Research on Topology optimization Method for Tanker Structures in

Research on Topology optmzaton Method for Tanker Structures n Cargo Tank Regon QIU WeqangMarc_quwq@163.com,GAO Chu,SUN L, LUO Renje Marne Desgn & Research Insttute of Chna (MARIC), Shangha, Chna Abstract In ths paper, the dfferences between the cargo tank structure topology optmzaton and structural members scantlng optmzaton are researched, such as the desgn varables, restrcton condtons, load-cases, and morbdty load-case due to the sgnfcant dfferences between the stran energy of dfferent loadcases. To carry out topology optmzaton of the cargo tank structure on the ordnary computer and to acheve clear topologcal confguraton, t s necessary to study on the approprate optmzaton objects, the property of elements, the topology methods and optmzaton software, the restrcton condtons, the percentage of volume constrcton and the weghtness factor of load-cases. The modelng methods of topology optmzaton fundament structures and analyss methods of topology optmzaton are recommended for the applcaton on the actual engneerng cases. For example, n the topology optmzaton of a VLCC wth one centerlne longtudnal bulkhead, the SIMP method (Sold Isotropc Mcrostructures wth Penalzaton) and BESO method (B-drectonal Evolutonary Structural Optmzaton Method) are appled to acheve clear topologcal confguratons of Prmary Support Members (PSM) n cargo tanks. 1 Introducton In the recent years, the theory and method of topology optmzaton are developng quckly, and applcaton on the car producton and spaceflght desgn step up day and day. But n the desgn of shp structures, the topology optmzatons are rarely used. As the key technque of modern structures, nnovatve desgn, the topology optmzatons are the expanson of tradtonal scantlng optmzatons and shape optmzatons. Am at the objects to develop new green, lower-energy, envronment safeguard and economc shps, the topology optmzatons are necessary means. At the present tme, the actual topology optmzaton cases of cargo tank structures are absent. The man reasons are the dsassocaton between the theory research and engneerng cases, and there are no methods to abstract the real structures from the topologcal confguratons of cargo tank structures. The man technque dffcultes are numerous and complcated, such as more desgn varables, more restrcton condtons, more load-cases, and morbdty load-cases due to the sgnfcant dfferences between the stran energy of dfferent load-cases. So these complcatons nduce optmzaton calculaton tmes that are nearly endless, the topologcal confguratons are not clear, and t s dffcult for abstracton of vable structure forms. In ths paper, the above technques are solved n general. Usng a VLCC wth one centerlne longtudnal bulkhead as an example, the SIMP method (Sold Isotropc Mcrostructures wth Penalzaton) and BESO method (B-drectonal Evolutonary Structural Optmzaton Method) are appled to acheve clear topologcal confguratons of Prmary Support Members (PSM) n cargo tanks based on the load-case condtons and boundary condtons from New Common Structural Rules (CSR). Page 1 of 18

2 Mathematcs and engneerng modelng of cargo tank structures TSCF 2016Shpbulders Meetng 2.1 Basc prncple of topology optmzaton and selecton of optmzaton methods The classcal crteron for optmzed desgn of shp structure s to: Mnmze the total steel weght of structures whch fulfll all the strength requrements. The most common strength requrement s the fourth strength theory. Dstortonal stran energy s the man state varable of materal. The dstortonal stran energy s to be descrbed as follow: e f l 2 (1 + m) s V 3E (2.1) e Dstortonal stran energy of elements; m Posson s rato; f Where, l s mnmum yeld strength of materal; V volume; E Young s modulus. When the materal s determnate, f the total element volume s to be mnmzed, every element stress should be maxmzed and be close to the mnmum yeld strength of materal. All the element stresses should be homogeneous, and the topology optmzaton s necessary. The topology optmzaton means that the optmal structure for most and optmal transfer route of loads n the apponted area. Two famlar topology optmzaton methods are ntroduced brefly, such as SIMP method (Sold Isotropc Mcrostructures wth Penalzaton) and BESO method (B-drectonal Evolutonary Structural Optmzaton Method). SIMP method s resoluton stablty s well and had been extensvely used n the software Hyperworks/Optstruct. BESO method s theory s easy to understand and program wth codes. The two methods can be verfed and supplement each other. 2.2 Mathematcs foundaton of topology optmzaton 2.2.1 Mathematcs foundaton of SIMP method SIMP method allows the densty of a materal to cover the complete range of values from 0 (vod) over ntermedate values (composte) to 1 (sold) and ths knd of composte materal model s called SIMP model. For the SIMP model, the elastcty tensor and the volume of a structure are gven by E jkl jkl p ( r ) r 0, 1, r ( ) E E p Volume x dx = jkl > = ò W (2.2) 0 Where r ( x), x ÎW, 0 r 1s a densty functon of the materal and E the elastcty tensor of jkl a gven sold sotropc reference materal. The densty functon r ( x) enters the stffness relaton n a power p > 1 whch has the effect of penalzng ntermedate denstes 0 r 1, shown that the use of the SIMP model wll force the topology desgn toward lmtng values r = 0 (vod) and r = 1 (sold). Restrctng to 2D plane elastcty, assume that SIMP model s composed of two materals. One s densty s 0(vod) and another densty s r.then the bulk modulus k and shear modulus m n elastcty tensor E of ths artfcal materal should satsfy the so-called Hashn-Shtrkman bounds jkl 0 0 0 0 rk m rk m 0 k, 0 m - k + - k + + k ( 1 r ) m ( 1 r )( 2m ) 0 0 0 0 0 (2.3) 0 Where, k and 0 m represents the bulk modulus and shear modulus of materal wth densty r. Page 2 of 18

