Soil Dynamics using Flac

7/21/2019 Soil Dynamics using Flac

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Dynamic Analysis with FLAC

by Peter Cundall

Itasca Consulting Group, Inc.

Sudbury, October 2003


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Why use FLAC or FLAC3D?

FLAC(3D)simulates the full, nonlinear response of asystem (soil, rock, structures, fluid) to excitation from

an external (e.g., seismic) source or internal (e.g.

vibration or blasting) sources.

Therefore it can reproduce the evolution of permanent

movements due to yield and the progressive

development of pore pressures (and their effect on

yield).

Equivalent- l inearmethods (as used in many

earthquake analyses) cannot do this directly.


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Topics

1. Review of equivalent linear method

2. Review of dynamic wave propagation

3. Boundary conditions

4. Damping
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The equivalent-linear method is common in earthquake engineering

for modeling wave transmission.

In this method (Seed and Idriss, 1969), multiple linear analyses are

performed (i terat ions), with averagedamping ratios and shear moduli ineach element determined from the elements maximum cyclic shear

strain in the previous iteration.

Laboratory-derived curves relate damping ratio and secant modulus to

amplitude of cycling shear strain. (See next slide).

Equivalent-Linear Method vs

Fully Nonlinear Method

Seed and Idriss (1969), Influence of Soil Conditions on Ground Motion

During Earthquakes, J. Soil Mech. Found., Div. ASCE, 95, 99-137


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0.0001 0.001 0.01 0.1 1

ShearStrainAmplitude(%)

0

10

20

30

40

50

Damp

ingRatio(

%)

0.0001 0.001 0.01 0.1 1

ShearStrainAmplitude(%)

0.00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.80.9

1.0

N

orma

lizedShearModulus,

G/Gm

ax

Mid-Range Sand Curve(Seed & Idriss, 1970)

Sand Fill Inland:Friction =32, hr=0.47, Go=440

Sand Fill under Rock Dike:Friction=30, hr=0.43, Go=440


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Only one run is done with a fu l ly non l inear methodsince non linearity is

followed directly by each element as the solution marches on in time.The dependence of damping and apparent modulus on strain level are

automatically modeled, pro vided that an app rop r iate nonl in ear law is

used.


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Characteristics of the

Equivalent-LinearMethod

1. Linear properties remain constant throughout the history of shaking.

During quiet periodsin the excitation history, elements will be over-

damped and too soft; during strongshaking, elements will be under-

damped and too stiff. However, there is a spatial variation in properties

that corresponds to different levels of motion at different locations.

2. The interference and mixing phenomena that occur between different

frequency components in a nonlinear material are missing from an

equivalent-linear analysis.

3. The method does not directly provide information on irreversibledisplacements and the permanent changes that accompany

liquefaction. These effects may be estimated empirically, however.


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Equivalent-LinearMethod (Cont.)

4. Plastic yielding, therefore, is modeled inappropriatelyno proper flow

rule.

5. The stress-strain curve is in the shape of an ellipsecannot be

changed.


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Fully NonlinearMethod

1. The method follows any prescribed nonlinear constitutive relation, and

the damping and tangent modulus are appropriate to the level of

excitation at each point in time and space.

2. Using a nonlinear material law, interference and mixing of different

frequency components occur naturally.

3. Irreversible displacements and other permanent changes are modeled

automatically.


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Fully NonlinearMethod (Cont.)

4. A proper plasticity formulation is used in all the built-in models,

whereby plastic strain increments are related to stresses.

5. The effects of using different constitutive models may be studied

easily.


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Boundary ConditionsThus far, all the examples have been for dynamic input at a

rigid base, and for zero damping (no energy absorption within

the material). We now introduce a quiet boundarythat absorbs

incident waves.

In a plane wave, stress is related to particle velocity:

n P nC v

S sC v

P-waves:

S-waves:

The coefficients are the acoustic impedances. If we apply these

impedances as boundary conditions, then incident waves that

approach in a normal direction will be perfectly absorbed.


