laboratoire de techniques aéronautiques et spatiales (asma)

Laboratoire de Techniques Aéronautiques et Spatiales (ASMA)

Milieux Continus & Thermomécanique

Université de Liège Tel: +32-(0)4-366-91-26 Chemin des chevreuils 1 Fax: +32-(0)4-366-91-41 B-4000 Liège – Belgium Email : l.noels@ulg.ac.be

Contributions aux algorithmes d’intégration temporelle conservant l’énergie

en dynamique non-linéaire des structures.

Travail présenté par

Ludovic Noels(Ingénieur civil Electro-Mécanicien, Aspirant du F.N.R.S.)

Pour l’obtention du titre de

Docteur en Sciences Appliquées3 décembre 2004

Introduction Industrial problems

Industrial context:– Structures must be able to resist to crash situations– Numerical simulations is a key to design structures– Efficient time integration in the non-linear range is needed

Goal: – Numerical simulation of blade off and wind-

milling in a turboengine – Example from SNECMA

Introduction Original developments

Original developments in implicit time integration:– Energy-Momentum Conserving scheme for elasto-plastic model

(based on a hypo-elastic formalism)– Introduction of controlled numerical dissipation combined with

elasto-plasticity– 3D-generalization of the conserving contact formulation

Original developments in implicit/explicit combination:– Numerical stability during the shift

– Automatic shift criteria

Numerical validation:– On academic problems

– On semi-industrial problems

Scope of the presentation

1. Scientific motivations

2. Consistent scheme in the non-linear range

3. Combined implicit/explicit algorithm

4. Complex numerical examples

5. Conclusions & perspectives

1. Scientific motivations– Dynamics simulation

– Implicit algorithm: our opinion

– Conservation laws

– Explicit algorithms

– Implicit algorithms

– Numerical example: mass/spring-system

1. Scientific motivationsDynamics simulations

Scientific context:– Solids mechanics

– Large displacements

– Large deformations

– Non-linear mechanics

extint FFxM

T JdVDF

: Cauchy stress; F: deformation gradient; f = F-1;

: derivative of the shape function; J : Jacobian

Spatial discretization into finite elements:– Balance equation

– Internal forces formulation

1. Scientific motivationsDynamics simulations

Temporal integration of the balance equation 2 methods:

– Explicit method

– Implicit method

1,1deduction

1ionapproximat intext1

nxnxFFnx

nxnxnx

)1,,1,,(1

)1,,1,,(1iterations

1ionextrapolat

extint

nxnxnxnxnxfnxnxnxnxnxnxfnx

nxnxnx

Very fast dynamics

Slower dynamics

• Non iterative• Limited needs in memory• Conditionally stable (small time step)

• Iterative• More needs in memory• Unconditionally stable (large time step)

1. Scientific motivationsImplicit algorithm: our opinion

If wave propagation effects are negligibleImplicit schemes are more suitable

– Sheet metal forming (springback, superplastic forming, …)– Crashworthiness simulations (car crash, blade loss, shock

absorber, …)

Nowadays, people choose explicit scheme mainly because of difficulties linked to implicit scheme:– Lack of smoothness (contact, elasto-plasticity, …)

convergence can be difficult– Lack of available methods (commercial codes)

Little room for improvement in explicit methods Complex problems can take advantage of combining explicit and implicit

algorithms

1. Scientific motivationsConservation laws

Conservation of linear momentum (Newton’s law)– Continuous dynamics

– Time discretization &

nnnodes nodes

nnnn Fxtxxxx ext2/12/111

nnnnnnodes

nnnn xxFDWWxxxx ][2

ext2/1

intintint111

nnodes

nn Ftxx ext2/11

Wint: internal energy;

Wext: external energy;

Dint: dissipation (plasticity …)

extFxt

intextint DWt

0int2/1

nodesnF

0int2/12/1

nodesnn Fx

nnnnn xxFDWW ][ 1int

2/1intintint

Conservation of angular momentum– Continuous dynamics

– Time discretization

Conservation of energy

– Continuous dynamics

– Time discretization &

1. Scientific motivations Explicit algorithms

Central difference (no numerical dissipation)

nMnnnM xFFMx 1 intext11

11 ² 2

nnnnn xtxtxtxx

11 1 nnnn xtxtxx

nnn FFMx

2/11 nnn xtxx

nnn xtxx 2/12/1

Hulbert & Chung (numerical dissipation) [CMAME, 1996]

