[PPT] - Extended Finite Element Method with Global Enrichment K. Agathos 1 PowerPoint Presentation

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Extended Finite Element Method with Global Enrichment

K. Agathos1
E. Chatzi2
S. P. A. Bordas3,4
D. Talaslidis1

1Institute of Structural Analysis and Dynamics of Structures

Aristotle University Thessaloniki

2Institute of Structural Engineering

ETH Z¨ urich

3Research Unit in Engineering Sciences

Luxembourg University

4Institute of Theoretical, Applied and Computational Mechanics

Cardiff University

2015

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Outline

Problem statement Governing equations Weak Form Global enrichment XFEM Motivation Related works Crack representation Tip enrichment Jump enrichment Point-wise matching Integral matching Displacement approximation Definition of the Front Elements Numerical examples 2D convergence study 3D convergence study Conclusions References

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Problem statement Governing equations

3D body geomery

u

Γ Γ

c

t ¯ t ¯

c

Γ

c t

Γ

t

Γ x y z

Γ = Γ0 ∪ Γu ∪ Γt ∪ Γc Γc = Γt

c ∪ Γ0 c

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Problem statement Governing equations

Governing equations

Equilibrium equations and boundary conditions: ∇ · σ + b = 0 in Ω u = ¯ u

n

Γu σ · n = ¯ t

n

Γt σ · n = 0

n

Γ0

c

σ · n = ¯ tc

n

Γt

c

Kinematic equations: ǫ = ∇su Constitutive equations: σ = D : ǫ

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Problem statement Weak Form

Weak form of equilibrium equations

Find u ∈ U such that ∀v ∈ V0

Ω

σ(u) : ǫ(v) dΩ =

Ω

b · v dΩ +

Γt

¯ t · v dΓ +

Γt

c

¯ tc · v dΓt

c

where : U =

u|u ∈
H1 (Ω)

3 , u = ¯

u on Γu

and

V =

v|v ∈
H1 (Ω)

3 , v = 0 on Γu

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Global enrichment XFEM Motivation

Motivation

◮ XFEM for industrially relevant (3D) crack problems

◮ Requires robust methods for stress intensity evaluation. ◮ Requires low solution times and ease of use.

◮ but standard XFEM leads to

◮ Ill-conditioning of the stiffness matrix for “large” enrichment domains. ◮ Lack of smoothness and accuracy of the stress intensity factor field

along the crack front.

◮ Blending issues close at the boundary of the enriched region. ◮ Problem size for propagating cracks (“old” front-dofs must be kept for

stability of time integration schemes).

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Global enrichment XFEM Motivation

Global enrichment XFEM

There exists different approaches to alleviate the above difficulties:

◮ Preconditioning (e.g. Mo¨

es; Menk and Bordas)

◮ Ghost penalty (Burman) ◮ Stable XFEM/GFEM (Banerjee, Duarte, Babuˇ

ska, Paladim, Bordas) - behaviour for realistic 3D crack not clear.

◮ Corrected XFEM/GFEM (Fries, Loehnert) ◮ SIF-oriented (goal-oriented) error estimation methods for SIFs

(R´

denas, Estrada, Ladev`

eze, Chamoin, Bordas)

◮ Restrict the variability of the enrichment within the enriched domain:

doc-gathering, cut-off XFEM (Laborde, Renard, Chahine, Sal¨ un and the French team ;-)

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Global enrichment XFEM Motivation

Global enrichment XFEM

An XFEM variant is introduced which:

◮ Extends dof gathering to 3D through global enrichment. ◮ Employs point-wise matching of displacements. ◮ Employs integral matching of displacements. ◮ Enables the application of geometrical enrichment to 3D.

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Global enrichment XFEM Related works

Related works

Similar concepts to the ones introduced herein can be found:

◮ In the work of Laborde et al.

→ dof gathering → point-wise matching

(Laborde, Pommier, Renard, & Sala¨ un, 2005)

◮ In the work of Chahine et al.

→ integral matching

(Chahine, Laborde, & Renard, 2011)

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Global enrichment XFEM Related works

Related works

◮ In the work of Langlois et al.

