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Modeling Holes and Inclusions by Level Sets
in the Extended Finite–Element Method
N. Sukumar, D. L. Chopp, N. Moës and T. Belytschko
Computer Methods in Applied Mechanics and Engineering,
190 (46–47), pp. 6183–6200, 2001
Abstract
A methodology to model arbitrary holes and
material interfaces (inclusions) without meshing the
internal boundaries is proposed. The numerical method
couples the level set method (Osher and Sethian, 1988) to
the extended finite element method (X-FEM) (Moës et al., 1999).
In the X-FEM, the finite element approximation is
enriched by additional functions through the notion of
partition of unity. The level set method
is used for representing the location of holes and material
interfaces, and in addition, the level set function is used
to develop the local enrichment for material interfaces.
Numerical examples in 2-dimensional linear elastostatics are
presented to demonstrate the accuracy and potential of the
new technique.
This paper is available in postscript format (26 pages, ~1.7MB) as well as in PDF.
Extended Finite Element Method and Fast
Marching Method for Three-Dimensional Crack Propagation
N. Sukumar, D. L. Chopp and B. Moran
Engineering Fracture Mechanics
70 (1), pp. 29–48, 2003
Abstract
A numerical technique for planar three-dimensional fatigue crack growth simulations is
proposed. The new technique couples the extended finite element method
(X-FEM) to the Fast Marching Method (FMM). In the X-FEM, a discontinuous
function and
the two-dimensional asymptotic crack-tip displacement fields are
added to the finite element approximation to account for the crack
using the notion of partition of unity. This enables the domain to be
modeled by finite elements with no explicit meshing of the crack
surfaces. The initial crack geometry is represented by level
set functions, and subsequently signed distance functions are used to
compute the enrichment functions that appear in the displacement-based
finite element approximation. The fast marching method in conjunction with the
Paris crack growth law is used to advance the crack front. Stress
intensity factors (SIFs) for planar three-dimensional cracks are
computed, and fatigue crack growth simulations for planar cracks are
presented. Good
agreement between the numerical results and theory is realized.
This paper is available in postscript format (35 pages, ~1.6MB) as well as in PDF.
Fatigue Crack Propagation of Multiple Coplanar
Cracks with the Coupled Extended Finite Element/Fast
Marching Method
D. L. Chopp and N. Sukumar
International Journal of Engineering Science
41 (8), pp. 845–869, 2003
Abstract
A numerical technique for modeling fatigue crack propagation of
multiple coplanar cracks is presented. The proposed method couples the
Extended Finite Element Method (X-FEM) to the Fast
Marching Method (FMM). The entire crack geometry,
including one or more cracks, is represented by a single signed
distance (level set) function. Merging of distinct cracks is handled
naturally by the fast marching method with no collision detection or
mesh reconstruction required. The fast marching method in conjunction
with the Paris crack growth law is used to advance the crack front.
In the X-FEM, a discontinuous function and the two-dimensional
asymptotic crack-tip displacement fields are added to the finite
element approximation to account for the crack using the notion of
partition of unity. This enables the domain to be
modeled by a single fixed finite element mesh with no explicit meshing
of the crack surfaces. In an earlier study
(Sukumar et al. 2003), the
methodology, algorithm, and implementation for three-dimensional crack
propagation of single cracks was introduced. In this paper,
simulations for multiple planar cracks are presented, with crack
merging and fatigue growth carried out without any user-intervention
or remeshing.
KEYWORDS: partition of unity, extended finite element method, fast marching
method, level set method, fatigue crack growth, coplanar cracks in 3-d
This paper is available in PDF.
Modeling Quasi-Static Crack Growth
with the Extended Finite Element Method. Part I: Computer Implementation
N. Sukumar and J.-H. Prévost
International Journal of Solids and Structures
40 (26), pp. 7513–7537, 2003
Abstract
The extended finite element method (X-FEM) is
a numerical method for modeling strong (displacement) as well
as weak (strain) discontinuities within a standard finite element
framework.
In the X-FEM, special functions are added to the finite element
approximation using the framework of partition of unity. For
crack modeling in isotropic linear elasticity, a discontinuous function and the
two-dimensional asymptotic crack-tip
displacement fields are used to account for the crack. This
enables the domain to be modeled by finite elements without
explicitly meshing the crack
surfaces, and hence quasi-static crack propagation
simulations can be carried out without remeshing.
In this paper, we discuss some of the key issues in the X-FEM
and describe its implementation within
a general-purpose finite element code. The finite element program
Dynaflow®
is considered in this study and
the implementation for modeling 2-d cracks in
isotropic and
bimaterial media is described. In particular, the
array-allocation for enriched degrees of freedom, use of
geometric-based queries for carrying out nodal
enrichment and mesh partitioning, and the
assembly procedure for the discrete
equations are presented. We place particular
emphasis on the design of a computer code
to enable the modeling of discontinuous phenomena
within a finite element framework.
KEYWORDS: strong discontinuities, partition
of unity, extended finite element,
finite element programming, crack modeling, singularity
This paper is available in postscript format as well as in PDF.
Modeling Quasi-Static Crack Growth
with the Extended Finite Element Method. Part II: Numerical Applications
R. Huang, N. Sukumar and J.-H. Prévost
International Journal of Solids and Structures
40 (26), pp. 7539–7552, 2003
Abstract
In Part I (Sukumar and Prévost, 2002),
we described
the implementation of the extended finite element method (X-FEM)
within Dynaflow®, a
standard finite element package. In our implementation,
we focused on 2-dimensional
crack modeling in linear elasticity.
For crack modeling in the X-FEM, a discontinuous function and
the near-tip asymptotic functions are added to the finite element
approximation using the framework of partition of unity.
This permits the crack to be represented without
explicitly meshing the crack surfaces and
crack propagation simulations can be
carried out without the need for any remeshing.
In this paper, we present numerical solutions for the
stress intensity factor for crack problems, and also
conduct crack growth simulations with the X-FEM.
Numerical examples are presented with a two-fold
objective: first to show the efficacy of the
X-FEM implementation in Dynaflow®
and secondly to demonstrate the accuracy and
versatility of the method to solve challenging
problems in computational failure mechanics.
KEYWORDS: strong discontinuities, partition
of unity, extended finite element, crack
propagation, bimaterial interface, mud-crack, channel-cracking,
thin films
This paper is available in postscript format as well as in PDF.
Partition of Unity Enrichment for
Bimaterial Interface Cracks
N. Sukumar, Z. Y. Huang, J.-H. Prévost and Z. Suo
International Journal for Numerical Methods in Engineering
59 (8), pp. 1075–1102, 2004
Abstract
Partition of unity enrichment techniques are developed for bimaterial
interface cracks.
A discontinuous function and the two-dimensional near-tip asymptotic
displacement functions are added to the finite element
approximation using the notion of partition of unity.
This enables the domain to be
modeled by finite elements without explicitly meshing the crack
surfaces. The crack-tip
enrichment functions are chosen as those that
span the asymptotic displacement fields
for an interfacial crack.
The concept of partition of unity
facilitates the incorporation of the oscillatory nature of
the singularity
within a conforming
finite element approximation.
The mixed mode (complex)
stress intensity factors for bimaterial interfacial
cracks are numerically
evaluated using the domain form of the interaction integral.
Good agreement between the numerical results and the reference
solutions for benchmark interfacial crack problems is realized.
KEYWORDS: partition of unity, extended finite element method, interface
crack, stress intensity factor, steady-state energy release rate
This paper is available in postscript format as well as in PDF.
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