A BRIEF STUDY OF XFEM.

Hotwani Vishal

and

Ashok V. Kumar*

Department of Mechanical and Aerospace Engineering

University of Florida, P.O. Box 116300,

Gainesville, FL 32611-6300

* Corresponding author. Tel.: (352) 392-0816; Fax: (352) 392-1071;

E-mail address: akumar@ufl.edu.

1

INDEX

I. ABSTRACT.

II. INTRODUCTION.

III. CRACK ENRICHMENT.

IV. BLENDING.

V. RAMPED STEP ENRICHMENT.

VI. IMPLICIT BOUNDARY METHOD AND SHIFTING IN XFEM.

VII. MODELING MATERIAL INTERFACE AND WEAK DISCONTINUITIES.

VIII. SUMMARY.

IX. REFRENCES.

2

ABSTRACT

This report deals with implementation of mesh independent extended finite element method

(XFEM)/generalized finite element method (GFEM) by applying concept of implicit boundary

method (IBM). Attention has been focused on utilizing current technique available to obtain

stress intensity factor (SIF) directly without any need for post processing of output variables as

practiced in early stages. A novel approach has been devised to avoid formation of unwelcomed

blending elements in the model, existence of which was inevitable by previous approaches. Also

ramped step enrichment for strong discontinuities is discussed which avoids formation of any

blending elements for step enrichment in case of an unshifted approximation. This new method

not only avoids labor of identify blending elements from implementation point of view but it also

reduces the size of global stiffness matrix hence saving computational effort. Shifting of

weighted approximation is being used to facilitate application of essential boundary condition

and improve performance of method. This paper explores application of IBM as a replacement to

shifting.

3

INTRODUCTION

Finite element analysis is finding application in almost all branches of engineering. Its diverse

application field has led to sophistication of method and ramifications focusing on a particular

set of problems. XFEM (extended finite element analysis) [6] is one such technique which

focuses on capturing weak and strong discontinuities in the solution field. Early approaches

involved use of polynomials as test functions but it required attention towards mesh refinement

for obtaining reasonable results. Improving mesh density has been termed as h-version while

using of higher order polynomials for test functions has been names p-version. J Fish[14,15]

tried to capture local phenomenon of high gradient in strain field by using technique names as s-

version. A similar attempt was made by J Fish and Belytscko[16] in which a sub domain of

interest with high gradient had a spectral approximation, and this corresponding mesh was

overlaid on regular finite element mesh. A more recent method MEPU [17] exploiting benefit of

s-version and PUM was demonstrated which accounts for coupling coarse scale domain with a

fine scale region without affecting sparsity of coarse field. Fish described methods such as

XFEM/GFEM as sparse global enrichment method SGEM and argued that MEPU is a decent

approach towards exploring use of FEM in nanotechnology. In last decade development in this

context has led to indentifying analytical solutions and using them directly as test function.

Enrichment was realized using partition of unity which was first explained by Babuska et al as

partition of unity method PUM [3]. This addition to regular FEM trial function has been called

‘extrinsic enrichment’ and it lead to additional variables in the weak formulation. Whereas an

alternative approach of replacing usual FEM shape functions with special functions to capture

local phenomenon has been termed as ‘intrinsic enrichment’. Global enrichment involving

enrichment of entire realm with such functions is also an option but it is computationally very

4

expensive technique. This approach of using analytical solution has helped to capture exact

solutions in models involving discontinuities. Following this path leads to division of domain

into three different regions which have been identified as ‘enriched domain’, ‘blending domain’

and usual ‘finite elements’ by most of the authors. Enriched domain being collection of all

elements that have all their nodes enriched. While blending elements are those that have some

but not all enriched nodes. A clear definition for the same could be found in later section of this

report. Typical example of such a model where XFEM is applied can be found in Fracture

mechanics where displacement field shows asymptotic change while strain could be singular at

the crack tip. Several other problems such as interface problems involving fluids, contact stresses

at joints or multi material problems, Shear bands and dislocation models have also been modeled

using this technique. Enriched elements have capability to reproduce exact solutions depending

on the type of enrichment used but blending elements do not have the same capability. Blending

elements serve to be a means of ensuring compatibility between enriched elements and one with

regular shape function. PUM is not satisfied in blending elements hence they are unable to

represent enriched function but instead they end up adding unwanted terms leading to impaired

accuracy and convergence properties as discussed by Chessa et al [13]. Fries presented a method

of global enrichment using a ramp function which was presented as corrected XFEM [1]. This

method was further studied by Ventura who referred to his proposition as ‘weight function

blending’ [2] which in spirit is same as Fries approach. A J Fawkes [4, 5] and colleagues tried to

solve problems involving crack tip singularities using finite element method in early approaches.

