ʹର͢Δ Lagrange ؔ਺Λ - 解答 - □von Neumann最大最小定理

□von Neumann最大最小定理

課題 2 解答

2.6 ʹର͢Δ Lagrange ؔ਺Λ

L (a, u, v) = f (u) + L

(a, u, v) = p · u + v · ( − K (a) u + p)

ͱ͓͘ɽ v ∈ U ͸ Lagrange ৐਺Ͱ͋Δɽ L ͷશඍ෼ʹରͯ͠ɼ

L

^′

(a, u, v) [b, u, ˆ v] ˆ

= − {

v ·

( ∂K (a)

∂a

u ∂K (a)

∂a

u )}

b

+ p · u ˆ − v · K (a) ˆ u (= 0 ⇐ K

(a) v = p ) + ˆ v · ( − K (a) u + p) (= 0 ⇐ K (a) u = p )

= g · b ∀ (b, u, ˆ v) ˆ ∈ Ξ × U × U (2.6)

͕੒Γཱͭɽ͜͜Ͱɼ g ͸୅ೖ๏ʹΑΔࣜ (2.4) ͱҰக͢Δɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

͞ΒʹɼͦΕΛ΋͏Ұ౓ a Ͱภඍ෼͢Ε͹ɼ

H =



 



∂

f ˜

∂a

₁

∂a

₁

∂

f ˜

∂a

₁

∂a

₂

∂

f ˜

∂a

₂

∂a

₁

∂

f ˜

∂a

₂

∂a

₂



 

 = l e



 



2 (p

₁

+ p

₂

)

a

³₁

0 0 2p

²₂

a

³₂



 

 (2.5)

ͱͳΔɽ a

, a

> 0 ͷͱ͖ɼ H ͸ਖ਼ఆ஋ͱͳΔɽ

21 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

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Ұํɼ j ∈ { 1, 2 } ^ʹରͯ͠

L

S(a,u)

(a, u, v) [b

, υ ˆ

] = v · {− (K

^′

(a) [b

]) u − K (a) ( ˆ υ

) } = 0

∀ (b

, υ ˆ

) ∈ T

(a, u) ΑΓɼ࣍ΛಘΔɽ

υ ˆ

= − K

⁻¹

(a) (K

^′

(a) [b

]) =



 

− u

₁

a

₁

0 − u

a

₁

− u

a

₂



  ( b

_i1

b

)

(2.8)

ࣜ (2.8) Λࣜ (2.7) ʹ୅ೖ͠ɼࣗݾਵ൐ؔ܎Λ༻͍Ε͹ɼ

h (a, u, v) [b

₁

, b

₂

]

= L

(a,u)(a,u)

(a, u, v) [(b

, υ ˆ

) , (b

, υ ˆ

)]

= b

· (Hb

) (2.9)

ͱͳΔɽ͜͜Ͱɼ H ͸୅ೖ๏ͰಘΒΕͨࣜ (2.5) ͱҰக͢Δɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

Hesse ߦྻ͸࣍ͷΑ͏ʹಘΒΕΔɽڐ༰ू߹ͱڐ༰ํ޲ू߹͋Δ͍͸઀໘Λ

S = { (a, u) ∈ Ξ × U | h (a, u) = 0

_U′

} ,

T

(a, u) = { (b, υ) ˆ ∈ Ξ × U | h

_au

(a, u) [b, υ] = ˆ 0

_Rⁿ

} . ͱ͓͘ɽ L ͷઃܭม਺ (a, u) ʹର͢Δ 2 ֊ภඍ෼͸ɼ

L

(a,u)(a,u)

(a, u, v) [(b

, υ ˆ

) , (b

, υ ˆ

)]

= ( L

(a, u, v) [b

₁

] + L

(a, u, v) [ ˆ υ

₁

])

[b

₂

] + (L

(a, u, v) [b

] + L

(a, u, v) [ ˆ υ

])

[ ˆ υ

]

= ( b

₂

ˆ υ

)

· (

H

( b

₁

ˆ υ

))

∀ (b

, υ ˆ

) , (b

, υ ˆ

) ∈ T

(a, u) (2.7) ͱͳΔɽ͜͜Ͱɼ࣍ͷΑ͏Ͱ͋Δɽ

H

=

( L

Saa

L

Sua

L

Suu

)

= −



 0

_R2×2

( v

K

v

K

_a₂

) ( K

_a^T₁

v K

_a^T₂

v )

0

_R2×2





23 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

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໰୊ 2.8 ( ੍໿͖ͭ໰୊ʹର͢Δޯ഑๏ )

ࢼߦ఺ x

∈ S ʹ͓͍ͯ f

(x

), f

i₁

(x

) = 0, · · · , f

i|^IA|

(x

) = 0, g

(x

), g

(x

), · · · , g

i|^IA|

(x

) Λط஌ͱ͢Δɽ·ͨɼ A ∈ R

^d^×^d

^{Λਖ਼ఆ஋࣮ରশߦྻɼ} c

Λਖ਼ͷఆ਺ͱ͢Δɽ͜ͷͱ͖ɼ

q (y

) = min

y∈X

{

q (y) = 1

2 y · (c

Ay) + g

₀

(x

) · y + f

₀

(x

)

� �

f

(x

) + g

(x

) · y ≤ 0 for i ∈ I

(x

) }

Λຬͨ͢ x

k+1

= x

+ y

ΛٻΊΑɽ

໰୊ 2.8 ͷ Lagrange ؔ਺Λ࣍ͷΑ͏ʹ͓͘ɽ

L

(y, λ

_k+1

) = q (y) + ∑

i∈I_A(x_k)

λ

_{i k+1}

(f

(x

) + g

(x

) · y)

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

■ ੍໿͖ͭ໰୊ʹ͓͚Δޯ഑๏ͱ Newton ๏ [1, 3.7 અ , 3.8 અ ]

͜ΕҎ߱ɼ࠷దઃܭ໰୊ʹ͓͚Δઃܭม਺ ϕ Λ x ͱ͔͘͜ͱʹ͢Δɽ

໰୊ 2.7 ( ੍໿͖ͭ࠷దԽ໰୊ )

X = R

^ͱ͢Δɽ f

₀

, · · · , f

∈ C

(X ; R ) ʹରͯ͠ɼ min

x∈X

{ f

(x) | f

(x) ≤ 0, · · · , f

(x) ≤ 0 } Λຬͨ͢ x ΛٻΊΑɽ

ෆ౳੍ࣜ໿͕ຬͨ͞ΕΔڐ༰ू߹ͱ༗ޮͳ੍໿ʹର͢Δఴ͑ࣈͷू߹Λ S = { x ∈ X | f

(x) ≤ 0, · · · , f

(x) ≤ 0 } ,

I

(x) = { i ∈ { 1, · · · , m } | f

(x) ≥ 0 } = {

i

₁

, · · · , i

_|_I_A_(x)_|

} ͱ͓͘ɽ

25 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

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໰୊ 2.8 ͷ࠷খ఺ y

ʹ͓͚Δ KKT ৚݅͸ɼ L

(y, λ

k+1

) = c

Ay

+ g

(x

) + ∑

i∈IA(xk)

λ

i k+1

g

(x

) = 0

X^′

, (2.10) L

Qλ_k+1

(y, λ

k+1

) = f

(x

) + g

(x

) · y

≤ 0 for i ∈ I

(x

) , (2.11) λ

i k+1

(f

(x

) + g

(x

) · y

) = 0 for i ∈ I

(x

) , (2.12)

λ

_{i k+1}

≥ 0 for i ∈ I

(x

) (2.13)

ͱͳΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

X

^�

x

R

g

₀

q

f

+g

· y

=0 f

+g

· y

=0

g

₁

y

ਤ 2.5: ੍໿͖ͭ໰୊ʹର͢Δޯ഑๏

27 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

(2) 状態決定問題を解いて，評価関数を計算する．

(4) 勾配法で設計変数の変動を求める．

(6) 設計変数を更新して ( k +1)，(2) をおこなう．

(7) 終了条件 Yes

No (3) 随伴問題を解いて，評価関数の勾配を計算する．

(5) Lagrange 乗数を求める．

(1) 初期設定 ( k =0)

(8) 終了

( k +1 → k )

ਤ 2.6: ੍໿͖ͭ࠷దԽ໰୊ʹର͢Δޯ഑๏ͷΞϧΰϦζϜ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

͜͜Ͱɼ y

_g0

, y

_gi₁

, · · · , y

_gi

|^IA|

Λ i ∈ I

(x

) ͝ͱʹޯ഑๏Λద༻ͨ͠ͱ͖ͷ ղͱ͢Δɽ͢ͳΘͪɼ

y

= − (c

A)

⁻¹

g

for i ∈ I

(x

) .

·ͨɼ λ

_k+1

∈ R

^|I^A^|

^Λະ஌ͷ Lagrange ৐਺ͱ͢Δɽ͜ͷͱ͖ɼ y

= y

(λ

_k+1

) = y

_g0

+ ∑

i∈I_A(x_k)

λ

_{i k+1}

y

_gi

͸ࣜ (2.10) Λຬͨ͢ɽ͞Βʹɼࣜ (2.11) ͸



 



g

i₁

· y

gi₁

· · · g

i₁

· y

gi|^IA|

.. . . .. .. . g

|^IA|

· y

_gi₁

· · · g

|^IA|

· y

_gi

|^IA|



 





 

λ

i₁k+1

.. . λ

i|^IA|k+1



  = −



 

f

i₁

+ g

i₁

· y

.. .

f

i|^IA|

+ g

i|^IA|

· y



 

ͱͳΔɽ͜ͷ࿈ཱҰ࣍ํఔࣜΑΓɼ λ

k+1

͕ܾఆ͞ΕΔɽ

29 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

(2) 状態決定問題を解いて，評価関数を計算する．

(8) 終了条件 Yes

No (4) 随伴問題を解いて，勾配と Hesse 行列を計算する．

(1) 初期設定 (k=0)

(9) 終了

(k+1→k) (5) Newton 法で設計変数の変動を求める．

(7) 設計変数を更新して (k+1)，(2) をおこなう．

(6) Lagrange 乗数を求める．

勾配法で設計変数の変動を求める．

随伴問題を解いて，勾配を計算する．

Lagrange 乗数を求める．

(3) 制約関数の Hesse 行列

Yes

ਤ 2.7: ੍໿͖ͭ࠷దԽ໰୊ʹର͢Δ Newton ๏ͷΞϧΰϦζϜ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

໰୊ 2.9 ( ੍໿͖ͭ໰୊ʹର͢Δ Newton ๏ )

ࢼߦ఺ x

∈ X ʹ͓͍ͯɼ λ

∈ R

^|I^A^|

^͸ KKT ৚݅Λຬͨ͢ͱ͢Δɽ·ͨɼ H

(x

) = H

₀

(x

) + ∑

i∈I_A(x_k)

λ

_ik

H

(x

) ͱ͓͘ɽ͜ͷͱ͖ɼ

q (y

) = min

y∈X

{

q (y) = 1

2 y · (H

(x

) y) + g

₀

(x

) · y + f

₀

(x

)

� �

f

(x

) + g

(x

) · y ≤ 0 for i ∈ I

(x

) }

Λຬͨ͢ x

k+1

= x

+ y

ΛٻΊΑɽ

ޯ഑๏ͷ A Λ H

ʹ͓͖͔͑Ε͹ Newton ๏ʹͳΔɽ

31 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

0 1

a(0)=(1/2,1/2)^T

0 1

bg0(0)

a₍₁₎

a=(2/3,1/3)^T a(2)

λ1(1)bg1(0)

λ1(2)bg1(1)

bg0(1)

a₍₅₎

0 1 2 3 4 5

0.90 0.95 1.00

Cost function

f0/f0 init

1+f1

Step number k

(a) ࢼߦ఺ͷਪҠ (b) ධՁؔ਺ͷཤྺ

ਤ 2.9: ޯ഑๏ʹΑΔ਺஋ղ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

■ ^਺஋ྫ

0 0

1 15

a

1 0

a

f

(a

,1–a

) f

(a

,a

)

˜

10 5

ਤ 2.8: ମੵ੍໿͖ͭฏۉίϯϓϥΠΞϯε࠷খԽ໰୊ͷ਺஋ྫ

33 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ؔ਺ղੳͷجૅ

§ 3 ؔ਺ղੳͷجૅ

■ ^ؔ਺ۭؒ [1, 4.3 અ ]

• W

^k,p

( Ω; R

)

: k ֊ඍ෼·Ͱ p ৐Մੵ෼ͳؔ਺ Ω → R

^ͷू߹

(Sobolev ۭؒ : ׬උͳϊϧϜۭؒ (Banach ۭؒ ) ͷੑ࣭Λ΋ͭ )

