長岡工業高等専門学校研究紀要
波動水槽による水位計測値を用いた水槽内における流れ場の推定解析
第51巻（2015）⺰ ᢥᵄേ᳓ᮏߦࠃࠆ᳓૏⸘᷹୯ࠍ↪޿ߚ᳓ᮏౝߦ߅ߌࠆᵹࠇ႐ߩផቯ⸃ᨆ㧙 ࠞ࡞ࡑࡦࡈࠖ࡞࠲ℂ⺰ࠍ↪޿ߚ᦭㒢ⷐ⚛⸃ᨆߦࠃࠆᵹࠇ႐ߩផቯࠕ࡞ࠧ࡝࠭ࡓߩ⚫੺ 㧙123ୖᯅ ⾆ᒾ ࡮ศ⨹ ᄥ৻ ࡮ⴡ⮮ ବᒾ1 㐳ጟᛛⴚ⑼ቇᄢቇ㧔Department of Mechanical Engineering, Nagaoka University of Technology㧕
2 㐳ጟᛛⴚ⑼ቇᄢቇ㧔Department of Mechanical Engineering, Nagaoka University of Technology㧕
3 ⅣႺㇺᏒᎿቇ⑼㧔Department of Civil Engineering, National Institute of Technology, Nagaoka College㧕 Estimation Analysis of Fluid Field in Wave Tank

Using Measurement Value of Water Elevation-Introduction of Estimation Algorithm of Flow Fieldby Finite Element Analysis Using Kalman Filter Theory123Takahiko KURAHASHI , Taichi YOSHIARA and Toshihiko ETO
Abstract

In this study, we present the flow field estimation analysis using the finite element method based on theKalman filter theory. The linear shallow water equation is employed as the governing equation. TheGalerkin method using the linear triangular element is applied to the discretization in space, and theselective lumping method is employed for the discretization in time. The flow field estimation, i.e.,estimation of distribution of the flow velocity and the water elevation, is carried out by the finite elementanalysis based on the Kalman filter theory. In addition, the practical observed water elevation is employedin the flow field estimation analysis.
Key Words : finite element method, Kalman filter theory, shallow water flow, observed water elevation㧝㧚ᐨ⺰ ᧄ⺰ᢥߢߪ㧘ታ㓙ߩ᳓૏⸘᷹୯ࠍ↪޿ߚᵹࠇ႐ߩផቯ⸘▚⚿ᨐߦߟ޿ߡ␜ߔ㧚৻⥸ߦᢙ୯⸃ᨆࠍⴕ߁႐ว㧘ኻ⽎ߣߔࠆ‛ℂࡕ࠺࡞ߦ㑐ߔࠆᡰ㈩ᣇ⒟ᑼߦ
ኻߒߡ㧘Ⴚ⇇᧦ઙࠍⷙቯߔࠆߎߣߦࠃࠅ⁁ᘒ㊂㧔ᵹㅦ㧘࿶ജ㧘᷷ᐲ╬㧕ߩಽᏓࠍ᳞߼ࠆߎߣߦߥࠆ߇㧘⸃ᨆኻ⽎㗔ၞߩႺ⇇ߦ߅޿ߡ⁁ᘒ㊂ߩ⸘᷹߇࿎㔍ߢ޽ࠆ႐ว߿㧘Ⴚ⇇ㇱಽ〒㔌߇㐳ߊᴛጊߩ⸘᷹ࠍⷐߔࠆ႐ว╬㧘Ⴚ⇇᧦ઙߩ⸳ቯ߇ᚻ㑆ߦߥࠆߎߣ߽޽ࠆ㧚ߎߩࠃ߁ߥ႐ว㧘᷹ⷰὐߩᖱႎ߆ࠄ㧘Ⴚ⇇᧦ઙ߿㗔ၞౝㇱߩ⁁ᘒ㊂ߩಽᏓࠍផቯߔࠆᣇᴺ߇޽ࠆ㧚ᄢ߈ߊ⏕ቯ⺰⊛ᚻᴺߣ⏕₸⺰⊛ᚻᴺ߇᜼ߍࠄࠇ㧘⏕ቯ⺰⊛ᚻᴺߢߪ㧘㓐઻ᄌᢙᴺ1)߿⋥ធᓸಽᴺ2)㧘߹ߚ⏕₸−1−倉橋貴彦・吉荒⺰⊛ᚻᴺߢߪ㧘ࠞ࡞ࡑࡦࡈࠖ࡞࠲ℂ⺰ߦၮߠߊᣇᴺ3)-5)߇᜼ߍࠄࠇࠆ㧚ਔᚻᴺ⺰ߦࠃࠅ⸃ᨆኻ⽎㗔ၞౝߦ߅ߌࠆ⁁ᘒ㊂ಽᏓࠍផቯߔࠆߎߣߪน⢻ߢ޽ࠆ߇㧘ᧄᬌ⸛ߢߪ㧘‛ℂࡕ࠺࡞ߦ㑐ߔࠆᡰ㈩ᣇ⒟ᑼ߿᷹ⷰ
୯ߦ⺋Ꮕࠍ⠨ᘦߒߚࠞ࡞ࡑࡦࡈࠖ࡞࠲ℂ⺰ߦၮߠߊᣇᴺߦࠃࠅ㧘⸃ᨆኻ⽎㗔ၞౝߦ߅ߌࠆ⁁ᘒ㊂ߩផቯࠍⴕ߁㧚 ᳓ᮏౝߩᵹࠇࠍ⴫ߔ‛ℂࡕ࠺࡞ߣߒߡߪ㧘ᵻ᳓㐳ᵄᣇ⒟ᑼߣߒ㧘ⓨ㑆ᣇะߦߪ᦭㒢ⷐ⚛ᴺ㧔ࠟ࡜࡯ࠠ
ࡦᴺ㧕㧘ᤨ㑆ᣇะߦኻߒߡߪ࠮࡟ࠢ࠹ࠖࡉ࡮࡜ࡦࡇ
ࡦࠣᴺߦࠃࠅ㔌ᢔൻࠍⴕ߁㧚ᢙ୯ታ㛎ߦࠃࠆᬌ⸛ߣ
ߒߡߪ㧘᳓ᮏౝߦ⸳⟎ߒߚ᳓૏ߩ⸘᷹࠺࡯࠲㧔࿑㧕ࠃࠅ㧘᳓ᮏౝߦ߅ߌࠆᵹㅦ߅ࠃ߮᳓૏ߩಽᏓߩផቯ♖ᐲߩᬌ⸽ࠍⴕ߁㧚太一・衛藤俊彦

No.3

No.2

No.1
Inflow direction

0.50m

y

0.25m

x

2.00m

2.00m
4.00m

: Observation point࿑-2  ᳓ᮏᐔ㕙ኸᴺ߅ࠃ᷹߮ⷰὐߩ૏⟎(a) 1࿁⋡࿑-1  ᵄേ᳓ᮏࠍ૶↪ߒߚ᳓૏⸘᷹ߩ᭽ሶ㧞㧚᳓૏⸘᷹ߦߟ޿ߡ᳓૏⸘᷹ߩ૏⟎㧔᷹ⷰὐ㧕ࠍ࿑ߦ␜ߔ㧚㕒᳓ᤨ
ߩ᳓ᷓߪ0.46mߢ޽ࠅ㧘ㅧᵄᤨߪ㧘ᝄ᏷ࠍ⴫ߔࠬ࡜
ࠗ࠳࡯ࡧࠜ࡝ࡘ࡯ࡓ୯ࠍ5.0㧘๟ᦼࠍ5.0secߣ⸳ቯߒߚ㧚᳓૏ߩ⸘᷹ߪ㕒᳓ᤨߩ᳓㕙߆ࠄߩ᳓૏ᄌേ㊂ࠍ⸘᷹ߔࠆߎߣߦߒ‫ᤨޔ‬㑆ಽ⸃⢻ࠍǍt=0.01sec.ߣߒ㧘10sec.ߩ⸘᷹ࠍ4࿁ⴕߞߚ㧚᳓૏ᵄᒻࠍ࿑ߦ␜ߔ㧚xゲߪᤨ㑆ࠍ␜ߒ㧘yゲߪ㕒᳓ᤨ߆ࠄߩ᳓૏ᄌൻ㊂ࠍ(b) 2࿁⋡␜ߒߡ޿ࠆ㧚ߤߩ⚿ᨐߦ߅޿ߡ߽㧘᷹ⷰὐߏߣߦᄙዋߩᝄ᏷ߩᄌൻ߽⷗ฃߌࠄࠇࠆ߇㧘ฦ᷹ⷰὐߦ߅ߌࠆᵄᒻࠃࠅ㧘ᵄ߇વ៝ߒߡ޿ࠆ᭽ሶ߇ಽ߆ࠆ㧚(c) 3࿁⋡−2−0.20m

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ᑼ(1)㨪(3)ߩ㔌ᢔൻߦኻߒߡ㧘ⓨ㑆ᣇะߩ㔌ᢔൻߦਃⷺᒻ৻ᰴⷐ⚛ࠍ↪޿ߚࠟ࡜࡯ࠠࡦᴺ㧘ᤨ㑆ᣇะߩ㔌ᢔൻߦኻߒߡ࠮࡟ࠢ࠹ࠖࡉ࡮࡜ࡦࡇࡦࠣᴺࠍ↪޿ࠆߣ㧘᦭㒢ⷐ⚛ᣇ⒟ᑼߣߒߡᑼ(4)߇ዉ߆ࠇࠆ㧚n 1^ `­K1n ½° n°®K 2 ¾°K n °¯ 3¿­v1n ½° n °㧘 n
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୯቟ቯᕈࠍ⏕଻ߔࠆߚ߼ߩࡄ࡜ࡔ࡯࠲ߢ޽ࠅ㧘0㨪1ߩ㑆ߢㆬቯߐࠇࠆ㧚ߎߩࡄ࡜ࡔ࡯࠲ߪ㧘৻⥸ߦ0.8⒟ᐲߩ୯߇↪޿ࠄࠇࠆ㧚Aeߪਃⷺᒻⷐ⚛ߩ㕙Ⓧࠍ␜ߔ㧚߹ߚ㧘ਃⷺᒻⷐ⚛ߦ߅ߌࠆᒻ⁁㑐ᢙࠍNi 㧔i=1
㨪3㧕ߣߔࠆߣ㧘bi ߪNi/x㧘ci ߪNi/yߦࠃࠅ⴫ߐ−3−倉橋貴彦・吉荒ࠇࠆ㧚⹦⚦ߪᢥ₂6)ࠍෳᾖߒߡ㗂ߊߎߣߦߔࠆ㧚ߎߎߢ㧘ᑼ(4)ߦ␜ߔ᦭㒢ⷐ⚛ᣇ⒟ᑼࠍᑼ(6)ߩ᭽ߦ◲⇛⊛ߦ⸥ㅀߔࠆ㧚^I ` >A@^I `n 1(6)

n太一・衛藤ߎߎߢ㧘ᑼ(9)ߦࠃߞߡ᳞߹ߞߚn+1ࠬ࠹࠶ࡊߩหൻ^ `೨ߩ⁁ᘒᄌᢙ Iˆ(n)1 ߣ㧘᷹ⷰ୯{zn+1}ߦࠃࠅ㧘หൻᓟߩn+1ࠬ࠹࠶ࡊ⋡ߩ⁁ᘒᄌᢙ߇᳞߹ࠆ߽ߩߣ઒ቯߒ㧘
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ࠬ࠹ࡓࡁࠗ࠭ߦࠃࠆࡌࠢ࠻࡞{qn}߇ਸ਼ߓࠄࠇߚࡌࠢ
࠻࡞ࠍ㧘ᑼ(6)ߦട߃ࠆߎߣߦࠃࠅ⌀୯ࠍ⴫ߔ߽ߩ
ߣߔࠆ㧚ࠪࠬ࠹ࡓࡁࠗ࠭ߪᐔဋ0㧘ಽᢔı2ߦࠃࠅ⴫ߒߚᱜⷙੂᢙߦࠃࠅ⴫ߐࠇࠆ߽ߩߣߔࠆߣ㧘⌀୯ߩᤨ㑆ㅴዷߩᣇ⒟ᑼߪ㧘ᑼ(7)ߩࠃ߁ߦਈ߃ࠄࠇࠆ㧚俊彦n 1()11n 1

