Implicit Newer

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LSDYNA IMPLICIT NOTES

Copyright 2000-2004All rights Reserved

Notes Developed by

Ala (Al) Tabiei, Ph.D.

[email protected]


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Mailing Address

Al Tabiei

4056 Granite Ct.Mason, OH 45040

USA

Tel: (513) 336-8075Fax: (513) 336-8075

EMAIL: [email protected]

ACKNOWELEDGMENTS

The author wishes to thank several people for valuable comments and

discussions on the notes. Firstly, Dr. J. Hallquist advance course notes wereof great help in constructing the lecture notes. Many thanks goes to Drs. B.Maker and R. Grimes for the valuable comments, suggestions, anddiscussions. Thanks also go to the LSTC staff for their support.


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TABLE OF CONTENTS

Introduction 1-5

Finite Element Equation & Time Integration 2-14

Equilibrium and Nonlinearity 3-25

Incremental Equations and Linearization 4-25Automatic Time Step ControlNonlinear Solution Methods for Implicit AnalysisLinear Equation Solver

Finite Element Modeling Techniques 5-661. Engineering a FEA Model:2. Element Selection3. Mesh Density4. Symmetry5. Modeling for Physical Phenomenon6. Ad Hoc Guidelines7. How to Tell If Your Results Are Correct

Contact and Friction 6-97Contact AlgorithmFriction

Material Nonlinearity 7-112

Damping 8-119

Reduce Integration & Hourglass Phenomenon 9-122

Reduced and Selective Reduce IntegrationHourglassing

Quasi-Static Initialization and Quasi-Static Simulations 10-107Quasi-Static InitializationQuasi-Static Simulations With Explicit FE


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Adaptive Meshing 11-132

Springback 12-136

Trouble Shooting 13-142

Workshop 14-145


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INTRODUCTION_______________________________________________________________________________________________

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LS-DYNA IMPLICIT NOTES 1-5

INTRODUCTION

Version 950 (1999) included the first release of implicit analysiscapability using LS-DYNA. Initially targeted at metal forming springbacksimulation, this new capability allows static stress analysis without interfacingto LSTC's implicit software package LS-NIKE3D.

The new implicit features in LS-DYNA were not simply created usingsubroutines from LS-NIKE3D. Instead, the existing element and materialroutines in LS-DYNA were augmented to add stiffness matrix calculationswhen the implicit mode is active. This approach has several advantages:

keyword input and arbitrary ID numbering are available in implicit mode existing optimized subroutines and data structures are used parallel processing capability is immediately available post-processing interfaces remain unchanged

This means that all of LS-DYNA's optimized programming and datastructures are used in the implicit mode, so high speed is preserved.Furthermore, pre- and post-processing interfaces are virtually unchanged, sothat existing support software can be used with LS-DYNA's new implicitfeatures. This also provides the first opportunity for LSTC customers to usean implicit code with keyword or structured format input, and fully arbitrary IDnumbering.

One disadvantage to this low-level, closely coupled implementation is that allof the functionality for the implicit is not immediately available in LS-DYNA.Instead, implicit stiffness matrix calculations must be added individually toactivate each LS-DYNA feature for implicit mode. Contact LSTC or your LS-DYNA distributor if you need a feature which is not shown, since this list isexpanding rapidly.


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INTRODUCTION_______________________________________________________________________________________________

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ACTIVATING IMPLICIT MODE

LS-DYNA's implicit mode may be activated in two ways. Using one ofthe new *CONTROL_IMPLICIT keywords, a simulation may be flagged torun completely in implicit mode. Alternatively, an explicit simulation may beseamlessly switched into implicit mode at a specific time using the*INTERFACE_SPRINGBACK_SEAMLESS feature or using a load curve.The seamless switching feature is intended to simplify metal formingspringback calculations, where the forming phase can be run in explicitmode, followed immediately by an implicit static springback simulation. Incase of difficulty, full restart capability is supported.Eight new keywords are available to support implicit analysis, along withcorresponding new control cards in the structured format input deck. Default

values are carefully selected to minimize input necessary for mostsimulations. These are summarized below:

Activating Implicit Analysis

several types of analyses can be performed fully explicit (default) fully implicit explicit followed by implicit ("seamless" springback)

switch between implicit and explicit eigenvalue analysis normal modes, etc.

New keywords have been added

*CONTROL_IMPLICIT_GENERAL*CONTROL_IMPLICIT_SOLVER*CONTROL_IMPLICIT_SOLUTION*CONTROL_IMPLICIT_AUTO*CONTROL_IMPLICIT_STABILIZATION*CONTROL_IMPLICIT_DYNAMICS*CONTROL_IMPLICIT_EIGENVALUE*CONTROL_IMPLICIT_MODES


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INTRODUCTION_______________________________________________________________________________________________

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not all features available in the explicit are in the implicit mode.

Implicit Keywords

*CONTROL_IMPLICIT_GENERAL (required for implicit)

activates implicit mode defines implicit time step size (standard LS-DYNA termination time is used

too) optional if *INTERFACE_SPRINGBACK_SEAMLESS is used

*CONTROL_IMPLICIT_SOLVER (optional)

parameters for linear equation solver, which inverts stiffness matrix:[K]{x}={f}

does NOT invoke a "linear" analysis

*CONTROL_IMPLICIT_SOLUTION (optional)

parameters for nonlinear equation solver (Newton-based methods) controls iterative equilibrium search, convergence "linear" analysis selected here (a special case where no iterations are

performed)

*CONTROL_IMPLICIT_AUTO (optional) activates automatic time step control

back up, try again if equilibrium iterations fail default is fixed time step size, error termination if any steps fail to

converge

*CONTROL_IMPLICIT_DYNAMICS (optional)

include inertia terms: nnextnnn MaffuKaM =+ +++ int111 problem "time" must now be real, physical time can improve convergence, especially when (even weak) rigid body modes

are present

*CONTROL_IMPLICIT_EIGENVALUE (optional) signals LS-DYNA to perform eigenvalue analysis, then stop number of eigenvalues/vectors, optional frequency shift

*CONTROL_IMPLICIT_MODES


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INTRODUCTION_______________________________________________________________________________________________

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to calculate constraint and/or attachment modes for later use in modalanalysis

computed modes are written in a binary file called d3mode

*CONTROL_IMPLICIT_STABILIZATION (optional, metal forming only) allows multi-step springback by unloading internal stresses over severalsteps

automatic springs attached to shell element nodes, slowly removed stabilization must be "completely removed" for accurate results

User's Manual contains helpful notes on each input parameter


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INTRODUCTION_______________________________________________________________________________________________

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A Note on Implicit Numerical Methods

Implicit governing equations contain two problems to solve

nn

ext

nnn

MaffuKaM =+++

+

int

111

Nonlinear Problem: *CONTROL_IMPLICIT_SOLUTION find displacements u which satisfy equilibrium fext=fint both K, fext and fint can be nonlinear functions of u iterative search employed using Newton-based method interactive switch " nlprint" toggles diagnostic output

Linearized Problem: *CONTROL_IMPLICIT_SOLVER

solve system of linear algebraic equations must solve during every nonlinear iteration great CPU and memory cost makes this problem important interactive switch " lprint" toggles diagnostic output

Implicit Eigenvalue Analysis

Extract "n" eigenvalues by subspace iteration lowest "n" frequencies optional frequency shift: "n" eigenvalues nearest to shift frequency initial stress influences frequencies when S/R Hughes-Liu shell is used

Two additional databases output d3eigv: binary plot database with each mode shape eigout: text file summarizing frequencies found

Simple input parameters

*CONTROL_IMPLICIT_GENERALactivates implicit analysis

*CONTROL_IMPLICIT_EIGENVALUEactivates eigenvalue solverinput number of eigenvalues, optional shift frequency


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INTRODUCTION_______________________________________________________________________________________________

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*CONTROL_IMPLICIT_BUCKLEImplicit buckling analysis is performed at the end of a standard implicitsimulation. You must perform the standard simulation first to applyload and create stress. Then we build material and geometric stiffnessmatricies using the model state at the termination time, and solve thebuckling eigenproblem. The resulting eigenvalues are multipliers which,when applied to the loading at the termination time, give the bucklingloads.

