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8/11/2019 262-NonconvergenceinCaesarII
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Nonconvergence in Caesar II
The equations in Caesar II (and all pipe flexibility analysis programs) are linear in nature
and therefore are readily solved using matrix operations on computers. Difficulties ith
numerical methods occur hen nonlinear boundary conditions exist. The computations
solve a system of linear algebraic equations simultaneously until the solution does notchange appreciably beteen successive iterations. !ut nonlinear boundary conditions
may cause a change in the actual equations themselves beteen successive iterations andis the basis for all convergence problems in this type of numerical routine.
"nce the iteration process begins the loads defined in the load case editor are applied tothe piping and the pipe moves according to these loads# the stiffness of the pipe# and ho
it is restrained. $hen gaps are closed then the stiffness changes from %ero to hatever
stiffness is specified for the restraint in input. This changes the equations that Caesar II
has to solve. "n each successive iteration Caesar II must chec& the status of the piping ateach nonlinear restraint location to determine hether the restraint is engaged or not
engaged and then change the equations accordingly. Convergence is not achieved untilevery restraint in the system retains the same engaged'notengaged state beteen severalsuccessive iterations (and of course the numerical solution also changes inappreciably at
each node in the system as ell).
The various types of nonlinear boundary conditions# the convergence difficulties they
may cause# and methods for obtaining convergence in Caesar II hen encountering these
difficulties are explained next.
Nonlinear estraints
Nonlinear restraints are defined as restraints hose stiffness is not uniform along the lineof action of the restraint. *nidirectional restraints (for example# pipe resting on a rac&)#
restraints ith a gap# and bilinear restraints are all nonlinear. $hereas fully bi
directional restraints are linear. The folloing illustration shos force vs. displacementcurves for several linear and nonlinear restraints in Caesar II. Note the slope of the curve
dictates the stiffness of the restraint (rigid restraints have an almost vertical slope). +ll
nonlinear restraints have more than one stiffness in their force vs. displacement curveshile all linear restraints have a constant stiffness.
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F
Y
k
Y R e s t r a i n t , l o w s t i f f n e s s
F
Z
k
Z R e s t r a i n t , h i g h s t i f f n e s s
Nonlinear Restraints
F
Z
k
- Z R e s t r a i n t
k = 0
F
X
k
X R e s t r a i n t w i t h g a p
k = 0g a p
k
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F
Z
Z 2 ( b i - l i n e a r ) R e s t r a i n t
k 1
F
- F
F
Y
! Y R e s t r a i n t w i t h g a p
k = 0
g a p
k
Note that rotational restraints may also be linear or nonlinear in a similar manner. The
stiffness in these cases relate a moment to rotation resulting in a stiffness in units of ,tlbs'degree (length-force'angle).
"iagnosing #on$ergen%e &roble's
Caesar II provides some good diagnostic tools for the user hen the solution ill not
converge. $hen the solution does not converge a indo called the Incore olver ill
be available (it alays comes up but in small converging systems this indo maysimply flash by on the screen) as in the illustration belo.
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/ressing the ,0 function &ey on the &eyboard ill free%e the solution so the user can
apply some diagnostics.
"n the upperleft portion of the Incore olver indo is some general information about
the solution. The most important of these is the Current Case and Total Cases. The
current case is the load case that is being solved for and the total cases is the total numberof native (noncombination) load cases in the load case editor. The solver performs the
analysis on the load cases in the order they are entered in the load case editor.
The middleright side of the indo gives information on the total number of nonlinear
restraints in the model (01 in the figure above) as ell as the number of these restraints
that did not converge on the previous iteration (2 in this case). "n some models thenumber of nonconverged restraints may fluctuate and it is recommended to attempt to
free%e the analysis hen this number is loest. "nce fro%en# pressing the Continue
button ill perform a single iteration of the analysis# so it is a simple matter to press
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Continue several times to obtain a minimum number of nonconverged restraints. Note
that pressing the ,0 function &ey (or the ,0 button in the Incore olver indo) ill
continue the analysis normally.
