Studies on Development of a Numerical Method for Fluid-Structure Interaction Analysis with Free Surface based on ALE Finite Element Method

म࢜࿦จཁࢫʢ2008^೥౓ʣ

ALE ༗ݶཁૉ๏ʹجͮࣗ͘༝ද໘Λ༗͢Δྲྀମ - ߏ଄࿈੒ղੳख๏ͷߏஙݚڀ

Studies on Development of a Numerical Method for Fluid-Structure Interaction Analysis with Free Surface based on ALE Finite Element Method

౔໦޻ֶઐ߈ɹ15߸ Տݪ࡚ɹ༤հ Yusuke KAWARASAKI

1.

௡೾΍౔ੴྲྀʹΑͬͯӡ͹Εͨ෺ମ͕అ๷΍Ո԰ʹি

ಥ͢Δͱɼ࣌ʹߏ଄෺͸ਙେͳඃ֐Λड͚Δɽ͜ͷΑ͏

ͳ໰୊͸ɼྲྀମͱߏ଄ͷڍಈ͕૬ؔ͢Δྲྀମ-ߏ଄࿈੒໰

୊Ͱ͋Γɼͦͷྗֶݱ৅ͷఆྔతͳ೺Ѳʹ͸ɼ࣌ʑࠁʑ ͱมԽ͢Δࣗ༝ද໘ͱߏ଄෺ͷԠ౴Λਫ਼౓ྑ͘ܭࢉ͢Δ ඞཁ͕͋Δɽྲྀମ-ߏ଄࿈੒ղੳख๏͸ɼք໘௥੻๏¹⁾ͱ ք໘ิ଍๏²⁾ʹେผ͞ΕΔɽք໘௥੻๏͸ղੳϝογϡ ͷڥքΛք໘ͱͯ͠௚઀తʹऔΓѻ͏ͨΊɼݻఆϝο γϡ্ͰϚʔΧʔ΍εΧϥʔؔ਺Λ༻͍ͯք໘Λؒ઀

తʹදݱ͢Δք໘ิ଍๏ʹൺ΂ɼք໘Λਫ਼౓ྑ͘ධՁ͢

Δ͜ͱ͕ՄೳͰ͋Δͱ͍͏௕ॴΛ༗͍ͯ͠Δɽ͔͠͠ͳ

͕Βɼք໘௥੻๏Ͱ͸େมܗΛ൐͏ࣗ༝ද໘΍ߏ଄෺ͷ

େҠಈΛ൐͏৔߹ɼղੳϝογϡ͕େ͖͘࿪Ή͜ͱʹΑ Γɼ਺஋తͳෆ҆ఆੑ͕ൃੜ͠ɼղੳΛਐΊΔ͜ͱ͕ࠔ

೉ͱͳΔ৔߹͕͋Δɽ͜ͷ໰୊఺Λࠀ෰͢ΔͨΊʹɼా

தΒ͸ɼόοΫάϥ΢ϯυϝογϡʹجͮ͘ϝογϡ࠶

ߏஙख๏Λಋೖͨ͠ALE༗ݶཁૉ๏ʹجͮ͘ྲྀମ–ߏ

଄࿈੒ղੳख๏³⁾ΛఏҊ͍ͯ͠Δɽ

ɹͦ͜ͰຊݚڀͰ͸ɼόοΫάϥ΢ϯυϝογϡʹجͮ

͘ϝογϡ࠶ߏஙख๏Λ༻͍ɼք໘௥੻๏ʹجͮࣗ͘༝

ද໘Λ༗͢Δྲྀମ-ߏ଄࿈੒໰୊Λਫ਼౓ྑ͘׌ͭϩόε τʹղੳՄೳͳख๏ͷߏஙΛ໨తͱ͍ͯ͠Δɽղੳख๏

ͱͯ͠ɼඇѹॖੑ೪ੑྲྀମΛԾఆͨ͠ྲྀମʹ͸҆ఆԽ༗

ݶཁૉ๏Λద༻͠ɼߏ଄෺͸߶ମΛԾఆ͠ɼ೚ҙܗঢ়ͷ औΓѻ͍͕ՄೳͱͳΔΑ͏վྑΛՃ͑ͨɽ·ͨɼ෺ମಉ

࢜΍෺ମͱน໘ͷ઀৮ʹ͸ݸผཁૉ๏ͷ઀৮ΞϧΰϦζ Ϝ⁴⁾Λಋೖͨ͠ɽ਺஋ղੳྫͱͯ͠ɼු༡෺ͷน໘িಥ

໰୊ΛऔΓ্͛ɼຊख๏ͷ༗ޮੑٴͼɼଥ౰ੑͷݕ౼Λ ߦ͏ɽ

2.

