[PPT] - The Element-Free Galerkin Method applied on Polymeric Foams Author: PowerPoint Presentation

SLIDE 1

The Element-Free Galerkin Method applied on Polymeric Foams

Author: Guilherme da Costa Machado Advisor: Marcelo Krajnc Alves, Ph.D. Co-Advisor: Rodrigo Rossi, Dr.Eng.

Florianópolis - Brazil September, 2006

Grupo de Mecânica Aplicada e Computacional

Federal University

f Santa Catarina

SLIDE 2



The Element-Free Galerkin Method



Moved List Square Approximation



Weight Function definition



Imposition of Essential Boundary Conditions



Celular Solids



Introduction



Characteristics



Mechanical Properties



Models examples



Methodology

Overview



Finite Strain Elastoplastic formulation



Kinematics of deformation



The concept of conjugated pairs



The Elastoplastic model



Strong, Weak and Incremental formulations



Solution Procedure



Examples (FEM)



Unilateral Contact and Friction Formulation



Problem definition



Imposition of the Contact and Friction terms



General Algorithm



Examples (EFG)



Conclusions

2

SLIDE 3

Cellular solids



Introduction



Typically used in energy absorption structures.



Applications areas:

Automotive/Transport industry;
Aerospace industry;
Packing industry;
Construction industry.



Mechanical X low density.

3

SLIDE 4

Cellular solids

4

SLIDE 5

Cellular solids



Relative density



Very low = 0,001;



Conventional = 0,05 to 0,20;



Transition value 0,3 treated as a solid with isolated pores.



Types



Polymeric



Metallic (aluminum, cooper, nickel, titanium and zinc)



Ceramic (carbon)



Natural (wood, cork and coral structures)

∗  /m

5

SLIDE 6

Sólidos celulares



Mechanical Properties

GIBSON & ASHBY(1997)

6

SLIDE 7



Mechanical Properties

GIBSON & ASHBY(1997)

Cellular solids

Strain Stress

Cellular walls/struts buckling plateau Linear elastic flexion Densification process

Elastoplastic foam behaviour under compression

1

lim



7

SLIDE 8

Cellular solids

 Modeling Approaches

 Periodical Models (GIBSON & ASHBY)

Dependence of the geometrical idealization;
Considers the local cell characteristics;
Limited applications;
Isotropy and cellular interaction, in the majority of the

cases, are not periodic phenomenon.

 Random models (ROBERTS,GARBOCZIA e BRYDON)

Voronoi tessellation and Gaussian random field;
Problems to define the RVE and the coordinate number;
Problems at the micro tomography correlation;
Border effects;
Self-contact densification process involves a huge

computational cost;

Explicit solver.

8

SLIDE 9

Cellular solids



Elastoplastic model for polymeric foams



Methodology



Finite Stains Algorithm;



Total Lagrange Description;



Rotated Kirchoff stress;



Hencky Logarithmic strain measure;



Volumetric hardening Law;



Finite Element Method - FEM;



Element Free Galerkin Method - EFG;



Contact formulation (Signorini hypothesis);



Friction formulation (regularized Coulomb’s law);

9

SLIDE 10



Required formulations

Finite Strain Elastoplastic Formulation



Stress/Strain elastic relation – an elastic law;



Yield function - indicating the stress level to start the plastic flow;



Stress/Strain plastic relation – material hardening/softening law;

10

SLIDE 11



Kinematics of deformation



Movement, strain gradient and the multiplicative decomposition



Polar decomposition, Cauchy-Green tensor and the log strain measure

( , ) ( , )

X e p

X t x X u x X t X j j = = + ¶ =  = ¶ =



       F F F F (

)

