senior_thesis_slides

Reformulation of Density-Independent Smoothed Particle
Hydrodynamics with Riemann Solver: Godunov DISPH
ᴷ ৽ SPH εΩʔϜ:Godunov DISPH ๏ʹ͍ͭͯ ᴷ
to be submitted to MNRAS
1 ౬ઙ୓޺
ࢦಋڭһ: 1 ৿ਖ਼෉
ஜ೾େֶ
ଔݚ಺༰֬ೝ, January 26, 2023
౬ઙ ୓޺ (ஜ೾େֶ) Godunov DISPH ଔݚ಺༰֬ೝ, January 26, 2023 1 / 45

View Slide

໨࣍
1 ֤छεΩʔϜʹ͍ͭͯ
Standard SPH ๏
SSPH with Artiﬁcial Conductivity ๏
Density-Independent SPH ๏
Godunov SPH ๏
·ͱΊ
2 Godunov DISPH ๏ಋग़
3 ςετܭࢉ
Riemann ໰୊
2D ѹྗฏߧ
఺ݯരൃ
2D Kelvin-Helmholtz ෆ҆ఆੑ
4 ݁࿦

View Slide

֤छεΩʔϜʹ͍ͭͯ Standard SPH ๏
Standard SPH (SSPH) ๏
SSPH ๏ಋग़ʹ༻͍Δඇ೪ੑஅ೤ѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
dv(r)
dt
= −
1
ρ(r)
∇P(r) + FAV (αAV ) (1)
du(r)
dt
= −
P(r)
ρ(r)
∇ · v(r) + GAV (αAV ) (2)
ཻࢠ i ͕࣋ͭ೚ҙͷ෺ཧྔ
F(ri) =
∫
F(r′
)W(|ri − r′
|, h)d3r′
+ O(h2)
=
∑
j
mj
ρj
FjW(|ri − rj|, h) + O(1)
(3)
ρi =
∑
j
mjWij(h) (4)

View Slide

• σϝϦοτ
▶ ྲྀମͷ઀৮ෆ࿈ଓ໘Λ͏·͘ѻ͑ͣɺද໘ுྗతͳޮՌ͕ݱΕͯ͠·͏
▶ িܸ೾Λଊ͑ΔͨΊʹਓ޻తͳࢄҳ߲͕ӡಈํఔࣜɺΤωϧΪʔํఔࣜʹඞཁɻ
ਓ޻೪ੑͷڧ͞Λௐઅ͢Δ೚ҙύϥϝʔλ α ΛਓؒͷखͰௐ੔͢Δඞཁ
▶ ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ

View Slide

઀৮ෆ࿈ଓ໘
• ີ౓͸Χʔωϧิ׬ʹΑͬͯٻΊΒΕΔ
ρ =
∑
j
mjWij(h) (5)
• ಺෦ΤωϧΪʔ͸࣌ؒੵ෼ͰٻΊΒΕΔ
• ઀৮ෆ࿈ଓ໘Ͱɺີ౓͸ͳΊΒ͔ɺ಺෦
ΤωϧΪʔ͸ٸܹʹมԽ
P = (γ − 1)ρu (6)
▶ P ʹෆ੔߹͕ੜ͡ɺͦͷ݁Ռද໘ுྗ͕
ൃੜ [Price, 2008]
▶ ཻࢠͷମੵཁૉ dV = m
ρ
͸ີ౓ෆ࿈ଓ
ͷͱ͜ΖͰਫ਼౓௿Լ
[Saitoh and Makino, 2013]

View Slide

• ߋʹɺಋग़தʹີ౓ͷඍ෼Λ࢖༻
−
∇P
ρ
= −∇
(
P
ρ
)
−
P
ρ2
∇ρ (7)
෺ཧతʹີ౓ෆ࿈ଓͱͳΔ઀৮ෆ࿈ଓ໘Ͱ͸ɺີ౓ͷۭؒඍ෼Λܦ༝ͯ͠ಋग़͞Εͨࣜ
͸ਖ਼͘͠ͳ͍͸ͣ [Saitoh and Makino, 2013]

