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poster_senior_thesis

 poster_senior_thesis

statictaku

May 20, 2023
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  1. Reformulation of Density-Independent Smoothed Particle Hydrodynamics
    with Riemann Solver: Godunov DISPH
    ౬ઙ୓޺1*ɾࢦಋڭ׭: ৿ਖ਼෉1
    1 ஜ೾େֶ ཧ޻ֶ܈ ෺ཧֶྨ, *[email protected]
    1. SPH๏ͱ͸
    • ཻࢠΛ༻͍ͯѹॖੑඇ೪ੑྲྀମ(ཧ૝ྲྀମ)ΛγϛϡϨʔγϣϯ͢
    ΔͨΊͷܭࢉख๏.֨ࢠ๏ͱൺ΂ͨϝϦοτ͕͍͔ͭ͋͘Δ.
    • ݩʑӉ஦෼໺ൃ঵.ۙ೥͸޻ۀ΍CG෼໺Ͱ΋༻͍ΒΕ͍ͯΔ.
    dv(r)
    dt
    = −
    1
    ρ(r)
    ∇P(r) + FAV
    (α) (1)
    du(r)
    dt
    = −
    P(r)
    ρ(r)
    ∇ · v(r) + GAV
    (α) (2)
    F(ri
    ) ≈

    F(r′
    )W(|ri
    − r′
    |, h)d3r′


    j
    mj
    ρj
    Fj
    W(|ri
    − rj
    |, h)
    (3)
    • ӡಈํఔࣜ(1),಺෦ΤωϧΪʔํఔࣜ(2)Λ,ࣜ(3)Λ༻͍ͯ཭ࢄԽ
    ͢Δ.W ͸Ψ΢γΞϯͷΑ͏ͳܗΛ͓ͯ͠Γ,पΓͷཻࢠͷӨڹΛ
    ॏͶ߹ΘͤΔΠϝʔδ(i,j ͸ཻࢠͷϥϕϦϯά). ཧ૝ؾମͷঢ়ଶ
    ํఔࣜΛ༻͍Δͱ͕ࣜด͡Δ.
    2.SPH๏ͷσϝϦοτ(઀৮ෆ࿈ଓ໘ͱিܸ೾)
    ઀৮ෆ࿈ଓ໘
    ྲྀମͷڥք໘Ͱ͋Δ઀৮ෆ࿈ଓ໘͸, खͰนΛԡ͢ͱ͖ʹͰ͖Δෆ࿈
    ଓ໘(खͱนͷ଎౓,ѹྗҰఆͰີ౓,ΤωϧΪʔ͸ෆ࿈ଓ)ͱࣅ͍ͯΔ.
    SPHͰ͸ີ౓͸ࣜ(1)ͰٻΊΔ.
    ρ =

