ESTABLISHMENT OF EFFECTIVE METAMODELS FOR SEAKEEPING PERFORMANCE IN MULTIDISCIPLINARY SHIP DESIGN OPTIMIZATION

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1 Journal of Marne Scence and Technology DOI: /JMST Ths artcle has been peer revewed and accepted for publcaton n JMST but has not yet been copyedtng, typesettng, pagnaton and proofreadng process. Please note that the publcaton verson of ths artcle may be dfferent from ths verson.ths artcle now can be cted as do: /JMST ESTABLISHMENT OF EFFECTIVE METAMODELS FOR SEAKEEPING PERFORMANCE IN MULTIDISCIPLINARY SHIP DESIGN OPTIMIZATION Dongqn L a, Phlp A. Wlson b, Xn Zhao a a School of Naval Archtecture & Ocean Engneerng, Jangsu Unversty of Scence and Technology, Zhengjang, Jangsu, Chna, 2123 b Flud Structure Interactons Research Group, Faculty of Engneerng and the Envronment, Unversty of Southampton, Southampton, UK, SO16 7QF Keywords: metamodel, Support Vector Machne, Desgn of Experment, seakeepng ABSTRACT Shp desgn s a complex multdscplnary optmzaton process to determne confguraton varables that satsfy a set of msson requrements. Unfortunately, hgh fdelty commercal software for the shp performance estmaton such as Computatonal Flud Dynamcs (CFD) and Fnte Element Analyss (FEA) s computatonally expensve and tme consumng to execute and deters the shp desgner s ablty to explore larger range of optmzaton solutons. In ths paper, the Latn Hypercube Desgn was used to select the sample data for coverng the desgn space. The percentage of downtme, a comprehensve seakeepng evaluaton ndex, was also used to evaluate the seakeepng performance wthn the short-term and long-term wave dstrbuton n the process of Multdscplnary Desgn Optmzaton (MDO). The fve motons of shp seakeepng performance contaned roll, ptch, yaw, sway and heave. Partcularly, a new effectve approxmaton modellng technque Sngle-Parameter Lagrangan Support Vector Regresson ( SPL-SVR ) was nvestgated to construct shp seakeepng metamodels to facltate the applcaton of MDO. By consderng the effects of two shp speeds, the establshed metamedels of shp seakeepng performance for the short-term percentage of downtme are satsfactory for seakeepng predctons durng the conceptual desgn stage; thus, the new approxmaton algorthm provdes an optmal and cost-effectve soluton for constructng the metamodels n MDO process. Ⅰ. INTRODUCTION An accurate and effectve predcton technque for seakeepng performance plays an mportant role n the hydrodynamc-based Multdscplnary Desgn Optmzaton (MDO) for shps. In order to obtan the accurate result of seakeepng predcton, frstly a hgh-precson calculaton method s requred n the prelmnary shp desgn stage, for example the strp theory rather than emprcal regresson models whch are wdely used n shp seakeepng predcton (Ö züm, 211). Secondly, the adopted calculaton method for seakeepng can be easly ntegrated nto the MDO process wthout any artfcal nterventon. Thrdly, perhaps the most mportant part s to mnmze the computatonal cost and complexty. In our prevous research, the smulaton codes for shp resstance and seakeepng performance were mplemented n the MDO (L et al., 212a; L et al., 212c; L et al., 213), unfortunately the calculaton were extremely expensve and tme-consumng. Although, hgh- performance computers are now correspondngly more powerful, the hgh computatonal cost and tme requrement stll

