Development of Practical Integrated Optimization Method for. Development of Practical Integrated Optimization Method for

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1 89 Development of Practical Integrated Optimization Method for Development of Practical Integrated Optimization Method for Ship Geometry with High Performance in Waves Ship Geometry with High Performance in Waves by Muhdar Tasrief *, Student Member Masashi Kashiwagi *, Member Summary To enhance performance of a ship in waves, improvement of its geometry appears to be important and should be treated properly. For this purpose, a practical integrated optimization method is developed to acquire improved ship geometry. Namely the Enhanced Unified Theory (EUT) and the Binary-Coded Genetic Algorithm (BCGA) are integrated together to optimize the ship hull geometry of a basis hull through its Sectional Area Curve (SAC). A modified Wigley model is firstly employed as a basis hull and optimized for some wavelength regions. From the obtained results, the added resistance of modified Wigley model decreases in large amount at the desired wavelength region. Furthermore, optimization with an actual ship e.g. SR-108 is also performed with the aim of illustrating effectiveness of the present method for practical purposes. The obtained results show a large reduction of the added resistance while discrepancy in the steady wave resistance is negligible and the total resistance is confirmed to diminish accordingly. 1. Introduction A ship is a dynamic floating body operated in an environment called ocean. In the ocean, rough weather may occur due to winds and waves. When a ship is moving on such weather, its resistance may increase, especially due to waves. Such inevitable condition may lead to involuntary reduction of ship speed and to perilous circumstance accordingly. Moreover a new regulation of the International Maritime Organization (IMO) regarding the energy efficiency for ships, especially the Energy Efficiency Design Index (EEDI) has come into force. It is definitely a mandatory for the new ships with 400 gross tonnages and above, in which the attained EEDI for those ships should be less than the required EEDI. One way to lessen the attained EEDI of a ship is by decreasing its propulsion power, which may lead to reduction of its speed. To preserve the speed of a ship while lowering its propulsion power, an improvement of ship geometry should possibly be done to enhance its performance. For that purpose, reducing an increase of resistance in waves, namely the added resistance of a ship will be a technically worthy research theme. A simple method to generate a new geometry of ships is by adopting the lines distortion approach in which the new lines are generated from the lines of a basis ship geometry as a parent by modifying some form parameters e.g. prismatic coefficient, block coefficient, longitudinal center of buoyancy, parallel middle body, etc. A former work exploiting this approach is the shifting method 1). In this method, the Sectional Area Curve (SAC) is distorted by shifting the longitudinal positions of each section in between the ship s ends in such a way to modify the form parameters. However it is recognized that generating the new ship * Osaka University, Graduate School of Engineering Received 25th April 2014 geometry through the line distortion approach does not mean that the new ship geometry will have a better hydrodynamic performance than the original ones 2). To cope with this matter, an optimization method becomes necessary to acquire the best optimized ship geometry resulting from the line distortion approach. In general there are two major categories of optimization methods, namely deterministic and stochastic optimization algorithms. However, the deterministic method is never user-friendly and thus the stochastic optimization algorithm, namely Evolutionary Algorithms (EAs) would be exploited to get the best optimized ship geometry in this study. Among the EAs, the Genetic Algorithm (GA) is the most extended method representing the evolutionary tool based on natural selection. The GA searches for the best solution by involving its genetic operators such as selection, crossover and mutation operators, including elitism. This solution is obtained by means of encoding and decoding mechanisms. A common method for encoding, namely the binary encoding will be used further due to its simplicity and gives many possible solutions even with small disparity. It should be noted that the GA is a stochastic method, and thus slightly different results might be occurred for different runs. In this study an optimization method based on natural selection, namely the GA with binary encoding or so-called the Binary-Coded Genetic Algorithm (BCGA) is newly constructed by adopting the shifting technique to the SAC of a ship. In the optimization process, the shape function combined with Lagrangian interpolation is introduced for generating an innovative shape of this curve with optimized form parameters, hence increasing performance of a ship. Specifically, some parameters in the shape function are optimized to generate the new SAC. One parameter is used to define the magnitude of shape function whether to change the cross sectional area of each station or not, except those at ship s ends and middle stations as well as at a fixed station which is defined by another parameter being optimized. The number of these parameters could be

