Ocean Engineering 160 (2018) 301–310

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Ocean Engineering

journal homepage: www.elsevier.com/locate/oceaneng

An alternative method for computing hydrostatic performances of a floatingbody with arbitrary geometrical configurations

Jie Tong a, Tsung-Yueh Lin b,*, Min-Mei Shih c, Jen-Shiang Kouh a

a Ocean College, Zhejiang University, Zhoushan, 316021, Chinab Research Department, CR Classification Society, Taipei, 10487, Taiwanc Department of Design, CSBC Corporation, 81234, Taiwan

A R T I C L E I N F O

Keywords:Hydrostatics and stabilityStatic pressure integrationSurface triangulationTrimaranFloating kuroshio turbine

* Corresponding author.E-mail address: tylin@crclass.org (T.-Y. Lin).

https://doi.org/10.1016/j.oceaneng.2018.03.036Received 26 September 2017; Received in revised

0029-8018/© 2018 Elsevier Ltd. All rights reserved

A B S T R A C T

In the fields of naval architecture and ocean engineering applications, floating platforms or multi-hull vessels withvery different geometrical appearance have been designed to meet their mission requirements. Many objects werebuilt by composing a number of sub-objects in an arbitrary way because of the high flexibility available for theconstruction work. Because the composition of the sub-objects normally are not arranged along a certain directionin a fixed sequence, it could be sometimes troublesome for computing the hydrostatic data of such objects. Inorder to facilitate this computation more easily and flexibly, a method has been developed in this paper, whichwas derived from the simple principles of an exact pressure integration over triangles for getting the totalbuoyancy force vector and the static equilibrium condition between the buoyancy force and the weight of thefloating object. The triangles thereby were generated by triangulation of the surfaces representing a wholefloating object. Finally, applications on a high-speed trimaran hull and a floating Kuroshio current turbine wereconducted for demonstrating the merit of this method. The placement of water compartments and free surfaceeffects were further analysed to evaluate the changing ballast conditions for hydrostatic and transitionalstabilities.

1. Introduction

Ship terminology defined a hull geometry in three orthogonal di-rections, bodyplan, waterplan, and sheerplan. These plans are related tostations, waterlines, and buttocks correspondingly, on which an offsettable is constructed. It is a common format to define a hull geometry andto analyse its hydrostatics and stability performance. The analysis in-volves area and volumetric properties and also static properties such asmoment, moment of inertia, and centre of buoyancy etc., so that a nu-merical integration method for processing geometries is needed. Justbefore the fast development of computer technology in recent decadesdifferent integration methods have been widely applied, such as trape-zoidal or Simpson's methods, which can be easily founded in a series ofwell-known naval architecture textbooks such as (Barrass and Derrett,2006), (Biran and Pulido, 2013), (Dudszus and Danckwardt, 1982),(Kobylinski and Kastner, 2003), (Lewis, 1988) and (Rawson and Tupper,2001). These methods are applicable to regular shapes, whose geomet-rical changes along a longitudinal direction are smooth. In addition, theshape has to be presented in a well-structured format. An offset table for

form 21 January 2018; Accepted

.

ships can meet these requirements. However, the main deficiency of thisintegration approach is that the volumetric properties are calculated byintegrating area properties, which are again by integrating line proper-ties. The double integral operation can accumulate errors from insuffi-cient resolution of the offset table and from the numerical methods. Somodern ocean structures involving abrupt geometrical changes, forinstance multi-hull vessels and ocean platforms, suffered difficulties toevaluate their performances in the traditional way. Nevertheless, mostoffshore units are structured by simple mathematical geometries, forexample rectangular box, cylinder, sphere and prism. Analytical solutionsfor area and volumetric properties (Bronshtein et al., 1997) exist for suchgeometrical components that the whole properties can be calculated bysummation or parallel axis theorem (Paul, 1979). But this is not the casefor other shapes beyond simple mathematical expressions.

Also in the case of determining the transverse statical stability, anapproach by making a distinction of small and large angles of heel isusually given in the aforementioned textbooks. The reason to take theapproach is due to the fact that the waterlines with different heel anglesmay cut the stepwise cross-sections in different manners. If the heel angle

11 March 2018

Fig. 1. Surface triangulation.

Fig. 2. Triangle-plane intersection.