n 1( )2n 11(10)n 1( )^ `⁁ᘒᄌᢙ Iˆ(n)1 ߣ㧘᷹ⷰ୯{zn+1}ߩ೨ߦ߆߆ࠆⴕ೉
[K1]ߣ[K2]ߪࠞ࡞ࡑࡦࠥࠗࡦⴕ೉ߣ๭߫ࠇ㧘ⴕ೉[K2]^I ` >A@^I ` >*@^q `n 1nn(7)ᑼ(7)ߪࠞ࡞ࡑࡦࡈࠖ࡞࠲ߩℂ⺰ߦ߅޿ߡࠪࠬ࠹ࡓ
ᣇ⒟ᑼߣ๭߫ࠇࠆ㧚߹ߚⴕ೉[A]ߪ⁁ᘒㆫ⒖ⴕ೉ߣ๭߫ࠇࠆ㧚ᰴߦ᷹ⷰ୯ߦߟ޿ߡߪᑼ(8)ߩࠃ߁ߦਈ
߃ࠄࠇࠆ㧚^z ` >H @^I ` ^r `n 1n 1n 1ߪࠞ࡞ࡑࡦࡈࠖ࡞࠲ℂ⺰ߦၮߠߊᑼዷ㐿ߦࠃࠅ㧘ⴕ
೉[K1]ࠍ૶ߞߡ⴫ߐࠇࠆߚ߼[f(K1)]ߣᦠ߈⴫ߒߡ޿
ࠆ㧚ᧄ⺰ᢥߢߪᑼዷ㐿ߩㆊ⒟߇㐳޿ߎߣ߆ࠄ㧘ᑼߩ⺃ዉߦߟ޿ߡߪᢥ₂7), 8)╬ࠍෳᾖߒߡ㗂ߊߎߣߦߔ^ `^ `ࠆ㧚ߎߎߢ㧘ផቯ୯ Iˆ(n)1 ߪ㧘⌀୯ I n 1 ߦផቯ⺋^ `

Ꮕ p (n)1 ࠍട߃ߚᒻߢ⴫ߐࠇࠆ߽ߩߣߔࠆ㧚(8)^Iˆ ` ^I ` ^p `n 1()n 1(11)n 1(){zn+1}ߪ᷹ⷰ୯ߦࠃࠅ᭴ᚑߐࠇࠆࡌࠢ࠻࡞ߢ޽ࠅ㧘⌀୯ I n 1 ߦ㧘᷹ⷰࡁࠗ࠭{rn+1}ࠍട߃ߚ୯ߦࠃࠅ⴫

(10)ᑼߣ(11)ᑼߦࠃࠅ㧘ផቯ⺋Ꮕߪࠞ࡞ࡑࡦࠥࠗࡦ
ⴕ೉[K1]ߦࠃࠅ⴫ߐࠇࠆߚ߼㧘ᑼ(12)ߩࠃ߁ߦࠞ࡞ߐࠇࠆ߽ߩߣߔࠆ㧚᷹ⷰࡁࠗ࠭{rn+1}߽ࠪࠬ࠹ࡓࡁ
ࠗ࠭ߣห᭽ߦ㧘ᐔဋ0㧘ಽᢔı2ߦࠃࠅ⴫ߒߚᱜⷙੂ
ᢙߦࠃࠅ⴫ߐࠇࠆ߽ߩߣߔࠆ㧚ⴕ೉[H]ߪ᷹ⷰⴕ೉
ߣ๭߫ࠇ㧘ฦ᷹ⷰ୯ߦኻߒߡኻᔕߔࠆ⌀୯ߩᚑಽߦ
1㧘ߘࠇએᄖߦߪ0߇౉ࠆⴕ೉ߣߔࠆ㧚ᑼ(7)㧘(8)ߪ

ࡑࡦࠥࠗࡦⴕ೉ߩ㑐ᢙߣߒߡ⴫ߔߎߣ߇ߢ߈ࠆ㧚ᑼ(10)ߩฝㄝ╙ੑ㗄ߩ㑐ᢙfߣ඙೎ߔࠆߚ߼ߦ㧘ߎߎߢ^ `ࠞ࡞ࡑࡦࡈࠖ࡞࠲ߩℂ⺰ߩᦨೋߦ⃻ࠇࠆᑼߢ޽ࠆ㧚 ߎߎߢ㧘᷹ⷰ୯ࠍ߽ߣߦ㧘⸃ᨆኻ⽎㗔ၞౝߦ߅ߌࠆ⁁ᘒ㊂ಽᏓߩផቯࠍⴕ߁ߎߣࠍ⠨߃ࠆ㧚᷹ⷰ୯ࠍ૶ߞߡ㧘ᑼ(6)ߦࠃࠆ⚿ᨐࠍ⵬ᱜߔࠆᠲ૞ࠍหൻߣ
๭߮㧘หൻ೨ࠍ(㧙)㧘หൻᓟࠍ(+)ࠍઃടߒߚᄌᢙߦ
ࠃࠅ⴫ߔ㧚߹ߚ㧘⌀୯ ^I`ߣ඙೎ߔࠆߚ߼ߦ㧘ផቯ^`୯ߦߪࡂ࠶࠻ࠍઃߌ Iˆ ߣ⴫␜ߔࠆ㧚หൻߐࠇߚnࠬߪ㧘㑐ᢙgߦࠃࠅᦠ߈⴫ߒߡ޿ࠆ㧚^p ` ^g K `n 1()(12)1ߎߎߢ㧘หൻᓟߩផቯ⺋Ꮕࠍᦨዊߣߔࠆࠃ߁ߥ㧘ࠞ࡞ࡑࡦࠥࠗࡦⴕ೉[K1]ࠍ᳿ቯߔࠆߚ߼ߩᑼዷ㐿ࠍⴕ߁㧚ߎߎߦߟ޿ߡ߽㧘ᑼ⺃ዉߩᚻ㗅ߪᢥ₂7), 8)╬ࠍෳᾖߒߡ㗂ߊߎߣߦߔࠆ߇㧘᷹ⷰ⺋Ꮕߣหൻ೨ߩផ
ቯ⺋Ꮕߪή⋧㑐ߣ޿߁઒ቯࠍ౉ࠇߚᑼዷ㐿ࠍⴕ߁ߚ>@߼㧘ផቯ⺋Ꮕߦኻߔࠆ౒ಽᢔⴕ೉ P(n) 1 㧔ᑼ(13)㧕

