Journal of Ship Research, Vol 56, No 3, September 2012, pp 129–145 http://dx.doi.org/10.5957/JOSR.56.3.100031 Journal of Ship Research Validation of Potential-Flow Estimation of Interaction Forces Acting upon Ship Hulls in Parallel Motion Serge Sutulo,* C Guedes Soares,* and Janne F Otzen† *Centre for Marine Technology and Engineering, Technical University of Lisbon Instituto Superior Te´cnico, Lisboa, Portugal { Force Technology, Kongens, Lyngby, Denmark The hydrodynamic interaction problem is of great importance for numerical ship handling simulators and at present, only a relatively simple potential double body panel method can be expected to supply estimates of interaction forces and moments in real time on commonly used hardware, without limitations on the hull shape, and on the mutual position and motion of the interacting bodies Such a code was developed on the basis of the classic Hess and Smith method and proved to be fast enough to model interaction in real time when a moderate number of panels is used In the present paper, results obtained with the potential code are validated against experimental data obtained in deep and shallow water towing tanks for the case of a tug operating near a larger vessel All the tests corresponded to the steady regime and only cases with parallel center planes were considered here The comparisons carried out for various discretization of the hulls provide useful information about natural limitations in breaking at too close lateral distances and about acceptable trade-off between the computational speed and accuracy In addition, influence of the nonzero sway and yaw velocities is investigated numerically Keywords: hydrodynamic interaction; potential flow; panel method; tank experimental data Introduction MARINE VEHICLES of any kind must often maneuver in the presence of various rigid objects such as other vessels, which can be moving, anchored, or moored; water basin boundaries; banks of various shape and length; piers and jetties; and floating production storage, and offloading units In all those cases, hydrodynamic interaction appears becoming sometimes very significant and jeopardizes navigation safety Hence, the ability to predict the corresponding hydrodynamic loads is of great practical value and this problem attracted attention of hydrodynamicists for more than 100 years Results of interaction studies can benefit the seamanship practice in the following ways: Qualitative understanding of the interaction phenomena, working out recommendations for ship operators; Manuscript received at SNAME headquarters May 16, 2010; revised manuscript received October 10, 2011 SEPTEMBER 2012 Estimation of the peak values of the loads, safe distances, and safe velocities in overtaking and encountering situations; Estimation of loads in the mooring lines caused by vessels passing by; Analysis of navigational accidents; Working out maneuvering standards; and Modeling of interaction effects in computerized maneuvering simulators Specifics of the last and the most recent application include that it requires a combination of reasonable accuracy with high computational efficiency and necessity to predict interaction forces and moments for arbitrary relative position and motion of all interacting bodies Of course, overtaking and encountering maneuvers continue to be the most critical from the viewpoint of possible accidents, as a too close approach on crossing courses is not likely At least, this is true for ships of more or less comparable size but, for instance, tugboats can approach the assisted ship in a more or less arbitrary way especially when they work in the pushing mode (van Hilten & Wulder 2006) This paper is mainly aiming to this kind of application 0022-4502/12/5603-0129$00.00/0 JOURNAL OF SHIP RESEARCH 129 The overall number of publications on the hydrodynamic interactions apparently exceeds 100 As an extensive review of earlier studies starting from first experimental results reported by Taylor in 1909 was presented by Norrbin (1975); the review presented here will mainly focus on more recent contributions Some of these studies consider specifically the ship-to-ship interactions while some others concentrate on bank-suction problems especially when the ship is sailing in a canal In many cases, however, both types of problems are solved with similar methods, which can be divided into theoretical, experimental, and numerical Of course the dividing line between “theoretical” and “numerical” approaches is often fuzzy as every numerical method always has certain theory behind it, whereas any theoretical solution must end in computations Typically, numerical methods are more straightforward, require better computing power, and are more versatile in applications while theoretical methods mostly are based on certain asymptotic assumptions about the shape of the bodies and their mutual location and motion A fundamental theoretical study of the ship-to-ship interaction was carried out by Abkowitz et al (1976) Their formulation was based on the so-called Havelock hypothesis according to which the potential flow model with rigid flat free-surface corresponding to the zero-Froude-number case is sufficient to explain the interaction phenomena This hypothesis looks quite reasonable at low Froude numbers and, although not quite accurate, was exploited in many studies including the present one Abkowitz et al (1976) introduced, however, additional assumptions: the ship hulls were supposed to be slender, they were approximated with bodies of revolution and these were approximated with axial source and dipole distributions Horizontal dipoles originated from lateral motions of the “own” ship (i.e., the ship of interest) with respect to water and from the induction of the other ship, and vertical dipoles were added in shallow water The strength of the dipoles was determined as result of a fast iteration process and the forces and moments were computed basing on the Lagally–Cummins formulae The agreement with experimental data was not very good and to make it acceptable, a certain effective beam of the equivalent body of revolution was to be introduced It was found that unsteadiness of the flow affects much more the yawing moment, than the sway force Beck (1977) applied the matched asymptotic expansions method to the problem of ship-to-bank interaction in a shallow channel In the outer region, the shallow water theory was applied, so the final results depended on the depth Froude number Besides sway force and yaw moment computed with the Blasius formulae, the sinkage force and trimming moment were estimated from the linearized Bernoulli integral A conclusion was drawn that the horizontal two-dimensional (2D) model was not satisfactory even at very small bottom clearances A similar method was much later used by Kijima (1997) who studied interaction between three bodies: two ships running parallel in the proximity of a pier simulated with a circular or oval cylinder In this study, the hydrodynamic part was combined with dynamics of the ships and time histories for forces and motion were obtained Interesting theoretical studies were undertaken by Yeung (1978) who applied the slender-body theory and the method of matched asymptotic expansions The method results in relatively fast algorithms and accounts for the Kutta condition on the aft part of the hull, but it heavily underestimates sway forces and yaw moments in the most interesting case of small separation between the hulls A similar approach was used by Yeung and Tan (1980) to study interaction with some fixed obstacles like a protruded wall, wedge, or embankment corner This study was later extended by Hsiung and Gui (1988) to a larger variety of obstacles Tuck and Newman (1974) performed one of the best asymptotic studies of the ship-to-ship interaction problem Two situations were investigated, both for the zero Froude number In the first case, the water depth was assumed to be infinite or finite but relatively large The so-called first-order theory was constructed, that is, each ship was just sailing in the flow field of its partner Then, the lateral loads on each of the slender hulls were computed using the generalized Taylor theorem for a body moving in an accelerated uniform flow The Kutta condition was accounted for on sterns of finite height The theory resulted in formulae for the sway force and yaw moment having a quadratic form structure Namely, for a ship in parallel uniform motion with the Nomenclature Ci = G() = H= m= mi = n= N= p= r= r= S= T= u= v= V= X= x, y, z = Y= g rts ; ers = 130 origin of the body axes for the ith body Green’s function depth of the fluid number of interacting bodies components of the generalized normal outer unity normal yaw moment pressure position vector velocity of yaw or distance rigid boundary in the fluid fluid kinetic energy velocity of surge velocity of sway velocity surge force coordinates in the body axes sway force Boltzmann’s symbols SEPTEMBER 2012 x, h, z = mij = pr = r= s= F= f= W= wi = Ñ= coordinates in the global fixed axes added mass coefficients a quasi-coordinate water density source density total velocity potential perturbation potential angular velocity a quasi-velocity the Hamilton operator Subscripts Ci = cur = e= i= I= p= r= related to the origin of the ith body current proper (eigen) number of the body (ship) induced (total) potential force/moment or “pressure” velocity relative JOURNAL OF SHIP RESEARCH SEPTEMBER 2012 + + + + + + + added masses Ship overtaking maneuvers were simulated and collision-dangerous situations were traced Finally, a Reynolds-Averaged Navier–Stokes Equation (RANSE) code was applied by Chen et al (2003) for tackling the problem of ship-ship interaction in shallow water Good agreement with experimental data was demonstrated except for the case of relatively small distance between the boards when insufficient fineness of the grid was suspected Pinkster (2004) focused on a rather specific interaction problem related to necessity of estimation