Introduction to the Extended Finite Element Method · • Extended Finite Element Method for Fracture Analysis of Structures by Soheil Mohammadi, Blackwell Publishing, 2008 • Extended
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Konstantinos Agathos Dipl. Civil Eng. Aristotle University of Thessaloniki
Prof. Dr. Eleni Chatzi, Chair of Structural Mechanics, IBK, D-BAUG
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Introduction to the Extended Finite Element Method
Method of Finite Elements II
The Extended Finite Element Method (XFEM) is a numerical method, based on the Finite Element Method (FEM), that is especially designed for treating discontinuities. Discontinuities are generally divided in strong and weak discontinuities.
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Introduction
Method of Finite Elements II
Strong discontinuities are discontinuities in the solution variable of a problem. In structures, the solution variable is usually the displacements so strong discontinuities are displacement jumps, e.g. cracks and holes.
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Strong and Weak discontinuities
Displacement jump
Cracked Bar
Method of Finite Elements II
Weak discontinuities are discontinuities in the derivatives of the solution variable. In structures such discontinuities would invole kinks in the displacements (jump in the strains), as for example in bimaterial problems.
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Strong and Weak discontinuities
Strain jump
E1 E2
Kink
Bimaterial Bar
Method of Finite Elements II
The biggest part of this presentation will be dealing with the modeling of strong discontinuities and more specificaly with cracks. All formulations will be derived for the 2D cracked domain case and in the end the corresponding formulations for weak discontinuities will be given. So some basic concepts of fracture mechanics will be briefly mentioned
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Fracture Mechanics
Method of Finite Elements II
• Problem Statement Determine the stress, strain and displacement distribution in structures in the presence of flaws such as cracks and small holes.
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Fracture Mechanics
Small Hole Crack
Method of Finite Elements II
• Problem geometry (cracked domain case)
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Fracture Mechanics
Y
X
Crack tip Crack tip
Crack plane
Crack front
y
z
x
2D Crack 3D Crack
Method of Finite Elements II
• Crack opening modes (i.e. how can the two crack surfaces deform)
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Fracture Mechanics
Mode I Mode II Mode III
Method of Finite Elements II
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Fracture Mechanics
Method of Finite Elements II
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Fracture Mechanics
Method of Finite Elements II
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Fracture Mechanics
Method of Finite Elements II
− Westergaard solution parameters
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Fracture Mechanics
X
Y
rθ
σ₀
α α
Method of Finite Elements II
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Fracture Mechanics
Stresses approach infinity near the crack tip
Method of Finite Elements II
In order to model the crack with FEM, the geometry has to be explicitly represented by the mesh, i.e. nodes have to be placed across the crack and on the crack tip. Example:
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FEM Solution
Crack tip 1
Crack tip 2
Crack
Cracked domain FEM mesh Crack tip Mesh refinement
Crack tip 1
Crack tip 2
Crack
Crack tip 3
Crack propagation Remeshing
Method of Finite Elements II
Remarks: • Mesh refinement is usually necessary near the crack tips in
order to represent the asymptotic fields asociated with the crack tips.
• As the crack propagates remeshing is needed which is computationally expensive especially in complex geometries and 3D domains.
• In some cases when remeshing, results need to be projected from one mesh to the other which further increases the computational cost.
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FEM Solution
Method of Finite Elements II
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Partition of Unity
Method of Finite Elements II
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Partition of Unity
Method of Finite Elements II
In the above equation two factors have to be determined: 1. The type of enrichment functions used (next section). 2. The parts of the approximation that are going to be enriched. In the case of cracks, the nature of the discontinuity is local since stress, strain and displacement fields are discontinuous or singular only near the crack tips or along the crack, so enrichment should be local too, i.e. only nodes near the crack are enriched. This matter is going to be addressed in more detail in a following section.