Implicit Output Databases

Binary Output Databases

plot, time history files in standard LS-DYNA format iteration plot database "d3iter" activated interactively " iter"

deformed geometry each nonlinear iterationgood for debugging convergence problems

ASCII Output Databases

NODOUT, ELOUT, GLSTAT, RCFORC, NODFOR, RBDOUT, etc same format as explicit

FUTURE DEVELOPMENTS

The list of features available for implicit analysis within LS-DYNA is growing.Customers are encouraged to notify either LSTC or LS-DYNA distributorsabout important features needed for their applications. Development prioritywill be strongly customer driven. LSTC anticipates that an important futureapplication area will be implicit stress initialization, followed seamlessly by anexplicit simulation. This will likely become the most common application ofimplicit LS-DYNA.


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INTRODUCTION_______________________________________________________________________________________________

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PROGRESS OF ITERATIONS

Control parameters for the nonlinear equation solver are input using the*control_implicit_solution keyword. By defaults, the progress of theequilibrium search is not shown to the screen. This output can be activatedusing the NLPRINT input parameter, or interactively toggled on and off byentering nlprint. The box below shows a typical iteration sequence,where the norms of displacement and energy are displayed. When thesenorms are reduced below user prescribed tolerances, equilibrium is reachedwithin sufficient accuracy, the iteration process is said to have converged,and the solution proceeds to the next time step.


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The next box depicts a typical print-out of the defult BFGS nonlinear equationsolver. Several automatic stiffness reformation are performed, initialy due todivergence, and later when the defult limit of 10 iterations is exceeded.


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IN DYNAMIC SIMULATION---TIME STEPSIN STATIC SIMULATION---LOAD STEPS

(multiple steps may be used to divide the nonlinear behavior intomanageable pices)


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FE EQUATIONS AND TIME INTEGRATION_______________________________________________________________________________________________

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FE EQUATIONS AND TIME INTEGRATION

The equations of equilibrium governing the dynamic response of the systemare:

[ ]{ } [ ]{ } [ ]{ } { })(tRuKuCuM ext=++ &&& (1)

where[M] is the mass matrix,[C] is the damping matrix,[K] is the stiffness matrix,{ }u&& , { }u& , and { }u are the nodal accelerations, velocities, and displacements

vectors respectively,{ })(tRext is the external forces vector.

Equation (1) can be rewritten in the form

{ } { } { } { })()()()( tRtRtFtF extintDI =++ (2)

where{ } [ ]{ }uMtFI &&=)(

{ } [ ]{ }uCtFD &=)( and

[ ]{ }uKtRint =)(

are the inertia force, the damping force, and the internal forces vectors all ofwhich are time dependent. Equations (1), which represent a system ofnonlinear second order differential equations, are solved with the help of theLSDYNA Explicit or Implicit finite element codes using direct time integrationtechniques.

Direct Time IntegrationWhen solving dynamic problems with the finite element method, the solutionis sought by dividing the total response time of the system into much smallertime intervals called time steps or time increments. The equilibrium equationsare solved and the values of the unknowns are determined at time t + tbased on knowledge of their values at time t (quasi-linearization).


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Using these values we continue solving the linearized differential equationsat t + 2t, and so on, for the entire response time of the system.When employing time integration, LSDYNA uses the implicit integrationoperator and the explicit integration operator.

The implicit integration operator definition is completed by the Newmarkformulae for displacement and velocity integration

( )[ ]( )[ ],1

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tttttt

ttttttt

uutuu

uututuu

++

++

++=

+++=

&&&&&&

&&&&&

(3)

where and are parameters of the system. Thus, expressing the velocitiesand accelerations at t + t in terms of the displacements at t + t andsubstitution into equations (1) yields:

[ ]{ } tttt FuK ++ = (4)

where[ ] [ ] [ ] [ ]( )[ ]tCMKKK = ,,, is the effective stiffness matrix, and

{ } [ ] [ ] { } { } { }( ){ }ttttttt

uuutCMFFF &&& ,,,,,, =++

is the effective load at time t + t.

The explicit dynamic analysis is based on integrating the equations of motionfor the system using the explicit central difference formula

( )

( )ttttt

tttttt

uut

u

uuut

u

+

+

=

+

=

2

1

21

2

&

&&

(5)

Thus equations (1) take the form

[ ]{ }ttt

FuM =+ 2/

(6)

where

[ ] [ ] [ ]( )[ ]tCMMM = ,, is the effective mass matrix, and{ } [ ] [ ] [ ] { } { }( ){ }

2/,,,,,,

tttttuutCMKFFF

= is the effective load at time t.


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Equations (4) are solved at time t + t and the corresponding solutionmethods are called implicit, while equations (6) are solved at time t, and thecorresponding methods are called explicit.

The explicit central difference operator is very convenient when lumped massmatrix can be assumed and velocity-dependent damping can be neglected.Then [ ]M is a diagonal matrix and the solution is achieved automaticallywithout having to solve the system of equations.

On the other hand the explicit operator is only conditionally stable, whichmeans that in order to obtain accurate results, the time step must be smallerthan a certain critical value which is defined by the mass and stiffnessproperties of the complete element assemblage.

Depending on the particular problem, the stable time increment of the modelmay be very small which would require too many steps to solve and thereforeextremely large CPU time. In such cases the implicit method, which isunconditionally stable, may provide better efficiency.


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EQUATION OF MOTION FOR TRANSLATIONAL VELOCITIES:

}{}{}{}{ int bodext RRRdt

dvM += , however in here we have additional terms as

follows:

}{}{}{}{}{}{ int conthgrbodext RRRRRdt

dvM +++=

=elements V

JI dVNNM is the mass matrix

=elements

extR }{ SdpNS

iI is the externally applied loads vector

=elements

R }{ int dVN

V j

I

ij

is the internal forces vector

dVbNR ielementsV

I

bod

=}{}

is the body forces vector

=elements

hgrhgr rR }{ is the hourglass resistant forces vector

=surfacescontact

contcontrR }{ is the contact forces vector

V and S are the actual volume and external surface


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Flowchart for Explicit Time Integration

Initial conditions );0(0 == t 0),0(21

===

ntuu &&

)]0([1

)0( int === tuCRRm

tu oext

o

ii

&&&

Update: Velocities nnn

utuu &&&& +=+

2

1

2

1

Displacements 21

1++ +=

nnn utuu &

Compute internal nodal forces

Loop over elements: E = 1 to NELE

Velocity-strains: 2/12/1 ++ = nEn uB &&

Stress rates: )( 21

2

1++

=nn

T &

Update stress: 2/11 ++ += nnn t &

Internal forces: ++ =V

nTn VdBr 11

int

Assemble: 1int+n

r into1

int

+nR

Compute external nodal forces 1+n

extR

Accelerations: ][1 2/11

int

11 ++++

=

nnn

extii

n uCRRm

u &&&

Output nn +1; go to 2


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Implicit Governing Incremental Equations

To obtain solution at load increment n+1 given solution at load increment n,the linearized equations are:

)()()(int

10

..

1

n

n

nextn

tn xRxRuxKuM =+ ++

M: lumped mass matrix

tK : tangent stiffness matrix based on geometry at next

R : external load based on applied load at step n+1, but geometry at nint

R : internal load based on displacement state and stress at load step n

Using Newmark method lead to the following equation

)()( 10n

n

nextxRxRuK

+

=

where

Mt

KK t 21

+=

])

2

1(

1[)(

...

int

nnn u

t

uMxRR

+

=

Solution of the above equation yields 0u . The coordinate vector is updatedusing

01 uxx nn +=+

The iteration for equilibrium now begins using:

ijQxRxRuxK

n

i

n

in

n

i

ext

i

n

jj ==

++

+

+ 11

1

1

)()()(

Where i denote the iteration number.

