࠻ࡐࡠࠫ࡯ᦨㆡൻᴺߦࠃࠆᦨㆡ⸳⸘⸘▚— ᵹ૕㑐ㅪಽ㊁ߢߩᔕ↪ߩ⹜ߺㄭ⮮ ⛮↵(ᩣ) ⼾↰ਛᄩ⎇ⓥᚲ

1

ߪߓ߼ߦ

CAE (Computer-Aided Engineering) ߣ޿߁⸒⪲߇ᐢߊቯ⌕ߒߡ޿ࠆߎߣ߇‛⺆ࠆࠃ߁ߦ‫ޔ‬᭽‫ࡁࡕߥޘ‬ߠߊࠅߩߥ߆ߢ‫ࡦ࡚ࠪ࡯࡟ࡘࡒࠪޔ‬ᛛⴚߪਇนᰳߩ߽ߩߣߥߞߡ޿ࠆ‫⥄ޕ‬േゞߦ߆߆ࠊࠆࡕࡁߠߊࠅߦ߅޿
ߡ߽‫ޔ‬᭴ㅧ࡮ᒝᐲ‫⓭ⴣޔ‬቟ో‫ޔ‬ᾲ࡮ᵹ૕ߣ޿ߞߚᯏ᪾♽ߩᛛⴚ㗔ၞࠍߪߓ߼ߣߒߡ‫ޔࠬࠢ࠾ࡠ࠻ࠢ࡟ࠛޔ‬ᖱႎ‫᧚ޔ‬ᢱߥߤࠍ฽߻޽ࠄࠁࠆᛛⴚಽ㊁ߢ‫ࡦ࡚ࠪ࡯࡟ࡘࡒࠪޔ‬ᛛⴚ߇ᵴ↪ߐࠇߡ޿ࠆ‫ߡߒߘޕ‬ᢙቇߪ‫ࠃߩߎޔ‬߁ߥࠪࡒࡘ࡟࡯࡚ࠪࡦᛛⴚࠍၮ␆߆ࠄᡰ߃ࠆ‫ޕ‬ࡕࡁߠߊࠅߦ߅ߌࠆ⸳⸘ߣߪ‫ޔ‬ᄢ㔀ᛠߦ⸒߃߫‫ޔ‬ᚲᦸߩᕈ⢻ࠍਈ߃ࠆࠃ߁ߦ⸳⸘ᄌᢙ (ౖဳ⊛ߦߪࡕࡁߩ
ᒻߔߥࠊߜ⸳⸘ኻ⽎ߩᒻ⁁߿ኸᴺ) ࠍቯ߼ࠆߎߣߢ޽ࠆ‫ࠆࠇࠄ޿↪ߢߎߘޕ‬ᚻᴺߪ‫߽߹޿ޔ‬ᄙߊߩ႐ว‫⹜ޔ‬૞ߣᕈ⢻⹏ଔࠍ➅ࠅ㄰ߔ⹜ⴕ㍲⺋⊛ߥ߽ߩߢ޽ࠆ‫ࡦ࡚ࠪ࡯࡟ࡘࡒࠪߢ߆ߥߩߘޕ‬ᛛⴚߪ‫ޔ‬ታ㛎ߣߣ߽ߦ (޽ࠆ޿ߪߘߩઍࠅߦ) ਥߦᕈ⢻⹏ଔߩᚻᲑߣߒߡ೑↪ߐࠇ‫ߩ↪⾌ޔ‬೥ᷫ߿ᤨ㑆ߩ⍴❗‫޿⦟ࠅࠃޔ‬ᕈ⢻ߩታ⃻ߥߤࠍㅢߓ‫ߦࠅߊߠࡁࡕޔ‬ᄙᄢߩ⽸₂ࠍߒߡ޿ࠆ‫ޕ‬ᦨㄭߢߪන৻ߩᛛⴚ㗔ၞߦߣߤ߹ࠄߕ‫ޟ߫߃଀ޔ‬᭴ㅧ࡮ᒝᐲ + ᾲ࡮ᵹ૕ + 㔚⏛᳇‫ⶄޔߤߥޠ‬ᢙߩ‛ℂ⃻
⽎ࠍหᤨߦᛒ߁ࡑ࡞࠴ࡈࠖࠫࠢࠬߩࠪࡒࡘ࡟࡯࡚ࠪࡦᛛⴚ (MP Sim) ߽㐿ᜏߐࠇ‫ޔ‬ㆡ↪ߢ߈ࠆ໧㗴ߩ▸࿐߇ᐢ߇ࠆߣߣ߽ߦ‫⃻ࠅࠃޔ‬ታߦㄭ޿໧㗴⸳ቯߩਅߢߩᕈ⢻⹏ଔ߇น⢻ߣߥߞߡ߈ߚ‫ޕ‬ട߃ߡ‫‛߫߃଀ޔ‬ᕈ߇ਇ᣿ߩ᧚ᢱࠍᛒ߁႐วߥߤ‫᧚ࠅࠃߦࡦ࡚ࠪ࡯࡟ࡘࡒࠪߩߢ࡞࡯ࠤࠬࡠࠢࡒޔ‬ᢱߩ․ᕈ (‛ᕈ) ࠍ᳞߼ߚ߁߃
ߢ‫ߢߎߘޔ‬ᓧࠄࠇߚ‛ᕈࠍ↪޿ߚࡑࠢࡠࠬࠤ࡯࡞ߢߩࠪࡒࡘ࡟࡯࡚ࠪࡦߦࠃࠅ⸳⸘ኻ⽎ߩᕈ⢻ࠍ⹏ଔߔࠆߣ
޿߁‫ࡦ࡚ࠪ࡯࡟ࡘࡒࠪߩ࡞࡯ࠤࠬ࠴࡞ࡑޔ‬ᛛⴚ (MS Sim) ߽↢߹ࠇߟߟ޽ࠆ‫ޕ‬ߎߩࠃ߁ߦ‫ࡦ࡚ࠪ࡯࡟ࡘࡒࠪޔ‬ᛛⴚߪࡕࡁߠߊࠅߩߥ߆ߢᣢߦᄢ߈ߥᓎഀࠍᜂ޿‫߽߹޿ߚ߹ޔ‬ㅴᱠࠍ⛯ߌߡ޿ࠆ‫੹ޕ‬ᓟߎࠇ߇ߐࠄߦ⊒ዷߒ‫ޔ‬ᚲᦸߩᕈ⢻ࠍታ⃻ߔࠆ⸳⸘ᄌᢙ (ࡕࡁߩᒻ) ࠍ⹜ⴕ㍲⺋ߦࠃࠆߎߣߥߊ
᳞߼ࠆߎߣ߇ߢ߈ࠆࠃ߁ߦߥࠇ߫‫ࡦ࡚ࠪ࡯࡟ࡘࡒࠪޔ‬ਥዉߩࡕࡁߠߊࠅ߇ታ⃻ߔࠆߎߣߦߥࠆ (࿑ 1)‫ߎߎޕ‬ߢᔅⷐߣߥࠆߩ߇ᦨㆡൻ (Opt) ߩᣇᴺ⺰ߢ޽ࠆ‫ᦨޕ‬ㆡൻࠍ฽߻ࠪࡒࡘ࡟࡯࡚ࠪࡦᛛⴚߦ㑐ߔࠆ⎇ⓥߪ‫ߘޔ‬ߩขࠅ⚵ߺో૕ߣߒߡߪ᳿ߒߡᣂߒ޿߽ߩߢߪߥ޿߇‫ޔ‬᭽‫ߦࠅߊߠࡁࡕߥޘ‬ㆡ↪ߢ߈ࠆ᳢↪⊛ߥᛛⴚߦ߹
ߢߪ⥋ߞߡ޿ߥ޿‫ᦨߢ߆ߥߩߘޕ‬ㄭ‫ᦨ࡯ࠫࡠࡐ࠻ޔ‬ㆡൻᴺߥߤߩᢙℂ⊛ᦨㆡൻᚻᴺߦၮߠߊࠪࡒࡘ࡟࡯࡚ࠪ
ࡦᛛⴚߦታ↪ߩน⢻ᕈ߇㜞߹ߞߡ߈ߚ‫ߛ߹ࠈߎߣߩ߹޿ޕ‬ㆡ↪ߢ߈ࠆ໧㗴ߩ▸࿐ߪ㒢ࠄࠇࠆ߇‫ߩߘޔ‬዁᧪ߦߪᄢ߈ߥᦼᓙ߇ᜬߚࠇࠆ‫ᧄޕ‬Ⓜߢߪ‫╩ޔ‬⠪ߩ๟ㄝߢᦨㄭⴕߞߚ‫ޔ‬ᵹ૕㑐ㅪಽ㊁ߢߩ࠻ࡐࡠࠫ࡯ᦨㆡൻᴺߩᔕ
↪ߩ⹜ߺࠍ⚫੺ߔࠆ‫ᧄ߅ߥޕ‬Ⓜߩౝኈߪ‫ޔ‬2010 ᐕ 9 ᦬ߦ㐿௅ߐࠇߚᣣᧄᢙቇળ⑺ቄ✚วಽ⑼ળߦ߅޿ߡ․೎᜗ᓙ⻠Ṷߣߒߡ⻠Ṷߒߚౝኈߩਥⷐㇱಽࠍ߹ߣ߼ߚ߽ߩߢ޽ࠆ‫ޕ‬

2࠻ࡐࡠࠫ࡯ᦨㆡൻᴺߩᵹ૕㑐ㅪ໧㗴߳ߩᔕ↪— ૐ࡟ࠗࡁ࡞࠭ᢙߩᵹࠇ
ߩߥ߆ߢ᛫ജ߇ᦨዊ޽ࠆ޿ߪ឴ജ߇ᦨᄢߣߥࠆ‛૕ᒻ⁁ߩត⚝‛૕ߩᒻ⁁ࠍߘߩ࠻ࡐࡠࠫ࡯ (ⓣߩ᦭ή߿㗔ၞߩㅪ⚿ᕈߥߤ) ߽฽߼ߡᦨㆡൻߔࠆߎߣ߇น⢻ߥ࠻ࡐࡠࠫ࡯

ᦨㆡൻᴺߪ‫ ޔ‬Bendsøe and Kikuchi [1] ߦࠃࠆవ㚟⊛ߥ⎇ⓥߦ┵ࠍ⊒ߔࠆ‫ߩߘޕ‬ᓟߎߩᣇᴺߪ⪺ߒ޿⊒ዷࠍㆀߍ‫ޔ‬᭴ㅧ࡮ᒝᐲ⸳⸘ߩಽ㊁ߢߪᣢߦ᏷ᐢߊᵴ↪ߐࠇߡ޿ࠆ߶߆‫ޔ‬વᾲ‫ޔ‬㖸㗀‫ޔ‬㔚⏛᳇ߥߤߩಽ㊁ߦ߽ߘߩᔕ↪߇ᐢ߇ߞߡ޿ࠆ [2]‫ߢ߆ߥߩߘޕ‬ᵹ૕㑐ㅪಽ㊁߳ߩዷ㐿ߪߥ߆ߥ߆ㅴ߹ߥ߆ߞߚ߇‫ޔ‬Ყセ⊛ᦨㄭߦߥߞ
ߡ‫ޔ‬Borrvall and Petersson [3] ߦࠃࠅߎߩಽ㊁ߩ໧㗴ࠍᛒ߁ᣇᴺ߇㐿ᜏߐࠇߚ‫ࠍࠇߎߡߒߘޕ‬ᄾᯏߣߒߡ‫ޔ‬

Stokes ᵹ߿‫࠭࡞ࡁࠗ࡟ޔ‬ᢙ߇ 100 ⒟ᐲએਅߩ㕖✢ᒻ☼ᕈᵹ (Navier-Stokes ᣇ⒟ᑼ) ࠍኻ⽎ߦ‫ޔ‬ᵹ〝ߩო㕙ᒻ⁁߿ᵹ૕ਛߩ‛૕ᒻ⁁ࠍᦨㆡ⸳⸘ߔࠆ໧㗴߳ߩᔕ↪߇⹜ߺࠄࠇࠆࠃ߁ߦߥߞߚ‫ߎߣߩߢ߹ࠇߎޔߒߛߚޕ‬ࠈ‫࠭࡞ࡁࠗ࡟ޔ‬ᢙ߇ 100 ⒟ᐲࠍ⿥߃ࠆ㕖✢ᒻ☼ᕈᵹࠍᛒߞߚ⎇ⓥ߿‫ޔ‬㕖✢ᒻ☼ᕈᵹਛߢߩ឴ജߩᦨᄢൻࠍ
⹜ߺߚ⎇ⓥߪ‫ޕ޿ߥࠇࠄ⷗ߤࠎߣ߶ޔ‬࿑ 1: ᢙℂᦨㆡ⸳⸘ࠪࠬ࠹ࡓ— MP/MS Sim + Opt ߦࠃࠆࡕࡁߠߊࠅ