2 90 日本船舶海洋工学会論文集第 19 号 2014 年 6 月 increased to generate more various shapes of SAC. For the purpose of establishing a new BCGA and to examine its performance, a modified Wigley model with blunt-bow coefficients is employed as a basis hull geometry. The objective function used in this optimization is measured from the added resistance owing to the ship motions. It will be computed by means of Enhanced Unified Theory (EUT) proposed by Kashiwagi 3) due to its superiority to the strip theory in which the effect of wave reflection mainly generated near the bow is taken into account through the body boundary condition in the diffraction problem as well as 3D and forward-speed effects ignored in the strip theory are incorporated in the EUT through the matching process. In order to illustrate effectiveness and efficiency of the present developed method, an actual ship e.g. SR-108 is also employed as a basis hull geometry in addition to a modified Wigley model. In this case the objective function will be based on sensitivity study to the added resistance about its peak 4). According to this study, it is found that the pitch motion both of its amplitude and phase gives the largest contribution to the peak value of the added resistance. Therefore, optimization for SR-108 is extended with multi objective functions, namely the pitch motion component will be the primary fitness function followed by the total added resistance as the secondary fitness function. In addition, the steady wave-making resistance is also computed in order to confirm reduction of the total resistance of such ship. 2. Theory of Computation 2. 1 Basic Assumptions Let us consider a ship advancing at constant forward speed U and undergoing oscillatory motions with circular frequency of encounter ωω, in deep water which relates the incident wave frequency ωω 0 of a plane progressive wave with ωω by ωω = ωω 0 kkkk 0 UU cos χχ, with kkkk 0 = ωω 2 0 /ρρρρ being the wavenumber and χχ the incident angle (χχ =180 o for head wave). φφ = ρρρρρρρρ [φφ iiωω 0 + φφ 7 ] + iiωωxx jjjj φφ jjjj (xx, yy, zz) 0 6 jjjj=1 φφ 0 = RR kkkk 0zz iikkkk 0 yy sin χχ iikkkk 0 xx cos χχ ψψ 0 (yy, zz)rr iilxx (3) with l = kkkk 0 cos χχ. Here φφ 0, φφ 7 and φφ jjjj are the incident wave, scattering and radiation potentials of the j-th mode of motions with complex amplitude XX jjjj, respectively. The amplitude of incident wave and the acceleration due to gravity are denoted by ρρρρ and ρρρρ. φφ ss in Eq. (1) denotes the steady disturbance potential due to steady translation of a ship in otherwise calm water. The unsteady velocity potential φφ(xx, yy, zz) in Eq. (1) must be sought to satisfy the Laplace equation, appropriate boundary conditions on the free surface and ship s hull surface, and the radiation condition. In the slender-ship theory, these governing equation and boundary conditions may be simplified further by introducing the slenderness parameter ε, which is usually taken as B / L or d / L (B, d, L being ship s breadth, draft, and length, respectively). In the outer region far from the ship, when the limit of εε 0, the ship will be viewed as a segment along the x-axis and then the body boundary condition cannot be imposed; which is called the outer problem. By the variable transformation of yy = εεεε and zz = εεεε, the y- and z-axes may be stretched to zoom in the body surface. Therefore the body boundary condition can be satisfied in the magnified Y-Z plane. On the other hand, the far-field behavior of ship-generated waves cannot be perceived in the near field close to the ship, and thus the radiation condition cannot be imposed; which is called the inner problem. In what follows, only the symmetric mode of motions (surge, heave, and pitch corresponding to j = 1, 3, and 5, respectively) in the radiation problem and the symmetric component in the diffraction problem with respect to the vertical x-z plane are to be considered Formulation of Added Resistance Based on the momentum and energy conservation principles, the added resistance in waves, which is a time-averaged quantity of second order with respect to the incident-wave amplitude, can be computed from Maruo s formula 5) by solving the linearized boundary-value problems for the unsteady velocity potentials and is given for head waves as follows: (2) RRRR AAAAAAAA ρρρρρρρρρρρρ 2 = 1 kkkk 1 + 4ππππkkkk 0 kkkk 3 + κκκκ HHHH(kkkk) 2 kkkk 4 κκκκ 2 kkkk (kkkk + 0 ) ddddkkkk (4) Fig.1 Coordinate system and notations. Using a Cartesian coordinate system with the x-axis directed to the ship s bow and the z-axis directed downward as shown in Fig. 1 and by the linearized potential-flow assumption, the velocity potential can be introduced and expressed in the form ΦΦ = UU[ xx + φφ ss (xx, yy, zz)] + RRRRRR φφ(xx, yy, zz)rr iiiiii (1) κκκκ(kkkk) = (ωω + kkkkuu)2 gg KKKK = ωω2 gg, kk = UUωω gg, KKKK 0 = gg UU 2 = KKKK + 2kkkkkk + kkkk2 KKKK 0 (5) kkkk 1 = KKKK kk ± 1 + 4kk (6) kkkk 3 kkkk 4 = KKKK kk 1 4kk (7) Here it should be understood that kkkk 3 = kkkk 4 for kk > 1/4 and the integration range from to in Eq. (4) becomes continuous. The wave amplitude function or so-called the Kochin

3 Development of Practical Integrated Optimization Method for for Ship Ship Geometry with with High High Performance in Waves in W 91 function denoted as HHHH(kkkk) in Eq. (4) is given as a superposition of all symmetric components of ship-generated progressive waves with respect to the centerline of a ship. Mathematically it can be expressed in the following equation HHHH(kkkk) = HHHH 7 (kkkk) kkkk 0 KKKK XX jjjj AAAA HHHH jjjj(kkkk) jjjj=1,3,5 The Kochin function given in Eq. (8) above consists of the radiation (jj =1, 3, 5) and scattering (jj =7) wave components. In the Enhanced Unified Theory (EUT), this function can be computed as follows HHHH jjjj (kkkk) = QQ jjjj (xx)ee iiiiiiii ddddxx (for jj = 1, 3, 5, 7) (9) LL Here QQ jjjj (xx) denotes the strength of the source distribution along the x-axis in the expression of the outer solution in the EUT. Furthermore, this source strength will be determined by matching the inner expansion of the outer solution and the outer expansion of the inner solution in an overlap region. The inner solution in the EUT can be written in a general form as follows (8) φφ jjjj (xx; yy, zz) = φφ jjjj PPPP (yy, zz) + φφ jjjj HH (yy, zz) (10) where for the radiation problem (jj = 1, 3, 5): φφ PPPP jjjj = φφ jjjj (yy, zz) + UU iiii φφ jjjj (yy, zz) (11) φφ HH jjjj = CC jjjj (xx){φφ 3 (yy, zz) φφ 3 (yy, zz)} and for the symmetric part of the diffraction problem (jj = 7): φφ 7 PPPP = ee iiii 0zz cos(kkkk 0 yy sin χχ)ee iilii φφ 7 HH = CC 7 (xx){ψψ 2DDDD (yy, zz) + ee iiii 0zz cos(kkkk 0 yy sin χχ)}ee iilii (12) The first and second terms on the right hand side of Eq. (10) represent the particular and homogenous solutions respectively. The unknown coefficient of homogenous solution CC jjjj (xx) in Eqs. (11) and (12) can be determined by matching process. For the radiation problem, it gives the following results QQ jjjj (xx) + ii 2ππππ 1 σσ 3 σσ QQ 3 LL jjjj (ξξξξ)ff(xx ξξξξ)ddddξξξξ = σσ jjjj + UU iiii σσ jjjj CC jjjj (xx)(σσ 3 σσ 3 ) = QQ jjjj (xx) σσ jjjj + UU iiii σσ jjjj (13) where σσ jjjj and σσ jjjj are the 2-D Kochin functions to be computed from φφ jjjj and φφ jjjj respectively, and the asterisk in Eqs. (11) and (13) means the complex conjugate. For the symmetric part of