Convergence is assumed if ed and is some defined values.


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Advantages as DT becomes smallmass contribution to stiffness matrix grows rapidly: MKK 21t

+=

stabilizing effect to equilibrium iterations


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Implicit Flow Chart

1. Set0

=n

,)0(),0(

..

00 ====tuutuu

select ,,t

2. Form 000int

0 ,,,, RandKKRM t

3. Calculate )( int001..

0 RRMuext=

4. Set 0=i

5. Solve for 0= iu using )()()(int

10

n

n

nextnxRxRuxK = +

6. Update coordinate vector 01

uxxnn +=+

7. Calculate )(, 11+

+n

n xR

8. Set 1+= ii

9. Solve for iu using

10. ijQxRxRuxKn

i

n

in

n

i

ext

i

n

jj ==++

++ 11

1

1

)()()(

11. If convergence is not attained and NO divergence

12. Set in

i

n

i uxx +=++ 11 and go to 8

13. Otherwise 1+ nn and

14. Set nin

i xx =+=

1

0 and )()(1

0

n

i

n

i xRxR+

= =

15. Calculate .nu and..

nu

16. Go to 4 if t


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Implicit vs. Explicit FEA

Explicit FEA codes have enjoyed much successl ease of usel robustnessl capacity

However, latest generation computers offer new opportunities for implicitFEA codesl huge memory is now common (RAM > 1.0 Gbyte)

Several crucial enhancements have been added, allowing implicit codes to

offer a credible alternative to explicitl material, contact algorithms (rigid tooling)l fast linear equation solverl robust nonlinear solverl keyword input, unified I/O databases

Summery

Conservation of momentum: fu += tt, LS-DYNA - Explicit time integration intnextnn ffMa = impact, penetration, high rate dynamics many small time steps Courant condition limits largest stable time step

l LS-DYNA - Implicit time integration nnext

nnn MaffuKaM =+ +++int

111

static, eigenvalue, low rate dynamic analyses few large time steps Linear equation solver - stiffness matrix Nonlinear equation solver - user chooses step size

Quasi-Static analysis: "time" represents a monotonically increasingparameter which characterizes the evolution of the loading.

LS-DYNA offers several implicit nonlinear solvers:


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Full Newton - reform K every iteration Modified Newton - reform K every j iterations Quasi-Newton - employ simple, approximate "update" to K

In every case, accuracy is maintained by re-evaluating

R each iteration


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EQUILIBRIUM AND NONLINEARITIES _______________________________________________________________________________________________

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EQUILIBRIUM AND NONLINEARITIES

The concept of mechanical equilibrium is: externalv

? RdB =

Geometric Nonlinearities

= externalvT RdVB

Geometric nonlinearities are caused by:

Nonlinear relationship between displacement increments and strainincrements

Integration over current volume

Effects: Stress- stiffening Bifurcation, buckling, and collapse Snap-through Necking Other large displacement effects

Material Nonlinearities

=v externalT RdVB

Material nonlinearities are caused by:

Dependence of stress on current strain

Effects: Plasticity Plastic hinge formation and plastic collapse Rubber nonlinear elasticity Other nonlinear material effects


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Boundary Nonlinearities

= externalvT RdVB

Boundary nonlinearities are caused by: Dependence of externalR on current displacements

Effects: Contact across body surfaces (by means of load transfer) Nonlinear external loads Pressure load nonlinearities

Follower forces

Linear Equilibrium

For each ,F there will always be only one solution u (existence and

uniqueness). IfF causes a displacement of u , then ( )aF causes a displacement ( )au

(scaling). If F causes a displacement u and P causes a displacement v then

PF+ causes a displacement vu + (superposition). In a linearized problem, the solution u is determined by the currentvalue of the load externalR .

externalRKu =

Nonlinear Equilibrium

For a particularF, there may be none, one, many, or an infinite numberof solutions u (non-existence and non-uniqueness).

IfF causes a displacement ofu , then ( )aF probably doesnt cause ( )au

(no scaling) If F causes a displacement u and P causes a displacement ,v then

( )PF+ probably doesnt cause ( )vu + (no superposition). In a real (nonlinear) problem, the unique solution is determined by the

entire load history of externalR


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=v externalT RdVB

Examples of non-unique solutions

Shallow spherical caps Plasticity

Linear vs. Nonlinear FEM

The FEM equation corresponding to linearized equilibrium is a matrixequation in dofN unknowns: extermalRKu =

The FEM equation corresponding to true (nonlinear) equilibrium is a

vector equation in dofN unknowns: =vT RdVB

The solution of a linear FEM model involves the solution of the lineardofdof NN matrix equation: externalRKu =

The solution of a nonlinear FEA model involves the simultaneous

solution of dofN nonlinear vector equations: =v externalT RdVB

There are no good, general methods for solving systems of more thanone nonlinear equation. There never will be any good, general methods.

Nonconvergence

What LS-DYNA Does:

Solves the equation iteratively by using some of the Newton methods (orvariations of these methods) to find an approximate solution that minimizesthe residuals!

What are residuals ?

Re-write the equilibrium equations as a function r ofu :

( ) = vT RdVBur

( )ur are the residuals (out-of-balance forces) at u . ( )ur is nonlinear! In general, ( ) 0ur , but, in equilibrium, ( ) 0=ur

Interpretation


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The residuals are just the distribution and the magnitude of extra externalforce at each DOF needed to bring the structure into equilibrium at u :

( ) = v externalT RdVBur 0 , u is not in equilibrium under externalR

( ){ } += v externalT urRdVB 0 , u is in equilibrium under ( )urRexternal+

Newton Method

Assume you are not in equilibrium at displaced position u so that 0)( ur Find u so that uu + is in equilibrium ( ) 0=+ uur Expand ( )ur in a Taylors series about current displacement u :

( ) ( ) 0... =+

+=+ uu

ruruur

u

Throw away higher order terms and solve the resulting equations for u :

( )uruu

r

u

=

Substitute the equilibrium equation into Newton-Raphson scheme :

( )uruu

r

u

=

+= v externalT

gent RdVBuK tan this equation is linear!

The solution of a linear FEA model involves single solution of the linear

dofdof NN matrix equation: externalFKu =

The solution of a nonlinear FEA model involves many solutions of the

linear dofdof NN matrix equation: externalv

T

gent RdVBuK += tan


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INCREMENTAL EQUATIONS & LINEARIZATION_______________________________________________________________________________________________

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INCREMENTAL EQUATIONS & LINEARIZATION

MATRIX EQUATIONS( ) ( ) 0,,)( =+ txPxxFtxM &&&

=,, &&& n dimensional acceleration, velocity, and geometry vectors= nn mass matrix=P body force and external load vector

=e V

e

t

e

dVBF

NONLINEARITIES

Nonlinearities are due to geometric affects and inelastic material behavior

( ) ( )=eV

e

t

e dVBxxF && ,,

Where and & are the strains and the strain rates. If linear

( ) xKxCxxFe += &&,

Where K, C and x are the stiffness matrix, damping and displacementvector. Nonlinearities also arise in P due to the geometry dependantloadings.

RESIDUAL VECTOR

Regardless of whether an implicit or explicit scheme is used, we require that

0=+= PFQ &&

If linear

0=++= PKCQ &&&


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LINEARIZATION

The residual vector becomes an implicit function of 1+nx only. We seek he

vector

1+nxsuch that

( ) 01 =+nxQ Assume an approximation 1+nkx to

1+nx for k=1,2,3, ..

