2022_autumn_asj

Transcript

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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