laboratoire de techniques aéronautiques et spatiales (asma) milieux continus & thermomécanique...
TRANSCRIPT
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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 : [email protected]
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
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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
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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
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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
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Scope of the presentation
1. Scientific motivations– Dynamics simulation
– Implicit algorithm: our opinion
– Conservation laws
– Explicit algorithms
– Implicit algorithms
– Numerical example: mass/spring-system
2. Consistent scheme in the non-linear range
3. Combined implicit/explicit algorithm
4. Complex numerical examples
5. Conclusions & perspectives
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1. Scientific motivationsDynamics simulations
Scientific context:– Solids mechanics
– Large displacements
– Large deformations
– Non-linear mechanics
extint FFxM
0
int0
V
T JdVDF
f
: Cauchy stress; F: deformation gradient; f = F-1;
: derivative of the shape function; J : Jacobian
0xD
Spatial discretization into finite elements:– Balance equation
– Internal forces formulation
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1. Scientific motivationsDynamics simulations
Temporal integration of the balance equation 2 methods:
– Explicit method
– Implicit method
1,1deduction
1ionapproximat intext1
nxnxFFnx
nxnxnx
M
)1,,1,,(1
)1,,1,,(1iterations
1
1
1ionextrapolat
extint
nxnxnxnxnxfnxnxnxnxnxnxfnx
FFx
nxnxnx
nxnxnx
M
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)
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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
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1. Scientific motivationsConservation laws
Conservation of linear momentum (Newton’s law)– Continuous dynamics
– Time discretization &
nodes
nnnodes nodes
nnnn Fxtxxxx ext2/12/111
MM
nodes
nnnnnnodes
nnnn xxFDWWxxxx ][2
1
2
11
ext2/1
intintint111
MM
nodes
nnodes
nn Ftxx ext2/11
MM
Wint: internal energy;
Wext: external energy;
Dint: dissipation (plasticity …)
extFt
x
M
extFxt
xx
M
intextint DWt
Wt
Kt
0int2/1
nodesnF
0int2/12/1
nodesnn Fx
nodes
nnnnn xxFDWW ][ 1int
2/1intintint
1
Conservation of angular momentum– Continuous dynamics
– Time discretization
&
Conservation of energy
– Continuous dynamics
– Time discretization &
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1. Scientific motivations Explicit algorithms
Central difference (no numerical dissipation)
nMnnnM xFFMx 1 intext11
11 ² 2
1²
nnnnn xtxtxtxx
11 1 nnnn xtxtxx
int1
ext1
11
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
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1. Scientific motivationsImplicit algorithms
-generalized family (Chung & Hulbert [JAM, 1993])
– Newmark relations:
nnnnn xtxtxx
tx ²
2
1
²
111
nnnnn xtxtxx
tx ²
2
1 1
11
01
1
1
1 extintext1
int11
nnF
Fnnn
F
Mn
F
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:
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1. Scientific motivationsNumerical example: mass/spring-system
=10 m /s.