2.1 جૅํఔࣜ

ALEهड़͞Εͨඇѹॖੑ೪ੑྲྀମͷӡಈํఔࣜٴͼ

࿈ଓࣜ͸ͦΕͧΕҎԼͷࣜ(1)ɼ(2)Ͱද͞ΕΔɽ ρ

∂u

∂t +¯u· ∇u−f

− ∇ ·σ(uɼp) = 0ɼon Ω_f (1)

∇ ·u= 0ɼ on Ω_f (2)

͜͜Ͱɼρ͸ີ౓ɼu͸ྲྀ଎ϕΫτϧɼ¯u͸૬ରྲྀ଎ϕ Ϋτϧɼf͸෺ମྗϕΫτϧΛද͍ͯ͠Δ.͜͜Ͱର৅ͱ

͢Δྲྀମ͸NewtonྲྀମͰ͋ΔͨΊɼมܗ଎౓ςϯιϧ

ਤ– 1 ղੳྖҬͱڥք

ε(u)͸ҎԼͷࣜ(3)Ͱද͞ΕΔɽ ε(u) = 1

2

∇u+ (∇u)^T

ɼ (3)

͜ͷԾఆʹΑΓɼԠྗςϯιϧσ^{͸࣍ࣜͰද͞ΕΔɽ}

σ=−pI+ 2με(u)ɼ (4)

͜͜Ͱɼp^͸ѹྗɼμ͸೪ੑ܎਺Ͱ͋Δɽ·ͨɼDirichlet ܕɼNeumannܕڥք৚݅͸ɼͦΕͧΕ࣍ࣜͰ༩͑Β ΕΔɽ

u=g on Γgɼ (5) n·σ=h on Γ_hɼ (6)

͜͜Ͱɼgɼh͸ͦΕͧΕྲྀ଎ɼτϥΫγϣϯͷط஌ྔ

Λࣔ͠ɼn͸֎޲͖๏ઢϕΫτϧΛࣔ͢ɽࣗ༝ද໘ʹ͓

͍ͯ͸ɼҎԼͷӡಈֶత৚݅Λຬͨ͢ඞཁ͕͋Δɽ

¯

u·n= 0 on Γfs, (7)

͜͜ͰɼΓfs͸ࣗ༝ද໘Ͱ͋Δɽࣗ༝ද໘্Ͱ͸ɼྗֶ

త৚݅ͱͯ͠stress-free৚͕݅ద༻͞ΕΔɽ

ɹ·ͨɼ߶ମΛԾఆͨ͠ߏ଄෺ͷڍಈ͸ҎԼͷӡಈํఔ

ࣜʹࢧ഑͞ΕΔɽ

m^su˙^s=f^s on Ω_s. (8)

͜͜Ͱɼm^sɼf^s͸෺ମͷ࣭ྔٴͼ׳ੑϞʔϝϯτɼ֎

ྗՙॏͰ͋Γɼu˙^s͸෺ମͷॏ৺ҐஔͰఆٛ͞Εͨฒਐ

੒෼ͱճస੒෼ͷՃ଎౓Λද͢ɽਤ-1ʹ͸ྖҬͱڥքͷ ఆٛΛࣔ͢. ਤதͷΓ_fsɼΓ_fsi͸ͦΕͧΕɼࣗ༝ද໘ٴ ͼɼྲྀମͱߏ଄ͷ࿈੒ڥքΛද͢ɽ

2.1.1 ^{҆ఆԽ༗ݶཁૉ๏}

ྲྀମͷجૅํఔࣜ(1)ɼ(2)ʹରͯ͠ɼSUPG/PSPG

๏ʹجͮ҆͘ఆԽ༗ݶཁૉ๏Λద༻͢ΔͱɼҎԼͷॏΈ

෇͖࢒ࠩํఔ͕ࣜಘΒΕΔɽ

Ω

w·ρ ∂u

∂t +¯u· ∇u−f

dΩ +

Ωε(w) :σdΩ +

Ωq∇ ·udΩ +

n_el e=1

Ω^eτm

¯

u· ∇w−1 ρ∇q

· ρ

∂u

∂t +u¯· ∇u−f

− ∇ ·σ dΩ

=

Γh

w·hdΓɼ (9)