Small time steps conservation conditions are approximated

Numerical oscillations may cause spurious plasticity

1. Scientific motivationsImplicit algorithms

-generalized family (Chung & Hulbert [JAM, 1993])

– Newmark relations:

nnnnn xtxtxx

1 extintext1

M FFFFxMxM

– M = 0 and F = 0 (no numerical dissipation)• Linear range: consistency (i.e. physical results) demonstrated

• Non-linear range with small time steps: consistency verified

• Non-linear range with large time steps: total energy conserved but without consistency (e.g. plastic dissipation greater than the total energy, work of the normal contact forces > 0, …)

– M 0 and/or F 0 (numerical dissipation)• Numerical dissipation is proved to be positive only in the linear range

– Balance equation:

1. Scientific motivationsNumerical example: mass/spring-system

=10 m /s.

l =10 m0

²U l² =15 N /m/

m = 2 kg

-20 -10 0 10 20

Example: Mass/spring system (2D) with an initial velocity perpendicular to the spring (Armero & Romero [CMAME, 1999])

– Newmark implicit scheme (no numerical damping)

– Chung-Hulbert implicit scheme (numerical damping)

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

-20 -10 0 10 20

Explicit methodtcrit ~ 0.72s;

1 revolution ~ 4s

1. Scientific motivations2. Consistent scheme in the non-linear range

– Principle– Dissipation property– The mass/spring system– Formulations in the literature: hyperelasticity– Formulations in the literature: contact– Developments for a hypoelastic model– Numerical example: Taylor bar– Numerical example: impact of two 2D-cylinders– Numerical example: impact of two 3D-cylinders

3. Combined implicit/explicit algorithm4. Complex numerical examples5. Conclusions & perspectives

2. Consistent scheme in the non linear range Principle

Consistent implicit algorithms in the non-linear range:

– The Energy Momentum Conserving Algorithm or EMCA

(Simo et al. [ZAMP 92], Gonzalez & Simo [CMAME 96]):

• Conservation of the linear momentum

• Conservation of the angular momentum

• Conservation of the energy (no numerical dissipation)

– The Energy Dissipative Momentum Conserving algorithm or

EDMC (Armero & Romero [CMAME, 2001]):

• Conservation of the linear momentum

• Conservation of the angular momentum

• Numerical dissipation of the energy is proved to be positive

2. Consistent scheme in the non linear range Principle

02/1int

nTnnn dVJDF

diss2/1nF

– EMCA:

• With and designed to verify

conserving equations• No dissipation forces and no dissipation velocities

xxxx nnnn

diss2/1

int2/1

ext2/1

nnnnn FFF

xx nnn

– Balance equation

Based on the mid-point scheme (Simo et al. [ZAMP, 1992])

– Relations displacements/velocities/accelerations

diss1nx

– EDMC: • Same internal and external forces as in the EMCA

• With and designed to achieve positive numerical dissipation without spectral bifurcation

2. Consistent scheme in the non linear range Dissipation property

Comparison of the spectral radius– Integration of a linear oscillator:

Low numerical dissipation High numerical dissipation

1 cosln

exp nnn ttt

: spectral radius;: pulsation

0 0.01 0.1 1 10 100 1000

CH implicitHHTNewmarkEDMC-1CH explicit

0 0.01 0.1 1 10 100 1000

CH implicitHHTNewmarkEDMC-1CH explicit

2. Consistent scheme in the non linear range The mass/spring system

-20 -10 0 10 20

Forces of the spring for any potential V

F n 1/ 2

int V (ln 1) V (ln )

ln 12 ln

– Without numerical dissipation (EMCA) (Gonzalez & Simo [CMAME, 1996])

– The consistency of the EMCA solution does not depend on t

– The Newmark solution does-not conserve the angular momentum

0 100 200 300 400 500

Time (s)

ConservingNewmark

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

-20 -10 0 10 20

2. Consistent scheme in the non linear range The mass/spring system

0 100 200 300 400 500

Time (s)

1diss1

diss2/1

llnlnl

– With numerical dissipation (EDMC 1st order ) with dissipation parameter 0< (Armero&Romero [CMAME, 2001]), here = 0.111

– Only EDMC solution preserves the driving motion:• The length tends towards the equilibrium length

• Conservation of the angular momentum is achieved

Length at rest

Equilibrium length

0 100 200 300 400 500Time (s)

