2023_spring_asj

Transcript

1. վྑ൛Balsara SwitchΛ༻͍ͨɺSPH๏ͷۭ࣋ͭؒθϩ
   ࣍ޡ͕ࠩ༠ൃ͢ΔِKelvin-Helmholtzෆ҆ఆͷ཈੍
   ౬ઙ୓޺, ৿ਖ਼෉
   ஜ೾େֶ
   ೔ຊఱจֶձय़ق೥ձ, March 15, 2023
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 1 / 13
   

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2. ໨࣍
   1 ֤छεΩʔϜʹ͍ͭͯ
   2 ِ Kelvin-Helmholtz ෆ҆ఆ
   3 ݁࿦
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 2 / 13
   

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3. ֤छεΩʔϜʹ͍ͭͯ
   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
   ਓ޻೪ੑͷڧ͞Λௐઅ͢Δ೚ҙύϥϝʔλ α ΛਓؒͷखͰௐ੔͢Δඞཁ
   SSPH DISPH GSPH(ࣗಈతʹద੾ͳ೪ੑ)
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 3 / 13
   

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4. ֤छεΩʔϜʹ͍ͭͯ
   Godunov DISPH ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ (࿦จ౤ߘ༧ఆ)
   mi
   dvi
   dt
   = −
   N
   ∑
   j
   [
   fgrad
   i
   P∗
   ij
   Ui
   Uj
   q2
   i
   ∇i
   Wij
   (hi
   ) + fgrad
   j
   P∗
   ij
   Ui
   Uj
   q2
   j
   ∇i
   Wij
   (hj
   )
   ]
   (3)
   dUi
   dt
   = fgrad
   i
   N
   ∑
   j
   P∗
   ij
   Ui
   Uj
   q2
   i
   vij
   · ∇i
   Wij
   (hi
   ). (4)
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 4 / 13
   

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5. ِ Kelvin-Helmholtz ෆ҆ఆ
   ۭؒθϩ࣍ޡࠩ
   F(r) =
   ∫
   F(r′
   )W(|r − r′
   |, h)d3r′
   + O(h2)
   =
   ∑
   j
   mj
   ρj
   FjW(|r − rj|, h) + O(1) + O(h2)
   (5)
   • ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ͢Δ͜ͱ͕஌ΒΕͯ
   ͍Δ
   ▶ ཻࢠ෼෍ͷඍົͳҧ͍Ͱ΋஋͕มԽ
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 5 / 13
   

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6. ِ Kelvin-Helmholtz ෆ҆ఆ
   • ਓ޻೪ੑ߲͸ຊདྷিܸ೾ྖҬͰͷΈಇ͍ͯ΄͍͕͠ɺγΞྖҬͰ΋ಇ͍ͯ͠·͏ɻ
   ▶ rij
   ͷϕΫτϧ্Ͱཻࢠ͕ޓ͍ʹ͍͍ۙͮͯΔ͔͕িܸ೾ͷ൑ఆ๏ʹͳ͍ͬͯΔ͔Β
   Balsara switch
   FAV (α) → 0.5(Fi + Fj)FAV (α), GAV (α) → 0.5(Fi + Fj)GAV (α)ɻ
   γΞྖҬͰ͸ Fi, Fj Λ 1 ΑΓখ͘͞ɺিܸ೾ྖҬͰ͸ Fi, Fj ͕ 1 ʹͳΔΑ͏ʹઃఆɻ
   Fi =
   |∇i · vi|
   |∇i · vi| + |∇i × vi| + 0.0001ci/hi
   (6)
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 6 / 13
   