͜͜Ͱɼwɼq͸ͦΕͧΕӡಈํఔࣜٴͼɼ࿈ଓࣜʹର

͢ΔॏΈؔ਺Λද͠ɼ·ͨɼτm͸SUPG/PSPG๏ͷ

҆ఆԽύϥϝʔλͰ͋ΓҎԼͷΑ͏ʹఆٛ͞ΕΔɽ

τm= 2 Δt

₂

+2||u||

he

₂ +4ν

he2

₂₋¹₂

ɼ (10)

͜͜Ͱɼν ^{͸ಈ೪ੑ܎਺ɼ}he͸ཁૉαΠζͰ͋Δɽࣜ

(9)ʹରͯ͠ɼۭؒํ޲ʹ཭ࢄԽΛߦ͏ͱҎԼͷ༗ݶཁ

ૉํఔ͕ࣜಘΒΕΔ.

(M+M_δ)u˙+ (A+A_δ)u

−(G−Gδ)1

ρp+νDu=F+Fδ, (11) C^Tu+Mε∂u

∂t +Kε(¯u)u−Fε+Cε1

ρp= 0ɼ(12)

͜͜ͰɼMɼKɼCɼS͸܎਺ߦྻɼF͸֎ྗϕΫτϧͰ

͋Γɼఴࣈδ^ɼε^͸ͦΕͧΕSUPG߲ɼPSPG߲ʹىҼ

͢Δ΋ͷΛදΘ͢ɽͳ͓ɼNeumannڥք৚݅͸τϥΫ γϣϯϑϦʔͱͯࣗ͠વڥք৚݅ͱͯ͠ߟྀ͍ͯ͠Δɽ 2.2 ྲྀମͱߏ଄ͷ࿈੒ղੳख๏

҆ఆԽΛࢪ͞Εͨࣜ(1)͸ҎԼͷࣜ(13)ͷΑ͏ʹॻ

͖׵͑Δ͜ͱ͕Ͱ͖Δɽ

M ˙˜u+Ku˜ −G˜p=F˜ (13)

ࣜ(13)ʹ͓͚ΔɼղੳྖҬશମͷઅ఺ʹؔ͢Δม਺ϕ ΫτϧuɼF˜ΛҠಈڥքΓfsi্ͱͦΕҎ֎ͷྲྀମྖҬ ʹ۠ผ͠ɼҠಈڥք্ͷزԿֶత࿈ଓ৚݅ٴͼɼฏߧঢ় ଶΛߟྀͨ͠ྲྀମͷӡಈํఔࣜɼ࿈ଓࣜٴͼɼߏ଄ͷӡ ಈํఔࣜ͸ҎԼͷΑ͏ʹද͞ΕΔɽ

M˜^αα M˜^αγ M˜^γα M˜^γγ

˙ u^α

˙ u^s∗

+

K˜^αα K˜^αγ K˜^γα K˜^γγ

u^α u^s∗

− G˜^α

G˜^γ

p

= F˜^α

F˜^γ

(14)

M˜^α_ε M˜^γ_ε

˙ u^α

˙ u^s∗

+

C˜^α C˜^γ u^α u^s∗

+G_εp=Fε (15) m^su˙^s=F^s_C−F^γ (16)

