1 Aerospace & Mechanical Engineering Department Continuum Mechanics & Thermomechanics Tel:...

1

Aerospace & Mechanical Engineering DepartmentContinuum Mechanics & Thermomechanics

Tel: +32-(0)4-366-48-26 Chemin des chevreuils 1 Fax: +32-(0)4-366-91-41 B-4000 Liège - Belgiume-mail: [email protected] www.ltas-mct.ulg.ac.be/who/noels/

A new energy-dissipative momentum-Conserving time integration scheme using the variational

updates framework

L. Noels, L. Stainier, J.-P. PonthotUniversity of Liège, LTAS-MCT, Belgium

Computational Methods in Structural Dynamics and Earthquake Engineering 2007

13 – 16 June 2007, Rethymno, Crete, Greece

2

Scope of the presentation

1. Scientific motivations

2. Conserving scheme in the non-linear range

3. New formulation using the variational updates approach

4. Numerical examples

5. Conclusions

3

1. Scientific motivationsDynamic simulations

Spatial discretization into finite elements Temporal integration of the balance equations 2 methods

– Explicit method

• Non iterative

• Small memory requirement

• Conditionally stable (small time step)

– Implicit method

• Iterative

• Larger memory requirement

• Unconditionally stable (large time step)

11intext1

1 ,deductionionapproximat

nnnnn

n

n

n

xxFFMx

x

x

x

),,,,(

),,,,(iterationsionextrapolat

111

111

extint

1

1

1

nnnnnn

nnnnnn

n

n

n

n

n

n

xxxxxfx

xxxxxfx

FFxM

x

x

x

x

x

x

Very fast dynamics

Slower dynamics

extint FFxM

4

1. Scientific motivationsNon-linear numerical example

Traditional methods can be unstable in the non-linear range Example (Armero & Romero [CMAME, 1999])

– Mass spring system with geometrical non-linearity– 1 revolution ~ 4s

Conservation laws are not satisfied leading to divergence, and motivating the development of conserving algorithms

Newmark implicit scheme (no numerical damping)

HHT implicit scheme

(numerical damping ∞= 0.9)

t=1s t=1.5s t=1s t=1.5s

-20 0 20-20

-10

0

10

20

X [m]

Y [

m]

Newmark; t = 1 s

-20 0 20-20

-10

0

10

20

X [m]

Y [

m]

Newmark; t = 1.5 s

-20 0 20-20

-10

0

10

20

X [m]

Y [

m]

HHT; t = 1 s

-20 0 20-20

-10

0

10

20

X [m]

Y [

m]

HHT; t = 1.5 s

5

2. Conserving scheme in the non-linear range Conservation laws

Conservation of linear momentum (Newton’s law )

– Time discretization: &

Conservation of angular momentum ( )

– Time discretization: &

Conservation of energy ( )

– Time discretization:

&

nodes

ext2/12/1

nodes nodes11 nnnnnn FxtxMxxMx

nodes

1ext

2/1intintint

1nodes

11 ][2

1

2

1nnnnnnnnn xxFDWWxxMxxM

nodes

ext2/1

nodes1 nnn FtxMxM

Wint: internal energy; Wext: external energy; Dint: dissipation (plasticity …)

extFt

xM

extFxt

xMx

intextint DWt

Wt

Kt

0nodes

int2/1 nF

0nodes

int2/12/1 nn Fx

nodes

1int

2/1intintint

1 ][ nnnnn xxFDWW

6

First Energy-Momentum Conserving Algorithm (EMCA) (Simo and Tarnow [ZAMP, 1992])

– Based on the mid point scheme:

– With and designed to verify conserving equations

– Example of the mass-spring system

2. Conserving scheme in the non-linear range EMCA

int2/1

ext2/1

1

nnnn FF

t

xxM

t

xxxx nnnn

11

2

nnnn

nnn xx

ll

lVlVF

122

1

1int2/1

)()(

ext2/1nF

EMCA, t=1s EMCA, t=1.5s

int2/1nF

-20 0 20-20

-10

0

10

20

X [m]

Y [

m]

EMCA; t = 1 s

-20 0 20-20

-10

0

10

20

X [m]

