I 'AO-4115 643 DAVID W TAYLOR NAVAL SHIP RESEARCH AND DEVELOPMENT CE--ETC F/G 20/4A LOCALIZED FINITE-ELEMENT METHOD FOR THREE DIMENSIONAL SHIP MO-ETC(U)

j MAT 02 K .J BAT

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A LOCALIZED FINITE-ELEMENT METHOD FOR THREE FormalDIMENSIONAL SHIP MOTION PROBLEMS . PERFORMING ORG. REPORT NUMBER

7. AUTHOR(s) 6. CONTRACT OR GRANT NUMBER(e)

Kwang June Bal

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASKAREA & WORK UNIT NUMBERS

David W. Taylor Naval Ship Research Task Area RRO140302and Development Center Work Unit 1542-018Bethesda, Maryland 20084

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Office of Naval Research (ONR-432) May 1982Arlington, Virginia 22217 13. NUMBER OF PAGES

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Finite-Element Method Natural FrequencyShip Motion EigenvaluesAdded Masses and Damping CoefficientsRestricted Water

20. ABSTRACT (Continue on reveree olde if neceeary and Identify by block number)

An application of the localized finite-element method to a three-

dimensional time-harmonic free surface flow in a canal is presented.

Boundary conditions on both the free surface and the body are linearized and

imposed on their equilibrium positions. By utilizing known set ofeigenfunctions, the computation domain is reduced to a very small local----.

domain where an eight-node linear three-dimensional element is used.

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--.-_Proper matching is also imposed between two sets of trial functions onthe truncated boundary. To be solved are the problems concerning: (1)six degree-of-freedom radiation and diffraction in three dimensions,(2) two dimensional motion corresponding to the local flow at midshipcross-sectional plane, and (3) related eigenvalues. Specifically, two rsets of results for two ship locations in a canal are presented. Inboth cases, the eigenvalues of the local cross-sectional plane areshown to play a significant role in the three-dimensional results.A remarkable similarity between exciting forces and moment and thedamping coefficients corresponding to the modes of their motions arealso observed in the results. The accuracy of the three-dimensionalresults presented here is also discussed by comparing two sets ofeigenvalues computed by using two different sizes of finite elements..

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TABLE OF CONTENTS

Page

LIST OF FIGURES......................... . ...... . .... . .. .. .. . ...

ABSTRACT .. ......................... ... . . .. .. .. ......

INTRODUCTION .. ........................ .. .. .. .. ......

MATHEMATICAL FORMULATION .. ............................ 2

LOCALIZED FINITE ELEMENT METHOD. .. ........................ 2

RESULTS AND DISCUSSIONS. .. ............................ 4

ACKNOWLEDGMENTS .. ...... .......................... 12

REFERENCES. ....... ............................. 12

LIST OF FIGURES

1 - Boundary Configuration of the Three Subdivided Fluid Domains .. ......... 3

2 - Cross-Section at x = 0. .. .......................... 5

3 - Added Masses, and Moments of Inertia of Added Mass, p iversus VB, for s/B = 0. .. .......................... 6

4 - Damping Coefficients, X ijversus VB, for s/B =0. .. .............. 7

5 - Coupling Hydrodynamic Coefficients p jand X ijversusvB, for s/B = 0 .. ............................... 8

6 - Magnitude of Excitation Forces and Moment versus VB. ........ ..... 9

7 - Purely Two-Dimensional Added Masses and Moment of AddedMass versus VB (Due to the Side walls, no WaveRadiation is Present) .. ........................... 9

8 - Added Masses and Moments of Inertia of Added Mass, pj~versus VB, for s/B - 0.125 (Off-Center) .. .................. 10

9 - Damping Coefficient, X ijversus for s/B = 0.125 .. .............. 11

10 - Some of Coupling Hydrodynamic Coefficients versusvB for s/B =0.125 .. ...... ...................... 12

Page

11 - Magnitudes of Wave Excitation Forces and Moments versus VB .......... ... 13

12 - Purely Two-Dimensional Added Masses and Moment of AddedMass versus VB for s/B = 0.125 (Due to the SideWalls, no Wave Radiation is Present) ......... .............. ... 14

iv

A LOCALIZED FINITE-ELEMENT METHOD FOR THREE DIMENSIONAL SHIP MOTION PROBLEMSO

Kwang June Bai

A David W. Taylor Naval Ship Research and Development CenterBethesda, Maryland 200S4

w

Abstract For two- and three-dimensional motions in a fluid of in-finite depth, or of finite but constant depth, the required

An application of the localized finite-element method singularities are well known, although of rather corn-to a three-dimensional time-harmonic free surface flow in a plicated analytical form, so that the approach described incanal is presented. Boundary conditions on both the free the foregoing corresponds to solving a Fredholm integralsurface and the body are linearized and imposed on their equation over the body surface, with a rather complicatedequilibrium positions. By utilizing known set of eigenfunc- kernel function. An eatensive list of literature on this sub-dons, the computation domain is reduced to a very small ject can be found in Wehausen1 .local domain where an eight -node linear three-dimensionalelement is used. Proper matching is also imposed between in this paper a numerically oriented as well as moretwo sets of trial functions on the truncated boundary. To versatile method is introduced as an alternative approach tobe solved are the problems concerning: (I) six degree-of. the solution of the problem. This alternative approach isrsdom radiation and diffraction in three dimensions, (2) based on a variational principle which is utilized to deter-

