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7/25/2019 05 Nonlinear Dynamics
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Code_Aster, SalomeMeca course materialGNU FDL licence (http://www.gnu.org/copyleft/fdl.html)
Nonlinear transient dynamics analysisDYNA_NON_LINE
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Introduction to nonlinear transient computing
for structural dynamicsDifferent kind of nonlinearities
Constitutive laws, large displacements, contact
Spatial descriptionDirect (physical DoF) or modal projection
Nonlinear direct dynamics in Code_Aster
Syntax of the DYNA_NON_LINE operator
Differences between STAT_NON_LINE and DYNA_NON_LINE
Damping representation
Some advices for a proper use of DYNA_NON_LINE
Numerical applications
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Different sort of nonlinearities (1)
ContactfrictionSingle DoF oscillator with perfect plasticity constitutive relation
Shock oscillator
Large transformations: pendulum
k
F < Fs
MM x F sign x k x Fext s. && ( ) * min( . , )=
Fext
+= gapxkxkFxM cext ..&& Fext
I M g M g. && . . sin . . (!
. . .)
= + 3 5
6 5
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Different sort of nonlinearities (2)
Nonlinear constitutive relations
Plasticity (steel)
Viscoplasticity (steel)Norton
Hoff  Rabotnov  Lematre
Damage (concrete)Rabotnov  Kachanov
Chaboche
d d de p = + d d
fp
=
( )TVGF
ipp ,,or
== &&
( ) ( )iext VDFfDDEE ,,with1~
== &
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Different sort of nonlinearities (3)
Nonlinear behavior for Civil Engineering dynamics computationsGlobal law for reinforced concrete shell elements: GLRC (R7.01.32)
Parameters identification: DEFI_GLRC (U4.42.06)
Compatible with excentered reinforcements finiteelements
Beams and columns: PMF (multifiber beam elements) with suitable constitutive relations(MAZARS,VMIS_CINE_GC)
( )TVGF
ipp ,,or
== &&
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Different sort of nonlinearities (4)
Large transformationsLarge displacements: GreenLagrange strain tensor (when > few %)
Several formulations (unlike small perturbations)Lagrangian, based on the GreenLagrange strain tensor ans the second PiolaKirchhoff stress tensor
Eulerian, based on the Almansi (strain) and the Cauchy (stress) tensors
Updated lagrangian formulation (for fast transient dynamics): simple but can be inaccurate (curvature effects inshells / membrane effects)
iji
j
j
i
k
i
k
j
u
x
u
x
u
x
u
x= + +
1
2.
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From linear to nonlinear analysis
Governing FE equations, at each time step:
Tangent stiffness operator KT (constitutive relations): nonlinear
Cutoff frequency can be difficult to defineNot only dependent of sollicitations and linear eigenmodes
Eigenfrequency are variablesNonlinear modal analysis
extFxxx =++ ... TKCM &&&
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Numerical methods for nonlinear analysis
Direct transient response with DYNA_NON_LINELocalized nonlinearities: shocks, friction
Limited size of the discretized system (< 500,000 DoF)
Nonlinear constitutive relations: plasticity, damage
Geometric nonlinearities: large displacements
Implicit time integrators: Newmark family, HHT, Krenk, method
Explicit time integration schemes: central difference, TchamwaWielgosz
Modal transient response withDYNA_VIBRAOnly localized nonlinearities (quasilinear system)
Low frequency responses
Modal reduction for fast computationExplicit time schemes: Euler, De Vogelre, adaptive
Implicit time schemes: Newmark
Nonlinearities as internal forces:full implicit representation
Nonlinearities in right hand terms ofequations: explicit representation
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Numerical methods for nonlinear analysis (1)
Modal decomposition (RITZ)Moderate and localized nonlinearities (most of the structure remains linear)
Low frequency phenomenon
Direct transient method (testcase sdld31a)
Implicit time schemes: often usedGlobal and strong nonlinearities
Medium to large size (available parallelized solvers) and medium frequency phenomenon
Explicit (fast transient dynamics): specific computations with Code_AsterGlobal and strong nonlinearities, except some contact algorithm
Medium sized problems (suboptimal code optimization for explicit) and high frequency phenomenon (wavepropagations)
When implicit solving does not converge
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Numerical methods for nonlinear analysis (2)
NEWMARK time integration family
Linear equilibrium equationDisplacement Speed Acceleration
=1/2
Unstable
zone
Conditional
stability
Unconditionalstability
=(+1/2)2/4
0
Average acc.
