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Arbitrary discontinuities in nite elements
Ted Belytschko, Nicolas Moës, S. Usui, Chandu Parimi
To cite this version:
Ted Belytschko, Nicolas Moës, S. Usui, Chandu Parimi. Arbitrary discontinuities in nite ele-
ments. International Journal for Numerical Methods in Engineering, Wiley, 2001, 50 (4), pp.993-1013.
�10.1002/1097-0207(20010210)50:43.0.CO;2-M�. �hal-01005275�
Arbitrary discontinuities in nite elements
T. Belytschko
∗;†; ‡
,N.Moes
§
, S. Usui
¶
and C. Parimi
Department of Mechanical Engineering; Northwestern University; 2145 Sheridan Road;
Evanston; IL 60208; U.S.A.
SUMMARY
A technique for modelling arbitrary discontinuities in nite elements is presented. Both discontinuities
in the function and its derivatives are considered. Methods for intersecting and branching discontinuities
are given. In all cases, the discontinuous approximation is constructed in terms of a signed distance
functions, so level sets can be used to update the position of the discontinuities. A standard displacement
Galerkin method is used for developing the discrete equations. Examples of the following applications
are given: crack growth, a journal bearing, a non-bonded circular inclusion and a jointed rock mass.
KEY WORDS: nite elements; fracture; fasteners; jointed rock
1. INTRODUCTION
This paper unies and extends the modelling of functions with arbitrary discontinuities and
discontinuous derivatives in nite elements rst proposed in References [1–4]. The disconti-
nuities are completely independent of the nite element mesh: they can cross elements in any
manner. This is particularly useful for evolution problems with moving discontinuities, such
as solidication, other phase changes, cracks, shear bands and joints in rock. In problems
involving the evolution and motion of discontinuities, it avoids the need for remeshing. It
also provides a powerful tool for modelling unusual problems in engineering, such as bolts,
joints, etc. In these problems, it avoids the need for sliding interfaces that conform to the
mating meshes.
The techniques for approximating discontinuities are based on the concepts described in
References [1; 2]. We emphasize modelling in nite element methods, but these methods also
apply to meshfree approximations such as the element-free Galerkin method, EFG [5].
∗
Correspondence to: Ted Belytschko. Mechanical Engineering Department, Northwestern University, 2145 N Sheri-
dan, Rm 224, Evanston, IL 60208-3111, U.S.A.
†
Walter P. Murphy, Professor
‡
E-mail: t-belytschko@northwestern.edu
§
Research Assistant Professor
¶
Graduate Student
Graduate Student
1
Figure 1. Illustration of nomenclature for two surfaces of discontinuity.
Other papers which address the issue of discontinuous elements are Oliver [6; 7], Armero
and Garikipati [8] and Duarte et al. [9]. The methods proposed here dier from the rst two
in that there are no incompatibilities in the element and the discontinuities can end within an
element. The method of Duarte et al. [9] provides an alternative method for discontinuous
functions.
The surfaces of discontinuity are dened by signed distance functions. This description is
not necessary for the application of these discontinuous approximations, but they are very
appealing because the methodology of level sets can then be applied to update these surfaces
for moving discontinuities, Sethian [10]. Other workers who have used level sets with nite
elements are Rao et al. [11] and Stolarski et al. [12].
The major appeal of these methods for incorporating discontinuities in nite elements is
that they do not require the mesh to conform to discontinuities in the approximating function
or its derivatives. They also avoid remeshing for moving discontinuities. Meshing, particularly
with triangles and tetrahedrons, has achieved a high level of robustness and speed, and many
are tempted to use it for everything. It is our belief that in many cases, methods that avoid
remeshing are preferable, for the costs of remeshing lie not only in the cost of creating a new
mesh, but the tremendous overhead associated with adapting visualization techniques and other
post-processing features, such as time histories of selected points, to sequences of meshes in
evolution problems.
