The XFEM using Crack Tip Enrichment with Large Support for ... · XFEM with Large Support for...
Transcript of The XFEM using Crack Tip Enrichment with Large Support for ... · XFEM with Large Support for...
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
The XFEM using Crack Tip Enrichment withLarge Support for Curved Cracks
Malak Baydoun and Thomas Peter Fries
AACHEN INSTITUTE FOR ADVANCED STUDYIN COMPUTATIONAL ENGINEERING SCIENCE
ECCM 2010, Paris
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XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Outline
1 Motivation
2 New Alternative
3 Other Alternatives
4 Studies
5 Conclusions
2
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
eXtended Finite Element Formulation
u(x,y) = ∑i∈I
Ni(x,y)ui︸ ︷︷ ︸Continuous
+ ∑j∈I?1
N?j (x,y) ·H(x,y)aj + ∑
k∈I?2
N?k (x,y) ·
(4
∑m=1
Bmbmk
)︸ ︷︷ ︸
Discontinuous
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t
!
t
!
t
3
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
eXtended Finite Element Formulation
u(x,y) = ∑i∈I
Ni(x,y)ui︸ ︷︷ ︸Continuous
+ ∑j∈I?1
N?j (x,y) ·H(x,y)aj + ∑
k∈I?2
N?k (x,y) ·
(4
∑m=1
Bmbmk
)︸ ︷︷ ︸
Discontinuous
!
t
!
t
!
t
!
t
!
t
!
t
3
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
eXtended Finite Element Formulation
u(x,y) = ∑i∈I
Ni(x,y)ui︸ ︷︷ ︸Continuous
+ ∑j∈I?1
N?j (x,y) ·H(x,y)aj + ∑
k∈I?2
N?k (x,y) ·
(4
∑m=1
Bmbmk
)︸ ︷︷ ︸
Discontinuous
!
t
!
t
!
t
!
t
!
t
!
t
• In XFEM, Optimal Convergence Rates with Fixed Radius forBranch Enrichments are achieved. [LABORDE ET AL.]
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XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Coordinate SystemsFor Curved Cracks/Crack Propagation, Different CoordinateSystems to evaluate the SIFs and/or Enrichments exist:
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Alternative 0• Easy to evaluate by Crack Tip
Information only.• Discontinuity follows a Straight Path.• Not Suitable for “Large” Radius
Enrichment.
4
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Coordinate SystemsFor Curved Cracks/Crack Propagation, Different CoordinateSystems to evaluate the SIFs and/or Enrichments exist:
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Alternative 0• Easy to evaluate by Crack Tip
Information only.• Discontinuity follows a Straight Path.• Not Suitable for “Large” Radius
Enrichment.
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Alternative 1• Discontinuity follows Curved Path.
• Bases e1 and e2.
• Drawbacks.
4
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
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5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
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5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
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5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
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5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
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5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Drawbacks of Alternative 1• Two Level Set Functions: φ and γ.
• ∀p ∈Ω: Two Signed Values and Bases.
• Bases are no longer Orthogonal.
• Inconvenient values away from the Tip.
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5
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Geometrical Reconstruction.
• Split the Domain into Triangles.
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6
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Geometrical Reconstruction.
• Split the Domain into Triangles.
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6
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Geometrical Reconstruction.
• Split the Domain into Triangles.
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6
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Geometrical Reconstruction.
• Split the Domain into Triangles.
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#!""$
#!"% '(#!"% &'(#!"% #'(
6
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Geometrical Reconstruction.
• Split the Domain into Triangles.
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#!"% '(#!"% &'(#!"% #'(
6
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• ∀p ∈ Triangle.• Interpolate Limiter Level Sets γ.• Interpolate Signed Distance for point p.• Find Signed Distance φ .
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7
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• ∀p ∈ Triangle.• Interpolate Limiter Level Sets γ.• Interpolate Signed Distance for point p.• Find Signed Distance φ .