In the neighborhood of 1+nkx we use the linear approximation to1+nxQ given

by:

( ) ( ) ( ) kn

kk

n

k

n xxJxxQxQ += ++

+ 11

1

1

1

k

n

k

n

k xxx +=+

+ 1

1

1

( ) 111 1| ++ =n

x

n

k kxQxJ

The Jacobian matrix is expressed as

x

P

x

F

x

x

x

F

x

xMJ

+

+

=&

&

&&

x

PK

x

xC

x

xMJ t

+

+

=&&&

where

x

FC

&

= is the tangent damping matrix

x

FKt

= is the tangent stiffness matrix.

STRAIN CALCULATION

To prevent locking during incompressible flow, we impose a constant

volumetric strain over the element. A consistent B matrix for the constantpressure assumption is employed in the tangent stiffness matrix. B is definedby Hughes in: T. R. Hughes, Generalization of selective IntegrationProcedures to Anisotropic and Nonlinear Media, Int. J. for NumericalMethods in Engineering, Vol. 15 no. 9, pp. 1413-1418 (1980).Based on midpoint geometry


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+

=

+

+

+

+

21

21

21

21

2

1n

i

n

j

n

j

n

iij

x

x

x

x &&&

this strain measure is a valid large strain measure , i.e., rigid body rotationsdo not cause straining

IMPLICIT ALGORITHM

( ) ( ) ( )nnnon

t xFxPuxK =+1

tK= Positive definite tangent stiffness matrix.

ou = Desired increment in displacements.

( )1+n

nxP = External load vector at n+1 based on the geometry at time n.( )nxF = Stress divergence vector at time n.

Update displacement vector

00

1

1 usxxnn +=+

and begin iterations for equilibrium

( ) ( ) 1111 ++++ == nin

i

nn

iit QxFxPuKj

where the subscript i denotes the iterate and j


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If convergence Is not attained , we update the displacement vector

ii

n

i

n

i usxx +=++

+11

1

and perform anther iteration. Lack of convergence within allowable number ofiteration (default=10) or divergence

1

1

1

0

++

+ < nin QQ

Causes tK to be reformed. Termination occurs if allowable number of

reformations is reached (default =15).The foregoing iteration method with tK held constant is called the modified-

Newton method. When convergence problems arise the stiffness matrix is

reformed.Four methods for updating the stiffness matrix are available:1. BFGS2. Broydens first method3. Davidon4. DFPThese are called quasi-Newton methods.

Quasi-Newton methods involve

Line searches Stiffness updates

Slightly more expensive than modified Newton but results in more stableprogram

QUASI-NEWTON SCHEMES

The secant matrices iK are found via the Quasi-Newton equations

iii QuK = 1

where)()( 111 iiiiiii FPFPQQQ ==


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Two classes of matrix updates that satisfy the Quasi-Newton equations areof interestRank one updates tii zxKK += 1

Rank two updates ttii vyzxKK ++= 1

Recall the Quasi-Newton equation iii uuK = 1 Subsitituting

t

ii zxKK += 1

we find

ii

t

ii QuzxuK =+ 111

By choosing

11

1/1

=

=

iii

i

t

uKQz

ux

we can satisfy the equation.Note that x is an arbitrary vector but is restricted such that

01 xut

i

Broyden set 1= iux and obtained the update

t

i

i

t

i

iiiii u

uu

uKQKK 111

111

+=

resulting in non symmetric secant matrices.Davidon chose

11 = iii uKQx

and obtained the updated formula

t

iii

i

t

iii

iiiii uKQ

uuKQ

uKQKK )(

)(11

111

111

+=

resulting in symmetric secant matrices.The inverse forms are found by the Scherman-Morrison formula:


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aAb

AabAAabA

t

tt

1

1111

1)(

+=+

where A is a nonsingular matrix

We can find the inverse form for Broydens update by letting

11

11

=

ii

iii

utu

uKQa

1= iub

in the Sherman-Morrison formula. Therefore,

ii

t

i

i

t

iiiiii

QKu

kuQKuKK

+=

1

11

111

1111

11 )(

The inverse form for Davidons method can likewise be found:

i

t

iii

t

iiiiiiii

QQKu

QKuQKuKK

+=

)(

))((111

111

1111

11

Again recall the Quasi-Newton equation iii QuK = 1

Substituting 11 ++= itt

ii uvyzxKK We find ii

t

i

t

ii QuvyuzxuK =++ 1111

Let

i

ii

i

t

i

t

Qv

uKz

uy

ux

==

=

=

11

1

1

/1

/1

Here x and y are vectors that are non-orthogonal to 1 iu i.e.,

01 xut

i and

01 yut

i and 01 yut

i

In the BFGS method


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11 = ii uKx

iQy =

Leading to the following updates formula

111

1111

1

1

+=ii

t

i

i

t

iii

i

t

i

t

iiii

uKu

KuuK

Qu

QQKK

That preserves the symmetry of the secant matrix. A double application ofthe Sherman-Morrison formula leads to the inverse form.Special product forms have been derived for the DFP and BFGS updatesand exploited by Matthies and Strang (1979).

)()(

1

1

1 t

iii

t

iii wvIKvwIK ++=

The primary advantage of the product form is that the determinant of iK and

therefore the change in condition number can be easily computed to controlupdates.Update vectorsLet,

iii

ii

QQ

u

=

=

1

1

ii

ii

ii

iiiii

w

Q

uQQv

=

+=

21

1

1 )(1

Determinant2

1 )1)(det()det( vwKKt

ii +=

Change in condition number

{ }

[ ]2

4

21

21

21

)1(4

)1(4)()(

wv

wvwvwvwwvv

ct

ttttt

+

+++

=

LINE SEARCHES


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With Quasi-Newton updates line searches are necessary to achieveconvergence.

After u is determined, a line search may be performed

search only if global response is hardening prevents divergence of Newton method find multiplier which gives best estimate of equilibrium

compute trial displacement 10, += susuu itr

evaluate out-of-balance force ( ) ( ) ( )trtrexttr ufufuR int=

search (iterate) to find s which minimizes out-of-balance force

ifs falls below 0.001, discard u and reform K

ii sdu =

where

iii QKd1=

iiii dsuu +=+1

Assume solution lies in direction id . The line search then component si such

that components of (P-F) in the search direction is zero

[ ] 0)()(. 11 = ++ iiii xFxPd

Broyden:

1

111111

1 )1)(1)....(1( +++=

ooookuwuwuwK ttiiii where

1

11

+

+= iiiiiiQKQKuw and )(/1

ii

t

iiwuu

= are the update factors.

Broydens Algorithm:

1. Solve iQKd1=

o

2. Do 1,1 = ki Recall du ti 1=


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11 += ii wdd

3.)(/1

/

1111

111

=

+=

kk

t

kk

kik

wuu

dsuuw

du tk 1=

4. 111 += kkk wdd

ARC-LENGTH METHOD

It is well known that the foregoing approach fails in the neighborhood of limitpoints. Geomtrically, the plane of =constant does not necessarily intersectthe load deflection curve. In the arc length method of Riks/Wempner aconstraint equation is added to limit the load step to a constant arc length,in load/displacement space.

Consider 0= pF

where is the proportional load factor. Newtons method leads to theincremental equations

FpxK iiji = )(

in classical approach = constant and iterations proceed until equilibrium is

reached.

