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A Numerical Solution Method for Three Dimensional Nonlinear Free Surface Problems C.-G. Kang, I.-Y. Gong (Ship Research Station, KIMM, Korea) ABSTRACT The nonlinear hydrodynamics of a three-dimen- sional body beneath the free surface is solved in the time domain by a semi-Lagrangian method. The boundary value problem is solved by using the bound- ary integral method. The geometries of the body and the free surface are represented by the curved panels. The surfaces are discretized into the small surface elements using a bi-cubic B-spline algorithm. The boundary values of ~ and ~ are assumed to be bi- linear on the subdivided surface. The singular part proportional to R are subtracted off and are inte- grated analytically in the calculation of the induced potential by singularities. The far field flow away from the body is rep- resented by a dipole at the origin of the coordinate system. The Runge-Kutta 4-th order algorithm is employed in the time stepping scheme. The three- dimensional form of the integral equation and the boundary conditions for the time derivative of the potential is derived. By using these formulas, the free surface shape and the equations of motion are calculated simultaneously. The free surface shape and the forces acting on a body oscillating sinu- soidally with large amplitude are calculated and com- pared with published results. Nonlinear effects on a body near the free surface are investigated. 1. INTRODUCTION The free surface effects are considered in the design of submerged bodies operating near free sur- face. The linearized theories were developed during many decades. Recently the nonlinear free surface problem is solved in the time domain by the semi- Lagrangian method Longuet-Higgins and Cokelet (1976) presented a mixed Eulerian-Lagrangian method for following the time-history of space-periodic irrotational sur C.G. Kang and I.Y. Gong, Korea Research Institute of Ships and Ocean Engineering 171 Jangdong Yuseonggu Daejeon, Korea 427 face waves. The only independent variables at the beginning of each time step were the coordinates and velocity potential of marked particles on the free surface. At each time-step an integral equation was solved for the new normal component of veloc- ity. This method was applied to a free, steady wave of finite amplitude, and was found to give excellent agreement with calculations based on Stokes's series. It was then extended to unsteady waves, produced by initially applying an asymmetric distribution of pressure to a symmetric, progressive wave. The re- sults showed the freely running wave then steepened and overturned. Using a technique similar to that of Longuet- Higgins and Cokelet (1976), Faltinsen (1977) solved a nonlinear two dimensional free surface problem in- cluding a harmonically oscillating body. The body intersected the free surface and was constrained to move in the vertical direction. The numerical cal- culations were reduced by representing the flow far away from the body as a dipole located at the center of the body. A formula to calculate the exact force on the body was presented. It was only necessary to know the velocity potential on the positions of the free surface and the wetted body surface. A numerical method for the time simulation of the nonlinear motions of two dimensional surface- piercing bodies of arbitrary shapes in water of finite depth was presented by Vinje & Brevig (1981~. Pe- riodicity in space was assumed. At each time step, Cauchy's integral theorem was applied to calculate the complex potential and its time derivative along the boundary. The solution was stepped forward in time by integrating the exact kinematic and dynamic free-surface boundary conditions as well as the equa- tion of motion for the body. They solved the problem of capsizing in beam seas, caused by extreme waves. Two-dimensional nonlinear free surface prob- lems by a dipole (vortex and source) distribution method were solved by Baker, et al. (1982) . The resulting Predholm integral equation of the second kind was solved by iteration which reduced storage