2 Journal of Marne Scence and Technology, Vol. **, No. * (215) lmt the use of MDO method n engneerng desgn and optmzaton. So far, a technque of metamodel (or surrogate model) can be adopted to solve ths problem n MDO (Lefsson and Kozel, 21), and s used to create a fast analyss module by approxmatng the exstng computer smulaton model n order to acheve more effcent analyss. The am of ths paper s to mprove a new smple and effectve algorthm of Support Vector Machnes (SVM) as a surrogate model to predct the shp seakeepng performance. Ⅱ. NECESSITY OF METAMODEL IN MDO Shp desgn essentally apples teraton to satsfy the relevant dscplnes, such as structural mechancs, economcs and hydrodynamcs, and may be nvestgated by dfferent teams of engneers wth dfferent smulaton codes. Due to these complextes, the MDO problem for shps s extremely hard to descrbe and compromse among several dscplnes. Furthermore, hgh-fdelty calculaton for shp performance wth CFD software n the MDO framework s lkely to be much more dffcult to acheve. One way of allevatng these burdens s by constructng approxmaton models, known as metamodels or surrogate models, whch are used to replace the specfc smulaton-based calculaton n MDO. A varety of metamodelng technques have been successvely developed, such as Artfcal Neural Network (Hoque et al., 211), Response Surface Method (Balabanov, 1997) and Krgng method (Zhang et al.,213), as surrogates of the expensve smulaton process n order to mprove the overall computaton effcency. They are then found to be a valuable tool to support a wde scope of actvtes n modern engneerng desgn, especally the shp desgn optmzaton. The accuracy of metamodels n the MDO wll greatly affect the result of optmzaton, so t s mportant to select a proper metamodelng approach especally when the sample szes become small and lmted. As a novel artfcal ntellgence approach, SVM specfcally target the ssue of lmted samples and acheve a good generalzaton performance as well as a global optmal extremum (Vapnk, 25). Hence, a new metamodelng technque whch we wll desgnate as Sngle-parameter Lagrangan Support Vector Regresson (SPL-SVR) has been developed and used for the constructon of metamodels of shp seakeepng performance n ths artcle. Ⅲ. DESIGN OF EXPERIMENTS Desgn of Experments (DOE) s a very powerful tool that can be utlzed n shp desgn. Ths technque enables desgners to determne nteractve effects of many factors that could affect the overall desgn varables, such as beam, draught, length and also provdes a full nsght of nteracton between desgn elements. 1. Latn Hypercube Desgns Latn Hypercube Desgn (Mckay, 1979) s chosen to gather the sample data whch wll be used to construct the metamodels. Ths method chooses ponts to maxmze the mnmum dstance between desgn ponts and mantan the even spacng between factor levels. The essence of Latn Hypercube Desgn s to control the poston of the samplng ponts and avod the problem of small neghbourhood concdence. The advantages are lsted as follows: (1) Columns and rows are all orthogonal. (2) Mutual exchange of columns or rows wll not change ther nature. (3) The number of samples (ponts) s not fxed. 2. Dstrbutons of shp samples The shp nformaton about the offshore supply vessels (OSV) was gathered from the shppng companes and desgn nsttutons. At the same tme, some desgn parameters were fxed to make the model smple and feasble. From these shps, we can tell that the OSV usually have 2 propellers and large block coeffcent. The dstrbutons of prncpal dmensons, length L pp, beam B, draught D,

3 D. L et al.: Metamodels for seakeepng performance n MDO depth T and deadweght tonnage DWT for these shps are shown n Fg. 1. velocty and wave angle were chosen as the desgn varables, whch can show specfc shape characterstcs of shp hull. The range of values for desgn varables are lsted n Table 1. Table 1 Range of desgn varables n DOE Desgn varables Length symbol Lower bound Upper bound Intal desgn L pp / m Breadth B / m Depth D / m Draught T / m Block coeffcent Prsmatc coeffcent Longtudn al centre of buoyancy Velocty Wave angle C b C p L CB / m V s /Kn 14.5 /14.5 / The Latn Hypercube Desgn n standard Model-based calbraton toolbox from commercal software Matlab was chosen to establsh the tranng data set, whch are used to construct and dscover a predctve relatonshp. Ffteen sets of shp tranng data were collected and the space dstrbuton s shown n Fg. 2. One shp hull of the tranng shps s shown n Fg. 3. Fg. 1 Dstrbuton of offshore supply vessels' prncpal dmensons wth DWT In fact, the shp seakeepng performance wll be affected by varous factors. However, the addton of more varables to the metamodel would hamper the result evaluaton and the methodology valdaton. Eventually, the length between perpendculars, breadth, depth, desgn draught, block coeffcent, longtudnal prsmatc coeffcent, longtudnal centre of buoyancy, shp Fg. 2 The space dstrbuton of tranng data set