એਅߢߪ‫ᦨ࡯ࠫࡠࡐ࠻ޔ‬ㆡൻᴺߩᵹ૕㑐ㅪ໧㗴߳ߩᔕ↪ߦ㑐ߒ‫ޔ‬໧㗴ߩቯᑼൻᴺࠍ◲නߦ⺑᣿ߔࠆߣߣ߽
ߦ‫ߩߎޔ‬ᣇᴺࠍ↪޿ߡૐ࡟ࠗࡁ࡞࠭ᢙߩᵹࠇߩߥ߆ߢ᛫ജ߇ᦨዊ޽ࠆ޿ߪ឴ജ߇ᦨᄢߣߥࠆੑᰴర޽ࠆ޿ߪ
ゲኻ⒓ߩ‛૕ᒻ⁁ࠍត⚝ߒߚ⚿ᨐࠍ␜ߔ‫ޕ‬ੑᰴర໧㗴ߦ㑐ߒߡߪ‫࠭࡞ࡁࠗ࡟ޔ‬ᢙ߇ 2,000 એਅߩᵹࠇࠍኻ⽎ߦ‫ޔ‬ᢿ㕙Ⓧ৻ቯߩ೙⚂᧦ઙߩਅߢ᛫ജ߇ᦨዊߣߥࠆੑᰴరᩇ⁁‛૕ߩᢿ㕙ᒻ⁁ࠍ᳞߼ߚ⚿ᨐߣ‫࠭࡞ࡁࠗ࡟ޔ‬ᢙ߇ 10 ߩᵹࠇࠍኻ⽎ߦ‫ޔ‬ᢿ㕙Ⓧ৻ቯߥࠄ߮ߦ᛫ജ਄㒢ߩ೙⚂᧦ઙߩਅߢ឴ജ߇ᦨᄢߣߥࠆੑᰴరᩇ⁁‛૕ߩᢿ㕙ᒻ⁁ࠍ᳞߼ߚ⚿ᨐࠍ⚫੺ߔࠆ‫ߚ߹ޕ‬ゲኻ⒓໧㗴ߦߟ޿ߡߪ‫࠭࡞ࡁࠗ࡟ࠅߪ߿ޔ‬ᢙ߇ 2,000 એਅߩᵹࠇ
ࠍኻ⽎ߦ‫ޔ‬૕Ⓧ৻ቯߩ೙⚂᧦ઙߩਅߢ᛫ജ߇ᦨዊߣߥࠆゲኻ⒓‛૕ߩሶඦ㕙ᒻ⁁ࠍ᳞߼ߚ⚿ᨐߣ‫ޔ‬Stokes ᵹਛߢ᛫ജ߇ᦨዊߣߥࠆゲኻ⒓‛૕ߩሶඦ㕙ᒻ⁁ࠍ᳞߼ߚ⚿ᨐࠍ⚫੺ߔࠆ‫ߪߡ޿ߟߦ⚦⹦ޕ‬ᢥ₂ [4]‫[ޔ‬5] ࠍෳᾖߐࠇߚ޿‫ᧄޔ߅ߥޕ‬Ⓜߢ⚫੺ߔࠆ⸘▚ߦߪ‫᦭ޔ‬㒢ⷐ⚛ᴺߦࠃࠆ᳢↪࠰ࡈ࠻࠙ࠚࠕ COMSOL ࠍ೑↪ߒߚ‫ޕ‬

2.1

ၮ␆ᣇ⒟ᑼ

‛૕ᒻ⁁ߩ⴫⃻㗔ၞ Ω ౝߩછᗧߩ૏⟎ 
x ߇ᵹ૕ߢභ߼ࠄࠇࠆ߆ุ߆ࠍ‫ޔ‬ᰴߦ⸥ߔࠃ߁ߥ㑐ᢙ γ(x) (ᵹ૕ಽ₸޽ࠆ޿ߪήᰴరᵹ૕ኒᐲߣ⸃㉼ߔࠆߎߣ߇ߢ߈ࠆ) ߢ⴫ߔ‫ ߡߒߘޕ‬γ(x) ߩⓨ㑆ಽᏓߦࠃࠅ‫ޔ‬ᵹ૕߇ሽ࿷ߔࠆ㗔ၞߩᒻࠍ⴫ߔߣߣ߽ߦ‫ߩߘޔ‬Ⴚ⇇ߣߥࠆო㕙߿‛૕⴫㕙ߩᒻ⁁ࠍ޽ࠊߖߡ⴫⃻ߔࠆ‫ᦨޕ‬ㆡൻ໧㗴ߩቯᑼൻࠍߔࠆ㓙‫ߩߎޔ‬㑐ᢙ γ(x) ߇⸳⸘ᄌᢙ (⸳⸘਄ᄌᦝߒᓧࠆ㊂) ߣߥࠆ‫ޕ‬γ(x) = 0
0 < γ(x) < 1γ(x) = 1
⾰㊂ߥࠄ߮ߦㆇേ㊂ߩ଻ሽೣ⎫࿕૕ㇱಽ ⎪
⎬
ਛ㑆ㇱಽ
ᵹ૕ㇱಽ

⎪
⎭

(1)

ቯᏱ‫ޔ‬㕖࿶❗ᕈߩ☼ᕈᵹ૕ (࠾ࡘ࡯࠻ࡦᵹ૕) ߩᵹࠇߪ‫ޔ‬ㅦᐲࡌࠢ࠻࡞ u(x)ߣ࿶ജ p(x) ࠍၮᧄᄌᢙߣߔࠆᰴߩࠃ߁ߥ⾰㊂ߥࠄ߮ߦㆇേ㊂ߩ଻ሽೣߦࠃࠅ⸥ㅀߐࠇࠆ‫ޕ‬ · u = 0∇ 2 u + f(x) · uuT
 · σ̃ + f(x) = −∇p + 1 ∇
∇
= ∇
Re

(2)
(3)

ߚߛߒߔߴߡߩᄌᢙߪᵹࠇ႐ߩઍ⴫㐳ߐ L ߣઍ⴫ㅦᐲ U ߥࠄ߮ߦᵹ૕ߩኒᐲ ρ ࠍ↪޿ߡήᰴరൻߐࠇߡ޿
ࠆ‫ޔ߅ߥޕ‬σ̃ ߪᵹ૕ਛߩᔕജ (࠹ࡦ࠰࡞)‫ ߚ߹ޔ‬f(x) ߪᵹࠇߩߥ߆ߦሽ࿷ߔࠆ‛૕ߩᓇ㗀ࠍ⴫ߔ૕Ⓧജ (ࡌࠢ࠻࡞) ߢ‫ߩߘޔ‬ᗧ๧ߣౕ૕⊛ߥᢙᑼ⴫⃻ߪߎߩ޽ߣߦ⺑᣿ߔࠆ‫ޕ‬
᭴ᚑᣇ⒟ᑼ

ᵹ૕ਛߩᔕജ σ̃ ߪ╬ᣇ⊛ߥ࿶ജ p ߣ‫ޔ‬ㅦᐲࡌࠢ࠻࡞ u ߩⓨ㑆൨㈩ߩ৻ᰴᑼ (࠾ࡘ࡯࠻ࡦᵹ૕ߩ႐ว) ߣߒߡ⴫⃻ߐࠇࠆ☼ᕈᔕജ (࠹ࡦ࠰࡞) τ̃ ߣ߆ࠄᚑࠆ‫ޕ‬

 T
1 
u
∇u + ∇σ̃ = −pI˜ + τ̃ ,
τ̃ =
Re

Re ≡

;

UL
ν

(4)

T

ߚߛߒ‫ޔ‬I˜ ߪ 2 㓏ߩන૏࠹ࡦ࠰࡞‫ⴕߪ޿ࠆ޽࡞࠻ࠢࡌߪ ) (ޔ‬೉ߩォ⟎‫ޔ‬ν ߪേ☼ᕈଥᢙ‫ޔ‬Re ߪ࡟ࠗࡁ࡞
࠭ᢙߢ޽ࠆ‫ޕ‬Ⴚ⇇᧦ઙ㗔ၞ Ω ߩႺ⇇ (⴫㕙) Γ = Γu ∪ Γp ߦ߅޿ߡ‫ޔ߫߃଀ޔ‬ਅ⸥ߩࠃ߁ߥႺ⇇᧦ઙ߇⺖ߖࠄࠇࠆ‫ޕ‬u = u∗ on Γu ,ㆇേࠛࡀ࡞ࠡߩ଻ሽೣ

p = p∗ on Γp(5)

⾰㊂ߥࠄ߮ߦㆇേ㊂ߩ଻ሽೣߣ⁛┙ߥ߽ߩߢߪߥ޿߇‫ޔ‬ᑼ (3) ߣㅦᐲࡌࠢ࠻࡞ uߣߩౝⓍࠍߣࠆߎߣߦࠃࠅ‫ޔ‬ᰴߩࠃ߁ߥㆇേࠛࡀ࡞ࠡߩ଻ሽೣ߇ዉ߆ࠇࠆ‫ޕ‬2

 · u u∇2 · σ̃ T u − Φ + f(x) · u=∇(6)ߚߛߒ‫ޔ‬Φ ߪ☼ᕈߦࠃࠅᵹ૕ౝㇱߢᢔㅺߔࠆࠛࡀ࡞ࠡߩ㊂ (න૏ᤨ㑆‫ޔ‬න૏૕Ⓧ޽ߚࠅ) ࠍ⴫ߔ‫ޕ‬u=Φ ≡ τ̃ : ∇ᄙሹ૕ࡕ࠺࡞

1
Re


 T
 u + ∇
u
∇

u=
: ∇

1
2Re


 T
 u + ∇
u
∇

2

(7)

ᵹࠇ႐ߩᦨㆡൻ໧㗴ߦ࠻ࡐࡠࠫ࡯ᦨㆡൻᴺࠍㆡ↪ߔࠆߦ޽ߚࠅ‫ޔ‬᳞߼ࠆߴ߈ᧂ⍮ߩ‛૕߿ოߩሽ࿷ࠍ⴫ߔߚ߼ߦᵹࠇ႐ߩߥ߆ߦ઒ᗐ⊛ߥᄙሹ૕߇޽ࠆ߽ߩߣ⠨߃ࠆ [3]‫ޕ‬ታߪవߦቯ⟵ߒߚ γ(x) ߪ‫ޔ‬ߎߩࠃ߁ߥᄙሹ૕ߩⓨ㑆ಽᏓࠍ⴫ߔ߽ߩߢ߽޽ࠆ‫ߥ߁ࠃߩߎޔߡߒߘޕ‬઒ᗐ⊛ߥᄙሹ૕߇ᵹࠇߦኻߒߡ෸
߷ߔᓇ㗀ࠍ⴫ߔߩ߇‫ޔ‬ㆇേ㊂଻ሽೣ (3) ߩฝㄝߦ⟎߆ࠇߚ૕Ⓧജ f(x) ߢ޽ࠆ‫ޕ‬Borrvall and Petersson [3]ߦߥࠄ޿‫ ߩߎޔ‬f(x) ࠍᵹࠇߩㅦᐲߦᲧ଀ߔࠆᛶ᛫ജߣߒߡ

f(x) = −α(x)u(x)(8)ߣߒߚ߁߃ߢ‫ߩߘޔ‬Ყ଀ଥᢙ α(x) ࠍਅ⸥ߩࠃ߁ߦ⸳⸘ᄌᢙ γ(x) ߣ㑐ଥઃߌࠆ‫ޕ‬α(x) = α (γ(x)) = αmax