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Quiet boundaries

The uniform-layer example is repeated, replacing the free surface with

a quiet boundary (not particularly useful, but it illustrates the effect).

Wave input

Quiet

boundary

Note: no

Amplitude-

doubling

Minimal

reflections


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Internal dynamic sourceA more useful case is that of an internal source (pressure

loading in tunnel), with quiet boundaries on three sides:

tunnelfree surface

What about staticconditions? A quiet boundary acts like

a dashpot, providing no resistance to long-term, static

loads.


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Initial static state

conf dyn ext=5

grid 200 50

model elas

gen circ 100 25 5

prop dens 1000 sh 1e7 bulk 2e7model null reg 100 26

set grav 10 dyn off

fix x i=1

fix x i=201

fix y j=1

ini syy -5e5 var 0 5e5 sxx -2.5e5 var 0 2.5e5

solve

set dyn on

apply xquiet yquiet long from 1,51 to 201,51

FLACsquiet command automatically applies existing reaction

forces in reverse at boundaries that are made quiet.

FLAC data file for the tunnel-problem

setup, and installation of quiet boundaries

Now in equilibrium, with quiet

boundaries in place

(vertical stress contours)


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Dynamic tunnel response

0.2 sec

0.4 sec

0.6 sec

0.8 sec

A pressure pulse is

applied inside the

tunnel. Contours of

velocity magnitude

are plotted, with the

same interval for all

(velocity-magnitude contours)


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Observations & cautions

The quiet boundary is a perfect absorber only for waves of

normal incidence, and for p- and s-waves only.

For oblique waves, and for Rayleigh waves, energy is

absorbed, but there is some reflected energy.

Therefore, boundaries should be placed far enough

away, so that material damping and/or geometric spreadingprevent significant boundary reflections from returning to

the area of concern. (Check with different boundary locations)


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External sources & quiet

boundaries

For external seismic sources through a compliant (soft) foundation, we

cant apply a velocity (or acceleration) condition at the boundary

because the quiet boundary needed for the soft foundation would be

nullified by the imposed velocity condition.

Therefore, we apply a stresscondition that is equivalent to the

velocity (or acceleration) in the incident wave. We use the formula

given previously: 2 S sC v

However, there is a factor of 2 because the input energy

divides into a downward- & upward-propagating wave.


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Input through quiet boundary

apply xquiet yquiet j=1

apply sxy -2e5 hist wave j=1

1000

100S

C 2

S sC v

equivalent to a velocity of 1.0

Quiet boundary

& stress input

Free surface


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Lateral boundaries, for

seismic input at baseIn a 2D case with seismic input at the base, how do we deal with

quiet lateral boundaries? Consider our tunnel example, with

stress-wave input at the base. At 0.2 sec .

quiet

quiet

quiet

Note that the lateral quiet boundaries distort the

incoming wave its no longer a plane wave there


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Free-field boundaries

To avoid the boundary distortion of the incident wave, we performtwo, 1D calculations for the free field, and use this data to

eliminate energy absorption if the main-grid motion is identical

to the free-field motion. (However, reflected waves are absorbed).


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Results with free-field

0.2 sec

0.4 sec

apply xquiet j=1

apply ff

apply sxy=1e5 hist=wave j=1

FLACcommands -

Note the

uniformconditions at

the lateral

boundaries


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Cautions with ff boundaries

All properties, conditions & variables are transferred fromfrom side columns in the main grid when APPLY FF is

given.

Things that are changed in the main grid afterwardsare

not s eenby the free field, apart from applied motion.

Interfaces and ATTACH lines cannot extend to the FF. Toemulate an interface, use a thin layer of weak zones. To

avoid boundary ATTACHes, use a wrap-around grid

This grid, with internal,attached fine grid, was

created by the GIICs

grid library feature.

Attach

line


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Further FF points

Any model or nonlinear behavior may exist in the free field, as

well as fluid coupling and flow (vertical only!).

However, the FF performs a small strain calculation, althoughthe main grid may be executing in large strain mode. In this

case, the results will be approximately correct if the

deformations near the FF boundaries are relatively small

(e.g., compared to grid dimensions).