>@࠹࠶ࡊߩ⁁ᘒᄌᢙࠍ૶޿㧘᦭㒢ⷐ⚛ᣇ⒟ᑼࠍ⸃ߊߣ㧘 ࠍ⸘▚ߒ㧘ⴕ೉ n 1 ߩ࠻࡟࡯ࠬࡁ࡞ࡓ㧔ᑼ(14)㧕
P(  )
ᑼ(9)ߩࠃ߁ߦหൻ೨ߩn+1ࠬ࠹࠶ࡊߩ୯߇᳞߹ࠆ㧚߇ᦨዊߣߥࠆ᧦ઙߦࠃࠅ㧘ࠞ࡞ࡑࡦࠥࠗࡦⴕ೉[K1]ߩ⸘▚ᑼࠍ⺃ዉߔࠆ㧚(9)Iˆ(n)1 >A@ Iˆ(n)^ `^ `−4−波動水槽による水位計測値を用いた水槽内における流れ場の推定解析^Iˆ ` >A@^Iˆ `1>P @ imax
¦ ^p ` ^p ` ^p ` ^p ` (13)imax 1n 1()n 0n 1()Tn 1()()n 1( )()7.

>

trace P(n) 1@ pn 1(  )1,1

 p (n)12, 2    p (n)13unx,3unx

3u nx¦pi 1(14)>w
trace P(n) 1 K 1w>K 1 @@ >0@ࠄ㧘ផቯ⺋Ꮕߦኻߔࠆ౒ಽᢔⴕ೉ߪ3nx㧘3nxߩ
ࠨࠗ࠭ߦߥࠆߚ߼㧘ᑼ(14)ߩ⸘▚ᑼߢߪផቯ⺋Ꮕߦኻߔࠆ౒ಽᢔⴕ೉ߩኻⷺᚑಽߩ୯ࠍi=1㨪3nx߹ߢ⿷ߔࠃ߁ߦ⸥ߒߡ޿ࠆ㧚߹ߚ㧘ᑼ(15)ߪ㧘ផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ࠍࠞ࡞ࡑࡦࠥࠗࡦⴕ೉ߩฦᚑಽߢᓸಽߒ㧘⚿ᨐߣߒߡᓧࠄࠇࠆⴕ೉߇㔖ⴕ೉ߦߥࠆߎߣࠍ␜ߒߡ߅ࠅ㧘ߎߩ᧦ઙᑼࠍḩߚߔࠃ߁ߦࠞ࡞ࡑࡦࠥࠗࡦⴕ೉[K1]ߩ⸘▚ᑼ߇ᓧࠄࠇࠆ㧚એ਄ߩ⠨߃ࠍ߽ߣߦ㧘ࠞ࡞ࡑࡦࡈࠖ࡞࠲ℂ⺰ߦࠃࠆ⁁ᘒផቯ⸘▚ߩࠕ࡞ࠧ࡝࠭ࡓߪએਅߩࠃ߁ߦ᭴▽ߢ߈ࠆ㧚[ࠞ࡞ࡑࡦࠥࠗࡦⴕ೉ߩ⸘▚]1. ೋᦼ୯ߩ౉ജ

2.

00()

0

3.

T()

ࠞ࡞ࡑࡦࠥࠗࡦⴕ೉ߩ⸘▚>K1 @ >P() @>H @T >H @>P( ) @>H @T  >R@T4.

ផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ⸘▚>P @ >P @  >K @>H @>P @()

5.

()

1()

1n 1n 1( )ᤨ㑆ࠬ࠹࠶ࡊߩ್ቯ㧦߽ߒ㧘n+1=imaxߥࠄ⸘▚⚳ੌ㧚ߘ߁ߢߥߌࠇ߫㧘
ࠬ࠹࠶ࡊ6߳㧚ߣߔࠆߣ㧘หൻ೨ᓟ߽฽߼ߡ⺋Ꮕ౒ಽᢔⴕ೉߇ᤨ㑆⊛ߦᄌൻࠍߒߥߊߥࠅ㧘⚿ᨐߣߒߡࠞ࡞ࡑࡦࠥࠗࡦ
ⴕ೉[K1]ߪᤨ㑆ߦኻߒߡᄌൻߒߥ޿ߎߣߦߥࠆ㧚਄⸥ߦ␜ߔ⸘▚ࠕ࡞ࠧ࡝࠭ࡓߪ㧘หൻ೨ᓟ߽฽߼ߡ⺋Ꮕ౒ಽᢔⴕ೉߇ᤨਇᄌߢ޽ࠆߎߣࠍ઒ቯߒ㧘⸳ቯߐࠇߚ߽ߩߢ޽ࠆ㧚߹ߚ㧘ࠞ࡞ࡑࡦࠥࠗࡦⴕ೉ߩ⸘▚ߦ␜ߒߚᄌᢙkߪផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ࠍᦨዊߦߔࠆ➅ࠅ㄰ߒ⸘▚ߦ߅ߌࠆ෻ᓳ࿁ᢙࠍ␜ߔ㧚ࠬ࠹࠶ࡊ5ߦ߅ߌࠆ෼᧤್ቯߦ↪޿ࠆផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ߪ㧘෻ᓳ࿁ᢙk=0ߦ߅ߌࠆផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ߦࠃࠅᱜⷙൻߒߚ୯ࠍ↪޿ߡ㧘෼᧤್ቯࠍⴕ߁ߎߣߦߔࠆ㧚

