statictaku
May 20, 2023
110

# poster_senior_thesis

May 20, 2023

## Transcript

1. ### Reformulation of Density-Independent Smoothed Particle Hydrodynamics with Riemann Solver: Godunov

DISPH ౬ઙ୓޺1*ɾࢦಋڭ׭: ৿ਖ਼෉1 1 ஜ೾େֶ ཧ޻ֶ܈ ෺ཧֶྨ, *[email protected] 1. SPH๏ͱ͸ • ཻࢠΛ༻͍ͯѹॖੑඇ೪ੑྲྀମ(ཧ૝ྲྀମ)ΛγϛϡϨʔγϣϯ͢ ΔͨΊͷܭࢉख๏.֨ࢠ๏ͱൺ΂ͨϝϦοτ͕͍͔ͭ͋͘Δ. • ݩʑӉ஦෼໺ൃ঵.ۙ೥͸޻ۀ΍CG෼໺Ͱ΋༻͍ΒΕ͍ͯΔ. dv(r) dt = − 1 ρ(r) ∇P(r) + FAV (α) (1) du(r) dt = − P(r) ρ(r) ∇ · v(r) + GAV (α) (2) F(ri ) ≈ ∫ F(r′ )W(|ri − r′ |, h)d3r′ ≈ ∑ j mj ρj Fj W(|ri − rj |, h) (3) • ӡಈํఔࣜ(1),಺෦ΤωϧΪʔํఔࣜ(2)Λ,ࣜ(3)Λ༻͍ͯ཭ࢄԽ ͢Δ.W ͸Ψ΢γΞϯͷΑ͏ͳܗΛ͓ͯ͠Γ,पΓͷཻࢠͷӨڹΛ ॏͶ߹ΘͤΔΠϝʔδ(i,j ͸ཻࢠͷϥϕϦϯά). ཧ૝ؾମͷঢ়ଶ ํఔࣜΛ༻͍Δͱ͕ࣜด͡Δ. 2.SPH๏ͷσϝϦοτ(઀৮ෆ࿈ଓ໘ͱিܸ೾) ઀৮ෆ࿈ଓ໘ ྲྀମͷڥք໘Ͱ͋Δ઀৮ෆ࿈ଓ໘͸, खͰนΛԡ͢ͱ͖ʹͰ͖Δෆ࿈ ଓ໘(खͱนͷ଎౓,ѹྗҰఆͰີ౓,ΤωϧΪʔ͸ෆ࿈ଓ)ͱࣅ͍ͯΔ. SPHͰ͸ີ౓͸ࣜ(1)ͰٻΊΔ. ρ = ∑ j mj Wij (h) (4) ͭ·Γ, ີ౓ͷۭؒ࿈ଓੑ͕Ծఆ͞Εͯ͠·͍ͬͯΔ. ·ͨ, ಋग़தͰ ͸ີ౓ͷۭؒඍ෼Λߦ͍ͬͯΔ. ͜ΕΒͷཧ༝͔Β,SPH Ͱ͸෺ཧత ʹີ౓ෆ࿈ଓͱͳΔ઀৮ෆ࿈ଓ໘͕͏·͘ѻ͑ͣʹඇ෺ཧతͳද໘ு ྗΛൃੜͤͯ͞͠·͏͜ͱ͕஌ΒΕ͍ͯΔ. ॲํᝦͱͯ͠,ີ౓ͷ୅ΘΓʹѹྗͷۭؒ࿈ଓੑ,ۭؒඍ෼ՄೳੑΛԾ ఆͨ͠DISPH๏͕ଘࡏ. ඇ෺ཧతͳද໘ுྗΛ཈͑Δ͜ͱ͕Ͱ͖ΔҰ ํ,িܸ೾ͰͷσϝϦοτ͸SPHͱಉ༷ʹଘࡏ͢Δ. িܸ೾ িܸ೾͸Ի೾ͱಉ༷, ৘ใͷ఻ൖʹΑΓى͜Δ. ෺ཧతͳڥքͰ͸ͳ ͍. ѹྗ,଎౓΋ෆ࿈ଓ. ਺஋ܭࢉͰিܸ೾Λଊ͑ΔͨΊʹ,ਓ޻೪ੑ ߲((1),(2)ͷ੺ࣈͷ߲)Λ࢖ͬͯিܸ೾Λ͋͑ͯ׈Β͔ʹ͢Δඞཁੑ͕ ͋Δɻਓ޻೪ੑ߲ͷڧ͞Λௐઅ͢Δύϥϝʔλα͸ਓ͕ؒखͰ,ܦݧ ౳Λ΋ͱʹܾΊ͍ͯΔ. ॲํᝦͱͯ͠,ཻࢠͷ૬ޓ࡞༻ܭࢉͷࡍʹRiemann solverͷղΛ༻͍Δ ͜ͱͰ,Ұ੾ͷύϥϝʔλͳ͠Ͱద੾ͳ೪ੑ͕িܸ೾ྖҬʹೖΔ Godunov SPH๏ʢGSPH๏ʣ͕ଘࡏ͢Δ. ͔͠͠,઀৮ෆ࿈ଓ໘ͷѻ͍ ͸SPHͱಉ༷ʹෆ׬શ. 3.Godunov DISPH๏ʹ͍ͭͯ DISPH ๏ͱ GSPH ๏Λ૊Έ߹Θͤͨ Godunov DISPH๏Λ։ൃ. ઀৮ෆ ࿈ଓ໘Λਖ਼͘͠ѻ͑,ద੾ͳ೪ੑ͕ύϥϝʔλͳ͠ͰೖΔ.ҎԼͰ͸ͦ ͷಋग़ʹ͍ͭͯܰ͘ड़΂Δ. ཻࢠ i ͷඍখ಺෦ΤωϧΪʔมԽ͸, ཻࢠ͕֎෦͔Βड͚ΔମੵมԽ Λ௨ͨ͡࢓ࣄʹΑΔ΋ͷ(ॏ৺ͷҐஔมԽΛ௨ͨ͡࢓ࣄ͸ӡಈΤωϧ Ϊʔʹ) dUi = WV olume i (5) DISPHͰ͸(SPHͰ΋) WV olume i = −Pi dVi (6) ͜Ε͸ཻࢠi͕dtͷؒʹ֎෦͔Βड͚Δѹྗ͸ɺPi + ϵͰ͋ΔͱԾఆ ͠ɺೋ࣍ͷඍখྔΛແࢹ͍ͯ͠Δ. (ྲྀମํఔࣜͷΤωϧΪʔͷࣜͰ ΋ಉ༷) ͜ͷԾఆͷԼͰ΋,ແݶݸͷཻࢠ,ແݶͷղ૾౓,ແݶখͷλΠ ϜεςοϓΛ࢖͑͹ਖ਼͍͠ղʹऩଋ͢Δ͸ͣ. ৽͍͠ߟ͑ํ Pix ΛɺཻࢠiΛத৺ͱͨ͋͠ΒΏΔํ޲΋,͔Βཻࢠi͕ड͚Δѹྗ ͷ͋Δछͷۭ࣌ؒؒฏۉྔͱͯ͠ WV olume i = −Pix dVi (7) ͕੒Γཱͭͱ͢Δɻͪ͜ΒͷԾఆͷԼͰ΋,ແݶݸͷཻࢠ,ແݶͷղ ૾౓,ແݶখͷλΠϜεςοϓΛ࢖͑͹ਖ਼͍͠ղʹऩଋ͢Δ͸ͣ. ۭ࣌ؒؒฏۉͷऔΓํʹ͸೚ҙੑ͕͋Δ. ࠓճ͸, ཻࢠ i ཻ͕ࢠ j ͔ Βड͚Δѹྗͷ࣌ؒฏۉΛ,ͦΕͧΕͷཻࢠͷ෺ཧྔΛೖྗ஋ͱͨ͠ Rieamnn solverͷղʹ͓͚Δ઀৮ෆ࿈ଓ໘্Ͱͷѹྗͱ͠,ۭؒฏۉ͸ ద౰ͳܗͰͱΔ. ͋ͱ͸ࣜ(7)͔ΒDISPH๏ͷಋग़ͱ΄΅ಉ͡΍Γํ ͰGDISPH๏ͷӡಈํఔࣜ,ΤωϧΪʔํఔࣜΛಋग़Մೳ. 4. ςετܭࢉ(GDISPHͷ઀৮ෆ࿈ଓ໘ͱিܸ೾ʹର͢Δੑೳ) ॳظ৚͕݅ѹྗฏߧͳͷͰ, ॳظ৚݅ͷ··มಈ͠ͳ͍ ͜ͱ͕෺ཧతͳղ. SPH, GSPH͸઀৮ෆ࿈ଓ໘ ͔Βͷද໘ுྗͷӨڹͰܗ ؙ͕͘ͳͬͯ͠·͏.Ӎཻ͕ ද໘ுྗͰؙ͘ͳΔݮগͱ ࣅ͍ͯΔ. DISPH, GDISPH Ͱ͸ඇ෺ཧతͳද໘ுྗͷ ޮՌ͕ͳ͘ͳ͍ͬͯΔ͜ͱ ͕Θ͔Δ.