q {1 − γ(x)}q + γ(x)(9)ߎߎߢ q (> 0) ߪ‫ޔ‬α(γ) ߩ㑐ᢙᒻ⁁ࠍ೙ᓮߔࠆࡄ࡜ࡔ࠲ߢ‫ ߫߃଀ޔ‬q = 0.1 ߣߔࠆ‫ ߚ߹ޕ‬αmax ߦߪ‫ޔ‬ᑼ(3) ਛߩઁߩ㗄ߩଥᢙߣᲧߴߡචಽᄢ߈޿୯ (ℂᗐ⊛ߦߪή㒢ߦᄢ߈޿୯) ࠍਈ߃ࠆᔅⷐ߇޽ࠅ‫⎇ᧄޔ‬ⓥߢߪ
ߎࠇࠍ
1
χ
(10)
αmax = 1 +Reߣߒߚ߁߃ߢ‫ޔ‬χ ߦචಽᄢ߈޿୯ (଀߃߫ χ = 104 ) ࠍਈ߃ࠆ‫ ߦ߁ࠃߩߎޕ‬γ ߩ㑐ᢙߣߒߡ⴫⃻ߐࠇߚ α(γ)ࠍㅢߓ‫ޔ‬ᵹࠇ႐ߩၮᧄᄌᢙߢ޽ࠆㅦᐲࡌࠢ࠻࡞ u ߣ࿶ജ p ߪ‫⸘⸳ޔ‬ᄌᢙ γ ߦଐሽߔࠆ㊂ߣߥࠆ‫ޕ‬2.2⋡⊛㑐ᢙ— ᵹࠇ߇‛૕ߦ෸߷ߔ᛫ജߥࠄ߮ߦ឴ജߩ⴫⃻ᦨㆡൻ໧㗴ߢߪ‫⸘⸳ޔ‬਄ᦨᄢ޽ࠆ޿ߪᦨዊߦߒߚ޿㊂ࠍ૗ࠄ߆ߩᒻߢ⸳⸘ᄌᢙ γ ߩ㑐ᢙߣߒߡ⴫⃻ߔࠆ‫ޕ‬ߎߩࠃ߁ߥ㑐ᢙ J = J(γ) ߪ⋡⊛㑐ᢙߣ๭߫ࠇࠆ‫ޕ‬ታ㓙ߦߪᛒ߁໧㗴ߦ⃻ࠇࠆၮᧄᄌᢙ (ߎߎߢߪㅦᐲࡌࠢ࠻࡞ u ߣ࿶ജ p) ߩ㑐ᢙߣߒߡ⴫⃻ߐࠇࠆߎߣ߇ᄙߊ‫ߩߘޔ‬႐ว J ߪ γ ߩ㒶⊛ߥ㑐ᢙߣߥࠆ‫ޕ‬ ߩ߁ߜ‫ ߩߘޔ‬x ᣇะᚑಽߢ޽ࠆ᛫ജ D ࠍᦨዊൻ‫ߘߪ޿ࠆ޽ޔ‬ߎߎߢߩ⋡⊛ߪ‫ޔ‬ᵹࠇ߇‛૕ߦ෸߷ߔജ Fߩ y ᣇะᚑಽߢ޽ࠆ឴ജ Y ࠍᦨᄢൻߔࠆߎߣߢ޽ࠆ‫ߩࠄࠇߎޕ‬ജߪ‫⋥߽ᦨޔ‬ធ⊛ߦߪ‛૕⴫㕙ߦߪߚࠄߊᔕജ (࿶ജߣ☼ᕈᔕജ) ߩⓍಽߣߒߡ⴫⃻ߔࠆߎߣ߇ߢ߈ࠆ‫ᦨ࡯ࠫࡠࡐ࠻ߒ߆ߒޕ‬ㆡൻᴺߢᄙሹ૕ࡕ࠺࡞ࠍ ߩ⴫⃻ߪ࿎㔍ߢ޽ࠆ‫ߎߘޕ‬↪޿ߡ‛૕๟ࠅߩᵹࠇࠍ⸘▚ߔࠆ႐วߦߪ‫‛ߥ߁ࠃߩߎޔ‬૕⴫㕙Ⓧಽߦࠃࠆ F ࠍ⴫⃻ߔࠆߎߣ߇ᔅⷐߣߥࠆ‫ߩߘޕ‬㓙‫ޔ‬వߦ⸥ߒߚਃߟߩ଻ሽೣࠍᵹࠇ႐ో૕ߢ
ߢߘࠇߣߪ⇣ߥࠆᒻߢ FⓍಽߒߚ߽ߩ߇ࠃࠅߤߎࠈߣߥࠆ‫ޕ‬ᵹࠇ႐ో૕ߢߩ଻ሽೣߩⓍಽ

⾰㊂ (2)‫ޔ‬ㆇേ㊂ (3)‫ߦ߮ࠄߥޔ‬ㆇേࠛࡀ࡞ࠡ (6) ߩ଻ሽೣࠍᵹࠇ႐ߩ㗔ၞో૕ Ω ߢⓍಽߔࠆߣ‫ޔ‬ᵹࠇ႐ో૕ߢߩ⾰㊂‫ޔ‬ㆇേ㊂ߥࠄ߮ߦㆇേࠛࡀ࡞ࠡߩ଻ሽࠍ⴫ߔᰴߩࠃ߁ߥᑼ߇ᓧ
ࠄࠇࠆ‫ޕ‬

Γ

Γ
2

p+
Γ

u · n dΓ =0u (u · n) dΓ =

u
2

(11)
f(x) dΩσ dΓ +
Γ(u · n) dΓ =ΓΩ(τ · u) dΓ +

Ω

(12)


f(x) · u − Φ dΩ(13)ߚߛߒ‫ޔ‬Γ ߪ㗔ၞ Ω ߩႺ⇇‫ޔ‬n ߪႺ⇇ Γ ਄ߢߩ (ᵹࠇߩ஥߆ࠄ⷗ߚ) ᄖะ߈න૏ᴺࡌࠢ࠻࡞‫ޔ‬σ = σ̃n = −pn +τߣ τ = τ̃n ߪ Γ ਄ߦߪߚࠄߊᔕജࡌࠢ࠻࡞ (ోᔕജ) ߣ☼ᕈᔕജࡌࠢ࠻࡞ߢ޽ࠆ‫ޕ‬
଻ሽೣߩⓍಽߦ㑐ߔࠆ⵬ഥ⊛ߥ㑐ଥᑼΓ ࠍછᗧߩቯࡌࠢ࠻࡞ߣߒߡ V ߣᑼ (12) ߣߩౝⓍࠍߣࠆߣ‫ޔ‬V (u · n) dΓ =u · VΓ dΓ + V ·σ · Vf(x) dΩ(14)Ωߣߥࠆ߇‫ߩߎޔ‬ᑼߪਅ⸥ߩࠃ߁ߦᦠ߈ᡷ߼ࠆߎߣ߇ߢ߈ࠆ‫ޕ‬


2


1
 ·
 2 − u − V dΓ − (−pn + τ ) · u dΓVu 2 + V(u · n) dΓ + σ · u − Vf dΩ =
2 Γ
Ω
Γ
Γ


 1
2
u2
 −
 (u · n) dΓ(u · n) − τ · u + σ · u − V
p+
(15)
=
u − V22Γߚߛߒᑼࠍᄌᒻߔࠆㆊ⒟ߢᑼ (11) ࠍ↪޿ߚ‫ߦࠄߐޕ‬ᑼ (13) ࠍ⠨ᘦߔࠆߣ‫ߩߎޔ‬ᑼߪᰴߩࠃ߁ߦᄌᒻߔࠆߎ
ߣ߽ߢ߈ࠆ‫ޕ‬

 ·
V
2.2.1

f(x) dΩ =ΩΩf(x) · u − Φ



 


2
 − 1 u − V
 (u · n)
σ · u − V
dΩ +
2
Γ

dΓ

(16)

ᵹࠇ߇‛૕ߦ෸߷ߔജ ߪ‫‛ޔ‬૕⴫㕙 ΓS ਄ߢߩᔕജ (ߩ෻ജ) −σ ߩⓍಽߣߒߡ⴫⃻ߐࠇࠆ‫ޕ‬ᵹࠇ߇‛૕ߦ෸߷ߔജ F
F = −ΓSσ dΓ = −(−pn + τ ) dΓ

(17)

ΓS

‛૕๟ࠅߩᵹࠇߩ໧㗴ࠍᛒ߁᥉ㅢߩ⸘▚ᴺߢߪ‫‛ޔ‬૕߇භ߼ࠆ㗔ၞࠍᄖ⇇ߣߒߡᛒ޿‫⴫ߩߘޔ‬㕙ߢ㕖ṖࠅႺ⇇᧦ઙ (u = 0) ࠍ⺖ߔ‫ߦ߁ࠃߩߎޕ‬ᵹࠇߦኻߔࠆ‛૕ߩᓇ㗀ࠍ㕖ṖࠅႺ⇇᧦ઙߦࠃࠅ⴫⃻ߔࠆ႐วߦߪ‫ޔ‬F ࠍ⹏ଔߔࠆᑼߣߒߡᑼ (17) ࠍߘߩ߹߹↪޿ࠆߎߣ߇ߢ߈ࠆ‫ޔߪ޿ࠆ޽ޕ‬ㆇേ㊂଻ሽೣߩⓍಽ (12) ࠍ↪޿ߡ (ߎߩࠃ߁ߥ⸘▚ᴺߩ႐ว૕Ⓧജ f ߪሽ࿷ߒߥ޿)‫ࠍࠇߎޔ‬㆙ᣇႺ⇇ Γ∞ ߢㆇേ㊂ᵹ᧤ −u (u · n) ߣᔕജσ ࠍⓍಽߒߚ㊂ߣߒߡ⴫ߔߎߣ߽ߢ߈ࠆ‫ޕ‬F = −

σ dΓ =
ΓSΓ∞{−u (u · n) + σ } dΓ

(18)৻ᣇ‫ᦨ࡯ࠫࡠࡐ࠻ޔ‬ㆡൻᴺߢ↪޿ࠄࠇࠆᄙሹ૕ࡕ࠺࡞ߩࠃ߁ߦ‫ޔ‬ᵹࠇߦኻߔࠆ‛૕ߩᓇ㗀ࠍⓨ㑆ߦಽᏓߔ
ࠆ૕Ⓧജ f ࠍㅢߓߡ⴫⃻ߔࠆ႐วߦߪ‫‛ޔ‬૕߇භ߼ࠆ㗔ၞߢ߽ㆇേ㊂଻ሽೣ߇ㆡ↪ߐࠇࠆ‫ޔ߈ߣߩߎޕ‬૕

Ⓧജ f ࠍ㗔ၞో૕ Ω ߢⓍಽߒߚ㊂ߪ‛૕߇ᵹࠇߦ෸߷ߔജߩ✚㊂ߦ╬ߒߊ‫ߩߘߡߞ߇ߚߒޔ‬෻ജߪᵹࠇ߇ ߦ╬ߒ޿ߪߕߢ޽ࠆ‫ޔߜࠊߥߔޕ‬F ࠍ૕Ⓧജߩ෻ജ −f ߩ㗔ၞⓍಽߣߒߡ⴫⃻ߔࠆߎߣ‛૕ߦ෸߷ߔജ F
߇ߢ߈ࠆ‫ޕ‬f dΩ =

F = −

Ωα(x)u dΩ

(19)

Ω

ߎߩ႐ว߽‫ޔ‬ታߪㆇേ㊂଻ሽೣߩⓍಽ (12) ࠍ↪޿ࠆߣߎߩ㊂ߪᑼ (18) ߩฝㄝߣ╬ߒߊ‫ޔ߽ࠄ߆ߣߎߩߎޔ‬ ࠍ⹏ଔߔࠆߎߣߩᅷᒰᕈ߇⏕߆߼ࠄࠇࠆ‫ޕ‬ᑼ (19) ߦࠃࠅ FfdΩ =

F = −

ΩΓ∞{−u (u · n) + σ } dΓ

(20)޽ࠆ޿ߪ߹ߚ‫ޔ‬−f ࠍⓍಽߔࠆ㗔ၞࠍ⚐ᵹ૕㗔ၞ ΩF (γ = 1 ߩㇱಽ) ߣ‫ࠇߘޔ‬એᄖߩ⚐࿕૕㗔ၞߣਛ㑆㗔ၞࠍ޽ࠊߖߚ‫ޟ‬ᡆૃ࿕૕㗔ၞ‫ޠ‬ΩS ∗ (0 ≤ γ < 1 ߩㇱಽ) ߩ߰ߚߟߦಽߌࠆߣ‫ޔ‬ΩF ߢߩⓍಽߪ f = 0 ߦࠃࠅᶖ
 · σ̃ − uuT ߦ⟎߈឵߃ߚ߁߃ߢ‫ࠟޔ‬Ṍߒ‫ ߚ߹ޔ‬ΩS ∗ ߢߩⓍಽߪㆇേ㊂଻ሽೣ (3) ࠍ߽ߣߦ f ࠍ f = −∇࠙ࠬߩ⊒ᢔቯℂࠍ೑↪ߒߡ ΩS ∗ ߩ⴫㕙 ΓS ∗ ߢߩⓍಽߦ⟎߈឵߃ࠆߣ

