THE APPLICATION OF MESHLESS CYLINDER CONTROL SURFACE IN RANKINE-KELVIN HYBRID METHOD

Transcription

1 odogadja/shipbuildig/ope aess Volume 67 umbe 3, 7 Hui Li Lihu Hao Huilog Re o Tia ISS 7-5X eiss THE APPLICATIO O MESHLESS CYLIDER COTROL SURACE I RAKIE-KELVI HYRID METHOD Summay UDC : :59.6 Oigial sietifi pape The solutio of the thee-dimesioal seaeepig poblem with fowad speed i the fequey domai still has some well-ow poblems. I this pape, a Raie-Kelvi hybid method beefitig the meits of both the Raie soue ad Kelvi soue is peseted ad demostated by studyig the wave diffatio/adiatio poblem with eo fowad speed as a example. A meshless ylidial sufae is seleted to be the otol sufae dividig the fluid domai ito two egios, ad the veloity potetial ad its omal deivative o the otol sufae ae epeseted by seies expasios. The peset Raie- Kelvi hybid method is validated by the added mass ad dampig assoiated with the liea adiatio foe, ad ompaiso is made with the doumeted aalytial solutio. All the wo i the pape sheds light o solvig the fowad speed hydodyami poblems. Key wods: Raie-Kelvi hybid method; seies expasio method; meshless ylidial otol sufae;. Itodutio As it is ow, whe solvig the hydodyami poblem with fowad speed, the implemetatio of the Kelvi soue method usig the Gee futio assoiated with a taslatig ad osillatig soue big alog seveal well-ow poblems oeig its wave ompoet. I patiula, it behaves with omplex sigulaities ad high osillatios whe both the field ad soue poits appoah to the fee sufae []. I the ealie wo, umeial esults wee i vey good ageemet fo immesed bodies lie sphee ad ellipsoid, o sufae pieig body of vey slede shape lie Wigley hull []. Ufotuately, fo the ealisti otaie ships, the esults peset lage disepay fom those of model tests. Te ad Che () [3] ame up with a ew method to solve the fowad speed poblem, ad 87

2 they ty to beefit the meits of the Raie soue method ad the Kelvi soue method by usig a hemisphee as a otol sufae. Howeve, the method made the alulatio of the itegatio ove the otol sufae omplex ad may lead to some toubles about sigulaities. I this wo, we use a ylide as a otol sufae istead of a semi-sphee i the same spiit as Liag ad Che s wo[4]. The Raie soue method is used i the iteal domai, ad the fudametal solutio is i the fom of, with stadig fo the distae betwee the field poit ad the soue poit. Paels ae distibuted ove the body sufae ad the fee sufae i the iteal domai. Kelvi soue method is used i the exteal domai ad the Gee futio whih satisfies the fee-sufae bouday oditio ad adiatio oditio is adopted. O the meshless ylidial otol sufae, the veloity potetial ad its omal deivative ae expaded ito ouie-laguee seies ad Lv et al. [5] have poved the effetiveess of appoximatio by Laguee seies fo diffatio waves o the ylide. The objetive of this wo is to solve the eo speed seaeepig poblem as a example to show the effetiveess of Raie-Kelvi hybid method with a meshless ylidial otol sufae. I the umeial implemetatio, we oside a hemisphee floatig at the fee sufae, ad the fowad speed is out uled at the momet. The added mass ad dampig oeffiiets ae alulated with the peset Raie-Kelvi hybid method, ad the ovegee test assoiated with the tems of Laguee futios ad ouie seies is made. I additio, the ifluee of diffeet otol sufae adius o the umeial esults is studied though ompaig with the aalytial solutio, ad evetually a satisfatoy auay is obtaied. Due to the fat that thee ae may ew defied oeffiiets oeig multi-fold itegals i this method, Chebyshev expasios ae utilied to appoximate the esultat itegal to impove omputatioal effiiey. The suessful appliatio of the Raie- Kelvi hybid method i eo speed poblem has established a foudatio fo solvig the fowad speed poblem.. The seies expasio method of veloity potetial ad its omal deivative I the Raie-Kelvi hybid method, as a meshless iula ylide sufae is seleted as otol sufae, the veloity potetial ad its omal deivative o the otol sufae eed to be expessed aalytially. I this pape, they ae expaded ito ouie-laguee seies. The veloity potetial o the otol sufae is a futio assoiated with the pola agle ad vetial oodiate,. positively dowwad, whih a be expessed as As the veloity potetial, deeases expoetially with the iease of, ad whe teds to ifiity, the veloity potetial teds to eo. The veloity potetial is expaded ito seies of Laguee as follows whee (), f L e f L is -th ode Laguee polyomial, ad L e is the -th ode Laguee futio. The othogoal popety of Laguee polyomials is osideed i Eq. () ad Eq. (3). L L e d () 88