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and computing time. Applications to breaking water waves over finite-bottom topography and interacting triads of surface and interracial waves were given. The semi-Lagrangian method was extended to vertically a~cisym~netric free surface flows by Dom- mermuth & Yue (1986) . Since they solved the fi- nite depth problem, a far field closure was imple- mented by matching the linearized solution outside a radiation boundary. The intersection line between the body and free surface was treated by extending Lin's(1984) method. The nonlinear hydrodynamics of an axisym- metric body beneath the free surface in the time do- main were solved by Kang & Troesch (1988~. The free surface shape and the forces acting on a sphere oscillating sinusoidally with large amplitude are cal- culated and compared with published results. The far field flow away from the body is represented by a dipole at the origin of the coordinate system similar to Faltinsen (1977~. This is only valid until waves arrive. Waves generated by the numerical error at the truncation boundary are not observed. In this paper, the method used for axisymmet- ric flows by Kang and Troesh(1988) is extended to three-dimensional free surface flows. The free surface shape and the forces acting on a three-dimensional body moving forward and oscillating sinusoidally with large amplitude are calculated and compared with published results. When the body motion is un- known, the time derivative of the potential on the body is needed for the time simulation. In two di- mensions, Vinje & Brevig (1982) derived the inte- gral equation and the boundary conditions for the time derivative of the potential and stream func- tion. However their formulas may not be extended to three-dimensional problems. The three-dimensional form is derived in this work. By using these for- mulas, the free surface shape and the equations of motion are calculated simultaneously. The Runge- Kutta 4-th order algorithm is employed in the time stepping scheme (See Appendix 1~. Numerical calculations are performed for the following cases: (a) A body oscillating vertically near the free sur- face origin located at the mean free surface. The fluid is assumed to be inviscid and incompressible and the Dow is assumed to be irrotational. The fluid domain is bounded with the following surfaces, the free sur- face, SF, the body, SB, and the surfaces at infinity, SOO (Fig. 1~. The surfaces, taken as a whole, will be denoted as S. The governing equation and the boundary conditions are as follows(Longuet-Higgins & Cokelet (1976) and Dommermuth & Yue (1986~: Laplace equation: V2 = 0 in the fluid domain (1) Kinematic free surface condition: D1; = VI on F(x,t) = 0 (2) Dynamic free surface condition D} = - 9Z + 2 V' Vie on F(x, t) = 0 (3) Body boundary condition: V.n~x,t) = V.n on B(x,t) = 0 (4) Radiation condition: ~ ~ O as Axe ~ oo,t < oo (5) where Din= a&t + V. V) iS the substantial deriva- tive, F(x, t) = 0 is the function representing the free surface geometry at time t, V includes both trans- lational and rotational velocities, and B(x,t) = 0 is the function representing the body surface geometry at time t. The Green function, G~xjy), satisfies the fol- lowing equation. (b) A body oscillating horizontally near the free V2GtX;Y) =-fix-Y) (6) 2. MATHEMATICAL FORMULATION Consider an ideal fluid below the surface given by F(x,t) = 0, where x(z,y,z) is a right-handed coordinate system with z positive upwards and the where x is the vector to the field point, y is the vector to the source point, and fix-y) is the Dirac delta function. Through the application of Green's second identity in the fluid domain, the potential is given as a~x,t)~(x,t) = i/t ~-~ Aids (7) where a is an included solid angle at x. In this prob- lem, a is 27r on the surface. The Green function that satisfies Eq. (6) is 428

