM, is described in [111]. Ref. [91] has a
bibliography of work on mesh generation and refinement up to 1993.
Error analysis with the BEM is presented in Chapter 2, while error analysis
and adaptivity in the context of the BCM and the BNM are discussed in Chapter
7, and Chapters 10, 11, respectively, of this book.
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![Chapter 1
BOUNDARY INTEGRAL
EQUATIONS
Integral equations, usual as well as hypersingular, for internal and boundary
points, for potential theory in three dimensions, are first presented in this chap-
ter. This is followed by their linear elasticity counterparts. The evaluation of
finite parts (FPs) of some of these equations, when the source point is an irreg-
ular boundary point (situated at a corner on a one-dimensional plane curve or
at a corner or edge on a two-dimensional surface), is described next.
1.1 Potential Theory in Three Dimensions
The starting point is Laplace’s equation in three dimensions (3-D) governing a
potential function u(x1, x2, x3) ∈ B, where B is a bounded region (also called
the body):
∇2
u(x1, x2, x3) ≡
∂2
u
∂x2
1
+
∂2
u
∂x2
2
+
∂2
u
∂x2
3
= 0 (1.1)
along with prescribed boundary conditions on the bounding surface ∂B of B.
1.1.1 Singular Integral Equations
Referring to Figure 1.1, let ξ and η be (internal) source and field points ∈ B
and x and y be (boundary) source and field points ∈ ∂B, respectively. (Source
and field points are also referred to as p and q (for internal points) and as P
and Q (for boundary points), respectively, in this book).
The well-known integral representation for (1.1), at an internal point ξ ∈ B,
is:
u(ξ) =
∂B
[G(ξ, y)τ(y) − F(ξ, y)u(y)]dS(y) (1.2)
3
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![4 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
B
x(P)
y(Q)
ξ(p)
r
(
ξ
,
y
)
η(q)
n(y)
n(x)
∂B
Figure 1.1: Notation used in integral equations (from [6])
An infinitesimal surface area on ∂B is dS = dSn, where n is the unit outward
normal to ∂B at a point on it and τ = ∂u/∂n. The kernels are written in terms
of source and field points ξ ∈ B and y ∈ ∂B. These are :
G(ξ, y) =
1
4πr(ξ, y)
(1.3)
F(ξ, y) =
∂G(ξ, y)
∂n(y)
=
(ξi − yi)ni(y)
4πr3(ξ, y)
(1.4)
in terms of r(ξ, y), the Euclidean distance between the source and field points
ξ and y. Unless specified otherwise, the range of indices in these and all other
equations in this chapter is 1,2,3.
An alternative form of equation (1.2) is:
u(ξ) =
∂B
[G(ξ, y)u,k(y) − Hk(ξ, y)u(y)]ek · dS(y) (1.5)
where ek, k = 1, 2, 3, are the usual Cartesian unit vectors, ek · dS(y) =
nk(y)dS(y), and:
Hk(ξ, y) =
(ξk − yk)
4πr3(ξ, y)
(1.6)
The boundary integral equation (BIE) corresponding to (1.2) is obtained by
taking the limit ξ → x. A regularized form of the resulting equation is:
0 =
∂B
[G(x, y)τ(y) − F(x, y){u(y) − u(x)}]dS(y) (1.7)
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![1.1. POTENTIAL THEORY IN THREE DIMENSIONS 5
with an alternate form (from (1.5)):
0 =
∂B
[G(x, y)u,k(y) − Hk(x, y){u(y) − u(x)}]ek · dS(y) (1.8)
1.1.2 Hypersingular Integral Equations
Equation (1.2) can be differentiated at an internal source point ξ to obtain the
gradient ∂u
∂ξm
of the potential u. The result is:
∂u(ξ)
∂ξm
=
∂B
∂G(ξ, y)
∂ξm
τ(y) −
∂F(ξ, y)
∂ξm
u(y)
dS(y) (1.9)
An interesting situation arises when one takes the limit ξ → x (x can even
be an irregular point on ∂B but one must have u(y) ∈ C1,α
at y = x) in
equation (1.9). As discussed in detail in Section 1.4.2, one obtains:
∂u(x)
∂xm
=
∂B
=
∂G(x, y)
∂xm
τ(y) −
∂F(x, y)
∂xm
u(y)
dS(y) (1.10)
where the symbol
= denotes the finite part (FP) of the integral. Equation (1.10)
is best regularized before computations are carried out. The regularized version
given below is applicable even at an irregular boundary point x provided that
u(y) ∈ C1,α
at y = x. This is:
0 =
∂B
∂G(x, y)
∂xm
u,p(y) − u,p(x)
np(y)dS(y)
−
∂B
∂F(x, y)
∂xm
u(y) − u(x) − u,p(x)(yp − xp)
dS(y) (1.11)
An alternative form of (1.11), valid at a regular boundary point x, [76] is:
0 =
∂B
∂G(x, y)
∂xm
τ(y) − τ(x)
dS(y)
− u,k(x)
B
∂G(x, y)
∂xm
nk(y) − nk(x)
dS(y)
−
∂B
∂F(x, y)
∂xm
u(y) − u(x) − u,p(x)(yp − xp)
dS(y) (1.12)
Carrying out the inner product of (1.12) with the source point normal n(x),
one gets:
0 =
∂B
∂G(x, y)
∂n(x)
τ(y) − τ(x)
dS(y)
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![6 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
− u,k(x)
B
∂G(x, y)
∂n(x)
nk(y) − nk(x)
dS(y)
−
∂B
∂F(x, y)
∂n(x)
u(y) − u(x) − u,p(x)(yp − xp)
dS(y) (1.13)
1.1.2.1 Potential gradient on the bounding surface
The gradient of the potential function is required in the regularized HBIEs (1.11
- 1.13). For potential problems, the gradient (at a regular boundary point) can
be written as,
∇u = τn +
∂u
∂s1
t1 +
∂u
∂s2
t2 (1.14)
where τ = ∂u/∂n is the flux, n is the unit normal, t1, t2 are the appropriately
chosen unit vectors in two orthogonal tangential directions on the surface of the
body, and ∂u/∂si, i = 1, 2 are the tangential derivatives of u (along t1 and t2)
on the surface of the body.
1.2 Linear Elasticity in Three Dimensions
The starting point is the Navier-Cauchy equation governing the displacement
u(x1, x2, x3) in a homogeneous, isotropic, linear elastic solid occupying the
bounded 3-D region B with boundary ∂B; in the absence of body forces:
0 = ui,jj +
1
1 − 2ν
uk,ki (1.15)
along with prescribed boundary conditions that involve the displacement and
the traction τ on ∂B. The components τi of the traction vector are:
τi = λuk,kni + µ(ui,j + uj,i)nj (1.16)
In equations (1.15) and (1.16), ν is Poisson’s ratio and λ and µ are Lamé
constants. As is well known, µ is the shear modulus of the material and is also
called G in this book. Finally, the Young’s modulus is denoted as E.
1.2.1 Singular Integral Equations
The well-known integral representation for (1.15), at an internal point ξ ∈ B
(Rizzo [141]) is:
uk(ξ) =
∂B
[Uik(ξ, y)τi(y) − Tik(ξ, y)ui(y)] dS(y) (1.17)
where uk and τk are the components of the displacement and traction respec-
tively, and the well-known Kelvin kernels are:
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![1.2. LINEAR ELASTICITY IN THREE DIMENSIONS 7
Uik =
1
16π(1 − ν)Gr
[(3 − 4ν)δik + r,ir,k] (1.18)
Tik = −
1
8π(1 − ν)r2
{(1 − 2ν)δik + 3r,ir,k}
∂r
∂n
+ (1 − 2ν)(r,ink − r,kni)
(1.19)
In the above, δik denotes the Kronecker delta and, as before, the normal n
is defined at the (boundary) field point y. A comma denotes a derivative with
respect to a field point, i.e.
r,i =
∂r
∂yi
=
yi − ξi
r
(1.20)
An alternative form of equation (1.17) is:
uk(ξ) =
∂B
[Uik(ξ, y)σij(y) − Σijk(ξ, y)ui(y)] ej · dS(y) (1.21)
where σ is the stress tensor, τi = σijnj and Tik = Σijknj. (Please note that
ej · dS(y) = nj(y)dS(y)). The explicit form of the kernel Σ is:
Σijk = Eijmn
∂Ukm
∂yn
= −
1
8π(1 − ν)r2
[ (1 − 2ν)(r,iδjk + r,jδik − r,kδij) + 3r,ir,jr,k ] (1.22)
where E is the elasticity tensor (for isotropic elasticity):
Eijmn = λδijδmn + µ[δimδjn + δinδjm] (1.23)
The boundary integral equation (BIE) corresponding to (1.17) is obtained
by taking the limit ξ → x. The result is:
uk(x) = lim
ξ→x
∂B
[Uik(ξ, y)τi(y) − Tik(ξ, y)ui(y)] dS(y)
=
∂B
= [Uik(x, y)τi(y) − Tik(x, y)ui(y)] dS(y) (1.24)
where the symbol
∂B
= denotes the finite part of the appropriate integral (see
Section 1.4).
A regularized form of equation (1.24) is:
0 =
∂B
[Uik(x, y)τi(y) − Tik(x, y){ui(y) − ui(x)}]dS(y) (1.25)
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![8 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
with an alternate form (from (1.21)):
0 =
∂B
[Uik(x, y)σij(y) − Σijk(x, y){ui(y) − ui(x)}]ej · dS(y) (1.26)
1.2.2 Hypersingular Integral Equations
Equation (1.17) can be differentiated at an internal source point ξ to obtain
the displacement gradient at this point:
∂uk(ξ)
∂ξm
=
∂B
∂Uik
∂ξm
(ξ, y)τi(y) −
∂Tik
∂ξm
(ξ, y)ui(y)
dS(y) (1.27)
An alternative form of equation (1.27) is:
∂uk(ξ)
∂ξm
=
∂B
∂Uik
∂ξm
(ξ, y)σij(y) −
∂Σijk
∂ξm
(ξ, y)ui(y)
ej · dS(y) (1.28)
Stress components at an internal point ξ can be obtained from either of
equations (1.27) or (1.28) by using Hooke’s law:
σij = λuk,kδij + µ(ui,j + uj,i) (1.29)
It is sometimes convenient, however, to write the internal stress directly.
This equation, corresponding (for example) to (1.27) is:
σij(ξ) =
∂B
[Dijk(ξ, y)τk(y) − Sijk(ξ, y)uk(y)] dS(y) (1.30)
where the new kernels D and S are:
Dijk = Eijmn
∂Ukm
∂ξn
= λ
∂Ukm
∂ξm
δij + µ
∂Uki
∂ξj
+
∂Ukj
∂ξi
= −Σijk (1.31)
Sijk = Eijmn
∂Σkpm
∂ξn
np = λ
∂Σkpm
∂ξm
npδij + µ
∂Σkpi
∂ξj
+
∂Σkpj
∂ξi
np
=
G
4π(1 − ν)r3
3
∂r
∂n
[(1 − 2ν)δijr,k + ν(δikr,j + δjkr,i) − 5r,ir,jr,k]
+
G
4π(1 − ν)r3
[3ν(nir,jr,k + njr,ir,k)
+(1 − 2ν)(3nkr,ir,j + njδik + niδjk) − (1 − 4ν)nkδij] (1.32)
Again, the normal n is defined at the (boundary) field point y. Also:
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![1.2. LINEAR ELASTICITY IN THREE DIMENSIONS 9
∂Uik
∂ξm
(ξ, y) = −Uik,m ,
∂Σijk
∂ξm
(ξ, y) = −Σijk,m (1.33)
It is important to note that D becomes strongly singular, and S hypersin-
gular as a source point approaches a field point (i.e. as r → 0).
For future use in Chapter 4, it is useful to rewrite (1.28) using (1.33). This
equation is:
uk,m(ξ) = −
∂B
[Uik,m(ξ, y)σij(y) − Σijk,m(ξ, y)ui(y)] nj(y)dS(y) (1.34)
Again, as one takes the limit ξ → x in any of the equations (1.27), (1.28) or
(1.30), one must take the finite part of the corresponding right hand side (see
Section 1.4.3). For example, (1.28) and (1.30) become, respectively:
∂uk(x)
∂xm
= lim
ξ→x
∂B
∂Uik
∂ξm
(ξ, y)σij(y) −
∂Σijk
∂ξm
(ξ, y)ui(y)
nj(y)dS(y)
=
∂B
=
∂Uik
∂xm
(x, y)σij(y) −
∂Σijk
∂xm
(x, y)ui(y)
nj(y)dS(y) (1.35)
σij(x) = lim
ξ→x
∂B
[Dijk(ξ, y)τk(y) − Sijk(ξ, y)uk(y)] dS(y)
=
∂B
= [Dijk(x, y)τk(y) − Sijk(x, y)uk(y)] dS(y) (1.36)
Also, for future reference, one notes that the traction at a boundary point
is:
τi(x) = nj(x) lim
ξ→x
∂B
[Dijk(ξ, y)τk(y) − Sijk(ξ, y)uk(y)] dS(y) (1.37)
Fully regularized forms of equations (1.35) and (1.36), that only contain
weakly singular integrals, are available in the literature (see, for example, Cruse
and Richardson [39]). These equations, that can be collocated at an irregular
point x ∈ ∂B provided that the stress and displacement fields in (1.38, 1.39)
satisfy certain smoothness requirements (see Martin et al. [93] and, also, Section
1.4.4 of this chapter) are:
0 =
∂B
Uik,m(x, y) [σij(y) − σij(x)] nj(y)dS(y)
−
∂B
Σijk,m(x, y) [ui(y) − ui(x) − ui,(x) (y − x)] nj(y)dS(y) (1.38)
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![10 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
0 =
∂B
Dijk(x, y) [σkp(y) − σkp(x)] np(y)dS(y)
−
∂B
Sijk(x, y) [uk(y) − uk(x) − uk,p(x)(yp − xp)] dS(y) (1.39)
An alternate version of (1.39) that can only be collocated at a regular point
x ∈ ∂B is:
0 =
∂B
Dijk(x, y)[τk(y) − τk(x)]dS(y)
− σkm(x)
∂B
Dijk(x, y)(nm(y) − nm(x))dS(y)
−
∂B
Sijk(x, y) [uk(y) − uk(x) − uk,m(x)(ym − xm)] dS(y) (1.40)
Finally, taking the inner product of (1.40) with the normal at the source
point gives:
0 =
∂B
Dijk(x, y)nj(x)[τk(y) − τk(x)]dS(y)
− σkm(x)
∂B
Dijk(x, y)nj(x)[nm(y) − nm(x)]dS(y)
−
∂B
Sijk(x, y)nj(x) [uk(y) − uk(x) − uk,m(x)(ym − xm)] dS(y) (1.41)
1.2.2.1 Displacement gradient on the bounding surface
The gradient of the displacement u is required for the regularized HBIEs (1.38
- 1.41). Lutz et al. [89] have proposed a scheme for carrying this out. Details
of this procedure are available in [27] and are given below.
The (right-handed) global Cartesian coordinates, as before, are (x1, x2, x3).
Consider (right-handed) local Cartesian coordinates (x
1, x
2, x
3) at a regular
point P on ∂B as shown in Figure 1.2. The local coordinate system is oriented
such that the x
1 and x
2 coordinates lie along the tangential unit vectors t1
and t2 while x
3 is measured along the outward normal unit vector n to ∂B as
defined in equation (1.14).
Therefore, one has:
x
= Qx (1.42)
u
= Qu (1.43)
where u
k, k = 1, 2, 3 are the components of the displacement vector u in the
local coordinate frame, and the orthogonal transformation matrix Q has the
components:
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![1.2. LINEAR ELASTICITY IN THREE DIMENSIONS 11
x
x
x
x
x
1
2
3
3
1
2
'
'
'
x
P
Figure 1.2: Local coordinate system on the surface of a body (from [27])
Q =
t11 t12 t13
t21 t22 t23
n1 n2 n3
(1.44)
with tij the jth
component of the ith
unit tangent vector and (n1, n2, n3) the
components of the unit normal vector.
The tangential derivatives of the displacement, in local coordinates, are
u
i,k , i = 1, 2, 3; k = 1, 2. These quantities are obtained as follows:
u
i,k ≡
∂u
i
∂sk
= Qij
∂uj
∂sk
(1.45)
where ∂u
i/∂sk are tangential derivatives of ui at P with s1 = x
1 and s2 = x
2.
The remaining components of ∇u in local coordinates are obtained from
Hooke’s law (see [89]) as:
∂u
1
∂x
3
=
τ
1
G
−
∂u
3
∂x
1
∂u
2
∂x
3
=
τ
2
G
−
∂u
3
∂x
2
∂u
3
∂x
3
=
(1 − 2ν)τ
3
2G(1 − ν)
−
ν
1 − ν
∂u
1
∂x
1
+
∂u
2
∂x
2
(1.46)
where τ
k, k = 1, 2, 3, are the components of the traction vector in local coordi-
nates.
The components of the displacement gradient tensor, in the local coordinate
system, are now known. They can be written as:
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![12 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
(∇u)local ≡ A
=
u
1,1 u
1,2 u
1,3
u
2,1 u
2,2 u
2,3
u
3,1 u
3,2 u
3,3
(1.47)
Finally, the components of ∇u in the global coordinate frame are obtained
from those in the local coordinate frame by using the tensor transformation
rule:
(∇u)global ≡ A = QT
A
Q =
u1,1 u1,2 u1,3
u2,1 u2,2 u2,3
u3,1 u3,2 u3,3
(1.48)
The gradient of the displacement field in global coordinates is now ready for
use in equations (1.38 - 1.41).