3 whee Thus, d (3) is Koee delta futio. is multiplied o the two sides of the Eq. () simultaeously ad we a itegate the both sides fom eo to ifiity with espet to seies expasio a be expessed as follows,, ad the oeffiiet of the f d (4) As the veloity potetial o the otol sufae is a peiodi futio about of whih the peiod is, so we have,,, f f oditio is satisfied o the peiod, amely, ad the Diihlet. The futio is otiuous o has fiite disotiuity poits of the fist id.. The futio has fiite exteme poits. So the veloity potetial o the otol sufae a be expaded ito ouie seies, ad its omplex expoetial fom is as follows f im (5) m me with im m f e d (6) The substitutio Eq. (5) ito Eq. () yields, m m im e (7) The substitutio Eq. (4) ito Eq. (6) yields im m, e dd (8) I the same way, the omal deivative of veloity potetial, of a abitay poit o the otol sufae a be also expaded ito Laguee-ouie seies as follows, m m im e (9) im m, e dd () I (7) ad (9), m ad m ae oeffiiets of the seies expasio ad defied by (8) ad (), espetively. 89

4 3. The appliatio of aie-elvi hybid method to eo speed poblem 3. Defiitio of paametes We oside the ship floatig i the sea of ifiite depth with eo speed ad use a ylide to sepaate the whole fluid domai ito two domais. I the exteal domai, the omal vetos of the bouday sufaes positively iwad to the domai. I the iteal domai, the omal vetos poit positively outwads of the domai. The omal vetos ad the oodiate system ae show i igue. ig. Defiitio of the oodiate system ad omal vetos It is oveiet to use the ylidial oodiate system as a ylidial sufae is hose to be the otol sufae. Hee is the tasfomatio betwee the two oodiates. whee h x hos y hsi () is the adius of the ylide,, vaies i the plae xoy 9,, a abitay poit a be expessed as P( h,, ) i the ylidial oodiate system. The bouday oditios i the iteal domai ae give as followigs g j j j,,,6 whee deotes the osillatio fequey. 3. Exteal poblem, ad the o the fee sufae () o the body sufae (3) I the exteal domai, the Gee futio used hee is give i the fom of [6] v (,).. ( G P v PV e J R d i e ) J R (4) P P I whih, g is the wave umbe, J () is the eoth ode essel futio of the fist id, PV.. is piiple-value itegal. o the veloity potetial at a abitay field poit P o the otol sufae i the exteal domai, appliatio of the Gee s seod idetity povides P G P, G P, ds (5)

5 Afte substitutig Eqs. (7) ad (9) ito Eq. (5), we obtai il il il l l l e [ G e G e ] ds (6) Appliatio of Eq. (8), we have il il im m l l (7) [ G e G e ] e dsdd We defie ew oeffiiets G il im m, G e e dsd d (8) H G e e dsdd il im m, So Eq. (7) a be ewitte as (9) m Hm, Gm, l l As the subsipts expessed as follow (),, ad m hage, we a get a system of equatios whih a be H E G () CC, CC, Hee H, G ae maties osisted of H, ad G, osisted of ad, m m, ad E is the uit matix of whih the dimesio is equal to, ae vetos CC, H. ially we a get the elatioship betwee the seies expasio oeffiiets of the veloity potetial ad its omal deivative o the otol sufae. C, C C, C G H E () Ad we defie the matix D as C, C C, C D G H E (3) 9