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G(x,y) = R = ~(8) where x is the position vector of a field point and y is that of a source point. The solution of Eq. (3) gives the potential ~ on the free surface F(x,t) = 0. Also in on the body is known from the body boundary condition, Eq. (4). The normal and tangential velocities on the free sur- face are needed to solve Eq. (3). The normal velocity on the free surface is a solution of Eq. (7). Details to calculate the tangential velocity is given in the sec- tion 3. Consequently, a Fredholm integral equation of the second kind on the body and of the first kind at the free surface may be solved. Far Field Condition The far field condition is important in the non- linear free surface problem. It can be resolved by using periodic boundary conditions if the physical problem has spatial periodicity (Longuet-Higgins & Cokelet, 1976~. Faltinsen (1977) and Kang (1988) assumed that the behavior of the potential is like that of a dipole at the origin of the coordinate sys- tem. A far field closure by matching the nonlinear computational solution to a general linear solution of transient outgoing radiated waves was used by Dommermuth & Yue (1986~. This method is math- ematically complete. A numerical radiation condition was posed so as no waves reflected from the truncated surface (Yang & Liu (198933. They found that the usual one- dimensional Sommer-feld condition gave reasonable results for an axisyrnmetric cylinder heaving in the still water. Also it was extended to 2-D case for the cylinder swaying in the still water. In this work, the far field closure similar to that used by Faltinsen (1977) and Kang (1988) is posed. It is simple and it works well until waves arrive at the truncation boundary. At the far field, the velocity potential, , and the wave elevation, a, are small from the radiation condition, Eq. (5~. For example, (z = q) = +(x = 0) + ~ .~ (z = 0) + H.O.T. (9) Assuming the behavior of the potential, , is like that of a dipole at the origin of the coordinate system, it follows that as r oo (z = 0) = 0 a~ (z = 0) ~ 3 a=/ ozfz=0)dt~~ 3dt (10) (Z = Q) ~ FIZZ = 0) ~ 6 where r is `~. This is only valid until waves arrive at the truncation boundary. If we take a large value of r, the potential on the free surface must be relatively small to the vertical velocity ~ . Therefore the effect of the po- tential at the far field can be neglected. The far field condition is approximately satisfied by including the effect of the vertical velocity ~ at the far field. 3. NUMERICAL IMPLEMENTATION The body surface and the free surface are dis- cretized into the small surface elements AS`j using a bicubic B-spline algorithm ( Barsky & Greenberg (1980) ) . The surfaces I\Sij(x,y,z) can be repre- sented by the parameters, u and v. Thus 1 1 ij(U, V) = ~ ~ bJ(U)Vi+J,j+tat(V) a= - 2 t= - 2 1 1 ydi(uiv)= ~ ~ bJ(U)Vi+s,j+tbt(V) (11) s=-2 t=-2 1 1 z,}(U,v)= ~ ~ bJ(U)ViX+s,j+tbt(V) a= - 2 t=-2 for 0 ~ u < 1 and 0 < v < 1 where bJ(u) and lot(u) are the uniform cubic B-spline basis functions and V`j are vertices (See Appendix 2). This allows the curved panels. The end condition should be imposed to get a complete B-spline approximation. There are sev- eral methods to impose end conditions according to the geometrical characteristics (Barsky(1982)). The derivative of B-spline interpolation at the end is set to get the tangent of the given geometry if the tan- gent is known. If the tangent is not known, the derivative at the end is set to be the slope between two vertices at the end obtained by using B-spline algorithm. The boundary values of ~ and ~ are assumed to be bilinear on the subdivided surface AS`j as shown below . = aO + alu + a2v + a3uv ~3 = ho + be + b2v + b3uv (12) for 0 ~ u < 1 and 0 < v < 1 To evaluate the integrals over the segments the two point Gaussian Quadrature formula (Perxiger 429

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(1981), Abrnmowitz ~ Stegun (19643) is used when the field point is not a corner of the pannel. In Eq. (7), Gn is not singular but G has (R-) type singu larity in the transformed u-v domain as the field and point approaches the source point. The singular ity is integrable and can be integrated by numeri cal quadrature. But since an accurate integration of the singularity requires a higher order quadrature formula, the method following Ferziger (1981) and Dommermuth & Yue (1986) can be used. The in tegral can be factored into the sum of the (R-~ type singular part which is integrable analytically and the where non-singular part which requires numerical quadra ture (Ferziger(1981~. Removal of (R-~ type singularity In Eq. (7), G has (R-~ type singularity as the field point approaches the source point. The (R) type singularity is integrable in the surface integra tion. /l-^sij R (13) First, consider the induced potential at (/Oo, 900, hoo) ,which is one of the corners of panel, by the source panel AS`j . Eq. (11) can alternatively be repre sented by the following equations: ~ 3 x' = ~ ~ fijuiv] i=0 j=0 ~ 5 Y' = ~ ~ 9iiuivi (14) `=o j=o z' = ~ ~ hijUiV] `=o j=o By using Eqs.