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Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

Chapter: Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology

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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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Suggested Citation:"Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology." National Research Council. 2003. Twenty-Fourth Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press. doi: 10.17226/10834.
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24th Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Multi Objective Optimization of Ship Hull Form Design by Response Surface Methodology Yoshimasa Minami, Munehiko Hinatsu (National Maritime Research Institute, Japan) ABSTRACT Recently multi objective design has been paid attention for the development of high performance ship in actual seas. So far the ship hull form has been mainly designed from the viewpoint of the propulsive performance. However it is important to evaluate the total performance of ship including the seakeeping performance and the propulsive performance in actual seas. In this study, we use CFD and strip theory to estimate the propulsive performance and seakeepin~ performance, respectively. INTRODUCTION r ~7 ~~ ¢ ~ ~ In optimization of ship hull form, we often optimize in such a way that the wave resistance takes the minimum value, because the wave resistance theory based on potential theory doesn't require much computational cost. However with great increase of performance of computers, CFD have been used to minimize viscous resistance or to optimize the nominal wake distribution by using nonlinear programming method [1,23. But there are few examples to apply multi objective design method using large-scale Recently since CFD techniques have been developed computation tool, because it still costs a lot. strikingly, CFD becomes one of most promising tools to This paper concerns the efficient multi-objective hull catch the difference of hydrodynamic property when a ship form design which multi object functions are propulsive hull form is slight changed. This is very important in the performance and seakeeping performance. The main optimization procedure of hull form. However, today CFD interest is development of high performance ship in actual still requires large amounts of computational cost. Hence sea. We improve the efficiency of multi objective design the iteration number of optimization has to be reduced in by applying the approximation method in optimization order to design the ship hull form efficiently. process. The RSM is used in the research of The gradient-based numerical method that is used configuration design for aerospace engineering. Giunta widely in optimization problem may search local optima for the numerous peaks and may require large computational cost to attain the optimum solution. Thus, we use a response surface methodology (RSM) as the optimization tool to search the global optimum for multi-objective functions. The RSM approximates the response surface by use of a polynomial function of design parameters, and the RSM can be evaluated without significant computational expense in the course of optimization process. ~ --a-- ---- =--------- ---- ------------ -- ----I In this study, Wigley model and Series 60 ship model form, it is important to choose suitable design parameters are used as the initial ship hulls and optimize their hull and how to express the ship hull form mathematically by form for multi-objective function. In the work, we choose use of the design parameters. We choose a cubic sine ship resistance, added wave resistance, and amplitude of function that varied under constant displacement condition. strip motion es the elements ofthe multi-objective function. As the design parameters are changed in optimization Then we show that RSM is an effective method for the process, the grids used to solve Navier-Stokes (NS) multi-objective design of ship hull form. equations have to be rearranged. This procedure is worked _ _ _ applied the RSM to design of the configuration of High Speed Civil Transport in order to estimate aerodynamics performance and total gross weight and he successfully showed the practical effectiveness of the RSM technique A. In this case, the design parameters are independent variables as principal dimension and wing configuration. In this paper, we apply the RSM to design of ship hull form using RSM and show the effectiveness for multi-objective design. Since the ship hull form design is carried out through the gradual modification of shin hull