6 3.3 Exteded bouday itegal equatio method ig. Shemati diagam of exteded bouday itegal equatio method As the appliatio of the fee-sufae Gee futio i the exteal domai poblem will esult i the ouee of iegula fequeies, exteded bouday itegal equatio method peseted i [7] is adopted to emove iegula fequeies, ad the iteio fee sufae is divided ito paels show i igue. The bouday itegal equatios ae give as follows P G ds Gds Gds 4 Gds Gds G ds PC P (4) (5) The dipole distibutio o eah pael of the iteio fee sufae is assumed ostat. Substitutig Eqs. (7) ad (9) ito Eqs. (4) ad (5) ad applyig Eq. (8), we a get the followig itegal equatios. Whe field poit is o otol sufae with (6) H G H m m, m, m, i l l i im Hm, G e dsdd i (7) G ad H m, have bee defied i Eqs. (8) ad (9). m, Whe field poit is o iteio fee sufae 4 H G H i,,, i l l i (8) with, il H G e ds (9), il G G e ds (3) 9

7 H, G ds (3) i I a simila way with Eq. (), we a get a matix fom C, C C, CC, H E H G, C,, C H H 4 E G (3) H m, Amog, H,, them, G,, H, C, H, C, H, G C,, H,, ad the veto is osisted of, H. ae sepaately osisted of. is the uit matix of whih the dimesio is equal to om Eq. (3), we a get the ew elatioship betwee the seies expasio oeffiiets of the veloity potetial ad its omal deivative o the otol sufae amog them D (33) D G H 4 E H G C, C C,,, C 4 H E H E H H 3.4 The validatio of the exteal domai C, C C,,, C To veify the elatioship betwee the veloity potetial ad its omal deivative obtaied fom the exteal domai, we have alulated the diffatio potetial of a ifiitely log vetial iula ylide. We oside a egula wave as the iidet wave, whih popagates alog the positive axis of x, ad the oespodig fist ode iidet wave potetial is as follows ga i x ga m m m i i m e e e i J R os m i E (34) (35) (whe m, m,om ) Amog them, A is the amplitude of the iidet wave, ad R,, is the ylidial oodiates of a abitay poit. Aodig to the body sufae oditio, we a get ga e i m J R os m (36) R iw 7 m m m Applyig Eqs.(9) ad (), the seies expasio oeffiiet of the omal deivative of diffatio potetial a be obtaied 93

8 whee ga m il e os m i J m R m e dd iw m ml ga iw m m l ml othes m mi Jm R ml (37) (38) Appedix a be efeed fo the alulatio details of Eq. (37) Substitute Eq. (37) ito Eq. () ad (33), we a get the seies expasio oeffiiets of the diffatio potetial, the umeial solutio a be obtaied based o Eq. (7). At the same time, the diffatio foe of a ifiitely log vetial iula ylide a be alulated based o the followig Eq. d wj iw 7 jds ( j,,3) (39) s Substitute Eq. (7) ito Eq. (39), the d im wj m j m iwr e dd (4) Substitute os, si, ito Eq. (4), It is obvious that w, w3, d w iwr,, d d d wj (4) Aalytial solutio of the diffatio potetial [6] is as follow m Jm R m m os (4) ga e i H R m 7 iw m Hm R Substitute Eq. (4) ito Eq. (39), aalytial solutio to the diffatio foe a be expessed as follow gr A J R i H R d w H R Compaiso of the umeial solutios fo the diffatio potetial ad diffatio foe with the aalytial oes ae peseted i Setio Iteal poblem I the iteal domai, Raie soue is adopted whih is give as follow P G (44) (43) 94