ARC-LENGTH METHOD OUTLINED

1. Nonlinear equilibrium equation0= pR

2. Linearized equation

)( iii uRpuK =

iii uxx +=+1 equivalently,

iii uuu +=+1

3. Constraint equation22 suu i

t

i =+


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4. Load level is split into 3 parts

iim ++=

m = load level representing an equilibrium state at the beginning of a load

incrementi = increment load level in iteration i

i = change in load level in iteration i

5. Iterative displacements

( ) ( )[ ]RPPKRPKu miii ++== 11

II

i

I

iii uuu +=

PKuI

i

1=

([ ]RPKu imII

i +=

)

1

Iu represents displacement due to constraint conditionIIu represents usual displacement increment

6. Linearization of the constraint equation

0),( 22 =+= suuuf it

i

0),( =+

+

iiii

ii uf

fu

u

f

or 0=++ iiiii fu

7. We can now solve for i

SubstitutingII

i

I

iii uuu +=

Into 0=++ iiiit fu

YieldsI

i

t

ii

II

i

t

iii

u

uf

++

=


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Input Parameters for Nonlinear Solver

*CONTROL_IMPLICIT_SOLUTION:

NSOLVR: select Newton method

.eq.1: linear aproximation (no equilibrium iterations) .eq.2: BFGS quasi-Newton method (DEFAULT)ILIMIT: set limits for number of iterations before re-evaluating

.eq.1: new each iteration = Full Newton Method .eq.10: use 10 inexpensive BFGS updates, reform if not yet converged

MAXREF: reformation limit before abandoning step if AUTO is active, dt will be reduced and step will be re-tried, so MAXREF

can be smaller (~5)

if AUTO not active, error termination occurs when MAXREF is reached soMAXRED should be larger (~15, default)

DCTOL, ECTOL: convergence tolerances

use NLPRINT=1 or " nlprint" to monitor progress of iterations

Nonlinear Solver Screen Dump

BEGIN implicit time step 3============================================================

time = 1.09990E+00

current step size = 3.67821E-01

Iteration: 1 *|du|/|u| = 1.0894498E-01 *Ei/E0 = 1.8731172E+00

DIVERGENCE (increasing residual norm) detected:|{Fe}-{Fi}| ( 1.0547507E+07) exceeds |{Fe}| ( 9.1389570E+06)

automatically REFORMING stiffness matrix...

Iteration: 2 *|du|/|u| = 3.8969724E-03 *Ei/E0 = 3.3420090E-02Iteration: 3 *|du|/|u| = 6.3582980E-03 *Ei/E0 = 3.3460971E-02Iteration: 4 *|du|/|u| = 1.3780216E-03 *Ei/E0 = 6.2154527E-03Iteration: 5 *|du|/|u| = 6.0081244E-03 *Ei/E0 = 7.7976128E-03Iteration: 6 *|du|/|u| = 1.4377093E-03 *Ei/E0 = 8.9132953E-03Iteration: 7 *|du|/|u| = 6.4089308E-03 *Ei/E0 = 1.7184228E-02Iteration: 8 *|du|/|u| = 1.8267103E-03 *Ei/E0 = 1.9337881E-03Iteration: 9 *|du|/|u| = 1.9491626E-03 *Ei/E0 = 2.3472405E-03Iteration: 10 *|du|/|u| = 2.2147158E-03 *Ei/E0 = 1.5075735E-03Iteration: 11 *|du|/|u| = 1.8921960E-03 *Ei/E0 = 1.9947323E-03Iteration: 12 *|du|/|u| = 1.5758326E-03 *Ei/E0 = 7.9428701E-04

DCTOL=1.E-03 ECTOL=1.E-02

ILIMIT=11


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ITERATION LIMIT reached, automatically REFORMING stiffness matrix...

Iteration: 13 *|du|/|u| = 7.1106170E-04 *Ei/E0 = 3.0991789E-03

Equilibrium convergence summary for time step 3 at time = 1.0999005E+00

Number of iterations to converge = 13Number of iterations to converge = 2


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Example: 1-D Nonlinear Spring

To illustrate the Newton method, consider a simple nonlinear spring:

Iteration #1:

( ) 5.2,40,0 410

10 ==== uku

If the nonlinear solution method is linear, stop here and assumesolution is correct. Do not bother to check for equilibrium.

In a nonlinear problem, evaluate equilibrium by determining the internal forceand the out-of balance force R:

( )

3

7,5.2

int1

11

==

==

FFR

uFu

ext

Check convergence of equilibrium iterationsby evaluating displacement, energy norms

300.0105.2

35.2

000.15.2

5.2

1

=

=

=

==

oo Ru

Ru

e

e

u

u

force,

F(u)

displacement, u1 32 4

4

8

12

5

force,

F(u

)


4

8

12

5

k=4k=2

k=1F=10

k(u)u


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Iteration #2: solve K?u=R, update u=u+?u( )

0.4,5.12

3

25.2,5.2

2123

2

=+====

==

uuuk

Ru

ku

1,9)( int22 === FFRuF ext

evaluate new R, convergence norms

060.0105.2

15.1

375.04

5.1

1

=

=

=

==

oo Ru

Ru

e

e

u

u

Iteration #3: solve K?u=R, update u=u+?u

( )

0.5,0.11

1

10.4,0.4

3233

2

=+====

==

uuuk

Ru

ku

.0,10)( int22 === FFRuFext

evaluate new R, convergence norms

000.010*5.2

.00.1

200.00.5

50.1

1

=

=

=

==

oo Ru

Ru

e

e

u

u

force,

F(u)


4

8

12

5

force,

F(u)


4

8

12

5


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Selecting Implicit Time Step Size

When should load be applied in more than one step? to reduce nonlinearity to manageable level

to accurately integrate material behavior to resolve high frequency response in dynamic problems input parameters: stepsize DT and termination time TERM

How many steps should be used? estimate nonlinearity in problem tradeoff:

few steps with many equilibrium iterations (unreliable)many steps with few iterations (too expensive)

observe progress of equilibrium searchreduce stepsize until norms decrease monotonicallyincrease stepsize if very few iterations are needed (


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AUTOMATIC TIME STEP CONTROL

If convergence fails, LSDYNA backs up to beginning of step with smaller DT.Solution is retried. If convergence again fails, solution is retried with a smallerstep and so on. Finally DT is set to DTMIN, if fails terminate.

An optional number of iterations, IOPT, per step is chosen. Let ILS be thenumber of iterations during the last increment. If

ILSIOPT

DT=DT-(DT-DTMIN) ])/(1[ 21

ILSIOPT

Where DT, DTMAX, and DTMIN are the current step size, the maximum steppermitted, and the minimum step size permitted, respectively.

Automatic time step control allows specification of the Optimum Numberof Equilibrium Iterations per step.

This indicated how hard LSDYNA should work in each time step.

If equilibrium is reached in fewer than optimum iterations, the size of thenext step is increased.

If equilibrium search requires more than the optimum number of iterations,then the next step size is decreased.

CONTROL_IMPLICIT_AUTOAutomatic Time Step Control

Automatic time step control adjusts stepsize during the simulation

very persistent, reliable


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After successful steps compare iteration count to target value ITEOPT increase/decrease size of next step if difference exceeds window ITEWIN

After failed steps decrease step size back up, repeat failed step with new DT

Exponential algorithm for adjusting step size increase stepsize by 1/5 decade until DTMAX is reached decrease stepsize by 1/3 decade until DTMIN is reached error termination if convergence fails when DT=DTMIN

Default time step control strategy

ITEOPT = 11, ITEWIN=5, DTMIN=0.001*DT0, DTMAX=10*DT0 decrease step if more than 16 iterations were required increase step if less than 6 iterations were required problem: stepsize continues to decrease even though every step

converges successfully

some models simply require more than 16 iterations per step

Aggressive step control strategy ITEOPT=200, conservative value for DTMAX stepsize always increased if convergence is successful agressively pushes stepsize toward DTMAX stepsize still decreases if convergence fails, but increases rapidly

thereafter

recommended for many problems


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NONLINEAR SOLUTION METHODS FORIMPLICIT ANALYSIS

*CONTROL_IMPLICIT_SOLUTION

Define control parameters for the implicit nonlinear equation solver:There are two groups for nonlinear solution. One group with four solutionmethods. The second group is the same as the first group, however, with thearclength method.