• W

^0,2

( Ω; R

)

= L

( Ω; R

)

• W

^k,2

( Ω; R

)

= H

( Ω; R

)

: H

( Ω; R

)

ͷ৔߹ɼͨͱ͑͹ɼ಺ੵ

(u, v)

_H1(Ω;R^d)

=

∫

Ω

{ u · v + (

∇ u

)

· (

∇ v

)}

dx ʹରͯ͠׬උͳ಺ੵۭؒ (Hilbert ۭؒ ) ͷੑ࣭Λ΋ͭɽ

• W

^0,^∞

( Ω; R

)

= L

^∞

( Ω; R

)

: ༗ք͔ͭՄੵ෼ͳؔ਺ͷू߹

• W

^1,∞

( Ω; R

)

: Lipschitz ࿈ଓͳؔ਺ͷू߹

• C

^0,σ

(Ω; R ) : σ ∈ (0, 1] ʹରͯ͠ɼ H¨older ࿈ଓͳؔ਺ͷू߹

(σ = 1 ͷͱ͖ɼ Lipschitz ࿈ଓʹରԠ͢Δ )

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

࠷దઃܭ໰୊ͷղ๏

0 1

a(0)=(0.5,0.5)^T

0 1

a(1)

bg0(0)

λ1(1)bg1(0)

λ1(2)bg1(1)

bg0(1)

a=(2/3,1/3)^T a(2)

a(3)

0 1 2 3

0.90 0.95 1.00

Cost function f0/f0 init

1+f1

Step number k

(a) ࢼߦ఺ͷਪҠ (b) ධՁؔ਺ͷཤྺ

ਤ 2.10: Newton ๏ʹΑΔ਺஋ղ

35 / 140

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■ ^{ؔ਺ۭؒͷแؚؔ܎}

ఆཧ 3.1 (Sobolev ͷຒଂఆཧ )

k ∈ { 1, 2, · · · } , p ∈ [1, ∞ ) ʹରͯ͠ɼ k + 1 − d/p ≥ k − d/q ͳΒ͹

W

^k+1,p

(Ω; R ) ⊂ W

^k,q

(Ω; R )

͕੒Γཱͭɽ͞Βʹɼ 0 < σ = k − d/p < 1 ͳΒ͹ɼ W

^k,p

(Ω; R ) ⊂ C

^0,σ

(Ω; R )

ͱͳΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ؔ਺ղੳͷجૅ

x f

(a) σ = 0.5 (b) σ = 1 (Lipschitz ࿈ଓ ) ਤ 3.1: H¨ older ࿈ଓͳؔ਺

37 / 140

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■ ^૒ରۭؒ [1, 4.4 અ ]

Banach ۭؒ X ʹରͯ͠ɼ

f (v) =

∫

Ω

uv dx = ⟨ u, v ⟩ ∀ v ∈ X

Λຬͨ͢Α͏ͳ༗քઢܗ൚ؔ਺ f ( · ) = ⟨ u, · ⟩ ^ͷू߹ (u ͷू߹ ) Λ X

^′

ͱ͔͍

ͯɼ X ͷ૒ରۭؒͱ͍͏ɽ·ͨɼ ⟨· , ·⟩ : X

^′

× X → R ^Λ ⟨· , ·⟩

_X^′_×X

^ͱ΋͔͍ͯɼ

૒ରੵͱ͍͏ɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ؔ਺ղੳͷجૅ

ఆཧ 3.2 (H¨ older ͷෆ౳ࣜ )

d ∈ N ^ʹରͯ͠ɼ Ω Λ R

^{্ͷՄଌू߹ͱ͠ɼ} p, q ∈ (1, ∞ ) ͸ 1

p + 1 q = 1

Λຬͨ͢ͱ͢Δɽ͜ͷͱ͖ɼ f ∈ L

(Ω; R ) ͱ g ∈ L

(Ω; R ) ʹରͯ͠

∥ f g ∥

_L¹_(Ω;_R₎

≤ ∥ f ∥

_L^p_(Ω;_R₎

∥ g ∥

_L^q_(Ω;_R₎

͕੒Γཱͭɽ

39 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ؔ਺ղੳͷجૅ

ࣜ (3.1) ͸ɼ

f (x + y

₁

) = f (x) + f

^′

(x) [y

₁

] + o ( ∥ y

₁

∥

) ͱ΋͔͚Δɽ͜͜Ͱɼ o ( ∥ y

₁

∥

) ͸

∥y₁

lim

∥_X→0

o ( ∥ y

∥

)

∥ y

₁

∥

= 0

ͷΑ͏ʹఆٛ͞ΕΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ؔ਺ղੳͷجૅ

ఆٛ 3.3 (k ֊ͷ Fr´ echet ඍ෼ )

X ͱ Y Λ R ^্ͷ Banach ۭؒͱ͢Δɽ x ∈ X ͷۙ๣ B ⊂ X ্Ͱ f : B → Y

͕ఆٛ͞Ε͍ͯΔͱ͢Δɽ೚ҙͷมಈϕΫτϧ y

∈ X ʹରͯ͠ɼ

∥y1

lim

∥X→0

∥ f (x + y

) − f (x) − f

^′

(x) [y

] ∥

∥ y

₁

∥

= 0 (3.1)

Λຬͨ͢༗քઢܗ࡞༻ૉ f

^′

(x) [ · ] ∈ L (X; Y ) ͕ଘࡏ͢Δͱ͖ɼ f

^′

(x) [y

] Λ f

ͷ x ʹ͓͚Δ Fr´echet ඍ෼ͱ͍͏ɽ͞Βʹɼ೚ҙͷ y

₂

∈ X ʹରͯ͠ɼ

∥y2∥

lim

X→0X

∥ f

^′

(x + y

₂

) [y

₁

] − f

^′

(x) [y

₁

] − f

^′′

(x) [y

₁

, y

₂

] ∥

∥ y

∥

= 0

Λຬͨ͢ f

^′′

(x) [y

, · ] ∈ L (X; L (X ; Y )) ͕ଘࡏ͢Δͱ͖ɼ f

^′′

(x) [y

, y

] Λ f ͷ x ʹ͓͚Δ 2 ֊ͷ Fr´echet ඍ෼ͱ͍͏ɽ L (X ; L (X; Y )) Λ L

(X × X; Y ) ͱ

͔͘ɽ k ∈ { 3, 4, . . . } ^֊ͷ Fr´echet ඍ෼ f

^(k)

΋ಉ༷ʹఆٛ͞ΕΔɽ

41 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ภඍ෼ํఔࣜͷڥք஋໰୊

U = {

u ∈ H

(

Ω; R

) �� u = 0

_R^d

on Γ

} , U (u

) = {

u ∈ H

(

Ω; R

) �� u = u

on Γ

} , a(u, v) =

∫

Ω

S (u) · E (v) dx, l (v) =

∫

Ω

b · v dx +

∫

ΓN

p

· v dγ.

໰୊ 4.2 ( ઢܗ஄ੑ໰୊ͷऑܗࣜ )

b ∈ L

( Ω; R

)

, p

∈ L

(

Γ

; R

)

, C ∈ L

^∞

(

Ω; R

^d^×^d^×^d^×^d

)

, u

∈ H

( Ω; R

) ͷͱ͖ɼ

a (u, v) = l (v) ∀ v ∈ U Λຬͨ͢ u ∈ U (u

) ΛٻΊΑɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ภඍ෼ํఔࣜͷڥք஋໰୊

§ 4 ภඍ෼ํఔࣜͷڥք஋໰୊

■ ^{ઢܗ஄ੑ໰୊} [1, 5.4 અ ]

Ω ⊂ R

^ͷڥք ∂Ω ͸ Lipchitz ࿈ଓͱ͢ΔɽͻͣΈͱԠྗΛ࣍ͷΑ͏ʹ͔͘ɽ

E (u) = 1 2

{ ∇ u

+ (

∇ u

)

}

, S (u) = CE (u)

Γ

p

b u

Ω

ਤ 4.1: ઢܗ஄ੑ໰୊

໰୊ 4.1 ( ઢܗ஄ੑ໰୊ )

ମੵྗ b : Ω → R

, ڥքྗ p

: Γ

→ R

, ط஌มҐ u

: Ω → R

^ʹରͯ͠ɼ

− ∇

S (u) = b

in Ω, S (u) ν = p

on Γ

, u = u

on Γ

Λຬͨ͢มҐ u : Ω → R

^ΛٻΊΑɽ

43 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ภඍ෼ํఔࣜͷڥք஋໰୊

■ ^{ղͷਖ਼ଇੑ} [1, 5.3 અ ]

໰୊ 4.1 ʹ͓͍ͯɼ֯఺ͱ Dirichlet ڥքͱ Neumann ڥքͷڥք Γ ¯

∩ Γ ¯

ͷ

ۙ๣Λ B ͱ͔͘͜ͱʹ͢Δɽ b ∈ L

(

Ω; R

)

ͳΒ͹ɼ u ∈ H

(

Ω \ B; ¯ R )

ΛಘΔɽ

α

Γ

x

₀

∈ Θ

B(x

,r

) θ Ω

ਤ 4.2: ֯Λ΋ͭ 2 ࣍ݩྖҬ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ภඍ෼ํఔࣜͷڥք஋໰୊

■ ପԁܕภඍ෼ํఔࣜͷऑղͷҰҙଘࡏ [1, 5.2 અ ] ఆཧ 4.3 (Lax-Milgram ͷఆཧ )

U Λ࣮ Hilbert ۭؒͱ͢Δɽ a ͸ڧѹత͔ͭ༗քͱ͢Δɽ·ͨɼ l ∈ U

^′

ͱ͢Δɽ

͜ͷͱ͖ɼ໰୊ 4.2 ͷղ u ∈ U ͸Ұҙʹଘࡏ͢Δɽ

ྫ୊ 4.4 ( ઢܗ஄ੑ໰୊ͷղͷҰҙଘࡏ )

໰୊ 4.2 ʹ͓͍ͯɼ | Γ

| > 0 ͷͱ͖ɼղ u ∈ U (u

) ͸Ұҙʹଘࡏ͢Δ͜ͱΛ

ࣔͤɽ

( ղ౴ ) ࣍ͷ͓͖͔͑ʹΑΓɼ Lax-Milgram ͷఆཧͷԾఆ͕੒Γཱͭ͜ͱ͕͔֬ΊΒΕΔɽ a ( ˜ u, v) = l (v) − a (u

, v) ˆ l (v)

□

45 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ภඍ෼ํఔࣜͷڥք஋໰୊

ఆཧ 4.6 ( ֯఺ۙ๣ʹ͓͚Δղͷਖ਼ଇੑ )

ݫີղ u ͸ x

ͷۙ๣Ͱ

Γ

ͱ Γ

͕ಉҰछڥքͳΒ͹ɼ α < π ͷͱ͖ɼ

Γ

₁

ͱ Γ

₂

͕ࠞ߹ڥքͳΒ͹ɼ α < π/2 ͷͱ͖ɼ W

^1,∞

(

B (x

, r

) ∩ Ω; R

)

ʹೖΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ภඍ෼ํఔࣜͷڥք஋໰୊

B(x0,r0) Ω

ΓN

ΓD

B(x0,r0) Ω

(a) ಉҰछڥքͰ։͖͕֯ α > π (b) ࠞ߹ڥքͰ։͖͕֯ α > π/2 ਤ 4.3: ಛҟੑ͕ݱΕΔ֯Λ΋ͭ 2 ࣍ݩྖҬ

ఆཧ 4.5 ( ֯఺ۙ๣ʹ͓͚Δղͷਖ਼ଇੑ [1, ఆཧ 5.3.2])

ݫີղ u ͸ x

ͷۙ๣Ͱ u ∈ H

(

B (x

, r

) ∩ Ω; R

)

ʹೖΔɽͨͩ͠ɼ

Γ

ͱ Γ

͕ಉҰछڥքͳΒ͹ɼ α ∈ [π, 2π) ͷͱ͖ s ∈ (3/2, 2]

Γ

₁

ͱ Γ

₂

͕ࠞ߹ڥքͳΒ͹ɼ α ∈ [π/2, π) ͷͱ͖ s ∈ (3/2, 2] ɼ͓Αͼ α ∈ [π, 2π) ͷͱ͖ s ∈ (5/4, 3/2] ͱͳΔɽ

47 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

■ ^{ઢܗۭؒͱڐ༰ू߹} [1, 8.1 અ ]

ઃܭม਺Λ θ ∈ D ⊂ X ͱ͓͘ɽͨͩ͠ɼઢܗۭؒ X ͱڐ༰ू߹ D ^Λ X = H

(D; R ) ,

D = X ∩ W

^1,^∞

(D; R ) ͱ͓͘ɽີ౓Λ

ϕ (θ) = 1

2 tanh θ + 1 2

ͱ͓͘ɽ͞Βʹɼঢ়ଶม਺ ( ঢ়ଶܾఆ໰୊ͷղ ) u ͷઢܗۭؒͱڐ༰ू߹Λ U = {

u ∈ H

(

D; R

) �� u = 0

_R^d

on Γ

} , U (u

) = {

u ∈ H

(

D; R

) �� u = u

on Γ

} , S = U ∩ W

^1,2q^R

(

D; R

)

, S (u

) = U (u

) ∩ W

^1,2q^R

(

D; R

) ͱ͓͘ɽͨͩ͠ɼ q

> d ͱ͢Δɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

§ 5 ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ [1, ୈ 8 ষ ]

φ(θ) D

ΓD

Γp

b uD

00 φ

�

φ^α

�

0 2 4 6

−2

−6 −4 1.0

0.5

θ φ

(a) ઢܗ஄ੑମ (b) ີ౓ ϕ ͱ߶ੑ ϕ

^α

∥ C ∥ (c) ີ౓ (tanh θ + 1) /2 ਤ 5.1: SIMP (solid isotropic material with penalization) Ϟσϧͱઃܭม਺ θ : D → R

49 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

■ ^ධՁؔ਺

ฏۉίϯϓϥΠΞϯεͱྖҬͷେ͖͞ʹର͢Δ੍໿ؔ਺Λ࣍ͷΑ͏ʹ͓͘ɽ

f

(θ, u) =

∫

b (θ) · u dx +

∫

Γ_N

p

· u dγ −

∫

Γ_D

u

· (ϕ

^α

(θ) S (u) ν) dγ, f

₁

(θ) =

∫

ϕ (θ) dx − c

₁

.