And the Young s modulus should satsfy re 3-2r 0 * 0 E E =, 0 r 1 (2.4) Isotropc materals have the relatonshp k E E =, m = 2 1 2 1 0 0 0 0 ( - u ) ( + u ) (2.5) Substtutng (2.5) nto (2.3), we have p 0 0 p 0 0 r E re r E re 0, 0 2 1 4 2 1 2 1 2 1 3 2 1 ( -u ) - ( + u ) r ( + u ) ( - r )( - u ) + ( + u ) (2.6) From (2.6), the penalzaton power p should satsfy * ì 2 4 ü p ³ p ( u ) = max í, ý î1- u 1+ u þ (2.7) For a Posson s ratou equal to 1/3, both for 2D and 3D problems a penalzaton power p > 3 n (2.7) ensures that the Hashn-Shtrkman bounds can be satsfed, so that the SIMP model can be realzed as ts stffness varyng wth r and FEM sngularty problem s avoded. Thus the SIMP model yeld relatvely nce results and ths smplcty of the penalzaton power p facltates the mplementaton of topology desgn n commercal fnte element codes. 2.2.2 Mathematcs foundaton of BESO method BESO s an evolutonary topology optmzaton method whose theory s relatvely smple and drect. Ths method s based on evolutonary strategy and ntroduces the ncluson rato (IR), rejecton rato (RR) and evolutonary volume rato (ER) whch are used to confrm the number of elements to be added and removed. And then by rankng the senstvty number of the elements (normally stran energy or von mses stress), removng and addng the certan elements. Smply speakng, utlzng a sutable scheme, to add more elements where the senstvty numbers are hgh and to remove more elements where senstvty numbers are low. The prncpal procedure of BESO s as follows: 1) Dscretze the desgn doman usng a fnte element mesh and assgn ntal property values (0 or 1) for the elements to construct an ntal desgn. 2) Perform fnte element analyss and then calculate the elemental senstvty number. 3) Accordng to Equaton 2.8 average the senstvty number applyng the flter scheme to mnmze the effect of Checkerboard and Mesh-dependency Problems Page 3 of 18

4) Average the senstvty number wth ts hstory nformaton usng Equaton 2.9 and then save the resulted senstvty number for next teraton. 5) Determne the target volume for the next teraton usng Equaton 2.10. 6) Determne the IR and RR based on V k+1, and then get and. Add the elements whose a > a and remove the elements whose t h add a < a t h del * 7) Repeat steps 2 6 untl the constrant volume V = MaxRat e V 0 s acheved and the convergence crteron for total senstvty number C s satsfed (Equaton 2.11). a = K å j = 1 K v å j = 1 v ( r j ) a ( r j ) n j (2.8) Where, K s the total number of nodes surroundng the target element wthn the crcle of radus r mn,. a a k + a 2 k -1 = (2.9) Where, s the element s ID and k s the current round number of the optmzaton. ( ) V = V 1 ± ER (2.10) k + 1 k er r or N å C k - + 1 k -N - + 1 = 1 = 1 = 1 - N C = t N C å å k - + 1 (2.11) Where, t s allowable convergence tolerance and N s an nteger number. Normally, N s selected to mply that the change n the mean complance C over the last N teratons s acceptably small. One advantage of BESO s that t could take account of the effect of the dstrbuton of senstvty number at desgn feld and non-desgn feld. In ths way the optmzed shape around the feld of boundary could be smoother. Nevertheless, snce the convergence crteron of BESO s volume fracton, t s hard for t to take account of dsplacement, bucklng and other structural desgn factors. Besdes, the effcency of BE- SO s relatvely lower than that of SIMP and the optmzed results reles greatly on the ntal shape. The ntal shape of the desgn feld plays a great role n determnng the fnal volume fracton and teraton effcency. If at the very begnnng, removng some redundant desgn feld based on prelmnary structural judgment, the optmzaton effcency and result could be much better. For dfferent optmzed object, t s hard to smply determne a sutable volume fracton. Ths value needs to be determned by repettve tests and experence. Although there are some negatve aspects of BESO, t s stll a good method to get an deal optmzed Page 4 of 18

