Introduction to the Extended Finite Element Method · • Extended Finite Element Method for Fracture Analysis of Structures by Soheil Mohammadi, Blackwell Publishing, 2008 • Extended

Konstantinos Agathos Dipl. Civil Eng. Aristotle University of Thessaloniki

Prof. Dr. Eleni Chatzi, Chair of Structural Mechanics, IBK, D-BAUG

Institute of Structural Engineering 1

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.


Introduction


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.


Strong and Weak discontinuities

Displacement jump

Cracked Bar


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.


Strong and Weak discontinuities

Strain jump

E1 E2

Kink

Bimaterial Bar


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


Fracture Mechanics


• Problem Statement Determine the stress, strain and displacement distribution in structures in the presence of flaws such as cracks and small holes.


Fracture Mechanics

Small Hole Crack


• Problem geometry (cracked domain case)


Fracture Mechanics

Y

X

Crack tip Crack tip

Crack plane

Crack front

y

z

x

2D Crack 3D Crack


• Crack opening modes (i.e. how can the two crack surfaces deform)


Fracture Mechanics

Mode I Mode II Mode III



Fracture Mechanics



Fracture Mechanics



Fracture Mechanics


− Westergaard solution parameters


Fracture Mechanics

X

Y

rθ

σ₀

α α



Fracture Mechanics

Stresses approach infinity near the crack tip


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:


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


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.


FEM Solution



Partition of Unity



Partition of Unity


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.


Partition of Unity



Near Tip Enrichment



Near Tip Enrichment


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


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



Jump Enrichment



Jump Enrichment



Jump Enrichment



Jump Enrichment

Jump enrichment terms for arbitrary crack orientation



Signed Distance Function

Xn

dXΓ

Bimaterial interface


Absolute value of the signed distance function:


Signed Distance Function

1D absolute value of the signed distance function

2D absolute value of the signed distance function for arbitrary discontinuity



XFEM Displacement Approximation


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.


Selection of enriched nodes

I I

Nodal support of external and internal node





Node enrichment examples:



tip enrichment jump enrichment


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.


Level Set Method



Level Set Method



Level Set Method



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


Non-regular discontinuity Non-regular level set function


FE Approximation of Level Sets

zero level set

FE approximation

FE approximation of level sets



Crack Representation with Level Sets

X

r

θ

ψ

ф

Crack tip

2D crack 3D crack



Weak Form and Discretization



Weak Form and Discretization


Gauss quadrature is not apropriate for the numerical integration of the discontinuous enrichment functions, so usually one of the following approaches is employed:


Numerical Integration

Division into subtriangles Division into subquads


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.


Post processing


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.


Extentions


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


References


Documents

Introduction to the Extended Finite Element Method · • Extended Finite Element Method for Fracture Analysis of Structures by Soheil Mohammadi, Blackwell Publishing, 2008 • Extended