Vol.187, pp , Λ ΛΛ ΛΛΛ Fair Surface Generation in Ship Hullform Design by Hiroshi Masuda, Member Hiroyuki Yamato, Member Yoshiyuki Furuk
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1 Vol.187, pp , Λ ΛΛ ΛΛΛ Fair Surface Generation in Ship Hullform Desin by Hiroshi Masuda, Member Hiroyuki Yamato, Member Yoshiyuki Furukawa, Student Summary This paper describes about the method for fair surface eneration in ship hullform desin aimin at the automatization of surface fairin. It enerates an approximatin surface from section curves accordin to the distribution of curvature. The enerated surface has dense parameters in hih curvature areas, and sparse parameters in low curvature areas. This method has three advantaes described below. First, it can suppress the distortions enerated by traditional methods. Second, it can use additional curves that cannot be used in traditional skinnin methods. Finally, it can do surface fairin efficiently without the reat loss of eometric feature CAD 3 CAD 3 3 CAD * ** *** 3 3
2 Fi.1 ( Fi.1 ) 5 Fi. 1 L,B,D,Cp curve,etc. Curve Desin Lines Surface Generation Surface OK 1 3 Fairin 2 Fairin Flow of surface eneration in ship hullform desin (L; B; D; C pcurve,etc.) (Lines) ( ) ( ) (Lines) 4 ( ) (Surface) ( ) ( ) ( ) ( )( ) ( ) ( ) 2.2 Harries Nowacki 1) Eida 2) 3 3 CAD B- spline NURBS 4.1
3 3 3) μ 9) 10)13) 14) μ 16) Bow's Freeform Surface Low Curvature Area Fi. 3 Example of section curves Global & Local Smoothness Hih Curvature Area Fi. 2 Characteristics of ship hullform E V ffe + fiv ( ff; fi ) 6) μ 9) Fi.2 E V 11) 12) Fi. 4 Distortion of surface enerated with skinnin Harries Nowacki 1) 1 Fi.3 Fi.4 Fi.4
4 4 Each Curve Approximation with Constraint Slidin Fairin Parameters by Connectin Constraint Arranin Constraint Points on Section Curves Generation of Crosssectional Curves Approximation of Cross-sectional Curves with Constraint Slidin Surface Generation Surface Fairin Each Curve Approximation with Constraint Slidin Fairin Parameters by Connectin Constraint Arranin Constraint Points on Additional Curves Fi. 6 Flow of this method 1. Fi. 5 Section curves that cannot be skinned Fi Fi.6
5 5 Fi.7 Fi B-spline B-spline U = fu 0;u 1;:::;u m p B-spline N i;k (k =0;:::;p) U N i;0(u) =n 1 if ui» u<ui+1 0 otherwise (1) Fi. 7 Surface eneration with constraint slidin usin distribution of curvature N i;p(u) = u ui N i;p 1(u) u i+p ui + u i+p+1 u u i+p+1 u i+1 N i+1;p 1(u) (2) C 1 u p U n P i C(u) = n 1 X i=0 N i;p(u)p i (3) 1. S 2 (u; v) (p; q) (U; V ) (u n) (v m) P ij S(u; v) = X X n 1 m 1 i=0 j=0 N i;p(u)n j;q(v)p ij (4) 2.