→ discretization along the crack front

(Langlois, Gravouil, Baieto, & R´ ethor´ e, 2014)

◮ In the s-finite element method

→ superimposed mesh

(Fish, 1992)

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Global enrichment XFEM Crack representation

Crack representation

Level set functions:

◮ φ (x) is the signed distance from the crack surface. ◮ ψ (x) is a signed distance function such that:

→ ∇φ · ∇ψ = 0 → φ (x) = 0 and ψ (x) = 0 defines the crack front

Polar coordinates: r =

φ2 + ψ2,

θ = arctan

φ

ψ

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Global enrichment XFEM Crack representation

Crack representation

crack surface crack extension crack front ) = 0 x ( ψ ) x ( ψ ) x ( φ x ) = 0 x ( φ r θ

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Global enrichment XFEM Tip enrichment

Tip enrichment

Enriched part of the approximation for tip elements: ute (x) =

K

Ng

K (x)

j

Fj (x) cKj Ng

K are the global shape functions to be defined.

Tip enrichment functions: Fj (x) ≡ Fj (r, θ) =

√r sin θ

2, √r cos θ 2, √r sin θ 2 sin θ, √r cos θ 2 sin θ

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Global enrichment XFEM Tip enrichment

Geometrical enrichment

◮ Enrichment radius re is defined. ◮ Nodal values ri of variable r are computed. ◮ The condition ri < re is tested. ◮ If true for all nodes of an element, the element is tip enriched.

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Global enrichment XFEM Jump enrichment

Jump enrichment

Jump enrichment function definition: H(φ) =

1

for φ > 0 − 1 for φ < 0 Shifted jump enrichment functions are used throughout this work.

1.0
0.5

0.0 0.5 1.0

1.0
0.5

0.0 0.5 1.0

1.0
0.5

0.0 0.5 1.0

1.0
0.5

0.0 0.5 1.0

1.0
0.5

0.0 0.5 1.0

1.0
0.5

0.0 0.5

1.0
0.5

0.0 0.5 1.0

1.0
0.5

0.0 0.5 1.0 0.0 0.5 1.0 1.5

)

1

H − ) φ ( H (

1

N ) φ ( H

1

N ) φ ( H

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Global enrichment XFEM Jump enrichment

Enrichment strategy

Motivation for an alternative enrichment strategy:

◮ Tip enrichment functions are derived from the first term of the

Williams expansion.

◮ Displacements consist of higher order terms as well. ◮ Those terms are represented by:

→ the FE part → spatial variation of the tip enrichment functions

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Global enrichment XFEM Jump enrichment

Enrichment strategy

◮ In the proposed method:

→ no spatial variation is allowed → higher order terms can only be approximated by the FE part

◮ Higher order displacement jumps can not be represented in tip

elements.

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Global enrichment XFEM Jump enrichment

Enrichment strategy

Proposed enrichment strategy:

e

r Tip enriched node Tip and jump enriched node Jump enriched node Tip enriched elements Jump enriched element crack front crack surface

Both tip and jump enrichment is used for tip elements that contain the crack.

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Global enrichment XFEM Point-wise matching

j

F

j

K

g

N

K

Kj

c )

1

x (

j

F

j

K

g

N

K

Displacement approximations of regular and tip elements:

ur (x) =

I

NI (x) uI +

J

NJ (x) aJ ut (x) =

I

NI (x) uI +

K

Ng

K (x)

j

Fj (x) cKj

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Global enrichment XFEM Point-wise matching

j

F

j

K

g

N

K

Kj

c )

1

x (

j

F

j

K

g

N

K

Displacements are matched by imposing the condition:

ur (xI) = ut (xI)

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Global enrichment XFEM Point-wise matching

j

F

j

K

g

N

K

Kj

c )

1

x (

j

F

j

K

g

N

K

Parameters aI are obtained:

aI =

K

Ng

K(XI)

j

Fj(XI)cKj

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Global enrichment XFEM Point-wise matching

j

F

j

K

g

N

K

Kj

c )

1

x (

j

F

j

K

g

N

K

Parameters aI can be expressed as:

aI =

K
j

T t−r

IKj cKj

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Global enrichment XFEM Point-wise matching

Tip and Jump Elements

Displacement approximations of tip and jump elements: uj (x) =

I

NI (x) uI +

J

NJ (x) aJ +

L

NL (x) (H (x) − HL)bL + +

M

NM (x) (H (x) − HM)bt

M ,

ut (x) =

I

NI (x) uI +

J

NJ (x) (H (x) − HJ)bJ + +

K

Ng

K (x)

j

Fj (x) cKj

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Global enrichment XFEM Point-wise matching