Y [

m]

EMCA; t = 1.5 s

0 100 200 300 400 500-250

-200

-150

-100

-50

0

Time [s]

Ang

ular

mom

entu

m [

kg m

2 /s]

EMCA; t = 1 sNewmark; t = 1 s

7

Introduction of numerical dissipation: Energy-Dissipative Momentum Conserving scheme (EDMC) (Armero & Romero [CMAME, 2001])

– Introduction of new terms in the EMCA:

– Same internal and external forces as in the EMCA

– With and designed to achieve positive numerical dissipation without

spectral bifurcation

– Example of the mass-spring system for first order accurate EDMC

2. Conserving scheme in the non-linear range EDMC

int2/1nF

ext

2/1nF

diss2/1nF

diss2/1

int2/1

ext2/1

1

nnnnn FFF

t

xxM

t

xxx

xx nnn

nn

1diss

11

2

diss1nx

nodes

1ext

2/1numintintint

1nodes

11 ][2

1

2

1nnnnnnnnn xxFDDWWxxMxxM

2

1

1

1diss1

nn

nn

nn

n

xx

xx

xxx

2

1

1

21''

diss2/1

1 nnll

nn

nlnl

n

xx

ll

VF nn

0 100 200 300 400 500-250

-200

-150

-100

-50

0

Time [s]

Ang

ular

mom

entu

m [

kg m

2 /s]

EDMC; = 0.9; t = 1 s

HHT; = 0.9; t = 1 s

0 100 200 300 400 5005

10

15

Time [s]

Spr

ing

leng

th [

m]

EDMC; = 0.9; t = 1 s

HHT; = 0.9; t = 1 s

9

2. Conserving scheme in the non-linear rangeFormulations in the literature

Elastic formulation– Saint Venant-Kirchhoff hyperelastic model (Simo et al. [ZAMP, 1992])

– General formulation for hyperelasticity (stress derived from a potential V)

(Gonzalez [CMAME, 2000])

with

– Classical formulation

Penalty contact formulation (Armero & Petöcz [CMAME, 1998-1999])

0xD

2

1

2

11

12/1

:2

22

2

nn

nnnn

nnn

O

VVV

V

CC

CC

CCC

CCC

CC

CS

F: deformation gradient; C: right Cauchy-Green strain; V: potential; : shape functions;

0

2int

V

dVDV

F

CF

)()( 2/12/1112/1dd

1 nnnnnnnnn uyuyxxngg

2/1dd

1

dd1cont

2/1

n

nn

nnn n

gg

gVgVF

0

2/11int

2/1 2V

nnn

n dVDF

SFF

10

2. Conserving scheme in the non-linear rangeFormulations in the literature

Elasto-plastic materials– Hyperelasticity with elasto-plastic behavior (Meng & Laursen [CMAME, 2001])

• energy dissipation of the algorithm corresponds to the internal dissipation of the material

• Isotropic hardening only

– Hyperelasticity with elasto-plastic behavior (Armero [CMAME, 2006])

• Energy dissipation from the internal forces corresponds to the plastic dissipation

• Modification of the radial return mapping

• Yield criterion satisfied at the end of the time-step

– Hypoelastic formulation• Stress obtained incrementally from a hardening law

• No possible definition of an internal potential!

• Idea: the internal forces are established to be consistent on a loading/unloading cycle

• Assumption made on the Hooke tensor

• Energy dissipation from the internal forces corresponds to the plastic dissipation

12

3. Variational updates approachPurpose of the work

Development of a general approach leading to conserving/dissipating algorithms for any material behavior!

