05 Nonlinear Dynamics

7/25/2019 05 Nonlinear Dynamics

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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


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2 - Code_Aster and Salome-Meca course material GNU FDL Licence

Introduction to non-linear 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 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


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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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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

== &&


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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 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 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 &&&


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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 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 (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 (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 (Hilber-Hughes-Taylor 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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Implicit direct transient dynamics

Numerical damping due to time integration scheme (test-casesdld31a)

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. (=0)

HHT (=0.05)

HHT (=0.01)


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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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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)

Tchamwa-Wielgosz: 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 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


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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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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 time-schemes like Newmark

( )( )C a K K i i

i

N

iT

=

=

1

mod


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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 (half-space media)Defined on some boundaries, usingAFFE_MODELE

MODELISATION = '3D_ABSO'


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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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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)

Tchamwa-Wielgosz (explicit):Parameter: (PHI = 1.05 as default): no dissipation if PHI = 1


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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


HHT (=0.1)


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 quasi-static artifices like excessively stiff materials(often used as simplified representation of reinforcements)


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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


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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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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


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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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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

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 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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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 out-of-core 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 (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&& + =


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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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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


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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

(push-over method)


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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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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


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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.code-aster.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)

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


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05 Nonlinear Dynamics