belytschko1999

PERGAMON
Computers and Structures 71 (1999) 173±195
Smoothing, enrichment and contact in the element-free
Galerkin method
T. Belytschko *, M. Fleming
Northwestern University, Department of Mechanical Engineering, Evanston, IL 60208, USA
Received 11 July 1997; received in revised form 8 September 1998; accepted 18 September 1998
Abstract
The element-free Galerkin (EFG) method belongs to the class of mesh-free methods, which are well-suited to
problems involving crack propagation due to the absence of any prede®ned element connectivity. However, the
original visibility criterion used to model cracks leads to interior discontinuities in the displacements. Three methods
for smoothing meshless approximations near nonconvex boundaries such as cracks are reviewed and compared: (1)
the di€raction method, which wraps the nodal domain of in¯uence a short distance around a point of discontinuity,
such as a crack tip; (2) the transparency method, which gradually severs the domains of in¯uence near crack tips;
and (3) the ``see-through'' method, or continuous line criterion. Two techniques for enriching the EFG
approximations near the tip of a linear elastic crack are also summarized and compared: extrinsic enrichment, in
which special functions are added to the trial function; and intrinsic enrichment, in which the EFG basis is
expanded by special functions. A contact algorithm based on a penalty method is also introduced for enforcing
crack contact in overall compressive ®elds. Several problems involving arbitrary crack propagation are solved to
illustrate the e€ectiveness of EFG for this class of problems. # 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Composite materials; Thermomechanical properties; Nonlinear constitutive law; Incremental mean ®eld method; Mori±
Tanaka method; Finite element method; Implicit implementation
1. Introduction
The element-free Galerkin (EFG) method is a meshfree method which is particularly useful for computing
arbitrary crack propagation. In EFG, an approximation
is written in terms of a set of nodes, with modi®cations
to account for the surfaces of the model. This class of
methods is often called meshless, gridless or particle
methods because of the absence of any prede®ned nodal
connectivity; we have recently adopted the cognomen
mesh-free because of its more positive ¯avor.
* Corresponding author. E-mail: [email protected].
Mesh-free methods were developed in the late 1970 s.
Lucy [28] introduced a particle method called smoothed
particle hydrodynamics (SPH) for modeling astrophysical phenomena and Gingold and Monaghan [15] and
Monaghan [30] used this method in problems without
boundaries such as rotating stars and dust clouds.
Libersky and Petschek [23] extended this method to
solve solid mechanics problems. Swegle et al. [40] noted
a tensile instability in SPH and proposed a stabilization
technique. Attaway et al. [2] coupled SPH to ®nite elements through a contact algorithm.
A separate branch of mesh-free methods arose from
the work of Nayroles et al. [31], who proposed a diffuse element method (DEM) using a basis function
0045-7949/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 4 5 ± 7 9 4 9 ( 9 8 ) 0 0 2 0 5 - 3
174
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
and a weight function to form a local approximation
based on a set of nodes. Belytschko et al. [7] recognized this approximation as the moving least squares
(MLS) approximation described in Lancaster and
Salkauskas [22] and developed a similar method called
the element-free Galerkin (EFG) method. It has proven very e€ective for fracture and crack
growth [4, 8, 27]. Krongauz and Belytschko [20]
recently showed that DEM is not consistent and developed a correction. Liu et al. [25] proposed a mesh-free
method called the reproducing kernel particle method
(RKPM) with an approximation based on a kernel.
The form of the kernel is similar to SPH, but it contains a correction function which enforces consistency.
Belytschko et al. [6] have shown that the discrete form
of the corrected kernel approximation is identical to
an MLS approximation.
A third viewpoint of mesh-free approximations is
based on partitions of unity. These methods include
hp-clouds [11] and the partition of unity ®nite element
method (PUFEM) [29]. In the hp-cloud method, a partition of unity based on moving least square approximations is constructed. Melenk and BabusÏ ka [29]
proposed an extrinsic enrichment using the concept of
the partition of unity.
Other mesh-free methods are the particle in cell
(PIC) method [39], the generalized ®nite di€erence
method [24], and the ®nite point method [34].
Belytschko et al. [6] provide a comprehensive review of
mesh-free methods and describe the relationship
between several of the methods.
Mesh-free methods such as EFG can provide an
excellent complement to ®nite element methods in situations where ®nite elements are not e€ective. A formulation for consistently coupling EFG and FE by
blending the approximations has been presented in
Belytschko [9]. This allows the speed and simplicity of
®nite elements to be exploited while allowing EFG to
be used in regions where a mesh-free method is
needed.
One class of problems which is inherently dicult
with ®nite element methods is crack propagation along
arbitrary paths. In ®nite element methods, two
approaches have been taken for crack growth modeling:
1. The so-called crack smearing models, where the
crack is represented by modi®cations of the constitutive equations, which we call crack ®tting.
2. The discrete crack model, where the crack is
restricted to element edges and arbitrary paths are
accommodated by remeshing, or crack tracking
models.
The situation is analogous to the treatment of shocks
in computational ¯uid dynamics, where two methods
have evolved:
1. Shock capturing, or shock ®tting models, in which
the shock is spread over several points and its orientation is independent of the mesh.
2. Shock tracking methods, where a distinct representation of the shock is incorporated in the model.
Shocks, like cracks, are discontinuities in the primary
dependent variable. However, while in shock problems
the material behavior is stable, in crack smearing
methods an unstable material model is used. Thus,
crack simulation by smearing methods is more dicult
and problematic.
One way to handle crack propagation by tracking
methods or discrete crack models is by remeshing the
geometry. Swenson and Ingra€ea [41] presented a local
remeshing technique in which elements ahead of the
crack tip in the propagation direction are removed and
the crack extended. The area around the crack tip is
triangulated to create a new local mesh. This method
has the advantage that mature ®nite element technology can be used. Drawbacks to this method include
diculties if the crack step size is too small, the need
for projection of state variables between meshes in
nonlinear and dynamic analyses and the substantial
cost of remeshing. Complex geometries and interacting
crack tips are dicult to treat.
Al-Ostaz and Jasiuk [1] among others, modeled fracture and crack growth with ®nite elements by deleting
elements which met a criterion. This approach is not
based on fracture mechanics. It requires a very ®ne
mesh to get an acceptable representation of a crack.
Other techniques for modeling crack growth include
spring network models in which the material is represented by a network of springs and crack propagation is simulated by breaking springs [37]. Boundary
element methods have also been used to model crack
propagation [14]. This method is attractive due to the
absence of a domain mesh, making crack extension
relatively simple in two dimensions. However, the need
for a Green's function limits the scope of this method
since Green's functions for anisotropic and nonlinear
problems are not readily available.
This paper describes recent advances and reviews
and compares earlier work in the element-free
Galerkin methods for computational fracture mechanics. In Section 2, the moving least squares methodology for EFG approximations is reviewed. The
elastostatic boundary value problem is presented along
with its associated weak form; nodal domains of in¯uence and integration of the weak form are discussed.
Some confusion invariably arises when the mesh-free
approximant is used in a Galerkin method because in-
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
tegration of the weak form is performed by Gauss
quadrature which requires integration cells. Although
this detracts from the ``mesh-free'' nature of the
method, the background cell structure by no means
destroys it. True mesh-free methods can be designed
by limiting quadrature to the nodal points, but this
detracts from the accuracy.
Smoothing of EFG approximation near nonconvex
boundaries is reviewed in Section 4. Without smoothing, EFG approximations near nonconvex boundaries
such as crack tips will be discontinuous. Three
methods for smoothing the approximant are described
and compared: (1) the di€raction method, which
smooths EFG approximations by wrapping the nodal
support a short distance around the point at which the
discontinuity would begin; (2) the transparency
method, which yields smooth approximations by
gradually enforcing the crack rather than abruptly;
and (3) the ``see-through'' method or continuous line
criterion for situations in which a continuous line can
be drawn between the node and a sampling point without leaving the domain of in¯uence. A modi®cation of
the di€raction and transparency methods which generalizes them to arbitrary, nonconvex boundaries is
described.
Section 5 summarizes and compares enrichment
techniques for the EFG method. These methods hinge
on knowledge of certain aspects of the solution and
are developed for linear elastic cracks. The enrichment
method can be classi®ed as: (1) extrinsic enrichment, in
which the approximation is enhanced by adding separate functions to the approximation; and (2) intrinsic
enrichment, in which the solution is added topthe
 basis.
For linear elastic cracks, enrichment by the r term or
the entire near-tip asymptotic solution is compared.
Section 6 introduces a contact algorithm for cracks
in sliding contact. Contact is enforced by a penalty
method and examples are presented to demonstrate its
usefulness for cracks in bodies subjected to compressive loading.