In the hite list box in the loerright section of the Incore olver is information on
individual nonconverged restraints. The node number of the restraint is given as is thedirection cosine of the line of action of the restraint. This provides easy identification of
hich restraint is causing the problem# hich is useful hen there are several nonlinear
restraints at the same node and only one or to of them are not converging. In theexample above# of the four restraints that can be seen ithout scrolling the list box# only
3 restraints are not converging (these are li&ely to be 43 restraints).
!elo the direction cosine of each nonconverged restraint is the state of the restraint."pen means that the restraint is not in contact ith the pipe and the stiffness at this
location is %ero. Closed indicates that the restraint is in contact ith the pipe and that its
stiffness is being used in the analysis. "ld tate is the previous iteration and Ne tate is
the current iteration. + state of liding'Not liding indicates the friction state of therestraint# hich is discussed further belo.
5a&e a note of all the nonconverged restraint node numbers and perhaps their direction
cosines and then clic& the Cancel button to terminate this analysis (limitedrun users are
not charged a run hen terminating the analysis in this manner).
Next# return to input and revie the nonconverged restraints onebyone. It is often
useful to revie the model graphically ith the node numbers turned on. In this ay it is
quic&ly determined hether the nonconverged restraints are clustered geometricallyclose to one another or if they are more evenly spread throughout the model. $hen there
are large numbers of nonlinear restraints in a model it can be very difficult to determine
the best ay to proceed. Determining the best method for obtaining convergence is asmuch art as it is science since e cannot &no exactly hat is happening numerically in
the solution of so many simultaneous equations.
ometimes# especially hen there are a small number of nonconverged restraints# a
simple change to the model may allo complete convergence. mall changes to
operating temperature# for example increasing it by 6 ,# may cause the system to
converge (this is if it is the operating case that is not converging. ,or sustained case nonconvergence the temperature ill not affect the solution).
#on$ergen%e &roble's ()e to Fri%tion
Numerical modeling of friction is a difficult process at best and it is notorious for causing
nonconvergence in flexibility analysis programs. ,riction alays acts in a plane that isnormal to the line of action of the restraint it is applied to. ,or example# on a 43 restraint
the friction# if any# ill act in the 78 plane.
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$hen pipe begins to move due to thermal expansion and other loads# static friction
opposes this motion and the force beteen the pipe and the restraint increases (see figure
belo). +s the load increases# usually due to heating of the pipe# it may eventually reacha point here the pipe ill brea& free from friction and begin sliding. This brea&aay
force is equal to the normal force (the force along the line of action of the restraint) times
the friction coefficient# 5u. In reality if the heating is gradual the pipe may slide a smallamount and relieve a small amount of the friction force# but then increase again until it
once again exceeds the brea&aay force. This may occur many times ith such
ratcheting repeating until the pipe temperature reaches steady state and the system is inequilibrium.
F
X - Z
! Y R e s t r a i n t w i t h f r i % t i o n
k
F n * +
- F n * +
k = 0
k = 0
+t this point# it is important to note that the friction force is still opposing further motion#even hen in equilibrium. In reality hen the pipe is sliding this force is equal to the
dynamic friction coefficient times the normal force# hich is slightly smaller than the
static friction coefficient times the normal force. It is a common misconception that once
the system reaches equilibrium that there is no further friction force on the system. Thisleads some to believe that friction is a transient phenomenon that only occurs during the
startup or shutdon cycle of a piping system. This is definitely not true9 ,rom the time
the friction brea&aay force as first exceeded until the system reaches equilibrium theforce opposing motion is beteen the static and dynamic brea&aay force# hich is a
fairly small range of force variability.
Caesar II must calculate three primary quantities to simulate friction at a given node
here a friction coefficient is defined on a restraint. ,irst# the direction of pipe motion
must be calculated. This is easy to do from the resultant displacements on a giveniteration. The angle that the resultant displacement vector in the friction plane ma&es
ith one of the axes in this plane is &non as the friction angle. Then a friction restraint
ith a stiffness (given by the friction stiffness in Configure'etup) is applied opposite this
direction of motion. The normal force is multiplied by the userdefined friction
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coefficient# (5u on the restraint definition) to determine the restraint brea&aay force.
