Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24th Symposium on Naval Hydrodynamics FuLuoka, JAPAN, 8-13 July, 2002 A Panel-E`ree Method for Time-Domain Analysis W. Qiu, H. Peng (Martec Limited, Canada) C.C. Hsiung (Dalhousie University, Canada) Abstract A panel-free method (PFM) has been developed to solve the radiation and the diffraction prob- lems of floating bodies in the time domain. The boundary integral equations in terms of source strength distribution were desingularized by remov- ing the singularity in the Rankine source of the time-dependent Green function. The geometry of a body surface was mathematically represented by NURBS surfaces. The integral equation was then globally discretized over the body surface by using Gaussian quadratures. No assumption is needed, as in the conventional panel methods, for the degree of approximation of distributed source strength on the body surface. The accuracy of PFM was demonstrated by its application to the radiation and diffraction prob- lems for a Wigley hull in the time domain. The computed force response functions, hydrodynamic coefficients, and the wave exciting forces were com- pared with published results. INTRODUCTION The predictions of wave-induced motions and load- ings are essential elements of ship design. Over the past few decades computational hydrodynam- ics has been developed as a powerful tool for both ocean engineers and naval architects. It allows eval- uation of preliminary designs or ship performance at a relatively low cost compared with experimental tests. Strip theory was applied as the first analyt- ical ship motion theory for computations and has been used as a practical prediction method, but it gives unsatisfactory predictions at low frequencies and at high forward speeds, and it is not applica- ble to ships of low length-beam ratios due to its 1 slender body assumptions. Also, the strip-theory approach is not able to compute the hydrodynamic pressure distribution over the hull surface except on sections. Some of the deficiencies of strip the- ory can be removed by the three-dimensional flow theory. Hess and Smith (1964) pioneered the panel method, in which a source distribution on a body surface was employed for flow computation. The body surface was subdivided into a number of flat quadrilaterals over which the source strength was assumed to be constant. Since the singularity so- lution of integration can be obtained by using pla- nar quadrilateral panels or triangles and constant source strength over a panel, the constant-source- flat-panel method has been used in a wide variety of problems both in the frequency domain and the time domain. It is often referred as a lower-order panel method. Normally, a large number of panels are required to achieve accurate results. H~gher-order panel methods have been devel- oped in various degrees to overcome the deficien- cies of the constant-source-flat-panel method. Most higher-order methods allow for linear or quadratic panels and first- or second-degree polynomial dis- tribution of source strength over a panel. It nor- mally requires more computational effort than the lower-order panel method. Maniar (1995) ap- plied B-splines and developed a higher-order panel method in which the potential and the geometry of a body are allowed any degree of continuity. Recently, Lee et al. (1998) and Danmeier (1999) have presented a geometry-independent higher- order method which separates the geometric and hydrodynamic representations. The body was represented by subdividing the body surface into patches. B-splines were employed to represent the geometry and the velocity potential on each patch. Lee and Newman (2001) used B-splines only to rep- resent the velocity potential. The accuracy of the solution can be refined by controlling the degree of B-Splines and/or subdividing patches. In the