4 Journal of Marne Scence and Technology, Vol. **, No. * (215) 2 adds b / 2 to the tem of confdence nterval at the same tme, and adopts the Laplace loss functon. Ths s termed the Sngle-parameter Lagrangan Support Vector Regresson (SPL-SVR). Hence the formula s stated as follows: Mn 1 T ( w w b 2 2 ) C l 1 Fg. 3 Transverse secton and 3D lay-out of shp hull Ⅳ. MATHEMATICAL FOUNDATION OF SPL-SVR The SVM (Vapnk, 1995; Smola et al., 24) s based on Statstcal Learnng Theory (SLT), whch has been recognzed as a powerful machne learnng technque. It offers a unted framework for the lmted-sample learnng problem and can solve the practcal problems such as model-choosng, multple dmensons, non-lnear problems and local mnma. By learnng from the tranng samples, the obtaned black box can descrbe the complcated mappng relaton wthout knowng the connecton between the dependent varables and ndependent varables. Classcal Support Vector Regresson (SVR) has been used to construct the metamodels n the Mult-objectve optmzaton (Yun et al., 29) and the result demonstrated that SVR can offer an alternatve and powerful approach to model the complex non-lnear relatonshps. In ths paper, we descrbe a new algorthm of SVR to establsh the metamodel of shp seakeepng n Multdscplnary Shp Desgn Optmzaton, whch was proposed n our prevous work (L et al., 212b) and recalled here for the readers convenence. Gven a tranng data set, x, ),, x, ), where x X, ( 1 y 1 ( l yl y R, l s the sze of tranng data. In order to reduce the overall complexty of the system, the new algorthm of SVR has only one parameter to control the errors nstead of * two parameters, n the classcal SVR, and s.t. T y w ( x ) b, 1,, l (1) The soluton of (1) can be transformed nto the dual optmzaton problem. are slack varables, b s the error bas, *, are Lagrange multplers, and C are coeffcents to control the VC dmenson of regresson functon. A Lagrange functon can be constructed and by the ntroducton of a kernel functon K x, x ) ( ( x ) ( x )), whch ( j corresponds to the dot product n the feature space gven by a nonlnear transformaton of the data vectors n the nput space. The problem posed n (1) can be transferred nto the dual optmzaton problem as follows. l 1 * * Mn ( )( ) K ( x x ) 1 l 1 2, j1 * j ( ) y ( ) s.t. * l 1 j * ( ) C,, (2) * The above formula can be stated n standard quadratc programmng problem. Suppose that: K K X *, H, 2l1 K K 1 A j y d y 1 l2l 2l1

