A More Robust Multi-Parameter Conformal Mapping Method for Geometry Generation of any Arbitrary Ship Section

Transcription

1 A More Robust Mult-Parameter Conformal Mappng Method for Geometry Generaton of any Arbtrary Shp Secton Mohammad Saleh a, Parvz Ghadm b*, Al Bakhshandeh Rostam c a Graduate Student at Dept. of Marne technology, Amrkabr Unversty of Technology, Teheran, Iran b Assocate Professor at Dept. of Marne technology, Amrkabr Unversty of Technology, Teheran, Iran c M.Sc. Graduate at Dept. of Marne technology, Amrkabr Unversty of Technology, Teheran, Iran Abstract The central problem of strp theory s the calculaton of potental flow around D sectons. One partcular method of solutons for ths problem s conformal mappng of the body secton to the unt crcle over whch soluton of potental flow s avalable. Here, a new mult-parameter conformal mappng method s presented that can map any arbtrary secton onto a unt crcle wth good accuracy. The procedure for fndng the correspondng mappng coeffcents s teratve. The suggested mappng technque s proved able to map any chne, bulbous, large and fne sectons, approprately. Several examples of mappng the symmetrcal as well as the nonsymmetrcal sectons have been demonstrated. For the symmetrcal and nonsymmetrcal sectons, the results of the current method are compared aganst other mappng technques and the currently produced geometres dsplay good agreement wth the actual geometres. Keywords: Mult-parameter Mappng; Arbtrary Shp Sectons; Geometry Generaton; Strp Theory; Potental Flow 1. Introducton Dynamc analyss of floatng and salng bodes s an mportant part of basc desgn of marne structures that helps the desgners to estmate the dmensons and forms of the bodes. Rgd body motons are defned by sx equatons and each equaton belongs to one degree of freedom. To obtan all body motons, these sx equatons are coupled and solved smultaneously. These dfferental equatons consst of mass, dampng, stffness terms and wave loads. Added mass or added nerta are taken nto account and s due to the acceleraton of flud that surrounds the body, whle the dampng term s due to the reducton n the energy level of the body whch generates the free surface waves durng the body motons. In order to compute the added mass and added nerta, dampng and wave loads, t s necessary to solve the potental flow problem around the body whch must satsfy some boundary condtons such as the body condton, free surface condton and radaton condton.

2 One method of soluton for calculatng the three dmensonal problems s to break down the geometry nto some two dmensonal sectons, solve each secton separately, and subsequently superpose the sectonal results to gan the response of the man geometry. Ths method s called strp theory whose smplcty and hgh accuracy has caused ts extensve applcaton n calculaton of the potental flows. Descrptons of the background of strp theory are provded by Salvesen et al [1], who also developed a detaled formulaton of the theory. Ther verson has been the most wdely used strp theores. One partcular scheme for applcaton of the strp theory n solvng the potental flow problems s conformal mappng. Ths s accomplshed by mappng any arbtrary shape to a smpler form lke a crcle for whch there s an analytcal soluton avalable for potental flow. By usng ths soluton and the nverse of the appled conformal mappng, the potental flow around the ntended arbtrary shape s determned. Ursell [] derved the potental flow around a crcular cylnder whch nvolved the heavng n the free surface. Ths potental flow satsfed the free-surface condton as well as the body boundary condton. Therefore, by usng the Ursel method to calculate the potental flow around a cylnder wth an arbtrary secton, one has to fnd the relaton between a crcle and the arbtrary secton. The relaton between the arbtrary secton and the crcle s establshed by some number of coeffcents and parameters. There are other types of conformal mappng that are used for flow analyses n the feld of hydrodynamcs. Schwartz-Chrstoffel method s one mappng scheme that has the superor benefts of mappng the sectons wth hgh curvature, lke the chne sectons, the rse of floors and the wedge shapes. Ths method maps the secton to a lne or a polygon. Ghadm et al [3] used Schwartz- Chrstoffel method to map the wedge secton onto a lne for solvng the water entry problem. They calculated the pressure dstrbuton for several entry angles n potental flow. Hashemnejad and Mohammad [4] mapped the ellptcal and crcular domans to rectangular doman to study the effect of ant-slosh baffles on free lqud oscllatons n partally flled horzontal crcular and ellptcal tanks. Modfed Schwarz-Chrstoffel mappngs, usng approxmate curve factors, were also presented by Andersson [5] that was able to map the upperhalf plane to a polygonal regon. A mappng technque that nvolves the transformaton of any arbtrary secton to a unt crcle s of mmense mportance. One good example of ths type of mappng scheme s the Remann mappng theory that was llustrated by Papamchael and Stylanopoulos [6]. In ths method, three real condtons must be mposed n order to make the conformal mappng unque. In other words, the problem of determnng the mappng n ths method has three degrees of freedom. Another example of transformng a rectangle to a unt crcle s called the KT-TG method. For a rectangular secton, the conformal transformatons combne successvely the Karmann Trefftz (KT) transformaton (see Halsey [7], for example) whch removes the corners and the Theodorsen Garrck (TG) [8] transformaton whch turns the ntermedate near crcle nto a perfect crcle. Ths method was used for computng the hydrodynamc forces actng on bodes of