໰୊ 5.2 ( ฏۉίϯϓϥΠΞϯε࠷খԽ໰୊ )

f

₀

ͱ f

₁

ʹରͯ͠ɼ

(θ,u−uD

min

)∈D×S(uD)

{ f

(θ, u) | f

(θ) ≤ 0, ໰୊ 5.1 } Λຬͨ͢ θ ΛٻΊΑɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

■ ^{ঢ়ଶܾఆ໰୊} ( ౳੍ࣜ໿ )[1, 8.8 અ ]

໰୊ 5.1 (θ ܕઢܗ஄ੑ໰୊ )

α > 1 Λఆ਺ͱ͢Δɽ θ ∈ D , b (θ), p

ʹରͯ͠ɼ

− ∇

(ϕ

^α

(θ) S (u)) = b

(θ) in D, ϕ

^α

(θ) S (u) ν = p

on Γ

,

u = u

on Γ

Λຬͨ͢ u ∈ S (u

) ΛٻΊΑɽ

51 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

L

ͷ Fr´echet ඍ෼ʹରͯ͠ɼ͕࣍੒Γཱͭɽ

L

₀^′

(θ, u, v

) [ϑ, u, ˆ v ˆ

]

= L

0θ

(θ, u, v

₀

) [ϑ]

+ L

(θ, u, v

) [ ˆ u] (= 0 ⇐ L

(θ, v

, u) = 0 ˆ ∀ u ˆ ∈ U ) + L

0v0

(θ, u, v

₀

) [ˆ v

₀

] (= 0 ⇐ L

(θ, u, v ˆ

₀

) = 0 ∀ v ˆ

₀

∈ U )

=

∫

{ b

^′

· (u + v

) − αϕ

^α⁻¹

ϕ

^′

S (u) · E (v

) } ϑ dx

= ⟨ g

₀

, ϑ ⟩ ∀ (ϑ, u, ˆ v ˆ

₀

) ∈ X × U × U Ұํɼ f

₁

(θ) ʹؔͯ͠͸ɼ͕࣍੒Γཱͭɽ

f

₁^′

(θ) [ϑ] =

∫

ϕ

^′

ϑ dx = ⟨ g

₁

, ϑ ⟩ ∀ ϑ ∈ X

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

■ ^ධՁؔ਺ͷ θ ඍ෼ [1, 8.8 અ ]

f

(θ, u) ͷ Lagrange ؔ਺Λ

L

(θ, u, v

₀

) = f

₀

(θ, u) + L

(θ, u, v

₀

)

ͱ͓͘ɽͨͩ͠ɼ L

͸໰୊ 5.1 ͷ Lagrange ؔ਺Ͱ࣍ͷΑ͏ʹ͓͘ɽ L

(θ, u, v

₀

)

=

∫

{− ϕ

^α

(θ) S (u) · E (v

₀

) + b (θ) · u } dx +

∫

ΓN

p

· u dγ +

∫

Γ_D

{ (u − u

) · (ϕ

^α

(θ) S (v

) ν) + v

· (ϕ

^α

(θ) S (u) ν) } dγ

53 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

ࣜ (5.1) ӈลͷ֤߲͸ɼ

L

0θθ

(θ, u, v

₀

) [ϑ

₁

, ϑ

₂

] =

∫

− (ϕ

^α

(θ))

^′′

S (u) · E (v

₀

) ϑ

₁

ϑ

₂

dx, (5.2) L

0θu

(θ, u, v

₀

) [ϑ

₁

, υ ˆ

₂

] =

∫

− (ϕ

^α

(θ))

^′

S ( ˆ υ

₂

) · E (v

₀

) ϑ

₁

dx, (5.3) L

0θu

(θ, u, v

) [ϑ

, υ ˆ

] =

∫

− (ϕ

^α

(θ))

^′

S ( ˆ υ

) · E (v

) ϑ

dx, (5.4)

L

0uu

(θ, u, v

) [ ˆ υ

, υ ˆ

] = 0 (5.5)

ͱͳΔɽͨͩ͠ɼ u − u

, v

₀

− u

, υ ˆ

₁

͓Αͼ υ ˆ

₂

͸ Γ

্Ͱ 0

_Rd

ͱͳΔ͜ͱΛ

༻͍ͨɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

■ ^ධՁؔ਺ͷ θ Hesse ܗࣜ

b ͸ θ ͷؔ਺Ͱ͸ͳ͍ͱԾఆ͢Δɽ (θ, u) Λઃܭม਺ͱΈͳ͠ɼͦͷڐ༰ू߹

ͱڐ༰ํ޲ू߹Λ

S = { (θ, u) ∈ D × S | L

(θ, u, v) = 0 for all v ∈ U } ,

T

(θ, u) = { (ϑ, υ) ˆ ∈ X × U | L

Sθu

(θ, u, v) [ϑ, υ] = 0 ˆ for all v ∈ U } ͱ͓͘ɽ͜ͷͱ͖ɼ L

ͷઃܭม਺ (θ, u) ʹର͢Δ 2 ֊ Fr´echet ภඍ෼͸

L

0(θ,u)(θ,u)

(θ, u, v

) [(ϑ

, υ ˆ

) , (ϑ

, υ ˆ

)]

= L

0θθ

(θ, u, v

₀

) [ϑ

₁

, ϑ

₂

] + L

0θu

(θ, u, v

₀

) [ϑ

₁

, υ ˆ

₂

] + L

0θu

(θ, u, v

) [ϑ

, υ ˆ

] + L

0uu

(θ, u, v

) [ ˆ υ

, υ ˆ

]

∀ (ϑ

, υ ˆ

) , (ϑ

, υ ˆ

) ∈ T

(θ, u) (5.1) ͱͳΔɽ

55 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

f

ͷ 2 ֊ θ ඍ෼͸ɼࣜ (5.8) ͱࣜ (5.2) Λࣜ (5.1) ʹ୅ೖ͢Δ͜ͱʹΑΓɼ

h

₀

(θ, u, v

₀

) [ϑ

₁

, ϑ

₂

] =

∫

{

2 (ϕ

^α

(θ))

^′2

ϕ

^α

(θ) − (ϕ

^α

(θ))

^′′

}

S (u) · E (v

₀

) ϑ

₁

ϑ

₂

dx

=

∫

β (α, θ) S (u) · E (v

) ϑ

ϑ

dx ͱͳΔɽͨͩ͠ɼ β (α, θ) ͸

β (α, θ) = α (α + 1) ( 1

2 tanh θ + 1 2

)

α−2

( sech

θ

2 )

− α ( 1

2 tanh θ + 1 2

)

α−1

( − sech

θ tanh θ )

ͱͳΔɽਤ 5.2 (a) ΑΓɼ β (α, θ) > 0 ͕֬ೝ͞ΕΔɽ͞Βʹɼࣗݾਵ൐ؔ܎Λ༻

͍Ε͹ɼ S (u) · E (v

) > 0 ͱͳΓɼ h

(θ, u, v

) [ · , · ] ͸ X ্ͷڧѹత͔ͭ༗ք ͳ͋Δ૒ 1 ࣍ܗࣜͱͳΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

Ұํɼ j ∈ { 1, 2 } ^ʹରͯ͠ɼ L

Sθu

(θ, u, v) [ϑ

, υ ˆ

]

=

∫

{ − (ϕ

^α

(θ))

^′

ϑ

S (υ) − ϕ

^α

(θ) S ( ˆ υ

) }

· E (v) dx

= 0 ∀ (ϑ

, υ ˆ

) ∈ T

(θ, u) (5.6)

ͱͳΔɽͨͩ͠ɼ v

ͱ υ ˆ

ͷ Dirichlet ڥք৚͕݅࢖ΘΕͨɽࣜ (5.6) ΑΓɼ

S ( ˆ υ

) = − (ϕ

^α

(θ))

^′

ϕ

^α

(θ) ϑ

S (u) in D (5.7)

ΛಘΔɽͦ͜Ͱɼࣜ (5.7) ͷ υ ˆ

Λࣜ (5.4) ͷ υ ˆ

ͱࣜ (5.3) ͷ υ ˆ

ʹ୅ೖ͢Ε

͹ɼ࣍ΛಘΔɽ

L

0θu

(θ, u, v

) [ϑ

, υ ˆ

] = L

0θu

(θ, u, v

) [ϑ

, υ ˆ

]

=

∫

(ϕ

^α

(θ))

^′2

ϕ

^α

(θ) S (u) · E (v

) ϑ

ϑ

dx (5.8)

57 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

■ H ¹ ޯ഑๏ͱ H ¹ Newton ๏ [1, 8.6 અ ]

໰୊ 5.3 (θ ܕ H ¹ ޯ഑๏ )

X ্ͷ༗ք͔ͭڧѹతͳ૒ 1 ࣍ܗࣜ a

: X × X → R ^ͱ g

(θ

) ∈ X

^′

͕༩͑Β Εͨͱ͖ɼ

a

(ϑ

_gi

, ψ) = − ⟨ g

(θ

) , ψ ⟩ ∀ ψ ∈ X Λຬͨ͢ ϑ

_gi

∈ X ΛٻΊΑɽ

ྫ͑͹ɼ

a

(ϑ, ψ) =

∫

( ∇ ϑ · ∇ ψ + c

ϑψ) dx ͱ͓͘ɽΞϧΰϦζϜ͸ɼਤ 2.6 ͕࢖ΘΕΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

Ұํɼ f

₁

(θ) ͷ 2 ֊ θ ඍ෼͸ɼ h

(θ) [ϑ

, ϑ

] =

∫

ϕ

^′′

(θ) ϑ

ϑ

dx =

∫

− sech

θ tanh θϑ

ϑ

dx ͱͳΔɽਤ 5.2 (b) ΑΓɼ h

(θ) [ · , · ] ͸ڧѹతʹ͸ͳΒͳ͍ɽ

0 1 2 3

θ

−1

−3 −2

α=3 β

α=2

−0.2

−0.4 0

θ

1 2 3

−1

−3 −2

φ

^��

0.4 0.2

(a) β (α, θ) (b) ϕ

^′′

(θ)

ਤ 5.2: ධՁؔ਺ͷ 2 ֊ θ ඍ෼ʹ͓͚Δ܎਺ؔ਺

59 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

ఆཧ 5.5 (θ ܕ H ¹ ޯ഑๏ [1, ఆཧ 8.4.2, ఆཧ 8.5.5])

θ ∈ D ^{ʹରͯ͠ɼ໰୊} 5.1 ͷղ u ͕ S (u

) ʹೖΔͱ͖ɼ i ∈ { 0, 1 } ^ʹରͯ͠ɼ g

∈ L

^q^R

(D; R ) ͱͳΔɽ͞Βʹɼ໰୊ 5.4 ͷऑղ ϑ

͸Ұҙʹଘࡏ͠ɼͦͷղ ϑ

_gi

͸ D \ B ¯ ্Ͱɼ W

^1,∞

ڃͱͳΔɽ·ͨɼ ϑ

_gi

͸ f ˜

(θ) ͷ߱Լํ޲Λ޲͍ͯ

͍Δɽ

( ূ໌ ) H¨older ͷෆ౳ࣜͳͲʹΑΓɼ g

∈ L

^q^R

(D; R ) ⊂ X

^′

ΛಘΔɽ͞Βʹɼ Lax-Milgram ͷఆཧΑΓɼ ϑ

_gi

͸Ұҙʹଘࡏ͠ɼ ϑ

_gi

͸ D \ B ¯ ্Ͱ W

^2,q^R

ڃΛಘ Δɽ͞Βʹɼ Sobolev ͷຒଂఆཧΑΓɼ

2 − d q

= 1 + σ > 1 (σ ∈ (0, 1)) ⇒ W

^2,q^R

(

D \ B, ¯ R )

⊂ W

^1,∞

(

D \ B, ¯ R )

͕੒Γཱͭɽ·ͨɼ͕࣍੒Γཱͭɽ

f ˜

(θ + ¯ ϵϑ

_gi

) − f ˜

(θ) = ¯ ϵ ⟨ g

, ϑ

_gi

⟩ + o ( | ¯ ϵ | )

= − ¯ ϵa

(ϑ

_gi

, ϑ

_gi

) + o ( | ¯ ϵ | ) ≤ − ϵα ¯

∥ ϑ

_gi

∥

²_X

+ o ( | ¯ ϵ | )

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

໰୊ 5.4 (H ¹ ಺ੵΛ༻͍ͨ θ ܕ H ¹ ޯ഑๏ )

θ ∈ D ^ʹରͯ͠ g

∈ X

^′

͕༩͑ΒΕͨͱ͖ɼ࣍Λຬͨ͢ ϑ

∈ X ΛٻΊΑɽ

− ∆ϑ

_gi

+ c

ϑ

_gi

= − g

in D,

∂

ϑ

= 0 on ∂D.