9 o the veloity potetial at a abitay field poit P o the bouday sufae of iteal domai osistig of otol sufae fee sufae ad body sufae, appliatio of the Gee s seod idetity yields S C P G G S ds C S S (45) The fee sufae ad body sufae ae divided by expessio (45) beomes S ad P G Gds C S paels, espetively. Thus, (46) Substitute the bouday oditios Eqs. () ad (3) ito Eq. (46), P G G ds G ds G ds g G ds G ds whee is the ompoet of the omal veto o. I Case, whee the field poit P is o the otol sufae S il e l il il [ G e G e ] ds l l g G ds G ds G ds G ds (47) (48) Appliatio of Eq. (8), oe a get 95

10 Deote il im m l il im l G e e dsdd G e e dsdd im G e dsdd im G e dsdd im G G e dsdd g il im G, m, G e e dsdd H G e e dsdd il im, m, im G, m, G e dsdd im H G e dsdd, m, im G, m, G e dsdd im H, m, G e dsdd (49) (5) Eq. (49) a the be ewitte as follows G H G m, m,, m,, m, l l H G H, m,, m,, m, g (5) I a simila way with Eq. (), we a get a matix fom as follows g C, C C, C C, C, C, C, G H G H G H (5) Hee CC, G, m,,, m, ad. CC, C, C, C, C, H, G, H, G, H ad G ae maties omposed of H,,, m H, G, m,, H, m, ad G, m,, ad, ae vetos osistig of The substitutio Eq. (33) ito Eq. (5) yields 96

11 C, C C, C C, C, C, C, E H G D H G H G g (53) I Case, whe the field poit P is o the body sufae o o the fee sufae the pael iludig the field poit P is umbeed by, the Eq. (47) beomes Deote il G e ds G e ds S il l l G ds G ds G G ds g,,,,,,,,,,,, il G G e ds il H G e ds G H G H G ds G ds G ds G ds S, (54) (55) The Eq. (54) a be ewitte as follows G H G H,,,,,, l l G H,,,, g (56) I a simila way with Eq. (), we a get a matix fom as follows E3 G H G H G H g, C, C, C,,,,, C,, Hee, G, H, G, H, G,, (57) ad H ae maties osisted of G,,, H,,, G,,, H,,, G,, ad H,, ad is veto osisted of ad, E3 is a uit matix with dimesios of, Substitutio of Eq. (33) ito Eq. (57) yields, C, C,,,, H G D H, G H E3 G g (58) 97

12 The the equatios fo the etie domai a be obtaied by ombiig Eq. (53) ad (58) C, C C, C C, C, C, E H G D, H, G H g C, G,, C, C,,, G H G D, H, G H E 3 g (59) Oe Eq. (59) is solved, oe a obtai the veloity potetial o the paels ad the seies expasio oeffiiets of the veloity potetial o the otol sufae. The method fo evaluatig the oeffiiets we have defied fo the multi-fold itegals is give i the appedies. om the expessio of the multi-fold itegals, the Chebyshev expasio is used to appoximate the itegatios ad impove the effiiey. 3.6 The solutio of added mass ad dampig oeffiiet whee, The added mass ad dampig oeffiiet a be solved as follows a ij a ij ibij is added mass, i j ds (6) S b b ij paels, Eq. (6) a be ewitte as is the dampig oeffiiet, as the body sufae is disetied ito ib ij i ij j j i S a ds ( i, j,,,6) (6) b The added mass ad dampig oeffiiets a be obtaied by substitutig the veloity potetial o the body sufae ito Eq. (6). 4. Results ad aalysis 4. Results fo the diffatio potetial umeial solutio has bee omputed with the ode of Laguee futio fom to ad the ode of ouie seies fom to, the adius of the ylide is 3. m, a fixed poit P (3..5.) is seleted expessed i ylidial oodiates o the iula ylide, the diffatio potetials at the fixed poit vayig with waveumbe ae show i igue 3, amog them, method efes to the oigial solutio method of the exteio domai i subsetio 3., Method efes to the exteded bouday itegal equatio method i subsetio

