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Code_Aster, Salome-Meca course materialGNU FDL licence (http://www.gnu.org/copyleft/fdl.html)
Non-linear transient dynamics analysisDYNA_NON_LINE
2 - Code_Aster and Salome-Meca course material GNU FDL Licence
Introduction to non-linear transient computing
for structural dynamics
Different kind of nonlinearitiesConstitutive 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
3 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different sort of non-linearities (1)
Contact-frictionSingle 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
4 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different sort of non-linearities (2)
Nonlinear constitutive relations
Plasticity (steel)
Viscoplasticity (steel)Norton
Hoff - Rabotnov - Lemaître
Damage (concrete)Rabotnov - Kachanov
Chaboche
d d de p d df
p
TVGF
ipp ,,or
iext VDFfDDEE ,, with 1~
5 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different sort of non-linearities (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 finite-elements
Beams and columns: PMF (multifiber beam elements) with suitable constitutive relations (MAZARS, VMIS_CINE_GC…)
TVGF
ipp ,,or
6 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different sort of non-linearities (4)
Large transformationsLarge displacements: Green-Lagrange strain tensor (when > few %)
Several formulations (unlike small perturbations)Lagrangian, based on the Green-Lagrange strain tensor ans the second Piola-Kirchhoff 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 in
shells / membrane effects)
ij
i
j
j
i
k
i
k
j
u
x
u
x
u
x
u
x
1
2.
7 - Code_Aster and Salome-Meca course material GNU FDL Licence
From linear to non-linear analysis
Governing FE equations, at each time step:
Tangent stiffness operator KT (constitutive relations): nonlinear
Cut-off frequency can be difficult to defineNot only dependent of sollicitations and linear eigenmodes
Eigenfrequency are variablesNonlinear modal analysis
extFxxx ... TKCM
8 - Code_Aster and Salome-Meca course material GNU FDL Licence
Numerical methods for non-linear analysis
Direct transient response with DYNA_NON_LINELocalized non-linearities: shocks, friction
Limited size of the discretized system (< 500,000 DoF)
Non-linear constitutive relations: plasticity, damage…
Geometric nonlinearities: large displacements
Implicit time integrators: Newmark family, HHT, Krenk, -method
Explicit time integration schemes: central difference, Tchamwa-Wielgosz
Modal transient response with DYNA_VIBRAOnly localized nonlinearities (quasilinear system)
Low frequency responses
Modal reduction for fast computation
Explicit time schemes: Euler, De Vogelære, adaptive
Implicit time schemes: Newmark
Nonlinearities as internal forces:
full implicit representation
Nonlinearities in right hand terms of
equations: explicit representation
9 - Code_Aster and Salome-Meca course material GNU FDL Licence
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 (test-case 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 (wave
propagations)
When implicit solving does not converge
10 - Code_Aster and Salome-Meca course material GNU FDL Licence
Numerical methods for nonlinear analysis (2)
NEWMARK time integration family
Linear equilibrium equation Displacement Speed Acceleration
g=1/2
Unstable
zone
Conditional
stability
Unconditional
stability
b=(g+1/2)2/4b
g0
Average acc.
Linear acc.
Fox Goodwin
Central diff.
g
b
g
b
.U1tUU
,U2
1tUtUU
avec
,UtUU
,UtUU
nnn
p
n
2
nnn
p
1nn
p
1n
1n
2
n
p
1n
b
b
b
.Ut
1FB
,t
t
UX
n
p
2
ext
1n
2
2
1n
M
KMA
g
g
b
.Ut
1UFB
,t
t
UX
n
p
n
pext
1n
2
1n
MK
KMA
b
.UFB
,t
UX
n
pext
1n
2
1n
K
KMA
11 - Code_Aster and Salome-Meca course material GNU FDL Licence
Numerical methods (3)
Implicit direct transient dynamics
HHT scheme (Hilber-Hughes-Taylor 1977)Numerical dissipation in high frequency domain, second order accuracy
Based 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 k ~ 2 . Q (no dissipation for k = 1, increasing with k)
3,00 4
12
b
g
2
1
ext
n
ext
1n
int
n
int
1n1n FF1FF1U M
.1
11
,1
: with
212
2
212
2
1
n
ext
nn
pext
n
n
pext
n
n
UFUt
FBt
tHHT
Ut
FBt
tNewmark
BU
KMKM
A
MKM
A
A
b
b
b
bb
b
12 - Code_Aster and Salome-Meca course material GNU FDL Licence
Numerical methods (4)
Implicit direct transient dynamics
Numerical damping due to time integration scheme (test-case
sdld31a)