higher-order panel methods, the singular 1/r term can be evaluated numerically in a variety of ways. For example, in the work of Danmeier (1999), an adaptive subdivision and triangulation scheme was used to evaluate the singularity of the Rankine term. Landweber and Macagno (1969) proposed a desingularized procedure which removed the singu- larity of 1/r before discretizing the integral equa- tion and applied it to the problem of uniform flow past an ellipsoid. The numerical solution then could be applied to the exact boundary geometry, and the integral equation could be discretized over the body surface by Gaussian quadratures. Theo- retically, this eliminates the errors due to both the geometrical approximation and the assumed degree of approximation of source strength distribution as in the panel method. Kouh and Ho (1996) further developed this method and applied it to solve prob- lems of uniform flow past a sphere, an ellipsoid and a Wigley hull in which geometries were represented by analytical expressions. Recently, Qiu and Hsiung (2001a,b,c) have de- veloped a so-called panel-free method for the time- domain analysis. In their work, the desingular- ized integral equation in terms of source strength distribution has been developed by removing the singularity due to the Rankine term in the time- dependent Green function. The Non-Uniform Ra- tional B-Splines (NURBS) were adopted to de- scribed the body geometry so that the desingular- ization method can be applied to arbitrary bod- ies. The integral equations were then globally dis- cretized over the body surface by Gaussian quadra- tures. They applied the panel-free method to the radiation and the diffraction problems of a hemi- sphere. Computed response functions, added mass and damping coefficients showed excellent agree- ment with published results by the analytical solu- tion and the panel method. The work presented in this paper is the contin- uing development of the panel-free method in the time domain. It has been extended to the radia- tion and the diffraction problems for a Wigley hull at zero speed. MATHEMATICAL FORMU- LATION It is assumed that the fluid is incompressible, invis- cid and free of surface tension and that the flow is irrotational. Consider the radiation and the diffrac- tion problems of a three-dimensional body at zero speed in a semi-infinite fluid with a free surface. The potential functions, ~k(P(x, y, z); t), for k =1,2, ...,6 and 7, satisfies the following governing equation, boundary conditions, far-field conditions and initial conditions: V2¢k = 0 d Vb + 9 0¢k = 0 Wok - V dn — ok V°k ~ O V¢k ~ o 0k = 0, ~k = 0 where Vnk = nk (k + mk (k, vn~c — arlk in Q (1) on z = 0 on Sb as R1 ~ x, on z = 0 as z ~ -x as t = to for k = 1,2, ,6 (2) for k = 7 (3) where 9 is the gravitional acceleration, (I denotes the incident wave potential, Q is the computa- tional domain, Sb is the mean wetted body surface, R1 = ~, t is time, to = 0 for the radiation problem and to =—x for the diffraction problem, (k iS the amplitude of unsteady motion in six de- grees of freedom, mk is m-terms and Ilk iS the unit inner normal vector pointing into the body surface. With the time-dependent Green function, the potential function can be obtained from the integral equation in terms of source distribution as follows: °k (P; t) = ~ dr ~ G(P, Q; t—T)ak (Q; 7)dS t0 Sb (4) where Q = Q(x', y', z') and (Jk iS the source strength which can be solved from 2 0¢k ~ p; t) = - - ~k (P; t)