result only f a sutable desgn feld and volume fracton has been selected. 2.3 Selecton of optmzaton objects and elements of FE analyss n ol tanker structures optmzaton 2.3.1 Select of research object for topology optmzaton As topology optmzaton s an teratve process, the calculaton data would be enormous and the calculaton process would be tme-consumng. And for a general large and medum sze ol tanker, the results of cargo hold analyss should meet the requrements of HCSR rules, thus t would consume a lot of tme completng rule computaton once. For those reasons, tme-consumng problem for the teratve process can t be completely solved even when the evolutonary algorthms are appled to solve the combnaton exploson problem. In order to complete the topology optmzaton calculaton on general mcrocomputers, a seres of non-essental structures are gnored n ths paper. And the research object s selected by the basc prncples of topology optmzaton, engneerng practce experences and engneerng constrant condtons, thus topology optmal structure wth good manufacturablty can be obtaned from a selected model wth reasonable calculaton amount. Accordng to basc theory of mechancs, when tensle stress or compressve stress s appled to fnte elements, stress and sze of the fnte element can be homogenzed smultaneously, and the topology optmal structure tends to mnmze shear stress n tself, whch can change the structure from shear bearng form nto tensle or compressve bearng form. But when lateral loads are appled to planar structures, large shear stress appears nevtably, and the shear stress can t be totally transformed nto tensle or compressve stress, whch may lead to uneven dstrbuton of stresses n structure. For the reason mentoned above, the planar bulkheads of deep tank don t conform to the mechansm of topology optmzaton. Therefore, on the premse of enough hold capacty, the area of planar bulkheads of deep tank should be mnmzed or the planar bulkheads should be changed nto curved bulkheads, whle the number of bulkheads of deep tank should meet the requrements of IMO rules. In fact, changng planar bulkheads nto curved bulkheads wll be rejected frmly by the shpyard, because t wll ncrease large amount of work. In the topology optmzaton of ol tank cargo hold structure process, the number of deep tank bulkheads should be as least as possble on the bass of topology optmzaton theory, and two measures can be taken to acheve that goal, such as deletng one of the two longtudnal bulkheads n VLCC, or reducng transverse bulkheads n some ol tankers on the premse of satsfacton of rules. But on the other hand, the knuckle lne of cargo hold, the heght of nner bottom, and the wdth of double hull depend more on requrements of general arrangement, cargo hold capacty, and calculaton of ol spll. Structural optmzaton desgn s one of the constrants, but t s not the most crucal one. The fnal decson s made by the general desgner. Structural desgners can submt an optmal project obtaned wth theory of classcal mechancs or calculated wth three dmensonal beam system to the general desgner, who wll consder dfferent stuatons to decde to adopt the project or not. Therefore, the longtudnal strength bulkheads of ol tanker were not selected as topology optmzaton object n ths paper. Sometmes the general arrangement plan may have dsadvantages for structural lghtweght desgn, and n these stuatons topology optmzaton has more sgnfcant effect. Structural desgners should make the best use of advanced desgn technology, and mnmze the adverse mpact of dfferent constrants. Consderng the requrements of smplfcaton process, the space between web frames or longtudnals should not be selected as topology optmzaton object ether. Otherwse, there wll be too many knds of longtudnals and plates, whch wll ncrease the cost of management and manufacture greatly. Wthn one sngle cargo hold, the optmal space of equally spaced longtudnals or web frames s convenent to obtan wth theory calculatons, so t s not necessary to optmze the space value. The desgn and arrangement of longtudnal strength structures can be completed wth rule computaton software, and t s not selected as optmzaton object ether. Above all, prmary supportng members n cargo holds, ncludng all the transverse webs and horzontal Page 5 of 18