~14) and (40), dS and R can be transformed and expanded into Taylor's series about u = 0, v = 0 as follows : dS = IJ~dudv jEG-F2dudv J ~ (~ ) + (39 ) + (68 ) ] '~(~) +(BY) +(aZ) 1 where , , ~ {3X' ~X' + ~y ~y + ~X ~Z ~ 2 ~ ~ d OCR for page 427
v respectively. Generally, the tangential vector to (fU,gu~hu) along u-axis is not perpendicular to the vector to (go) along v-axis. Therefore Gram- Schmidt orthogonalization is needed to get the or- thogonal tangential vectors on the surface. 4. CALCULATION OF THE TIME DERIVA TIVE OF POTENTLY For the time simulation, ~ should be known to calculate the forces and moments acting on the body. In two dimensions Vinje & Brevig(1982) derived an integral equation and boundary condition for ~ by using the ~ and ~ formulation. However their results can not be extended to the three-dimensional case. Since `~&n(~) can not be calculated by using the given motion, a boundary value problem for A, the time derivative of the potential in body fixed coordinates, is derived as follows: dt On = n dtV' + V' d-t = n ldtV'+ (y. V)V] + V) ~ (w x n) = n ~ [V dt + V(V V) + w x V+] + V} (w x n) an dt which can be expressed as .g ~ dt ~ = n ~ ~ mitt + w x r-w x VT) (23) Following the nomenclature of Vinje & Brevig (1982), the operator `~` is ~ 98' + V V), V = V T + w x _, VT is the translational velocity of the center of mass of the body, _ is the position vector to the boundary from the center of mass of the body, and w is the angular velocity vector of the body. Eq. (23) is useful in that most quantities of interest are expressed in the body coordinate system rather than a fixed, inertial one. Since V Vie satisfies Laplace equation, the time derivative of the potential, A, can be calculated by using Green's theorem. The limiting behavior of V. V' at red oo can also be checked, or V V' = 0(r) = 0((~)) as r ~ oo (24) Applying Green's theorem for ~ instead of in Eqs. (1), (6), and (7), the following equation can be obtained: R.H.S. of Eq. (23) may be represented as follows: n . ( ,9fT + W X _-~ X VT) = ~ anti (26) where n = (n~,n2,n~,) and _ x n = (n`,n5,n6) . The time derivative of the potential, A, can be decomposed as follows: die = Thai deli + d7 (27) The auxiliary terms, A; and "7, are solutions of the following boundary value problems: deli) = ni and and 6i = 0 and dt ~n( dt ) = 0 on B(x,t) = 0 (28) d7 = V Vie-2 V' V'-gz on F(x,t) = 0 (29) The time derivative of the potential on the free surface,"7, is calculated by using solutions of the integral equation, Eq. (7). 5. THE PRESSURES AND FORCES Once the time derivative of the potential is known, the pressures are found by applying Bernoulli's equation. Bernoulli's equation is derived for the vari- ables relative to an inertial coordinate system. How- ever, it is convenient for the purpose of solving the boundary value problem to use body fixed coordi- nates. Under these circumstances, spatial differen- tiation is invariant with coordinate transformation, but temporal differentiation is not. Bernoulli's equa- tion can be expressed as - = - a' - 1 V' V' - go dt + V V' -2V) V.d)-go. (30) The term ~ in the above equation is calculated by Eq. (27). With the pressure known, the force and moment become F = mV _ _ at lli'~n(~t) - (at) ]GdS (25) = llpodS-milk 431