13 - Code_Aster and Salome-Meca course material GNU FDL Licence
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(Xi
t+dt). X = R
Xi+lt+dt = Xi
t+dt + X
Itérations à matrice constante Itérations à matrice tangente
Non convergence des itérations à Convergence des itérations à
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(Xi
t+dt)
Xi+lt+dt = Xi
t+dt + X
14 - Code_Aster and Salome-Meca course material GNU FDL Licence
Numerical methods (6)
Explicit direct transient dynamics
Time integration schemes and solving method
Central difference (from Newmark family with b= 0 and g = 1/2):
no dissipationConditional stability: critical time step (CFL condition)
Tchamwa-Wielgosz: HF dissipation (like HHT)Parameter: f = 1.05 (default value)
= 1 : no damping (but not equivalent to central difference)
Acceleration is the primal unknown for equilibrium resolutionSolving operator = mass matrix (lumped for numerical efficiency)
tcrit = 2 / wmax with wmax : 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 / r)
15 - Code_Aster and Salome-Meca course material GNU FDL Licence
DYNA_NON_LINE operator syntaxLike STAT_NON_LINE, DYNA_NON_LINE arguments are non-assembled 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
16 - Code_Aster and Salome-Meca course material GNU FDL Licence
Damping representation (1)
Rayleigh damping (for each material)
C = .K + b.M (well suited for linear modal dynamics)Defined with DEFI_MATERIAU and AMOR_ALPHA = , AMOR_BETA = b
Relation with modal damping coefficient: 2 x (w) = / w + b . W
Parameters can be fitted with 2 methods
1. Mean value xbetween w1 and w1
2. Enforcing xvalue at w1 and w2
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
17 - Code_Aster and Salome-Meca course material GNU FDL Licence
Damping representation (2)
Modal damping:
with: ai = 2 xi / (ki/wi)
• Syntax AMOR_MODAL(
MODE_MECA = mode, (eigenmodes)
AMOR_REDUIT = l_amor, [l_R] (xi list)
REAC_VITE = ‘OUI’, [DEFAUT] (update at each Newton’s iteration, or not)
)
Remarks
• Modal analysis on the linear system needed before NL calculation
• Modal damping terms are explicited in equilibrium equations: time step
may have to be reduced in order to insure stability, even with an
implicit time-schemes like Newmark
C a K Ki i
i
N
iT
1
mod
18 - Code_Aster and Salome-Meca course material GNU FDL Licence
Damping representation (3)
Localized dampers: dashpotsAffected on discrete elements (POI1 or SEG2): in AFFE_MODELE
MODELISATION = ‘DIS_T’ / ‘DIS_TR’
Damping values with AFFE_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 DNL only if AMOR_ALPHA is defined
(even if its value is 0), except in NEW11 release (11.2.7 version or above)
Absorbing boundaries (half-space media)Defined on some boundaries, using AFFE_MODELE
MODELISATION = '3D_ABSO'
19 - Code_Aster and Salome-Meca course material GNU FDL Licence
Damping representation (4)
Numerical damping due to time integration scheme
Newmark: modified average acceleration (HHT and
MODI_EQUI='NON')
g12/4;d1/2
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 with MODI_EQUI='OUI')Same parameter: (ALPHA = -0.3 as default value)
= 0: average acceleration (NEWMARK): undamped and second order
Damping increases when decreases and remains second order
20 - Code_Aster and Salome-Meca course material GNU FDL Licence
Damping representation (5)
Numerical damping due to time integration scheme (cont.)
Q-method or Krenk (THETA_METHODE or KRENK)Parameters: k ~ 2 . Q (undamped if k = 1, damping increasing with k)
Well suited for nonregular problems (first order accuracy)
Tchamwa-Wielgosz (explicit): Parameter: f (PHI = 1.05 as default): no dissipation if PHI = 1
21 - Code_Aster and Salome-Meca course material GNU FDL Licence
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…)
22 - Code_Aster and Salome-Meca course material GNU FDL Licence
Some advice for a proper use of DYNA_NON_LINE
Read Reference documentations (at least R5.05.05)
Read documentation U2.06.13
If loadings are time dependent (for instance: accelerograms in
seismic analysis)Sufficiently regular functions:
Sampling is correct (small time steps)
C2, or at least C1
Avoid some quasi-static artifices like excessively stiff materials
(often used as simplified representation of reinforcements)
23 - Code_Aster and Salome-Meca course material GNU FDL Licence
Some advice for a proper use of DYNA_NON_LINE (cont.)