out in design system. Furthermore in order to estimate the seakeeping performance, we construct the interface module that make the input data for the strip method by use of the grid for NS computation. The multi-objective optimizer is adopted the Genetic Algorithm. GA is used to search global optimum from many response surfaces simultaneously and effectively. The flow of our design system is shown in Fig.1. Multi-objective design of ship hull is done along this flow. RESPONCE SURFACE METHODLOGY The application of the RSM to design optimization is aimed at reducing the cost of expensive analysis method like CFD. The RSM make approximate Unction of response y predicted by variable xi . y = f(X}'X7 .~. An ) + £ (1) In the expression of RS, the quadratic polynomials model I often used because high order polynomials have many coefficients and requires large computational cost to approximate the polynomials. A quadratic model in m variables has the form y = cO + I< ~ C jX j + I< j irk< C jkX jXk (2) Where y is the response,x is the design variable and c is the tuning parameters. The quadratic polynomial model has (m + b(m + 2) /2 coefficients in m variables. Estimating the unknown coefficient requires nS analyses. nS should be more than the number of total coefficients. This polynomial model may be written in matrix notation as Ty=c X Where y = [yell y~2~)~ ... yens'] x= : (I) xl ( no ) . . xl (XnV ) c = LCo Cl Cn~ ] When c is substituted for c into Equation (3), values of the response can be predicted at any location. In most application of the RSM, the range of parameters is defined from the lower and upper bounds on the design variables. If n design parameters have 2n vertices, it becomes impractical to evaluate the designs at all of the vertices on the high order. Therefore we use D-optimal design that can minimize the error of predicted value by using the RSM on the lower order t43. The D-optimal design states that the best set of points in calculation data is selected to maximize the determinant |XTX| of equation (4~. So "D" stands for the determinant of the (XTX)-~ matrix associated with the model. The D-optimal design leads to minimize maximum variance of the predicted response surface of this model. OPTIMIZATION SCHEME FOR MULTI OBJECTIVE FUNCTION The optimizer in multi-objective design must search some candidate regions around the optimum point on each response surface widely. The genetic algorithm is commonly used to search a global optimum in wide region. GA leads to reduce calculation time and increase search efficiency of searching process for multi objective functions. GA usually became worse convergence as the population size becomes large. So we use Micro genetic algorithm (,u GA) to optimize multi-objective functions. ,u GA has recently been used for faster convergence, since this algorithm is small population size and less time requirement in evaluating fitness [53. In recent researches, it has been reported that,u GA is useful and prospective for the optimal design A. Multi-objective design needs generally trade-off function to satisfy each performance for the designer or owner's demand. In this study trade-off function is normalize by the minimum desired level as follows, 0j(X) = Wi x | fi( t) ~' | : Minimize Where O is normalized optimum value vector, fi(x) is approximate function of each response surface, wi is the weight function determined by designer and f is vector of standard level which designer desire for each objection functions. The optimizer for Multi-objective function minimizes the norm of normalized optimum vector . n, = (m + D(m + 2) nV : number of design variables 2 From a statistical point view, the unique least squares MODIFICATION FUNCTION OFHULLFORM AND solution c to Equation (3) is denoted as; CALCULATION GRID c = (XTX) 1XTy (4) In the optimization processes, it is important how to

express the ship hull form with suitable parameters. Generally, the constant displacement condition is commonly used as the constraint condition in the ship hull form optimization. In order to improve the total performance, a modification function should not only changes ship hull form locally but also changes globally. We newly devise the modification function under the constant displacement condition. This function is based on a cubic sine function. The cubic sine function has keep continuity in 1st and 2nd longitudinal and transverse derivatives at the boundaries of modification of hull form. In the ship hull form design, 5 parameters are used in this study. Taking the Cartesian coordinate system such that the origin is on the cross point of the water plane and symmetry plane and midship section. X-axes is positive toward ship stern, where x=-0.5 and 0.5 correspond to AP and FP. y-axis is positive toward starboard. z-axis positive vertically downward. The modification function is defined as; y(x, y) = yO (x, z) W(x, y) (6) Where, Adz) express the original hull surface. w(X y) is the modification function to change hull form in transverse-direction. We consider the modification method of hull form in two stages. First stage, the prismatic curve Cp is deformed. The modification functions in Cp curve are defined at front part and apt part. They are: :ACpf (x) = Cf · Am sins (xf ) ,~.5 < x < of l /iCPa (x) = Ca Am sin (xa ) xa < x < o.s Where x' = x x' =_ x f (O.S+xr) a (0.5-xa) tCp (X) = Cp (X)orignal + ~Cpf (X) _ 0.5 S x < Of l Cp (X) = Cp (x)Orjgnal + pupa (X) xa < x < 0.5 Of and xa are the fore and aft boundaries of hull modification area in Cp curve. Am=dmxB is midship area. In order to keep displacement of ship constant, the coefficient of modification function on Cp curve is define as; (0.5 -Xa) =- cS (0.5+ Xf ) In Cp curve, the fore part of modification area becomes equal to the aft part of modification area by using this equation. Second Stage, the hull form is modified in breadthwise direction at yz-plane. In order to keep smoothness at the boundary of modification range, the modification function in breadthwise direction is expressed with the cubic sine function. Corresponding to the fore part of modification in Cp curve, ' _z z' = z ~ dcr dm +df ~ Z'=_ z2dcf ad 3 W(x,z) = ACpf (x) sin (fizz') 4tr (10) Corresponding to the aft part of modification in Cp curve, ad 3 W(X, Z) = ACPa (X) sin (fizz ) 4;r _z — z Idea d +d m ca (11) z =— z2dCa ca Where dmis design draft, dcf ~ dCa is point of maximum in above cubic sine function. The constant 3 d/4 fez in this function is used to correspond the integral of this function to the modified cross section area by equation (7). This modification system of hull form is illustrated in Fig.2. Hence, the design parameters become five that is Lf ~La'Cf ,dcf ,dca - For optimization of ship hull, it is important to rearrange grid system in every optimization step. In this study, we applied flexible grid method 173. The Initial grid is generated by the use of a grid generation program. As the design parameters change, the hull form (hull surface grid) is changed through the above modification method. The outer grid is rearranged in proportion to value which normal vector of modification on hull surface is multiplied into the inverse ratio of the length between the grid point and outer boundary, if the grids at outer boundary are fixed. These modification systems of grid are illustrated in Fig.3. K is defined the number of grid in the normal direction. (8) K is determined l at hull surface and kmaX at outer boundaries. XkneW = Xk°ld + wk (X1 ew _ X1 id ) (12) Where Xk are the grids in yz-plane, dk w, =. __ ~dk=~mE31Xm Arm 11 if k>2 (1 dkmax O if k=2 (9) dk is the length from grid point (K=k) to outer boundary (K=kmax). In order to estimate the seakeeping performance along with the propulsive performance, the data for the strip method has to be obtained automatically from the gird for NS computation and we devise an interface module in our