+I dial ~G(P,Q;t—I) (Q;~)dS to sb Nap where the time-dependent Green function G(P, Q; t - r) for the infinite water depth (Wehausen and Laiton, 1960) can be written as G(P, Q; tar) = Go(P, Q)~(t-~)+H(t-~)F(P, Q; to-) (6) with Go(P,Q) = -4 (- -—) (7) 5(t—or) is the Dirac delta function and H(t - r) is the Heaviside unit step function, and F(P,Q;t—7) = ~ coo . 2 ~ J N/~ sin[~/~(t—7)]ek(Z+Z ) JO (kR)dk with r = `/(x—X')2 + (y _ y/)2 + (z _ Z')2 (9) rat = >/(x—X,)2 + (y _ y')2 + (Z + Z')2 (10) R = v/(x - X')2 + (y _ y')2 (11) and JO is the Bessel function of the zeroth order. Based on Gauss's flux theorem, Eq. (5) can be desingularized as And =—(I; t) + ~ Ark (Q; t) ~ —irk (P; t) ~ ~ lids +2 ~ ark (Q; t) In dS +J~ dri ( ~Q; )ak(Q;~)dS (12) where Go (P. Q) and G2 (P. Q) are defined as GAP, Q) = - 4 (- +—) (13) G2(P, Q) = 4 — (14) After the source strength is solved from Eq. (12), the velocity potential then can be obtained from the following non-singular integral equation: (5) 0k(P;t) = ~ Gt (P. Q) Ark (Q; t) - )(Q) -~t py ] dS +2 Js~ ark (Q; t)G2 (P. Q)dS + To ~ ~ ; t) + J dri ak(Q; r)F(P, Q; t—r)dS (15) to Sb dSb L where y(P) is the source distribution on Sb which makes the body surface an equipotential surface of potential on and satisfies the homogeneous integral equation )(p) = -I )(Q) on ids (16) (8) Equation (16) can be desingularized in a similar way to Eq. (12), and y(P) can be solved by find- ing the eigenfunction of ~K(P, A)/ p associated with the eigenvalue equal to -1, where K(P, Q) = 2Gi(P,Q). The potential, ho' is constant in the interior of the equipotential surface, and can be computed at the origin by No=- Jo )(Q)(~Q~ + ~Q,~)dS (17) where Aid and If denote distances between Q and the origin, and Q', the image of Q. and the origin, respectively. Since the initial boundary value problems for radiation and diffraction has been linearized, the response function can be used to describe the re- sponse of the linear system. According to the work of Liapis and Beck (1985), the radiation force can be expressed as: . . . Fik (t) =—~ jk Ok (t)—~ jk (k (t)—Ok (k (t) rt J K'Rk(t—T)(k(~)dT, j, k = 1, 2, ,6 o (18) where Ok is the added-mass; Ok is the hydrody- namic damping coefficient; Ok is the coefficient of the hydrodynamic restoring force; and K~Rk(t) is the force response function which can be solved by a direct solution scheme (Cong, et al., 1998) from Eq. (18) using a non-impulsive input velocity (k(t) = ~/57iexp(-oet2~. Here or is an arbitrary number. 3

The impulse response function KjD(t) for the diffraction problem can be solved from the follow- sing equation: Ioo KjD (t—r)r10 (~)dr =—gj7 (t)—hj7 (t) (19) —00 where gj7 (t) = p ~ o7 (t)r? jdS (20) sb hj7 (t) = - p l o7 (t)mjdS (21) sb The non-impulsive incident wave ~10 and the corre- sponding derivatives of the incident wave potential were given by King et al (1988). In this work, mj's are all zero for zero speed case. NUMERICAL IMPLEMEN- TATION While many mathematical representations have been adopted to describe the body surface, non- uniform rational B-Splines (NURBS) have be- come the preferred method (Farin, 1991). The widespread acceptance and popularity of NURBS are because they provide a general and flexible de- scription for a large class of free-form geometric shape. Their intrinsic characteristics of local con- trol, low memory requirement, coupled with a sta- ble and efficient generating algorithm, make them a powerful geometric tool for surface description, especially for complicated body geometry. In the panel-free method, NURBS were adopted to de- scribe the body surface mathematically. It is assumed that Np patches are used to describe a body surface. Each patch can be represented by a NURBS surface. Let P(x~u,v),y~u,v),ztu,v)) be a