A Study on the uncertainty and sensitivity in numerical simulation of parametric roll

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1 Downloaded from orbt.dtu.dk on: Nov, 8 A Study on the unertanty and senstvty n numeral smulaton of parametr roll Cho, Ju Hyuk; Nelsen, Ulrk Dam; Jensen, Jørgen Junher Publshed n: Proeedngs of the th Internatonal Symposum on Pratal Desgn of Shps and Other Floatng Strutures (PRADS 6) Publaton date: 6 Doument Verson Peer revewed verson Lnk bak to DTU Orbt Ctaton (APA): Cho, J., Nelsen, U. D., & Jensen, J. J. (6). A Study on the unertanty and senstvty n numeral smulaton of parametr roll. In U. Dam Nelsen, & J. Junher Jensen (Eds.), Proeedngs of the th Internatonal Symposum on Pratal Desgn of Shps and Other Floatng Strutures (PRADS 6) Tehnal Unversty of Denmark (DTU). General rghts Copyrght and moral rghts for the publatons made aessble n the publ portal are retaned by the authors and/or other opyrght owners and t s a ondton of aessng publatons that users reognse and abde by the legal requrements assoated wth these rghts. Users may download and prnt one opy of any publaton from the publ portal for the purpose of prvate study or researh. You may not further dstrbute the materal or use t for any proft-makng atvty or ommeral gan You may freely dstrbute the URL dentfyng the publaton n the publ portal If you beleve that ths doument breahes opyrght please ontat us provdng detals, and we wll remove aess to the work mmedately and nvestgate your lam.

2 Proeedngs of PRADS6 th 8 th September, 6 Copenhagen, Denmark A Study on the unertanty and senstvty n numeral smulaton of parametr roll Ju-hyuk Cho, Jørgen Junher Jensen,, Ulrk Dam Nelsen, Department of Mehanal Engneerng, Tehnal Unversty of Denmark Centre for Autonomous Marne Operatons and Systems (AMOS), NTNU, Norway Abstrat Unertantes related to numeral modellng of parametr roll have been nvestgated by usng a 6-DOFs model wth nonlnear dampng and roll restorng fores. At frst, unertanty on dampng oeffents and ts effet on the roll response s evaluated. Seondly, unertanty due to the effetve (equvalen wave onept n alulaton of restorng moment s studed. Fnally, unertanty to roll response from dfferent methods of GZ alulaton has been heked. It s found that the equvalent wave onept s suffently aurate for the purpose of GZ alulaton. Two dfferent GZ approxmatons gve a good agreement wth dret alulaton method f relevant oeffents have been properly found n the fttng. Keywords Unertanty; Senstvty; nonlnear dampng; nonlnear restorng; GZ approxmaton; 6-DOFs model, Effetve (Equvalen wave; Introduton For modern ontaner arrers and ar arrers, the parametr roll phenomenon beomes mportant n desgn and operatonal onsderaton for the safety of lfe, argo and the shp. The phenomenon and mehansm of parametr roll are well known n shppng and marne ndustry thanks to many prevous studes. However, there are stll reports of serous adents from parametr roll of ontaner and ar arrers. Therefore, there s a need for on-board deson support systems to enhane safety aganst parametr roll. Several smulaton tools, onsderng 6-DOFs motons and nstantaneous wetted surfae n the alulaton of wave extng moments and restorng fores, suh as LAMP (Frane et al., and Shn et al.,), WASIM (Vada,99) and WISH (Km and Km, 8) an aurately deal wth the parametr roll dynams. However these elaborate tools are nadequate n onboard system beause of ther large omputatonal tme. Therefore a smplfed model, whh s suffently aurate and fast for a probablst approah, s neessary. For the applaton of on-board system, the numeral framework from Jensen (7) an be a promsng anddate. Ths framework s omposed of a.