Application of Gaussian Curvature Method in Development of Hull Plate Surface

Transcription

1 Marine Engineering Frontiers (MEF) Volme 2, 204 Application of Gassian Crvatre Method in Development of Hll Plate Srface Xili Zh, Yjn Li 2 Dept. of Infor., Dalian Univ., Dalian 6622, China 2 Dept. of Naval Archit., Dalian Univ. of Technol, Dalian 6023, China floralam_200@sina.com Abstract Line-heating process as the main method to form the doble crvatre hll plate, can be considered as the inverse process of the hll plate srface development, moreover, the processing parameter can be determined by calclating deformation degrees generated by the plate srface development. In this paper, we present a new method for development of hll plate srface based on its Gassian Crvatre. According to the property of the offset srface and BP neral network model, the anglar distortion of the plate srface development is analyzed. The plate srface is adaptively segmented based on Gassian crvatre, and the transverse shrinkage of the plate be calclated. A new method for srface development has been established to calclate the anglar distortion and the transverse shrinkage simltaneosly in plate srface development, which can provide gidance to determine the reheating information for nformed plates. Keywords Srface Development; Gassian Crvatre; Line-heating; Anglar Distortion; Transverse Shrinkage Introdction The process of line-heating is the main method to form the doble crvatre hll plate. To obtain the desired doble-crved shape, the plate is shrnk on the partial areas. Line-heating process considers the inverse process of hll plate srface development. While the shrinkage of the plate srface development considers the deformation of line-heating process. The parameters of the line-heating process are determined merely by the deformation prodced by the plate srface development. The focs is how to calclate the deformation prodced by the plate srface development [JiZhoshang 200]. Related Work It is well known in mathematics that the ndeveloped srface can be developed into its corresponding plane region by approximation. Many nmerical methods have been sed to develop these srfaces, sch as stretching and withot deformation. Chen [Chen- Dongren 2003] has presented a novel optimal method to flatten complex srfaces based on mesh edges and spring-mass model. Li has given the calclation method of shrink distortion for plate srface [LiYjn 995]. Bt above mentioned researchers have not considered the thickness of plate, and the anglar distortion has been ignored for the nit thick plate. At present, the anglar distortion of the plate srface development has not been nderstood clearly [Zh- Xili 2009]. The transverse shrinkage of the plate did not show entirely the crvatre of the thick plate. The anglar distortion and the transverse shrinkage of the plate are both the decisive factors for the parameters of the line-heating process. The plate srface can be adaptively divided into regions based on gass crvatre and then the transverse shrinkage of the plate can be calclated. According to the property of the offset srface, a new method for srface development has been established to calclate the anglar distortion and the transverse shrinkage simltane development, which can provide gidance to determine the reheating information for nformed plates. This stdy is the key technology of comprehensive atomatic processing for line-heating, which can improve or shipyard's competitiveness in international markets and has important application vale and academic significance. Theories abot Srface Development Definition of Offset Srface Spport P be a point on the srface S, and P be the new point on the srface S λ. S λ is called the offset srface of S, when the point P can be obtained by removing λ distance along the normal direction of P. The fnction of S is S= Sv (, ). Then the fnction of S λ is