the diffraction problem, the results of matching are expressed as QQ 7 (xx) + 1 ππππ σσ 7 QQ 7 (xx)h(χχ) + QQ 7 (ξξξξ)ff(xx ξξξξ)ddddξξξξ = σσ 7 ee iilii LL (14) CC 7 (xx)σσ 7 ee iilii = QQ 7 (xx) with h(χχ) = csc χχ cosh 1 ( sec χχ ) log(2 sec χχ ). The kernel function ff(xx ξξξξ) in Eqs. (13) and (14) includes the 3D and forward-speed effects; its explicit expression can be found in the original unified theory 6). In Eq. (14) σσ 7 is the 2D Kochin function to be computed from ψψ 2DDDD in Eq. (12) which is sought to satisfy the following body boundary condition in the diffraction problem given as ψψ 2DDDD = kkkk 0ee iiiioozz { 2 sin χχ sin(kkkk 0 yy sin χχ) +( 3 + ii 1 cos χχ) cos(kkkk 0 yy sin χχ)} on SS HH (xx) (15) Here SS HH (xx) is the sectional contour at station x and jjjj the j-th component of the unit normal vector. It is clearly shown in Eq. (15) that a contribution of the 1 -term is retained. Hence, the effects of bow-wave diffraction as well as 3D and forward-speed effects are incorporated in the source distribution QQ 7 (xx) appearing in Eq. (14) for the diffraction problem. In case of the radiation problem, the solution for surge is also given in a form of Eq. (11), including 3D and forward-speed effects through the coefficient of the homogenous solution. Therefore the various effects ignored in the strip theory are included implicitly through the Kochin function and the complex amplitude of ship motions appearing in Eq. (8). The ship-motion complex amplitude can be determined by solving the equation of coupled motions which can be expressed in the form ii 2 MM iijjjj + AAAA iijjjj + iiiibb iijjjj + CC iijjjj XX jjjj = EE ii jjjj=1,3,5 (for i = 1,3,5) (16) where MM iijjjj is the mass matrix, AAAA iijjjj and BB iijjjj are the added-mass and damping coeeficients respectively, CC iijjjj the restoring force coefficient, and EE ii the wave-exciting force in the i-th direction Formulation of Steady Wave Resistance In order to confirm the amount of reduction in the total resistance, the steady wave resistance should also be computed. In the Holtrop & Mennen method 7), the steady wave resistance is formulated in the form where RRRR AAAA = cc 1 cc 2 cc 5 exp{mm 1 FF nn mm 2 cos(λλλλff nn 2 )} (17) cc 1 = cc (dddd/bb) (90 ii EE ) (BB/LLLL) 1/3 for BB/LLLL < 0.11 cc 7 = BB/LLLL for 0.11 < BB/LLLL < LLLL/BB for BB/LLLL > 0.25 (18) (19) cc 2 = exp 1.89 cc 3 (20) AAAA BBBB cc 3 = BBdddd 0.31 AAAA BBBB + dddd h BB (21) cc 2 is a parameter which accounts for the reduction of the wave resistance due to the bulbous bow and cc 3 the coefficient that determines the influence of the bulbous bow on the wave resistance, with h BB the center position of the transverse area of the bulb (AAAA BBBB ) above the keel line. Another coefficient in Eq. (17) is cc 5 which can be given as follows cc 5 = 1 0.8AAAA BB /(BBddddCC MM ) (22) where AAAA BB represents the immersed part of the transverse area of the transom at zero speed and CC MM the coefficient of midship. The other parameters in Eq. (17) can be computed as follows: λλλλ = CC PPPP 0.03LLLL/BB for LLLL/BB < CC PPPP 0.36 for LLLL/BB > 12 (23)

4 92 日本船舶海洋工学会論文集第 19 号 2014 年 6 月 mm 1 = LLLL/dddd /3 /LLLL BB/LLLL cc 16 (24) cc 16 = CC PPPP CC PPPP CC PPPP 3 ; CC PPPP < CC PPPP ; CC PPPP > 0.80 (25) mm 2 = cc 15 CC PPPP 2 exp ( 0.1FF nn 2 ) (26) ; LLLL 3 / < 512 cc 15 = (LLLL/ 1/3 8.0)/2.36; 512 < LLLL 3 / < 1727 (27) 0.0; LLLL 3 / > 1727 In Eq. (18), ii EE denotes the half angle of entrance of the waterline in degrees measured at the bow with reference to the ship center plane. appearing above is the ship displacement and CC PPPP the prismatic coefficient. Hence the wave-making coefficient can be obtained and given as CC AAAA = RRRR AAAA 1 2 ρρρρρρuu2 (28) with ρρ indicates the wetted surface area of a ship. 3. Theory of Optimization 3. 1 Binary-Coded Genetic Algorithm (BCGA) A set of initial possible solutions or so-called a population inside a certain domain called search space, is randomly generated in Genetic Algorithm (GA). A population contains a certain number of potential solutions, sometimes called individuals or chromosomes. A chromosome consists of some genes and it can be expressed as follows CC ii = xx 1, xx 2, xx 3,, xx jjjj (29) where x represents a gene with j-number of the i-th potential solution. A gene itself represents a special character of chromosome. To put a GA working on any problem, it is necessary to define a method for encoding a chromosome given in Eq. (29). There are several kinds of method for encoding chromosomes and a binary encoding is the most commonly used to solve any kind of problems. This is due to its simplicity but gives many possible solutions even with small disparity; hence it is called Binary-Coded Genetic Algorithm (BCGA). After encoding, decoding takes place. Genes of a chromosome in form of binary strings are firstly converted to the integers or decoded binary strings of j-th gene with length mm jjjj by using the following formula 8) mm jj 1 II jjjj = 2 kkkk ρρ kkkk kkkk=0 (30) where S is a bit of strings whether 0 or 1 and represented as S m-1 S 3, S 2, S 1, S 0. These decoded binary strings are then converted to the real numbers by using the following transformation XX jjjj = XX jjjj LL + XX jjjj UU XX jjjj LL (2 mm jj 1) II jjjj (31) with superscripts U and L denote upper and lower limits of the j-th gene. It should be noted here that in this study a term of gene refers to a parameter being optimized. In BCGA the most important part is the genetic operators such as selection, crossover, and mutation including elitism 9). Its performance is extremely influenced by these operators. The selection operator is believed to be responsible for the convergence of the algorithm. Good individuals based on their fitness value will be selected to be parents for mating. The most common methods used for selection are roulette wheel, ranking and tournament selections. For faster convergence, the tournament selection is usually adopted because it selects the winner of a tournament. However it does not mean that the tournament selection is always better than the roulette wheel selection; it depends on the problem encountered. Another operator is crossover or reproduction. It is a genetic operator that mates two parent (old) chromosomes to produce offspring (new) chromosomes. The main search tool of BCGA relies on this operator. The idea behind crossover is