-dofs equaton of moton supplemented wth the frst order relablty method (FORM) for probablst alulatons. Vd-Perunov (9) showed that a reasonably good agreement s obtaned between FORM and Monte-Carlo method, and moreover FORM requres by far the shortest omputatonal tme whh would be very benefal for applaton to on-board deson support system. However, for the robustness of the system the auray of the numeral model needs to be mproved. Therefore, ths study fouses on unertantes n the numeral model and ther onsequene on the roll response. It s expeted that the major unertantes ome from nonlnear effets assoated wth the dampng and restorng fores. On the other hand, unertanty from the randomness of waves are exluded from the study, beause a determnst wave form only dependent on the spetral shape of the waves s provded by the FORM alulaton (Jensen, 7). The study s dvded nto three parts, ) Unertanty to roll response from dampng oeffents modeled by a ombnaton of a lnear, a quadrat and a ub varaton n the roll veloty. ) Unertanty due to the effetve (equvalen wave onept - one of the most wdely used approxmaton n alulaton of nonlnear restorng moments. ) Unertanty to roll response from dfferent methods of GZ alulaton. Equaton of shp moton.-dofs equaton of moton n the work by Jensen (7) s extended to the 6-DOFs equaton of moton wth mpulse response funton (IRF) approah formulated by Cummns (96): ( M F jfk a ( F ) x ( k jdff t K ( F jc t d ( F jdamp ( F jsprng () ( The radaton mpulse response funton K and nfntefrequeny added mass a are alulated from preomputed hydrodynam oeffents: K t b ostd () a a ( ) K t sntdt where a (ω) and b (ω) are the added mass and dampng oeffents, respetvely. The vsous dampng F jdamp s modeled as:

3 F t jdamp( ) n () where β, β and β are non-dmensonal dampng oeffents, and ω n s the roll natural frequeny whh an be taken from the (lnearzed) frequeny doman soluton. Empral soft sprng fores F jsprng are appled for surge, sway and yaw. Exept for the roll mode, the wave extng fores, F jfk and F jdff, and the restorng fores, F jc are alulated assumng lnearty by the onverted IRF from the frequeny doman responses and lnear restorng oeffents. In the roll mode, the nonlnear extaton s onsdered by usng three dfferent methods: ) A dret alulaton where the pressure on the exat wetted surfae s ntegrated by onsderng the atual wave elevaton and 6-DOFs motons; ) and ) Two smplfed methods (ntrodued n later setons). n and the 9 perent onfdene nterval of the roll responses are plotted n Fg.. The standard devaton of roll ampltude n the steady state s alulated as.97rad (.6deg), and the 9 perent onfdene nterval s alulated as.7rad (.deg). Satter plots n Fg. provde a vsual desrpton of the trend of outputs, y, wth respet to sampled parameters. A lear orrelaton was found between roll response and β whh means that the β has the hghest mpat on the maxmum roll angle for ths example. Table : Input for samplng parameter Mean Std. β..6 β.. β.. Unertanty of the dampng oeffents Unertanty and senstvty analyss for dampng oeffents have been made to provde useful nformaton for mprovng robustness of the system. Generally the roll dampng oeffents n the numeral alulaton are determned from free roll deay tests and used as determnst values. However, those values mght have some degree of unertanty beause of the proess of measurng and evaluaton. Therefore the unertanty n roll response from the dampng oeffents should be heked f the quanttatve haratersts of parametr roll are of nterest. Some useful nformaton about the errors mght be provded also from the free roll deay tests. However, n ths study, arbtrary errors ntrodued n a statstal sense are gven n the dampng oeffents to hek the effet on the roll angle. Monte-Carlo smulaton wth Eq.