卒業研究発表会資料

Transcript

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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