F = −fdΩ =
Ω

ΩS ∗



 · σ̃ − uuT dΩ =∇ΓS ∗{σ − u (u · n)} dΓ =ΓS ∗{−pn + τ − u (u · n)} dΓ(21)ߣߥࠆ‫ ߢߎߎޕ‬ΓS ∗ ߪ ΩF ߣ ΩS ∗ ߣߩ⇇㕙ߢ޽ࠆ߇‫ߪࠇߎޔ‬ᡆૃ⊛ߥ‛૕⴫㕙ߣ޿ߞߡ߽ࠃߊ‫ߩߢߎߘޔ‬ㆇേ㊂ᵹ᧤ u (u · n) ߪ࠯ࡠߦㄭ޿ߣ⠨߃ࠄࠇࠆ‫ߩߎߡߞ߇ߚߒޕ‬ᑼߪF = −

ΩfdΩ ≈

σ dΓ =
ΓS ∗(−pn + τ ) dΓ

(22)

ΓS ∗

ߣߥࠅ‫ޔ‬ೋ߼ߦ⸥ߒߚᑼ (17) ߣห╬ߩ߽ߩߣߥࠆ‫ޔ߅ߥޕ‬ΓS ∗ ߪᡆૃ࿕૕㗔ၞ ΩS ∗ ߩ⴫㕙ߢ޽ࠅ‫ߞ߇ߚߒޔ‬ߡ n ߪ (ᵹࠇߩ஥߆ࠄߢߪߥߊ) ࿕૕ߩ஥߆ࠄ⷗ߚᄖะ߈න૏ᴺࡌࠢ࠻࡞ߢ޽ࠆ‫߼ߚߩߘޕ‬ᑼ (17) ߣߪ╓ภ߇෻ኻߦߥߞߡ޿ࠆ‫ޕ‬

2.2.2᛫ജᦨዊൻ໧㗴ߩ⋡⊛㑐ᢙᵹࠇ߇‛૕ߦ෸߷ߔ᛫ജ D ߪ‫ޔ‬ᑼ (19) ޽ࠆ޿ߪ (20) ߩ x ᣇะᚑಽߣߒߡਅ⸥ߩࠃ߁ߦ⴫ߔߎߣ߇ߢ߈ࠆ‫ޕ‬

D

=
Ω

D

α(x)u dΩ ≡ J1

(23){−u (u · n) − pnx + τx } dΓ ≡ J2a(24){u (u · n) + pnx } dΓ ≡ J2b

(25)

=
Γ∞

≈

−

Γ∞

ߚߛߒ‫ޔ‬u ߪㅦᐲࡌࠢ࠻࡞ u ߩ x ᣇะᚑಽߢ޽ࠆ‫ޔߪ޿ࠆ޽ޕ‬㗔ၞో૕ Ω ߢߩߺߞߟߩ଻ሽೣߩⓍಽ⴫⃻ ࠍ৻᭽ᵹߩㅦᐲ U ߦ╬(11)–(13) ‫ߦ߮ࠄߥޔ‬వߦዉ޿ߚ⵬ഥ⊛ߥ㑐ଥᑼ (16) ߦ߅޿ߡછᗧߩቯࡌࠢ࠻࡞ V

 =U
 = [U, 0, 0]) ߣߒߚ߽ߩࠍ޽ࠊߖߡ↪޿ࠆߎߣߦࠃࠅ‫ޔ‬D ࠍਅ⸥ߩࠃ߁ߦᦠ߈ᡷ߼ࠆߎߣ߽ߢߒ޿ (V߈ࠆ (⹦ߒ޿⺑᣿ߪᢥ₂ [4] ෳᾖ)‫߅ߥޕ‬ήᰴరߩ⸘▚ߢߪㅢᏱ U = 1 ߢ޽ࠆߚ߼‫ޔ‬ଥᢙ 1/U ߪ⋭⇛ߒߡ߽
ࠃ޿‫ޕ‬

D

=

≈ −
D

=
≈


u2
− p+
2
Γ∞

1
U

1
U
1
U

1
U


u2
2




2
 + τ · u + 1 u − U
 (u · n)(u · n) − σ · u − U2(u · n) dΓ ≡ J3b
 


2


 − 1 u − U
 (u · n)
σ · u − U
Φ + α(x)u2 dΩ −
2
Ω
Γ∞


Φ + α(x)u2 dΩ ≡ J4b
p+

Γ∞

Ω

dΓ ≡ J3a

(26)
(27)


dΓ ≡ J4a

(28)
(29)

࠻ࡐࡠࠫ࡯ᦨㆡൻᴺߦࠃࠆ᛫ജᦨዊൻ໧㗴ߦ߅޿ߡߪ‫⊛⋡ޔ‬㑐ᢙ J (ߎߩ႐วߪᦨዊൻߔࠆ㊂) ߦߎࠇࠄߩ
߁ߜߩ޿ߕࠇ߆ߩ⴫⃻ࠍਈ߃ࠇ߫ࠃ޿‫ޕ‬J = one of {J1 , J2a , J2b , J3a , J3b , J4a , J4b }
2.2.3

(30)

឴ജᦨᄢൻ໧㗴ߩ⋡⊛㑐ᢙᵹࠇ߇‛૕ߦ෸߷ߔ឴ജ Y ߪ‫ޔ‬ᑼ (19) ޽ࠆ޿ߪ (20) ߩ y ᣇะᚑಽߣߒߡਅ⸥ߩࠃ߁ߦ⴫ߔߎߣ߇ߢ߈ࠆ‫ޕ‬

Y

=
Ω

Y

α(x)v dΩ ≡ K1

(31){−v (u · n) − pny + τy } dΓ ≡ K2a(32){v (u · n) + pny } dΓ ≡ K2b

(33)

=
Γ∞

≈ −

Γ∞

ߚߛߒ‫ޔ‬v ߪㅦᐲࡌࠢ࠻࡞ u ߩ y ᣇะᚑಽߢ޽ࠆ‫ᦨ࡯ࠫࡠࡐ࠻ޕ‬ㆡൻᴺߦࠃࠆ឴ജᦨᄢൻ໧㗴ߦ߅޿ߡߪ‫ޔ‬⋡⊛㑐ᢙ J (ߎߩ႐วߪᦨᄢൻߔࠆ㊂) ߦߎࠇࠄߩ߁ߜߩ޿ߕࠇ߆ߩ⴫⃻ࠍਈ߃ࠇ߫ࠃ޿‫ޕ‬J = one of {K1 , K2a , K2b }(34)ߥ߅‫ޔ‬᛫ജ D ޽ࠆ޿ߪ឴ജ Y ࠍ⴫ߔฦᑼߦ߅޿ߡㄭૃ╓ภ (≈) ߩ޽ߣߦ⸥ߐࠇߚ⴫⃻ߪ‫‛ޔ‬૕߆ࠄචಽ )‫☼ߚ߹ޔ‬ᕈᔕജ߇ච
㔌ࠇߚߣߎࠈߦ⸳ߌࠄࠇߚ㆙ᣇႺ⇇ Γ∞ ߢߪㅦᐲ߇৻᭽ᵹߩㅦᐲߣ߶߷╬ߒߊ (u≈Uಽߦዊߐ޿ (τ ≈ 0) ߣ޿߁ㄭૃߦၮߠߊ߽ߩߢ޽ࠆ‫ޕ‬

2.3

೙⚂᧦ઙ

৻⥸⊛ߥᦨㆡൻ໧㗴ߦ߅޿ߡߪ‫ޔ‬૗ࠄ߆ߩ೙⚂ࠍ⸳ߌߚ߁߃ߢ⋡⊛㑐ᢙࠍᦨዊൻ޽ࠆ޿ߪᦨᄢൻߔࠆߎ
ߣ߇ᄙ޿‫⎇ᧄޕ‬ⓥߦ߅ߌࠆ᛫ജᦨዊൻ޽ࠆ޿ߪ឴ജᦨᄢൻߩ໧㗴ߦ߅޿ߡ߽‫ޔ‬ᵹࠇ႐ߩၮᧄᄌᢙߢ޽ࠆㅦ
ᐲu(x) ߣ࿶ജ p(x)‫‛ߪ޿ࠆ޽ޔ‬૕ߩᄢ߈ߐ߿ᒻ⁁‫ޔ‬㈩⟎ (⸳⸘ᄌᢙ γ(x) ߇ߘࠇࠄࠍ⴫ߔ) ߦ㑐ߔࠆ೙⚂ࠍ⸳ߌࠆߎߣ߇⠨߃ࠄࠇ‫ࠍࠇߎޔ‬ᰴߩࠃ߁ߦᦠ߈⴫ߔߎߣߦߔࠆ‫ޕ‬G (u(x; γ(x)), p(x; γ(x)); γ(x)) = 0(35)ౕ૕⊛ߦߪ‫‛߫߃଀ޔ‬૕ߩ೨ᣇᛩᓇ㕙Ⓧ߿૕Ⓧ (ੑᰴర໧㗴ߢ޽ࠇ߫‛૕ߩᢿ㕙Ⓧ) ࠍ৻ቯߣߔࠆ೙⚂᧦ઙߥߤ߇⠨߃ࠄࠇࠆ (ታߪߎߩࠃ߁ߥ೙⚂ࠍ⺖ߐߥ޿ߣ‫‛߫߃଀ޔ‬૕ߩ૕Ⓧ߿ᢿ㕙Ⓧࠍዊߐߊߔࠆߎߣߦࠃࠅ
᛫ജࠍ޿ߊࠄߢ߽ዊߐߊߔࠆߎߣ߇ߢ߈ࠆ)‫ޕ‬ᓟߢ␜ߔ⸘▚଀ߦ߅޿ߡߪ‫ޔ‬ੑᰴరᩇ⁁‛૕ߩᢿ㕙Ⓧࠍ৻ቯ(ήᰴరᢿ㕙Ⓧ = 1) ޽ࠆ޿ߪゲኻ⒓‛૕ߩ૕Ⓧࠍ৻ቯ (ήᰴర૕Ⓧ = 1) ߣߔࠆਅ⸥ߩࠃ߁ߥ೙⚂᧦ઙࠍਈ߃ߡ޿ࠆ‫ޕ‬G (γ(x)) :=Ω{1 − γ(x)} dΩ − 1 = 0(36)ߥ߅‫ޔ‬ήᰴరߩᢿ㕙Ⓧ߿૕Ⓧࠍ 1 ߦ╬ߒ޿ߣߔࠆߩߪ‫ޔ‬㐳ߐߩၮḰ㊂ L ࠍ‛૕ᢿ㕙Ⓧ A ߩᐔᣇᩮ (A/L2 = 1)޽ࠆ޿ߪ‛૕૕Ⓧ V ߩਃਸ਼ᩮ (V /L3 = 1) ߣߔࠆߚ߼ߢ޽ࠆ‫ޕ‬

2.4

⸘▚ᚻ㗅

᛫ജᦨዊൻ໧㗴ߦ߅޿ߡߪ‫߫߃଀ޔ‬ᑼ (36) ߩࠃ߁ߥ೙⚂᧦ઙߩਅߢ‫ޔ‬ᑼ (23) ߩࠃ߁ߥ⋡⊛㑐ᢙ J =J1 (u(x; γ(x)); γ(x)) ࠍᦨዊൻߔࠆ⸳⸘ᄌᢙ γ(x) ߩಽᏓࠍ⸘▚ߦࠃࠅ᳞߼ࠆ‫ߩߘޕ‬ᚻ㗅ߩ᭎⇛ࠍਅߦ⸥ߔ‫ޕ‬
1. ߪߓ߼ߦ‫ޔ‬γ(x) ߩಽᏓࠍㆡᒰߦ઒ቯߔࠆ (᦭㒢ⷐ⚛ᴺߦࠃࠆታ㓙ߩ⸘▚ߢߪ‫᦭ޔ‬㒢ߥ n ୘ߩ㔌ᢔ⊛ߥ૏⟎ 
xi ߢߩ γ ߩ୯ {γi = γ(xi ); i = 1, · · · , n} ࠍਈ߃ࠆ)‫ޔ߫߃଀ޕ‬ੑᰴరᩇ⁁‛૕ߩೋᦼᒻ⁁ߣߒߡේὐߦ⟎߆ࠇߚන૏㕙Ⓧߩ౞ᩇࠍ⸳ቯߔࠆ߽ߩߣߒߡ‫ߩߘޔ‬୯߇౞ᩇߩౝㇱߢ 0‫ޔ‬ᄖㇱߢ 1 ߣߥࠆࠃ߁ߥ γ(x) ߩಽᏓࠍਈ߃ࠆ‫ޕ‬⴫ 1: ᢿ㕙Ⓧ৻ቯߩ೙⚂᧦ઙਅߢ᛫ജ߇ᦨዊߣߥࠆੑᰴరᩇ⁁‛૕ߩᛶ᛫ଥᢙ