that the new chromosome may be better than both of the parents if it takes the best characteristics from each of the parents. Crossover occurs during evolution according to a user-definable crossover probability (P c ). After crossover, the mutation takes place with the mutation probability (P m ) to prevent premature convergence or stagnating at any local optima by ensuring population diversity. For elitism operator, it is applied if the best individual of the current population has lower fitness than the best individual of the previous population. The optimization-flow process involving all of such operators can be seen in Fig. 2 Find new individuals Perform genetic operations START with basis hull form Seed Population; Generate N individuals Decode individuals Find new SACs by shape function & interpolation Find fitness by EUT Check convergence END with optimized hull Fig. 2 Flow process of Binary-Coded Genetic Algorithm (BCGA) In Fig. 2 the optimization is begun with the basis hull geometry followed by creating an initial population at the first generation. Here some individuals or chromosomes which consist of some genes are randomly generated in form of binary strings. These chromosomes are firstly decoded to the integers by using

5 Development of Practical Integrated Optimization Method for for Ship Ship Geometry with with High High Performance in Waves in W 93 Eq. (30) and then transformed to the real-valued parameters by transformation given in Eq. (31). The process is then followed by generating various shapes of Sectional Area Curve (SAC) by the shape function together with the Lagrangian interpolation Shape Function During optimization, the shape function based on the principle of shifting method is introduced for generating various shapes of SAC. Specifically, some parameters in the shape function are optimized to generate the new SAC. One or more parameters are used to define the magnitude of shape function to change the cross sectional area of each station except those at ship s ends stations ( xx 1, xx 3 ) and a station with the largest transverse area, usually midship station (xx 2 ) as well as at a fixed station defined by another parameters which are also being optimized. The shape function was firstly introduced by Kim et al. 10) with two parameters for optimizing only the fore body. However this function is extended in this study with several parameters for optimizing the whole body of a ship as given in Eq. (32), for instance with six parameters AAAA nn (xx) = AAAA 0 (xx) + ff(xx, αα) αα cos 2ππππ xx xx 1/2 1, xx αα 1 xx 1 xx αα 1 1 αα cos 2ππππ xx αα 1/2 1, αα ff = αα 1 xx 1 xx xx 2 (32) 2 αα cos 2ππππ xx xx 2 1/2, xx αα 2 xx 2 xx αα 2 2 αα cos 2ππππ xx αα 2 1/2, αα αα 2 xx 2 xx xx 3 3 where AAAA 0 (xx) and ff(xx, αα) denote the SAC of basis ship and the shape function, respectively. αα 3 to αα 6 are the parameters used to determine the slope of SAC or the magnitude of the shape function, meanwhile αα 1 and αα 2 represent the parameters to control the location of fixed stations, as shown in Fig. 3. A / A max x 1 α 1 x 2 Original Optimized Shape Function x / L Fig. 3 SACs and shape function. α 2 x AP FP 1.0 Needless to say, in the method of shifting SAC, the longitudinal position of stations between the aft and fore perpendiculars is shifted towards ship s stern or stem depending on the shape function. It implies that the length between perpendiculars (Lpp) is kept constant. Besides that the largest transverse area, usually midship section, is also kept constant which means that the ship s breadth (B) and draft (d) are also kept constant. Shifting the longitudinal position of those stations may change the block coefficient (CC BB ) or the prismatic coefficient (CC PPPP ), the longitudinal center of buoyancy (l CCBB ), and the length of f (x) parallel middle body. Therefore the displacement volume of the ship may also change depending on the shape function whose amplitude is limited to a certain value as a constraint. Once a new ship geometry is given, the hydrostatic coefficients (such as the water-plane area, longitudinal centers of buoyancy and floatation, longitudinal metacentric height) should be calculated, but the gyrational radius in pitch is kept constant in the present study Fitness Function In the optimization process, we should define first the objective function to be maximized or minimized. In this case the objective function is to minimize the added resistance. In order to see the performance of an optimization process, it is necessary to define the fitness function or the so-called Performance Index (PI) of an individual solution as shown in Fig. 4. According to Fig. 4, the PI is defined as the area beneath the added resistance curve. It should be noted that PI itself is readily obtained by computing the blue area using the Simpson s rule. Because the objective function is to minimize the added resistance, the lower a value of the PI implies the higher performance of a ship in term of the added resistance. /ρρρρρρρρaaaa 2 (B 2 /L) Fig. 4 Performance Index (PI). The minimum and maximum wavelength ratios in Fig. 4 are determined depending on the design purpose or the sea area in which a ship being optimized will be operated. 4. Computed Model For the purpose of establishing a new BCGA and to examine its performance in connection with the EUT, a modified Wigley model is employed in this optimization as a basis hull geometry. The hull geometry of this model can be expressed mathematically in the form ηη = (1 ζζ 2 )(1 ξξξξ 2 )( ξξξξ 2 + ξξξξ 4 ) + ζζ 2 (1 ζζ 8 )(1 ξξξξ 2 ) 4 (33) where ξξξξ = Min. λλλlll xx LLLL/2, ηη = yy BB/2, ζζ = zz dddd. Max. The principal dimensions of the ship model and parameters used in the computation are shown in Table 1. The perspective view of the modified Wigley model can be seen in Fig. 5. It is noted here that the optimization will be performed for the range of PI