3 arbtrary shape n vscous flow by Scolan and Etenne [9]. Carmona and Fedorovsky [10] studed some propertes of Carathéodory domans and ther conformal mappngs onto the unt dsk. Brown [11] presented a two parameter conformal mappng from a dsk to a crcular-arc quadrlateral, symmetrc wth respect to the coordnate axes. Brown and Porter [1] developed a conformal mappng from the crcular-arc quadrlaterals wth four rght angles onto a unt dsk. Lu et al [13] proposed a representaton of general D domans wth arbtrary topologes usng conformal geometry. A natural metrc can be defned on the proposed representaton space, whch gves a metrc to measure dssmlartes between objects. The man dea was to map the exteror and nteror of the doman conformally onto unt dsks and crcular domans. As a general comment, although the use of Green functons wth or wthout forward speed, produces more accurate results for the shp hull forms, but due to ther smplcty, conformal mappng technques are stll sutable for seakeepng calculatons. As mentoned earler, there are many types of conformal mappngs that are used for the analyss of flow around marne structures, but the most mportant and nterestng mappng technque s that whch can map any shp sectons onto a unt crcle, whch s the focus of the current study. In ths regard, the frst and most famous conformal mappng method s two-parameter Lews conformal mappng whch was presented by Lews [14]. The base of ths method was the smlarty of the breadth and draft and area of the secton and those of the mapped secton. Ths method s able to map a crcle to a conventonally arbtrary shape but does not produce good results for any arbtrary sectons. Thereafter, three-parameter conformal mappng was ntroduced by Tasa [15]. They extended two-parameter Lews conformal mappng to three-parameter conformal mappng, by takng nto account the frst order moments of half of the real body cross sectonal surface and that of the mapped secton about the x and y-axes. In 1969, von Kerczek and Tuck [16] for the frst tme presented mult-parameter conformal mappng based on Least Square Method by usng an teratve procedure to solve nonlnear system of equaton whch resulted n fndng the mappng coeffcents. Ths method was later smplfed by equatng the vertcal poston of the centrod of the real body cross secton and that of the mapped secton by Athanassouls and Loukaks [17]. The result of ths method was proved to be better than that of Lews method, but ths method was stll not able to map any or all sectons. In the meantme, Anghel and Cobanu [18] showed how to extend the Lews transformaton for obtanng the contour of the shp's cross secton of dfferent types of shps. Moreover, they demonstrated how a more accurate transformaton of the cross sectonal hull shape can be obtaned by usng a larger number of parameters. Subsequently, a mult-parameter conformal mappng was presented by Tasa [19, 0] whch was further developed by de Jong [1]. Rosén and Palmqust [] used the de Jong and Lews mappng method to compute the added mass and dampng coeffcents of a shp, for studyng the parametrc rollng n waves. Westlake

4 and Wlson (WW) [3] developed a technque for carryng the mult-parameter conformal mappng of the shp sectons onto a unt crcle. The basc dea was to calculate one unknown mappng parameter after another when the startng known mappng coeffcent was calculated for =0, where was the number of mappng coeffcents. Also, n ther mappng method, the correspondng angle n the unt crcle s calculated by nterpolaton, usng the lne length method correspondng to each defned pont. Another mult-parameter conformal mappng s the method that was presented by Journée and Adegeest [4], but they only studed the symmetrcal sectons. In the currently proposed mult-parameter mappng method, the least square scheme that was presented by de Jong [1], has been used to compute the mappng coeffcents, but a new technque has been ntroduced that s drastcally dfferent n detal and can map any arbtrary symmetrcal and non symmetrcal sectons onto a unt crcle n a much qucker way. However, contrary to the method suggested by Westlake and Wlson [3], the value of the frst mappng coeffcents based on de Jong s [1] general mappng equatons, s assumed to be one, all the mappng parameters are calculated smultaneously, and by makng a smple assumpton, an equaton s obtaned and solved for the correspondng angles n the unt crcle plane. Furthermore, n the proposed method, the ntal guess for the mappng coeffcents for the nonsymmetrcal sectons, wll be assumed as the average of the two mappng coeffcents found based on the approach adopted for the symmetrcal sectons and for the rght and left (half) dstance from the centerlne on the waterlne. The newly ntroduced schemes make the proposed mult-parameter mappng technque more robust, shorten the process of mappng and decrease the computng tme. To recaptulate the man deas of ths paper, a more robust mult-parameter conformal mappng method s presented that s able to map any arbtrary sectons onto a unt crcle. Ths method, shortens the process of mappng, decreases the computng tme, and has the ablty to map the extreme bulbous sectons, the non-symmetrcal sectons, the large area coeffcent sectons, sectons wth hard chne and fne sectons among others. The process of mappng s outlned and detaled clearly and many practcal examples of the proposed method have been offered whch demonstrate ts great accuracy. Results of the current mappng technque, n the case of symmetrcal sectons, have been plotted and compared aganst the results of the Lews mappng method whch ndcate better agreement of the current results wth the ponts on the real sectons. To further ndcate the abltes of the current mappng method n the cases of rectangular, large bulbous and fne sectons for both symmetrcal and non-symmetrcal form, comparsons are also made between the current mappng results and the results of Westlake and Wlson [3] Mappng scheme. These comparsons demonstrate that n symmetrcal cases, the current mappng method s more robust whle n non-symmetrcal cases, the results are very smlar to that by Westlake and Wlson [3] mappng technque. For a more clear observaton of the relatons between the real secton ponts and ther correspondng angles n the mapped unt crcle plane, correspondng angles for the case of symmetrc rectangular secton has been dsplayed n a fgure.