2. Mesh-free approximations by MLS
A mesh-free approximation for a discrete system is
one which is written entirely in terms of the parameter
values at nodesÐno prede®ned connectivities between
nodes are established, in contrast to ®nite elements or
®nite di€erence method. Instead, the connectivity is
established during the construction of the approximation. A smooth, monotonically decreasing weight
function is de®ned at each node such that in two
dimensions the whole domain is covered by the support of at least three distinct functions. Common
shapes, often called weight function supports, in two
dimensions are circles and rectangles, and are spheres
175
Fig. 1. A computational model for a mesh-free method showing the boundary, nodes and circular supports.
and bricks in three dimensions. When a weight function intersects an internal boundary such as a crack or
a hole, in the visibility criterion method the support of
the weight function is truncated (see Fig. 1).
Two commonly used weight functions are the
Gaussian and the quartic spline given in Eqs. (1a) and
(1b). For the circular weights shown in Fig. 1, the
weight functions are:
Gaussian : w dI †
exp ÿdI =c†2 † ÿ exp ÿ dmI =c†2 †
dI RdmI
1 ÿ exp ÿ dmI =c†2 †
ˆ
0
dI > dmI
1a†
quartic spline : w dI †
2 3 4
dI
1ÿ6
‡8 dI ÿ3 dI
dmI
dmI
ˆ
dmI
0
dI RdmI
dI > dmI
1b†
where dI = kxÿ xIk is the distance from a sampling
point x to a node xI, and dmI is the domain of in¯uence or support of a node, i.e. the area over which the
weight function is nonzero. The variable c in the
Gaussian weight is used to control the dilation of the
weight function. It is useful to de®ne a characteristic
nodal spacing, cI, which is a distance such that a node
possesses a minimum set of neighbors sucient for
regularity of the equations used to determine the
approximant. The weight function parameters are
de®ned in terms of cI
dmI ˆ dmax cI
c ˆ acI
where dmax and a are constants.
2†
176
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
For the Gaussian weight function in Eq. (1a), the
parameter a is kept constant while dmax is increased,
the shape of the weight function will not change and
the e€ective domain of in¯uence will be smaller than
the actual domain of in¯uence. It is recommended that
the ratio dmax/ar4.0 to avoid poorly formed shape
functions. In addition, a > 0.5 is needed for smooth
shape functions and derivatives. In this paper, the
Gaussian weight with dmax = 2.5, a = 0.625 (=dmax/4)
is used unless otherwise stated. The characteristic
nodal spacing, cI, is chosen as the distance to the second nearest neighbor for regularly spaced nodes and
the distance to the third nearest neighbor for irregularly spaced nodes.
The approximation u h(x) at any point x in the
domain O is written
uh x† ˆ pT x†a x†
4†
where p(x) is a basis (usually polynomial) and a(x) are
unknown coecients. We have used
pT x† ˆ ‰1; x; yŠ
linear basis
5†
In Section 5 it is shown that other functions can be
added to the basis when it is desirable to enrich the
solution.
To ®nd the approximation of the ®eld variable by
Eq. (4), it is necessary to determine the coecients
a(x). The moving least squares (MLS) methodology is
used [22]. Given a set of nodes with coordinates xI at
which the ®eld variable uI is known, a weighted L2
norm can be written
Jˆ
n
X
w x ÿ xI †‰pT xI †a x† ÿ UI Š2
6†
Iˆ1
where w(xÿ xI) is the weight function of node I at
point x, and n is the number of neighbors of point x,
i.e. nodes with w(xÿ xI) > 0.
The minimum of J with respect to a(x) leads to a set
of linear equations
A x†a x† ˆ C x†u
7†
where
A x† ˆ
n
X
w x ÿ xI †p xI †pT xI †
8a†
Iˆ1
which can be substituted into Eq. (4) to yield an approximation in terms of the nodal coecients
uh x† ˆ
n
X
pT x†Aÿ1 x†CI x†uI
9†
Iˆ1
where CI(x) is the Ith column of C(x) and uI is the
nodal coecient for the Ith neighbor of x. De®ning
the shape function fI(x) as
fI x† ˆ pT x†Aÿ1 x†CI x†
10†
allows the approximation to be written as
uh x† ˆ
n
X
Iˆ1
fI x†uI
11†
which is a form familiar to those with a ®nite element
background. These approximations retain the same
continuity as the weight function. The weight function
is generally a C 1 function, so the approximation is
also C 1, i.e. continuously di€erentable.
The spatial derivatives of the shape functions, computed by the chain rule, are
fI;i x† ˆ pT;i x†Aÿ1 x†CI x† ‡ pT x†‰Aÿ1
;i x†CI x†
‡ Aÿ1 x†CI;i x†Š
12†
where A,iÿ 1 =ÿ A ÿ 1A,iA ÿ 1. Note that the second term
in Eq. (12) is expensive to compute because of the
term A,iÿ 1. Nayroles et al [31] in DEM computed only
the ®rst term of the derivatives which results in the inability of their approximation to satisfy the patch test.
Krongauz and Belytschko [18] have shown that DEM
can be rendered convergent by a Petrov±Galerkin formulation.
2.1. Fast shape function and derivative computation
The number of operations required to form shape
functions and their derivatives can be reduced by the
procedure in Belytschko et al. [5] and Fleming et
al. [12]. The shape function in Eq. (10) can be written
as
fI x† ˆ pT x†Aÿ1 x†CI x† ˆ g T x†CI x†
13†
with corresponding derivatives
fI;i x† ˆ g T;i x†CI x† ‡ g T x†CI;i x†
C x† ˆ ‰w x ÿ x1 †p x1 †; w x ÿ x2 †p x2 †; . . . ;
w x ÿ xn †p xn †Š
8b†
u ˆ ‰u1 ; u2 ; . . . ; un Š
8c†
The matrix A(x) is often called the moment matrix.
Eq. (7) can be solved for a(x) to yield
a x† ˆ Aÿ1 x†C x†u
14†
Comparing the underlined terms in Eq. (13) leads to
the relationship
A x†gg x† ˆ p x†
15†
The coecients g(x) can be obtained by an LU decomposition of A(x) and backsubstitution, which
requires fewer computations than a full inversion of
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
177
A(x) which is required to form the shape functions
with Eq. (10).
The derivatives of g(x) are obtained by taking the
derivative of Eq. (15):
The discrete form of Eq. (19) for a mesh-free method
can be obtained using the approximation from Eq. (11)
as approximations for u and du. The resulting system
of discrete equations can be written
A;i x†gg x† ‡ A x†gg ;i x† ˆ p;i x†
Ku ˆ fext
16†
and rearranging the terms which are known to the
right-hand side leads to
A x†gg;i x† ˆ p;i x† ÿ A;i x†gg x† ˆ r x†
17†
Using the LU decomposition of A(x), which is available form solving Eq. (15), g,i(x) can be computed with
only a backsubstitution. Higher order derivatives can
be easily obtained by repeating the procedure. While
this procedure for computing shape functions and derivatives is theoretically identical to directly evaluating
Eqs. (10) and (12), the number of computations is
reduced.
3. Elastostatics
This paper focuses on fracture of linear elastic
media. The elastostatic boundary value problem is
reviewed, the variational form is given and numerical
approximations by mesh-free methods are shown;
enforcement of essential boundary conditions is also
discussed.
Consider a two-dimensional domain O bounded by
G. The equation of equilibrium is
r s u† ‡ b ˆ 0 in
O
18†
where s(u) is the stress tensor, u is the displacement
®eld, and b is the body force. The boundary conditions
are
u ˆ u
s n ˆ t
on Gu
on Gt
where the superposed bar indicates prescribed values
and n is the unit normal vector to G.
The variational (or weak) form for Eq. (18) can be
written
dW u† ˆ
ÿ
O
rs du : s u†; dO ÿ
O
du b dO
Gt
du t; dG ÿ dWu u† ˆ 0 8du 2 H
where Hs is the symmetric gradient operator. The term
dWu(u) is required for enforcing the essential boundary
conditions in a mesh-free method and will be discussed
in the subsequent section.
For linear elasticity, the strain±displacement
equation and the stress±strain law are
E ˆ 12 ru ‡ ru†T †;
sˆD:E
where the sti€ness matrix K $ Rneqneq (neq is the number of equations) and the external force vector
fext $ Rneq de®ned by
KIJ ˆ
fext
I ˆ
20†
BI DBJ dO
O
Gt
fI t dG ‡
22a†
O
fI b dO
22b†
nsd
where KIJ $ Rnsdnsd , fext
(nsd is number of spatial
I $R
dimensions), D is the elasticity matrix and BI is a
matrix of shape function derivatives
2
3
0
fI;x
fI;y 5
BI ˆ 4 0
23†
fI;y fI:x
3.1. Enforcement of essential boundary conditions
One drawback of MLS approximations is that they
are not interpolants, i.e. fI(xJ)$ dIJ, and consequently
shape functions from nodes on the interior of the
domain are nonzero on the boundary. Therefore, essential boundary conditions cannot be satis®ed directly.
