EOIRS OF SHONAN INSTITUTE OF TECHNOLOGY Vol. 37, No., 2003 * A Geneal Numeical Solution to Ditibuted Shape Optimization Poblem in Stuctual Deign aatohi SHIODA* In thi pape, a numeical hape optimization method of continua i peented fo typical tength, igidity and vibation poblem in tuctual deign. A the tength poblem, the minimization poblem of maximum te and the hape detemination poblem that achieve a given deied te ditibution ae fomulated. The igidity poblem involve the minimization poblem of extenal wok and the hape detemination poblem that achieve a given deied diplacement ditibution. Alo, the vibation poblem involve maximization of eigen fequency with mode tacking. Each poblem i fomulated and enitivity function ae deived uing the Lagangian multiplie method and the mateial deivative method. The taction method, which i a hape optimization method, i employed to find the optimal domain vaiation that educe the objective functional. The popoed numeical analyi method make it poible to deign optimal tuctue efficiently. Example of computed eult ae peented to how the validity and pactical utility of the popoed method. GUI (Gaphical Ue Inteface) * 4 0 CAD ) 7
37 2 2. CAD 2.2 600 Galileo 973 Zienkievictz and Campbell 2) 994 3) 4) 7) 3 8) W R n, n 2, 3 T (X): X W x W Q T D (X) T (X) DV O( D ) () V T (X) Eule V C Q {V C (W: R n ) V 0 in Q} (2) O( D ) lim O( D )/ D 0 D Æ0 J f J W f dx Eule J () (3) 8
J f dx f n dg n W G (4) È ( x ) max Í, (8) x ŒW Î a n n n i V i n G W Eintein ( ),i ( )/ x i ( ) Eule ( ) Lagange 8) J f a KS 9) G J f dg (5) W ( x) KS( ( x)) ÏÔ Ê ˆ Ô ln Ìexp dx Á Ë ÓÔ a Ô (9) Eule J J { (, ini ) } d f f f k n G G k 2 3 4 4. (6) KS (x) 5 200 KS l p 4..2 Fig. W G W V W G W G deign f f i, i=,, n P P i W G (9) Given W, f in W, P on G, o R (0) find W (o V ) () {( xx yy) ( yy zz) ( zz xx) 2 2 2 2 2 2 2 2 6( )} xy yz xz (7) 4.. Fig.. Domain vaiation of continuum. 9
37 that minimize ln W ÏÔ Ê exp ( x ) ˆ Ô Ì Á dx Ë ÓÔ a Ô (2) A a W Ê expá Ë a ˆ dx (9) ubject to 0 0 (3) a(v,w) l(w) fo all w U (4) a(v,w) l(w) () (2) v v i w w i U, 0 R W a( vw, ) eijklvk, lwi, jdw l( w) f w dw Pw dg i i i i W G (5) (6) e e ijkl L w(x), L ÏÔ Ê ( x ) ˆ Ô L( W, vw,, L) ln Ìexp dx Á W Ë ÓÔ a Ô l( w) a( v, w) L( ) 0 (7) n L v,w L a (v,w ) l (w ) fo all w U (20) Ê ˆ a( v, w) exp ijdx A Á Ë A vdx fo all v ŒU k W W Ê expá Ë (2) L ( 0 ) 0, 0 0, L 0. (22)(23)(24) (20) v (2) w (22) (24) Kuhn Tucke L v, w L L l G (V ) (25) V l G (V ) a a ˆ ij ij ij k (n l V l 0 o G ) (f 0) L V L l( w ) a( v, w ) A W Ê ˆ exp Á ijdx a (, ) Ë v w a ÏÔ Ê ˆ Ô Ì eijklvk, lwi, j fiwi exp A Á L G ÓÔ Ë a Ô nvd l l G L ( 0 ), V ŒCQ (8) ij G i i G l ( V ) GVdG (26) Ï ÔÊ Ê ˆ ˆ Ô G ÌÁ eijklvk, lwi, j fiwi exp n A Á L a ÓÔ Ë Ë Ô on G deign G \G fix (27) G v (20) L (22) (24) V 20
4.2 (x) (x) (28) W C W C (2) ij a( v, w ) 2 { ( x) ( x) } vdx k v Wc { } W ( x) ( x) dx C fo all v U (29) (26),(27) V l G (V ) G, G 2 (30) (32) W G 2 l ( V ) G VdG G VdG G i i i i c 2 G { ( x) ( x) } n G 2 ( e v w f w ) n ijkl k, l i, j i i L (30) (3) (32) 5 5. v (6) l(v) l(v) V l G (V ) 2 ij k G (33) (34) (35) a( v, w) l( v ) fo all v ŒU G ( 2 fv e v v L) n on G G \ G i i ijkl k, l i, j deign fix (33) (34) (35) (33) (20) v w G 2 5.2 4.2 G D G i i G l ( V ) GVdG {( v ( x) v ( x)}{( v ( x) v ( x)} dg i i i i (36) G D ṽ i (37) a( v, w ) 2 { v ( x) v ( x)} v dg i i i WC (37) V l G (V ) G G i i G l ( V ) GVdG G ( e v w f w ) n ijkl k, l i, j i i L (38) (39) 2