Extended finite element method presented by Belytscko [6, 7, 8] uses analytical solutions for

such solution fields near crack tip which has been area of development in past decade.

Nagashima along with colleagues tried to model interface cracks between dissimilar materials

using XFEM and proved its effectiveness for stress analysis and stress intensity factor

calculation in bi-material fracture problems [8]. Earlier papers [6-8, 9] used the leading or the

5

lower order terms of asymptotic solution and calculations of desired stress intensity factor were

carried out using output results of analysis using complicated post processing. Karihaloo and

Xiao [10, 11] made an attempt to incorporate higher order terms in enrichment function to

improve accuracy of solution and also to develop technique to obtain SIF directly without any

post processing. Results obtained by [10,11] shows it is highly accurate method compared to

previous approaches and if we further limit the additional degrees of freedom due to crack tip

enrichment to be equal, the solution assumes the asymptotic field and coefficients are required

SIF. In our future discussion we would try to incorporate this suggested displacement

approximation by [10, 11] in IBM [12]. This will help us simplify the approximation by avoiding

shifting of the approximation as the essential boundary conditions are satisfied and functions

posses kronecker-δ property.

6

7

CRACK ENRICHMENT

XFEM was first introduced by Belytscko [6] with its application field concentrated on fracture

mechanics. Lot of effort has been made in this direction in order to capture discontinuities in

solution field in the given domain which in our case would be across the crack in a given

material under loading. Discontinuities could be in form of a jump which is referred as ‘strong’

discontinuity or in form of a kink referred as ‘weak’ discontinuity. Solution field as well its

gradient is discontinuous in case of strong discontinuities while only gradient is discontinuous in

case of kinks. This method is also useful to reduce stiffness of approximation in vicinity of high

gradients which helps in improving accuracy without much effort towards mesh refinement. This

reduction in sophistication of mesh as well as computational effort is also a motivation towards

XFEM. Regular FEM approximates the solution field by piece wise interpolation with help of

usual finite elements with polynomial shape functions. But in several cases these polynomial

functions cannot approximate jumps and kinks resulting in stiff solution and degradation of

accuracy. Examples of such events would be modeling of cracks for calculation of SIF, Shear

bands, dislocations, material and phase interface. In materials possessing cracks we have a jump

in displacement field across the crack while crack tip has a high gradient in stress intensity. This

has been modeled using several different techniques in past using following general

approximation which was first introduced by Belytscko.

u=∑i∈ I

.

N i X i+∑j∈I ¿

.

N j¿φa j

Where

N i Regular finite element shape functions.

N ¿j Partition of unity shape functions which may be same asN i.

φ Enrichment function.

8

I Entire domain.

I ¿ Enriched domain.

a j Degrees of freedom associated with I ¿ due to enrichment.

This was general structure given to XFEM approximation in [6] whereas definition of

enrichment function φ and I ¿ depends on specific application.

Also, ∑ N ¿=1

This helps us to maintain partition of unity over entire domain which gives it capability to

represent any enrichment function in domainI ¿as explained by Babuska [3]. But this

approximation looses the Kronecker – δ property and application of essential boundary

conditions becomes difficult. A shifting of enrichment function was proposed Belytscko [7]

which assumes following approximation of the displacement field.

u=∑i∈ I

.

N i X i+∑j∈ I ¿

.

