Chapter VI Extended Natural Neighbours Method

Chapter VI  Page 143 

Chapter VI

Extended Natural Neighbours Method

Content

Summary 144

VI.1. Introduction                    146 

VI.2. Domain decomposition 148

VI.3. Discretization 150

VI.3.1. Discretization of ε_ij,Σ_ij,r_i in cells of type O 150

VI.3.2. Discretization of ε_ij,Σ_ij,r_i in cells of type H 150

VI.3.3. Discretization of ε_ij,Σ_ij,r_i in cells of type C 151

VI.3.4. Discretization of the displacements 153

VI.3.5. Definition of the crack function C(X) 154 

VI.4. Discretization of the FdV variational principle 156

VI.5. Solution of the equations 159

VI.5.1. Stresses in terms of displacement discretization parameters 159

VI.5.2. Equilibrium equations in terms of displacement discretization parameters 160 VI.5.2.1. Equilibrium equations associated with the degrees of freedom u_i^J of the cells of type O 160

VI.5.2.2. Equilibrium equations associated with the degrees of freedom a_i^J of the cells of type H and C 163

VI.5.3. Summary of the equations 165

VI.5.4. Final equation system 171

VI.6. Applications 172

VI.6.1. Mode 1 tests 172

VI.6.2. Mode 2 tests 180

VI.6.3 Single edge crack 182

VI.6.4 Nearly incompressible case 186

VI.7. Conclusion 187

Chapter VI  Page 144 

Summary

In chapter V, the natural neighbours method was extended to the domain of 2D linear elastic fracture mechanics (LEFM) based on the FRAEIJS de VEUBEKE variational principle which allows assumptions on the displacement field, the stresse field, the strain field and the surface distributed support reactions field.

The basic idea is to take advantage of the flexibility of the FdV variational principle and to introduce, near the crack tips, stress and strain assumptions inspired from the analytical solution provided by the LEFM [WESTERGAARD, H.M. (1939)].

This has been done by introducing a node at each crack tip and by introducing these stress and strain assumptions in the corresponding Voronoi cell. In this approach, the crack lip has to be conformed to the edges of the Voronoi cell.

The present chapter develops an eXtended Natural nEighbours Method (XNEM) to solve 2D problems of LEFM.

In the proposed method, the crack is represented by a line that does not conform to the nodes or to the edges of the Voronoi cells.

The 2D domain contains N nodes (including nodes on the domain contour) but there is no node at the crack tips. The N Voronoi cells corresponding these nodes are built.

Then the discretization of the displacement field is enriched with Heaviside functions but not with crack tip functions since the experience of chapter V (section V.7) has shown that such an enrichment does not significantly improve the solution.

We have 3 types of cells:

• cells of type O that do not contain a crack;

• cells of type H that are divided into 2 parts by a crack;

• cells of type C that contain a crack tip.

We admit the discretization hypotheses separately in the 3 different types of cells.

For ordinary cells:

1. The assumed stresses are constant over an ordinary cell I.

2. The assumed strains are constant over an ordinary cell I.

3. The assumed support reactions are constant over each edge of ordinary Voronoi cells

on which displacements are imposed.

For cells of type H that are cut into 2 parts by a crack:

1. The assumed stresses are constant over each part of the cell.

2. The assumed strains are constant over each part of the cell.

3. The assumed support reactions are piecewise constant over the edges on which displacements are imposed

For cells of type C that contain a crack tip, the stress and strain fields are enriched in a similar way as the LFMVCs of chapter V, i.e. by introducing in the stress and strain discretizations the solutions of the LEFM [WESTERGAARD, H.M. (1939)] for mode 1 and mode 2.

Chapter VI  Page 145  Introducing all the assumptions and enrichments in the FdV variational principle produces the set of equations governing the discretized solid.

As same as in the method developed in chapter V, the stress intensity coefficients are obtained as primary variables of the solution in the present method.

These equations are recast in matrix form and it is shown that the discretization parameters associated with the assumptions on the stresses and on the strains can be eliminated at the Voronoi cell level so that the final system of equations only involves the nodal displacements, the assumed support reactions and the stress intensity coefficients.

These support reactions can be further eliminated from the equation system if the imposed support conditions only involve displacements imposed as constant (in particular displacements imposed to zero) on a part of the solid contour.

