The Element-Free Galerkin Method applied on Polymeric Foams · applied on Polymeric Foams Author: Guilherme da Costa Machado Advisor: Marcelo Krajnc Alves, Ph.D. Co-Advisor: Rodrigo

The Element-Free Galerkin Method applied on Polymeric Foams

Author: Guilherme da Costa MachadoAdvisor: Marcelo Krajnc Alves, Ph.D.

Co-Advisor: Rodrigo Rossi, Dr.Eng.

Florianópolis - Brazil

September, 2006

Grupo de Mecânica Aplicada e Computacional

Federal University of Santa Catarina

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

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

Cellular solids

4

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

Sólidos celulares

Mec

han

ical

Pro

per

ties

G

IBSO

N &

ASH

BY(

1997

)

6

M

ech

anic

al P

rop

erti

es

GIB

SO

N &

ASH

BY(

1997

)

Cellular solids

Strain

Str

ess

Cellular walls/strutsbuckling plateau

Linear elastic flexion

Densification process

Elastoplastic foam behaviourunder compression

1lim

7

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

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

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

Kinematics of deformation Movement, strain gradient and the

multiplicative decomposition

Polar decomposition, Cauchy-Greentensor and the log strain measure

( , )

( , )X

e p

X t x X u

xX t

X

j

j

= = +

¶= =

¶=

F

F F F

( )

F R U F R U

U C

C F F

E Uln

T

p p p e e e

e e

e e e

e e

= =

=

=

=


11

Hill’s Principle: work rate invariability

1 1 1 1 1,

2e

o o o or r r r r= ⋅ = ⋅ = ⋅ = ⋅ = ⋅ D D P F S C Et t s

3

1

3

1

3

1

( )

( )

1ln( )( )

2

Te e e

ei i i

i

ei i i

i

ei 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


12

Kirchoff rotated stress

e T e= ( ) ( ) = det( )R R Ft t t swhere

Hyperelastic Hencky’s model

( )

[ ]

( )( )

22

3det

12

e

o

ijkl ik jl il jk

ij klijkl

K

r

r m r r m r

r r

d d d d

d d

*

* * * *

* *

= ( )

æ ö÷ç( ) = ( ) + ( )- ( ) Ä÷ç ÷çè ø

=

= +

Ä =

t

D I

I

E

I I

F

I I ( ) ( ) ME c Egr r* *=


GIBSON & ASHBY(1997)

13

22 2( , ) 0

2 2

o oc t c tp p p p

q p q pa aæ öé ù é ù- +÷ç= + - - £ê ú ê ú÷ç ÷÷ç ê ú ê úè øë û ë û

22 2( )( , )

pq p q pnb= +

( )pp vH e

( ),p pv aa e e

t cp pV=

( ) ( ) lno p p pc c p v vp p H Je e= + =-( ) ( ) ln

pp o p p

y a y a ao

LH

Lt e t e e

öæ ÷ç ÷= + =- ç ÷ç ÷çè ø

14Y

ield

Fu

nct

ion

an

d t

he

Pla

stic

Flo

w P

oten

tial

( )

( )

( )

1 1

1 1

1 1

1 1

2

1 1

1 1

,

3,

10

1 0

0, ,

in terms of

where 0

testen n

testen n

n n

teste testen 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 ùïïê ú+ -ïê úïë û é ùïï ê úï é ùï ê úï ê ú+ - = ê úí ê úï ê úë ûï ê úïï ê úë ûï Dïïïïî

D

D >

G

G

Elastic Prediction

p =F 0

1

testep pn n+ =F F

( )

[ ]

( )( )

1

1 1

11

1 1 1

1 1

1 1

1 1 1

1 1 1

det

1ln

2

( )

32

teste

teste teste teste

teste teste

teste

teste

teste teste

e pn n n

on

n

Te e en n n

e en n

en n

teste en n n

teste D Dn n n

p K tr

q

rr

r

r

-+ +

**+

+

+ + +

+ +

*+ +

*+ + +

+ + +

=

=

=

=

= ( )