View Slide

Riemann ໰୊
{
ρ = 1, P = 1, v = 0 x < 0 ͷͱ͖
ρ = 0.125, P = 0.1, v = 0 x > 0 ͷͱ͖
(8)
ີ౓ ѹྗ ଎౓

View Slide

Riemann ໰୊
ڧ͍িܸ೾͕ൃੜ͢Δ໰୊
{
ρ = 1, P = 1000, v = 0 x < 0 ͷͱ͖
ρ = 1, P = 0.1, v = 0 x > 0 ͷͱ͖
(9)
ີ౓ ѹྗ ଎౓

View Slide

SSPH
িܸ೾؅໰୊ ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)

View Slide

2D ѹྗฏߧ
ॳظ৚݅ͷ··มಈ͠ͳ͍ͷ͕෺ཧతͳղ ѹྗ͸શྖҬͰҰఆɹ
• Ի଎ Cs = 1.02 ͰܭࢉྖҬΛԣ੾Δ࣌ؒ໿ 1.0 ͷ 8 ഒͰ͋Δ t = 8.0 ·Ͱܭࢉ

View Slide

• ௿ີ౓ྖҬͷԻ଎ Cs = 1.02, ߴີ౓ྖҬͷԻ଎ Cs = 2.04. ࠷΋஗͍Ի଎͕ܭࢉྖҬΛ
ԣ੾Δ࣌ؒ໿ 1.0 ͷ 8 ഒͰ͋Δ t = 8.0 ·Ͱܭࢉ
• ॳظ৚݅Ͱ͸ѹྗ͸શཻࢠҰఆͰ଎౓ 0 ີ౓ͷෆ࿈ଓ෦෼͕ଘࡏ.
▶ Կ΋ى͜Βͳ͍ͷ͕ਖ਼ղ
• ද໘ுྗͷޮՌͰؙ͘ͳͬͯ͠·͏

View Slide

ਓ޻೪ੑ
ඞཁੑͱ໰୊఺
• ඞཁੑ
▶ ඇ೪ੑѹॖੑྲྀମͰ͸িܸ೾͕ൃੜɻিܸ೾෦෼Ͱ͸෺ཧྔ͸ෆ࿈ଓʹมԽ. ͦΕΛ਺஋ܭ
ࢉͰଊ͑ΒΕΔΑ͏ʹ͢ΔͨΊʹਓ޻೪ੑͰ׈Β͔ʹ͢Δ
• ໰୊఺
▶ ਓ޻೪ੑʹ͸ύϥϝʔλ͕ଘࡏ. ύϥϝʔλͷ஋ʹΑͬͯγϛϡϨʔγϣϯ݁Ռ͕มԽ
▶ িܸ೾ྖҬҎ֎Ͱ΋ਓ޻೪ੑ͕༗ޮʹͳΔ (γΞྖҬͳͲ)ɻͨͩ͠γΞྖҬͰͷਓ޻೪ੑ
Λ཈͑Δॲํᝦ͸ଘࡏ (Balsara switch [Balsara, 1995])

View Slide

఺ݯരൃ
ڧ͍িܸ೾͕ൃੜ͢Δ 3 ࣍ݩͷ໰୊
0 < x, y, z < 1 ͷྖҬͷਅΜதʹ߹ܭ 1 ͷΤω
ϧΪʔΛׂΓৼΔɻີ౓͸ 1ɺ଎౓͸ 0. ֎ଆͷ
ΤωϧΪʔ͸΄΅ 0
಺෦ΤωϧΪʔϓϩϑΝΠϧ

View Slide

SSPH
• ਓ޻೪ੑ܎਺খ͍͞ͱϊ
Πζ͕େ͖͘ͳΔɻେ͖
͗͢Δͱղ͕ಷΔɻ
• ͓͓ΉͶղੳղͱҰக͠
͍ͯΔ͕ɺਓ޻೪ੑڧ͘
ͯ͠΋଎౓Ͱৼಈ͕ൃੜ
• ѹྗʹ΋ৼಈ͕͋Δ
• ௿ີ౓ྖҬͷѹྗʹେ͖
ͳޡࠩ
• িܸ೾ޙ໘Ͱͷີ౓͕ղ
ੳղͱඍົʹҰக͍ͯ͠
ͳ͍