strngers were selected as topology optmzaton objects n ths paper. 2.3.2 Topology optmzaton elements of FE analyss It s dfferent from the 3D sold elements topology optmzaton,the elements of three cargo hold FE analyss and topology optmzaton are 2D shell elements and 1D beam elements. The shp structures are be smulated use shell and beam elements n general. Though they are 3D structure n mnature, the macrostructures are 2D type. In tradtonal FEA, the web plate of prmary support members are smulated use shell elements, the flange plates of prmary support members (ab. PSM) and bucklng stffeners on PSM s web plate are smulated use beam/rod elements. The elements of topology optmzaton are also to be lke that, otherwse the topology optmzaton calculaton tmes wll be too long to endure. The authors also tres to add some rod elements n the shell elements edges to smulate the flange plates of PSM and bucklng stffeners n the topology analyss process. But the result are unsuccessful most of shell elements are deleted and rod elements are reserved, also the topologcal confguraton s unvable for shpyard workman. In any case, the flange plates of PSM and bucklng stffeners are to be smulated as addtonal thckness on the shell elements n some equvalent manners, such as equvalent secton areas. The thcknesses of topology basc structures are to be the PSM s web plate thckness multply by magnfed factor. The factor should be dfferent for dfferent shp types and structure areas. 2.4 Selecton of topologcal feld n topology optmzaton The topology basc structures n topologcal feld should be defned frstly, and nclude the entre possble topologcal confguraton. Accordng to the preamble, all the transverse webs and horzontal grders n cargo hold are to be taken as the topology objects. And because the longtudnal members are not taken as the topology objects, the stress restrctons on them wll not be take nto account to reduce the calculaton tme. In ths case, The PSM to support them may be lost somewhere n optmzaton procedure because they may not be on the man load transfer route whch s n accordance wth the mechansm of topology optmzaton. See fgure 1. The vertcal grder s lost after the topology optmzaton, but ths result can t be accepted by desgner n practce. Consequently, the vertcal grders, deck webs and horzontal grders wth mnmum depth wll be kept nevtably, and they wll nflect the dstrbuton of the entre cargo hold structures dstortonal stran energy. So we need try to take the vertcal grders, deck webs and horzontal grders wth certan depths as the unalterable elements. What s more, these depths should be calculated based on HCSR and terated mprovement several tmes untl that wll not affect the topologcal confguratons. The other areas except the unalterable elements should be taken as topologcal feld and fulflled wth shell elements as followng fgure 2. Page 6 of 18

Fg.1 Topologcal confguraton when no topologcal feld restrcton Fg. 2 Topologcal felds n transverse webs For the general tanker structures, the topologcal felds dstrbute dscretely on the 6~8 transverse webs and 2~3 horzontal grders. The topologcal confguratons of them are to be related together accordng to certan rules or not? In the start of topology optmzaton, to research every nch, all the topologcal felds should be unattached to each other. But to reduce the workmanshp of shpyards, t also be permtted for the conjuncton of dfferent desgn regons accordng to certan rules. As shown n fgure 3, symmetrcal to the centerlne of cargo hold n long drecton, the two smlar transverse webs are assembled together wth same topologcal confguraton. The topologcal confguratons of horzontal grders can also be assembled together or dvded nto two topologcal felds. Fg. 3 Assemble relatons between dfferent topologcal felds 2.5 Optmzaton models of ol tanker cargo structures topology optmzaton 2.5.1 Optmzaton models of SIMP method The general formulaton to solve a shp structure optmzaton problem means to fnd a combnaton of Page 7 of 18

