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卒業研究発表会資料

 卒業研究発表会資料

statictaku

May 20, 2023
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  1. Reformulation of Density-Independent Smoothed Particle
    Hydrodynamics with Riemann Solver: Godunov DISPH
    ᴷ ৽ SPH εΩʔϜ:Godunov DISPH ๏ʹ͍ͭͯ ᴷ
    1 ౬ઙ୓޺
    ࢦಋڭһ: 1 ৿ਖ਼෉
    ஜ೾େֶ
    ଔۀݚڀൃදձ, January 31, 2023
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 1 / 17

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  2. ໨࣍
    1 ֤छεΩʔϜʹ͍ͭͯ
    SPH ๏
    Density-Independent SPH ๏,
    GSPH ๏
    2 Godunov DISPH ๏ಋग़
    3 ςετܭࢉ
    2D ੩ਫѹฏߧ
    ఺ݯരൃ
    4 ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 2 / 17

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  3. ֤छεΩʔϜʹ͍ͭͯ SPH ๏
    SPH ๏ಋग़ʹ༻͍Δѹॖੑඇ೪ੑஅ೤ྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
    dv(r)
    dt
    = −
    1
    ρ(r)
    ∇P(r) + FAV (α) (1)
    du(r)
    dt
    = −
    P(r)
    ρ(r)
    ∇ · v(r) + GAV (α) (2)
    • σϝϦοτ
    ▶ ྲྀମͷ઀৮ෆ࿈ଓ໘Λ͏·͘ѻ͑ͣɺඇ෺ཧతͳද໘ுྗ͕ൃੜ.
    ີ౓ͷۭؒ࿈ଓੑΛԾఆ͠ɺ࿈ଓؔ਺Ͱۙࣅ͍ͯ͠Δ͔Βɻ
    ▶ িܸ೾Λଊ͑ΔͨΊʹਓ޻తͳࢄҳ߲͕ӡಈํఔࣜɺΤωϧΪʔํఔࣜʹඞཁɻ
    ਓޱ೪ੑͷڧ͞Λௐઅ͢Δ೚ҙύϥϝʔλ α ΛਓؒͷखͰௐ੔͢Δඞཁ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 3 / 17

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  4. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏, GSPH ๏
    DISPH ๏ɺGSPH ๏ͱ͸
    • [Saitoh and Makino, 2013] Ͱ DISPH ๏͕ɺ[Inutsuka, 2002] Ͱ GSPH ๏͕։ൃ͞Εͨɻ
    DISPH ๏
    ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͳѹྗΛ࿈ଓؔ਺Ͱ
    ۙࣅ (ѹྗͷ࿈ଓੑɺۭؒඍ෼ՄೳੑΛԾ
    ఆ) ͠ɺྲྀମͷඍ෼ํఔࣜΛղ͍͍ͯΔɻ
    িܸ೾Λଊ͑ΔͨΊʹɺਓ޻೪ੑύϥ
    ϝʔλ͕ඞཁɻ
    GSPH ๏
    ཻࢠಉ࢜ͷ૬ޓ࡞༻ͷܭࢉͷࡍɺܭࢉࣜ
    தͷѹྗͱ଎౓ʹ Riemann solver Λ༻͍
    Δɻࠓճ͸ [Cha and Whitworth, 2003] ʹ
    ΑΔ Case3 ͷ GSPH Λ࢖༻. িܸ೾ͷͨ
    Ίͷύϥϝʔλઃఆඞཁͳ͠.
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 4 / 17

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  5. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏, GSPH ๏
    Riemann Solver ʹ͍ͭͯ
    • ॳظ৚݅
    W = (ρ, P, v)
    • ղͷλΠϓ
    P∗
    L
    = P∗
    R
    , v∗
    L
    = v∗
    R
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 5 / 17

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  6. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏, GSPH ๏
    2D ѹྗฏߧ
    ॳظ৚݅ͷ··มಈ͠ͳ͍ͷ͕෺ཧతͳղ ѹྗ͸શྖҬͰҰఆɹ
    • Ի଎ Cs = 1.02 ͰܭࢉྖҬΛԣ੾Δ࣌ؒ໿ 1.0 ͷ 8 ഒͰ͋Δ t = 8.0 ·Ͱܭࢉ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 6 / 17