Initial conditionsNon-regular loadings: artificial oscillations
Computing from an initial static pre-stressed state (effect of gravity)
First step: quasi-static calculation with STAT_NON_LINE
Time step value: very strong physical meaningCriterion based on desired cut-off frequency
Criterion based on prescribed conditions
Criterion based on wave propagation: Courant conditionStability condition for explicit time schemes
24 - Code_Aster and Salome-Meca course material GNU FDL Licence
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)
25 - Code_Aster and Salome-Meca course material GNU FDL Licence
Some advice for a proper use of DYNA_NON_LINE (cont.)
Time integration scheme choice (test-case 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 Tchamwa-Wielgosz
26 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
27 - Code_Aster and Salome-Meca course material GNU FDL Licence
Some advice for a proper use of DYNA_NON_LINE (cont.)
Chaining operators
STAT_NON_LINE DYNA_NON_LINE: OK
DYNA_NON_LINE implicit explicit: OK
DYNA_NON_LINE explicit implicit: MACRO_BASCULE_SCHEMA (cf. sdnv100j) because
a specific balancing algorithm has to be used, in order to avoid artificial numerical
oscillations
Post-processing 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 100
Implicit 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)
28 - Code_Aster and Salome-Meca course material GNU FDL Licence
Some advice for a proper use of DYNA_NON_LINE (cont.)
Optimizations for large problemsParallel computing could be used
Scalability (speedup) is usually less efficient than in quasistatic problems because of the
very 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 CPU
Archiving 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 out-of-core behavior (writing memory data
on filesystem)
Using .mess informations (after each operator):
Statistiques mémoire(Mo): 15521./8920./1603./248. (VmPeak/VmSize/Optimum/Minimum)
Documentation U1.03.03 for details about memory management
29 - Code_Aster and Salome-Meca course material GNU FDL Licence
Straight transient solution: Friction problem
Friction block with a spring (test-case SDND100)
Ft=10, Fn=1, U0=8.5E-4, Xloc=Z
k
m U0
gapkgMF nn
F Ft n
M X K X FT T N
30 - Code_Aster and Salome-Meca course material GNU FDL Licence
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
31 - Code_Aster and Salome-Meca course material GNU FDL Licence
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)
Soil-structure interactionDeconvolution
Raft uplift
Fluid-structure interactionAdded masses
Sloshing effects
Nonlinear soil behavior
32 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different NL transient studies (3)
Large steel tanks: buckling with FSI using DYNA_NON_LINE
Steel 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. water
level
=15,7 m
1
2
3
4
5
6
7
Bolts for
fastening
Ring 1
Ring 2
Conical top
8
Buckling mode
(push-over method)
33 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different NL transient studies (4)
Large steel tanks: buckling with FSI using DYNA_NON_LINE
FSI coupled model (u,p,f) formulation (Ohayon)
Pressure values on deformed shape (amplified) of fluid domain
T = 0,1 s T = 1 s T = 3 s
34 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different NL transient studies (5)
Circular tunnel excavationSoil constitutive relation: Drucker-Prager
Convergence is very difficult with an implicit method (STAT_NON_LINE or
DYNA_NON_LINE)
Explicit pseudo-dynamic 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
35 - Code_Aster and Salome-Meca course material GNU FDL Licence
Different NL transient studies (6)
Circular tunnel excavationSome results
Quasi-static method: convergence until 94-96 % of deconfinement
Explicit method: 99 % deconfinement reached
0,0E+00
1,0E+05
2,0E+05
3,0E+05
4,0E+05
5,0E+05
6,0E+05
7,0E+05
8,0E+05
9,0E+05
0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1
taux de déconfinement
tem
ps C
PU
co
nso
mm
é (
s)
M1 : HHT
M4 : HHT
M1 : STAT
M4 : STAT
M4 : DIFF_CENT en poursuite de STAT
M4 : DIFF_CENT en poursuite de HHT
M2c : DIFF_CENT en poursuite de STAT
M2c : STAT : M2c
Shear bands
CPU time comparison
36 - Code_Aster and Salome-Meca course material GNU FDL Licence
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)
37 - Code_Aster and Salome-Meca course material GNU FDL Licence
Evolutions in Code_Aster
Computations control: energetic balance analysis
Damping enhancementsRayleigh (other stiffness matrix choice: secant…)
Dissipation control during transient computation…
Time step adaptationEvent driven
CFL updating during computation
Distributed computations (at solver level with MUMPS, PETSc,
FETI, or with transient subdomain methods)
38 - Code_Aster and Salome-Meca course material GNU FDL Licence
For more information
http://www.code-aster.org
Other Code_Aster trainingNonlinear 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)
Test-cases (names beginning with sdn of fdn…)
BookNon-Linear Finite Element Analysis - A Short course taught by T. J. R. HUGHES and T.
BELYTSCHKO - Zace Services Ltd - ICE Division
39 - Code_Aster and Salome-Meca course material GNU FDL Licence
End of presentation
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