    j
    mj
    Wij
    (h) (4)
    ͭ·Γ, ີ౓ͷۭؒ࿈ଓੑ͕Ծఆ͞Εͯ͠·͍ͬͯΔ. ·ͨ, ಋग़தͰ
    ͸ີ౓ͷۭؒඍ෼Λߦ͍ͬͯΔ. ͜ΕΒͷཧ༝͔Β,SPH Ͱ͸෺ཧత
    ʹີ౓ෆ࿈ଓͱͳΔ઀৮ෆ࿈ଓ໘͕͏·͘ѻ͑ͣʹඇ෺ཧతͳද໘ு
    ྗΛൃੜͤͯ͞͠·͏͜ͱ͕஌ΒΕ͍ͯΔ.
    ॲํᝦͱͯ͠,ີ౓ͷ୅ΘΓʹѹྗͷۭؒ࿈ଓੑ,ۭؒඍ෼ՄೳੑΛԾ
    ఆͨ͠DISPH๏͕ଘࡏ. ඇ෺ཧతͳද໘ுྗΛ཈͑Δ͜ͱ͕Ͱ͖ΔҰ
    ํ,িܸ೾ͰͷσϝϦοτ͸SPHͱಉ༷ʹଘࡏ͢Δ.
    িܸ೾
    িܸ೾͸Ի೾ͱಉ༷, ৘ใͷ఻ൖʹΑΓى͜Δ. ෺ཧతͳڥքͰ͸ͳ
    ͍. ѹྗ,଎౓΋ෆ࿈ଓ. ਺஋ܭࢉͰিܸ೾Λଊ͑ΔͨΊʹ,ਓ޻೪ੑ
    ߲((1),(2)ͷ੺ࣈͷ߲)Λ࢖ͬͯিܸ೾Λ͋͑ͯ׈Β͔ʹ͢Δඞཁੑ͕
    ͋Δɻਓ޻೪ੑ߲ͷڧ͞Λௐઅ͢Δύϥϝʔλα͸ਓ͕ؒखͰ,ܦݧ
    ౳Λ΋ͱʹܾΊ͍ͯΔ.
    ॲํᝦͱͯ͠,ཻࢠͷ૬ޓ࡞༻ܭࢉͷࡍʹRiemann solverͷղΛ༻͍Δ
    ͜ͱͰ,Ұ੾ͷύϥϝʔλͳ͠Ͱద੾ͳ೪ੑ͕িܸ೾ྖҬʹೖΔ
    Godunov SPH๏ʢGSPH๏ʣ͕ଘࡏ͢Δ. ͔͠͠,઀৮ෆ࿈ଓ໘ͷѻ͍
    ͸SPHͱಉ༷ʹෆ׬શ.
    3.Godunov DISPH๏ʹ͍ͭͯ
    DISPH ๏ͱ GSPH ๏Λ૊Έ߹Θͤͨ Godunov DISPH๏Λ։ൃ. ઀৮ෆ
    ࿈ଓ໘Λਖ਼͘͠ѻ͑,ద੾ͳ೪ੑ͕ύϥϝʔλͳ͠ͰೖΔ.ҎԼͰ͸ͦ
    ͷಋग़ʹ͍ͭͯܰ͘ड़΂Δ.
    ཻࢠ i ͷඍখ಺෦ΤωϧΪʔมԽ͸, ཻࢠ͕֎෦͔Βड͚ΔମੵมԽ
    Λ௨ͨ͡࢓ࣄʹΑΔ΋ͷ(ॏ৺ͷҐஔมԽΛ௨ͨ͡࢓ࣄ͸ӡಈΤωϧ
    Ϊʔʹ)
    dUi
    = WV olume
    i (5)
    DISPHͰ͸(SPHͰ΋)
    WV olume
    i
    = −Pi
    dVi (6)
    ͜Ε͸ཻࢠi͕dtͷؒʹ֎෦͔Βड͚Δѹྗ͸ɺPi
    + ϵͰ͋ΔͱԾఆ
    ͠ɺೋ࣍ͷඍখྔΛແࢹ͍ͯ͠Δ. (ྲྀମํఔࣜͷΤωϧΪʔͷࣜͰ
    ΋ಉ༷) ͜ͷԾఆͷԼͰ΋,ແݶݸͷཻࢠ,ແݶͷղ૾౓,ແݶখͷλΠ
    ϜεςοϓΛ࢖͑͹ਖ਼͍͠ղʹऩଋ͢Δ͸ͣ.
    ৽͍͠ߟ͑ํ
    Pix
    ΛɺཻࢠiΛத৺ͱͨ͋͠ΒΏΔํ޲΋,͔Βཻࢠi͕ड͚Δѹྗ
    ͷ͋Δछͷۭ࣌ؒؒฏۉྔͱͯ͠
    WV olume
    i
    = −Pix
    dVi (7)
    ͕੒Γཱͭͱ͢Δɻͪ͜ΒͷԾఆͷԼͰ΋,ແݶݸͷཻࢠ,ແݶͷղ
    ૾౓,ແݶখͷλΠϜεςοϓΛ࢖͑͹ਖ਼͍͠ղʹऩଋ͢Δ͸ͣ.
    ۭ࣌ؒؒฏۉͷऔΓํʹ͸೚ҙੑ͕͋Δ. ࠓճ͸, ཻࢠ i ཻ͕ࢠ j ͔
    Βड͚Δѹྗͷ࣌ؒฏۉΛ,ͦΕͧΕͷཻࢠͷ෺ཧྔΛೖྗ஋ͱͨ͠
    Rieamnn solverͷղʹ͓͚Δ઀৮ෆ࿈ଓ໘্Ͱͷѹྗͱ͠,ۭؒฏۉ͸
    ద౰ͳܗͰͱΔ. ͋ͱ͸ࣜ(7)͔ΒDISPH๏ͷಋग़ͱ΄΅ಉ͡΍Γํ
    ͰGDISPH๏ͷӡಈํఔࣜ,ΤωϧΪʔํఔࣜΛಋग़Մೳ.
    4. ςετܭࢉ(GDISPHͷ઀৮ෆ࿈ଓ໘ͱিܸ೾ʹର͢Δੑೳ)
    ॳظ৚͕݅ѹྗฏߧͳͷͰ,
    ॳظ৚݅ͷ··มಈ͠ͳ͍
    ͜ͱ͕෺ཧతͳղ.
    SPH, GSPH͸઀৮ෆ࿈ଓ໘
    ͔Βͷද໘ுྗͷӨڹͰܗ
    ؙ͕͘ͳͬͯ͠·͏.Ӎཻ͕
    ද໘ுྗͰؙ͘ͳΔݮগͱ
    ࣅ͍ͯΔ. DISPH, GDISPH
    Ͱ͸ඇ෺ཧతͳද໘ுྗͷ
    ޮՌ͕ͳ͘ͳ͍ͬͯΔ͜ͱ
    ͕Θ͔Δ.

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