第8章位相最適化問題

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1 8 February 25, 2009

2 1 (topology optimizaiton problem) ( ) H 1

3 2 2.1 ( ) V S χ S : V {0, 1} S (characteristic function) (indicator function) 1 (x S ) χ S (x) = 0 (x S ) 2.1 ( ) Lipschitz D R d χ Ω (Ω D) Ω 0 Ω 1, Ω 2, D

4 ( ) Lipschitz D R d Y Y a : D R m (m N) ( a = (a 1, a 2, θ)) k : D R (homogenization method) ( ) k = k H (a), k H : R m R a W a 0 > 0 1 = (1, 1,, 1) T W = { a W 1, (Ω; R m ) a0 a 1 } D Y a 1 a 2 θ W

5 2.3 ( ) Lipschitz D R d φ : D R k : D R k = k H (φ), k H : R R φ W φ 0 > 0 W = { φ W 1, (Ω; R) φ0 φ 1 } D Á 0 Á 1, Á 2,

6 2.4 (SIMP ) 2.3 SIMP (solid isotropic material with penalization) p > 1 Á p 1 k = k H (φ) = φ p 0 0 Á0 1 Á (level set method)

7 3 SIMP SIMP SIMP 3.1 (SIMP BV (φ)) 2.4 (SIMP ) f : Ω R u : Ω R (φ p u) = φ f in Ω u = 0 on Γ Á D

8 ( J (0) (φ, u), J (φ, u) ) = ( J (0) (φ, u), ( J (l) (φ, u) ) R l) m+1 g (l) : R 2 R J (l) (φ, u) = g (l) (φ, u) dx Ω 3.2 (SIMP ) 2.4 (SIMP ) J(φ, u) φ W BV (φ) u φ W { J (0) (φ, u) } J(φ, u) 0 min φ W

9 3.1 ( ) W φ k W (k = 0, 1, 2, ) ρ : D R U φ ɛ = φ ɛρ ρ U ɛ > 0 P W W U = W 1, (D; R) = C 0,1 ( Ω k ; R ) φ ɛ = φ + ɛρ in D, ρ U φ k+1 = P W (φ ɛ )

10 φ k+1 W φ 0 W DV(φ) ρ U { φ k} k SIMP ( DV(φ)) W, U, J(φ, u) φ W ɛ > 0 BV(φ ɛ ) u ɛ ρ U J ( (0) φ ɛρ, u ɛρ ) { = J (0) (φ ɛ, u ɛ ) J(φ ɛ, u ɛ ) 0 } min ρ U, ρ =1

11 3.2 (u u ) u u ɛ BV (φ) BV (φ ɛ ) ρ U u Gâteaux u u = lim ɛ 0 u ɛ u ɛ 3.1 (u ) φ W ρ U u Gäteaux u (pφ p 1 ρ u ) (φ p u ) = ρ f u ρ in Ω u = 0 in Γ

12 3.4 ( AD (l) (φ)) g (l) / u = g (l),u v (l) : Ω R (φ p v (l)) = g (l),u in Ω v (l) = 0 on Γ

13 (J (l) (φ, u) ) φ W BV (φ) AD (l) (φ) u, v (l) J (l) (φ ɛ, u) Fréchet J (l) (φ ɛ, u ɛ ) = J (l) (φ, u) + ɛ G (l) ρ dx + o(ɛ) G (l) = g (l),φ pφp 1 v (l) u + f v (l) G (l) (density gradient) Ω

14 ( ) J (l) (φ, u) = Ω ( g (l),φ (φ, u)ρ + g(l),u (φ, u)u ) dx = g (l),φ (φ, u), ρ Ω + g (l),u (φ, u), u Ω a ( pφ p 1 ρ, u, v ) + a ( φ p, u, v ) = ρ f, v Ω u V v V a ( φ p, u, v (l)) = g (l),u (φ, u), u v (l) V Ω where a (φ, u, v) = φ u v dx V = H0 1 (Ω; R) 1 v = v (l) Ω u V J (l) (φ, u) = g (l),φ (φ, u), ρ Ω a ( pφ p 1 ρ, u, v (l)) + ρ f, v (l) Ω

15 3.1 (J (l) (φ, u) ) DV(φ), f L 2 (Ω), g (l) W 1, ( R 2 ; R ). 3.2 G (l) G (l) W 1,1 (Ω) W 1, (Ω; R). 3.1 (J (l) (φ, u) ) G (l) W 1, (Ω; R) G (l) ρ