The following properties, already obtained for the method of chapter V, remain valid:

In the cells of types O and H:

1. In the absence of body forces, the calculation of integrals over the area of the domain is avoided: only integrations on the edges of the Voronoi cells are required.

2. The derivatives of the Laplace interpolation functions are not required.

In the cells of type C:

1. Some integrations on the area of these cells are required but they can be calculated analytically.

2. The other integrals are integrals on the edges of these cells

3. The derivatives of the Laplace interpolation functions are not required

Applications to the mode 1 test, mode 2 test and single edge crack test are performed. The results validate this method and nearly incompressibility locking is avoided.

Chapter VI  Page 146 

VI.1. Introduction

In chapter V, a new approach to solve problems of Linear Elastic Fracture Mechanics (LEFM) has been developed.

The basic idea was to take advantage of the flexibility of the FdV variational principle and to introduce, near the crack tips, stress and strain assumptions inspired from the analytical solution provided by the Theory of Elasticity [WESTERGAARD, H.M. (1939)].

This has been done by imposing a node at each crack tip and by introducing these stress and strain assumptions in the corresponding Voronoi cell called Linear Fracture Mechanics Voronoi Cell (LFMVC).

In the present chapter, a different approach is proposed in which it is not necessary to locate nodes at the crack tips.

It is somehow inspired from the eXtended Finite Element Method (XFEM) that was initially developed by [MOES N. et al. (1999), DOLBOW J.E. (1999)] for a single crack in 2D problems.

The XFEM is an approach based on the virtual work principle and, consequently, the primary unknowns are the displacements.

The fundamental idea is to define the crack by a line that does not conform to the finite element edges (figure VI.1) and to enrich the discretization of the displacement field.

ordinary nodes nodes of set λ_H nodes of set λ_C Figure VI.1. Fundamental idea of the XFEM

crack

Chapter VI  Page 147  Let λ_H be the set of nodes the shape function support of which is cut by the crack.

Let λ_C be the set of nodes shape function support of which contains the crack tips.

The displacement field is discretized by:

∑ ∑

∑

∑

= ∈λ ∈λ = ⎥⎥

⎦

⎤

⎢⎢

⎣ Φ ⎡ +

Φ +

Φ

=

C

H J k

J k k J J

i J J

N J

iJ J

i u H(X ) b c F (X )

u ⁴

1 1

(VI.1) where

ΦJ is the usual finite element shape function associated with node J ; )

X (

H is a Heaviside function :

⎩⎨

⎧

−

= +

crack the of side other on the 1

crack the of side one on ) 1

X (

H (VI.2)

) X (

F_k is a set of 4 functions inspired by the displacement field near a crack tip for mode 1 and mode 2:

1(r,θ)= r sin2θ

F (VI.3.a)

2 2θ

=

θ) r cos ,

r (

F (VI.3.b)

θ θ

=

θ) r sin cos ,

r (

F³ 2 (VI.3.c)

θ θ

=

θ) r cos sin ,

r (

F⁴ 2 (VI.3.d)

with (r,θ) the polar coordinates at the crack tip (see chapter II, section II.4.5).

In addition to the usual degrees of freedomu_i^J(the nodal displacements),

• the b_i^J are new displacement parameters taking account of the displacement discontinuity through the crack;

• the c_k^Jare new displacement parameters taking account of the particular form of the displacement field near the crack tips.

This type of enrichment is called a “topological enrichment”.

This elegant approach simplifies the meshing operations since it is not necessary to make the crack lips coincide with element edges and, consequently, is well suited for the modelling of crack propagation without remeshing.

Furthermore, it takes account of the particular form of the displacements near the crack tips which improves the quality of the results.

The method has been extended to 3D cases [SUKUMAR N. et al. (2000)], to the modelling of cracks in plates [DOLBOW J. et al. (2000-a) and (2000-b)], to the case of branched and intersecting cracks [DAUX C. et al. (2000)], to the case of higher order elements [STAZI F.L.

et al. (2003)], to the case of bimaterials interfaces [SUKUMAR N. et al. (2004)], …

In [STOLARSKA M. et al. (2001)], it has been coupled with the “level set method” [OSHER S.A. (1999)] for representing and propagating the crack.