the vectors of desgn varables X = [ x1, x2, L, x ] T N defnng the structure whch optmzes the objectve functon. An essental task of the shp structure optmzaton s to reduce the hull structure weght, consequently the mnmum weght s assumed as an objectve functon. The desgn varables should also meet the complex set of constrants mposed on ther values. Therefore the mathematcal formulaton of the optmzaton problems reads: T ì Fnd X = [ x1, x2, L, xn ] ï ïmnmze f ( X) = W ( X) ï subject K ( X) U( X) = F ï í σ ( X) -[ σ] 0 ï ï V ( X) f ï V0 ï ïî 0 < xmn xe xmax 1 (2-12) Wth the SIMP method, the materal densty s used as the desgn value, and vares between x mn and x max. X s a vector of N desgn varables, K, U and F are the stffness matrx, the global dsplacement vector and the load vector. K ( X) U( X) = F expresses that the desgn varable should satsfy the well-known lnearly elastc equlbrum equaton. von Mses stress constrants σ ( X) -[ σ] 0 for the model are consdered based on the HCSR rule. Volume fracton V ( X) / V 0 = desgn volume at current teraton/ntal desgn volume whch descrbes a fracton of ntal desgn space s also constraned. In practce, too small the upper lmtaton f makes the optmzed structure looks lke a slender beam whch can hardly fulfll the bucklng crtera n HCSR rule, on the other hand, too large the f wll sgnfcantly ncreases the calculaton tme and usually makes the optmzed structure fuzzy and ndstnct. Therefore f s assumed based on tral calculatons and be selected between 0.13~0.5 case by case to acheve sharp and clear results. 2.5.2 Optmzaton steps of BESO method Based on the theory descrbed n 2.2.2, BESO method s appled by the followng steps: 1) Matlab s used as a control platform for data processng, teraton montor and result judgment; 2) Combne Msc.Patran and PCL to acheve: a) Automatc FEM analyss results and other parameters; b) Based on the method of BESO, calculate senstvty number of each element and judge whch should be added and removed; c) Update the FEM model and generate analyss fle(*.bdf); 3) Apply Nastran to analyze and get FEM result. Evolutonary volume rato(er) can be taken about 5% when the volume fracton s above 40% (dfferent due to scope of ntal topology feld) and 1% when the volume fracton s below 40%. It should be noted that the MaxRate s the only parameter and constrant of BESO for the control of optmzaton result. Dfferent MaxRate wll drectly lead to dfferent result. However, t s hard to determne ts value just based on experence for dfferent desgn feld, structural type. Based on a lot of plot calculatons, the MaxRate should be under 20%. Nevertheless, ts specfc value could only be determned by enumerated calcula- Page 8 of 18

ton for dfferent value step by step and then compared by the desgner manually. 3 Defnton of Structural Model, boundary condtons and load condtons n cargo tank FEA 3.1 Structural Model and boundary condtons Ths secton meets the requrements of HCSR fully. Structural Model s to see Fgure 4. The Boundary Constrants at Model Ends are to see followng Table 1. Fg. 4 Typcal 3-Cargo Hold Models wthn Mdshp Regon of Ol Tankers Locaton Table 1 Boundary Constrants at Model Ends Translaton Rotaton δ x δ y δ z θ x θ y θ z Crosssecton - Rgd Lnk Aft End Rgd Lnk Rgd Lnk - - Independent pont - Fx Fx M T - end - - Cross secton End beam Fore End Crosssecton - Rgd Rgd Rgd Lnk Lnk Lnk - - Independent pont - Fx Fx Fx - - Intersecton of CL&IB Fx - - Cross secton End beam Where: - no constrant appled (free) Page 9 of 18

3.2 Load condtons and assembled coeffcent Ths secton s manly due to the requrements of HCSR. Table 2 Load combnatons for FE analyss for one centerlne ol-tght bulkheads ol tankers Page 10 of 18