Re

0.1

1

10

20

50

100

200

500

1,000

2,000

CD

90.8

12.1

2.69

1.81

1.10

0.76

0.53

0.33

0.22

0.15

⴫ 2: ᢿ㕙Ⓧ৻ቯߥࠄ߮ߦ᛫ജ਄㒢ߩ೙⚂᧦ઙਅߢ឴ജ߇ᦨᄢߣߥࠆੑᰴరᩇ⁁‛૕ߩ឴ജଥᢙߣ឴᛫Ყ

(Re = 10)β ≡ CD /CDmin

1

1.1

1.2

1.5

2

CL

0

1.89

2.84

5.27

8.72

CL /CD

0

0.64

0.88

1.31

1.62

2. ᵹࠇߩၮ␆ᣇ⒟ᑼ (2) ߣ (3) ࠍ⸃߈‫ޔ‬ㅦᐲಽᏓ u(x) ࠍ᳞߼ࠆ‫ޕ‬
3. ⋡⊛㑐ᢙ J = J1 (u(x; γ(x)); γ(x)) ߩ୯ࠍ⹏ଔߔࠆ‫ޔߚ߹ޕ‬㓐઻ᄌᢙᴺࠍ↪޿ߡ‫⸘⸳ޔ‬ᄌᢙ γi ߩᄌൻߦኻߔࠆ⋡⊛㑐ᢙߣ೙⚂᧦ઙߩᄌൻ₸‫ߜࠊߥߔޔ‬ᗵᐲ ∂J/∂γi ߣ ∂G/∂γi ࠍ⹏ଔߔࠆ‫ޕ‬㓐઻ᄌᢙᴺߦ
ࠃࠆᗵᐲ⸃ᨆߦߟ޿ߡߪઃ㍳ A ෳᾖ‫ޕ‬
4. ਄⸥ᗵᐲ߇චಽߦዊߐߌࠇ߫‫ᦨޔ‬ㆡ⸃߇ᓧࠄࠇߚ߽ߩߣߒߡ⸘▚ࠍ⚳ੌߔࠆ‫ ߩ߈ߣߩߎޕ‬γ(x) = 0 ߩ㗔ၞ߇᛫ജ߇ᦨዊߣߥࠆ‛૕ߩᒻ⁁ࠍ⴫ߔ‫ޕ‬
5. ᗵᐲ߇චಽߦዊߐߊߥߌࠇ߫‫ޔ‬ㆡᒰߥᦨㆡൻߩᚻᴺ (଀߃߫ㅙᰴੑᰴ⸘↹ᴺ [6]) ࠍ↪޿‫ޔ‬᳞߼ࠄࠇߚ
਄⸥ᗵᐲࠍ߽ߣߦ‫ޔ‬೙⚂᧦ઙࠍḩߚߒߟߟ⋡⊛㑐ᢙࠍዊߐߊߔࠆᣇะߦ γ(x) ߩಽᏓ (ታ㓙ߦߪ γ ߩ㔌ᢔ୯ γi ) ࠍᄌൻߐߖࠆ‫ޕ‬
6. ᚻ㗅 2 ߦᚯࠆ‫ޕ‬

2.5

ᔕ↪଀

2.5.1

ᵹࠇߩߥ߆ߦ⟎߆ࠇߚੑᰴరᩇ⁁‛૕ߩ᛫ജᦨዊൻߣ឴ജᦨᄢൻ

Navier-Stokes ᵹߦ߅ߌࠆ᛫ജᦨዊੑᰴరᩇ⁁‛૕ x ᣇะߩ৻᭽ߥᵹࠇ (ߘߩㅦᐲ U ࠍㅦᐲߩၮḰ㊂ߣߔࠆ) ߩߥ߆ߦ⟎߆ࠇߚੑᰴరᩇ⁁‛૕ߦ㑐ߒ‫ߩߘޔ‬ᢿ㕙Ⓧ A ࠍ৻ቯߣߔࠆ೙⚂᧦ઙߩਅߢ‫ޔ‬᛫ജ߇ᦨዊߣ
ߥࠆᢿ㕙ᒻ⁁ࠍ᳞߼ࠆ‫ᦨޕ‬ㆡൻߩㆊ⒟ߢ‛૕ߩ૏⟎߇ᄢ߈ߊ⒖േߔࠆߎߣࠍ㒐ᱛߔࠆߚ߼‫ߩߘޔ‬㊀ᔃ૏⟎
ࠍቯὐ (ේὐ) ߦ࿕ቯߔࠆ೙⚂᧦ઙࠍ޽ࠊߖߡ⺖ߔ‫⊛⋡ޕ‬㑐ᢙ j ߩ⴫⃻ߪᑼ (23) ߦࠃࠆ‫߅ߥޕ‬㐳ߐߩၮḰ㊂√√
L ࠍ‛૕ᢿ㕙Ⓧ A ߩᐔᣇᩮߔߥࠊߜ L = A ߣߒ‫࠭࡞ࡁࠗ࡟ޔ‬ᢙ Re ࠍ Re ≡ U A/ν ‫ߚ߹ޔ‬᛫ജ D ߩήᰴర㊂ߢ޽ࠆᛶ᛫ଥᢙ CD ࠍ CD ≡ 2D/ ρU 2 A ߣቯ⟵ߔࠆ‫࠭࡞ࡁࠗ࡟ޕ‬ᢙ Re ߇ 0.1 ߆ࠄ 2,000 ߩ▸࿐ߩ᭽‫ߥޘ‬୯ߩᵹࠇߦኻߒߡᓧࠄࠇߚ⚿ᨐࠍ࿑ 2 ߣ⴫ 1 ߦ␜ߔ‫ޕ‬

Navier-Stokes ᵹߦ߅ߌࠆ឴ജᦨᄢੑᰴరᩇ⁁‛૕ ᛫ജᦨዊൻߣห᭽ߩ᧦ઙߩਅߢ‫ޔ‬឴ജ߇ᦨᄢߣߥࠆ
ੑᰴరᩇ⁁‛૕ߩᢿ㕙ᒻ⁁ࠍ᳞߼ࠆ‫ޕ‬ታߪ឴ജᦨᄢൻ໧㗴ߢߪ‫‛ޔ‬૕ߩᢿ㕙Ⓧࠍ৻ቯ (ήᰴరᢿ㕙Ⓧ=1) ߣߔࠆ೙⚂᧦ઙࠍ⺖ߒߡ߽‫߫߃଀ޔ‬ㄫ߃ⷺࠍ߽ߟᚳᐔߥᒻ⁁ߩ‛૕ߩෘߐࠍߤࠎߤࠎ⭯ߊߒߡ޿ߊߎߣߦࠃࠅ޿
ߊࠄߢ߽឴ജࠍᄢ߈ߊߔࠆߎߣ߇ߢ߈ࠆ‫ߢߎߘޕ‬೙⚂᧦ઙߣߒߡ‫ޔ‬ᢿ㕙Ⓧߥࠄ߮ߦ㊀ᔃ૏⟎ߦ㑐ߔࠆ೙⚂᧦ઙߦട߃‫ޔ‬ᛶ᛫ଥᢙ CD ߩ਄㒢ࠍⷙቯߔࠆ೙⚂᧦ઙࠍ޽ࠊߖߡ⺖ߔ (⚿ᨐߣߒߡ CD ߪ਄㒢୯ߣߥࠆ)‫ޕ‬ታ㓙ߦߪ࡟ࠗࡁ࡞࠭ᢙ Re ࠍ 10 ߦ࿕ቯߒ‫࠭࡞ࡁࠗ࡟ߩߎޔ‬ᢙߢߩ᛫ജᦨዊᒻ⁁ߩᛶ᛫ଥᢙ (CDmin = 2.69) ߣߩᲧࠍ⴫ߔ β ≡ CD /CDmin ߦ 4 ߟߩ୯ β = {1.1, 1.2, 1.5, 2} ࠍ⸳ቯߒ‫ޔ‬឴ജ߇ᦨᄢߣߥࠆੑᰴరᩇ⁁‛૕ߩᢿ㕙ᒻ⁁ࠍ᳞߼ߚ‫⚿ߩߘޕ‬ᨐࠍ࿑ 3 ߦ␜ߔ‫ߚ߹ޕ‬ฦᛶ᛫ଥᢙᲧ β ߢߩ឴ജᦨᄢᒻ⁁ߩ឴ജଥᢙ CL ≡ 2L/(ρU 2 A)ߣ឴᛫Ყ (CL /CD ) ࠍ⴫ 2 ߦ␜ߔ‫⊛⋡߅ߥޕ‬㑐ᢙ J ߩ⴫⃻ߪᑼ (31) ߦ⸥ߒߚ૕Ⓧജߩ y ᣇะᚑಽߩⓍಽߦࠃࠆ‫ޕ‬(a) Re = 0.1(f) Re = 100(b) Re = 1(g) Re = 200(c) Re = 10(h) Re = 500(d) Re = 20(i) Re = 1, 000(e) Re = 50(j) Re = 2, 000

࿑ 2: ᢿ㕙Ⓧ৻ቯߩ೙⚂᧦ઙਅߢ᛫ജ߇ᦨዊߣߥࠆੑᰴరᩇ⁁‛૕ߩᢿ㕙ᒻ⁁ߣߘߩ๟ࠅߩㅦᐲಽᏓ

(a) CD /CDmin = 1(᛫ജᦨዊᒻ⁁, CDmin = 2.69)(d) CD /CDmin = 1.5(e) CD /CDmin = 2(b) CD /CDmin = 1.1(c) CD /CDmin = 1.2
࿑ 3: ᢿ㕙Ⓧ৻ቯߥࠄ߮ߦ᛫ജ਄㒢ߩ೙⚂᧦ઙਅߢ឴ജ߇ᦨᄢߣߥࠆੑᰴరᩇ⁁‛૕ߩᢿ㕙ᒻ⁁ߣߘߩ๟ࠅ
ߩㅦᐲಽᏓ (Re = 10)

(a) Re = 0.1(f) Re = 100(b) Re = 1(g) Re = 200(c) Re = 10(h) Re = 500(d) Re = 20(i) Re = 1, 000(e) Re = 50(j) Re = 2, 000