6 6 4.2 Fi.8 O ri in a l C o n stra in t P o in ts ^Cs l C s l (u k;l )= ^C s l (u k;l ) (6) ^C s l (u) = n 1 X i=0 N i;p(u) ^Ps il (7) u k;l C c k(k =0;:::;n 1) C c k(v) = m 1 X j=0 N j;q(v)p c kj (8) C c k(v l;i )= ^Ps il (9) q m V i C c k ^C c k(v) = m 1 X j=0 N j;q(v) ^Pc kj (10) A r r a n i n N e w C o n stra in t P o in ts ^C c k(μv l;i )=C c k(μv l;i ) (11) q m V S Fi. 8 C urve R econstruction Curve approximation with constraint slidin S(u; v) = X X n 1 m 1 i=0 j=0 N i;p(u)n j;q(v) ^Pc ij (12) u v p q U V C s l (l =0; :::; m 1) C s l (u) = n l 1 X i=0 N i;pl (u)p s il (5) p l n l U l 4.3
7 7 Fi. 9 Points distributed with arc lenth [u min;u max] C(u)»(u) [a; b] Z b V d (a; b) = j _C(u)jdu (13) V c (a; b) = a Z b»(u)du (14) a Fi. 10 Distribution of curvature n div fu iji =0;:::;ndiv 1(u 0 = u min;u ndiv 1 = u max) V d (u i 1;u i)+k c V c (u i 1;u i) = V d (u min;u max)+k c V c (u min;u max) n div (15) Fi. 11 Points distributed with arc lenth and curvature 17) k c = w c V d (u min;u max) V c (u min;u max) (16) w c w c Fi.9 Fi.10 (15) Fi Fi.12 (15)
8 8 μu i;j = km i;j 1u i;j 1 + k m i;ju i;j + k m i;j+1u i;j+1 k m i;j 1 + km i;j + km i;j+1 (17) ki;j m k i;j m fμu i;j ji =0;:::;ndiv 1;j =0;:::;m Fi. 12 Network of points distributed with arc lenh and curvature parameterization Additional Curves u i, j u i, j 1 u i, j u i, j+1 Normal Section Curves Fi. 14 Additional curves Fi. 13 Fairin parameters with connectin constraint Fi.13 m (15) fu i;j ji = 0;:::;n div 1;j =0; :::; m 1 Fi.14 [u l min;u l max] C a c(u l ) [u min ;u max] C n a(u ) C n b (u ) u s;u e
9 9 C ( a u u min ) u s C l ( l c u ) f c ( uc, i l ) u a, i u e u max Farin C ( b u ) u b, i 5. Fi. 15 Arranin constraint on an additional curve f!l (u )=u l min + u u s (u l max u l min) (18) u e u s C n a [u s : u e] u a;i C n b u b;i u c;i Fi. 16 Arranin constraint points on section curves ka a = jc n b (u b;i ) Ca c ( f (u a;i ))j (19)!l kb a = jc n a(u a;i ) Ca c ( f (u a;i ))j (20)!l u c;i = ka au a;i + ka b u b;i k a a + k a b (21) 4.6 9) Farin 10) Fi. 17 Generation of cross-sectional curves Fi. 18 Arranin constraint points on cross-sectional curves
10 10 6. Fi. 19 Generated surface oriinal points enerated sections Fi. 20 Comparin enerated sections with oriinal points Fi.5 (16) w c =1 (17) ki;j m =1(i = 0;:::;n div 1;j =0; :::; m 1) Fi.16 Fi.17 Fi.18 Fi.19 Fi.20 1) S.Harries and H.Nowacki Form parameter approach to the desin of fair hull shapes Proceedins of 10th ICCAS Vol pp ) Y.Eida Y.Nishikido and Y.Sasaki Development of practical 3-D lines fairin system for ship hull form (MELFAS) and its application to CIMS Proceedins of 10th ICCAS Vol pp ) J.Kjellander Smoothin of bicubic parametric surfaces CAD 15(5) (1983) pp ) N.G.Fo Creative definition and fairin of ship hulls usin a B-spline surface CAD 16(4) (1984) pp ) D.F.Roers and N.G.Fo Constrained B- spline curve and surface fittin CAD 21(10) (1989) pp ) H.Haen and G.Schulze Automatic smoothin with eometric surface patches CAGD 4 (1987) pp
11 11 7) D.Feruson P.Frank and A.Jones Surface shape control usin constrained optimization on the B-spline representation CAGD 5 (1988) pp ) N.J.Lott and D.I.Pullin Method for fairin B- spline surface, CAD 20(10) (1988) pp ) ) G.Farin and N.Sapidis Curvature and the Fairness of Curves and Surfaces IEEE Computer Graphics and Applications March (1989) pp ) G.Farin G.Rein N.Sapidis and A.Worsey Fairin cubic B-spline curves CAGD 4 (1987) pp ) N.Sapidis and G.Farin Automatic fairin alorithm for B-spline curves CAD 22(2) (1990) pp ) S.Hahmann and S.Knoz Knot-removal surface fairin usin search strateies CAD 30(2) (1998) pp ) R.Klass Correction of local surface irreurarities usin reflection lines CAD 12(2) (1980) pp ) E.Kaufmann and R.Klass Smoothin surfaces usin reflection lines for families of splines CAD 20(6) (1988) pp ) C.Zhan and F.Chen Removin local irreurarities of NURBS surfaces by modifyin hihliht lines CAD 30(12) (1998) pp ) L.Piel and W.Tiller The NURBS Book 2nd edition, Spriner ) 2000
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