J. Tong et al. Ocean Engineering 160 (2018) 301–310

is small, the resulting intersection point of a waterline and the ship sidecan be assumed to be vertical wall-sided. In this case, a simplified rela-tionship can be easily derived between the transverse stability and theheeling angle. For large angles of heel, precise results can also be gottenby more elaborately formulated equations. This method shows however adisadvantage with two separate steps, which is kind of annoying. Someanalytical formulation for arbitrary 2D shape and implementations inMATLAB (Wu, 2005; Duan et al., 2015) have been done and can provideaccurate results, However a more general three-dimensional methodwith single integrated computing process is inevitable to calculate hy-drostatic properties of an arbitrary configuration of geometry.

On the other hand the determination of floating states, for bodies,subject to external loads in some scenarios requires a root finding pro-cess. In other words, this process finds the position and orientation of awatertight body, so that all forces and moments are balanced. Usually thematrix methods was used to calculate the floating states (Zhao & Lin,1985; Kopecky, 2007). For each iteration, the waterplan area, centre offloatation, moment of inertia, displacement volume and centre of buoy-ancy are calculated and filled in the Jacobian matrix. Not only thecomputational workload is high but also those geometrical propertiesmight suffer inaccuracies in the traditional approach as described.

Apart from the gradient-based solver, another method of nonlinearprogramming used to calculate the floating state of ships was proposed(Ma et al., 2003, 2007). It established the absolute value of the totalrecovery arm as the objective function to ensure that the displacement isequal to the weight as the constraint condition, and the optimization ofthe mathematical model for the design variables is the draft. Comparedwith the traditional matrix method, the method in each iteration does notneed to calculate the surface properties, instead just calculating the tiltsof displacement volume and centre of buoyancy, and hence reducingamount of calculation greatly, but the nonlinearity of hydrostatic prop-erties requires some remeshing techniques (Lee & Lee, 2016). Otherstreated this optimization problem by the genetic arithmetic (Lu et al.,2005, 2006; Jin et al., 2007), in which the free float calculation is sum-med up as a multi-objective constraint optimization problem. Accordingto the free floating condition of ships, since it is based on the calculationof surface expression, it does not need a given initial iteration point, onlythe total weight and centre of gravity of the ship are need. In addition, ituses the draught, pitch angle and roll angle as the design variablesdirectly without the calculation of the tangent value of the dip angle.Compared with other iterative methods, it is more accurate than themethods based on two-dimensional representation.

To summarize the drawbacks of current geometrical process for hy-drostatics are the erroneous double integration of offset tables andlimited analytical solutions for mathematical shapes. The former furtherreduces the performance of determining floating states. The presentmethod derives the analytical solution to hydrostatic pressure for trian-gulated surfaces, which avoids numerical integration and breaks thelimitation of mathematical shapes. The overhead of the proposed methodis the additional triangulation operation which is another topic aboutsurface grid preparing, and to present large numbers of triangles of abody requires relatively larger files to store and also higher memoryusage.

2. Methodology

2.1. Surface triangulation and intersection

In computer-aided geometry modelling, a surface is usually presentedin parametric form, i.e. u and v direction in unit domains (Piegl andTiller, 2013), (Gallier, 2000). Isoparametric curves are easy to extractfrom a surface by holding one parameter constant. Simply interleave theisoparametric curves in u and v directions a structure grid can be ob-tained. Triangulating the structured grid into triangle panels is con-necting diagonal vertexes of each cell, as shown in Fig. 1. Advancedcontrols such as aspect ratio, maximum edge length, and maximum

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deviation from surface are on demand. The closed hull surface andcompartment geometry are triangulated with normal directions pointinto fluids.

Geometrically speaking, the waterplans of a floating body and the freesurface inside a compartment in calm water are intersections of a flatplane and 3D closed surfaces. Since these surfaces are triangulated, it isthe problem that intersects triangles with a plane. Four conditions isillustrated in Fig. 2 and an intersection exist in case II and case III, whereh represents the immersed or above depth in/out of water. An intersectedsegment must be a straight line starts and ends on the edges of the tri-angle, which are also straight lines. So the calculation of the end points ofthis line is simply by calculating an intersection of a plane and line seg-ments of the triangle edges. After obtaining the end points on the edges incase III, the immersed triangle is a special case of case I that one edge lieson the plane. For case II the immersed part is a quadrilateral, which canbe further divided into two sub-triangles, both reduced to case I. So theimmersed part of a triangulated surface can be represented as a set oftriangles, which will be resolved hydrostatic pressures. A special casethat a triangle lies perfectly on the plane, the case V in Fig. 2, andgenerate a jump of waterplan change will be specifically treated in thefollowing section. This condition does not effect on the pressure since itswater head is zero.