3┨ߦ␜ߒߚ⸘▚ࠕ࡞ࠧ࡝࠭ࡓߦᓥ޿㧘2┨ߦ␜ߒ>P @ >A@>P @>A@  >*@>Q@>*@Tn 1( )ߎߎߦ㧘ⴕ೉[Q]߅ࠃ߮[R]ߪ㧘ࠪࠬ࠹ࡓ⺋Ꮕ߅ࠃ᷹߮ⷰ⺋Ꮕߦࠃࠆ౒ಽᢔⴕ೉ࠍ␜ߔ㧚ߥ߅㧘ⴕ೉[A]㧘
[ǌ]㧘[Q]㧘[H]㧘[R]߇ᤨ㑆ߦኻߒߡᄌൻߒߥ޿߽ߩ੍᷹⺋Ꮕ౒ಽᢔⴕ೉ߩ⸘▚()^Iˆ ` ^Iˆ ` >K @ ^z ` >H @^Iˆ `㧠㧚᳓૏⸘᷹୯ࠍ૶ߞߚᵹࠇ႐ߩផቯ⸘▚ߦ㑐ߔࠆᬌ⸛

>P @ >Pˆ @, ^I ` ^Iˆ `0()

8.

(15)

ߎߎߦ㧘imaxߪ⸃ᨆኻ⽎ᤨ㑆ࠍ␜ߒ㧘nxߪ⸃ᨆኻ⽎㗔ၞߩ✚▵ὐᢙࠍ⴫ߔ㧚ᧄ⺰ᢥߢߪ㧘x,yᣇะᵹㅦu,v߅ࠃ߮᳓૏ᄌേ㊂Ș߇⸃ᨆኻ⽎ᄌᢙߢ޽ࠆߎߣ߆หൻᓟߩᦨㆡផቯ୯ߩ⸘▚n 1()(  ) iin()1ߚታ㛎ࡕ࠺࡞ߦኻߔࠆᵹࠇ႐ߩផቯ⸘▚ࠍⴕ߁㧚⸃ᨆߦ↪޿ࠆ᦭㒢ⷐ⚛ࡔ࠶ࠪࡘ࿑ߪ࿑ߦ␜ߔ㧚㧔ᵹࠇߩᣇะߪ㧘᳓ᮏታ㛎ߣߩⷞὐࠍวࠊߖࠆߚ߼ߦ㧘ฝ஥ߢ⊒↢ߒߚᵄ߇Ꮐ஥߳વ᠞ߔࠆ߽ߩߣߒߡ⸃ᨆࡕ࠺࡞㧔᦭㒢ⷐ⚛ࡔ࠶ࠪࡘ࿑㧕ࠍ૞ᚑߒߡ޿ࠆ㧚㧕߹ߚ⸃ᨆߦ↪޿ࠆ⸘▚᧦ઙߪ⴫ߦ␜ߔ㧚ߎߎߦ␜ߔᢙ୯⸃ᨆ଀ߢߪ㧘࿑ߦ߅ߌࠆ᷹ⷰὐNo.1㧘No.2ߦ߅ߌࠆ᳓૏ߩ⚻ᤨᄌൻߩ୯ࠍ↪޿㧘᷹ⷰὐNo.3ߦ
߅ߌࠆ᳓૏ߩ⚻ᤨᄌൻߦߟ޿ߡᦨㆡផቯ୯ߣ᷹ⷰ୯ࠍᲧセߒ㧘ផቯ♖ᐲߦ㑐ߔࠆ⠨ኤࠍⴕ߁㧚᷹ⷰ୯ߣផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ߦࠃࠆ෼᧤್ቯ㧦߽ߒ㧘|trace[P(+)](k+1) 㧙trace[P(+)](k)|<İߥࠄࠬ࠹࠶ࡊ 6߳㧚 ߒߡߪ㧘࿑ߦ߅ߌࠆ1࿁⋡ߩ᷹ⷰ୯ࠍណ↪ߒ㧘ᤨ㑆ಽ⸃⢻Ǎt=0.01sec.ߢ෼㍳ߐࠇߚ᷹ⷰ୯㑆ࠍߐߘ߁ߢߥߌࠇ߫ࠬ࠹࠶ࡊ2߳㧚ࠄߦ10╬ಽߒ㧘ᤨ㑆Ⴧಽ㊂ࠍǍt=0.001sec.ߣߒߚ߽
[ࠞ࡞ࡑࡦࠥࠗࡦⴕ೉ߩ⸘▚ᓟߩᦨㆡផቯ୯ߩ⸘▚]ߩࠍ↪޿ࠆ㧚
6. ᡰ㈩ᣇ⒟ᑼߩ᦭㒢ⷐ⚛ᣇ⒟ᑼߦࠃࠆࠬ࠹࠶ࡊߩ
ᦝᣂ−5−倉橋貴彦・吉荒太一・衛藤俊彦