࿑ 4: ૕Ⓧ৻ቯߩ೙⚂᧦ઙਅߢ᛫ജ߇ᦨዊߣߥࠆゲኻ⒓‛૕ߩሶඦ㕙ᒻ⁁ߣߘߩ๟ࠅߩㅦᐲಽᏓ

2.5.2

ᵹࠇߩߥ߆ߦ⟎߆ࠇߚゲኻ⒓‛૕ߩ᛫ജᦨዊൻ

Navier-Stokes ᵹߦ߅ߌࠆ᛫ജᦨዊゲኻ⒓‛૕ x ᣇะߩ৻᭽ߥᵹࠇ (ߘߩㅦᐲ U ࠍㅦᐲߩၮḰ㊂ߣߔࠆ)ߩߥ߆ߦ⟎߆ࠇߚゲኻ⒓‛૕ߦ㑐ߒ‫ߩߘޔ‬૕Ⓧ V ࠍ৻ቯߣߔࠆ೙⚂᧦ઙߩਅߢ᛫ജ߇ᦨዊߣߥࠆሶඦ㕙ᒻ⁁ࠍ᳞߼ࠆ‫ޕ‬ੑᰴరᩇ⁁‛૕ߩ႐วߣห᭽ߦ‫ޔ‬㊀ᔃ૏⟎ࠍቯὐ (ේὐ) ߦ࿕ቯߔࠆ೙⚂᧦ઙࠍ޽ࠊߖߡ⺖ߔ‫ߚ߹ޕ‬⋡⊛㑐ᢙ j ߩ⴫⃻ߪᑼ (23) ߦࠃࠆ‫߅ߥޕ‬㐳ߐߩၮḰ㊂ L ࠍ‛૕ߩ૕Ⓧ V ߩਃਸ਼ᩮߔߥࠊߜ L = V 1/3 ߣߒ‫ޔ‬࡟ࠗࡁ࡞࠭ᢙ Re ࠍ Re ≡ U V 1/3 /ν ‫ߚ߹ޔ‬᛫ജ D ߩήᰴర㊂ߢ޽ࠆᛶ᛫ଥᢙ CD ࠍ CD ≡ 2D/ ρU 2 V 2/3ߣቯ⟵ߔࠆ‫࠭࡞ࡁࠗ࡟ޕ‬ᢙ߇ 0.1 ߆ࠄ 2,000 ߩ▸࿐ߩ᭽‫ߥޘ‬୯ߩᵹࠇߦኻߒߡᓧࠄࠇߚ⚿ᨐࠍ࿑ 4 ߣ⴫ 3 ߦ␜ߔ‫ޕ‬⴫ 3: ૕Ⓧ৻ቯߩ೙⚂᧦ઙਅߢ᛫ജ߇ᦨዊߣߥࠆゲኻ⒓‛૕ߩᛶ᛫ଥᢙ

Re

0.1

1

10

20

50

100

200

500

1,000

2,000

CD

230

25.5

3.96

2.42

1.30

0.83

0.55

0.32

0.21

0.14

Stokes ᵹߦ߅ߌࠆ᛫ജᦨዊゲኻ⒓‛૕ ਄ߦ␜ߒߚࠃ߁ߥ Navier-Stokes ᵹߦ߅ߌࠆ᛫ജᦨዊൻߩ⚿ᨐ߇ᱜߒ޿ߎߣࠍᬌ⸽ߔࠆߎߣߪ࿎㔍ߢ޽ࠆ‫࠭࡞ࡁࠗ࡟ޔߒ߆ߒޕ‬ᢙ߇࠯ࡠߩᭂ㒢ߦ޽ߚࠆ Stokes ᵹߦ㑐ߒߡߪ‫ޔ‬Pirroneau [7] ߦࠃࠅℂ⺰⊛ߥ੍᷹߇ߥߐࠇߡ޿ࠆߩߢ‫⎇ᧄޔ‬ⓥߢ߽ߎࠇߣหߓ໧㗴ࠍᛒߞߚ‫⚿ߩߘޕ‬ᨐࠍ࿑ 5 ߦ␜ߔ‫⚿ߩ▚⸘ޕ‬ᨐᓧࠄࠇߚ᛫ജᦨዊ‛૕ߩሶඦ㕙ᒻ⁁ߪ‫ޔ‬೨✼ߣᓟ✼ߩ㗂ⷺ߇ߣ߽ߦ⚂ 120 ᐲߢ޽ࠆߎߣࠍ฽߼‫ޔ‬Pironneau ߦࠃࠆ੍᷹ߣ㕖Ᏹߦ⦟ߊ৻⥌ߔࠆ‫ޕ‬Navier-Stokes ᵹߦ߅޿ߡ߽ᱜߒ޿⚿ᨐࠍਈ߃ࠆߣ޿߁଻⸽ߪߥ޿߇‫⎇ᧄޔ‬ⓥߢ↪޿ࠆᣇᴺ (․ߦ⋡⊛㑐ᢙߩ⴫⃻) ߩᅷᒰᕈ߇‫ޔ‬ㇱಽ⊛ߦߣߪ޿߃‫ޔ‬ᬌ⸽ߐࠇߚߣ޿ߞߡࠃ޿‫ޕ‬

(a) ᦨㆡൻ⸘▚ߩ⚿ᨐ(b) ᛫ജᦨዊゲኻ⒓‛૕ߩሶඦ㕙ᒻ⁁࿑ 5: Stokes ᵹਛߩ᛫ജᦨዊゲኻ⒓‛૕ (૕Ⓧ৻ቯ೙⚂)3߅ࠊࠅߦ
ᢙℂ⊛ᦨㆡൻᚻᴺߦၮߠߊ⸳⸘⸘▚ߩ଀ߣߒߡ‫ޔ‬ᵹ૕㑐ㅪಽ㊁ߢߩ࠻ࡐࡠࠫ࡯ᦨㆡൻᴺߩᔕ↪ߩ⹜ߺ (ૐ࡟ࠗࡁ࡞࠭ᢙᵹࠇߦ߅ߌࠆ᛫ജߩᦨዊൻߣ឴ജߩᦨᄢൻ) ࠍ⚫੺ߒߚ‫ޕ‬వⴕߔࠆ᭴ㅧ࡮ᒝᐲ߿࿕૕ᾲવዉߩ
ಽ㊁ࠍ฽߼ߡ߽‫ߛ߹ࠈߎߣߩ߹޿ޔ‬ㆡ↪ߢ߈ࠆ໧㗴ߩ▸࿐ߪ㒢ࠄࠇࠆ߇‫ࡦ࡚ࠪ࡯࡟ࡘࡒࠪޔ‬ਥዉߩࡕࡁߠߊ
ࠅߩታ⃻ߦะߌ‫ߥ߁ࠃߩߎޔ‬ᛛⴚߩ዁᧪ߦߪᄢ߈ߥᦼᓙ߇ᜬߚࠇࠆ‫ࠍࠇߎޔߢ߆ߥߩߘޕ‬ᡰ߃ࠆၮ⋚ߣߒߡ‫ޔ‬ᢙቇߩᓎഀߪࠃࠅ㊀ⷐߥ߽ߩߣߥࠈ߁‫ޕ‬ෳ⠨ᢥ₂[1] M. P. Bendsøe and N. Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, Vol. 71, No. 2, pp.
197–224, 1988.
[2] M. P. Bendsøe and O. Sigmund. Topology Optimization: Theory, Methods and Applications. Springer,
second edition, 2003.
[3] T. Borrvall and J. Petersson. Topology optimization of ﬂuids in Stokes ﬂow. International Journal
for Numerical Methods in Fluids, Vol. 41, pp. 77–107, 2003.
[4] ㄭ⮮, ᧻᫪, Ꮉᧄ. ૐ࡟ࠗࡁ࡞࠭ᢙᵹࠇߦ߅ߌࠆᦨㆡ‛૕ᒻ⁁ߩត⚝ (࠻ࡐࡠࠫ࡯ᦨㆡൻᴺߦࠃࠆ᛫ജᦨ
ዊൻߣ឴ജᦨᄢൻ). ᣣᧄᯏ᪾ቇળ⺰ᢥ㓸 (B ✬), Vol. 76, No. 768, pp. 1152–1161, 2010.
[5] ㄭ⮮, ᧻᫪, Ꮉᧄ. ࠻ࡐࡠࠫ࡯ᦨㆡൻᴺߦࠃࠆゲኻ⒓‛૕ߩ᛫ജᦨዊൻ. ᣣᧄᔕ↪ᢙℂቇળ 2010 ᐕᐲ ᐕળ ⻠Ṷ੍Ⓜ㓸, pp. 79–80, 2010.
[6] P. E. Gill. SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM Journal on
Optimization, Vol. 12, No. 4, pp. 979–1006, 2002.
[7] O. Pironneau. On optimum proﬁles in Stokes ﬂow. Journal of Fluid Mechanics, Vol. 59, pp. 117–128,
1973.

A

ᗵᐲ⸃ᨆߦߟ޿ߡ

A.1

ᗵᐲ⸃ᨆߣߘࠇࠍ↪޿ߚᦨㆡ⸳⸘

⸳⸘ߣߪᚲᦸߩᕈ⢻ࠍታ⃻ߔࠆ (ᕈ⢻ࠍᦨ⦟ൻߔࠆ) ࠃ߁ߦ⸳⸘ᄌᢙࠍቯ߼ࠆߎߣߣ޿ߞߡࠃ޿‫⸳ࠆ޽ޕ‬⸘᩺߇ሽ࿷ߔࠆ߽ߩߣߒߡ‫⸘⸳޿ࠃࠅࠃࠍࠇߎޔ‬᩺ߣߔࠆߚ߼ߦߪ‫⸘⸳ޔ‬ᄌᢙ ρ ߦኻߔࠆᕈ⢻ j ߩᄌൻ₸ߔߥࠊߜ‫ޟ‬ᗵᐲ‫ޠ‬δj/δρ ߦၮ߈‫ ޔ‬j ߇ࠃࠅࠃߊߥࠆᣇะߦ ρ ࠍᄌᦝߔࠇ߫ࠃ޿‫ߩߎޕ‬ᗵᐲࠍ᳞߼ࠆᣇᴺߩ߭
ߣߟߪᛛⴚ⠪ߩ⋥ᗵߦ㗬ࠆߎߣߢ޽ࠈ߁ (࿑ 6)‫ޕ‬ᾫ✵ᛛⴚ⠪ߩ⋥ᗵߪ‫ޔ߫ߒ߫ߒޔ‬㕖Ᏹߦࠃ޿♖ᐲߩᗵᐲࠍ
ਈ߃ࠆ߽ߩߣ⠨߃ࠄࠇࠆ‫ߦࠇߎޕ‬ઍࠊࠅ‫ޔ‬ᢙℂ⊛ߥᣇᴺ⺰ߦၮ߈ᗵᐲࠍ᳞߼ࠃ߁ߣ޿߁ߩ߇ᗵᐲ⸃ᨆߢ޽ࠆ
߇‫ߩߘޔ‬㓙‫ޔ‬‫ޟ‬ਥᄌᢙ‫ߜࠊߥߔޠ‬ᒰ⹥໧㗴ࠍ⸥ㅀߔࠆၮᧄ⊛ߥᄌᢙ (ၮᧄ‛ℂ㊂) y ߩ⸳⸘ᄌᢙ ρ ߦኻߔࠆᄌൻ₸ dy/dρ ߇ᔅⷐߣߥࠆ (࿑ 7)‫ޔߒ߆ߒޕ‬ᄙߊߩ႐ว‫ߩߎޔ‬㊂ (dy/dρ) ߪ◲නߦߪ᳞߹ࠄߥ޿‫ޕ‬ታߪ㓐઻
ᄌᢙᴺߣ๭߫ࠇࠆ㕖ᏱߦᏁᅱߥᣇᴺ߇޽ࠅ‫ޔ߫ࠇࠃߦࠇߎޔ‬dy/dρ ࠍ↪޿ࠆߎߣߥߊᗵᐲࠍ⹏ଔߔࠆߎߣ߇
ߢ߈ࠆ (࿑ 8)‫ޕ‬࿑ 6: ᛛⴚ⠪ߩ‫⋥ޟ‬ᗵ‫ߦޠ‬ၮߊ⸳⸘࿑ 7: ᢙℂ⊛ߥᣇᴺ⺰ (ᗵᐲ⸃ᨆ) ߦၮߊ⸳⸘

࿑ 8: 㓐઻ᄌᢙᴺࠍ↪޿ߚᗵᐲ⸃ᨆߦၮߊ⸳⸘

A.2

㓐઻ᄌᢙᴺߦࠃࠆᗵᐲ⸃ᨆ— ᭎ᔨ㓐઻ᄌᢙᴺߦࠃࠆᗵᐲ⸃ᨆߦߟ޿ߡߪ‫ࠅ߹޽ޔ‬ᐢߊߪ⍮ࠄࠇߡ޿ߥ޿ߣᕁࠊࠇࠆߩߢ‫ޔ‬ᢙቇ⊛ߦ෩ኒߣߪ
޿߃ߥ޿ౝኈߢ޽ࠆߎߣߪߏኈ⿔޿ߚߛߊߣߒߡ‫ߩߘޔ‬᭎ᔨࠍ⺑᣿ߔࠆ‫ޕ‬⸳⸘ᄌᢙᗵᐲ⸃ᨆߣߪ‫⸘⸳ޔ‬ኻ⽎ߩᕈ⢻߿․ᕈ߇⸳⸘਄೙ᓮߢ߈ࠆ㊂ (⸳⸘ᄌᢙ) ߣߣ߽ߦߤߩࠃ߁ߦᄌ