5 D. L et al.: Metamodels for seakeepng performance n MDO K( x1, x1) K x x K ( 2, 1) K( xl, x1) K( x1, x2) K( x2, x2) K( xl, x2) K( x1, xl ) K( x x 2, l ) K( xl, xl ) ll (3) The dual problem can then be expressed n the followng standard quadratc programmng form. Mn s.t. 1 T T X HX d X 2 AX C X (4) Thus, the approxmaton functon s calculated as follows: l f ( x) 1 * ( ) K( x x) 1 (5) The complexty of the above functon only depends on the number of support vectors (SVs). In ths paper, the Radal Bass Functon (RBF) s used. K( x x ) exp( x x ) (6) j 2 j Where () denotes the nner product n the space, a feature space of possbly dfferent dmensonalty such that :X and b R. From the above formulas, t can be seen that the algorthm proposed n ths paper s smpler than the classcal SVR. Moreover, there s no need to compute the bas b whch wll mprove the effcency and accuracy of the calculaton. Ths new algorthm has been demonstrated to be n farly good agreement wth expermental measurements (L et al., 212b). Therefore, ths new algorthm s sutable for constructon of metamodels for the shp seakeepng predcton n the prelmnary desgn process. Ⅴ. CALCULATION OF SHIP SEAKEEPING PERFORMANCE AND ESTABLISHMENT OF METAMODEL Before establshng the metamodels of shp seakeepng performance n MDO, an effcent smulaton method for the shp seakeepng performance for OSV should be chosen together wth the typcal wave condtons that the shp desgn s lkely to be operatng wthn. 1. The actual wave condtons The wave spectrum attempts to descrbe the ocean wave condtons after a wnd wth constant velocty blowng for a long tme. A typcal ocean wave spectrum wll be much more complcated and varable. The JONSWAP spectrum whch s sutable for deep water areas such as the North Sea and South Chna Sea s capable of gvng the safe analyss results of shp motons. It s descrbed as follows: S ( ) H 1 / 3 T p exp 1.25( TP) exp ( T 1) P 2 / 2 S () spectral densty functon( m 2 s ); wave frequency; H 1/3 sgnfcant wave heght( m ); T p peak wave perod ( s ); p spectral peak frequency, peak enhancement factor. 2 p 1/ T ; 2. Wave Scatter Table for South Chna Sea p (7) Consderng the actual wave nfluence n desgn, the long term trends for the maxmum wave parameters such as sgnfcant wave heght and modal perod of the waves wll be needed. In order to create a long-term forecast, the Wave Scatter Table needs to be known whch represents the jont probablty dstrbuton of the sgnfcant wave heght H 1/ 3 and zero up-crossng perod T Z. The wave nformaton for South Chna Sea (Hogben et al., 1986) s lsted n Table 2, where the area ranges s east longtude,.5-23 north lattude. The shp seakeepng performance wll be nfluenced by varous factors. The allowed values of seakeepng crtera or probabltes for shp motons were estmated from the nformaton whch were gleaned from OSV operators and lsted n Table 3.

6 Journal of Marne Scence and Technology, Vol. **, No. * (215) H 1/ 3 (m) Table 2 Wave Scatter Table for South Chna Sea (Annual) T Z < > Total Table 3 Allowed values of seakeepng crtera or Seakeepng crtera probabltes for OSV Unt (s) 3. Comprehensve evaluaton ndex for seakeepng performance Value or probablty kn Weght 14.5kn Roll Ptch Heave m Deck wetness % Slam % Propeller emergence % Vertcal acceleraton at bow g.4g.2.1 It s necessary to decde a proper comprehensve evaluaton ndex for shp seakeepng performance whch wll be used n the process of MDO. Typcally two ndces are used: the frst s the percentage of workng tme, the second s the percentage of desred speed. Here a new ndex of long-term forecast percentage of downtme s proposed and t s an effectve evaluaton ndex for seakeepng qualty, whch shows the ablty of shps workng n the prescrbed condtons (envronment and tme). Specfc calculaton steps are lsted as follows: Step 1: Choose the navgaton and workng veloctes of OSV accordng to ts requrement. Calculate the frequency response transfer functons under the specfc velocty V s and wave angle m n regular wave condtons. Step 2: Select the ocean wave spectrum, predct the shp moton responses and acceleratons by the sgnfcant wave heght n rregular wave condton. Step 3: Gather nformaton about the actual envronmental condton and wave statstcal probablty dstrbuton P H, T ) (Scatter ( j Dagrams), to establsh the seakeepng crtera group C k. Step 4: Calculate the moton Response Ampltude Operators (RAO) ( r a ) 1/ 3 for varous seakeepng crtera factors under the specfc velocty wave perod T j V s, wave angles m and. Then calculate the lmtng wave heght H smjk and the percentage of downtme POT smk for dfferent seakeepng crtera factors k under the specfc velocty V and wave angle. s m Step 5: Based on the weghtng coeffcent k of each seakeepng crtera factor k n the seakeepng crtera group C k, calculate the comprehensve evaluaton ndex POT k whch s named the short-term percentage of downtme. Ths ndex ndcates the ultmate workng capacty of shps under the gven velocty and wave angle. POT k POT k k k smk k Step 6: Consderng the velocty frequency dstrbuton f ) and wave angle frequency v ( V s dstrbuton f ) n the real voyage, the ( m comprehensve evaluaton ndex POT for seakeepng performance whch s named the long-term percentage of downtme can be calculated as below: POT fv( Vs ) f ( m) POTk s m

