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2022_autumn_asj

 2022_autumn_asj

statictaku

May 09, 2023
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  1. Density-Independent SPH๏ٴͼGodunov SPH๏΁ͷ
    Integral Approachͷద༻ͱͦͷੑೳ
    ᴷ ৽ SPH εΩʔϜ:Godunov DISPH ๏ʹ͍ͭͯ ᴷ
    ౬ઙ୓޺, ৿ਖ਼෉
    ஜ೾େֶ
    January 6, 2023
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 1 / 16

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  2. ໨࣍
    1 ֤छεΩʔϜʹ͍ͭͯ
    SPH ๏
    Density-Independent SPH ๏, GSPH ๏
    Godunov DISPH ๏
    Integral Approach
    2 ςετܭࢉ
    ࣮ݧͷ໨త
    Riemann ໰୊
    2D ੩ਫѹฏߧ
    2D Kelvin-Helmholtz ෆ҆ఆੑ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 2 / 16

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  3. ֤छεΩʔϜʹ͍ͭͯ SPH ๏
    SPH ๏ͱ͸
    ѹॖੑྲྀମΛϥάϥϯδϡతʹղ͘਺஋ܭࢉख๏ɻཧ޻ֶͷଟذʹ౉Δ෼໺Ͱ޿͘༻͍ΒΕ
    ͍ͯΔɻ͋ΔҐஔͰͷྲྀମͷ෺ཧྔΛɺपΓͷཻࢠ͔ΒͷӨڹͷ଍͠߹ΘͤͰදݱɻ
    • ϝϦοτ
    ▶ 1 ࣍ݩͷίʔυΛɺ΄΅ͦͷ·· 3 ࣍ݩͷίʔυʹద༻Մೳ
    ▶ ߴີ౓ྖҬ͕ࣗಈతʹߴղ૾౓ͱͳΔ
    ▶ ॏྗ૬ޓ࡞༻ͱͷ૊Έ߹Θ͕ͤ༰қ
    • σϝϦοτ
    ▶ ྲྀମͷ઀৮ෆ࿈ଓ໘Λ͏·͘ѻ͑ͣɺͦͷͨΊྲྀମͷෆ҆ఆੑͷ੒௕Λ๦͛Δ
    ▶ িܸ೾Λଊ͑ΔͨΊʹਓ޻తͳࢄҳ߲͕ӡಈํఔࣜɺΤωϧΪʔํఔࣜʹඞཁ
    ▶ ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 3 / 16

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  4. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏, GSPH ๏
    DISPH ๏ɺGSPH ๏ͱ͸
    • Saitoh & Makino (2013) Ͱ DISPH ๏͕ɺInutsuka (2002) Ͱ GSPH ๏͕։ൃ͞Εͨɻ
    DISPH ๏
    ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͳѹྗΛΧʔωϧਪ
    ఆ͠ɺྲྀମͷඍ෼ํఔࣜΛղ͍͍ͯΔɻ
    ਓޱ೪ੑ͕ඞཁɻ
    GSPH ๏
    ཻࢠಉ࢜ͷ૬ޓ࡞༻ͷܭࢉͷࡍʹ
    Riemannsolver Λ༻͍ΔɻࠓճͷൃදͰ͸
    Cha & Whitworth (2003) ʹΑΔ Case3 ͷ
    GSPH Λ࢖༻ɹਓޱ೪ੑඞཁͳ͠ɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 4 / 16

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  5. ֤छεΩʔϜʹ͍ͭͯ Godunov DISPH ๏
    ࣗ෼͕։ൃͨ͠ GDISPH ๏ʹ͍ͭͯ
    • DISPH ๏ͱಉ͘͡ɺີ౓ͷ୅ΘΓʹ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͳѹྗΛΧʔωϧਪఆ͠ɺ
    GSPH ๏ͷಋग़ʹ฿ͬͯີ౓ʹཅʹґଘ͠ͳ͍ SPH ํఔࣜΛ࡞੒͍ͯ͘͠ɻ
    ѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
    dv(r)
    dt
    = −
    1
    ρ(r)
    ∇P(r) (1)
    du(r)
    dt
    = −
    P(r)
    ρ(r)
    ∇ · v(r) (2)
    GDISPH ๏ಋग़ʹ༻͍Δࣜ
    q(r) = ΣjmjujW(|rj − r|, h(r)) =
    P(r)
    γ − 1
    (3)
    F(r) ∼

    F(r′)W(|r − r′|, h(r′))d3r′ (4)
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 5 / 16