͜͜ͰɼF^s_C ͸઀৮ྗΛද͠ɼu˙^s∗,u^s∗͸ߏ଄෺ͷॏ৺

ͱͦͷද໘ͷ֤અ఺ؒͷزԿֶతͳؔ܎Λߟྀͨ͠ߏ଄

෺ͷ෺ཧྔͰ͋Δɽࣜ(14),(15),(16)ͷ࣌ؒํ޲ͷ཭

ࢄԽͱͯ͠ɼΫϥϯΫɾχίϧιϯ๏Λ༻͍ɼ࿈ཱҰ࣍

ํఔࣜͷղ๏ʹ͸ɼGMRES๏Λ༻͍Δɽ 2.3 ղੳϑϩʔνϟʔτ

ຊݚڀʹ͓͚ΔɼղੳϑϩʔνϟʔτΛҎԼʹࣔ͢ɽ ղੳϑϩʔνϟʔτதͷྲྀ଎ɾѹྗɾߏ଄෺଎౓ͷٻղ ෦෼Ͱ͸ɼڧ࿈੒๏ʹΑΓྲྀମɼߏ଄ͦΕͧΕͷ෺ཧྔ

Λಉ࣌ʹٻΊ͍ͯΔɽҎ߱ɼղੳϑϩʔνϟʔτͷϝο

࠺࡯࠲౉ജ

ࡔ࠶ࠪࡘౣ᭴▽

࡮᭴ㅧ‛Ⴚ⇇ߩ⴫⃻

࡮⥄↱⴫㕙ᒻ⁁ߩ⴫⃻

࡮‛ℂ㊂ߩౣ㈩⟎

ធ⸅್ቯ

ᵹㅦ࡮࿶ജ࡮᭴ㅧ‛

ㅦᐲߩ᳞⸃

▵ὐᄌ૏ߩ᳞⸃

࠺࡯࠲಴ജ

Newton-Raphson෻ᓳ Time Loop

ਤ– 2 ղੳϑϩʔνϟʔτ

γϡ࠶ߏஙख๏ٴͼɼ઀৮൑ఆͷৄࡉΛड़΂Δɽ 2.4 όοΫάϥ΢ϯυϝογϡʹجͮ͘ϝογϡ࠶ߏ

ஙख๏

ຊݚڀͰ͸ɼେมܗ͢Δࣗ༝ද໘Λ༗͢Δ໰୊ΛऔΓ ѻ͏ͨΊʹɼόοΫάϥ΢ϯυϝογϡʹجͮ͘ϝο γϡ࠶ߏஙख๏Λಋೖ͢Δɽྲྀମ-ߏ଄࿈੒໰୊ͷͨΊ ͷϝογϡ࠶ߏஙख๏Ͱ͸ɼ͋Β͔͡Ίߏ଄෺ɼྲྀମ͕

Ҡಈ͢ΔͰ͋Ζ͏ྖҬશମʹόοΫάϥ΢ϯυϝογϡ Λ഑ஔ͢Δɽͦͯ͠ɼͦΕΛ༻͍ͯߏ଄෺ͷڥքΓfsiɼ

ࣗ༝ද໘ܗঢ়ΓfsΛਖ਼֬ʹදݱͨ͠ղੳྖҬͱϝογϡ Λߏங͠ɼ෺ཧྔͷ࠶഑ஔΛߦ͍ɼղੳΛਐΊΔ͜ͱͱ ͳΔ³⁾ɽ

2.5 ઀৮ͷ൑ఆٴͼ઀৮ྗͷධՁ

ߏ଄෺ͱน໘ͱͷ઀৮Λ൑ఆ͢ΔͨΊʹɼߏ଄෺ද໘ ٴͼɼน໘্ʹ઀৮൑ఆԁΛઃஔ͢Δɽ઀৮൑ఆԁͷ௚

ܘ͸น໘্ͷ࠷খϝογϡ෯ͱͨ͠ɽ·ͨɼ൑ఆԁͷઃ

ஔ͸ɼਤ–3ʹࣔ͢Α͏ʹɼน໘্ٴͼߏ଄෺্ʹ֤ԁͷ

͓͓Αͦ൒ܘ෼͕ॏͳΔΑ͏ʹ഑ஔͨ͠ɽ઀৮൑ఆ͸ɼ ߏ଄෺্ͷ֤ԁͱน໘্ͷ֤ԁͷத৺ڑ཭͕ɼ઀৮൑ఆ ԁͷ௚ܘҎԼͱͳͬͨ৔߹Λ઀৮ͱ͢Δɽ

ɹ઀৮ྗͷධՁʹ͸ɼݸผཁૉ๏Ͱ༻͍ΒΕΔํ๏Λಋ

ೖͨ͠ɽ઀৮͍ͯ͠Δͱ൑ఆ͞Εͨ൑ఆԁΛɼਤ–4 ʹ

‛૕

ធ⸅್ቯ౞

ਤ– 3 ڥք্ͷ઀৮൑ఆԁ

ਤ– 4 ๏ઢɾ઀ઢํ޲Voigt_Ϟσϧ

ࣔ͞ΕΔVoigtϞσϧʹΑΔ๏ઢํ޲ٴͼɼ઀ઢํ޲ͷ όωͱμογϡϙοτͰ࿈݁͢Δɽ͜͜Ͱɼ઀৮൑ఆԁ ʹൃੜͤ͞ΒΕͨ๏ઢํ޲όωʹΑΔྗ͸ɼҎԼͷࣜͰ ද͞ΕΔɽ