5 . Basc formulaton and assumptons for conformal mappng The tranformaton of any arbtrary secton (n Cartesan plane) to a unt crcle plane, usng a mult-parameter conformal mappng s consdered to be [1]: z F n0 a n1 n1 (1) where z x y represents a Cartesan plane (as shown n Fg.1), x s the real part of z, y s the magnary part of z, e e represents a unt crcle, F s the scale factor, a 1 and 1 a n 1 n 1,..., are the mappng coeffcents, s the correspondng angle n the mapped unt crcle plane, β s consdered another scalng factor ndcatng how close any mapped secton s from the mapped shp secton, and s the number of mappng parameters. Fg.1. Actual coordnate system for the arbtrary secton (rght) and the transformed unt crcle plane (left). Based on the dagrams presented n Fg.1, the x and y coordnates of the shp secton n Eq.1 can be rewrtten n the form of x F y F n0 n0 n (n1) a e snn 1 n1 1 n (n1) a e cosn 1 n1 1 () (3) As shown n Fg.1, β=0 results n the mappng of the boundary of shp secton n whch case, Eqs. and Eq.3 become x 0 F ) n0 n an 1 sn(n 1 1 (4)

6 y 0 F ) n0 n an 1 cos(n 1 1 (5) Draft and breadth of each secton are defned as follows: D s F b wth b n0 n ( 1) a (6) n1 B s F a wth a n0 a, n1 B s F (7) a Usually, the arbtrary sectons are defned by some dscrete ponts such as ( x, y )( 0,1,,..., I). There s also an angle n each unt crcle plane correspondng to each pont of the arbtrary secton. By applyng the approprate mappng coeffcents and the correspondng values of n Eqs.4 and 5, ponts on the arbtrary secton wll be produced. Therefore, the conformal mappng problem reduces to that of fndng the mappng coeffcent and the values of. The number of mappng coeffcents,.e., s an arbtrary value. To determne the exact breadth at the water lne and the draught, there are two ponts for whch ther correspondng angles n the unt crcle are known. These two ponts are found from Eqs.8 and 9 and are gven below. B x s, y 0 x 0, Ds y 0 (8) (9) The method starts by guessng the ntal values for mappng coeffcents and subsequently usng these coeffcents to fnd the correspondng angles ( ) n the unt crcle for each pont of the arbtrary secton. ext, the new values of the mappng coeffcents are found by usng the computed angles ( ). Ths computaton s contnued untl the summaton of squared errors between the mapped and exact ponts of the secton becomes less than a desgnated error tolerance value. 3. The Proposed Mult-Parameter Conformal Mappng Technque Ths method can be descrbed and utlzed for both symmetrcal and nonsymmetrcal sectons. In ths secton, ntally the symmetrcal secton conformal mappng and subsequently the conformal mappng for non-symmetrcal sectons wll be descrbed Conformal Mappng of the Symmetrcal sectons

7 The results of conformal mappng of ( x, y ) are assumed to be ( x0, y0 )( 0,1,,..., I). Frst, by usng the two-parameter Lews conformal mappng method, the ntal values of conformal mappng coeffcents are defned. Then, for mnmzaton of the square of dfferences between ( x, y ) and ( x0, y0 )( 0,1,,..., I), two condtons or constrants should be satsfed whch are as follows: a) Frst condton or constrant: Ponts ( x0, y0 ) of the transformed contour le on the normal of the arbtrary secton. Ths normal passes through the ponts ( x, y ) of the secton contour, as shown n Fg.. By satsfyng ths condton, the unknown values of ( 0,1,,..., I) wll be obtaned. b) Second condton or constrant: The summaton of square of dfferences of ( x, y ) and ( x0, y0 )( 0,1,,..., I) must be less than an arbtrary desgnated value. By satsfyng ths condton and utlzng the obtaned values of found from the frst condton, the values of mappng coeffcents wll be produced. These determned coeffcents are used as ntal values of mappng coeffcents. Computng the unknown values of s repeated wth these new ntal mappng coeffcents and the mappng coeffcents are updated by usng the obtaned values of. Ths procedure s repeated untl the summaton of square of dfferences of ( x, y ) and ( x0, y0 ) becomes less than an arbtrary desgnated error tolerance value. Fg.. ormal lne to the arbtrary secton that passes through x, y ) and x, y ). ( ( 0 0