of the mooring line loads originated from vessels passing by Specifics of the situation are that although the passing vessels sail at deeply subcritical Froude numbers even in small water depth and the Havelock hypothesis must hold, these vessels generated low-frequency seiches in the harbor area whose influence dominates the near-field interaction Hence, in Pinkster’s algorithm the near-field interaction was completely neglected as well as the ship-waves wash The moving ship was considered as a wave generator and then a diffraction problem was first solved in the frequency domain and further converted to the time domain by means of the fast Fourier transform An interesting case where the free-surface interaction effects turned out decisive was studied numerically by So¨ding and Conrad (2005) A real collision case happened on the Elba River and was studied post factum A larger vessel with dimensions L B T ¼ 264.0 39.9 11.4 m was running at 15 kn and overtaking a smaller ship with particulars L B T ¼ 101.0 18.4 6.3 m cruising at 13.5 kn speed The water depth was about 14.5 m making the depth-based Froude number equal to 0.65, that is, large enough for most shallow-water wavemaking effects to show up The distance between the centerplanes of the ships was 150 m, which makes the near-field interaction negligible while the waveorigin interaction was so strong that the smaller ship was captured by depression of the water surface, lost its steering capability, and finally collided with the overtaking vessel Two different Rankine source nonlinear free surface potential flow codes were used and results were compared with experimental data giving very good agreement for the yaw moment, a satisfactory one for the surge force, and a somewhat worse agreement for the sway force One of the codes was run with and without vortex distribution aiming at satisfying the Kutta condition The difference was only significant for the sway force, where presence of vortices improved the agreement A large group of studies was based on experimental methods An approximate empirical method for predicting forces in overtaking maneuvers was developed by Brix (1993) Extensive automated tests with scaled models of four different shapes were carried out by Vantorre et al (2002) covering only parallel motion, that is, overtaking, passing by, and encountering situations in shallow water A special two-carriage experimental layout was used and time histories of the surge and sway interaction forces and of the yaw moment for various lateral distances and speed values were obtained On the basis of the huge amount of assembled data a simple approximate empiric method for estimating peak values of the interaction forces was proposed Later, Vantorre et al (2003) also studied interaction of a moving ship with banks of various configurations Results were finally processed to the form of regressions for the sway force and yaw moment, which were found to depend on the under-keel clearance, distance between the ship and the bank, Froude number, propeller slipstream velocity, and a few hull form parameters + other the formula for the loads had the structure aV1 V2 þ bV22 , where a, b are constant coefficients, V1 is the speed of the “own” vessel upon which the estimated force is acting, and V2 is the speed of another or “target” ship Here, one of the limitations of the theory is obvious: when the target ship is at rest, no interaction is observed The second studied case is related to the pair of ships moving with the same speed in extremely shallow water Here the flow was considered far-field 2D in the horizontal plane and was treated by matched asymptotic expansions Threedimensional (3D) effects were accounted for by means of the so-called blockage coefficient The paper also explains why the double-body potential formulation makes sense in the interaction and maneuvering problems: unlike what is happening in seakeeping and wave resistance problems, this approximation still remains nontrivial and captures the main effect It was also supposed that viscous effects would be more significant than those related to the wavemaking King (1977) focused on unsteady effects accompanying interaction of two bodies using a 2D formulation, that is, the water depth was assumed to be close to the draft The wavemaking was completely neglected and the linearized airfoil theory was applied The lateral distances had to be comparable to the ship lengths The agreement with experimental data was only judged to be satisfactory for the sway force on the stationary (not moving) body King and Tuck (1979) investigated a somewhat complementary problem of the steady motion of slender ships parallel to banks It was shown that in the inner region near a beach with the constant slope at zero Froude number the inner problem becomes isomorphic to the axial flow problem Gui et al (1992) discussed application of the Schwarz–Christoffel transformation to the 2D problem of a maneuvering ship interacting with polygonal boundaries A simplified theoretical study was undertaken by Mastushkin (1977), who studied the case of parallel (in the same or opposite directions) motion of two ships Within the double-body approach each hull was approximated with an ellipsoid with the axes equal to the length of the ship, its breadth, and doubled draft The velocity potential was defined approximately through the iteration process Analytic velocity potentials for isolated ellipsoids were taken as the first approximation The cited article does not contain any validation against experiments or independent numerical results That validation was, however, done by the author in his earlier studies related to the case of zero drift angles The main conclusion drawn from these works is that even a very crude approximation of the ship hull can bring good estimates for the interaction forces at least in some situations One of the first applications of a numerical method belongs likely to Ivanov (1981) who successfully applied the potential flow algorithm by Hess and Smith to the zero –Froude number interaction problem However, all the computations were performed offline as the then available computing hardware was too slow A more advanced but still offline application of another version of the 3D panel potential code was undertaken by Korsmeyer et al (1993) who studied body-to-body interaction in a rectangular canal Only parallel motion was studied for two spheroids, two Mariner ships, and two Panamax bulk-carriers Comparisons with experimental data showed a fairly good agreement A direct 3D potential-flow formulation was also exploited by Yasukawa (2003) who used an unspecified implementation of the panel method for determining the flow potential and the interaction forces which were expressed through a set of generalized JOURNAL OF SHIP RESEARCH 131 Ch’ng et al (1993) carried out bank-interaction experiments with MarAd and S-175 models equipped with propellers but without rudders The varying factors were: the Froude number, relative bottom clearance, lateral distance, bank slope angle, and the thrust Linear regressions were devised and it was demonstrated that the propeller thrust may influence substantially the interaction sway force and yaw moment Li et al (2003) developed another empirical model for predicting interacting forces with banks of three standard shapes: vertical, sloping, and flooded (stepped) The obtained regressions were validated against independent tests and used in simulations A more specialized experimental study focusing on inland vessels was performed by Gronarz (2006) Interesting details about this study include original measurement records showing uncertainty levels, investigation of the influence of the propeller, and convenient approximations for the forces Ankudinov et al (2006) outlined an integrated approach to bank/ obstacle interaction problem Although no specifics are given, it can be classified as a kind of data assimilation method based, among other things, on certain undisclosed theoretical considerations, although solution of the full hydrodynamic interaction problem was likely avoided Lee and Kijima (2003) applied some previously developed interaction estimation algorithm to analyze the behavior of interacting ships in restricted waterways and under wind action Varyani et al (2003) studied mooring-lines loads originating from the hydrodynamic interaction between the moored vessel and the passing by ship The zero –Froude number theory was used to estimate the flow field around the moving ship and the pressure integration resulted in estimates for the surge and sway forces and the yaw moment acting on the moored ship Although the final formulae not look obvious and the back interaction from the moored ship was completely neglected, comparisons with experimental results showed fair agreement Later, this and other similar studies resulted in a generic set of approximations of interaction sway forces and yaw moments proposed by Varyani et al (2004) It seems that lately more focus has been given to experimental studies and attempts to develop some simple, regression-type, algorithms for fast calculation of interaction forces This is, likely, caused by real-time and accelerated-time simulation requirements In-loop numerical methods can then seem inconvenient However, the drawback of all available empirical approximations (and also most analytic solutions) is that they can only cover a limited number of situations For instance, most of the ship-ship interaction studies were performed for the case of zero yaw velocity and for ships on parallel courses and with parallel center planes These may be sufficient to predict the most dangerous situations in overtaking maneuvers but are not enough for correct description of the interaction in all possible situations, which is highly desirable for bridge simulators For example, when simulating maneuvers of a large ship assisted by tugs working also in the pushing mode (van Hilten & Wulder 2006) it would be highly undesirable to