View Slide

ۭؒθϩ࣍ޡࠩ
• ཭ࢄԽͷࡍʹཻࢠ෼෍ͷඇ౳ํੑ͔Β͘Δۭؒθϩ࣍ͷޡ͕ࠩൃੜ
F(r) =
∫
F(r′
)W(|r − r′
|, h)d3r′
+ O(h2)
=
∑
j
mj
ρj
FjW(|r − rj|, h) + O(?) + O(h2)
(10)
Fj Λ ri
ͷपΓͰల։͢ΔͱɺFj ∼ Fi − ∇iFi · rij + O(h2) ͱͳΔɻ͜ΕΛ F(ri) = Fi ʹ୅
ೖ͢Δͱ
Fi ∼ FiΣj
mj
ρj
Wij(h) − ∇iFi · Σj
mj
ρj
rijWij(h) + O(h2) (11)
ࣜ (11) ͕ O(h2)(཭ࢄԽલͷޡࠩͱಉ͡) ʹͳΔʹ͸ɺ
Σj
mj
ρj
Wij(h) = 1, Σj
mj
ρj
rijWij(h) = 0 (12)
ͱͳΔඞཁ͕͋Δ͕ɺ࣮ࡍͷܭࢉͰ͸ඞͣ͠΋͜͏ͳΔͱ͸ݶΒͳ͍ɻಉ༷ͳٞ࿦͔Βɺ
SPH ͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ΋ۭؒθϩ࣍ͷޡ͕ࠩଘࡏ͢Δ͜ͱ͕ಋ͖ग़͞ΕΔɻ

View Slide

֤छεΩʔϜʹ͍ͭͯ SSPH with Artiﬁcial Conductivity ๏
SSPH with ArtCond ๏
• [Price, 2008] ͕ߟҊ
• ΤωϧΪʔ֦ࢄ߲ΛೖΕɺ಺෦ΤωϧΪʔΛ׈Β͔ʹ͢Δ͜ͱͰѹྗδϟϯϓΛղফ͢
Δ͜ͱΛ໨ࢦ͍ͯ͠Δ
• ಋग़ͷࡍʹີ౓ͷۭؒඍ෼͸ೖͬͨ··
SSPH with ArtCond ಋग़ʹ༻͍ΔѹॖੑྲྀମͷӡಈํఔࣜɺΤωϧΪʔํఔࣜ
dv(r)
dt
= −
1
ρ(r)
∇P(r) + FAV (αAV ) (13)
du(r)
dt
= −
P(r)
ρ(r)
∇ · v(r) + GAV (αAV ) + HAC(αu) (14)

View Slide

SSPH with Artiﬁcial Conductivity

View Slide

• ਓ޻೤఻ಋ͕͋ͬͯ΋ܗ͸େ͖͘มܗ
• ೤఻ಋͷޮՌͰෆ࿈ଓ෦෼͕΅΍͚Δ
• ਓ޻೤఻ಋ͸ɺࠓߟ͍͑ͯΔ෺ཧ (ΦΠϥʔํఔࣜ) ʹ͸ೖ͍ͬͯͳ͍ޮՌΛೖΕͯ͠
·͍ͬͯΔ

View Slide

֤छεΩʔϜʹ͍ͭͯ Density-Independent SPH ๏
DISPH ๏
• [Saitoh and Makino, 2013] ʹΑͬͯߟҊ͞Εͨख๏
• ઀৮ෆ࿈ଓ໘Ͱ࿈ଓͱͳΔѹྗΛ༻ཻ͍ͯࢠͷମੵཁૉΛۙࣅ͠ɺѹྗͷۭؒඍ෼Λ
࢖༻
▶ ઀৮ෆ࿈ଓ໘ͷѻ͍͕ྑ͘ͳΔ
SPH ͷ෺ཧྔ
F(ri) ≈
∫
F(r′
)W(|ri − r′
|, h)d3r′
≈
∑
j
mj
ρj
FjW(|ri − rj|, h)
(15)
ρi =
∑
j
mjWij(h) (16)
ಋग़தͰ ∇ρ Λ࢖༻
DISPH ͷ෺ཧྔ
F(ri) ≈
∫
F(r′
)W(|ri − r′
|, h)d3r′
≈
∑
j
Uj
qj
FjW(|ri − rj|, h)
(17)
qi =
∑
j
UjWij(h) =
Pi
(γ − 1)
(18)
ಋग़தͰ ∇ρ ͸࢖Θͣɺ∇q Λ࢖༻. ີ౓
ʹཅʹґଘ͠ͳ͍