The outcome of the intersection of a plane and a closed meshedsurface is one or several planar polygons, as shown in Fig. 3. The polygoncontour is constructed by N vertices and N linear segments, and the x andy are the coordinates of vertices. The area properties, including enclosingarea A, centre of polygon Cx and Cy, and moments of inertia about thepolygon centre in X and Y directions Ix and Iy, are analytically calculatedby Eqs. (1)–(3). The values in Y direction are obtained by swapping theterms in the first parenthesis: x to y in Eq. (2) and y to x in Eq. (3). Theseequations are derived from 2D triangle properties, which constitutes thebasic element of a polygon.

A ¼ 12

�����XN�1

i¼0

ðxiyiþ1 � xiþ1yiÞ����� (1)

Fig. 3. Simple planar polygon.

J. Tong et al. Ocean Engineering 160 (2018) 301–310

Cx ¼ 1 XN�1

ðxi þ xiþ1Þðxiyiþ1 � xiþ1yiÞ (2)

6A i¼1

Ix ¼ 112

Xi¼N�1

i¼1

�y2 i þ yiyiþ1 þ y2 iþ1

�ðxiyiþ1 � xiþ1yiÞ (3)

2.2. Hydrostatic pressure

The overall pressure of the floating body is by summing up pressure ofeach immersed triangle j, which can be formulated in its parametric formin 3D space. A position vector r on the triangle is expressed as Eq. (4),where u and v are parameters defined in [0,1] and [0,1-u] domains, r0, r1,and r2 are the vertices of the triangle, r01 and r02 are vectors from r0 tothe other two vertices. The pressure acting on r is proportional to thewater depth from free surface, of which elevation is zero in the presentsetup. The water density is ρ. So the water head is the Z component of r,denoted a scalar rz. Integrating the pressure over the triangle domain andget Eq. (5). The pressure force can be calculated simply by multiplyingthe depth of the centroid of the triangle rc and triangle area. We can gothrough a lengthier derivation of the moment and enforcing it zero tofind the position where an equivalent point pressure force acting on, as inAppendix. This result is written in Eq. (6), and the terms in the bracketsare defined as the force centre rf as shown in Fig. 4.

r!ðu; vÞ ¼ r!0 þ r!01uþ r!02v (4)

P!

j ¼ ∬A

�ρgrðu; vÞz

�n!j dAj ¼ ρgrcz n!jAj (5)

Fig. 4. Parametric definition and force centres.

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M!

j ¼ ρgAj⋅1

12rcz

�9rcz r

!c þ r0z r

!0 þ r1z r

!1 þ r2z r

!2

�� n!j

¼ ρgAj⋅1

12rczr!f � n!j (6)

2.3. Statical stability

The calculation of hydrostatic data of a ship is necessary to obtain itshydrostatic performance. The calculations include areas, centroids, mo-ments of inertia of areas, volumes, and centres of volume. These prop-erties are plotted as functions of draught in curves, which aredisplacement V, longitudinal centres of floatation LCF and buoyancy LCB,waterplan area Aw, vertical centre of buoyance VCB, and metacentre KM,KML. Displacement is calculated by Archimedes' principle, which thedisplaced fluid weight is equal to the total pressure force, calculated byEq. (5). Area properties of floatation plane is the intersection contour ofthe geometry set of triangles and a flat plane. Eq. (7) is used to calculatemetacentre radii BMT and BML respectively in longitudinal and trans-verse directions, where Ix and Iy are the moments of inertia of floatationplane already defined in Eq. (3).

BMT ¼ IxV; BML ¼ Iy

V(7)

However, the above equation is derived based on the assumptions ofsmall angle inclination and on the vertical wall assumption. Fig. 5 definesthe inclination angle ө, wall tangent angle φ and half breadth y0 at thewaterplan intersection. For arbitrary geometries, the vertical wall con-dition is seldom met, so it is necessary to derive the applicability of Eq.(7). For non-vertical wall condition, i.e. φ is not a right angle, it is goingto prove the area of the shaded triangle approaches zero for small angleinclination, ө tends to0. In the enlarged view of this shaded triangle inFig. 5 the sine rule states that the triangle area is:

Area ¼ 12:y0tanθ:S:sinð90� þ θÞ ¼ 1

2⋅y0tanθ⋅y0tanθ

sinð90� � φÞsinðφ� θÞ ⋅sinð90� þ θÞ

¼ 12⋅y02tan2θ

cos θ cos φsin φ cos θ � cos φ sin θ

(8)

For small angle inclination ө, tan(ө)→ ө, sin(ө)→ ө, and cos(ө)→1, so

Area→12⋅y02θ2

cos φsin φ� θ⋅cos φ

(9)

The vertical wall condition, φ¼ 90�, the area directly becomes zero,which is the basis of deriving equation (7). Also the fraction is finite tojustify the area approaches zero, except φ¼ 0�, the horizontal tangent. If

Fig. 5. Non-vertical wall condition.

J. Tong et al. Ocean Engineering 160 (2018) 301–310

we take this condition in the above equation:

Area→12⋅y02θ2

1�θ

¼ �12⋅y02θ (10)

So even at the extreme, the area will not be singular; instead it be-comes zero when applying the small angle inclination. To conclude,equation (7) is applicable at any occasion as the waterplan exists.

This implies that the waterplan properties should always be singlevalued, and hence the volumetric properties are first-order differentiable.If all the triangles do not perfectly lie on the horizontal plane at anyoccasions, the described requirement is satisfied, and then the waterplanproperties can be calculated. Otherwise the program will give a warningmessage and calculate the average values of both conditions of existenceand nonexistence of perfectly on-plane triangles. The averaging treat-ment guarantees the applicability of Euler's method for solving balancecondition.

2.4. Free surface effect of internal fluids

To deal with partially filled compartments two problems should beaddressed. The compartment model is the same as the triangulated sur-face but in the normal direction inwards, so that the hydrostatic pro-cedure is identical as prescribed except the internal waterplan elevationand the containing fluid density ρin are separately configured accordingto filled rate of each compartment. When compartments are filled withcertain amounts of fluids, the total weight and centre of mass arechanged, so that a new equilibrium should be found. This is done in aniterative manner since the centres of mass of compartments are alsofunctions of the trim angle. The iteration algorithm is by solving Euler'smethod, updating sinkage and trim step by step. Within each step, bi-section method is used to find the fluid level of each compartment fora given volume and trim angle.

After the equilibrium is achieved, the free surface effect is then cor-rected. During the procedure to find the internal fluid levels, the inter-section contour of an inclined plane and compartment geometry, also aset of triangles, is found. Then the moments of inertia of this small freesurface plane about transverse ix and longitudinal axis iy are calculated.The virtual centre of gravity is calculated according to Eq. (11). Thelocation of a compartment affects the total centre of gravity, while thefree surface correction only depends on the shape of the free surface inthe compartments.

GVT ¼ ρinixρV

;GVL ¼ ρiniyρV

(11)

3. Computational examples

3.1. Validation with trimaran

In the presented study a 100-m high-speed trimaran is selected as avalidation example. The benchmark was conducted by using commercialsoftware Orca3D on Rhinoceros CAD platform to evaluate its hydrostaticsagainst various draft and stability curves at design condition. The prin-cipal dimensions of this vessel is listed in Table 1. At its design draft,which submerges the side hulls partially, the ship weight is 1400 tons.The baseline of side hulls is 2.5m above the keel line and the bottomstern shape of the main hull is a wedge. The two geometrical features willmake the hydrostatic curves bent very sharply alone draught. The origin

Table 1Principal dimensions Of trimaran.

Principal Dimensions DesignDraft

DesignSpeed

LOA BOA LWL GM Weight Depth

102m 27.5 m 98.7m 7.2 m 1400tons

13m 3.3m 40 kn

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is set at the transom stern of the main hull towards bow. A horizontalcylinder compartment at midship in the main hull, shown in Fig. 6. Thevolume of the cylinder is 317m3. The whole hull form is triangulated into7150 vertexes and 11854 panels, and the compartments are done in thesame way as well. The cylinder compartment (blue) uses 1225 vertexesand 2256 triangles.

The trimaran's hydrostatic properties are calculated at its even keelcondition. The draught is from 0.8m to 7.0m, with 0.1m stepped. Thelongitudinal metacentric height KML, waterplan area Aw, and displace-ment V are drawn in dashed curves in Fig. 7, reading the upper scaleunits, while the other four, LCF, LCB, KB, and KM are drawn in solidcurves, reading the lower scale units. Black squares are the three dashedvalues computed by Orca3D and black crosses are four solid values. Tworesults agree with each other very well. The differences across the entiredraft domain are less than 1%, except the displacement is slightly over-predicted at deep draught. This might be due to the different methodscomputing the displacement: Orca3D follows integrating waterplan areasrespect to draft, which is prone to accumulate errors to deeper draught.On the other hand, the present method uses analytical solution to obtainbuoyant volume.