point on a patch; x, y and z denote the Cartesian coordinates; and u and v are two parameters for the surface defini- tion. On a NURBS surface, P(u, v) can be defined as follows: =0 j=o ij C.,j. i q (v) P(u,v) = ~i~-o~j ow',jN` ~(u)Njqtv) (22) Since Eq. (12) is singularity free, it can be dis- cretized by directly applying the Gaussian quadra- ture and the trapezoidal time integration scheme. The Gaussian quadrature points are arranged in the computational space, rs, then their correspond- ing coordinates, normals and Jacobian in the phys- ical space can be obtained based on Eq. (22~. Therefore, Eq.~12) can be written as 0~a~p;; t) = -ok (Pi; t) Up Nj Mj +~ ,Wj fakfQ~S;t)VpG~(Pi, QUASI npi j=1 r=1 s=1 —ark (Pi; t) VQG 1 (Pi, Q.:' S) · ngr~ ] Np Nj Mj ~ ~ ~ Wj ak f Q; ; t) V pG2 (Pi Qrs) n j=1 r=1 s=1 Np Nj Mj rs rs F(P, Qj; t) +!5t[2 ~ OWE (JktQj ;to) one ~=1 r=1 s=1 kin—1 Np Nj Mj F p rs t—t + £ ~ `£ ~ Wjrs ~ i' Qj; k) k=1 j=1 r=1 s=1 for i = 1,2,...,Np where Wjrs = Wrw5J's' Nj and Mj are the num- ber of Gaussian quadrature points in the u- and v- directions on the jth patch. Pi = Pifu~,vm),n = 1,...Ni,m = 1,...Mi and Ads = Qj~ur~vs) are the position vectors of Gaussian quadrature points on the ith and jth patches in the physical space, re- spectively; npi and nets are the corresponding unit normals; wr and ws are the weighting coefficients in the a- and v-directions; Jjrs is the Jacobian of QjrS; t is the time; to is the lower limit of time; and At is the time step, to = to+ki\t and t = to+k~\t, where k and kit are the time constants at any in- stant and for the total time, respectively. It can be seen that the algorithm can be easily controlled by changing the number and the arrangement of Gaussian quadrature points. NUMERICAL RESULTS The panel-free method was applied to a Wigley hull where wij are the weights; Ci,j form a network of at zero speed. The hull geometry is defined by: control points; and Ni,ptu) and Nj~qtv) are the nor- malized B-spline basis functions of degrees p and q 7' ~1 _ `2 y ~1 _ <2' t1 + 0 2~2) + `2 (1 - (~) (1 _ tE2~4 in the u- and v-directions, respectively. (24) cJk chars; tk)] (23) 4

where the nondimensional variables are given by = 2x/L, ~1 = 2y/B, and ~ = z/T, where L is the ship's length, B is the beam, and T is the draft. The hull has L/B = 10, L/T = 16 and a block coefficient of 0.5606. NURBS and analytical rep- resentations were both used for the computation. In the NURBS expression, the Wigley hull was de- scribed by a 13 x 13 control net (Np=1) with degrees 3 in both u- and v-directions. Figure 1 shows the distribution of 6 x21 Gaussian quadrature points and the control net, where the Gaussian points are denoted by "+" and the solid lines represents the control net. Yet Figure 1: Control net and Gaussian point distribu- tion (6x21) for a Wigley hull The computations were carried out for the ra- diation problem. Figure 2 shows the computed heave and pitch response functions at a time step dt = 0.25s using 6x21 and 6x25 Gaussian points on the hull represented by NURBS surfaces. In order to investigate the convergence of the compu- tation, lOx30 Gaussian points were also applied on the analytically represented surface. It was found that the computation was not sensitive to time step and the number of Gaussian points. In these figures, the heave and pitch response func- tions, K33 and K55, are nondimensionalized as K33 / ~ pgV /L) /7[ and K55 / ~ pgV ~ x:, respec- tively. The time t is nondimensionalized as t~/37~. The heave and pitch added mass for the Wigley hull at zero speed was computed from the response functions. The heave and pitch added mass and the frequency are nondimensionalized as A33/(pV), A33/(pVL2) and