design system for this purpose. Firstly the interface module extracts the strip data from the grid data corresponding to the strip section point in lengthwise direction. This module provides the data for the strip method at the following 25 sections. As = (FP,—,—,—,1,1—, - - ,8—,9,9—,9—,9—, AP) 4 2 4 2 2 4 2 4 This module naturally provides the data for strip methods as the ship hull is modified in the course of optimization. CALCULATION OF RESISTANCE We use CFD code to estimate the resistance of ship. The CFD code in this study is NEPTUNE code developed in our institute t83. The governing equations are the Navier-Stokes equations with the artificial compressibility assumption. The coordinate system in this study is shown where in Fig.4. They can be given in conservative form as: 1 R(Q) = V ~(H-HV) = 0 (20) 'ilk Faces A cell-centered finite volume approach for spatial discretization is adopted. q and v, are placed at the center of each grid cell and the grid cell is treated as a control volume. Integration of the governing equations (14) over a cell, yields |,~V + ~ {(e + ev)dSx + ( f + fV)dSv + (g + gv)dSz } At v (18) Where the integration of flux is conducted by using the Gauss integration theorem. His the volume of the cell and TV is its boundary. Sx, Sy and Sz are the area vector components of cell boundaries. The governing equations (18) are discretized as follows: Balk +R(Q) 0 (19) id +~(e+eV)+~(f+fv)+~(g+gv) =0 (14) at do by Liz The dependent variables q, the inviscid e, f ,g and the viscous fluxes ev, fV, gv are written as [ev fv gV]=-(R +V! e U2+p TV vU v2+p we WV - o 2uX Ox + Vx uy + vx 2vy uz + wX vz + wy O O (16) In Eq. (2), u, v, and w are the Cartesian components of the fluid velocity. p is the modified pressure defined by * z P=P + 2 F. n Where p is the original pressure and Fn is the Froude number based on the ship length and Re is the Reynolds number and v, the non-dimensional kinematic eddy viscosity determined from a turbulence model. ,6 is a positive constant of artificial compressibility. Turbulence model in this study is used Baldwin-Lomax model. In the above equation, note i, j and k are the cell numbering. H and H' are inviscid flux and viscous flux on the cell's boundary. The change of the volume is expressed as the sum of the six boundary faces of cell. The inviscid flux H is defined using contravariant velocity U as (15) H - SxE + SyF + SzG = ,l]U uU +PSx vU + PSy wU + pSz (21) U = Sxu+Syv+Szw (22) The inviscid flux ~ is evaluated by MUSCL method based on the flux differencing splitting scheme. The viscid flux Hv cab be also expressed as; Hv -SXEV +SyFV +SzGv (23) Where the gradient of velocity on the boundary faces of (17) cell is conducted using the Gauss integration theorem Time integration is used by backward Euler formula as follows; Q(n+~) ink + R(n+~) = 0 (24) At Here, the subscript /\t is time step. Equation (24) is solves by an approximate Newton relaxation method as