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  6. ֤छεΩʔϜʹ͍ͭͯ Godunov DISPH ๏
    GDIPH ๏ʹ͓͚Δ Case3 ܕͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
    ˙
    vi = −ΣjmjujP∗
    ij
    u∗
    ij
    [
    1
    q2
    i
    ∇iW(ri − rj, hi) +
    1
    q2
    j
    ∇iW(ri − rj, hj)
    ]
    ˙
    ui = −ΣjmjujP∗
    ij
    u∗
    ij
    [
    v∗
    ij
    − vi
    ]
    [
    1
    q2
    i
    ∇iW(ri − rj, hi) +
    1
    q2
    j
    ∇iW(ri − rj, hj)
    ] (5)
    ఴ͑ࣈʹ*ͷ෇͍ͨ෺ཧྔʹ͸ࣗ༝౓Λ࣋ͨͤΔɻGSPH ๏Ͱ͸ɺriemann solver ͷղ
    ྫͱͯ͠ɺu∗
    ij
    = ui, v∗
    ij
    = vi+vj
    2
    ͱ͠ɺP∗
    ij
    ͸ riemann solver ͔ΒٻΊΔύλʔϯΛ
    GDISPH Case3 2 2 ͱ໊͚ͮΔɻ(ຊൃදͰ͸͜Ε͕ॏཁͳ GDISPH ͷεΩʔϜ)
    Χʔωϧͷۭؒඍ෼ʹ͸ɺGarcia-Senz et al. (2012) ͕ߟҊͨ͠ IA Λ࢖༻͢Δ͔൱͔ͷࣗ༝
    ౓͕ଘࡏɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 6 / 16

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  7. ֤छεΩʔϜʹ͍ͭͯ Integral Approach
    Integral Approach(IA) ͱ͸
    • ೤֦ࢄ߲౳ʹݱΕΔ 2 ֊ඍ෼߲ͷܭࢉʹ࢖༻͞Ε͍ͯͨํ๏ (Brookshaw 1985) ΛҰ֊
    ඍ෼ʹԠ༻
    • 1 ֊ඍ෼ͰݱΕΔΧʔωϧͷۭؒඍ෼Λผͷ΋ͷʹม͑Δ͚ͩͰΑ͍ɻ
    ∇iW(ri − rj, hi) ∼ τ−1
    i
    (rj − ri)Wij(hi)
    ∇iW(ri − rj, hj) ∼ τ−1
    j
    (rj − ri)Wij(hj)
    τ(r) =
    [∫
    (r′ − r) ⊗ (r′ − r)W(|r − r′|, h(r))
    ]
    d3r′
    (6)
    • IA ʹΑΓ෺ཧྔͷ 1 ֊ඍ෼ͷਫ਼౓͕ྑ͘ͳΓɺۭؒ 0 ࣍ͷޡࠩ΋খ͘͞ͳΔ͜ͱ͕ใࠂ
    ͞Ε͍ͯΔɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 7 / 16

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  8. ςετܭࢉ ࣮ݧͷ໨త
    ࣮ݧͷ໨త
    • SPH ๏ɺDISPH ๏
    ▶ Χʔωϧͷۭؒඍ෼Λ IA ʹ͢Δ͔൱͔
    • Case3 ͷ GSPH ๏
    ▶ Χʔωϧͷۭؒඍ෼Λ IA ʹ͢Δ͔൱͔
    ▶ riemann solver ͷߴਫ਼౓Խ͢Δ͔൱͔ɺ͢ΔͳΒྲྀ଎੍ݶؔ਺ΛԿʹ͢Δ͔
    • Case3 ͷ GDISPH ๏
    ▶ Χʔωϧͷۭؒඍ෼Λ IA ʹ͢Δ͔൱͔
    ▶ ఴ͑ࣈʹ*ͷ෇͍ͨ෺ཧྔΛ riemann solver Ͱղ͔͘൱͔ɺղ͔ͳ͍ͳΒԿʹ͢Δ͔
    ▶ riemann solver ͷߴਫ਼౓Խ͢Δ͔൱͔ɺ͢ΔͳΒྲྀ଎੍ݶؔ਺ΛԿʹ͢Δ͔
    riemann solver ͷߴਫ਼౓Խ͸ Cha & Whitworth (2003) ͷ΍ΓํͰߦͬͨɻ
    ֤εΩʔϜʹର༷ͯ͠ʑͳ૊Έ߹Θͤͷύλʔϯ͕ଘࡏɻ͜ͷதͰ࠷΋༏ΕͨεΩʔϜΛબ
    ఆ͢ΔͨΊɺ໿ 150 ݸͷύλʔϯʹؔͯ͠ riemann ໰୊,2D ੩ਫѹฏߧ,2D Kelvin-Helmholtz
    ෆ҆ఆੑͷܭࢉΛߦͬͨɻͦͷதͰ͍͔ͭ͘ͷεΩʔϜΛϐοΫΞοϓ͠ɺͦͷಛੑΛൺֱ
    ݕ౼͢Δɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 8 / 16