If NSOLVR is equal to:

EQ 2: BFGS updates

EQ 3: Broyden updatesEQ 4: DFP updatesEQ 5: Davidon updatesEQ 6: BFGS updates + arclengthEQ 7: Broyden updates + arclengthEQ 8: DFP updates + arclengthEQ 9: Davidon updates + arclength

Standard Newton Method:The nth load increment is applied-then using the tangential stiffness matrixthe iterative displacent are found and hence the residual forces.The newtangential stifness is then evaluated and the new iterative displacement andredidual forces are found. The process is repated until convergence isobtained. The figure below depicts the algorithm.


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Equilibrium is reached when iterations converge: 0,0 )( iRu

Modified Newton-Raphson Method

The difference here between this method and the one before is that there isno update of the tangential stiffness matrix after the initial iteration. Thefigure below depints the algorithm.

( ) ( ))(int)()( iextii uffRuuK ==

Forc

e

Norm

Displacemnt

Norm

un u(n+1)1 u(n+1)2 u(n+1)3 u(n+1)

fn+1ext

fnext

R1R2Ku1()

Kun( )

fint


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_______________________________________________________________________________________________


BFGS Method

Also called the quasi-newton method (QN). In here an approximation to the

local tangent matrices are found. No factorization is required. Theapproximation to the local tangent matrix was defined in the previous section.The figure below depicts the algorethim.

Snap-through and Collapse

In many collapse problems the equilibrium path could look like the figurebelow. In this type of behavior the BFGS method or in general the Newtonmethod has deficulties with the limit points. For the ascending branch of theequilibrium curve, load control is greatly the most efficient method. As theplateau in the response is reached (point A), then difficulties withconvergence are experienced (with the BFGS method the solutiion will jumpto point A). If displacement control is adopted then when point B is reachedagain there will be convergence difficulties (solution could jump to point B).To overcome this problem an algorithen is added to LS-DYNA that providethe correct structural response. This algorithm is called the Arc-LengthMethod.


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Arc-Length Method

The arc-length method basic idea is that a constaint equation is added so

that load level is modified at each iteration rather thann holding it constant.The method is graphicaly depicted in the figure below. Two arc-lengthmethods are available in LS-DYNA. These methods are called the Rammmethod (updated normal plane method) and the Crisfield method (sphericalpath method)


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The Ramm MethodIn here as oppose to forcing the iteration process to follow a planeperpendicular to the initial tangent for each load step, the iteration process isconstrained to be normal to the current arc. The method is graphicly depicted

in the figure below.

The Crisfield MethodIn here the iteration process is constrained to follow a spherical path centredat point o with radius equal to the arc length. The method is graphicly

depicted in the figure below.


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Linear Equation Solver

*CONTROL_IMPLICIT_SOLVER

Sparse Direct Linear Equation Solvers

LS-DYNA has 3 options for direct solution of the sparse systems oflinear equations.

All 3 options are based on the multifrontal algorithm. Multifrontal is a member of the current generation of sparsity preserving

factorization algorithms that are also have very fast computationalrates.

That is multifrontal works with a sparsity preserving ordering to reducethe overall size of the direct factorization.

Sparsity Preserving Orderings

In LS-DYNA 970 there are two ordering algorithms for preserving thesparsity of the direct factorization.

The algorithms are Multiple Minimum Degree (MMD) and METIS. MMD computes the ordering using locally based decisions and a

bottom-up approach. It is inexpensive and very effective for small problems, that is problems

with fewer than 100,000 rows. METIS computes the ordering from a top down approach. While

METIS usually takes more time than MMD to compute the ordering, theMETIS ordering reduces the work for the factorization enough torecover the additional ordering cost.

METIS is especially effective for large problems, especially those thatare modeling three-dimensional solids.

The user can specify either algorithm using

*CONTROL_IMPLICIT_SOLVER.

Multifrontal Algorithm

The multifrontal algorithm factors a sparse matrix in a way that vastlyreduces the amount of work required to compute the factorization


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_______________________________________________________________________________________________


compared to methods such as the frontal, profile, skyline, and variableband.

These older methods counted on clustering the nonzero entries of thefactorization close to the diagonal to keep the size of the factorization

and the amount of work required to compute the factorization to aminimum.

Three Solver Options

In Version 970 of LS-DYNA there are three direct linear solver options. FromVersion 960 to 970 LSTC removed the two older (and less efficient) options.The three options are

Solver

No.

Method

4 Real*4 implementation of multifrontal which includes automatic out-of-memory capabilities as well as distributed memory parallelism.Can use either MMD or METIS orderings. Default method.

5 Real*8 implementation of Solver No. 4.6 Multifrontal solver from BCSLIB-EXT. Uses Real*8 arithmetic with

extensive capabilities for large problems and some Shared MemoryParallelism.Can use either MMD or METIS orderings. If the other solvers

cannot factor the problem in the allocated memory, try using thissolver.

Solver 6 on a single processor computer should be comparable toSolver 5 but has more extensive capabilities for solving very largeproblems with limited memory.

Solvers 4 and 5 should be used for distributed memory parallelimplementations of LSDYNA.

Solver 6 can be used in shared memory parallel. In an installation of LSDYNA where both integer and real numbers are

stored in 8 byte quantities, then Solvers 4 and 5 are equivalent.


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Sparse Eigensolver

LSDYNA now includes the Block Shift and Invert Lanczos eigensolverfrom BCSLIB-EXT.

This eigensolver is used in LSDYNA to compute the normal modes andmode shapes for the vibration analysis problem The Lanczos algorithm iteratively computes a better and better

approximation to the extreme eigenvalues and the correspondingeigenvectors.

BCSLIB-EXT uses a sophisticated logic to chose a sequence of shiftsto enable the computation of a large number of eigenvalues andeigenvectors.

The implementation of BCSLIB-EXT in LSDYNA includes a sharedmemory implementation. However only limited parallel speed-up is

available for most problems. The wall clock time for the eigensolver is as much a function of the

speed of the I/O subsystem on the computer as the CPU time. Parallelism can only speed up the CPU time and does nothing to

speed-up the I/O time.

The user can request how many and which eigenvalues to computeusing the keyword *CONTROL_IMPLICIT_EIGENVALUE.

Via the parameters on this keyword, the user can request any of the following

problems:

Compute the lowest 50 modes (that is nearest to zero) Compute the 20 modes nearest to 30 Hz. Compute the lowest 20 modes between 10 Hz and 50 Hz. Compute all of the modes between 10 Hz and 50 Hz. Compute all of the modes below 50 Hz. Compute the 30 modes nearest to 30 Hz between 10 Hz and 50 Hz.

For example, in running a 240,000 node car body model, one need 1400 Mwto extract 30 eigenvalues, but only 750 Mw to perform a linear analysis.


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During each nonlinear iteration, the linear system RuK = is solved.