13 Aalytial Re{ } Method Re{ } Aalytial Im{ } Method Im{ } ig. 3 The ompaiso esults of diffatio potetials I igue 3, the umeial solutio agees well with aalytial solutio fo waveumbe vayig fom. to.6 exept fo is.8. As we a see, umeial solutio has a flutuatio whe waveumbe is.8, ad this wave umbe is egaded as iegula fequey. Diffeet poits o the iula ylide ae hose with h =3. ad with diffeet iumfeetial loatios of the poits ae show i igue 4 =.. The diffatio potetials at the iegula fequey vayig Aalytial Re{ } Method Re{ } Method Re{ } Aalytial Im{ } Method Im{ } Method Im{ } ig. 4 The ompaiso esults of diffatio potetials at iegula fequey om igue 4, we a see that the umeial solutio obtaied fom exteded bouday itegal equatios i method is i good ageemet with aalytial solutio, whih illustates the elatioship betwee the veloity potetial ad its omal deivative o otol sufae is ight ad exteded bouday itegal equatio method is apable of emovig the iegula fequey. 99

14 Aalytial Re Aalytial Im Method Re Method Im oe(x 3 ) ig. 5 The ompaiso esults of diffatio foe Diffatio foe i the dietio of solutio shows a satisfied auay. x axis is give i igue 5, ad the umeial 4. The esults of added mass ad dampig oeffiiets of a hemisphee A hemisphee is hose as the example fo alulatio, the umeial solutios about the added mass ad dampig oeffiiet ae ompaed with the aalytial solutios give by Hulme[8]. o the oveiee, we have defied the followig oditios show i table. Table list of oditios Paamete's ame otatio Coditio Radius of the hemisphee m m m m m Radius of the otol sufae R 3m 3m 3m 4m 6m umbe of paels of hemisphee umbe of paels o fee sufae Ode of Laguee futio 5 Ode of ouie seies M 5 The ovegee test assoiated with the ode of ouie-laguee seies has bee made though oditio, ad 3. The esults ae show i the followig figues, amog them, L is the haateisti legth whih is equal to the adius of hemisphee.

15 Aalytial Solutio Coditio Coditio Coditio Aalytial Solutio Coditio Coditio Coditio 3 a.4 a L L ig. 6 Suge added mass oeffiiets with diffeet ode of ouie-laguee seies ig. 7 Heave added mass oeffiiets with diffeet ode of ouie-laguee seies.4.3 Aalytial Solutio Coditio Coditio Coditio 3.3 Aalytial Solutio Coditio Coditio Coditio 3 d. d L ig. 8 Suge dampig oeffiiets with diffeet ode of ouie-laguee seies L ig. 9 Heave dampig oeffiiets with diffeet ode of ouie-laguee seies om the esults, we a see that the umeial esults ae i good ageemet with the aalytial solutios, i igue 6 ad igue 8. The esults of suge motio seem sesitive to the ode of ouie-laguee seies at a high wave fequey, ad the disepay betwee aalytial solutios ad umeial esults i oditio ae lage tha that i oditio ad oditio 3. om igue 7 ad igue 9, the esults of heave added mass ad dampig oeffiiets have show a good peisio i oditio. I olusio, the umeial esults show a good ovegee with the iease of the ode of ouie-laguee seies. At the same time, the ifluee of diffeet otol sufae adius to umeial esults has also bee studied though oditio 3, 4 ad 5. The esults ae show i the followig figues.