φ

^α

(θ) D

−

g

c

ਤ 5.3: H

ۭؒͷ಺ੵΛ༻͍ͨ H

ޯ഑๏

61 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

■ ^਺஋ྫ

¡D0 ¡p0

0 200 400 600

評価関数

f0/f0 init

1+f1/c1

1.0

0 0.2 0.4 0.6 0.8 1.2

H¹勾配法 H¹Newton 法

ステップ数 k

(a) ॳظີ౓ͱڥք৚݅ (b) ධՁؔ਺ͷཤྺ

(c) H

ޯ഑๏ʹΑΔ࠷దີ౓ (d) H

Newton ๏ʹΑΔ࠷దີ౓

ਤ 5.4: 2 ࣍ݩઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ໰୊ʹର͢Δ਺஋ྫ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷີ౓ܕҐ૬࠷దԽ

໰୊ 5.6 (θ ܕ H ¹ Newton ๏ )

ࢼߦ఺ θ

∈ X ʹ͓͍ͯɼ λ

∈ R

^|I^A^|

^͸ KKT ৚݅Λຬͨ͢ͱ͢Δɽ·ͨɼ h

(θ

) [ϑ

, ϑ

] = h

(θ

) [ϑ

, ϑ

] + ∑

i∈IA(θk)

λ

h

(θ

) [ϑ

, ϑ

] ∀ ϑ

, ϑ

∈ X ͱ͓͘ɽ a

: X × X → R ^Λ X ্ͷ༗ք͔ͭڧѹతͳ૒ 1 ࣍ܗࣜͱ͢Δͱ͖ɼ

c

h

(θ

) [ϑ

, ψ] + c

a

(ϑ

, ψ) = − ⟨ g

(θ

) , ψ ⟩ ∀ ψ ∈ X Λຬͨ͢ φ

_gi

∈ X ΛٻΊΑɽ

ਤ 2.7 ͷΞϧΰϦζϜ͕࢖ΘΕΔɽ

63 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

■ ^{ঢ়ଶܾఆ໰୊} ( ౳੍ࣜ໿ )[1, 9.11 અ ]

໰୊ 6.1 ( ઢܗ஄ੑ໰୊ )

ϕ ∈ D , b (ϕ), p

(ϕ), u

(ϕ) ͓Αͼ C (ϕ) ʹରͯ͠ɼ

− ∇

S (ϕ, u) = b

(ϕ) in Ω (ϕ) , S (ϕ, u) ν = p

(ϕ) on Γ

(ϕ) , S (ϕ, u) ν = 0

_R^d

on Γ

(ϕ) \ Γ ¯

(ϕ) , u = u

(ϕ) on Γ

(ϕ)

Λຬͨ͢ u ∈ S (u

) ΛٻΊΑɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

§ 6 ઢܗ஄ੑମͷܗঢ়࠷దԽ [1, ୈ 9 ষ ]

■ ^{ઢܗۭؒͱڐ༰ू߹} [1, 9.1 અ ]

ઃܭม਺Λ ϕ ∈ D ⊂ X ͱ͓͘ɽͨͩ͠ɼઢܗۭؒ X ͱڐ༰ू߹ D ^Λ X = {

ϕ ∈ H

(

R

; R

) �� ϕ = 0

_Rd

on Ω ¯

_C0

} , D = {

ϕ ∈ X ∩ W

^1,∞

(

R

; R

) �� શ୯ࣹͷ৚݅ } .

ͱ͓͘ɽͨͩ͠ɼ Ω ¯

⊂ Ω ¯

͸ઃܭ্ͷݻఆྖҬ͋Δ͍͸ڥքͱ͢Δɽ ঢ়ଶม਺ ( ঢ়ଶܾఆ໰୊ͷղ ) u ͷઢܗۭؒͱڐ༰ू߹Λ

U = {

u ∈ H

(

R

; R

) �� u = 0

_Rd

on Γ

(ϕ) } , U (u

) = {

u ∈ H

(

R

; R

) �� u = u

on Γ

(ϕ) } , S = U ∩ W

^1,∞

(

R

; R

) , S (u

) = U (u

) ∩ W

^1,^∞

(

R

; R

) ͱ͓͘ɽ

65 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

■ ^{ධՁؔ਺ͷܗঢ়ඍ෼} [1, 9.11 અ ]

f

₀

(ϕ, u) ͷ Lagrange ؔ਺Λ

L

(ϕ, u, v

) = f

(ϕ, u) + L

(ϕ, u, v

)

ͱ͓͘ɽͨͩ͠ɼ L

͸໰୊ 6.1 ͷ Lagrange ؔ਺Ͱ࣍ͷΑ͏ʹ͓͘ɽ

L

(ϕ, u, v) =

∫

Ω(ϕ)

( − S (u) · E (v) + b · v) dx +

∫

Γ_p(ϕ)

p

· v dγ +

∫

ΓD(ϕ)

{ (u − u

) · (S (v) ν) + v · (S (u) ν) } dγ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

■ ^ධՁؔ਺ [1, 9.11 અ ]

ฏۉίϯϓϥΠΞϯεͱྖҬͷେ͖͞ʹର͢Δ੍໿ؔ਺Λ࣍ͷΑ͏ʹ͓͘ɽ

f

(ϕ, u) =

∫

Ω(ϕ)

b · u dx +

∫

Γ_N(ϕ)

p

· u dγ −

∫

Γ_D(ϕ)

u

· (S (u) ν) dγ,

f

(ϕ) =

∫

Ω(ϕ)

dx − c

.

໰୊ 6.2 ( ฏۉίϯϓϥΠΞϯε࠷খԽ໰୊ )

f

ͱ f

ʹରͯ͠ɼ

(ϕ,u−

min

uD)∈D×S

{ f

₀

(ϕ, u) | f

₁

(ϕ) ≤ 0, ໰୊ 6.1 } Λຬͨ͢ Ω (ϕ) ΛٻΊΑɽ

67 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

͕੒Γཱͭɽͨͩ͠ɼ G

Ω0

= 2S (u) (

∇ u

)

, g

_Ω0

= − S (u) · E (u) + 2b · u, g

= 2κ (p

· u) ν,

g

∂p0

= 2 (p

· u) τ Ͱ͋Δɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

b (ϕ), p

(ϕ), u

(ϕ) ͓Αͼ C (ϕ) ͸෺࣭ݻఆͰ͋ΔͱԾఆ͢Δɽ͜ͷͱ͖ɼ L

₀^′

(ϕ, u, v

₀

) [φ, u, ˆ v ˆ

₀

]

= L

0ϕ^′

(ϕ, u, v

) [φ]

+ L

(ϕ, u, v

₀

) [ ˆ u] (= 0 ⇐ L

(ϕ, v

₀

, u) = 0 ˆ ∀ u ˆ ∈ U ) + L

0v₀

(ϕ, u, v

) [ˆ v

] (= 0 ⇐ L

(ϕ, u, v ˆ

) = 0 ∀ v ˆ

∈ U )

= ⟨ g

, φ ⟩ ( ⇐ [1, ໋୊ 9.3.4 ͱ໋୊ 9.3.7] )

=

∫

Ω(ϕ)

( G

_Ω0

· ∇ φ

+ g

_Ω0

∇ · φ ) dx

+

∫

Γ_p(ϕ)

g

· φ dγ +

∫

∂Γ_p(ϕ)∪Θ(ϕ)

g

∂p0

· φ dς

∀ (φ, u, ˆ v ˆ

) ∈ Ξ × U × U

69 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

Ұํɼ f

(ϕ) ͷܗঢ়ඍ෼͸ɼ f

₁^′

(ϕ) [φ] = ⟨ g

, φ ⟩ =

∫

Ω(ϕ)

∇ · φ dx ͱͳΔɽ͋Δ͍͸ɼ

f

₁^′

(ϕ) [φ] = ⟨ g ¯

, φ ⟩ =

∫

∂Ω(ϕ)

ν · φ dγ ͱͳΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

͋Δ͍͸ɼ b, p

, u

͓Αͼ C ͸ۭؒݻఆͷؔ਺Ͱ͋ΔͱԾఆͯ͠ɼ u ͱ v

͸ q

> d ʹରͯ͠ W

^2,2q^R

(

R

; R

)

ͱԾఆͱ͖ɼ L

0ϕ^∗

(ϕ, u, v

₀

) [φ]

= ⟨ g ¯

₀

, φ ⟩ ( ⇐ [1, ໋୊ 9.3.9 ͱ໋୊ 9.3.12] )

=

∫

∂Ω(ϕ)

g ¯

∂Ω0

· φ dγ +

∫

Γp(ϕ)

g ¯

· φ dγ +

∫

∂Γp(ϕ)∪Θ(ϕ)

g ¯

∂p0

· φ dς

+

∫

ΓD(ϕ)

g ¯

· φ dγ

ͷΑ͏ʹ͔͔ΕΔɽ͜͜Ͱɼ

¯

g

_∂Ω0

= ( − S (u) · E (u) + 2b · u) ν,

¯

g

= 2 (∂

+ κ) (p

· u) ν, g ¯

∂p0

= 2 (p

· u) τ ,

¯

g

_D0

= 2 { ∂

_ν

(u − u

) · (S (u) ν) } ν .