l =10 m0
²U l² =15 N /m/
m = 2 kg
x0
-20
-10
0
10
20
-20 -10 0 10 20
X (m)
Y (
m)
-20
-10
0
10
20
-20 -10 0 10 20
X (m)
Y (
m)
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
-20 -10 0 10 20
X (m)
Y (
m)
-20
-10
0
10
20
-20 -10 0 10 20
X (m)
Y (
m)
Explicit methodtcrit ~ 0.72s;
1 revolution ~ 4s
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Scope of the presentation
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
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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
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2. Consistent scheme in the non linear range Principle
0
02/1
02/1
02/1int
2/1V
nTnnn dVJDF
f ext
2/1nF
diss2/1nF
– EMCA:
• With and designed to verify
conserving equations• No dissipation forces and no dissipation velocities
t
xxxx nnnn
11
2
diss2/1
int2/1
ext2/1
1
2
nnnnn FFF
xxM
t
xxx
xx nnn
nn
1diss
11
2
– 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
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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
11
2
1 cosln
exp nnn ttt
x
: spectral radius;: pulsation
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.01 0.1 1 10 100 1000
t
Sp
ectr
al r
adiu
s
CH implicitHHTNewmarkEDMC-1CH explicit
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.01 0.1 1 10 100 1000
t
Sp
ectr
al r
adiu
s
CH implicitHHTNewmarkEDMC-1CH explicit
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2. Consistent scheme in the non linear range The mass/spring system
-20
-10
0
10
20
-20 -10 0 10 20
X (m)
Y (
m)
Forces of the spring for any potential V
F n 1/ 2
int V (ln 1) V (ln )
ln 12 ln
2
x n 1
x n
– 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
-250
-200
-150
-100
0 100 200 300 400 500
Time (s)
An
gu
lar
mo
me
ntu
m (
kg
m²/
s)
ConservingNewmark
EMCA, t=1s EMCA, t=1.5s
-20
-10
0
10
20
-20 -10 0 10 20
X (m)
Y (
m)
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2. Consistent scheme in the non linear range The mass/spring system
-250
-200
-150
-100
-50
0
0 100 200 300 400 500
Time (s)
An
gu
lar
mo
men
tum
(kg
m²/
s)
EDMC
HHT
2
1
1
1diss1
nn
nn
nn
n
xx
xx
xxx
2
''1
1
21
diss2/1
1 nn
nn
llnlnl
n
xx
ll
VF
nn
– 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
8
9
10
11
12
0 100 200 300 400 500Time (s)
Len
gth
(m
)
EDMC
HHT
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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]):
0xD
00),,(
2
100
2
100int
2/1
20
10
VdVD
O
pvmVf
nnV
nn
nF
nn
GLGL
GLGL
GL
FF
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:
00
int
VdVD
VF
GLF
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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
n
t
g<0
Fcont
Computation of the classical contact force:
– Penalty method
– Augmented Lagrangian method
– Lagrangian method
ngkF N
cont
ngknF Nk
)(cont
nF
cont
kN: penalty
: Lagrangian
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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
2/1dd
1
dd1cont
2/1
n
nn
nnn n
gg
gVgVF
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
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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
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2. Consistent scheme in the non linear rangeDevelopments for a hypoelastic model
Incremental strain tensor:
E nn 1
1
2ln F
nn 1T
Fnn 1
E: natural strain tensor; F: deformation gradient
Elastic incremental stress:
nn 1 H : E n
n 1
: 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:
0
10
1int1 0V
Tnnn JdVDF
f
f = F-1; D derivative of the shape function; J : Jacobian
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2. Consistent scheme in the non linear rangeDevelopments for a hypoelastic model
00**1*1
4
1int2/1
110
10
VdVJDn
nJDnnnF nTnnnTnn
fCfIfCFI
111
10
int*
:
:/
n
nnn
nn
nplnn
nJDGL
GLGL
GLC
111
1110
int**
:
:/
n
nnn
nn
nplnn
nJDA
AA
AC
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
1
2
nnnn FF
xxM
– Balance equation
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
0int
2/12/1 nodes
nn Fx
cycle nodes
nnn xxFD ][ 1int
2/1int
&
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2. Consistent scheme in the non linear rangeDevelopments for a hypoelastic model