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-pun(d)-V. V++ 2Ve V++gZ)dS s -mgk M = |ip_ x ndS. s For three-dimensional bodies the force and mo- ment are rearranged as follows: F = F. + F2 + (pgV-mg~k (32) M = M~+M2 where V in Eq. (32) is the displaced volume of the sphere, F. = -p || n d,~;7dS SB F2 = -p AL n(v . Ad-2 Vet . V) dS, (33) SB Ml = -p If r x n dads, and SB M2 = -p || _ x n(V Vet-2V' V+) d S. SB 6. NUMERICAL CALCULATION IIeave motion To demonstrate the usefulness of the technique shown in the previous section, the force acting on a sphere oscillating beneath the free surface is de- termined. The motion of a sphere is given by z = -h + a cos at for t greater than zero. Initially the potential and wave elevation at the free surface are zero. The number of elements on the body is 200 and that on the free surface is 40 x 40. The truncation boundary is the position from the origin of the cm ordinate system where waves reaches in four periods of the body motion ~ -16 < x/R < 16, - 16 < y/R ~ 16~. So, it depends on the group velocity of wave. Even spacing is used on the body and free surface. The typical time interval is approximately 0.05 period of motion for the time simulation of the sphere. The mean depth of immersion for the center of the sphere, h, is h/R=2.0. The time history of the force acting on the oscillating sphere with a large ra- tio of motion amplitude, a, to radius, R. (a/R=0.5) was calculated and is compared with the results of the axisymmetric free surface problem (Kang & Troesch (19883) in Fig. 2. They show good agree- ment. In the reference~l3], the calculation results for the sphere oscillating vertically with small am- plitude showed good agreement with those given by Ferrant(19873. This means the AD algorithm in this (31) paper works well. In Fig. 3, the time history of force components which consist of F. and F2 is shown. The forces are nondimensionalized by pgKaR3, where K is a wave number, w2/g, and 9 is the gravitational constant and the time t is nondimensionalized by ,/i~. The harmonic distributions of the total force are shown in Table 1. The second order amplitude of the force is 6.5% of the first order one. And the mean force is 1.5% of the first order. Fig. 4 shows the three di- mensional wave profiles at four different times. All the wave profiles are exaggerated by factor 50 in z- direction. In the figures T is a period of the motion. Surge motion The surge motion of the sphere is given by x = a cos wt for t greater than zero. The mean depth of immersion for the center of the sphere is h/R = 2.0. The amplitudes of the motion is a/R = 0.5. The nondimensionalized wave number, KR, is equal to 1.0. The time histories of the forces acting on the sphere are shown in Fig. 5-6. The harmonic distri- butions of the horizontal and vertical forces are given in Table 2. The three dimensional wave profiles at 4 different times are shown in Fig. 7. In case of surge motion, the first order surge force is dominant unlike the heave motion. However nonlinear effects appear only in the vertical force. The first order vertical force is negligible, but the mean and second order vertical forces are not. The mean vertical force is important for a submerged body to keep a constant depth. Advancing motion Saw-tooth instability is not observed in the com- putation of the oscillatory motions. But it seems to be inevitable and break down the numerical time stepping in case of advancing sphere. It may be due to short waves generated by the body. The length of short waves is less than the mesh size in this compu- tation. The numerical error does not die out but was accumulated continuously. Eventually the numerical scheme breaks down. Thus a simple numerical filter- ing scheme are tried to avoid break down, but still does not work well. Fig. 8 shows the wave profile before breakdown. All the calculations were carried out on CRAY2S and each solution time was approximately 50000 sec onds. 432

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7. CONCLUSION In this paper the nonlinear hydrodynamics of a three-dimensional body beneath the free surface is solved in the time domain. The free surface shape and forces acting on a sphere oscillating sinusoidally with large amplitude are calculated and compared with published results. The far field flow away from the body is represented by a three dimensional dipole at the origin of the coordinate system. This is only valid until waves arrive at the truncation bound- ary. Any numerical instability is not observed in the computation of the oscillatory motions. The inter gral equation and boundary conditions to calculate the time derivative of the potential on the body are derived. By using these formulas, the free