impose any restrictions on the mutual motion and position of the involved vessels On the other hand, the computing power of even average personal computers has grown during late years to such an extent that certain numerical algorithms can be certainly used online without hampering the simulation, although this is still not true for viscous-flow and/or free-surface codes 132 SEPTEMBER 2012 In general, the hydrodynamic interaction phenomenon can be viewed as being composed of five simpler phenomena: Near-field potential interaction without a free surface; Interaction related to the free-surface effects or wavemaking interaction; Boundary layer and viscous wake interaction; Interaction caused by longitudinal trailing vortices; and Action of propeller slipstreams and of the thrusters jets It is still difficult to draw definite and precise conclusions about the relative contribution of each component but certain experience and data accumulated so far indicate that the near-field double body potential interaction can be the most important component in many cases, that is, the Havelock hypothesis is pretty much justified On the other hand, this is one of the few hydrodynamic models which can be treated numerically online in real time at least when the accuracy requirements are eased and the number of approximating panels is reduced An alternative could be a vortexlattice method, which has the potential advantage of the possibility of accounting for the Kutta condition when it is thought indispensable The latter is caused by viscosity but once formulated can be applied to nonviscous flows requiring, however, presence of vortices which cannot appear in the perfect fluid While the Kutta condition can be easily formulated in the case of lifting bodies and surfaces with sharp trailing edges it turned out impossible to create an adequate formulation for most slender ship hulls and it is mostly accounted for through neglecting part of the predicted transverse load on the aft part of the body when it goes about proper forces and moments It is much less clear how it should be formulated in the three-dimensional interaction case and the literature data are, in general, inconclusive and not definitely indicate the importance of the corresponding contribution It is much easier to apply the Kutta condition in the twodimensional models when it is sufficient to impose a circulation (Dand 1976), but such a model is poorly applicable even at extremely low water depth That is why, sticking with Havelock hypothesis mentioned above, Sutulo and Guedes Soares (2008) have developed a code based on the classic Hess and Smith method suitable for online computations of interaction forces for an arbitrary number of ship hulls with arbitrary relative position and motion, primarily in deep water This method was later fused with the maneuvering simulation code and an uncontrolled and controlled motion of two vessels in overtaking maneuver was actually simulated (Sutulo & Guedes Soares 2009) After those publications, the code was extended to cover the flat bed shallow water case All validations of the code were so far either internal (comparisons with analytic solutions, convergence studies) or numerical results were only compared against approximate empirical methods In the present paper, the code is validated against experimental results obtained in towing tanks for a specific tugboat operating near a typical large vessel Although the code works at any combination of position and motion of the involved vessels, only results related to parallel motion with equal speeds are validated here They are also complemented with demonstration of the influence of the sway and yaw motion obtained numerically The exposure of numerical results is preceded by description of the method and of the experimental setup and followed by conclusions The general consistency of the approach followed is also confirmed by the fact that a similar online numerical method for estimating JOURNAL OF SHIP RESEARCH the interaction forces was recently and independently applied by Xiang and Faltinsen (2010) Computation of Interaction Forces by means of the Potential-Flow Code 2.1 Problem statement and governing equations Let Oxhz be the Earth-fixed right-handed Cartesian frame of reference with the origin located on the undisturbed free surface of the perfect fluid and with the z-axis directed downwards as illustrated in Figure The fluid is either infinitely deep or limited with a horizontal flat rigid bottom located at z ¼ H The direction of the x-axis can be chosen arbitrarily, depending on the nature of the specific problem There are in this fluid some bodies described by their wetted surfaces Si, i ¼ 1, , m, which can be submerged completely, float on the free surface, or be somehow fixed to the bottom In the general case, the actual surface may depend on the time t at least (for rigid bodies) in terms of their position These bodies can represent ships, submersibles, pieces of banks, any structures, islands, etc The number m of these bodies is arbitrary but finite and in fact most situations are covered by m Then, attached to each body is the frame, Ci xiyizi , which can coincide with the Earth axes at some time moment At any time moment t, the position of each body can be described with the vector ri or with the coordinates: advance xCi, transfer hCi, and submergence (heave) z Ci, as well as with the standard Euler angles: heading y i, pitch qi, and roll wi The instantaneous motion of each body is described with the vectors of the velocity of the origin VCi and of the angular velocity Wi Alternatively used are the linear and angular quasi-velocities of surge, sway, heave, yaw, pitch, and roll: ui, vi, wi, pi, qi, ri Within the formulation followed here, the scalar parameters related to the position and motion in the plane other than horizontal are identically equal to zero It is supposed that the fluid is perfect and the flow is irrotational, so it is fully described with the potential F(x, h, z, t) at z ! 0, above the bottom and outside the bodies Also, it is worthwhile to suppose that a horizontal uniform currently described by the velocities Vxcur and Vhcur or by the vector Vcur is present The total potential can then be represented as F ¼ Vxcur x þ Vhcur h þ f; ð1Þ where f(x, h, z, t) is the perturbation potential and the perturbation (induction) fluid velocity will then be VI ¼ Ñf At each time moment, the perturbation potential satisfies the governing Laplace equation Df ¼ 0; ð2Þ and the nonpenetration boundary condition on each body ¶f ¼ Vr Á n ð3Þ ¶n where n is the outer unity normal to each body and the relative local velocity is Vr ¼ V À Vcur ; ð4Þ where V is the absolute local velocity of a point on the body surface, depending in the common way on the parameters VCi and Wi The low –Froude number assumption yields the following condition on the free surface: ¶f ¼0 ¶z ð5Þ and the same condition must be applied on the flat horizontal bottom In the case of deep fluid, the perturbation potential must vanish at infinity As the governing equation is elliptic, the time only enters in the formulation as parameter but in the case of moving bodies the boundary is unsteady and the Neumann problem must be resolved at any time moment 2.2 Solution 2.2.1 General theory The formulated problem is standard and a common method of solution is to distribute a single layer of sources with density s on the entire wetted surface Then, the following Fredholm integral equation of the second kind holds: ð 2ps ðMÞ þ s ðPÞ S ¶GðM; PÞ dSðPÞ ¼ f ðMÞ; ¶nM ð6Þ where M(x, y, z) and P(x ¢, y ¢, z ¢) are respectively the field (observation) and the source points belonging here to the surface S which consists of all the wetted surfaces present in the current problem The Green function G() in the case of deep fluid and rigid free surface takes the form Fig Frames of reference in the case of two ships SEPTEMBER 2012 1 ð7Þ Gðx; y; z; x¢; y¢; z¢Þ ¼ þ ― ; r r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ ðx À x¢Þ þ ðy À y¢Þ þ ðz À z¢Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ― r ¼ ðx À x¢Þ þ ðy À y¢Þ þ ðz þ z¢Þ Application of the same mirror-image principle to the finite depth fluid case results in a more complicated formula for the Green function:   1 Gðx; y; z; x¢; y¢; z¢Þ ¼ ( þ― ; ð8Þ i¼À1 ri ri JOURNAL OF SHIP RESEARCH 133 ð VI ðMÞ ¼ s ðPÞÑM GðM; PÞdSðPÞ; ð9Þ S ð fðMÞ ¼ s ðPÞGðM; PÞdSðPÞ: S The pressure distribution is then calculated through the unsteady Bernoulli equation which in moving frames takes the form (Lamb 1968) ! ¶f  ð10Þ p ¼ r À þ V2r À V2p ; ¶t where r is the fluid density and Vp ¼ VI À Vr : ð11Þ Then, the potential force Fpi and moment Mpi acting on ith body will be: ð ð Fpi ¼ À pn dS; Mpi ¼ À pr ´ n dS: ð12Þ Si Si The potential hydrodynamic load can be also represented in the standard component form, that is, in the plane motion problem, in terms of the surge force Xp, sway force Yp, and yaw moment Np The thus defined forces and moments will, however, include the proper inertial hydrodynamic loads which are already accounted for in all consistent maneuvering mathematical models So, to obtain the pure interaction forces, proper potential loads must be also calculated and subtracted from the loads defined by (12) The pressure Equation (10) includes the local derivative ¶f ¶t , which vanishes for steady flows When the pressure and loads in the unsteady flow are determined with the local derivative dropped, the corresponding solution is called quasi-steady 2.2.2 Determination of proper inertial forces and moments The most convenient way to calculate the proper inertia loads is to apply the added-mass formalism associated with the equations devised by Thomson, Tait, and Kirchhoff (Lamb 1968) The starting point is the following representation of the kinetic energy of the fluid in the presence of a moving rigid body: where wj are the quasi-velocities of the body in concern (i.e., w1 º u—velocity of surge, w º v—velocity of sway, ., w6 º r—angular velocity of