2022_autumn_asj

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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
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
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
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
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
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
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
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
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
10. ςετܭࢉ Riemann ໰୊ ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 10 / 16
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
12. ςετܭࢉ 2D ੩ਫѹฏߧ ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 12 / 16
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
14. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 14 / 16
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
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
17. ςετܭࢉ 2D Kelvin-Helmholtz ෆ҆ఆੑ Reference I ౬ઙ ୓޺ (ஜ೾େֶ) DISPH, Godunov SPH, Godunov DISPH, IA January 6, 2023 16 / 16