Iterative Methods

(preconditioned) conjugate gradient

potentially low operation count convergence difficult for some problems promising future developments

x x

x

x x x x x x

x x x x x

x x x x

x x x x x x

x x x x x

x x x x

x x x

x x

x

x x x x x x

x x x x x

x x x x

x x x

x x

x

x x x

x x

x

Direct Methods

gaussian elimination inexpensive backsolve (quasi-Newton) The sparse direct solver looks promising for large linear systems substantial storage savings huge operation count reduction

Example: Beam Tower Model (Reverse Engineering, Ltd.)

beam elements 43,590 degrees of freedom


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Linear Equation Solver Dataskyline sparse

memory (Mw) 18


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_______________________________________________________________________________________________


LSOLVR = 14: Lanczos method LSOLVR = 15: Lanczos with Jacobi preconditioning LSOLVR = 16: Lanczos with Incomplete Choleski preconditioning

All iterative solvers use the sparse matrix storage scheme eliminates all zero entries inside bandwidth minimizes total storage requirement Boeing Harwell format for portability

Iterative solvers outperform direct solvers in some problems solid, massive structures narrow range of natural frequencies (good numerical condition

number)


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_______________________________________________________________________________________________


Example:

20x20x20 brick elements, 26,460 equationsSolver type DIRECT ITERATIVE

LSOLVR 0 (default) 16Memory 120 Mb 9 MbCPU 400 sec 21 sec

Linear Equation Solver Memory and CPU

An optional print-out is available to show memory and CPU requirements ofthe linear equation solver: [K]{x}={f}

activated by keyword input *CONTROL_IMPLICIT_SOLVER: lprint activated interactively lprint memory estimate printed before attempting to allocate change memory using command line option ls-dyna3d i=inputfile.k memory=8000000

units are words1 word stores 1 floating point numbersingle precision = 4 bytes per word

double precision = 8 bytes per word

F


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_______________________________________________________________________________________________


Linear Equation Solver Memory and CPU

BEGIN implicit time step 42983==========================================================

time = 1.55684E+02current step size = 7.78421E+01

SPARSE LINEAR EQUATION SOLVER STORAGE data (Mwords)( 225972 degrees of freedom)pointer arrays: initial = 11.523

actual = 6.413stiffness coefficients = 6.187

Factorization Workspace (estimated)symbolic = 14.015numeric = 18.335

Final Storage Requirements (10% for pivoting)incore out-of-core

symbolic factorization = 5.276 5.276numeric factorization = 69.772 5.292

numeric solution = 65.561 3.145TOTAL = 87.648 23.168

TOTAL available = 98.196 98.196

an INCORE solution will be performedInitialization CPU = 7.220E+00 secondsSymbolic Factorization CPU = 1.065E+01 secondsNumeric Factorization CPU = 8.539E+02 seconds

Forward/Backward CPU = 5.060E+00 seconds


BEGIN implicit time step 42983============================================================

time = 1.55684E+02current step size = 7.78421E+01

SPARSE LINEAR EQUATION SOLVER STORAGE data (Mwords)( 225972 degrees of freedom)

pointer arrays: initial = 11.523actual = 6.413

stiffness coefficients = 6.187Factorization Workspace (estimated)

symbolic = 14.015numeric = 18.335

Final Storage Requirements (10% for pivoting)incore out-of-core

symbolic factorization = 5.276 5.276numeric factorization = 69.772 5.292

numeric solution = 65.561 3.145TOTAL = 87.648 23.168

TOTAL available = 98.196 98.196

an INCORE solution will be performed

Initialization CPU = 7.220E+00 secondsSymbolic Factorization CPU = 1.065E+01 secondsNumeric Factorization CPU = 8.539E+02 seconds

Forward/Backward CPU = 5.060E+00 seconds


memory required

number of equations

solution method

CPU timings

begin factorization here

memory available

1 Mword =4 Mbytes (single)

1 Mword = 8 Mbytes (double)


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REMARKS

In nonlinear problems equilibrium iterations must always be permitted.Reformation of the stiffness matrix at the beginning of every time step

often speeds convergence of the equilibrium search. Stiffness matrixreformation normally occurs once at the beginning of each step, andwithin a time step if either the equilibrium iteration limit is reached or thesolution diverges.

To suppress the stiffness reformation at the beginning of each step, setthe parameter ISTIF, in *CONTROL_IMPLICIT_SOLUTION, to a value,which exceeds the total number of steps in the problem. This is ofteneconomical for contact dominated problems, where the solution divergesimmediately after the first equilibrium iteration, causing a (second)stiffness reformation.

To suppress equilibrium iterations during each step, set the parameterNSOLVR in *CONTROL_IMPLICIT_SOLUTION to a one. A single linearsolution will be performed each step. This solution may not represent anequilibrium state, hence results obtained may be inaccurate.

In problems dominated by contact it may be cost effective to limit theBFGS iterations to five or less between stiffness reformations. This allowsLSDYNA/IMPLICIT to better account for rapidly changing contact areas.Enter a value of one for ILIMIT in *CONTROL_IMPLICIT_SOLUTION toobtain Full Newton Method with Line Search.

The convergence strategy for DNORM=2, which is the defult, in*CONTROL_IMPLICIT_SOLUTION compares the current displacementincrement to the total displacement over the analysis. This criterionbecomes less strict as the analysis proceeds. By setting DNORM=1 thestrategy considers the total displacement over the step, hence is moreconsistent as the deformation evolves.

In problems where there is much rigid body motion the displacementtolerance may be insufficient, and it may be advisable, in some problems,to tighten the energy tolerance to 10 -3.

The arc length solver is intended for use with buckling problems where theload-displacement curve is not monotonic, a problem class wheretraditional displacement or energy norm driven method fail. In this method,the solution is advanced in increments of constant arc length on the load-displacement curve, regardless of its path. This may require loadreduction or unloading as the problem passes a bifurcation point.


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_______________________________________________________________________________________________


The user may define a single node whose load-deflection history controlsthe solution. If no node is defined, norms of the global force anddisplacement vectors are used in the generalized arc length method.

*CONTROL_IMPLICIT_GENERALIGS Geometric nonlinearity is introduced when displacements become large

(example: tip deflection of a cantilever beam is greater than the beamthickness), or rotation become large (example: beam rotation theta islarge enough that the approximation sin(theta)=theta is not reasonable).

A second geometric nonlinearity is due to finite rigid body rotation. Both ofthese nonlinearity are always accounted for in LSDYNA-Implicit.

The flag IGS is an option to include an additional term in the stiffnessmatrix which reflects a change in apparent stiffness due to rotation of

stress. This flag does not change the degree of nonlinearity included incalculation of internal or external forces. IGS does not alter the equilibriumsolution. However, the path taken during the iteration process to reach thefinal solution depends on the stiffness matrix. In some cases, adding thisterm will accelerate convergence and in other cases this term will create asingular stiffness matrix and halt the iteration process. For instanceincluding this term will speed convergence for transversely loaded plateswith clamped boundaries, where transverse deflection generates sharpincrease in membrane stress. It is required by hyperelastic materials.

Two cases where you can benefit from the geometric stiffness are:

1) cantilevered beam type problems where either both ends are clamped(lots of membrane force), or where the tip deflection is very large so that theslope of the tip becomes say 30 degrees or more. Here the geometricstiffness will improve convergence rate of the nonlinear iteration process.

2) Modal analysis where you wish to see the effect of stress on the frequencyresponse of the structure. This is the guitar string problem.

*CONTROL_IMPLICIT_GENERALCNSTNThe consistent tangent flag CNSTN applies only to material type 3 for 2D and3D solid elements, and to material type 115 for 3D solid elements. In thesematerials, an option is available to use an alternative procedure forcomputing the material tangent stiffness matrix. In general, the "consistent"


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_______________________________________________________________________________________________


tangent stiffness will be softer than the default stiffness. This may in somecases accelerate convergence of the nonlinear equilibrium iterations, but inother cases may lead to divergence. Smaller load steps may be necessaryto realize a benefit from the consistent tangent stiffness. Since the net benefitof this feature is unclear, it is not active by default.

*CONTROL_IMPLICIT_SOLVERLSOLVR The LSOLVR "linear solver" provides the solution to the linear system of

algebraic equations Kx=f. There are several methods to compute thisexpensive solution. LSOLVR selects between them.

What is the best and what situations for both direct and iterative.

Several options are available using parameter LSOLVR on*CONTROL_IMPLICIT_SOLVER. For most simulations, the defaultsolver should be used. When using a double precision executable(recommended), the single and double precison solver options areequivalent.