1.3 Nearly Singular Integrals in Linear Elastic-
ity
It is well known that the first step in the BEM is to solve the primary problem
on the bounding surface of a body (e.g. equation (1.25)) and obtain all the
displacements and tractions on this surface. The next steps are to obtain the
displacements and stresses at selected points inside a body, from equations
such as (1.17) and (1.30). It has been known in the BEM community for many
years, dating back to Cruse [37], that one experiences difficulties when trying
to numerically evaluate displacements and stresses at points inside a body that
are close to its bounding surface (the so-called near-singular or boundary layer
problem). Various authors have addressed this issue over the last 3 decades.
This section describes a new method recently proposed by Mukherjee et al.
[104].
1.3.1 Displacements at Internal Points Close to the Bound-
ary
The displacement at a point inside an elastic body can be determined from
either of the (equivalent) equations (1.17) or (1.21). A continuous version of
(1.21), from Cruse and Richardson [39] is:
uk(ξ) = uk
ˆ
(x)+
∂B
[ Uik(ξ, y)σij(y) − Σijk(ξ, y){ui(y) − ui(x̂)} ] nj(y)dS(y)
(1.49)
where ξ ∈ B is an internal point close to ∂B and a target point x̂ ∈ ∂B is close
to the point ξ (see Fig. 1.3). An alternative form of (1.49) is:
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![1.3. NEARLY SINGULAR INTEGRALS IN LINEAR ELASTICITY 13
B
∂B
ξ
z
z
v
x
v
y
Figure 1.3: A body with source point ξ, field point y and target point x̂
(from [104])
uk(ξ) = uk(x̂)+
∂B
[ Uik(ξ, y)τi(y) − Tik(ξ, y){ui(y) − ui(x̂)} ] dS(y) (1.50)
Equation (1.49) (or (1.50)) is called “continuous” since it has a continuous
limit to the boundary (LTB as ξ → x̂ ∈ ∂B) provided that ui(y) ∈ C0,α
(i.e.
Hölder continuous). Taking this limit is the standard approach for obtaining
the well-known regularized form (1.26) (or (1.25)).
In this work, however, equation (1.49) (or (1.50)) is put to a different, and
novel use. It is first observed that Tik in equation (1.50) is O(1/r2
(ξ, y)) as
ξ → y, whereas {ui(y) − ui(x̂)} is O(r(x̂, y)) as y → x̂. Therefore, as y → x̂,
the product Tik(ξ, y){ui(y) − ui(x̂)}, which is O(r(x̂, y)/r2
(ξ, y)), → 0 ! As a
result, equation (1.50) (or (1.49)) can be used to easily and accurately evaluate
the displacement components uk(ξ) for ξ ∈ B close to ∂B. This idea is the
main contribution of [104].
It is noted here that while it is usual to use (1.17) (or (1.21)) to evaluate
uk(ξ) when ξ is far from ∂B, equation (1.49) (or (1.50)) is also valid in this
case. (The target point x̂ can be chosen as any point on ∂B when ξ is far from
∂B). Therefore, it is advisable to use the continuous equation (1.49) (or (1.50))
universally for all points ξ ∈ B. This procedure would eliminate the need to
classify, a priori, whether ξ is near to, or far from ∂B.
1.3.2 Stresses at Internal Points Close to the Boundary
The displacement gradient at a point ξ ∈ B can be obtained from equation
(1.34) or the stress at this point from (1.30). Continuous versions of (1.34) and
(1.30) can be written as [39]:
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![14 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
uk,n(ξ) = uk,n(x̂) −
∂B
Uik,n(ξ, y) [σij(y) − σij(x̂)] nj(y)dS(y)
+
∂B
Σijk,n(ξ, y) [ui(y) − ui(x̂) − ui,(x̂) (y − x̂)] nj(y)dS(y) (1.51)
σij(ξ) = σij(x̂) +
∂B
Dijk(ξ, y)[τk(y) − σkm(x̂)nm(y)]dS(y)
−
∂B
Sijk(ξ, y)[uk(y) − uk(x̂) − uk,(x̂)(y − x̂)] dS(y) (1.52)
The integrands in equations (1.51) (or (1.52)) are O(r(x̂, y)/r2
(ξ, y)) and
O(r2
(x̂, y)/r3
(ξ, y)) as y → x̂. Similar to the behavior of the continuous BIEs
in the previous subsection, the integrands in equations (1.51) and (1.52) → 0
as y → x̂. Either of these equations, therefore, is very useful for evaluating the
stresses at an internal point ξ that is close to ∂B. Of course (please see the
discussion regarding displacements in the previous section), they can also be
conveniently used to evaluate displacement gradients or stresses at any point
ξ ∈ B.
Henceforth, use of equations (1.17), (1.21), (1.30) or (1.34) will be referred
to as the standard method, while use of equations (1.49), (1.50), (1.51) or (1.52)
will be referred to as the new method.
1.4 Finite Parts of Hypersingular Equations
A discussion of finite parts (FPs) of hypersingular BIEs (see e.g. equations
(1.9 -1.11)) is the subject of this section. The general theory of finite parts
is presented first. This is followed by applications of the theory in potential
theory and in linear elasticity. Further details are available in Mukherjee [102].
1.4.1 Finite Part of a Hypersingular Integral Collocated
at an Irregular Boundary Point
1.4.1.1 Definition
Consider, for specificity, the space R3
, and let S be a surface in R3
. Let the
points x ∈ S and ξ /
∈ S. Also, let Ŝ and S̄ ⊂ Ŝ be two neighborhoods (in S) of
x such that x ∈ S̄ (Figure 1.4). The point x can be an irregular point on S.
Let the function K(x, y) , y ∈ S, have its only singularity at x = y of
the form 1/r3
where r = |x − y |, and let φ(y) be a function that has no
singularity in S and is of class C1,α
at y = x for some α 0.
The finite part of the integral
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![1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 15
S
S
x
S
y S
ξ
S
S
Figure 1.4: A surface S with regions Ŝ and S̄ and points ξ, x and y (from [102])
I(x) =
S
K(x, y)φ(y)dS(y) (1.53)
is defined as:
S
= K(x, y)φ(y)dS(y) =
SŜ
K(x, y)φ(y)dS(y)
+
Ŝ
K(x, y)[φ(y) − φ(x) − φ,p(x)(yp − xp)]dS(y)
+ φ(x)A(Ŝ) + φ,p(x)Bp(Ŝ) (1.54)
where Ŝ is any arbitrary neighborhood (in S) of x and:
A(Ŝ) =
Ŝ
= K(x, y)dS(y) (1.55)
Bp(Ŝ) =
Ŝ
= K(x, y)(yp − xp)dS(y) (1.56)
The above FP definition can be easily extended to any number of physical
dimensions and any order of singularity of the kernel function K(x, y). Please
refer to Toh and Mukherjee [168] for further discussion of a previous closely re-
lated FP definition for the case when x is a regular point on S, and to Mukherjee
[101] for a discussion of the relationship of this FP to the CPV of an integral
when its CPV exists.
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![16 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
1.4.1.2 Evaluation of A and B
There are several equivalent ways for evaluating A and B.
Method one. Replace S by Ŝ and Ŝ by S̄ in equation (1.54). Now, setting
φ(y) = 1 in (1.54) and using (1.55), one gets:
A(Ŝ) − A(S̄) =
ŜS̄
K(x, y)dS(y) (1.57)
Next, setting φ(y) = (yp − xp) (note that, in this case, φ(x) = 0 and
φ,p(x) = 1) in (1.54), and using (1.56), one gets:
Bp(Ŝ) − Bp(S̄) =
ŜS̄
K(x, y)(yp − xp)dS(y) (1.58)
The formulae (1.57) and (1.58) are most useful for obtaining A and B when
Ŝ is an open surface and Stoke regularization is employed. An example is the
application of the FP definition (1.54) (for a regular collocation point) in Toh
and Mukherjee [168], to regularize a hypersingular integral that appears in the
HBIE formulation for the scattering of acoustic waves by a thin scatterer. The
resulting regularized equation is shown in [168] to be equivalent to the result of
Krishnasamy et al. [75]. Equations (1.57) and (1.58) are also used in Mukherjee
and Mukherjee [99] and in Section 3.2 of [102].
Method two. From equation (1.57):
A(Ŝ) − A(S̄) =
ŜS̄
K(x, y)dS(y) = lim
ξ→x
ŜS̄
K(ξ, y)dS(y) (1.59)
The second equality above holds since K(x, y) is regular for x ∈ S̄ and
y ∈ ŜS̄. Assuming that the limits:
lim
ξ→x
Ŝ
K(ξ, y)dS(y), lim
ξ→x
S̄
K(ξ, y)dS(y)
exist, then:
A(Ŝ) = lim
ξ→x
Ŝ
K(ξ, y)dS(y) (1.60)
Similarly:
Bp(Ŝ) = lim
ξ→x
Ŝ
K(ξ, y)(yp − xp)dS(y) (1.61)
Equations (1.60) and (1.61) are most useful for evaluating A and B when
Ŝ = ∂B, a closed surface that is the entire boundary of a body B. Examples
appear in Sections 1.4.2 and 1.4.3 of this chapter.
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![1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 17
Method three. A third way for evaluation of A and B is to use an auxiliary
surface (or “tent”) as first proposed for fracture mechanics analysis by Lutz et
al. [89]. (see, also, Mukherjee et al. [110], Mukherjee [105] and Section 3.2.1 of
[102]. This method is useful if S is an open surface.
1.4.1.3 The FP and the LTB
There is a very simple connection between the FP, defined above, and the
LTB approach employed by Gray and his coauthors. With, as before, ξ /
∈ S,
x ∈ S (x can be an irregular point on S), K(x, y) = O(|x − y|−3
) as y → x
and φ(y) ∈ C1,α
at y = x, this can be stated as:
lim
ξ→x
S
K(ξ, y)φ(y)dS(y) =
S
= K(x, y)φ(y)dS(y) (1.62)
Of course, ξ can approach x from either side of S.
Proof of equation (1.62). Consider the first and second terms on the right-
hand side of equation (1.54). Since these integrands are regular in their respec-
tive domains of integration, one has:
SŜ
K(x, y)φ(y)dS(y) = lim
ξ→x
SŜ
K(ξ, y)φ(y)dS(y) (1.63)
and
Ŝ
K(x, y)[φ(y) − φ(x) − φ,p(x) (yp − xp)]dS(y)
= lim
ξ→x
Ŝ
K(ξ, y)[φ(y) − φ(ξ) − φ,p(ξ)(yp − ξp)]dS(y) (1.64)
Use of equations (1.60, 1.61, 1.63 and 1.64) in (1.54) proves equation (1.62).
1.4.2 Gradient BIE for 3-D Laplace’s Equation
This section is concerned with an application of equation (1.54) for collocation
of the HBIE (1.9), for the 3-D Laplace equation, at an irregular boundary point.
A complete exclusion zone, Ŝ = ∂B is used here. An application of a vanishing
exclusion zone, for collocation of the HBIE for the 2-D Laplace equation, at an
irregular boundary point, is presented in Mukherjee [102].
Using equations (1.4) and (1.6), equations (1.9) and (1.10) are first written
in the slightly different equivalent forms:
∂u(ξ)
∂ξi
=
∂B
[Di(ξ, y)τ(y) − Si(ξ, y)u(y)] dS(y) (1.65)
∂u(x)
∂xi
=
∂B
= [Di(x, y)τ(y) − Si(x, y)u(y)] dS(y) (1.66)
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![18 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
where:
Di(x, y) = −G,i(x, y) , Si = −Hk,i(x, y)nk(y) (1.67)
Use of (1.54) in (1.66), with S = Ŝ = ∂B, results in:
u,i(x) =
∂B
Di(x, y)
u,p(y) − u,p(x)
np(y)dS(y)
−
∂B
Si(x, y)
u(y) − u(x) − u,p(x)(yp − xp)
dS(y)
− Ai(∂B)u(x) + Cip(∂B)u,p(x) (1.68)
where, using method two in Section 1.4.1.2:
Ai(∂B) = lim
ξ→x
∂B
Si(ξ, y)dS(y) (1.69)
Cip(∂B) = lim
ξ→x
∂B
[Di(ξ, y)np(y) − Si(ξ, y)(yp − ξp)] dS(y) (1.70)
It is noted here that the (possibly irregular) boundary point x is approached
from ξ ∈ B, i.e. from inside the body B.
The quantities A and C can be easily evaluated using the imposition of
simple solutions. Following Rudolphi [143], use of the uniform solution u(y) = c
(c is a constant) in equation (1.65) gives:
∂B
Si(ξ, y)dS(y) = 0 (1.71)
while use of the linear solution:
u = u(ξ) + (yp − ξp)u,p(ξ)
τ(y) =
∂u
∂yk
nk(y) = u,p(ξ)np(y) (with p = 1, 2, 3) (1.72)
in equation (1.65) (together with (1.71)) gives:
∂B
[Di(ξ, y)np(y) − Si(ξ, y)(yp − ξp)] dS(y) = δip (1.73)
Therefore, (assuming continuity) Ai(∂B) = 0, Cip(∂B) = δip, and (1.68)
yields a simple, fully regularized form of (1.66) as:
0 =
∂B
Di(x, y)[u,p(y) − u,p(x)]np(y)dS(y)
−
∂B
Si(x, y)[u(y) − u(x) − u,p(x)(yp − xp)]dS(y) (1.74)
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![1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 19
which is equivalent to equation (1.11).
A few comments are in order. First, equation (1.74) is the same as Rudol-
phi’s [143] equation (20) with (his) κ = 1 and (his) S0 set equal to S and
renamed ∂B. (See, also, Kane [68], equation (17.34)). Second, this equation
can also be shown to be valid for the case ξ /
∈ B, i.e. for an outside approach
to the boundary point x . Third, as noted before, x can be an edge or corner
point on ∂B (provided, of course, that u(y) ∈ C1,α
at y = x - Rudolphi
had only considered a regular boundary collocation point in his excellent paper
that was published in 1991). Finally, as discussed in the Section 1.4.3, equation
(1.74) is analogous to the regularized stress BIE in linear elasticity - equation
(28) in Cruse and Richardson [39] .
1.4.3 Stress BIE for 3-D Elasticity
This section presents a proof of the fact that equation (1.39) is a regularized
version of (1.36), valid at an irregular point x ∈ ∂B, provided that the stress
and displacement fields in (1.39) satisfy certain smoothness requirements. These
smoothness requirements are discussed in Section 1.4.4. The approach is very
similar to that used in Section 1.4.2.
The first step is to apply the FP equation (1.54) to regularize (1.36). With
S = Ŝ = ∂B, the result is:
σij(x) =
∂B
Dijk(x, y) [σkp(y) − σkp(x)] np(y)dS(y)
−
∂B
Sijk(x, y) [uk(y) − uk(x) − uk,p(x)(yp − xp)] dS(y)
− Aijk(∂B)uk(x) + Cijkp(∂B)uk,p(x) (1.75)
where, using method two in Section 1.4.1.2:
Aijk(∂B) = lim
ξ→x
∂B
Sijk(ξ, y)dS(y) (1.76)
Cijkp(∂B) = lim
ξ→x
∂B
EmkpDijm(ξ, y)n(y)dS(y)
− lim
ξ→x
∂B
Sijk(ξ, y)(yp − ξp)dS(y) (1.77)
with E the elasticity tensor (see (1.23)) which appears in Hooke’s law:
σm = Emkpuk,p (1.78)
Simple (rigid body and linear) solutions in linear elasticity (see, for example,
Lutz et al. [89], Cruse and Richardson [39]) are now used in order to determine
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![20 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS
the quantities A and C. Using the rigid body mode uk = ck (ck are arbitrary
constants) in (1.30), one has:
0 =
∂B
Sijk(ξ, y)dS(y) (1.79)
while, using the linear solution:
uk(y) = (yp − ξp)uk,p(ξ), uk,m(y) = uk,m(ξ),
τk(y) = σkm(y)nm(y) = Ekmrsur,s(ξ)nm(y) (1.80)
in equation (1.30) gives:
σij(ξ) = uk,p(ξ)
∂B
[EmkpDijm(ξ, y)n(y) − Sijk(ξ, y)(yp − ξp)] dS(y)
(1.81)
Taking the limit ξ → x of (1.79), using continuity of the integral and com-
paring with (1.76), gives A = 0. Taking the limit ξ → x of (1.81) and comparing
with (1.77), one has:
σij(x) = Cijkpuk,p(x) (1.82)
Comparing (1.82) with (1.78) yields C(∂B) = E.
Therefore, equation (1.75) reduces to the simple regularized equation (1.39).
Equation (1.39) is equation (28) of Cruse and Richardson [39] in the present
notation. As is the case in the present work, Cruse and Richardson [39] have
also proved that their equation (28) is valid at a corner point, provided that
the stress is continuous there.
It has been proved in this section that the regularized stress BIE (28) of
Cruse and Richardson [39] can also be obtained from the FP definition (1.54)
with a complete exclusion zone.
1.4.4 Solution Strategy for a HBIE Collocated at an Ir-
regular Boundary Point
Hypersingular BIEs for a body B with boundary ∂B are considered here. Regu-
larized HBIEs, obtained by using complete exclusion zones, e.g. equation (1.74)
for potential theory or (1.39) for linear elasticity, are recommended as starting
points.