yaw) and mjk are the added mass coefficients defined for the ith body as ð mjk ¼ Àr fj mk dS ð14Þ Si where m1, m2, m3 are the projections of the normal n onto the body axes x, y, and z respectively and m4, m5, m6 are similar projections of r n Each of the potentials fj is obtained as solution to the Equation (6) with the right-hand side f (M) ¼ mj (M) Applying the third Newton law, it is possible to obtain from the Euler–Lagrange equations (Lurie 2002) the following representation for the generalized hydrodynamic reactions Ps, s ¼ 1, , 6:   6 d ¶T ¶T ¶T ¶T þ ( g rts wt þ ( esr À ; ð15Þ ÀPs ¼ r;t¼1 r¼1 dt ¶ws ¶wr ¶p r ¶p s + pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ri ¼ ðx À x¢Þ þ ðy À y¢Þ þ ðz À z¢ þ 2iH Þ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ― that is considered as r i ¼ ðx À x¢Þ þ ðy À y¢Þ þ ðz þ z¢ þ 2iHÞ a series of double hulls with the distance 2H between the waterplanes In practical computations, the series are truncated as farther mirror images have weak influence on the pressure distribution on the central body In this context, the point coordinates can be taken in any suitable frame of reference, that is, one can use x instead of x, etc The right-hand side of the equation is f(M) ¼ Vr (M)Án(M) when the boundary condition (3) is in effect After the Equation (6) is solved, the induction velocity and potential distributions can be expressed through the already known single-layer density s(M): where g rts and esr are the Boltzmann symbols whose definitions and methods of evaluation are given by Lurie (2002) and p s are the quasi-coordinates, that is, p_ s ¼ ws It can be shown that in the case of a body in the unbounded fluid the fluid kinetic energy does not depend on the generalized coordinates and on the quasi-coordinates Then, for the horizontal components of the proper hydrodynamic inertial forces Xe, Ye, and Ne, which are most important for the majority of applications, it can be obtained after some evaluations: Xe ¼ Àm11 uÁ þ m22 vr; ð16Þ Ye ¼ Àm22 vÁ À m26 rÁ À m11 ur; N ¼ Àm vÁ À m rÁ þ ð m À m Þuv À m ur: e 26 66 11 22 26 The pure interaction forces will then be XI ¼ Xp À Xe ; YI ¼ Yp À Ye ; NI ¼ Np À Ne : ð17Þ Most of the proper force corrections, except for the Munk moment, are only present in unsteady and/or curvilinear motion In the most popular case of overtaking in parallel paths all the corrections will be zero if only the motion with zero drift angle is presumed The elegancy of the added mass approach to evaluating potential forces on an isolated body in unbounded fluid stimulated several attempts to extend it to the case of interacting bodies through introduction of generalized inertial coefficients similar to added masses but accounting for the interaction as done, for instance, by Yasukawa (2003) However, apparently this approach makes little sense as these generalized parameters will inevitably depend on the instantaneous configuration of the hydrodynamic system and, in the general case, must be recomputed at every refreshment step The main advantage of the added mass formalism consisting in the possibility to describe completely inertial hydrodynamic properties of any rigid body with only 21 constant parameters is therefore lost and the resulting algorithms seem to be even less economical than the direct force calculation Similar complications occur in the hydrodynamics of deformable bodies Another approach based on the application of the momentum conservation theorem (Korsmeyer et al 1993) is more promising but close analysis showed that it does not have serious advantages over the direct pressure integration either T ¼ 12 (m jk j; k¼1 134 SEPTEMBER 2012 wj wk ; ð13Þ 2.2.3 Numerical algorithm Most of the developed methods of solution to the Equation (6) and further calculations of the JOURNAL OF SHIP RESEARCH integrals (8) are based on the discretization of the surface S into a n S number of primary panels, that is, S ¼ Sj , further approxima- Table Main particulars of tug and tanker model Tug (Ship 1) j¼1 + tion of each primary panel with a simpler computational panel Then, certain assumptions about the distribution of the source density are adopted, which make possible the computation of the influence functions from one panel to another analytically or with some efficient integration formulae The nonpenetration boundary condition is usually satisfied at some collocation point on each panel, which results in a n n linear algebraic set of equations that typically happens if the source density is assumed constant over each panel If the distribution over each panel is described by more than one parameter, the order of the linear set can be higher than the number of panels in consideration As the resulting linear set is practically always well-conditioned with a dominant diagonal, iteration methods of solution are normally preferred which gives the computation time approximately proportional to n2 The most popular and, historically, the first potential flow method developed by Hess and Smith (1964, 1967) uses flat quadrilateral panels and constant source density on each panel A number of improved higher-order methods have appeared later Examples of these are the method by Kough and Ho (1996) and—one of the latest developments—a method based on the spline approximation of the surface, not using explicit panels (Ni et al 2005) A thorough inspection of publications on potential flow calculations indicates, however, that while higher-order methods have sensible advantage in accuracy and efficiency for smooth bodies (spheres, ellipsoids etc.), this advantage nearly vanishes for practical ship forms which can include knuckles, regions of rapidly changing curvature, transom, outer keels, etc Also, in the interaction problems grid refinement at small gaps may be required anyway Bratu (1975) indicated that inappropriate use of a higherorder method can even decrease the accuracy A similar opinion was expressed by Bertram (2000, p 218) Most often, from the viewpoint of overall efficiency the classic Hess and Smith method with flat panels appears to be the best option especially bearing in mind that this method is perfectly and flawlessly described in the easily available publications cited above That is why in the present study this method was selected and implemented anew, with adaptation to multibody computations Experimental Determination of Interaction Forces + + + All model tests were conducted in the towing tanks at FORCE Technology The tests were conducted in both the shallow tank with the horizontal dimensions (25m 8m) and with the depth that can be varied from to 0.25 m, and the deep water towing tank (240m 12m 5.5m) 3.1 Geometry and conditions The hulls of the tug and the assisted ship were considered in model scale 1:25 The assisted ship was a conventional tanker and the tug was a Svitzer M-class ASD tug Both models were tested unpropelled The main particulars of the ships are listed in Table The used coordinate system and sign convention of the forces acting on the tug is as illustrated in Figure SEPTEMBER 2012 Main Particulars Unit Full scale Model Length overall, LOA Length between perpendiculars, L Breadth, B Draft, T Displacement, Ñ m m 29.5 25.6 1.18 1.024 m m m3 11.0 4.6 649 0.44 0.184 0.0416 Tanker (Ship 2) Full scale Model 189.5 186.2 7.58 7.44 31.6 10.3 49197 1.26 0.41 3.15 3.2 Model test setup The model of the tanker was attached underneath the carriage, fixed in all degrees of freedom, as seen in Figure The tug model was placed with various heading angles (however, only results corresponding to the zero heading angle are discussed here) and positions around the tanker The model of the tug was connected to the carriage via two strain gauges positioned fore and aft inside the model, which was free to heave, roll, and pitch The longitudinal and transverse loads measured by the gauges were transformed in the obvious way into the surge and sway force components and the yaw moment acting on the tug The tanker model was fully fixed during the tests and no forces acting upon it were measured as being considered not interesting in the case of a much smaller interacting craft 3.3 Test Program The applied test program included variations of longitudinal and transverse relative positions between tug and tanker for three different water depths and two speeds One of the depths corresponded to the deep-water tank and could be considered as infinite The dimensional and dimensionless values of the remaining two depths H1 and H2 are given in Table The tests were performed at two towing speeds corresponding to kn and kn in full scale More detailed data about these speed regimes are presented in Table All data was acquired as time series using a sampling frequency of 45 Hz The time series were only recorded after the carriage had reached steady velocity The force contribution due to the presence of the tanker, that is, interaction forces, was evaluated by subtracting the forces measured in the tests with the tug alone from the forces measured in the tests with both tug and tanker 3.4 Uncertainty estimation The assumed value of the relative mean square estimate of the error of one measurement based on multiple uncertainty assessment carried out for different conditions and models at the FORCE Technology tanks was estimated to be 5% Assuming that any possible bias had been eliminated, the observed error was treated as random and Gaussian As the interaction forces and moment were always obtained as differences between the corresponding components measured on the tug model in presence and absence of the tanker model, the resulting variance was calculated as doubled variance of one measurement The resulting absolute mean pffiffiffi square was only larger than the original error by the factor 2, although the relative error could be substantially larger when JOURNAL OF SHIP RESEARCH 135 Fig Model test setup Numerical Results Table Depth conditions 4.1 Body surface discretization Depth Relative to: Absolute Depth in Full Scale Draft of the tug T1 