Three-Dimensional Crack Propagation with Global Enrichment XFEM and - - PowerPoint PPT Presentation

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Three-Dimensional Crack Propagation with Global Enrichment XFEM and Vector Level Sets K. Agathos 1 G. Ventura 2 E. Chatzi 3 S. P. A. Bordas 4 , 5 1 Institute of Structural Analysis and Dynamics of Structures Aristotle University Thessaloniki 2

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Three-Dimensional Crack Propagation with Global Enrichment XFEM and Vector Level Sets

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

1Institute of Structural Analysis and Dynamics of Structures

Aristotle University Thessaloniki

2Department of Structural and Geotechnical Engineering

Politecnico di Torino

3Institute of Structural Engineering

ETH Z¨ urich

4Research Unit in Engineering Science

Luxembourg University

5Institute of Theoretical, Applied and Computational Mechanics

Cardiff University

September 9, 2015

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Outline

1

Global enrichment XFEM Definition of the Front Elements Tip enrichment Weight function blending Displacement approximation

2

Vector Level Sets Crack representation Level set functions Point projection Evaluation of the level set functions

3

Numerical Examples Edge crack in a beam Semi circular crack in a beam

4

Conclusions

5

References

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Global enrichment XFEM

An XFEM variant (Agathos, Chatzi, Bordas, & Talaslidis, 2015) is introduced 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.

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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.

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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. Each element is defined by two nodes. A good starting point for front element thickness is h.

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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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Front element shape functions

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

1 − ξ

2 1 + 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.

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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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Tip enrichment functions

Tip enrichment functions used: 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

  • Tip enriched part of the displacements:

ut (x) =

  • K∈N s

Ng

K (x)

  • j

Fj (x) cKj where Ng

K are the global shape functions

N s is the set of superimposed nodes

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Weight functions

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

i

r

e

r a) b) ) x ( ϕ ) x ( ϕ

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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

Ng

K (x)

  • j

Fj (x) − −

  • T∈N t

NT (x)

  • K∈N s

Ng

K (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.

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Weight functions

Enrichment strategies used for tip and jump enrichment.

e

r crack front crack surface

i

r

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

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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).

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Crack front

Crack front at time t: Ordered series of line segments ti Set of points xi

1 2 i i+1

1

x

2

x

i

x

+1 i

x

i

t

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Crack front advance

Crack front at time t + 1: Crack advance vectors st

i at points xi

New set of points xt+1

i

= xt

i + st i 1 2 i i+1

1

x

2

x

i

x

+1 i

x

i

t

i t

s

+1 i t

s

+1 i +1 t

x

i +1 t

x

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Crack surface advance

Crack surface advance: Sequence of four sided bilinear segments. Vertexes: xt

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

+1 i t

s

+1 i +1 t

x

i +1 t

x

i

x

+1 i

x

i t

s

i

t

i +1 t

t

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Kink wedges

Discontinuities (kink wedges) are present: Along the crack front (a). Along the advance vectors (b).

kink wedge kink wedge crack front advance vector crack front advance vector crack front crack front a) b)

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Level set functions

Definition of the level set functions at a point P: f distance from the crack surface. g distance from the crack front.

g crack front

i t

s

+1 i t

s f

i +1 t

s

+1 i +1 t

s P crack surface

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Point projection

u v

i t

s

+1 i t

s

i +1 t

x

i

x

+1 i +1 t

x

i +1 t

x

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

    

φx = g1 (u, v) xt

i + g2 (u, v) xt i+1 + g3 (u, v) xt+1 i+1 + g4 (u, v) xt+1 i

φy = g1 (u, v) yt

i + g2 (u, v) yt i+1 + g3 (u, v) yt+1 i+1 + g4 (u, v) yt+1 i

φz = g1 (u, v) zt

i + g2 (u, v) zt i+1 + g3 (u, v) zt+1 i+1 + g4 (u, v) zt+1 i

where gi (u, v), u, v ∈ [−1, 1] are linear shape functions.

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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).

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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.

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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 on that element. If u / ∈ [−1, 1] for all elements then the projection lies on the advance vector. 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

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Evaluation of the level set functions

Level set function f: f = P − P′ where P′ is either: Projection to an element of the crack surface Closest point projection to a kink wedge Level set function g: g = P − P′ where P′ is a closest point projection to the crack front

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Edge crack in a beam

L H d a α

Geometry: L = 1 unit H = 0.2 units d = 0.1 units a = 0.05 units α = 45◦ Mesh: Far from the crack h = 0.02 units In the vicinity of the crack h = 0.005 units

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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

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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

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Semi circular crack in a beam

L H d

a

α

Geometry: L = 1 unit H = 0.2 units d = 0.1 units a = 0.025 units α = 45◦ Mesh: Far from the crack h = 0.02 units In the vicinity of the crack 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.

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Bibliography

Agathos, K., Chatzi, E., Bordas, S., & Talaslidis, D. (2015). A well-conditioned and optimally convergent xfem for 3d linear elastic

  • fracture. International Journal for Numerical Methods in

Engineering. Fries, T. (2008). A corrected XFEM approximation without problems in blending elements. International Journal for Numerical Methods in Engineering. Fries, T., & Baydoun, M. (2012). Crack propagation with the extended finite 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 for description of propagating cracks in finite elements. International Journal for Numerical Methods in Engineering. Ventura, G., Gracie, R., & Belytschko, T. (2009). Fast integration and weight function blending in the extended finite element method. International journal for numerical methods in engineering.

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