An irregular collocation point x for 3-D problems is considered next. Let
∂Bn, (n = 1, 2, 3, ..., N) be smooth pieces of ∂B that meet at an irregular point
x ∈ ∂B. Also, as before, let a source point, with coordinates xk, be denoted by
P, and a field point, with coordinates yk, be denoted by Q.
Martin et al. [93] state the following requirements for collocating a regular-
ized HBIE, such as (1.39) at an irregular point P ∈ ∂B. These are:
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![1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 21
(i) The displacement u must satisfy the equilibrium equations in B.
(ii) (a) The stress σ must be continuous in B.
(b) The stress σ must be continuous on ∂B.
(iii) |ui(Qn) − ui
L
(Qn; P)| = O(r
(1+α)
n ) as rn → 0, for each n.
(iv) [σij(Qn) − σij(P)]nj(Qn) = O(rα
n) as rn → 0, for each n.
Box 1.1 Requirements for collocation of a HBIE at an irregular point
(from [93]).
In the above, rn = |y(Qn) − x(P)|, Qn ∈ ∂Bn, and α 0. Also,
uL
i (Qn; P) = ui(P) + ui,j(P)[yj(Qn) − xj(P)] (1.83)
There are two important issues to consider here.
The first is that, if there is to be any hope for collocating (1.39) at an
irregular point P, the exact solution of a boundary value problem must satisfy
conditions (i-iv) in Box 1.1. Clearly, one should not attempt this collocation
if, for example, the stress is unbounded at P (this can easily happen - see an
exhaustive study on the subject in Glushkov et al. [50]), or is bounded but
discontinuous at P (e.g. at the tip of a wedge - see, for example, Zhang and
Mukherjee [183]). The discussion in the rest of this book is limited to the class
of problems, referred to as the admissible class, whose exact solutions satisfy
conditions (i - iv).
The second issue refers to smoothness requirements on the interpolation
functions for u, σ and the traction τ = n · σ in (1.39). It has proved very
difficult, in practice, to find BEM interpolation functions that satisfy, a priori,
(ii(b)-(iv)) in Box 1.1, for collocation at an irregular surface point on a 3-D
body [93]. It has recently been proved in Mukherjee and Mukherjee [111],
however, that interpolation functions used in the boundary contour method
(BCM - see, for example, Mukherjee et al. [109], Mukherjee and Mukherjee
[99]) satisfy these conditions a priori. Another important advantage of using
these interpolation functions is that ∇u can be directly computed from them
at an irregular boundary point [99], without the need to use the (undefined)
normal and tangent vectors at this point. In principle, these BCM interpolation
functions can also be used in the BEM.
The BCM and the hypersingular BCM (HBCM) are discussed in detail
in Chapter 4 of this book. Numerical results from the hypersingular BCM,
collocated on edges and at corners, from Mukherjee and Mukherjee [111], are
available in Chapter 4.
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![Chapter 2
ERROR ESTIMATION
Pointwise (i.e. that the error is evaluated at selected points) residual-based error
estimates for Dirichlet, Neumann and mixed boundary value problems (BVPs)
in linear elasticity are presented first in this chapter. Interesting relationships
between the actual error and the hypersingular residuals are proved for the first
two classes of problems, while heuristic error estimators are presented for mixed
BVPs. Element-based error indicators, relying on the pointwise error measures
presented earlier, are proposed next. Numerical results for two mixed BVPs
in 2-D linear elasticity complete this chapter. Further details are available in
[127].
2.1 Linear Operators
Boundary integral equations can be analyzed by viewing them as linear equa-
tions in a Hilbert space. A very readable account of this topic is available in
Kress [73]. Following Sloan [155], it is assumed here that the boundary ∂B is
a C1
continuous closed Jordan curve given by the mapping:
z : [0, 1] → ∂B, z ∈ C1
, |z
| = 0
where z ∈ C, the space of complex numbers. The present analysis excludes
domains with corners. It is also assumed that any integrable function v on ∂B
may be represented in a Fourier series:
v ∼
∞
k=−∞
v̂(k)e2πikx1
= a0 +
∞
k=1
(ak cos(2πkx1) + bk sin(2πkx1) (2.1)
where i ≡
√
−1 and:
v̂(k) =
1
0
e−2πikx1
v(x1)dx1, k ∈ Z (2.2)
23
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- 5. MECHANICAL ENGINEERING A Series of Textbooks and Reference Books Founding Editor L. L. Faulkner Columbus Division, Battelle Memorial Institute and Department of Mechanical Engineering The Ohio State University Columbus, Ohio 1. Spring Designer’s Handbook, Harold Carlson 2. Computer-Aided Graphics and Design, Daniel L. Ryan 3. Lubrication Fundamentals, J. George Wills 4. Solar Engineering for Domestic Buildings, William A. Himmelman 5. Applied Engineering Mechanics: Statics and Dynamics, G. Boothroyd and C. Poli 6. Centrifugal Pump Clinic, Igor J. Karassik 7. Computer-Aided Kinetics for Machine Design, Daniel L. Ryan 8. Plastics Products Design Handbook, Part A: Materials and Components; Part B: Processes and Design for Processes, edited by Edward Miller 9. Turbomachinery: Basic Theory and Applications, Earl Logan, Jr. 10. Vibrations of Shells and Plates, Werner Soedel 11. Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni 12. Practical Stress Analysis in Engineering Design, Alexander Blake 13. An Introduction to the Design and Behavior of Bolted Joints, John H. Bickford 14. Optimal Engineering Design: Principles and Applications, James N. Siddall 15. Spring Manufacturing Handbook, Harold Carlson 16. Industrial Noise Control: Fundamentals and Applications, edited by Lewis H. Bell 17. Gears and Their Vibration: A Basic Approach to Understanding Gear Noise, J. Derek Smith 18. Chains for Power Transmission and Material Handling: Design and Appli- cations Handbook, American Chain Association 19. Corrosion and Corrosion Protection Handbook, edited by Philip A. Schweitzer 20. Gear Drive Systems: Design and Application, Peter Lynwander 21. Controlling In-Plant Airborne Contaminants: Systems Design and Calculations, John D. Constance 22. CAD/CAM Systems Planning and Implementation, Charles S. Knox 23. Probabilistic Engineering Design: Principles and Applications, James N. Siddall DK3139_series.qxd 1/20/05 11:17 AM Page 1 © 2005 by Taylor & Francis Group, LLC
- 6. 24. Traction Drives: Selection and Application, Frederick W. Heilich III and Eugene E. Shube 25. Finite Element Methods: An Introduction, Ronald L. Huston and Chris E. Passerello 26. Mechanical Fastening of Plastics: An Engineering Handbook, Brayton Lincoln, Kenneth J. Gomes, and James F. Braden 27. Lubrication in Practice: Second Edition, edited by W. S. Robertson 28. Principles of Automated Drafting, Daniel L. Ryan 29. Practical Seal Design, edited by Leonard J. Martini 30. Engineering Documentation for CAD/CAM Applications, Charles S. Knox 31. Design Dimensioning with Computer Graphics Applications, Jerome C. Lange 32. Mechanism Analysis: Simplified Graphical and Analytical Techniques, Lyndon O. Barton 33. CAD/CAM Systems: Justification, Implementation, Productivity Measure- ment, Edward J. Preston, George W. Crawford, and Mark E. Coticchia 34. Steam Plant Calculations Manual, V. Ganapathy 35. Design Assurance for Engineers and Managers, John A. Burgess 36. Heat Transfer Fluids and Systems for Process and Energy Applications, Jasbir Singh 37. Potential Flows: Computer Graphic Solutions, Robert H. Kirchhoff 38. Computer-Aided Graphics and Design: Second Edition, Daniel L. Ryan 39. Electronically Controlled Proportional Valves: Selection and Application, Michael J. Tonyan, edited by Tobi Goldoftas 40. Pressure Gauge Handbook, AMETEK, U.S. Gauge Division, edited by Philip W. Harland 41. Fabric Filtration for Combustion Sources: Fundamentals and Basic Technology, R. P. Donovan 42. Design of Mechanical Joints, Alexander Blake 43. CAD/CAM Dictionary, Edward J. Preston, George W. Crawford, and Mark E. Coticchia 44. Machinery Adhesives for Locking, Retaining, and Sealing, Girard S. Haviland 45. Couplings and Joints: Design, Selection, and Application, Jon R. Mancuso 46. Shaft Alignment Handbook, John Piotrowski 47. BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid Flow, and Heat Transfer, V. Ganapathy 48. Solving Mechanical Design Problems with Computer Graphics, Jerome C. Lange 49. Plastics Gearing: Selection and Application, Clifford E. Adams 50. Clutches and Brakes: Design and Selection, William C. Orthwein 51. Transducers in Mechanical and Electronic Design, Harry L. Trietley 52. Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena, edited by Lawrence E. Murr, Karl P. Staudhammer, and Marc A. Meyers 53. Magnesium Products Design, Robert S. Busk 54. How to Integrate CAD/CAM Systems: Management and Technology, William D. Engelke DK3139_series.qxd 1/20/05 11:17 AM Page 2 © 2005 by Taylor & Francis Group, LLC
- 7. 55. Cam Design and Manufacture: Second Edition; with cam design software for the IBM PC and compatibles, disk included, Preben W. Jensen 56. Solid-State AC Motor Controls: Selection and Application, Sylvester Campbell 57. Fundamentals of Robotics, David D. Ardayfio 58. Belt Selection and Application for Engineers, edited by Wallace D. Erickson 59. Developing Three-Dimensional CAD Software with the IBM PC, C. Stan Wei 60. Organizing Data for CIM Applications, Charles S. Knox, with contributions by Thomas C. Boos, Ross S. Culverhouse, and Paul F. Muchnicki 61. Computer-Aided Simulation in Railway Dynamics, by Rao V. Dukkipati and Joseph R. Amyot 62. Fiber-Reinforced Composites: Materials, Manufacturing, and Design, P. K. Mallick 63. Photoelectric Sensors and Controls: Selection and Application, Scott M. Juds 64. Finite Element Analysis with Personal Computers, Edward R. Champion, Jr. and J. Michael Ensminger 65. Ultrasonics: Fundamentals, Technology, Applications: Second Edition, Revised and Expanded, Dale Ensminger 66. Applied Finite Element Modeling: Practical Problem Solving for Engineers, Jeffrey M. Steele 67. Measurement and Instrumentation in Engineering: Principles and Basic Laboratory Experiments, Francis S. Tse and Ivan E. Morse 68. Centrifugal Pump Clinic: Second Edition, Revised and Expanded, Igor J. Karassik 69. Practical Stress Analysis in Engineering Design: Second Edition, Revised and Expanded, Alexander Blake 70. An Introduction to the Design and Behavior of Bolted Joints: Second Edition, Revised and Expanded, John H. Bickford 71. High Vacuum Technology: A Practical Guide, Marsbed H. Hablanian 72. Pressure Sensors: Selection and Application, Duane Tandeske 73. Zinc Handbook: Properties, Processing, and Use in Design, Frank Porter 74. Thermal Fatigue of Metals, Andrzej Weronski and Tadeusz Hejwowski 75. Classical and Modern Mechanisms for Engineers and Inventors, Preben W. Jensen 76. Handbook of Electronic Package Design, edited by Michael Pecht 77. Shock-Wave and High-Strain-Rate Phenomena in Materials, edited by Marc A. Meyers, Lawrence E. Murr, and Karl P. Staudhammer 78. Industrial Refrigeration: Principles, Design and Applications, P. C. Koelet 79. Applied Combustion, Eugene L. Keating 80. Engine Oils and Automotive Lubrication, edited by Wilfried J. Bartz 81. Mechanism Analysis: Simplified and Graphical Techniques, Second Edition, Revised and Expanded, Lyndon O. Barton 82. Fundamental Fluid Mechanics for the Practicing Engineer, James W. Murdock 83. Fiber-Reinforced Composites: Materials, Manufacturing, and Design, Second Edition, Revised and Expanded, P. K. Mallick 84. Numerical Methods for Engineering Applications, Edward R. Champion, Jr. DK3139_series.qxd 1/20/05 11:17 AM Page 3 © 2005 by Taylor & Francis Group, LLC
- 8. 85. Turbomachinery: Basic Theory and Applications, Second Edition, Revised and Expanded, Earl Logan, Jr. 86. Vibrations of Shells and Plates: Second Edition, Revised and Expanded, Werner Soedel 87. Steam Plant Calculations Manual: Second Edition, Revised and Expanded, V. Ganapathy 88. Industrial Noise Control: Fundamentals and Applications, Second Edition, Revised and Expanded, Lewis H. Bell and Douglas H. Bell 89. Finite Elements: Their Design and Performance, Richard H. MacNeal 90. Mechanical Properties of Polymers and Composites: Second Edition, Revised and Expanded, Lawrence E. Nielsen and Robert F. Landel 91. Mechanical Wear Prediction and Prevention, Raymond G. Bayer 92. Mechanical Power Transmission Components, edited by David W. South and Jon R. Mancuso 93. Handbook of Turbomachinery, edited by Earl Logan, Jr. 94. Engineering Documentation Control Practices and Procedures, Ray E. Monahan 95. Refractory Linings Thermomechanical Design and Applications, Charles A. Schacht 96. Geometric Dimensioning and Tolerancing: Applications and Techniques for Use in Design, Manufacturing, and Inspection, James D. Meadows 97. An Introduction to the Design and Behavior of Bolted Joints: Third Edition, Revised and Expanded, John H. Bickford 98. Shaft Alignment Handbook: Second Edition, Revised and Expanded, John Piotrowski 99. Computer-Aided Design of Polymer-Matrix Composite Structures, edited by Suong Van Hoa 100. Friction Science and Technology, Peter J. Blau 101. Introduction to Plastics and Composites: Mechanical Properties and Engineering Applications, Edward Miller 102. Practical Fracture Mechanics in Design, Alexander Blake 103. Pump Characteristics and Applications, Michael W. Volk 104. Optical Principles and Technology for Engineers, James E. Stewart 105. Optimizing the Shape of Mechanical Elements and Structures, A. A. Seireg and Jorge Rodriguez 106. Kinematics and Dynamics of Machinery, Vladimír Stejskal and Michael Valásek 107. Shaft Seals for Dynamic Applications, Les Horve 108. Reliability-Based Mechanical Design, edited by Thomas A. Cruse 109. Mechanical Fastening, Joining, and Assembly, James A. Speck 110. Turbomachinery Fluid Dynamics and Heat Transfer, edited by Chunill Hah 111. High-Vacuum Technology: A Practical Guide, Second Edition, Revised and Expanded, Marsbed H. Hablanian 112. Geometric Dimensioning and Tolerancing: Workbook and Answerbook, James D. Meadows 113. Handbook of Materials Selection for Engineering Applications, edited by G. T. Murray DK3139_series.qxd 1/20/05 11:17 AM Page 4 © 2005 by Taylor & Francis Group, LLC
- 9. 114. Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and Reinhard Hanselka 115. Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven M. Lepi 116. Applied Computational Fluid Dynamics, edited by Vijay K. Garg 117. Fluid Sealing Technology, Heinz K. Muller and Bernard S. Nau 118. Friction and Lubrication in Mechanical Design, A. A. Seireg 119. Influence Functions and Matrices, Yuri A. Melnikov 120. Mechanical Analysis of Electronic Packaging Systems, Stephen A. McKeown 121. Couplings and Joints: Design, Selection, and Application, Second Edition, Revised and Expanded, Jon R. Mancuso 122. Thermodynamics: Processes and Applications, Earl Logan, Jr. 123. Gear Noise and Vibration, J. Derek Smith 124. Practical Fluid Mechanics for Engineering Applications, John J. Bloomer 125. Handbook of Hydraulic Fluid Technology, edited by George E. Totten 126. Heat Exchanger Design Handbook, T. Kuppan 127. Designing for Product Sound Quality, Richard H. Lyon 128. Probability Applications in Mechanical Design, Franklin E. Fisher and Joy R. Fisher 129. Nickel Alloys, edited by Ulrich Heubner 130. Rotating Machinery Vibration: Problem Analysis and Troubleshooting, Maurice L. Adams, Jr. 131. Formulas for Dynamic Analysis, Ronald L. Huston and C. Q. Liu 132. Handbook of Machinery Dynamics, Lynn L. Faulkner and Earl Logan, Jr. 133. Rapid Prototyping Technology: Selection and Application, Kenneth G. Cooper 134. Reciprocating Machinery Dynamics: Design and Analysis, Abdulla S. Rangwala 135. Maintenance Excellence: Optimizing Equipment Life-Cycle Decisions, edited by John D. Campbell and Andrew K. S. Jardine 136. Practical Guide to Industrial Boiler Systems, Ralph L. Vandagriff 137. Lubrication Fundamentals: Second Edition, Revised and Expanded, D. M. Pirro and A. A. Wessol 138. Mechanical Life Cycle Handbook: Good Environmental Design and Manufacturing, edited by Mahendra S. Hundal 139. Micromachining of Engineering Materials, edited by Joseph McGeough 140. Control Strategies for Dynamic Systems: Design and Implementation, John H. Lumkes, Jr. 141. Practical Guide to Pressure Vessel Manufacturing, Sunil Pullarcot 142. Nondestructive Evaluation: Theory, Techniques, and Applications, edited by Peter J. Shull 143. Diesel Engine Engineering: Thermodynamics, Dynamics, Design, and Control, Andrei Makartchouk 144. Handbook of Machine Tool Analysis, Ioan D. Marinescu, Constantin Ispas, and Dan Boboc DK3139_series.qxd 1/20/05 11:17 AM Page 5 © 2005 by Taylor & Francis Group, LLC