Chapter VI  Page 148  It has also been proposed [BECHET E. et al. (2005)] to replace the “topological” enrichment, in which only the elements touching the crack are enriched, by a “geometrical” enrichment in which a given domain size is enriched. This improved the quality of the results.

It is also worth mentioning that in [DOLBOW J. and DEVAN A. (2004)], an assumed strain enrichment is proposed in which the 2 parts of the finite elements cut by the crack are separately enriched by an assumed strain field. The development of the method relies on the HU-WASHIZU variational principle (chapter II, section II.3.1). In this paper, the enrichment of the displacement filed is limited to the first 2 terms of (VI.1), which means that the enrichment of the nodes of the set λ_C is not included.

The present chapter develops an eXtended Natural nEighbours Method (XNEM) to solve 2D problems of LEFM.

In the proposed method, the crack is also represented by a line that does not conform to the nodes or to the edges of the Voronoi cells.

Then the discretization of the displacement field is enriched with Heaviside functions as (VI.2) but not with crack tip functions of the type (VI.3) since the experience of chapter V shows that such an enrichment does not significantly improve the solution (see section V.7).

On the other hand, the stress and strain assumptions in the cells cut by the crack are enriched by considering, in each of the two parts, different constant stresses and constant strains.

Finally, for the cells containing the crack tips, the stress and strain fields are enriched in a similar way as the LFMVC cells of chapter V, i.e. by introducing in the stress and strain discretizations the solutions of the Theory of Elasticity [WESTERGAARD, H.M. (1939)] for mode 1 and mode 2.

Since the purpose is to explore a new numerical method, the options retained for the development are the simplest ones:

• there is a single crack with 1 or 2 tips;

• it does not propagate;

• the “topological” enrichment rather than the “geometrical” enrichment is chosen These limitations could obviously be eliminated taking advantage of the existing techniques used for the XFEM, but this is outside the scope of the present thesis.

VI.2. Domain decomposition

In the natural neighbours method, the domain contains N nodes (including nodes on the domain contour) and the N Voronoi cells corresponding to these nodes are built.

The area of the domain is:

∑

=

= ^N

I AI

A

1

(VI.4) with A_Ithe area of Voronoi cell I.

If cracks are present in the domain, we have 3 types of cells:

ƒ cells of type O (figure VI.3) that do not contain a crack;

ƒ cells of type H (figure VI.4) that are divided into 2 parts by a crack;

ƒ cells of type C (figure VI.5) that contain a crack tip.

Chapter VI  Page 149  Figure VI.2. Domain decomposition Figure VI.3. Cell of type O

Figure VI.4. Cell of type H Figure VI.5. Cell of type C X

X2 I

Ordinary cell

0 , _L >

L θ α

X

X2 L

Y1

Y2

r_L

θ

L

α

L

Cell of type C Crack tip

Cell of type H part A

Cell of type H part B

Chapter VI  Page 150  Note that there may be several cells of type C if there are several cracks or even a single crack the 2 tips of which are inside the domain.

We could also have a cell containing several crack tips (type C+C) or a cell divided into several parts by several cracks (type H+H) or a cell divided in 2 parts by a crack and containing a crack tip (type H+C).

Here, we only consider cells of types O, H and C.

In addition, for simplicity, we consider that there is only 1 crack and that, in each cell, the crack is straight.

Let N_O,N_H,N_C be the numbers of cells of types O, H and C respectively with

C H

O N N

N

N = + + (VI.5)

the total number of cells.

VI.3. Discretization

VI.3.1. Discretization of ε_ij,Σ_ij,r_i in cells of type O We admit the following discretization hypotheses for ordinary cells:

1. The assumed stresses Σ_ij are constant over an ordinary cell I:

I ij ij =Σ

Σ I =1,N_O (VI.6)

2. The assumed strains ε_ij are constant over an ordinary cell I:

I ij ij ε

ε = I =1,N_O (VI.7)

3. The assumed support reactions t_i are constant over each edge K of ordinary Voronoi cells on which displacements are imposed

iK

i r

r = (VI.8)

VI.3.2. Discretization of ε_ij,Σ_ij,r_i in cells of type H

As shown on figure VI.4, the cell is divided into 2 parts A and B by the crack.