~ s performed for a panamax ontanershp as gven n Jensen (7). The Latn Hyperube Samplng desrbed by MKay and Bekman (979) s used and t s assumed that the nput unertanty follows a normal dstrbuton wthout orrelaton among nput parameters. The mean values are taken from Bulan () and standard devaton s hosen as % of the mean value. The nput parameters are presented n Table and the samplng result s shown n Fg.. The result of the Monte-Carlo smulaton s presented n Fg.. A regular wave s hosen wth the enounter frequeny lose to twe the roll frequeny so that parametr roll easly an our. To study the unertanty, maxmum roll angles, y, at one partular nstant wthn the steady state perod, are seleted as output of the system. Then the omplete representaton of the unertanty n y s possble usng the Cumulatve Dstrbuton Funton (CDF), whh provdes the probablty that Y s less than or equal to y: () In ths study, the empral CDF s estmated and shown n Fg. usng Kaplan-Meer method from a ommeral software pakage (MATLAB a). Mean values Fg. : Latn Hyperube Samplng for dampng oeffents (number of sample: ) Fg. Result of M-C smulaton wth unertanty of dampng oeffents (, a/l=.) Prob x<x.7.. mean=.69rad t=.s 97.%=.89rad.%=.rad Roll (rad) Fg. Empral CDF from the output y at t=.s

4 Roll angle, (rad) Fg. Mean and 9% onfdene nterval of the roll response (zoomed n near the maxmum peak) Fg. : Wave approxmaton Senstvty vsualzaton of three parameters The Improved Grm effetve wave, Bulan (8), and a smlar onept alled the equvalent wave proedure from Kroeger (986) are wdely used n numeral studes of parametr roll n rregular waves beause of ther smplty and effeny n the alulaton of roll restorng moment. In both methods, the rregular sea surfae s approxmated as a regular wave whh has one effetve wave length, but varyng wave heght and rest poston. Ths enables the nstantaneous restorng moment n roll to be obtaned n a smpler way; for example, from ether a polynomal expresson wth ftted oeffents, or nterpolaton from dret alulaton. To help the hoe between the two dfferent approahes and also to hek ther mplatons on the unertanty to the GZ alulaton, sample alulatons are performed for three dfferent wave expressons, ) atual wave surfae, ) Improved Grm effetve wave () and ) Equvalent wave proedure (Equv.). A large ontaner vessel wth a modern hull desgn gven n Cho et al. () s seleted for the alulaton (length L=m, breadth B=8.m, draft T=.m, speed V=8. m/s). Statonary sea ondtons are assumed and spefed by a JONSWAP wave spetrum. Sea states 6 and 7 n the North Atlant seas are seleted as wave ondtons, wth mean wave lengths equal to the shp length.68l and L, respetvely. Table : Wave ondtons Sea state H s (m) T p (s) T z (s) The atual wave surfae, Z atual (x, for the statonary stohast long-rested wave s expressed as: Z Roll (rad) atual 9% Confdene nterval mean Tme (s) n Roll angle, (rad) ( x, h, x, t () Roll angle, (rad)... h, x, t au os( kx e u sn( kx e, a S d, (6) where S() s the wave spetrum as funton of wave frequeny, and u, u are statstal ndependent and standard normal dstrbuted varables. The Improved Grm effetve wave, Bulan (8), s gven by the expressons: Z eff f ( x, t x t x ( )os s ( )sn () ( C ( s Qsn Q sn Q ( k), f ( ) S k Q Q n Q os, S( ) d g a n n a f ( k )os t u u C S, tan e, a f ( k )sn t e, ( u / u ) where k s the wave number. It s noted that the flutuatng mean value a( n the orgnal work of Bulan s negleted. On the other hand, the Equvalent wave, Kroeger (986), s alulated aordng to: Z equv h( ( h(, x ) os ( x x) a( H ( X ( x,, os x dx L e b( H ( X ( x,, sn x dx L e X ( x, ( x Vos h( a ( b ( a( os f b( h( x ( a( os f b( h( () (6) (7) where the wave rest poston x s measured relatve to the aft end of the vessel. In both approahes, the effetve wave length L e s hosen to be the shp s length L. The alulated wave surfaes are frst vsualzed n Fg. 6 and Fg. 7 for sea state 7. As shown n the fgures, the dfferenes from the approxmatons are neglgble and suessfully follow the overall shape of the atual wave surfae retanng a regular wave form, exept that the short waves, whh are far from the seleted L e, are fltered out.