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Multistepping

As mentioned, FLACstime step is determined by the smallestzone with the highest stiffness. For models with a few stiff

elements (e.g., a concrete tunnel liner in soft soil), the calculation

can be very inefficient.

Multisteppingtakes account of the natural time step of each

zone. Zones with large natural time steps are only updated

infrequently, compared to those with small natural time steps.

Significant savings in calculation time are obtained, if the systemcontains objects with great contrasts in stiffness. The smallest

time step is still taken, but there is far less work per step.


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Multistepping (2)In this example (from Unterberger et

al, 1997), the tunnel liner is concreteand the surrounding material is soft

soil. Dynamic loading was applied to

the rail bed, and vibrations at ground

surface monitored.

The use of multistepping reduced

calculation time by a factor of 5 times.

Provided that the wavelength limit is

respected, tests have shown that

errors of less than 1% are introducedby multistepping.


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Material Models and Damping

Ideally, a comprehensive model for soil would account for all thephysical effects that occur during cyclic loading, such as energy

dissipation, volume changes and stiffness degradation.

An ideal model does not exist, so we need to compromise, andaccount for some important aspects (such as damping and

cyclic volume changes) separately.

First, we consider the important attributes of soil, and then howto capture their effects in a FLACmodel.


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Soil characteristics

1. Continuously nonlinear; apparent modulus degrades with strain.2. Hysteresis for all levels of cyclic strain, resulting in an increasing

level of damping with cyclic amplitude. Damping is rate-

independent.

3. Hysteresis for superimposed mini-cycles; damping for all

components of a complex waveform.

4. Appropriate volume strain induced by shear strain; in particular,

volume-strain accumulation with cycles of shear strain.

5. Volume strain associated with neutral loading (constant shear

stress, but varying angles of principal axes).


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Elastic/plastic modelsThe built-in models in FLACconsist of various

elastic/perfectly-plastic relations. There is onlyhysteresis for cyclic excursions that involve yielding.

strain

stress(Note that even this crude model produces

continuous damping and modulus

relations, for excursions above yield)

There may be volume

changes during yield

but normally they aredilatant (not such as

to cause liquefaction)

Oth h ELM*


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Other approachesELM*SHAKE, and other frequency-domain schemes, use viscous

damping, but scaled with frequency so that the dissipation

appears to be independent of frequency. However, this impliesunrealistic stress/strain curves:

Note that the material

anticipates the impending

change in direction of shear

strain increment (since the

response curves downwards

before the reversal point). This

is not possible generally it

violates causality!

*Equivalent Linear Method

U i l i / l i d l


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Using elastic/plastic modelsIf we use an elastic/perfectly-plastic model, we may need to

account for additional factors, such as:

1. damping, for stress cycles below the yield limit;

2. volume-strain accumulation, as a function of number of

cycles and their amplitude;

3. modulus degradation, by using tables based on averaged

strain levels (not normally done).

We will consider damping and volume-change formulations

shortly, but note that the elastic/plastic modelin spite of its

simplicityis good in many situations, particularly those in

which the accumulated plastic deformation (slumping, partialslip) is required to be estimated. The model is not so good for

estimating amplification factors of acceleration, for low-level

shaking.


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Damping overview

Note that - even without explicit material damping - energy

may be absorbed in FLACsimulations:

by geometric spreading of waves;

by absorption at quiet boundaries;

by plastic flow in yield models;

by 3D radiation damping.


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Rayleigh damping

Rayleigh damping may be used in FLACas an approximation to

hysteretic (frequency-independent) damping. Two viscous

elements are used to make up the damping matrix:

The mass-proportional term is like a dashpot connecting each

gridpoint to ground. The stiffness-proportional term is like a

dashpot connected across each zone (responding to strainrate).

Although both dashpots are frequency-dependent, an

approximately frequency-independent response can be obtainedover a limited frequency range, by the appropriate choice of

coefficients.