View Slide

DISPH

View Slide

• ද໘ுྗͷޮՌ͕ͳ͘ͳ͍ͬͯΔ

View Slide

DISPH
• DISPH ͸ڧ͍িܸ೾ʹର
ͯ͠ෆಘҙͳ͸͕ͣͩɺ
ద੾ͳ೪ੑ͕͋Ε͹े෼
ͳੑೳ
• ద੾ͳ೪ੑͰ͋Ε͹ɺج
ຊతʹ SPH ͱࣅͨΑ͏
ͳ݁Ռ
• SPH ͱͷҧ͍ͱͯ͠͸௿
ີ౓ྖҬͰͷѹྗͷޡࠩ
͕ൺֱత཈͑ΒΕ͍ͯΔ

View Slide

֤छεΩʔϜʹ͍ͭͯ Godunov SPH ๏
Godunov SPH ๏
• [Inutsuka, 2002] ʹΑͬͯߟҊ͞Εͨख๏ɻ
• ཻࢠͷ૬ޓ࡞༻ܭࢉͷࡍʹ Riemann solver Λ࢖༻
• ͨͩ͠௨ৗͷ SPH ΑΓ΋ܭࢉίετߴ, ίϯύΫτͰ͸ͳ͍Ψ΢γΞϯΧʔωϧΛ࢖͏
ඞཁੑ͋Γ
▶ ຊൃදͰ͸ [Cha and Whitworth, 2003] ʹΑͬͯఏҊ͞Ε͍ͯΔ GSPH Case3 Λ࢖༻
GSPH Case3 ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
dvi
dt
= −Σjmjp∗
ij
[
1
ρ2
i
∇iWij(hi) +
1
ρ2
j
∇iWij(hj)
]
(19)
dui
dt
= −Σjmjp∗
ij
(v∗
ij
− vi) ·
[
1
ρ2
i
∇iWij(hi) +
1
ρ2
j
∇iWij(hj)
]
(20)

View Slide

͋Δ෺ཧྔ F ʹରͯ͠ɺͦͷॏΈ͖ͭฏۉΛ F∗
ij
Λ
∫
F(r)
ρ2(r)
W(|r − ri|, h(r))W(|r − rj|, h(r))d3r
= F∗
ij
∫
1
ρ2(r)
W(|r − ri|, h(r))W(|r − rj|, h(r))d3r
(21)
Λຬͨ͢Α͏ʹ༩͑ΒΕΔͱ͢Δɻ͜ΕΛྑ͍ਫ਼౓Ͱຬͨ͢Α͏ͳ F∗
ij
Λ༻͍Δɻ
GSPH Case3-2 ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
dvi
dt
= −Σjmjp∗
ij
[
1
ρ2
i
∇iWij(hi) +
1
ρ2
j
∇iWij(hj)
]
(22)
dui
dt
= −Σjmjp∗
ij
(
vi + vj
2
− vi) ·
[
1
ρ2
i
∇iWij(hi) +
1
ρ2
j
∇iWij(hj)
]
(23)

View Slide

Riemann Solver ʹ͍ͭͯ
• ॳظ৚݅
W = (ρ, P, v)
• ղͷλΠϓ
P∗
L
= P∗
R
, v∗
L
= v∗
R

View Slide

• 1 ࣍ݩ Riemann solver ͷೖྗ஋ͷܾΊํ͸༷ʑɻԿΛग़ྗ஋ʹ͢Δ͔༷ʑɻGodunov ๏
(Grid ๏) Ͱ༻͍ΒΕ͍ͯΔ MUSCL ๏Λ૊ΈࠐΜͩΓ΋Ͱ͖Δ
• ຊൃදͰ͸ɺGSPH ಺Ͱ༻͍Δ Riemann solver ͷೖྗ஋ͱཻͯ͠ࢠ i ͱཻࢠ j ͷ෺ཧྔ
Λɺग़ྗ஋ͱͯ͠ star region Ͱͷ஋Λ༻͍Δɻ