N j¿(φ−φ j)a j

Where φ iis the enrichment function evaluated at the nodes. This approximation helps us apply

essential boundary conditions though the enrichment might still be non zero on some part or

boundary other than nodes. For crack enrichment Belytscko [6] first suggested to use following

functions as enrichment for capturing the displacement close to crack tip,

{φi (r , θ ) }={√ r cos (θ2 ) , √ r sin( θ2 ), √ r sin( θ2)sinθ ,√ r cos (θ

2)sinθ }

Karihaloo and colleagues [18, 12] tried to used higher order terms in enrichment which can be

written as follows

{u(x )v (x )}=∑i∈ I N i {u i( x)v i(x)}+∑j∈ I ¿ N j {utip (x)vtip (x)

}

9

{utip ( x )v tip ( x )}=∑

n=1

N

[ f 11 n f 12n

f 21n f 22n]{ K ¿

K IIn}

Where K ¿ , K IIn are respective coefficients of enrichment function. The coefficient of first terms

represent mixed mode SIF in Isotropic and homogenous materials i.e. K I 1represents Mode I and

K II 1represents Mode II SIFs.

[ f 11n

f 12n

f 21n

f 22n]= r

n2

2 μn√2π [ (k+( n2 )+(−1 )n)cosn2θ−n

2cos ( n

2−2)θ

(k+( n2 )−(−1 )n)sinn2θ−n

2sin (n

2−2)θ

(k−( n2 )−(−1 )n)sinn2θ+ n

2sin (n

2−2)θ

−(k−( n2 )+ (−1 )n)cosn2θ−n

2cos ( n

2−2)θ

]These functions when plotted assume following shape.

a) Function F11, b) Function F12 c) Overall assumed U(x)

10

d) Function F21 e) Funtion F22 f) Overall assumed U(v)

Thus overall approximation used by them can be summarized as

{u(x )v (x )}=∑i∈ I N i {u i( x)v i(x)}+ ∑m∈I ¿∗¿NmH ( x ){b1m

b2m}+∑j∈ I ¿

N j∑n=1

N

[ f 11n f 12n

f 21 n f 22n ]{ K¿

K IIn}¿

¿

H (x ) is a Heaviside or sign distance function required to introduce a strong discontinuity in

displacement field which enables crack to pass through a element having such an enrichment. N

is the number truncated terms from the analytical solution filed as shown in [12]. The domains I,

I ¿ and I ¿∗¿ ¿ will be explained in detail in later section of this report.

As it could be noticed that coefficient of enrichment functions are required SIFs, this approach

reduces the extra effort required to calculate computationally costly J-integrals. Also all the

coefficient are restricted to be equal and regular degrees of freedom associated with finite

element shape functions are assumed to be zero. Thus the approximation is same as theoretical

solution filed expected at crack tip. This helps in achieving higher accuracy compare to previous

methods which do not incorporate higher order terms associated with analytical solution into

approximation field.

11

Figure 1 model domain with structured grid, yellow nodes indicating step enrichment and red nodes indicating crack tip enrichment

12

BLENDING

One of the chief advantages of XFEM is that the enrichment is localized and is only effective in

domains where the discontinuity is expected or known. At the borders between the enriched and

normal FE domain there is incompatibility due to additional degrees of freedom associated with

the enriched domain. This leads to existence of special elements referred as blending elements

earlier in this report that help in smooth transition between two different types of domains. The

problems associated with blending have been discussed in past [19, 20, and 2]. Blending

elements do not satisfy partition of unity and hence they are unable to represent enriched

function but instead they act against and affect the convergence and accuracy. Most of the

approaches developed in past to handle this problem are specific to certain application and do not

apply to arbitrary enrichment functions in general. A more general approach ‘Corrected XFEM’

was adopted in by Fries [20] wherein all the nodes in blending are enriched but the enrichment is

forced to be zero at the interface. This path leads to suppression of all parasitic terms associated

with blending elements although they would not be able to represent enrichment function. A

ramp function was multiplied with the enrichment which was called as weigh function by [2]

which was referred to as weight function blending. This weight function has compact support

and has a value one at enriched nodes and goes down to zero in regular finite element part of

model with help of a C0 polynomial function. Results obtained Ventura [2] show that weight

function blending has better accuracy and convergence properties. Gracie [21] tried to avoid

formation of blending elements using Discontinuous Galerkin (DG) by coupling two regions and

imposing continuity between the two domains using penalty approach. This method was

compared to Assumed Strain (AS) approach discussed by Chessa et al [19]. In the present

approach there are no blending elements present as the coefficients of the enrichment function

are confined to be zero along the boundary of enriched domain.