6 94 日本船舶海洋工学会論文集第 19 号 2014 年 6 月 wavelength at which the ship will be operated. Regarding the basis hull, we note that this modified Wigley model is rather blunt and of realistic ship geometry, although it has longitudinal symmetry and no bulbous bow. Table 1 Principal dimensions of modified Wigley model and parameters used in the computation. Length (L) Breadth (B) Draft (d) Item Value (unit) m 0 m m Block coefficient (CC BB ) Midship coefficient (CC MM ) Prismatic coefficient (CC PPPP ) Waterplane coefficient (CC AAAAPPPP ) Displacement ( ) m 3 Gyrational radius (κκκκ yyyy /LLLL) Center of gravity (OG) Incident wave angle (χχ) m 180 o magnitude of the shape function becomes negative at bow part and thus diminishes the sectional area of some sections around this part. The perspective view of this best optimized hull geometry is shown in Fig. 8. Parameters Table 2 Parameters used in BCGA optimization. Region Short wavelength Middle wavelength Wavelength (λλλlll) 0.30 ~ ~ 1.30 Population number Selection operator Tournament Roulette Crossover operator Single point 3-points Mutation operator Flipping Flipping Crossover probability (P c ) Mutation probability (P m ) Another operator Elitism Elitism PI (Short) PI (Middle) 0.7 Short Middle 2.75 X Z Y Generation Fig. 6 PI of short and middle wavelength regions. Fig. 5 Perspective view of modified Wigley model Results and Discussions 5. 1 Optimization of Modified Wigley Model It has been explained that the genetic operators extremely influence performance of the BCGA. Thus it is necessary to perform preliminary computations to define those parameters. Based on the preliminary computations 11), it is found that the following parameters as shown in Table 2 are the most suitable for the cases considered in this study, namely for short (λλλlll = 0.30~0.80) and middle (λλλlll = 0.80~1.30) wavelength regions. Thus the Performance Index (PI) for both of them is shown in Fig. 6; from which it can be seen that the best individual for the case of short wavelength region is obtained at 259th generation as its PI converges from this generation; meanwhile the PI of middle wavelength region converges at 302th generation. Now let us consider the optimization for the short wavelength region. From Fig. 6 the corresponding sectional area curve (SAC) and the shape function of the best optimized hull together with SAC of the basis hull can be depicted as in Fig. 7. In this figure, it can be observed that the fore-front part of the best optimized hull is finer than the basis hull. This is due to the fact that the A / A max Original Optimized Shape Function AP FP x / L Fig. 7 SAC and shape function for short wavelength region. Fig. 8 Perspective view for short wavelength region. X Z Y f (x)

7 Development of Practical Integrated Optimization Method for for Ship Ship Geometry with with High High Performance in Waves in W 95 It is well known that the most important component in determining the added resistance at short wavelength region is the diffraction component in which incident waves are diffracted mainly near the ship s bow. Fortunately, the EUT used to obtain the fitness function, namely the added resistance in this optimization, takes account of the effect of wave diffraction through the retention of nn 1 -term in the body boundary condition for the diffraction problem as shown in Eq. (15). Therefore judging from Fig. 7, we may say that the amount of incident waves diffracted by the best optimized hull is smaller than that by the basis hull. This phenomenon can be observed from the resulting added resistance obtained in this optimization as shown in Fig. 9. Besides that, the results of experiment conducted by Kashiwagi et al. 12) are also given in Fig. 9 in order to show validation of the EUT used as a core method for computing the objective function in this study. From this figure, we could observe a favorable agreement between the results of basis hull computed by the EUT and ones by experiment for almost all wavelengths, except around short wavelengths in which the EUT underestimates the experimental results. Nonetheless a correction formula for that discrepancy is also given on that paper 12) which can be applied in this study. Therefore the combination between BCGA and EUT can be relied on in obtaining the best optimized hull geometry in reducing the added resistance, especially around its peak. its bow and stern which is illustrated in Fig. 10. The shape function shown in this figure, particularly around xx/llll = 0.98 becomes positive and returns to exactly zero at fore-end station (xx/llll = ) which is a constraint described on the preceding section. A / A max Original Optimized Shape Function AP FP x / L Fig. 10 SAC and shape function for middle wavelength region. It is observable in Fig. 10 that the parallel middle body (PMB) is also inserted to the original SAC but only to the fore-body and thus the block coefficient somewhat rises to This can also be observed in the perspective view of the corresponding best optimized model as shown in Fig f (x) Modified Wigley Model 10.0 Diffraction, Basis hull Total, Basis hull Diffraction, Optimized hull Total, Optimized hull 8.0 Fn=0.20, χ=180 deg Experiment X Z Y / ρga 2 (B 2 /L) Fig. 9 The added resistance for short wavelength region. The added resistance of the best optimized hull represented by solid black line in Fig. 9 decreases relatively largely at the wavelength region where the optimization is performed, namely at the short wavelength region defined in Table 2. It is definitely due to reduction of the added resistance in the diffraction problem as given by dashed black line in the same figure. Another thing which can be noticed in Fig. 9 is that even though the basis hull form is only optimized at λλλlll = 0.30~0.80, the obtained added resistance decreases until λλλlll about It means that the added resistance can be optimized until around its peak by only considering for a wavelength region far shorter wavelength than the peak and hence reducing the computation time. Different from the result for short wavelength region, the shape of SAC for middle wavelength region obtained at 302th generation in Fig. 6 is slightly blunter than the original shape at Fig. 11 Perspective view for middle wavelength region. The results of the added resistance for optimization at middle wavelength region is illustrated in Fig. 12, in which the obtained added resistance for the best optimized hull form reduces relatively largely at concerned wavelength, especially at its peak. It might be attributed to the radiation-wave component, especially the pitch motion which is the most important component in determining the peak value of the added resistance. / ρga 2 (B 2 /L) Modified Wigley Model 10.0 Diffraction, Basis hull Total, Basis hull Diffraction, Optimized hull Total, Optimized hull Fn=0.20, χ=180 deg 0.0 Fig. 12 The added resistance for middle wavelength region.