The term dWu(u) in Eq. (19) is used to enforce the essential boundary conditions. Some forms which have been
suggested are: (1) Lagrange multipliers [7] where
dWu u† ˆ
Gu
dl u ÿ u † dG ‡
Gu
du l dG;
24†
where l is a Lagrange multiplier; (2) a modi®ed variational principle in which the Lagrange multiplers are
replaced by their physical meaning, the traction [26],
where
dWu u† ˆ
Gu
dt u ÿ u † dG ‡
Gu
du t dG
25†
where t = sn; and (3) a penalty method [4], where
Wu u† ˆ
1
21†
b
2
Gu
jju ÿ u jj2 dG
26†
where b is a penalty parameters.
Another method of enforcing essential boundary
conditions in mesh-free methods is by coupling with
®nite elements [9, 17]. In this method, a row of ®nite elements is placed along the essential boundaries; the
®nite element shape functions are blended with the
EFG shape functions. The boundary conditions can
then be enforced by prescribing the values at the
nodes.
178
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
3.2. Quadrature issues
Computing the sti€ness matrix and force vector,
Eq. (22a) and (22b), requires quadrature over the
domain O, which requires a subdivision of the domain.
Since mesh-free methods have no intrinsic subdivision
like ®nite elements, it is necessary to introduce a subdivision of the domain. Two approaches are used:
1. Background elements, as shown in Fig. 2a, which
are constructed by a ®nite element mesh generator.
The vertices of this background mesh are often used
as the initial array of nodes for the EFG model;
however, additional nodes may be added where
desired such as the nodes at the crack tip in the
model shown.
2. Cell quadrature, in which an octree cell structure,
independent of the domain, is used. Each integration point is checked as to whether or not it lies
inside the domain; points which lie outside are discarded.
Cell schemes are often criticized because signi®cant
errors are expected when surfaces pass through the
cell. However, we have found the errors to be small
and even insigni®cant. The surprisingly small errors
can be shown by the model shown in Fig. 3. A crack
is placed between rows of nodes and the near crack tip
displacement ®eld is applied to the boundary; for this
problem, the asymptotic near-tip ®eld is the exact solution. Gauss quadrature is performed using background cells with vertices at the nodes. The problem is
also solved by subdividing the cells through which the
crack passes so that the cell boundaries coincide with
the crack. When the crack and integration cells are not
coincident, the error in strain energy is only 1%; the
error in the stress intensity factor is 0.1%.
A nodal integration technique was proposed by
Beissel and Belytschko [33] in an e€ort to make EFG
completely mesh-free. However, the method requires
stabilization and the accuracy is inferior to the method
with background integration. Hegen [16] proposed subdividing the cells through which a crack passes by a
triangulation technique to avoid integration errors.
In this paper, element quadrature is used with 44
Gauss quadrature points in each cell. For cells near a
crack tip, the quadrature order is increased to 99.
Cells surrounding a crack are not subdivided to align
cell boundaries with the crack.
4. EFG approximations near nonconvex boundaries
The smoothness which is inherent in meshless
methods is a two-edged sword. On one hand, it provides approximations which are smooth. However,
when a discontinuity occurs in either the geometry or
the material, this higher order smoothness leads to dif®culties. An interface between two materials leads to
Fig. 2. Two integration methods for integrating the weak
form with a mesh-free method. (a) Element quadrature. (b)
Cell quadrature.
Fig. 3. Discrete model for near-tip crack problem.
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
discontinuous strains. This situation is modeled by
®nite elements by placing element edges coincident
with the interface. In EFG, the e€ects of the interface
must also be modeled. Cordes and Moran [10] treated
the individual materials as separate bodies and joined
them together with Lagrange multipliers. Krongauz
and Belytschko [19] have developed jump nodes which
represent the discontinuity in the derivative. These
jump nodes must be placed on the material interface
just like FE edges must be placed on the interface.
It is also necessary to treat discontinuities in the approximation function. In this section, techniques for
modeling discontinuities in the function will be discussed. The visibility criterion will be presented ®rst.
To overcome some of the limitations of the visibility
criterion, the di€raction, transparency and ``seethrough`` methods have been developed and are
reviewed. Numerical examples will be given to illustrate the behavior of these alternative techniques and
show when they are necessary. We also introduce a
modi®cation of the di€raction and transparency
methods which disables it where they are not needed.
4.1. Visibility criterion for discontinuous approximations
The visibility criterion, which was used by
Belytschko et al. [7], de®nes the domain of in¯uence of
a node as the ®eld of vision at the node. All boundaries, internal and external, are considered to be opaque so that the ®eld of vision is interrupted when a
boundary is encountered. A diculty with the visibility
criterion arises for nodes near the tip of a crack, such
as node I in Fig. 4. The ®eld of vision is cut by the
crack along line AB, which extends into the domain.
This leads to a discontinuity in the weight function as
well as the shape function along this line as shown in
Fig. 5. It should be noted that the visibility criterion
leads to discontinuities in shape functions for nodes
near nonconvex boundaries such as holes. This occurs
Fig. 4. Domain of in¯uence by the visibility criterion for a
node near a crack; the shaded area is eliminated from the
domain of in¯uence by the visibility criterion. Note interior
discontinuity for node I along AB.
179
Fig. 5. Contour plot of the shape function fI(x) as determined by the visibility criterion for a node adjacent to a line
of discontinuity due to a crack.
when a ray from a node near the hole grazes the
boundary.
The presence of discontinuities within the domain is
undesirable in a Galerkin method and must be handled
with care. The length and size of the discontinuities
depends on the nodal re®nement near a nonconvex
boundary, i.e. as the nodal spacing goes to zero, the
lengths of the discontinuities tend to zero. Using this
argument and the theory for nonconforming ®nite elements, Krysl and Belytschko [21] showed that the discontinuous approximations generated by the visibility
criterion converge.
4.2. Continuous approximations
4.2.1. Di€raction and transparency methods
Continuous and smooth approximations can be constructed near nonconvex boundaries by the di€raction
method [5, 35, 36]. The nodal support is wrapped
around nonconvex boundaries similar to the way light
di€racts around sharp corners. This method, which
has also been called the wrap-around method, is quite
general and can be used for cracks or smooth boundaries such as interior holes.
Consider Fig. 6a, where the ray between the node xI
and a sampling point x intersects a crack and the tip is
within the domain of in¯uence of the node. The weight
function distance dI is modi®ed (lengthened) by
s1 ‡ s2 x† l
dI ˆ
s0 x†
27†
s0 x†
where s1 = kxI ÿxck, s2(x) = kxÿ xck, s0(x) = kxÿ xIk,
and xI is the node, x is the sampling point, and xc is
the crack tip. The parameter l is used to adjust the
distance of the support on the opposite side of the
crack. It was found that l = 1, 2 performs well. A contour plot of a shape function by the di€raction method
is shown in Fig. 7a.
180
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
Fig. 6. The di€raction (wrap-around) and transparency
methods for constructing smooth weight functions around
nonconvex boundaries. (a) Di€raction method. (b)
Transparency method.
The di€raction method also works well for general
nonconvex boundaries. The tangent point between the
node and the nonconvex boundary is used as the
wrap-around point, xc, and Eq. (27) is used to compute the weight function distance, dI.
Another technique for constructing continuous approximations is the transparency method [6, 36], which
will be described here for cracks. In this method, the
transparency of the crack varies so that it is completely
transparent at the tip and becomes completely opaque
a short distance from the tip. In this way, the ®eld of
vision for a node near the crack tip is not abruptly
truncated when it reaches the crack tip, but rather
diminishes smoothly to zero a short distance from the
tip of the crack.
When a ray passes between a node xI and a
sampling point x, and crosses the crack as shown in
Fig. 6b, the distance parameter dI in the weight func-
Fig. 7. Shape function contours associated with node A near
a crack tip constructed using the di€raction and transparency
methods. The quartic weight function in Eq. (1b) was used
with dmax = 2.01. (a) Shape function for di€raction (l = 2).
(b) Shape function for transparency (k = 0.5).
tion is modi®ed (lengthened) by the following:
dI x† ˆ s0 x† ‡ dmI
sc x† l
;
sc
lr2
28†
where s0(x) = kxÿ xIk, dmI is the radius of support for
node I, and sc(x) is the intersection distance behind the
crack tip. The parameters sc sets the distance behind
the crack tip at which complete opacity occurs:
sc ˆ kh
29†
where h is the nodal spacing and k is a constant,
usually 0 < k < 1.