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7. ِ Kelvin-Helmholtz ෆ҆ఆ
   GDISPH ΁ͷ Balsara switch ͷద༻
   • GSPH,GDISPH ͸ rij ͷϕΫτϧ্Ͱ 1 ࣍ݩ Riemann solver Λղ͘ɻೖྗ஋Λཻࢠ i ͱ j
   ͷ෺ཧྔͱ͠ɺग़ྗ஋Λ௨ͯ͠೪ੑ͕ೖΔ
   ▶ γΞྖҬͱিܸ೾ྖҬͷ۠ผ͸͔ͭͳ͍ɻ
   • DISPH ͸ (ඇਓ޻೪ੑ߲)+(ਓ޻೪ੑ߲) ʹཅʹ෼͔Ε͍ͯΔ͕ɺGDISPH ͸ 1 ͭͷ߲ʹ
   ·ͱ·͍ͬͯͯ෼཭͍ͯ͠ͳ͍ɻ
   • ཧ૝͸ GDISPH தͷ P∗
   ij
   Λ (ඇ೪ੑ߲)+(೪ੑ߲) ʹཅʹ෼͚Δ͜ͱ
   ▶ ·ͩ͏·͍͍ͬͯ͘ͳ͍ɻBalsara switch ͷద༻ྫ΋ͳ͍ɻ
   ࣮ݧతʹࢼͯ͠ΈΔ
   • (DISPH ͷඇਓ޻೪ੑ߲)+0.5(Fi + Fj)(GDISPH ͷ߲ ʔ DISPH ͷඇਓ޻೪ੑ߲) Λɺ
   GDISPH ͷࣜͱͯ͠༻͍Δɻ
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 7 / 13
   

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8. ِ Kelvin-Helmholtz ෆ҆ఆ
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 8 / 13
   

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9. ِ Kelvin-Helmholtz ෆ҆ఆ
   ݪҼͷ༧૝
   ϊΠζͷਖ਼ମ
   ཻ֤ࢠʹ͓͚Δɺۭؒ 0 ࣍ͷޡࠩ΍
   O(h2) ͷޡࠩͷ͔͔Γํͷҧ͍ʹΑΔ΋
   ͷɻཻ֤ࢠؒͷඍົͳ਺஋ͷͣΕ͕ɺ
   ඇৗʹ೾௕ͷখ͍͞ઁಈͱͳΔɻ
   ϊΠζ͕େ͖͘੒௕͢Δཧ༝
   • Balsara swtich ͷಋೖʹΑͬͯγΞྖ
   ҬͰͷ೪ੑ͕΄΅ 0 ʹͳΔɻ
   • ڪΒ͘෺ཧతʹ͸ɺྲྀମͷ೪ੑ͕େ
   ͖͘ͳΔͱɺKH ෆ҆ఆੑ͕ى͜Δ
   ࠷খͷ೾௕΋େ͖͘ͳΔɻ
   ೪ੑ͕ 0 ͳΒ͹ɺ
   ͲͷΑ͏ͳ೾௕ʹରͯ͠΋ෆ҆ఆੑ੒௕
   ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 9 / 13
   

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10. ِ Kelvin-Helmholtz ෆ҆ఆ
    վળࡦ
    • ͲͷΑ͏ͳঢ়گʹ͓͍ͯ΋ɺ෺ཧతͳઁಈʹΑΔෆ҆ఆੑͷ੒௕͸ڐ͠ɺ਺஋తͳϊΠ
    ζʹΑΔઁಈͷ੒௕͸཈͑Δɻ
    → Balsara swtich ͷڧ͞Λௐઅ͢Δ͜ͱͰ࣮ݱΛ໨ࢦ͢ (γΞʹ͋͑ͯ೪ੑΛՃ͑Δ)ɻ
    Balsara switch
    • FAV (α) → 0.5(Fi + Fj)FAV (α), GAV (α) → 0.5(Fi + Fj)GAV (α)ɻ
    Fi =
    |∇i · vi|
    |∇i · vi| + |∇i × vi| + 0.0001ci/hi
    (7)
    զʑͷఏҊख๏
    • Fi, Fj ͷ୅ΘΓʹ F′
    i
    , F′
    j
    Λ༻͍Δɻ
    • β ͸ [0, 1] ͷ࣮਺ΛऔΔ೚ҙύϥϝʔλɻβ = 1 Ͱ Balsara swtich Φϯɹ β = 0 Ͱ
    Balsara swtich Φϑ
    F′
    i
    = 1 + β(Fi − 1) (8)
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 10 / 13
    