- 10. 145. Implementing Concurrent Engineering in Small Companies, Susan Carlson Skalak 146. Practical Guide to the Packaging of Electronics: Thermal and Mechanical Design and Analysis, Ali Jamnia 147. Bearing Design in Machinery: Engineering Tribology and Lubrication, Avraham Harnoy 148. Mechanical Reliability Improvement: Probability and Statistics for Experimental Testing, R. E. Little 149. Industrial Boilers and Heat Recovery Steam Generators: Design, Applications, and Calculations, V. Ganapathy 150. The CAD Guidebook: A Basic Manual for Understanding and Improving Computer-Aided Design, Stephen J. Schoonmaker 151. Industrial Noise Control and Acoustics, Randall F. Barron 152. Mechanical Properties of Engineered Materials, Wolé Soboyejo 153. Reliability Verification, Testing, and Analysis in Engineering Design, Gary S. Wasserman 154. Fundamental Mechanics of Fluids: Third Edition, I. G. Currie 155. Intermediate Heat Transfer, Kau-Fui Vincent Wong 156. HVAC Water Chillers and Cooling Towers: Fundamentals, Application, and Operation, Herbert W. Stanford III 157. Gear Noise and Vibration: Second Edition, Revised and Expanded, J. Derek Smith 158. Handbook of Turbomachinery: Second Edition, Revised and Expanded, edited by Earl Logan, Jr. and Ramendra Roy 159. Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, George A. Antaki 160. Turbomachinery: Design and Theory, Rama S. R. Gorla and Aijaz Ahmed Khan 161. Target Costing: Market-Driven Product Design, M. Bradford Clifton, Henry M. B. Bird, Robert E. Albano, and Wesley P. Townsend 162. Fluidized Bed Combustion, Simeon N. Oka 163. Theory of Dimensioning: An Introduction to Parameterizing Geometric Models, Vijay Srinivasan 164. Handbook of Mechanical Alloy Design, edited by George E. Totten, Lin Xie, and Kiyoshi Funatani 165. Structural Analysis of Polymeric Composite Materials, Mark E. Tuttle 166. Modeling and Simulation for Material Selection and Mechanical Design, edited by George E. Totten, Lin Xie, and Kiyoshi Funatani 167. Handbook of Pneumatic Conveying Engineering, David Mills, Mark G. Jones, and Vijay K. Agarwal 168. Clutches and Brakes: Design and Selection, Second Edition, William C. Orthwein 169. Fundamentals of Fluid Film Lubrication: Second Edition, Bernard J. Hamrock, Steven R. Schmid, and Bo O. Jacobson 170. Handbook of Lead-Free Solder Technology for Microelectronic Assemblies, edited by Karl J. Puttlitz and Kathleen A. Stalter 171. Vehicle Stability, Dean Karnopp DK3139_series.qxd 1/20/05 11:17 AM Page 6 © 2005 by Taylor & Francis Group, LLC
- 11. 172. Mechanical Wear Fundamentals and Testing: Second Edition, Revised and Expanded, Raymond G. Bayer 173. Liquid Pipeline Hydraulics, E. Shashi Menon 174. Solid Fuels Combustion and Gasification, Marcio L. de Souza-Santos 175. Mechanical Tolerance Stackup and Analysis, Bryan R. Fischer 176. Engineering Design for Wear, Raymond G. Bayer 177. Vibrations of Shells and Plates: Third Edition, Revised and Expanded, Werner Soedel 178. Refractories Handbook, edited by Charles A. Schacht 179. Practical Engineering Failure Analysis, Hani M. Tawancy, Anwar Ul-Hamid, and Nureddin M. Abbas 180. Mechanical Alloying and Milling, C. Suryanarayana 181. Mechanical Vibration: Analysis, Uncertainties, and Control, Second Edition, Revised and Expanded, Haym Benaroya 182. Design of Automatic Machinery, Stephen J. Derby 183. Practical Fracture Mechanics in Design: Second Edition, Revised and Expanded, Arun Shukla 184. Practical Guide to Designed Experiments, Paul D. Funkenbusch 185. Gigacycle Fatigue in Mechanical Practive, Claude Bathias and Paul C. Paris 186. Selection of Engineering Materials and Adhesives, Lawrence W. Fisher 187. Boundary Methods: Elements, Contours, and Nodes, Subrata Mukherjee and Yu Xie Mukherjee DK3139_series.qxd 1/20/05 11:17 AM Page 7 © 2005 by Taylor & Francis Group, LLC
- 12. DK3139_title 1/20/05 11:12 AM Page 1 Boundary Methods Elements, Contours, and Nodes Boca Raton London New York Singapore A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc. Subrata Mukherjee Cornell University Ithaca, New York, U.S.A. Yu Xie Mukherjee Cornell University Ithaca, New York, U.S.A. © 2005 by Taylor & Francis Group, LLC
- 13. Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2599-9 (Hardcover) International Standard Book Number-13: 978-0-8247-2599-0 (Hardcover) Library of Congress Card Number 2004063489 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Mukherjee, Subrata. Boundary methods : elements, contours, and nodes / Subrata Mukherjee and Yu Mukherjee. p. cm. -- (Mechanical engineering ; 185) ISBN 0-8247-2599-9 (alk. paper) 1. Boundary element methods. I. Mukherjee, Yu. II. Title. III. Mechanical engineering (Marcel Dekker, Inc.) ; 185. TA347.B69M83 2005 621'.01'51535--dc22 2004063489 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Taylor & Francis Group is the Academic Division of T&F Informa plc. DK3139_discl Page 1 Wednesday, January 19, 2005 9:05 AM © 2005 by Taylor & Francis Group, LLC
- 14. iii To our boys Anondo and Alok and To Yu’s teacher, Professor Zhicheng Xie of Tsinghua University, a distinguished scholar who has dedicated himself to China. © 2005 by Taylor & Francis Group, LLC
- 15. v PREFACE The general subject area of concern to this book is computational science and engineering, with applications in potential theory and in solid mechanics (linear elasticity). This field has undergone a revolution during the past several decades along with the exponential growth of computational power and memory. Problems that were too large for main frame computers 15 or 20 years ago can now be routinely solved on desktop personal computers. There are several popular computational methods for solving problems in po- tential theory and linear elasticity. The most popular, versatile and most commonly used is the finite element method (FEM). Many hundreds of books already exist on the subject and new books get published frequently on a regular basis. Another popular method is the boundary element method (BEM). Compared to the FEM, we view the BEM as a niche method, in that it is particularly well suited, from the point of view of accuracy as well as computational efficiency, for linear problems. The principal advantage of the BEM, relative to the FEM, is its dimensionality advantage. The FEM is a domain method that requires discretization of the entire domain of a body while the BEM, for linear problems, only requires discretization of its bounding surface. The process of discretization (or meshing) of a three-dimensional (3-D) object of complex shape is a popular research area in computational geometry. Even though great strides have been made in recent years, meshing, for many applications, still remains an arduous task. During the past decade, mesh-free (also called mesh- less) methods have become a popular research area in computational mechanics. The main purpose here is to substantially simplify the task of meshing of an object. Advantages of mesh-free methods become more pronounced, for example, for prob- lems involving optimal shape design or adaptive meshing, since many remeshings must be typically carried out for such problems. One primary focus of this book is a marriage of these two ideas, i.e. a discussion of a boundary-based mesh-free method - the boundary node method (BNM) - which combines the dimensionality advantage of the BEM with the ease of discretization of mesh-free methods. Following an introductory chapter, this book consists of three parts related to the boundary element, boundary contour and boundary node methods. The first part is short, in order not to duplicate information on the BEM that is already available in many books on the subject. Only some novel topics related to the BEM are presented here. The second part is concerned with the boundary contour method (BCM). This method is a novel variant of the BEM in that it further reduces the dimensionality of a problem. Only one-dimensional line integrals need to be numerically computed when solving three-dimensional problems in linear elasticity by the BCM. The third part is concerned with the boundary node method (BNM). The BNM combines the BEM with moving least-squares (MLS) approximants, thus producing a mesh-free boundary-only method. In addition to the solution of 3-D problems, Part II of the book on the BCM presents shape sensitivity analysis, shape optimization, and error estimation and adaptivity; while Part III on the BNM includes error analysis and adaptivity. © 2005 by Taylor & Francis Group, LLC
- 16. vi This book is written in the style of a research monograph. Each topic is clearly introduced and developed. Numerical results for selected problems appear through- out the book, as do references to related work (research publications and books). This book should be of great interest to graduate students, researchers and practicing engineers in the field of computational mechanics; and to others inter- ested in the general areas of computational mathematics, science and engineering. It should also be of value to advanced undergraduate students who are interested in this field. We wish to thank a number of people and organizations who have contributed in various ways to making this book possible. Two of Subrata’s former graduate students, Glaucio Paulino and Mandar Chati, as well as Yu’s associate Xiaolan Shi, have made very significant contributions to the research that led to this book. Sin- cere thanks are expressed to Subrata’s former graduate students Govind Menon and Ramesh Gowrishankar, to one of his present graduate students, Srinivas Telukunta, and to Vasanth Kothnur, for their contributions to the BNM. Earlin Lutz, Anan- tharaman Nagarajan and Anh-Vu Phan have significantly contributed to the early development of the BCM; while Subrata’s just-graduated student Zhongping Bao has made excellent contributions to the research on micro-electro-mechanical sys- tems (MEMS) by the BEM. Sincere thanks are expressed to our dear friend Ashim Datta for his help and encouragement throughout the writing of this book. Much of the research presented here has been financially supported by the Na- tional Science Foundation and Ford Motor Company, and this support is gratefully acknowledged. Most of the figures and tables in this book have been published before in journals. They were all originally created by the authors of this book, together with their coauthors. These items have been printed here by permission of the original copyright owner (i.e. the publishers of the appropriate journal), and this permission is very much appreciated. The original source has been acknowledged in this book at the end of the caption for each item. Subrata and Yu Mukherjee Ithaca, New York October 2004 © 2005 by Taylor & Francis Group, LLC
- 17. Contents Preface v INTRODUCTION TO BOUNDARY METHODS xiii I SELECTED TOPICS IN BOUNDARY ELEMENT METHODS 1 1 BOUNDARY INTEGRAL EQUATIONS 3 1.1 Potential Theory in Three Dimensions . . . . . . . . . . . . . . . 3 1.1.1 Singular Integral Equations . . . . . . . . . . . . . . . . . 3 1.1.2 Hypersingular Integral Equations . . . . . . . . . . . . . . 5 1.2 Linear Elasticity in Three Dimensions . . . . . . . . . . . . . . . 6 1.2.1 Singular Integral Equations . . . . . . . . . . . . . . . . . 6 1.2.2 Hypersingular Integral Equations . . . . . . . . . . . . . . 8 1.3 Nearly Singular Integrals in Linear Elasticity . . . . . . . . . . . 12 1.3.1 Displacements at Internal Points Close to the Boundary . 12 1.3.2 Stresses at Internal Points Close to the Boundary . . . . . 13 1.4 Finite Parts of Hypersingular Equations . . . . . . . . . . . . . . 14 1.4.1 Finite Part of a Hypersingular Integral Collocated at an Irregular Boundary Point . . . . . . . . . . . . . . . . . . 14 1.4.2 Gradient BIE for 3-D Laplace’s Equation . . . . . . . . . 17 1.4.3 Stress BIE for 3-D Elasticity . . . . . . . . . . . . . . . . 19 1.4.4 Solution Strategy for a HBIE Collocated at an Irregular Boundary Point . . . . . . . . . . . . . . . . . . . . . . . . 20 2 ERROR ESTIMATION 23 2.1 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Iterated HBIE and Error Estimation . . . . . . . . . . . . . . . . 25 2.2.1 Problem 1 : Displacement Boundary Conditions . . . . . 25 2.2.2 Problem 2 : Traction Boundary Conditions . . . . . . . . 28 2.2.3 Problem 3 : Mixed Boundary Conditions . . . . . . . . . 30 2.3 Element-Based Error Indicators . . . . . . . . . . . . . . . . . . . 32 2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 33 vii © 2005 by Taylor & Francis Group, LLC
- 18. viii CONTENTS 2.4.1 Example 1: Lamé’s Problem of a Thick-Walled Cylinder under Internal Pressure . . . . . . . . . . . . . . . . . . . 34 2.4.2 Example 2: Kirsch’s Problem of an Infinite Plate with a Circular Cutout . . . . . . . . . . . . . . . . . . . . . . . 36 3 THIN FEATURES 39 3.1 Exterior BIE for Potential Theory: MEMS . . . . . . . . . . . . 39 3.1.1 Introduction to MEMS . . . . . . . . . . . . . . . . . . . . 39 3.1.2 Electric Field BIEs in a Simply Connected Body . . . . . 41 3.1.3 BIES in Infinite Region Containing Two Thin Conducting Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.4 Singular and Nearly Singular Integrals . . . . . . . . . . . 46 3.1.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 49 3.1.6 The Model Problem - a Parallel Plate Capacitor . . . . . 50 3.2 BIE for Elasticity: Cracks and Thin Shells . . . . . . . . . . . . 54 3.2.1 BIES in LEFM . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.2 Numerical Implementation of BIES in LEFM . . . . . . . 60 3.2.3 Some Comments on BIEs in LEFM . . . . . . . . . . . . . 61 3.2.4 BIEs for Thin Shells . . . . . . . . . . . . . . . . . . . . . 62 II THE BOUNDARY CONTOUR METHOD 65 4 LINEAR ELASTICITY 67 4.1 Surface and Boundary Contour Equations . . . . . . . . . . . . . 67 4.1.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . 67 4.1.2 Interpolation Functions . . . . . . . . . . . . . . . . . . . 68 4.1.3 Boundary Elements . . . . . . . . . . . . . . . . . . . . . 71 4.1.4 Vector Potentials . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.5 Final BCM Equations . . . . . . . . . . . . . . . . . . . . 74 4.1.6 Global Equations and Unknowns . . . . . . . . . . . . . . 76 4.1.7 Surface Displacements, Stresses, and Curvatures . . . . . 76 4.2 Hypersingular Boundary Integral Equations . . . . . . . . . . . . 78 4.2.1 Regularized Hypersingular BIE . . . . . . . . . . . . . . . 78 4.2.2 Regularized Hypersingular BCE . . . . . . . . . . . . . . 78 4.2.3 Collocation of the HBCE at an Irregular Surface Point . . 80 4.3 Internal Displacements and Stresses . . . . . . . . . . . . . . . . 82 4.3.1 Internal Displacements . . . . . . . . . . . . . . . . . . . . 82 4.3.2 Displacements at Internal Points Close to the Bounding Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.3 Internal Stresses . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.4 Stresses at Internal Points Close to the Bounding Surface 84 4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.1 Surface Displacements from the BCM and the HBCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 © 2005 by Taylor & Francis Group, LLC
- 19. CONTENTS ix 4.4.2 Surface Stresses . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4.3 Internal Stresses Relatively Far from the Bounding Surface 90 4.4.4 Internal Stresses Very Close to the Bounding Surface . . . 90 5 SHAPE SENSITIVITY ANALYSIS 93 5.1 Sensitivities of Boundary Variables . . . . . . . . . . . . . . . . . 93 5.1.1 Sensitivity of the BIE . . . . . . . . . . . . . . . . . . . . 93 5.1.2 The Integral Ik . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1.3 The Integral Jk . . . . . . . . . . . . . . . . . . . . . . . . 96 5.1.4 The BCM Sensitivity Equation . . . . . . . . . . . . . . . 98 5.2 Sensitivities of Surface Stresses . . . . . . . . . . . . . . . . . . . 99 5.2.1 Method One . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.2 Method Two . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.3 Method Three . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.4 Method Four . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3 Sensitivities of Variables at Internal Points . . . . . . . . . . . . 101 5.3.1 Sensitivities of Displacements . . . . . . . . . . . . . . . . 101 5.3.2 Sensitivities of Displacement Gradients and Stresses . . . 103 5.4 Numerical Results: Hollow Sphere . . . . . . . . . . . . . . . . . 106 5.4.1 Sensitivities on Sphere Surface . . . . . . . . . . . . . . . 107 5.4.2 Sensitivities at Internal Points . . . . . . . . . . . . . . . 108 5.5 Numerical Results: Block with a Hole . . . . . . . . . . . . . . . 110 5.5.1 Geometry and Mesh . . . . . . . . . . . . . . . . . . . . . 110 5.5.2 Internal Stresses . . . . . . . . . . . . . . . . . . . . . . . 112 5.5.3 Sensitivities of Internal Stresses . . . . . . . . . . . . . . . 112 6 SHAPE OPTIMIZATION 115 6.1 Shape Optimization Problems . . . . . . . . . . . . . . . . . . . . 115 6.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2.1 Shape Optimization of a Fillet . . . . . . . . . . . . . . . 116 6.2.2 Optimal Shapes of Ellipsoidal Cavities Inside Cubes . . . 118 6.2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7 ERROR ESTIMATION AND ADAPTIVITY 125 7.1 Hypersingular Residuals as Local Error Estimators . . . . . . . . 125 7.2 Adaptive Meshing Strategy . . . . . . . . . . . . . . . . . . . . . 126 7.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.3.1 Example One - Short Clamped Cylinder under Tension . 127 7.3.2 Example Two - the Lamé Problem for a Hollow Cylinder 130 III THE BOUNDARY NODE METHOD 133 8 SURFACE APPROXIMANTS 135 8.1 Moving Least Squares (MLS) Approximants . . . . . . . . . . . 135 8.2 Surface Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 139 © 2005 by Taylor & Francis Group, LLC