ቯ⚿ᨐߩᅷᒰᕈࠍ࠴ࠚ࠶ࠢߔࠆߚ߼ߦ㧘ផቯὐࠍ
No.3ߩ᷹ⷰὐ㧔᷹ⷰ୯ߪ଻᦭ߒߡ޿ࠆ߇㧘޽߃ߡ૶↪ߒߥ޿᷹ⷰὐ㧕ߣߒ㧘ࠞ࡞ࡑࡦࡈࠖ࡞࠲ℂ⺰ߦࠃࠆ᳓૏ᄌേ㊂ߩផቯ୯ߣታ㓙ߦᓧࠄࠇߚ᷹ⷰ୯ߩᲧセࠍ࿑ߦ␜ߔ㧚⚿ᨐߣߒߡ㧘 No.3ߩὐߦ߅ߌࠆ
᷹ⷰ୯ߣߩ㑆ߦߪᏅ߇↢ߓߡ޿ࠆߎߣࠍ⏕⹺ߢ߈ࠆ㧚߹ߚ㧘࿑ߪ2.0sec.ߏߣߦ߅ߌࠆ᳓૏ᄌേ㊂߅ࠃ߮ᵹㅦߩᦨㆡផቯ୯ߩಽᏓࠍ␜ߒߡ޿ࠆ㧚ታ㛎ߢߪ᳓〝᏷ᣇะߦ߅޿ߡ৻᭽ߦ᳓૏ࠍᄌൻߐߖߡ޿ࠆߚ߼㧘᳓〝ゲᣇะߦኻߒߡ1ᰴరᵹࠇߦߥࠆߎߣ߇ℂᗐߢ޽ࠆ߇㧘᳓〝᏷ᣇะߦኻߒߡ߽᳓૏߇৻᭽ߢߪߥ޿⚿ᨐ߇ᓧࠄࠇߡ޿ࠆ㧚߹ߚ㧘ᵹㅦߩಽᏓߦߟ޿ߡߪ㧘ᵄߩવ᠞ߔࠆᣇะߦᵹࠇ߇↢ߓߡ޿ࠆ⚿ᨐߣߥߞߡ߅ࠅ㧘ᤨೞߏߣߦᵹㅦࡌࠢ࠻࡞ߩᄢ߈ߐ߽ᄌࠊߞߡ޿ࠆߎߣࠍ⏕⹺ߢ߈ࠆ㧚ߎߩ⚿ᨐࠃࠅ㧘ᵹࠇߩะ߈߿ᄢ߈ߐߦߟ޿ߡߪ㧘ㆡಾߦᓧࠄࠇߡ޿ࠆ߽ߩߣ⠨߃ࠄࠇࠆ㧚ផቯὐNo.3ߦ߅޿ߡߩᦨㆡផቯ୯ߣ᷹ⷰ୯ߦᏅ߇↢ߓߡ޿ࠆߎߣ߿㧘᳓〝ゲᣇะߦኻߔࠆᵄᒻߩಽᏓ߇ㆡಾߦᓧࠄࠇߡ޿ߥ޿ὐߪ㧘ᵹࠇ႐ࠍ⴫
ߔࡕ࠺࡞ߣߒߡ⸳ቯߒߡ޿ࠆᡰ㈩ᣇ⒟ᑼ㧘߹ߚ᷹ⷰὐߩᢙ߿㧘᷹ⷰὐ૏⟎ߦଐࠆߣߎࠈ߇ᄢ߈޿ߣ⠨߃ࠄࠇ㧘ࠃࠅផቯ♖ᐲࠍ޽ߍࠆߚ߼ߦߪㆡಾߥᡰ㈩ᣇ
⒟ᑼ߅ࠃ᷹߮ⷰὐᢙߩ⸳ቯ㧘߹ߚ᷹ⷰὐߩ㈩⟎ࠍ⷗⋥ߔᔅⷐ߇޽ࠆ߽ߩߣ⠨߃ࠄࠇࠆ㧚᦭㒢ⷐ⚛ࡔ࠶ࠪࡘో૕࿑ 㧔࿑-4-1㧕

᦭㒢ⷐ⚛ࡔ࠶ࠪࡘ᜛ᄢ࿑ 㧔࿑-4-2㧕
࿑-4  ᦭㒢ⷐ⚛ࡔ࠶ࠪࡘ࿑
⴫-1 ⸘▚᧦ઙ✚▵ὐᢙ✚ⷐ⚛ᢙᤨ㑆Ⴧಽ㊂Ǎt㧘sec.
ᤨ㑆ࠬ࠹࠶ࡊ
࡜ࡦࡇࡦࠣࡄ࡜ࡔ࡯࠲e㊀ജടㅦᐲg㧘m/sec2
㕒᳓ᤨߩ᳓ᷓh, m㚟േⴕ೉[ǌ]ࠪࠬ࠹ࡓ⺋Ꮕ౒ಽᢔⴕ೉[Q]᷹ⷰ⺋Ꮕ౒ಽᢔⴕ೉[R]ផቯ⺋Ꮕ౒ಽᢔⴕ೉[P(+)]ߩ
ೋᦼ୯⁁ᘒᄌᢙࡌࠢ࠻࡞{Ǿ0 (+)}ߩ
ೋᦼ୯෼᧤್ቯቯᢙǭ

99
160
0.001
1000
0.80
9.8
0.46
න૏ⴕ೉
0.01න૏ⴕ೉
0.1න૏ⴕ೉
1.0න૏ⴕ೉
㔖ࡌࠢ࠻࡞
10-4

࿑-5  ផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ߩ෼᧤ ⸃ᨆ⚿ᨐࠍએਅߦ␜ߔ㧚߹ߕ㧘෻ᓳ࿁ᢙk=0ߦ߅ߌࠆផቯ⺋Ꮕ౒ಽᢔⴕ೉ߩ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ߦࠃࠅᱜⷙൻࠍⴕߞߚផቯ⺋Ꮕ౒ಽᢔⴕ೉ߦ㑐ߔࠆ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ߩ෼᧤ጁᱧࠍ࿑ߦ␜ߔ㧚⚿ᨐࠃࠅ㧘෻ᓳ⸘▚߇ㅴ߻ߦᓥ޿㧘ᱜⷙൻߐࠇߚ࠻࡟࡯ࠬࡁ࡞ࡓߩ୯ߪᷫዋߒ෼᧤ߒߡ޿ࠆߎߣ߇ࠊ߆ࠆ㧚ᰴߦ㧘
᷹ⷰὐNo.1㧘No.2ߦ߅ߌࠆ᷹ⷰ୯ߣᦨㆡផቯ୯ߩᲧセࠍ࿑ߦ␜ߔ㧚ฦ᷹ⷰὐߦ߅޿ߡ㧘ផቯ୯ߪ᷹ⷰ୯ߣ߅߅߻ߨ৻⥌ߒߡ޿ࠆߎߣ߇ࠊ߆ࠆ㧚߹ߚ㧘ផ