16 Aalytial Solutio Coditio 3 Coditio 4 Coditio Aalytial Solutio Coditio 3 Coditio 4 Coditio 5 a.4 a L L ig. Suge added mass oeffiiets fo diffeet otol sufae adius ig. Heave added mass oeffiiets fo diffeet otol sufae adius.5.4 Aalytial Solutio Coditio 3 Coditio 4 Coditio Aalytial Solutio Coditio 3 Coditio 4 Coditio 5.3. d. d L ig. Suge dampig oeffiiets fo diffeet otol sufae adius L ig. 3 Heave dampig oeffiiets fo diffeet otol sufae adius om the esults we a see, added mass ad dampig oeffiiets of suge motio ae moe sesitive to otol sufae adius tha that of heave motio. I igue ad igue 3, the umeial esults show a good ageemet with the aalytial solutio i diffeet oditios. I igue ad igue, the ifluee of the otol sufae adius to the umeial esult is small at a low wave fequey, ad with the iease of the waveumbe, the umeial esults i oditio 3 ae moe auate tha the othe oditios oveall. The mai easo is moe paels ae eeded o the fee sufae with the iease of the otol sufae adius. Cosequetly, osideig the hose of otol sufae adius, we pefe to a smalle oe. 5. Colusio The Raie-Kelvi hybid method has bee applied to solve eo speed seaeepig poblems suessfully. om this pape, we a daw the followig olusios. The appliatio of Kelvi soue i the exteal domai will esult i iegula fequey, ad a exteded bouday itegal equatio method has bee used to elimiate the iegula fequeies.. The esult is oveget with the iease of the ode of ouie-laguee seies, ad a bette auay a be ahieved whe a smalle adius of the otol sufae is hose with the same ode of ouie-laguee seies. 3. All the wo i this pape has laid the foudatio fo solvig the fowad speed hydodyami poblem.

17 ACKOWLEDGMETS This eseah is fuded by Chia MOST 973 pla (C373). The disussios with D. Liag Hui ad Che Xiao-o of ueau Veitas ae appeiated. REERECES [] Che X.., Wu G.X. (). O sigula ad highly osillatoy popeties of the Gee futio fo ship motios. Joual of luid Mehais, Vol 445, [] Che X.., Diebold L., Douteleau Y. (). ew Gee-futio method to pedit wave-idued ship motios ad loads. I Twety-Thid Symposium o aval Hydodyamis. Val de Reuil, ae, Septembe 7. [3] Te I., Che X.. (). A oupled Raie-Gee futio method applied to the fowad-speed seaeepig poblem. 5th IWWW, Habi, Chia, May 9. [4] Liag H., Che X.. (6). A multi-domai method fo the omputatio of wave loads. 3th IWWW, Plymouth, MI, USA, Apil 3 6. [5] Lv M.D., Li H., Re H.L., Che X.. (5). Expasio by Laguee utio fo Wave Diffatio aoud a Ifiite Cylide. I Poeedigs of Computatioal Mathematis ad Appliatios Cofeee, [6] Wehause J.V., Laitoe E.V. (96). Sufae Waves, i lügge S., Tuesdell C., eds., Eylopaedia of Physis (Spige Velag), Vol 9: [7] Zhu X. (994). Iegula fequey emoval fom the bouday itegal equatio fo the wave-body poblem (Dotoal dissetatio), Massahusetts Istitute of Tehology. [8] Hulme A. (98). Wave foes atig o a floatig hemisphee udegoig foed peiodi osillatios. Joual of luid Mehais, Vol (), [9] Che X.. (4). Pesoal otes o ouie-laguee expasio o a ifiite vetial ylide. 3

18 APPEDIX I this pat, some itegals about Laguee futio will be alulated followig his otes [9]. The alulatio about the itegatio of Laguee futio. e L d e L L d e L e L e L d e L d (.) Eq. (.) a be tasfomed ito eusio Eq. (.) e L d e L L e L d ( ) Chage the ode of Laguee polyomial fom get the followig esult i ito i i i (.) ad euse Eq. (.), we a e L d e L L e L d e L L e ially, the itegatio about Laguee futio a be solved efeig to Eq. (.3) (.3) ( ) e d e L d i e L i L i e i (.4). e L d e L L d e L e L e L d e L d (.5) Eq. (.5) a be tasfomed ito eusio Eq. (.6) e L d e L e L e L d e L i L e i i i 4 (.6) ially, the itegatio about Laguee futio a be solved efeig to Eq. (.6)

