8/8/2019 Computational Fracture Mechanics - Dr. Paul Wash
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Computational Fracture MechanicsDr. Paul “Wash” Wawrzynek
September 27 - October 1, 2010
Università degli Studi di Brescia
A short course on
Lecture 5: Extended Finite Element
Methods (XFEM)
8/8/2019 Computational Fracture Mechanics - Dr. Paul Wash
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Conventional and Extended FE
[ ] [ ] [ ][ ] dV J B D Bk
T
∫ =
[ ] [ ][ ] dV J B B D B Bk k
k k ec
T
eceeec
cecc
∫ =
=
0
p
a
u
K K
K K
eeec
cecc
∑= i ii u N u
∑∑ Φ+= j j ji ii au N u
element stiffness matrix
[ ]{ } { } puK =
Conventional finite elements:
Extended finite elements:
displacement interpolation
global system
element stiffness matrix
displacement interpolation
global system
auxiliary basis
functions
auxiliary nodal
variables
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Element Enrichment for Cracks
∑ ∑∑∑
Ψ
++=
i j ji ji
ii ii ii
cr N
b H N u N u
),(),,(
),,(),,(),,(),,(
θ ζ η ξ
ζ η ξ ζ η ξ ζ η ξ ζ η ξ
=Ψ θ
θ θ
θ θ θ sin
2
cos,sin
2
sin,
2
cos,
2
sin r r r r
elements that contain a crack front (tip) areenriched with √r branch functions
elements completely crossed by a crack are
enriched with the heavy side step function
Multiplying the enrichment functions by the conventional shape functions binds the
modeled crack geometry to the original conventional mesh! In theory, this is not
strictly necessary, but it simplifies the implementation and is done in “all” published
implementations of the method.
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Branch Cut Transition ElementsEnriched crack front (tip) elements are not compatible, in a finite element sense,
with adjacent conventional elements, e.g., the patch test will fail.
∑ ∑∑ Ψ+=
i j ji jiii ii cr N u N u ),(),,(),,(),,( θ ζ η ξ α ζ η ξ ζ η ξ
Transition elements are used to remedy this
0=α 1=α where for nodes and for nodes
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Crack Geometry by Level Sets
0>ϕ
0<ϕ
0>ψ 0<ψ
0==ψ ϕ
0,0 >= ψ ϕ
crack front
crack face
Published XFEM implementations store crack geometry using a “level set” approach.
Two nearly orthogonal spatially varying sign/magnitude functions are used. One is zero
on the crack surface. The other is zero at the crack front.
Values for the level set functions are stored a FE node points and interpolated with the
standard shape functions.
∑∑ == i iii ii N N ψ ψ ϕ ϕ ,
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Integration Of Step Enriched ElementsElements that have been enriched with the step function must be integrated by dividing
the elements into subdomains.
Published implementations use triangular subdomains but other schemes are posssible
Conventional integration:
These two elements would give the same
stiffness matrix
This element would behave as if no
crack was present
Subdomain integration:
Elements divided into triangular subdomains for integration
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Integration Of Step Enriched Elements
Determining integration subdomains in 2D is relatively straightforward. It can become
considerably more difficult in 3D, especially for unstructured meshes (remeshing light)
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Integration Of Crack Front Elements
[ ] [ ][ ] dV J B B D B Bk k
k k ec
T
eceeec
cecc
∫ =
∑ ∑
Ψ=
i j ji jie cr N u ),(),,(),,( θ ζ η ξ ζ η ξ
=Ψ θ θ θ θ θ θ sin2
cos,sin2
sin,2
cos,2
sin r r r r
The expression for the stiffness matrices for enriched elements is
c N N ue
∂
Ψ∂+Ψ
∂
∂=
∂
∂
ξ ξ ξ
The matrix contain terms related toξ ∂
∂ eue B
Informally,
with
so will contain (singular) termse Br
1
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Integration Of Crack Front Elements
Zhang, Cui, and Liu, “A set of symmetric quadrature rules on triangles and tetrahedra,” Journal of Computational Mathematics, Vol.27,No.1, 2009, 89–96
It is very difficult to perform numerical integration of singular functions robustly. This
is particularly true in 3D where the singularity is along a “line”.
Most XFEM papers do not comment on how the singular integrations are performed.
The few that mention it seem to indicate that high order (e.g. 10) Gauss quadrature is
used.
In essence, Guass quadrature fits a polynomial through the values at the integration
points and integrates the polynomial exactly. Polynomials, regardless the order, can
never model a singularity and integrated values can become sensitive to small changes
in geometry for poorly shaped integration domains.
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Limitations on the Representation of Crack
Front GeometryBecause published XFEM implementations use the standard element shape functions to
interpolate the enrichment functions and the level set crack geometry, the geometry of
the portion of a crack that passes through the element is limited by those shape functions
(e.g., a nearly straight line crack front in a linear element).
A “real” crack what XFEM can
model
This means that for 3D: 1) the crack geometries that can be represented are implicitly
tied to the local finite element mesh, and 2) for reasonable approximations of curved
crack fronts the dimension of the local finite elements should be much smaller than the
crack size.
crack front
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Computing SIF’s for XFEM ModelsTwo methods have been proposed for extracting stress intensity factors from 3D XFEM
models: 1) Displacement Correlation, and 2) J -Integral
The experience with conventional finite elements is that locally refined meshes are
required for accurate SIF’s using displacement correlation.The natural domain of integration for evaluating the J-integral for a portion of the
crack front is the element containing that portion of the front. This can lead to very
irregular integration domains. In J value is independent shape of the domain of
integration (path independence). In practice this may not be the case, especially for
near degenerate cases.
“real” crack front
XFEM crack front
locations where J
will be computed
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Duflot, Wyart, Lani, and Martiny, “Application of XFEM to multi-site crack propagation”, ???
Remeshing for XFEM
In most practical applications, the mesh will likely need to be refined in order to
place enough elements along the crack to adequately approximate the crack
geometry and to give a reasonable number of SIF values along the crack front.
Question: If one needs to remesh anyways, why not remesh with a mesh that is
optimized to produce accurate SIF’s (e.g., singular crack-front elements with
concentric “rings’ of surrounding elements.
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Summary• XFEM can be used to model crack growth through a FEM mesh without the
requirement of remeshing.
• Elements that contain cracks are enriched with additional shape functions
(and degrees of freedom).
• Special crack-front “transition” elements are required to maintain finite
element compatibility.
• Though not required, level set methods are usually used to track crack
geometry.• Special care must be taken to integrate enriched elements, and very high
order integration may be required for crack-front elements.
• Crack geometry representation may be limited by the original mesh
refinement.
• The accuracy in computed SIF’s may be limited by the original mesh
refinement.
• Remeshing may be required for accurate results!