F_n=Knd_n+ηnu_n (17)

͜͜ͰɼKnɼηn͸ɼͦΕͧΕɼ๏ઢํ޲ͷόωఆ਺ɼ μογϡϙοτͷ೪ੑ܎਺Ͱ͋Δɽd_n͸઀৮൑ఆԁͷ

ॏ߹ڑ཭Ͱ͋Γɼu_n͸൑ఆԁத৺Ͱͷ૬ର଎౓ͷ๏ઢ

ํ޲੒෼Ͱ͋Δɽ·ͨɼ઀ઢํ޲όωʹΑΔྗ͸ɼҎԼ ͷࣜͰද͞ΕΔɽ

F_t=Ktd_t+ηtu_t (18)

͜͜ͰɼKtɼηt͸๏ઢํ޲ͷόωఆ਺ɼμογϡϙοτ ͷ೪ੑ܎਺Ͱ͋Δɽd_tɼu_t ͸ͦΕͧΕɼ઀৮൑ఆԁͷ

๏ઢํ޲มҐٴͼɼ૬ର଎౓ͷ઀ઢํ޲੒෼Λද͢ɽ·

ͨɼߏ଄෺ʹ࡞༻͢Δྗ͸ɼ֤઀৮൑ఆԁͰͷόωʹΑ ΔྗF_nɼF_tΛߏ଄෺ਤ৺ҐஔͰධՁ͢Δɽͳ͓ɼόω ఆ਺KnɼKt͸ఆ਺ͱͯ͠༩͑ͨɽ·ͨɼμογϡϙο τͷ೪ੑ܎਺͸ɼ࣍ࣜʹΑΓܾఆͨ͠ɽ

η=−2√ mKlne

√π²+ lne (19)

͜͜Ͱɼm͸ߏ଄෺ͷ࣭ྔͰ͋Γɼe͸൓ൃ܎਺Ͱ͋Δ. 2.6 ೚ҙܗঢ়෺ମͷ׳ੑϞʔϝϯτ

೚ҙܗঢ়ͷ෺ମΛऔΓѻ͏৔߹ɼߏ଄ͷӡಈํఔࣜ

(8)தͷ࣭ྔߦྻʹ͓͚Δ׳ੑϞʔϝϯτΛٻΊΔඞཁ

͕͋ΔɽҎԼʹͦͷखॱΛࣔ͢ɽ·ͣɼਤ–5(ӈ্)ͷΑ

͏ʹ෺ମʹରͯ͠෼ׂΛߦ͏ɽཁૉ෼ׂΛߦ͏͜ͱͰɼ

෺ମΛ༗ݶݸͷ࣭఺ͷू߹ͱԾఆ͢Δɽଓ͍ͯɼ෺ମͷ

ॏ৺఺ΛٻΊΔͨΊʹɼਤ–5(Լ)ͷΑ͏ʹ֤ཁૉͷॏ৺

‛૕

ਤ– 5 ෺ମʹର͢Δཁૉ෼ׂͱॏ৺఺

఺ΛٻΊɼ֤࣭఺ͱ෺ମશମͷྗͷ௼߹͍ͷؔ܎ΑΓ෺

ମͷॏ৺఺ΛٻΊΔɽ࠷ऴతʹҎԼͷࣜΑΓɼ෺ମͷ׳

ੑϞʔϝϯτIΛٻΊΔɽ

I= miR²i (20)

͜͜ͰɼR²i ͸֤ཁૉͷॏ৺఺͔Β෺ମͷॏ৺఺·Ͱͷ ڑ཭Λද͢ɽҎ্ͷखॱʹΑΓಘΒΕͨ׳ੑϞʔϝϯτ Λܭࢉʹ࢖༻͢Δɽ

3.