two dimensional motion corresponding to the local flow at mine the velocity potential throughout the fluid domain.the midship cross-sectional plane, and (3) related eigen- The present work is a direct extension of the earlier workvalues. Specifically, two sets of results for two ship loca- by Ba and Yeung 2 for a general three.dimensional bodydons in a canal are presented. In both cases, the eigen- geometry in a canal. In this procedure, the known solutionvalues of the local cross-sectional plane are shown to play a space in certain subdomains is made use of in order tos gnificant role in the thwee-dimensional results. A reduce the 'computation box' to a size as small as ponible.remarkable similarity between exciting forces and moment We call this drastically-reduced computation box theand the damping coefficients corresponding to the modes localized finite-element domain. The phrase "localizedof their motions are also observed in the results. The ac- finite-clement method" is used to denote the finite elementcuracy of the three-dimensional results presented here is method applied only in this localized subdonsain. In thisalso discussed by comparing two sets of eigenvalues com- particular problem, i.e., in a free-surface flow in a canal,puted by using two different sizes of finite elements, the solution space is represented by the complete set of

eigenfunctions in the subdomain. It is essential to make the

I. Introduction domain of computation as small as possible because thepractical success or failure of a numerical method, in

Steady-state time-harmonic motions of an inviscid, in- general, mainly depends upon the size of the computationcompressible fluid with free surface in the presence of a box. In the present procedure, the reduction of the original

infinite fluid domain to a small localized subdomain isbody go bodies in it are described by a boundary-value pro- achieved by replacing the conventional radiation conditionble governed by the Laplace equation with appropriate by the matching condition. This requires that the velocitybodary conditions in the past, problms of this e potential and its normal derivative (i.e., normal velocity)were generally solved by distributing sources (and/ordipoles) on the body boundary and using Green's theorem represented by two sets of different trial functions, defined

to obtain an integral equation for the strength of these in the adjacent subdomains, be continuous along the fic-

boundary singularities; or, alternatively, by using sources titious interface juncture boundary surface. Then we ap-

and higher-order multipole expansions at an interior point proximate the potential in the localized finite-element do-

within the body, the strengths of these singularities being main by piecewise polynomial trial functions defined in

determined so as to satisfy the body boundary condition. each finite element. Thus, in effect, an integral equation

In all cas it is conventional to utilize the singularities over a body surface with a complicated kernel is replaced

which are solutions of the boundary-value problem stated by a system of equations over a much larger fluid domain,

above, except for the body boundary condition which is in- but with a much simpler kernel.

yoked separately to determine the singularity distribution. Previous studies of the finite element method applied

to time-harmonic ship motion problems have been made by

_ai34 , Berkhoff

7, Smith

8 , Chern and Meig, Seto andYamamoto1

0 Yue et al t Euvrard et aI 2, andChowdbury d).

OM work was supported by the Numerical Naval Hydro-dynamics Program at the David W. Taylor Naval Ship Additional numerial results of the related two-R&D Center. This Program is jointly supported by dimensional boundary-value and eigenvalue problems at theNSRDC and the Office of Naval Research. Part of the midship cross-sectional plane are also presented. Two dif-present work was done while the author was a visiting ferent sizes of the finite elements in the midship cvoss-professor at the Ecole Nationale Suprieure de Techniques sectional plane are used in these additional computations.Avances (ENSTA) in 1980, and partially supported by These results are used as reference in discussing the ac.the Ministry of Defense, France. curacy of the present three-dimensional results.

11. Mathematical Formulation with

Considered here is steady-state time-harmonic free- (ai, a2 , n3 ) - (nx. nY, ndsurface flow in the presence of a body floating or submerg.ed in a canal with a rectangular uniform cross-section. In4, n5.n6) - j-" c) aHowever, a local variation of the bottom and side-wallgeometry is present and a uniform rectangular cross-section where Wis the unit normal vector into the body with com-of each side (not necessarily the same on both sides), can ponenhs (n. , ny, n),-c is the vector from the origin to thebe similarly treated by the present method. The coordinate center of rotation, andsystem is right-handed and rectangular. The y-axis isdirected opposite to the force of gravity, and the xz-plane r - (x, y. z).coincides with the undisturbed free surface. The bottom of _8+1the canal is in the y = -H plane and the side walls are in For m = 7. f(s) - - (7)the z - :t W/2 planes. We neglect surface tension, andassume that the fluid is inviscid and incompressible and where +Iis the spatial potential associated with the incidentthat the motion is irrotaional. Furthermore, we assure that whee sys The ipatia v potential o wit the litdethe motion of the body and, consequently, the generated wave system. The incident wave potential of unit amplitudewave. to be snall in some sense, so that the boundary con- incoming frod a - _m is given bydition on the body and on the free surface can be lineariz- i cosh m o (.y + H) eimox (8)ed and satisfied at the mean equilibrium positions instead *l (x, y, z) - a o s Hof at their instantaneous positions. cosh m. H

Let * (x, y, z. t) be the velocity potential describing where m o is wave number. Finally, to make the solution ofthe flow field. The continuity equation requires 0 to satisfy this problem unique, we impose the radiation condition re-Laplace's equation. Let a be the angular frequency of the quiring that the generated waves must be outgoing.time-harmonic solution. Then, introducing the usual timeand spatial decomposition we have Ill. Localized Finite Element Method