Not all the load condtons wll affect the cargo hold structures scantlng sgnfcantly. To reduce the calculaton tme, the most advanced exstng shp should be analyzed for all the load condtons before the topology optmzaton. It should be estmated whch ones are the key load condtons to determne the scantlngs, whch ones are secondary and whch ones are unconsdered. But f the topologcal confguratons are dstnctly dfferent from the exstng shp structural form, No load condtons are unconsdered at the start of topology optmzaton unless they are proven to be replaced by other load condtons. Because the topology optmzaton s a bran-new, tme-consumng work, extensve test calculatons are necessary. A few key load condtons are selected to carry out senstvty analyss wth some optmzaton parameters (szng grd, elements thckness, scopes of unalterable elements, stress restrcton, assemble forms of topologcal confguratons) changed n certan ranges, so the most approprate values of optmzaton parameters are confrmed before the all load condton calculatons. It s dfferent for SIMP method and BESO method n the selectons and assembled coeffcents of load condtons. They wll be ntroduced as follows. 3.2.1 Load combnatons for topology optmzaton usng SIMP method For Altar Hyperworks/Optstruct solves topologcal optmzaton problems usng SIMP method, the hull structure topology optmzaton s frstly performed usng ths commercal code. On ths software platform, multple sub-cases (loadng patterns) can be consdered, by default, they have the same calculaton weght and cannot be modfed. In order to keep stable convergence, the optmzaton procedure tends to mnmze the global complance and retan the smallest number of stress constrants usng constrant screenng algorthm automatcally, thus some strange structures may appear n the results e.g., long slender beam-lke structures wth low stress level. Consder the loadng patterns of a double hull ol tanker wth one longtudnal bulkhead, some loadng pattern wth cargo ol alternatvely loaded along the shp length may have large stran energy whle the maxmum von Mses stress n transverse structure members s relatvely low, on the contrary, some loadng pattern wth cargo ol loaded asymmetrcally may have small stran energy but the stress level n the transverse structural members s hgh. Although the objectve functon s set to mnmze the hull weght, a slender vertcal beam-lke structure throughout the entre cargo hold depth s usually generated n the optmzaton results when the loadng patterns wth large stran energy are ncluded n the optmzaton subcases. Nether can ths knd of slender structure be easly produced nor can t easly meet the bucklng strength requrement of the HCSR rule. Through multple calculatons, t s found that the vertcal slender beam-lke structures are easly generated n loadng patterns B3 and B11 whether the optmzaton object s set to mnmze the weght or the global complance. For loadng patterns B6,B7 and B8, the slender beam-lke structures appears when the objectve functon s set to mnmze the global complance, but dsappeared when the objectve functon s set to mnmze the weght. Loadng patterns B3 and B11 may be the domnant loadng patterns of bottom shell and longtudnal bulkhead plates to meet the bucklng strength and shear strength crtera, they are most probably the domnant loadng patterns for longtudnal structural members, thus, n transverse structural member topology optmzaton they are consdered as nterference subcases and have to be abandoned. The nfluences of these load patterns wll be consdered n shape and scantlng optmzaton later. 3.2.2 Load combnatons for topology optmzaton usng BESO method BESO has an obvous advantage when dealng wth multple load case for ts smple theory and flexble adjustment to the program by the desgner accordng to dfferent requrements. Page 11 of 18

Applyng a weghtng factor to dfferent load cases s a usual method for ths problem. However, t s hard to determne whch load case s more crtcal than the others. Whle applyng the equal weghtng factor wll gnore some specal cases especally when the senstvty numbers of these cases are relatvely lower than the others for asymmetrc load reason. For example, as asymmetrc load cases, the hgh transformng or hgh stress area of B1 and B2 are almost half of those for B3 and B6, the symmetrc cases. Thus, n order to mnmze ths unfar average, the followng equaton for determnng the weghtng factor shall be used: l = n Max( e å = 1 Max( e case. sumfe _ P, e case. sumfe _ P case. sumfe _ S, e ) case. sumfe _ S ) Where, case. e sumfe _ P and case. e sumfe _ S are the subtotal of senstvty number for port or starboard sde of elements for the th load case. 4 Bref ntroductons of topology optmzaton results n a VLCC wth one centerlne longtudnal bulkhead and applcatons on others ol tanker On the bass of topology optmzaton theory, SIMP method and BESO method were appled separately to obtan PSM s topologcal confguraton n a VLCC wth one centerlne longtudnal bulkhead, and then shape optmzaton and sze optmzaton were carred out. All the structural drawngs n cargo holds and calculaton reports had been approved by ABS, whch means topology optmzaton technology had acheved success n structural optmum desgn. The new structures desgned by topology optmzaton method are much lghter than tradtonal structures, and much easer to manufacture. Also an nventon patent s acheved accordng the research concluson. The parameters are shown n followng table 3. Table 3 Typcal parameter comparsons between VLCC wth One centerlne BHD and tradtonal VLCC Compared subjects Shp type VLCC wth One centerlne bulkhead Tradtonal VLCC Comparsons (%) Number of CH 12 15 80% Number of transverse BHD 6 5 120% Number of wash-bhd 0 4 Number of transverse webs 38 41 92.7% Total effectve cross-sectonal area of longtudnal strength members(m 2, only based on HCSR, wthout FEAbased modfcatons) Total length of the longtudnal plate cross secton (m) 9.5285 10.555 90.3% 341.2 377.8 90.3% Total number of longtudnals 374 415 90.1% Page 12 of 18