View Slide

িܸ೾؅໰୊
GSPH Case3 GSPH Case3-2 (v∗
ij
= vi+vj
2
)

View Slide

ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)
GSPH Case3 GSPH Case3-2 (v∗
ij
= vi+vj
2
)

View Slide


View Slide

GSPH Case3
• ύϥϝʔλͳ͠ͰɺSPH,DISPH ͰݟΒΕ
ͨিܸ೾ޙ໘Ͱͷ଎౓ͷৼಈ΍ີ౓ͷղ
ੳղͱͷͣΕ͕ͳ͘ͳ͍ͬͯΔ
• ௿ີ౓ྖҬͷѹྗ͸ SPH ͱಉ༷ޡ͕ࠩ
͋Δ
• ௿ີ౓ྖҬͷ଎౓ʹൺֱతେ͖ͳޡࠩ

View Slide

֤छεΩʔϜʹ͍ͭͯ ·ͱΊ
·ͱΊ
• DISPH ͸઀৮ෆ࿈ଓ໘ͷѻ͍͕ྑ͍ (઀৮ෆ࿈ଓ໘ͷͨΊͷ௥Ճͷࢄҳ߲ɺύϥϝʔλ
ͳ͠)
• ద੾ͳਓ޻೪ੑύϥϝʔλΛબ୒͢Ε͹ɺڧ͍িܸ೾΍ٸܹͳѹྗޯ഑ʹରͯ͠΋े෼
ͳੑೳ
• GSPH ͸িܸ೾ΛҰ੾ͷύϥϝʔλͳ͠Ͱѻ͑Δ (ѹྗʹ Riemann solver ͷղΛ༻͍Δ
͜ͱͰɺద੾ͳ೪ੑ͕෇Ճ͞ΕΔ)
▶ Riemann Solver Λ DISPH ʹ૊ΈࠐΉ (ѹྗ෦෼Λ Riemann solver ͷղΛ༻͍Δ) ͜ͱͰɺ
ύϥϝʔλͳ͠Ͱ઀৮ෆ࿈ଓ໘, িܸ೾Λѻ͑ΔεΩʔϜΛ࡞Δ

View Slide

Godunov DISPH ๏ಋग़
Godunov DISPH ๏ಋग़ (೤ྗֶୈҰ๏ଇΛى఺)
ཻࢠ i ͷඍখ಺෦ΤωϧΪʔมԽ͸ɺཻࢠ͕֎෦͔Βड͚ΔମੵมԽΛ௨ͨ͡࢓ࣄʹΑΔ΋
ͷ (ॏ৺ͷҐஔมԽΛ௨ͨ͡࢓ࣄ͸ӡಈΤωϧΪʔʹ)
dUi = WV olume
i
(24)
DISPH Ͱ͸ (SPH Ͱ΋)
WV olume
i
= −PidVi (25)
ཻࢠ i ͕ dt ͷؒʹ֎෦͔Βड͚Δѹྗ͸ɺPi + ϵ Ͱ͋ΔͱԾఆ͠ɺೋ࣍ͷඍখྔΛແࢹͯ͠
͍Δ. (ྲྀମํఔࣜͷΤωϧΪʔͷࣜͰ΋ಉ༷) ແݶݸͷཻࢠ, ແݶখͷ smoothing length, ແ
ݶখͷλΠϜεςοϓΛ࢖͑͹ਖ਼͍͠͸ͣ
৽͍͠ߟ͑ํ
Pix Λɺཻࢠ i Λத৺ͱͨ͋͠ΒΏΔํ޲͔Βཻࢠ i ͕ड͚Δѹྗͷ͋Δछͷۭ࣌ؒؒฏۉྔ
ͱͯ͠
WV olume
i
= −PixdVi (26)
͕੒Γཱͭͱ͢Δɻ

View Slide

ཻࢠ i ͷମੵཁૉ
Vi =
Ui
qi
(27)
ཻࢠ i ͷӨڹ൒ܘ಺ͷཻࢠ਺ΛҰఆʹ͢Δͱ͍͏৚݅
qi
Ui
hD
i
= const (28)
ΤωϧΪʔํఔࣜ͸
dUi
dt
= fgrad
i
N
∑
j
PixUiUj
q2
i
vij · ∇iWij(hi). (29)
fgrad
i
=