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Partition of Unity
Method of Finite Elements II
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Near Tip Enrichment
Method of Finite Elements II
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Near Tip Enrichment
Method of Finite Elements II
A simple example is considered first in order to demostrate the concept, the results will the be generalized to more complex cases: The objective is to represent mesh 1 using mesh 2 plus some enrichment terms
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Jump Enrichment
1 2 3
9 4 5
6 7 8
10
Y
X
1 2 3
4 5
6 7 8
11
Y
X
Mesh 1 Mesh 2
Method of Finite Elements II
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Jump Enrichment
Method of Finite Elements II
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Jump Enrichment
Method of Finite Elements II
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Jump Enrichment
Method of Finite Elements II
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Jump Enrichment
Jump enrichment terms for arbitrary crack orientation
Method of Finite Elements II
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Signed Distance Function
Xn
dXΓ
Bimaterial interface
Method of Finite Elements II
Absolute value of the signed distance function:
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Signed Distance Function
1D absolute value of the signed distance function
2D absolute value of the signed distance function for arbitrary discontinuity
Method of Finite Elements II
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XFEM Displacement Approximation
Method of Finite Elements II
In order to select the nodes to be enriched the following definition is necessary: The support of a node is the set of elements that contain that node.
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Selection of enriched nodes
I I
Nodal support of external and internal node
Method of Finite Elements II
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Selection of enriched nodes
Method of Finite Elements II
Node enrichment examples:
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Selection of enriched nodes
tip enrichment jump enrichment
Method of Finite Elements II
In order to facilitate the evaluation of the enrichment functions and their derivatives, which is necessary for the calculation of the stiffness matrices, the level set method (LSM) is employed in most XFEM implementations. The level set method is also a powerful tool for tracking moving interfaces, which makes it’s use very common in problems such as crack propagation.
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Level Set Method
Method of Finite Elements II
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Level Set Method
Method of Finite Elements II
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Level Set Method
Method of Finite Elements II
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Level Set Method
Circular level set function Elliptical level set function -2
-1.5-1
-0.50
0.51
1.52
-2-1.5
-1-0.5
00.5
11.5
20
0.5
1
1.5
2
2.5
3
-2-1.5
-1-0.5
00.5
11.5
2
-2-1.5
-1-0.5
00.5
11.5
20
0.5
1
1.5
2
2.5
Method of Finite Elements II
Non-regular discontinuity Non-regular level set function
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FE Approximation of Level Sets
zero level set
FE approximation
FE approximation of level sets
Method of Finite Elements II
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Crack Representation with Level Sets
X
r
θ
ψ
ф
Crack tip
2D crack 3D crack
Method of Finite Elements II
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Weak Form and Discretization
Method of Finite Elements II
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Weak Form and Discretization
Method of Finite Elements II
Gauss quadrature is not apropriate for the numerical integration of the discontinuous enrichment functions, so usually one of the following approaches is employed:
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Numerical Integration
Division into subtriangles Division into subquads
Method of Finite Elements II
Apart from stresses strains and displacements, one quantity of interest when post processing XFEM results is the stress intensity factors. Their calculation is based on the evaluation of an integral (interaction integral) over an area around the crack tip. The procedure is similar to the one for the FE case. Stress intensity factors are necessary for the calculation of the stress fields around the crack tip as well as for determining the direction of crack propagation.
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Post processing
Method of Finite Elements II
The method can be extended in a very straightforward manner to more general and complex problems such as: • Crack propagation • Branched and intersecting cracks • Plastic enrichment • Nonlinear finite elements • Dynamic problems
So a wide variety of applications can be treated.
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Extentions
Method of Finite Elements II
References/recomended reading: • Extended Finite Element Method for Fracture Analysis of
Structures by Soheil Mohammadi, Blackwell Publishing, 2008 • Extended Finite Element Method for Crack Propagation by
Sylvie Pommier, John Wiley & Sons, 2011 • Elastic Crack Growth in Finite Elements by T. Belytschko and T.
Black, International Journal for Numerical Methods in Engineering, 1999
• A Finite Element Method for Crack Growth without Remeshing by N. Moës, J. Dolbow and T. Belytschko, International Journal for Numerical Methods in Engineering, 1999
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References
Method of Finite Elements II
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