We admit the following discretization hypotheses for cells of type H:

1. The assumed stresses Σ_ijare constant over each part of the cell:

I , ijA

ij Σ

Σ = in part A of cell I I =1,N_H (VI.9-a)

I , B ij

ij Σ

Σ = in part B of cell I I =1,N_H (VI.9-b) 2. The assumed strains ε_ij are constant over each part of the cell:

I , A ij ij ε

ε = in part A of cell I I =1,N_H (VI.10-a)

Chapter VI  Page 151 

I , Bij ij ε

ε = in part B of cell I I =1,N_H (VI.10-b) 3. The assumed support reactions r_i are piecewise constant over the edges K on which

displacements are imposed

iK

i r

r = on the whole edge K if it is not cut by the crack (VI.11-a)

K , iA

i r

r = on part A of edge K if it is cut by the crack (VI.11-b)

K , iB

i r

r = on part B of edge K if it is cut by the crack (VI.11-c)

VI.3.3. Discretization of ε_ij,Σ_ij,r_i in cells of type C

It was recalled in Chapter II, section II.4.5. that, in LEFM, the stress fields near the crack tip (figure VI.5) are given by:

for mode 1:

⎪⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

θ

− θ

θ + θ

θ

− θ θ

= π

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ τ τ τ

2 3 2

2 3 1 2

2 3 1 2

2 2

1 12

22 11

L L

L L

L L

L L

cos sin

sin sin

sin sin

r cos

K (VI.12)

for mode 2:

⎪⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

θ

− θ θ

θ θ

θ

θ + θ

− θ

= π

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧ τ τ τ

) sin sin

( cos

cos sin

cos

) cos cos

( sin r

K

L L

L

L L

L

L L

L

L

2 3 1 2

2

2 3 2

2

2 3 2 2

2 2

2 12

22 11

(VI.13)

These stresses are expressed in the local frame (Y1,Y2) attached to the crack tip (figure VI.5).

K1 and K₂ are the stress intensity factors of modes 1 and 2 respectively.

These stresses are valid for both plane stress and plane strain conditions.

In order to have a better representation of the stresses in the vicinity of the crack tip, we admit the following discretization hypotheses for cells of type C:

1. The assumed stresses are given in the local frame (Y1,Y2)L attached to the crack tip by:

{ } { } { }

C L

O L L L

L

O O O L

N L P P

P P

P P

P P P P

P P

, 1 ,

12 22 11

12 22 11

12 22 11

= +

=

⎪ ⇔

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎪ +

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎪ =

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

Σ Σ

Σ Σ

(VI.14)

with

{ }

P^{Σ L} = f(r_L)

[

K_Σ^L₁

{

g₁(θ_L)

}

+K_Σ^L₂

{

g₂(θ_L)

}

^L

]

=K_Σ^L₁

{ }

H₁ ^L +K_Σ^L₂

{ }

H₂ ^L (VI.15)

Chapter VI  Page 152 

{ } { }

2 ) sin3 sin2 1 2( cos

2 cos3 sin2 cos2

2 ) cos3 cos2 2 2( sin )

(

; 2 cos3 sin2

2 sin3 sin2 1

2 sin3 sin2 1 cos2 )

( ₂

1

⎪⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

−

− +

=

⎪⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

− +

−

=

θ θ

θ

θ θ

θ

θ θ

θ θ

θ θ

θ θ

θ θ

θ θ g

g ;

r ) π

r (

f 2

= 1

(VI.16)

{ }

H₁ ^L = f(r_L)

{

g₁(θ_L)

}

;

{ }

H₂ ^L = f(r_L)

{

g₂(θ_L)

}

(VI.17) K_ΣL₁ and K_Σ^L₂ , L=1,NC are stress parameters associated with the cells of type C.

Of course, the stresses Σ_ij^L in the global frame (X1, X2) can be obtained by tensor rotation.

2. The assumed strains are interpolated by:

{ }

^L

{ } { }

^{O L L}

L

O O O L

γε

γ γ γ

γ γ γ

γ γ γ

γ γ

+

=

⎪ ⇔

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎪ +

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎪ =

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

Σ Σ Σ

2

2 2

L

12 22 11

12 22 11

12 22 11

(VI.18-a)

{ }

γ^{Σ L} =

[ ]

D ^L

{

K_Σ^L1

{ }

H1 ^L +K_Σ^L2

{ }

H2

}

(VI.18-b) with

[ ]

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

+

−

−

=

) 1 ( 2 0 0

0 1

0 1 1

*

*

*

*

L L

L

L L

D E

ν ν

ν

(VI.19)

state stress plane in

state strain plane in ν

1 ²

⎪⎪

⎩

⎪⎪

⎨

⎧

= −

L L L

* L

E E

E

state stress plane in ν

state strain plane in ν 1

ν

⎪⎪

⎩

⎪⎪

⎨

⎧

= −

L L L

*

vL (VI.20)

These strains are also given in the local frame (Y1,Y2)L attached to the crack tip and can be rotated into the global frame if necessary.