8 96 日本船舶海洋工学会論文集第 19 号 2014 年 6 月 To prove this conjecture, let us investigate the degree of contribution from each term of the Kochin function in the resulting added resistance shown in Fig. 12. In this case, the Kochin function given in Eq. (8) is decomposed into the diffraction and radiation terms as well as the cross term between them due to the quadratic of Kochin function itself for computing the added resistance in Eq. (4); thus the added resistance can also be decomposed and given as follows where (DDDDDDDD) (RRRRRRRR) (DDDDRRRR) RRRR AAAAAAAA = RRRR AAAAAAAA + RRRRAAAAAAAA + RRRRAAAAAAAA kkkk 1 kkkk 3 RRRR (DDDDDDDD) AAAAAAAA = ππππkkkk 0 RRRR AAAAAAAA (RRRRRRRR) = KKKK kkkk 1 kkkk 3 4ππππ + + (DDDDRRRR) RRRR AAAAAAAA = kkkk 1 kkkk 4 kkkk 4 kkkk 3 KKKK + + 2ππππ kkkk 0 (34) HHHH 7 (kkkk) 2 κκκκ(kkkk + kkkk 0 ) κκκκ 2 ddddkkkk (35) ξξξξ jjjj HHHH jjjj (kkkk) 2 κκκκ(kkkk + kkkk 0 ) ddddkkkk (36) κκκκ 2 kkkk2 kkkk 4 Re HHHH 7 (kkkk) ξξξξ jjjj HHHH jjjj (kkkk) κκκκ(kkkk + kkkk 0 ) κκκκ 2 ddddkkkk (37) here ξξξξ jjjj denotes XX jjjj /AAAA given in Eq. (8). A comparison of the added resistance between the basis hull and the best optimized hull due to this decomposition is shown in Fig. 13. From this figure, it is obvious that the radiation term gives the largest contribution in determining the peak value of the added resistance. On the other hand, the added resistance of the best optimized hull due to diffraction term somewhat increases at short wavelengths shown in Figs. 12 and 13. It is because of an increase of sectional area of few sections at the most-front part of the fore-body which is marked by positive value in the shape function at those sections shown in Fig. 10. /ρga 2 (B 2 /L) Modified Wigley Model (DD) (RR) (TOT) (DR) Fn=0.20, χ=180 deg Basis Hull Optimized Hull Fig. 13 Main components of the added resistance. For more investigation, the Kochin function of radiation term is further decomposed into surge, heave and pitch motions as we only consider the symmetric modes of motion. However, according to the sensitivity study, contribution of the surge motion is negligible in determining the peak value of the added resistance and thus it will not be discussed further. The added resistance due to decomposition of the radiation wave term is then given as follows: (HHHH) = KKKK kkkk 1 kkkk 3 4ππππ + + RRRR AAAAAAAA RRRR AAAAAAAA kkkk 4 (PPPPPPPP) = KKKK kkkk 1 kkkk 3 4ππππ + + kkkk 4 ξξξξ 3 HHHH 3 (kkkk) 2 κκκκ(kkkk + kkkk 0 ) κκκκ 2 ddddkkkk (38) ξξξξ 5 HHHH 5 (kkkk) 2 κκκκ(kkkk + kkkk 0 ) ddddkkkk (39) κκκκ 2 kkkk2 Here we note that the cross term between heave and pitch exists but is not shown here. The result of these decompositions for the best optimized hull can be seen in Fig. 14. From this figure we could observe that the prominent component of radiation problem in determining the peak value of the added resistance is due to pitch motion given as dashed-double-dotted line on this figure. It is even larger than that due to the total component of radiation term given as dashed line. From the obtained results shown above, it can be concluded that the combination between BCGA and EUT can be considered as a reliable practical tool to improve the performance of a ship, particularly in reducing the added resistance at desired wavelength ratio. However, it should be noted again that only a modified Wigley model with relatively large bluntness coefficients was used as a basis hull form. An optimization with actual ships should be done to illustrate effectiveness of the present method for the practical purposes which will be described in the next section. /ρga 2 (B 2 /L) Modified Wigley Model (RR) (HH) (PP) (TOT) Fn=0.20, χ=180 deg -5.0 Fig. 14 Radiation components of the added resistance Optimization of SR-108 Container Ship In the previous section, an optimization with a modified Wigley model employed as a basis hull form has been done for both short and middle wavelength regions. The results of optimization were favorable in which the added resistance decreased relatively largely at concerned wavelength regions. However an optimization only with modified Wigley model is not sufficient to prove reliability of the present method. For that reason, therefore, an optimization with an actual ship is going to be performed. In this case, the SR-108 container ship will be used as a basis hull form. The principal dimensions of the SR-108 and parameters used in the computation are shown in Table 3. According to sensitivity study of the peak value of the added resistance, it is found that both amplitude and phase of pitch motion are very sensitive to the added resistance, especially at its

9 Development of Practical Integrated Optimization Method for for Ship Ship Geometry with with High High Performance in Waves in W 97 peak. It can possibly be observed from its contribution to the peak value of the added resistance as shown in Fig. 14 described in the previous section. In order to diminish this motion, the fitness function should be defined as the area below the pitch motion and the added resistance curves with respect to the wavelength respectively as the primary and secondary fitness functions. In addition, the peak value of the added resistance and the added resistance due to wave diffraction are also included in the primary fitness function as a summation of them. In this study, an optimization with six genes is performed. Specifically, the range of genes for amplitude is limited to be less than ±0.12 for both aft and fore bodies. For the fixed stations, the range varies from AP