Tip and Jump Elements

Jump enriched element

1 t

b )

1

H − )

c

x ( H (

2 t

b )

2

H − )

c

j

F

j

K

g

N

K

Kj

c )

2

x (

j

F

j

K

g

N

K

Kj

c )

c

x (

j

F

j

K

g

N

K

Kj

c )

c

x (

j

F

j

K

g

N

K

Point-wise matching condition:

uj (xn) = ut (xn)

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Global enrichment XFEM Point-wise matching

Tip and Jump Elements

The condition is imposed:

◮ at nodes → parameters aI are obtained ◮ at additional points → parameters bt I are obtained:

(H(Xl) − HI) bt

I =

K

Ng

K(Xl)

j

Fj(Xl)cKj −

I

NI(Xl)aI Parameters bt

I can be reformulated as:

bt

I =

K
j

T t−j

IKj cKj

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Global enrichment XFEM Point-wise matching

Selection of additional points

jump node jump element tip element tip node crack tip 1 2 3 4 5 6 a b points point-wise matching

The condition is imposed at the points where the crack intersects element edges or faces.

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Global enrichment XFEM Point-wise matching

Selection of additional points

3D case:

1 2 3 4 5 6 1 2 1 2 3 4 5 6 1 2 3 4 tip node a b c d e f a b c d a b c d a b e f crack surface a) Point-wise matching at an edge b) Point-wise matching at a face c) Point-wise matching at several faces d) Point-wise matching at several faces points matching point-wise

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Global enrichment XFEM Point-wise matching

Selection of additional points

Special case:

jump node jump element tip element tip node crack tip 1 2 3 4 5 6 7 8 a b points point-wise matching

◮ Edge 3-4 does not belong to a

tip element.

◮ Evaluating the tip enrichment

functions at 3-4 leads to errors.

◮ The values obtained from edge

4-7 will be used for 3-4.

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Global enrichment XFEM Point-wise matching

Selection of additional points

In order to implement the above procedure:

◮ Point-wise matching elements are looped upon prior to the assembly. ◮ Parameters bt i are computed and stored.

Parameters bt

i can be computed for all nodes.

The whole procedure is computationally inexpensive.

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Global enrichment XFEM Integral matching

Integral matching

Motivation:

◮ For P1 elements and topological enrichment a loss of accuracy occurs. ◮ The effect is more pronounced for mode I loading. ◮ This is attributed to the displacement jump between regular and tip

elements.

◮ A possible solution is the addition of one layer of tip elements.

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Global enrichment XFEM Integral matching

Hierarchical functions

The addition of hierarchical blending functions is proposed. Those functions:

◮ Eliminate the displacement jump in a weak sense. ◮ For linear quadrilateral elements assume the form:

Nh (ξ1, ξ2) = (1 − |ξ1|) (1 + ξ2) 2

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Global enrichment XFEM Integral matching

Integral matching

Displacements along the edges of regular and jump elements: ur (ξ1, ξ2) =

I

NI (ξ1, ξ2) uI +

J

NJ (ξ1, ξ2) aJ + Nh (ξ1, ξ2) ah ut (ξ1, ξ2) =

I

NI (ξ1, ξ2) uI +

K

Ng

K (x)

j

Fj (x) cKj Integral matching condition:

S

(ur − ut) dS = 0 Coefficients ah are obtained as: ah

i =

K
j

T h

iKjcKj

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Global enrichment XFEM Integral matching

Integral matching-Mode I

Mode I, hierarchical functions are used to eliminate displacement jumps in a weak sense:

displacement jump tip element regular element

h

N

Kj

c ) x (

j

F

j

)

x (

K g

N

K

1

2 3 4 1 2

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Global enrichment XFEM Integral matching

Integral matching-Mode II

Mode II, displacement jumps almost vanish in a weak sense:

regular element tip element displacement jump

Kj

c ) x (

j

F

j

)

x (

K g

N

K

1

2 3 4 1 2

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Global enrichment XFEM Integral matching

Integral matching

Imposition of integral matching condition: tip node

1 2 3 4 5 6 7 8 9 10 11

matching integral tip element crack surface

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Global enrichment XFEM Displacement approximation

Displacement approximation

Displacement approximation for the whole domain: u (x) =

I∈N

NI (x) uI +

J∈N j

NJ (x) (H (x) − HJ)bJ + +

K∈N s

Ng

K (x)

j

Fj (x) cKj + upm (x) + uim (x) upm (x) =

I∈N t1

NI (x)

K
j

T t−r

IKj cKj +

+

J∈N t2

NJ (x) (H (x) − HJ)

K
j

T t−j

IKj cKj

uim (x) =

I∈N h

Nh

I (x)

K
j

T h

IKjcKj

Nodal sets: N set of all nodes in the FE mesh. N j set of jump enriched nodes. N s set of superimposed nodes which will be described next. N t1 set of transition nodes between tip and regular elements. N t2 set of transition nodes between tip and jump elements. N h set of edges where the blending functions are added.