2. Consistent scheme in the non linear rangeFormulations in the literature: hyperelasticity

Hyperelastic material (stress derived from a potential V):– Saint Venant-Kirchhoff hyperelastic model (Simo et al. [ZAMP, 1992])

– General formulation for hyperelasticity (Gonzalez [CMAME, 2000]):

00),,(

100int

F: deformation gradient

GL: Green-Lagrange strain

V: potential

: shape functions

– Hyperelasticity with elasto-plastic behavior: energy dissipation of the algorithm corresponds to the internal dissipation of the material (Meng & Laursen [CMAME, 2001])

– Classical formulation:

2. Consistent scheme in the non linear rangeFormulations in the literature: contact

Description of the contact interaction:n: normal

t: tangent

g: gap

Fcont: force

Computation of the classical contact force:

– Penalty method

– Augmented Lagrangian method

– Lagrangian method

ngkF N

ngknF Nk

)(cont

kN: penalty

: Lagrangian

2. Consistent scheme in the non linear rangeFormulations in the literature: contact

– Computation of a dynamic gap for slave node x projected on master surface y(u)

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

1 nnnnnnnnn uyuyxxngg

dd1cont

Penalty contact formulation (normal force proportional to the penetration “gap”) (Armero & Petöcz [CMAME, 1998-1999]):

– Normal forces derived from a potential V

Augmented Lagrangian and Lagrangian consistent contact formulation (Chawla & Laursen [IJNME, 1997-1998]):– Computation of a gap rate

)()( 2/12/1112/1r

2/1 nnnnnnnn uyuyxxngt

2. Consistent scheme in the non linear rangeDevelopments for a hypoelastic model

Hypoelastic model:– 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

The EMCA or EDMC for hypoelastic constitutive model:– Valid for hypoelastic formulation of (visco) plasticity– Energy dissipation from the internal forces corresponds to the

plastic dissipation

Incremental strain tensor:

E nn 1

E: natural strain tensor; F: deformation gradient

Elastic incremental stress:

nn 1 H : E n

: Cauchy stress; H: Hooke stress-strain tensor

Plastic stress corrections:(radial return mapping: Wilkins [MCP, 1964], Maenchen & Sack [MCP, 1964], Ponthot [IJP, 2002])

scnn 1

f ( vm, p )

sc: plastic corrections; vm: yield stress; p: equivalent plastic strain

Final Cauchy stresses: (final rotation scheme: Nagtegaal & Veldpaus [NAFP, 1984], Ponthot [IJP, 2002])

n 1 Rnn 1 n n

n 1 sc Rnn 1T

R: rotation tensor;

Classical forces formulation:

1int1 0V

Tnnn JdVDF

f = F-1; D derivative of the shape function; J : Jacobian

00**1*1

1int2/1

VdVJDn

nJDnnnF nTnnnTnn

fCfIfCFI

Dint: internal dissipation due to the plasticity; A: Almansi incremental strain tensor (A = Apl + Ael); GL: Green-Lagrange incremental strain (GL = GLpl + GLel)

EMCA (without numerical dissipation):

– New internal forces formulation:

– Correction terms C* and C**: (second order correction in

the plastic strain increment)

int2/1

ext2/1

nnnn FF

F: deformation gradient; f: inverse of F; D derivative of the shape function; J : Jacobian = det F; : Cauchy stress

– Verification of the conservation laws

0int2/1

nodesnF

2/12/1 nodes

cycle nodes

nnn xxFD ][ 1int

2/1int

1diss2/1

nDnnnF

TnTn fDfIfDFI

01el1el

GLGLGL

1el1el**

EDMC (1st order accurate with numerical dissipation):

– New dissipation forces formulation:

– Dissipating terms D* and D**:

diss2/1

int2/1

ext2/1

nnnnn FFF

– Verification of the conservation laws

0diss2/1

nodesnF

2/12/1 nodes

][][ 1diss

2/11diss

2/1num

nnnnodes

nnn xxxxxFD M

Dnum: numerical dissipation

2. Consistent scheme in the non linear range Numerical example: Taylor bar

Impact of a cylinder :– Hypoelastic model

– Elasto-plastic hardening law

– Simulation during 80 µs

2. Consistent scheme in the non linear rangeNumerical example: Taylor bar

Time step size (µs)

Newmark

EMCA with corrections (C*, C**)