5 [m] at x=7m [m] at x=m [m] at x=7m - Fg. 6: [m] [m] Fg. 7: Atual Equv. - - tme (s) 6 - Tme hstores of wave elevaton at three dfferent poston (F n =.9, H s =7.m, T z =.6s, followng seas) t = s Atual Equv. - -L/ L/ -6 - x [m] 6 t = 9s Atual Equv L/ L/ -6 - x [m] [m] [m] - -L/ L/ -6 - x [m] 6 t = 7s Atual Equv. Examples of nstantaneous wave surfae along the vessel (F n =.9, H s =7.m, T z =.6s, followng seas) The nonlnear restorng moments and roll responses for the three dfferent wave expressons are alulated. The restorng moments are obtaned from pressure ntegraton on the atual wetted surfae onsderng 6-DOFs shp motons. The alulated restorng moment and orrespondng roll response for sea state 7 are presented n Fg. 8. It s seen that, only around t=s, the roll restorng moments from the effetve wave approxmatons are underestmated. Exept for ths partular devaton, the results from the approxmatons are n good agreement wth the results based on the atual wave profle. Table summarzes the statsts for the same wave ondton but wth an nreased tme wndow of, seonds. Dfferenes n statsts for the roll moton are not sgnfant and, as expeted from the omparson of wave profle, the dfferent wave approxmatons gve almost the same results. Another alulaton has been done for sea state 6 to hek f the approxmatons are reasonable for a wave ondton whh has a smaller mean wave length (λ.68l for sea state 6 n Table ). Fg. 9 ompares the results obtaned from the dfferent wave expressons. Although there s some dsrepany at t=8~9s, there s a good math along the atual wave surfae and the two dfferent wave approxmatons t = s Atual Equv. - -L/ L/ -6 - x [m] The statstal results n Table show that the error level s slghtly nreased ompared wth the ase for sea state 7, but stll n the aeptable level as the absolute values of error are less than. degrees. Fg. 8: Computed roll restorng moment and roll response (F n =.9, H s =7.m, T z =.6s, JON- SWAP spetrum, followng seas) Table : Statstal results of roll response (Fn=.9, Hs=7.m, Tz=.6s, JONSWAP spetrum, followng seas, duraton:,s) Roll Atual Equv. (rad) wave RMS.7.8 (.%) Maxmum.6.66 (.%) Mnmum (.9%) Restorng moment (kn/m) Restorng moment (kn/m). x 9 Fg. 9: Atual Equv. -. tme (s) (.%).66 (.%) -.8 (.%). x 9 Atual Equv. -. tme (s) Computed roll restorng moment and roll response (Fn=.9, Hs=.m, Tz=8.79s, JON- SWAP spetrum, followng seas) Table : Statstal results of roll response (Fn=.9, Hs=.m, Tz=8.79s, JONSWAP spetrum, followng seas, duraton:,s) Roll Atual Equv. (rad) wave RMS (.%) Maxmum (.%) Mnmum (-.9%).7 (.8%).896 (.8%) -.78 (-.%)

6 GZ approxmaton In ths seton, the onsequene related to unertanty from the GZ alulaton s dealt wth. For the GZ alulaton, two dfferent smplfed methods are seleted. The frst method, (A), s smlar to that by Jensen (7), whle the seond method, (B), resembles that by e (6). GZ approxmaton method (A) In the frst method, smlar to that by Jensen (7), the GZ urve s approxmated as a funton of the roll angle, the wave ampltude and the wave rest poston along the shp: h( GZ (, GZ sw( ) GZ (, x ( ) GZ sw( ).L (8) where h( s the nstantaneous wave heght alulated