w = w~/57~, respectively. The computed heave added mass by PFM was com- pared with those of TiMIT and WAMIT. TiMIT and WAMIT are two panel-method codes from MIT for time-domain and frequency-domain wave analysis, respectively. Note that results of TiMIT and WAMIT used here were taken from the work of gingham (1994~. Figure 3 shows the compari- son. The irregular frequencies are shown at ~ ~ 5.8 and w ~10 for both TiMIT and WAMIT in which the half-hull was discretized by 144 panels. PFM shows an oscillation around w = 5.8, but its behav- ior is different from those of TiMIT and WAMIT. The curve around the irregular frequencies tends smoother as more Gaussian points are distributed. The computed heave and pitch damping co- efficients were compared with those by Bing- ham (1994) in Figures 4. The heave and pitch damping coefficients are nondimensionalized as B33 / (pVcv) 47; and B55 / (pVLai) x:, respec- tively. The wave exciting forces were also determined for the Wigley hull. The heave and pitch response functions for the Froude-Krylov forces were com- puted according to the work by King et al. (1988~. Then the exciting forces were compared with re- sults from King (1987), where 120 panels were used to represent the half-hull. In the computation of diffraction problem, the time step was chosen as 0.25s. The computed heave and pitch response functions due to the diffracted waves in compari- son with the results of King (1987) are presented in Figure 5, where the heave and pitch response functions, K37 and K57, are nondimensionalized as K37 / (P9V /L) ~ and K57 / ~ pgV ~ /7i, respec- tively. The oscillation of the curve shown in results by the panel method is not presented in the results by PFM. Applying Fourier transform to the diffraction and Froude-Krylov response functions, we were able to obtain the frequency-domain wave exciting forces. The forces and phases were compared with those results from King (1987) and the strip the- ory results taken from his work in Figures 6 and 7. To be consistent with the presentation of King, the frequency is nondimensionalized as ~ = kL, and the nondimensional heave and pitch exciting forces are given as F3 7 / ~ pgV / L) and F5 7 / ~ pgV ), respec- tively. The results by PFM show a good agreement with those by the panel method and by the strip theory. With the number of Gaussian points in- creased for NURBS surface, the computed results 5

by PFM approached to solutions that were com- puted by the analytical hull expression and large number of Gaussian points. CONCLUSIONS A panel-free method (PFM) has been developed to solve the radiation and the diffraction problems in the time domain. In PFM, the integral equation in terms of source strength is desingularized be- fore it is discretized. The singularity-free integral equation allows for application of Gaussian quadra- ture globally over the exact body geometry. The body geometry can be either described in an ana- lytical definition or by a parametric representation. The complex body geometry can be accurately de- scribed by NURBS surfaces which have been widely used in the field of computer aided design. In general, compared with the panel method, PFM involves less numerical manipulation, since panelization of a body surface is not needed. Pro- gramming of the PFM is easier than the panel method. It is more accurate, since the assumption for the degree of approximation of source strength distribution as in the panel method is no longer needed, and Gaussian quadrature can be directly and globally applied to the body surface with a mathematical description. The accuracy of the so- lution can be easily enhanced and controlled by changing the number and distribution