Draft of the tanker T2 2.66 3.38 1.19 1.51 H1 ¼ 12.25m H2 ¼ 15.57m Table Speed regimes data Speed Value in Full Scale Froude Number Based on: Knots m/s Length of the tug L1 Length of the tanker L2 Depth H1 Depth H2 2.06 3.09 0.13 0.195 0.048 0.072 0.188 0.282 0.167 0.25 the interaction load was obtained as a small difference On some of the plots in the next section, calculated error bars are shown As the described estimation procedure presumed access to the loads on the model measured both in free run and in presence of the interacting ship and these data were not always available, the error bars are not shown on all the plots but they should be expected to be of similar length 136 SEPTEMBER 2012 The two ship forms, a tug (Ship 1) and a tanker (Ship 2), described in the previous section were used for all the calculations presented in this article The number of offsets in the delivered hull shape description files made possible creation of a very large number of panels for each vessel without interpolation However, the resulting grid was impractical and a special file management program was developed, which first, transformed the hull offsets into the format required by the panel code, then recreated vertices of the sections in conformity with the given ship draft and, finally, the data were thinned along the sections and along the vertices on each section to obtain rougher but, expectedly, more economical form representations A parametric cubic spline interpolation was used to redistribute the vertices on a section appropriately The summary of the number of panels used for computations performed for the present study is given in Table and graphical representation of the ship surfaces with various numbers of panels is shown in Figures to 4.2 Forces and moments: numerical results and comparisons Most of the computations were performed for validating the algorithm against the experimental data and for estimating possible error at various numbers of panels Hence, reproduced here were situations supported by experimental data As in course of the experiment both models were always towed with the same JOURNAL OF SHIP RESEARCH Table Number of panels in computations Total Number of Panels 3506 936 570 356 292 Number of Panels on Ship Number of Panels on Ship 1392 378 210 128 120 2114 558 360 228 172 velocity V, the calculations were carried out in the quasi-steady mode The origin of the body axes associated with Ship was placed in the origin of the global frame, so the global ordinate h of Ship also served, with the inversed sign, as the side distance from Ship and the corresponding abscissa x as the longitudinal stagger of Ship The heading angle y was always kept zero for both vessels as well as the drift angle b in case of parallel rectilinear towing The following nondimensional parameters were used to represent the results: the longitudinal dimensionless stagger x¢s ¼ 2x=L2 , dimensionless lateral offset h¢s ¼ 2h=B2 , ¢ ¼ 2X1;2 =ð rAV Þ, dimensionless dimensionless surge force X1;2 ¢ yaw sway forces Y1;2 ¼ 2Y1;2 À =ð rAV Þ, Á and the dimensionless pffiffiffiffiffiffiffiffiffiffiffi ¢ ¼ 2N1;2 = rAL1;2 V , where A ¼ Ñ1 Ñ2 is the moment N1;2 reference area Results for the deep water, full-scale speed kn and x s¢ ¼ 0:014 that is, with the both ships approximately abreast are shown in Figure It can be seen that the agreement for the surge force is rather poor: the experiment gives definitely negative values or, in other words, an additional interaction resistance while the most precise computation with the finest grid gives this force very close to zero which, however, is expectable from the theoretical viewpoint as the velocity of the perfect fluid in a narrow gap may go to infinity resulting in an unlimited negative pressure Especially large Fig Typical computational configuration of two ships with 378 þ 558 ¼ 936 panels discrepancies happen at small distance which can be explained with local free surface deformations resulting in additional increased pressure in the bow region which is not accounted for by the double-body model and which is not counterbalanced by any similar effects in the stern region Another neglected effect could be the additional friction resistance due to the accelerated flow between the sidewalls, but this rather explains additional resistance observed at larger distances At very small lateral distances, when the boundary layers on both sidewalls merge together, it results in a kind of additional hydraulic drag increasing the form resistance As a result of the two described phenomena, the suction first drops and at closer distances even transforms into repulsion However, as comparative computations showed (Fonfach et al 2011), in the present case and, likely, for all surface displacement ships, freesurface effects dominate: despite low Froude numbers based on the Fig Representation of hull for Ship 1: left—1392 panels; center—378 panels; right—120 panels Fig Representation of hull for Ship (compressed in longitudinal direction by factor 3): left—2114 panels; center—558 panels; right—172 panels SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 137 Copyright of Journal of Ship Research is the property of Society of Naval Architects & Marine Engineers and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use Journal of Ship Research, Vol 56, No 3, September 2012, pp 170–182 http://dx.doi.org/10.5957/JOSR.56.3.100022 Intelligent Quaternion Ship Domains for Spatial Collision Risk Assessment Ning Wang Marine Engineering College, Dalian Maritime University, Dalian, China In this article, a novel ship domain model termed intelligent quaternion ship domain (IQSD) has been proposed by effectively considering ship maneuverability and human factors Unlike previous ship domains, the proposed IQSD is much more dependable and flexible for navigators to make decisions The key characteristics are as follows: 1) the domain size is determined by the quaternion including radii, i.e., fore, aft, starboard, and port, which sufficiently take factors affecting the domain (i.e., ship maneuverability, speeds and courses, etc.) into account; 2) the domain shape is modeled by another parameter k, which makes the IQSD more flexible because the ship boundary could be not only linear or nonlinear, but also thin or fat; 3) furthermore, the navigator states including skill ability and physical and mental states have been effectively modeled by a fuzzy system, which determines the shape parameter, k To reasonably relate the IQSD to potential applications, i.e., collision risk assessment, collision avoidance and trajectory planning, etc., a generalized IQSD (GIQSD) with fuzzy boundaries has been developed by using fuzzy sets As a consequence, the GIQSD would become more practical and convenient for applications because uncertainty and fuzzy information has been merged into the GIQSD Furthermore, concepts of longitudinal and lateral risk based on the GIQSD have been defined to estimate the spatial collision risk (SCR) for ships encountered Finally, comprehensive simulations have been conducted on various encounter situations and comparative studies with typical ship domains have been intensively analyzed Simulation results demonstrate that the IQSD is more effective and flexible than previous ship domains, and the intelligent SCR based on the GIQSD is reasonable and dependable Keywords: quaternion ship domain; spatial collision risk; collision avoidance; human factors Introduction THE SHIP domain has been widely applied to the marine traffic engineering since the concept was first proposed by Fujii and Tanaka (1971) and Goodwin (1975) Afterward, some researchers (Davis et al 1980, 1982; Coldwell 1983; Zhao et al 1993; Kijima & Furukawa 2001, 2003; Smierzchalski 2001; Wilson et al 2003; Pietrzykowski & Uriasz 2004, 2006, 2009; Pietrzykowski 2008; Zhu et al 2001) have presented various ship domains with different shapes and sizes It is well known that ship domains play a very important Manuscript received at SNAME headquarters March 25, 2010; revised manuscript received June 30, 2012 170 SEPTEMBER 2012 role in risk assessment (Gucma & Pietrzykowski 2006; Szlapczynski 2006a, 2006b; Pietrzykowski 2008), collision avoidance (Hwang 2002; Pedersen & Inoue 2003; Wilson et al 2003; Kao et al 2007), marine traffic simulations (Lisowski et al 2000), and the optimal trajectory planning (Smierzchalski & Michalewicz 2000), etc Note that there are no systematic and flexible ship domain models so far although various ship domains have been proposed over the past 30 years Usually, the definition of ship domains is that described by Goodwin (1975): “the surrounding effective waters which the navigator of a ship wants to keep clear of other ships or fixed objects.” In light of this descriptive concept of ship domains, various geometrical ship domains with different shapes and sizes have been presented in related literatures 0022-4502/12/5603-0170$00.00/0 JOURNAL OF SHIP RESEARCH (Wang et al 2009) However, the main limitations of these models are as follows: • the previous ship domains lack an analytical presentation making its use in applications difficult; • these ship domains are considered as objective zones in the marine traffic rather than subjective areas, which navigators voluntarily want to keep because they are usually derived from VTS statistical data; • these geometrical ship domains are essentially stationary spatial models because they just make use of temporal information (i.e., the distance and relative bearing between the own and target ships) to judge whether the target ship violates the predefined area at that time; • and these models cannot be reasonably used for collision risk assessment, collision avoidance, and VTS systems, etc., because there are no analytical or systematic definitions of collision risk and decision-making, etc., on the basis of ship domains Another problem is that existing ship domains are mostly described in geometrical manners, which is easy to understand but difficult to be