The surge, heave and pitch motion can be obtained by solution of ship motion equation. Ship motion equation is the complex linear system as follows; j=~3 5 [ (dip + Aid) + i~Bij + City pj = Ei (~27) Where * (n+l),m I + { bR ~ ~Q(n+l),m ~ Q'(~- ) - Q~,k 1~ R(n+~),m ~ (25) `~Q(n+l),m = Q(n+l),m+1 _ Q(n+l),m (26) In the evaluation of R , the inviscid flux in R is approximated by using the first order upwind differencing for the inviscid flux in R . la is 4X4 identify matrix except the first diagonal element which is zero in order to satisfy the incompressible continuity equation. The subscript m denotes iteration number on each time step. Equation (24) starts from On+ = On at each time step and after reaching the convergence, Qn+~ is set to Qn+i m+ . At each step of the Newton iteration, a large sparse linear system of equations has to be solved. A symmetric Gauss-Seidel relaxation approach is used. In order to accelerate the convergence of the equation, a multigrid scheme (V-cycle) and a local time stepping method are employed. The full multigrid strategy (FMG) is used. In the FMG, the solution procedure starts with the coarsest grid and after some cycles in the coarsest grid, the solution is interpolated into the next finer grid. This time, the solution is obtained with three-level full multigrid method. The procedure repeats until the solution of the finest grid is obtained. CALCULATION OF SEAKEEPING PERFOMANCE The seakeeping performance of the ship is evaluated by use of a strip theory (Salvesen-Tuck-Faltinsen method) in this design system t93. The Strip theory is usually used to evaluate seekping performance in primarily design for shipbuilders. This theory is practical design tool for seakeeping performance. Consider a ship advancing with constant speed U and in the incident wave with small-amplitude motions of angular frequency ~ in deep water. The longitudinal coordinate system is shown in Fig.4. The x-axis is positive toward bow, and z-axis is positive downward. X is the angle between the direction of ship course and the direction of incident waves. The wave condition in this study is only one case. The direction of wave is in head sea, the ratio of ship length and wave height is 1/50 and the ratio of wave length and ship length is 1.0. for i = 1,3,5 Where Mjj denotes the generalized mass matrix end A,j, Bij and Cij are the added mass, damping and restoring coefficients. These are evaluated independent of the hydrodynamic analysis. The wave excitation forces and moments are determined in accordance with the STF method, and nonzero term among there are ME ~ = M33 = pV M55 = It l C33 = Maw C35 = C53 = -,~AWXW C = ,~VBML J Here V is the displacement, IN the moment of inertia about the y-axis, Aw the water plane area, xwthe center of the water plane area, and BM, the longitudinal metacentric height. The added resistance is calculated from Mauro's formula t103: Ma to [ ~0 ~ ~ :iC(k)| + |S(k)| ~ x vie) {k - ko cos X) do (~28) jv (k) - k Where v(k) = (~+kU2)1g = k+2~+k2 /Ko k i=- 20 (I+2T+~). k i= 20 (1-2r+~) K=~/, r=Uw/g, Ko=~2, a is wave height Here C(k) and S(k) denotes the symmetric and antisymmetric parts of the Kochin function respectively, and they can bee written as: C(k) = C7 (k) ~ Cj (k) S(k) = S7 (k) -—Ad,—S j (k) | (29) where j denotes the mode of ship motion. The details of the solution method for these velocity potentials aren't described in this paper.