- 20. x CONTENTS 8.3 Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.4 Use of Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . 142 8.4.1 Hermite Type Approximation . . . . . . . . . . . . . . . . 142 8.4.2 Variable Basis Approximation . . . . . . . . . . . . . . . . 143 9 POTENTIAL THEORY AND ELASTICITY 151 9.1 Potential Theory in Three Dimensions . . . . . . . . . . . . . . . 151 9.1.1 BNM: Coupling of BIE with MLS Approximants . . . . . 151 9.1.2 HBNM: Coupling of HBIE with MLS Approximants . . . 155 9.1.3 Numerical Results for Dirichlet Problems on a Sphere . . 156 9.2 Linear Elasticity in Three Dimensions . . . . . . . . . . . . . . . 165 9.2.1 BNM: Coupling of BIE with MLS Approximants . . . . . 165 9.2.2 HBNM: Coupling of HBIE with MLS Approximants . . . 167 9.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . 168 10 ADAPTIVITY FOR 3-D POTENTIAL THEORY 175 10.1 Hypersingular and Singular Residuals . . . . . . . . . . . . . . . 175 10.1.1 The Hypersingular Residual . . . . . . . . . . . . . . . . . 175 10.1.2 The Singular Residual . . . . . . . . . . . . . . . . . . . . 176 10.2 Error Estimation and Adaptive Strategy . . . . . . . . . . . . . . 177 10.2.1 Local Residuals and Errors - Hypersingular Residual Ap- proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.2.2 Local Residuals and Errors - Singular Residual Approach 178 10.2.3 Cell Refinement Criterion . . . . . . . . . . . . . . . . . . 179 10.2.4 Global Error Estimation and Stopping Criterion . . . . . 179 10.3 Progressively Adaptive Solutions: Cube Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 10.3.1 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . 181 10.3.2 Initial Cell Configuration # 1 (54 Surface Cells) . . . . . 181 10.3.3 Initial Cell Configuration # 2 (96 Surface Cells) . . . . . 182 10.4 One-Step Adaptive Cell Refinement . . . . . . . . . . . . . . . . 188 10.4.1 Initial Cell Configuration # 1 (54 Surface Cells) . . . . . 190 10.4.2 Initial Cell Configuration # 2 (96 Surface Cells) . . . . . 191 11 ADAPTIVITY FOR 3-D LINEAR ELASTICITY 193 11.1 Hypersingular and Singular Residuals . . . . . . . . . . . . . . . 193 11.1.1 The Hypersingular Residual . . . . . . . . . . . . . . . . . 193 11.1.2 The Singular Residual . . . . . . . . . . . . . . . . . . . . 194 11.2 Error Estimation and Adaptive Strategy . . . . . . . . . . . . . . 194 11.2.1 Local Residuals and Errors - Hypersingular Residual Ap- proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 11.2.2 Local Residuals and Errors - Singular Residual Approach 195 11.2.3 Cell Refinement Global Error Estimation and Stopping Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 11.3 Progressively Adaptive Solutions: Pulling a Rod . . . . . . . . . 195 © 2005 by Taylor & Francis Group, LLC
- 21. CONTENTS xi 11.3.1 Initial Cell Configuration . . . . . . . . . . . . . . . . . . 197 11.3.2 Adaptivity Results . . . . . . . . . . . . . . . . . . . . . . 197 11.4 One-Step Adaptive Cell Refinement . . . . . . . . . . . . . . . . 198 Bibliography 203 © 2005 by Taylor & Francis Group, LLC
- 22. INTRODUCTION TO BOUNDARY METHODS This chapter provides a brief introduction to various topics that are of interest in this book. Boundary Element Method Boundary integral equations (BIE), and the boundary element method (BEM), based on BIEs, are mature methods for numerical analysis of a large variety of problems in science and engineering. The standard BEM for linear problems has the well-known dimensionality advantage in that only the two-dimensional (2-D) bounding surface of a three-dimensional (3-D) body needs to be meshed when this method is used. Examples of books on the subject, published dur- ing the last 15 years, are Banerjee [4], Becker [9], Bonnet [14], Brebbia and Dominguez [16], Chandra and Mukherjee [22], Gaul et al. [47], Hartmann [62], Kane [68] and Parı́s and Cañas [121]. BEM topics of interest in this book are finite parts (FP) in Chapter 1, error estimation in Chapter 2 and thin features (cracks and thin objects) in Chapter 3. Hypersingular Boundary Integral Equations Hypersingular boundary integral equations (HBIEs) are derived from a differ- entiated version of the usual boundary integral equations (BIEs). HBIEs have diverse important applications and are the subject of considerable current re- search (see, for example, Krishnasamy et al. [76], Tanaka et al. [162], Paulino [122] and Chen and Hong [30] for recent surveys of the field). HBIEs, for exam- ple, have been employed for the evaluation of boundary stresses (e.g. Guiggiani [60], Wilde and Aliabadi [173], Zhao and Lan [185], Chati and Mukherjee [24]), in wave scattering (e.g. Krishnasamy et al. [75]), in fracture mechanics (e.g. Cruse [38], Gray et al. [54], Lutz et al. [89], Paulino [122], Gray and Paulino [58], Mukherjee et al. [110]), to obtain symmetric Galerkin boundary element formulations (e.g. Bonnet [14], Gray et al. [55], Gray and Paulino ([56], [57]), to xiii © 2005 by Taylor & Francis Group, LLC
- 23. xiv INTRODUCTION TO BOUNDARY METHODS evaluate nearly singular integrals (Mukherjee et al. [104]), to obtain the hyper- singular boundary contour method (Phan et al. [131], Mukherjee and Mukherjee [99]), to obtain the hypersingular boundary node method (Chati et al. [27]), and for error analysis (Paulino et al. [123], Menon [95], Menon et al. [96], Chati et al. [27], Paulino and Gray [125]) and adaptivity [28]. An elegant approach of regularizing singular and hypersingular integrals, us- ing simple solutions, was first proposed by Rudolphi [143]. Several researchers have used this idea to regularize hypersingular integrals before collocating an HBIE at a regular boundary point. Examples are Cruse and Richardson [39], Lutz et al. [89], Poon et al. [138], Mukherjee et al. [110] and Mukherjee [106]. The relationship between finite parts of strongly singular and hypersingular in- tegrals, and the HBIE, is discussed in [168], [101] and [102]. A lively debate (e.g. [92], [39]), on smoothness requirements on boundary variables for collocating an HBIE on the boundary of a body, has apparently been concluded recently [93]. An alternative way of satisfying this smoothness requirement is the use of the hypersingular boundary node method (HBNM). Mesh-Free Methods Mesh-free (also called meshless) methods [82], that only require points rather than elements to be specified in the physical domain, have tremendous potential advantages over methods such as the finite element method (FEM) that require discretization of a body into elements. The idea of moving least squares (MLS) interpolants, for curve and surface fitting, is described in a book by Lancaster and Salkauskas [78]. Nayroles et al. [117] proposed a coupling of MLS interpolants with Galerkin procedures in order to solve boundary value problems. They called their method the diffuse element method (DEM) and applied it to two-dimensional (2-D) problems in potential theory and linear elasticity. During the relatively short span of less than a decade, great progress has been made in solid mechanics applications of mesh-free methods. Mesh-free methods proposed to date include the element-free Galerkin (EFG) method [10, 11, 12, 13, 67, 174, 175, 176, 108], the reproducing-kernel particle method (RKPM) [83, 84], h − p clouds [42, 43, 120], the meshless local Petrov-Galerkin (MLPG) approach [3], the local boundary integral equation (LBIE) method [152, 188], the meshless regular local boundary integral equation (MRLBIE) method [189], the natural element method (NEM) [158, 160], the general- ized finite element method (GFEM) [157], the extended finite element method (X-FEM) [97, 41, 159], the method of finite spheres (MFS) [40], the finite cloud method (FCM) [2], the boundary cloud method (BCLM) [79, 80], the boundary point interpolation method (BPIM) [82], the boundary-only radial basis function method (BRBFM) [32] and the boundary node method (BNM) [107, 72, 25, 26, 27, 28, 52]. © 2005 by Taylor & Francis Group, LLC
- 24. xv Boundary Node Method S. Mukherjee, together with his research collaborators, has recently pioneered a new computational approach called the boundary node method (BNM) [26, 25, 27, 28, 72, 107]. Other examples of boundary-based meshless methods are the boundary cloud method (BCLM) [79, 80], the boundary point interpolation method (BPIM) [82], the boundary only radial basis function method (BRBFM) [32] and the local BIE (LBIE) [188] approach. The LBIE, however, is not a boundary method since it requires evaluation of integrals over certain surfaces (called Ls in [188]) that can be regarded as “closure surfaces” of boundary elements. The BNM is a combination of the MLS interpolation scheme and the stan- dard boundary integral equation (BIE) method. The method divorces the tra- ditional coupling between spatial discretization (meshing) and interpolation (as commonly practiced in the FEM or in the BEM). Instead, a “diffuse” interpo- lation, based on MLS interpolants, is used to represent the unknown functions; and surface cells, with a very flexible structure (e.g. any cell can be arbitrarily subdivided without affecting its neighbors [27]) are used for integration. Thus, the BNM retains the meshless attribute of the EFG method and the dimen- sionality advantage of the BEM. As a consequence, the BNM only requires the specification of points on the 2-D bounding surface of a 3-D body (including crack faces in fracture mechanics problems), together with surface cells for in- tegration, thereby practically eliminating the meshing problem (see Figures i and ii). The required cell structure is analogous to (but not the same as) a tiling [139]. The only requirements are that the intersection of any two surface cells is the null set and that the union of all the cells is the bounding surface of the body. In contrast, the FEM needs volume meshing, the BEM needs surface meshing, and the EFG needs points throughout the domain of a body. It is important to point out another important advantage of MLS inter- polants. They can be easily designed to be sufficiently smooth to suit a given purpose, e.g. they can be made C1 or higher [10] in order to collocate the HBNM at a point on the boundary of a body. The BNM is described in Chapters 8 and 9 of this book. Figure i: BNM with nodes and cells (from [28]) Figure ii: BEM with nodes and elements (from [28]) © 2005 by Taylor & Francis Group, LLC
- 25. xvi INTRODUCTION TO BOUNDARY METHODS Boundary Contour Method The Method The usual boundary element method (BEM), for three-dimensional (3-D) lin- ear elasticity, requires numerical evaluations of surface integrals on boundary elements on the surface of a body (see, for example, [98]). [115] (for 2-D linear elasticity) and [116] (for 3-D linear elasticity) have recently proposed a novel approach, called the boundary contour method (BCM), that achieves a further reduction in dimension! The BCM, for 3-D linear elasticity problems, only re- quires numerical evaluation of line integrals over the closed bounding contours of the usual (surface) boundary elements. The central idea of the BCM is the exploitation of the divergence-free prop- erty of the usual BEM integrand and a very useful application of Stokes’ the- orem, to analytically convert surface integrals on boundary elements to line integrals on closed contours that bound these elements. [88] first proposed an application of this idea for the Laplace equation and Nagarajan et al. gen- eralized this idea to linear elasticity. Numerical results for two-dimensional (2-D) problems, with linear boundary elements, are presented in [115], while results with quadratic boundary elements appear in [129]. Three-dimensional elasticity problems, with quadratic boundary elements, are the subject of [116] and [109]. Hypersingular boundary contour formulations, for two-dimensional [131] and three-dimensional [99] linear elasticity, have been proposed recently. A symmetric Galerkin BCM for 2-D linear elasticity appears in [119]. Recent work on the BCM is available in [31, 134, 135, 136, 186]. The BCM is described in Chapter 4 of this book. Shape Sensitivity Analysis with the BCM and the HBCM Design sensitivity coefficients (DSCs), which are defined as rates of change of physical response quantities with respect to changes in design variables, are useful for various applications such as in judging the robustness of a given design, in reliability analysis and in solving inverse and design optimization problems. There are three methods for design sensitivity analysis (e.g. [63]), namely, the finite difference approach (FDA), the adjoint structure approach (ASA) and the direct differentiation approach (DDA). The DDA is of interest in this work. The goal of obtaining BCM sensitivity equations can be achieved in two equivalent ways. In the 2-D work by [130], design sensitivities are obtained by first converting the discretized BIEs into their boundary contour version, and then applying the DDA (using the concept of the material derivative) to this BCM version. This approach, while relatively straightforward in principle, becomes extremely algebraically intensive for 3-D elasticity problems. [100] offers a novel alternative derivation, using the opposite process, in which the DDA is first applied to the regularized BIE and then the resulting equations © 2005 by Taylor & Francis Group, LLC
- 26. xvii are converted to their boundary contour version. It is important to point out that this process of converting the sensitivity BIE into a BCM form is quite challenging. This new derivation, for sensitivities of surface variables [100], as well as for internal variables [103], for 3-D elasticity problems, is presented in Chapter 5 of this book. The reader is referred to [133] for a corresponding derivation for 2-D elasticity Shape Optimization with the BCM Shape optimization refers to the optimal design of the shape of structural com- ponents and is of great importance in mechanical engineering design. A typical gradient-based shape optimization procedure is an iterative process in which iterative improvements are carried out over successive designs until an optimal design is accepted. A domain-based method such as the finite element method (FEM) typically requires discretization of the entire domain of a body many times during this iterative process. The BEM, however, only requires surface discretization, so that mesh generation and remeshing procedures can be carried out much more easily for the BEM than for the FEM. Also, surface stresses are typically obtained very accurately in the BEM. As a result, the BEM has been a popular method for shape optimization in linear mechanics. Some examples are references [33], [145], [178], [144], [169], [177], [161] and the book [184]. In addition to having the same meshing advantages as the usual BEM, the BCM, as explained above, offers a further reduction in dimension. Also, surface stresses can be obtained very easily and accurately by the BCM without the need for additional shape function differentiation as is commonly required with the BEM. These properties make the BCM very attractive as the computational engine for stress analysis for use in shape optimization. Shape optimization in 2-D linear elasticity, with the BCM, has been presented by [132]. The corre- sponding 3-D problem is presented in [150] and is discussed in Chapter 6. Error Estimation and Adaptivity A particular strength of the finite element method (FEM) is the well-developed theory of error estimation, and its use in adaptive methods (see, for example, Ciarlet [34], Eriksson et al. [44]). In contrast, error estimation in the boundary element method (BEM) is a subject that has attracted attention mainly over the past decade, and much work remains to be done. For recent surveys on error estimation and adaptivity in the BEM, see Sloan [155], Kita and Kamiya [70], Liapis [81] and Paulino et al. [124]. Many error estimators in the BEM are essentially heuristic and, unlike for the FEM, theoretical work in this field has been quite limited. Rank [140] proposed error indicators and an adaptive algorithm for the BEM using tech- niques similar to those used in the FEM. Most notable is the work of Yu and Wendland [171, 172, 181, 182], who have presented local error estimates based © 2005 by Taylor & Francis Group, LLC