implemented in practical applications To circumvent this problem, a unified analytical framework for describing ship domain models has been presented by Wang, Meng, Xu, and Wang (2009) However, these transformed analytical ship domains are still inflexible Therefore, a reasonable and dependable ship domain is urgently required for systematic navigational safety Wang (2010) has proposed a novel ship domain model termed quaternion ship domain (QSD), in which the ship maneuverability is used to identify the ship domain scale However, human uncertainties and traffic densities are not considered in the QSD framework, whereby the shape parameter should be predefined by users In this article, we propose a novel ship domain model termed intelligent quaternion ship domain (IQSD) whereby the size is determined by four radii (i.e., fore, aft, starboard, and port) and the shape is defined by another index parameter, k, which could effectively model the navigator states In this case, these parameters could sufficiently take into account factors affecting the IQSD ship domains, i.e., maneuvering capability, human being factors, dimensions, courses and speeds of encountered ships, and COLREGS rules, etc Based on the IQSD, a generalized IQSD (GIQSD) with fuzzy boundaries is developed by using fuzzy sets, which can deal with uncertainty and fuzzy information in the real-life world The GIQSD is therefore a much more flexible ship domain construct Furthermore, an intelligent spatial collision risk (SCR) based on the GIQSD is presented by defining the longitudinal collision risk (LonCR) and the lateral collision risk (LatCR) Finally, computer simulations on several encounter situations are presented and comparative studies are comprehensively conducted in a uniform framework Simulation results demonstrate the validation and superiority of the proposed IQSD, GIQSD, and SCR based on the GIQSD The intelligent quaternion ship domain Note that previous results make it impossible to definitely and systematically identify the shape and size of a ship domain The main reasons are restated as follows: 1) factors considered in a certain ship domain are usually partially emphasized because it is difficult to consider all the factors that would affect the ship domain; and 2) the ship domain determination is usually obtained SEPTEMBER 2012 by different methods (i.e., statistic and analytical methods) based on various statistical data To circumvent these foregoing problems, an IQSD is proposed in this article to make it feasible for the practical application of the ship domain in the process of collision risk assessment and collision avoidance at sea, particularly in shipboard navigational decision support systems Unlike previous ship domains defined by circles, ellipses, and polygons, the proposed IQSD is identified by the quaternion consisting of four elements Rfore, Raft, Rstarb, and Rport, where Rfore and Raft indicate longitudinal radius Rlon of the IQSD in fore and aft directions, respectively, and Rstarb and Rport indicate lateral radius Rlat of the IQSD in starboard and port directions, respectively Consequently, the IQSD is an area defined by a closed curve joining these four elements and the inner zone, which can be described as follows: È È ÉÉ IQSD ¼ ðx; yÞj f ðx; y; QÞ 1; Q ¼ Rfore ; Raft ; Rstarb ; Rport ð1Þ where f(.) is the function defining the boundary of the IQSD and Q ¼ {Rfore, Raft, Rstarb, Rport} is the quaternion Intuitively, these four parameters make it more straightforward and more effective for a navigator onboard to establish a simple and feasible ship domain quickly As a consequence, the IQSD can be well used in collision risk assessment and avoidance, etc To be precise, the function f(.) in Eq (1) can be implemented in linear or nonlinear manners Note that polygons and ellipses are frequently used in previous developed ship domains Without loss of generality, the IQSD boundary can be illustrated in two shapes, i.e., the quadrangle and the combined ellipse, shown in Fig Vertices of the quadrangle and semiaxes of the combined ellipse are defined by longitudinal and lateral radii in the quaternion, respectively That is, Ri, i Î{fore, aft, starb, port} denote corresponding radii of the IQSD, shown in Fig To be specific, the flexible shape of the IQSD model could be parameterized by the shape index k According to Eq (1), the overall description of the IQSD is presented as follows: È È É É IQSDk ¼ ðx;yÞj fk ðx;y;QÞ 1;Q ¼ Rfore ;Raft ;Rstarb ;Rport ;k !  fk ðx;y;QÞ ¼ k 2x ð1 þ sgn xÞRfore À ð1 À sgn xÞRaft  k 2y þ ð1 þ sgn yÞRstarb À ð1 À sgn yÞRport where the sign function sgn(.) is defined as follows: & 1;x ! sgn x ¼ À1;x < ð2Þ ð3Þ ð4Þ Note that the convex boundary the IQSD is preserved for ship domains that navigators onboard and observers onshore build To effectively incorporate the navigator’s effect on ship domains, the navigator’s states, i.e., skill ability, physical, and mental states, are considered to determine the shape index k of the IQSD model To be specific, the shape index k could be intuitively defined by the following fuzzy system, Rule i: IF s1 is Ai1 ; s2 is Ai2 and s3 is Ai3 ; THEN k ¼ ki ð5Þ where, s1, s2, and s3 denote the navigator’s states (i.e., skill ability, physical, and mental states); Aij , i ¼ 1,2, ,n, j ¼ 1,2,3, are the jth fuzzy set of the ith fuzzy rule in the whole n fuzzy rules; and ki is JOURNAL OF SHIP RESEARCH 171 Fig (a) Quadrangle ship domain, (b) Combined ellipse ship domain the corresponding output It follows that the overall output of the shape index k could be described as follows: n kðsÞ ¼ ( fi ðsÞki i¼1 n ( fi ðsÞ ð6Þ i¼1 where, s ¼ [s1, s2, s3]T is the navigator state vector and wi is the firing strength of the ith fuzzy rule given by fi ðsÞ ¼ Y À Á Aij sj ð7Þ j¼1 In this case, it is obvious that the shape index k could be well identified by the navigator’s states and therefore determines the shape of the IQSD, whereas the quaternion Q (i.e., the radii of the IQSD Rfore, Raft, Rstarb, Rport) identifies the ship domain size As a consequence, we can reasonably model the IQSD by the parameters Q and k Estimation formulae for parameters of the blocking area (Kijima & Furukawa 2003) are used to estimate longitudinal and lateral radii in Q, which can be given by  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > þ ðk =2Þ2 L > ¼ þ 1:34 kAD R > fore DT > > >  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < þ ðk =2Þ2 L Raft ¼ þ 0:67 kAD DT > > > > Rstarb ¼ ð0:2 þ kDT ÞL > > : Rport ¼ ð0:2 þ 0:75kDT ÞL ð8Þ where L is the own ship length, kAD and kDT represent gains of the advance, AD, and the tactical diameter, DT, respectively, and can be calculated as follows: & kAD ¼ AD =L ¼ 100:3591lgVown þ0:0952 ð9Þ kDT ¼ DT =L ¼ 100:5441lgVown À0:0795 172 SEPTEMBER 2012 where Vown is the own ship speed represented in knots It should be noted that the IQSD reasonably takes into account the ship maneuvering capability identified by the advance, AD, and the tactical diameter, DT Besides, the ship speed, Vown, is used to determine the instantaneous maneuvering capability, which would be responsible for a reasonable ship domain In fact, AD and DT of the own ship can be easily obtained, whereas it is difficult to know these parameters of other ships encountered Therefore, for the sake of simplicity, kAD and kDT in Eq (8) can be alternatively predefined by navigators according to specifications onboard Intelligent quaternion ship domain-based collision risk assessment The boundary of the abovementioned IQSD divides the water area around a ship into two zones, dangerous and safe—ship domain and an area outside the domain, respectively However, the human being tends to distinguish the area around the ship to be more than two zones, which can be described in linguistic terms, very safe, safe, less safe, dangerous, very dangerous, etc Depending on these fuzzy terms, the navigator tries to maintain certain zones clear of other ships and objects, whereas these fuzzy terms can be modeled by fuzzy membership functions Therefore, the IQSD model with a clear boundary, which is ideal and limited in applications, can be extended to fuzzy ship domains with series of boundaries that indicate different navigational safeties On the other hand, the ship collision risk can be assessed by fuzzy ship domains because series of fuzzy boundaries can be used to identify the level of navigational safety or danger 3.1 The generalized intelligent quaternion ship domain Based on the IQSD, the GIQSD can be formulated as follows: GIQSDk ðr Þ ¼ fðx; yÞj fk ðx; y; QðrÞÞ 1; k ! 1g ð10Þ JOURNAL OF SHIP RESEARCH where, where,  k 2x fk ðx; y; Qðr ÞÞ ¼ ð1 þ sgn xÞRfore ðr Þ þ ð1 À sgn xÞRaft ðr Þ  k 2y þ ð1 þ sgn yÞRstarb ðr Þ þ ð1 À sgn yÞRport ðr Þ ð11Þ È É Qðr Þ ¼ Rfore ðrÞ; Raft ðrÞ; Rstarb ðr Þ; Rport ðrÞ ; < r < Ri ðr Þ ¼ ln 1r ln r10 ð12Þ !1k Ri ; i Îf fore; aft; starb; portg ð13Þ Here, r Î(0, 1) is used to indicate the collision risk of the corresponding fuzzy boundary of the GIQSD and k is the shape index similar to that of IQSD It should be noted that we get GIQSDk(r0) ¼ IQSDk because Q(r0) ¼ Q Usually, we set r0 ¼ 0.5 In other words, the IQSD is a special case of the GIQSD, which consists of series of fuzzy boundaries determined by collision risk levels 3.2 The generalized intelligent quaternion ship domain -based collision risk assessment According to the GIQSD, fuzzy set theory can be used to define a novel model of ship collision risk (or navigational safety) assessment To be precise, definitions of longitudinal and lateral collision risk based on the GIQSD are first presented to define the SCR, which combines longitudinal and lateral collision risk 1) LonCR: Based on the GIQSD, the longitudinal collision risk is presented to measure the collision risk in the longitudinal direction of the own ship It can be defined by the following fuzzy set where the membership function is given by the asymmetric Gaussian function as follows: si ¼  Ri 1=k ; i Îfstarb; portg ð17Þ ln r10 3) SCR: As a consequence, the combined SCR can be composed by the LonCR and the LatCR and is described as follows: SCRk ðx; yÞ ¼ LonCRk ðxÞ Á LatCRk ðyÞ  k 2x ¼ exp À ð1 þ sgn xÞsfore þ ð1 À sgn xÞsaft  k ! ! 