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in Elasticity Pipe using CIP method

Furuta Nobuyasu

Scho ol of Information Science,

Japan Advanced Institute of Science and Technology

February 13, 1998

Keywords: CIP, Moving b oundary.

Background and Purpose

Euler's method and Lagrange's method are used as a means to solve the equation of

motion of the uid sofar. Euler's method thinks about the volume elementin the mesh

wherethespaceinsideisxedanditisatechniquebywhichthe velo cityand thepressure

of the uid whichcrosses in the mesh momentarily there are calculated. Asagainst this,

lagrange's methodisatechniqueby whichthemovementispursuedconsideringthe uid

to bea meeting of the particlewithout xing the ow area. In aword, the mesh is xed

to grid point inthe spaceas for Euler's method, the other hand side, the mesh moves in

Lagrange's metho d according to the movement of grid point.

Fromthe dierence of such acharactor, in the problem of the calculating interaction

and uid and solid area like the moving boundary, Lagrange's method is used. The

compatibility of the Lagrange's method with the moving boundary problem is b etter,

because itis dragged tomovinguid and solid and mesh istransformed.

However,whenthe problem ofthe movingboundaryis solvedby Lagrange'smethod,

repapering the mesh of each time step is needed. In addition, there is a problem that

the mesh collapses when a big transformation of the uid and the solid is caused and

arithmetic precision decreases.

It is easy to treat because the grid is xed to the space for this in Euler's method.

However, it is dicult to know co ordinates of the boundary of the uid and the solid

accuracy go od. When the pursuit of the boundary of uid and solid is calculated, the

decreasein accuracy because ofnumericdiusion is caused.

Copyrightc 1998byFurutaNobuyasu

TheCIPmethodproposedbyYab eTakashiisatechniquethataccuracygoo dcanbevery

solvedas forthe advection. Ifwecalculate the p ositionof the movingboundarybyusing

the CIP mehto d,wecome tobe able to takeup the problem of the movingb oundary by

usingEuler's mesh. The problem wherethe boundary ofthe uidarea and the solidarea

moveswill beable to takeupit easily.

Thereisabigcharacteristicintheinterp olationmethodofeachcalculationgridinthe

CIP method. Basically, this interpolation uses the third spline interp olation. However,

interp olationformulaisnotdecidedlikethethirdsplineinterp olationofnaturalaccording

to continuity of the rst or second dierentiation. In the CIP metho d, the prole of the

value and the intersticshas eachgrid pointand interpolation formulais easily requested

fromthe prole. AsfortheCIPmethod,itisconciseandnumericdiusionisalso alittle.

The characteristicof it isthat the expansion intomulti-dimensional iseasy.

Experiment

An one-dimensional advection equationwassolved asa preliminary experimentbyusing

the CIP metho d in this research. It was conrmed to evaluate the accuracy of the CIP

method,and toobtain enoughaccuracy. However,aover-sho ot wasseen inthe gridwith

the discontinuities. Therefore, we do about the inclination of the discontinuities right

and left dividing. As a result, we can conrm where the over-sho ot was not caused in

discontinuitinous.

Next, a two-dimensional advection equation was solved by the CIP method. Enough

accuracy onpractical use wasobtained.

The CIP method was applied to the method of the Navier-Stokes equation. The

Navier-Stokesequationwasseparated atthe advectiveterm andnon-advectiveterm,and

advective term was solved by using the CIP method and non-advective term was solved

by using the dierence metho d. The cavity ow was ualitatively approximated well by

this technique with the one whichhad been calculatedby apast dierencemethod.

Next, the calculation where the solid movedthe piston upand down in the uid was

done. Here, the CIP methodwasused tocalculate the positionof the solid whichmoved

intheuid. Forthemovingb oundaryproblemweexaminedhowbythemovingboundary

to give the boundary condition. By calculating this model the voltexes occurredwhen a

solid deceleraters. Afterof the stationaryof a solid, that voltexes was observed.

Finally,owofthe vibratingpipewasanalyzed. Inthiscomputationmodel, Thispip e

repeatesvibration whichisthat centerpart inastraighttubeslowlyreturnstostricture.

The stricture part of a pipe was transformed by using the CIP method. The velo city

of stricture was set in the grid of neighb orho od in contraction area. The p osition of the

movingboundaryiscalculatedbysolvingtheadvection. When theCIPmethodwasused,

movingboundaryproblem issolvedand weare conrmedtobeable tosolvethe problem

of the moving boundary tobe accompanied by acomplicated transformation.

The content shown by this researchis shown ab ove.

1. The CIP method was appliedtothe Navier-Stokesequation.

2. TheCIPmethodwasappliedtothecalculationoftheboundaryinmovingboundary,

and how by the moving b oundary togivethe boundary conditionwas shown.

3. It was shown tobeable to treat the movement and the transformation of the solid

area easilyby using the CIP method.

4. Asthisapplicationwecan think aproblem wheresolidiscausedcomplicatedtrans-

formation.