8 In order to satsfy the frst condton, the vector passng through the mapped pont of ( x0, y0 ) and normal to the secant lne connectng two ponts of ( x 1, y 1) and ( x 1, y 1),.e. h, must be parallel to the vector passng through the ponts ( x 0, y0 ) and ( x, ) y,.e. v. Therefore, ther cross product must be zero. The unt normal vector passng through the mapped pont of ( x0, y0 ) and normal to the secant lne connectng two ponts of ( x 1, y 1) and ( x 1, y 1) on the real secton s defned as h sn cos j (10) where s the angle between the vertcal lne that passes through pont ( x0, y0 ) and the unt normal vector. The angle s constant throughout the teraton because the normal lne s constant,.e. h s constant, but angle vares n each teraton and converges to the exact correspondng angle n the unt crcle plane. The cause of ths varaton s the change n vector v. The vector connectng the two ponts of x, y ) and x, y ) s gven as v ( 0 0 x x ) ( y y ) j ( ( 0 0 (11) As mentoned earler, the two vectors n Eqs. (10) and (11) are parallel. Accordngly, ther cross product s zero whch leads to the followng equaton: ( x x0 0 )cos ( y y )sn 0 (1) Based on Fg., can be found from the equatons (13) and (14) gven as: cos sn x x 1 1 (13) ( x 1 x 1) ( y 1 y 1) y y 1 1 (14) ( x 1 x 1) ( y 1 y 1) Substtutng Eqs.4, 5, 13, 14 nto Eq.1, would yeld n x cos F cos ( F sn ( n0 ( 1) n a n0 n1 ( 1) n a n1 cos(n 1) ) 0 sn(n 1) ) y sn (15)

9 Solvng Eq.(15) for each pont of the arbtrary secton,.e. ( x, y )( 0,1,,..., I), results n the correspondng values of n the unt crcle plane. Ths nonlnear equaton s solved by mplementng the bsecton method. Applyng the second condton, would result n E I e 0 E (16) Where E Is desgnated error tolerance value And e s e (17) ( x x0 ) ( y y0 ) Substtutng Eqs.4 and 5 n Eq.17, would yeld n E I 0 x n0 n n ( 1) Fa n1 sn(n 1) y ( 1) Fan 1 cos(n 1) (18) n0 Therefore, the new values of Fa n1 should be computed n such a way that the value of E, by applyng least square method, becomes mnmum as n E Fa j1 0 for j 0,...,. (19) Ths mnmzaton results n 1equatons as n n0 0 I x sn( j 1) y cos( j 1) 0 I ( 1) n Fa n1 cos ( j n) For: j 0,.... (0) For a better and faster convergence n solvng the system and obtanng the exact breadth and draft, the last two equatons n the system of equatons (.e. for j=-1 and j= n Eq.0) are replaced by the related equatons of breadth and draft as n Eqs. and 3. Consequently, Eq.0 can now be wrtten as Eqs.1,, and 3 as follows: n0 0 I x sn( j 1) y cos( j 1 0 I ( 1) n Fa n1 cos ( j n) For: j 0,... (1)

10 n0 n ( 1) Fa D () n1 s n0 Fa n1 B s (3) Ths system of equatons can be rewrtten n matrx form AX=B as n (4) (5) and

11 (6) The new values of ( Fa n1 ; n 0,1,..., ) wll be computed by solvng the system of equatons n (1), () and (3). The system of equatons can be solved by usng any drect or teratve numercal methods. In ths artcle, the soluton has been performed usng the LU decomposton method. The new values of can be obtaned by usng the new values of ( Fa n1 ; n 0,1,... ) n Eq.15. Applyng the newly obtaned values of 0,1,,..., I n Eqs.1,, and 3 and solvng the system of equatons provdes new values of ( Fa n1 ; n 0,1,... ). Ths repeatng procedure s contnued untl the value of that s computed from Eq.18 becomes less than a desgnated value. Fnally, snce a 1 1, the value of the scale factor F equals to Fa 1. Other mappng coeffcents wll be obtaned by dvdng the computed values of ( Fa n1 ; n 0,1,... ) by F. 3-. Conformal Mappng of the on-symmetrcal sectons The procedure for conformal mappng of the non-symmetrcal sectons s smlar to the symmetrcal sectons mappng, wth the exceptons of the followng three tasks:

12 1. The correspondng angles n the unt crcle plane for two ponts of the secton that le on the waterlne must be found, whereas n the case of symmetrcal sectons they were known to be 0 and. Based on Fg.3, the angle s defned by ts sne and cosne as x x 1 cos, (7) ( x x ) ( y y ) 1 1 y y 1 sn, (8) ( x x ) ( y y ) 1 1 Fg.3. ormal lne to the secton at the ntersecton wth the waterlne(wl). Therefore, the two values of, mentoned above, can be obtaned by placng the results of Eqs.7 and 8 nto Eq.15 and solvng the resultng equaton.. As mentoned before, for determnng the exact breadth and draft n the mappng of the symmetrcal sectons, two condtons are mposed n the form of two equatons on the system of equatons (1), but n the non-symmetrcal secton, ths has been proved mpractcal because soluton convergence s not acheved, as soluton always dverges. Ths omsson of equatons may lead to some error, but ths dscrepancy only exsts between the two ponts on the waterlne and ther correspondng ponts on the mapped