ൻߔࠆ߆ (ᗵᐲ) ࠍ᳞߼ࠆߎߣߢ޽ࠆ‫⸘⸳ߪߢߎߎޕ‬ᄌᢙࠍ◲නߦ ρ ߣ⴫ߔ߇‫ ߫߃଀ޔ‬ρ ࠍήᰴర᧚ᢱኒᐲ
ߣߔࠇ߫‫ߩߘޔ‬ⓨ㑆ಽᏓߦࠃࠅ‛૕ᒻ⁁ࠍ⴫⃻ߔࠆߎߣ߇ߢ߈ࠆ‫ޕ‬ਥ໧㗴⸳⸘ኻ⽎ߩᝄ⥰޽ࠆ޿ߪ⁁ᘒࠍ⴫ߔᄌᢙ (⁁ᘒᄌᢙߣ๭߫ࠇࠆߎߣ߇ᄙ޿߇‫ߪߢߎߎޔ‬ਥᄌᢙߣ๭߱) ࠍ y ߣߔࠆ‫ ߡߒߘޕ‬y ߩᝄ⥰ࠍ⸥ㅀߔࠆᢙቇ⊛ߥ⴫⃻ (ၮ␆ᣇ⒟ᑼ) ࠍਥ໧㗴ߣ๭߱ߎߣߦߔࠆ‫ߚ߹ޕ‬ߎߎߢߪਥ໧㗴߇ਅ⸥ߩࠃ߁ߥᑼ (ઍᢙᣇ⒟ᑼ޽ࠆ޿ߪᓸಽᣇ⒟ᑼ) ߣߒߡ⴫⃻ߐࠇࠆ߽ߩߣߔࠆ‫ߩߎޕ‬ᑼߦߪ⸳⸘ᄌᢙ ρ ߇ᔅߕ฽߹ࠇߡ޿ࠆߪߕߢ޽ࠅ‫ ࠆ޽ߢ⸃ߩߘߡߞ߇ߚߒޔ‬y ߪ ρ ߩ㑐ᢙ (᳢㑐ᢙ) ߣߥࠆ‫ޕ‬
⸳⸘ᄌᢙ

ၮ␆ᣇ⒟ᑼ

ρ
e.g. ήᰴర᧚ᢱኒᐲ

⇒f [y; ρ] = 0
y: ਥᄌᢙ

⸃⇒y = y[ρ]଀߃᷷߫ᐲ߇ਥᓎߣߥࠆ⸳⸘໧㗴ߢ޽ࠇ߫‫⁁ޔ‬ᘒᄌᢙ y ߪ᷷ᐲಽᏓ‫ޔ‬ၮ␆ᣇ⒟ᑼߪᾲવዉᣇ⒟ᑼߢ޽ࠆ‫ޕ‬⋡⊛㑐ᢙ⸳⸘ߩ⋡⊛ߪ‫ޔ‬ᄙߊߩ႐ว‫⸘⸳ޔ‬ኻ⽎ߩᕈ⢻߿․ᕈࠍᦨᄢൻ޽ࠆ޿ߪᦨዊൻߔࠆߎߣߢ޽ࠆ‫ߘޕ‬ߎߢᦨᄢൻ޽ࠆ޿ߪᦨዊൻߔߴ߈㊂ࠍ⋡⊛㑐ᢙ j ߣߒߡ⴫⃻ߔࠆ‫ޕ‬j ߪ৻⥸⊛ߦߪ ρ ߣ y ߩ㑐ᢙߢ޽ࠆ߇‫ޔ‬y = y[ρ] ߢ޽ࠆߎߣ߆ࠄ‫⚿ޔ‬ዪ‫ޔ‬j ߪ⸳⸘ᄌᢙ ρ ߩߺߩ㑐ᢙߣߺߥߔߎߣ߇ߢ߈ࠆ‫ޕ‬j = j [y[ρ], ρ]

(37)

⋡⊛㑐ᢙ

ߥ߅ਥ໧㗴߇Ⴚ⇇୯໧㗴 (ᓸಽᣇ⒟ᑼ) ߣߒߡ⸥ㅀߐࠇࠆ႐วߦߪ‫⊛⋡ޔ‬㑐ᢙ j ߪ㗔ၞⓍಽ޽ࠆ޿ߪႺ⇇Ⓧಽߣߒߡ⴫⃻ߐࠇࠆߎߣ߇ᄙ޿‫ޕ‬ᰴߦ ρ ߇ᄌൻߒߚߣ߈ߦߘߩ㑐ᢙߢ޽ࠆ j ߇ߤߩࠃ߁ߦᄌൻߔࠆ߆ (j ߩ ρ ߦኻߔࠆᄌൻ₸‫ߜࠊߥߔޔ‬ᗵᐲ) ࠍ᳞߼ࠆߚ߼‫ޔ‬j ߩ ρ ߦኻߔࠆᄌಽ (ోᓸಽ) ࠍߣࠆ‫ޕ‬δj =∂j [y[ρ], ρ]∂j [y[ρ], ρ]
δy +
δρ =
∂y
∂ρ

∂j [y[ρ], ρ] dy ∂j [y[ρ], ρ]
+
∂y
dρ
∂ρ

δρ

(38)

ߘߒߡߎࠇࠍ (ᒻᑼ⊛ߦ) ਅ⸥ߩࠃ߁ߥᓸಽ (᳢㑐ᢙᓸಽ) ߢᦠ߈⴫ߔߎߣߦߔࠆ‫ޕ‬δj∂j [y[ρ], ρ] dy ∂j [y[ρ], ρ]
=
+
δρ
∂y
dρ
∂ρ

j ߩ ρ ߦኻߔࠆᄌൻ₸ (ᗵᐲ)(39)ߣߎࠈߢߎߎߢਥ໧㗴ߩ⸃߇⸃ᨆ⊛ߦ᳞߼ࠄࠇߥ޿㒢ࠅ‫ޔ‬ਥᄌᢙ y ߩ ρ ߦኻߔࠆᄌൻ₸ dy/dρ ࠍ᳞߼ࠆߎ
ߣߪ࿎㔍ߥߚ߼‫ߩߎޔ‬ᑼ߆ࠄᗵᐲࠍ᳞߼ࠆߎߣߪ㔍ߒ޿‫ޔߢߎߘޕ‬છᗧߩᄌᢙ λ ࠍዉ౉ߒ‫ޔ‬ਅ⸥ߩᑼߩࠃ߁
ߦਥ໧㗴ߩᏀㄝ (ᱷᏅ) f ߣ λ ߣߩⓍ1 ࠍ⋡⊛㑐ᢙ j ߦട߃ߚ㊂ߣߒߡ‫ޔ‬᜛ᒛ⋡⊛㑐ᢙ J ࠍቯ⟵ߔࠆ‫ޕ‬J ߪ ρߣ y ߥࠄ߮ߦ λ ߩ㑐ᢙߣߥࠆ‫߅ߥޕ‬ၮ␆ᣇ⒟ᑼࠃࠅ f = 0 ߢ޽ࠆߚ߼‫ޔ‬ታߪ J = j ߢ޽ࠆ‫ޕ‬J = j [y[ρ], ρ] + λ f [y[ρ], ρ] = J [y[ρ], ρ; λ] = j [y[ρ], ρ]᜛ᒛ⋡⊛㑐ᢙ

(40)ߘߒߡవߣห᭽ߦ J ߩ ρ ߦኻߔࠆᄌಽࠍߣࠅ‫ޔ‬ᓸಽߢᦠ߈⴫ߔߣ‫ޔ‬ᰴߩࠃ߁ߦߥࠆ‫ޕ‬δJ∂J [y[ρ], ρ; λ] dy ∂J [y[ρ], ρ; λ]
δj
=
+
=
δρ
∂y
dρ
∂ρ
δρ




J ߩ ρ ߦኻߔࠆᄌൻ₸ (ᗵᐲ)(41)≡g[λ;y[ρ],ρ]ߎߎߢ dy/dρ ߩଥᢙࠍ g ߣߔࠆ‫ޕ‬g ߽ ρ ߣ y ߥࠄ߮ߦ λ ߩ㑐ᢙߣߥߞߡ޿ࠆ‫ޕ‬㓐઻໧㗴ᰴߦߎߩᑼߩߥ߆ߢߩ dy/dρ ߩᓇ㗀ࠍߥߊߔߎߣࠍ⠨߃ࠆ‫ߜࠊߥߔޕ‬છᗧߢ޽ߞߚ λ ࠍㆡಾߦㆬ߮‫ޔ‬g = g [λ; y[ρ], ρ] ߇࠯ࡠߣߥࠆࠃ߁ߦߔࠆ‫ޕ‬
ၮ␆ᣇ⒟ᑼ

g [λ; y[ρ], ρ] = 0
λ: 㓐઻ᄌᢙ

⸃⇒λߎߩࠃ߁ߥ⋡⊛ߢዉ౉ߐࠇߚᄌᢙ λ ߪ㓐઻ᄌᢙߣ๭߫ࠇࠆ‫ߡߒߘޕ‬㓐઻ᄌᢙ λ ߇ḩߚߔߴ߈ߎߩ᭽ߥၮ␆
ᣇ⒟ᑼ (ઍᢙᣇ⒟ᑼ޽ࠆ޿ߪᓸಽᣇ⒟ᑼ) ߪ㓐઻ᣇ⒟ᑼ‫ࠅࠃߦࠇߘߚ߹ޔ‬ቯ⟵ߐࠇࠆ໧㗴ߪ㓐઻໧㗴ߣ๭߫ࠇࠆ‫ޕ‬
1 ਥ໧㗴߇Ⴚ⇇୯໧㗴 (ᓸಽᣇ⒟ᑼ) ߣߒߡ⸥ㅀߐࠇࠆ႐วߦߪ‫ޔ‬‫ޟ‬ਥ໧㗴ߩᱷᏅ f ߣ λ ߣߩⓍ‫ߪޠ‬ታ㓙ߦߪ‫ޟ‬ਥ໧㗴ߩᱷᏅ f ߣ λߣߩⓍߩ㗔ၞⓍಽ‫ޕࠆ޽ߢޠ‬ᗵᐲλ ߇㓐઻໧㗴ߩ⸃ߢ޽ࠆߥࠄ߫‫⊛⋡ޔ‬㑐ᢙ j (޽ࠆ޿ߪ᜛ᒛ⋡⊛㑐ᢙ J) ߩ⸳⸘ᄌᢙ ρ ߦኻߔࠆᄌൻ
₸ߔߥࠊߜᗵᐲࠍ‫ޔ‬dy/dρ ࠍ↪޿ࠆߎߣߥߊ‫ޕࠆ߈ߢ߇ߣߎࠆߔ⃻⴫ޔ‬δjδJ∂J [y[ρ], ρ; λ]
=
=
δρ
δρ
∂ρ

(42)

ߎߩᑼߩฝㄝߪ⸃ᨆ⊛޽ࠆ޿ߪᢙ୯⊛ߦ⹏ଔߔࠆߎߣ߇น⢻ߢ޽ࠅ‫ޔࠄ߆ࠇߎޔ‬ਥ໧㗴ߩ⸃ y ߣ㓐઻໧㗴ߩ⸃ λ (ߥࠄ߮ߦ⸳⸘ᄌᢙ ρ) ߇ਈ߃ࠄࠇࠆߣᗵᐲ߇᳞߼ࠄࠇࠆߎߣߦߥࠆ‫ޕ‬ᦨㆡ⸃ߩត⚝

⸳⸘ኻ⽎ߩᕈ⢻߿․ᕈࠍᦨ⦟ൻߔࠆߚ߼ߦ‫ޔ‬਄ߢᓧࠄࠇߚᗵᐲࠍ߽ߣߦ૗ࠄ߆ߩᣇᴺߢ j ࠍࠃࠅࠃߊߔࠆᣇะߦ⸳⸘ᄌᢙ ρ ࠍᦝᣂߔࠆ‫ߩߘޕ‬ᣇᴺߣߒߡᦨ߽৻⥸⊛ߥߩߪᢙℂ⸘↹ᴺ (ㅙᰴੑᰴ⸘↹ᴺ
ߥߤ‫ߥࠈ޿ࠈ޿ޔ‬ᣇᴺ߇޽ࠆ) ߢ޽ࠆ߇‫߽ߦ߆߶ߩߘޔ‬ᡆૃ⊛ߥᤨ㑆 τ ࠍዉ౉ߒ‫⸘⸳ޔ‬ᄌᢙ ρ ࠍ τ ߩ㑐ᢙߣߒߡ‫ޔ‬ρ(τ ) ߇ᗵᐲߦၮ߈ᤨ㑆⊛ߦᄌൻߔࠆ⃻⽎ࠍ⴫ߔᤨ㑆⊒ዷᣇ⒟ᑼߩ෼᧤⁁ᘒߣߒߡ ρ ߩᦨㆡ⸃ࠍ᳞߼
ࠆߣ޿ߞߚߎߣ߽ߢ߈ࠆ‫ޕ‬