The elastic parameters E_L,ν_L can be different in different cells if necessary.

K_ΣL₁ and K_Σ^L₂ constitute an additional set of displacement discretization parameters associated with the cells of type C.

3. The assumed support reactions r_i are piecewise constant over the edges K on which displacements are imposed:

iK

i r

r = on the whole edge K if it is not cut by the crack (VI.21-a)

Chapter VI  Page 153 

K A i

i r

r = ^, on part A of edge K if it is cut by the crack (VI.21-b)

K B i

i r

r = ^, on part B of edge K if it is cut by the crack (VI.21-c) The situation corresponding to (VI.21-b) and (VI.21-c) is illustrated on figure VI.6.

On parts A and B of the edge cut by the crack, it is possible to impose different displacements u~_i^Aand u~_i^B.

Figure VI.6. Crack cutting the domain contour  

VI.3.4. Discretization of the displacements Let us define the following sets.

• λ^H: the set of nodes corresponding to all the cells of type H (i.e. cut by the crack).

This set has NH elements.

• λ^C: the set of nodes corresponding to all the cells of type C (containing a crack tip).

This set has NC elements. In the present case, since we consider a single crack, NC =1 or 2.

• Λ : the union of the sets λ^H and of the sets λ^C. Λ is the labelled as the set of crack nodes.

C

H λ

λ

Λ = ∪

The discretization of the displacements is composed of two contributions

J i J J

N

J

J i J

i u C X a

u =

∑

Φ +

∑

Φ

Λ

∈

=

) (

1

(VI.22) where:

• u_i^J is the displacement of node J (corresponding to Voronoi cell J)

• Φ_J is the corresponding Laplace interpolant

• C(X ) is a function, labelled as the crack function, that will be defined precisely in section VI.3.5

Part B of the edge cut by the crack Part A of the edge cut by the crack

Chapter VI  Page 154 

• a_i^J is an additional set of displacement discretization parameters attached to the nodes belonging to the set Λ (the set of crack nodes)

The first term is the usual interpolation of the natural neighbours method.

The second term will be designed to allow a displacement discontinuity through the crack.

VI.3.5. Definition of the crack function C(X) 

The idea is to make C(X) equal to +1 on one side of the crack and to -1 on the other side.

However, care must be taken for its definition ahead of the crack tips where it can take different values.

Let C₁−C₁^* and C₂ −C^*₂ be 2 straight lines having the same directions as the 2 crack ends (figure VI.7). They define 2 sub-domains A_A and A_B in which a function H(X ) is given by:

AA

X if )

X (

H =+1 ∈

AB

X if )

X (

H =−1 ∈

Let A₁−B₁ and A₂ −B₂ be 2 straight lines perpendicular to the 2 crack ends (figure VI.8).

Let A₁^* −B₁^* and A^*₂ −B^*₂ be 2 straight lines parallel to A₁−B₁ and A₂−B₂ respectively.

They are such that their points P₁ and P₂ are located on the crack at a user defined distance from the corresponding crack tips (figure VI.8). It is assumed that the crack is straight on

1 P1

C − as well as on C₂−P₂ .

In the example of figure VI.8, points P₁ and P₂ are located at the intersection of the crack with the edge of the cell of type C.

Figure VI.9 shows a zoom on crack tip 1. The subscript 1 is omitted for simplicity.

Line A−B is oriented by the abcissa s such that s=0 at point Cand s>0 as indicated on the figure (at 90° counterclockwise with respect to the direction of the crack tip).

Any point X inside the domain belongs to a line C−X −Q intersecting A^* −B^* at point Q. This line is oriented by the abcissa ξ such that ξ =0 at point Cand ξ =1 at point Q.

Hence, point Xcan be referred to by the skew coordinates (s,ξ) attached to the crack tip.