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7
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• ∀p ∈ Triangle.• Interpolate Limiter Level Sets γ.• Interpolate Signed Distance for point p.• Find Signed Distance φ .
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!!$ %&
7
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• ∀p ∈ Triangle.• Interpolate Limiter Level Sets γ.• Interpolate Signed Distance for point p.• Find Signed Distance φ .
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7
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• ∀p ∈ Triangle.• Interpolate Limiter Level Sets γ.• Interpolate Signed Distance for point p.• Find Signed Distance φ .
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7
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• ∀p ∈ Triangle.• Interpolate Limiter Level Sets γ.• Interpolate Signed Distance for point p.• Find Signed Distance φ .
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7
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Convenient Values γ.
• Bases are almost Orthogonal.
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8
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Convenient Values γ.
• Bases are almost Orthogonal.
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8
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Convenient Values γ.
• Bases are almost Orthogonal.
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8
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Convenient Values γ.
• Bases are almost Orthogonal.
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8
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Convenient Values γ.
• Bases are almost Orthogonal.
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8
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Alternative 2• Convenient Values γ.
• Bases are almost Orthogonal.
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8
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Other Alternatives
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Alternative 2– a• Applicable if Radius includes
more than One Increment.
9
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Other Alternatives
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Alternative 2– b• Fits Straight Cracks and Small
Angle Increments: Quadrilateral.
9
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Other Alternatives
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Alternative 2– b• Fits Straight Cracks and Small
Angle Increments: Quadrilateral.
9
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Other Alternatives
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Alternative 2– b• Fits Straight Cracks and Small
Angle Increments: Quadrilateral.
9
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Other Alternatives
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Alternative 2– b• Fits Straight Cracks and Small
Angle Increments: Quadrilateral.
9
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Other Alternatives
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more than One Increment.
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Angle Increments: Quadrilateral.
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Alternative 2– c• Mid Angle of Orthogonal Limiter
Level Sets.
9
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Other Alternatives
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more than One Increment.
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Alternative 2– b• Fits Straight Cracks and Small
Angle Increments: Quadrilateral.
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#!"% &'#!"% (&'#!"% #&'
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Alternative 2– c• Mid Angle of Orthogonal Limiter
Level Sets.
9
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Signed Distance
0
2
4
6
01
23
45
67
−4
−3
−2
−1
0
1
2
3
4
Alternative 1• Signed Distance is Tangent to the
Crack at the Tip.
10
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Signed Distance
0
2
4
6
01
23
45
67
−4
−3
−2
−1
0
1
2
3
4
Alternative 1• Signed Distance is Tangent to the
Crack at the Tip.
0
2
4
6
02
46
−4
−3
−2
−1
0
1
2
3
4
5
6
Alternative 2–b• Signed Distance is Tangent to the
Crack Path.
10
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Signed Distance
0
2
4
6
01
23
45
67
−4
−3
−2
−1
0
1
2
3
4
Alternative 1• Signed Distance is Tangent to the
Crack at the Tip.
0
2
4
6
01234567−4
−3
−2
−1
0
1
2
3
4
5
6
Alternative 2–b• Signed Distance is Tangent to the
Crack Path.
10
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Derivatives of Level SetsDerivatives of Level Sets at the Integration points are Required.
• Option 1:1 Find the Signed Distance at the Integration Points.
2 Not easy to evaluate the Derivatives.
• Option 2:1 Find the Signed Distance at the Nodes.
2 Evaluate the Derivatives by using the Shape Functions.
• Comparing both Options for ∇xφ :
11
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Derivatives of Level SetsDerivatives of Level Sets at the Integration points are Required.
• Option 1:1 Find the Signed Distance at the Integration Points.
2 Not easy to evaluate the Derivatives.
• Option 2:1 Find the Signed Distance at the Nodes.