A contour plot of a shape function near a crack tip
constructed by the transparency method is shown in
Fig. 7b. Note that the function is continuous at the
crack tip.
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
Fig. 8. Surface plot for shape function using the transparency
method (k = 1) when nodes are placed too close to the crack
surface.
One drawback of the transparency method is that it
does not work well when nodes are placed too close to
the crack surface. Fig. 8 shows a surface plot of a
shape function constructed by the transparency
method when nodes are placed along the crack surface.
Note the trough which appears in the shape function
ahead of the crack. This trough appears because,
although the crack tip is transparent for this node, the
transparency changes rapidly with the angle, i.e. sc(x)
in Eq. (28) increases rapidly. There is no discontinuity
in the shape function, only a small dip. To circumvent
this diculty in the transparency method, a restriction
must be imposed on the position of the nodes: all
nodes should be placed so that the normal distance
from the node to the crack surface is greater than
roughly h/4, where h is the nodal spacing.
4.2.2. ``See-through'' method
Terry [42] proposed a ``see-through`` method for constructing continuous approximations near nonconvex
boundaries. In this method, all or part of the boundary
is made completely transparent such that discontinuities are eliminated. Terry [42] found that better accuracy was obtained for a problem with an interior hole
when the boundary of the hole was not strictly
enforced by the visibility criterion.
Duarte and Oden [11] and Krysl and Belytschko [21]
suggested a smoothing technique in which the crack
was completely transparent if the crack tip is within
the domain of in¯uence of a node. This is also called
the continuous line criterion: if a continuous line connecting the node to a point lies entirely within the
domain of in¯uence of the node, the point is visible.
While this technique is easy to implement and provides
smooth approximations, when used with cracks it
e€ectively shortens the crack and leads to inaccurate
181
Fig. 9. Crack opening displacement and Mises stress contours
when using the ``see-through`` method (dmax = 2). The dots are
the nodal locations and the contours are generated using
values at the integration points.
solutions (see Fig. 9). This method does work for
cracks when the enrichment techniques from Section 5
are used. In this case, the approximation function near
the crack tip overcomes the limitations of the linear
approximation generated by the continuous line criterion to yield correct crack opening displacements.
4.2.3. Mixed criteria
The methods described in this section for dealing
with nonconvex boundaries can be used in combination, depending on the smoothness of the boundary.
Nonconvex boundaries can be categorized as smooth
or strongly discontinuous, with smooth boundaries due
to internal holes and strongly discontinuous boundaries, which arise due to cracks and notches. The stress
concentrations due to smooth boundaries can be accurately computed by increasing the nodal resolution,
while the stress singularities which arise from sharp
boundaries require in®nite resolution or enrichment.
The see-through method works well for nonconvex
boundaries when nodal re®nement is adequate. In this
case, nodal domains of in¯uence which extend across
the boundary are not harmful. The di€raction method
can also be used for such boundaries, but its added
complications have been found to be super¯uous.
Both the di€raction and transparency methods can
be used to construct approximations which are continuous within the domain but discontinuous across
the crack. The see-through method should generally
not be used for cracks, but for larger wedge angles the
singularity is usually not of interest.
A criterion for introducing discontinuities selectively
can be based on the angle of the wedge. This con-
182
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
dition, written in terms of the surface normals, is
when nA nB Rb;
domain of influence is cut
30†
where nA and nB are the surface normals to the wedge
(see Fig. 10) and b is a cuto€ value. A minimum cuto€
value of b = 0 is recommended; this corresponds to a
wedge angle of o = 908. If the wedge angle exceeds
this value, the method can be used.
5. Enrichment of EFG for crack tip ®elds
We review and compare to methods for enhancing
EFG approximations: (1) intrinsic enrichment, where
the enrichment functions are included in the EFG
basis; and (2) extrinsic enrichment, where the approximation is enriched by adding functions externally to
the EFG basis.
5.1. Extrinsic MLS enrichment
In extrinsic enrichment of a meshless approximation,
a function closely related to the solution is added to
the approximation in Eq. (4) [12]. For example, in linear elastic fracture mechanics, the near tip asymptotic
®eld or its constituents can be added. The approximation takes the form
uhi x† ˆ pT x†ai x† ‡
nc
X
jˆ1
aj Qji
x†
i ˆ 1; 2
31†
where u hi (x) denotes the approximation for ui(x), p(x)
is a complete polynomial basis in the spatial coordinates, nc is the number of cracks in the model, ai(x) is
the coecients of the polynomial basis; a j is a global
unknown associated with crack j.
The functions Qi(x), which describe the near-tip displacement ®eld for a mode 1 elastic crack, are [44]:
r
1
r
y
y
Q1 x† ˆ
cos
k ÿ 1 ‡ 2 sin2
32a†
2m 2p
2
2
r 1
r
y
y
Q2 x† ˆ
sin
k ‡ 1 ÿ 2 cos2
32b†
2m 2p
2
2
where r is the distance from the crack tip, y is the
angle from the tangent to the crack path at the crack
tip (see Fig. 11), m is the shear modulus and k the
Kolosov constant de®ned as
k ˆ 3 ÿ 4n
k ˆ 3 ÿ n†= 1 ‡ n†
plane strain
plane stress
Using the moving least squares methodology outlined in Section 2 leads to an approximation of the
form
uhi x† ˆ
n
X
Iˆ1
‡
fI x†uIi
nc
X
jˆ1
"
j
a
Qji
x† ÿ
n
X
Iˆ1
#
fI x†Qji
xI †
33†
where fI(x) is the shape function de®ned in Eq. (10).
5.2. Extrinsic PU enrichment
Extrinsic enrichment of meshless methods can also
be carried out using partition of unity (PU)
methods [6, 11, 29]. In this method, the approximation
is augmented by enrichment functions added extrinsically to the existing EFG approximation from Eq. (11).
This basis can consist of higher order polynomials or,
for linear elastic fracture problems, terms from the
asymptotic near tip ®eld can be used. The extrinsic
Fig. 10. Domain of in¯uence near a wedge-shaped nonconvex
boundary. The boundary is enforced if nAnBRb.
Fig. 11. Local coordinate system at crack tip.
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
basis is smoothly added to the approximation through
the partition of unity.
The essential element of this method is the construction of a partition of unity. A partition of unity f(x) is
a local approximation for which
n
X
Iˆ1
fI x† ˆ 1
34†
It can easily be seen that MLS approximations are
partitions of unity since Eq. (34) is the reproducing
condition for a constant, which MLS approximations
must satisfy [5].
Enriched approximations based on partitions of
unity take the form
uh x† ˆ
n
X
Iˆ1
fkI x†uI ‡
me
n X
X
Iˆ1 iˆ1
f0I x†bIi qi x†
35†
where uI and bIi are nodal coecients, and n is the
number of neighbors of point x. A superscript is
added to the shape functions fI in the approximation
to denote the order of the polynomial order of the
basis used in forming the partition of unity. The vector
q(x) is called the extrinsic basis. In linear
p elastic fracture problems, this basis can contain r or the full
span of Eqs. (32a) and (32b). It is sometimes convenient to construct the partitions of unity using
Shepard functions (i.e.. k = 0) which satisfy constant
consistency and add the terms to satisfy linear and
higher order consistency to the extrinsic PU basis, but
the conditioning of the discrete system equations is
then impaired.
The partition of unity, fI(x), can be formed from a
linear basis, which yields linear consistency. The approximation can be enriched locally by adding the
known form of the solution to the extrinsic basis, q(x),
where needed. It should be noted that the enrichment
should be added to each node whose domain of in¯uence extends into the region to be enriched.
5.3. Intrinsic basis enrichment
Meshless approximations can be intrinsically
enriched by including the enrichment functions in the
basis. For example, in fracture mechanics, one can
include the asymptotic near-tip displacement ®eld in
Eqs.
p(32a) and (32b), or an important ingredient, such
as r in the basis p(x). The choice of basis functions
depends on the coarse-mesh accuracy desired. For
higher accuracy, the full asymptotic ®eld from Eqs.
(32a) and (32b) should be included whilepfor
 higher
speed at some cost of accuracy, only the r function
should be included in the basis. Both methods are
described in the subsequent sections.
183
5.3.1. Full enrichment
In full intrinsic enrichment of EFG approximations
for fracture problems, the entire near-tip asymptotic
displacement ®eld is included in the basis. Following
some trigonometric manipulation, it can be shown [12]
that all the functions in Eqs. (32a) and (32b) are
spanned by the basis
pT x† ˆ
p
p
y p
y p
y
y
1; x; y; r cos ; r sin ; r sin sin y; r cos sin y
2
2
2
2
36†
(the linear terms are not related to the near-tip ®elds
and are represented through the linear completeness of
the EFG approximant). This basis can be used in
Eq. (4) and leads to approximations of the form
uh x† ˆ
n
X
pT x†Aÿ1 x†CI x† uI
|‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚}
Iˆ1
fI x†
37†
where fI(x) is the enriched EFG shape function.