1 +
hi
Dqi
N
∑
j
Uj
∂Wij(hi)
∂hi


−1
. (30)

View Slide

Pix ͷఆٛ
Pix ΛɺP∗
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), (31)
͕੒Γཱͭͱఆٛ͢Δɻ
• P∗
ij
Λཻࢠ i ཻ͕ࢠ j ͔Βड͚Δѹྗͷ࣌ؒฏۉͱ͠ɺۭؒฏۉ͸೚ҙੑ͕ଘࡏ͢Δɻ

View Slide

Godunov DISPH ͷӡಈํఔࣜ, ΤωϧΪʔํఔࣜ
ΤωϧΪʔํఔࣜ͸
dUi
dt
= fgrad
i
N
∑
j
P∗
ij
UiUj
q2
i
vij · ∇iWij(hi). (32)
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)
]
(33)
Pix = Pi ͷͱ͖ (DISPH Ͱ࢖༻͞Ε͍ͯΔ) ͸, γάϚͷத਎ͷୈҰ߲ͷѹྗ͸ Piɺୈೋ߲ͷ
ѹྗ͸ Pj

View Slide

ςετܭࢉ Riemann ໰୊
Godunov DISPH

View Slide

ςετܭࢉ 2D ѹྗฏߧ
• ද໘ுྗʹΑΔޮՌ͸ݟΒΕͳ͍
• ෺ཧతͳղͱͳ͍ͬͯΔ

View Slide

ςετܭࢉ ఺ݯരൃ
GDISPH • GSPH ͱಉ༷ʹ,SPH,DISPH ͱൺֱ͠
ͯύϥϝʔλͳ͠Ͱ଎౓ͷৼಈΛ཈͑Β
Ε͍ͯΔ
• SPH,DISPH ͱҧͬͯ, িܸ೾ޙ໘Ͱͷີ
౓ͷ஋΋ύϥϝʔλͳ͠ͰղੳղͱҰக
• SPH,GSPH ͱҧͬͯ, ௿ີ౓ྖҬͷѹྗ
ޡ͕ࠩ཈͑ΒΕ͍ͯΔ
• ௿ີ౓ྖҬͰ଎౓ɺ಺෦ΤωϧΪʔʹৼ
ಈ͕ൃੜ
GDISPH ͸িܸ೾ޙ໘ͷੑೳ͸ GSPH ͱಉ༷
(SPH,DISPH ʹൺ΂ͯྑ͍݁Ռ). ௿ີ౓ྖҬ
Ͱ͸ GDISPH ݻ༗ͷ໰୊͕ൃੜɻͨͩ͠ɺଞ
ͷεΩʔϜ΋௿ີ౓ྖҬͰݻ༗ͷ໰୊͋Γɻ

View Slide

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໿ 4 ഒͷ t = 4.0 ·Ͱܭࢉ
Balsara switch Λ෇͚ͯܭࢉ
ॳظ৚݅ ༩͑ͨઁಈ

View Slide

Balsara switch ͱ͸
• [Balsara, 1995] ʹΑͬͯఏҊ͞Εͨख๏
• ਓ޻೪ੑ߲͸ຊདྷিܸ೾ྖҬͰͷΈಇ͍ͯ΄͍͕͠ɺγΞྖҬͰ΋ಇ͍ͯ͠·͏ɻ
• ͦΕΛ๷͙ͨΊɺBalsara switch ͕Α͘࢖ΘΕ͍ͯΔɻ
Balsara switch
ਓ޻೪ੑ߲ʹ 0.5(Fi + Fj) Λ͔͚ΔɻγΞྖҬͰ͸ Fi, Fj Λ 1 ΑΓখ͘͞ɺিܸ೾ྖҬͰ͸
Fi, Fj ͕ 1 ʹͳΔΑ͏ʹઃఆɻ
Fi =
|∇i · vi|
|∇i · vi| + |∇i × vi| + 0.0001ci/hi
(34)