2 Marine Engineering Frontiers (MEF) Volme 2, 204 S = Sv (, ) + eλnv (, ) () λ Where n denotes the nit normal vector on S. Let λ be pls constant, and λ 0, e = ±. Assme S λ be the oter offset srface of S when e =, whereas S λ be the inner offset srface of S when e =. Let n be the nit normal vector of any point on the srface S. Then S( v, ) Sv( v, ) n = (2) S( v, ) S( v, ) Denote the partial derivatives of S as S( v, ) and Sv (, v ), the velocities along and v direction. According to the definition of offset srface, S( v, ) and Sv ( v, ) shold be calclated before S λ. Let S be an niform bicbic B-spline srface. P i, j ( i = 0,,, m; j = 0,,, n) is the control points on the srface. Then, S ( v, ) and Sv ( v, ) are as following n m Pi +, j Pij S(,) v = 4 Bi,3 Bj,4() v (3) v i= 0 j= 0 i+ 4 i+ P P S v B B v (4) v v n m i, j+ ij v(,) = 4 i,4 () j,3() i= 0 j= 0 i+ 4 i+ Then, the oter offset srface S λ can be calclated according to Fnction () [WeiYe-chn 2005]. Definition of Gass Crvatre Sppose P be a point on the srface S, k, k2 are the max and min vales of normal crvatre on the direction of tangent, calling the main crvatre on the point of P in the srfaces S. The direction t, t 2 is called the main direction. K is the prodct of two main crvatres on the point of P in the reglar srface, which is defined as the gass crvatre or the sm crvatre. H is the vale of arithmetic mean on the point of P in the reglar srface, which is called the mean crvatre on the point of P. The srface is the plane if kn 0 at anywhere on the srface. Generically, the vale between k and k2 isn t zero at least in a srface. Analysis of Srface Development Developable srface is rled srface, where all points of the same rling line share a common tangent plane. The rlings are principal crvatre lines with vanishing normal crvatre and the Gassian crvatre gets vanished at all srface points. The developable srface can be nfolded into a plane by bending along the perpendiclarity rling direction. For the ndeveloped srface, we need to consider gass crvatre on every point as follows [ChenXiaopeng 2009]. TAUBIN algorithm is adopted to compte the main crvatre of every points, moreover, to solve the gass crvatre on the scatter points. The method is high effective on the times and the complexity degree. Analysis of Development for the Hll Plate Srface There are two parameters for the partial deformation of the line-heating process simlation. One is the transverse shrinkage along the heating line, and the other is the anglar distortion along the thickness of the plate. The transverse shrinkage is prodced between two consective processes which are the high temperatre pressed and the refrigeration strained. The anglar distortion is prodced by the temperatre grads along the thickness of plate, and the primary reason is the different deformation between the top-side srface and the back-side srface. The formed plate and the partial deformation have the specific corresponding relationship. Then, the whole shape of the plate ensres the partial deformation. In mathematics, most of the srface can be regarded as the stretchy srface, which is free pll and shrnk. Bt the plate is regarded as the rigid srface in engineering application. A lot of cracks are prodced when the rigid plate srface is developed into a plane. These cracks are the deformation on the heating-line process which contain the transverse shrinkage and the anglar distortion. Transverse Shrinkage The thickness of plate can be ignored for the analysis of transverse shrinkage. The crack mst be presented when sch plate srface is developed into the plane. The breadth of those cracks is the transverse shrinkage. There are many methods to calclate the transverse shrinkage for sch srface, sch as gide strip, rled approach and area mapping etc [ChenXiao-peng 2009]. All above method of developing parametric srface are the region segmentation along the same directions, which has the same range of gass crvatre. Every regional segmentation can be approached by the rled srface that can be segmented by triangle gridding. According to the geometric characteristics of the complex srface of hll plate, the srface can be fitted firstly to obtain the parameter gridding. The vale of 2

3 Marine Engineering Frontiers (MEF) Volme 2, 204 gass crvatre is needed to be compted at every point, which is the inhesion property of these points for refreshing data. The comptation of power vale for every point on the scattered srface is the basic fondation for the regional segmentation [WeiYe-chn 2005]. The property of srface development lies on the gass crvatre entirely, then the triangle srface with the same range of gass crvatre is classified in one region. The srface is finally divided into some regions of development. The whole plate srface is segmented based on the breadth first sing search algorithm. Then every region is developed into plane based on the eqidistant mapping method. At the end, the transverse shrinkage is calclated by this way. FIG. THE DEFORMATION OF THE DEVELOPMENT FOR THE THICK PLATE SURFACE the anglar distortion for the thick plate srface development is as following: l = θ ε δ (5) Where ε denotes the netrosphere displacement coefficient. And δ denotes the plate thickness. Algorithm of Development for the Hll Plate Srface On the research of the thick plate srface development, the transverse shrinkage of the plate srface development can be calclated by the netrosphere srface. The thickness of plate is denoted as δ. The fitting top-side srface is S. The netrosphere srface S εδ is the oter offset srface S. We se experiment data to train BP network, and get the ε vale of different thickness plate [ZhXi-li 2005]. Then S εδ can be developed by the above region segmentation algorithm based on its qasi-rling. The transverse shrinkage of the crack can be calclated. The vale of θ is given by the nip angle of the normal vector on the netrosphere srface, and it is eqal to on the top-side srface plate. The anglar distortion l can be calclated throgh Eqn.(5). The sm of the transverse shrinkage and the anglar distortion are the deformation at the same crack on the srface plate, flow chart of which is shown in Figre 2. Anglar Distortion The top-side of srface splits while the back-side of srface shrinks dring the process of the developing plate srface. In the middle of the thickness of the steel plate layer that mst be scalable invariant of a certain layer, here it is defined as netral layer of [WeiYe-chn 2005]. The netral layer plate can be regarded as the pper srface of the offset srface. As shown in fig., the plate thickness is denoted as δ. The points of A and B on the top-side srface separately correspond to the points of E and F on the netrosphere, and those corresponding points have the parallel normal vector according to the definition and the property of offset srface. Then the points of A and E have the sperposition normal; so do B and F. GH denotes the crack prodced by the split of AB. The dotted line of EF is called the netrosphere. ε is the displacement coefficient of the netrophere. Then the distant is ε δ between the top-side srface and the netrophere srface. The breadth of crack GH is the anglar distortion denoted as l. The angle GOH is the nip angle on the crack denoted as θ. The vale of θ can be determined by the nip angle of the two points E and F s normal vectors. The formla of θ = tgθ exists when the vale of θ is very small. Ths, the formla of FIG. 2 THE FLOW CHART OF THE DEFORMATION FOR THE PLATE SURFACE DEVELOPMENT Case Stdy of Development for the Hll Plate Srface Case Stdy of Anglar Distortion Inpt parameters are the thickness of the plate, nip 3

