Extended Finite Element Method

Topic:

How to Determine Displacement and Stress of Cracked Plate
by Extended Finite Element Method
(XFEM)
OUTLINE
INTRODUCTION
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BASIS THEORY
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CALCULATION EXAMPLES
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CONCLUSIONS
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REFERENCES
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1. INTRODUCTION
Cracked structure
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1. INTRODUCTION
How to calculate when cracks appear in structure?
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1. INTRODUCTION
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Some models for simulation of a crack problem
a) Smeared crack element
b) Discrete crack element
c) Singular element
d) Enriched element

1. INTRODUCTION
Consider a tension plate
fixed
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Crack
Enriched element

2. BASIS THEORY
Consider a tension plate
fixed
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2. BASIS THEORY
If the plate has no crack => use finite element method (FEM) to calculate
(1)
Consider a tension plate
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2. BASIS THEORY
Assume a plate has a small crack => add tip enrichment elements
(1)
(2)
FEM + enrichment functions => XFEM
Tip enrichment node
Standard node
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Crack

2. BASIS THEORY
If a plate has a large crack => add tip enrichment + edge enrichment elements
FEM + tip enrichment and edge enrichment => XFEM
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Crack

2. BASIS THEORY
Enrichment functions in XFEM [1]
+ Heaviside function of edge element:
level set function [1]
With
+ Crack tip enrichment functions:
B­αk = [B1, B2, B3, B4]
With:
polar coordinates at the tip of the crack
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2. BASIS THEORY
Enrichment functions in XFEM [1]
+ Crack tip enrichment functions:
B­αk = [B1, B2, B3, B4]
With:
polar coordinates at the tip of the crack
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2. BASIS THEORY
Enrichment functions in XFEM [1]
+ Heaviside function of edge element:
level set function [1]
With
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2. BASIS THEORY
Displacement equation in XFEM for element “e”
In which, the global stiffness matrix ( ) in XFEM is defined as follows:
u: standard element,
a: Edge enrichment element,
b: Tip enrichment element
(1)
(2)
(3)
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(4)
(5)

2. BASIS THEORY
If the plate has no crack => use finite element method (FEM) to calculate
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Displacement equation in XFEM for element “e”
(1)
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u: standard element
(2)
Displacement vector in XFEM:
Force vector in XFEM:
Stiffness Matrix in XFEM:
(3)
(4)

2. BASIS THEORY
If the plate has small crack => use Extended finite element method (XFEM) to calculate
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Displacement equation in XFEM for element “e”
(1)
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(5)
Displacement vector in XFEM:
Force vector in XFEM:
Stiffness Matrix in XFEM:
u: standard element,
b: Tip enrichment element
(6)
(7)

2. BASIS THEORY
If the plate has large crack => use Extended finite element method (XFEM) to calculate
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Displacement equation in XFEM for element “e”
(1)
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(8)
Displacement vector in XFEM:
Force vector in XFEM:
Stiffness Matrix in XFEM:
(9)
(10)
u: standard element,
a: Edge enrichment element,
b: Tip enrichment element
PLATE
THEORY
EXTENDED FINITE
ELEMENT METHOD
Calculate Displacement
and Stress plate
Analyze the results
+
Using by
Matlab
program
Comparing
the results with
previous studies

2. BASIS THEORY
The calculation procedure
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa, a = cracked length, homogeneous material

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3. EXAMPLE
σ = 1000
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa, a = cracked length

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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3
σ = 1000 MPa
FEM (Abaqus)
Uy_max = 0.757mm
XFEM
Uy_max= 0.761mm
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, homogeneous material, Poisson ν = 0.3, σ = 1000MPa, a = cracked length

σ = 1000
Comparing max displacement between XFEM and FEM
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa , a = crack length = 20mm
XFEM
FEM (Abaqus)
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa , a = crack length = 20mm
XFEM
FEM (Abaqus)
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa , a = crack length = 20mm
XFEM
FEM (Abaqus)
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa
XFEM
FEM (Abaqus)
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa , homogeneous material


Comparing max displacement between XFEM and FEM
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa

XFEM
FEM (Abaqus)
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa

XFEM
FEM (Abaqus)
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3. EXAMPLE
Data: W = 40 mm, H = 80 mm, E=117.103 Mpa, Poisson ν = 0.3, σ = 1000MPa

XFEM
FEM (Abaqus)
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4. CONCLUSIONS
The analysis result of cracked plate by XFEM is suitable with the previous studies.
As the cracks increase, the displacement increase.
The displacement and stress depend on the position of the cracked plate.
The greater the crack length, the more the displacement changes.
XFEM is easy to simulate cracks without having to re-mesh the structure.
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4. CONCLUSIONS
σ
σ
=200
IN THE FUTURE
Propagation analysis of the craked plate using XFEM.
Develop the problem with many crack using XFEM.
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[1] Moes N, et al., “A finite element method for crack growth without remeshing,” International Journal for numerical methods in Engineering, 46, 131-150, 1999.
[2] Soheil Mohammadi, eXtended Finite Eement Method, School of Civil Engineering University of Tehran Tehran, Iran, 2008.
[3] Sukumar, et al., “Modeling quasi-static crack growth with the extended finite element method,” Part I: Computer implementation. International Journal of Solids and Structures, 40, 7513–7537, 2003.
[4] Sukumar, et al., “Extended finite element method and fast marching method for three – dimensional fatigue crack propagation,” Engineering Fracture Mechanics, 70, 29 – 48. 2003a.
[5] Dolbow, et al., “An extended finite element method for modeling crack growth with frictional contact,” Finite Elements in Analysis and Design, 36 (3) 235–260. 2000c.
[6] Dolbow , et al., “ An extended finite element method for modeling crack growth with frictional contact,” Computer Methods in Applied Mechanics and Engineering, 190, 6825–6846, 2001.
5. REFERENCES
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THANK YOU SO MUCH!