13 secton. However, t has been proved, based on the results of the examples provded n ths paper, that the reported error s so mnmal that t s defntely gnorable. Therefore, for computng the new values of ( Fa n1 ; n 0,1,... ), the prevously computed values of should be used n the system of equatons (6) as n n0 0 I x sn( j 1) y cos( j 1 0 I ( 1) n Fa n1 cos ( j n) For: j 0,... (9) 3. Also, as opposed to the conformal mappng of the symmetrcal sectons for whch an ntal guess for the mappng coeffcents was produced by the Lews method, for the case of non-symmetrcal mappng, the ntal guess s obtaned n a more complex procedure: a. Frst, by followng the same procedure for the symmetrcal sectons and assumng that ts left (half) dstance from the centerlne on the waterlne s ( B sl ), we fnd the mappng coeffcents. b. Then, by followng the same procedure for the symmetrcal sectons and assumng that ts rght (half) dstance from the centerlne on the waterlne s ( B sr ), we fnd the mappng coeffcents. c. Fnally, the ntal guess for the mappng coeffcents of the non-symmetrcal sectons wll be assumed as the average of the two mappng coeffcents found n parts (a) and (b). Fg.4. Schematc of a non-symmetrcal secton. In the process of mappng the nonsymmetrcal sectons at the stage of computng the values, t s possble that Eq.15 doesn t have a soluton. Therefore, n order to avod ths dffculty and to contnue wth the computatons, the value of can be assumed n the form of Eq.30.

14 ( ) (30) 1 1 For better understandng of the current method, a flowchart of the mappng procedure has been plotted n Fg.5. Fg.5 The mappng procedure flowchart.

15 3-3. Fndng the Optmum Values of Error (E) and umber of Coeffcents () In the mappng process, based on a desgnated value of such as =5, ether the program (shown n Fg.5) after a few lmted teratons, reaches an optmum E (desgnated a-pror) or t contnues for a large number of teratons wthout reachng a desrable E. Therefore, a maxmum number of 00 teratons s allowed n the mappng process for the symmetrc sectons and a maxmum number of 300 teratons s assgned for the non-symmetrc sectons n the current computatons. Hence, for every value of, there s a partcular value of E mn for whch the program does not have the ablty of reachng a lesser error, anymore. The mnmum value of Emn found for =5 s used for settng the desgnated error tolerance value for =6 and the process contnues. To summarze the complete process of fndng the optmum value of E leadng to an optmum value of, the followng steps are pursued: 1) We start wth =5 and E=10 as desgnated error tolerance value. ) Execute the program accordng to the flowchart gven n Fg.5. a. If t does not reach the desgnated error tolerance of E=10, we move on to =6 and E=10 as desgnated error tolerance value. b. If t reaches the desgnated error tolerance of E=10 before 00 teratons for symmetrcal cases and 300 teratons for nonsymmetrcal cases; then, a program whch s wrtten for fndng an optmum error for any n Fg.6, s executed. We set equal to 5 and fnd the best value of E or so called E mn correspondng to =5. Suppose ths value s E mn =1.0. Snce ths value s 1.0 and greater than an arbtrary set number of 0.1, we subtract 0.1 from 1.0 whch results n 0.9. Ths s now consdered as the desgnated error tolerance value for =6 n the next round of teraton. 3) We contnue based on the flowchart n Fg.6 wth =6 and E=0.9 as desgnated error tolerance value. a. If t does not reach the desgnated error tolerance of E=0.9 before 00 teratons for symmetrcal cases and 300 teratons for nonsymmetrcal cases (as shown n Fg.5), we move on to =7 and E=0.9 b. If t reaches the desgnated error tolerance of E=0.9 before 00 teratons for symmetrcal cases and 300 teratons for nonsymmetrcal cases (as shown n Fg.5); then, the program n Fg.6, s executed. We set equal to 6 and fnd the best value of E or so called E mn correspondng to =6. Suppose ths value s found to be E mn =0.01. Snce ths value (0.01) s less than 0.1, we dvde 0.01 by 10 whch results n Ths s now consdered as the desgnated error tolerance value for =7 n the next round of teraton. 4) The mappng process gets repeated for all rangng from 5 to 100 accordng to the flowchart n Fg.7. The last value of E and ts correspondng for whch flowchart n

16 Fg.6 s executed, are the sought optmum values of E and for the consdered geometrcal secton.

17 Fg.6. The flowchart of fndng the best values of E for each.

18 Fg.7. The flowchart of fndng the best values of E and n the mappng process. 4. Examples and dscussons The results of mappng dfferent sectons are presented by showng the best values of E that has been obtaned from Eq.18 for dfferent values of number of mappng coeffcents (). These values of E have been based on and are the smallest possble values of E for whch a convergent soluton s not attanable. In other words, the most accurate mappng profle s acheved by ths value of E. Sectons that have been presented as examples n ths secton nclude rectangular secton, large bulbous secton, fne secton and chne secton. In each of these examples, the results of mappng