from the equvalent wave proedure, whh was valdated n the prevous seton. GZ sw s the GZ urve n stll water, and t s alulated ether by nterpolaton from measurements data or n analyt form usng: GZ sw ( ) A sn A A A (9) The nstantaneous GZ n a regular wave wth ampltude.l wave s estmated from the empral approxmaton: GZ, x ( C sn C C C x os x D D D D sn sn () The oeffents (C, C, C,C,D,D,D,D and L e ) have been found by the least square method wth the data obtaned from dret alulaton for a regular wave for dfferent rest postons x, wth a wave length equal to the length L of the shp and a wave heght equal to.l. Eq. 8 then assumes a lnear varaton of GZ wth wave ampltude. The same ontaner vessel as used n the prevous seton s onsdered. It was found that negatve values of GZ are obtaned when the wave rest s loated amdshps (x =L/) for ths modern hull form, beause the enter of gravty KG s hgh whle BM has small value wth the mnmum waterplane area (GM=KB+BM-KG), see Fg.. In ths approah four GZ urves (rest loatons at AP, AP+L/, AP+L/, AP+L/ and FP) should be ftted by usng one analyt approxmaton, therefore any dependent output, suh as the roll response, nherently wll have unertantes. On the other hand, f the GZ urves at x = and x =L/ (the maxmum and mnmum ase whh are rual for ths phenomenon) are well-ftted, the model gves satsfatory agreement wth the dret alulaton (Vd-Perunov, ). However, for ths sample alulaton, the negatve GZ urves nrease the unertanty as an be seen from Fg.. Z(m) - - X(m) D Plot - Y(m) - X(m) Fg. An example of the ase for unstable ondton wth negatve GZ (h=.l, x =L/, ϕ=deg) Y(m) (rad) X= X=L/ X=L/ X=L/ X= X=L/ X=L/ X=L/ (rad) Fg. Unertantes n GZ urves wth h=.l m (symbol o represents the results from dret pressure ntegraton) - a) fttng wth four dfferent x, b) fttng wth x at, L/ and L/, ) fttng wth x at, L/ and L/, d) fttng wth x at and L/ From the prevous seton the maxmum roll angle was found as. rad for sea state 7 (See Table ). Therefore, among the four dfferent ases (a-d) n Fg., the frst ase was seleted as the best one onsderng only roll angle up to. rad. For the tme doman smulaton, the Froude-Krylov and the restorng moments n the roll mode, F FK and F C n the Eq. are substtuted by Eq.8. The result from the tme doman smulaton s presented n Fg.. There s a sgnfant dfferene n restorng moment between the ase of GZ approxmaton and the dret alulaton of GZ. The fttng wth roll angle mght not be the reason for ths beause t was satsfatory n the range of roll angle up to. rad. A more plausble reason mght be that the wave heght of perent of the shp length (h=.l m) used n ths approxmaton s muh hgher than the sgnfant wave heght used n the tme doman smulaton (H s =7.m). If h=.l s used n the alulaton of GZ urves and n Eq. 8, the nonlnearty from hull shape an be very severe as shown n Fg.. However, as the present ondton does not lead to so hgh wave heghts, a sgnfant unertanty may arse from the exessve nonlnear effet. Therefore t s sug- - - Waterplane area X= X=L/ X=L/ X=L/ (rad) X= X=L/ X=L/ X=L/ (rad)