View Slide

GDISPH ΁ͷ Balsara switch ͷద༻
• GSPHɺGDISPH ʹ Balsara switch ͷద༻Λߦͬͨྫ͸ͳ͍ɻ
• DISPH ͸ (ඇਓ޻೪ੑ߲)+(ਓ޻೪ੑ߲) ʹཅʹ෼͔Ε͍ͯΔ͕ɺGDISPH ͸ 1 ͭͷ߲ʹ
·ͱ·͍ͬͯͯ෼཭͍ͯ͠ͳ͍ɻ
• ཧ૝͸ GDISPH தͷ P∗
ij
Λ (ඇ೪ੑ߲)+(೪ੑ߲) ʹཅʹ෼͚Δ͜ͱ
զʑͷఏҊख๏
• (DISPH ͷඇਓ޻೪ੑ߲)+0.5(Fi + Fj)(GDISPH ͷ߲ - DISPH ͷඇਓ޻೪ੑ߲) Λɺ
GDISPH ͷࣜͱͯ͠༻͍Δɻ

View Slide


View Slide

ݪҼͷ༧૝
ϊΠζͷਖ਼ମ
ཻ֤ࢠʹ͓͚Δɺۭؒ 0 ࣍ͷޡࠩ΍
O(h2) ͷޡࠩͷ͔͔Γํͷҧ͍ʹΑΔ΋
ͷɻཻ֤ࢠؒͷඍົͳ਺஋ͷͣΕ͕ɺ
ඇৗʹ೾௕ͷখ͍͞ઁಈͱͳΔɻ
ϊΠζ͕େ͖͘੒௕͢Δཧ༝
• balsara switch ͷಋೖʹΑͬͯγΞྖ
ҬͰͷ೪ੑ͕΄΅ 0 ʹͳΔɻ
• ڪΒ͘෺ཧతʹ͸ɺྲྀମͷ೪ੑ͕େ
͖͘ͳΔͱɺKH ෆ҆ఆੑ͕ى͜Δ
࠷খͷ೾௕΋େ͖͘ͳΔɻ
೪ੑ͕ 0 ͳΒ͹ɺ
ͲͷΑ͏ͳ೾௕ʹରͯ͠΋ෆ҆ఆੑ੒௕

View Slide

݁࿦
݁࿦
• Godunov DISPH
▶ DISPH ʹ Riemann solver Λ૊ΈࠐΜͩ Godunov DISPH Λ࡞Δ͜ͱͰɺ઀৮ෆ࿈ଓ໘΋ি
ܸ೾΋ύϥϝʔλͳ͠Ͱѻ͑ΔΑ͏ʹͳͬͨ
▶ Godunov DISPH ΋௨ৗͷ SPH ͱಉ༷ɺγΞྖҬͰ΋೪ੑΛ͔͚ͯ͠·͏໰୊఺͕͋Δɻ
ͦͷ໰୊఺ղফͷͨΊɺGDISPH ʹ Balsara switch Λ૊ΈࠐΉͨΊͷख๏ΛߟҊɻKH ෆ҆
ఆੑͷܭࢉͰ DISPH ͱಉ༷ͳ݁Ռ
• Kelvin-Helmholtz ෆ҆ఆੑ
▶ ॳظ৚݅ʹΑͬͯ͸ϊΠζʹΑΔઁಈ΋੒௕͢Δ
▶ ͲͷΑ͏ͳঢ়گͰ΋ɺཻࢠؒڑ཭ఔ౓ͷ೾௕ͷઁಈͷ੒௕Λ཈͑ΒΕΔํ๏Λݟ͚͍ͭͨɻ
ࠓޙͷ՝୊

View Slide

݁࿦
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/)

View Slide

݁࿦
Reference I
[Balsara, 1995] Balsara, D. S. (1995).
Von neumann stability analysis of smoothed particle hydrodynamicssuggestions 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.

View Slide

݁࿦
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.