(X, y, 2. t) - Re[+(-) (x, y,z) S*(m)(t)) (1) The fluid domain defined in the foregoing formulationis unbounded along the x-axis. For numerical computa-

m I, ... 7 tions, it is highly desirable that the computation box bemade as small as possible. The goal of reducing the

where *(') ) + i(2M) (2) original infinite fluid domain to a manageable finite do-main is achieved by making use of the known solution

is the complex-valued spatial potential, and a(m)(t) is the space in -the truncated infinite subdomains which will becomplex time-harmonic motion amplitude of the body cor- defined later. As a result, the computation domain isreponding to the m-th mode. i.e.. reduced to a very local subdomain. called a localized finite

element domain, which may barely include any source ofRe[(-)(t)] disturbance in the fluid. The present numerical method is

(3) called a localized finite element method because the finiteRe M+ element numerical computations are made only for a local

domain. Throughout this section, the subscript m in the

where om(t) is the body motion amplitude. For m - 7, the potential, defined in the Mathematical Formulation sectiondiffraction problem corresponding to an incoming wave is omitted with the understanding that + will always dependsystem of unit amplitude, one simply sets of)I = 0 and a 7) upon m.- I/o in Equation (3). The potential function * (x, y, z)

must satisfy Let us draw two imaginary vertical planes Jt and J2which separate the original fluid domain D into the three

a2 a2 d2 0 () subdomains: D o.D I, D 2 as shown in Figure 1. We assurex2+ e

/(x, y, z)( that DO includes the ship (and/or any other sources of

disturbance). The boundary surfaces OD. . 8D1 , and aD 2

in the fluid, are denoted, respectively, as

011SF (a) 8Do - SFo + SWo + S o + J! + J2

+ So+y~+IF e(9)

#,IS. - Vn - f(s) (5b) SDI -SA +Sw, + Sa +Jj + Spj i = 1,2

*aI - 0 (3c) where SFI. Sai. and Swi, denote, respectively, the free sur-353 face, the bottom, and the canal side walis. in the subdo-

-.0 (3d) mains Di (for i - 0, 1, 2) and where Spj (for i = 1, 2) arethe boundaries at infinity. The ship hull surface is denoted

where v a *2 /g by SO.

g - acceleration due to gravitySF - undisturbed free surface, y = 0, outside of the body Let 40. 41. and 42 denote the velocity potentials definedSo - bottom surface in the subdomains Do. DI, and D2, respectivejy. Then we%; - side walls have, from Equations (4) and (5).So - body surface below the mean free surface.

The normal velocity f(s) depends upon the mode index m.For m - I, 2 .... 6. corresponding to the sway, heave. on - 0, - 0 on SF,surge, roll, pitch, and yaw modes of motion, respectively. (10)f(s) is given by *o - f(s) on So

I)(s) = ni (6) * --0 onl SaoJ SWo

6k2

9 B1 *2 I

D 1 2

J2 ~Figre -Bouday Cnfiurtios f Csc hre Sbdvidd Fui Doai

-r bit

Sol Si. D3K(.. 1 51 JD

FiigudrsodnE uar I tha Boa r roperirad ation sothTreSudvedFidDm n

and 0fr =I. i 1.2

4on + in - 0 OnSI(1

Sin - Si0 o - oni i (33 -ffsf)ods +ffij l(#.I'o +.d, (1d)

Ry the unertur oo d ion in quation s (12), thtpo e soluationsfof Equtionis (mpose tg (nfniie ar un1u and +.In deitiont +ff 41S14 9t+ 2 d 3

omuato setin (Thi ca be show by a2lying

whres thenorema vtoi tdifeene oftherdsouioo thefli#o142- foaiiaqai defined inr tah peti, ecto, aF4h etn h is aito f ihrfntoa ob eo ~.

souios I. nd de1x ne ion the4hre bon an) J K1 od +4. Kh~h. +~ -L 0 t+16*

B ey saseta the jueneral solutions of Equations 1) h ouin

th oeestof eEquauntions (1r Ghog(2 reqend iduntica on of4 .+-2 221the solems.n The the gnfr s tep n toar the contructin l Istesm a ovnf Eutos(1)truh 1)2tio h r priate unctioni an for sour bayaioaplyeuines oso hsb aigte1w rainudb sn

Gevent tore the uldpiadfre retialseuationsrenstoem.Teieraepesos of the fscoerinin (0)atrough def 2)e is the choviof traeuctiontinaithevoleetinl the subdraton oeirfnct wihal l t eoie.

tsilfutions o $i and , aered cnhefo thr e solutions ad fitKmn domai to -be ) smll thntedo anoeLpae(th know genueal solutinrlsotens e Eantinwhh eingrlhaeobecmudwlllobes l.

ofthe po rinto functionalfrorvaitoaleuto Os tohow thr hadbneht tak e vaytons (imyusi-

eqivlnttote opld atildifeetileqatos rens hofm . heiterl xpesin o tef3c

deie n(0 hog 1)i h hieo ra ucio inl nov nytesboanD hc ecl h

bons) to represent the trial solutions +1 and 42 in the corn- reduces to a set of linear algebraic equations. (In this pro-putation of the approximate solutions, and vice versa. cedure, only the coefficients are subject to variation.)