They can be deduced from its cartesian coordinates (X₁,X₂) by simple geometrical considerations developed in annex 4.

The same can also be done for crack tip 2.

So, any point (X₁,X₂) in the domain has skew coordinates (s₁,ξ₁) with respect to crack tip 1 and (s₂,ξ₂) with respect to crack tip 2.

Chapter VI  Page 155  Figure VI.7. The two sides of a crack. Figure VI.8. The different zones for C(X).

Figure VI.9. Zoom on a crack tip +1

=

⇒ H(X) A_A

−1

=

⇒H(X ) A_B

C1

C2

C*₂

C₁*

C1

C2

C*₂

C₁*

B1

B2

B*₂

A₁*

B₁*

P1

P2

A2

A*₂

A1

B A

A* _C

P

B*

>0 s

<0 s

=0 s

Q

0 ξ=

1 ξ = 1

ξ >

0 ξ<

X

Chapter VI  Page 156  With the help of these skew coordinates, the crack function C(X ) is defined as follows:

2 1 1

2

2 1

2

1 2

1 2 1

2 1

1 0

1 0

1 1

0

1 1

0

1 1

0 0

0

ξ ξ ξ

ξ

ξ ξ

ξ

ξ ξ

ξ ξ ξ

ξ ξ

H C and

if

H C and

if

H C and

if

H C and

if

C or

if

=

⇒

<

<

<

<

=

⇒

≥

<

<

=

⇒

≥

<

<

=

⇒

≥

≥

=

⇒

<

<

(VI.23)

Examples of the crack function are given in annex 4.

VI.4. Discretization of the FdV variational principle

We start from the FdV variational principle introduced in chapter II and we recall its expression for completeness (we keep the equation numbers of chapter II).

dA X )

u X ( u dA

u ) X F

( dA ) (

A

i ij j j ij i

i A

j i ji A

ij ij

ij

∫ ∫

∫

_⎥^⎥

⎦

⎤

⎢⎢

⎣

⎡ −ε

∂ +∂

∂ Σ ∂

δ + δ

∂ + Σ

− ∂ δε Σ

− σ

= Π

δ 2

1

∫

∫

^{Σ − δ + Σ − δ}

+

t

u S

i i ji j S

i i ji

j r ) u dS (N T ) u dS

N

( =0 (II.36)

or

6 0

5 4 3 2

1+ Π + Π + Π + Π + Π =

Π

=

Π δ δ δ δ δ δ

δ (II.31)

with the different terms

= ∫

= ∫

A ij ij AδW(εij )dA σ δε dA Π

δ ₁ (II.25)

dA X )

u X

( u

A ij

i j j

ij i

∫ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

∂ +∂

∂

= ∂δ δ δε

2 Σ 1 Π

δ ₂ (II.26)

dA X )

u X ( u

A ij

i j j ij i

∫ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

∂ +∂

∂

= ∂ ε

2 Σ 1 δ Π

δ ₃ (II.27)

dA u F

A∫ i i

−

= δ

Π

δ ₄ (II.28)

−∫

=

StTiδuidS Π

δ ₅ (II.29)

dS u r dS ) u u~

(

r _i

S i S

i i i

u u

δ

−

− δ

= Π

δ ⁶

∫ ∫

(II.30)

For the linear elastic case, the stresses are given by:

ij ij

ij ε

ε σ W

∂

= ∂ ( )

(II.34)

and

kl ij ijkl

ij C ε ε

ε

W 2

) 1

( = (II.35)

Chapter VI  Page 157  where Cijkl is the classical Hooke’s tensor.

In (II.29) and (II.30), the integrals are computed along the domain contour. This contour is the union of some of the edges of the exterior Voronoi cells. These edges are denoted by S_K and we have

∑

=

= ^M

K

SK

S

1

;

∑

=

= ^Mu

K K

u S

S

1

;

∑

=

= ^Mt

K K

t S

S

1

; S =S_u∪S_t ⇒M =M_u +M_t (VI.24) where M is the number of edges composing the contour, Mu the number of edges on which displacements u~_i are imposed and Mt the number of edges on which surface tractions T_i are imposed.

