Three-Dimensional Crack Propagation with GlobalEnrichment XFEM and Vector Level Sets

K. Agathos1 G. Ventura2 E. Chatzi3 S. P. A. Bordas4,5

1Institute of Structural Analysis and Dynamics of StructuresAristotle University Thessaloniki

2Department of Structural and Geotechnical EngineeringPolitecnico di Torino

3Institute of Structural EngineeringETH Zurich

4Research Unit in Engineering ScienceLuxembourg University

5Institute of Theoretical, Applied and Computational MechanicsCardiff University

September 9, 2015K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 1 / 29

Outline

1 Global enrichment XFEMDefinition of the Front ElementsTip enrichmentWeight function blendingDisplacement approximation

2 Vector Level SetsCrack representationLevel set functionsPoint projectionEvaluation of the level set functions

3 Numerical ExamplesEdge crack in a beamSemi circular crack in a beam

4 Conclusions5 References

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 2 / 29

Global enrichment XFEM

An XFEM variant (Agathos, Chatzi, Bordas, & Talaslidis, 2015) isintroduced which:

Enables the application of geometrical enrichment to 3D.

Extends dof gathering to 3D through global enrichment.

Employs weight function blending.

Employs enrichment function shifting.

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 3 / 29

Front elements

A superimposed mesh is used to provide a p.u. basis.

Desired properties:

Satisfaction of the partition of unity property.

Spatial variation only along the direction of the crack front.

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 4 / 29

Front elements

tip enriched elements crack front

FE mesh

front element boundaries front element node

front element

A set of nodes along the crackfront is defined.

Each element is defined by twonodes.

A good starting point for frontelement thickness is h.

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 5 / 29

Front elements

Volume corresponding to two consecutive front elements.

crack front

crack surface

boundaryfront element

Different element colors correspond to different front elements.

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 6 / 29

Front element shape functions

Linear 1D shape functions are used:

Ng (ξ) =[1− ξ

21 + ξ

2

]

where ξ is the local coordinate of the superimposed element.

Those functions:

form a partition of unity.

are used to weight tip enrichment functions.

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 7 / 29

Front element shape functions

Definition of the front element parameter used for shape functionevaluation.

ξ1x

2x

in

+1inie

mx 0x 1−=ξ5.0−=ξ

= 0ξ5.= 0ξ

= 1ξboundaryfront element

nodefront element

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 8 / 29

Tip enrichment functionsTip enrichment functions used:

Fj (x) ≡ Fj (r , θ) =[√

r sin θ2 ,√

r cos θ2 ,√

r sin θ2 sin θ,√

r cos θ2 sin θ]

Tip enriched part of the displacements:

ut (x) =∑

K∈N sNg

K (x)∑

jFj (x) cKj

where

NgK are the global shape functions

N s is the set of superimposed nodesK. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 9 / 29

Weight functionsWeight functions for a) topological (Fries, 2008) and b) geometricalenrichment (Ventura, Gracie, & Belytschko, 2009).

irer

a) b)

)x(ϕ )x(ϕ

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 10 / 29

Displacement approximation

u (x) =∑I∈N

NI (x) uI + ϕ (x)∑

J∈N j

NJ (x) (H (x)− HJ)bJ+

+ ϕ (x)

∑K∈N s

NgK (x)

∑j

Fj (x)−

−∑

T∈N tNT (x)

∑K∈N s

NgK (xT )

∑j

Fj (xT )

cKj

where:

N is the set of all nodes in the FE mesh.

N j is the set of jump enriched nodes.

N t is the set of tip enriched nodes.

N s is the set of nodes in the superimposed mesh.K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 11 / 29

Weight functionsEnrichment strategies used for tip and jump enrichment.

er

crack front

crack surface

ir

crack front

crack surface

Topological enrichment Geometrical enrichment

a) b)

Jump enriched element

Tip enriched node Tip and jump enriched node Jump enriched node

Tip enriched element Blending element

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 12 / 29

Vector Level Sets

A method for the representation of 3D cracks is introduced which:

Produces level set functions using geometric operations.

Does not require integration of evolution equations.

Similar methods:

2D Vector level sets (Ventura, Budyn, & Belytschko, 2003).

Hybrid implicit-explicit crack representation (Fries & Baydoun, 2012).

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 13 / 29

Crack front

Crack front at time t:

Ordered series of line segments ti

Set of points xi

1

2

i

i+1

1x

2x

ix

+1ix

it

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Crack front advance

Crack front at time t + 1:

Crack advance vectors sti at points xi

New set of points xt+1i = xt

i + sti

1

2

i

i+1

1x

2x

ix

+1ix

it

its

+1its +1i

+1tx

i+1tx

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 15 / 29

Crack surface advance

Crack surface advance:

Sequence of four sided bilinearsegments.