It is interesting that the stationary value of both func- A complete set of eigenfunctions (the resonance fre-dons ate quencies are excluded) are given in Wehausen 14 as

e± i Knx cosh n.(y + H) Cos nx z-)KI 04g. #1. 42) If - t f(s) *ds (17a) fe(Z-12 H ~,,_

K2 {40. +1. 42) I~f f(s) 40 ds t'~ (17bY+)cs~z-9

.so where W is the width of the taqk and the upper and lowersigns are to be taken in the subdomains D I and D2, respec-

i; 4. 4h. and +2 are the exact solutions. This stationary tively, and where'value is simply (-1/2e) times the added mass and (-l/2 Qo) 1Ltimes the damping coefficients. Ko- [-z. i) 2](4)

Either functional, defined it, (14) or (15), will give a (I) (20)variational equation mathematically equivalent to the + I2 + 2 1)original boundary-value problem at hand. The use of the Kap p Wfunctional given in (14) is slightly advantageous because thenormal derivative of #0 is not involved in the coupling in- and mo and mp (p = 1.2 .... ) ae the real roots oftgal terms. In the present computation, the functionalgiven in Equation (14) is used. The equivalence of the dif- s0 tanh moHI - vferential equation to the variational problem is basic to the (21)choice of the computational scheme. One significant dif. mp tan mPH = -vference between the functional method and the differentialequation is the fact that the expressions for the associated The exponent of the first term in Equation (19) becomesfmctionals in (14) and (15) involve no second derivatives, real when nt/W >mo , resulting in a local disturbance, andowing to the integration by parts used to construct these if n = 0, it becomes purely a two-dimensional case. Thefunctionals, i.e.. Green's theorem is used here. It follows case when mo - nn/W (n -I, 2. . . .), i.e., the case ofthat the functionals will be well defined if only the first resonance, is left out in the present study. Here we con-derivative of the function, rather than the second, is re- sidered only the case of mo 0 nn/W.quired to be bounded. Therefore. the class of admissiblefunctions, in the problem to frnd the stationary point, is IV. Results and Discussionsenlarged to a space bigger than that for the original dif-ferential equation. We now have the advantage, while sear- After the potential +(i). defined in the Mathematicalching for the stationary point of the functional, of being Formulation section, has been computed, thepermitted to try functions outside the class of those hydrodynamic coefficients can be computed byoriginally admissible. In practice, this means that we canmow try continuous functions whose first derivatives areonly piecewise continuous. In other words, the first l+- -ij= #(-)nj d, (22)derivative can have, finite discontinuities at the juncture 19fYS6boundary between adjacent elements. It is very easy to con-stics basis functions that satisfy the previous re- where j7, are the added masses for i, j = 2. 3, momentsquirements. of added mass for i (or j) = 1, 2, 3 and j (or i) = 4. 5, 6,

and moments of inertia of added mass for i, = 4, 5, 6,In the localized finite element domain Do , the basis for and where Aii (i. j I ..... 6) are the damping coeffi-

the trial function is chosen from a polynomial basis. cients. In presenting our results, the nondimensionalSpecifically. eight-node isoparametric, linear three- hydrodynamic coefficients, 1i and Ai are defined by usingdimensional elements were used in the present numerical the ship displacement V and the ship length L as folows:calculations. As mentioned earlier, the trial functions in D1and D2 will be chosen from a subspace of the solutionspace which satisfies the Laplace equation with the free- i - 1,/2.surface condition, the side-wall condition, the bottom con- - !, 2, 3dition, and the radiation condition at infinity. The cigen- 10 - i/°Vfunctions or the Green functions of the above problem canrepresent the solution space. However, we will choose theuigen-function space in this paper for its simplicity. P4 - Aii/eVL, i - 1, 2, 3, j 4. 3, 6

The variational equation (16a) goes into operational j/VL or - 4, . 6, (2)

form in the following way. Let qij li - 0, i. 2. and j -I. 2, . . . Mi) be the basis for the trial functions in eachsubdomain Di 0 0. 1, 2). Then the solution is assumedto be 5,ij - Piq/eVL2.

Mi Ai.j - 4.5,6

1 - qj q (18) ?j - I where V - L B T is the ship displacement volume. The

in Di ( - 0. 1, 2). where ( ij are coefficients to be deter- results of wave excitation forces and moments are alsomined. By substituting Equation (il) in the functional similarly nondimensionalized by using the unit amplitudedefined in Equation (14), the variational equation (16a) 4

of the incoming wave, Y = I. and the ship length L as was selected partly due to the limitation of numerical ac.curacy without taking very small finite elements, while the

F1 Fi/4?gYL 2 1first mode of canal resonance, i.e., when m. W/w - I (seei - 1, 2, 3 (24) the text following Equation (21)), is included. This first

M M/QsYL3

) mode of the canal resonance, mo W/I - I, corresponds toB - 1.44066 in the present ship-canal scometry.

where oi (( ) + ++ 1 ) ni ds Considering the wave number range to be tested andIfs the computation time, we decided to use the 507-node

(25) model throughout the present computations. In subdividingifS the localized finite-element domain into a number of finite

- (+(7) + #1) ni + 3 ds elements, we used four equal elements along the depth andJfs, 0 four equal elements across the width of the canal. Five

equal elements were used along both sides of the xaxjs andIn the present computations, a rectangular barge which five more elements were used further along both sides of

floats parallel to the tank walls. i.e., the angle between the the x-axis. The ship replaces the middle ten elements alongcenter plane of the barge and the xy-p.lane is zero, is the x4ids and the middle two elements along the z4xis andtreated for its simplicity in data preparation. However, any the two elements along the depth from the'free surface.arbitrry angle and ship geometry can be handled by the When the ship is at an off-center location in a canal, wepresent computer program. The midship section plane is simply stretch the elements on one side and shrink those onassumed to coincide with the yz-plane for the present test the other side mainly for the ease of data preparation,model. The cross-section of the canal at the origin, i.e., the especially for the case wshen the off-center distance is notyz-plane is given in Figure 2. We denote the distance bet- too great. Any complicated situation can be accommodatedween the ship center-plane and the xy-plane by s. as shown by the present method but only through more tedius datain Figure 2. The moment is computed with respect to the preparation.cintroid of the ship's %,ater-plane area. i.e.,T € = (0. 0. s)in Equation (6). In presenting our numerical results, the For all cases treated here, the computations were madenondimensional wave number, vB = o 2B/g is plotted along for 23 values of vB between 0.2 and 2.4 with a constant in-the abscissa. crement of 0.1 throughout the present computations. Then

the 23 computed values of the hydrodynamic coefficientswere plotted by a computer using a straight-lineinterpolation.