Back to the characteristics of this trimaran, there are two sharp cuspson the KML curve at 1.5 m draught. This is the critical draught depth thatbreaks the waterplan into two contours, as shown in Fig. 8. The stern areaof the wedge abruptly occurs and also increases waterplan area sharply.Another effect is the longitudinal centre of floatation moves toward stern,which also makes a bent on LCF. The other unsmooth position is at 2.5m,where the side hulls begin to touch water. Because the side waterplansare placed away from center line, small areas can create large moment ofinertias transversally. On the transverse metacentre curve, KM, the valuedramatically increases after 2.5m draught and shows a trimaran's char-acteristics of high stability at its maximum of 19m. The volumetricproperties, KB, LCB, and V, in contrast, are smooth.

The second validation metric is the statical stability curve at designcondition. The initial displacement is 1400 tons, LCG was set as LCB tokeep even keel position. The heel angle was calculated from 0� to 90�,stepped 2.5�. During the evaluation process displacement and LCG werekept constant, so that sinkage and trim were dynamically balanced undervarious heel angles. Fig. 9 compares the results of present method (solidand dashed curves) and Orca3D (crosses and squares). The righting leverin meters were plotted on the left axis, and the trim angle in degrees wereon the right. We can see very high accuracy of the present method: themaximum difference of righting lever is less than 0.7%, and of trim angleis 1.5% lower than Orca3D only at high heel angles. The trim angle is theoutcome of balanced of LCG and LCB, for the latter which is also a

Fig. 6. Trimaran layouts.

Fig. 7. Hydrostatic curves.

Fig. 8. WATERPLANE at 1.5 m DRAUGHT.

Fig. 9. Statical stability curve at design draft.

Fig. 10. LCG, trim, sinkage.

Fig. 11. Vertical centres and virtual VCG.

J. Tong et al. Ocean Engineering 160 (2018) 301–310

volumetric property. The present method uses analytical solution toresolve center of buoyant force rf without erroneous integrationapproach.

Fig. 12. Fkt components.

3.2. Ballast conditions

The design ship centre of gravity is set at (35.78m, 0.0m, 7.5m)according to its design specifications at even keel condition. The lightship weight is 1400t. Configurations of the compartment filled ratio isfrom zero (empty) to 1.0 (full), 0.1 stepped. Because of a known totalvolume of this compartment, water level inside is solved at a given vol-ume of fluid and then updating the overall centre of gravity to find theequilibrium trim angle until the pitch moment converges to zero.

Fig. 10 plots trim angle, sinkage, and overall longitudinal centre ofgravity versus filling ratio. Since the compartment is located in front ofmidship, the ballast water causes LCG moved forward and further trim-med by bow. Sinkage increased due to the increment of the ballast.Fig. 11 plots VCG, VCB, KM, and free-surface effect KGv. The ballast alsolowed the centre of gravity down due to the location of the compartment.Increment in sinkage resulted in the raise of buoyancy centre. Themaximum free surface effect occurred at about half-filled ratio, where the

305

free surface area is maximum. The GMeff though is decreased by theballast, but the overall effect of ballast increased GM.

3.3. Floating Kuroshio Turbine

To demonstrate the capability of the present method for highlyirregular multibody, the floating Kuroshio turbine (FKT) is an incentivedesign of the offshore renewable energy program in Taiwan. FKT isdesigned for avoiding environmental loads during the typhoon season.Fig. 12 shows its construction components and Table 2 lists its principaldimensions. FKT is composed of four parts: nacelle (white), rotors (or-ange), buoy (blue), and struts (yellow and red). FKT is moored at the tipsof the nacelle to the seabed, which means the rotors are downwind. The

Table 2Principal dimensions of fkt.

Principal Dimensions RatedPower

DesignSpeed

LOA Span H Disp. Weight Diameter

4.0 m 8.0 m 4.0 m 24m3 16 tons 5.0 m 20 kW 1.5m/s

Fig. 14. Force diagram under operation.

Table 3Hydrodynamic force of fkt.