2 Evaluate the Derivatives by using the Shape Functions.
• Comparing both Options for ∇xφ :
11
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Derivatives of Level SetsDerivatives of Level Sets at the Integration points are Required.
• Option 1:1 Find the Signed Distance at the Integration Points.
2 Not easy to evaluate the Derivatives.
• Option 2:1 Find the Signed Distance at the Nodes.
2 Evaluate the Derivatives by using the Shape Functions.
• Comparing both Options for ∇xφ :
11
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Radius Enrichment
0 1 2 3 4 5 6
−2
−1
0
1
2
Alternative 0• Enrichment does not conform to
the Discontinuity.
12
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Radius Enrichment
0 1 2 3 4 5 6
−2
−1
0
1
2
Alternative 0• Enrichment does not conform to
the Discontinuity.
0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
2
Alternative 1• Enrichment conforms to the
Discontinuity under someRestrictions away from the Tip.
12
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Radius Enrichment
0 1 2 3 4 5 6
−2
−1
0
1
2
Alternative 0• Enrichment does not conform to
the Discontinuity.
0 1 2 3 4 5 6
−1.5
−1
−0.5
0
0.5
1
1.5
2
Alternative 1• Enrichment conforms to the
Discontinuity under someRestrictions away from the Tip.
0 1 2 3 4 5 6
−2
−1
0
1
2
Alternative 2–b• Enrichment conforms to the
Discontinuity.
12
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Stress Intensity Factors• Inclined Center Crack: β = 40.
around the crack tip, where 9! 9 Gauss quadrature wasused. Quadratic XFEM seems to perform as well as the EFGmethod. While the EFG method tends to overestimate thestress intensity factors, quadratic XFEM tends to under-estimate the result.
6.3Edge crack under shear stressA plate is clamped on the bottom and loaded by a sheartraction s " 1:0 psi over the top edge. The material pa-rameters are 3! 107 psi for Young’s modulus and 0.25 forPoisson’s ratio. The reference mixed mode stress intensityfactors as given in [35] and [31] are:
KI " 34:0 psi!!!!!
inp
KII " 4:55 psi!!!!!
inp
The equivalent stress intensity factor Keq obtained fromthe J-integral for plain strain problem is:
J " 1# m2
EK2eq $46%
The equivalent stress intensity factor is compared to!!!!!!!!!!!!!!!!!!!!!
$K2I & K2
II%p
using an elliptic criterion described in Bazant[1]. In Fig. 17, we can see that the quadratic elementconverges slightly faster that the linear element and ismore accurate. In Fig. 18, KI and KII is seen to exhibit thesame behavior.
6.4Mixed mode crack in infinite bodyThe problem of an angled center crack in a body wasconsidered as shown in Fig. 19. The plate is subjected to afar-field state of stress r equal to unity. The crack is oflength 2a and is oriented with an angle b with respect tothe x-axis. The material parameters are 3! 107 psi forYoung’s modulus and 0.25 for Poisson’s ratio. The stressintensity factors KI and KII are given in terms of the angleb by Yau et al. [35] and Dolbow et al. [11].
KI " r!!!!!!!!!
$pa%p
cos2$b% $47%
KII " r!!!!!!!!!
$pa%p
sin$b% cos$b% $48%where a is the half crack length. For the computations bwas chosen to be 41:9872'.
Figure 20 shows the convergence for KI and KII .Good accuracy is obtained for a reasonable number ofnodes. The stress intensity factors are computed by an
Table 1. Stress intensity factors computed by quadratic XFEMcompared to EFG
Cracklength
KI XFEM(linear)
KI XFEM(quadratic)
KI EFG byBelytschkoet al. [7]
KI exact
0.21 1.0616 1.1243 1.1401 1.13410.22 1.1000 1.1691 1.1779 1.18160.23 1.1321 1.2187 1.2487 1.23030.24 1.1558 1.2707 1.2807 1.27880.28 1.3783 1.4760 1.5036 1.49350.50 3.1299 3.5064 3.5512 3.5423
Fig. 17. Convergence for edge crack under shear. K is computedfrom the J-integral
Fig. 18. Convergence for edgecrack under shear. KI and KII
computed by the interactionintegral
Fig. 19. Discretization used for angled crack in a plate underuniaxial tension
45
Alternative 2–b• SIFs Match.