3.1 ඬྲྀ෺ͷน໘িಥ໰୊

ຊख๏ͷ༗ޮੑΛݕ౼͢ΔͨΊʹɼඬྲྀ෺ͷিಥ໰୊

5)Λߦͬͨɽਤ-6ʹղੳϞσϧΛࣔ͢. ӈนΑΓ0.4m ͷҐஔʹஔ͔Εͨ෺ମʹରͯ͠ɼԼྲྀଆͷήʔτΛٸ։

͢Δ͜ͱʹΑΓੜ੒͞Εͨஈ೾ʹΑΓɼ෺ମ͕น໘ʹি

ಥ͢Δ໰୊Ͱ͋Δɽ෺ମ͸ɼ௚ܘ0.08mͷԁܗͱ͠ɼີ

౓͸500kg/m³Ͱ͋Δɽ·ͨɼྲྀମ͸ਫΛԾఆͨ͠ɽ༗

ݶཁૉ෼ׂʹ͸ɼ࠷খϝογϡ෯:0.007mͷඇߏ଄֨ࢠ ΛόοΫάϥ΢ϯυϝογϡͱͯ͠༻͍ͨɽඍখ࣌ؒ૿

෼ྔ͸ɼ1.0×10⁻³sͱ͠ɼ઀৮ྗΛධՁ͢Δόω܎਺

͸ɼ10⁴N/mͱͨ͠ɽ

ɹਤ-7 ʹӈนʹ࡞༻͢Δྗͷ࣌ࠁྺΛࣔ͢ɽ͜ͷਤΑ Γɼຊղੳ݁Ռ͸ɼ࣮ݧ஋⁵⁾ͱ֓Ͷྑ͍ҰகΛ͍ࣔͯ͠

Δɽਤ-8ʹ֤࣌ࠁʹ͓͚Δ෺ମҐஔͱྲྀମྖҬΛࣔ͢ɽ

͜ͷਤΑΓɼ೾ʹΑΓ෺ମ͕Լྲྀଆʹӡ͹Εɼน໘ʹΑ Γ௓Ͷฦ͞ΕΔ༷ࢠ͕ଊ͑ΒΕ͍ͯΔ͜ͱ͕ղΔɽ

ࠥ࡯࠻

ࠥ࡯࠻

ਤ– 6 ղੳϞσϧ

2 2.5 3

0 2 4 6

ታ㛎୯. (ᳰ㊁ࠄ, 2004)

ᧄ⸃ᨆ ฝო

ߦ૞

↪ߔ ࠆ᳓ ᐔജ

[N/cm ]

Time[s]

ਤ– 7 ӈนʹ࡞༻͢Δྗͷ࣌ࠁྺ

ਤ– 8 ֤࣌ࠁʹ͓͚Δ෺ମҐஔͱྲྀମྖҬ

3.2 ෳ਺ු༡෺ͷ஄ੑߏ଄෺িಥ໰୊

ຊख๏ͷ೚ҙܗঢ়෺ମ΁ͷద༻ͱͯ͠ɼෳ਺෺ମͷ ߏ଄෺িಥ໰୊ΛऔΓ্͛ͨɽ·ͨɼຊղੳͰ͸ɼྲྀ

ମྗٴͼɼ઀৮ྗ͕ߏ଄෺ʹ༩͑ΔӨڹΛߟྀ͢Δͨ

Ίʹɼߏ଄෺Λ஄ੑମͱԾఆ͠ɼԠྗղੳʹ͍ͭͯ΋

ߦͬͨɽਤ–9ʹղੳϞσϧΛࣔ͢ɽ஄ੑߏ଄෺͸ີ

౓:2.5×10³kg/m³ɼϠϯά཰:2.0×10¹⁰N/m²ɼϙΞι ϯൺ:0.2ͱͨ͠ɽͳ͓ɼຊղੳͰ͸ɼߏ଄෺ͷมܗ͕ඍ গͰ͋ΔͱΈͳ͠ɼมܗ͸ྲྀମଆͰ͸ߟྀ͍ͯ͠ͳ͍ɽ