7 gested that the wave ampltude should not be fxed as h=.l but determned aordng to a sgnfant wave heght of the target wave ondton. Restorng moment (kn/m). -.. x 9 Fg. : Calulated restorng moment and roll angle from GZ approxmaton wth h=.l (F n =.9, H s =7.m, T z =.6s, followng seas) In order to verfy ths presumpton, the wave heght n the dret alulaton s modfed to h=.l=7m whh s nearly smlar to a sgnfant wave heght of 7.m. It s noted that the denomnator n the seond term n Eq. 8 s also modfed as.l. The oeffents n Eq. are updated and the result s shown n Fg.. The GZ urve for x =L/ stll has the negatve values n some range, but the new result s muh mproved ompared wth the prevous one. The updated results from tme doman smulaton are shown n Fg.. Interestngly, a good agreement wth the result from dret alulaton s now obtaned both n the restorng moment and roll response. It s seen that an unexpeted phase-hange oasonally happens, e.g. around t=,s. The reason for ths s not lear and may be due to the unertanty from the GZ approxmaton. Table shows that the agreement n the statstal haratersts of roll moton between the dret method and the approxmaton wth h=.m s qute aeptable. Fg. : Approxmated GZ urves wth h=.l 7m GZ approxmaton method (B) The seond method of GZ alulaton appled n ths study s suggested by e (6): GM ( GM sw GZ(, GZ sw( ) f () GM sw GZ Dret GZ Approxmaton -. tme [s].. X= X=L/ X=L/ X=L/ (rad) Restorng moment (kn/m). -.. x 9 Fg. Calulated restorng moment and roll angle from GZ approxmaton wth h=.l (F n =.9, H s =7.m, T z =.6s, followng seas) Table : Comparson of statstal results of roll angle for dfferent wave ampltude n GZ urve fttng (H S =7.m, T Z =.6s, duraton: s) Method GZ dret GZ approx. GZ approx. (h=.l) (h=.l) RMS roll (rad) Max. roll (rad).8.. Mn. roll (rad) where f(ϕ) s the GZ fator funton expressed as: GZ Dret GZ Approxmaton -. tme (s) sn f GM sw sn sn () max The advantage of ths model s that the fator α an be adjusted to represent nonlnearty n the shape of GZ urves and t has been shown that the results from ths model are promsng ompared wth the method of Belenky and Umeda, see IMO SLF (). In the present study, α s determned to be 6. from a global optmzaton method by usng the results from dret alulaton shown n Fg.. The obtaned GZ urves from the approxmaton are presented n Fg.. The optmzaton foused on the range.~. rad of roll angles. The alulated urves seem n far agreement wth data from the dret alulaton. The alulated tme hstores for restorng moment and roll angle are shown n Fg. 6. As t an be seen n the restorng moment, the mprovement s obvous n the detals of shape and the phase-hange. The reason for ths s that the GM( n Eq. s dretly alulated for the nstantaneous wave elevaton, whle n Eq. 8, GZ s assumed to have a lnear varaton wth the wave elevaton. If the GM( s approxmated, n some way, the auray should be heked as t may play a major role n the total onsequene due to unertanty n the model. It s noted that the unertanty from approxmatng GM s not dealt wth n ths study, and further researh should be undertaken where assoated results/effets of several models, e.g., a flutuaton of GM by lnear transfer funton ntrodued by Song et al. (), are onsdered.