2y þ ð1 þ sgn yÞsstarb þ ð1 À sgn yÞsport ð18Þ Comparing Eq (18) with Eq (11), we can find that Eq (11) can be deduced from Eq (18) if we set SCRk(x, y) ¼ r (0 < r < 1) In other words, the fuzzy boundaries of the GIQSD fk(x, y; Q[r]) ¼ indicate the corresponding spatial collision risk level SCRk(x, y) ¼ r It follows that the SCR could be steadily assessed on the basis of the proposed GIQSD model with ship maneuverability and navigator states effectively considered Simulation studies In this section, simulation studies on performance evaluation of the IQSD, GIQSD, and collision risk assessment based on GIQSD are comprehensively presented in two phases, i.e., IQSD and GIQSD validation, and GIQSD-based risk collision assessment 4.1 Intelligent quaternion ship domain and generalized intelligent quaternion ship domain validation To demonstrate relationships between the IQSD radii, Rfore, Raft, Rstarb, Rport, and the own ship speed, Vown, a numerical simulation has been conducted and results are illustrated in  k ! Fig 2, where the parameters are simply selected as follows: the 2x ship length L ¼ 160 m and the ship speed Vown varies from to LonCRk ðxÞ ¼ exp À ð1 þ sgn xÞsfore þ ð1 À sgn xÞsaft 20 knots As shown in Fig 2, it is evident that all the IQSD radii, Rfore, ð14Þ Raft, Rstarb, Rport, increase with the ship speed Compared with the where, radii of Fujii ship domain (Fujii & Tanaka 1971), which is considered as a simple and powerful ship domain all the time, the fore Ri and the port radii of the IQSD, Rfore and Rport, is similar to those of si ¼  ; i Î fore; af t ð15Þ f g 1=k Fujii domain when the ship speed is approximately 15 knots, ln r10 whereas the aft radius, Raft, is much less than that of the Fujii domain and the starboard radius, Rstarb, is slightly larger than that of the Fujii domain because the ship maneuvering capability and 2) LatCR: Similar to the LonCR, the lateral collision risk is encounter situations defined in COLREGS have been reasonably defined to measure the collision risk in the lateral direction of considered in the IQSD the own ship It is also defined by an asymmetric Gaussian For the validation of shape identification (shape index k) for membership function given by, IQSD and GIQSD, a fuzzy system, according to Eqs (5) through (7), is realized by considering two fuzzy sets defined by Gaussian  k ! membership functions for each navigator state, i.e., Aij Î{high, 2y LatCRk ðxÞ ¼ exp À low}, shown in Fig Eight fuzzy rules contributing to the resulð1 þ sgn yÞsstarb þ ð1 À sgn yÞsport tant fuzzy system for the estimation of the shape index k are listed ð16Þ in Table SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 173 Fig Comparisons of the intelligent quaternion ship domain (IQSD) radii, Rfore, Raft, Rstarb, and Rport, with longitudinal and lateral radii of Fujii ship domain Rlon(Fujii) and Rlat(Fujii) for various ship speed Fig Fuzzy sets of navigator states Table Fuzzy rules for identification of shape index k by considering navigator states Rule No s1 (physical state) s2 (mental state) s3 (skill ability) ki High High High High Low Low Low Low High High Low Low High High Low Low High Low High Low High Low High Low 4 174 SEPTEMBER 2012 To visibly illustrate the shape index k dependent on navigator states, we depict the output with respect to any two states as input variables and another state set to be 0.5, which is shown in Fig It can be seen that the outputs for shape index k are reasonable because the higher k coincides with conservative ship domain resulting from poor navigator states Based on the foregoing validation for the size and shape identification of ship domains, the evaluation of the overall IQSD and GIQSD models is conducted on various ship maneuverability and human being factors Without loss of generality, three typical navigator states, i.e., the worst, moderate, and best ones (s ¼ [0 0]T, s ¼ [0.5 0.5 0.5]T, and s ¼[1 1]T), are considered to determine the shape of IQSD and GIQSD, whereas the ship domain size varies JOURNAL OF SHIP RESEARCH Fig The output of shape index k with respect to skill ability s3 and physical state s1 of the navigator (mental state s2 set to be 0.5) with the typical ship speed 10, 15, and 25 knots The results are shown in Fig It can be seen that the ship domain size evidently becomes larger with increasing ship speed, whereas the shape of the IQSD is critically determined by the navigator states The results in Fig indicate that higher ship speed contributes to larger size of IQSD and worse navigator states would result in much more conservative domain shape, and vice versa To demonstrate the visible validation of the GIQSD, we simply fix the navigator states as the moderate level s ¼ [0.5 0.5 0.5]T and evaluate the performance varying with different typical ship Fig Validation results of the intelligent quaternion ship domain (IQSD) with different ship speed and navigator states SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 175 Fig Validation results of the generalized intelligent quaternion ship domain (GIQSD) with different ship speed and predefined navigator states s ¼ [0.5 0.5 0.5]T speed, 10, 15, and 25 knots The results are shown in Fig 6, from which we find that the GIQSD consists of series of IQSDs with sequential collision risk levels It is evident that the GIQSD with collision risk as 0.5 coincides with the corresponding IQSD, whereby the size and the shape also increase with increasing ship speed and decreasing navigator states Table The principal dimensions of own and target ships L (m) B (m) Cb Own Ship Target Ship 175.0 25.4 0.57 325.0 53.0 0.83 4.2 Generalized intelligent quaternion ship domain-based risk collision assessment In this phase, several encounter situations of own and target ships in risk of collision are made to demonstrate the effective and superiority of the GIQSD-based collision risk assessment method It is well accepted that encounter scenarios between ships include three types of situations, i.e., head-on, crossing, and overtaking, whereas crossing situations can be divided into before and abaft the beam All of the mentioned situations are considered in this article that the validity of the GIQSD can be comprehensively examined For comparative studies with other ship domains, Fujii and Tanaka (1971) and Goodwin (1975) models, numerical simulations are presented in the uniform simulation circumstance whereby the principal dimensions of own and target ships are listed in Table and initial conditions of ships in encounter situations are listed in Table 3, Table The initial conditions for own and target ships in encounter situations Own Ship Target Ship Scenarios Position (x, y) (n.m.) Speed (knots) Course ( ) Position (x, y) (n.m.) Speed (knots) Course ( ) Head-on Crossing before the beam Crossing abaft the beam Overtaking (0, 0) (0, 0) (0, 0) (0, 0) 15 15 15 20 0 0 (3, 0.1) (3, 1) (0, 2) (1.2, 0.1) 12 12 20 12 180 220 320 176 SEPTEMBER 2012  JOURNAL OF SHIP RESEARCH respectively For clear comparisons, the representative Fujii and Tanaka (1971) and Goodwin (1975) ship domains are considered in this article Note that Fujii ship domain features ellipsoidal boundary and Goodwin ship domain consists of three fan-shaped sectors with different radii Nevertheless, the size and shape of them are fixed with a specific ship regardless of ship speed and navigator states varying 4.2.1 Head-on situation The target ship comes nearly toward the own ship from the front While these two ships approach closer and closer, different ship domains violated sequentially are shown in Fig It is evident from Fig that the Goodwin and Fujii models are the most conservative and risky ship domains, respectively, whereas the GIQSD model provides much more appropriate and reasonable ship domains than other models Moreover, Fig 8a shows in detail the points where various ship domains would be invaded while the distance between two ships decreases It also shows that the GIQSD model is moderate among these considered ship domains Therefore, the GIQSD model can be potentially used for collision risk assessment, collision avoidance, and others alike As shown in Fig 8b , the SCR combined by LonCR and LatCR can be obtained based on the GIQSD We can see that the own ship SCR and the target SCR are identical, whereas the target SCR is much larger than the own SCR because the target ship is larger than the own ship In other words, the SCR varies reasonably with different ships because the maneuvering capability is considered in the GIQSD 4.2.2 Crossing situation before the beam On this situation, the target ship is positioned before the beam of the own ship at the beginning and crosses the own ship trajectory shown in Fig 9, from which simulation results are similar to those of the head-on situation Goodwin, GIQSD, and Fujii ship domains are invaded sequentially in the process of crossing-type encounter situation It is evident that the GIQSD model provides the most reasonable ship domains compared with Fujii and Goodwin models Furthermore, Fig 10a shows in detail the points where various ship domains would be invaded, whereas the target ship approaches to the own ship on this situation We can draw the same conclusions that GIQSD models are moderate among these considered ship domains Simulation results of corresponding SCR-based various ship domains are shown in Fig 10b, whereby the GIQSD-based SCR performs in a continuous manner rather than discrete one In this case, the GIQSD-based SCR would preserve flexible and reasonable risk assessments 4.2.3 Crossing situation abaft the beam In this situation, the target ship is positioned abaft the beam of the own ship at the beginning and crosses the own ship trajectory Simulation results are shown in Figs 11 and 12 , respectively It is evident that the superiority of the GIQSD to Fujii and