4 Marine Engineering Frontiers (MEF) Volme 2, 204 angle and anglar distortion, while otpt parameters are the netrosphere displacement, the model of BP nerve network designed. The nmber of nit intermediate hidden layer is 2. The model otpt layer is the sigmoid fnction nerve cell. The stdy velocity is 0.0, and the error target is 0.0. Using the training samples, training of the BP network model, in Figre 3 for the network model in the sm of sqare error crve of corresponding different training times, in the network model trained by 7 times, the error model can be seen to achieve error index. After learning the training after the sccess of fixed weights and thresholds as shown in Tab. [ZhXi-li 2005], the BP net model trains the settled power vale or threshold vale, given the netrosphere ε = The region segment threshold vale is By sing the new method, the comptational reslts of the development distortion for the plate srface are shown in Table 2. TAB. 2 COMPUTATIONAL RESULTS OF THE DEVELOPMENT FOR PLATE SURFACE The deforming of Heat-Ling is the sm of the transverse shrinkage and the anglar distortion. The nmerical reslts are obtained to show the efficiency of this method by contrast with the data in actorial paper [ZhXi-li 2009]. It is discovered that the religion calclated vale error is small within 0.03 mm. Conclsion FIG. 3 THE ERROR SQUARE SUM CORRESPONDING THE DIFFERENT TRAINING OF THE NETWORK MODEL TAB. THE VALUE OF RIGHT AND THRESHOLD FOR THE SUCCEED TRAINING The deformation of the development for hll plate srface can be sed to forecast the deformation of Heat-Ling. In this paper, we have presented a new method to analyze the anglar distortion of the plate development based on the BP neral network model, and the common region segmentation algorithm gets improved, by the adoption of the gass crvatre for the calclation of the transverse shrinkage of the developing plate. This method offers the reasonable line-heating parameter for the plane plate deforming directly to the srface plate. REFERENCES Using above settled power vale or threshold vale in table, the netrosphere srface displacement ε of the thick plate is compted corresponding to these inpt parameters. Then the anglar distortion l can be calclated throgh Eqn.(5). By contrast with the metrical vale, the compted vale of anglar distortion erroris small within 0.00 mm. It seems that the BP net model is reliable. Case Stdy of Transverse Shrinkage The thickness of the deformed plate is 20 mm, length of which is 3000 mm and the breadth is 400 mm. By sing the fairing algorithm to fit sch plate srface Chen Dongren, Wang Gojin, A complex srface adaptive segment and development algorithm based on Its Qasi- Rlings, Jornal of Software, vol.4 no.3, pp , 2003 Chen Xiao-peng, Research on Srface Compartment Techniqe Based on Gass Crvatre, Jiangs Machine Bilding & Atomation, pp, 4-7, 2009 JI Zhoshang, Li Yjn, Development of stdy of ship hll plate bending by line heating, Jornal of Dalian University of Technology, vol. 4no.5, pp , 200 Li Yjn, Ji Zhoshang, Dong Shof, Approximate calclation of contraction of formed camber, Jornal of 4

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