A.3

ᵹ૕໧㗴ߦ߅ߌࠆ㓐઻ᄌᢙᴺߦࠃࠆᗵᐲ⸃ᨆ— ᛫ജᦨዊൻ໧㗴ᵹࠇߩߥ߆ߢ᛫ജ߇ᦨዊߣߥࠆ‛૕ߩᒻ⁁ࠍ᳞߼ࠆ໧㗴ࠍ଀ߣߒߡ‫ޔ‬㓐઻ᄌᢙᴺߦࠃࠅᗵᐲࠍ⹏ଔߔࠆ
ᣇᴺߦߟ޿ߡ⺑᣿ߔࠆ‫ޕ‬⸳⸘ᄌᢙᵹࠇߩߥ߆ߦ⟎߆ࠇߚ‛૕ߩᒻ⁁ࠍήᰴరᵹ૕ኒᐲ ρ ߩಽᏓߢ⴫⃻ߔࠆ߽ߩߣߒ (0 ≤ ρ ≤ 1: ‛૕ㇱಽߢ ρ = 0‫ޔ‬ᄙሹ૕ㇱಽߢ 0 < ρ < 1‫ޔ‬ᵹ૕ㇱಽߢ ρ = 1)‫ᦨࠍࠇߎޔ‬ㆡൻ໧㗴ߩ⸳⸘ᄌᢙߣߔࠆ (ᧄᢥਛߢߪ ρ ࠍ γ ߣ⸥ߒߚ)‫‛ޔߡߒߘޕ‬૕߇ᵹࠇ߆ࠄฃߌࠆᛶ᛫ജ (᛫ജ) ߇ᦨዊߣߥࠆ ρ ߩಽᏓࠍ᳞߼ࠆ‫ޕ‬ਥ໧㗴ቯᏱ࡮㕖࿶❗ᕈߩᵹࠇࠍᛒ߁߽ߩߣߔࠆ‫ޕ‬ਥᄌᢙߪㅦᐲࡌࠢ࠻࡞ u(x) ߣ࿶ജ p(x)‫ޔ‬ਥ໧㗴 (ߩၮ␆ᣇ⒟ᑼ) ߪਥᄌᢙ߇㗔ၞ Ω ߢḩߚߔߴ߈ㅪ⛯ᑼߣ Navier-Stokes ᣇ⒟ᑼߢ޽ࠆ‫ޕ‬Ω ߩႺ⇇ Γ ߪ‫ޔ‬ㅦᐲࡌࠢ࠻࡞ 
u = u∗ ߇ⷙቯߐࠇࠆႺ⇇ Γu ߣᔕജࡌࠢ࠻࡞ σ = σ∗ ߇ⷙቯߐࠇࠆႺ⇇ Γσ ߆ࠄᚑࠆ߽ߩߣߔࠆ‫ޕ‬ · u = 0,∇
 · uuT − σ̃ − f = 0 in Ω∇ㅪ⛯ᑼߣ Navier-Stokes ᣇ⒟ᑼ

u = u∗ on Γu , σ = σ∗ on Γσ Ⴚ⇇᧦ઙ

 T
2
1 
u
∇u + ∇, σ = σ̃n = −pn + τ , τ = τ̃nσ̃ ≡ −pI˜ + τ̃ , τ̃ =ε̃, ε̃ ≡
Re
2

(43)
(44)
(45)

ߎߎߢ Navier-Stokes ᣇ⒟ᑼਛߩ f ߪᵹࠇߩߥ߆ߦ⟎߆ࠇߚ‛૕ߩሽ࿷ࠍ⴫ߔ૕Ⓧജߢ‫ޔ‬ਅ⸥ߩࠃ߁ߦ⴫ߐࠇࠆ (ᄙሹ૕ࡕ࠺࡞ [3]‫ᧄޔ‬ᢥෳᾖ)‫ޕ‬q (1 − ρ)f = −αu, α = αmax
, αmax =
q+ρ

1+

1
Re

χ, χ = 104 e.g., q = 0.1 e.g.

(46)ߎࠇࠃࠅ‫⸘⸳ޔ‬ᄌᢙߢ޽ࠆήᰴరᵹ૕ኒᐲ ρ ߦଐሽߔࠆ α = α(ρ) ࠍㅢߓ‫ޔ‬u ߣ p ߪ ρ ߦଐሽߔࠆ㊂ߣߥࠆ‫ޕ‬⋡⊛㑐ᢙ⸳⸘ߩ⋡⊛ (ᦨᄢൻ޽ࠆ޿ߪᦨዊൻߔߴ߈㊂) ࠍ⋡⊛㑐ᢙ j ߣߒߡ⴫⃻ߔࠆ‫ޕ‬᛫ജᦨዊൻ໧㗴ߩ⋡⊛㑐ᢙߪ‫ޔ‬ਅ⸥ߩࠃ߁ߦ⴫ߔߎߣ߇ߢ߈ࠆ (ᧄᢥෳᾖ)‫ޕ‬j=Ω

Lj dΩ +

Γj dΓ;

Lj = αu, j = 0(47)છᗧߩࠬࠞ࡜࡯ᄌᢙ ν ߣࡌࠢ࠻࡞ᄌᢙ λ ࠍዉ౉ߒ‫ޔ‬−ν (ଢቱ਄ࡑࠗ࠽ࠬߩ╓ภࠍߟߌࠆ)ࠍㅪ⛯ᑼߩᏀㄝߦ‫ ߚ߹ޔ‬λ ࠍ (ౝⓍߣߒߡ) Navier-Stokes ᣇ⒟ᑼߩᏀㄝߦਸ਼ߓߚ߁߃ߢ㗔ၞ Ω ߢⓍಽߔࠆ‫ޕ‬᜛ᒛ⋡⊛㑐ᢙ

ߘߒߡߎࠇࠍ⋡⊛㑐ᢙ j ߦട߃ߚ㊂ߣߒߡ᜛ᒛ⋡⊛㑐ᢙ J ࠍቯ⟵ߔࠆ‫ޕ‬

J ≡j−

Ω



 · u dΩ +
ν ∇

Ω




λ · ∇
 · uuT − σ̃ − f dΩ = j(48)

j ߦട߃ࠄࠇߚߩߪᧄ᧪࠯ࡠߣߥࠆ㊂ߢ޽ࠆߚ߼‫ޔ‬᜛ᒛ⋡⊛㑐ᢙ J ߪ߽ߣߩ⋡⊛㑐ᢙ j ߣ╬ߒ޿‫ޕ‬Gaußߩ⊒ᢔቯℂࠍ೑↪ߒߡߎߩᑼࠍᄌᒻߔࠆߣ‫ޔ‬ਅ⸥ߩࠃ߁ߥ⴫⃻߇ዉ߆ࠇࠆ‫ޕ‬
 

 


 λ u − p ∇ · λ − ∇ · π̃ · u − λ · f dΩ
J =Lj − uT ∇
Ω


≡L




j + (u · n) u · λ − λ · σ + π · u dΓ
+
Γ



(49)

≡

ߎߎߢ㗔ၞⓍಽߩⵍⓍಽ㊂ࠍ L‫ޔ‬Ⴚ⇇ⓍಽߩⵍⓍಽ㊂ࠍ  ߣߔࠆ‫ޕ‬L ߩߥ߆ߢਥᄌᢙ u ߣ p ߦߪᓸಽṶ▚߇߆߆ߞߡ޿ߥ޿ (ߘߩࠃ߁ߦ J ߩ⴫⃻ࠍᄌᒻߒߚ)‫ ߅ߥޕ‬π̃ ‫ޔ‬κ̃‫ޔ‬π ‫ޔ‬κ ߪਅ⸥ߢቯ⟵ߐࠇࠆ㊂ߢ޽ࠆ‫ޕ‬
2
1      T
∇λ + ∇λπ̃ ≡ −ν I˜ + κ̃, κ̃ =, π = π̃n = −νn + κ, κ = κ̃n(50)
˜, ˜ ≡Re2ߎߎߢߎࠇ߹ߢછᗧߢ޽ߞߚࠬࠞ࡜࡯ᄌᢙ ν ߣࡌࠢ࠻࡞ᄌᢙ λ ࠍ㓐઻ᄌᢙ (㓐઻࿶ജߣ㓐઻ㅦᐲࡌࠢ࠻࡞ߣ๭߱) ߣ⠨߃ࠆ‫ޕ‬㓐઻ᄌᢙߣߒߡߩ ν(x) ߣ λ(x) ߩḩߚߔߴ߈ၮ␆ᣇ⒟ᑼߪ‫ޔ‬dL/dp = 0 ߥࠄ߮㓐઻໧㗴ߦ dL/du = 0 ࠃࠅዉ߆ࠇࠆ‫ޔ‬㗔ၞ Ω ߢᚑࠅ┙ߟߴ߈ᓸಽᣇ⒟ᑼ

 · λ = 0,∇  T · λuT + π̃ − ∇ λ u − g = − [α, 0, 0]T in Ω;−∇g = −αλ(51)ߣ‫ޔ‬d/dp = 0 ߥࠄ߮ߦ d/du = 0 (޽ࠆ޿ߪ d/du = 0 ߥࠄ߮ߦ d/dσ = 0) ࠃࠅዉ߆ࠇࠆ‫ޔ‬Ⴚ⇇Γ = Γu ∪ Γσ ߢᚑࠅ┙ߟߴ߈Ⴚ⇇᧦ઙλ = 0 on Γu , π + (u · n) λ + u · λ n = 0 on Γσ(52)ߣߢ᭴ᚑߐࠇࠆ‫߇ࠄࠇߎޕ‬㓐઻໧㗴ࠍቯ⟵ߔࠆ‫ޕ‬ᗵᐲ᜛ᒛ⋡⊛㑐ᢙ J ߩᗵᐲ‫ ߜࠊߥߔޔ‬J ߩ⸳⸘ᄌᢙ ρ ߦኻߔࠆᄌൻ₸ࠍ᳞߼ࠆ‫⋡ߩߣ߽ߣ߽ޔߪࠇߎޕ‬⊛㑐ᢙ j ߩᗵᐲߢ߽޽ࠆ‫ޕ‬ᗵᐲߪ‫ޔߪߦ⊛⥸৻ޔ‬㗔ၞⓍಽߣႺ⇇Ⓧಽߩ๺ߣߒߡ⴫ࠊߐࠇࠆ‫ޕ‬δj = δJ =ΩSΩ δρ dΩ +ΓSΓ δρ dΓ ޽ࠆ޿ߪ (ᒻᑼ⊛ߦ)

δj
δJ
=
=
δρ
δρ

Ω

SΩ dΩ +

Γ

SΓ dΓ

(53)

ߎߎߢ㗔ၞⓍಽߩⵍⓍಽ㊂ SΩ ߪ‫ޔ‬㗔ၞౝߩฦ૏⟎ߢߩዪᚲ⊛ߥᗵᐲࠍ⴫ߒ‫ޔ‬᛫ജᦨዊൻ໧㗴ߢߪᰴߩࠃ߁
ߦ⴫ࠊߐࠇࠆ‫ޕ‬SΩ

=∂L dL dα
+
=
∂ρ
dα dρ



∂L ∂L df·+∂α∂ f dα



 dα
dα 
= u + λ · u
dρ
dρ

<0

߹ߚႺ⇇ⓍಽߩⵍⓍಽ㊂ SΓ ߪ‫ޔ‬Ⴚ⇇਄ߩฦ૏⟎ߢߩዪᚲ⊛ߥᗵᐲࠍ⴫ߒ‫ޔ‬᛫ജᦨዊൻ໧㗴ߢߪਅ⸥ߩࠃ߁
ߦ࠯ࡠߣߥࠆ‫ޕ‬

SΓ

=

∂
d dα
+
=
∂ρ dα dρ∂∂ df
·+∂α ∂ f dα



dα
=0
dρ

(54)