−6−ጁᱧ㧔෻ᓳ࿁ᢙ k=0 ߦ߅ߌࠆ୯ߦࠃࠅᱜⷙൻߒߚ୯㧕波動水槽による水位計測値を用いた水槽内における流れ場の推定解析

No.1(x , y) = (4 , 0.2) 㧔࿑-6-1㧕(a-1) t=2.0sec.ߦ߅ߌࠆ᳓૏ಽᏓ 㧔࿑-8-1㧕

No.2(x , y) = (2 , 0.2) 㧔࿑-6-2㧕

࿑-6  ᷹ⷰὐ No.1㧘No.2 ߦ߅ߌࠆ᷹ⷰ୯ߣᦨㆡផቯ୯ߩᲧセ (a-2) t=2.0sec.ߦ߅ߌࠆᵹㅦಽᏓ 㧔࿑-8-2㧕

No.3(x , y) = (0 , 0.2)

࿑-7  ផቯὐ No.3 ߦ߅ߌࠆ᷹ⷰ୯ߣᦨㆡផቯ୯ߩᲧセ(b-1) t=4.0sec.ߦ߅ߌࠆ᳓૏ಽᏓ 㧔࿑-8-3㧕

−7−倉橋貴彦・吉荒太一・衛藤俊彦(b-2) t=4.0sec.ߦ߅ߌࠆᵹㅦಽᏓ 㧔࿑-8-4㧕(d-1) t=8.0sec.ߦ߅ߌࠆ᳓૏ಽᏓ 㧔࿑-8-7㧕(c-1) t=6.0sec.ߦ߅ߌࠆ᳓૏ಽᏓ 㧔࿑-8-5㧕(d-2) t=8.0sec.ߦ߅ߌࠆᵹㅦಽᏓ 㧔࿑-8-8㧕(c-2) t=6.0sec.ߦ߅ߌࠆᵹㅦಽᏓ 㧔࿑-8-6㧕(e-1) t=10.0sec.ߦ߅ߌࠆ᳓૏ಽᏓ 㧔࿑-8-9㧕

−8−波動水槽による水位計測値を用いた水槽内における流れ場の推定解析⻢ㄉ㧦ᧄ⺰ᢥࠍၫ╩ߔࠆߦ޽ߚࠅ㧘⑼ቇ⎇ⓥ⾌⵬ഥ㊄㧔ၮ⋚(C)㧕15K05786ߩេഥࠍฃߌߚ㧚ߎߎߦ⻢ᗧࠍ⴫ߔ㧚߹ߚ㧘ᧄ⺰ᢥߪ㧘ਛᄩᄢቇℂᎿቇㇱㇺᏒ
ⅣႺቇ⑼ ฬ⹷ᢎ᝼ Ꮉේ⌬ੱ వ↢ߩ࠯ࡒ㧔᦭㒢ⷐ⚛
ᴺ㧘ㅒ⸃ᨆℂ⺰㧕ߩౝኈࠍਅߦ⎇ⓥࠍⴕߞߚ߽ߩߢ
޽ࠆ㧚ᧄ⺰ᢥਛߦ␜ߒߚᢙ୯⸘▚ࠍⴕ߁ߦ޽ߚࠅ㧘਻Ꮊᄢቇᖱႎၮ⋚⎇ⓥ㐿⊒࠮ࡦ࠲࡯ߩ㜞ᕈ⢻Ṷ▚ࠨ࡯ࡃࠪࠬ࠹ࡓPRIMERGY CX400ࠍ೑↪ߐߖߡ㗂޿ߚ㧚߹ߚ㧘ࠞ࡞ࡑࡦࡈࠖ࡞࠲ߩℂ⺰ߦߟ޿ߡߪ㧘㐳
ጟᛛⴚ⑼ቇᄢቇᄢቇ㒮 ᯏ᪾ഃㅧᎿቇኾ᡹ ಎᢎ᝼ ዊᨋᵏ⑲వ↢߆ࠄ߽⾆㊀ߥᗧ⷗ࠍ㗂޿ߚ㧚Ꮉේ⌬ੱవ↢㧘ዊᨋᵏ⑲వ↢ߥࠄ߮ߦ਻Ꮊᄢቇᖱႎၮ⋚⎇ⓥ㐿
⊒࠮ࡦ࠲࡯ߩࠬ࠲࠶ࡈߩᣇ‫ߦޘ‬ኻߒߡ߽⻢ᗧࠍ⴫ߔ㧚(e-2) t=10.0sec.ߦ߅ߌࠆᵹㅦಽᏓ 㧔࿑-8-10㧕

ෳ⠨ᢥ₂1)

T.Kurahashi and M.Kawahara, Examinations for Terminal
Condition of Lagrange Multiplier for Heat Transfer

࿑-8  2.0sec.ߏߣߦ߅ߌࠆᵹㅦ㧘᳓૏ᄌേ㊂ߩಽᏓControl Problems, International Journal for Numerical
Methods in Engineering, Vol.73, pp.982-1009, 2008.
2)