What we want– No assumption on the material behavior

– Material model unchanged compared to the standard approach:• From a given strain tensor, the outputs of the model are the same

• Use of the same material libraries

– Expression of the internal forces for the conserving algorithm remains the same as in the elastic case

– Yield criterion satisfied at the end of the time step

Solution derives from the variational formulation of visco-plastic updates [Ortiz & Stainier, CMAME 1999] – Definition of an energy, even for complex material behaviors

Introduction of numerical dissipation via Deff

nnn

n

DCC

CS ,2 1

1

eff

1

C: right Cauchy-Green strain; S: second Piola-Kirchhoff stress; Deff: incremental potential

13

3. Variational updates approachUse of an incremental potential

Description of the variational updates for elasto-plasticity– Multiplicative plasticity

– Plastic flow

– Functional increment

– Effective potential:• Minimization with respect to pl satisfies yield criterion

• Minimization with respect to N satisfies radial return mapping

– Usual stress derivation:

Introduction of numerical dissipation

– Modification of potential

– Dissipation function

plelFFF

plplpl1 )exp( nn FNF

ttWW

WWD

nnnn

nnnnnnnnnn

plpl1*plplpl

1pl

pl1plelpl1

1pl11

elplpl11 ),(),(),,,,(

NFFNFFNFF

),,,,(min pl1

pl11

,1

eff

pl1

NFFFN

nnnnn DDn

1

1eff

1 2

n

nn

D

C

FS

Wel: reversible potential; Wpl: dissipation by plasticity; : dissipation by viscosity

N: plastic flow direction

ttDD nn

nnnnnn

elel1disspl

1pl

11,

1eff ),,,,(min

pl1

CCNFFF

N

elelel

el2el

elel1diss ::

2C

CCC

CC

V

ttnn

14


Internal forces directly obtained from Gonzalez elastic formulation [CMAME 2000]

with

Dissipation forces are part of

C

C

CCC

CCC

CC

CS

2

1eff

1eff

1eff

2/1

:2

,2

22

nnnn

nnn

DD

D

2

1

2

11

12/1

:2

22

2

nn

nnnn

nnn

O

VVV

V

CC

CC

CCC

CCC

CC

CS

0

2/11int

2/1 2V

nnn

n dVDF

SFF

int2/1nF

diss

2/1nF

15


Modification to quasi-incompressible form– Definition of a mean volumetric deformation on the element e.

– Definition of distortion gradient

– Split of the stress tensor

– Volume part

– Deviatoric part

with , and resulting from the minimization of

dGSS 2/1dev

2/12/1 2 nnn p

CC

CC

CdG

2

2/1

2/1:n

n

JJJ

e

e

ee

en

ene

nen

e

en

en

n

DD

Dp

2

1eff

1eff

1eff

2/1

22,

22

C

C

CCC

CC

C

CS

2

el2/1

effelel

1eff

el2/1

effdev

2/1

ˆˆ,ˆ

2ˆ

2e

DD

D

nnn

e

nn

FF 3/1ˆ J

T

nenn

nne

n DJ

D pl2/1

elel2/1

eff1pl

2/13/22/1

el2/1

eff

ˆ

ˆ,ˆDEV

ˆF

C

CCF

C

C

n

n

nnnD pl2/1pl1 ,,,,2

CNCCel

2/1ˆ

nCpl2/1nF2/1nJ

16


Properties:– 2 material configurations computed

• Mid configuration trough

• Final configuration trough

– Material model unchanged

– Yield criterion verified at configuration n+1

– Conservation of linear and angular momentum

– Numerical dissipation of energy (first order accurate)

nnD CC ,1eff

21

effnnD CC

C

dVttWWWWxxFV

nnnnnodes

nnn

0

diss*plpl1

elel11

int2/1 ][

dVV

xxmxxxMdVtDV

nnnnn

V

00

elelel

el2el2

1diss

11dissnum ::

22C

CCC

nodes

1ext

2/1numintintint

1nodes

11 ][2

1

2

1nnnnnnnnn xxFDDWWxxMxxM

dVWWV

0

elint

dVtWWDV

nn

0

*plpl1

int

17

4. Numerical ExamplesSimulation of a tumbling beam

Tumbling beam– Initial symmetrical loads (t < 10s)– Elasto-perfectly-plastic hyperelastic material

EMCA, t=0.5 s EDMC, t=0.5s, ∞= 0.7

18


Time evolution of the results

0 50 100 150 2000

5

10

15

20

25

Time [s]

Ene

rgy

[J]

Dint

W int

Dint + W int

Fint . x

0 50 100 150 2000

20

40

60

Time [s]