of Gaussian quadrature points. The robustness and accuracy of PFM has been demonstrated by its applications to the radiation and the diffraction problems of a Wigley hull at zero speed in the time domain. The computed radi- ation and diffraction response functions, hydrody- namic coefficients and wave exciting forces for the Wigley hull agree well with published results. The oscillatory error of the memory function at large time tends to be reduced by PFM in comparison with the panel method. The evaluation of the waterline integral by PFM will be developed in the near future for the steady forward speed case. The concept of the panel-free method can also be applied to the frequency-domain computation. ACKNOWLEDGMENT This work was supported by the Natural Sciences and Engineering Research Council of Canada. REFERENCES gingham, H.B. (1994). Simulation Ship Motions in the Time Domain. Ph.D. Thesis, Massachusetts Institute of Technology, Massachusetts. Cong, L.Z., Huang, Z.J., Ando, S. and Hsiung, C.C. (1998~. Time-Domain Analysis of Ship Motions and Hydrodynamic Pressures on a Ship Hull in Waves. Proceedings 2nd International Conference on Hydroelasticity in Marine Technology, Fuknoka, Japan, pp. 485-495. Danmeier, D.G. (1999). A Higher-Order Panel Method for Large-Amplitude Simulation of Bodies in Waves. Ph.D. Thesis, Massachusetts Institute of Technology, Massachusetts. Farin, G.E. (1991). CURBS for Curve and Surface Design. SIAM Activity Group on Geometric De- . sign. Hess, J.L. and Smith, Calculation of Nonlifting about Arbitrary Three-Dimensional Journal of Ship Research, Vol. 8, No. 22-44. A.M.O. (1964). Potential Flow Bodies. 3, pp. King, B.K. (1987). Time-Domain Analysis of Wave Exciting Forces on Ship and Bodies. Ph.D. Thesis, The University of Michigan, Michigan. King, B.K., Beck, R.F. and Magee, A.R. (1988). Seakeeping Calculations With Forward Speed Using Time Domain Analysis. Proceedings 17th _ymposium on Naval Hydrodynamics, The Hague, Netherlands, pp. 577-596. Kouh, J.S. and HO, C.H. (1996). A High Order Panel Method Based on Source Distribution and Gaussian Quadrature. Schiffstechnik, Bd. 43, pp. 38-47. Landweber, L. and Macagno, M. (1969). Irrota- tional Flow about Ship Forms. IIHR Report, No. 123, The University of Iowa, Iowa City, Iowa. 6

Lee, C.H. and Newman, J.N. (2001). Solu- tion of Radiation Problems with Exact Geometry. Proceedings 16th International Workshop on Water Waves and Floating Bodies, Hiroshima, Japan, pp. 93-96. Lee, C.H., Farina, L. and Newman, J.N. (1998~. A Geometry-Independent Higher-Order Panel Method and Its Application to Wave-Body Inter- actions. Proceedings of Engineering Mathematics and Applications Conference, Adelaide. Liapis, S.J. and Beck, R.F. (1985~. Seakeeping Computations Using Time Domain Analysis. Proceedings 4th International Conference on Numerical Ship Hydrodynamics Washington , , D.C., pp. 34-54 Maniar, H.D. (1995~. A Three Dimensional Higher Order Panel Method Based on B-Splines. Ph.D. Thesis, Massachusetts Institute of Technology, Massachusetts. Qiu, W. and Hsiung, C.C. (2001a). A Panel-Free Method for Time-Domain Analysis of Radiation Problem. Proceedings 16th International Workshop on Water Waves and Floating Bodies, Hiroshima, Japan, pp. 129-132. Qiu, W. and Hsiung, C.C. (2001b). A Panel-Free Method for Time-Domain Analysis of Radiation Problem. Ocean Engineering, accepted for publi- cation. Qiu, W. and Hsiung, C.C. (2001c). Time- Domain Analysis of Diffraction Problem by a Panel-Free Method. Proceedings 6th Canadian Hydromechanics and Marine Structure Conference. Vancouver, Canada, pp. 15-21. Wehausen, J.V. and Laitone, E.V. (1960~. Sur- face Waves. Handbuch der Physik (ea. S. Flugge), Springer-Verlag, Vol. 9. 7