8 Fg. Calulated GZ urves from the seond GZ approxmaton (for h=.l, α=6.) Restorng moment (kn/m) (rad) Fg. 6 Calulated restorng moment and roll angle from the seond GZ approxmaton (h=.l, F n =.9, H s =7.m, T z =.6s, followng seas) Conlusons Parameter estmaton for GZwave. x 9 GZ Dret GZ Approxmaton -. tme (s) Ths study has nvestgated the phenomenon of parametr roll and, spefally, the onsequene of unertantes n the nput data and governng ondtons of the numeral model has been studed. The purpose of the study s to a large extent related to on-board deson support systems, where a omputatonal effent framework, suh as that onsdered by Jensen (7), s requred. In ontrast to the.-dofs equaton of moton studed by Jensen (7), the present study onsders an extended varant wth all 6-DOFs to better quanttatvely evaluate the effet of the ntrodued unertantes. The 6-DOFs equaton of moton wth an IRF approah s seleted as the governng equaton, and a nonlnear model s appled for the roll dampng and roll restorng moment. The unertanty and senstvty from dampng oeffents are evaluated by usng Monte-Carlo smulaton. The unertanty from the effetve (equvalen wave onept to the roll restorng moment and roll response s heked by omparson wth the results from the atual wave profle. Fnally, smplfed GZ alulaton proedures are appled, and assoated unertantes are nvestgated. The results of unertanty of and senstvty to dampng oeffents mght provde useful nformaton for the robustness of an on-board system. Based on expermental data, the measured errors from free roll deay tests should be used n the samplng of Monte-Carlo smulaton. It s found that the effetve (equvalen wave onept s suffently aurate n the alulaton of nonlnear restorng moment and the dfferene between two smplfed methods ( and Equv. ) s neglgble. Smlarly, two dfferent GZ approxmatons gve aeptable agreement wth the results from a dret alulaton. It s suggested that the rato of wave ampltude to length, a/l, n the fttng proedure should be seleted aordng to the atual wave ondtons, otherwse the raton may ause a sgnfant unertanty. Referenes Bulan G (). Nonlnear parametr rollng n regular waves a general proedure for the analytal approxmaton of the GZ urve and ts use n tme doman smulatons. Oean Eng, Vol, pp 9. Bulan G (8). On an mproved Grm effetve wave. Oean Eng, Vol, pp 88. Cho, JH, Jung, BH, and Hwang, JH, (). Evaluaton of Sprngng-ndued Fatgue Damage for Ultralarge Contaner Carrer. Pro rd Paf/Asa Offshore Meh Symp, ISOPE, Anhorage, Alaska, June -July. Frane, WN, vadou, M, Treakel, TW, Paullng, JR, Mhel, RK and Moore, C (). An Investgaton of Head Sea Parametr rollng and Its Influene on Contaner lashng Systems. Marne Tehnology, Vol, No., pp 9. IMO SLF/INF.and/WP.(). Development of New Generaton Intat Stablty Crtera. IMO doument, London. Jensen, JJ (7). Effent estmaton of extreme nonlnear roll motons usng the frst-order relablty method (FORM). J. Marne Sene and tehnology Vol No. pp 9-. Km, KH and Km, YH (8). WISH JIP projet reort and manual. Marne Hydrodynam Laboratory, Seoul Natonal Unversty, Korea. Kroeger, HP (986). Rollsmulaton von Shffen m Seegang. Shffstehnk, Vol, pp876. e, JH and Km, YH (6). Development of a semanalytal method for the effent predton of parametr roll. Oean Eng, Vol, pp. MKay, MD, Bekman, RJ and Conover, WJ (97). A omparson of three methods for seletng values of nput varables n the analyss of output from a omputer ode. Tehnometrs,, pp 9. MATLAB and Statsts Toolbox Release a, The MathWorks, In., Natk, Massahusetts, Unted States. Shn, YS, Belenky, VL, Paullng, JR, Weems, KM and Ln,WM (). Crtera for Parametr Roll of Large Contanershps n Head Seas. Transatons of SNAME, Vol, pp -7. Song, KH, Km,YH and Park, DM (). Quanttatve and qualtatve analyses of parametr roll for shp desgn and operatonal gudane. Pro. the Insttuton of Mehanal Engneers, Part M: J. of Engneerng for the Martme envronment, Vol 7 (), pp Vada, T (99). WASIM - Theory and Numeral

9 Methods. Det Norske Vertas Internal Report. No Vd-Perunov, J(9). Estmaton of parametr rollng of shps-omparson of dfferent probablst mehtods. Pro. MARSTRUCT 9, nd Internatonal Conferene on Marne Strutures-Analyss and Desgn of Marne Strutures, Lsbon, Portugal, Marh 68. Vd-Perunov, J (). Influene of the GZ alulaton method on parametr roll predton. Oean Eng, Vol 8, pp 9-.