é ù= - ê úë û

= ⋅

t

t t

F F F

F

C F F

E C

E

E

D

Plastic Corrector

( ) ( )1 1teste

n n+ +⋅ = ⋅

yes

no

11

expp pn n

n

l++

æ ö¶ ÷ç ÷= Dç ÷ç ÷ç ¶è øtF FG

End

11 1( , , ) 0p teste

n

teste testen n vq p e

++ + £

Return Mapping Algorithm

15

The global problem value

Strong formulation: reference configuration

• For each , determine that it is solution of

div ( , ) ( ) ( , ) 0

( , ) ( , ) ( , )

( , ) ( )

o o

to

uo

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 ∈ t o , t f


16

The global problem value

Weak formulation: reference configuration

• Find such that

where( )1; 0nu u ud d+ = " Î

F

( ) ( )

( ){ }( ){ }

1 1 1 1

1

1

;

,

, 0

to o o

n n o o n o n oX

ui p o

ui p o

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

1nu + Î


17

Local Linearization (Newton’s Method)

• Considers as being enough regular

and expanding by Taylor in terms of , we obtain

( ),⋅ ⋅F

( ) ( )11 1 1; ; 0k k k

n n nu u u u u ud d d++ + += +D = " Î

F F

( ) ( ) ( )( ) ( ) ( )

( )1

1 1 1 1 1

1 1 1 1

1 1 11

; ; ;

;o

kn

k k k k kn n n n n

k k k kn n n n oX X

ij ipkn jp ip jk lpijkl

kl klu

u u u u u D u u u

D u u u u u u d

Pu F F F

F F

d d d

d d

tt

+

+ + + + +

+ + + +W

- - -+

é ù+ D + Dë û

é ùD = ⋅ D ⋅ Wë û

¶ ¶é ù = = -ê úë û ¶ ¶

ò

F F F

F

1knu +


18

Global Equilibrium

Find , that satisfies the residual equilibrium equation1hnu +

( ) ( )int1 1 1 0h h ext

n n nr u f u f+ + += - =

( ) ( )( ) ( ) ( )

1

1 1 1

1 1 1 1 1

0k k k

k k k k k

h h hn n n

h h h h hn n n n n

r u r u u

r u u r u Dr u u

+

+ + +

+ + + + +

= +D =

é ù+ D + Dê úë û

( ) ( ) ( ) ( ) ( )11 1 1 t n

o o o

T T Tint h g h ext g gn n o n o o o o of u P u d f b d t dAr

++ + +W W G

= W = W +ò ò ò G

Global Linearization (Newton’s Method)

[ ] ( )K 1 1

1for

k k

k

g hn n

gn

u r u

u

+ +

+

D =-

D

( )1 1 1

kk k gn n nDr u u u+ + +

é ùD = Dë û K

( )o

Tg god

W= WòK G AG


19

While

End

Global Equilibrium AlgorithmStart

( )

( )( )

( ) ( )( )

( )

F I

F F F

C F F

C

U C

R F U

E U

1122

1 1

1

1 1

1 1 1

1

1 1

1

1 1 1

1 1ln

teste

teste teste teste

teste

teste teste

teste teste teste

teste teste

n nX

e pn n n

Te e en n n

en i i i

e en n i i i

e e en n n

e en n

u

l l

l l

l

l

+ +

-+ +

+ + +

+

+ +

-

+ + +

+ +

= +

=

=

= Ä

= = Ä

=

=

å

å

( ) ( )0 00 int1 1 1

0

h h extn n nr r u f u f

error r

+ + += = -

=

max

kr tol

k k

>

<

( )1

khnu +=K K

1

1 1 1

k k kg g gn n nu u u

+

+ + += +D

[ ] ( )1 1

k kg hn nu r u+ +D = - K

01 1

0

n n

k

u u+ +

¬

=

1kerror r +=

1

1

k kr r

k k

+¬

¬ +

Computing at the integration points

yes

no

20

Examples Axisymmetric Uniaxial Compression

0

0

30

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

*

=

=

=

=

=

=

=

=

=

30u mm= -


21TRI6

Uniaxial Compression (Axisymmetric)