19 e d e L d i e L i L i e i Whe (.7) e d L d L L (.8) Whe the itegatio is omputed ea appoximatio, we have doe the followig e / The we have m m m m! (.9) m / e L d L d m m! m m! m!!! m m (.).3 e d e d We use the esult of Eq. (.4), the i i e d e d e L L i L i d e L d (.) Refeig to Eq. (), the aalytial solutio of Eq. (.) a be obtaied as follow e d e d 4 (.) 5

20 .4 e d e d Simila to Eq. (.), we a get the followig esult e d e d 4 (.3) APPEDIX This pat itodues the expessio of Gee futio i the ylidial oodiate system. Kelvi soue a be witte as Eq. (.) G( P,) G G G G w v.. ( v PV e J R d i v e ) J vr v P P (.) amog them, v,,.., ( G G Gw v PV e J R d G ) i v e J v R. v P G P is the Raie soue whih is assoiated with the distae betwee the field poit P( x, y, ) ad the soue poit (,, ) : G x y R Z (.) P, Amog them R x y Z Eq. (.) a be ewitte as Z R Z e J R d (.3) I the ylidial oodiate system, the field poit P ad the soue poit a be witte as P( h,, ), ( h', ', ) Usig the idetity ip' J R e J h J h (.4) We a get p p p 6

21 ip G e e J h J h d (.5) p p p I a simila way ip G e e J h J h d (.6) p p p ip G w v e PV.. e J p h J p h d v (.7) p ( ) ip G i e e J h J h (.8) p p p APPEDIX 3 The omputatioal methods of the itegals defied i the exteal domai. G m, a be divided ito the followig fou pats G il im m, G e e dsd d (3.) G il im m, G e e dsd d (3.) G il im m, 3 G w e e dsd d (3.3) G il im m, 4 G e e dsd d (3.4) The we a alulate the fou pats sepaately with the fomulas metioed i appedixes ad. Substitute Eq. (.5) ito Eq. (3.), use the othogoally of ouie seies, we a get Eq. (3.5) il im Gm, G e e dsdd p p pm pl h J h J h d e d e d p p p pm pl h J h J h d e d e d p Substitute Eq. (.) ad (.3) ito Eq. (3.5), we a obtai the followig esults Whe m, 4 p p pm pl p (3.5) (3.6) G h J h J h d 7

22 Othewise 4 m, p p pm pl p G h J h J h d I a simila way m, 8 p p pm pl p G h J h J h d (3.7) (3.8) m, 3 6 pl pm.. p p (3.9) G h p v J h d To mae the alulatio of umeial method auate ad effiiet, we a deal with Eq. (3.9) as follows m, 3 6 pl pm.. p p G h p v J h d pl pm p p 6 h p. v. J h d 6 h p. v. J h J h pl pm p p p pl pm p p 6 h J h d d (3.) Substitute Eq. (.8) ito (3.4), we a get il im Gm 4 i e J R e e dsdd hi pm pl J p h J p h p 6 The omal deivative of Kelvi soue o the otol sufae a be witte as G G G G G G h h w (3.) (3.) The ip G e e J h J h d p p p (3.3) 8

23 ip G e e J h J h d p p p (3.4) ip G w p. v. e e J p h Jp h d p (3.5) ip G i e e J p h Jp h (3.6) p pats Simila to the alulatio of G m,, H m, il im m, a also be divided ito the followig fou H G e e dsdd (3.7) H G e e dsdd il im m, (3.8) H G e e dsdd il im m, 3 w (3.9) H G e e dsdd il im m, 4 The esult a be witte as (3.) m, 4 p p pm pl p fo H h J h J h d 4 m, p p pm pl p (3.) fo H h J h J h d m 8 pm pl p p p H h J h J h d (3.) (3.3) (3.4) H 6 h p. v. J h J h d m 3 pl pm p p p m 6 pm pl p p ' p H h i J h J h (3.5) 4 9