00
114
1diss2/1
10
**0
*
VdVDn
nDnnnF
TnTn fDfIfDFI
111
01el1el
*
:
::2
nnn
nnn
nnn
nn JH
GLGLGL
EED
111
10
1el1el**
:
::/2
nnn
nnn
nnn
nn JH
AAA
EED
EDMC (1st order accurate with numerical dissipation):
– New dissipation forces formulation:
– Dissipating terms D* and D**:
diss2/1
int2/1
ext2/1
1
2
nnnnn FFF
xxM
– Balance equation
– Verification of the conservation laws
0diss2/1
nodesnF
0diss
2/12/1 nodes
nn Fx
][][ 1diss
2/11diss
2/1num
nnnnodes
nnn xxxxxFD M
&
Dnum: numerical dissipation
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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
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2. Consistent scheme in the non linear rangeNumerical example: Taylor bar
-0.6
-0.4
-0.2
0
0.2
0.1 1
Time step size (µs)
Inte
rnal
en
erg
y (J
)
Newmark
EMCA with corrections (C*, C**)
EMCA without corrections
0.1%
1.0%
10.0%
100.0%
1000.0%
0.1 1Time step size (µs)
Inte
rnal
en
erg
y er
ror
NewmarkEMCA with corrections (C*, C**)EMCA without correctionsSecond order
0
500
1000
1500
2000
2500
0.1 1
Time step size (µs)
New
ton
-Rap
hso
n i
tera
tio
ns
Newmark
EMCA with corrections (C*, C**)
EMCA without corrections
Simulation without numerical dissipation: final results
50
52
54
56
58
60
0.1 1Time step size (µs)
To
tal
ener
gy
(J)
NewmarkEMCA with corrections (C*, C**)EMCA without correctionsInitial energy
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2. Consistent scheme in the non linear rangeNumerical example: Taylor bar
-0.6
-0.4
-0.2
0
0.2
0.1 1
Time step size (µs)
Inte
rnal
en
erg
y (J
)
NewmarkEDMC-1HHTCH
-0.6
-0.4
-0.2
0
0.2
0 0.2 0.4 0.6 0.8 1
Spectral radius at infinity pulsation
Inte
rnal
en
erg
y (J
)
NewmarkEDMC-1HHTCH
0
200
400
600
800
1000
1200
1400
0 0.2 0.4 0.6 0.8 1
Spectral radius at infinity pulsation
New
ton
-Rap
hso
n i
tera
tio
ns
NewmarkEDMC-1HHTCH
Simulations with numerical dissipation: final results
0.1%
1.0%
10.0%
100.0%
1000.0%
0.1 1
Time step size (µs)
Err
or
on
th
e in
tern
al e
ner
gy
Newmark EDMC-1HHT CHFirst order
– Constant spectral radius at infinity pulsation = 0.7
– Constant time step size = 0.5 µs
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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
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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
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2. Consistent scheme in the non linear rangeNumerical example: impact of two 2D-cylinders
Results evolution comparison
0
2
4
6
8
10
0 2 4Time (s)
En
erg
y p
last
ical
ly
dis
isp
ated
(J)
Newmark
EMCA
Meng & Laursen0
1
2
3
4
5
6
0 2 4Time (s)
En
erg
y p
last
ical
ly
dis
sip
ated
(J)
Newmark
EMCA
Meng & Laursen
-0.50
0.51
1.52
2.53
0 1 2 3 4Time (s)
Wo
rk o
f th
e co
nta
ct
forc
es (
J)Newmark
EMCA
-0.1
-0.05
0
0.05
0.1
0 1 2 3 4Time (s)
Wo
rk o
f th
e c
on
tac
t f
orc
es
(J)
Newmark
EMCA
t=15 mst=1.875 ms
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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
xz
y
YX xx 00 10
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2. Consistent scheme in the non linear rangeNumerical example: impact of two 3D-cylinders
– At the end of the simulation:
Results comparison with a reference (EMCA; t=0.5µs):– During the simulation:
25
50
75
100
0 0.0025 0.005
Time (s)
X-l
inea
r m
om
entu
m o
f ri
gh
t cy
lin
der
(kg
m/s
)
Reference
EDCM-1
HHT
0
50
100
150
200
0 0.0025 0.005
Time (s)
Z-a
ng
ula
r m
om
entu
m o
f ri
gh
t cy
lin
der
(kg
m²/
s)
Reference
EDCM-1
HHTx
zy x
zy
-5000
-4500
-4000
-3500
-3000
1 10Time step size (µs)
Wo
rk o
f fr
icti
on
al
con
tact
fo
rces
(J)
Reference
EDMC-1
HHT-0.1
0.0
0.1
0.2
0.3
1 10Time step size (µs)
En
erg
y n
um
eric
ally
d
issi
pat
ed (
%)
Reference
EDMC-1
HHT
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Scope of the presentation
1. Scientific motivations
2. Consistent scheme in the non-linear range
3. Combined implicit/explicit algorithm– Automatic shift– Initial implicit conditions– Numerical example: blade casing interaction
4. Complex numerical examples
5. Conclusions & perspectives
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3. Combined implicit/explicit algorithmAutomatic shift