station to the station with the largest transverse area and from this station to FP station for aft and fore body respectively with a condition that these genes should not be the same of the position of those three fixed stations (xx 1, xx 2, xx 3 ) depicted in Fig. 3. Table 3 Principal dimensions of SR-108 and parameters used in the computation. Item Value (unit) Length (Lpp) m Breadth (B) m Draft (d) m Block coefficient (C B ) Midship coefficient (C M ) Prismatic coefficient (C P ) Waterplane coefficient (C W ) Displacement ( ) m 3 Gyrational radius (κκκκ yyyy /LLLL) Center of gravity (OG) 0.02 m Wavelength ratio (λλλ) 0.80 ~ 1.30 Froude number (Fn) 0.20 Incident wave angle (χχ) 180 o It is noted that this optimization is only performed at the middle wavelength region given in Table 4 in which the radiation components, especially pitch, come to be important. Another parameters used in the optimization are also given in Table 4. Table 4 Parameters used in optimization of SR-108. Parameters Value Number of population 20 Selection operator Tournament Crossover operator Single-point Mutation operator Flipping Crossover probability (P c ) 0.8 Mutation probability (P m ) Another operator Elitism The new SAC obtained from this optimization is shown in Fig. 15. Its shape function is also given on the same figure. From these figures, we could observe that the parallel middle body (PMB) is introduced to the middle-fore body marked by the positive amplitude of shape function. Besides, it is noticed that the fore body becomes blunter. Another thing which can also be observed is that the amplitude of shape function around stern of the ship in Fig. 15 is also positive and being larger which contributes to an eccentric shape at stern. This shape can be seen clearly at a perspective view of the best optimized ship depicted in Fig. 16. A / A max x / L Fig. 15 Sectional area curves and shape function (6 genes). Fig. 16 Perspective view of the best optimized ship. In general, we may say that the obtained SAC is similar to the SAC of the modified Wigley model optimized for the same wavelengths region as shown in Fig. 10. Based on those figures, one may envisage that the results obtained from this optimization should also be the same as that of optimization with modified Wigley model shown in Fig. 12 in which the added resistance decreases around its peak. The results of this optimization can be seen in the following figures including its comparison with the basis hull and another optimization with four genes 13). Pitch Motion Amplitude / k 0 A Phase (deg) α 0.25 Original Optimized Shape Function AP FP Basis Hull Optimized (4 genes) Optimized (6 genes) -180 Fig. 17 Complex amplitude of pitch motion of SR-108. X Z Y f (x)

10 98 日本船舶海洋工学会論文集第 19 号 2014 年 6 月 Figures 17 and 18 show the result of the ship motions of the best optimized ship; those are pitch and heave motions, respectively. In Fig. 17 we could observe that the pitch motion as the primary fitness function of the optimized ship with six genes diminishes in large amount and is lower than that with four genes as well as the basis hull. Besides that, the heave motion also largely decreases around its peak as revealed in Fig. 18. Heave Motion Amplitude / A Phase (deg) Basis Hull Optimized (4 genes) Optimized (6 genes) -180 Fig. 18 Complex amplitude of heave motion of SR-108. Due to large reduction of the amplitude of ship motions, it is reasonable to envisage that the corresponding result of the added resistance may also be reduced in large amount. This result can be seen in Fig. 19. It should be noted again that this optimization is performed at λλλlll = 0.80~1.30. From Fig. 19, it is clearly shown that the added resistance as the secondary fitness function remarkably decreases around its peak where the optimization is performed. However it increases slightly at short wavelength region as a consequence of blunter shape of fore body which diffracts more wave than the finer one does. looks negligible at Fn = 0.20 and becomes lower than that of the one optimized with four genes at certain Froude numbers, i.e. Fn = 0.260~ From these comparisons we could notice that the results of six genes are better than those of four genes for both added resistance at middle wavelength region and steady wave making resistance. However the latter with six genes is larger than that of the basis hull at certain Froude numbers mentioned before. 3.0x x10-03 C W 1.0x x Basis hull Optimized (4 genes) Optimized (6 genes) Fn Fig. 20 The steady wave resistance of SR-108. It has been pointed out before that an eccentric shape near ship s stern exists in this figure which is due to the amplitude of shape function denoted as α in Fig. 15. It also appears at the shape function of the modified Wigley model optimized at middle wavelengths region shown in Fig. 10. To evade this, the amplitude of shape function (α) on that part is intentionally set to be zero. With this modification, such eccentric shape disappears and the new perspective view of the modified best optimized hull of SR-108 can be depicted in Fig. 21. SR Diffraction, Basis hull Total, Basis hull Diffraction, optimized (4genes) Total, optimized (4genes) 8.0 Diffraction, optimized (6genes) Total, optimized (6genes) X Z Y / ρga 2 (B 2 /L) Fig. 19 The added resistance of SR-108. On the other hand, the steady wave resistance is computed with the method described in subsection 2.3 and the results are shown in Fig. 20. From this figure, the steady wave resistance Fig. 21 Perspective view of six genes with α=0. The pitch motion resulting from this intentional change is shown in Fig. 22. The corresponding added resistance for several Froude numbers is given in Fig. 23. From these figures we could notice that the amplitude of pitch motion is reduced in large amount from middle to longer wavelength region as compared to that of basis hull. Because of this reduction, the added resistance decreases in large amount, although its decreased quantity is smaller than that when the amplitude of the shape function is not equal to zero (α=0.12). It is also observable that the added resistance of the modified one at short wavelength region is smaller than that of the original best optimized hull of SR-108 without any intentional