The computational results for the case when the ship iss located at the center of a canal are presented in Figures 3

through 6. Figure 7 shows the hydrodynamic coefficientsZ Z.- I )Z / computed for the midship section, i.e., the yz-plane as a

H/ purely two dimensional problem with no wave radiationpresent. The result for a two-dimensional model provides

B the resonance mode which plays a significant role in thelocal flow field in three-dimensional problem.

-In Figure 3, the computed values of added mass andmoment of inertia of added mass are shown. The surge

v12 added-mass coefficient, MI, shows an oscillatory behavior,being approximately zero at vB = 0.9 and reaching a localmaximum at vB - 1.4. The heave added-mass coefficientF22 is negative approximately between vB - 0.20 and 1.20.Both sway added-mass coefficients P33 and roll moment of

In the present computations, two locations of the body inertia of added mass jj44 change abruptly from positivein a canal are computed; first, the ship is located at the peak values to negative peak values between vB = 0.9 andcenter of the canal, i.e., s/B = 0, and second, the ship is 1.0. A similar behavior is also observed in the yaw momentat an off-center location in a canal. In both cases, the ship of inertia of added mass M35 between vB , 1.2 and 1.3.is parallel to the canal walls, as mentioned earlier. The pitch moment of inertia of added mass A6 isSpecifically, the computations are made for W/B - H/T minimum around vB = 1.3.- W/H - B/T = 2 and Lr'T = 10 for both values of

s/B - 0 and 0.125. Before we compute these two cases, we In Figure 4, the six diagonal damping coefficients, A.have done computations for a two-dimensional rectangular (i = 1, 6), are shown for s/B - 0. The value of AII de-cylinder which is uniform between two walls, i.e., the creases with increasing value of vB. and is approximatelycylinder ies normal to the xy-plane. These test results were zero at vB - 1.2 and reaches a local maximum at 1.8. Thecompared with those computed by the results of our values of A.2 decrease monotonically and also rapidly toprevious wok in two-dimensional problems. The com- zero at vB - I. The values of A33 and AM are zero betweenoarison was good. vB - 0.2 and 1.4, then sharply increase to local maxima at

2.0, and finally decrease. The value of A35 shows a behaviorSeveral sets of finite element subdivisions were initially similar to A33 and 144. but becomes zero again at vB - 2.0

tested in the present study. The total number of nodes and increases again. The pitch damping coefficient A66 de-creases rather rapidly to zero between iB ,- i1.0 and 1.8. Ittested are 180. 350, 507. 1404, and 1782 (the corresponding is itert to e th t ee O , A.0 and A.$ Iris of interest to note that the values of 133. 1,44, and ASS arenumbers of nodes on the snip boundary are, 44. 81, 81, zero for vBS 1.4. This is because the flow fields for the249, and 249. respectively). The indices n in the eigenfunc- sway, roll, and yaw motions are purely due to local distur-Lion (Equation (19)) were taken betmeen 6 and 10, and bances when sB< w/2 tanh R/2 - 1.4406 --(this valuewhile the indices p were taken between 6 and 10 for n = 0. corresponds to the first mode over the canal resonance).the values of p were redu,:ed as n increased. The present (Note that the motions are asymmetric with respect to thecomputations were made for the range of non-dimensional xy-plane while the only admissible far-field wave solution iswave numbers, 2n/moL - 0.52 through 2.72. This range pure two-dimensional).

I~t~~llm emu~w-,m* .. ., 5

0.01 1 -- -4-

0.10-

0.0-.- - -0.05-O.O3- f-

2 -0.01-

-1- 1- -- -

0- -

-. - - 01551333

-2 0

0.05--

0 - - - - t6

0.10

_,,0 -, 00,,

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5vB vB

I-"

Figure 3 - Added Masses, and Moments of Inertia of Added Mass, aij verus vB, for s/B 0

Due to the symmetry of the body and the canal excitation force F2 1 decreases monotonically from its valuegeometry with respect to the xy-plane and the yz-plane. at vB = 0.2 to zero at 1.2 and then reaches a local max-there are only four nonzero coupling hydrodynamic coeffi- imum at 1.8. It is of interest to note that the zero value ofcients. These are shown in Figure 5. The pitch moment of the heave excitation force would be at YO - 1.0683. whichsurge added mass M"16 reaches a maxm um at YB .I.I and gives an incident wavelength equal to the ship length, if thethen decreases. The roll-sway coupling term 014 changes Froude-Krylov approximation is made; this value is not tooabruptly from a negative peak to a positive peak between far from the computed result of viB = 1.2. The magnitudevs - 0.9 and 2.0. The surge-pitch coupling damping coef- of the pitch excitation moment decreases from a local max-ricient A,, is a maximum at vB , 1.4 and slowly decreases imum of zero at viB 1.8 and slowly increases to the nextto a positive constant. The sway-roll coupling damping coef- local maximum. The rest of the other excitation forces andricient A34 is zero, as discussed earlier, between sB = 0.2 moments, not shown here, are all zero, due to symmetry inand 1.4, reaches a negative peak sharply, and then increases, the problem.

