soil dynamics using flac
TRANSCRIPT
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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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Characteristics of the
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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Characteristics of the
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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Characteristics of the
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.