㧡㧚⚿⺰

᧻ᧄ ᢅ㇢, ↰ਛ ᱜ㓉, ችᎹ ቞, “Ⴚ⇇ⷐ⚛ᴺࠍ↪޿ߚ⋥ធᓸಽᴺߦၮߠߊ㊄ဳߩਃᰴరቯᏱᾲવዉ⸳⸘ᗵᧄ⺰ᢥߢߪ㧘ᵄേ᳓ᮏߦ߅ߌࠆ᳓૏⸘᷹୯ࠍ↪޿ߚᵹࠇ႐ߩផቯ⸃ᨆࠍⴕߞߚ㧚ᵹࠇ႐ߩ⸃ᨆߦኻߒߡߪ㧘ᡰ㈩ᣇ⒟ᑼߣߒߡᵻ᳓㐳ᵄᣇ⒟ᑼࠍዉ౉ߒ㧘ⓨ㑆ᣇะߩ㔌ᢔൻߦኻߒߡߪ㧘ਃⷺᒻ৻ᰴⷐ⚛ࠍ↪
޿ߚࠟ࡜࡯ࠠࡦᴺ㧘ᤨ㑆ᣇะߩ㔌ᢔൻߦߪ࠮࡟ࠢ࠹
ࠖࡉ࡮࡜ࡦࡇࡦࠣᴺࠍ↪޿ߚ㧚߹ߚ㧘ᵹࠇ႐ߩផቯ⸃ᨆߦߟ޿ߡߪ㧘ࠞ࡞ࡑࡦࡈࠖ࡞࠲ߩℂ⺰ࠍㆡ↪ߒߚ㧚⚿ᨐߣߒߡ㧘᷹ⷰ୯ࠍ૶↪ߒߚὐߦ߅޿ߡߪ㧘ផቯ୯ߪ᷹ⷰ୯ߣ߅߅߻ߨ৻⥌ߒߡ޿ࠆߎߣࠍ⏕⹺ߢ߈ߚ㧚ߒ߆ߒ㧘᷹ⷰὐࠍ᦭ߔࠆNo.3ߩὐߦ߅޿ߡ㧘᷹ⷰ୯ߣᦨㆡផቯ୯ࠍᲧߴߚߣߎࠈ㧘Ꮕ߇↢ߓߡ޿ࠆߎߣࠍ⏕⹺ߢ߈㧘ផቯߐࠇߚ᳓૏ಽᏓߣᵹㅦಽᏓߦߟ޿ߡߪ㧘᳓૏ಽᏓߪ㧘᷹ⷰὐࠍ㒰߈ㆡಾߥ᳓૏ߩ୯߇ᓧࠄࠇߡ޿ߥ޿߇㧘ᵹㅦಽᏓߪ᳓〝ౝߩᵹࠇߩᣇะߦᴪߞߚࡌࠢ࠻࡞ಽᏓ߇ᓧࠄࠇߡ޿ࠆߎߣࠍ⏕⹺ߢ߈ߚ㧚ᧄᬌ⸛ߦ߅޿ߡ૶↪ߒߡ޿ࠆᡰ㈩ᣇ⒟ᑼߦ߅޿ߡ㧘☼ᕈߩᓇ㗀߿᳓〝ᐩ㕙ߩ៺ᡂߩലᨐࠍ㒰෰ߒߚᑼࠍ૶↪ߒߡ޿ࠆߎߣ߿㧘⸳ቯߔࠆ᷹ⷰὐߩᢙ߿᷹ⷰὐߩ㈩⟎૏⟎ߦࠃߞߡ߽ផቯ♖ᐲ߇ᄌࠊࠆ߽ߩߣ⠨߃ࠄࠇࠆߎߣ߆ࠄ㧘੹ᓟߪ㧘⸃ᨆߦ૶↪
ߔࠆᡰ㈩ᣇ⒟ᑼ߿᷹ⷰὐᢙ㧘᷹ⷰὐ૏⟎߇ផቯ♖ᐲߦਈ߃ࠆᓇ㗀ߦߟ޿ߡ⠨ኤࠍߔࠆ੍ቯߢ޽ࠆ㧚

ᐲ⸃ᨆᴺ”, ᣣᧄᯏ᪾ቇળ⺰ᢥ㓸 A ✬, Vol. 59, No. 567,
pp.2750-2757, 1999.
3)

R.E.Kalman㧘 A New Approach to Linear Filtering andPrediction Problems㧘Transactions of the ASME-Journalof Basic of Engineering, Vol.82, No.1, pp.35-45, 1960.

4)

R.E.Kalman and R.S.Bucy㧘 A New Approach to LinearFiltering and Prediction Problems㧘 Transactions of theASME-Journal of Basic of Engineering, Vol.82, No.1,
pp.35-45, 1960.

5)

K.Yonekawa and M.Kawahara, Application of Kalman
Filter Finite Element Method and AIC, International
Journal of Computational Fluid Dynamics, Vol. 17㧘No. 4㧘
pp.307-317, 2003.

6)┻ౝ ೣ㓶, ᮯጊ ๺↵, ኹ↰ ⾫ੑ㇢㧘”⸘▚ജቇ ᦭㒢ⷐ⚛ᴺߩၮ␆”㧘᫪ർ಴7)⿷┙ ୃ৻㧘ਣ↰ ৻㇢, “ࠞ࡞ࡑࡦࡈࠖ࡞࠲ߩၮ␆”, ᧲੩㔚ᯏᄢቇ಴8)−9−, pp.175, 2003.

ዪ, pp.95-121, 2012.

᧛਄ ┨, ⊓ဈ ትᅢ, ၳ ቬᦶ, ㋈ᧁ ⺈, ̌᦭㒢ⷐ⚛ᴺ࡮Ⴚ⇇ⷐ⚛ᴺߦࠃࠆㅒ໧㗴⸃ᨆ – ࠞ࡞ࡑࡦࡈࠖ࡞࠲ߣ╬ଔ੺࿷‛ᴺߩᔕ↪ –̍, ࠦࡠ࠽␠, pp.90-94, ࠦࡠ࠽␠, 2002.
(2015. 9. 7 ฃઃ)