In Figure 6, the magnitudes of the wave excitation It should be noted in comparing the dr.mping coeffi-forces and moment are shown. The magnitude of the surge cients At, A22, and A in Figure 4, the excitation forceselcitation force IFuI reaches a local minimum at vB = 1.2 IFiI and Fi21. and the moment IM31, in Figure 6, respec-and a local maximum at 1.6. The magnitude of the heave tively. There exists a close similarity in the behavior of the

6

1.0- -

02- 0.8-

0.1 - N 0.4- -

0.144

3- - 0.03-

0.02-

0- -,

0.10

0.10-

0.051 -

0.05-

0 ... ~ ~ i33 O-Tr-r

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.IS 2O 2.5siB VB

Figure 4 - Damping Coefficients, Aij versus vB, for s/] - a

damping coefficients and that of wave excitations cor- may be expected to provide some qualitative check becauseresponding to each mode of motion in the range of the the ship is long compared with the length scale of the mad-values vB = 0.2 through 1.44, This is not a coincidence ship cross-section in our problem at hand. In nondimen-because there exist, for vB < 1.44, onl) a purely to- sionalizing our two-dimensional hydrodynamic coefficients,dimensional radiating wave (see the text following Equation we assume that the two-dimensional problem has a unit(21)). On the other hand, if the wavelength becomes short length so that the nondimensionalization of theand if there exist many modes of the three.dimensional hydrodynamic coefficients will be consistent with thewave system, then this similarity may no longer hold. definition given in Equations (23).

In Figure 7, the hydrodynamic coefficients computed As expected. comparisons of P33 and M" in Figure 3from a purely two-dimensional problem in the yz-plane (x and A34 in Figure 5. with y33. p4, and P34 in Figure 7.

0 0) are presented as mentioned earlier. These computa- show remarkable similarities between them. except that thelions are made because the added mass and moment of in- locations of the spikes in two- and three-dimensions arecrtia of added-mass coefficients are mainly dependent upon somewhat different: the spikes in the values O! P33, P44,the local flow field. Therefore, a two-dimensional result and p.4 occur at sA - 0.794 in two-dimensions and at v

7

41160.-0.1' -.

415

0 -

P34 s

444-

-.,. . ./.

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

vSs

Figure S - Coupling Hydrodynamic Coefflclents, ju and AU versu vD, for sI/ - 0

- 0.9 through 1.0 in three-dimensions. Comparison of the boundary-value problem at hand by retaining the matrixheave added-mass coefficient 122 in Figure 3 with that in elements corresponding to the nodes at the midship cross-Figure 7 also shows a similarity in their qualitative behavior sectional plane. In the second set, in which a coarse meshwhen vB >0.5. However, when vB <0.5. the three- subdivision (total of 25 nodes) is used. four eigenvaluesdimensionaliy effect seems to be more significant, because and eigenvectors are obtained.the value of 0'22 increases in three-dimensions, while itdecreases more rapidly in two-dimensions, as vB decreases The lowest nonzero cigenvalue is YB - 0.78410 in thefrom 0.5 to (0.2. first set which has fine mesh subdivisions and vB ,

0.82912 in the second coarse mesh subdivisions. The higherA dlose examination of the local flow field at the mid- modes of eigenvalues we computed are out of the range

ship cross-section corresponding to both positive and considered here. However, the location of the spikes occur-negative peaks, for example, in the sway added mass coef- ring in our three-dimensional results, for example 133 inflcient 133 in Figures 3 and 7, shows that the relative fluid Figure 3, is between vB - 0.9 and 1.0 not at vB -motion under the ship is out of phase for the positive and 0.82912. If we had solved a three-dimensional eigenvaluein phase for the negative values of added mass. In order to problem corresponding to the three-dimensional boundary-have a close examination on the location of these spikes, in value problem treated in the present paper, we could haveaddition, we also treated an eigenvalue problem correspon- obtained eigenvalues all of which have a nonzero imaginaryding to the foregoing two-dimensional boundary-value pro- part. However, the three-dimensional eigenvalue problem isblem by a finite element method. When the finite element not treated here. The eigenvalues of the two.dimensionalmethod is applied to the eigenvalue problem, the (original) problem treated here are all real. From the comparisonseigenvalue problem defined in a partial differential equa- among the two lowest eigenvalues, vB - 0.79410 andzion reduces to a general matrix eigenvalue problem of a 0.82912, ar. J the location of the spikes in the present three-type JAI V - ,iC] R, where (A] and [C] are matrices and dimensional numerical results (033, y4.4, and 134 i.e., vB isXis an eigenvector. In the present computations two sets between 0.9 and 1.0), we can conclude that the differenceof eigenvalue problems are solved. In the first set, [A] and in the locations of the spikes in P3) in Figures 3 and 7 is|C) are identical to the matrices used in solving the caused partially by the inaccuracy in our three-dimensionalboundary-value problem to obtain the results given in computations (due to the use of not-fine-enough mesh sub-Figure 7. To obtain this set of matrices, 96 quadrilateral divisions) and partially by the nature of three.elements with a total of 345 nodes in the fluid and IS dimensionality.nodes on the free surface are used. As a result, 18 igen-values and eigenvectors are obtained in the first set. In the It should be kept in mind that in the values of Ma2, insecond set, the submatrices are taken for JAI and (CI from Figure 7, if %e had computed M122 at v1 - 0.7841. then wethe matrices constructed for our three-dimensional could have noticed a singular behavior like the spikes