Linear acc.
Fox Goodwin
Central diff.
( )
+=
++=
+=
+=
++
++
.U1tUU
,U2
1
tUtUUavec
,UtUU
,UtUU
nnn
p
n
2
nnn
p
1nn
p
1n
1n
2
n
p
1n
&&&&
&&&
&&&&
&&
+
+
+
+
.Ut
1
FB
,t
t
UX
n
p
2
ext
1n
2
2
1n
M
KMA
+
+
+
+
.Ut
1UFB
,t
t
UX
n
p
n
pext
1n
2
1n
&
&
MK
KMA
+
+
+
.UFB
,t
UX
n
pext
1n
2
1n
K
KMA
&&
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Numerical methods (3)
Implicit direct transient dynamicsHHT scheme (HilberHughesTaylor 1977)
Numerical dissipation in high frequency domain, second order accuracyBased on a Newmark scheme: modified average acceleration (first order)
Modification of equilibrium equation: average between tn and tn+1 :
Linear equilibrium solving:
Krenk scheme: similar to method (order 1)
Parameter ~ 2 . (no dissipation for = 1, increasing with )
3,00 ( )4
12
+= +=21
( ) ( ) extnext
1n
int
n
int
1n1nFF1FF1U +=++ +++
&&M
( ) ( )
+
+=
+=
+=
+=
=
+
+
+
.111
,1
:with
212
2
212
2
1
n
ext
nn
pext
n
n
pext
n
n
UFUt
FBt
tHHT
Ut
FBt
tNewmark
BU
KMKMA
MKM
A
A
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Numerical methods (4)
Implicit direct transient dynamics
Numerical damping due to time integration scheme (testcasesdld31a)
t
21.81.61.41.210.80.60.40.20
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
Acc. moy. mod. (=0.05)
HHT (=0.1)
Acc. moy. mod. (=0.01)
Acc. moy. (=0)
HHT (=0.05)
HHT (=0.01)
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Numerical methods (5)
Implicit direct transient dynamics
Solving with nested loops algorithmTime loop (like linear case)
NEWTION iterations (like STAT_NON_LINE operator)
K (Xit+dt) + K(Xit+dt). X = R
Xi+lt+dt = Xit+dt + X
Itrations matrice constante Itrations matrice tangente
Non convergence des itrations Convergence des itrations
matrice constante matrice tangente
Equilibrium verified at each time step
Explicit scheme: the NEWTON loop disappears
Constitutive relation solving (local loops: at each Gauss point)
Repeat until convergence
K(Xit+dt). X = R K(Xit+dt)
Xi+lt+dt = Xit+dt + X
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Numerical methods (6)
Explicit direct transient dynamicsTime integration schemes and solving method
Central difference (from Newmark family with = 0 and = 1/2):
no dissipationConditional stability: critical time step (CFL condition)
TchamwaWielgosz: HF dissipation (like HHT)Parameter: = 1.05 (default value)
= 1 : no damping (but not equivalent to central difference)
Acceleration is the primal unknown for equilibrium resolution
Solving operator = mass matrix (lumped for numerical efficiency)
tcrit = 2 / max with max : higher eigen pulsation of the discretized system
Other interpretation: tcrit ~ l min EF/ c with:
l min FE : smallest caracteristic length
c : wave celerity (traction : c2 = E / )
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DYNA_NON_LINE operator syntaxLike STAT_NON_LINE, DYNA_NON_LINE arguments are nonassembled matrices and vectors,unlike linear dynamics operators (DYNA_VIBRA)
Before DYNA_NON_LINE
Boundary conditions definition
Constitutive relation: INCREMENT (COMP_INCR)
Strain tensor: PETIT / PETIT_REAC / GROT_GDEP / SIMO_MIEHE /
GDEF_HYPO_ELAS / GREEN_REAC / GDEF_LOG
Initial conditions
Options for the Newton algorithm
Convergence criterion
Solver choice (direct or iterative / sequential or parallel)
Time integration scheme
(Eigenvalues calculation on tangent updated matrices)
Options for results storage
After DYNA_NON_LINE
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Damping representation (1)
Rayleigh damping (for each material)
C = .K + .M (well suited for linear modal dynamics)Defined with DEFI_MATERIAU andAMOR_ALPHA = ,AMOR_BETA =
Relation with modal damping coefficient: 2 () = / + .