71 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

ࣜ (6.1) ͷӈลୈ 2 ߲͸ɼ L

0ϕ^′u

(ϕ, u, v

) [φ

, υ ˆ

]

=

∫

Ω(ϕ)

[{ S ( ˆ υ

₂

) (

∇ v

^T₀

)

+ S (v

₀

) (

∇ υ ˆ

₂^T

)

}

· ∇ φ

^T₁

− (S( ˆ υ

₂

) · E (v

₀

)) ∇ · φ

₁

]

dx (6.2)

ͱͳΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

■ ^{ධՁؔ਺ͷܗঢ়} Hesse ܗࣜ [1, 9.11 અ ]

b = 0

_R^d

ΛԾఆ͢Δɽઃܭม਺ (ϕ, u) Λ (ϕ, u) ͷڐ༰ू߹ͱڐ༰ํ޲ू߹Λ S = { (ϕ, u) ∈ D × S | L

(ϕ, u, v) = 0 for all v ∈ U } ,

T

(ϕ, u) = { (φ, υ) ˆ ∈ X × U | L

Sϕu

(ϕ, u, v) [φ, υ] = 0 ˆ for all v ∈ U } ͱ͓͘ɽ͜ͷͱ͖ɼ L

ͷ (ϕ, u) ʹର͢Δ 2 ֊ Fr´echet ภඍ෼͸ɼ

L

0(ϕ^′,u)(ϕ^′,u)

(ϕ, u, v

₀

) [(φ

₁

, υ ˆ

₁

) , (φ

₂

, υ ˆ

₂

)]

= (L

0ϕ^′

(ϕ, u, v

) [φ

] + L

(ϕ, u, v

) [ ˆ υ

])

_ϕ

[φ

] + (L

0ϕ^′

(ϕ, u, v

) [φ

] + L

(ϕ, u, v

) [ ˆ υ

])

[ ˆ υ

]

= L

0ϕ^′ϕ^′

(ϕ, u, v

) [φ

, φ

] + L

0ϕ^′u

(ϕ, u, v

) [φ

, υ ˆ

] + L

0ϕ^′u

(ϕ, u, v

₀

) [φ

₂

, υ ˆ

₁

] + L

0uu

(ϕ, u, v

₀

) [ ˆ υ

₁

, υ ˆ

₂

]

∀ (φ

, υ ˆ

) , (φ

, υ ˆ

) ∈ T

(ϕ, u) (6.1) ͱͳΔɽࣜ (6.1) ͷӈลୈ 1 ߲ͷܭࢉ͸লུ͢Δɽ

73 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

೚ҙͷ v ∈ U ʹରͯࣜ͠ (6.3) ͕੒Γཱͭ͜ͱ͔Βɼ S ( ˆ υ

) (

∇ v

)

= {(

∇ φ

^T_j

)

S (u) + C (

∇ φ

^T_j

∇ u

)

− ∇ · φ

S (u) } (

∇ v

)

(6.4)

͕ಘΒΕΔɽ·ͨɼࣜ (6.3) ͸ L

Sϕ^′u

(ϕ, u, v) [φ

, υ ˆ

]

=

∫

Ω(ϕ)

[ ∇ v

S (u) ∇ φ

^T_j

+ S (v) {(

∇ u

)

((

∇ φ

^T_j

)

− ∇ · φ

) − (

∇ υ ˆ

_j^T

)

}]

· I dx

= 0 ͱ΋͔͚Δɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

Ұํɼ j ∈ { 1, 2 } ^ʹରͯ͠ɼ L

Sϕ^′u

(ϕ, u, v) [φ

, υ ˆ

]

=

∫

Ω(ϕ)

{ S (u) · (

∇ φ

^T_j

∇ v

)

+ S (v) · (

∇ φ

^T_j

∇ u

)

− (S (u) · E (v)) ∇ · φ

− S ( ˆ υ

) · E (v) } dx

=

∫

Ω(ϕ)

[{( ∇ φ

^T_j

)

S (u) + C (

∇ φ

^T_j

∇ u

)

− S (u) ∇ · φ

− S ( ˆ υ

) } (

∇ v

)

]

· I dx

= 0 ∀ (φ

, υ ˆ

) ∈ T

(ϕ, u) (6.3)

ͱͳΔɽͨͩ͠ɼ v ͱ υ ˆ

ͷ Dirichlet ڥք৚͕݅࢖ΘΕͨɽ

75 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

Ҏ্ͷ݁ՌʹՃ͑ͯɼࣗݾਵ൐ؔ܎ͱ

h

₀

(ϕ, u, v

₀

) [φ

₁

, φ

₂

] = h

₀

(ϕ, u, v

₀

) [φ

₂

, φ

₁

]

͕੒Γཱͭ͜ͱΛ༻͍Ε͹ɼ f

₀

ͷܗঢ় Hesse ܗࣜ͸࣍ͷΑ͏ʹಘΒΕΔɽ h

(ϕ, u, u) [φ

, φ

]

=

∫

Ω(ϕ)

[ S (u) · E (u) {(

∇ φ

^T₂

)

· ∇ φ

^T₁

+ ( ∇ · φ

) ( ∇ · φ

) } + (

∇ u

S (u) )

· {

∇ φ

^T₁

(

∇ φ

^T₂

)

+ ∇ φ

^T₂

(

∇ φ

^T₁

)

}

− 2 (S (u) E (u)) · {

∇ φ

^T₂

( ∇ · φ

) + ∇ φ

^T₁

( ∇ · φ

) }]

dx

∀ φ

₁

, φ

₂

∈ X

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

ͦ͜Ͱɼ S (v) (

∇ υ ˆ

^T_j

)

= ∇ v

S (u) ∇ φ

^T_j

+ S (v) (

∇ u

)

{(

∇ φ

^T_j

)

− ∇ · φ

}

(6.5) ΛಘΔɽࣜ (6.2) ʹࣜ (6.4) ͱࣜ (6.5) Λ୅ೖ͢Ε͹ɼࣜ (6.1) ͷӈลୈ 2 ߲͕

ܭࢉ͞ΕΔɽಉ༷ʹɼࣜ (6.1) ͷӈลୈ 3 ߲͸ɼӈลୈ 2 ߲ͷ݁Ռʹ͓͍ͯ φ

ͱ φ

₂

Λ͍Ε͔͑ͨ΋ͷͱͳΔɽࣜ (6.1) ͷӈลୈ 4 ߲͸ 0 ͱͳΔɽ

77 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

■ H ¹ ޯ഑๏ͱ H ¹ Newton ๏ [1, 9.8 અ ]

໰୊ 6.3 ( ྖҬมಈܕ H ¹ ޯ഑๏ )

X ্ͷ༗ք͔ͭڧѹతͳ૒ 1 ࣍ܗࣜ a

: X × X → R ^ͱ g

∈ X

^′

͕༩͑ΒΕͨ

ͱ͖ɼ

a

(φ

_gi

, ψ) = − ⟨ g

, ψ ⟩ ∀ ψ ∈ X Λຬͨ͢ φ

∈ X ΛٻΊΑɽ

ྫ͑͹ɼ

a

(φ, ψ) =

∫

Ω(ϕ)

{( ∇ φ

)

· (

∇ ψ

)

+ c

_Ω

φ · ψ } dx ͱ͓͘ɽΞϧΰϦζϜ͸ɼਤ 2.6 ͕࢖ΘΕΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

Ұํɼ f

(ϕ) ͷܗঢ় Hesse ܗࣜ͸࣍ͷΑ͏ʹͳΔɽ h

(ϕ) [φ

, φ

]

=

∫

Ω(ϕ)

{ − (

∇ φ

^T₂

)

· ∇ φ

^T₁

+ ( ∇ · φ

) ( ∇ · φ

) }

dx ∀ φ

, φ

∈ X

79 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

໰୊ 6.4 ( ྖҬมಈܕ H ¹ Newton ๏ )

ࢼߦ఺ ϕ

∈ X ʹ͓͍ͯɼ λ

∈ R

^|^I^A^|

^͸ KKT ৚݅Λຬͨ͢ͱ͢Δɽ·ͨɼ h

(ϕ

) [φ

, φ

] = h

(ϕ

) [φ

, φ

] + ∑

i∈IA(xk)

λ

h

(ϕ

) [φ

, φ

]

∀ φ

, φ

∈ X

ͱ͓͘ɽ a

: X × X → R ^Λ X ্ͷ༗ք͔ͭڧѹతͳ૒ 1 ࣍ܗࣜͱ͢Δɽ͜ͷ ͱ͖ɼ

c

h

(ϕ

) [φ

, ψ] + c

a

(φ

, ψ) = − ⟨ g

(ϕ

) , ψ ⟩ ∀ ψ ∈ X Λຬͨ͢ φ

∈ X ΛٻΊΑɽ

ਤ 2.7 ͷΞϧΰϦζϜ͕࢖ΘΕΔɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

ΓD(φ)

Γp(φ) pN(φ)

uD(φ)b(φ)

Ω(φ)

ΓD(φ)

Γp(φ)=Γηi(φ) (φ)

−gpi−gηi

−g∂pi−g∂ηi

−gΩi

cΩ

−GΩi

(Á+') (Á)

x '(x)

ঢ়ଶܾఆ໰୊ H

ޯ഑๏ ܗঢ়ߋ৽

(b, p, u

) → u g

→ φ

Ω (ϕ) → Ω (ϕ + ϵφ

) ਵ൐໰୊

(ζ

, η

) �→ v

ਤ 6.1: H

ޯ഑๏ʹΑΔܗঢ়ߋ৽

81 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

0 20 40 60 80 100 120 0.4

0.6 0.8

Cost function

1.0 f

/f

0 init

1+f

/c

H

grad. meth.

H

Newton meth.

Iteration number k

ਤ 6.3: 2 ࣍ݩઢܗ஄ੑମʹର͢Δ਺஋݁Ռ : ධՁؔ਺ͷཤྺ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ઢܗ஄ੑମͷྖҬมಈܕܗঢ়࠷దԽ

■ ^਺஋ྫ

ΓD0

Γp0

ΓD0

(a) ॳظܗঢ় (b) H

ޯ഑๏ (c) H

Newton ๏ ਤ 6.2: 2 ࣍ݩઢܗ஄ੑମʹର͢Δ਺஋݁Ռ : ܗঢ়ൺֱ

83 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ^ઃܭม਺

• L = { 1, 2, · · · , |L|} : ϦϯΫʹ෇͚ΒΕͨ൪߸ͷू߹

• Ω

_0l

⊂ R

: ϦϯΫ l ∈ L ^ͷ࣌ࠁ t = 0 ͷͱ͖ͷॳظྖҬ

• Ω

0 def

= { Ω

}

_l∈L

, ∂Ω

0 def

= { ∂Ω

}

_l∈L

• X = {

ϕ ∈ H

(

R

; R

) �� ϕ = 0

_Rd

on Ω ¯

_C0

}

• D = {

ϕ ∈ X ∩ W

^1,^∞

(

R

; R

) �� _{શ୯ࣹͷ৚݅} }

• ϕ

∈ D ^ʹରͯ͠ɼ Ω

(ϕ

) = { (i + ϕ

) (x) | x ∈ Ω

_0l

}

• ϕ

^def

= { ϕ

}

i∈L

∈ D

^|L|

, Ω (ϕ)

^def

= { Ω

(ϕ

) }

l∈L

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

§ 7 ϦϯΫػߏͷܗঢ়࠷దԽ [3]

Ω

b

Ω

x

p

Γ

p03

Ω

(φ

)

Ω

(φ

)

ਤ 7.1: t = 0 ʹ͓͚ΔϦϯΫ݁߹͞Εͨ߶ମͷྖҬมಈ

85 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

• u

(q

, x) = x

(t) − x

(0) + θ

(t) e

× (x − x

(0)) : q

͔Β Ω

(ϕ

) ্ ͷ೚ҙͷ఺ x ͷมҐ

• Ω ˜

(ϕ

, q

) = { x + u

(q

, x) | x ∈ Ω

(ϕ

) } : ߶ମӡಈ͢ΔྖҬ

• Γ ˜

(q

) = {

x + u

(q

, x) � � x ∈ ∂Ω

∩ Γ

} : ߶ମӡಈ͢Δඇಉ࣍

Neumann ڥք

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ^ঢ়ଶม਺

• x

_Gl

: (0, t

) → R

: Ω

(ϕ) ͷॏ৺ͷҐஔ

• θ

: (0, t

) → R : x

_Gl

पΓͷճస

• q

(t) = (

(x

(t) − x

(0))

, θ

(t) )

: (0, t

) → R

^d^F

: Ω

(ϕ) ͷ߶ମӡಈ (d

= 3)

• q = (

q

₁^T

, · · · , q

_|L|^T

)

: (0, t

) → R

^d^F^|L|

: શମܥͷ߶ମӡಈ

Q = {

q ∈ H

(

(0, t

) ; R

^d^F^|L|

) ��

� �

( q ˙ (0) q (0)

)

= ( 0

0 ) }

, Q

=

{

q ∈ H

(

(0, t

) ; R

^d^F^|L|

) ��

� �

( q ˙ (0) q (0) )

= ( q

₁

q

) } , Q

(q) =

{

r ∈ H

(

(0, t

) ; R

^d^F^|L|

) ��

� �

( r ˙ (t

) r (t

)

=

( q ˙ (t

) q (t

)

) }

87 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ҰൠԽ࣭ྔͱҰൠԽ֎ྗ

ϦϯΫػߏͷҰൠԽ࣭ྔ

M (ϕ) = diag (

m

₁

(ϕ

₁

) , m

₁

(ϕ

₁

) , j

_G1

(ϕ

₁

) , . . . , m

_|L|

(

ϕ

_|L|

)

, m

_|L|

( ϕ

_|L|

)

, j

_G|L|

( ϕ

_|L|

))

ͨͩ͠ɼ

m

(ϕ

) =

∫

Ω_l(ϕ_l)

ρ

dx, j

_Gl

(ϕ

) =

∫

Ωl(ϕl)