transverse webs space 5.95 5.66 105.1% Surface areas of typcal transverse webs (m 2 ) Averaged weght of typcal transverse webs (ton, except for wash BHD) SIMP 575.8 78.7% 731.8 BESO 643.3 87.9% SIMP 100.8 70.4% 143.14 BESO 111.3 77.8% maxmum desgned stll water bendng moment (kn*m) 8,450,000 7,600,000 111.2% Structural weght per meter n cargo hold (ton) SIMP 111.75 87.8% 127.3 BESO 112.55 88.4% Accordng to real shp desgn experence, the optmal structures obtaned wth SIMP method are smpler to manufacture and lghter n weght. When the structural drawngs and calculaton reports were submtted to ABS, the classfcaton socety was n agreement wth ths statement. On the other hand, there stll exst some unsolved problems whch wll be mentoned n the next secton. The success of shp structural topology optmzaton depends on whether the result s convergent or not, whether the result s n accordance wth the basc prncples of classcal structural mechancs or not, and whether the result s smlar to mature shp structures or not. Accordng to the defnton of topology optmzaton, for the topology results that converge to low volume fracton tend to be lke frame structures and truss structures. But for the structures wth hgh ntal stran energy, the optmal topology results are manly composed of contnuous structures whch contan many holes. The topology optmzaton results of structures of VLCC wth one centerlne longtudnal bulkhead are n accordance wth these mechancal rules. The optmal topology structure n cargo hold seems lke the captal letter K, whle the optmal topology structures n bottom and sde structures s contnuous n geometry except for some holes. The typcal transverse webs, horzontal strngers and correspondng structural drawngs obtaned wth SIMP method are shown n Fgure 5 to Fgure 8. The typcal transverse webs and correspondng structural drawngs obtaned wth BESO method are shown n Fgure 9 to Fgure 10. The comparsons of outlnes of typcal webs and horzontal strngers between tradtonal VLCC and VLCC wth one centerlne longtudnal bulkhead are shown n Fgure 11 to Fgure 12. Fg.5 Typcal topologcal confguraton of web frame based on SIMP method Fg.6 Drawng of typcal topology optmal of web frame based on SIMP method Page 13 of 18

Fg.7 Typcal topologcal confguraton of horzontal strnger based on SIMP method Fg.8 Drawng of typcal topology optmal of horzontal strnger based on SIMP method Fg.9 Typcal topologcal confguraton of web frame based on BESO method Fg.10 Drawng of typcal topology optmal of web frame based on BESO method Page 14 of 18

Fg.11 Comparson of typcal web outlnes between tradtonal VLCC and VLCC wth one centerlne longtudnal bulkhead Fg.12 Comparson of typcal horzontal strnger outlnes between tradtonal VLCC and VLCC wth one centerlne longtudnal bulkhead Topology optmzaton technology can also be appled to new shp development of other knds of ol tankers. Based on the basc theory of topology optmzaton and experences acheved from development of a VLCC wth one centerlne longtudnal bulkhead, mproved desgns of cargo hold structures n Panamax ol tanker, Aframax ol tanker and Suezmax ol tanker were completed by MARIC, and the new developed structures satsfy HCSR, meanwhle are about 1.1 percent to 3.6 percent lghter n weght than tradtonal CSR structures. The comparsons of typcal web frames between Aframax ol tanker satsfed wth topology optmzaton theores and tradtonal ol tanker are shown n Fgure 13. The knuckle lne of nner hull bulkhead was adjusted slghtly accordng to topology optmzaton theores, the ends of vertcal Page 15 of 18

grders and deck transverses tend to lke truss structures, and the holes n sde double hull structures are more and bgger. But n general, the dfferences of structures between new developed Aframax ol tanker and tradtonal Aframax ol tanker are not enormous. Fg.13 Comparson of typcal web frame between Aframax ol tanker developed based on topology optmzaton and tradtonal Aframax ol tanker 5 Questons to be resolved n the future 5.1 The model test of bran-new structures In the case that the topologcal confguratons are smlar to the exstng shp structures except that there are more openngs wth large areas are n the low stress regon and the shape of prmary support member of cargo hold s more close to shape of K see Fgure 13. The model test of structure could be avoded. However, f the topologcal confguratons are completely dfferent from the exstng shp structure, the reduced-scale model test of local structures should be conducted as far as possble except that a seres of detaled and ntact calculaton s done to persuade the shp-owners of a bran-new desgn. see fgure11 and fgure 12. 5.2 Applcaton of sold elements on topology optmzaton It s reasonable that 2D shell elements are used n topology felds n the start of topology optmzaton to mprove the effcency of topologcal operatons. The flange plates of PSM are very mportant for PSM to bear the lateral pressure, but the shell elements cannot descrbe the postons and dmensons of the flange plates n topology felds. Therefore, before the shape optmzaton, t s necessary to carry out a topology optmzaton of typcal transverse webs usng sold elements. It s an essental extenson of a 2D topology optmzaton n general. The authors had pcked out a typcal prmary support member on port sde to perform a topology optmzaton by usng the mxed shell and sold elements. In order to smulate the flange, the sze of sold element s set to about 100x100x50mm(LxBxT), as a result, the optmzaton procedure s tme consumng and the topologcal confguraton after convergence s also not so deal, see fgure 14, but t also can gve some useful hnts to the desgner, lke as the settng routes and scantlngs of flange plates. Because the calculaton numbers of samples are too less to extensvely represent the entre load cases. The mesh sze s also not so accurate to satsfy the requrement of engneerng nstantaton. Ths s one of mportant problems whch s to be resolved n the future. Page 16 of 18