1 0.8 a: ~ 0.6 o ._ u: ._ o z 0.4 0.2 o -0.2 s 4 vet us ~ 2 o ._ I;; ._ 1 a: a z o 1.2 I \ PFM (NURBS' 6x25) . \ I PFM (analy.,lOx30) ~ | . ~ W_ ~ 0 1 2 3 4 5 6 Nondimensional time l 3~1~ I ~ PFM (NURBS' 6X25) \ I PFM (analy.,lOx30) 1 \ ~ ==~ -2 - 0 1 2 3 4 5 6 Nondimensional time Figure 2: Heave and pitch radiation force response functions for a Wigley hull 8

1.5 ce o ._ cat ._ o He us id o ._ 1 0.5 o 0.06 0.05 0.04 0.03 0.02 0.01 o 2 \ C ~ >~ Z4.5 5 5.5 6 6.5 .,~ PFM (NURBS, 6x21) PFM (analy., lOx30) - TiMIT WAMIT 0 , , , 1 0 2 4 6 8 Nondimensional frequency 10 12 14 16 l l l l l l | PFM (NURBS, 6x21) I PFM (analy., lOx30) ~ | ~ \ 1 , 0 2 4 6 8 10 12 14 16 Nondimensional frequency Figure 3: Heave and pitch added mass for a Wigley hull 9

0.08 us m~ 0.06 - o A: 0.04 z o 1 0.8 0.6 o ._ a: cot ._ O 0 4 0.2 PFM (NURBS, 6x21) ;~ PFM (analy., lOx30) Y \, TiMIT WAMIT 0 1 ~ Z ~ ~ 0 51 ~ Ha 5 5.5 6 6.5 7 7.5 t ~~ .,, . .... l I o 0.1 2 4 6 8 10 12 14 16 Nondimensional frequency PFM (NURBS, 6x21) PFM (analy., lOx30) ~ TiMIT ....... 1 ~ 0.02 / ~ J ~~ :. . O :z 4 ~ A r TO 12 Nondimensional frequency Figure 4: Heave and pitch damping coefficients for a Wigley hull 10 14 16

The impulse response function KjD(t) for the diffraction problem can be solved from the follow- sing equation: Ioo KjD (t—r)r10 (~)dr =—gj7 (t)—hj7 (t) (19) —00 where gj7 (t) = p ~ o7 (t)r? jdS (20) sb hj7 (t) = - p l o7 (t)mjdS (21) sb The non-impulsive incident wave ~10 and the corre- sponding derivatives of the incident wave potential were given by King et al (1988). In this work, mj's are all zero for zero speed case. NUMERICAL IMPLEMEN- TATION While many mathematical representations have been adopted to describe the body surface, non- uniform rational B-Splines (NURBS) have be- come the preferred method (Farin, 1991). The widespread acceptance and popularity of NURBS are because they provide a general and flexible de- scription for a large class of free-form geometric shape. Their intrinsic characteristics of local con- trol, low memory requirement, coupled with a sta- ble and efficient generating algorithm, make them a powerful geometric tool for surface description, especially for complicated body geometry. In the panel-free method, NURBS were adopted to de- scribe the body surface mathematically. It is assumed that Np patches are used to describe a body surface. Each patch can be represented by a NURBS surface. Let P(x~u,v),y~u,v),ztu,v)) be a point on a patch; x, y and z denote the Cartesian coordinates; and u and v are two parameters for the surface defini- tion. On a NURBS surface, P(u, v) can be defined as follows: =0 j=o ij C.,j. i q (v) P(u,v) = ~i~-o~j ow',jN` ~(u)Njqtv) (22) Since Eq. (12) is singularity free, it can be dis- cretized by directly applying the Gaussian quadra- ture and the trapezoidal time integration scheme. The Gaussian quadrature points are arranged in the computational space, rs, then their correspond- ing coordinates, normals and Jacobian in the phys- ical space can be obtained based on Eq. (22~. Therefore, Eq.~12) can be written as 0~a~p;; t) = -ok (Pi; t) Up Nj Mj +~ ,Wj fakfQ~S;t)VpG~(Pi, QUASI npi j=1 r=1 s=1 —ark (Pi; t) VQG 1 (Pi, Q.:' S) · ngr~ ] Np Nj Mj ~ ~ ~ Wj ak f Q; ; t) V pG2 (Pi Qrs) n j=1 r=1 s=1 Np Nj Mj rs rs F(P, Qj; t) +!5t[2 ~ OWE (JktQj ;to) one ~=1 r=1 s=1 kin—1 Np Nj Mj F p rs t—t + £ ~ `£ ~ Wjrs ~ i' Qj; k) k=1 j=1 r=1 s=1 for i = 1,2,...,Np where Wjrs = Wrw5J's' Nj and Mj are the num- ber of Gaussian quadrature points in the u- and v- directions on the jth patch. Pi = Pifu~,vm),n = 1,...Ni,m = 1,...Mi and Ads = Qj~ur~vs) are the position vectors of Gaussian quadrature points on the ith and jth