ρ

∥ x − x

_Gl

(0) ∥

²_R^d

dx

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ^ӡಈ੍໿

• J = { 1, 2, · · · , |J |} : ϦϯΫ݁߹੍໿ʹ෇͚ΒΕͨ൪߸ͷू߹

• i ∈ J ^ʹରͯ͠ɼ u

(q

, x

_Jil

) − u

(q

, x

_Jim

) = 0

_R²

: ϦϯΫ݁߹੍໿

• T = { 1, 2, · · · , |T |} : ฒਐӡಈ੍໿ʹ෇͚ΒΕͨ൪߸ͷू߹

• i ∈ T ^ʹରͯ͠ɼ u

Til

(q

, x

) = (x

Glj

(t) − x

Glj

(0)) · e

= 0 : ฒਐӡಈ

੍໿

• R = { 1, 2, · · · , |R|} : ճసӡಈ੍໿ʹ෇͚ΒΕͨ൪߸ͷू߹

• i ∈ R ^ʹରͯ͠ɼ θ

(t) = 0 : ճసӡಈ੍໿

• ψ (q) = 0

_R|C|

( C

^def

= ( { 1, 2 } × J ) ∪ T ∪ R ) : ͢΂ͯͷӡಈ੍໿

89 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ^{ঢ়ଶܾఆ໰୊}

໰୊ 7.1 ( ϦϯΫӡಈ )

ϕ ∈ D

^|L|

͸༩͑ΒΕͨͱ͢Δɽ·ͨɼ q

, q

∈ R

^d^F^|L|

^{͕༩͑ΒΕͨͱͯ͠ɼ} ( ψ

^′

(q

) [q

]

ψ (q

₀

) )

= ( 0

0 )

Λຬͨ͢ͱ͢Δɽ͞Βʹɼ rank ψ

_qT

(q

₀

) = |C| ^{ͱ͢Δɽ͜ͷͱ͖ɼ} ( M (ϕ) (

ψ

_qT

(q) )

ψ

_q^T

(q) 0

) ( q ¨

− µ )

=

( s

− ψ

^′′

(q) [ ˙ q, q] ˙ )

in (0, t

) , ( q ˙ (0)

q (0) )

= ( q

₁

q

)

Λຬͨ͢ (

q

, µ

)

: (0, t

) → R

^d^F^|L|⁺^|C|

^ΛٻΊΑɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

ϦϯΫ l ∈ L ^{ʹ࡞༻͢Δମੵྗ} b

∈ L

(

(0, t

) ; L

^∞

(

R

; R

))

ͱඇྵͷڥք

ྗ p

∈ L

(

(0, t

) ; L

^∞

(

Γ

p0l

; R

))

ʹର͢ΔҰൠԽ֎ྗ

s

= (

s

^T_Fl

, s

_Ml

)

∈ R

^d^F

ͨͩ͠ɼ

s

_Fl

=

∫

Ωl(ϕl)

b

(t) dx +

∫

Γp0l

p

_Nl

(t) dγ,

s

e

=

∫

Ωl(ϕl)

b

(t) × (x − x

(t)) dx +

∫

Γp0l

p

(t) × (x − x

(t)) dγ

શମܥͷҰൠԽྗ

s = (

s

^T₁

, . . . , s

^T_|L|

)

91 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

µ

(q)

µ

(q) x

(q)

x

(q)

x

(q) x

(q) µ

e

₃

Ω

µ

e

₂

(a) ϦϯΫ݁߹ྗ (b) ฒਐ੍໿ྗͱճస੍໿Ϟʔϝϯτ ਤ 7.2: ӡಈ੍໿ʹର͢Δ੍໿ྗ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

໰୊ 7.1 ʹ͓͍ͯɼ µ ͸ɼӡಈ੍໿ʹର͢Δ Lagrange ৐਺Ͱɼӡಈ੍໿Λຬ

ͨͨ͢Ίʹ࡞༻੍ͨ͠໿ྗΛද͢ɽͦ͜Ͱɼ µ ͷཁૉΛ

µ = (

µ

^T_J1

, . . . , µ

^T_{J|J |}

, µ

^T_T1

, . . . , µ

^T_{T|T |}

, µ

^T_R1

, . . . , µ

^T_R|R|

)

∈ E ͱ͔͘͜ͱʹ͢Δɽ͜͜Ͱɼ࣍ͷΑ͏ʹ͓͘ɽ

E = L

(

(0, t

) ; R

^|C|

)

■ ੍໿ྗͱ੍໿Ϟʔϝϯτ

• ϦϯΫ݁߹ྗ : x

∈ Ω

(ϕ) ʹ͓͍ͯ µ

: (0, t

) → R

^ɼ x

∈ Ω

(ϕ) ʹ

͓͍ͯ − µ

(0, t

) → R

( ਤ 7.2 (a))

• ฒਐӡಈ੍໿ʹର͢Δ੍໿ྗ : x

∈ Ω

(ϕ) ʹ͓͍ͯ µ

e

( ਤ 7.2 (b))

• ճసӡಈ੍໿ʹର͢Δ੍໿Ϟʔϝϯτ : µ

e

on x

∈ Ω

(ϕ) ( ਤ 7.2 (b))

93 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

໰୊ 7.2 ( ϦϯΫӡಈͷऑܗࣜ )

໰୊ 7.1 ͷԾఆ͕੒Γཱͭͱ͖ɼ

L

Sr,η

(q, µ, r, η) [ˆ r, η] = 0 ˆ ∀ (ˆ r, η) ˆ ∈ Q

(0) × E Λຬͨ͢ (q, µ) ∈ Q

× E ΛٻΊΑɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

໰୊ 7.1 ͷ Lagrange ؔ਺͸ɼ (q, µ, r, η) ∈ Q

× E × Q

(q) × E ʹରͯ͠

L

(ϕ, q, µ, r, η)

=

∫

( − q ¨

M (ϕ) ˙ r + µ

ψ

_q^T

(q) ˙ r + ¨ q

(

ψ

_q^T

(q) )

η + ψ

^′′

(q) [ ˙ q, q] ˙ · η + η

ψ

_qT

(q) ˙ q + ¨ r

(

ψ

_qT

(q) )

µ + ψ

^′′

(q) [ ˙ r, r] ˙ · µ + s · r ˙ ) dt + 1

2 q ˙

(t

) M (ϕ) ˙ q (t

) (7.1)

ͱ͓͘ɽ·ͨɼ࣍ͷΑ͏ʹ͔͘ɽ L

Sr,η

(ϕ, q, µ, r, η) [ˆ r, η] ˆ

=

∫

( − q ¨

M (ϕ) ˙ˆ r + µ

ψ

_q^T

(q) ˙ˆ r + ¨ q

(

ψ

_q^T

(q) )

ˆ

η + ψ

^′′

(q) [ ˙ q, q] ˙ · η ˆ + ˆ η

ψ

_q^T

(q) ˙ q + ¨ ˆ r

(

ψ

_q^T

(q) )

µ + ψ

^′′

(q) [

˙ˆ r, r ˙ˆ ]

· µ + s · r ˙ˆ ) dt

∀ (ˆ r, η) ˆ ∈ Q

(0)

95 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ^ܗঢ়ඍ෼

f ˜

₀^′

(ϕ) [φ] = ∑

l∈L

∫

Ωl(ϕl)

g

_M0l

∇ · φ

dx = ∑

l∈L

⟨ g

_0l

, φ

⟩ = ⟨ g

₀

, φ ⟩

͋Δ͍͸ɼ

f ˜

₀^′

(ϕ) [φ] = ∑

l∈L

∫

∂Ωl(ϕl)

g

_M0l

ν · φ

dγ = ∑

l∈L

⟨ g ¯

_0l

, φ

⟩ = ⟨ g ¯

₀

, φ ⟩

ͨͩ͠ɼ

g

M0l

= −

∫

t_T 0

ρ

u

( ¨ q

) · u

( ˙ r

) dt

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ^{ܗঢ়࠷దԽ໰୊}

f

₀

(ϕ, q, µ) = −

∫

t_T 0

s · q ˙ dt, f

₁

(ϕ) = c

₁

− ∑

i∈L

∫

Ωl(ϕl)

dx

໰୊ 7.3 ( ମੵ੍໿͖ͭ֎ྗ࢓ࣄ࠷େԽ໰୊ )

࣍Λຬͨ͢ Ω (ϕ) ΛٻΊΑɽ

(ϕ,q,µ)∈D

min

^|L|×Q₀×E

{ f

(ϕ, q, µ) | f

(ϕ) ≤ 0, ໰୊ 7.2 }

97 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

0 100 200

0.5 1.0 1.5 2.0

(1{f

/c

)/2 f

/f

0 init

Iteration number of reshaping

C ost f u n ct ion

50 150

ਤ 7.4: ܗঢ়मਖ਼ʹର͢ΔධՁؔ਺ͷཤྺ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

■ ^਺஋ྫ

֎ྗ p

_N3

= ( − 0.2, 0)

N/mm ɼॳظ࢟੎ θ

₁

(0) = 60

^◦

ɼ θ

₂

(0) = 150

^◦

ɼॳظ

֯଎౓ 0.2618rad/s ɼऴ୺࣌ࠁ θ

(t

) = 73

^◦

( ॳظܗঢ়ͷͱ͖ )

ॳظܗঢ় ࠷దܗঢ়

ਤ 7.3: ܗঢ়ൺֱ

99 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ

§ 8 ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ [4]

ϒϨʔΩ໐͖͸ɼϩʔλͱύουؒͷຎࡲʹΑΔࣗྭৼಈݱ৅Ͱ͋Δɽಛʹɼ

ෳૉݻ༗஋ͷ࣮෦͕ਖ਼ͱͳΔݻ༗ৼಈͱߟ͑ΒΕ͖ͯͨɽ

ਤ 8.1: ं྆ͷϒϨʔΩ෦඼

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϦϯΫػߏͷܗঢ়࠷దԽ

ਤ 7.5: όολͷ଍ʢʁʣ

101 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ

໰୊ 8.1 ( ݻ༗ৼಈ )

ϕ ∈ D ^{͕༩͑ΒΕͨͱ͖ɼ} k ∈ { 1, 2, · · · } ^{ʹରͯ͠ɼ࣍Λຬͨ͢}

(s

, u ˆ

) = (s

, u ˆ

) ∈ C × S ^ΛٻΊΑɽ s

²_k

ρ

u ˆ

− ( ∇ · S ( ˆ u

))

= 0

_R^d

in Ω

, s

ρ

u ˆ

− ( ∇ · S ( ˆ u

))

= 0

_R^d

in Ω

(ϕ) , S ( ˆ u

) ν

= 0

_Rd

on (

∂Ω

_R0

\ Γ ¯

_R0

) , S ( ˆ u

) ν

= 0

_R^d

on (

∂Ω

(ϕ) \ Γ ¯

) ,

S ( ˆ u

) ν

= Re [α { ( ˆ u

− u ˆ

) · ν

} ν

] on Γ

, S ( ˆ u

) τ

= Re [µα { ( ˆ u

− u ˆ

) · ν

} τ

] on Γ

_R0

, S ( ˆ u

) ν

= α { ( ˆ u

− u ˆ

) · ν

} ν

on Γ

, S ( ˆ u

) τ

= − µα { ( ˆ u

− u ˆ

) · ν

} τ

on Γ

ˆ

u

= ˆ u

on (Γ

_R0

∪ Γ

_P0

) , u ˆ = 0

_Rd

on Γ

_D0

.

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ

P0 R0

¡P0, ¡R0

Coulomb friction

¡D0 ¡P0

¡R0

®, ¹ ºP

¿P ºP

¿R R0 P0

¡P0 P0 P(Á)=(i+Á)( _P0)

¡D0

(i+Á)(x)

(a) σΟεΫͱύου (b) Coulomb ຎࡲ (c) ύουͷྖҬมಈ ਤ 8.2: ϒϨʔΩϞσϧ

■ ^{ঢ়ଶܾఆ໰୊}

ݻ༗ৼಈϞʔυ ( มҐ ) u ˆ ͷ Fourier ม׵ͷͨΊͷઢܗۭؒͱڐ༰ू߹

U = { ˆ

u ∈ H

(

R

; C

) �� u ˆ = 0

_Cd

on Γ

_D0

} , S = U ∩ W

^2,2q^R

(

R

; C

)

103 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ

■ ^਺஋ྫ

ද 8.1: ෳૉݻ༗஋ͷมԽ

ॳظܗঢ় ࠷దܗঢ়

k Re Im

1 -1.692E+01 8.022947E+03 2 -1.444E+01 9.438261E+03 3 8.613E+00 1.249724E+04 4 -2.944E+01 1.437360E+04 5 -5.783E+01 1.629984E+04 6 -5.356E+01 2.168113E+04 7 -5.195E+01 2.394771E+04 8 -6.593E+01 2.573753E+04 9 -6.325E+01 2.711726E+04 10 -6.896E+01 2.893466E+04

k Re Im

1 -1.647E+01 7.745197E+03 2 -1.765E+01 1.027973E+04 3 -1.163E+01 1.110440E+04 4 -3.048E+01 1.503565E+04 5 -4.185E+01 2.092213E+04 6 -5.070E+01 2.186379E+04 7 -6.588E+01 2.671747E+04 8 -7.522E+01 2.756015E+04 9 -7.540E+01 3.137934E+04 10 -7.658E+01 3.320161E+04

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ

■ ^{ܗঢ়࠷దԽ໰୊}

Ϟʔυ࣍਺ k ͸༩͑ΒΕΔͱԾఆ͢ΔɽධՁؔ਺Λ࣍ͷΑ͏ʹఆٛ͢Δɽ f

₀

(ϕ, s

) = 2Re [s

] = s

+ s

^c_k

,

f

(ϕ) = −

∫

ΩP(ϕ)

dx + c

.