2. APPROXIMATION FOR DISCONTINUOUS FUNCTIONS
2.1. Discontinuities in functions
We consider a domain with boundary as shown in Figure 1. We rst describe the method
for the approximation of a scalar variable u(x) but the method is easily extended to vector
elds.
The surface discontinuities in the dependent variable u(x) are denoted by
;=1 to m,
where m is the number of discontinuities. We rst consider the construction of an approx-
imation that is itself discontinuous on
, which is often called a strong discontinuity. The
mesh is completely independent of the geometry or location of the discontinuity. We denote
2
the shape functions at node I by N
I
(x) and the corresponding nodal values of the dependent
variable by u
I
.
The approximations will be of the following form:
u(x)=
I
N
I
(x)(u
I
+ a
I
I
(x)) =
I
I
(x;a
I
;u
I
) (1)
where u
I
are nodal values and a
I
are additional degrees of freedom associated with the
enrichment
I
(x) for the discontinuity. We will sometimes use the last form because the
enrichment varies from node to node and many nodes require no enrichment. It can be seen
that this is an application of the partition of unity concept [13].
Although the surfaces can be represented by any technique, for convenience we describe the
surfaces of discontinuity
by signed distance functions f
(x). The signed distance function
is dened by
f
(x) = min
x∈
x −
x sign(n
+
· (
x − x)) (2)
where
x is a point on the surface of discontinuity
and n
+
is a unit normal to the surface of
discontinuity from the subdomain where the distance function is positive. As is well known,
the point
x is the closest point projection of x on
, which is the orthogonal projection for
a continuously dierentiable surface; see Figure 1 for a depiction of the ingredients of the
above equation at a typical point. We usually approximate the distance function by a nite
element or meshless approximation
f
(x)=
I
f
I
N
I
(x) (3)
where N
I
(x) are the shape functions. When N
I
(x) are the standard C
0
nite element shape
functions, the surface (or line in 2D) of discontinuity is C
0
, i.e. piecewise continuously
dierentiable. If smoother representations of the surface are desired, moving least square
approximations such as those described for meshless methods in Reference [5] can be used,
even when the solution approximation is based on nite elements. The representation of the
discontinuity by Equation (3) enables it to be described completely by nodal data.
The approximation at a node I depends on whether the support of N
I
(x) (i.e. the domain
on which N
I
(x) is non-zero) is bisected (i.e. cut completely in two) by the discontinuity or
the discontinuity ends within the support of N
I
(x). The two cases are illustrated in Figure 2.
The support of N
I
(x) generally includes the domains of all elements which share node I .
When any element in the support of N
I
(x) is bisected by the discontinuity, we call it a
bisected support. If the discontinuity only partially cuts the support, we call it a slit support.
On a bisected support, the enrichment for representing a discontinuity in the function is
given by
I
(x)=H (f
(x)) (4)
where H (x) is a step function given by
H (x)=
0 for x¡0
+1
for x¿0
(5)
3
Figure 2. Bisected and slit supports.
Comparing Equation (4) with Equation (1) we can see that the approximation function at
an enriched node is
I
(x;u
I
;a
I
)=N
I
(x)(u
I
+ a
I
H (f
(x))) (6)
The above enrichment introduces the step function along the curve f
(x) = 0. The coe-
cients a
I
are additional unknowns in the discrete equations and govern the magnitude of the
discontinuity in the domain of the support of the shape function N
I
(x).
The above can be viewed as an enrichment with a windowed step function, where N
I
(x)
is the window function. The window function localizes the enrichment so that the discrete
equations will be sparse.
For a slit support, the approximation for a node is given by
I
(x;u
I
;a
I
)=N
I
(x)(u
I
+
a
I
b
(x)) (7)
where b
(x) are branch functions around the discontinuity. The branch functions are con-
structed in terms of the geometry of the surface of the discontinuity.
Consider for example the discontinuity shown in Figure 3. The virtual extension of the
surface of discontinuity is constructed by
∇f
· (x − x
A
)=0 (8)
and the signed distance function is extended on the basis of this virtual extension.
4