The equation for the given node is then changed to reflect the direction and stiffness of
the friction restraint for the next iteration.
,or the node to be converged# the friction brea&aay force# the state of the friction
restraint# and the direction of the friction restraint must all be ithin certain tolerancesbeteen the current and last iteration. ometimes# especially ith many restraints ith
friction# the solution ill :bounce; beteen a situation here the pipe is sliding (remove
the friction restraint and replace it ith the friction force opposing motion) and notsliding (put in a friction restraint opposing motion). This is a discrete change in the
boundary condition# hich can result in an infinite loop since the equations for the node
simply change bac& and forth repeatedly.
,btaining #on$ergen%e on ste's with Fri%tion
,ortunately# there are several simple settings that can affect the ay that friction is
modeled in Caesar II. $hen a solution ill not converge there are three items that can bechanged that ill determine hether or not the solution ill be completed. +ll of these
items may be accessed in the Caesar II Configuration ,ile (shon belo).
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noted that the choice of friction coefficient on the restraint definition can be difficult to
accurately determine. The friction coefficient is a function of the surface roughness of
both the restraint and pipe# hich can have large variations due to moisture# corrosion#pitting and other defects# and manufacturing processes.
Determination of the system>s sensitivity to friction can be determined using Caesar IIand ill help the engineer determine# at least sub
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#on$ergen%e &roble's ()e to Nonlinear Restraints
Initially# the piping system is said to be in a neutral state# hich is simply the geometricalrepresentation of ho the system as input. This means that for bidirectional restraints
ith gaps the pipe is resting ith an equal gap on each side and is essentially
unrestrained. ,or unidirectional restraints the pipe is initially in contact ith therestraint# but ith %ero load since no loads are being applied to the piping system. ,or a
unidirectional restraint ith a gap the pipe is resting here the restraint ould be if it
had no gap# but is not in contact ith the restraint.
"ccasionally# a restraint ill be in a numerical situation here the solution ill change
bac& and forth due to the continual changes in the equations. +t this point the solution is
in an infinite loop and ill never converge. The analysis must be terminated and the inputsomeho changed to avoid this numerical methods problem.
.ineari/ation of Non-%on$erge( Restraints
+ very effective method for obtaining convergence is the changing of one or more
restraints from nonlinear to linear. ?inear restraints do not have convergence issues.
If the restraint is bidirectional ith a gap it is often useful to either remove the gap
(lineari%ing the restraint) or change the gap dimension (remaining nonlinear# butchanging the solution domain). "ften# engineers model restraints ith very small gaps to
accommodate the diameter groth of pipe due to thermal expansion. +lthough you
should not change the design of such gaps# often these small gaps are unneccessary hen
modeling a system for pipe flexibility analysis and the resultant error in the results shouldbe tolerably small. In such cases removal of gaps# at least on nonconverged restraints#
ill allo the analysis to converge.
If the restraint is unidirectional then it can be lineari%ed by changing it to bidirectional.
In this ay a 43 restraint can be changed to a 3 restraint# a B8 restraint can be changed
to an 8 restraint# and so on.
!ilinear restraints are rarely nonconvergent# but if they are# a small change in the
brea&aay force# ,y# or brea&aay torque (on rotational bilinears) may accomplish
convergence. +ltering the and 0 values of stiffness may also help. ince bilinearrestraints are only used in unusual circumstances to model a very specific type of
behavior (usually soil modeling or hinge t or lineari%ation of to restraints near the opposite
edges of the area of nonconvergence ill often do the tric&. =xperimentation ith the
location of lineari%ation of restraints ill sometimes be required before the solution can
be obtained and in such systems there is no simpler method.
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It should be mentioned at this point that if the model also contains friction at the locations
of nonlinear restraints# then the friction problem should be dealt ith first# prior tolineari%ing restraints (see discussion that follos).
It is important to reali%e that any time a nonlinear restraint is lineari%ed to obtainconvergence that error ill be introduced into the results# unless of course the design is
changed to accommodate the use of the ne linear restraint. It is prudent therefore to
assess# at least sub