surface shape and forces are calculated simultaneously. A Rung - Kutta 4-th order scheme is employed in the solution method. Nonlinear effects on the oscillating body submerged in infinite water depth are studied. ACKNOWLEDGMENT This work was supported by the Basic Research Program, contract ED469 of the Korea Research In- stitute of Ships and Ocean Engineering (KRISO). Acknowledgement is also given to the Ministry of Science and Technology of Korea (MOST). The au- thors should appreciate Dr. Choung Mook Lee for his encouragements. REFERENCES [1] Abramowitz, M. & Stegun, I.A., Handbook of Mathematical Functions, Government Printing Office, Washington, 1964. [2) Baker, G.R., Meiron, D.I. & Orsag, S.A., "Gen- eralized Vortex Methods for Free-Surface Flow Pro- blems," Journal of Fluid Mechanics, 123, pp477- 501, 1982 . [3] Barsky, B.A. & Greenberg, D.P., Determining a Set of B-Spline Control Vertices to Generate an Interpolating Surface," Computer Graphics and Image Process 14, pp203-226, 1980. . [4~ Barsky, B.A., End Conditions and Boundary Conditions for Uniform B-Spline Curve and Sur- face Representations," Computers in Industry 3, pp. 17-29, 1982. [51 Dommermuth, D.G. & Yue, D.K.P. 1986, "Study of Nonlinear Axisymmetric Body-Wave Interac- tions," In Proc. 16th Symp.on Naval Hydrodyna- mics, Berkeley. [6] Dommermuth, D.G., Numerical Methods for Solving Nonlinear Water-Wave Problems in the Time Domain," Ph.D. Thesis, MIT, 1987. [7) Faltinsen, O.M., "Numerical Solution of Tran- sient Nonlinear Free-Surface Motion Outside or Inside Moving Bodies," Proc. 2nd Intl. Conf. on Num. Ship Hydra., U.C. Berkeley, 1977. [8) Ferrant, P., "Sphere immergee en mouvement de pilonnement de grande amplitude," Premiers Journes De L'hydrodynamique, Nantes, 1987. [9I Ferziger, J.H., Numerical Methods for Engineer- ing Application, John Wiley and Sons,Inc., 1981. [10] Forbes, L.K., "An Algorithm for 3-Dimensional Free-Surface Problems in Hydrodynamics, " Journal of Computational Physics 82, pp.33() 347, 1989. [11] Gradsteyn, I.S. and Ryzhik, I.M., Table of Inter grails, Series, and Products, Academic Press, 1980. [12) Kang, C.-G., Bow Flare Slarruning and Non- linear Free Surfac - Body Interaction in the Time Domain," Ph.D. Thesis, Univ. of Michigan, 1988. [13] Kang, C.-G.& Troesch, A.W., "Nonlinear In- teration betwwen Axisyrnmetric Bodies and The EYee Surface-Body in Water of Infinite Depth," Proc. the Seminar on Ship Hydrodynamics to - honor Prof. J.H.Hwang, Seoul, Korea, 1988. . [14] Kaplan,W., Advanced Mathematics for Engine e , Addison-Wesley Publishing Co. [15] Lin, W.M., "Nonlinear Motion of the Free Sur- face near a Moving Body," Ph.D. Thesis, M.I.T., Dept. of Ocean Engineering, 1984. [16] Lin, W.M., Newman, J.N. & Yue, D.K.P., "Non- linear Forced Motions of Floating Bodies," Proc. 15th Symp. on Naval Hydra., Hamburg, 1984. _ . . [17] Longuet-Higgins, M.S. & Cokelet, E.D., UThe Deformation of Steep Surface Waves on Water, I. A Numerical Method of Computation," Proc. R. Sac. Land. A., 350, ppl-26, 1976 [18] Newman, J.N., Transient Axisymmetric Mo- tion of a Floating Cylinder," J. Fluid Mech., 157, ppl7-33, 1985. [19] Vinje, T. & Brevig, P., Nonlinear Ship Mo- tions," Proc. 3rd Intl. Conf. on Numerical Ship Hydrodynamics, Paris, 1981. [20] Vinje, T. & Brevig, P., ~Nonlinear, Two Di- mensional Ship Motions," Seminar on the Norwei- gan Ships in Rough Seas ISIS) Project, 1982. 21] Yang,C. & Liu,Y.Z., ~ Tim - Domain Calcu- lation of the Nonlinear Hydrodynamics of Wavy Body Interaction," 5 th Intl. Conf. on Num. Ship 433

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Hydro., Hiroshima, 1989. Table 1 Harmonic Distributions of the Total Force for Heave Motion (a/R=0.5,h/R=2.0,KR=l.o) Heave Force F/pgKaR~ I-TH COS SIN o 1 2 3 -0. 2850707E-O1 0. OOOOOOOE+OO 0. 1843049E+01 0. 2665849E+OO -0. 1033978E+00 0.6141333E-O1 0. 6449880E-01 0. 8355246E-02 -0. 511481 lE-02 -0. 1019468E-O1 0. 2418429E-01 0. 2957444E-02 Table 2 Harmonic Distribution of the Total Force for Surge Motion (a/R=O.5,h/R=2.0,KR= 1.O) Surge Force F/pgKaR3 I-TH COS SIN o 2 3 4 5 -0. 1308621E-02 0. OOOOOOOE+OO 0.1927655E+01 0. 1237885E+OO 0.2138726E-03 -0. 1002954E-02 0. 3005543E-01 0.1583521E-02 -0. 1381397E-04 -0. 4408678E-03 0. 2590462E-01 -0. 4266882E-03 Heave Force F/pgKaR3 I-TH COS SIN -0. 1274143E-01 0.0000000E+OO -0. 4001656E-03 0. 4211906E-04 0. 2037149E-01 0. 2325045E-O1 -0. 3133378E-03 0. 8355940E-04 -0. 3846344E-03 -0. 2575103E-03 -0. 2204935E-03 0. 9557988E-04 t SF y ( ) Fig. 1 Coordinate System 1~ ' 1 Fig. 2 Comparison of Heave Force Acting on the Sphere by 3-D and Axisymmetric Solutions (a/R=0.5, h/R=2.0, KR=1.O) Fig. 3 Time History of the Heave Force Compo- nents Acting on the Heaving Sphere (a/R=0.5, h/R=2.0, KR=1.O) 434