໰୊ 8.2 ( ମੵ੍໿͖ͭෳૉݻ༗஋࣮෦࠷খԽ໰୊ )

࣍Λຬͨ͢ Ω

(ϕ) ΛٻΊΑɽ

(ϕ,ϕ,sk

min

)∈D×C×S

{ f

₀

(ϕ, s

) | f

₁

(ϕ) ≤ 0, ໰୊ 8.1 }

105 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ

(a) ॳظܗঢ় (b) ࠷దܗঢ়

ਤ 8.4: 3 ࣍ݻ༗ৼಈϞʔυͷൺֱ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ϒϨʔΩ໐͖ݱ৅ʹର͢Δܗঢ়࠷దԽ

(a) ༗ݶཁૉϞσϧ

(a) ॳظܗঢ় (b) ࠷దܗঢ়

ਤ 8.3: ܗঢ়ൺֱ

107 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

M

Á( µ )º

º( µ ) M ( µ)

x

t

h

¡

p

b

D

»

¹

( »)

» x

x

¹

ਤ 9.2: Ϗʔυ෇͖ͷγΣϧϞσϧ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

§ 9 γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏ [5]

γΣϧߏ଄ʹઃ͚ΒΕͨখ͞ͳߴ͞ͷತԜ͸ϏʔυͱΑ͹ΕΔɽ

(a) ࣗಈंͷϗϫΠτϘσΟ (b) Ϗʔυߏ଄

ਤ 9.1: Ϗʔυ෇͖ͷγΣϧϞσϧ

109 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

ઃܭม਺ θ ͷઢܗۭؒͱڐ༰ू߹

X (θ) = {

θ ∈ H

(D; R ) � � θ = 0 on D ¯

} , D = X ∪ W

^1,∞

(D; R )

ঢ়ଶܾఆ໰୊ͷղ ( ঢ়ଶม਺ ) u ˆ ͷઢܗۭؒͱڐ༰ू߹

U = { ˆ

u ∈ H

(

M (θ) ; R

⁵

) �� u ˆ = 0

_R⁵

on Γ

(θ) } S = U ∩ W

^2,2q^R

(

Ω (θ) ; R

⁵

)

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

■ ^{ঢ়ଶܾఆ໰୊}

ॳظͷγΣϧʹର͢Δதཱ໘ͱྖҬ M

= {

µ

(ξ) ∈ R

� � ξ ∈ D } , Ω

= {

x + ξ

ν

(x) ∈ R

� � x ∈ M

, ξ

∈ ( − t/2, t/2) }

ઃܭม਺ θ : D → R ʹରͯ͠ɼϏʔυͷߴ͞Λ࣍ͷΑ͏ʹ͓͘ɽ ϕ (θ) = h

π tan

⁻¹

θ + h 2 มಈޙͷதཱ໘ͱྖҬ

M (θ) = {

(µ

+ ϕ (θ) ν

◦ µ

) (ξ) � � ξ ∈ D } Ω (θ) = {

x + ξ

₃

ν (x) ∈ R

� � x ∈ M (θ) , ξ

₃

∈ ( − t/2, t/2) } .

111 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

■ ^{ܗঢ়࠷దԽ໰୊}

f (θ, u) = ˆ

∫

M(θ)

( ¯ b · v

+ ¯ p

z − m ¯ · r ) dx +

∫

Γp(θ)∩∂M(θ)

( ¯ p

· v

+ ¯ p

z − m ¯ · r) dγ

໰୊ 9.2 ( ฏۉίϯϓϥΠΞϯε࠷খԽ໰୊ )

࣍Λຬͨ͢ θ ΛٻΊΑɽ

(θ,u)ˆ

min

∈D×S

{ f (θ, u) ˆ | ^໰୊ 9.1 }

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

໰୊ 9.1 (Mindlin-Reissner ͷ൘ཧ࿦ʹΑΔઢܗ஄ੑ໰୊ )

θ ∈ D ^{ʹରͯ͠ɼ࣍Λຬͨ͢} u ˆ ∈ S ^ΛٻΊΑɽ

−∇

^TM

S ˆ

( ˆ u) = ¯ b

−∇

· m ( ˆ u) = ¯ p

_N3

−∇

^TM

M ( ˆ u) + m ( ˆ u) = ¯ m

 



  in M (θ) , S ˆ

( ˆ u) ν

= ¯ p

m ( ˆ u) · ν

= ¯ p

_N3

M ( ˆ u) ν

= ¯ m

 



  on Γ

(θ) ∩ ∂M (θ) , S ˆ

( ˆ u) ν

= 0

_R²

m ( ˆ u) · ν

= 0 M ( ˆ u) ν

= 0

_R²

 



  on (

Γ

(θ) \ Γ ¯

(θ) )

∩ ∂M (θ) , u ˆ = 0

_R⁵

on Γ

(θ) .

113 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

0.0Mpa 4.0Mpa

(a) ࠷దܗঢ় (b) Mises Ԡྗ

ਤ 9.4: Ͷ͡ΓՙॏΛ͏͚Δย࣋ͪγΣϧͷ࠷దܗঢ়

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

γΣϧߏ଄ʹ͓͚ΔϏʔυͷઃܭ๏

■ ^਺஋ྫ

¡

p

0.0Mpa 7.0Mpa

(a) ڥք৚݅ (b) Mises Ԡྗ

ਤ 9.3: Ͷ͡ΓՙॏΛ͏͚Δย࣋ͪγΣϧͷॳظܗঢ়

115 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

■ ^{ঢ়ଶܾఆ໰୊}

ఆৗྲྀͷྲྀ଎ͱѹྗͷઢܗۭؒͱڐ༰ू߹

U = {

u ∈ H

(

R

; R

) �� u = 0

_R^d

on ∂Ω (ϕ) } , U (u

) = {

u ∈ H

(

R

; R

) �� u = u

on ∂Ω (ϕ) } , S = U ∩ W

^1,^∞

(

R

; R

) , S (u

) = U (u

) ∩ W

^1,∞

(

R

; R

) , P =

{

q ∈ L

(

R

; R ) � � � �

∫

Ω(ϕ)

q dx = 0 }

, Q = P ∩ L

^∞

(

R

; R )

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

§ 10 ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊ [6]

அ໘ੵ͕֊ஈঢ়ʹมԽ͢Δ Poiseuille ྲྀΕ৔ʹ͓͍ͯɼఆৗྲྀ͔Βඇఆৗྲྀʹ ਪҠ͢Δྟք Reynolds ਺ Re

͸ 40 ͋ͨΓͰ͋Δͱใࠂ͞Ε͍ͯΔɽ

ਤ 10.1: ஈ෇͖ Poiseuille ྲྀΕ৔

117 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

͞Βʹɼ u ͱ p ͷ͔͘ཚΛ x ∈ Ω (ϕ) ͱ τ ∈ [0, ∞ ) ʹରͯ͠ɼ

u (τ, x) = u (0, x) + e

^sτ

u ˆ (x) + e

^s^c^τ

u ˆ

(x) = u (0, x) + 2Re [e

^sτ

u ˆ (x)] , p (τ, x) = p (0, x) + 2Re [e

^sτ

p ˆ (x)]

ͱԾఆ͢Δɽͨͩ͠ɼ s ∈ C ^ɼ ( · )

͸ෳૉڞ໾Λද͢ɽ u ˆ ͱ p ˆ ʹର͢Δઢܗۭؒ

ͱڐ༰ू߹Λ࣍ͷΑ͏ʹ͓͘ɽ

U ˆ = { ˆ

u ∈ H

(

R

; C

) �� u ˆ = 0

_Rd

on ∂Ω (ϕ) } , S ˆ = ˆ U ∩ W

^1,^∞

(

R

; C

) , P ˆ =

{ ˆ q ∈ L

(

R

; C ) �

� �

�

∫

Ω(ϕ)

ˆ

q dx = 0 }

, Q ˆ = ˆ P ∩ L

^∞

(

R

; C )

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

໰୊ 10.1 ( ఆৗ Navier-Stokes ໰୊ )

ϕ ∈ D ^ʹରͯ͠ b, u

, µ ͓Αͼ ρ ͕༩͑ΒΕͨͱ͖ɼ ρ (u · ∇ ) u

− ∇

(

µ ∇ u

)

+ ∇

p = b

in Ω (ϕ) ,

∇ · u = 0 in Ω (ϕ) , u = u

on ∂Ω (ϕ) ,

∫

Ω(ϕ)

p dx = 0

Λຬͨ͢ (u, p) ∈ S (u

) × Q ^ΛٻΊΑɽ

119 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

■ ^{ܗঢ়࠷దԽ໰୊}

ධՁؔ਺Λ͔͘ཚݻ༗஋࣮෦ f

(s

) = s

+ s

^c_r

= 2Re [s

]

ͱ͓͘ɽͨͩ͠ɼ r ͸ Re[s

] ͕࠷େͱͳΔϞʔυ࣍਺ͱ͢Δɽ

໰୊ 10.3 ( ͔͘ཚݻ༗஋࣮෦ͷ࠷খԽ໰୊ )

࣍Λຬͨ͢ Ω (ϕ) ΛٻΊΑɽ min

(ϕ,u,p,s˜ r,ˆur,ˆpr)∈D×S×Q×C×S×ˆ Qˆ

{ f

(s

) � � _໰୊ 10.1, ໰୊ 10.2 }

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

໰୊ 10.2 ( ͔͘ཚݻ༗஋໰୊ )

ϕ ∈ D ^ʹରͯ͠ (u, p) ͕༩͑ΒΕͨͱ͖ɼ i ∈ { 1, 2, · · · } ^ʹରͯ͠

ρs u ˆ

^T_i

+ ρ (u · ∇ ) ˆ u

^T_r

+ ρ ( ˆ u

· ∇ ) u

− ∇

(

µ ∇ u ˆ

^T_i

)

+ ∇

p ˆ = 0

^T_Rd

in Ω (ϕ) ,

∇ · u ˆ

= 0 in Ω (ϕ) , ˆ

u

= 0

_Rd

on ∂Ω (ϕ) ,

∫

Ω(ϕ)

ˆ

p dx = 0,

∫

Ω(ϕ)

ρ u ˆ

· u ˆ

^c_i

dx = 1

Λຬͨ͢ s

∈ C ^ͱ ( ˆ u

, p ˆ

) ∈ S × ˆ Q ˆ ^ΛٻΊΑɽ

121 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

¡0.006

¡0.004

¡0.002 0 0.002 0.004 0.006 0.008

35 40 45 50 55

Real part of the eigenvalue

Reynolds number Re Initial domain

Optimized domain

ਤ 10.3: Reynolds ਺ʹର͢Δݻ༗஋ͷ࣮෦࠷େ஋ͷਪҠ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ྲྀΕ৔ͷ҆ఆੑ޲্ͷͨΊͷܗঢ়࠷దԽ໰୊

■ ^਺஋ྫ

(a) ॳظܗঢ়

(b) ࠷దܗঢ়

ਤ 10.2: ஈ෇͖ Poiseuille ྲྀΕ৔ ( ॳظྖҬʹ͓͍ͯ Re = 45)

123 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

■ ^{ঢ়ଶܾఆ໰୊}

ઃܭม਺ ϕ ͷઢܗۭؒͱڐ༰ू߹

X = {

ϕ ∈ H

(

D

; R

) �� ϕ = 0

_R^d

on ∂D

∪ ∂F

∪ ∂G

∪ Γ

∪ Ω ¯

} , D = {

ϕ ∈ X ∩ W

^1,^∞

(

R

; R

) �� _{શ୯ࣹͷ৚݅} } .