F R U F R U U C C F F E U ln

T

p p p e e e e e e e e e e

= = = = =

Finite Strain Elastoplastic Formulation

11

SLIDE 12



Hill’s Principle: work rate invariability

1 1 1 1 1 , 2

e

r

r r r r = ⋅ = ⋅ = ⋅ = ⋅ = ⋅     D D P F S C E t t  s

3 1 3 1 3 1

( ) ( ) 1 ln( )( ) 2

T

e e e e i i i i e i i i i e i i i i

l l l l l l l l l

= = =

= = Ä = Ä = Ä

å å å

      C F F C U E



Spectral decomposition

Finite Strain Elastoplastic Formulation

12

SLIDE 13



Kirchoff rotated stress

e T e

= ( ) ( ) = det( ) R R F t t t s

where



Hyperelastic Hencky’s model

( ) [ ]

( )

( ) 2 2 3 det 1 2

e

ijkl

ik jl il jk ij kl ijkl

K r r m r r m r r r d d d d d d

* * * * * * *

= ( ) æ ö ÷ ç ( ) = ( ) + ( )- ( ) Ä ÷ ç ÷ ç è ø = = + Ä = t  D I I E I I F I I

( ) ( )

M

E c E

g

r r

* *

=

Finite Strain Elastoplastic Formulation

GIBSON & ASHBY(1997)

13

SLIDE 14

2 2 2

( , ) 2 2

c

t c t

p p p p q p q p a a æ ö é ù é ù

+

÷ ç = +

£

ê ú ê ú ÷ ç ÷ ÷ ç ê ú ê ú è ø ë û ë û 

2

2 2 ( )

( , )

p

q p q p

n

b = + 

( )

p p v

H e

( )

,

p p v a

a e e

t c

p p V =

( ) ( )

ln

p

p p c c p v v

p p H J e e = + = -

( ) ( )

ln

p p

p

p y a y a a

L

H L t e t e e ö æ ÷ ç ÷ = + = - ç ÷ ç ÷ ç è ø

14



Yield Function and the Plastic Flow Potential

SLIDE 15

( ) ( )

( )

1 1 1 1 1 1 1 1 2 1 1 1 1

, 3 ,

1 1 , , in terms of where

teste n n teste n n n n teste teste n n n n n n

p q p q

p p q q q p p q

kb l m l

l l l

+ + + + + + + + + + + +

D D

ìé ù ï ïê ú +

ïê

ú ïë û é ù ï ï ê ú ïé ù ï ê ú ïê ú +

= ê ú

íê ú ï ê ú ë û ï ê ú ï ï ê ú ë û ï D ï ï ï ï î D D >

G G

 Elastic Prediction

p =

 F

1

teste

p p n n +

= F F

( )

[ ]

( ) ( )

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

det 1 ln 2 ( ) 3 2

teste teste teste teste teste teste teste teste teste teste

e p n n n

n

n T e e e n n n e e n n e n n teste e n n n teste D D n n n

p K tr q r r r r

+

+ * * + + + + + + + * + + * + + + + + +

= = = = = ( ) é ù = - ê ú ë û = ⋅ t t t F F F F C F F E C E E D

Plastic Corrector ( ) ( )

1 1 teste n n + +

⋅ = ⋅

yes no

1 1

exp

p p n n n

l

+ +

æ ö ¶ ÷ ç ÷ = D ç ÷ ç ÷ ç ¶ è ø t F F G

End

1

1 1

( , , )

p teste n

teste teste n n v

q p e

+

+ +

£ 

Return Mapping Algorithm

15

SLIDE 16



The global problem value



Strong formulation: reference configuration

For each , determine that it is solution of

div ( , ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( )

t
u
X t

X b X t em X t N X t t X t em u X t u X em r

=

W = G = G              P P

( , ) u X t  

t ∈ to,tf

Finite Strain Elastoplastic Formulation

16

SLIDE 17



The global problem value



Weak formulation: reference configuration

Find such that

where

( )

1; n

u u u d d

+

= " Î     F

( )

{ }

( )

{ }

1 1 1 1 1 1

; , ,

t

n

n

n
n
X

u i p

u

i p

u

u u u d b u d t u dA u u W u u em u u W u em d d r d d d d d

+ + + + W W G

= ⋅  W - ⋅ W - ⋅ = Î W = G = Î W = G

ò ò ò



             F P  

1 n

u + Î  

Finite Strain Elastoplastic Formulation

17

SLIDE 18



Local Linearization (Newton’s Method)