8

... - " " . ' -- -,w rl ' L . e - - , . . . . .. *o " - . - ,2 " : -

- -- - 'v -. .;..,.- ,r

0.04- 2--

0.02' -

0'--

-50 --

0.2 - -

0-

0.02 -

0.0 0.6 1.0 1.5 2.0 2.5v8 10"

Figure 6 - Magnitudes of Excitation Forces and 513Moment versus &,B

shown in the other curves, i.e.. the resulting matrix equa- 0- --

tion for the cigenvalues being singular. It seems that thesingular behavior of y.)2 is much more localized than theother caefficien's in Figure '.

It is of interest to note that one can predict the lowest -10-

nonzero eigenvalue in the foregoing problem by a simpleone-dimensional analysis as if treating a U-shaped pipe fill-ed with water. This simple analysis gises the lowest eigen-value, vB - 2/3, for the present case, which is not toobad. One can improve this %alue by introducing a few

elementary potential functions, i.e.. x, y, and '2 - y2. in a 0.0 0.6 1.0 1.5 20 2

few finite elements, and performing simple integrations, as vewas done in Bai

6 . This irnprosed approach gives the lowesteigenvalue, vB - 0.75. A more detailed finite element Figure 7 - Purely Two-Dimensional Added Massas andmethod applied to hydrodynamic eigenvalue problems will Moment of Added Mass versus PS (Due to the sidebe reported in a separate paper in the near future. walls, no wave radiation is peMIa)

' " 9

We anticipated some singular behavior near vB - 0.9 and 1.0 in the same figure. Of interest is that the values1.44. which is the first mode of the canal resonance. So we of pl1, p2, and 1+66 in Figure 8 and jal6. 116 . and A34 inmade additional computations around this value (excluding Figure 10 show spikes which were not shown in thethis value as discussed earlier) by taking finer intervals in previous case (s/B = 0). This can be interpreted in theYB. Our computed results do not show any abnormality sense that the range of the influence of the local eigen-like spikes, contrary to our anticipation, values is larger in the case of s/B - 0.125 than in the case

of s/B - 0.In Figures $ through 30, the hydrodynamic coeffi-

cients, pig and tij. are shown for the case of an offcenter Similar spikes are also observed in the damping coeffi-location of a ship in a canal, s/B = 0.125. The excitation cients in Figure 9. The damping coefficients 433 and A44forces and moments for this case are shown in Figure II. reach a maximum peak at vB 1 1.0 and for A 5 at vB -The general behavior of all results for an off-center case is 1.2. Some of the coupling hydrodynamic coefficients aresimilar to those previously shown, except that the ap- shown in Figure 10. The magnitudes of wave excitationpearance of more sudden spikes is observed. The values of forces and moments are shown in Figure 11. The behaviorj II' MS$. and 1M in Figure 8 show spikes at vB = 1.2 and of wave excitation forces and moments are very similar to1.3. The values of Fz. 1133. and p" have spikes at vB - the damping coefficients corresponding to motion which

has already been discussed in the previous case of s/B - 0.

0.10 0.01-o. .IP40. 01 i ' 0.0.05 N-0.01- - --

0' vvvi ~ ivvF--------- -0.02- -,, u--- -ri-ii- -v--u--u--v

02 , 1.0 - 550./

-2- -1.0 - -- -

20- 0.15-

1033 ~60.1-0- -.

0.05- -

-20- --0-

-40- ..v. .. . .- -0.050.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

vs v

Ipre I - Added Mams and Moments of Inertia of Added Mass, pij versus vil, for s/ - 0.125 (Off-Cester)

10

2 AWL-.. .~~. -. . -.t .+ ++,l , + - , :*:+ .-

. + •. : : J+.+ • ,,

x10-- 0.6- - - - - -

0.2- - - __ '40.- --0.1- -0.

0.-- -

A2 02

o0. .o .0.3-

1'22 Ass)'5

4- - -0- -

0.1-

A 6A'33 '6

.--- -o0.10 -

0.5- -- 0.- - V

0-- -0- --

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

vB VB

FIgure 9 - Damping Coefficients, Ajj versus vB, for s/B - 0,125

In Figure 12, the six hydrodynamic coefficients were Most of the discussions given in the previous case ofcomputed by treating a purely to -dimensional restricted s/B - 0 hold for the present case of s/B = 0.125. Fromwater problem (ship located at off-center) with no radiating the results and discussions presented here, it seemswave present. as discassed ear!icr in connection with Figure necessary to make more refinements in future investiga-7. We computed two sets of eigenvalues as in the previous tions. In the present investigation, we concentrated mainlyCase: the smallest (nonzero) eigen.alue for this case is vB on the application of the localized finite element method to- 0.81140 computed %ith the fine mesh subdivisions, and solve a three-dimensional ship-motion problem by taking aIs vB - 0.85796 computed with coarse mesh subdivisions simple geometry with coarse mesh subdivisions. However,which give submatrix corresponding nodes at the midship- we tested two slightly different methods or solving thesection plane at x - 0 taken from the three.dimensional matrix equation. In the first method the final matrix whichproblem. When we compare these cigensalues with those is complex banded and symmetric is solved by a Gaussianobtained earlier for s/B = 0. the smallest eigensalue in- elimination. In the second method, we solve the real, band.creases when a ship at the canal's center moves away from ed symmetric submatrix equation first and then solve athe center, rather small complex full-matrix equation. In the second

11

0.05

0..1

0 -0.150

,4.1 ,~~-,-i- T---r -sii.-r- v s-- 420' .'ri . u~r .I'' .,,- .*. . .. .