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Global enrichment XFEM Definition of the Front Elements

Front elements

A superimposed mesh is used to provide a basis for weighting tip enrichment functions. Desired properties:

◮ Satisfaction of the partition of unity property. ◮ Spatial variation only along the direction of the crack front.

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Global enrichment XFEM Definition of the Front Elements

Front elements

Special elements are employed which are both:

◮ 1D → shape functions vary only along one dimension ◮ 3D → they are defined in a three-dimensional domain

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Global enrichment XFEM Definition of the Front Elements

Front elements

tip enriched elements crack front FE mesh front element boundaries front element node front element

◮ A set of nodes along the crack

front is defined.

◮ Such points are also required for

SIF evaluation.

◮ A good starting point for front

element thickness is h.

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Global enrichment XFEM Definition of the Front Elements

Front elements

Volume corresponding to two consecutive front elements.

crack front crack surface boundary front element

Different element colors correspond to different front elements.

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Global enrichment XFEM Definition of the Front Elements

Open crack fronts

Front element definition:

◮ Unit vectors ei are defined parallel to the element directions:

ei =

xi+1−xi |xi+1−xi|. ◮ For every nodal point i a unit vector ni is defined: ni = ei+ei−1 |ei+ei−1|. ◮ A plane is defined that passes through the node: ni · (x0 − xi) = 0. ◮ The element volume is defined by the planes corresponding to its

nodes.

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Global enrichment XFEM Definition of the Front Elements

Open crack fronts

Vectors associated with front elements.

1 − i

n 1 − i

1 − i

e

i

n

i

e

+1 i

n i + 1 i front element

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Global enrichment XFEM Definition of the Front Elements

Closed crack fronts

a) Application of the method used for open crack fronts to closed crack fronts → front elements overlap. b) Method used for closed crack fronts → overlaps are avoided.

a) b)

c

x

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Global enrichment XFEM Definition of the Front Elements

Closed crack fronts

Element definition using an additional point (xc):

◮ Vectors ei are defined for every element. ◮ Point xc is defined as: xc =

n

i=1

xc n . ◮ Vectors nci joining points i to the internal point xc are defined:

nci = xc − xi.

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Global enrichment XFEM Definition of the Front Elements

Closed crack fronts

◮ Vectors nni normal to vectors ei and nci are defined: nni = ei × nci. ◮ Vectors ni are defined: ni = nti×nci |nti×nci|. ◮ Planes normal to the vectors ni are defined: ni · (x0 − xi) = 0. ◮ Element volumes are defined as in the open crack front case.

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Global enrichment XFEM Definition of the Front Elements

Closed crack fronts

Vectors used in the definition of front elements.

x

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Global enrichment XFEM Definition of the Front Elements

Closed crack fronts

Discretization of a non-planar closed crack front using an additional point xc.

c

x front element plane projection

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Global enrichment XFEM Definition of the Front Elements

Front element parameter

A function similar to the level sets is defined which varies along the crack front.

= 1 η = 2 η = 3 η = 4 η = 5 η = 6 η = 7 η front node front element

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Global enrichment XFEM Definition of the Front Elements

Front element parameter

Evaluation of the parameter for a point x0: Plane equations corresponding to the nodes of each element are evaluated: fi(x0) = ni · (x0 − xi) fi+1(x0) = ni+1 · (x0 − xi+1)

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Global enrichment XFEM Definition of the Front Elements

Front element parameter

Once fi and fi+1 are obtained:

◮ If fi < 0 or fi+1 > 0 the point lies outside the element ◮ If fi = 0 or fi+1 = 0 the point lies on the plane corresponding to node

i or i + 1: η = i or η = i + 1

◮ If fi > 0 and fi+1 < 0 the point lies inside the element

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Global enrichment XFEM Definition of the Front Elements