Considers as being enough regular

and expanding by Taylor in terms of , we obtain

( ) , ⋅ ⋅ F

( ) ( )

1 1 1 1

; ;

k k k n n n

u u u u u u d d d

+ + + +

= + D = " Î        F F

( ) ( ) ( ) ( )

( ) ( ) ( )

1

1 1 1 1 1 1 1 1 1 1 1 1 1

; ; ; ;

k

n

k k k k k n n n n n k k k k n n n n

X

X ij ip k n jp ip jk lp ijkl kl kl u

u u u u u D u u u D u u u u u u d P u F F F F F d d d d d t t

+

+ + + + + + + + + W

+

é ù + D + D ë û é ù D = ⋅  D ⋅  W ë û ¶ ¶ é ù = =

ê

ú ë û ¶ ¶

ò

  

                F F F F  

1 k n

u + 

Finite Strain Elastoplastic Formulation

18

SLIDE 19



Global Equilibrium



Find , that satisfies the residual equilibrium equation

1 h n

u + 

( ) ( )

int 1 1 1 h h ext n n n

r u f u f

+ + +

=

=

    

( ) ( ) ( ) ( ) ( )

1

1 1 1 1 1 1 1 1

k k k k k k k k

h h h n n n h h h h h n n n n n

r u r u u r u u r u Dr u u

+

+ + + + + + + +

= + D = é ù + D + D ê ú ë û               

( )

( ) ( )

1

1 1 1

t n

T

T T int h g h ext g g n n

n
f

u P u d f b d t dA r

+

+ + + W W G

= W = W +

ò ò ò

       G  



Global Linearization (Newton’s Method) [ ]

( )

K

1 1 1

for

k k k

g h n n g n

u r u u

+ + +

D = - D    

( )

1 1 1

k

k k g n n n

Dr u u u

+ + +

é ù D = D ë û     K

( )

T

g g

d

W

= W

ò

K G AG

Finite Strain Elastoplastic Formulation

19

SLIDE 20

While

End

Global Equilibrium Algorithm

Start ( )

( )

( ) ( )

F I F F F C F F C U C R F U E U

1 1 2 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

ln

teste teste teste teste teste teste teste teste teste teste teste teste

n n X e p n n n T e e e n n n e n i i i e e n n i i i e e e n n n e e n n

u l l l l l l

+ +

+

+ + + + + + +

+

+ + + +

= +  = = = Ä = = Ä = =

å å

 

   

( ) ( )

int 1 1 1 h h ext n n n

r r u f u f error r

+ + +

= =

=

      

max k

r tol k k > < 

( )

1

k

h n

u + =  K K

1

1 1 1

k k k

g g g n n n

u u u

+

+ + +

= + D    [ ]

( )

1 1

k k

g h n n

u r u

+ +

D = -    K

1 1 n n

k u u

+ +

¬ =  

1 k

error r

+

= 

1

k k

r r k k

+

¬ ¬ +  

Computing at the integration points

yes no 20

SLIDE 21



Examples



Axisymmetric Uniaxial Compression

3

0,082034 0,040470 0,10 928,092 0,049 / 0,00 0,25 1,54 0,30

y c m p

Mpa p MPa E MPa Kg m c t V r n n g

*

= = = = = = = = =

30 u mm = -

Finite Strain Elastoplastic Formulation

21

TRI6

SLIDE 22



Uniaxial Compression (Axisymmetric)

Experimental data ZHANG, J. at al (1998)

Finite Strain Elastoplastic Formulation

22

SLIDE 23



Multiaxial Compression – Axisymmetric Cone Slice

3

0,082 0,040 0,01 928,092 0,049 / 0,00 0,25 1,54 0,30

y
c

m

p

Mpa p MPa E MPa Kg m c t V r n n g

*

= = = = = = = = =

80 u mm =

Finite Strain Elastoplastic Formulation

23

TRI6

SLIDE 24

Note: Calculated with TRI6 visualized as TRI3 element!