7 D. L et al.: Metamodels for seakeepng performance n MDO 4. Establshment of shp seakeepng metamodel The hydrodynamc desgn of shps nvolves several stages, from prelmnary or early-stage desgn to late-stage and fnal-stage desgn. As the purpose of ths study s to develop practcal metamodels for seakeepng performance evaluaton n the hydrodynamc-based MDO at the early shp desgn stage. a practcal calculaton tool, based on the strp theory called Seakeepng Manager from the commercal software NAPA, s used to calculate the shp motons, heave, ptch, roll, sway and yaw n rregular wave condtons. (a) Roll (b) Ptch (c) Heave (d) Deck wetness (e) Propeller emergence (f) Slam

8 Journal of Marne Scence and Technology, Vol. **, No. * (215) (g) Vertcal acceleraton at bow Fg.4 Response functon for the 7 seakeepng crtera ( V s = knots) The workng speed of OSV s knots and navgaton speed s 14.5 knots. The chosen wave seakeepng crtera can be used to evaluate the overall seakeepng performance of shps. angles are, 3, 6, 9, 12, 15 and 18. The Sngle-parameter Lagrangan Support One of those tranng shps s chosen as an example. As mentoned above, seven seakeepng Vector Regresson (SPL-SVR), an approxmaton method presented above s used to construct the crtera ncludng roll, ptch, slam, heave, metamodels of shp seakeepng performance at propeller emergence, deck wetness and vertcal acceleraton at bow, are predcted to evaluate the shp seakeepng performance. The response functons of these seakeepng crtera are shown n Fg.4. The percentages of downtme at dfferent wave angles are shown n Fg.5, whch ndcate the early desgn stage, wthout runnng expensve model tests or tme-consumng CFD smulatons. The computer code s set up wth the Matlab software,and t s based on the theory of Support Vector Machne and ntergrated n the process of MDO. that the comprehensve evaluaton ndex for seven (a) (b) 3 (c) 6 (d) 9

9 D. L et al.: Metamodels for seakeepng performance n MDO (e) 12 (f) 15 (g) 18 Fg.5 The percentage of downtme at dfferent wave angles ( V s =14.5 knots) 4.1 The ffteen shp types as the test data set was adopted and ts kernel parameters should be In ths stuaton, those ffteen shp types consdered carefully. The nne parameters lsted collected wth DOE method are selected as n Table 1 were chosen as the desgn varables, tranng data set and also as test data set. In the and the shp short-term percentage of downtme algorthm SPL-SVR, the RBF kernel functon as the output varable. (a) (b) 3 (c) 6 (d) 9

10 Journal of Marne Scence and Technology, Vol. **, No. * (215) (e) 12 (f) 15 The SPL-SVR method were compared wth Seakeepng Manager ANN and classcal SVR. The results wth knots and 14.5 knots were shown n Fg.6 and Fg.7. The Relatve Error (RE) and Mean relatve error (MRE) are appled y as performance ndexes: y RE *1%, y (g) 18 Fg.6 Fttng curves of 15 shp types ( V s = knots) * n * 1 y y MRE n y 1 and. Here, y s the real value y * s the predcted one. The results for the navgaton speed wth wave angle 12 s taken as an example, lsted n Table 4. (a) (b) 3 (c) 6 (d) 9