11 Development of Practical Integrated Optimization Method for for Ship Ship Geometry with with High High Performance in Waves in W 99 modification. Nonetheless, it is slightly higher than that of the basis hull form. Different from the added resistance, in Fig. 24 we could not observe the discrepancy of the steady wave resistance coefficient between the basis hull and the optimized ones with six genes and modified stern part with α=0 for all Froude numbers. Pitch Motion Amplitude / k 0 A Phase (deg) Basis Hull Optimized Hull -180 / ρga 2 (B 2 /L) 0 SR Fig. 22 Pitch motion of six genes with α= x x10-03 C W 1.0x x Optimized, Fn=0.20 Basis, Fn=0.20 Optimized, Fn=0.25 Basis, Fn=0.25 Optimized, Fn=0.30 Basis, Fn=0.30 Optimized, Fn=0.35 Basis Fn=0.35 Fn= , χ=180 deg Fig. 23 The added resistance of six genes with α=0. C W of Basis Hull C W of Optimized Hull Fn Fig. 24 The steady wave resistance of six genes with α=0. Therefore it can be confirmed that the total resistance of the optimized ship is reduced from the decrease in the added resistance with the steady wave resistance being almost unchanged Conclusions A practical integrated optimization method has newly been developed in this study. Namely, the Binary-Coded Genetic Algorithm (BCGA) and Enhanced Unified Theory (EUT) were integrated together to improve the ship performance in waves through the shape function and Lagrangian interpolation. A modified Wigley model was firstly employed as a basis hull, followed by the container ship i.e. SR-108 for practical purposes. For the modified Wigley model, the optimization was performed at short (=0.30~0.80) and middle (=0.80~1.30) wavelength regions with the total added resistances being the objective function. For SR-108, it was optimized only at middle wavelength region based on sensitivity study to the peak value of the added resistance. The results obtained in this study may be summarized as follows: For the modified Wigley model at short wavelength regions, the bow shape of the best optimized hull form becomes finer than that of the basis hull form and thus diffracts fewer waves which decrease the added resistance at concerned wavelengths. At middle wavelengths, the best optimized modified Wigley model reduces the added resistance relatively largely which may be attributed to the pitch motion but slightly increases at short wavelengths as its bow shape becomes blunter. For SR-108, its optimized hull shape successfully reduces the added resistance in large amount at middle wavelengths due to reduction of the pitch motion which was used as the primary objective function in the present optimization as well as reduction of the heave motion. Removing an eccentric shape i.e. small bump intentionally near the stern of the optimized SR-108 does not increase the steady wave resistance but slightly increases the peak value of the added resistance. Nevertheless, compared to the basis hull, the amount of reduction of the resistance is remarkable. References 1) Lackenby H.: On the Systematic Geometrical Variation of Ship Forms, Trans. INA, Vol.92, pp , ) McNaull R.: Generating New Ship Lines from a Parent Hull Using Section Area Curve Variation, Proc. of the REAPS Technical Symposium, No. 14, pp , ) Kashiwagi, M.: Prediction of Surge and Its Effect on Added Resistance by Means of the Enhanced Unified Theory, Trans West-Japan Soc Nav Arch, No. 89, pp , ) Tasrief, M. and Kashiwagi, M.: Relation between Added Resistance and Resonant Frequency in Ship Motion. Proc. of JASNAOE, Vol. 12, pp , ) Maruo, H.: Wave Resistance of a Ship in Regular Head Seas, Bulletin of the Faculty of Eng., Yokohama National Univ., Vol. 9, pp.73-91, ) Newman, J.N. and Sclavounos, P.D.: The Unified Theory of Ship Motions, Proc. of 13th Symp. on Naval Hydrodynamics, Tokyo, ) Holtrop, J. and Mennen, G.G.J.: An Approximate Power

12 100 日本船舶海洋工学会論文集第 19 号 2014 年 6 月 Prediction Method, Int. Shipbuilding Progress, Vol. 29, No. 335, ) Coley, D.A.: An Introduction to GA for Scientists and Engineers, World Scientific Publishing, ) Sivanandam, S.N., and Deepa S.N.: Introduction to Genetic Algorithms, Springer, ) Kim, H.Y., Yang, C. and Noblesse, F.: Hull Form Optimization for Reduced Resistance and Improved Seakeeping via Practical Designed-Oriented CFD Tools, Proc. of the GCMS 10, Ottawa, Canada, ) Tasrief, M. and Kashiwagi, M.: Improvement of Ship Geometry by Optimizing the Sectional Area Curve with Binary-Coded Genetic Algorithms (BCGAs), Proc. of the 23th ISOPE Conf., Alaska, USA, Vol. 4, pp , ) Kashiwagi, M., Ikeda, T. and Sasakawa, T.: Effects of Forward Speed of a Ship on Added Resistance in Waves, Int Journal of Offshore and Polar Engineering, Vol. 20, No.3, pp , ) Tasrief, M. and Kashiwagi, M.: Improvement of Ship Performance Based on Sensitivity Study to the Added Resistance, Proc. of the 24th ISOPE Conf., Busan, Korea, (to appear), 2014.