13

RAMPED STEP ENRICHMENT

When a strong discontinuity is to be applied in solution field functions such as level set function,

sign distance function or Heaviside function have been used successfully in form of step

enrichment. This report will use Heaviside function for the same as we need to assign ability to

elements to let crack pass though them. If step enrichment is shifted it normally vanishes outside

the enriched element and hence there are no blending elements. Even if approximation is not

shifted, it does not affect much as the enrichment is constant in blending elements. We have

avoided using shifting as boundary conditions are applied using IBM so in order to avoid

blending element we present a weight function which when multiplied by Heaviside function

gives a ramped step function that is zero along all edges of element and hence there are no

blending elements present in the model. The weight function helps the enrichment to die down

smoothly at the edges. This is also very easy to implement as only elements that are cut by the

discontinuity are to be enriched. This method can also used for elements that are cut by interface

in a model involving two different material properties such as contact problems.

Red line represents discontinuity and yellow indicate nodes enriched by Heaviside function.

Let ∂(x ) be the equation of discontinuity then Heaviside function can be defined as follows,

φ=H (∂ ( x ) )=1 :∂ ( x )>0

¿0 :∂ ( x )≤0

This function can be represented as follows in a isoparametric four node element with

discontinuity along X=0 axis

14

Figure 2 showing Heaviside function for strong discontinuity along Y=0

Ramped step enrichment can be summarized as follows

φ=H (∂ ( x ) )N

H is a Heaviside function which is described above andN is the weighting function which can be

chosen as follows.

N =(1−∑k=1

n

N k)

Where k is set of all nodes in the enriched element such that value of Heaviside function is unity

at that node. It can be represented as follows.

H (∂ (xk ))=1

Figure 3 showing ramped step function with discontinuity along Y=0

15

IMPLICIT BOUNDARY METHOD AND SHIFTING IN XFEM

This technique is used to apply boundary conditions in FEM unconditionally, whether we have

nodes on the boundary of model or not. IBM [13] uses implicit equations of boundary to force

satisfaction of essential boundary condition using structured grid. Advantage of using this

technique is that structured grid generation is easy and elements are uniform which significantly

improves computational as well solution accuracy. IBM was used to use B spline functions as

interpolation scheme which supplies higher order continuity i.e. at least C1 continuous, hence

improving convergence properties. Satisfying Kronecker-δ property is experienced to be difficult

with higher order test and trial functions. B spline approximations necessitate use of structured

grids and so in order for solution to posses Kronecker-δ property to satisfy the Dirichlet

boundary conditions we need to represent boundary as implicit equations. A similar problem is

experienced in general XFEM approximations where solution lacks Kronecker-δ property and a

method known as shifting is used to cope up with this challenge. Shifting satisfies boundary

conditions on nodes but it may not be satisfied along the element edge. Shifting has been

discussed in detail by Ventura et al [14] and two different types of shifting methods studied

which was something as follows

1. Shifted product of weight and enrichment function.

u=∑i∈ I

.

N i X i+∑j∈I ¿

.

N j¿(φω−φ jω j)a j

2. Applying weight to shifted enrichment function

u=∑i∈ I

.

N i X i+∑j∈ I ¿

.

N j¿ω (φ−φ j)a j

ω is the weight function which has compact support and is applied in order have smooth

blending between regular finite elements and enriched elements.

16

Ventura showed that type 2 has better accuracy and optimal convergence rate. This report

presents a method to avoid necessity of shifting and the approximation assumed by this approach

has equally good accuracy compared when compared to shifted approximation. This approach

simplifies the approximation making it easier to implement as well reduces computational effort

required.

u ( x )=D ( x )∗U (x )+a

u(x )=D(x )¿

The Function D(x) represents step function used as weighting function that enforces boundary

conditions. This method is effective even if shape functions used do not posses Kronecker-δ

property.

D ( x )={ 0 : x ≤0

1−(1− xδ)k

:0≤ x≤δ

1 : x≥δ

Where Function D(x) is a step function which tends to be Heaviside function aslim δ→0

17

IMPLEMENTATION

An effort was made to incorporate the changes suggested in JAVA code IBFEM. It is based on

meshfree method discussed in IBM [13] and uses a structured grid for analysis. A new class of

enriched element was generated for XFEM which includes ramped step enriched and crack tip

enriched elements. The mathematical formulation for both the element type was discussed in

previous section as the present topic deals with stiffness matrix associated with the elements. In

order to use the previously available code to test the suggested changes following scheme was

adapted as in the elements were super imposed. Instead of having an element with regular and

enriched degrees of freedom, two different elements having same global coordinates were used.