11 D. L et al.: Metamodels for seakeepng performance n MDO (e) 12 (f) 15 (g) 18 Fg.7 Fttng curves of 15 shp types ( V s =14.5 knots) Table 4 Results wth Relatve Error for downtme POT wth wave angle 12 ( V s =14.5 knots) Seakeepng Shp type Manager ANN SVR SPL-SVR number Value (%) Value (%) Relatve Error Value (%) Relatve Error Value (%) Relatve Error % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % short 4.2 The fve shp types as the test data set Smlarly, shp types 1 to 1 were selected as tranng data set and shp types 11 to 15 as test data set. The results wth knots and 14.5 knots were shown n Fg.8 and Fg.9. Due to space lmtatons, the result for the workng speed wth wave angle 3 s taken as an example, lsted n Table 5.

12 Journal of Marne Scence and Technology, Vol. **, No. * (215) (a) (b) 3 (c) 6 (d) 9 (e) 12 (f) 15 (g) 18 Fg.8 Fttng curves of shp type 11 to 15 ( V s = knots) From these metamodels for seakeepng workng speed knots and 6.24% for the performance whch are establshed wth the proposed SPL-SVR method, the total Mean Squared Error s 2.92% for the workng speed knots and 3.11% for the navgaton speed 14.5 knots when usng 15 shp types as test data set; the total Mean Squared Error s 3.3% for the navgaton speed 14.5 knots when usng fve shp types as test data set. Consderng the above two crcumstances, ths new algorthm SPL-SVR s sutable for the nonlnear approxmaton problem both n terms of reduced calculaton tme and accuracy. If the

13 D. L et al.: Metamodels for seakeepng performance n MDO tranng data set, the kernel parameters and the smulaton method for seakeepng performance are chosen properly, metamodels wth hgh precson can be generated for shp seakeepng performance and used to calculate the short-term seakeepng performance POT k nstead of a CFD method at the prelmnary shp desgn stage. Wth the shp speeds and wave angles n the real voyage, the comprehensve evaluaton ndex POT for the long-term seakeepng performance can be evaluated n the multdscplnary shp desgn optmzaton. (a) (b) 3 (c) 6 (d) 9 (e) 12 (f) 15 (g) 18 Fg.9 Fttng curves of shp type 11 to 15 ( V s =14.5 knots)

14 Journal of Marne Scence and Technology, Vol. **, No. * (215) Table 5 Results wth Relatve Error for downtme POT wth wave angle 3 ( V s = knots) short Shp type Seakeepng Manager ANN SVR SPL-SVR number Value Value Relatve Value Relatve Value Relatve (%) (%) Error (%) Error (%) Error % % % % % % % % % % % % % % % 4.3 The optmzaton results of the OSV An optmzaton platform s establshed here wth the professonal software Optmus. Dfferent modules of the offshore supply vessel are ntegrated n Optmus to demonstrate the applcaton of MDO n shp desgn. The exact soluton framework s shown n Fg.1. The MDO method s used to get the optmum results whch are shown n Table 6 and the optmzed hull lnes of OSV s shown n Fg.11. Fg.1 The framework for optmzaton n Optmus Table 6 The optmzaton result of OSV Varables Symbol Intal desgn Optmum Length L / m Breadth B / m Depth D / m Draught T / m Block coeffcent C b