Ene

rgy

[J]

K Dint + W int

Dint + W int + K Fext . x

EMCA, t=0.5 s EDMC, t=0.5s, ∞= 0.7

0 50 100 150 2000

5

10

15

20

25

Time [s]

Ene

rgy

[J]

Dint

W int

Dint + W int

(Fint. x-V t diss)

0 50 100 150 2000

20

40

60

Time [s]

Ene

rgy

[J]

K

Dint + W int

Dint + W int + K Fext . x

).( 0V

dV txF dissint

19


Influence of time step size (∞= 0.7)

Influence of ∞ (t = 0.5)

0 50 100 150 2000

20

40

60

Time [s]

Din

t + W

int +

K [

J]

EDMC; t = 0.125 sEDMC; t = 0.25 sEDMC; t = 0.5 sEDMC; t = 0.75 sNewmark; t = 0.75 s

0 50 100 150 2000

5

10

15

20

25

Time [s]

Din

t [J]

EDMC; t = 0.125 sEDMC; t = 0.25 sEDMC; t = 0.5 sEDMC; t = 0.75 sNewmark; t = 0.75 s

0 50 100 150 2000

20

40

60

Time [s]

Din

t + W

int +

K [

J]

EDMC; = 0.95

EDMC; = 0.9

EDMC; = 0.8

EDMC; = 0.7

Newmark; = 0.7

0 50 100 150 2000

5

10

15

20

25

Time [s]

Din

t [J]

EDMC; = 0.95

EDMC; = 0.9

EDMC; = 0.8

EDMC; = 0.7

Newmark; = 0.7

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4. Numerical ExamplesImpact of two 2D-cylinders

Two cylinders (Meng & Laursen)

– Left one has an initial velocity (initial kinetic energy 14J)

– Elasto-perfectly-plastic hyperelastic material

22


Results comparison at the end of the simulation

0 0.099 0.197 0.296 0.394Equivalent plastic strain

EDMC, t=1.5 ms, ∞= 0.7


EDMC, t=15 ms, ∞= 0.7


CH, t=1.5 ms, ∞= 0.7

0 0.189 0.379 0.568 0.758

CH, t=15 ms, ∞= 0.7

Equivalent plastic strain

23


Results evolution comparison

0 1 2 3 40

5

10

15

Time [s]

Din

t [J]

EDMC; t = 1.5 msEDMC; t = 15 msCH; t = 1.5 msCH; t = 15 ms

0 1 2 3 40

5

10

15

Time [s]

K [

J]


0 1 2 3 410

12

14

16

18

20

Time [s]

Din

t +W

int +

K [

J]


0 1 2 3 4

0

1

2

3

4

Time [s]

F

ext .

x

[J]


24

4. Numerical ExamplesImpact of a square tube

Impact (50 m/s) of a square tubes on a rigid punch– Saturated hardening law (aluminum)– Masses added at tube extremities (to simulate structure inertia)

EDMC, ∞= 0.8, no point mass EDMC, ∞= 0.8, point masses = 100% of tube mass

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4. Numerical ExamplesImpact of a square tube

Time evolution of the results

0 0.5 1 1.5 20

5

10

15

20

Time [ms]

Ene

rgy

[J]

Dint

W int

K

0 0.5 1 1.5 2-0.2

-0.15

-0.1

-0.05

0

Time [ms]

F

ext .

x

[J]

0 0.5 1-0.2

-0.15

-0.1

-0.05

0

Time [ms]

F

ext .

x

[J]

EDMC, ∞= 0.8, no point mass EDMC, ∞= 0.8, point masses = 100% of tube mass

0 0.5 10

2

4

6

8

10

12

Time [ms]

Ene

rgy

[J]

Dint

W int

K

26

Developed a visco-elastic formulation leading to a conserving time integration scheme

Use of the variational updates formulation:– The formulation derives from a energy potential

– The formulation is general for any material behavior

– The formulation includes numerical dissipation

The internal forces expression remains the same as for elasticity The momenta are conserved Numerical dissipation of energy The yield criterion is satisfied at the end of the time step Numerical examples demonstrate the robustness

5. Conclusions

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