- 27. xviii INTRODUCTION TO BOUNDARY METHODS on a linear error-residual relation that is very effective in the FEM. More re- cently, Carstensen et al. [18, 21, 19, 20] have presented error estimates for the BEM analogous to the approach of Eriksson [44] for the FEM. There are numerous stumbling blocks in the development of a satisfactory theoretical analysis of a generic boundary value problem (BVP). First, theoretical analy- ses are easiest for Galerkin schemes, but most engineering codes, to date, use collocation-based methods (see, for example, Banerjee [4]). Though one can view collocation schemes as variants of Petrov-Galerkin methods, and, in fact, numerous theoretical analyses exist for collocation methods (see, for example, references in [155]), the mathematical analysis for this class of problems is difficult. Theoretical analyses for mixed boundary conditions are limited and involved (Wendland et al. [170]) and the presence of corners and cracks has been a source of challenging problems for many years (Sloan [155], Costabel and Stephan [35], Costabel et al. [36]). Of course, problems with corners and mixed boundary conditions are the ones of most practical interest, and for such situations one has to rely mostly on numerical experiments. During the past few years, there has been a marked interest, among mathe- maticians in the field, in extending analyses for the BEM with singular integrals to hypersingular integrals ([21, 19, 156, 45]. For instance, Feistauer et al. [45] have studied the solution of the exterior Neumann problem for the Helmholtz equation formulated as an HBIE. Their paper contains a rigorous analysis of hypersingular integral equations and addresses the problem of noncompatibil- ity of the residual norm, where additional hypotheses are needed to design a practical error estimate. These authors use residuals to estimate the error, but they do not use the BIE and the HBIE simultaneously. Finally, Goldberg and Bowman [51] have used superconvergence of the Sloan iterate [153, 154] to show the asymptotic equivalence of the error and the residual. They have used Galerkin methods, an iteration scheme that uses the same integral equation for the approximation and for the iterates, and usual residuals in their work. Paulino [122] and Paulino et al. [123] first proposed the idea of obtaining a hypersingular residual by substituting the BEM solution of a problem into the hypersingular BEM (HBEM) for the same problem; and then using this residual as an element error estimator in the BEM. It has been proved that ([95], [96], [127]), under certain conditions, this residual is related to a measure of the local error on a boundary element, and has been used to postulate local error estimates on that element. This idea has been applied to the collocation BEM ([123], [96], [127]) and to the symmetric Galerkin BEM ([125]). Recently, residuals have been obtained in the context of the BNM [28] and used to obtain local error estimates (at the element level) and then to drive an h-adaptive mesh refinement process. An analogous approach for error estimation and h- adaptivity, in the context of the BCM, is described in [111]. Ref. [91] has a bibliography of work on mesh generation and refinement up to 1993. Error analysis with the BEM is presented in Chapter 2, while error analysis and adaptivity in the context of the BCM and the BNM are discussed in Chapter 7, and Chapters 10, 11, respectively, of this book. © 2005 by Taylor & Francis Group, LLC
- 28. Part I SELECTED TOPICS IN BOUNDARY ELEMENT METHODS 1 © 2005 by Taylor & Francis Group, LLC
- 29. Chapter 1 BOUNDARY INTEGRAL EQUATIONS Integral equations, usual as well as hypersingular, for internal and boundary points, for potential theory in three dimensions, are first presented in this chap- ter. This is followed by their linear elasticity counterparts. The evaluation of finite parts (FPs) of some of these equations, when the source point is an irreg- ular boundary point (situated at a corner on a one-dimensional plane curve or at a corner or edge on a two-dimensional surface), is described next. 1.1 Potential Theory in Three Dimensions The starting point is Laplace’s equation in three dimensions (3-D) governing a potential function u(x1, x2, x3) ∈ B, where B is a bounded region (also called the body): ∇2 u(x1, x2, x3) ≡ ∂2 u ∂x2 1 + ∂2 u ∂x2 2 + ∂2 u ∂x2 3 = 0 (1.1) along with prescribed boundary conditions on the bounding surface ∂B of B. 1.1.1 Singular Integral Equations Referring to Figure 1.1, let ξ and η be (internal) source and field points ∈ B and x and y be (boundary) source and field points ∈ ∂B, respectively. (Source and field points are also referred to as p and q (for internal points) and as P and Q (for boundary points), respectively, in this book). The well-known integral representation for (1.1), at an internal point ξ ∈ B, is: u(ξ) = ∂B [G(ξ, y)τ(y) − F(ξ, y)u(y)]dS(y) (1.2) 3 © 2005 by Taylor Francis Group, LLC
- 30. 4 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS B x(P) y(Q) ξ(p) r ( ξ , y ) η(q) n(y) n(x) ∂B Figure 1.1: Notation used in integral equations (from [6]) An infinitesimal surface area on ∂B is dS = dSn, where n is the unit outward normal to ∂B at a point on it and τ = ∂u/∂n. The kernels are written in terms of source and field points ξ ∈ B and y ∈ ∂B. These are : G(ξ, y) = 1 4πr(ξ, y) (1.3) F(ξ, y) = ∂G(ξ, y) ∂n(y) = (ξi − yi)ni(y) 4πr3(ξ, y) (1.4) in terms of r(ξ, y), the Euclidean distance between the source and field points ξ and y. Unless specified otherwise, the range of indices in these and all other equations in this chapter is 1,2,3. An alternative form of equation (1.2) is: u(ξ) = ∂B [G(ξ, y)u,k(y) − Hk(ξ, y)u(y)]ek · dS(y) (1.5) where ek, k = 1, 2, 3, are the usual Cartesian unit vectors, ek · dS(y) = nk(y)dS(y), and: Hk(ξ, y) = (ξk − yk) 4πr3(ξ, y) (1.6) The boundary integral equation (BIE) corresponding to (1.2) is obtained by taking the limit ξ → x. A regularized form of the resulting equation is: 0 = ∂B [G(x, y)τ(y) − F(x, y){u(y) − u(x)}]dS(y) (1.7) © 2005 by Taylor Francis Group, LLC
- 31. 1.1. POTENTIAL THEORY IN THREE DIMENSIONS 5 with an alternate form (from (1.5)): 0 = ∂B [G(x, y)u,k(y) − Hk(x, y){u(y) − u(x)}]ek · dS(y) (1.8) 1.1.2 Hypersingular Integral Equations Equation (1.2) can be differentiated at an internal source point ξ to obtain the gradient ∂u ∂ξm of the potential u. The result is: ∂u(ξ) ∂ξm = ∂B ∂G(ξ, y) ∂ξm τ(y) − ∂F(ξ, y) ∂ξm u(y) dS(y) (1.9) An interesting situation arises when one takes the limit ξ → x (x can even be an irregular point on ∂B but one must have u(y) ∈ C1,α at y = x) in equation (1.9). As discussed in detail in Section 1.4.2, one obtains: ∂u(x) ∂xm = ∂B = ∂G(x, y) ∂xm τ(y) − ∂F(x, y) ∂xm u(y) dS(y) (1.10) where the symbol = denotes the finite part (FP) of the integral. Equation (1.10) is best regularized before computations are carried out. The regularized version given below is applicable even at an irregular boundary point x provided that u(y) ∈ C1,α at y = x. This is: 0 = ∂B ∂G(x, y) ∂xm u,p(y) − u,p(x) np(y)dS(y) − ∂B ∂F(x, y) ∂xm u(y) − u(x) − u,p(x)(yp − xp) dS(y) (1.11) An alternative form of (1.11), valid at a regular boundary point x, [76] is: 0 = ∂B ∂G(x, y) ∂xm τ(y) − τ(x) dS(y) − u,k(x) B ∂G(x, y) ∂xm nk(y) − nk(x) dS(y) − ∂B ∂F(x, y) ∂xm u(y) − u(x) − u,p(x)(yp − xp) dS(y) (1.12) Carrying out the inner product of (1.12) with the source point normal n(x), one gets: 0 = ∂B ∂G(x, y) ∂n(x) τ(y) − τ(x) dS(y) © 2005 by Taylor Francis Group, LLC
- 32. 6 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS − u,k(x) B ∂G(x, y) ∂n(x) nk(y) − nk(x) dS(y) − ∂B ∂F(x, y) ∂n(x) u(y) − u(x) − u,p(x)(yp − xp) dS(y) (1.13) 1.1.2.1 Potential gradient on the bounding surface The gradient of the potential function is required in the regularized HBIEs (1.11 - 1.13). For potential problems, the gradient (at a regular boundary point) can be written as, ∇u = τn + ∂u ∂s1 t1 + ∂u ∂s2 t2 (1.14) where τ = ∂u/∂n is the flux, n is the unit normal, t1, t2 are the appropriately chosen unit vectors in two orthogonal tangential directions on the surface of the body, and ∂u/∂si, i = 1, 2 are the tangential derivatives of u (along t1 and t2) on the surface of the body. 1.2 Linear Elasticity in Three Dimensions The starting point is the Navier-Cauchy equation governing the displacement u(x1, x2, x3) in a homogeneous, isotropic, linear elastic solid occupying the bounded 3-D region B with boundary ∂B; in the absence of body forces: 0 = ui,jj + 1 1 − 2ν uk,ki (1.15) along with prescribed boundary conditions that involve the displacement and the traction τ on ∂B. The components τi of the traction vector are: τi = λuk,kni + µ(ui,j + uj,i)nj (1.16) In equations (1.15) and (1.16), ν is Poisson’s ratio and λ and µ are Lamé constants. As is well known, µ is the shear modulus of the material and is also called G in this book. Finally, the Young’s modulus is denoted as E. 1.2.1 Singular Integral Equations The well-known integral representation for (1.15), at an internal point ξ ∈ B (Rizzo [141]) is: uk(ξ) = ∂B [Uik(ξ, y)τi(y) − Tik(ξ, y)ui(y)] dS(y) (1.17) where uk and τk are the components of the displacement and traction respec- tively, and the well-known Kelvin kernels are: © 2005 by Taylor Francis Group, LLC
- 33. 1.2. LINEAR ELASTICITY IN THREE DIMENSIONS 7 Uik = 1 16π(1 − ν)Gr [(3 − 4ν)δik + r,ir,k] (1.18) Tik = − 1 8π(1 − ν)r2 {(1 − 2ν)δik + 3r,ir,k} ∂r ∂n + (1 − 2ν)(r,ink − r,kni) (1.19) In the above, δik denotes the Kronecker delta and, as before, the normal n is defined at the (boundary) field point y. A comma denotes a derivative with respect to a field point, i.e. r,i = ∂r ∂yi = yi − ξi r (1.20) An alternative form of equation (1.17) is: uk(ξ) = ∂B [Uik(ξ, y)σij(y) − Σijk(ξ, y)ui(y)] ej · dS(y) (1.21) where σ is the stress tensor, τi = σijnj and Tik = Σijknj. (Please note that ej · dS(y) = nj(y)dS(y)). The explicit form of the kernel Σ is: Σijk = Eijmn ∂Ukm ∂yn = − 1 8π(1 − ν)r2 [ (1 − 2ν)(r,iδjk + r,jδik − r,kδij) + 3r,ir,jr,k ] (1.22) where E is the elasticity tensor (for isotropic elasticity): Eijmn = λδijδmn + µ[δimδjn + δinδjm] (1.23) The boundary integral equation (BIE) corresponding to (1.17) is obtained by taking the limit ξ → x. The result is: uk(x) = lim ξ→x ∂B [Uik(ξ, y)τi(y) − Tik(ξ, y)ui(y)] dS(y) = ∂B = [Uik(x, y)τi(y) − Tik(x, y)ui(y)] dS(y) (1.24) where the symbol ∂B = denotes the finite part of the appropriate integral (see Section 1.4). A regularized form of equation (1.24) is: 0 = ∂B [Uik(x, y)τi(y) − Tik(x, y){ui(y) − ui(x)}]dS(y) (1.25) © 2005 by Taylor Francis Group, LLC
- 34. 8 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS with an alternate form (from (1.21)): 0 = ∂B [Uik(x, y)σij(y) − Σijk(x, y){ui(y) − ui(x)}]ej · dS(y) (1.26) 1.2.2 Hypersingular Integral Equations Equation (1.17) can be differentiated at an internal source point ξ to obtain the displacement gradient at this point: ∂uk(ξ) ∂ξm = ∂B ∂Uik ∂ξm (ξ, y)τi(y) − ∂Tik ∂ξm (ξ, y)ui(y) dS(y) (1.27) An alternative form of equation (1.27) is: ∂uk(ξ) ∂ξm = ∂B ∂Uik ∂ξm (ξ, y)σij(y) − ∂Σijk ∂ξm (ξ, y)ui(y) ej · dS(y) (1.28) Stress components at an internal point ξ can be obtained from either of equations (1.27) or (1.28) by using Hooke’s law: σij = λuk,kδij + µ(ui,j + uj,i) (1.29) It is sometimes convenient, however, to write the internal stress directly. This equation, corresponding (for example) to (1.27) is: σij(ξ) = ∂B [Dijk(ξ, y)τk(y) − Sijk(ξ, y)uk(y)] dS(y) (1.30) where the new kernels D and S are: Dijk = Eijmn ∂Ukm ∂ξn = λ ∂Ukm ∂ξm δij + µ ∂Uki ∂ξj + ∂Ukj ∂ξi = −Σijk (1.31) Sijk = Eijmn ∂Σkpm ∂ξn np = λ ∂Σkpm ∂ξm npδij + µ ∂Σkpi ∂ξj + ∂Σkpj ∂ξi np = G 4π(1 − ν)r3 3 ∂r ∂n [(1 − 2ν)δijr,k + ν(δikr,j + δjkr,i) − 5r,ir,jr,k] + G 4π(1 − ν)r3 [3ν(nir,jr,k + njr,ir,k) +(1 − 2ν)(3nkr,ir,j + njδik + niδjk) − (1 − 4ν)nkδij] (1.32) Again, the normal n is defined at the (boundary) field point y. Also: © 2005 by Taylor Francis Group, LLC
- 35. 1.2. LINEAR ELASTICITY IN THREE DIMENSIONS 9 ∂Uik ∂ξm (ξ, y) = −Uik,m , ∂Σijk ∂ξm (ξ, y) = −Σijk,m (1.33) It is important to note that D becomes strongly singular, and S hypersin- gular as a source point approaches a field point (i.e. as r → 0). For future use in Chapter 4, it is useful to rewrite (1.28) using (1.33). This equation is: uk,m(ξ) = − ∂B [Uik,m(ξ, y)σij(y) − Σijk,m(ξ, y)ui(y)] nj(y)dS(y) (1.34) Again, as one takes the limit ξ → x in any of the equations (1.27), (1.28) or (1.30), one must take the finite part of the corresponding right hand side (see Section 1.4.3). For example, (1.28) and (1.30) become, respectively: ∂uk(x) ∂xm = lim ξ→x ∂B ∂Uik ∂ξm (ξ, y)σij(y) − ∂Σijk ∂ξm (ξ, y)ui(y) nj(y)dS(y) = ∂B = ∂Uik ∂xm (x, y)σij(y) − ∂Σijk ∂xm (x, y)ui(y) nj(y)dS(y) (1.35) σij(x) = lim ξ→x ∂B [Dijk(ξ, y)τk(y) − Sijk(ξ, y)uk(y)] dS(y) = ∂B = [Dijk(x, y)τk(y) − Sijk(x, y)uk(y)] dS(y) (1.36) Also, for future reference, one notes that the traction at a boundary point is: τi(x) = nj(x) lim ξ→x ∂B [Dijk(ξ, y)τk(y) − Sijk(ξ, y)uk(y)] dS(y) (1.37) Fully regularized forms of equations (1.35) and (1.36), that only contain weakly singular integrals, are available in the literature (see, for example, Cruse and Richardson [39]). These equations, that can be collocated at an irregular point x ∈ ∂B provided that the stress and displacement fields in (1.38, 1.39) satisfy certain smoothness requirements (see Martin et al. [93] and, also, Section 1.4.4 of this chapter) are: 0 = ∂B Uik,m(x, y) [σij(y) − σij(x)] nj(y)dS(y) − ∂B Σijk,m(x, y) [ui(y) − ui(x) − ui,(x) (y − x)] nj(y)dS(y) (1.38) © 2005 by Taylor Francis Group, LLC
- 36. 10 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS 0 = ∂B Dijk(x, y) [σkp(y) − σkp(x)] np(y)dS(y) − ∂B Sijk(x, y) [uk(y) − uk(x) − uk,p(x)(yp − xp)] dS(y) (1.39) An alternate version of (1.39) that can only be collocated at a regular point x ∈ ∂B is: 0 = ∂B Dijk(x, y)[τk(y) − τk(x)]dS(y) − σkm(x) ∂B Dijk(x, y)(nm(y) − nm(x))dS(y) − ∂B Sijk(x, y) [uk(y) − uk(x) − uk,m(x)(ym − xm)] dS(y) (1.40) Finally, taking the inner product of (1.40) with the normal at the source point gives: 0 = ∂B Dijk(x, y)nj(x)[τk(y) − τk(x)]dS(y) − σkm(x) ∂B Dijk(x, y)nj(x)[nm(y) − nm(x)]dS(y) − ∂B Sijk(x, y)nj(x) [uk(y) − uk(x) − uk,m(x)(ym − xm)] dS(y) (1.41) 1.2.2.1 Displacement gradient on the bounding surface The gradient of the displacement u is required for the regularized HBIEs (1.38 - 1.41). Lutz et al. [89] have proposed a scheme for carrying this out. Details of this procedure are available in [27] and are given below. The (right-handed) global Cartesian coordinates, as before, are (x1, x2, x3). Consider (right-handed) local Cartesian coordinates (x 1, x 2, x 3) at a regular point P on ∂B as shown in Figure 1.2. The local coordinate system is oriented such that the x 1 and x 2 coordinates lie along the tangential unit vectors t1 and t2 while x 3 is measured along the outward normal unit vector n to ∂B as defined in equation (1.14). Therefore, one has: x = Qx (1.42) u = Qu (1.43) where u k, k = 1, 2, 3 are the components of the displacement vector u in the local coordinate frame, and the orthogonal transformation matrix Q has the components: © 2005 by Taylor Francis Group, LLC
- 37. 1.2. LINEAR ELASTICITY IN THREE DIMENSIONS 11 x x x x x 1 2 3 3 1 2 ' ' ' x P Figure 1.2: Local coordinate system on the surface of a body (from [27]) Q = t11 t12 t13 t21 t22 t23 n1 n2 n3 (1.44) with tij the jth component of the ith unit tangent vector and (n1, n2, n3) the components of the unit normal vector. The tangential derivatives of the displacement, in local coordinates, are u i,k , i = 1, 2, 3; k = 1, 2. These quantities are obtained as follows: u i,k ≡ ∂u i ∂sk = Qij ∂uj ∂sk (1.45) where ∂u i/∂sk are tangential derivatives of ui at P with s1 = x 1 and s2 = x 2. The remaining components of ∇u in local coordinates are obtained from Hooke’s law (see [89]) as: ∂u 1 ∂x 3 = τ 1 G − ∂u 3 ∂x 1 ∂u 2 ∂x 3 = τ 2 G − ∂u 3 ∂x 2 ∂u 3 ∂x 3 = (1 − 2ν)τ 3 2G(1 − ν) − ν 1 − ν ∂u 1 ∂x 1 + ∂u 2 ∂x 2 (1.46) where τ k, k = 1, 2, 3, are the components of the traction vector in local coordi- nates. The components of the displacement gradient tensor, in the local coordinate system, are now known. They can be written as: © 2005 by Taylor Francis Group, LLC
