new hybrid explicit implicit description of cracks in xfem 0 · 2011. 7. 5. · hybrid...

Hybrid Explicit –Implicit Description ofCracks in XFEM

Malak Baydoun & Thomas P. Fries

AACHEN INSTITUTE FOR ADVANCED STUDYIN COMPUTATIONAL ENGINEERING SCIENCE

WCCM/APCOM 2010Sydney, 19th July

Outline

Motivation

Hybrid Explicit–Implicit description of cracksExplicit description of the crackExplicit–Implicit descriptionExplicit/Geometric updateStudySummary

Conclusions and Outlook

1

Motivation eXtended Finite Element Method for cracks

eXtended Finite Element formulation for cracksu(x) = ∑

i∈INi(x)ui︸︷︷︸

Continuous

+ ∑j∈Icut

N?j (x) ·H(x)aj + ∑

k∈Ibranch

N?k (x) ·

(4

∑m=1

Bmbmk

)︸︷︷︸

Discontinuous

!

t

4

!

t

1

2

Motivation eXtended Finite Element Method for cracks

eXtended Finite Element formulation for cracksu(x) = ∑

i∈INi(x)ui︸︷︷︸

Continuous

+ ∑j∈Icut

N?j (x) ·H(x)aj + ∑

k∈Ibranch

N?k (x) ·

(4

∑m=1

Bmbmk

)︸︷︷︸

Discontinuous

!

t

4

!

t

1

• Branch enrichment functions: B = f (r,θ).

• Stress Intensity Factors’ evaluation:

Interaction integral: add the XFEM fields and auxiliary fields from puremodes of fracture–AuxiliaryFields = g(r,θ).

2

Motivation Background & Goals

Background & Goals : XFEM for crack modelingCrack surface can be described explicitly or implicitly.

3

Motivation Background & Goals

Background & Goals : XFEM for crack modelingCrack surface can be described explicitly or implicitly.

Explicit Implicit

Enriched nodes/cut elements – +

Polar coordinates(enrichments, SIFs) – +

Crack propagation + –

3

Hybrid Explicit–Implicit description of cracks Explicit description of the crack 2D

Explicit description of the crack

• Polyline crack representation constructed by segments.• Independent of the main mesh.

4

Hybrid Explicit–Implicit description of cracks Explicit–Implicit description 2D

Explicit–Implicit description• Build level sets based on the explicit crack.

5


Explicit–Implicit description• Build level sets based on the explicit crack.• Treat each segment separately.

• φ1 =

{shortest distance to the segment in the segment’s domainshortest distance to the tips outside the segment’s domain.

• φ1 : signed distance.

5


Explicit–Implicit description• Build level sets based on the explicit crack.• φ1 : signed distance.• φ2 : distance to the last segment tip.

5


Explicit–Implicit description• Build level sets based on the explicit crack.• φ1 : signed distance.• φ2 : distance to the last segment tip.• ‖φ1‖ = ‖φ2‖ : away from the crack.

5


Explicit–Implicit description• Evaluation of the polar coordinates (r,θ).

6


Explicit–Implicit description• Evaluation of the polar coordinates (r,θ).• r = φ2.

6


Explicit–Implicit description• Evaluation of the polar coordinates (r,θ).• r = φ2.• φ ext

1 = φ1 before the tip.

6


Explicit–Implicit description• Evaluation of the polar coordinates (r,θ).• r = φ2.• φ ext

1 = φ1 before the tip.• φ ext

1 : shortest distance to the extension of last segment.

6



• θ =

π− sin−1 φ ext1φ2

before the tip

sin−1 φ ext1φ2

after the tip

6

Hybrid Explicit–Implicit description of cracks Explicit/Geometric update 2D

Explicit/Geometric update of the crack• Old line mesh.

7


Explicit/Geometric update of the crack• Old line mesh.• Propagation angle from the SIFs.

7


Explicit/Geometric update of the crack• Old line mesh.• Propagation angle from the SIFs.• Add segment/increment to old line mesh.

7

Hybrid Explicit–Implicit description of cracks Study 2D

Crack under Bending Areais & Belytschko [2005]• Asymmetric bending.

8

Hybrid Explicit–Implicit description of cracks Summary 2D

Summary: 2DImplicit crack description

• Crack described implicitly by two level sets.

Hybrid explicit –implicit crack description

• Crack described explicitly independent of the mesh.

9





• Crack described explicitly independent of the mesh.• Included in the domain by level sets.