Goodwin ship domains is well demonstrated in this scenario because the GIQSD is not only nonconservative, but also secure for navigators at sea or VTS onshore To the contrary, the Fujii and Goodwin ship domains tend to be much more risky and conservative, respectively Similarly, the SCR based on the GIQSD in this situation is also reasonable for risk assessment and collision avoidance because the GIQSD provides not only flexible risk assessment, but also differences between the SCR of own and target ships The main reason is that the maneuverability and COLREGS with respect to different ships has been incorporated into the GIQSD 4.2.4 Overtaking situation In this situation, the target ship overtakes the own ship from the stern Simulation results are Fig The generalized intelligent quaternion ship domain (GIQSD) model compared with other ship domains on head-on encounter situation SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 177 Fig (a) The points where ship domains are invaded while distance between ships decreases; (b) the spatial collision risk on the head-on encounter situation Fig The generalized intelligent quaternion ship domain (GIQSD) model compared with other ship domains on the encounter situation of crossing before the beam 178 SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH Fig 10 (a) The points where ship domains are invaded while distance between ships decreases, (b) the spatial collision risk on the encounter situation of crossing before the beam Fig 11 The generalized intelligent quaternion ship domain (GIQSD) model compared with other ship domains on the encounter situation of crossing abaft the beam SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 179 Fig 12 (a) The points where ship domains are invaded while distance between ships decreases, (b) the spatial collision risk on the encounter situation of crossing abaft the beam Fig 13 The generalized intelligent quaternion ship domain (GIQSD) model compared with other ship domains on the overtaking encounter situation 180 SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH Fig 14 (a) The points where ship domains are invaded while distance between ships decreases, (b) the spatial collision risk on the overtaking encounter situation shown in Figs 13 and 14 , respectively It can be seen that simulation results coincide with the actual situations Note that the GIQSD is reasonable for this scenario According to the COLREGS, the target ship does not need to take actions unless critical situations appear Therefore, the GIQSD is appropriate and advisable for this situation From Fig 14, we can see that the target SCR is very identical to the own SCR before the maximum SCR It means that the own and target ship possess equal responsibility for collision avoidance in the overtaking situation, which coincides with practical engineering It should be highlighted that the mentioned advantages profit from the flexibility of the GIQSD with asymmetric radii, which consider many factors (i.e., encounter situations, maneuvering capability, and COLREGS rules, etc.) Note that the target SCR is much larger than that of the own SCR when the own ship overtakes the target ship because the aft domain of the own ship is much smaller than the fore domain of the target ship Actually, there is no risk in collision for these two ships at that time because the own ship speed is much larger than that of the target ship Conclusions In this article, a novel IQSD has been proposed to realize a universal and reasonable ship domain model for potential applications to collision risk assessment, collision avoidance at sea, and VTS systems, etc Unlike various forms of ship domains defined SEPTEMBER 2012 by circles, ellipses, and polygons, the proposed IQSD model is parameterized by the quaternion Q and the index k, where parameters Q and k determine the size and ship of the IQSD It should be noted that some factors (i.e., maneuvering parameters, dimensions, courses and speeds of encountered ships, and COLREGS rules, etc.) affecting the ship domain have been well taken into account by four elements, i.e., ship domain radii in the directions of fore, aft, starboard and port, Rfore, Raft, Rstarb, Rport, in the quaternion Q, whereas the parameter k enhances the flexibility of the IQSD shape To effectively consider the influence of navigator states on ship domains, the shape index k is modeled by a fuzzy system with skill ability and physical and mental states of navigators as inputs Moreover, a GIQSD with fuzzy domain boundaries has been presented by using fuzzy sets As a consequence, the GIQSD is much more practical than the IQSD because uncertainty and fuzzy information can be well used in the GIQSD Furthermore, LonCR and LatCR based on the GIQSD have been proposed to implement the SCR, which can be used for risk assessment, collision avoidance, and trajectory planning, etc Comprehensive simulations on different encounter situations have been presented for the validation of the IQSD and GIQSD models and the analysis of comparative studies with other typical ship domains Simulation results show that the IQSD and the GIQSD models are effective and superior to other ship domains It follows that these simulation studies indicate that the IQSD, GIQSD, and SCR based on the GIQSD can be well used in collision risk and avoidance Note that collisions are the major cause of shipping JOURNAL OF SHIP RESEARCH 181 losses Therefore, there is no doubt that research results in this work would be widely applied to ship navigation in the future However, there are still some problems to be dealt with in this framework of ship domains for collision risk and avoidance Future work needs to determine the collision avoidance characteristics based on the proposed SCR and further estimate a safe trajectory of the ship Acknowledgments The author thanks the Editor-in-Chief, 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Vol 3, RC-31-1RC-31-9 YEUNG, R W 1978 On the interaction of slender ships in shallow water, Journal of Fluid Mechanics, 85, part 1, 143159 YEUNG, R W., AND TAN, W T 1980 Hydrodynamic interactions of ships with fixed obstacles, J Ship Research, 24, 5059 JOURNAL OF SHIP RESEARCH 145 Copyright of Journal of Ship Research is the property of Society of Naval Architects & Marine Engineers and its content may... potential flow, Ship Technology Research, 40, 176188 SOăDING, H., AND CONRAD, F 2005 Analysis of overtaking manoeuvres in a narrow waterway, Ship Technology Research, 52, 189193 SUTULO, S., AND GUEDES SOARES, C 2002 An algorithm for optimized design of maneuvering experiments, Journal of Ship Research, 46, 214227 JOURNAL OF SHIP RESEARCH SUTULO, S., AND GUEDES SOARES, C 2008 Simulation of the hydrodynamic... computation loop of a typical bridge simulator JOURNAL OF SHIP RESEARCH Fig 9 Interaction forces and moment on Ship 1 in shallow water: depth H1, FnH ẳ 0.188, xs ẳ 0.014 (a) surge force, (b) sway force, (c) yaw moment; left, influence of the number of panels; right, influence of the number of reflections SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH 141 Fig 10 Interaction forces and moment on Ship 2 in shallow... for individual use Journal of Ship Research, Vol 56, No 3, September 2012, pp 183196 http://dx.doi.org/10.5957/JOSR.56.3.110018 Near-Field Hydrodynamics of a Surface-Effect Ship Lawrence J Doctors The University of New South Wales, Australia A consistent near-field linearized potential-flow analysis of a surface-effect ship traveling at a steady speed in laterally restricted water of finite depth is... 2 17ị 2.6 Influence of seals A good introduction to the behavior of bow seals, consisting of a set of fingers, was published by Malakhoff and Davis (1981) Their description of the mechanics of deflection under the impact of the water flow is particularly helpful Figure 3a illustrates an idealization of the bow seal Most of the complexities of the distorted three-dimensional shape of these fingers, when... J Hydronautics, 13, 4, 113119 ZALEK, S F., AND DOCTORS, L J 2010 Experimental Study of the Resistance of Surface-Effect Ship Seals, Proceedings, 28th Symposium on Naval Hydrodynamics, Pasadena, California, 17 pp, September 2010 JOURNAL OF SHIP RESEARCH Copyright of Journal of Ship Research is the property of Society of Naval Architects & Marine Engineers and its content may not be copied or emailed... speed of 20 knots, (c) speed of 30 knots, (d ) speed of 40 knots, (e) speed of 50 knots 6 Predictions for wave profiles The final set of results is shown as the five parts of Fig 11, corresponding to nominal speeds of 10 to 50 knots These are all for the medium displacement of 1888 t In each case, the following SEPTEMBER 2012 curves are plotted: the measured free-surface elevation, z, as a series of symbols,... expressed for the management of this project by Prof Armin W Troesch and Dr Kevin J Maki in the Department of Naval Architecture and Marine Engineering at the University of Michigan References BARRATT, M J 1965 The wave drag of a hovercraft, Journal of Fluid Mechanics, 22, Part 1, 3947 JOURNAL OF SHIP RESEARCH 195 BHUSAN, S., STERN, F., AND DOCTORS, L J 2011 Verification and validation of URANS wave resistance... force, (b) sway force, (c) yaw moment; left, influence of the number of panels; right, influence of the number of reflections 142 SEPTEMBER 2012 JOURNAL OF SHIP RESEARCH Fig 11 Interaction forces and moment on Ship 1 in shallow water: depth H2, FnH ẳ 0.25, xs ẳ 0.61 (a) surge force, (b) sway force, (c) yaw moment SEPTEMBER 2012 Fig 12 Influence of elementary maneuvering motions on interaction forces... investigation of ship- bank interaction forces, Proceedings, International Conference on Marine Simulation and Ship Maneuverability, August 2528, Kanazawa, Japan, Vol 3, RC-31-1RC-31-9 VANTORRE, M., VERZHBITSKAYA, E., AND LAFORCE, E 2002 Model based formulations of ship- ship interaction forces, Ship Technology Research, 49, 124140 SEPTEMBER 2012 VARYANI, K S., KRISHNANKUTTY, P., AND VANTORRE, M 2003 Prediction of