In order to generate global stiffness matrix suitable connectivity was used so that terms add up at

their respective places during assembly stage.

In general for enriched element

X=∑i∈ I

.

N iU i+∑j∈I ¿

.

N j¿φa j

In our case we have chosen N i =N j¿.

Now since,

u=∑i∈ I

.

N iU i+∑j∈I ¿

.

N j¿φa j

ε=∇s N iU i+∇sN j

¿φa j

σ=[C ]ε

σ=[C ][∇s N i X i+∇sN j

¿φa j]

B1=∇s N iB2=∇sN j

¿φ

18

Using this approximation in weak form of principle of virtual work we get,

∫Ω

.

ϵT σ=δX F

{δU i

δ a j }∫Ω.

[B1

B1] [C ] [B1 X i+B2a j ]={δU i

δ a j }[FuFa]∫Ω

.

¿¿

Thus

[K ] [ Xe ]=[F ]

[K ]=¿

For 2D elasto static problem the displacement field assumed can be described as,

U=[uv ]=[N1 0 N 2 0 N 3 0 N4 00 N1 0 N2 0 N3 0 N 4

][u1

v1

u2

v2

u3

v3

u4

v4

]But for enriched element this assumed field becomes,

19

U=[uv ]=[N1 0 N 2 0 N 3 0 N4 0 φN 1 0 φ N2 0 φ N3 0 φN 4 00 N1 0 N2 0 N3 0 N 4 0 φ N1 0 φN 2 0 φ N3 0 φ N 4][

u1

v1

u2

v2

u3

v3

u4

v4

a1

b1

a2

b2

a3

b3

a4

b4

]Strain in case of plain stress conditions can be written as follows for an enriched element.

ε=[∂u∂ x∂v∂ y

∂v∂ x

+∂u∂ y

]ε=[

∂ N i

∂x0

0∂N i

∂ y∂ N i

∂ y∂N i

∂ x]

⏟

[X i ]+[∂φ N i

∂ x0

0∂φ N i

∂ y∂φ N i

∂ y∂φ N i

∂ x]

⏟

[ai]

B1 B2

ε=ε1+ε2

20

ε 1=[∂N 1

∂ x0

∂ N2

∂x0

∂ N3

∂ x0

∂N 4

∂ x0

0∂ N1

∂ y0

∂N 2

∂ y0

∂ N3

∂ y0

∂ N4

∂ y∂N 1

∂ y

∂ N1

∂x

∂N2

∂ y

∂ N2

∂x

∂N3

∂ y

∂ N3

∂x

∂N 4

∂ y

∂N 4

∂x][u1

v1

u2

v2

u3

v3

u4

v4

]ε 2=[

∂φN 1

∂ x0

∂φN 2

∂x0

∂φN 3

∂ x0

∂φN 4

∂ x0

0∂φN 1

∂ y0

∂φN 2

∂ y0

∂φN 3

∂ y0

∂φN 4

∂ y∂φ N1

∂ y∂φN 1

∂ x∂φ N2

∂ y∂φ N2

∂ x∂φ N3

∂ y∂φ N3

∂ x∂φN 4

∂ y∂φN 4

∂ x] [.a1

b1

a2

b2

a3

b3

a4

b4

]Whereas [K ]=¿

Which will be a 16 x 16 matrix in this case can be very easily derived using the above definition

of B1 and B2 matrix. This procedure can be extended to other elements and K matrix can be

derived accordingly.