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11. ِ Kelvin-Helmholtz ෆ҆ఆ
    β = 0.5
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 11 / 13
    

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12. ِ Kelvin-Helmholtz ෆ҆ఆ
    β = 0.85
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 12 / 13
    

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13. ݁࿦
    ݁࿦
    • Kelvin-Helmholtz ෆ҆ఆੑ
    ▶ ϊΠζʹΑΔઁಈ΋੒௕͢Δ. ղܾࡦͱͯ͠ɺ͋͑ͯγΞྖҬʹ೪ੑΛՃ͑Δ͜ͱΛఏҊ.
    ํ޲ੑ͸ਖ਼ͦ͠͏. ΍Γํʹؔͯ͠͸վળͷ༨஍͋Γɻ
    ▶ ͲͷΑ͏ͳঢ়گͰ΋ɺཻࢠؒڑ཭ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ཈͑ΒΕΔํ๏Λݟ͚͍ͭͨɻ
    ࠓޙͷ՝୊
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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14. ݁࿦
    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೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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15. ݁࿦
    Reference I
    Balsara, D. S. (1995).
    Von neumann stability analysis of smoothed particle hydrodynamicsŠsuggestions for
    optimal algorithms.
    Journal of Computational Physics, 121(2):357–372.
    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, 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.
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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16. ݁࿦
    Reference II
    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, S.-i. (2002).
    Reformulation of smoothed particle hydrodynamics with riemann solver.
    Journal of Computational Physics, 179.
    Price, D. J. (2008).
    Modelling discontinuities and kelvin–helmholtz instabilities in sph.
    Journal of Computational Physics, 227(24):10040–10057.
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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17. ݁࿦
    Reference III
    Saitoh, T. R. and Makino, J. (2013).
    A DENSITY-INDEPENDENT FORMULATION OF SMOOTHED PARTICLE
    HYDRODYNAMICS.
    The Astrophysical Journal, 768(1):44.
    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೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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18. ݁࿦
    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೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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19. ݁࿦
    Godunov DISPH ๏ʹ͍ͭͯ (࿦จ౤ߘ༧ఆ)
    [Inutsuka, 2002] ͷํ๏ͱ͸ҧͬͨ, զʑͷΦϦδφϧͷํ๏Ͱ Riemann solver Λ DISPH ʹ૊
    ΈࠐΉ
    Godunov DISPH ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