TRI6 TRI3

24

SLIDE 25

25

SLIDE 26



The discretization of the differential equations system uses the weak Galerkin form;



Do not required a finite element mesh;



The discretization is based on a set of nodes (discrete data);



The connection in terms of nodes interactions may be change constantly, and modeling fracture (ability to simulate crack growth), free surfaces, large deformations, etc. is considerably simplified;



Accuracy can be controlled more easily, since in areas where more refinement is needed, nodes can be added easily;



Meshfree discretization can provide accurate representation of geometric object;



A special strategy to impose the essential boundary conditions.



Main characteristics BELYTSCHKO et al. (1994)

Element-Free Galerkin Method (EFG)

26

SLIDE 27

Element-Free Galerkin Method (EFG)

Comparison of the deformations at different time stages for a block of hyperelastic material under compression by using:

EFG FEM

LI & LIU(2002)

27

SLIDE 28

Element-Free Galerkin Method (EFG)



Introduction



Enables the construction of an approximate function that fits a discrete set of data



Moving Least Square Approximation (MLSA). LANCASTER & SALKAUSKAS (1981);

( )

h

u X

{ }

, I=1,...,n

I I

u u =

28

SLIDE 29



Simple Least Squares Method



Finding the best-fitting curve to a given set of discrete points (or nodes)

( )

{ }

( )

{ }

[ ] [ ]

( ) ( ) ( )

A A

2 2 1 1 1 1

( ) , arg min .

n n h I I I I I m n I I I n I I I I I

J a u x u p a u a J a a a u b p x p x b p x

* * = = * * = =

=

=
=

" = é ù = Ä = ë û

å å å å

            R

where and



ALL values uI , I=1,..n exerts influence to determinate ALL the aj coefficients, for the n discrete points.

Element-Free Galerkin Method (EFG)

{ }

( ) ( ) ( )

1

, I=1,...,n ,

m h I I j j j

u u u x p x a p x a

=

= = =

å

    

discrete points approximation function

29

SLIDE 30

Element-Free Galerkin Method (EFG)

( ) ( ) ( ) ( )

( )

[ ]

( ) ( ) ( ) ( ) ( )

( )

[ ]

( ) ( ) ( )

[ ]

2 1 1 1 1 1

,

n I I I I n I I I I I I I n I I I n I I I

J a w x x p x a x u x w x x p x p x b w x x p x x a x u b a x x u b

= = =

=

é ù =

ê

ú ë û é ù =

Ä

ë û =

=

ì ü ï ï ï ï = í ý ï ï ï ï î þ

å å å å

               A A A

( ) ( ) ( ) ( )

[ ]

( )

A

1 1 1

( ) , ,

n n h I I I I I I

u x p x a x u p x x b u x

=

=

= = = F

å å

   

intrinsic base functions global shape functions moment matrix

( )

2 2 1

1, , , , , ,..., ,... ,

T k k k k

p x x y x xy y x xy y Co

é

ù = ê ú ë û  

whit consistency order ( ) ( ) ( )

[ ] A

1

,

I I

x p x x b

F

=   The Moving Least Square Approximation (MLSA)

30

SLIDE 31



Influence Domain definition

( )

I

w x x







Weight function – Quartic Spline

( )

max max

2 3 4

1 6 8 3 para 1; para 1; max ,

I I I I I i I I i

r r r r w r r x x r r r s r r x x i L ì - +

£

ï ï ï = í ï > ï ï î

=

= ⋅ =

Î

    

rI max

x1 xI x2 x3 x4 x5

Influence factor Associated list Parameterized radius

Element-Free Galerkin Method (EFG)

31

SLIDE 32



Stability Conditions LIU et al. (1996)



The choice of the size of the support to assure



The consistency order of the intrinsic base

( )

{ }

( )

card dim

i i

x x x é ù F ¹ ³ ë û    A

( )

1

x

é

ù ë û  A

( )