RESULTS OF DESIGN USING RESPONSE SURFACE METHODLOGY In the ship hull form design, we use 5 parameters as Of ,Xa,Cf ,dcf Edna . The Constraint conditions in this study are fixed displacement and keep space at local position (e.g. engine room) and fixed principal dimensions ~ L, B. dm ). The object functions are resistance, added wave resistance, amplitude of ship motion. a) Wigley model (case of two design parameters) The case of two design parameters is carried out and the result can be visualized in order to recognize the effect of design parameters easily. Wigley model is used for simplicity to handle the hull form. The principal dimensions of this model are Length 5.0 m, Breadth 0.5m and draft 0.3125m. The two design parameter are Cf and xa . The other parameter is fixed in optimization process. The range of parameters is shown in table 1. The ranges of design parameters are determined to consider geometry constraint conditions roughly. Computational conditions are as follows, Rn = 5.57 x 106 end Fn = 0.25 in the present work. The only one of incident wave condition is studied. The wave condition is in head sea and wavelength ratio is 1.0 and wave height ratio is 1/50. The numbers of grid system for Wigley model is 105X33X33 and H-O grid system is used. Although the present grid is relatively coarse, it is sufficient to grasp the relation between the solutions and hull deformation. The hull data used in Strip theory is generated through interface grid module from this grid. First of all, the benchmark test of an optimization problem is done to check the optimization efficiency of the RSM. We have optimized a single objective function putting weighting factor on multi objective functions of resistance, added wave resistance and ship motion for Wigley model. The simulation condition is above. Fig.5 shows the convergences of Optimization process using the RSM. The results of Successive Quadratic Programming and Genetic Algorithm have been shown in Fig.5. Thus, the RSM has be able to search the optimum value by 12 calculations. On the other hand, the SQP has searched optimum value almost by almost 50 calculations and the GA have not converged by 150 calculations yet. We have understood the optimization efficiency of the RSM from these results Next, the approximated response surface using RSM compare with response surface using calculation data for design parameters to investigate effectiveness of the RSM. In order to express actual response surface is used the results of calculation for design parameters (20 level in each design parameter). Thus the number of calculations is 400 (20 X 20~. The components of multi objective functions in this study are the resistance coefficient c' as the propulsive performance, the added wave resistance coefficient Craw, the nondimensional amplitude of heave motion z' and the nondimensional amplitude of pitch motion B' as the seakeeping performance. The 400 calculation conditions are eliminated from consideration any infeasible design which exceeded any of the geometric constraint. At least 12 calculations of estimation are needed to fit a quadratic polynomial in two variables. Note that preliminary research indicates that the number of calculations in RSM process needs at least 1.5 to 2.5 times than the number of unknown coefficients [11~. So we choose 24 calculation conditions twice as many as the number of coefficients in RSM model. The 24 parameters are selected in order to minimize (XTX)~' by D-optimal design method. The 24 configuration of ship hull modified by using these parameters are evaluated using CFD tool and strip theory. The RSM can construct response surface by 24 results of calculation. The response surface made in the RSM calls ARS (approximated Response Surface by using the RSM. Also, in order to understand actual effect of the difference of hull form, we constructed the response surface of numerical experiment (RSNE) by interpolating the results of calculations for all levels in each design parameters. The correlation coefficient of the ARS and the RSNE is almost 1.0. As one example, the correlation of resistance is shown in Fig.6. So approximate values of RSM agree with calculation results well. The ARS and RSNE in each objective function are shown in Fig.7. We have understood that the results of the ARS agree well with the feature of response surface configuration and the position of minimum from the view of response surface. For cat and 0' in Fig.7, we understand that the RSNE and the ARS are almost quadric surface for the change of design parameters. On the other hand, for era,,, and din Fig.7 the minimum of added wave resistance and the amplitude of heave motion change linearly as the design parameters change. The design parameter Cf seems to influence on added wave resistance and amplitude of heave motion. A linear term of RS model is predominant for values of this estimation. From the response surface, we recognize that enlarging the front area of the ship minimize these estimation values. To minimize multi-object functions by using ARS, Genetic Algorism is applied as one of optimization methods. The weights of objective functions are equal to each estimation value. The constraint condition isn't needed to decide in this optimization process, because the upper bound and the lower bound of the design parameters have already been decided under the geometrical constraints conditions. The optimized solutions can be obtained in 200 calculation cycle by using GA. This optimization calculation takes 1 minute on Personal Computer. The results of optimization are shown in % as

compared to that of original hull form in Fig.8. The design parameters become (-0.1, -0.01, 0.1, -0.04, -0.01~. resistance is reduced in about 5%. The coefficient of The optimized solutions can be obtained in 500 calculation Added wave resistance is reduced in about 1%. Other the cycle by using GA. This optimization calculation takes a amplitude of ship motion is reduced a little. We understand few minutes on Personal Computer. We evaluate the that this design system is successful and useful for multi performances of two designed hulls. The multi objective objective optimization. Fig.9 shows the comparison of functions of two design hull forms are shown in % as optimized hull form and original hull. The optimized compared to that of original hull form in Fin. 11. hull form is to reduce the fore area for original hull form and become V type at fore part. The stern of designed ship expands slightly. b) Series 60 ship model (case of five design parameters) Next, the five design parameters of Of if ~xa~dcf and dCa are used in order to apply for practical problem. Using five design parameters become more flexible. The ranges of parameters are shown in table 2. The bounds of design parameters are determined to consider geometry constraint. Multi objective functions are the resistance coefficient c`, the added wave resistance coeff~cientcr=,,, the nondimesional amplitude of heave motion z' and nondimensional amplitude of pitch motion B' . The principal dimensions of Series 60 model in this study are Length 7.0 m, Breadth 0.933m and draft 0.373m. The computational conditions are Rn = 1.49 x 107 and Fn = 0.25 . The only one of incident wave condition is studied in this case. The wave condition is in head sea and wavelength ratio is 1.0 and wave height ratio is 1/50. The numbers of grid system for Series 60 model is 105X57X33 and H-O grid system is used. Firstly we choose 3125 design parameters (5 level in , I, c7 Firstly in case of optimization of total performance, the resistance (in calm water) is reduced in about 2.0%. The coefficient of Added wave resistance decrease slightly. Also the amplitude of pitch motion is reduced in about 1.0%. The amplitude of heave motion decreases slightly. The resistance is slightly improved. Since Series 60 model is thin hull form, the little modification of hull don't influence greatly on the resistance. Fig.12 shows the body plan of original and designed ship. The aft part of modified hull form change near U type. The aft part is smooth compared with original hull form. For optimization of seakeeping performances, the amplitude of pitch motion is reduced in around about 5%. The amplitude of heave motion increases slightly. The resistance and the coefficient of Added wave resistance and increase slightly. For the increase of the added wave resistance, we think that the sensitivities of weight functions for the added wave resistance and amplitude of ship motion aren't the same order. In this case, we have found that the amplitude of heave motion is related to the added wave resistance compared with the amplitude of pitch motion. From the body plan of designed ship in Fig.12, we understand that the fore and aft part of hull form become near V type to enlarge damping force. Thus, above simulation results show that the designer each design parameter). The quadratic model in five analyzes the trade-off of multi-object functions easily variables needs 21 coefficients. So we select 42 design ~~ parameters twice as many as number of coefficient by using D-optimal design. These 42 configurations are evaluated using by CFD tool and strip theory. The response surfaces are constructed from 42 results of calculation. The each correlation coefficient of ARS and RSNE in multi objective functions is almost 1.0 respectively. Fig.10 show the correlation of the ARS and the RSNE in c,. So we conclude that ARS can presume an actual response for design parameters in this problem enough. We minimize multi-object functions of ARS by using optimization method GA. The constraint conditions aren't considered in optimization process, since the bounds of design parameters are determined under geometry constraint. The weight of multi objective function is used two cases. In one case, the weights are set to (1,1,1,1,1) for c,, craw, z' and 0' to minimize totally objective functions. As the results of optimization, design parameters become (-0.35, -0.01, 0,35, -0.025, -0.040~. In another case, the weights are (1,10,10,10,10) to minimize the seakeeping performance. As the results of optimization, without large computational cost. CONCLUSION This paper concerns ship hull design for multi-objective function. The main focus is placed on the development of optimization scheme. The effectiveness of the RSM can be shown in comparison with the RSNE and the ARS for two design parameters using Langley model. In this study, He quadratic model have sufficient accuracy to express response surfaces of multi-objective functions for hull form design. The RSM model with 5 variables can be applied to design Series 60 ship model for multi-objective functions. As weight functions of multi objective function are changed, we can design 2 ship configurations for op~ni~ion of total estimation of performance and seakping performance. We show that the desired ship hull can be estimated easily by changing weight coefficient of a multi-objective function