Experimental data ZHANG, J. at al (1998)


22

Multiaxial Compression – Axisymmetric Cone Slice

3

0, 082

0, 040

0, 01

928, 092

0, 049 /

0, 00

0,25

1,54

0, 30

oyoc

m

o

p

Mpa

p MPa

E MPa

Kg m

c

t

V

rnng

*

=

===

=====

80u mm=


23

TRI6

Note: Calculated with TRI6 visualized as TRI3 element!

TRI6

TRI3

24

25

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


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

EFG FEM

LI & LIU(2002)27


Introduction

Enables the construction of an approximate function

that fits a discrete set of data

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

( )hu X

{ }, I=1,...,nI Iu u=

28

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

( ) { } { }( ){ }

[ ]

[ ] ( ) ( ) ( )

A

A

22

1 1

1

1

( ) ,

argmin

.

n nh

I I II I

m

n

I II

n

I I I II

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.


{ } ( ) ( ) ( )1

, I=1,...,n ,m

hI I j j

j

u u u x p x a p x a=

= = =å

discrete pointsapproximation function

29

Element-Free Galerkin Method (EFG) ( ) ( ) ( ) ( )

( )[ ] ( ) ( ) ( )

( ) ( )

( )[ ] ( )

( ) ( )[ ]

2

1

1

1

1

1

,n

I I II

n

I I II

I I I

n

I II

n

I II

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 úë û

é ù= - Äë û

= -

=

ì üï ïï ï= í ýï ïï ïî þ

å

å

å

å

A

A

A

( ) ( ) ( ) ( )[ ] ( )A 1

1 1

( ) , ,n n

hI I I I

I I

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

= =

= = = Få å

intrinsic base functionsglobal shape

functions

moment matrix

( ) 2 2 11, , , , , ,..., ,... ,T k k k

k

p x x y x xy y x xy y

Co

-é ù= ê úë û

whit consistency order

( ) ( ) ( )[ ]A 1,I Ix p x x b-F =

The Moving Least Square Approximation (MLSA)

30

Influence Domain definition

( )Iw x x-

Weight function – Quartic Spline

( )

max

max

2 3 41 6 8 3 para 1;

0 para 1;

max ,

II

I I

I i I Ii

r r r rw r

r

x xrr

r s r

r x x i L

ì - + - £ïïï= íï >ïïî

-= =

⋅

= - Î

rI max

x1

xI

x2

x3 x4

x5Influence

factor

Associated list

Parameterized radius


31

Stability Conditions LIU et al. (1996)

The choice of the size of the support to assure

The consistency order of the intrinsic base

( ){ } ( )card 0 dimi ix x xé ùF ¹ ³ ë û A

( )1

x-é ùë û

A

( ) 21, ,T

p x x y xé ù= W Î " Î Wê úë û

R


rI max

x1

xI

x2

x3 x4

x5

32


( ) 2 2 11, , , , , ,..., ,... ,T k k kp x x y x xy y x xy y-é ù= ê úë û


33



34

Imposition of the Essential Boundary Conditions using The Augmented Lagrange Method


( )I J IJx dF ¹

( )( )( ) ( )

1

1 1

1 1 1

1 1

1

k ikh

n

k i k i k kh g

n n

u h un u n n o

u

h g g h g g gn n u u

Q u u em

u u u u

le

d d l l

+

+ +

+ + +

+ +

é ùê ú= - + - Gê úë û

= = =

N

( )( ) ( ) ( )

1 1 1 1,kk i k i k i

u

uo

u h h u h u g u gn n o n nu u Q u d f u f u

ld d d d+ + + +G

= - ⋅ G = ⋅ + ⋅ò

F

35

New Problem Definition

Find that

1 1 1 1( , ) ( , ) ( , ) ( , ) 0k k u k c kn n n nu u u u u u u u ud d d d d+ + + += + + = " Î F F F F