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Rayleigh dampingcont.

frequency

ratio of damping to critical

combined

stiffness-proportional only

mass-proportional only

Note 3:1 frequency range over which

combined damping is almost constant


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A Suggested Procedure for

Selecting Rayleigh Damping

Parameters

1 Estimate Material Damping from Cyclic


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1 - Estimate Material Damping from Cyclic

(triaxial or shear) Tests1- 3

A A

D

B

CC

A

C

D = 1 (w)

4 w

where:

D = fraction of critical damping

w = energy dissipated during cycle, and

w = stored energy at peak

w = 1 [(D- B) (A + C)]

2

w = 1 |A| A

2


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Typical Results from Cyclic

Triaxial Tests

2 C t D i R ti f


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2- Compute Damping Ratio for

Elastic/Plastic Model

Damping Ratio = D = 2 (

-m)

m

G


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3 - Perform dynamic analysis

of dam assuming elastic

material behavior

Collect histories of cyclic shear strain for

representative elements (i.e., elements that

represent the behavior of different materials

and positions within dam).


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4 - Determine required

damping ratio

For each group of elements determine

required damping ratio based on differencebetween lab damping and model damping in

the range of expected shear strains.

Typical damping ratios are 5% or less.


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5 - Perform FFT analysis of shear strain time

histories to determine central frequency for

each group of elements

The central objective of Rayleigh damping isto supply constant damping (independent offrequency) over a wide a frequency range aspossible.

Rayleigh damping approximates hystereticdamping over a 3-to-1 frequency range.


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Rayleigh dampingcont.

The drawbacks with Rayleigh damping are that:

1. The center frequency must be chosenfrom

sometimes conflicting data (e.g., the site resonance or

the earthquake average frequency)2. The stiffness-proportional term causes the time step to

be reduced as the damping ratio (lambda), at the

highest natural frequency, is increased:


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Other approaches - bilinear

It is tempting to use a simple hysteretic formulatione.g., a bi-linearlawbut this can lead to unrealistic effects, such as the conversion

of low-frequency energy to high frequency energy (see above).

Oth h


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Other approaches

continuous functions

The particular form of the stress/strain relation is very important.

Using a smooth law (similar to a bounding-surface law) leads to

much more realistic spectra:


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Fully nonlinear FLACmodels

FLACcan import material modelseither written in the FISH

language or in C++, as DLLs (dynamic link libraries), loaded as

needed. The latter feature is relatively new.

Itasca has recently implementedas a DLLthe modeldescribed by Wang, Dafalias & Shen (ASCE J. Eng.

Mech.,1990). This is known as the bounding surface

hypoplasticty model for sand (referred to as the Wangmodel).

The model displays all the soil characteristics noted earlier. Someresults from FLACsingle-zone tests illustrate these.

The Wang model some


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The Wang modelsome

FLACresultsThe following plot shows the results of a shearbox test, in whichthere are several mini-cycles of strain within the main cycle.

Note that the

response is

continuous, andthat there is

dissipation for

small sub-cycles

shear strain

shear stress

( )


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Wang model results (2)

An undrained, cyclic triaxial test is simulated, giving the following

results for a loose material.

The effective mean

stress reduces,

causing the shear

stress-difference todecrease, due to

material yield.

effective mean stress

stress difference


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Wang model results (3)Further results from the undrained, cyclic triaxial test:

axial strain

stress differenceThe results show

a progressive

degradation in

modulus, with

increasing cycles

W d l


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Wang model useIn principle, the Wang model is capable of reproducing many

important aspects of soil behavior, including those that are involved

in cyclic loading.

However, the model needs up to 15 material parameters (although

several of these have default values). The calibration procedure is

therefore more complicated than that of a simple model, such asMohr Coulomb.

The Itasca-implemented Wang model is currently being tested, and

will be made available on the web site when its operation seems to

be correct. It also works with FLAC3D.

The model is an example of a C++ User Defined Model.