Trim Drag (kN) Lift (kN) Trim Moment (kN-m)

�20� 11.73 �41.49 �64.80�10� 4.83 �21.59 �25.58�5� 2.99 �11.99 �9.58�2.5� 2.48 �6.07 �2.280� 2.31 0.00 0.002.5� 2.48 6.32 10.725� 3.00 12.42 6.0710� 4.79 22.52 23.8020� 11.58 43.94 26.04

Table 4Compartment filling ratios of fkt ballast control.

Ballast Condition (a%/b%/c%) Weight (t) LCG KG Trim (deg)

0%/0%/0% 16.0 0.000 0.000 0.2610%/0%/0% 17.6 �0.156 0.337 0.6820%/0%/0% 19.2 �0.287 0.642 1.1210%/0%/10% 19.2 �0.073 0.623 5.090%/0%/20% 19.2 0.142 0.642 8.370%/0%/10% 17.6 0.077 0.342 4.33

J. Tong et al. Ocean Engineering 160 (2018) 301–310

shape of the buoy is a large foil and has nine compartments, as shown inFig. 13. By controlling seawater pumping in/out/across of the compart-ments, FKT is allowed to adjust its immersion depth and trim angle, sothat it not only keeps even position but dives deeply to prevent wavesinduced by typhoons and floats for maintenance operation. The foil shapeof the buoy also provides hydrodynamic lift forces under uneven posi-tion. The proposed hydrostatic method was extended further for FKT,which is subjected to mooring and hydrodynamic forces. In additional tothe operation condition, the transitional stability during the launchingprocess is evaluated in the following sections.

Under normal operation and steady Kuroshio current, FKT is expectedto align with the current direction without yaw and roll due to itsdownwind design. This simplified the degree of motion to 2D vertical-plane motion, i.e. surge, heave, and pitch. Steady position and trimangle are then the result of balance of the total force and pitchingmoment, demonstrated in Fig. 14. The four forces in the diagram are:gravity force G, buoyancy force B, mooring force T, and hydrodynamicforce F. Without the free surface effect under normal operation, the hy-drodynamic force F of the whole body and rotors are functions of trimangle and were evaluated by time-consuming CFD software in advance.This term is treated as an external force, interpolated from Table 3. Thebuoyancy force is calculated by the present method. While we are goingto obtain a steady solution, the mooring force T is assumed always thesame magnitude of the sum of the other three but in opposite direction,so that the total force is zero. So the remaining unknown, the trim angle,is solved iteratively for the total moment reaches zero as well. It is notgoing to discuss the complex mooring mechanics, but simply assume themooring line is a rigid rope without bending or stretching and pinnedonto the seabed at depth 200m. As long as the length of the rope is given,the relative position of FKT to the pin is just a trigonometric operation. Inthe non-filled ballast conditions, the center of gravity of ballast water isalso a function of trim, treated as the same way as in the trimaran case.

The ballast control follows the steps: all empty compartments, thenpump water into the fore compartments until full, then transfer theballast from fore compartments to the aft, finally pump the ballast outfrom the aft compartments until empty and return to the initial condition.Table 4 lists the results, where a%b%c% stands for the filling ratio of thecompartments. It can show the variations of total weight and centre ofgravity during each step. The ballast water increased the total weight and

Fig. 13. Compartments of bouy.

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raised KG. In the transferring step, trim angle was dramatically increased,which provided an angle of attack of the buoy; hence an upward lift forcepulled the body closer to sea surface. For the deepest immersion, the forecompartments were full. This is physically correct since the most ballastpressed on the fore part and thus made the buoy generating a negative liftand forced the body down. It is clearly shown in Fig. 15 that to avoid thewavy surface in typhoon, the ballast should be moved forward, while for

Fig. 15. Submergence under different ballasts.

Fig. 16. Definition of various centers of fkt.

Fig. 17. Compartment filling rate for even keel.

Fig. 18. Transversal transition stability.

Fig. 19. Longitudinal transition stability.

J. Tong et al. Ocean Engineering 160 (2018) 301–310

the maintenance it is moved backward then pumped out.The other scenario of FKT is the launching process, which should be

conducted at calm sea condition without current. The rotors are locked inposition and the mooring line is loose. So there are no hydrodynamic andmooring forces. Initially FKT with empty compartments is floating evenlyon the sea surface. By pumping the ballast in the fore and aft compart-ments simultaneously but with different rates, FKT will sink evenly untilfully submerge. During this diving process FKT should acquire sufficienthydrostatic stability in both longitudinal and transverse directions, sothat a disturbance would not flip the body. This is termed as the transi-tional stability, which is critical for submarines. So there are one ballastcontrol way for even position, which is solved first. Then the determi-nation of its transitional stability, i.e. GM versus submergence, is con-ducted. The free-surface effect caused by the partially filledcompartments during the process is counted the same way as in thetrimaran case. Fig. 16 defines the centres of gravity KG, buoyancy KB,and metacentres KML and KMT.