Front element parameter

For points lying inside front elements:

◮ Integer Part: ηi = i ◮ Fractional part:

1

x

2

x x

i

e

i

n

+1 i

n

1

t

2

t

i

e

x = x0 + tei t ∈ R t1 = ni · (x0 − xi) ni · ei t2 = ni+1 · (x0 − xi+1) ni+1 · ei x1 = x0 + t1ei x2 = x0 + t2ei

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Global enrichment XFEM Definition of the Front Elements

x1 = x0 + t1ei x2 = x0 + t2ei x10 = x0 − x1 x12 = x2 − x1 ηf = |x10| |x12| Finally: η = ηi + ηf

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Global enrichment XFEM Definition of the Front Elements

Front element shape functions

Linear 1D shape functions are used: Ng (ξ) =

1 − ξ

2 1 + ξ 2

◮ ξ is the local coordinate of the superimposed element.

◮ Those functions are used to weight tip enrichment functions.

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Global enrichment XFEM Definition of the Front Elements

Front element shape functions

Definition of the front element parameter used for shape function evaluation.

ξ

1

x

2

x

i

n

+1 i

n

i

e

m

x x

1 − = ξ 5 . − = ξ = 0 ξ 5 . = 0 ξ = 1 ξ

boundary front element node front element

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Global enrichment XFEM Definition of the Front Elements

Front element shape functions

The evaluation of ξ is almost identical to the evaluation of ηf : ξ = 2 x12 · xm0 |x12|2 where x12 = x2 − x1 xm0 = x0 − xm xm = x1 + x2 2

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Numerical examples 2D convergence study

2D convergence study

◮ An L × L square domain with an edge crack of length a is considered. ◮ Boundary conditions are provided by the Griffith problem. ◮ Both topological and geometrical enrichment are used. ◮ The alternative jump enrichment strategy is not used.

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Numerical examples 2D convergence study

2D convergence study

u

Γ L L a

node where boundary conditions are applied

◮ Dimensions of the problem: L = 1 unit, a = 0.5 units. ◮ Material parameters: E = 100 units and ν = 0.0. ◮ Mesh consists of n × n linear quadrilateral elements,

n = 11, 21, 41, 61, 81, 101 .

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Numerical examples 2D convergence study

2D convergence study

Acronyms used for the 2D convergence study

Acronym Description FEM The FE part of the approximation XFEM Standard XFEM (with shifted enrichment functions) XFEMpm1 XFEM using dof gathering and point-wise matching XFEMpm2 XFEMpm1 with the additional p.m. condition of subsection GE-XFEM XFEMpm2 with integral matching (Global Enrichment XFEM)

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Numerical examples 2D convergence study

L2 and energy norms

11 21 41 61 81101 10

−4

10

−2

n error 11 21 41 61 81101 10

−4

10

−2

n error XFEM, E XFEM, L2 XFEMpm1, E XFEMpm1, L2 XFEMpm2, E XFEMpm2, L2 GE−XFEM, E GE−XFEM, L2 Mode I, re=0.00 Mode II, re=0.00 11 21 41 61 81101 10

−5

10

−3

10

−1

n error 11 21 41 61 81101 10

−4

10

−2

n error XFEM, E XFEM, L2 XFEMpm1, E XFEMpm1, L2 XFEMpm2, E XFEMpm2, L2 GE−XFEM, E GE−XFEM, L2 Mode I, re=0.12 Mode II, re=0.12

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Numerical examples 2D convergence study

L2 and energy norms

Convergence rates

re = 0.00 re = 0.12 Mode I Mode II Mode I Mode II XFEM E 0.491 0.493 1.030 0.982 XFEM L2 0.908 0.928 1.980 1.955 XFEMpm1 E 0.483 0.489 1.243 1.211 XFEMpm1 L2 1.044 0.984 2.355 1.773 XFEMpm2 E 0.483 0.479 1.245 1.179 XFEMpm2 L2 1.022 1.414 2.311 2.151 GE-XFEM E 0.477 0.476 1.156 1.140 GE-XFEM L2 1.326 1.446 2.086 2.100

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Numerical examples 2D convergence study

Stress intensity factors

r = 0.00 r = 0.12 Mode I Mode II Mode I Mode II XFEM 1.071 1.005 2.195 2.021 GE-XFEM 0.759 1.246 2.545 2.029

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SLIDE 62

Numerical examples 2D convergence study

Conditioning

Condition numbers of the system matrices produced by XFEM and GE-XFEM.