16 G (l) ( W 1, (Ω; R) ) W ( ) 3.1 (DV(φ) ) DV(φ), f L 2 (D; R), g (l) W 1, ( R 2 ; R ) DV(φ) ρ

17 H 1 V = U, X = H 1 (Ω) 3.5 (H 1 ) G (l) (l = 0, 1, 2,, m) 3.2 ρ (l) G X = H1 (Ω; R) a ( ρ (l) G, v) = ( G (l), v ) L 2 (Ω) v X a(, ) X c 1 a (y, z) = (y, z) X = ( y z + c 1 yz) dω Ω

18 3.2 (DV H 1 ).. W, U, X = H 1 (Ω), p > 1, f L 2 (Ω), g (l) W ( 2, R 2) φ W 3.2 G (l) X H ρ (l) G X U ρ(l) G J(l) (φ, u) U ρ(l) n ρ (l) G X (n { } ) ρ (l) n U n 1 ρ (l) G 2 ρ (l) G 3.2 (H 1 ) 3.2 Galerkin ρ (l) Gh U h U

19 4 SIMP 4.1 ( BV (φ)) Lipschitz Ω R d (d = 2, 3) d Ω Γ Γ ( ) E = ( E i jkl )i jkl Rd d d d 2.4 (SIMP ) Γ 0 0 ( ) f : Ω R d ( ) p : Γ \ Γ 0 R d u : Ω R d f, p u p (φ p Eε(u)) = φ f in Ω φ p Eε(u)ν = p on Γ \ Γ 0 u = 0 on Γ 0 ξ = ( ξ i j / x j )i (ξ Rd d sym ) Γ 0 Ω f

20 ( J (0) (φ, u), J (1) (φ) ) R 2 J (0) (φ, u) = Ω φ f u dx + p u dγ J (1) (φ) = φ dx Γ\Γ 0 Ω SIMP 3.2 J(φ, u) = ( J (0) (φ, u), J (1) (φ) ) 3.1 ( ) DV(φ) ρ U 4.2 ( DV(φ)) W, U, J(φ, u) φ W ɛ > 0 BV(φ ɛ ) u ɛ ρ U J ( (0) φ ɛρ, u ɛρ ) { = J (0) (φ ɛ, u ɛ ) J (1) (φ ɛ ) 0 } min ρ U, ρ =1

21 u Gâteaux u = lim ɛ 0 (u ɛ u)/ ɛ 4.1 (u ) φ W ρ U u Gäteaux u ( pφ p 1 ρeε(u) ) ( φ p Eε(u ) ) = ρ f ( pφ p 1 ρeε(u) + φ p Eε(u ) ) ν = 0 on Γ \ Γ 0 in Ω u = 0 on Γ 0

22 4.3 ( AD (0) (φ)) f, p v (0) : Ω R d ( φ p Eε(v (0) ) ) = φ f in Ω φ p Eε(v (0) )ν = p on Γ \ Γ 0 u = v (0) v (0) = 0 on Γ 0

23 4.1 AD (l) (φ) 4.2 (J (l) (φ, u) ) φ W BV(φ) u J (0) (φ ɛ, u) Fréchet J (0) (φ ɛ, u ɛ ) = J (0) (φ, u + ɛ G (0) ρ dx + o(ɛ) G (0) = pφ p 1 (Eε(u)) ε(u) + 2 f u G (0) (density gradient) Ω

24 ( ) J (0) (φ, u) = Ω ( ρ f u + φ f u ) dx + p u dγ Γ\Γ 0 = ρ f, u Ω + φ f, u Ω + p, u Γ\Γ AD (l) (φ) a ( pφ p 1 ρ, u, v ) + a ( φ p, u, v ) = ρ f, v Ω u V v V a ( φ p, u, v (0)) = φ f, u Ω + p, u Γ\Γ 0 v (0) V u V where a (φ, u, v) = φ (Eε(u)) ε(v) dx Ω V = {v H ( 1 Ω; R d) } v = 0 on Γ 0 1 v = v (0) J (0) (φ, u) = a ( pφ p 1 ρ, u, v (0)) + ρ f, u + v (0) Ω

25 5 ( ) H 1

26 [1] M. P. Bendsøe and O. Sigmund. Topology optimization : theory, methods and applications. Springer, Berlin ; Tokyo, [2] J. Haslinger and R. A. E. Mäkinen. Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM, [3] H. Azegami and S. Kaizu. Smoothing gradient method for non-parametric shape and topology optimization problems. In Proceedings of the 7th World Congress on Structural and Multidisciplinary Optimization (WCSMO-7)(CD-ROM), pp. 1 10, [4], , pp , 2007.

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)

第8章 位相最適化問題

第8章位相最適化問題