Fig. 17 plots the immersed depth and LCG in meters on the left axisagainst the total ballast water filling ratio, from empty 0%–95% justreaches neutral balance of buoyancy and gravity forces. As the thicknessof the bouy is not large, an additional 0.5m draft from its initial positioncaused full submergence. The blue curve is the fore to aft compartmentfilling ratio on the right axis, and it indicates that ballast pumped into theaft compartments should be faster than into the fore at the beginning, foran even keel position. Until 40% total ballast moment, the pumping ratesinverse. Under this control, FKT retains even keeled during its divingprocess.

Then the longitudinal and transversal stabilities were calculated ateach sinkage position. For a full submerged condition the floatation planevanishes, i.e. BM and BML are zeros, shown in Figs. 18 and 19 that themetacentres collide with buoyancy centres at 95% total filling ratio. Alsoas the increment of submergence by pumping ballast into the above buoy,the centres of gravity and buoyancy were both raised. An interesting noteabout the free surface effect is that the virtual centres of gravity weremaximized at half ballast condition. This appears to be caused by themaximum waterplan area inside the buoy at it 50% filling ratio. Last butnot least, the transversal stability is considerably larger than longitudinalone not only due to the aspect ratio of the buoy is relative low, but alsocontribution of the transversal bulkheads, which reduce the moment ofwaterplan area by nine times.

4. Conclusions

The study presents a general method to calculate hydrostatic perfor-mance based on the analytical solution of static pressure on a triangle.Arbitrary surface configurations only require closed volume are trian-gulated and intersected with different calm water levels. The waterplanarea properties are analytically treated as planar polygons. Multi-hullship is demonstrated and validated by its hydrostatic performance atdifferent drafts and statical stability at design condition. Compartment isalso modelled in the same way as hull surface but only flip its surfacenormal direction inwards. Ballast fluid can be controlled via a filled ratio.The level inside the compartment and a new equilibrium trim andsinkage are solved iteratively. Finally the free surface effect is correctedand stated as a virtual centre of gravity that is above the original one andhence reduced stability.

The other demonstrating example, the floating Kuroshio Turbine, theirregular body shape is composed of different parts. FKT is subjected totwo additional forces, the hydrodynamic forces and mooring line. Theproposed method showed its capability of dealing with such a complexproblem to determine its balanced position and trim angle, which arecritical references in the preliminary design stage. An extensive usage ofthe method is to calculate the transition stability during a launchingprocess of FKT. Like a submarine, FKT is controlled by a buoyancy en-gine, which pumps ballast water in/out of the compartments. The con-trolling procedure for keeping FKT even keel is obtained, then its

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stabilities during the diving process are calculated.

Appendix

Mathematical derivations for the analytical solution of hydrostatic pressure on a fully submerged triangle are described. Three non-colinear po-sitional vectors r0, r1, and r2 present the vertices of a triangle, as Fig. A.1 below.

Fig. A.1. Parametric representation of a triangle.

Then define two vectors r01 and r02 from r0 to the other two vertices

r!01 ¼ r!1 � r!0 (A.1)

r!02 ¼ r!2 � r!0 (A.2)

Then the triangle can be presented in parametric form as Eq. (A.3) and differentiate r respect to u and v for later use, Eq. (A.4).

r!ðu; vÞ ¼ r!0 þ r!01uþ r!02v ; for 0 � u � 1 and 0 � v � ð1� uÞ (A.3)

∂ r!ðu; vÞ∂u ¼ r!u ¼ r!01;

∂ r!ðu; vÞ∂v ¼ r!v ¼ r!02 (A.4)

And we use subscript z represents the z-component of a position vector. the water head is the Z component of r, denoted a scalar rz. The pressure forceP on the triangle is the integration of every infinitesimal element over the entire parametric domain as:

P!¼ ∫ P

!dA ¼ ∫ u¼1

u¼0∫v¼ð1�uÞv¼0

�ρgrðu; vÞz

�n!ðu; vÞj r!01 � r!02jdvdu (A.5)

where the normal of the triangle is:

n!¼ r!01 � r!02

j r!01 � r!02j(A.6)

By substituting variables with triangle area we have:

P!¼ 2ρg n!A

�∫ ∫ r0zdvduþ ∫ ∫ r01zudvduþ ∫ ∫ r02z vdvdu

�(A.7)

integrating the variables u and v of each term:

∫ u¼1u¼0∫

v¼ð1�uÞv¼0 r0zdvdu ¼ 1

2r0z

∫ u¼1u¼0∫

v¼ð1�uÞv¼0 r01zudvdu ¼ 1

6r01z

∫ u¼1u¼0∫

v¼ð1�uÞv¼0 r02z vdvdu ¼ 1

6r02z (A.8)

So the total pressure vector is:

P!¼ 2ρg n!A

�12r0z þ 1

6r01z þ 1

6r02z

�(A.9)

Now we can define a position vector for the centre of this triangle rc as:

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J. Tong et al. Ocean Engineering 160 (2018) 301–310

r!c ¼ r!0 þ 13r!01 þ 1

3r!02 (A.10)

So rewrite (A.9) and simplify as:

P!¼ ρgrcz n!A (A.11)

By the similar formulations of the total pressure, the moment vector induced by the hydrostatic pressure is:

M!¼ rðu; vÞ* p!¼ ρg∫ u¼1

u¼0∫v¼ð1�uÞv¼0 r!ðu; vÞ � �

rðu; vÞzð r!01 � r!02Þ�dvdu

¼ 2ρgA∫ ∫

�r!0 � n!j

�ð r!0 þ r!01uþ r!02vÞzdvduþ ∫ ∫�u r!01 � n!j

�ð r!0 þ r!01uþ r!02vÞzdvduþ ∫ ∫�v r!02 � n!j

�ð r!0 þ r!01uþ r!02vÞzdvdu (A.12)

Integrating each terms we have:

M!¼ ρgA

��r!0 þ 1

3r!01 þ 1

3r!02

�r0z þ 1

3

�r!0 þ 1

3r!01 þ 1

3r!02

�r01z þ 1

3

�r!0 þ 1

3r!01 þ 1

3r!02

�r02z �

�136

r!02 � 118

r!01

�r01z �

�136

r!01 � 118

r!02

�r02z

�� n!

(A.13)

By substituting with rc vector:

M!¼ ρgA

�r!c⋅rcz þ 1

18

�r!01 � 1

2r!02

�r01z þ

�r!02 � 1

2r!01

�r02z

�� n! (A.14)

Using the vector operation shown in Fig. A.2 to define vector A and B, Eqs. (A.15) and (A.16) , and then substitute into Eq. (A.14) for furtherrewriting as Eq. (A.17) :

A!¼ 1

2r!02 þ r!21 ¼ r!01 þ 1

2r!02 � r!02 ¼ r!01 � 1

2r!02 (A.15)

B!¼ 1

2r!01 þ r!12 ¼ r!02 þ 1

2r!01 � r!01 ¼ r!02 � 1

2r!01 (A.16)

M!¼ ρgA

�r!c⋅rcz þ 1

18

�A!r01z þ B

!r02z

��� n!¼ ρgA�rcz r!c þ 1

36

�r01z r

!01 þ r12z r

!12 þ r20z r!20

��� n! (A.17)

The next step is to derive centre of pressure force rf, so that the total pressure acting on this point will produce an equivalent moment. That is:

M!¼ r!f � P

!(A.18)

By substituting P and M vectors derived above, and eliminate common terms on both sides:

r!f ¼ 1rcz

�rcz r!c þ 1

36

�r01z r

!01 þ r12z r

!12 þ r20z r!20

��(A.19)

It is able to expand each vector to its primitive form as:

r!f ¼ 1rcz

rcz r!c þ 1

36

�3�r0z r

!0 þ r1z r

!1 þ r2z r

!2

�� �r0z þ r1z þ r2z

�ð r!0 þ r!1 þ r!2Þ��

(A.20)

So finally the moment is simplified as:

M!¼ ρgA⋅

112rcz

�9rcz r

!c þ r0z r

!0 þ r1z r

!1 þ r2z r

!2

�� n! (A.21)

Fig. A.2. Definition of Vectors A and B.

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J. Tong et al. Ocean Engineering 160 (2018) 301–310

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