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  7. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏, GSPH ๏
    • SPH ͱ GSPH Ͱ͸ඇ෺ཧ
    తͳද໘ுྗͷޮՌͰܗ
    ͕େ͖͘มܗ
    • DISPH Ͱ͸ඇ෺ཧతͳද
    ໘ுྗͷޮՌ͕ͳ͘ͳͬ
    ͍ͯΔ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 7 / 17

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  8. ֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏, GSPH ๏
    ͜Ε·Ͱͷ·ͱΊ
    • DISPH ͸઀৮ෆ࿈ଓ໘ͷѻ͍͕ྑ͍ (઀৮ෆ࿈ଓ໘ͷͨΊͷ௥Ճͷࢄҳ߲ɺύϥϝʔλ
    ͳ͠)
    • GSPH ͸িܸ೾ΛҰ੾ͷύϥϝʔλͳ͠Ͱѻ͑Δ (ѹྗʹ Riemann solver ͷղΛ༻͍Δ
    ͜ͱͰɺద੾ͳ೪ੑ͕෇Ճ͞ΕΔ)
    ຊݚڀͷ໨త
    Riemann Solver Λ DISPH ʹ૊ΈࠐΉ͜ͱͰɺύϥϝʔλͳ͠Ͱ઀৮ෆ࿈ଓ໘, িܸ೾Λѻ͑
    ΔεΩʔϜΛ࡞Δ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 8 / 17

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  9. Godunov DISPH ๏ಋग़
    Godunov DISPH ๏ಋग़ (೤ྗֶୈҰ๏ଇΛى఺)
    ཻࢠ i ͷඍখ಺෦ΤωϧΪʔมԽ͸ɺཻࢠ͕֎෦͔Βड͚ΔମੵมԽΛ௨ͨ͡࢓ࣄʹΑΔ΋
    ͷ (ॏ৺ͷҐஔมԽΛ௨ͨ͡࢓ࣄ͸ӡಈΤωϧΪʔʹ)
    dUi = WV olume
    i
    (3)
    DISPH Ͱ͸ (SPH Ͱ΋)
    WV olume
    i
    = −PidVi (4)
    ཻࢠ i ͕ dt ͷؒʹ֎෦͔Βड͚Δѹྗ͸ɺPi + ϵ Ͱ͋ΔͱԾఆ͠ɺೋ࣍ͷඍখྔΛແࢹͯ͠
    ͍Δ.
    ৽͍͠ߟ͑ํ
    Pix Λɺཻࢠ i Λத৺ͱͨ͋͠ΒΏΔํ޲͔Βཻࢠ i ͕ड͚Δѹྗͷ͋Δछͷۭ࣌ؒؒฏۉྔ
    ͱͯ͠
    WV olume
    i
    = −PixdVi (5)
    ͕੒Γཱͭͱ͢Δɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 9 / 17

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  10. Godunov DISPH ๏ಋग़
    F(ri) ≈

    j
    Uj
    qj
    FjW(|ri − rj|, h) (6)
    qi =

    j
    UjWij(h) =
    Pi
    (γ − 1)
    (7)
    DISPH ͷಋग़๏ͱ΄΅ಉ͡Α͏ʹɺ্ͷࣜΛ༻͍ͯΤωϧΪʔํఔࣜ͸
    dUi
    dt
    = fgrad
    i
    N

    j
    PixUiUj
    q2
    i
    vij · ∇iWij(hi). (8)
    fgrad
    i
    =

    1 +
    hi
    Dqi
    N

    j
    Uj
    ∂Wij(hi)
    ∂hi


    −1
    . (9)
    ͱͳΔɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 10 / 17