13
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Stress Intensity Factors• Inclined Center Crack: β = 40.
around the crack tip, where 9! 9 Gauss quadrature wasused. Quadratic XFEM seems to perform as well as the EFGmethod. While the EFG method tends to overestimate thestress intensity factors, quadratic XFEM tends to under-estimate the result.
6.3Edge crack under shear stressA plate is clamped on the bottom and loaded by a sheartraction s " 1:0 psi over the top edge. The material pa-rameters are 3! 107 psi for Young’s modulus and 0.25 forPoisson’s ratio. The reference mixed mode stress intensityfactors as given in [35] and [31] are:
KI " 34:0 psi!!!!!
inp
KII " 4:55 psi!!!!!
inp
The equivalent stress intensity factor Keq obtained fromthe J-integral for plain strain problem is:
J " 1# m2
EK2eq $46%
The equivalent stress intensity factor is compared to!!!!!!!!!!!!!!!!!!!!!
$K2I & K2
II%p
using an elliptic criterion described in Bazant[1]. In Fig. 17, we can see that the quadratic elementconverges slightly faster that the linear element and ismore accurate. In Fig. 18, KI and KII is seen to exhibit thesame behavior.
6.4Mixed mode crack in infinite bodyThe problem of an angled center crack in a body wasconsidered as shown in Fig. 19. The plate is subjected to afar-field state of stress r equal to unity. The crack is oflength 2a and is oriented with an angle b with respect tothe x-axis. The material parameters are 3! 107 psi forYoung’s modulus and 0.25 for Poisson’s ratio. The stressintensity factors KI and KII are given in terms of the angleb by Yau et al. [35] and Dolbow et al. [11].
KI " r!!!!!!!!!
$pa%p
cos2$b% $47%
KII " r!!!!!!!!!
$pa%p
sin$b% cos$b% $48%where a is the half crack length. For the computations bwas chosen to be 41:9872'.
Figure 20 shows the convergence for KI and KII .Good accuracy is obtained for a reasonable number ofnodes. The stress intensity factors are computed by an
Table 1. Stress intensity factors computed by quadratic XFEMcompared to EFG
Cracklength
KI XFEM(linear)
KI XFEM(quadratic)
KI EFG byBelytschkoet al. [7]
KI exact
0.21 1.0616 1.1243 1.1401 1.13410.22 1.1000 1.1691 1.1779 1.18160.23 1.1321 1.2187 1.2487 1.23030.24 1.1558 1.2707 1.2807 1.27880.28 1.3783 1.4760 1.5036 1.49350.50 3.1299 3.5064 3.5512 3.5423
Fig. 17. Convergence for edge crack under shear. K is computedfrom the J-integral
Fig. 18. Convergence for edgecrack under shear. KI and KII
computed by the interactionintegral
Fig. 19. Discretization used for angled crack in a plate underuniaxial tension
45
Alternative 2–b• SIFs Match.
13
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Stress Intensity Factors• Curved Center Crack: β = 30
interaction integral as described in Sect. 5. For coarsediscretization, the values for KI are more accurate thanthe values for KII . This is probably due to the fact thatthe computations were made in a finite body. If a largermodel relative to the crack length were used this differ-ence would have been less noticeable. The results givenin Table 2 also show very good symmetry in the behaviorat the two tips.
6.5Center crack in a finite plateThe problem of a finite plate with a center crack wasstudied [10]. The geometry of the plate is described inFig. 21. The analytical solution to this problem is given inSuo and Combescure [10]. The stress intensity factor isgiven by:
KI ! r
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
pa secpa2w
" #" #
r
"49#
where a is the half crack-length and w ! W=2 is the halfwidth of the plate, and r is the tensile load applied at thetop of the plate.