37 6 v () l () l () (4) a(v (),w) l () b(v (),w), fo all w U (40) b(v (),w) ( ) i i W b( v, w ) v w dx (4) w Lagange L v () w L a(v (),w ) l () b(v (),w ), fo all w U (42) a(v (),w) l () b(v (),w), fo all v () U (43) b(v (),w) (44) L ( 0 ) 0, 0 0, L 0. (45)(46)(47) (42) (44) v () (43) w v () w (45) (47) (22) (24) Kuhn Tucke v () w L Lagange L V G G i i G l ( V ) GVdG G ( e v v l v L) n ijkl ( ) k, l ( ) i, j ( ) ( ) i (48) (49) 7 2 4 6 7. V (50) a(v,w) l(w) fo all w C Q (50) (50) G V (50) V L L DL l G (DV ) O( D ) (5) (50) (5) e ijkl a(v,w) $ a 0 : a( x, x) a x fo all x ŒU (52) D DL a(dv,dv ) 0 (53) (50) V L 0) 7.2 FE 2 22
FE FE (50) D G G deign Fig. 2 7.3 v v () w (2) (25) (54) [B] T [D] w 2{ (x) (x)} ij T [ B] [ D][ B] dx{ x} W W Ï T Ô Ê ˆ [ B] [ D] Ì exp dx A Á ÓÔ Ë a Ô ij Ô Fig. 2. Schematic flow chat of hape optimization ytem. 2{v i (x)-ṽ i (x)} w 7.4 PID PID ) L G L 8 23
37 Fig. 3. Solid am poblem. Fig. 5. Iteation hitoie. Fig. 6. Notched plate poblem. Fig. 4. Initial & optimal hape and te ditibution (Pa). Fig. 7. Taget hape. 4 8 8. 8.. Fig. 3 Fig. 3(a) P Fig. 3(b) Fig. 4 Fig. 5 30% 8..2 Fig. 6 W C (x) Fig. 7 Fig. 6(a) 24
Fig. 8. Shape identification eult. Fig. 0. Iteation hitoie. Fig.. Toion am poblem. Fig. 9. Compaion of te ditibution along A B. Fig. 6(b) Fig. 8 Fig. 9 A B Fig. 0 Fig. 7 8.2 8.2. Fig. Fig. (a) Fig. (b) Fig. 2 Fig. 3 2 Fig. 4 Fig. 5 8.2.2 Fig. 6 G D y ṽ 2 Fig. 6(a) Fig. 6(b) Fig. 7 Fig. 8 25
37 Fig. 4. Iteation hitoie. Fig. 2. Initial hape. Fig. 5. Optimal hape fo min-max te poblem. Fig. 3. Optimal hape. Fig. 9 26
Fig. 6. Table uppot poblem. Fig. 9. Iteation hitoie. Fig. 7. Initial hape and defomation mode. 8.3 Fig. 20(a) (Fig. 20(b)) PC (ulti Point Containt) Fig. 2(a) (b) 20% 9 Fig. 8. Shape identification eult. 27
37 Fig. 20. Engine mount backet poblem. Fig. 2. Initial & optimal hape and vibation mode. AI ) 60-574, A(994), 479. 2) Zienkiewicz, O. C. and Campbell, J., in Optimal Stuctual Deign, Gallaghe, R. H. and Zienkiewicz, O. C. (Ed.), John Wiley, New Yok (973). 3) Cea, J., in Optimization of Ditibuted Paamete Stuctue, Haug, E. J. and Cea, J., Vol. 2, Sijthoff & Noodhoff, Alphen aan den Rijn (98). 4), 62-604, A (996), 283 2837 5), 63-607, A (997), 60 67. 6), 6-587, C (995), 49 54.. 7) Ihaa, H., Azegami, H. and Shimoda,., in Compute Aided Optimum Deign of Stuctue VI, WIT Pe, 28
Southampton, (999), 87 95. 8) Sokolowki, J. and Zoleio, J. P., in Intoduction to Shape Optimization, Shape Senitivity Analyi, Spinge-Velag, New Yok, (99). 9) Keielmeie, G. and Steinhaue, R., in Sytematic Contol Deign by Optimizing a Vecto Pefomance Index, Intenational Fedeation of Active Contol Sympoium on Compute-Aided Deign of Contol Sytem, Zuich, Switzeland, Augut 29 3, (979), 3 7. 0) Vol., 996 5 ). ) Shimoda,., Azegami, H., Sakuai, T., in Poceeding of 9 th. Intenational Confeence on Vehicle Stuctual echanic and CAE, Toy, Apil 4 6, SAE (995), 223. 29