੩ిϙςϯγϟϧͷઢܗۭؒͱڐ༰ू߹

U = {

u ∈ H

(D

; R ) � � u = 0 on ∂E (ϕ) ∪ ∂G (ϕ) } , U (u

) = {

u ∈ H

(D

₀

; R ) �

� u = u

on ∂E (ϕ) ∪ ∂G (ϕ) } , S = U ∩ W

^1,^∞

(D

₀

; R ) ,

S (u

) = U (u

) ∩ W

^1,∞

(D

; R ) ,

ͱ͓͘ɽͨͩ͠ɼ u

∈ H

(D

; R ) ∩ W

^1,∞

(D

; R ) ͸ ∂E (ϕ) ্Ͱ u = α ɼ

∂G (ϕ) ্Ͱ u = 0 Λຬͨؔ͢਺ͱ͢Δɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

§ 11 ੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊ [7]

੩ి༰ྔࣜηϯα͸ࢦͷݕग़ͳͲʹ࢖ΘΕΔɽ

G0 G0 G0

∂D0

Ω₀

ਤ 11.1: ॳظ੩ి৔ Ω

= D

\ ( E ¯

∪ G ¯

)

125 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

໰୊ 11.2 ( ݕग़ମ͕͋Δ੩ి৔ )

ϕ ∈ D ^ͱਖ਼ఆ਺ α ͕༩͑ΒΕͨͱ͖ɼ

− ∇ · e (u

) = 0 in Ω (ϕ) \ F

= D

\ (E (ϕ) ∪ G (ϕ) ∪ F

) ,

∂

u

= 0 on ∂D

(ϕ) , u

= α on ∂E (ϕ) , u

= 0 on ∂G (ϕ) ∪ F ¯

Λຬͨ͢ u

∈ S (u

) ΛٻΊΑɽ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

໰୊ 11.1 ( جຊ੩ి৔ )

ϕ ∈ D ^{ʹରͯ͠ɼਖ਼ఆ਺} α ͕༩͑ΒΕͨͱ͖ɼ

− ∇ · e (u) = 0 in Ω (ϕ) = D

₀

\ (E (ϕ) ∪ G (ϕ)) ,

∂

u = 0 on ∂D

(ϕ) , u = α on ∂E (ϕ) , u = 0 on ∂G (ϕ)

Λຬͨ͢ , u ∈ S (u

) ΛٻΊΑɽ

127 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

■ ^਺஋ྫ

G0 0

@D0

¡C0

(a) ໰୊ઃఆ (b) ༗ݶཁૉϞσϧ

ਤ 11.2: ಥ͖ग़ͨ઀஍ిۃͷ͋Δ 2 ࣍ݩ੩ి৔

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

■ ^{ܗঢ়࠷దԽ໰୊}

f

(ϕ, u, u

) = − ∥ u − u

∥

²_H¹_(Ω(ϕ)\F₀_(ϕ);_R₎

= −

∫

Ω(ϕ)\F₀

{ (u − u

)

+ (e (u) − e (u

)) · (e (u) − e (u

)) } dx, f

(ϕ) =

∫

E(ϕ)

dx − s

໰୊ 11.3 ( ମੵ੍໿͖ͭ੩ి৔ؒͷޡࠩϊϧϜ࠷খԽ໰୊ )

࣍Λຬͨ͢ Ω (ϕ) ΛٻΊΑɽ

(ϕ,u,uF)∈D×S

min

(uD)×S(uD)

{ f

₀

(ϕ, u, u

) | f

₁

(ϕ) ≤ 0, ໰୊ 11.1, ໰୊ 11.2 }

129 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

0 2 4 6 8 10 12

0.6 0.7 0.8 0.9 1 1.1

Costfunctions

Number of Iteration

1+f

/s

2+f

/jf

0 init

j

0 2 4 6 8 10 12

0 20 40 60 80

Sencing capacitance [pF/m] Increasing rate of capacitance

Number of Iteration

0 0.1 0.2 0.3 0.4

with finger without finger increasing rate

(a) ධՁؔ਺ (b) ੑೳ

ਤ 11.4: ܗঢ়มಈʹର͢Δ܁ฦཤྺ

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

(a) ॳظ (b) ࠷దԽޙ ( ະऩଋ )

ਤ 11.3: ྖҬมಈલޙͷܗঢ়

131 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

(a) ॳظ (b) ࠷దԽޙ ( ऩଋ )

ਤ 11.6: ྖҬมಈલޙͷܗঢ় (E (ϕ) ⊂ D

\ Ω ¯

)

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

@D0

D0n C0 0

(a) ໰୊ઃఆ (b) ༗ݶཁૉϞσϧ

ਤ 11.5: ಥ͖ग़ͨ઀஍ిۃͷ͋Δ 2 ࣍ݩ੩ి৔ (E (ϕ) ⊂ D

\ Ω ¯

)

133 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

FreeFEM++Λ༻͍࣮ͨश

12 FreeFEM++ Λ༻͍࣮ͨश

H

ޯ഑๏ / H

Newton ๏

2 ࣍ݩઢܗ஄ੑମ ( ฏۉίϯϓϥΠΞϯε࠷খԽ໰୊ , hook):

9.11.5_shape_elastic/2d-hook/domain_integral/grad/

9.11.5_shape_elastic/2d-hook/domain_integral/Newton/

༗ݶཁૉͷ࣍਺ : 1 ࣍ཁૉ / 2 ࣍ཁૉ

3 ࣍ݩઢܗ஄ੑମ ( ฏۉίϯϓϥΠΞϯε࠷খԽ໰୊ , cantilever):

9.11.5_shape_elastic/3d-cantilever/boundary_integral/grad/

main.edp ϑΝΠϧΛςΩετΤσΟλͰ։͖ɼ

fespace Vh(Th,[P2,P2,P2]);//Finite element space Λ

fespace Vh(Th,[P1,P1,P1]);//Finite element space ʹมߋ͢Δɽ

ಛҟ఺ͷڍಈ : ڥքੵ෼ܕܗঢ়ඍ෼ / ྖҬੵ෼ܕܗঢ়ඍ෼

2 ࣍ݩઢܗ஄ੑମ ( ฏۉίϯϓϥΠΞϯε࠷খԽ໰୊ , L-shape):

9.11.5_shape_elastic/2d-L-shape/boundary_integral/grad/

9.11.5_shape_elastic/2d-L-shape/domain_integral/grad/

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

੩ి༰ྔࣜηϯαͷܗঢ়࠷దԽ໰୊

0 5 10 15 20

0.6 0.7 0.8 0.9 1 1.1

Costfunctions

Number of Iteration

1+f

/s

2+f

/jf

0 init

j

0 5 10 15 20

0 20 40 60 80

0 0.1 0.2 0.3 0.4

Sencing capacitance [pF/m] Increasing rate of capacitance

Number of Iteration with finger without finger increasing rate

(a) ධՁؔ਺ (b) ੑೳ

ਤ 11.7: ܗঢ়มಈʹର͢Δ܁ฦཤྺ (E (ϕ) ⊂ D

\ Ω ¯

)

135 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ࢀߟจݙ

[1] ൞্ ल޾ . ܗঢ়࠷దԽ໰୊ .

৿๺ग़൛ , ౦ژ , 10 2016.

[2] H. Azegami.

Second derivatives of cost functions and h1 newton method in shape optimization problems.

In Patrick van Meurs, Masato Kimura, and Hirofumi Notsu, editors, Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Proceedings of the International Conference CoMFoS16, Mathematics for Industry 30, pages 61–72.

Springer Singapore, 12 2017.

[3] H. Azegami, L. Zhou, K. Umemura, and N. Kondo.

Shape optimization for a link mechanism.

Structural and Multidisciplinary Optimization, 48(1):115–125, 2 2013.

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

·ͱΊ

13 ·ͱΊ

࠷దઃܭ໰୊͸ɼঢ়ଶܾఆ໰୊͕౳੍ࣜ໿ͱͯ͠ՃΘͬͨෆ౳͖ࣜͭඇઢܗ

࠷దԽ໰୊ͷΫϥεͱΈͳ͢͜ͱ͕Ͱ͖Δɽͦͷղ๏͸ޯ഑๏΍ Newton

๏ʹج͍ͮͯߏ੒͞ΕΔɽ

࿈ଓମͷܗঢ়࠷దԽ໰୊΁ͷ֦ு͸ؔ਺ղੳʹΑ࣮ͬͯݱ͞ΕΔɽ

੡඼ઃܭʹ໾ཱͭΑ͏ͳ໰୊Λద੾ʹߏ੒Ͱ͖Ε͹ɼͦΕΒΛղ͘͜ͱ͸Մ

ೳͱͳ͍ͬͯΔɽ

137 / 140

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ࢀߟจݙ

ࢀߟจݙ (cnt.)

[7] Masayoshi Satake, Noboru Maeda, Shinji Fukui, and Hideyuki Azegami.

Shape optimization of electrostatic capacitive sensor.

In Proceedings of the 10th World Congress on Structural and Multidisciplinary Optimization (WCSMO-10) (USB), pages 1–10, 5 2013.

ܗঢ়࠷దԽཧ࿦ͱ੡඼ઃܭ΁ͷԠ༻

ࢀߟจݙ

ࢀߟจݙ (cnt.)

[4] K. Shintani and H. Azegami.

Shape optimization for suppressing brake squeal.

Structural and Multidisciplinary Optimization, 50(6):1127–1135, 5 2014.

[5] K. Shintani and H. Azegami.

A design method of beads in shell structure using non-parametric shape optimization method.

In Proceedings of the the ASME 2014 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETCCIE 2014) (eBook), pages 1–8, 8 2014.

[6] T. Nakazawa and H. Azegami.

Shape optimization of ﬂow ﬁeld improving hydrodynamic stability.

Japan Journal of Industrial and Applied Mathematics Japan Journal of Industrial and Applied Mathematics, 10 2015.

Accepted.

139 / 140

MI レクチャーノートシリーズ刊行にあたり

　本レクチャーノートシリーズは、文部科学省 21 世紀 COE プログラム「機能数理学の構築と展開」（H.15-19 年度）において作成した COE Lecture Notes の続刊であり、文部科学省大学院教育改革支援プログラム「産業界が求める数学博士と新修士養成」（H19-21 年度）および、同グローバル COE プログラム「マス・フォア・インダストリ教育研究拠点」（H.20-24 年度）において行われた講義の講義録として出版されてきた。平成 23 年 4 月のマス・フォア・

インダストリ研究所（IMI）設立と平成 25 年 4 月の IMI の文部科学省共同利用・

共同研究拠点として「産業数学の先進的・基礎的共同研究拠点」の認定を受け、

今後、レクチャーノートは、マス・フォア・インダストリに関わる国内外の研究者による講義の講義録、会議録等として出版し、マス・フォア・インダストリの本格的な展開に資するものとする。

平成 26 年 10 月

ドキュメント内 Vol.79 MI Lecture Notes|九州大学マス・フォア・インダストリ研究所 math 5acd67e59a821 (ページ 48-108)

ʹର͢Δ Lagrange ؔ਺Λ

□von Neumann最大最小定理

課題 2 解答

2.6 ʹର͢Δ Lagrange ؔ਺Λ

L (a, u, v) = f (u) + L

(a, u, v) = p · u + v · ( − K (a) u + p)

ͱ͓͘ɽ v ∈ U ͸ Lagrange ৐਺Ͱ͋Δɽ L ͷશඍ෼ʹରͯ͠ɼ

L

(a, u, v) [b, u, ˆ v] ˆ

= − {

v ·

( ∂K (a)

∂a

u ∂K (a)

∂a

u )}

b

+ p · u ˆ − v · K (a) ˆ u (= 0 ⇐ K

(a) v = p ) + ˆ v · ( − K (a) u + p) (= 0 ⇐ K (a) u = p )

= g · b ∀ (b, u, ˆ v) ˆ ∈ Ξ × U × U (2.6)

͕੒Γཱͭɽ͜͜Ͱɼ g ͸୅ೖ๏ʹΑΔࣜ (2.4) ͱҰக͢Δɽ

͞ΒʹɼͦΕΛ΋͏Ұ౓ a Ͱภඍ෼͢Ε͹ɼ

H =



 



∂

f ˜

∂a

∂a

∂

f ˜

∂a

∂a

∂

f ˜

∂a

∂a

∂

f ˜

∂a

∂a



 

 = l e



 



2 (p

+ p

)

a

0

0 2p

a



 

 (2.5)

ͱͳΔɽ a

, a

> 0 ͷͱ͖ɼ H ͸ਖ਼ఆ஋ͱͳΔɽ

Ұํɼ j ∈ { 1, 2 } ʹରͯ͠

L

(a, u, v) [b

, υ ˆ

] = v · {− (K

(a) [b

]) u − K (a) ( ˆ υ

) } = 0

∀ (b

, υ ˆ

) ∈ T

(a, u) ΑΓɼ࣍ΛಘΔɽ

υ ˆ

= − K

(a) (K

(a) [b

]) =



 

Ұํɼ j ∈ { 1, 2 } ^ʹରͯ͠