2

1, ,

T

p x x y x é ù = W Î " Î W ê ú ë û    R

Element-Free Galerkin Method (EFG)

rI max

x1 xI x2 x3 x4 x5

32

SLIDE 33



Stability Conditions LIU et al. (1996)

( )

2 2 1

1, , , , , ,..., ,... ,

T k k k

p x x y x xy y x xy y

é

ù = ê ú ë û  

Element-Free Galerkin Method (EFG)

33

SLIDE 34



Stability Conditions LIU et al. (1996)

Element-Free Galerkin Method (EFG)

34

SLIDE 35



Imposition of the Essential Boundary Conditions using The Augmented Lagrange Method

Element-Free Galerkin Method (EFG)

( )

I J IJ

x d F ¹ 

( )

( ) ( ) 1 1 1

1 1 1 1 1

1

k i k h n k i k i k k h g n n

u h u n u n n

u

h g g h g g g n n u u

Q u u em u u u u l e d d l l

+ + +

+ + + + +

é ù ê ú = - +

G

ê ú ë û = = =             N

( )

( ) ( )

1 1 1 1

,

k k i k i k i u u

u

h h u h u g u g n n

n

n

u u Q u d f u f u

l

d d d d

+ + + + G

= - ⋅ G = ⋅ + ⋅

ò

        F

35

SLIDE 36



New Problem Definition



Find that

1 1 1 1

( , ) ( , ) ( , ) ( , )

k k u k c k n n n n

u u u u u u u u u d d d d d

+ + + +

= + + = " Î            F F F F

1 k n

u + Î  

( )

P

1 1

1 1 1 1 1 1 1 1 1 1

; ( , ) ( , , ) , ( , , )

t

k

u n

k

c n

n

n

n
n
X

u k u k n n n u u

c

k c k n n n

u

u u u d b u d t u dA u u Q u u dA u u Q u u dA

n n

d d r d d d e l d d e l d

+ +

+ + + + W W G + + + G + + + G

= ⋅ W - ⋅ W - ⋅ = - ⋅ = - ⋅

ò ò ò ò ò



                   F F F

where

Unilateral Contact with Friction

36

SLIDE 37

( , ) ( , ) ( , ) g X u X t Y X t X t

n

n é ù = - +

⋅

ê ú ë û       

[0,1]

arg min ( , ) ( )

l

l X u X t Y l

* Î

= +



  

37

Unilateral Contact with Friction

SLIDE 38



Imposition of the Normal Contact and Friction Terms



Normal and Tangential Works

( )

1 1 1

, , ,

c c k i c k i n n T n

u u u u u u

n

d d d

+ + +

= +       F F F

( )

( ) ( )

( )

( ) 1 1 1 1 1

1 1 1 1 1

, , , , , ,

c k c n n

c

c k i c n n n

c

k i k i k i c n n n

c

k i k i c T n T n T T

u

u Q u u d u u Q u Q e u d

n n n n n

d e l n d d e d

+ + + + +

+ + + G + + G

= - ⋅ G = - ⋅ G

ò ò

          F F

38

Unilateral Contact with Friction

SLIDE 39



Normal Contact



Augmented Lagrange Method

( )

1 1 1

1 1

1 , ,

c k k n n n

k i k i n n

Q u g u

n n n n n

e l l e

+ + +

+ +

= +  

( )

( ) 1 1

1 1

,

k i k i k i c c c n n

c

h h k i h c g n n

u

u Q u d f u

n n n

d n d d

+ +

+ + G

= - ⋅ G = ⋅

ò

      F

39

Unilateral Contact with Friction

SLIDE 40

 Tangential Contact



Regularized Coulomb's Law (Penality Method)



Stick condition



Slip condition

( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1

, 1

k i k i k i k i c c c c n n n n k i c k i c k i n k i n n c n

T T f T T T

Q Q Q c Q Q Q u Q

n n n n

g e g g

+ + + + + + + +

¡ =

æ

ö ÷ ç ÷ ç ÷ = - + ç ÷ ç ÷ ç ÷ ç è ø ³ ¡ =     

¡ £ ¡ >

40

Unilateral Contact with Friction

SLIDE 41

QTn1

cki  cfQ n1 cki QTn1

ctesteki

QTn1

ctesteki



Frictional Algorithm

( ) ( ) 1 1 k i c c k i n n n n

T T T

Q Q u u

n +

+

( ) ( )