11 21 41 61 81 101 10

4

10

8

10

12

n condition number FEM (slope=0.889) XFEM, re=0.00 (slope=1.019) XFEM, re=0.12 (slope=8.952) GE−XFEM, re=0.00 (slope=1.000) GE−XFEM, re=0.12 (slope=2.004)

Condition numbers of the FE part are also plotted.

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SLIDE 63

Numerical examples 3D convergence study

3D convergence study

A benchmark problem is proposed which:

◮ Includes the full solution for the whole crack. ◮ Involves variation of the SIFs along the crack front. ◮ Involves a curved crack front.

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SLIDE 64

Numerical examples 3D convergence study

3D convergence study

◮ A penny crack in an infinite solid is considered. ◮ Evaluation of L2 and energy norms is possible. ◮ An Lx × Ly × Lz parallelepiped domain with a penny crack of radius a

is used.

◮ Analytical displacements are imposed as boundary conditions. ◮ A uniform normal and shear load is applied at the crack faces.

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SLIDE 65

Numerical examples 3D convergence study

3D convergence study

x

L

y

L

z

L a

c t

Γ x y z

u

Γ

node where boundary conditions are applied ◮ Uniform normal and shear loads of magnitude 1 are applied at Γt c. ◮ Problem dimensions: Lx = Ly = 2Lz = 0.4 units and a = 0.1 unit. ◮ Material parameters: E = 100 units and ν = 0.3. ◮ Mesh consists of nx × ny × nz hexahedral elements,

nx = ny = 2nz = n and n ∈ {21, 41, 61, 81, 101}.

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Numerical examples 3D convergence study

3D convergence study

Acronyms used for the 3D convergence study

Acronym Description XFEM Standard XFEM (with shifted enrichment functions) GE-XFEM The proposed method (Global Enrichment XFEM) GE-XFEM1 The proposed method without the new enrichment strategy

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SLIDE 67

Numerical examples 3D convergence study

L2 and energy norms

Influence of the crack front mesh density in the energy (E) and L2 norms.

4 8 16 32 64 128 10

−3

10

−2

10

−1

nf error GE−XFEM, E, re=0.00 GE−XFEM, L2, re=0.00 GE−XFEM, E, re=0.02 GE−XFEM, L2, re=0.02

nf is the number of elements along the front.

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Numerical examples 3D convergence study

L2 and energy norms

Influence of the enrichment radius (re) in the energy (E) and L2 norms for the 31 × 61 × 61 mesh. 1 2 3 4 10

−3

10

−2

10

−1

re/h error GE−XFEM, E GE−XFEM, L2 GE−XFEM1, E GE−XFEM1, L2 The proposed enrichment strategy improves the behavior of the solution.

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Numerical examples 3D convergence study

L2 and energy norms

n error XFEM, E XFEM, L2 GE−XFEM1, E GE−XFEM1, L2 GE−XFEM, E GE−XFEM, L2 re=0.04

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SLIDE 70

Numerical examples 3D convergence study

L2 and energy norms

Convergence rates

re = 0.00 re = 2.2h re = 0.02 re = 0.04 XFEM E 0.492 0.686 0.911 1.015 XFEM L2 1.009 1.405 1.824 1.976 GE-XFEM1 E

1.016

0.706 GE-XFEM1 L2

1.481

0.289 GE-XFEM E 0.558 0.850 1.057 0.988 GE-XFEM L2 1.535 1.594 1.753 1.448

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SLIDE 71

Numerical examples 3D convergence study

Stress intensity factors

Mode I, II and III stress intensity factors for the 21 × 41 × 41 mesh.

10 20 30 40 50 60 70 80 90 2% 4% θ KI 10 20 30 40 50 60 70 80 90 3% 6% θ KII 10 20 30 40 50 60 70 80 90 5% 10% θ KIII 10 20 30 40 50 60 70 80 90 1.5% 3% θ KI 10 20 30 40 50 60 70 80 90 2% 4% θ KII 10 20 30 40 50 60 70 80 90 4% 8% θ KIII XFEM GE−XFEM re=0.00 re=0.02

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SLIDE 72

Numerical examples 3D convergence study

Stress intensity factors

Mode I, II and III stress intensity factors for the 41 × 81 × 81 mesh.

