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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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    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 - 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 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

    == &&

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    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 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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    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 &&&

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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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    Numerical methods (4)

    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. 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)

    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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    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 time-schemes 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 (half-space 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)

    Tchamwa-Wielgosz (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 quasi-static 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 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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    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 (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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    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

    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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    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 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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    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

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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

    (push-over 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: 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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    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.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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    End of presentation

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