eXtended Finite Element Method (XFEM) for material

Introduction Discontinuities The method Cracks Bibliografia

eXtended Finite Element Method (XFEM)for material modelling.

Application to cracks

Michele Ruggeri

May 10, 2010

Michele Ruggeri

eXtended Finite Element Method (XFEM) for material modelling. Application to cracks


Introduction

The eXtended Finite Element Method (XFEM) is a versatile toolfor the analysis of problems characterized by discontinuities,singularities, localized deformations and complex geometries.

Michele Ruggeri



This method can simplify the solution of many problems inmaterial modeling, such as

I the propagation of crack,

I the evolution of dislocations,

I the modeling of grain boundaries,

I the evolution of phase boundaries.

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Non-smooth solutions properties: discontinuities andsingularities

A discontinuity may be defined as a rapid change of a fieldquantity over a length which is negligable compared to thedimensions of the observed domain. In the real world,discontinuities are frequently found.

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I (a) In solids stresses and strains are discontinuous acrossmaterial interfaces.

I (b) In solids displacements are discontinuous at cracks.I (c) Tangential displacements are discontinuous across shear

bands.I (d) In fluids, velocity and pressure fields may involve

discontinuities at the interface of two fluids.I (e)(f) Shocks and boundary layers can be interpreted as

discontinuities.Michele Ruggeri



Classification of discontinuities

I weak discontinuities (kink),

I strong discontinuities (jump).

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The advantages of this method is that the finite element mesh canbe completely independent of the morphology of these entities.

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The level-set method

The decription of discontinuities in the context of the XFEM isoften realized by the level-set method. A level-set function is ascalar function f within the domain whose zero-level is interpretedas the discontinuity. As a consequence, the domain Ω is dividedinto two subdomains Ω+ and Ω− on either side of the discontinuitywhere the level-set function is positive or negative, respectively.

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Example

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Often, the signed distance function is used as a particular level-setfunction

f (x) = ±minxΓ∈Γ‖x − xΓ‖

It is noted, that level-set functions are typically defined by discretevalues at the nodes in the domain. They are then interpolated inthe element interiors by standard finite element shape functions.

f h(x) =∑i∈I

fiNi (x)

con fi = f (xi )

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Overview

XFEM is a numerical method that enables a local enrichment ofapproximation spaces.The enrichment is realized through the partition of unity concept.The method is useful for the approximation of solutions withpronounced non-smooth characteristics in small parts of thecomputational domain, for example near discontinuities andsingularities.

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General formulation

XFEM = FEM + enrichment

Finite element approximation: uFEM

uFEM(x) =∑i∈I

Ni (x)ui

with

I I set of all nodes in the domain,

I Ni standard FE function of node i ,

I ui unknown of the standard FE part at node i .

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Partition of unity

The foundation of this method is the partition of unity concept forenriching finite element approximation.A global partition of unity in a domain Ω is a set of functions ϕisuch that ∑

i

ϕi (x) = 1, per ogni x ∈ Ω

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General formulation

uXFEM(x) = uFEM + uenriched

uXFEM(x) =∑i∈I

Ni (x)ui +∑i∈I∗

ϕi (x)Ψ(x)qi

with

I I ∗ ⊂ I nodal subset of the enrichment,

I ϕi partition of unity function of node i ,

I Ψ global enrichment functions,

I qi unknown parameter at node i .

Mi (x) = ϕi (x)Ψ(x) local enrichment function of node i

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Classification of elements

I reproducing elements,

I blending elements (!).

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Choice of enriched nodes

For weak and strong discontinuities, the nodal subset I ∗ is builtfrom all nodes of elements that are cut by the discontinuity.

cut elements: mini∈I el

(fi ) ·maxi∈I el

(fi ) < 0

uncut elements: mini∈I el

(fi ) ·maxi∈I el

(fi ) > 0

with

I f level-set function,

I I el set of element nodes.

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Michele Ruggeri



Global enrichment functions

I weak discontinuities: abs-functions of the level set-function

Ψ(x) = |f (x)|

I strong discontinuities: Heaviside-function of the levelset-function

Ψ(x) = H(f (x))

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Application to cracks modelling

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uh(x) =∑i∈S

Ni (x)ui +∑i∈SH

Ni (x) [H(f (x))− H(f (xi ))] q0i

+n∑

j=1

∑i∈SC

Ni (x)[Ψj(x)−Ψj(xi )

]qji

I S set of all nodes of finite element mesh,

I SC set of nodes of elements around the crack tip,

I SH set of nodes of elements cut by the crack but not in SC .

I

Ψj

j=1,...,nset of enrichment functions (approximation of

near tip beheviour)

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Example: cracks in elastic materials

Ψj4

j=1=√

r cos(θ/2), sin(θ/2), sin(θ/2)sin(θ), cos(θ/2) sin(θ)

I based on asymptotic solution of Williams

I r , θ polar coordinates

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Michele Ruggeri



Michele Ruggeri



Michele Ruggeri



Bibliografia

http://www.xfem.rwth-aachen.de/

Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P.,Meshless methods: an overview and recent developments.Computer Methods in Applied Mechanics and Engineering,Vol. 139, 1996.

Belytschko T., Gracie R., Ventura G., A review of theeXtended/Generalized Finite Element Methods for materialmodelling. Modelling and Simulation in Materials Science andEngineering, Vol. 17, 2009.

Sukumar N., Moes N., Moran B., Belytschko T., Extendedfinite element method for three dimensional crack modelling.International Journal for Numerical Methods in Engineering,Vol. 48, 2000.

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eXtended Finite Element Method (XFEM) for material