    ΤωϧΪʔํఔࣜ͸
    dUi
    dt
    = fgrad
    i
    N
    ∑
    j
    P∗
    ij
    UiUj
    q2
    i
    vij · ∇iWij(hi). (9)
    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)
    ]
    (10)
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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20. ݁࿦
    ઀৮ෆ࿈ଓ໘
    • ີ౓͸Χʔωϧิ׬ʹΑͬͯٻΊΒΕΔ
    ρ =
    ∑
    j
    mjWij(h) (11)
    • ಺෦ΤωϧΪʔ͸࣌ؒੵ෼ͰٻΊΒΕΔ
    • ઀৮ෆ࿈ଓ໘Ͱɺີ౓͸ͳΊΒ͔ɺ಺෦
    ΤωϧΪʔ͸ٸܹʹมԽ
    P = (γ − 1)ρu (12)
    ▶ P ʹෆ੔߹͕ੜ͡ɺͦͷ݁Ռද໘ுྗ͕
    ൃੜ [Price, 2008]
    ▶ ཻࢠͷମੵཁૉ dV = m
    ρ
    ͸ີ౓ෆ࿈ଓ
    ͷͱ͜ΖͰਫ਼౓௿Լ
    [Saitoh and Makino, 2013]
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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21. ݁࿦
    • ߋʹɺಋग़தʹີ౓ͷඍ෼Λ࢖༻
    −
    ∇P
    ρ
    = −∇
    (
    P
    ρ
    )
    −
    P
    ρ2
    ∇ρ (13)
    ෺ཧతʹີ౓ෆ࿈ଓͱͳΔ઀৮ෆ࿈ଓ໘Ͱ͸ɺີ౓ͷۭؒඍ෼Λܦ༝ͯ͠ಋग़͞Εͨࣜ
    ͸ਖ਼͘͠ͳ͍͸ͣ [Saitoh and Makino, 2013]
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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22. ݁࿦
    2D Kelvin-Helmholtz ෆ҆ఆੑ
    Ґஔ y = 0.25, y = 0.75 ͷͦΕͧΕʹɺy ࣠ํ޲ͷ଎౓ʹ 2 ೾௕෼ͷઁಈΛ༩͍͑ͯΔ
    ෆ҆ఆੑͷ੒௕ͷλΠϜεέʔϧ τKH = 1.06 ͷ໿ 4 ഒͷ t = 4.0 ·Ͱܭࢉ
    Balsara switch Λ෇͚ͯܭࢉ
    ॳظ৚݅ ༩͑ͨઁಈ
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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23. ݁࿦
    վળࡦ 2
    • γΞʹ೪ੑͰ͸ͳ͘ΤωϧΪʔ֦ࢄ͕ಇ͘Α͏ʹͯ͠ΈΔɻࠓճ͸ਓ޻೤఻ಋ߲ΛՃ
    ͑Δɻ
    • [Price, 2008] ͕઀৮ෆ࿈ଓ໘Ͱͷඇ෺ཧతͳද໘ுྗΛ཈͑Δख๏ͱͯ͠ߟҊ
    • ਪ঑஋ͷ αu = 1 Λ࢖༻
    SSPH with ArtCond ಋग़ʹ༻͍ΔѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
    dv(r)
    dt
    = −
    1
    ρ(r)
    ∇P(r) + FAV (αAV ) (14)
    du(r)
    dt
    = −
    P(r)
    ρ(r)
    ∇ · v(r) + GAV (αAV ) + HAC(αu) (15)
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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24. ݁࿦
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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25. ݁࿦
    ೪ੑ͕ྑͦ͞͏ͳཧ༝
    • ΤωϧΪʔ֦ࢄ߲͸ີ౓͕ಉ͡ʹͳΔΑ͏ಇ͖ɺ೪ੑ߲͸଎౓͕ಉ͡ʹͳΔΑ͏ʹಇ
    ͘ɻγΞྖҬͰ͸ͳΔ΂͘ಇ͔ͳ͍Α͏ʹ͍ͨ͠ɻ
    ▶ গͳ͍֦ࢄ or ೪ੑͰɺཻࢠؒڑ཭ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ཈͍͑ͨ
    ▶ ΤωϧΪʔ֦ࢄ߲෇͖ or ೪ੑ߲෇͖ͷྲྀମํఔࣜͷઢܗղੳΛߦ͏͜ͱͰௐ΂͍ͨɻࠓޙ
    ͷ՝୊
    • KH ෆ҆ఆੑͷλΠϜεέʔϧΛݟΔͱɺີ౓ͷۉ࣭ԽΑΓ΋γΞ଎౓ͷۉ࣭Խͷ΄͏
    ͕ KH ෆ҆ఆੑʹ༩͑ΔӨڹ͕ڧͦ͏
    ▶ ೪ੑͷ΄͏͕ΑΓޮՌతʹϊΠζͷ੒௕Λ཈͑ΒΕΔͱ༧૝Ͱ͖Δ
    τkh =
    λ(ρh + ρl)
    √
    ρhρl|vx,h − vx,l|
    (16)
    ౬ઙ ୓޺ (ஜ೾େֶ) Balsara switch, Kelvin-Helmholtz ೔ຊఱจֶձय़ق೥ձ, March 15, 2023 13 / 13
    

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