without using large-scale calculation tools. As next step, we'll try to add propeller effect and short-term prediction to design system multi objective function propeller effect estimate sea-margin (needed engine power) in actual seas. On the other hand, the easy and practicable modification method is used in this study. We need to devise much more suitable modification method of ship hull form for multi-objective design in the future. In additions as the coefficient of RSM model increases to express complex configuration, we have to develop a design technique that approximate the response surface with quick response for change of design parameters. ACKNOWLEDGEMENT Optimal Design of lifting Bodies", Journal of the Kansai Society of Naval Architect, No. 235, pp.1-8, 2001 to] Burgreen, GW. and Baysal, O.: "Three-Dimensional Aerodynamics Shape Optimization of Wings Using Sensitivity Analysis" ,ALAA Paper 94-0094, 1994. t8] Hirata7 N. and Hino, T.:"An Efficient Algorithm for Simulating Free-Surface Tublent Flows around an Advancing Ship", Journal of the society of Naval Architects of Japan7 Vol.185, pp.1-8, 1999 [9] Salvesen7 N., E.O. Tuck and O. Faltinsen: "Ship motion and sea load ", TSNAME, Vol. 78, 1970 We would like to thank Dr. Todoroki of Tokyo Institute of t10] Mango, H.:'~Wave resistance of a ship in Regular Head Technology for his Response Surface Methodology Software. Sea", Bulletin ofthe Faculty of Eng., Yokohama National (http://floridames.titech.ac jp/todo-j.htrnl). REFERENCES [1] Hinn T. Kodama Y and Hirers N. . "nv`1r~1vnamin ~ ma, ^., ^~7 ^., FAX ~ ~ ~7 , ~ ~ Eva ~ `~ - Shape Optimization of Ship Hull Forms Using CFD,7 Prceedings 3rd Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design Osaka Japan May 25-27, 1998, pp. 533-541. [2] graham Y., Himeno, Y., and Tsu~har~ T.,: An Application of Computational Fluid Dynamics to Tanker Hull Form Optimization Problem,77 Proceedings 3 rd Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, Osaka, Japan, May 25-27, 1998, pp.515-531. [3] A. A. Giunta, V. Balabanov, S. Burgee, B. Geossman, R.T.Haftka, W.H. Mason, and L. T. Watoaon, "Multidisciplinary optimization of a super sonic transport using design of experiments theory and response surface modeling", Aeronautical J., vol.101 (1008), pp.357-356, 1997. [4] Myera,R.H. and Montgomery,D.C. "Response Surface Methodology, Process and Product Optimization Using Design Experimets", pp.1-141, 279-401,462-480,John Willey & Sons, New York, NY (1995) [5] Kriah~rnar, K.: "Micro-Genetic Algorithms for Stationary and Non-Stationa~y Function Optimization", SPA: Intelligent Control `and Adaptive System, Vol. 1196, Philadelphia, PA, 1989. [6] Md. Mashud KARlM, M. Ikehat`a, K Suzuki `and H. Kai: "Application of Micro-Genetic Argorithm ~ ,u GA) to the Univ., Vol.9, (1960), pp.73-91 t11] Guinta, A.A., Dudley, J.M., Narducci, R. Grossman, B., Havana, R. T., Mason, W. H., and Watson, L. T.: "Noisy Aerodynamic Response and Smooth Approximations in HSCT Design" , Proceeding of He 5~ A AA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, pp. 1117-1128,Panama City Branch, FL, AIAA Paper 944376, 1994 | Initial ship configuration ~ I Desineen parameter set I Modification ofhullform ~ DisplacementisfL'ced Sinusoidal function t~ ) Generate grid system | ~ . ~ ~ . Calculation grid / I Interface of grid system 1l ~ StriD data at each section | Calculation of Resistance | [ Calculation of Seakeeping ~ ~ , CFD \ REM modeling Search | Multi objective optimizer |optimum ship hull form | ,/ Strip Theory Fig.1 The flow of design system for multi objective problem