In case of applying enrichment for singularity using Karihaloo approach we find that the

displacement in any direction is dependent on degrees of freedom in perpendicular direction. The

formulation for displacement as mentioned above is given by

{u(x )v (x )}=∑i∈ I N i {u i( x)v i(x)}+∑j∈ I ¿ N j {utip (x)vtip (x)

}

φ={utip ( x )v tip ( x )}=∑

n=1

N

[ f 11n f 12n

f 21n f 22n]{ K ¿

K IIn}

Thus implementation of this displacement field in regular FEA code creates difficulties. Also the

coefficients are equal so in order to achieve that collapsing of nodes was used wherein all the

21

nodes were mapped to one node in global stiffness matrix. Thus any enriched element added

would add its respective stiffness i.e. B2 in the global stiffness matrix. In this case B2 can be

summarized as follows. As we know,

ε=[∂ N i

∂x0

0∂N i

∂ y∂ N i

∂x∂N i

∂ y]

⏟

[X i ]+[∂φ N i

∂ x0

0∂φ N i

∂ y∂φ N i

∂ x∂φ N i

∂ y]

⏟

[ai]

B1 B2

Thus

B2=[∂φN 1

∂ x0

∂φN 2

∂x0

∂φN 3

∂ x0

∂φN 4

∂ x0

0∂φN 1

∂ y0

∂φN 2

∂ y0

∂φN 3

∂ y0

∂φN 4

∂ y∂φ N1

∂ x∂φN 1

∂ y∂φ N 2

∂x∂φ N2

∂ y∂φ N3

∂ x∂ φN 3

∂ y∂φN 4

∂ x∂φN 4

∂ y]

In this case enrichment is a vector as stated above so the corresponding stiffness matrix becomes

B2=[∂ F11N i

∂ x

∂ F11N i

∂ x

∂φN 2

∂ x0

∂φN 3

∂ x0

∂φN 4

∂x0

∂ F11N i

∂ y∂ F11N i

∂ y0

∂φN 2

∂ y0

∂φN 3

∂ y0

∂φN 4

∂ y∂φ N1

∂ x∂φN 1

∂ y∂φ N2

∂ x∂φN 2

∂ y∂φ N3

∂x∂φ N3

∂ y∂φN 4

∂x∂φN 4

∂ y]

B2=[ ∂ F11N i

∂ x

∂ F11N i

∂ x⋮

∂ F11N i

∂ y¿ ⋮ ]

22

MODELING MATERIAL INTERFACE AND WEAK DISCONTINUITIES

This section deal with modeling of weak discontinuities which may occur at material interfaces.

The earliest approaches involved as in [7] was using a level set function where the enrichment

was absolute value of level set function

φ i=|(∅ (x i ))|=|∅ (x i)|

But this approximation creates well known blending problems and gives sub optimal results.

Another interesting approximation was suggested by Moes [22] which can be described as

follows.

φ i=∑i=I

|∅ (x i)|N i−|∑i=I ∅ (x i )N i|

23

This enrichment vanishes outside elements intersected by weak discontinuities as it can be seen

from above plot and hence there is no need of special treatment to blending elements.

SUMMARY

First section of this report describes fundamentals of XFEM with a short brief history of research

in this direction. Application of XFEM was studied in various fields and particular focus has

been given to application in fracture mechanics. Later sections deals with different methods of

crack enrichment and problem of blending. This report deals with ways studying various

methods suggested and obtaining SIF for Mix mode fracture. Of all the various approaches

Karihaloo [11] method of using analytical solution directly as approximation is most appealing

because of it advantages. It helps us in evaluating SIF directly without any computationally

expensive post processing but its implementation in current FEA code presents a challenge.

Higher order terms were used and a general approximation for mixed mode fracture mechanics

was studied. A general problem associated with localized enrichment is of blending was

addressed and ways around it for step and crack tip enrichment are developed which gives us a

chance to further explore their application. Step enrichment does not pose a problem when we

have a shifted approximation but it should be avoided when the approximation is not shifted. The

need of shifting can be avoided using IBM which is essentially an easier and effective way to

apply boundary conditions in case of mesh independent models. A ramped step enriched

24

presented in this report gives a reliable way to bring the enrichment function to zero in the same

element that has discontinuity. The same problem in case of weak discontinuities was addressed

by Moes [22] which takes care of problem of blending. In case of crack tip enrichment blending

problem could be avoided by forcing the solution field to be zero at the domain boundary.

Derivation of stiffness matrix has also been studied for better understanding of the method and

its compatibility with IBM which uses Dirichlet functions in solution structure. All cases studied

give us a positive chance to explore its application and so they should be implemented for

checking their feasibility for application in a existing FEA code.

25

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