19 the symmetrcal sectons, the results of mappng non-symmetrcal sectons, and ther comparson aganst the results of Westlake and Wlson [3] have been presented, respectvely. However, t must be noted that comparson of the current results n the case of chne sectons wth that of Westlake and Wlson [3] has not been presented because ther study dd not nclude ths partcular secton. For comparson of the obtaned results for the symmetrcal sectons aganst that of the Lews mappng, they are plotted next to the ponts of the real sectons n Fgs.8 (a), 10 (a), 14 (a), and 15 (a). As evdenced n these fgures, the produced mapped ponts by the current method perfectly match the ponts of the real secton, whle the ponts produced by the Lews mappng have consderable dfference wth the real secton ponts. For a better vew of the relatons between the mapped secton ponts and ther correspondng angles n the unt crcle plane, the partcular case of symmetrc rectangular secton and ts transformaton n the unt crcle plane are demonstrated n Fg.9. In ths fgure, some ponts of the real secton and ther correspondng angles n the unt crcle plane have been marked wth the same numberng (or dentfer). As mentoned earler, to demonstrate the capabltes of the current mappng method n the cases of rectangular, large bulbous and fne secton for both symmetrcal and non-symmetrcal form, comparson of current mappng results and Westlake and Wlson Mappng [3] results s presented. The results ndcate that n symmetrcal cases, current mappng method s more accurate and robust, whle n non-symmetrcal cases, the results are smlar to that by Westlake and Wlson mappng method [3] Rectangular Secton Cross secton of most of the barges, the floatng breakwaters and the floatng pontoons s rectangular. Because of hard chne n the corners, the mappng of these sectons has more dffculty than the common sectons. In other words, the mapped secton and the real secton wll not agree wth each other. However, mappng of these sectons by the present method has produced very good results and shows hgh compatblty wth the real sectons. In ths secton, two examples are presented; one for the symmetrcal and the other for the non-symmetrcal rectangular secton. The non-symmetrcal secton s a rectangular secton at 15 degrees of heel. To better understand the relaton between the secton ponts and ther correspondng angles n unt crcular plane, the secton ponts and ther correspondng angles are dsplayed n Fg.8 for the case of symmetrc rectangular secton.

20 Fg.8. Illustraton of a Symmetrc rectangular secton (left) and ts correspondng angles n the transformed unt crcle plane (rght). The results of mappng for the symmetrcal and non-symmetrcal rectangular sectons are shown n Fgs.9 and 11. The result of Lews mappng technque s also demonstrated n Fg.9 for the case of symmetrcal secton. The Mnmum values of E whch are obtaned for dfferent values of n mappng of symmetrc rectangular secton and non-symmetrcal secton have been shown n Fgs.10 and 1 Fg.9. Results of conformal mappngs for symmetrc rectangular secton.

21 Fg.10. Mnmum value of E whch s obtaned for dfferent values of n mappng of symmetrc rectangular secton. Fg.11 Results of conformal mappngs for non-symmetrc rectangular secton.

22 Fg.1. Mnmum values of E whch are obtaned for dfferent values of n mappng of non-symmetrc rectangular secton. For the purpose of comparng the current results aganst those by Westlake and Wlson [3], the current scheme has been appled to the same examples whch are solved n Westlake and Wlson s artcle [3]. Accordngly, accuracy of the proposed method has been examned for two symmetrcal and non-symmetrcal sectons. Coordnates of the ponts related to the examned sectons along wth the ponts resultng from the Westlake and Wlson s mappng [3] and ther fnal mapped ponts are respectvely plotted n Fgs.13 and 14.

23 Fg.13. WW [3] offsets and fnal mapped pont by usng 1 transform parameters for symmetrc rectangular secton. Therefore, error from the results of Westlake and Wlson s mappng [3] for =1 s equal to E WW I ( x x0 ) ( y y0 ) Results of the proposed technque for the current secton are gven n table 1, whch for =1 s equal to 3.0E-9. Ths ndcates greater accuracy compared to Westlake and Wlson s [3] results. Table 1. Mnmum values of E that are obtaned for dfferent values of by usng the current mappng of symmetrc rectangular secton E 8.0 E E-8 3.0E-9 It s qute apparent that, n order to reach the same level of accuracy as n the Westlake and Wlson s [3] mappng wth =1 parameters, t s suffcent to use =6 parameters n the proposed method, whch s ndcatve of robustness and fastness of the current technque. ext, comparson s made for the case of a non-symmetrc rectangular secton wth heel angle of 15 degrees whch s examned n the work by Westlake and Wlson [3]. Geometry of the secton and fnal mapped ponts s plotted n Fg.14.

24 Fg.14. WW [3] offsets and fnal mapped pont by usng 1 transform parameters for non- symmetrc rectangular secton at 15 degrees of heel. Here, error from the results of Westlake and Wlson s mappng [3] for =1 s equal to E WW I ( x x0 ) ( y y0 ) Results of the proposed technque for the current secton are gven n table. Table. Mnmum value of E whch s obtaned for dfferent values of by usng current mappng for the nonsymmetrc rectangular secton at 15 degree of heel E It s clear that, error for =1 n the current case s almost the same as the error from Westlake and Wlson s [3] resutls and that, n order to ncrease the accuracy n comparson wth WW s results, has to be slghtly ncreased further,.e. = Bulbous sectons Almost all of the more novel commercal shps have bulbous sectons at ther bows. Sectons wth large bulb can not be mapped easly. The two examples that are dscussed here for the symmetrcal as well as non-symmetrcal bulbous sectons ndcate the great ablty of the present mappng method for mappng such large bulbous sectons. The non-symmetrcal secton s a bulbous secton at 15 degrees of heel.