I — O ri~n al H u 11 fo rm /0.8 0.6 ~ 0.4 _' 0.2 O ~ . _ . ~ ~ m o difie d h u 11 fo rm . - . N_ 5.2 5.0 ~0 5 Xa O Xf 0.5 `' 4.8 X 0 0.1 0.2 0.3 o -0.05 -0.1 N -0.1 5 -0.2 -0.25 -0.3 -0.35 1 dcf' dca ............. ............... 7~. l Fig. 2 Modification system of ship hull form I Hull Surface/ K ~. Outer Boundary |K- kmaX Fig. 3 Rearrangement of calculation and corresponding to ~e modification of hull fo~m ~r O1 ~ Z ~_ ~ X1 X Fig.4 Coordinate system ~ _.~OP 1 ~GA · RSM p~ ~_1- ~U 4.6 U~U 1 4.4 0 50 100 150 200 iteration no. Fig. 5 Companson of convergence histo~y of Objective f unction 8.0E-04 cn c, 4.0E-04 ,< O.OE+OO / O.OE+OO 4.0E-04 8.0E-04 Ct RSNE Fig.6 Correlation of ~e RSNE and ~eARS Table 1 Design parame~rs in ~lgley model case Design Variables Range or Value Xf 0.3 (fixed) —0.40 < xa < 0.10 Cf -0.10 < CS < 0.10 dcf -0.03125 (Fixed) dca -0.03125 (Fixed)

7 z o. o.~ - O.` . o.2 -- 0.3 - xa 0.4 RSNE z x Y z _ in, - u. 'O. 1 If z O. ARS 7 z Fig.7 Comparison of response surface of multi objective functions between forge RSNE and the ARS

Orignal Ct Craw z' v~ I I :: : _ 1 ~ _ 1 Cl . . Yll~ I ~ _ ~ R ~ ; ~ _ _~ _ =— - _ . . . ~ __ = l l l - - _~ ~_ 0.9 0.92 0.94 0.96 0.98 1 1.02 Fig.8 Objective function for optimizes results as compared to the value of original hull form (Wigley model case) i rllllrltmllllr -0.04 -0.02 0 0.02 0.04 y a) original - D! -0.04 -0.02 0 0.02 0.04 y b) modified Fig.9 Comparison of original hull form and optimized hull form 5.9E-04 ~' 1 ~ k'! ! 1 5.7E-04 5.7E-04 5.9E-04 6.1 E-04 6.3E-04 Ct RSNE Fig. 10 Correlation ofthe RSNE and the ARS Table2 Design parameters in Series60 model case Design Variables Xf xn I Cf I L dCf I dca l | Range or Value 1 o.lo<xf<0.40 ~0.40 < Xa < -0.10 -0.01<Cf <0.01 1 -0.04 ~ dcf ~ -0.01 I -0.04 < dCa ~ -0.01 1 i ~ 1 1 1 1 O· I l~ 1 rlgna' _T I ~' 1~1 1 ~raw _— I 1 1 Z 1 1 F I I I I I I ~ql ~ 1 1 v ~ I I 1 1 1 0.9 0.92 0.94 0.96 0.98 1 1.02 a) Optimization for the total performance Orignal Ct Craw v~ 1 1 i 1 i ~ 1 ~ 1 -F I I I I i 1 1 ~ r I I T T I 1 l ~ I I I I 1 1 -___~ 1 ~ 1 ~ i ~ 1 _1 1 1 1 I ~ 1 1 1 0.9 0.92 0.94 0.96 0.98 1 1.02 b) Optimizaiion for the Seakeeping performance Fig.11 Objective function for optimizes results as cor~pared to the value of original hull form (Series 60 model case)