Fg.14 topologcal confguraton of mxed shell elements and sold elements 5.3 Problems n the cargo structures topology optmzaton 1. When software Hyperworks/Optstruct (SIMP method) s used n the cargo structures topology optmzaton, the stress restrctons are not convenent to select. The defnton of stress constrants can only be lmted to sngle von Mses permssble stress and stress constrants cannot apply to 1D element. It s also dffcult to freely select the elements satsfed wth stress restrctons n the unalterable regon. The stress restrctons are to be solved n the shape optmzaton, and the local geometrcal shape parameters n the hgh stress areas are to be adjusted to reduce the peak stress. 2. Some key parameters n software Hyperworks/Optstruct cannot be changed, whch s not convenent for the shp structures topology optmzaton. Such as the assembled coeffcent of the load cases, the overrun rato of element stress, etc. In the topology optmzaton, some load cases whch nduce hgh stran energy but lower peak stress value are abandoned, and thus the result s applcablty s lmted. 3. It s not recommended to use mass of the entre model as an objectve nether n SIMP optmzaton method nor n BESO method, for too large the global mass makes the objectve functon less senstve towards desgn modfcatons made n the optmzaton. Also both the yeld strength and bucklng strength cannot be taken as restrcton at the same tme. These problems are to be solved n the phase of scantlng optmzaton. 6 Conclusons In ths paper, the advanced structure topology optmzaton theores, methods and software have been adopted to acheve the optmzaton objectve.e. mnmzng the weght of hull structures. The calculaton s n accordance wth HCSR, durng whch the dffcultes such as huge calculaton model of topology optmzaton, numerous loadng condtons for calculaton and the nterference of ll-condtoned load cases have been overcome. In the end a topology optmzaton structure of tanker s cargo hold whch can be appled to practcal engneerng desgn s provded.takng the VLCC wth one centerlne longtudnal bulkhead as a specmen shp, the topology optmzaton and desgn for typcal web frames and horzontal strngers n cargo hold has been done, and the methods also has been appled to developng new shp types for other large crude ol tanker whch s n complance wth HCSR. Although there are some defects n the software and process of topology optmzaton n some measure, relatvely deal topologcal confguratons through the two topology optmzaton methods are obtaned.untl now, no numercal optmzaton procedure can guarantee that global optmum result for Page 17 of 18

large-scale optmzaton problem could be acheved, accordngly the structure confguratons derved by topology optmzng may not be the most optmal soluton usng these methods, but they are thought to be very close to the most optmal soluton. Reference lterature [1] Bendsoe M P, Kkuch N. Generatng Optmal Topology n Structural Desgn Usng a Homogenzaton Method[J]. Computer Methods n Appled Mechancs and Engneerng, 1988,71:197~224 [2] YM Xe, GP Steven. Shape and layout optmzaton va an evolutonary procedure[c]. In Proceedngs of Internatonal Conference on Computatonal Engneerng Scence. Hong Kong Unversty of Scence and Technology, 1992: 17-22. [3] Bendsoe M P, Sgmund O. Topology optmzaton: Theory, Methods and Applcatons[M]. Sprnger Verlag: Berln, 2003. [4] Retz A. Suffcency of a fnte exponent n SIMP (power law) methods[j]. Structural & Multdscplnary Optmzaton, 2001, 21(2):159-163. [5] Wang M Y,Wang X M, Guo D M. A level set method for structural topology optmzaton [J]. Computer Methods n Appled Mechancs and Engneerng.2003, 192: 227-246. [6] O.M.Quern, V.Young, G.P.Steven, et al. Computatonal Effcency and Valdaton of B-drectonal Evolutonary Structural Optmzaton. Computer Methods n Appled Mechancs and Engneerng. 2000, 189:559~573 [7] Huang X, Xe M. Evolutonary topology optmzaton of contnuum structures: methods and applcatons[m]. John Wley & Sons, 2010. [8] Bendose M.P. and Sgmund O. Materal nterpolatons n topology optmzaton. Archve of Appled Mechancs, 1999,69:635~654 Page 18 of 18