( )

1 1

1

k i teste c c k i T n n n n

T T T T

Q Q u u

e

+ +

=

( )

( ) 1 1 k i k i teste c c n n

T T

Q Q

+ +

=

SLIP CONDITION

no yes

STICK CONDITION

( ) ( ) 1 1

( , )

k i teste k i c c n n

T

Q Qn

+ +

¡ £ Start End

41

Unilateral Contact with Friction

SLIDE 42



Examples using EFG



Uniaxial Compression (with the same d.o.f of the FEM example) 0,082034 0,040470 0,50 928,09288 0,049 0,00 0,25 1,54 0,30.

y
c

m

p

Mpa p MPa E MPa c t V r n n g

*

= = = = = = = = =

6 1

10 tol

=

ū  −30mm

6

1,5 10

u

s e

=

= 100 7 load steps pintg

42

Unilateral Contact with Friction

SLIDE 43

 Uniaxial compression and the densification process

ZHANG, J. at al (1998)

Unilateral Contact with Friction

43

SLIDE 44

 Axisymmetric Cone Slice

(with the same d.o.f of the FEM example)

6 1 6

10 1,5 10

u

tol s e

=

= =

1000 7 load steps pintg

80 u mm = -

Unilateral Contact with Friction

44

SLIDE 45

45

SLIDE 46

B A

46

SLIDE 47

EFG FEM

10

8,74.10-

( )

71,28 15%



0,00 61,22

47

Note: Calculated with TRI6 visualized as TRI3 element!

TRI6 TRI3

SLIDE 48



Indentation Test

6 1 6 2

10 10 tol tol

=

=

4 3 6

1,5 0,10 10 10 10

f T u

s c

n

e e e

=

= = = = 1000 7 steps pintg

15 u mm = -

Unilateral Contact with Friction

48

SLIDE 49

3%

0.5489 0.5659 49

SLIDE 50

Conclusions

 The constitutive model  The features  a single-surface yield criteria;  a non-associated plastic flow law;  the relative density dependence ;  Show a good prediction for the responses of rigid polymeric foams under simple and complex monotonic loading conditions.  The yield surface parameters are simple and can be obtained from two independent experiments

 Uniaxial compression (ISO 844 / ASTM D1621-04a);  Hydrostatic compression;

 The relative density dependence has as a consequence a correction in the tangent modulus.

r* ( ) 

( ) ( )

= r r

* *

æ ö æ ö ¶ ¶ ¶ ÷ ç ÷ ç + ÷ ç ÷ ç ÷ ÷ ç ç ÷ è ø ¶ ¶ ¶ è ø

e e

E E F F F   t

50

SLIDE 51

 EFG x FEM  EFG method showed to be:

 Able to withstand the analysis of very large deformation processes, without remeshing and breaking up;  More robust to capture high deformation levels and deformation gradients;  More expensive with respect the computational aspect (more integration points);  In another hand, load step size bigger than the FEM. (small number of interactions!)

 The choice of the penalties values (in both cases EBC and Contact) implies in changes at the convergence rate;

 Difficulties to find the “right” value for each problem.

Conclusions

51

SLIDE 52

 Numerical Aspects  The Returning Map Algorithm do not capture critical points (limit or path bifurcation).  For this reason, strategies was performed to improve the condition number of the local tangent modulus at the plateaus. (matrix is singular if the condition number is infinite)  Compression causes stress bottom up due to foam consolidation.  Problems to insure convergence (Turning causes low convergence rates);  Further, the hardening laws are interpolated by a polynomial fitting. After 60% of the logarithm strain, this fitting don’t represent the experimental data.  For this reason, the divergence between the numerical and experimental data is plausibly.  At overall, the numerical procedure showed to be adequate to describes the rough non-linearities (material and contact with friction).

Conclusions

52