Figure 22 shows the improved accuracy and conver-gence of the quadratic element over the linear element.Again, we can observe symmetric behavior at the twocrack tips.
Fig. 20. Stess intensity factorserror for the angled crack ininfinite plate. KI (left) and KII
(right) computed by the inter-action integral; ‘‘tip 1’’ and ‘‘tip2’’ refer to the two crack tips
Table 2. Stress intensity factors for angled center crack byquadratic elements
NumNodes
KIKanaI
tip 1 KIKanaI
tip 2 KIIKanaII
tip 1 KIIKanaII
tip 2 Mesh
1661 0.6619 0.6647 0.2207 0.2205 11 · 212377 0.6916 0.6940 0.7279 0.7278 13 · 253449 1.0464 1.0491 1.1171 1.1172 15 · 314193 1.0260 1.0288 1.0670 1.0673 17 · 335293 1.0224 1.0251 1.0512 1.0515 19 · 37
Fig. 21. Finite plate containing a centered crack
Fig. 22. Stress intensity factor error for a centered crack in afinite plate
Fig. 23. Curved crack in an infinite plate
46
• SIFs Convergence.
14
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Stress Intensity Factors• Curved Center Crack: β = 30
interaction integral as described in Sect. 5. For coarsediscretization, the values for KI are more accurate thanthe values for KII . This is probably due to the fact thatthe computations were made in a finite body. If a largermodel relative to the crack length were used this differ-ence would have been less noticeable. The results givenin Table 2 also show very good symmetry in the behaviorat the two tips.
6.5Center crack in a finite plateThe problem of a finite plate with a center crack wasstudied [10]. The geometry of the plate is described inFig. 21. The analytical solution to this problem is given inSuo and Combescure [10]. The stress intensity factor isgiven by:
KI ! r
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
pa secpa2w
" #" #
r
"49#
where a is the half crack-length and w ! W=2 is the halfwidth of the plate, and r is the tensile load applied at thetop of the plate.
Figure 22 shows the improved accuracy and conver-gence of the quadratic element over the linear element.Again, we can observe symmetric behavior at the twocrack tips.
Fig. 20. Stess intensity factorserror for the angled crack ininfinite plate. KI (left) and KII
(right) computed by the inter-action integral; ‘‘tip 1’’ and ‘‘tip2’’ refer to the two crack tips
Table 2. Stress intensity factors for angled center crack byquadratic elements
NumNodes
KIKanaI
tip 1 KIKanaI
tip 2 KIIKanaII
tip 1 KIIKanaII
tip 2 Mesh
1661 0.6619 0.6647 0.2207 0.2205 11 · 212377 0.6916 0.6940 0.7279 0.7278 13 · 253449 1.0464 1.0491 1.1171 1.1172 15 · 314193 1.0260 1.0288 1.0670 1.0673 17 · 335293 1.0224 1.0251 1.0512 1.0515 19 · 37
Fig. 21. Finite plate containing a centered crack
Fig. 22. Stress intensity factor error for a centered crack in afinite plate
Fig. 23. Curved crack in an infinite plate
46
Alternative 2–b Alternative 2–c
• SIFs Convergence.
14
XFEM with LargeSupport for
Curved Cracks
M. BaydounT.P. Fries
Motivation
New Alternative
OtherAlternatives
StudiesSigned Distance
Derivatives of LevelSets
Radius Enrichment
Stress IntensityFactors
Conclusions
Conclusions
• New Bases System (φ ,γ): Triangles and a Quadrilateral.
• Improved Definition of γ.
• Improved Stress Intensity Factors Convergence Rates.
• Improved Evaluation of Enrichment Functions.
• Flexibility and Multiplicity of Alternatives.
15