- 38. 12 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS (∇u)local ≡ A = u 1,1 u 1,2 u 1,3 u 2,1 u 2,2 u 2,3 u 3,1 u 3,2 u 3,3 (1.47) Finally, the components of ∇u in the global coordinate frame are obtained from those in the local coordinate frame by using the tensor transformation rule: (∇u)global ≡ A = QT A Q = u1,1 u1,2 u1,3 u2,1 u2,2 u2,3 u3,1 u3,2 u3,3 (1.48) The gradient of the displacement field in global coordinates is now ready for use in equations (1.38 - 1.41). 1.3 Nearly Singular Integrals in Linear Elastic- ity It is well known that the first step in the BEM is to solve the primary problem on the bounding surface of a body (e.g. equation (1.25)) and obtain all the displacements and tractions on this surface. The next steps are to obtain the displacements and stresses at selected points inside a body, from equations such as (1.17) and (1.30). It has been known in the BEM community for many years, dating back to Cruse [37], that one experiences difficulties when trying to numerically evaluate displacements and stresses at points inside a body that are close to its bounding surface (the so-called near-singular or boundary layer problem). Various authors have addressed this issue over the last 3 decades. This section describes a new method recently proposed by Mukherjee et al. [104]. 1.3.1 Displacements at Internal Points Close to the Bound- ary The displacement at a point inside an elastic body can be determined from either of the (equivalent) equations (1.17) or (1.21). A continuous version of (1.21), from Cruse and Richardson [39] is: uk(ξ) = uk ˆ (x)+ ∂B [ Uik(ξ, y)σij(y) − Σijk(ξ, y){ui(y) − ui(x̂)} ] nj(y)dS(y) (1.49) where ξ ∈ B is an internal point close to ∂B and a target point x̂ ∈ ∂B is close to the point ξ (see Fig. 1.3). An alternative form of (1.49) is: © 2005 by Taylor Francis Group, LLC
- 39. 1.3. NEARLY SINGULAR INTEGRALS IN LINEAR ELASTICITY 13 B ∂B ξ z z v x v y Figure 1.3: A body with source point ξ, field point y and target point x̂ (from [104]) uk(ξ) = uk(x̂)+ ∂B [ Uik(ξ, y)τi(y) − Tik(ξ, y){ui(y) − ui(x̂)} ] dS(y) (1.50) Equation (1.49) (or (1.50)) is called “continuous” since it has a continuous limit to the boundary (LTB as ξ → x̂ ∈ ∂B) provided that ui(y) ∈ C0,α (i.e. Hölder continuous). Taking this limit is the standard approach for obtaining the well-known regularized form (1.26) (or (1.25)). In this work, however, equation (1.49) (or (1.50)) is put to a different, and novel use. It is first observed that Tik in equation (1.50) is O(1/r2 (ξ, y)) as ξ → y, whereas {ui(y) − ui(x̂)} is O(r(x̂, y)) as y → x̂. Therefore, as y → x̂, the product Tik(ξ, y){ui(y) − ui(x̂)}, which is O(r(x̂, y)/r2 (ξ, y)), → 0 ! As a result, equation (1.50) (or (1.49)) can be used to easily and accurately evaluate the displacement components uk(ξ) for ξ ∈ B close to ∂B. This idea is the main contribution of [104]. It is noted here that while it is usual to use (1.17) (or (1.21)) to evaluate uk(ξ) when ξ is far from ∂B, equation (1.49) (or (1.50)) is also valid in this case. (The target point x̂ can be chosen as any point on ∂B when ξ is far from ∂B). Therefore, it is advisable to use the continuous equation (1.49) (or (1.50)) universally for all points ξ ∈ B. This procedure would eliminate the need to classify, a priori, whether ξ is near to, or far from ∂B. 1.3.2 Stresses at Internal Points Close to the Boundary The displacement gradient at a point ξ ∈ B can be obtained from equation (1.34) or the stress at this point from (1.30). Continuous versions of (1.34) and (1.30) can be written as [39]: © 2005 by Taylor Francis Group, LLC
- 40. 14 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS uk,n(ξ) = uk,n(x̂) − ∂B Uik,n(ξ, y) [σij(y) − σij(x̂)] nj(y)dS(y) + ∂B Σijk,n(ξ, y) [ui(y) − ui(x̂) − ui,(x̂) (y − x̂)] nj(y)dS(y) (1.51) σij(ξ) = σij(x̂) + ∂B Dijk(ξ, y)[τk(y) − σkm(x̂)nm(y)]dS(y) − ∂B Sijk(ξ, y)[uk(y) − uk(x̂) − uk,(x̂)(y − x̂)] dS(y) (1.52) The integrands in equations (1.51) (or (1.52)) are O(r(x̂, y)/r2 (ξ, y)) and O(r2 (x̂, y)/r3 (ξ, y)) as y → x̂. Similar to the behavior of the continuous BIEs in the previous subsection, the integrands in equations (1.51) and (1.52) → 0 as y → x̂. Either of these equations, therefore, is very useful for evaluating the stresses at an internal point ξ that is close to ∂B. Of course (please see the discussion regarding displacements in the previous section), they can also be conveniently used to evaluate displacement gradients or stresses at any point ξ ∈ B. Henceforth, use of equations (1.17), (1.21), (1.30) or (1.34) will be referred to as the standard method, while use of equations (1.49), (1.50), (1.51) or (1.52) will be referred to as the new method. 1.4 Finite Parts of Hypersingular Equations A discussion of finite parts (FPs) of hypersingular BIEs (see e.g. equations (1.9 -1.11)) is the subject of this section. The general theory of finite parts is presented first. This is followed by applications of the theory in potential theory and in linear elasticity. Further details are available in Mukherjee [102]. 1.4.1 Finite Part of a Hypersingular Integral Collocated at an Irregular Boundary Point 1.4.1.1 Definition Consider, for specificity, the space R3 , and let S be a surface in R3 . Let the points x ∈ S and ξ / ∈ S. Also, let Ŝ and S̄ ⊂ Ŝ be two neighborhoods (in S) of x such that x ∈ S̄ (Figure 1.4). The point x can be an irregular point on S. Let the function K(x, y) , y ∈ S, have its only singularity at x = y of the form 1/r3 where r = |x − y |, and let φ(y) be a function that has no singularity in S and is of class C1,α at y = x for some α 0. The finite part of the integral © 2005 by Taylor Francis Group, LLC
- 41. 1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 15 S S x S y S ξ S S Figure 1.4: A surface S with regions Ŝ and S̄ and points ξ, x and y (from [102]) I(x) = S K(x, y)φ(y)dS(y) (1.53) is defined as: S = K(x, y)φ(y)dS(y) = SŜ K(x, y)φ(y)dS(y) + Ŝ K(x, y)[φ(y) − φ(x) − φ,p(x)(yp − xp)]dS(y) + φ(x)A(Ŝ) + φ,p(x)Bp(Ŝ) (1.54) where Ŝ is any arbitrary neighborhood (in S) of x and: A(Ŝ) = Ŝ = K(x, y)dS(y) (1.55) Bp(Ŝ) = Ŝ = K(x, y)(yp − xp)dS(y) (1.56) The above FP definition can be easily extended to any number of physical dimensions and any order of singularity of the kernel function K(x, y). Please refer to Toh and Mukherjee [168] for further discussion of a previous closely re- lated FP definition for the case when x is a regular point on S, and to Mukherjee [101] for a discussion of the relationship of this FP to the CPV of an integral when its CPV exists. © 2005 by Taylor Francis Group, LLC
- 42. 16 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS 1.4.1.2 Evaluation of A and B There are several equivalent ways for evaluating A and B. Method one. Replace S by Ŝ and Ŝ by S̄ in equation (1.54). Now, setting φ(y) = 1 in (1.54) and using (1.55), one gets: A(Ŝ) − A(S̄) = ŜS̄ K(x, y)dS(y) (1.57) Next, setting φ(y) = (yp − xp) (note that, in this case, φ(x) = 0 and φ,p(x) = 1) in (1.54), and using (1.56), one gets: Bp(Ŝ) − Bp(S̄) = ŜS̄ K(x, y)(yp − xp)dS(y) (1.58) The formulae (1.57) and (1.58) are most useful for obtaining A and B when Ŝ is an open surface and Stoke regularization is employed. An example is the application of the FP definition (1.54) (for a regular collocation point) in Toh and Mukherjee [168], to regularize a hypersingular integral that appears in the HBIE formulation for the scattering of acoustic waves by a thin scatterer. The resulting regularized equation is shown in [168] to be equivalent to the result of Krishnasamy et al. [75]. Equations (1.57) and (1.58) are also used in Mukherjee and Mukherjee [99] and in Section 3.2 of [102]. Method two. From equation (1.57): A(Ŝ) − A(S̄) = ŜS̄ K(x, y)dS(y) = lim ξ→x ŜS̄ K(ξ, y)dS(y) (1.59) The second equality above holds since K(x, y) is regular for x ∈ S̄ and y ∈ ŜS̄. Assuming that the limits: lim ξ→x Ŝ K(ξ, y)dS(y), lim ξ→x S̄ K(ξ, y)dS(y) exist, then: A(Ŝ) = lim ξ→x Ŝ K(ξ, y)dS(y) (1.60) Similarly: Bp(Ŝ) = lim ξ→x Ŝ K(ξ, y)(yp − xp)dS(y) (1.61) Equations (1.60) and (1.61) are most useful for evaluating A and B when Ŝ = ∂B, a closed surface that is the entire boundary of a body B. Examples appear in Sections 1.4.2 and 1.4.3 of this chapter. © 2005 by Taylor Francis Group, LLC
- 43. 1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 17 Method three. A third way for evaluation of A and B is to use an auxiliary surface (or “tent”) as first proposed for fracture mechanics analysis by Lutz et al. [89]. (see, also, Mukherjee et al. [110], Mukherjee [105] and Section 3.2.1 of [102]. This method is useful if S is an open surface. 1.4.1.3 The FP and the LTB There is a very simple connection between the FP, defined above, and the LTB approach employed by Gray and his coauthors. With, as before, ξ / ∈ S, x ∈ S (x can be an irregular point on S), K(x, y) = O(|x − y|−3 ) as y → x and φ(y) ∈ C1,α at y = x, this can be stated as: lim ξ→x S K(ξ, y)φ(y)dS(y) = S = K(x, y)φ(y)dS(y) (1.62) Of course, ξ can approach x from either side of S. Proof of equation (1.62). Consider the first and second terms on the right- hand side of equation (1.54). Since these integrands are regular in their respec- tive domains of integration, one has: SŜ K(x, y)φ(y)dS(y) = lim ξ→x SŜ K(ξ, y)φ(y)dS(y) (1.63) and Ŝ K(x, y)[φ(y) − φ(x) − φ,p(x) (yp − xp)]dS(y) = lim ξ→x Ŝ K(ξ, y)[φ(y) − φ(ξ) − φ,p(ξ)(yp − ξp)]dS(y) (1.64) Use of equations (1.60, 1.61, 1.63 and 1.64) in (1.54) proves equation (1.62). 1.4.2 Gradient BIE for 3-D Laplace’s Equation This section is concerned with an application of equation (1.54) for collocation of the HBIE (1.9), for the 3-D Laplace equation, at an irregular boundary point. A complete exclusion zone, Ŝ = ∂B is used here. An application of a vanishing exclusion zone, for collocation of the HBIE for the 2-D Laplace equation, at an irregular boundary point, is presented in Mukherjee [102]. Using equations (1.4) and (1.6), equations (1.9) and (1.10) are first written in the slightly different equivalent forms: ∂u(ξ) ∂ξi = ∂B [Di(ξ, y)τ(y) − Si(ξ, y)u(y)] dS(y) (1.65) ∂u(x) ∂xi = ∂B = [Di(x, y)τ(y) − Si(x, y)u(y)] dS(y) (1.66) © 2005 by Taylor Francis Group, LLC
- 44. 18 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS where: Di(x, y) = −G,i(x, y) , Si = −Hk,i(x, y)nk(y) (1.67) Use of (1.54) in (1.66), with S = Ŝ = ∂B, results in: u,i(x) = ∂B Di(x, y) u,p(y) − u,p(x) np(y)dS(y) − ∂B Si(x, y) u(y) − u(x) − u,p(x)(yp − xp) dS(y) − Ai(∂B)u(x) + Cip(∂B)u,p(x) (1.68) where, using method two in Section 1.4.1.2: Ai(∂B) = lim ξ→x ∂B Si(ξ, y)dS(y) (1.69) Cip(∂B) = lim ξ→x ∂B [Di(ξ, y)np(y) − Si(ξ, y)(yp − ξp)] dS(y) (1.70) It is noted here that the (possibly irregular) boundary point x is approached from ξ ∈ B, i.e. from inside the body B. The quantities A and C can be easily evaluated using the imposition of simple solutions. Following Rudolphi [143], use of the uniform solution u(y) = c (c is a constant) in equation (1.65) gives: ∂B Si(ξ, y)dS(y) = 0 (1.71) while use of the linear solution: u = u(ξ) + (yp − ξp)u,p(ξ) τ(y) = ∂u ∂yk nk(y) = u,p(ξ)np(y) (with p = 1, 2, 3) (1.72) in equation (1.65) (together with (1.71)) gives: ∂B [Di(ξ, y)np(y) − Si(ξ, y)(yp − ξp)] dS(y) = δip (1.73) Therefore, (assuming continuity) Ai(∂B) = 0, Cip(∂B) = δip, and (1.68) yields a simple, fully regularized form of (1.66) as: 0 = ∂B Di(x, y)[u,p(y) − u,p(x)]np(y)dS(y) − ∂B Si(x, y)[u(y) − u(x) − u,p(x)(yp − xp)]dS(y) (1.74) © 2005 by Taylor Francis Group, LLC
- 45. 1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 19 which is equivalent to equation (1.11). A few comments are in order. First, equation (1.74) is the same as Rudol- phi’s [143] equation (20) with (his) κ = 1 and (his) S0 set equal to S and renamed ∂B. (See, also, Kane [68], equation (17.34)). Second, this equation can also be shown to be valid for the case ξ / ∈ B, i.e. for an outside approach to the boundary point x . Third, as noted before, x can be an edge or corner point on ∂B (provided, of course, that u(y) ∈ C1,α at y = x - Rudolphi had only considered a regular boundary collocation point in his excellent paper that was published in 1991). Finally, as discussed in the Section 1.4.3, equation (1.74) is analogous to the regularized stress BIE in linear elasticity - equation (28) in Cruse and Richardson [39] . 1.4.3 Stress BIE for 3-D Elasticity This section presents a proof of the fact that equation (1.39) is a regularized version of (1.36), valid at an irregular point x ∈ ∂B, provided that the stress and displacement fields in (1.39) satisfy certain smoothness requirements. These smoothness requirements are discussed in Section 1.4.4. The approach is very similar to that used in Section 1.4.2. The first step is to apply the FP equation (1.54) to regularize (1.36). With S = Ŝ = ∂B, the result is: σij(x) = ∂B Dijk(x, y) [σkp(y) − σkp(x)] np(y)dS(y) − ∂B Sijk(x, y) [uk(y) − uk(x) − uk,p(x)(yp − xp)] dS(y) − Aijk(∂B)uk(x) + Cijkp(∂B)uk,p(x) (1.75) where, using method two in Section 1.4.1.2: Aijk(∂B) = lim ξ→x ∂B Sijk(ξ, y)dS(y) (1.76) Cijkp(∂B) = lim ξ→x ∂B EmkpDijm(ξ, y)n(y)dS(y) − lim ξ→x ∂B Sijk(ξ, y)(yp − ξp)dS(y) (1.77) with E the elasticity tensor (see (1.23)) which appears in Hooke’s law: σm = Emkpuk,p (1.78) Simple (rigid body and linear) solutions in linear elasticity (see, for example, Lutz et al. [89], Cruse and Richardson [39]) are now used in order to determine © 2005 by Taylor Francis Group, LLC
- 46. 20 CHAPTER 1. BOUNDARY INTEGRAL EQUATIONS the quantities A and C. Using the rigid body mode uk = ck (ck are arbitrary constants) in (1.30), one has: 0 = ∂B Sijk(ξ, y)dS(y) (1.79) while, using the linear solution: uk(y) = (yp − ξp)uk,p(ξ), uk,m(y) = uk,m(ξ), τk(y) = σkm(y)nm(y) = Ekmrsur,s(ξ)nm(y) (1.80) in equation (1.30) gives: σij(ξ) = uk,p(ξ) ∂B [EmkpDijm(ξ, y)n(y) − Sijk(ξ, y)(yp − ξp)] dS(y) (1.81) Taking the limit ξ → x of (1.79), using continuity of the integral and com- paring with (1.76), gives A = 0. Taking the limit ξ → x of (1.81) and comparing with (1.77), one has: σij(x) = Cijkpuk,p(x) (1.82) Comparing (1.82) with (1.78) yields C(∂B) = E. Therefore, equation (1.75) reduces to the simple regularized equation (1.39). Equation (1.39) is equation (28) of Cruse and Richardson [39] in the present notation. As is the case in the present work, Cruse and Richardson [39] have also proved that their equation (28) is valid at a corner point, provided that the stress is continuous there. It has been proved in this section that the regularized stress BIE (28) of Cruse and Richardson [39] can also be obtained from the FP definition (1.54) with a complete exclusion zone. 1.4.4 Solution Strategy for a HBIE Collocated at an Ir- regular Boundary Point Hypersingular BIEs for a body B with boundary ∂B are considered here. Regu- larized HBIEs, obtained by using complete exclusion zones, e.g. equation (1.74) for potential theory or (1.39) for linear elasticity, are recommended as starting points. An irregular collocation point x for 3-D problems is considered next. Let ∂Bn, (n = 1, 2, 3, ..., N) be smooth pieces of ∂B that meet at an irregular point x ∈ ∂B. Also, as before, let a source point, with coordinates xk, be denoted by P, and a field point, with coordinates yk, be denoted by Q. Martin et al. [93] state the following requirements for collocating a regular- ized HBIE, such as (1.39) at an irregular point P ∈ ∂B. These are: © 2005 by Taylor Francis Group, LLC
- 47. 1.4. FINITE PARTS OF HYPERSINGULAR EQUATIONS 21 (i) The displacement u must satisfy the equilibrium equations in B. (ii) (a) The stress σ must be continuous in B. (b) The stress σ must be continuous on ∂B. (iii) |ui(Qn) − ui L (Qn; P)| = O(r (1+α) n ) as rn → 0, for each n. (iv) [σij(Qn) − σij(P)]nj(Qn) = O(rα n) as rn → 0, for each n. Box 1.1 Requirements for collocation of a HBIE at an irregular point (from [93]). In the above, rn = |y(Qn) − x(P)|, Qn ∈ ∂Bn, and α 0. Also, uL i (Qn; P) = ui(P) + ui,j(P)[yj(Qn) − xj(P)] (1.83) There are two important issues to consider here. The first is that, if there is to be any hope for collocating (1.39) at an irregular point P, the exact solution of a boundary value problem must satisfy conditions (i-iv) in Box 1.1. Clearly, one should not attempt this collocation if, for example, the stress is unbounded at P (this can easily happen - see an exhaustive study on the subject in Glushkov et al. [50]), or is bounded but discontinuous at P (e.g. at the tip of a wedge - see, for example, Zhang and Mukherjee [183]). The discussion in the rest of this book is limited to the class of problems, referred to as the admissible class, whose exact solutions satisfy conditions (i - iv). The second issue refers to smoothness requirements on the interpolation functions for u, σ and the traction τ = n · σ in (1.39). It has proved very difficult, in practice, to find BEM interpolation functions that satisfy, a priori, (ii(b)-(iv)) in Box 1.1, for collocation at an irregular surface point on a 3-D body [93]. It has recently been proved in Mukherjee and Mukherjee [111], however, that interpolation functions used in the boundary contour method (BCM - see, for example, Mukherjee et al. [109], Mukherjee and Mukherjee [99]) satisfy these conditions a priori. Another important advantage of using these interpolation functions is that ∇u can be directly computed from them at an irregular boundary point [99], without the need to use the (undefined) normal and tangent vectors at this point. In principle, these BCM interpolation functions can also be used in the BEM. The BCM and the hypersingular BCM (HBCM) are discussed in detail in Chapter 4 of this book. Numerical results from the hypersingular BCM, collocated on edges and at corners, from Mukherjee and Mukherjee [111], are available in Chapter 4. © 2005 by Taylor Francis Group, LLC