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  11. Godunov DISPH ๏ಋग़
    Pix ͷఆٛ
    Pix ʹ͸೚ҙੑ͕ଘࡏ (ۭ࣌ؒؒฏۉྔͷऔΓํͷ೚ҙੑ)ɻ
    P∗
    ij
    Λཻࢠ i ͱཻࢠ j ͷ෺ཧྔΛೖྗ஋ͱͨ͠ࡍͷ Star region Ͱͷѹྗͷ஋ͱͯ͠
    Pix
    N

    j
    UiUj
    q2
    i
    vij · ∇iWij(hi) =
    N

    j
    P∗
    ij
    UiUj
    q2
    i
    vij · ∇iWij(hi), (10)
    ͕੒Γཱͭͱఆٛ͢Δɻ
    • P∗
    ij
    Λཻࢠ i ཻ͕ࢠ j ͔Βड͚Δѹྗͷ࣌ؒฏۉྔͱ͢Δɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 11 / 17

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  12. Godunov DISPH ๏ಋग़
    Godunov DISPH ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
    ΤωϧΪʔํఔࣜ͸
    dUi
    dt
    = fgrad
    i
    N

    j
    P∗
    ij
    UiUj
    q2
    i
    vij · ∇iWij(hi). (11)
    ӡಈํఔࣜ͸ɺ࡞༻൓࡞༻Λຬͨ͠ΤωϧΪʔอଘ΋ຬͨ͞ͳ͚Ε͹ͳΒͳ͍ͱ͍͏৚͔݅
    ΒٻΊΒΕΔɻ
    mi
    dvi
    dt
    = −
    N

    j
    [
    fgrad
    i
    P∗
    ij
    UiUj
    q2
    i
    ∇iWij(hi) + fgrad
    j
    P∗
    ij
    UiUj
    q2
    j
    ∇iWij(hj)
    ]
    (12)
    Pix = Pi ͷͱ͖ (DISPH Ͱ࢖༻͞Ε͍ͯΔ) ͸, γάϚͷத਎ͷୈҰ߲ͷѹྗ͸ Piɺୈೋ߲ͷ
    ѹྗ͸ Pj
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 12 / 17

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  13. ςετܭࢉ 2D ੩ਫѹฏߧ
    • ඇ෺ཧతͳද໘ுྗʹΑΔޮՌ͸ݟΒΕͳ͍
    • ෺ཧతͳղͱͳ͍ͬͯΔ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 13 / 17

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  14. ςετܭࢉ ఺ݯരൃ
    ఺ݯരൃ
    ڧ͍িܸ೾͕ൃੜ͢Δ 3 ࣍ݩͷ໰୊
    0 < x, y, z < 1 ͷྖҬͷਅΜதʹ߹ܭ 1 ͷΤω
    ϧΪʔΛׂΓৼΔɻີ౓͸ 1ɺ଎౓͸ 0. ֎ଆͷ
    ΤωϧΪʔ͸΄΅ 0
    ಺෦ΤωϧΪʔϓϩϑΝΠϧ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 14 / 17

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  15. ςετܭࢉ ఺ݯരൃ
    SSPH
    • ਓ޻೪ੑ܎਺খ͍͞ͱϊ
    Πζ͕େ͖͘ͳΔɻେ͖
    ͗͢Δͱղ͕ಷΔɻ
    • ͓͓ΉͶղੳղͱҰக͠
    ͍ͯΔ͕ɺਓ޻೪ੑڧ͘
    ͯ͠΋଎౓Ͱৼಈ͕ൃੜ
    • ѹྗʹ΋ৼಈ͕͋Δ
    • ௿ີ౓ྖҬͷѹྗʹେ͖
    ͳޡࠩ
    • িܸ೾ޙ໘Ͱͷີ౓͕ղ
    ੳղͱඍົʹҰக͍ͯ͠
    ͳ͍
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 15 / 17