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  9. ςετܭࢉ Riemann ໰୊
    Riemann ໰୊
    ڧ͍িܸ೾͕ൃੜ͢Δ໰୊
    {
    ρ = 1, P = 1000, v = 0 x < 0 ͷͱ͖
    ρ = 1, P = 0.1, v = 0 x > 0 ͷͱ͖
    (7)
    ີ౓ ѹྗ ଎౓
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 9 / 16

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  10. ςετܭࢉ Riemann ໰୊
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 10 / 16

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  11. ςετܭࢉ 2D ੩ਫѹฏߧ
    2D ੩ਫѹฏߧ
    ॳظ৚݅ͷ··มಈ͠ͳ͍ͷ͕෺ཧతͳղ ѹྗ͸શྖҬͰҰఆɹ
    • Ի଎ Cs = 1.02 ͰܭࢉྖҬΛԣ੾Δ࣌ؒ໿ 1.0 ͷ 8 ഒͰ͋Δ t = 8.0 ·Ͱܭࢉ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 11 / 16

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  12. ςετܭࢉ 2D ੩ਫѹฏߧ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 12 / 16

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  13. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    2D Kelvin-Helmholtz ෆ҆ఆੑ
    Ґஔ y = 0.25, y = 0.75 ͷͦΕͧΕʹɺy ࣠ํ޲ͷ଎౓ʹ 2 ೾௕෼ͷઁಈΛ༩͍͑ͯΔ
    ෆ҆ఆੑͷ੒௕ͷλΠϜεέʔϧ τKH = 1.06 ͷ໿ 2 ഒͷ t = 2.0 ·Ͱܭࢉ
    ॳظ৚݅
    ༩͑ͨઁಈ ༗ݶମੵ๏Ͱղ͍ͨ৔߹
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 13 / 16

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  14. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 14 / 16

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  15. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    ·ͱΊ
    • GDISPH Case3 2 2
    ▶ Riemann ໰୊ɺ੩ਫѹฏߧ໰୊ʹ͓͍ͯ DISPH ͱಉ༷ͳ݁Ռ,Kelvin-Helmholtz ෆ҆ఆੑʹ
    ؔͯ͠͸ GSPH ͱಉ༷ͳ݁ՌɻSPH ͷεΩʔϜͱͯ͠ɺѹॖੑྲྀମͷ໰୊Λ͖ͪΜͱղ͚
    ͍ͯΔɻ
    ▶ ߴਫ਼౓Խͨ͠৔߹ɺ೪ੑ͕খ͘͞ͳΔ͜ͱ͕ݪҼͰڧ͍িܸ೾ԼͰͷৼಈ͕ൃੜͯ͠͠
    ·͏ɻ
    • IA
    ▶ Kelvin-Helmholtz ෆ҆ఆੑͰ͸ɺIA ԽΑΓߏ଄͕ΑΓେ͖͘੒௕ɻ͔͠͠ɺ੩ਫѹฏߧ໰୊
    ͔Β DISPH ܥͷεΩʔϜͰ͸ IA ԽʹΑΓϊΠζΛर͍΍͘͢ͳ͍ͬͯΔɻ
    • SSPH,DISPH
    ▶ Kelvin-Helmholtz ෆ҆ఆੑʹ͓͍ͯɺਓ޻తͳϊΠζ͕ෆ҆ఆੑʹΑΓ૿௕͞Εͯ͠·ͬͯ
    ͍Δɻt=2.0 Ͱͷ Kelvin-Helmholtz ෆ҆ఆੑͷ༷ࢠΛݟΔͱɺਅΜத͔Βݟͯࠨଆʹ͋Δେ
    ͖ͳߏ଄ͱӈଆʹ͋Δେ͖ͳߏ଄͕߹ಉͳܗʹͳ͍ͬͯͳ͍ɻ
    খ͞ͳϊΠζ͕࠷ऴతʹେ͖͍ӨڹΛ༩͍͑ͯΔɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 15 / 16

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  16. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    ग़య
    Figure: ”Fancy SPH convolution scheme (verbose, modified colors scheme)” created by Jlcercos is
    licenced under CC BY-SA 4.0(https://creativecommons.org/licenses/by-sa/4.0/)
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 15 / 16

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  17. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ
    Reference I
    ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 16 / 16

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