9





• Crack described explicitly independent of the mesh.• Included in the domain by level sets.• φ1 and φ2 to locate enriched nodes.

9





• Crack described explicitly independent of the mesh.• Included in the domain by level sets.• φ1 and φ2 to locate enriched nodes.• φ ext

1 and φ2 to evaluate (r,θ).

9


Summary: 2D –Update crack description in propagationImplicit crack description

• Crack described implicitly by two level sets.• Geometric update of level sets.




9



• Crack described implicitly by two level sets.• Geometric update of level sets.• Level set update using transport equations.




9



• Crack described implicitly by two level sets.• Geometric update of level sets.• Level set update using transport equations.



1 and φ2 to evaluate (r,θ).• Geometric update of the crack

9



• Planar straight crack.

10



• Planar straight crack.• Surface mesh crack representation constructed by triangles.• Independent of the main mesh.

10



• Planar straight crack.• Surface mesh crack representation constructed by triangles.• Independent of the main mesh.• Flexible mesh depending of purpose.

10



• Complex crack representation.• Non–planar crack.

10



• Complex crack representation.• Non–planar crack.• Surface mesh crack representation constructed by triangles.• Flexible mesh depending of purpose.

10


Explicit–Implicit description• Build level sets based on the explicit crack.

11


Explicit–Implicit description• Build level sets based on the explicit crack.• Treat each triangle/patch separately.

• φ1 =

{shortest distance inside the domain of the patch.shortest distance to the edges outside the domain of the patch.

• φ1: signed distance.

11


Explicit–Implicit description• Build level sets based on the explicit crack.• φ1: signed distance.• Treat each front segment separately.• φ2: distance to the crack front segment.

11


Explicit–Implicit description• Build level sets based on the explicit crack.• φ1: signed distance.• φ2: distance to the crack front segment.

11


Explicit–Implicit description• Build level sets based on the explicit crack.• φ1: signed distance.• φ2: distance to the crack front segment.• ‖φ1‖ = ‖φ2‖ : away from the crack.

11


Explicit–Implicit description• Evaluation of the polar coordinates (r,θ).

12



12


Explicit–Implicit description• Evaluation of the polar coordinates (r,θ).• r = φ2.• Virtually extend my crack till the end of the domain.

12


Explicit–Implicit description• Evaluation of the polar coordinates (r,θ).• r = φ2.• Virtually extend my crack till the end of the domain.• φ ext

1 = φ1 before the crack front.• φ ext

1 : shortest distance to the extension of front patches.

12



• θ =

π− sin−1 φ ext1φ2

before the front

sin−1 φ ext1φ2

after the front

12


Explicit/Geometric update of the crack• Old triangular mesh.

13


Explicit/Geometric update of the crack• Old triangular mesh.• Propagation angles from the SIFs at the front.

13


Explicit/Geometric update of the crack• Old triangular mesh.• Propagation angles from the SIFs at the front.• Extend tips.

13


Explicit/Geometric update of the crack• Old triangular mesh.• Propagation angles from the SIFs at the front.• Extend tips.• Add triangles to old triangular mesh.

13

Hybrid Explicit–Implicit description of cracks Study 3D

Crack under Bending• Symmetric bending.

14





• Crack described explicitly independent of the mesh.

15





• Crack described explicitly independent of the mesh.• Convenient triangular mesh.

15





• Crack described explicitly independent of the mesh.• Convenient triangular mesh.• Included in the domain by level sets.

15





• Crack described explicitly independent of the mesh.• Convenient triangular mesh.• Included in the domain by level sets.• φ1 and φ2 to locate enriched nodes.

15





• Crack described explicitly independent of the mesh.• Convenient triangular mesh.• Included in the domain by level sets.• φ1 and φ2 to locate enriched nodes.• φ ext


15



• Crack described implicitly by two level sets.• Level set update using transport equations.




15



• Crack described implicitly by two level sets.• Level set update using transport equations.



1 and φ2 to evaluate (r,θ).• Geometric update of the crack

15


Conclusions

• Explicit/Geometric crack description.• Crack is independent of the main mesh.• Crack can be meshed flexibly based on shape.• Hybrid explicit–implicit crack description by building level sets.• New definition of level set functions.• Avoid transport equations.• Flexibility in propagation.

Outlook• Robust scheme for 3D crack propagation.

16


THANK YOU FOR YOUR ATTENTION !

17

new hybrid explicit implicit description of cracks in xfem 0 · 2011. 7. 5. · hybrid...

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