EMCA without corrections

100.0%

1000.0%

0.1 1Time step size (µs)

NewmarkEMCA with corrections (C*, C**)EMCA without correctionsSecond order

Newmark

EMCA with corrections (C*, C**)

EMCA without corrections

Simulation without numerical dissipation: final results

0.1 1Time step size (µs)

NewmarkEMCA with corrections (C*, C**)EMCA without correctionsInitial energy

2. Consistent scheme in the non linear rangeNumerical example: Taylor bar

NewmarkEDMC-1HHTCH

0 0.2 0.4 0.6 0.8 1

Spectral radius at infinity pulsation

NewmarkEDMC-1HHTCH

0 0.2 0.4 0.6 0.8 1

Spectral radius at infinity pulsation

NewmarkEDMC-1HHTCH

Simulations with numerical dissipation: final results

100.0%

1000.0%

Newmark EDMC-1HHT CHFirst order

– Constant spectral radius at infinity pulsation = 0.7

– Constant time step size = 0.5 µs

2. Consistent scheme in the non linear range Numerical example: impact of two 2D-cylinders

Impact of 2 cylinders (Meng&Laursen) :– Left one has a initial velocity (initial kinetic energy 14J)

– Elasto perfectly plastic hypoelastic material

– Simulation during 4s

2. Consistent scheme in the non linear rangeNumerical example: impact of two 2D-cylinders

Results comparison at the end of the simulationNewmark(t=1.875 ms)

Newmark(t=15 ms)

EMCA (with cor., t=1.875 ms)

EMCA (with cor., t=15 ms)

0 0.089 0.178 0.266 0.355Equivalent plastic strain Equivalent plastic strain

Equivalent plastic strain

0 0.090 0.180 0.269 0.359

0 0.305 0.609 0.914 1.22Equivalent plastic strain

0 0.094 0.187 0.281 0.374

Results evolution comparison

0 2 4Time (s)

Newmark

Meng & Laursen0

0 2 4Time (s)

Newmark

Meng & Laursen

0 1 2 3 4Time (s)

J)Newmark

0 1 2 3 4Time (s)

Newmark

t=15 mst=1.875 ms

2. Consistent scheme in the non linear range Numerical example: impact of two 3D-cylinders

Impact of 2 hollow 3D-cylinders:– Right one has a initial

velocity (

– Elasto-plastic hypoelastic material (aluminum)

– Simulation during 5ms

– Use of numerical dissipation

– Frictional contact

YX xx 00 10

– At the end of the simulation:

Results comparison with a reference (EMCA; t=0.5µs):– During the simulation:

0 0.0025 0.005

Time (s)

Reference

EDCM-1

0 0.0025 0.005

Time (s)

Reference

EDCM-1

1 10Time step size (µs)

Reference

EDMC-1

HHT-0.1

1 10Time step size (µs)

Reference

EDMC-1

3. Combined implicit/explicit algorithm– Automatic shift– Initial implicit conditions– Numerical example: blade casing interaction

3. Combined implicit/explicit algorithmAutomatic shift

Shift from an implicit algorithm to an explicit one:

– Evaluation of the ratio r*stepexplicit1

stepimplicit1*

maxexpl

intoldimpl

newimpl 2

impltr

– Explicit time step size depends on the mesh

– Implicit time step size depends on the integration error (Géradin)

– Shift criterion

b: stability limit;

max: maximal eigen pulsation

eint: integration error; Tol: user tolerance

: user security

3int e

inodesi

3. Combined implicit/explicit algorithmAutomatic shift

Shift from an explicit algorithm to an implicit one:

– Evaluation of the ratio r*

maxexpl

impl trt

– Explicit time step size depends on the mesh

– Implicit time step size interpolated form a acceleration difference

– Shift criterion

b: stability limit;

max: maximal eigen pulsation

Tol: user tolerance

: user security

stepexplicit1

stepimplicit1*

explrefimpl

teTolt

inodesi

3. Combined implicit/explicit algorithmInitial implicit conditions

– Dissipation of the numerical modes: spectral radius at bifurcation equal to zero.

diss2/*

int2/*

ext2/*

2 rnrnrn

nrn FFFxx

Stabilization of the explicit solution:

– Consistent balance of the r* last explicit steps:

tn tn+r* tn+ r*+ 1

texp l

explic it im plic it

t im p l tim p l

tn -r*

d issipa tion balance

3. Combined implicit/explicit algorithmNumerical example: blade casing interaction

Blade/casing interaction :– Rotation velocity

3333rpm

– Rotation center is moved during the first half revolution

– EDMC-1 algorithm

– Four revolutions simulation

3. Combined implicit/explicit algorithmNumerical example: blade casing interaction

Final results comparison:

Explicit part of the combined method

0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

1.0E+06

1.2E+06

0 1 2 3 4Round

Implicit (EDMC-1)

Combined

Explicit

100150200250300350400

CPU (min)

Implicit (EDMC-1) Combined

Explicit

4. Complex numerical examples– Blade off simulation

– Dynamic buckling of square aluminum tubes

4. Complex numerical examplesBlade off simulation

Numerical simulation of a blade loss in an aero engine

casing

blades

bearing

flexible shaft

Front view

Back view

0 680 1360

Von Mises stress (Mpa)

Blade off :– Rotation velocity

5000rpm

– EDMC algorithm

– 29000 dof’s

– One revolution simulation

– 9000 time steps

– 50000 iterations (only 9000 with stiffness matrix updating)

200000

400000

600000

800000

0.0 0.2 0.4 0.6 0.8 1.0

Implicit (EDMC-1)

Combined

Explicit

CPU time comparison before and after code optimization:

CPU (days)

Implicit (EDMC-1)CombinedExplicit

Before optimization After optimization

1500000

3000000

4500000

0.0 0.2 0.4 0.6 0.8 1.0

Implicit (EDMC-1)

Combined

Explicit

CPU (days)

4. Complex numerical examplesDynamic buckling of square aluminum tubes

98.27 m/s 64.62 m/s 25.34 m/s 14.84 m/sImpact velocity :

Absorption of 600J with different impact velocities : – EDMC algorithm– 16000 dof’s / 2640 elements– Initial asymmetry– Comparison with the

experimental results of Yang, Jones and Karagiozova [IJIE, 2004]

4. Complex numerical examplesDynamic buckling of square aluminum tubes

Time evolution for the 14.84 m/s impact velocity:Explicit part of the combined method

0.0E+00 4.0E-03 8.0E-03 1.2E-02

Time (s)

Implicit (EDMC-1)

Combined

Explicit

1.0E-08

1.0E-07

1.0E-06

1.0E-05

0.0E+00 4.0E-03 8.0E-03 1.2E-02

Time (s)

Implicit (EDMC-1) Combined Explicit

14.84 25.34 64.62 98.27

Impact velocity (m/s)

14.84 25.34 64.62 98.27

Impact velocity (m/s)

Implicit (EDMC-1)CombinedExplicitExperimental (Yang)Jones et al (shells, explicit)

– Improvements

– Advantages of new developments

– Drawbacks of new developments

– Futures works

Original developments in consistent implicit schemes:– New formulation of elasto-plastic internal forces

– Controlled numerical dissipation

– Ability to simulate complex problems (blade-off, buckling)

5. Conclusions & perspectivesImprovements

Original developments in implicit/explicit combination:– Stable and accurate shifts

– Automatic shift criteria

– Reduction of CPU cost for complex problems (blade-off, buckling)

Advantages of the consistent scheme:– Conservation laws and physical consistency are verified for each

time step size in the non-linear range

– Conservation of angular momentum even if numerical dissipation is introduced

Advantages of the implicit/explicit combined scheme:– Reduction of the CPU cost– Automatic algorithms– No lack of accuracy– Remains available after code optimizations

5. Conclusions & perspectivesAdvantages of new developments

Drawbacks of the consistent scheme:– Mathematical developments needed for each element, material…

– More complex to implement

5. Conclusions & perspectivesDrawbacks of new developments

Drawback of the implicit/explicit combined scheme:– Implicit and explicit elements must have the same formulation

5. Conclusions & perspectivesFuture works

Development of a second order accurate EDMC scheme Extension to a hyper-elastic model based on an

incremental potential Development of a thermo-mechanical consistent scheme Modelization of wind-milling in a turbo-engine ...

Laboratoire de Techniques Aéronautiques et Spatiales (ASMA)

Milieux Continus & Thermomécanique

Université de Liège Tel: +32-(0)4-366-91-26 Chemin des chevreuils 1 Fax: +32-(0)4-366-91-41 B-4000 Liège – Belgium Email : l.noels@ulg.ac.be

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