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  16. ςετܭࢉ ఺ݯരൃ
    GDISPH
    • SPH ͱҧͬͯύϥϝʔλͳ͠Ͱ଎౓ͷৼ
    ಈΛ཈͑ΒΕ͍ͯΔ
    • SPH ͱҧͬͯ, িܸ೾ޙ໘Ͱͷີ౓ͷ஋
    ΋ύϥϝʔλͳ͠ͰղੳղͱҰக
    • SPH ͱҧͬͯ, ௿ີ౓ྖҬͷѹྗޡ͕ࠩ
    ཈͑ΒΕ͍ͯΔ
    • ௿ີ౓ྖҬͰ଎౓ɺ಺෦ΤωϧΪʔʹৼ
    ಈ͕ൃੜ
    GDISPH ͸িܸ೾ޙ໘ͷੑೳ͸ GSPH ͱಉ༷
    (SPH ʹൺ΂ͯྑ͍݁Ռ). ௿ີ౓ྖҬͰ͸
    GDISPH ݻ༗ͷ໰୊͕ൃੜɻͨͩ͠ɺଞͷε
    ΩʔϜ΋௿ີ౓ྖҬͰݻ༗ͷ໰୊͋Γɻ
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 16 / 17

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  17. ݁࿦
    ݁࿦
    • Godunov DISPH
    ▶ DISPH ʹ Riemann solver Λ૊ΈࠐΜͩ Godunov DISPH Λ࡞Δ͜ͱͰɺ઀৮ෆ࿈ଓ໘΋ি
    ܸ೾΋ύϥϝʔλͳ͠Ͱѻ͑ΔΑ͏ʹͳͬͨ
    ▶ ࠓޙ͸ςετܭࢉͰ͸ͳ͍ɺ࣮ࡍͷܭࢉͰͷੑೳΛଞͷεΩʔϜͱൺֱ͍͖ͯ͠,GDISPH
    ͷ࣮༻ੑʹ͍ͭͯݕূ͍͖͍ͯͨ͠
    ࣌ؒͷ౎߹্ࡌͤΒΕͳ͔ͬͨ࿩ (઀৮ෆ࿈ଓ໘ͷѻ͍͕ SPH ΑΓ΋ྑ͍ͱ͞Ε͍ͯΔε
    ΩʔϜͷൺֱ,Godunov DISPH ๏Ͱ Kelvin-Helmholtz ෆ҆ఆੑͷܭࢉ, ଞͷλΠϓͷ Godunov
    DISPH ಋग़) ͕ଔ࿦ʹࡌ͍ͬͯ·͢
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 17 / 17

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  18. ݁࿦
    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೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 17 / 17

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  19. ݁࿦
    Reference I
    [Balsara, 1995] Balsara, D. S. (1995).
    Von neumann stability analysis of smoothed particle hydrodynamicsŠsuggestions for optimal algorithms.
    Journal of Computational Physics, 121(2):357–372.
    [Brookshaw, 1985] Brookshaw, L. (1985).
    A method of calculating radiative heat diffusion in particle simulations.
    Publications of the Astronomical Society of Australia, 6(2):207 r 210.
    [Cha and Whitworth, 2003] Cha, S.-H. and Whitworth, A. P. (2003).
    Implementations and tests of godunov-type particle hydrodynamics.
    Monthly Notices of the Royal Astronomical Society, 340(1):73–90.
    [Garc´
    ıa-Senz et al., 2012] Garc´
    ıa-Senz, D., Cabez´
    on, R. M., and Escart´
    ın, J. A. (2012).
    Improving smoothed particle hydrodynamics with an integral approach to calculating gradients.
    Astronomy & astrophysics, 538:A9.
    [Inutsuka, 2002] Inutsuka, S.-i. (2002).
    Reformulation of smoothed particle hydrodynamics with riemann solver.
    Journal of Computational Physics, 179.
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 17 / 17

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  20. ݁࿦
    Reference II
    [Price, 2008] Price, D. J. (2008).
    Modelling discontinuities and kelvin–helmholtz instabilities in sph.
    Journal of Computational Physics, 227(24):10040–10057.
    [Saitoh and Makino, 2013] Saitoh, T. R. and Makino, J. (2013).
    A DENSITY-INDEPENDENT FORMULATION OF SMOOTHED PARTICLE HYDRODYNAMICS.
    The Astrophysical Journal, 768(1):44.
    [Wadsley et al., 2017] Wadsley, J. W., Keller, B. W., and Quinn, T. R. (2017).
    Gasoline2: a modern smoothed particle hydrodynamics code.
    Monthly Notices of the Royal Astronomical Society, 471(2):2357–2369.
    ౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔۀݚڀൃදձ, January 31, 2023 17 / 17

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