~;~ 0.05 . . . . _ -0.05 0 y a) Original _ ~~\\\A.P. ~ p I_ ~ 1\\\\\ 11 1 ~ _ ~ ' ~ ~ , ., 1,, , . _ ~ ., , ~ 05 0 0.05 y 0.025 o -0.025 -0.05 0.025 o -0.025 -0.05 N N by Optimized hull form for minimization of resistance F.P n non _~: ~ . . ., ., ~ 1- 0 05 ~ ~ 0.1 35 y O -0.025 -0.05 N c) Op~ni~d hull form for minimization of seakeeping Fig. 12 Comparison of body plan between for Me original hull forth and optimized hull form

DISCUSSION L.J. Doctors University of New South Wales, Australia The reviewer would like to congratulate the two authors on a most practical and interesting piece of research. Thus, skipping momentarily to the results, one can see in Figure 7 and Figure 10, for example, how the performance of the ship has been improved by means of the described optimization process, with respect to its total resistance, added resistance in waves, and motions in waves. Furthermore, it is noteworthy to observe the optimized hull form in Figure 11 displaying a bulbous bow, when seakeeping is considered, as one might expect. My first question relates to the fact that the resistance of the ship is computed on the basis of a CFD code, in which one presumably solves the Navier-Stokes (viscous-flow) equations with nonlinear free-surface conditions. The only essential approximation here would be the particular turbulence model employed by the computer program. On the other hand, the authors have used the Salvesen, Tuck, and Faltinsen (1970) theory for computing the ship motions. This method is very practical and well behaved but, nevertheless, assumes that the motions are small and that the ship is slender. The Maruo (1960) formula in Equation (31) is similar in that linearized free-surface conditions and a thin ship is assumed. To what degree do the authors believe this inconsistency in the different theories to be a concern? That is, perhaps, one could utilize a simpler, thin-ship, theory for the ship resistance, as was done by Day and Doctors (1997a and 1 997b). Secondly, I would appreciate the authors indicating a typical number of required ship- performance evaluations during the optimization process. Thus, in the work of Day and Doctors, in which the efficient Genetic Algorithm was utilized, the vessel performance had to be evaluated thousands of times. Only by using the various features of the abovementioned linear theories, could the computation be kept reasonable. It would therefore seem, on the other hand, that the optimization process using a CFD solver would be extremely time-consuming. Thirdly, could Equation (30) be clarified? The standard Salvesen, Tuck, and Salvesen theory predicts ship motions in five degrees of freedom (indexes 2, 3, 4, 5, and 6~. Alternatively, for simple motions in head or stern seas, only two degrees of freedom are considered (indexes 3 and 5~. Have the authors here (indexes 1, 3, and 5) decided to include the surge degree of freedom? Once again, I would like to express my appreciation to the authors for a most informative paper. AUTHORS' REPLY Thank you for your interest in our work and your questions. For first question, we think that the inconsistency of these theories is a difference where designer gives priority in design process. Each evaluation function of performances is calculated parallel and independently in optimization process of the RSM. The propulsive performance is influenced by viscous flow around hull. Moreover, this performance influences greatly on economy of ship. So we have wanted to estimate this performance strictly by using CFD. CFD tool can simulate a viscous flow in high accuracy and get information of flow field around hull for hull design or propeller design etc. On the other hand, ship motion can be estimated practically in normal wave condition by strip method excluding rough sea condition. We have checked the relative improvement of seakeeping performance for the original using by strip method. For second question, we have wanted to examine whether the Genetic algorithm (GA) is effective to the multi-objective optimization problem in hull design. In this study, a multi-objective function has been converted into a single object function putting the weighting factor. The GA can search the optimum of a single response surface in several minutes. But the searching time of the GA isn't shorter compared with that of other optimization method when searching the optimum of the response surface. One of reasons is that the multi-peak of response surface doesn't appear by using simple Response Surface model and simplification of multi-objective function.

We understand that general optimization method is enough for this optimization case. In the future, we have wanted to investigate the optimization method for searching the optimum of high-order response surface model. For third question, the equation of general longitudinal motion is described in this thesis. But I'm sorry, I have not been describing clearly in this thesis that the influence of the surge motion is not considered in optimization process in order to simplify the calculation of ship motion.

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