In contrast to the extrinsic methods presented in
Section 5.1 and 5.2, this method involves no additional
unknowns. However, because of the increased size of
the basis, additional computational e€ort is required
to invert the moment matrix, A(x). In addition, the
domain of in¯uence must be enlarged to achieve regularity of A(x). For multiple cracks, four additional
terms would have to be added to the basis for each
crack; a method for coupling an enriched basis with a
linear basis which avoids this diculty is presented in
Section 5.4.
Using an enriched basis can lead to an ill-conditioned moment matrix A(x). While this generally
does not a€ect the ®nal solution, it is troublesome. It
has been found that the e€ects of ill-conditioning can
be mitigated by reducing the number of computations
by the procedure in Section 2.1. The e€ects of ill-conditioning can also be reduced by diagonalizing the
moment
matrix
by
Gram±Schmidt
orthogonalization [26].
When an enriched basis is used at any node of a
mesh, it must be used at all nodes of the mesh or a
technique described in Section 5.4 must be used to
blend it to nodes with a di€erent basis. Simply deleting
functions from the basis results in discontinuities in the
approximation.
5.3.2.pRadial
enrichment

In r enrichment, we use the basis
p
pT x† ˆ ‰1; x; y; rŠ
38†
where r is the radial distance from the crack tip. This
enrichment is useful because the angular variation
184
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
around the crack tip is smooth, but the radial variation
is singular in the stress.
p
The advantage of r enrichment is that the intrinsic
basis is only expanded by one term and inverting the
moment matrix, A(x), to form the shape functions is
much cheaper than for full enrichment. In addition, it
does not seem to be necessary to use the smoothing
techniques in Section 4 because there are no discontinuities in the radial direction; the discontinuities in the
angular direction lead to a noticeable loss of accuracy
when full enrichment is used. However, this enrichment does not contain the discontinuity behind the
crack tip, so its convergence is much slower.
the coupling region (r = r1) and equal to zero on the
linear boundary of the coupling region (r = r2) (see
Fig. 12). Some suggested polynomial ramp functions
are
5.4. Coupling enriched and linear approximations
uh x† ˆ
Enriching the approximation for the entire domain
of a problem is generally unnecessary and increases
computational expense. For example, the crack-tip
singular ®eld is local; it extends 00.1a from the crack
tip, where a is the length of the crack. Two techniques
are presented here for coupling enriched and linear approximations, one uses a consistent coupling to maintain C 0 continuity, the other does not.
The ®rst technique involves coupling the approximation over a transition region as a linear basis combination of the enriched linear approximation (this is
similar to the way Belytschko et al. [9] coupled EFG
to ®nite elements). The approximation is written
uh x† ˆ Ruenr x† ‡ 1 ÿ R†ulin x†
39†
where u enr(x) is the enriched approximation and u lin is
the linear basis approximation; R is a ramp function
which is equal to unity on the enriched boundary of
Fig. 12. Schematic for the coupling of enriched and linear approximations.
Rˆ1ÿx
3
4
5
R ˆ 1 ÿ 10x ‡ 15x ÿ 6x
linear ramp
quintic ramp
40†
where x = (rÿ r1)/(r2 ÿ r1), r is the radial distance from
the crack tip.
The coupled approximation for the enriched intrinsic
basis from Section 5.3 is written
n
X
Iˆ1
f~ i x†uI
41†
where
f~ I x† ˆ Rfenr
x† ‡ 1 ÿ R†flin
I
I x†
42†
and f enr
I (x) is the shape function formed from the
enriched basis of Eq. (36) and f lin
I (x) is the shape
function formed from a linear basis. This method
ensures a compatible displacement ®eld. The continuity
in the strain ®eld depends on the continuity of the
ramp function, R (i.e. the linear ramp will yield continuous displacements, but discontinuous strains at
r = r1, r2; both displacements and strains will be continuous and smooth with the quintic ramp).
Through numerical experiments, it was found that if
the area of enrichment about a crack tip is circular,
the inner radius of the ramp, r1, can begin at the crack
tip. However, the outer radius, r2, needs to be outside
the singularity-dominated zone to obtain good accuracy.
It is possible to con®ne the enrichment to the crack
tip region by simply changing from an enriched to an
unenriched basis, but the approximation then is not
continuous. While this is not strictly permissible in a
Galerkin method, the errors are small if the transition
occurs outside the singular-stress dominated region
(>0.1a) so that jump is very small.
Mixing the enriched and linear approximations was
found to work better for the intrinsically enriched
basis in Section 5.3 than for the extrinsic MLS enrichment in Section 5.1. The extrinsic MLS enrichment is
sensitive to discontinuities and some loss of accuracy
was noticed.
Another method of localizing the enrichment to the
crack tip region is to use extrinsic PUM enrichment
from Section 5.2. The extrinsic basis is then added
only to those nodes which are near the crack tip. It is
necessary to enrich all nodes whose domains of in¯uence include the crack tip; otherwise, the enrichment is
incomplete and the solution is poor.
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
6. Contact Enforcement in EFG
In general fracture problems, contact between two
crack faces often develops. To model contact of crack
surfaces in EFG, a two surface model of the crack is
needed. This section presents the formulation and
methodology for contact in a mesh-free method [13]. A
penalty method is used to treat the unilateral constraint on the crack surface.
6.1. Contact surface description
Consider Fig. 13a, where two bodies, body A and
body B, are in contact. Body A has domain OA with a
boundary GA, while body B has domain OB with
boundary GB. The contact surface, denoted G c, is the
portion of the boundaries of the two bodies which
touch, i.e. G c = GA\GB.
A cracked body loaded in compression will lead to
contact along the crack surface (see Fig. 13b). This is
an example of self-contact, i.e. a body in contact with
itself. In this case, the one surface will be de®ned as
GA and the other surface as GB; the contact surface is
de®ned to be G c = GA\GB. We can conform to the
usual model of contact if we split the body into two as
shown by the dashed line in Fig. 13b.
It is convenient to formulate the equations of contact in terms of a local coordinate system at the contact surface. Consider the two-dimensional contact
problems shown in Fig. 13. A local coordinate system
is set up at each point on the contact surface, with eÃ1
tangent to the surface at that point and eÃ2 normal to
185
the surface. The normal to body A is then
nA ˆ e^2
43†
and the normal to body B at that point on the contact
surface is equal and opposite in sign
nB ˆ ÿnA
44†
6.2. Interpenetration condition
Interpenetration occurs when two material points
occupy the same spatial point. If GA is de®ned as the
master surface and GB is de®ned as the slave surface,
the interpenetration condition is written as the motion
of surface B relative to surface A. For a point P on GB
which has penetrated the master surface GA (see
Fig. 14), the interpenetration is de®ned as
gN ˆ jjx A ÿ xP jj2 ˆ min jjxA ÿ xP jj
xA 2GA
gN ˆ jjx A ÿ xP jj2 ˆ 0 if
xA ÿ xP † nA < 0
45†
where x A is the minimizer of the distance from point P
to GA. As shown in Fig. 14, x A is the point at which
the normal projection from GA intersects GB at point
P [45].
For small displacement elastostatics, the interpenetration function in Eq. (45) can be written in terms of
the displacements as
gN ˆ u ÿ uP † n A
A
46†
A
where x = u(x A ) and x is the outward normal to GA
Fig. 13. Contact of two bodies and self-contact along a crack surface. In both cases, the contact surface is G c = GA\GB. (a) Two
bodies in contact. (b) Self-contact along a crack.
186
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
6.4. Weak form discretization
The interpenetration function in Eq. (46) is a function of the displacements along the contact surface. In
a meshless method such as EFG, these displacements
are computed using the nodal values and the EFG
shape functions as in Eq. (11). This allows the interpenetration function to be written as
X
X
fI x†uIi n A
fJ x†uJj n Bj
50†
gN ˆ
i ‡
I2BA
J2BB
X
ˆ
I2BA [BB
fI n ai †uIi
51†
where
Fig. 14. Interpenetration of two bodies.
at xA. The solution to the contact problem is then a
displacement ®eld which satis®es Eq. (18) and
gN ˆ 0
47†
6.3. Penalty method for contact
The contact enforcement can be written in two basic
ways, Lagrange multipliers and penalty methods. A
Lagrange multiplier method results in a system of
equations with an inequality which must be satis®ed
for contact to be enforced. In a penalty method, a penalty is placed on the amount of interpenetration. In
this section, the penalty method is described and implemented for small displacement elastostatics.