Using the domain decomposition, these terms become:

( )

∑ ∫

₌ ^−Σ

=

Π ^N

I A

I ij ij ij

I

dA

1

1 σ δε

δ (VI.25)

∑ ∫

_{= ⎥}^⎥

⎦

⎤

⎢⎢

⎣

⎡

∂ +∂

∂ Σ ∂

=

Π ^N

I I

A i

j j

i

ij dA

X u X

u

1 I

2 ( )

2

1 δ δ

δ (VI.26)

∑ ∫ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

∂ +∂

∂

= ∂

= N

I I

A ij

i j j

ij i ) ε dA

X u X

( u δ

δ

1 I

3 2

Σ 1

Π (VI.27)

∑ ∫

=

−

=

Π ^N

I

I A

i i u dA F

1 I

4 δ

δ (VI.28)

∑ ∫

−

= = _K

t

S i i K

M

K T δu dS

Π δ

5 1 (VI.29)

∑ ∫ ∫

= ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ δ − − δ

= Π

δ ^u

K K

M

K

K i S

i S

K i i

i(u~ u )dS r u dS

r

1

6 (VI.30)

Introducing the discretizations of section VI.3 in the FdV variational principle (II.31) leads to the equations of table VI.1.

The details of the calculations are given in annex 5.

The next section is devoted to the solution of these equations and will be concluded by a complete summary of the variables and notations used in this chapter.

In particular, the different symbols and matrices appearing in table VI.1 are defined in tables VI.5 and VI.6.

In these equations only the matrices

[ ]

^{IH L and}

[

^HDH

]

^Lrequire an integration on the area of the cells of type C but these integrations can be performed analytically as in chapter V.

Chapter VI  Page 158  Table VI.1. Discretized equations for XNEM

Stress identification: the assumed stresses in cells of types O, H and C are equal to the constitutive stresses deduced from the assumed strains in those cells.

{ } { }

^{Σ I = σ I =}

[ ]

^{C I}

{ }

^{ε I, I =1,N}_{O (VI.31-a)}

{ }

Σ ^A,J =

{ }

σ ^A,J =

[ ]

C ^J

{ }

ε ^A,J, J∈λ_H (VI.31-b)

{ }

Σ ^B,J =

{ }

σ ^B,J =

[ ]

C ^J

{ }

ε ^B,J, J∈λ_H (VI.31-c)

{ }

^{P0 L =}

[ ]

^{C L}

{ }

^{γ0 L, L∈λ}_{C (VI.31-d)}

{ } { }

^KΣ ^{L = K}Σ ^L, ^L∈λ_C (VI.31-e) Equilibrium equations associated with the degrees of freedom u_i^Jof the cells of type O

[ ] { } [ ] { } [ ] { } [ ] [ ]

{ }

^{[ ] { }}

{ }

^F~

{ }

^{T~ B} { }^{r B} { }^{r B} { }^{r J ,N}

K V P W R A

A A

O u

B u A

u t

C H

H O

M K

M K

) K ( KJ B M

K

) K ( KJ A KJ K

M K N IJ

I IJ

L L JL

JL T , L L I , B B IJ I I, A A IJ I I N IJ

I

1

1 1 1

1 1

0 1

= +

+ +

+

=

⎥⎦⎤

⎢⎣⎡ +

+ Σ +

Σ +

Σ

∑ ∑ ∑

∑

∑

∑

∑

∑

∑

= = =

=

=

∈ Σ

∈

∈

= λ λ λ

(VI.32)

Equilibrium equations associated with the degrees of freedom a_i^J of the cells of type H, C [ ] { }

[ ]

{ }

[ ]

{ } [ ] [ ]

[ { }

[ ] { }

]

{ }

+

{ }

+ { } + { } + { } ∈Λ

=

+ +

Σ +

Σ +

Σ

∑ ∑ ∑

∑

∑

∑

∑

∑

∑

= = =

=

=

∈ Σ

∈

∈

=

J r

B r

B r

B T

F

K V P W R A

A A

O u

B u A

u t

C H

H O

M

K

M

K

K KJ B C M

K

K KJ A C KJ K

C KJ C M

K IJ C N

I

L JL C JL L

C T L L I B IJ B C I I A IJ A C I I IJ C N

I

~ ,

~

1 1

) ( 1

) ( 1

1

, 0 ,

, ,

,

1 λ λ λ

(VI.33) Compatibility equations in the cells of type O

{ } [ ] { } [ ] { }

_O

J

J T IJ C J

T N IJ

J I

I A u A a I N

A ^{, ,} , 1,

1

= +

=

∑ ∑

Λ

∈

=

ε _(VI.34-a)

Compatibility equations in the cells of type H.