Parameters can be fitted with 2 methods
1. Mean value between 1 and 12. Enforcing value at 1 and 2
Choice for the stiffness matrix K:AMOR_RAYL_RIGI = 'TANGENTE' (default) or 'ELASTIQUE'
ELASTIQUE: for elastic matrix: keeps constant Rayleigh damping (best choice for GLRC)
TANGENTE: for tangent matrix: Rayleigh damping decreases when nonlinearities appear, due to constitutive
relation
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Damping representation (2)
Modal damping:
with: ai = 2 i/ (ki/i)
SyntaxAMOR_MODAL(
MODE_MECA = mode, (eigenmodes)
AMOR_REDUIT = l_amor, [l_R] (i list)REAC_VITE = OUI, [DEFAUT] (update at each Newtons iteration, or not)
)
Remarks
Modal analysis on the linear system needed before NL calculation
Modal damping terms are explicited in equilibrium equations: time stepmay have to be reduced in order to insure stability, even with animplicit timeschemes like Newmark
( )( )C a K K i i
i
N
iT
=
=
1
mod
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Damping representation (3)
Localized dampers: dashpotsAffected on discrete elements (POI1 or SEG2): inAFFE_MODELE
MODELISATION = DIS_T / DIS_TR
Damping values withAFFE_CARA_ELEM, keyword: DISCRET
CARA = A_T_D_N / A_TR_D_N / A_T_D_L
Remark
Those dashpots are taken into account in DNLonly if AMOR_ALPHA is defined
(even if its value is 0), except in NEW11 release (11.2.7 version or above)
Absorbing boundaries (halfspace media)Defined on some boundaries, usingAFFE_MODELE
MODELISATION = '3D_ABSO'
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Damping representation (4)
Numerical damping due to time integration scheme
Newmark: modified average acceleration (HHT andMODI_EQUI='NON')
= (1)2/4 ; = 1/2
Parameter: (ALPHA = 0.3 as default value)
= 0: average acceleration (NEWMARK): undamped and 2nd order accuracy
Damping increases when decreases and only first order accuracy
Full HHT (HHT withMODI_EQUI='OUI')Same parameter: (ALPHA = 0.3 as default value)
= 0: average acceleration (NEWMARK): undamped and second orderDamping increases when decreases and remains second order
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Damping representation (5)
Numerical damping due to time integration scheme (cont.)
method or Krenk (THETA_METHODE or KRENK)Parameters: ~ 2 . (undamped if = 1, damping increasing with )
Well suited for nonregular problems (first order accuracy)
TchamwaWielgosz (explicit):Parameter: (PHI = 1.05 as default): no dissipation if PHI = 1
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Damping representation (6)
Numerical damping due to time integration scheme (cont.)