The weak form for two-dimensional, small strain,
linear elastic problems is given in Eq. (19). When contact constraints are added, the resulting weak form is
dW ‡ dWc ˆ 0
48†
where dW is de®ned in Eq. (19). The contribution of
the contact forces to the virtual work, dWc, is found
by enforcing the contact constraint such that the interpenetration is zero. For a penalty method,
b
dWc ˆ
d g 2N † dG ˆ
Gc 2
Gc
bdgN gN dG
49†
where b is a large number called the penalty parameter
and can be thought of physically as a sti€ spring
placed between the contact surfaces.
n ai ˆ n A
i
for I 2 BA
n ai ˆ n Bi
for I 2 BB
52†
and BA refers to nodes which are neighbors to a point
on surface GA and BB refers to nodes which are neighbors to the corresponding point on GB. It should be
noted that, because EFG shape function do not satisfy
the Kronecker delta condition, some nodes not on the
contact surface will be used for computing the interpenetration in Eq. (50).
The weak form for contact in Eq. (49) can be discretized using Eq. (50) to yield
dWc ˆ
Gc
bdd TIi fI n ai †T fJ n aj †dJj dG ˆ ddT fc
53†
where fc is the force due to contact. The contact force
for an elastic material can be written in terms of the
displacements as
fc ˆ K c d
54†
where Kc is the contact sti€ness
KcIJ ˆ
Gc
b fI n ai †T fJ n aj † dG
55†
Note that KcIJ $ R22 . Using a penalty method only
requires that the sti€ness matrix be modi®ed and
requires no additional unknowns as with Lagrange
multiplier methods. The resulting system of equations
becomes
K ‡ Kc †d ˆ fext
56†
The contact sti€ness Kc is, in general, a function of
the displacements because it is nonzero only when the
interpenetration function gN in Eq. (46) is positive.
Eq. (56) is a nonlinear system of equations and must
be solved using Newton's method.
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
7. Numerical results
Several solutions are described to illustrate the e€ectiveness of enriching the EFG formulation for fracture
problems. Solutions are given for single and mixed
mode problems.
Two of the problems illustrate the performance of
the smoothing techniques for nonconvex boundaries.
The EFG method is used for numerical computations
and a background element mesh is used for integrating
the weak form.
7.1. In®nite plate with a hole
An in®nite plate with a hole subjected to a remote
unit traction in the x-direction is solved and the solutions are compared for the visibility, di€raction and
``see-through'' methods. The solution of this problem
is given in Timoshenko and Goodier [43] as:
a2 3
3a 4
sxx ˆ 1 ÿ 2
cos 2y† ‡ cos 4y† ‡ 4 cos 4y† 57a†
r 2
2r
2
a 1
3a 4
syy ˆ ÿ 2
cos 2y† ÿ cos 4y† ÿ 4 cos 4y†
57b†
r 2
2r
a2 1
3a 4
sin 2y† ‡ sin 4y† ‡ 4 sin 4y†
sxy ˆ ÿ 2
57c†
r 2
2r
187
graded with additional re®nement around the hole up
to a radius of 2a. The integration cells coincide with
the nodal arrangement; 99 Gauss quadrature is used
in the graded region, with 55 quadrature in the
remaining quadrature cells; for the simulations by the
di€raction method, l = 2.
The error in energy as a function of the domain of
in¯uence is shown in Fig. 16, where the exact solution
is computed from the solution given in Timoshenko
and Goodier [43], and the energy norm is computed by
energy norm ˆ
1
2
O
1=2
s ÿ s h † : E ÿ E h † dO
58†
The results show that for smaller nodal supports, there
are no signi®cant di€erences in accuracy between the
discontinuous and the smooth approximation; however, as the domain of in¯uence increases, the discontinuous approximation is less accurate.
7.2. Near-tip crack problem
where a is the radius of the hole.
A discrete model of the problem is constructed by
applying the exact tractions corresponding to Eqs.
(57a)±(57c) on the boundaries; due to symmetry only a
quarter of the plate is modeled (see Fig. 15). The
dimensions used are a = 1.0 in., l = 5.0 in.; plane
strain, linear elastic conditions are assumed with
Young's modulus and Poisson's ratio, E = 30106 psi
and n = 0.3, respectively. The EFG models were
A closed form solution for a crack can be constructed by using the well-known near-tip ®eld in a
domain about the crack tip and prescribing the displacements along the boundary according to this ®eld.
This can be considered a patch test for singular ®elds.
A square patch with sides of length 2a and a crack of
length a is used. This problem is used to study the
e€ects of smoothing EFG approximations for crack
problems and compare the performance of full enrichment (extrinsic and intrinsic), radial enrichment, and a
linear basis. The displacement ®eld for a mode 1 crack
is [44]
Fig. 15. Typical mesh for the in®nite plate with a hole problem. Due to symmetry, only a quarter mesh was modeled.
Fig. 16. Error in energy versus the support size dmax for the
in®nite plate with a hole problem.
188
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
r
k1 r
y
y
cos
ux x† ˆ
k ÿ 1 ‡ 2 sin 2
2
2
2m 2p
r k1 r
y
y
uy x† ˆ
sin
k ‡ 1 ÿ 2 cos 2
2
2
2m 2p
59a†
59b†
where r is the distance from the crack tip and y is the
angle measured from the line of the crack. The stresses
resulting from this displacement ®eld satisfy equilibrium so the solution is exact if the displacements
from Eqs. (59a) and (59b) are prescribed on the outer
boundaries.
One drawback of the smooth approximations is that
when a linear basis is used, the computed crack opening pro®le is not parabolic at the tip. The shape functions wrap around the crack tip, so the crack is
e€ectively shortened if the smoothing e€ect is too large
or the mesh is too coarse, leading to the crack opening
displacement (COD) shown in Fig. (17a). The maximum stress is then not at the crack tip, but is shifted a
Fig. 17. Crack opening displacement and Mises stress contours for a mode 1 fracture problem using the di€raction method (l = 1).
The dots are nodal locations and the contours are generated using values at the integration points. (a) Uniform mesh (linear basis).
(b) Re®nement at crack tip (linear basis). (c) Uniform mesh (enriched basis).
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
small distance that depends on the mesh re®nement.
This e€ect can be reduced by increasing l in the diffraction method, or decreasing k in the transparency
method. When the mesh is re®ned locally as in Fig. 17b
or an enriched approximation is used as in Fig. 17c,
the maximum stress shifts to the crack tip and the
COD pro®le becomes parabolic.
As stated previously, in spite of the discontinuities
due to the visibility criterion, the approximation is still
convergent. Fig. 18 shows four levels of increasing
nodal re®nement surrounding the crack tip. Fig. 19a
shows the error in energy using a linear basis plotted
as the number of nodes at the crack tip region is changed; Fig. 19b shows the corresponding stress intensity
factors for those levels of re®nement. It is readily seen
that under these circumstances the discontinuous approximations do not impair the accuracy compared to
the smooth approximations; both the error in energy
and the stress intensity factor are quite accurate.
Moreover, in the smooth approximations generated by
the ``see-through'' the error is quite large, leading to
the conclusion that these approximations should not
be used in conjunction with sharp nonconvex boundaries such as cracks.
It should also be noted that over-re®nement of the
crack tip region can actually increase the error if the
189
mesh away from the crack tip is not re®ned. Fig. 19a
shows that for the highest level of re®nement in
Fig. 18d, the error actually increases slightly over the
previous re®nement. As a general rule for mesh-free
methods, a sharp gradient in nodal spacing leads to
error. A likely source of this error is that in the coarse
region the domains of in¯uence extend into the re®ned
region while the converse is not true.
Without enrichment, the EFG method requires considerable nodal re®nement near the crack tip to capture the singular stress ®eld with reasonable accuracy.
The model used to illustrate this is shown in Fig. 20.
A regular grid of nodes is used throughout the domain
with a radial array of nodes around the crack tip. The
stress ®eld ahead of the crack tip computed with a linear basis and the visibility criterion is shown in Fig. 21.
The radial array aids in capturing the singular stress
®eld, but the stresses are oscillatory. These oscillations
are typical in approximating a singular ®eld by a
smooth function, and they lead to moderate domain
dependence for the J integral.
Enriching the EFG trial function aids in capturing
the singular stress ®eld around the crack tip; when full
enrichment is used the oscillations are almost completely eliminated. The stress along y = 08 from the crack
tip computed with extrinsic MLS enrichment is shown
Fig. 18. Four levels of nodal re®nement using a star-shaped nodal array at the crack tip. (a) Two rings, ®ve nodes/ring. (b) Four
rings, nine nodes/ring. (c) Six rings, 13 nodes/ring. (d) Eight rings, 17 nodes/ring.