25 The secton ponts and results of current mappng technque for symmetrc and non-symmetrc mappng are llustrated n Fgs.15 and 17, whle the errors are plotted n Fgs.16 and 18. The result of Lews mappng technque s also ncluded n Fg.15 for the case of symmetrcal secton. Fg.15. Results of conformal mappngs for symmetrc bulbous. Fg.16. Mnmum values of E whch can be obtaned for dfferent values of n mappng of symmetrc large bulb secton.

26 Fg.17. Results of conformal mappngs for non-symmetrc bulbous secton. Fg.18. Mnmum values of E whch are obtaned for dfferent values of n mappng of non-symmetrc Bulbous sectons at 15 o of heel. Results of the current mappng technque for the symmetrc Bubous secton (n the cases of =19 and =31) are llustrated and compared wth the results of the Westlake and Wlson [3] (n the case of =48) n Fg.19. As evdenced n these fgures, the proposed technque offers

27 more accurate results (compared to the real sectons) wth a fewer mappng parameters than the Westlake Wlson s [3] results. The mappng of non-symmetrc Bulbous secton wth 15 degrees heel, whch was also done by Westlake and Wlson [3], has been done by the current mappng technque, too. Results of mappng of ths type of secton for =16, 0, 8, 36, and 48 parameters have been graphcally presented n Fg.0. In these plots, the currents results as well as those by Westlake and Wlson [3] are llustrated and compared. As evdenced n these plots, there s not much dfference between the results of the two technques and both methods posses smlar error at varous. Therefore, one can easly conclude that the current mappng technque, eventhough offers compatble results wth those by Westlake and Wlson [3] n the case of non-symmetrc mappng bulbous secton, but n the case of symmetrc mappng, offers more accurate results n a shorter tme perod. Fg.19. Results of conformal mappngs for symmetrc bulbous secton for a) =19, compared wth the results of WW [3] mappng for =48 b) =31, compared wth the results of WW [3] mappng for =48.

28 Fg.0. Comparson of the current results and that of WW [3] mappng for non- symmetrc bulbous secton for a) =16, b) =0, c) =8, d) =36, and e) =48.

29 4-3. Fne secton Mappng of the fne sectons s also a cumbersome task and most of the mappng methods do not have the ablty to produce the mapped sectons wth good accuracy. The aft and fore sectons of the shp, frgate and destroyer shp sectons are consdered as fne sectons. The followng examples for symmetrcal and non-symmetrcal sectons llustrate the proper accuracy of results of the current mappng method. The non-symmetrcal secton s a fne secton at 0 degree of heel. Results of mappng a symmetrc and non-symmetrc fne sectons are demonstrated n Fgs.1 and 3. In ths fgure, the results of Lews mappng for the case of symmetrc mappng s also presented. The resultng errors for these mappngs have been dsplayed n Fgs. and 4. Fg.1. Results of conformal mappngs for the symmetrc fne secton.

30 Fg.. Mnmum values of E whch are obtaned for dfferent values of n mappng of symmetrc fne secton. Fg.3. Results of conformal mappngs for non-symmetrc fne secton.

31 Fg.4. Mnmum values of E whch can be obtaned for dfferent values of n mappng of non-symmetrc fne secton. Coordnates of the symmetrc and non-symmetrc fne sectons analyzed earler by WW [3] are plotted n Fgs.5 and 7. Fg.5. WW [3] offsets and fnal mapped pont by usng 0 transform parameters for symmetrc fne secton. Error resultng from WW [3] method for =0 s equal to E WW I ( x x0 ) ( y y0 ) The results of the current mappng technque for symmetrc fne secton have been shown n Fg.6.

32 Fg.6. Mnmum values of E whch can be obtaned for dfferent values of n mappng of symmetrc fne secton. In the case of non-symmetrc fne secton, coordnates of the secton and the results of WW [3] technque are gven n Fg.7.

33 Fg.7. WW [3] offsets and fnal mapped pont by usng 0 transform parameters for non symmetrc fne secton. Error resultng from the WW [3] mappng for =0 s equal to E WW I ( x x0 ) ( y y0 ) Results of the current method for the nonsymmetrc fne secton are tabulated n table 3. Table 3. Mnmum value of E whch s obtaned for dfferent values of n mappng of non-symmetrc fne secton E It s clear that there s not much dfference betwee the current results and that those by WW [3] technque for non-symmetrc mappng of fne secton Chne sectons Plannng boats, catamarans and naval shps have chne to decrease the resstance and power consumpton. Dynamc analyss of these vessels plays the man role n the desgn states. The followng examples demonstrate the great mappng ablty of the current method for the fne sectons. The non-symmetrcal example s a chne secton at 0 0 of heel angle.

34 Results from the symmetrc as well as non-symmetrc chne sectons are llustrated n Fgs.8 and 30. Also, the result of Lews mappng for the case of symmetrc mappng s presented n Fg.8. Fg.8. Results of conformal mappngs for symmetrc chne. Fg.9. Mnmum values of E whch are obtaned for dfferent values of n mappng of symmetrc chne secton. Coordnates of the non-symmetrc chne secton and result of current mappng are plotted n Fg.30, whle the results of mappng are presented n Fg.31.