- 48. Chapter 2 ERROR ESTIMATION Pointwise (i.e. that the error is evaluated at selected points) residual-based error estimates for Dirichlet, Neumann and mixed boundary value problems (BVPs) in linear elasticity are presented first in this chapter. Interesting relationships between the actual error and the hypersingular residuals are proved for the first two classes of problems, while heuristic error estimators are presented for mixed BVPs. Element-based error indicators, relying on the pointwise error measures presented earlier, are proposed next. Numerical results for two mixed BVPs in 2-D linear elasticity complete this chapter. Further details are available in [127]. 2.1 Linear Operators Boundary integral equations can be analyzed by viewing them as linear equa- tions in a Hilbert space. A very readable account of this topic is available in Kress [73]. Following Sloan [155], it is assumed here that the boundary ∂B is a C1 continuous closed Jordan curve given by the mapping: z : [0, 1] → ∂B, z ∈ C1 , |z | = 0 where z ∈ C, the space of complex numbers. The present analysis excludes domains with corners. It is also assumed that any integrable function v on ∂B may be represented in a Fourier series: v ∼ ∞ k=−∞ v̂(k)e2πikx1 = a0 + ∞ k=1 (ak cos(2πkx1) + bk sin(2πkx1) (2.1) where i ≡ √ −1 and: v̂(k) = 1 0 e−2πikx1 v(x1)dx1, k ∈ Z (2.2) 23 © 2005 by Taylor Francis Group, LLC
- 49. Discovering Diverse Content Through Random Scribd Documents
- 50. PLAN No. 472. GETTING A START WITH INK POWDERS A young man whose ambition was to build up a permanent business from a small beginning, as he was practically without capital, concluded to start on one item at first, and gradually add others as he could afford it, so he chose inks—not one, but several kinds of inks. These inks he purposed to put up in the form of powders, leaving only the hot water to be added by the customer. But the different formulas were all so good that from anyone of them an enterprising man could work into a good-paying business, and they are therefore submitted herewith as separate plans. Here is the formula he used for producing a powder for a writing fluid that is equal to the best inks on the market and better than most of them. And the beauty of it was that he could sell enough of the powder for 10 cents to make a full pint of the very best ink, and realize a very good profit on it over that price: Nigrosin, 1 ounce; soluble blue or water blue anilin, 2 ounces; salicylic acid, 15 grains; dextrin, 11 ⁄2 ounces. This will make from one to two gallons of ink, when dissolved in hot water, according to the shade desired—the more powder the darker the ink. Fine for either ordinary or fountain pen, and sold well.
- 51. PLAN No. 473. BLUE INK POWDER Many people prefer blue ink, and for them he made powders of an excellent quality as follows: Water-blue anilin, 1 dram; dextrin, 5 drams; or according to the following formula: Soluble Prussian blue, 1 dram; dextrin, 2 drams. Dissolve the powder in hot water, varying the intensity of the blue shade as desired, by using more or less powder. This was a popular and profitable seller.
- 52. PLAN No. 474. GREEN INK POWDER Green ink is a novelty, and for that reason many people like to use it. He made the powders for green ink as follows: Green anilin, 1 dram; dextrin, 4 drams. To use, dissolve in hot water, using more or less of the powder as darker or lighter shades of the green are desired. Very easy and cheap to make; very easy and profitable to sell.
- 53. PLAN No. 475. RED INK POWDER Red ink is always in demand, but many of the red inks on sale at stationery and other stores are of a very inferior quality. Red ink made from the following formula, as this man made it, gives universal satisfaction in all cases where red ink is required: Red anilin, 1 dram; dextrin, 1 dram. To use, dissolve the powder in hot water. These various ink powders are usually put up in packages of a sufficient quantity to make a pint of ink, and this requires from a teaspoonful to a tablespoonful of the powder. Having no capital, the young man began with the direct selling plan, canvassing from house to house and from store to store, and selling to his acquaintances whenever possible. From the profits these sales brought him, he was soon able to take up the trust scheme, sending twenty-four packages of the powder, put up in small envelopes, to boys and girls whose names he obtained in various ways, offering them a premium of a watch, a camera, roller skates, silver spoons, or other articles he could buy cheap in quantities, when each one had sold and remitted for the twenty-four packages. Later he inserted 25-word ads. in various papers, and made a large number of sales direct by mail from that source. Today he owns the largest and best patronized stationery store in his town.
- 54. HOW SEVEN BOYS EARNED MONEY Seven boys, from 12 to 15 years of age, all pupils at the same city school, and all close chums, adopted seven different ways of earning a little money during vacation, and it is pleasing to know that all seven succeeded. Here are the plans they followed, one boy to each plan: PLAN No. 476. CANCELED POSTAGE STAMPS One boy went to the large business houses and collected all the canceled stamps he could find on envelopes received through the mails. Many of these were from foreign countries and brought good prices when offered to dealers or boys making stamp collections, while the domestic stamps he sold for 25 cents per thousand. During the vacation period that year he made over $50. PLAN No. 477. BOUGHT A PRINTING PRESS Another boy induced his father to help him buy a small printing press, and cards of various sizes. He then took orders for the printing of these cards for other boys and for men needing the cheaper grade, charging 75 cents per hundred and cleared up nearly $40 above expenses, besides paying for his printing press. PLAN No. 478. PARLOR MAGIC The next boy with a taste for entertaining, and being clever at sleight-of-hand tricks, bought a book on parlor magic, and gave entertainments at his own home and the homes of other boys, charging 10 cents admission. He performed these tricks so well that everyone felt that he or she had received full value for the dime paid
- 55. at the door, and the youthful entertainer realized a net profit of almost $60 during the three months of his summer vacation. PLAN No. 479. DID SCROLL-SAW WORK The fourth boy, being of a mechanical turn of mind, bought a scroll-saw, with which he made a great variety of very pretty things, and for these the neighbors were glad to pay good prices, especially where he made any special design to order. He was very skilful in his work, and was kept busy most of the time, so that his net earnings during vacation were $37. PLAN No. 480. A LEMONADE STAND The fifth boy had a taste for merchandising, and set up a lemonade stand in the front yard of his home, where many people passed every day. He had various-sized glasses in which he put his lemonade, properly made and tastefully displayed, and sold his product at 1 to 5 cents a glass, according to size. He also had some very good ice cream which he sold in small dishes at 2 to 5 cents a dish. Children were his principal customers, but even at these low prices, he made a good profit on his sales, and the business netted him a little more than $30 altogether. PLAN No. 481. DOING ODD JOBS The sixth boy did odd jobs wherever he could find them, such as carrying satchels or parcels from stores, or to and from trains, pushing baby carriages in the parks, running errands for neighbors, and anything else that came handy. He was always on the lookout for work and was very seldom idle. His earnings were $23.75, and he was very well satisfied with that.
- 56. PLAN No. 482. COLLECTING OLD MAGAZINES FOR SALE The seventh boy went from house to house, collecting all the old magazines that people were willing to give away, and sold these to dealers at a good price per pound, as anything made of paper was in good demand. This boy was more successful and his earnings were $70 during that three months of vacation.
- 57. SUGGESTIONS FOR THE FOLLOWING PLANS A few of the following plans, are mere outlines containing suggestions which may be worked out in more detail by those who wish to make use of them. New features may be added as they suggest themselves to each person adopting one or more of the plans as a means of making a living. In giving so many under one heading, space will not permit a separate method for handling each plan. In order to determine the best selling plan, or the best method of profitably handling any of the ways outlined, it would be well for a person to read as many of the plans set forth as possible, and become familiar with the various means employed by others to obtain the best possible results. Selling plans for produce named in this book are of various kinds, and include personal solicitation by a house-to-house canvass, the employment of agents to sell on a commission basis, placing the article on sale with druggists and dealers, mail order, advertising in suitable mediums, giving away of coupons to dealers, who in turn give them to their customers; the trust plan, or sending a certain number of articles or packages to children, to be sold by them at a certain low price, and paying a premium either in merchandise or cash; filling orders by parcel post; placing of general advertising through a reputable advertising agency, that will not only help to prepare the proper kind of advertising, but also be able to select the best mediums for that particular product; selling of certain items of information direct to the customer, telling him how to make practical use of certain ideas of which he had no previous knowledge. All the above selling plans are set forth in various parts of this book, in connection with the statement of how certain plans were successfully worked by individuals who adopted them as a livelihood, and the testimony of these persons should prove a valuable guide to others seeking similar results.
- 58. PLAN No. 483. CADET OFFICER FOR U. S. SEE PLAN No. 217
- 59. PLAN No. 484. LITTLE “TINKERING” JOBS Replenishing and replacing batteries for doorbells, mending kitchen-ware, and replacing various articles about the house will often give a very good income in a small place where experts from large establishments are not within reach. Many an elderly man, who could not do anything else, has made a comfortable living by doing these little “odd jobs.”
- 60. PLAN No. 485. CARPENTER FOR U. S. SEE PLAN No. 217
- 61. PLAN No. 486. DESIGNER-LANDSCAPE. SEE PLAN No. 217
- 62. PLAN No. 487. THE “HOKEY-POKEY” SUMMER SELLER One of the most delicious confections, and one that scores the largest number of sales during the summer season, is made as follows: One can condensed milk; 2 tablespoonfuls cornstarch; a little cold milk. Put the remainder of the milk in a double boiler, and when hot add the cornstarch. Cook five minutes, then add the condensed milk, and set aside to cool; then add the vanilla, and freeze. Cut into squares or sticks and pack closely in a wooden pail, and it will sell readily for 5 or 10 cents a stick. A splendid seller at fairs, picnics, parties, etc., and a popular delicacy in the city at soft-drink stands and confectionery stores. Yields an unusually large profit.
- 63. PLAN No. 488. A SHOE POLISH IN POWDER FORM Shoe polishes always sell, and it is only a question as to which is the best one. The following is not excelled: Take powdered gum arabic, 5 pounds; sugar, 11 ⁄4 pounds; analine black, 3 ounces. Powder these and mix well. Then divide into ten packets, each of which will produce a pint of polish, or into twenty packets that will make a half-pint each, though more may be made from, a packet, as it is rather thick, especially for kid or glaze leathers. It can be used with either water or vinegar, or these combined, in which to dissolve the powder. Apply with a brush, and continue the friction until the superfluous fluid dries and the polish appears. To make this a tan polish, use 1 ounce of chrysodine, instead of the analine black. A fine polish and a good profit in this preparation.
- 64. PLAN No. 489. LETTER CARRIERS FOR U. S. SEE PLAN No. 217
- 65. PLAN No. 490. METAL POLISHING BLOCKS These are made of precipitated chalk, 2 pounds; powdered tripoli, 1 ⁄2 pound; jewelers’ rouge, 1 ounce. Mix into a stiff paste, with 1 ounce of glycerine and a pint of water, previously mixed, and pour on just enough of the liquid to work the powders to the consistency of fresh dough. Then place in little wooden butter molds to shape them and set aside to dry, then force out and fill again. The blocks are used with a soft cloth and a few drops of water, which will give metal articles a fine polish. You can sell all you can make of these, and realize a profit on them that will surprise you.
- 66. PLAN No. 491. CEMENT WORKER FOR U. S. SEE PLAN No. 217
- 67. PLAN No. 492. CERAMICS FOR U. S. SEE PLAN No. 217
- 68. PLAN No. 493. SOAP LEAVES FOR TRAVELERS’ USE These are made by passing sheets of paper over rollers and through a hot solution of liquid soap, then passing it over drying cylinders, and cutting it into sheets of the desired size. They are so convenient and cheap that travelers will buy them and there is a good profit in making and selling them.
- 69. PLAN No. 494. HAVING THE BUTTER YELLOW IN WINTER Just a little secret, but it is worth a good deal to buttermakers and housewives who pride themselves upon the color of their butter, and will pay something to know just how to obtain it: Just before you finish churning, put the yolk of one or more eggs into the churn, and you’ll have just the color you desire—a rich yellow.
- 70. PLAN No. 495. REMOVING FOUL AIR FROM WELLS To determine whether or not the air at the bottom of the well is foul, place a lighted torch or lamp in a bucket and lower it into the well. If it continues to burn when the bucket rests on the water, it is safe to descend. If it is extinguished, the air is foul. To remove this, lower a pail filled with burning straw, or by dropping two or three quarts of freshly slaked lime down the well. But test with the light again before descending. Plenty of people who have wells would gladly pay a small sum to have this information mailed to them.
- 71. PLAN No. 496. A QUICK FATTENING FOR FOWLS Fowls will quickly fatten if given a mixture of ground rice, well scalded with milk, to which some coarse sugar has been added, making it rather thick. Feed several times a day, but not too much at a time. An ad. in poultry journals, offering to tell how this is done, for 25 cents, should bring excellent results.
- 72. PLAN No. 497. ARM AND BUST DEVELOPER Regarding it as every woman’s duty to look her best at all times, a young lady in Denver prepared a most effective arm and bust developer from the following formula: Lanolin, 2 ounces; cocoa butter, 2 ounces; olive oil, 2 ounces. These she melted in a double boiler, and heat until cold, when it was ready to put up in 2-ounce jars that sold for 40 cents each, and proved so satisfactory that she received hundreds of orders each month, through a few ads. judiciously placed, besides having a good sale through drug stores. The directions she gave were to first bathe the parts with hot water, to open the pores, and then rub in the cream very thoroughly at bedtime for a number of nights.
- 73. PLAN No. 498. REMEDY FOR BRITTLE NAILS Women who are annoyed by having brittle nails are always glad to learn of some effective way to make and keep them soft. This prompted a young lady in St. Paul to utilize the following formula: White petroleum, 1 ounce; powdered castile soap, 1 dram; oil of bergamot, a few drops. This softens the nails, cures hang-nails and renders the cuticle around the nails soft and pliable, so that it can be easily removed with a towel or orange stick. One small ad. in a leading magazine brought a great many orders, and by repeating the ad. in other periodicals, the young lady earned $1,500 clear profit the first year.
- 74. PLAN No. 499. BATH POWDER The delights of the bath are greatly multiplied by adding a well prepared bath powder, and one of the very best of these was put up by this lady, as follows: Borax, 10 ounces; tartaric acid, 10 ounces; starch, 5 ounces. Mix the ingredients together, and perfume with lavender water. Two teaspoonfuls of the powder to a tub of water will soften and perfume the same making it at the same time more cleansing and delightful. She put this powder up in 8-ounce paper boxes, and sold it for 25 cents a box. It proved a good seller all the year round and the profits were exceptionally large. The drug stores carried it in stock, as it assisted greatly in making other sales, owing to the demand for it.
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