The BCS-EXT solver (LSOLVR=6) is superior for very large modelswhich must run out-of-core. (Activate the linear solver print flagLPRINT=1 or interactively type " lprint" to toggle printing of

memory information to the screen, which will indicate memoryrequirements for in-core and out-of-core execution.) The BCS-EXT solver is more versatile in exporting blocks of the

stiffness matrix to disk files, and can run with the least core memory ofany of the direct solvers. This solver also utilizes a more complexpivoting strategy, producing superior solutions in cases where negativeeigenvalues nearly exist.

The iterative solvers (LSOLVR=10 to 16) are generally poor forautomotive applications. However, these solvers can dramaticallydecrease memory and CPU requirements for large brick element

models, suchas those found in civil engineering soil models. If your model is largelycomprised of brick elements, it is probably worthwhile to test aniterative solver. LSOLVR=10 activates what we feel to be currentlythe best of the iterative solver options. It is presently set to activateiterative solver 16, but this may change as improved iterative solversare added.


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*CONTROL_IMPLICIT_SOLVERORDER.

Matrix reordering occurs during the symbolic factorization phase of thelinear equation solution procedure. During reordering, matrix rows and

columns are shifted to minimize the number of non-zero entries in theinverted stiffness matrix. This is similar to the bandwidth minimizationoperation which occurred with the old skyline solvers. Because ofreordering, the node numbering scheme chosen by the user isarbitrary.

Two reordering options are available, selected with the ORDERparameter on *CONTROL_IMPLICIT_SOLVER. The Multiple MinimumDegree (MMD) method (ORDER=1) is the least expensive method, butmay not produce an optimal reordering, leading to more cost during thenumeric factorization (matrix inversion) phase.

The METIS reordering option (ORDER=2) is the most expensivemethod, and usually produces the best reordering and fastest numericfactorization phase. Presently, MMD is used by default for smallmodels, and METIS is used for large models.

The obvious goal in selecting a reordering method is to minimize thecombined cost of the symbolic plus numeric factorization phases of thesolve. To judge these costs, use LPRINT=1 or type " lprint" totoggle printing of linear solver information to the screen. Pay particularattention to the CPU costs:

Symbolic Factorization CPU = ...

Numeric Factorization CPU = ...

In general, the cost of symbolic factorization should never exceed 10%of the numeric factorization cost, and is usually 1% or less. If thesymbolic cost is large, try activating the MMD reordering, Beware thatthe numeric factorization cost will probably grow, so the net gain usingMMD may not be positive.

*CONTROL_IMPLICIT_SOLUTIONNSOLVR In contrast, the nonlinear solver controls the iteration process as we seek

to find the displacements which balance internal and external forces.


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There are several nonlinear solvers to choose from, many of which arebased on Newton iteration. The parameter NSOLVR is used to make thischoice.

The simplest nonlinear solver option is "NSOLVR=1", which is to skip theequilibrium iterations and accept the linear approximation to the solutions.

This choice is commonly referred to as a "linear" solution (not to beconfused with the linear equation solver LSOLVR above).

Screen Messages and Interactive Commands It helps to have a quick look at the progress of a run. An input parameters

is added to activate printing this information to the screen. The defaultvalues prevent printing. These two parameters are:

NLPRT on *CONTROL_IMPLICIT_SOLUTION and LPRINT on *CONTROL_IMPLICIT_SOLVER. We recommend *****ALWAYS***** activating NLPRT, and also

recommend activating LPRINT the first time you run a large problem toget a sense of how much memory and cpu the stiffness matrixfactorization will require.

Interactive switches are added to activate each type of printing. Theswitches have the same name as the input parameters:

Typing " nlprt" activates the nonlinear print flag. Typing " lprint" activates the linear print flag.

Typing the switch a second time deactivates the flag, so you can turn themon and off at will. One caution is that you must enter these interactivecommands during the nonlinear iteration phase of a time step. If you happento issue the command during another solution phase, say while the plot file isbeing printed, then the command will be ignored.

*CONTROL_IMPLICIT_GENERALIMFLAG=6This is used for explicit and implicit analysis with intermittent eigen-value

extraction. Dynamic explicit analysis can produce stress oscillations and as aresults intermittent eigen-value extraction may lead to miss-leading resultsbecause of the stress oscillations.

There are three methods which can be used to account for stress effects ineigenvalue analysis. Shell type 6 may be used, but is not required.


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1) Perform a standard eigenvalue analysis:

Use IMFLAG=1 and DT0=1.0 on *CONTROL_IMPLICIT_GENERAL.Enter the number of eigenvalues to be computed using NEIGV on

*CONTROL_IMPLICIT_EIGENVALUE. Introduce stress using the *INITIAL_STRESS keywords (probablycomputed in a previous simulation which used*INTERFACE_SPRINGBACK_LSDYNA to output a "dynain" file).

Activate the geometric stiffness terms using IGS=1 on*CONTROL_IMPLICIT_GENERAL.

2) Perform an implicit intermittent eigenvalue analysis. In this method, eigenvalues will be extracted at one or more times

during an implicit analysis.

Set up an implicit analysis to apply load to the model.Enter a negative integer value, NEIGV=(-n) on *CONTROL IMPLICITEIGENVALUE. This indicates that curve ID=n will give NEIGV as afunction of time.

Define curve ID=n. Define a point at each time when eigenvalues willbe computed. For each curve point, the curve value gives the numberof eigenvalues to extract at that time.

The geometric stiffness terms will be automatically activated duringeigenvalue calculations. Parameter IGS on

*CONTROL_IMPLICIT_GENERAL can be selected as needed to givethe best convergence behavior during the implicit loading analysis.

3) Perform an explicit intermittent eigenvalue analysis. In this method, eigenvalues will be extracted at one or more times

during an explicit analysis.

Beware that stress oscillations which normally occur during explicitdynamic analysis can introduce geometric stiffness effects which mayappear counter-intuitive.

Set up an explicit analysis to apply load to the model. Select IMFLAG=6 on *CONTROL_IMPLICIT_GENERAL to activateexplicit intermittent eigenvalue extraction.

Enter a negative integer value, NEIGV=(-n) on*CONTROL_IMPLICIT_EIGENVALUE. This indicates that curve ID=nwill give NEIGV as a function of time.


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Define curve ID=n. Define a point at each time when eigenvalues willbe computed. For each curve point, the curve value gives the numberof eigenvalues to extract at that time.

An eigenvalue analysis is a linear analysis, so double precision must be usedfor accuracy. Linear also means infinitesimal deformation, so contact gapswhich are "open" stay "open", and gaps which are "closed" stay "closed".With our penalty type contact, we introduce stiff penalty springs betweenparts wherever they contact. So, in an eigenvalue analysis, stiff springs areintroduced wherever contact interface gaps are "closed". This effectivelybonds or ties the parts together. No springs are introduced in areas whereinterface gaps are "open", so in these areas the parts are completely free.So, you could say that the contact was "ignored" in areas where gaps are"open".


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FINITE ELEMENT MODELING TECHNIQUES

1. Engineering a FEA Model2. Element Selection3. Mesh Density4. Symmetry5. Modeling for Physical Phenomenon6. Ad Hoc Guidelines7. How to Tell If Your Results Are Correct

1. ENGINEERING A FFA MODEL

PRIMARY CONSIDERATIONS

Design Situation Physical Nature of Design Concerns Information Required Most Effective Level of Simulation Detail

DESIGN SITUATION

New Design or Existing Problem

PHYSICAL NATURE OF DESIGN CONCERN (S) Does previous product knowledge exist that can be used to pinpoint high

risk area

What is the nature of the predominant physical phenomena that must besimulated

What assumption can be made and what are their effects on simulationresults

INFORMATION REQUIRED

Full scale tests, material characterization, geometry, etc.

MOST EFFECTIVE LEVEL OF SIMULATION DETAIL

Purpose of Analysis/Evaluation Focus Requirements for time Responsiveness


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Cost/Benefit Trade off Confidence Level of Model and Loading Assumption

Implicit Newer

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