10 20 30 40 50 60 70 80 90 0.75% 1.5% θ KI 10 20 30 40 50 60 70 80 90 2% 4% θ KII 10 20 30 40 50 60 70 80 90 8% 16% θ KIII 10 20 30 40 50 60 70 80 90 0.75% 1.5% θ KI 10 20 30 40 50 60 70 80 90 0.75% 1.5% θ KII 10 20 30 40 50 60 70 80 90 3% 6% θ KIII XFEM GE−XFEM re=0.00 re=0.02

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SLIDE 73

Numerical examples 3D convergence study

Conditioning

◮ Conditioning of the proposed method is compared to XFEM. ◮ The number of iterations required by the solver is used as an estimate. ◮ A comparison of the time needed to solve the resulting systems of

equations is also provided.

◮ A CG solver with a diagonal preconditioner is used.

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SLIDE 74

Numerical examples 3D convergence study

Conditioning

Influence of the crack front mesh density in the number of iterations for the 31 × 61 × 61 mesh.

4 8 16 32 64 128 200 250 300 350 400 450 nf iterations GE−XFEM, re=0.00 (slope=0.021) GE−XFEM, re=0.02 (slope=0.119)

nf is the number of elements along the front.

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SLIDE 75

Numerical examples 3D convergence study

Conditioning

Number of iterations required for three different enrichment radii.

21 41 61 81 101 10

2

10

3

10

4

n iterations XFEM, re=0.00 (slope=0.045) XFEM, re=0.02 (slope=2.206) XFEM, re=0.04 (slope=2.552) GE−XFEM, re=0.00 (slope=0.547) GE−XFEM, re=0.02 (slope=0.894) GE−XFEM, re=0.04, (slope=1.248)

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Numerical examples 3D convergence study

Conditioning

Performance of the PCG solver.

400 800 10

−50

10

−40

10

−30

10

−20

10

−10

10 iterations error 400 800 1200 1600 10

−50

10

−40

10

−30

10

−20

10

−10

10 iterations error XFFEM, re=0.00 XFEM, re=0.02 GE−XFEM, re=0.00 GE−XFEM, re=0.02 21x41x41 mesh 41x81x81 mesh

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SLIDE 77

Numerical examples 3D convergence study

Number of additional dofs

Total number of enriched dofs

Mesh FE dofs XFEM dofs XFEM dofs XFEM dofs GE-XFEM (re = 0.00) (re = 0.02) (re = 0.04) dofs 11 × 21 × 21 17,424 2,232 2,232 5,856 696 21 × 41 × 41 116,424 5,376 11,904 42,288 1,920 31 × 61 × 61 369,024 9,456 37,752 137,280 4,464 41 × 81 × 81 847,224 14,424 84,696 320,664 7,512 51 × 101 × 101 1,623,024 20,376 162,528 620,184 11,544

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SLIDE 78

Conclusions

A method was introduced which:

◮ Employs point-wise and integral matching. ◮ Uses a novel enrichment strategy. ◮ Generalizes and extends the dof gathering approach to 3D. ◮ Is applicable to general 3D problems.

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SLIDE 79

Conclusions

A benchmark problem was proposed which:

◮ Involves a curved crack front. ◮ Enables the computation of L2 and energy norms for the 3D case.

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SLIDE 80

Conclusions

Advantages of the method:

◮ It improves accuracy almost in every case. ◮ Enables the application of geometrical enrichment in 3d applications. ◮ Reduces the number of additional dofs. ◮ Reduces computational cost in every case.

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Conclusions

Possible disadvantages:

◮ When the enrichment radius exceeds a certain value, the L2 norm

increases.

◮ The method is not straightforward to implement in existing XFEM

codes.

◮ The additional point wise-matching constraints are complex to

implement for higher order elements.

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SLIDE 82

Conclusions

Bibliography

Chahine, E., Laborde, P., & Renard, Y. (2011). A non-conformal eXtended Finite Element approach: Integral matching Xfem. Applied Numerical Mathematics. Fish, J. (1992). The s-version of the finite element method. Computers & Structures. Laborde, P., Pommier, J., Renard, Y., & Sala¨ un, M. (2005). High-order extended finite element method for cracked domains. International Journal for Numerical Methods in Engineering. Langlois, C., Gravouil, A., Baieto, M., & R´ ethor´ e, J. (2014). Three-dimensional simulation of crack with curved front with direct estimation of stress intensity factors. International Journal for Numerical Methods in Engineering.

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