1knu + Î

( ) ( )

( )

P

1

1

1 1 1 1

1 1 1

1 1 1

;

( , ) ( , , )

, ( , , )

to o o

k

u no

k

c no

n n o o n o n oX

u k u kn n n u u o

c k c kn n n o

u u u u d b u d t u dA

u u Q u u dA

u u Q u u dAn 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

( , ) ( , ) ( , )g X u X t Y X t X tn né ù= - + - ⋅ê úë û

[0,1]argmin ( , ) ( )

ll X u X t Y l*

Î= + -

37


Imposition of the Normal Contact and Friction Terms Normal and Tangential Works

( ) ( )( ) ( )( )1 1 1, , ,c c k i c k in n T nu u u u u und d d+ + += +

F F F

( )( ) ( )( ) ( )

( )( ) ( )( ) ( )

1 1

1 1 1

1 1 1

1 1

, , ,

, , ,

c k

c n no

c c k i

c n n no

c k i k i k i cn n n o

c k i k i cT n T n T T o

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


Normal Contact

Augmented Lagrange Method

( )( ) ( )( )1 1 11 1

1, ,c k k

n n n

k i k in nQ u g un n n n

n

e l le+ + ++ += +

( )( ) ( ) ( ) ( )

1 11 1,k i k i k ic c

c n no

c h h k i h c gn n ou u Q u d f un n nd n d d

+ ++ +G

= - ⋅ G = ⋅ò

F

39


Tangential Contact Regularized Coulomb's Law (Penality Method)

Stick condition

Slip condition

( ) ( )( ) ( ) ( )

( ) ( )

( )

( )

1 1 1 1

1

1 1

1

,

1

0

0

k i k i k i k ic c c c

n n n n

k ic

k ic k i n

k in n c

n

T T f

T TT

Q Q Q c Q

QQ u

Q

n n

n

n

ge

g

g

+ + + +

+

+ +

+

¡ = -

æ ö÷ç ÷ç ÷= - +ç ÷ç ÷ç ÷çè ø³

¡ =

0¡ £

0¡ >

40


QTn1cki cfQ n1

cki QTn1c teste

ki

QTn1c teste

ki

Frictional Algorithm

( ) ( )

1 1

k ic c k i

n n n nT T TQ Q u un + +

( )( )( )

1 1

1k itestec c k i

Tn n n nT T T TQ Q u ue+ +

= - -

( ) ( )

1 1

k ik i testec c

n nT TQ Q+ +

=

SLIP CONDITION

no

yes

STICK CONDITION

( ) ( )

1 1( , ) 0

k iteste k ic c

n nTQ Qn+ +¡ £

Start

End

41


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.

oy

oc

m

o

p

Mpa

p MPa

E MPa

c

t

V

r

n

n

g

*

=

=

=

=

=

=

=

=

=

61 10tol -=

ū −30mm

6

1,5

10u

s

e -

=

=

100

7

load steps

pintg

42


Uniaxial compression and the densification process

ZHANG, J. at al (1998)


43

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

61

6

10

1,5

10u

tol

s

e

-

-

=

=

=

1000

7

load steps

pintg

80u mm= -


44

45

BA

46

EFG

FEM

108,74.10-

( )71,28 15%-

0,00

61,22- 47

Note: Calculated with TRI6 visualized as TRI3 element!

TRI6 TRI3

Indentation Test6

1

62

10

10

tol

tol

-

-

=

=

4

3

6

1,5

0,10

10

10

10

f

T

u

s

c

ne

e

e

-

-

-

=

=

=

=

=

1000

7

steps

pintg

15u mm=-


48

3%

0.5489

0.5659

49

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*( )

( )( )=

rr

**

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

ee EE

F F F t

50

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

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

Documents

The Element-Free Galerkin Method applied on Polymeric Foams · applied on Polymeric Foams Author: Guilherme da Costa Machado Advisor: Marcelo Krajnc Alves, Ph.D. Co-Advisor: Rodrigo