190
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
Fig. 20. Mesh used for near-tip crack problem (added rings of
nodes at r = 0.03a, 0.09a, 0.18a).
larity for a polynomial basis of any order [21, 38]. The
rate of convergence for the linear basis in Fig. 24 is
0.53, which is just slightly higher than the theoretical
value of 1/2. For the fully enriched approximations
(extrinsic or intrinsic), the exact solution is contained
in the trial function and there should be no error due
to approximation. The errors are apparently due to
quadrature error and discretization of the essential
boundary conditions.
7.3. Plate with a hole and two cracks
Fig. 19. Energy error and stress intensity factors with four
crack tip nodal re®nements. All calculations were made with
a linear basis and Gaussian weight function (dmax = 2.5,
a = 0.625); k = 1.00 is the exact solution. (a) Error in energy.
(b) Stress intensity factors.
in Fig. 22. These results were calculated using the
nodal mesh shown in Fig. 20 without the re®nement at
the crack tip. The di€raction method with l = 1
(unless otherwise speci®ed) is used with the enriched
approximations to provide continuous shape functions
near the crack tip. It can be seen that the enriched
EFG method is able to capture the singularity and
eliminate oscillations at the crack tip without extra
re®nement in the crack tip region. The stress pro®les
for extrinsic and complete intrinsic basis captures the
singular stress much better than the linear basis with
no extra nodes, but it tends to underestimate the stress
®eld, as shown in Fig. 23.
The relative error in strain energy is shown in
Fig. 24, and it can be seen that enrichment dramatically increases the absolute accuracy. For a linear
basis, the slope of the line represents the rate of convergence of the approximation for uniform nodal
re®nement. For problems with a singularity, the rate
of convergence is controlled by the order of the singu-
A plate with a center hole and two symmetric cracks
is subjected to uniaxial tension of s = 1 applied at the
top and bottom surfaces. The dimensions of the plate
are h = 20 in, W = 10 in, R = 1.25 in. The discrete
p
model is shown in Fig. 25. A linear basis with r
enrichment is used for all computations.
Fig. 21. Stresses ahead of the crack tip (y = 0, r > 0) for the
near-tip crack problem.
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
Fig. 22. Stresses ahead of the crack tip for the near-tip crack
problem with extrinsic enrichment Eq. (33).
This problem illustrates the use of a mesh-free
method with both smooth and strongly discontinuous
boundaries. The strong boundary discontinuity of the
crack is modeled by the visibility criterion, while the
see-through method is used for the hole.
Results for the non-dimensional stress intensity factors as a function of crack length are presented in
Fig. 26. The EFG results compare well with the reference solution [33], which was computed by a boundary
collocation method.
191
Fig. 24. Convergence of the error in energy for the mode 1
crack problem. For the di€raction method, l = 2.
and propagates in the direction of the applied load.
Nemat-Nasser and Horii [32] have presented experimental results in which thin slits were cut in glass
plates and resin plates. They found that the cracks initiated secondary cracks which grew towards the loading direction. If no lateral loads were applied, the
fracture tended to be stable and the propagating cracks
arrest if the load is not increased suciently.
Numerical results by the EFG method are presented
for a slanted array of cracks loaded in compression in
the vertical direction. This loading leads to stable
7.4. Compression-loaded cracks
Results are presented for crack growth in a compressive ®eld. This situation is common in fracture of geomaterials, such as rocks under large geotectonic states
of stress. A condition known as axial splitting occurs
in which the initial crack in a compressive ®eld turns
Fig. 23. Stresses ahead of the crack tip (y = 0, r > 0) for the
near-tip crack problem.
Fig. 25. Geometry of plate with a hole and two cracks. EFG
nodes are included to illustrate symmetric model used.
192
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
Fig. 26. Normalized stress intensity factors versus crack
length for a tension-loaded plate with a hole and two symmetric cracks. The normalized hole radius is 2R/W = 0.25.
crack growth, i.e. the stress intensity factors decreases
as the crack length increases. To continue the propagation of the cracks, the applied load is increased 30%
in each step. In these models, contact between crack
faces was modeled. Fig. 27 shows three stages of crack
Fig. 28. Progression of crack propagation of a scattered
arrangement of eight cracks in an overall compressive ®eld.
(a) Step 1; (b) step 6; (c) step 10.
growth for three internal cracks (six crack tips) in a
compressive ®eld. The outermost cracks propagated in
the direction of the applied load. The inner cracks
interacted with each other and in one case the cracks
bridged. Fig. 28 shows results for a scattered arrangement of eight cracks in an overall compressive ®eld.
The outer cracks grew towards the boundaries, but
one of the innermost cracks curled around such that it
bridged with itself.
An interesting phenomenon is that, under the compressive loading, the crack direction tends to become
unstable as the crack nears arrest. This is shown in
Fig. 29, where the crack path deviates from the smooth
path as the simulation progresses. A plot of the stress
intensity factor reveals that as the mode 1 stress intensity factor begins to decrease, the ratio of mode 2 to
mode 1 stress intensity factor increases. This ratio is
essential in determining the crack growth direction by
the maximum principal stress criterion. When this
ratio becomes too large, the crack path is susceptible
to oscillation.
8. Discussion
Fig. 27. Crack propagation of a slanted array of six cracks in
an overall compressive ®eld. (a) Step 1; (b) step 4; (c) step 7.
Three aspects of crack modeling by mesh-free
methods have been reviewed: the construction of
smooth approximations at the crack tip; enrichment of
the approximation and contact on crack faces. Three
methods for the smoothing of approximations were
considered: the di€raction, transparency, and ``seethrough'' or continuous line methods. These methods
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
Fig. 29. Crack path oscillation as cracks near arrest.
eliminate discontinuities around the ends of a discontinuity within the domain while still maintaining the
necessary discontinuities across internal boundaries
such as cracks. Results show that for a linear basis the
rate of convergence for the visibility criterion and the
smooth approximations is comparable. However, the
discontinuities within the domain due to the visibility
criterion can adversely a€ect convergence when enrichment is used. Accuracy also deteriorates when the visibility criterion is used with large domains of in¯uence.
In the di€raction method, continuity of the approximation within the domain is attained by wrapping the
domain of in¯uence partially around a nonconvex
boundary. The method is robust and yields accurate
solutions for smooth boundaries and sharp corners
such as cracks. The derivatives of the approximation
will not be continuous at a crack tip, but this poses no
diculties because no quadrature points are placed at
that point.
193
In the transparency method, the approximation is
smoothed by adding a small region to the crack which
has a varying measure of transparency. The method
has not been generalized to arbitrary nonconvex
boundaries. A drawback of this method is that when
nodes are too close to the crack surface, troughs occur
in the shape function ahead of the crack. This e€ect is
due to the rapid spatial variation in the transparency
for rays between nodes close to the crack surface and
points on the opposite side of the crack. In general,
the normal distance between a node and the crack surface should be at least h/4, where h is the nodal spacing, when the transparency method is used.
In the ``see-through'' method, which is also known
as the continuous line criterion, the boundary is
ignored whenever a continuous line between a node
and a sampling point can be drawn within the domain
of in¯uence without crossing the boundary. For a
smooth boundary such as a hole, the entire boundary
is transparent; whereas, for a crack, the boundary is
only transparent for nodes with the crack tip in their
domain of in¯uence. This method works well for
smooth boundaries, but when applied to a crack, the
resulting approximation shortens the crack by the
radius of the domain of in¯uence. Therefore, large
errors occur in fracture solutions unless enrichment or
signi®cant re®nement is used.
Smooth approximations are more mathematically
palatable in a Galerkin method, but from a numerical
standpoint they do not seem to be necessary in all
cases. When using a low order polynomial basis (e.g. a
linear basis) to solve a fracture problem, a signi®cant
amount of nodal re®nement is needed at the crack tip,
which decreases the lengths of the interior discontinuities created by the visibility method [21]. Smooth
shape functions are recommended when enriched approximations are used. When the exact solution is
included in the trial function or the basis, interior discontinuities degrade the accuracy of the solution.
Enrichment of EFG provides a means to increase
accuracy and reduce computational e€ort in linear
elastic fracture mechanics. It is also applicable to other
problems for which local solutions are known. In
extrinsic MLS enrichment, the trial functions are augmented with the near-tip asymptotic displacement ®eld.
In intrinsic enrichment, the EFG basis is augmented to
include the terms from the near-tip displacement ®eld
which account for the presence of the crack. Full
radial and angular enrichment as well as only radial
enrichment are presented.
These methods have been shown to provide similar
accuracy and work well even with coarse models. An
advantage of extrinsic MLS enrichment is that it can
be used for problems with multiple cracks with little
additional expense. Drawbacks of this method include
the computer programming, which is rather involved.
194
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
Furthermore, the unknowns corresponding to the
stress intensity factors seem to be quite sensitive to
perturbations and can not be used to evaluate stress
intensity factors. Instead, the J integral must be used
for calculating stress intensity factors.
