Nonconvergence_in_Caesar_In_Caesar_II

8/14/2019 Nonconvergence_in_Caesar_In_Caesar_II

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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 and

is 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 to

the piping and the pipe moves according to these loads# the stiffness of the pipe# and hoit 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 at

each 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 several

successive iterations (and of course the numerical solution also changes inappreciably ateach node in the system as ell).

The various types of nonlinear boundary conditions# the convergence difficulties theymay 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. displacement

curves for several linear and nonlinear restraints in Caesar II. Note the slope of the curvedictates 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 curves

hile 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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+s previously discussed# the friction stiffness is set in the Configuration file and is seen


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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

Nonconvergence_in_Caesar_In_Caesar_II

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