View Slide

݁࿦
SPH GDF ๏
• Geometric Density Average Force (GDF)([Wadsley et al., 2017])
• Gasoline2 Ͱ༻͍ΒΕ͍ͯΔ͓ΓɺSPH ๏ΑΓ΋઀৮ෆ࿈ଓ໘ͷѻ͍͕ྑ͍
GDF ๏ͷӡಈํఔࣜಋग़
• ྲྀମͷӡಈํఔࣜΛมܗ͢Δ
−
∇P
ρ
= −
1
ρ2−σ
∇
(
P
ρσ−1
)
−
P
ρσ
∇ρσ−1 (35)
• σ = 1 ͷͱ͖ (GDF ಋग़Ͱ༻͍Δ, ີ౓ͷۭؒඍ෼͕ͳ͍)
−
∇P
ρ
= −
1
ρ
∇P −
P
ρ
∇1 (36)
• σ = 2 ͷͱ͖ (௨ৗͷ SPH Ͱ࢖༻, ີ౓ͷۭؒඍ෼͕͋Δ)
−
∇P
ρ
= −∇
(
P
ρ
)
−
P
ρ2
∇ρ (37)

View Slide

݁࿦
• ͜ΕΛ F(r) =
∑
j
mj
ρj
FjW (|r − rj|, h(r)) Ͱ཭ࢄԽ͢Δ (ࣜ (3)) ͱ
dvi
dt
= −
∑
j
mj
(
Pi
ρσ
i
ρ2−σ
j
+
Pj
ρ2−σ
i
ρσ
j
)
∇iWij(h) (38)
• σ = 1 Λ୅ೖ (Α͘༻͍ΒΕ͍ͯΔ standard SPH ͸ σ = 2 ͷ࣌ͱಉ༷ͷܗ)
dvi
dt
= −
∑
j
mj
(
Pi + Pj
ρiρj
)
∇iWij(h) (39)
ΤωϧΪʔํఔࣜ΋ಉ༷ͷܗͷ΋ͷΛ࢖༻
dui
dt
=
∑
j
mj
(
Pi
ρiρj
)
vij · ∇iWij(h) (40)

View Slide

݁࿦
িܸ೾؅໰୊
ڧ͍িܸ೾ (ॳظ৚݅Ͱ 104 ͷѹྗࠩ)

View Slide

݁࿦

View Slide

݁࿦
γΞ଎౓Λ׈Β͔ʹભҠͤ͞Δ
R(y) =
1
1 + exp [−2(y − 0.25)/0.025]
1
1 + exp [2(y − 0.75)/0.025]
(41)
vx(y) = vx,l + R(y)[vx,h − vx,l] (42)
ॳظ৚݅
x ࣠ํ޲ͷ଎౓

View Slide

݁࿦

View Slide

݁࿦
վળࡦ
• ͲͷΑ͏ͳঢ়گʹ͓͍ͯ΋ɺ෺ཧతͳઁಈʹΑΔෆ҆ఆੑͷ੒௕͸ڐ͠ɺ਺஋తͳϊΠ
ζʹΑΔઁಈͷ੒௕͸཈͑Δɻ
→ balsara switch ͷڧ͞Λௐઅ͢Δ͜ͱͰ࣮ݱΛ໨ࢦ͢ɻ
Balsara switch
• ਓ޻೪ੑ߲ʹ 0.5(Fi + Fj) Λ͔͚Δɻ
Fi =
|∇i · vi|
|∇i · vi| + |∇i × vi| + 0.0001ci/hi
(43)
զʑͷఏҊख๏
• Fi, Fj ͷ୅ΘΓʹ F′
i
, F′
j
Λ༻͍Δɻ
• β ͸ [0, 1] ͷ࣮਺ΛऔΔ೚ҙύϥϝʔλɻβ = 1 Ͱ balsara switch Φϯɹ β = 0 Ͱ balsara
switch Φϑ
F′
i
= 1 + β(Fi − 1) (44)

View Slide

݁࿦
೪ੑΛ༻͍Δཧ༝
• 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|
(45)

View Slide

݁࿦

View Slide

݁࿦
Riemann ໰୊
{
ρ = 1, P = 0.4, v = −2 x < 0 ͷͱ͖
ρ = 1, P = 0.4, v = 2 x > 0 ͷͱ͖
(46)
ີ౓ ѹྗ ଎౓

View Slide

݁࿦

View Slide

senior_thesis_slides

senior_thesis_slides

statictaku

More Decks by statictaku

Other Decks in Research

Featured

Transcript