Fully-nonlinear with Mohr


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Fully-nonlinear with Mohr

Coulomb?Finally, if many elements can be used, it is possible to approximatecontinuous yielding with the M-C model. For example, a shear box is

set up with a Gaussian distribution of friction angle

grid 40 20

m m

prop dens 2000 sh 1e8 bu 2e8 fric 25 rdev 10 tens 1e10

def qqq

loop i (1,izones)

loop j (1,jzones)

friction(i,j) = max(0.0,friction(i,j))

endLoop

endLoop

end

qqq

Contours of friction angle


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For a load-unload-load cycle with uniform-strength material-

Shear displacement

Shear stress

With the non uniform strength material we get the following response:


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With the non-uniform strength material, we get the following response:

Cyclic strain = 4 units Cyclic strain = 5 units

Note (a) the continuously-nonlinear response, and (b) the

larger specific energy loss for greater cyclic strain.


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This approach would is useful if the model can contain large

numbers of elements. The distribution of strengths needs to

be chosen so that the laboratory results for damping &

modulus versus strain are matched.

The advantages are: (a) that the response is similar to that of a

fully nonlinear model, and (b) the time-step is unaffected (recall

that Rayleigh damping causes the timestep to reduce).

A d i f l ti


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A new damping formulation

For the next releases of FLACand FLAC3D, an optional

hysteretic damping will be available for dynamic simulations. Thedamping is independent of the material models, and consists of a

strain-dependent multiplier on the tangent shear modulus.

0

0.2

0.4

0.6

0.8

1

1.2

0.0001 0.001 0.01 0.1 1 10

Cyclic strain %

Modulusreductionfactor

If the secant modulus is given by

a degradation curve, then thetangent modulus can be derived:

s

s

t s

M

dMdM M

d d

secant modulus

tangent modulus

shear stress

shear strain

s

t

M

M

From Seed & Idris (1970)


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Given a particular modulus-degradation function, the resulting

tangent-modulus is used to multiply the apparent shear modulus

(G) provided by the constitutive model:t

G M G

The apparent strain is the

deviatoric strain

accumulated since the

previous reversal point.Such reversal-points are

kept in a stack so that

embedded cycles within a

main cycle may be

followed.

FLAC (Version 4.00)

LEGEND

12-Feb-03 15:39

step 3700

HISTORY PLOT

Y-axis :

Ave. SXY ( 1, 1)

X-axis :

X displacement( 1, 2)

-40 -20 0 20 40

(10 )-05

-2.000

-1.000

0.000

1.000

2.000

(10 )+04

JOB TITLE :

Itasca Consulting Group, Inc.

Minneapolis, Minnesota USA

Thus, energy is dissipated for mini-loops as well as the

main hysteresis loop.

The new damping formulation has three advantages


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The new damping formulation has three advantages.

1. Standard G/Gmax degradation curves used in

equivalent-linear analyses may be used directly in FLAC

& FLAC3D, to perform fully nonlinear simulations with thesame material response.

2. The damping does not affect the time step (in contrast to

Rayleigh damping, which may profoundly reduce the time

step).

3. The damping may be used with any material model, andwith any of the other damping schemes (optionally)

active.

One disadvantage is that published degradation curves seemto be inconsistenti.e., a hysteretic model that conforms to the

G/Gmax curve does not necessarily conform to the associated

damping curve

1 2 60


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0

0.2

0.4

0.6

0.8

1

1.2

0.0001 0.001 0.01 0.1 1 10

Seed data

FLAC - Sig3 fit

0

10

20

30

40

50

60

0.0001 0.001 0.01 0.1 1 10

Seed data

FLAC - Sig3 fit

0

0.2

0.4

0.6

0.8

1

1.2

0.0001 0.001 0.01 0.1 1 10

Seed data

FLAC - de fault model

0

10

20

30

40

50

60

0.0001 0.001 0.01 0.1 1 10

Seed data

FLAC - default model

Good fit to Seed & Idris data for G/Gmax (sigmoidal 3-parameter

function)note inconsistent damping result.

Approximate fit to both G/Gmax and damping curves (default FLAC

2-parameter model)

G/Gmax

G/Gmax

D - % of

critical

D - % of

critical


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So far, no serious simulations have been performed using

the new formulation. Comparisons with similar equivalent-

linear analyses will be made before the new feature isreleased.

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