patches in the physical space, re- spectively; npi and nets are the corresponding unit normals; wr and ws are the weighting coefficients in the a- and v-directions; Jjrs is the Jacobian of QjrS; t is the time; to is the lower limit of time; and At is the time step, to = to+ki\t and t = to+k~\t, where k and kit are the time constants at any in- stant and for the total time, respectively. It can be seen that the algorithm can be easily controlled by changing the number and the arrangement of Gaussian quadrature points. NUMERICAL RESULTS The panel-free method was applied to a Wigley hull where wij are the weights; Ci,j form a network of at zero speed. The hull geometry is defined by: control points; and Ni,ptu) and Nj~qtv) are the nor- malized B-spline basis functions of degrees p and q 7' ~1 _ `2 y ~1 _ <2' t1 + 0 2~2) + `2 (1 - (~) (1 _ tE2~4 in the u- and v-directions, respectively. (24) cJk chars; tk)] (23) 4

18 16 14 LO 12 cot o c: 10 c' ._ O 8 z 6 4 2 o n 2 4 150~ 100 t 50L L) is 0 I: AL -50~ lain -150 l _'\L _ 'I? \^ 1 - - 1 PFM (NURBS, 6x21) PFM (analy., lOx30) King (1987) Strip theory A) ., PFM (NURBS, 6x21) PFM (analy., lOx30) King (1987) Strip theory i''/ ,, -200 ~ ~ ~ ~ _ 0 2 4 6 kL 10 14 10 12 14 Figure 6: Heave wave exciting forces and phases for a Wigley hull 12

2.5 2 l l l /~' \ PFM (NURBS, 6x21) _ ,y \ PFM (analy., lOx30) at ~ King (1987) ~ ILL 15 | ~ . ~ I Strip theory ~ l E ¢: ~ to 1 _ ~ \ 0.5 _/ \" 11 , . . . . . o 150 100 50 cam ct us l 4 6 8 10 12 14 kL l o -50 -100 l l l PFM (NURBS, 6x21) PFM (analy, lOx30) King (1987) Strip theory ~ _ ; I ~ ~ . -I- ~ ~ ~ l l l /: f , . . . 2 4 6 8 kL 10 12 14 Figure 7: Pitch wave exciting forces and phases for a Wigley hull 13

DISCUSSION J. Tao VBD-European Development of Centre for Inland and Coastal Navigation, Germany I would like to express my appreciation for the author who presented a new method which seems to have a large development potential for many time domain problems. From your preliminary results, I have the impression that this panel-free method works well for radiation problems. We can see the good agreement force radiation force response functions as well as for added mass and damping. However, there is a large difference in predicting diffraction-force response functions, particularly in Figure 5 for the pitch mode. Could you explain this result? I would expect that a validation for your results would clarify the advantage of the new method. AUTHORS' REPLY There are some differences between the computed pitch force response functions by the panel-free method and by the conventional panel method. We, unfortunately, were not able to find the experimental results of wave exciting forces of the Wigley hull at zero speed for comparison. DISCUSSION L.J. Doctors The University of New South Wales, Australia In my own panel-code for ship motions, I use a "lid" on the internal free surface. This is an extremely simple and effective way of totally eliminating the problem of the irregular frequencies. Can you implement a similar idea in your method in order to deal with the matter of the irregular frequencies? AUTHORS' REPLY The irregular frequencies shown in the computed obtained by Fourier transform of the response function computed in the time domain. They were due to the oscillation of the response added mass and damping coefficients in the frequency domain were function in the time domain, particularly at large time. In the panel- free method, this could be improved by increasing the number of Gaussian points and/or changing the distribution of Gaussian points. The "lid" may be useful for the frequency- domain computation, but we wonder if it could improve the time-domain computation.