15 D. L et al.: Metamodels for seakeepng performance n MDO Prsmatc coeffcent Longtudnal centre of buoyancy Speedablty C p L CB / m C SP/(1-3) Seakeepng Seakeepng Manoeuvrng Manoeuvrn g Cost Cost / $(1 7 ) Object F Fg. 11 The optmzed hull lnes of OSV Ⅵ. CONCLUSIONS In ths paper, a new SVR algorthm was proposed to establsh the metamodels for predctng the shp seakeepng performance of OSV. The valdty and relablty of the proposed approach has been evaluated n several dfferent ways. Comparng t to ANN and the classcal SVR, the proposed SPL-SVR can acheve more accurate results. At the meantme, usng metamodels n place of computatonally expensve computer models and smulatons can drastcally reduce the desgn tme and enable desgners and decson makers to explore larger range of feasble desgn solutons. Further research wll focus on the constructon of metamodels of shp performance ncludng shp resstance and manoeuvrng for dfferent commercal shps at the prelmnary desgn stage, also together wth the ntegraton method n Multdscplnary Desgn Optmzaton. In the future, we beleve that metamodel-based optmzaton wll have numerous potental applcatons n the feld of marne engneerng and shp desgn. ACKNOWLEDGMENTS The authors wsh to thank the Natonal Natural Scence Foundaton of Chna (Grant No ), the Natural Scence Foundaton of Jangsu Provnce of Chna (Grant No. BK212696, No. BK29722)and the Project funded by the Prorty Academc Program Development of Jangsu Hgher Educaton Insttutons (PAPD) for ther fnancal support. REFERENCE Balabanov, O. (1997). Development of approxmatons for HSCT wng bendng materal weght usng response surface methodology. Ph.D Thess, Vrgna Polytechnc Insttute and State Unversty, Blacksburg,VA. Hogben, N., N. Dacunha and G. Olver (1986). Global wave statstcs. Brtsh Martme Technology, Hoque, S., B. Farouk and C.N. Haas (211). Development of artfcal neural network based metamodels for nactvaton of anthrax spores n ventlated spaces usng computatonal flud

16 Journal of Marne Scence and Technology, Vol. **, No. * (215) dynamcs. Journal of the Ar & Waste Management Assocaton 61(9), Lefsson, L. and S. Kozel (21). Mult-fdelty desgn optmzaton of transonc arfols usng physcs-based surrogate modelng and shape-preservng response predcton. Journal of Computatonal Scence 1(2), L, D., Y. Guan and Y. L (212a). Multdscplnary desgn collaboratve optmzaton for the Offshore Supply Vessel. Shp Engneerng 34(6), L, D., Y. Guan, Q. Wang and Z. Chen (212b). Support vector regresson-based multdscplnary desgn optmzaton for shp desgn. Proceedngs of the 31th Internatonal Conference on Ocean, Offshore and Arctc Engneerng, Ro de Janero, Brazl. L, D., Z. Jang and Y. Yang (212c). Multdscplnary and mult-objectve desgn optmzaton based on adaptve weghted-sum method. Shpbuldng of Chna 53(4), L, D., Z. Yao and Y. Guan (213). Improved Multdscplnary Desgn Collaboratve Optmzaton for Offshore Supply Vessel. Journal of Jangsu Unversty of Scence and Technology (Natural Scence Edton) 27(1), McKay, M. D., W. J. Conover and R. J. Beckman (1979). A comparson of three methods for selectng values of nput varables n the analyss of output from a computer code. Technometrcs 21(2), Ö züm, S., B. Sener and H. Ylmaz (211). A parametrc study on seakeepng assessment of fast shps n conceptual desgn stage. Ocean Engneerng 38(13), Smola, A. J. and B. Schölkopf (24). A tutoral on support vector regresson. Statstcs and Computng 14, Vapnk, V. (1995). The Nature of Statstcal Learnng Theory. Sprnger-Verlag, Vapnk, V. (25). Unversal learnng technology: support vector machnes. NEC Journal of Advanced Technology 2(2), Yun, Y., M. Yoon and H. Nakayama (29). Mult-objectve optmzaton based on meta-modelng by usng support vector regresson. Optmzaton Engneer1, Zhang, G., G. Wang., X. L and Y. Ren (213). Global optmzaton of relablty desgn for large ball mll gear transmsson based on the Krgng model and genetc algorthm. Mechansm and Machne Theory 69,