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~=1.99T t=1 . 99T Fig. 4 Wave Profiles for Heave Motion (a/R=0.5, h/R=2.0, KR=1.0) t=3 2 4T Fig. 7 Wave Profiles for Surge Motion (a/R=0.5, h/R=2.0, KR=1.0) 435

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~rat --------- r2 Appendix 1. The 4th order Runge-Kutta method [9] When y' = fishy) is nonlinear, this can be solved by Runge-Kutta methods. The most com- monly used Runge-Kutta methods are fourth order accurate and there are a number of these. The best known such method (sometimes called the fourth or- der Runge-Kutta method) is Fig. 5 Time History of the Surge Force Compm Yn+l/2 = Yn + 2J(Zn,Yn) Dents Acting on the Surging Sphere (Euler predictor - half step) (a/R=0.5, h/R=2.0, KR=1.0) * h Yn;~/2 = Yn + 2 /(Xn+~/2, Yn+~/2) Z-Force Fig. 6 Time History of the Heave Force Compo- nents Acting on the Surging Sphere (a/R=0.5, h/R=2.0, KR=1.0) Fig. 8 Wave Profile Generated by an Advancing Sphere (U = 2.557m/sec for 2.557 < t, U = t for O < t < 2.557, h/R = 2.0, t = 3.6sec) (Backward Euler corrector - half step) Yn;1 = Yn + h f (fin+ 1/2 ~ An+ 1/2 ~ (34) (Midpoint rule predictor - full step) Yn+1 = Yn + 6 [fern, Yn) + 2f~xn+l/2,yn+ll2) +2f~=n+ll2,yn+ll2) + f~xn+l~yn+l)] (Simpson's rule corrector- full step) Looking at this method one can see that derivation of such methods is not an easy task. An analysis of it for the general case is also difficult. It is not too dif- ficult to analyze, however, when applied to y' = ay. We find that Yn+1 = (1 + ah + 2 + 6 24 ) (35) so that the method is indeed fourth order accurate and the error is of order (ah)5/120 . It is interesting to note that the steps that comprise this method are of order one,one,two, and four, respectively, and the method has inherited the accuracy of the final corrector. Appendix 2. Parametric Uniform B-spline Surface Representation [3] 436

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A B-spline surface is defined in a piecewise man- ner, where each piece is a segment of the surface called a surface patch. The entire surface is a mo- saic of these patches sewn together with appropriate continuity (Fig. 93. A bicubic B-spline surface con- sists of patches which are cubic in each of the two parametric directions and it is everywhere continu- ous along with its first and second derivative vec- tors, in both directions. This continuity constraint reduces to requiring continuity of the first and sec- ond parametric derivative vectors across the borders of adjacent patches. The B-spline surface is defined by, but does not interpolate, a set of points called control vertices. These control vertices form a two- dimensional array. Although the vertices actually exist in three-dimensional x-y-z space, they are or- ganized as a two-dimensional graph. Each vertex is either an interior vertex or a boundary vertex. This notion can be formalized quite elegantly by drawing on graph theory. The set of control vertices can be considered as a graph V, E whose vertices form the set V={V`j ~ i=O,.~.,m;j=O,.~.,n) and with the set of edges E = {(V`;, V`,j+~) I = 0, , m-1; j = 0, . ., n3 U{(V`;, Vi+~,j) Pi = 0, , m-1; j = 0, . . , nil The interior vertices are the vertices Vij, where 1 < i < m-1 and 1 < j < n-1 And the boundary vertices are Voj, j = 0, ...,n-1, Vi",i = 0, ...,m-1, Vmj, j = 1, ..., n, and V`O, i = 1, ..., m . To emphasize this graph-theoretic interpretation, the term control graph to describe the set of control vertices is chosen (Fig. 10). A major advantage of the B-spline formu- lation is that it is a local representation. A bicubic B-spline surface patch is controlled by 16 control ver- tices and is unaffected by all other control vertices. I\ An interior conrro! vertex /\ / \ / \/ ~ A boundary - ~1 corporal ve rcex Fig. 10 A B-Spline Control Graph Conversely, a given control vertex exerts influence over only 16 surface patches and has no effect on the remaining patches. This means that the effects of moving a control vertex are limited to 16 patches. A point on the (i,j) th uniform bicubic B-spline sur- face patch is a weighted average of the 16 vertices V`+rj+,, ~ ~ = - 2, - 1,0,1 and s = - 2, - 1,0,1 . The mathematical formulation for the patch Qij(u,v) is then 1 1 Qij(U,V) = ~ ~ bbre(U,V)Vi+r,j+e r= - 2 A= - 2 for O < a, v < 1 (36) The set of bivariate uniform basis functions is the tensor product of the set of univariate uniform basis functions. That is, bbr,,,(u,v) = b,(U)be(V) for r = - 2, - 1, O. 1, and s = - 2, - 1, O. 1 (37) ('`, -I = ~ ~ b,(.)V`+,,,+,b,(~) for O < u,v < 1 (38) Fig. 9 A B-Spline Surface is a Mosaic of Surface Patches. Thus, the formulation for the patch Qij(u,v) can be rewritten as The univariate uniform cubic B-spline basis func- tions are graphed in Fig. 11, and can be written in matrix form as 437

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[ b-2(U) b_l(U) loo(U) bl(u) 1 --1 3 -3 1 = [u3u2ul]1/6 3 -6 3 0 (39) 1 4 1 0 1, ~ b1 (u) loo(u) b ~ (u) b (us Fig. 11 Graphs of the Univariate Uniform Cubic B-Spline Basis Functions Appendix 3 Calculation of the Surface Integral and the Normal Vector The surface integral can be calculated as fol- lows (Kaplan, 1981~: [; Hair, y, x) dir s sat.. - || Of (u, v) ~ g(u, v), h(u, v)]~/EG-F2dudv (40) where x = f(u,v), y=g(u,v), z=h(u,v) P1 = xu~+y~+zuk P2 = xv~+yuj+zvk - E - IP112 = (0~)2 + (~Y)2 + (~Z)2 XX + Byway + ~Z~Z MU TV MU TV MU TV = ( 3~)2 + ( ~Y)2 + ( ~Z)2 F = P1 P2 G = lP2 12 = The normal vector on the surface S is calcu- lated by using the following formula. n= Apt pa (4~) DISCUSSION Tor Vinje Norwegian Contractors, Norway It seems to me that you have treated the computation of the forces in a way that is much more complicated than necessary. The splitting up of the B/3t values, according to Vinje & Breirg, does not make sense for forced motion. In this case it can be computed by means of a central difference scheme at a later stage. If you, on the other hand, are dealing with problems where the dynamic equations of motion of the body have to be integrated in time, you have to split the forces (and B/3t) in the way you did. This is due to the occurrence of numerical instability if you apply, for instance, backward differences in time to compute B/8t, and thus the forces acting on the body. AUTHORS' REPLY Even if the calculation examples in this paper are restricted to the forced motion, this motion can be directly applied to the problems where the body motion is unknown. The calculation method of the forces in this paper is not much complicated because the integral equation for the potential (7) has the same influence coefficients that the integral equation for the time derivative of the potential (25). Also, this method gives more accurate results for the calculation of the forces than a central difference scheme. DISCUSSION Pierre Ferrant Sirehna, S.A., France Your computations were run only for w = 1.0. According to my computations on the submerged heaving sphere, w = 1.0 is a local minimum for the higher harmonic terms in the force. Furthermore, this frequency is too high to make higher harmonics appear in the wave field, and your result for the wave field is purely harmonic and certainly very close to the result of a fully linearized models as demonstrated in my paper presented in this conference. For this geometry and for heaving motion, major nonlinear effects appear especially low frequency, say about w = 0.4. Future runs of your code should be situated in this frequency range in order to obtain significant nonlinear body-wave interaction. AUTHORS' REPLY I agree with you. I have some experiences for the computation on a submerged heaving sphere by using the axisymmetic code (13). The nonlinear effects appear strongly at low frequencies and small submerged depth (21). Reference: (21) Kang, C.<, Nonlinear Free Surface Flows for an Axisymmetric Submerged Body, submitted to J. of the Society of Naval Architects of Korea. 438