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)



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



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.



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



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 ,



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



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)



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



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.



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



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



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.



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!

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Computational Fracture Mechanics - Dr. Paul Wash