Damping representation for a 1 DoF linear systemIdea: HF damping and no damping in LF range
Complementary to structural damping (Rayleigh)
t
21.81.61.41.210.80.60.40.20
0.02
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
Acc. moy. mod. (=0.05)
HHT (=0.1)
Acc. moy. mod. (=0.01)
Acc. moy. (=0)
HHT (=0.05)
HHT (=0.01)
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Some advice for a proper use of
DYNA_NON_LINERead Reference documentations (at least R5.05.05)
Read documentation U2.06.13
If loadings are time dependent (for instance: accelerograms inseismic analysis)
Sufficiently regular functions:
Sampling is correct (small time steps)
C2, or at least C1
Avoid some quasistatic artifices like excessively stiff materials(often used as simplified representation of reinforcements)
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Some advice for a proper use of
DYNA_NON_LINE (cont.)
Initial conditionsNonregular loadings: artificial oscillations
Computing from an initial static prestressed state (effect of gravity)
First step: quasistatic calculation with STAT_NON_LINE
Time step value: very strong physical meaningCriterion based on desired cutoff frequency
Criterion based on prescribed conditions
Criterion based on wave propagation: Courant conditionStability condition for explicit time schemes
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Some advice for a proper use of
DYNA_NON_LINE (cont.)
Damping/dissipation: in this order
Material (behavior) dissipation
Dissipation in links, joints and assemblies (friction)
Modal damping (values from experiments)
Rayleigh damping (often fitted on modal damping values)
HF damping from the time scheme (if needed)
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Some advice for a proper use of
DYNA_NON_LINE (cont.)Time integration scheme choice (testcase sdld31a)
Implicit: often the best choice in Code_AsterLow to medium frequency problems
Tight respect of equilibrium (can leads to convergence problems)
Almost all the quasistatic methods are available (except continuation)
Different time schemes: Newmark family, Full HHT, Krenk, method
Explicit: fast dynamics (wave propagations): CPU time consumingHigh frequency problems
Stability conditions (named Courant or CFL): T ~ l min EF/ c
No convergence problem (if CFL is insured) but the numerical solution accuracy has to be checked
HF dissipation: HHT or TchamwaWielgosz
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Some advice for a proper use of
DYNA_NON_LINE (cont.)Specificities of explicit computations
Stability condition (CFL): T ~ l min EF/ c
Solving using acceleration (unlike displacement used with implicit schemes)Mass matrix inversion: lumping is recommended
Displacement boundary conditions (Dirichlet) are expressed in acceleration terms
Rayleigh dampingOnly proportional to the mass matrix (stiffness proportional terms tends to lower the CFL condition value)
Computational efficiency is low with Code_Aster(compared to specialized codes like LS
DYNA or EUROPLEXUS), except with modal reduction
No exact contact algorithm: only penalization method
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Some advice for a proper use of
DYNA_NON_LINE (cont.)Chaining operatorsSTAT_NON_LINE DYNA_NON_LINE: OK
DYNA_NON_LINE implicit explicit: OKDYNA_NON_LINE explicit implicit:MACRO_BASCULE_SCHEMA(cf. sdnv100j) because
a specific balancing algorithm has to be used, in order to avoid artificial numericaloscillations
Postprocessing and results storageSyntax: like STAT_NON_LINE (with the addition of velocity and acceleration fields)
Feature: large number of time steps: filtering of storageArchiving: storage of full fields only for some time steps
Explicit time schemes: archiving time step / computation time step = 10 to 100Implicit time scheme: archiving time step / computation time step = 1 to 10
Observation: storage of evolutions on some nodes of the model (at each time step)
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Some advice for a proper use of
DYNA_NON_LINE (cont.)