Intrinsic basis enrichment is easy to implement in
EFG as it only requires the modi®cation of the basis.
One drawback of this method is that is becomes expensive for multiple cracks because enrichment terms
must be added for each crack tip and the size of the
moment matrix which must be inverted at each point
depends on the number of terms in the basis.
However, this drawback can be ameliorated by limiting
the enrichment zone to the crack tip region as
described in Section 5.4.
These methods provide signi®cant reductions in the
number of unknowns required to obtain an accurate
solution for fracture mechanics problems by EFG. The
methods are easily able to treat arbitrary crack growth,
which makes them promising for this class of dicult
problems.
Numerical results were also presented for crack
propagation under compressive loading with crack surface contact. We were able to replicate experimental
results which show that for an initial crack under compressive load with contact, the crack growth is in the
direction of the applied loads. If the applied loads
remain constant, the fracture is stable and the crack
arrests after some growth. It is found that as the crack
nears arrest, the crack path has a tendency to oscillate
because the mode 2 stress intensity factors becomes
large along the path, which may indicate that the
crack path is unstable.
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
Acknowledgements
The support of the Oce of Naval Research and the
Army Research Oce is gratefully acknowledged.
[16]
[17]
References
[1] Al-Ostaz A, Jasiuk I. Damage initiation and propagation
in an elastic-brittle material with randomly distributed
holes. In: 1995 Winter Annual Meeting. ASME, New
York, 1995.
[2] Attaway SW, Heinstein MW, Swegle JW. Coupling of
smooth particle hydrodynamics with the ®nite element
method. Nuclear Engineering and Design 1994;150:199±
205.
[3] Beissel S, Belytschko T. Nodal integration of the element-free Galerkin method. Computer Methods in
Applied Mechanics and Engineering 1996;139:49±74.
[4] Belytschko T, Gu L, Lu YY. Fracture and crack growth
by
element-free
Galerkin
methods.
Modelling
[18]
[19]
[20]
[21]
Simulations in Materials Science and Engineering
1994;2:519±34.
Belytschko T, Krongauz Y, Fleming M, Organ DJ, Liu
WK. Smoothing and accelerated computations in the element free Galerkin method. Journal of Computational
and Applied Mathematics 1996;74:111±26.
Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl
P. Meshless methods: An overview and recent developments. Computer Methods in Applied Mechanics and
Engineering 1996;139:3±47.
Belytschko T, Lu YY, Gu L. Element-free Galerkin
methods. International Journal for Numerical Methods
in Engineering 1994;37:229±56.
Belytschko T, Lu YY, Gu L, Tabbara M. Element-free
Galerkin methods for static and dynamic fracture.
International Journal of Solids and Structures
1995;32:2547±70.
Belytschko T, Organ D, Krongauz Y. A coupled ®nite element±element-free Galerkin method. Computational
Mechanics 1995;17:186±95.
Cordes LW, Moran B. Treatment of material discontinuity in the element-free Galerkin method. Computer
Methods in Applied Mechanics and Engineering
1996;139:.
Duarte CAM, Oden JT. An hp adaptive method using
clouds. Computer Methods in Applied Mechanics and
Engineering 1996;139:237±62.
Fleming M, Chu Y, Moran B, Belytschko T. Enriched element-free Galerkin methods for crack tip ®elds.
International Journal for Numerical methods in
Engineering 1997;40:1483±504.
Fleming MA. The element-free Galerkin method for fatigue and quasi-static fracture. Ph.D. thesis, Northwestern
University, 1997.
Gallego R, Dominguez J. Dynamic crack propagation
analysis by moving singular boundary elements. ASME,
Journal of Applied Mechanics 1992;59:S158±S162.
Gingold RA, Monaghan JJ. Kernel estimates as a basis
for general particle methods in hydrodynamics. Journal
of Computational Physics 1982;46:429±53.
Hegen D. An element-free Galerkin method for crack
propagation in brittle materials. Ph.D thesis, Eindhoven
University of Technology., 1997.
Krongauz Y, Belytschko T. Enforcement of essential
boundary conditions in meshless approximations using
®nite elements. Computer Methods in Applied
Mechanics and Engineering 1996;131:133±45.
Krongauz Y, Belytschko T. Consistent pseudo-derivatives in meshless methods. Computer Methods in
Applied Mechanics and Engineering 1997 (in press).
Krongauz Y, Belytschko T. EFG approximation with
discontinuous derivatives. International Journal for
Numerical methods in Engineering 1998;41:1215±33.
Krongauz Y, Belytschko T. A Petrov±Galerkin di€use
element method (PG DEM) and its comparison to EFG.
Computational Mechanics 1997c;19:327±33.
Krysl P, Belytschko T. Element-free Galerkin: convergence of the continuous and discontinuous shape functions. Computer Methods in Applied Mechanics and
Engineering1996(submitted).
T. Belytschko, M. Fleming / Computers and Structures 71 (1999) 173±195
[22] Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Mathematics of Computation
1981;37:141±58.
[23] Libersky LD, Petschek AG. Smoothed particle hydrodynamics with strength of materials. In The Next Free
Lagrange Conference. Trease H, Fritts J, Crowley W,
editors. 248±257, 1991.
[24] Liszka T, Orkisz J. The ®nite di€erence method at arbitrary irregular grids and its application in applied mechanics. Computers and Structures 1980;11:83±95.
[25] Liu WK, Jun S, Zhang YF. Reproducing kernel particle
methods. International Journal for Numerical methods
in Engineering 1995;20:1081±106.
[26] Lu YY, Belytschko T, Gu L. A new implementation of
the element free Galerkin method. Computer Methods in
Applied Mechanics and Engineering 1994;113:397±414.
[27] Lu YY, Belytschko T, Tabbara M. Element-free
Galerkin methods for wave propagation and dynamic
fracture. Computer Methods in Applied Mechanics and
Engineering 1995;126:131±53.
[28] Lucy LB. A numerical approach to the testing of the ®ssion
hypothesis.
The
Astronomical
Journal
1977;82(12):1013±24.
[29] Melenk JM, BabusÏ ka I. The partition of unity ®nite element method: Basic theory and applications. Computer
Methods in Applied Mechanics and Engineering
1996;139:289±314.
[30] Monaghan JJ. Why particle methods work. SIAM
Journal of Scienti®c and Statistical Computing
1982;3(4):422.
[31] Nayroles B, Touzot G, Villon P. Generalizing the ®nite
element method: di€use approximation and di€use elements. Computational Mechanics 1992;10:307±18.
[32] Nemat-Nasser S, Horii H. Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. Journal of Geophysical Research
1982;87:6805±21.
[33] Newman J.C. Jr An improved method of collocation for
the stress analysis of cracked plates with various shaped
boundaries. Technical Report TN D-6376, NASA, 1971.
195
[34] OnÄate E, Idelsohn S, Zienkiewicz OC, Taylor RL, Sacco
C. A stabilized ®nite point method for analysis of ¯uid
mechanics problems. Computer Methods in Applied
Mechanics and Engineering 1996;139:315±46.
[35] Organ DJ. Numerical solutions to dynamic fracture problems using the element-free Galerkin method. Ph.D.
thesis, Northwestern University., 1996.
[36] Organ DJ, Fleming M, Belytschko T. Continuous meshless approximations for nonconvex bodies by di€raction
and
transparency.
Computational
Mechanics
1996;18:225±35.
[37] Schlangen E, Mier JGM. Experimental and numerical
analysis of micromechanisms of fracture of cement-based
composites.
Cement
&
Concrete
Composites
1992;14:105±18.
[38] Strang G, Fix G. An Analysis of the Finite Element
Method. Englewood Cli€s, NJ: Prentice-Hall, 1973.
[39] Sulsky D, Zhou SJ, Schreyer HL. Application of a particle-in-cell method to solid mechanics. Computers and
Physics Communications 1995;87:236±53.
[40] Swegle JW, Hicks DL, Attaway SW. Smoothed particle
hydrodynamics
stability
analysis.
Journal
of
Computational Physics 1995;116:123±34.
[41] Swenson DV, Ingra€ea AR. Modeling mixed-mode
dynamic crack propagation using ®nite elements: Theory
and applications. Computational Mechanics 1988;3:381±
97.
[42] Terry TG. Fatigue crack propagation modeling using the
element free Galerkin method. Master's thesis,
Northwestern University., 1994.
[43] Timoshenko SP., Goodier J.N. Theory of Elasticity
(Third ed.). New York: McGraw-Hill, 1970.
[44] Williams ML. On the stress distribution at the base of a
stationary crack. Journal of Applied Mechanics
1957;24:109±14.
[45] Wriggers P, Miehe C. Contact constraints within coupled
thermomechanical analysisÐa ®nite element model.
Computer Methods in Aplied Mechanics and
Engineering 1994;113:301±19.