Shift from an implicit algorithm to an explicit one:
– Evaluation of the ratio r*stepexplicit1
stepimplicit1*
CPU
CPUr
maxexpl
bt
5.21
intoldimpl
newimpl 2
Tol
e
t
t
expl
*
impltr
t
– 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
ref
2impl
3int e
xtxte
inodesi
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3. Combined implicit/explicit algorithmAutomatic shift
Shift from an explicit algorithm to an implicit one:
– Evaluation of the ratio r*
maxexpl
bt
expl*
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*
CPU
CPUr
52
explrefimpl
2
x
teTolt
inodesi
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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
M
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
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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
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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
Co
nta
ct f
orc
e (N
)
Implicit (EDMC-1)
Combined
Explicit
050
100150200250300350400
CPU (min)
Implicit (EDMC-1) Combined
Explicit
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Scope of the presentation
1. Scientific motivations
2. Consistent scheme in the non-linear range
3. Combined implicit/explicit algorithm
4. Complex numerical examples– Blade off simulation
– Dynamic buckling of square aluminum tubes
5. Conclusions & perspectives
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4. Complex numerical examplesBlade off simulation
Numerical simulation of a blade loss in an aero engine
casing
blades
disk
bearing
flexible shaft
Front view
Back view
0 680 1360
Von Mises stress (Mpa)
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4. Complex numerical examplesBlade off simulation
Blade off :– Rotation velocity
5000rpm
– EDMC algorithm
– 29000 dof’s
– One revolution simulation
– 9000 time steps
– 50000 iterations (only 9000 with stiffness matrix updating)
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4. Complex numerical examplesBlade off simulation
Final results comparison:
0
200000
400000
600000
800000
0.0 0.2 0.4 0.6 0.8 1.0
Round
Fo
rce
on
bea
rin
g (
N)
Implicit (EDMC-1)
Combined
Explicit
CPU time comparison before and after code optimization:
0
5
10
15
20
25
30
35
CPU (days)
Implicit (EDMC-1)CombinedExplicit
Before optimization After optimization
0
1500000
3000000
4500000
0.0 0.2 0.4 0.6 0.8 1.0
Round
Fo
rce
on
cas
ing
(N
)
Implicit (EDMC-1)
Combined
Explicit
0
5
10
15
20
25
30
35
CPU (days)
Implicit (EDMC-1)CombinedExplicit
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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]
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4. Complex numerical examplesDynamic buckling of square aluminum tubes
Final results comparison:
Time evolution for the 14.84 m/s impact velocity:Explicit part of the combined method
-6000
-4500
-3000
-1500
0
0.0E+00 4.0E-03 8.0E-03 1.2E-02
Time (s)
Fo
rce
on
1/4
of
the
tub
e (N
)
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)
Tim
e st
ep (
s)
Implicit (EDMC-1) Combined Explicit
0
10
20
30
40
50
60
14.84 25.34 64.62 98.27
Impact velocity (m/s)
CP
U (
day
s)
Implicit (EDMC-1)CombinedExplicit
0
20
40
60
80
14.84 25.34 64.62 98.27
Impact velocity (m/s)
Cru
shin
g d
ista
nce
(m
m)
Implicit (EDMC-1)CombinedExplicitExperimental (Yang)Jones et al (shells, explicit)
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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
– Improvements
– Advantages of new developments
– Drawbacks of new developments
– Futures works
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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)
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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
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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
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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 ...
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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 : [email protected]
Merci de votre attention