Vertexes: xti , xt

i+1, xt+1i+1 , xt+1

i

+1its

+1i+1tx

i+1tx

ix

+1ix

its

iti+1tt

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 16 / 29

Kink wedges

Discontinuities (kink wedges) are present:

Along the crack front (a).Along the advance vectors (b).

kink wedge kink wedge

crack frontadvance vector

crack frontadvance vector

crack front

crack front

a) b)

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 17 / 29

Level set functions

Definition of the level set functionsat a point P:

f distance from the crack surface.

g distance from the crack front.

g

crack front

its

+1its

fi+1ts

+1i+1ts

P

crack surface

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 18 / 29

Point projection

uv

its

+1its

i+1tx

ix

+1i+1tx

i+1tx

Element parametric equations φ (u, v), u, v ∈ [−1, 1]:

φx = g1 (u, v) x t

i + g2 (u, v) x ti+1 + g3 (u, v) x t+1

i+1 + g4 (u, v) x t+1i

φy = g1 (u, v) y ti + g2 (u, v) y t

i+1 + g3 (u, v) y t+1i+1 + g4 (u, v) y t+1

iφz = g1 (u, v) z t

i + g2 (u, v) z ti+1 + g3 (u, v) z t+1

i+1 + g4 (u, v) z t+1i

where gi (u, v), u, v ∈ [−1, 1] are linear shape functions.K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 19 / 29

Point projection

Equation of the tangent plane Π0 at (u0, v0):

det

x − φx (u0, v0) y − φy (u0, v0) z − φz (u0, v0)φx ,u (u0, v0) φy ,u (u0, v0) φz,u (u0, v0)φx ,v (u0, v0) φy ,v (u0, v0) φz,v (u0, v0)

= 0

Normal vector to the parametric surface at (u0, v0):

n (u0, v0) = (A,B,C)

where A,B,C are the minors of the previous matrix at (u0, v0).

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 20 / 29

Point projection

Point P can be expressed as:

P = P′ (u, v) + λn (u, v)

where:

P′ the projection of the point to the surface.λ unknown parameter.

The above is solved for u, v and λ to obtain the projection.

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 21 / 29

Evaluation of the level set functions

At each step t:

For each point all crack advance segments are tested.

If for a certain element u, v ∈ [−1, 1] then the point is projected onthat element.

If u /∈ [−1, 1] for all elements then the projection lies on the advancevector.

If v /∈ [−1, 1] for all elements then the projection lies either:→ at a previous crack advance segment→ at the crack front at time t − 1 or t

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 22 / 29

Evaluation of the level set functions

Level set function f:

f = P− P′

where P′ is either:

Projection to an element of the crack surfaceClosest point projection to a kink wedge

Level set function g:

g = P− P′

where P′ is a closest point projection to the crack front

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 23 / 29

Edge crack in a beam

L

H

d

aα

Geometry:

L = 1 unitH = 0.2 unitsd = 0.1 unitsa = 0.05 unitsα = 45◦

Mesh:

Far from the crackh = 0.02 unitsIn the vicinity of thecrack h = 0.005 units

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 24 / 29

Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29

Edge crack in a beam

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29

Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

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Edge crack in a beam

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29

Semi circular crack in a beam

L

H

da

α

Geometry:

L = 1 unitH = 0.2 unitsd = 0.1 unitsa = 0.025 unitsα = 45◦

Mesh:

Far from the crackh = 0.02 unitsIn the vicinity of thecrack h = 0.005 units

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Semi circular crack in a beam

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Conclusions

A method for 3D fracture mechanics was presented which:

Enables the use of geometrical enrichment in 3D.Eliminates blending errors.

A method for the representation of 3D cracks was presented which:

Avoids the solution of evolution equations.Utilizes only simple geometrical operations.

The methods were combined to solve 3D crack propagation problems.

K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 28 / 29

Bibliography

Agathos, K., Chatzi, E., Bordas, S., & Talaslidis, D. (2015). Awell-conditioned and optimally convergent xfem for 3d linear elasticfracture. International Journal for Numerical Methods inEngineering.

Fries, T. (2008). A corrected XFEM approximation without problems inblending elements. International Journal for Numerical Methods inEngineering.

Fries, T., & Baydoun, M. (2012). Crack propagation with the extendedfinite element method and a hybrid explicit-implicit crack description.International Journal for Numerical Methods in Engineering.

Ventura, G., Budyn, E., & Belytschko, T. (2003). Vector level sets fordescription of propagating cracks in finite elements. InternationalJournal for Numerical Methods in Engineering.

Ventura, G., Gracie, R., & Belytschko, T. (2009). Fast integration andweight function blending in the extended finite element method.International journal for numerical methods in engineering.K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 29 / 29