35 Fg.30. Results of conformal mappngs for non-symmetrc chne secton. Fg.31. Mnmum value of E whch are obtaned for dfferent values of n mappng of non - symmetrc chne secton. Results of the current mappng technque for symmetrc and non-symmetrc chne secton deomonstrated n Fgs.8 and 30 ndcate great match wth the real secton and sutable accuracy of the scheme Computatonal Tmes and Accuracy Results of current mappng method were lsted earler for each of the geometrcal sectons. In ths secton, n addton to presentng the optmum values of error (E) and number of mappng coeffcents () for each of these sectons, computatonal tmes for fndng these values. Also, effort has been made to show the accuracy assocated wth the proposed technque for

36 determnng these quanttes. Ths s accomplshed by usng the ash-sutclffe model effcency coeffcent (E) ash-sutclffe model effcency coeffcent The ash-sutclffe model effcency or accuracy coeffcent (E) s commonly used to assess the predctve power of hydrologcal dscharge models [5]. However, t can also be used to quanttatvely descrbe the accuracy of outputs for other physcal models. Here, It s utlzed for assessng the accuracy of the appled mappng and s defned as E E x y 1 1 I 1 I 1 I ( X 1 I ( X 1 ( Y re, re, re, ( Y re, X X Y Y ma, re re ) ma, ) ) ) where Ex s the ash-sutclffe model effcency coeffcent n the x-drecton (for assessng the x-coordnate of the mapped ponts) and E y s the ash-sutclffe model effcency coeffcent n the y-drecton (for assessng the x-coordnate of the mapped ponts). Also, X, s the x- coordnate of the real secton ponts, mean of x-coordnates of the real secton ponts, ma re X ma, s the x-coordnate of the mapped ponts, re X s the Y re, s the y-coordnate of the real secton ponts, Y, s the y-coordnate of the mapped ponts, Yre s the mean of y-coordnates of the real secton ponts. Essentally, the closer the effcency coeffcent s to 1, the more accurate the mappng s. Table 4 dsplays the best values of error (E), number of mappng coeffcents (), the requred tme for fndng the optmum values of E and and the correspondng values of ash-sutclffe model effcency coeffcents. As evdenced n ths table, great accuracy has been acheved n the appled mappng technque for all the symmetrc and non-symmetrc sectons, snce the effcency of mappng for both x and y coordnates of ponts ranges from to 1. Table 4. Optmum values of E and, the requred computatonal tme, and the mappng effcency. Sectons Symmetrc rectangular on-symmetrc rectangular symmetrc bulbous The best value of E.6875E The best value of Log E mn Tme to fndng the best and E(sec) E x E y

37 on-symmetrc bulbous Symmetrc fne on- symmetrc fne Symmetrc chne on- symmetrc chne E Conclusons A new generaton of mult-parameter conformal mappng method has been proposed. Usng ths method, t has been demonstrated that any arbtrary secton such as large bulbous, fne, rectangular and chne sectons and sectons wth abrupt curvatures can be easly mapped to a unt crcle wth hgh accuracy and mnmum error. Contrary to the prevous generaton of multparameter mappng technques, the suggested method s drastcally dfferent n detal and can map any arbtrary symmetrcal and non symmetrcal sectons onto a unt crcle n a much qucker way. It uses least square scheme to compute the mappng coeffcents, consders a dfferent value for the frst mappng coeffcent, solves all the mappng parameters smultaneously, and by makng a smple assumpton, an equaton s obtaned whch s solved for the correspondng angles n the mapped unt crcle plane. Moreover, t makes use of the results of the symmetrcal case for assgnng the ntal guess for the mappng coeffcents n the case of non-symmetrcal sectons. In ths case, the ntal guess s assumed to be the average of the two mappng coeffcents found based on the approach adopted for the symmetrcal sectons and for the rght and left (half) dstance from the centerlne on the waterlne. The newly ntroduced technques make the mult-parameter mappng technque more robust, shorten the process of mappng and decrease the computng tme. Several examples of mappng the symmetrcal as well as non-symmetrcal sectons have been demonstrated wth hgh accuracy. Results of the proposed mappng technque, n the case of symmetrcal sectons, have been sketched and compared aganst the results of the Lews mappng. In the case of symmetrcal sectons, comparson between the current mappng method and the method of Westlake and Wlson [3] llustrates better agreement of the current results wth the ponts of the real sectons. Another sgnfcant pont to be mentoned s the fact that, n order to acheve a desgnated accuracy, the current method needs lesser value that makes the proposed mappng technque more accurate and faster. On the other hand, n the case of nonsymmetrcal mappng, comparson of the results of the current mappng method and Westlake and Wlson scheme ndcated smlarty of the results or neglgble dfferences. To better dsplay the relatons between the ponts on the real secton and ther correspondng angles n the mapped unt crcle plane, the correspondng angles for the symmetrc rectangular secton have been plotted n a fgure. Based on the suggested mult-parameter mappng technque, a unque program has been produced that can be joned wth any two dmensonal potental problem solvers for the hydrodynamcs analyss of flow around dfferent marne structures.

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