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