Optimizations for large problems
Parallel computing could be usedScalability (speedup) is usually less efficient than in quasistatic problems because of thevery large number of time steps
Mumps solver is more robust (direct solver)
Petsc solver (iterative solver) offers a better speedup but it can fail
Speedups are correct with 4 to 8 CPUArchiving of result (with OBSERVATION)
Splitting long transient simulations with POURSUITE
Storing base in /scratch
In some cases, maxbase value has to be increased
Memory allocation should be large in order to avoid outofcore behavior (writing memory dataon filesystem)
Using .mess informations (after each operator):Statistiques mmoire(Mo): 15521./8920./1603./248. (VmPeak/VmSize/Optimum/Minimum)
Documentation U1.03.03 for details about memory management
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Straight transient solution: Friction problem
Friction block with a spring (testcase SDND100)
Ft=10, Fn=1, U0=8.5E4, Xloc=Z
k
m U0
gapkgMF nn ==
F Ft n=
M X K X FT T N&& + =
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Different NL transient studies (1)
PipingPlasticity (on beam or pipe FE)
Raft upliftShocks with friction
Initial state computed with STAT_NON_LINE
Cables pinchingShocks and large displacements
Initial state computed with STAT_NON_LINE
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Different NL transient studies (2)
Concrete buildings or dams
Damage in concrete (ENDO_ISOT_BETON) and plasticity in steel
Global laws like GLRC (shell elements)
Soilstructure interactionDeconvolution
Raft uplift
Fluidstructure interactionAdded masses
Sloshing effects
Nonlinear soil behavior
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Different NL transient studies (3)
Large steel tanks: buckling with FSI using DYNA_NON_LINESteel plasticity + orthotropic carbon fiber reinforcements
Acoustic fluid (compressible and inviscid) with free surface
Large displacements and structural instability: buckling
Rnom.=5,7 m
2 m
2 m
2 m
2 m
2,005 m
10,12 m
16 m
2,005 m
2,005 m
1,985 m
Max. waterlevel
=15,7 m
1
2
3
4
5
6
7
Bolts forfastening
Ring 1
Ring 2
Conical top
8
AMPLITUDE
67.
Buckling mode
(pushover method)
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Different NL transient studies (4)
Large steel tanks: buckling with FSI using DYNA_NON_LINEFSI coupled model (u,p,) formulation (Ohayon)
Pressure values on deformed shape (amplified) of fluid domain
T = 0,1 s T = 1 s T = 3 s
Temps
Acclration(g)
l
l i i
l

l i
 1.0
 0.8
 0.6
 0.4
 0.2
0. 0
0. 2
0. 4
0. 6
0. 8
0 1 2 3 4 5 6 7 8 9
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Different NL transient studies (5)
Circular tunnel excavationSoil constitutive relation: DruckerPrager
Convergence is very difficult with an implicit method (STAT_NON_LINE or
DYNA_NON_LINE) Explicit pseudodynamic method (increased mass terms)
Loading: deconfinement
2D mesh: from 1,500 to 60,000 nodes
R = 3m 57 m
57 m
X
YKeystone
Right foot
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Different NL transient studies (8)
Structure in large rotationsPivot link in the up and left corner
Subject to gravity
No damping
2 schemes are comparedCentral difference (explicit)
Average acc. (implicit)
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Evolutions in Code_Aster
Computations control: energetic balance analysis
Damping enhancements
Rayleigh (other stiffness matrix choice: secant)
Dissipation control during transient computation
Time step adaptation
Event drivenCFL updating during computation
Distributed computations (at solver level with MUMPS, PETSc,
FETI, or with transient subdomain methods)
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For more information
http://www.codeaster.org
Other Code_Aster training
Nonlinear methods (focused on constitutive relations, XFEM, contact)Dynamics
Documentations: R5.05.05, U4.53.01, U2.06.13, U2.06.10,
R7.01.32 (GLRC)
Testcases (names beginning with sdn of fdn)
BookNonLinear Finite Element Analysis  A Short course taught by T. J. R. HUGHES and T.BELYTSCHKO  Zace Services Ltd  ICE Division
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End of presentation
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