෺ମͷີ౓͸500kg/m³Ͱ͋Γɼྲྀମ͸ਫΛԾఆͨ͠ɽ

༗ݶཁૉ෼ׂʹ͸ɼ࠷খϝογϡ෯:0.003mͷඇߏ଄֨

ࢠΛόοΫάϥ΢ϯυϝογϡͱͯ͠༻͍ͨɽඍখ࣌ؒ

૿෼ྔ͸ɼ5.0×10⁻⁴sͱ͠ɼ઀৮ྗΛධՁ͢Δόω܎

਺͸ɼ10⁴ N/mͱͨ͠ɽਤ–10ʹ֤࣌ࠁʹ͓͚Δྲྀମ

ྖҬܗঢ়ͱߏ଄෺಺෦ͷԠྗ෼෍Λࣔ͢ɽߏ଄෺ࠨଆͷ ਫҐ্ঢͱڞʹߏ଄෺಺෦ͷԠྗ΋্ঢ͠ɼ෺ମ͕઀৮

ͨ͠ࡍʹ͸ہॴతͳԠྗ͕ੜ͍ͯ͡Δɽ͜ͷ݁ՌΑΓྲྀ

ମྗٴͼɼ෺ମͷ઀৮ྗʹΑΔɼߏ଄෺಺෦ʹੜ͡ΔԠ

ྗΛࢉఆ͢Δ͜ͱ͕Մೳͱͳͬͨɽ

4.

ຊݚڀͰ͸ɼόοΫάϥ΢ϯυϝογϡΛ༻͍ͨք໘

௥੻๏ʹجͮࣗ͘༝ද໘Λ༗͢Δྲྀମ-ߏ଄࿈੒໰୊Λ ਫ਼౓ྑ͘ϩόετʹղੳՄೳͳख๏ͷߏஙΛߦ͍ɼ਺஋

ղੳྫΛ௨ͯ͠ҎԼͷ݁࿦Λಘͨɽ

• ຊख๏ʹΑΔղੳ݁Ռ͸࣮ݧ஋ͱྑ͍ҰகΛࣔ

͠ɼຊख๏ͷ༗ޮੑΛࣔ͢͜ͱ͕Ͱ͖ͨɽ

• ೚ҙܗঢ়ͷ׳ੑϞʔϝϯτΛߟྀ͢Δ͜ͱͰɼ೚

ҙܗঢ়Λ༗͢Δ෺ମಉ࢜ͷ઀৮໰୊ΛऔΓѻ͏͜

ͱ͕Մೳͱͳͬͨɽ

• ຊख๏ʹΑΓಘΒΕͨྲྀମྗٴͼ෺ମͷ઀৮ྗʹ ΑΔɼ஄ੑମߏ଄෺಺෦ʹੜ͡ΔԠྗΛࢉఆ͢Δ

͜ͱ͕Մೳͱͳͬͨɽ

ɹࠓޙ͸ɼղੳ݁Ռͷৄࡉͳݕ౼ɼ઀৮ྗࢉఆͷͨΊͷ όωఆ਺ͷݕ౼Λߦ͏༧ఆͰ͋Δɽ

ࢀߟจݙ

1) Nomura, T. : ALE finite element computations of fluid- structure interaction problems, Compter Methods in Applied Mechanics and Engineering,112, pp.291-308, 1994.

2) Hirt, C.W., Nichols, B.D. : Volume of fluid(VOF) method for the dynamics of free boundaries, Journal

ᒢᕈ૕

೰૕

ᒢᕈ૕

೰૕

ਤ– 9 ղੳϞσϧ

࿶ജ ᔕജ

ਤ– 10 ֤࣌ࠁʹ͓͚ΔྲྀମྖҬܗঢ়ͱ஄ੑମͷԠྗ෼෍

of Computational Physics,39, pp.201-225, 1981.

3) ాத੟ࡾ, ֽࢁ࿨உ: όοΫάϥ΢ϯυϝογϡʹجͮ͘

ϝογϡ࠶ߏஙख๏Λ༻͍ͨࣗ༝ද໘Λ༗͢Δྲྀମ–ߏ଄

࿈੒໰୊ͷͨΊͷALE_{༗ݶཁૉ๏},_{Ԡ༻ྗֶ࿦จू},_౔໦

ֶձ,8, pp.295-302, 2005.

4) P.A.Cundall and O.D.L.Strack,:A discrete numer- ical model for granular assemblies, Geotechnique ,29,No.1,pp47-65,1979.

5) ஑໺ਖ਼໌ɼాத׮޷: ஈ೾௡೾ʹΑΔඬྲྀ෺ͷিಥྗʹؔ

͢Δ࣮ݧతݚڀ,ిྗதԝݚڀॴใࠂ,ݚڀใࠂ:2004.