0.5-.0-

01 114.01' -

4.50.0 2.

Fngure 10 - Some of Coupling Hydrodynamic Coefficients versus yR for s/B - 0.125

method. the main core memory space can be reduced by 3. Bai, K.J.. 1972, "A Variational Method in Potentialhalf roughly. The Central Processor Unit Time in the CDC Flows with a Free Surface," Ph.D. dissertation. Dept.6600 was approximately 40 stconds to compute six motions of Naval Architecture, Univ. of California, Berkeley.and one diffraction problem for one wa'.e number by solv-ing the complex matrix directly. By the second method, ap- 4. Bai, KiJ.. 1975, "Diffraction of Oblique Waves by anproximately 20 percent of the total CPU time was saved. In Infinite Cylinder." J. Fluid Mechanics, Vol. 68,the present computer program. a considerable amount of pp. 513-535.time is used for the input-output operation due to out-of-con storage use. S. Bai, KiJ.. 1976, "The Added Mass and Damping

Coefficients of and the Excitation Forces on FourAxisymmetric ocean Platforms," David W. TaylorNaval Ship R&D Center, Bethesda. SPD-670.

Acknowledgements 6. Bai, K.J., 1977. "The Added Mass of a RectangularCylinder in a Rectangular Cansal." J. Hydronautics.

The author is grateful to Mr. Fredd Thrasher for his Vol. 11, pp. 29-32.help in obtaining computer plots in this paper. The authoris also grateful to Professor Daniel Euvrard, Drs. A. Jami, 7. Berkhof, .J.C.W., 1972, "Computation of CombinedM. Lenoir, and D. Martin at ENSTA for many helpful Reffraction - Diffraction," Proc. 13th Coastaldiscussions. Engineering Conference, Vol. 2, pp. 471-490.

8. Smith. D.A.. 1974, "Finite Element Analysis of theForcrd Oscillation of Ship Hull Forms," M.S. Thesis,

References Naval Postgraduate School. Monterey.

1. Wehausen, J.V. and Laitone, E.V., 1960, "Surface 9. Chen, H.S. and Mlei, C.C.. 1974. "Oscillations andWaves," Handbuch der Physik, Vol. IX, Springer- Wave Forces in a Man-Mlade Harbor in the OpenVerlag. Berlin, pp. 446-778. Sea," The Tenth Symposium on Naval

Hydrodynamics, Office of Naval Research, held at2. Dai, K.J., and Yeung. R.W., 1974, "Numerical Solu- MIT, Cambridge.

tions to Free Surface Flow Problems," The TenthSymposium on Naval Hydrodynamics. Office ofNaval Research, held at MIT, Cambridge.

12

...- .. . . ......-

0.0--77 - - 77 - - 1j-

~X10-20.040

0.02- S -

0 .. .I . V V .. 0" 1 .0.06 -

0.2 IF21 M2I1M1

0.04-- --LVI' \' 0.---

.E 100L . 1.9iii--ii-,-O.i,-, -r-5v -- yy ivii- TT

0.04- -

a.. -I 31 I M31

0.04- \

0.0 0.5 1.0 1.6 2.0 2.5 0.0 0.6 1.0 1.5 2.0 2.5

vs vs

Figure I I - Magnitudes of Wave Excitation Forces and Moments versus vB

10. Seto, H. and Yamamoto, Y.. 1975, "Finite Element 12. Euvrard, D.. Jami. A.. Morice. C.. and Ousset, Y..Analysis of Surface Wave Problems by a Method of 1977. "Calcul Numerique des Oscilations d'un NavireSuperposition," The First International Conference Engendrees par Ia Houle," Part I & II, J. Mecanique.on Numerical Ship Hydrodynami:s, DTNSRDC, Vol. 16, No. 2. pp. 289-326. No. 3, pp. 327-394.Bethesda, pp. 49-70.

13. Chowdbury. P.C.. 1972, "Fluid Finite Elements forIi. Yue, D.K.P., Chen. H.S.. and Mei, C.C.. 1976, Added-Mass Calculations." Int. Shipbuilding Progress,

"Three.Dimensional Calculations of Wave Forces by Vol. 19, No. 217, pp. 302-309.a Hybrid Element Method." The Eleventh Symposiumon Naval Hydrodynamics, Office of Naval Research.pp. 325-332.

13

A- .' -

-. 7 - _U7 777

65. 10-

P~22 JA23

/_0---000,

.520-20" - _ _ __ _

-10- -r TTirTT 1--rrv--r -30- i-s---- ,-rr--r-r ,T-r-r-r-----10-. , . . . .35-

P33 P124100- -I 0--

50- -6-

-60- 1

30- 60-

20 P44 4034

10- 20-

0 - ---

-0" -0- -

10 v-v-v-i- T~s-T TTT i~~v-Tr -20 ,rT- -,-r- -- r--,r,- -- v--0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

Figure 12 - Purel) Tao-Dimensional Added Masses and Moment of Added Mass versus va for $/8 - 0.125(Due to the side wails, no wave radiation is present.)

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