24 I the exteded bouday itegal equatio method, whe the field poit is o the otol sufae ad the soue poit o the iteio fee sufae, the omal deivative of Kelvi soue o the fee sufae a be witte as G G G G G G w (3.6) G G (3.7) H m, ip G w p. v. e e J p h J p h d p (3.8) ip G i e e J p h J p h (3.9) p defied i Eq. (7) a be divided ito fou pats H H H H H (3.3) m, m, m, m, 3 m, 4 whee H H (3.3) m, m, Gw im Hm, 3 e dsdd i (3.3) G im Hm, 4 e dsdd i Substitute Eq. (3.8) ito Eq. (3.3) (3.33), 3 im m i m m H S e J ( h) J ( h) d (3.34) Substitute Eq. (3.9) ito Eq. (3.33) 4 im H m, 4 J m h Jm h S e i i (3.35) G, Whe the field poit is o the fee sufae ad the soue poit o the otol sufae, defied i Eq. (3) is divided ito fou pats G G G G G (3.36),,,, 3, 4, il G G e ds (3.37)

25 , ' il G G e ds (3.38), 3 w il G G e ds (3.39), 4 il G G e ds (3.4) Amog them, G, G,, ad the deivatio of Appedix 4, il v, 3 4 l l G, is give i Eq. (4.4) i G h v e J vh J vh dv (3.4) v v v il v, 4 8 l l v G v h e J v h J v h (3.4) I a simila way, ad H, H, H,, ad the esult of defied i Eq. (9) a also be divided ito fou pats, H, will be give i Eq. (4.5) of Appedix 4, efeig to Eq. (3.5) ad Eq. (3.6), the oe a obtai the followig esults il + v v, l l H v he p v J vh J vh dv (3.43) v v v ip, 4 8 l l v v H v h e J v h J v h (3.44) Whe both the field poit ad soue poit ae o the iteio fee sufae, simila to the way of dealig with H m,, H, defied i Eq. (3) a be divided ito fou pats, ad it is obviously that H,, H,, H G ds v p v e vj vr dv S v, 3 w.. i i v v (3.45) H G ds v J v R ds v J v R S (3.46), 4 i i i APPEDIX 4 The alulatio of the itegals defied i Eqs. (5) ad (55). G, m,, H, m, ae the fist pats of Gm,, H m, espetively, whih have bee solved i Appedix 3, whe the soue poit is o the body sufae o o the fee sufae, the soue o eah pael is ostat.

26 im G, m, G e dsdd im e S J m h Jm h d e d e d (4.) whee S epesets the aea of pael o the body sufae. Substitute the esult of. ad. i Appedix ito Eq. (4.), the G, m, a be solved fo. The omal deivative of Kelvi soue o the body sufae a be witte as G Gos si G os si G h h 3 (4.) the im H G e dsdd, m, im im im os si m m S e J h J h d e d im os si S e J m h Jm h d e d h m m S e J h J h d e d (4.3) Whe the soue poit is o the fee sufae, im G, m, e S J m h Jm h d (4.4) im, m, m m H e S J h J h d (4.5) Whe the field poit is o the body sufae ad the soue poit o the otol sufae,, il G G e ds ip il e e J p h J p h d e ds (4.6) p il l l e h J h J h d e d il H G e ds,, il l l he J h J h d e d (4.7) Whe the field poit is o the fee sufae ad the soue poit is the otol sufae

27 il G,, he J l h J h d v (4.8) il H,, h e J l vh Jl h d (4.9) Coespodig autho: Hui Li, College of Shipbuildig Egieeig, Habi Egieeig Uivesity, Heilogjiag 5,Chia. Huili@hbeu.edu., Tel.: Submitted:.8.5. Aepted: Hui Li, huili@hbeu.edu. Lihu Hao Huilog Re o Tia College of Shipbuildig Egieeig, Habi Egieeig Uivesity, Heilogjiag 5,Chia. 3