{ } [ ] { } [ ] { }

_H

J

J T IJ A C J

T IJ A N

J I A I

A A u A a I

A ε =

∑

+

∑

∈λ

Λ

∈

=

, ,

, ,

1 ,

, (VI.34-b)

{ } [ ] { } [ ] { }

_H

J

J T IJ B C J

T IJ B N

J I B I

B A u A a I

A ε =

∑

+

∑

∈λ

Λ

∈

=

, ,

, ,

1 ,

, (VI.34-c)

Compatibility equations in the cells of type C.

{ } [ ] [ ] { } [ ] { } [ ] { }

_C

I

L I T IL C N

I

L I T IL L

L L L

L D IH K W v W b L

A γ + =

∑

+

∑

∈λ

Λ

∈

Σ = ^{, ,} ,

1

, 0 ,

(VI.34-d)

[ ] { } [ ] { } [ ] { } [ ] { }

_C

I

L I T IL C N

I

L I T IL L

L L T

L HDH K V v V b L

IH γ + =

∑

+

∑

∈λ

Λ

∈

=

Σ ^{, ,} ,

1

, 0 ,

, (VI.34-e)

Chapter VI  Page 159  Compatibility equations with the imposed displacements on the boundaries on the edges of

cells of type O.

{ } { } { }

u^O

N K J

KJ J C N

J

KJ u J B a U K M

B ~ , 1,

1

=

= +

∑

∑

= ∈Λ

(VI.35-a) Compatibility equations with the imposed displacements on the boundaries on the edges of

cells of type H.

{ } { } { }

u^A

K N A

J KJ J C N

J

KJ u J B a U K M

B ~ ^{( )}, 1,

1

=

= +

∑

∑

= ∈Λ

(VI.35-b)

{ } { } { }

u^B

K N B

J KJ J C N

J

KJ u J B a U K M

B ~ ^{( )}, 1,

1

=

= +

∑

∑

= ∈Λ

(VI.35-c)

VI.5. Solution of the equations

VI.5.1. Stresses in terms of the displacement discretization parameters Introducing (VI.34-a,b,c) in (VI.31-a,b,c), we get:

{ } [ ] ^{[ ] { } [ ] { }}

O

J

J T IJ C J

T N IJ

J

I C I A ^, u A ^, a , I 1,N

1

* =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ +

=

Σ

∑ ∑

Λ

∈

=

(VI.36-a)

{ } [ ] ^{[ ] { }} [ ] { }

_H

J

J T IJ A C J

T IJ A N

J I I

A C A u A a I∈λ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ +

=

Σ

∑ ∑

Λ

∈

=

, ,

, ,

1

, * (VI.36-b)

{ } [ ] ^{[ ] { }} [ ] { }

_H

J

J T IJ B C J

T IJ B N

J I I

B C A u A a I∈λ

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ +

=

Σ

∑ ∑

Λ

∈

=

, ,

, ,

1

, * (VI.36-c)

Then, we use (VI.31-d,e) to rewrite (VI.34-d,e):

[ ] { } [ ] [ ] { } [ ] { } [ ] { }

_C

I

L I T IL C N

I

L I T IL L

L L L

L

L D P D IH K W v W b L

A + =

∑

+

∑

∈λ

Λ

∈

Σ = ^{, ,} ,

1

,

0 , (VI.36-d)

[ ] [ ] { } [ ] { } [ ] { } [ ] { }

_C

I

L I T IL C N

I

L I T IL L

L L L T

L D P HDH K V v V b L

IH + =

∑

+

∑

∈λ

Λ

∈

Σ = ^{, ,} ,

1

, 0 ,

, (VI.36-e)

with

[ ]

I

^{[ ]}

^I

I C

C* = A1 , I =1,N (VI.37)

From (VI.36-d,e), we can deduce

{ }

K_Σ ^L and

{ }

^{P0 L:}

{ } [ ] { } [ ] { }

_C

I

L , I L I b K N

I

L , I L I v K

L M v M b , L

K =

∑

+

∑

∈λ

Λ

∈ Σ

= Σ

Σ

1

(VI.38)

with