t Prt t P - - PowerPoint PPT Presentation

❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s

▲❛✉r❡♥t ❈❤✐r♦♥

▲❍❊❊❆ ▲❛❜ ✲ ❊❝♦❧❡ ❈❡♥tr❛❧❡ ◆❛♥t❡s

✶❡r ❥✉✐❧❧❡t ✷✵✶✺

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✸ ✴ ✷✺

❙♣❤✲✢♦✇ s❝♦♣❡s

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✹ ✴ ✷✺

❙P❍ ❜❛❝❦❣r♦✉♥❞

❈♦♥s❡r✈❛t✐♦♥ ❧❛✇

∂tU + ∂xf(U) = 0 ∂xf(U) →        ❋✐♥✐t❡ ❞✐✛❡r❡♥❝❡s ❋✐♥✐t❡ ✈♦❧✉♠❡s ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ❙P❍ ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❡r♠ ✐♥ ❙P❍ ❄

❈♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t ♣r♦♣❡rt✐❡s ✐s ❛♣♣r♦❛❝❤❡❞ ❜② ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✺ ✴ ✷✺

❙P❍ ❜❛❝❦❣r♦✉♥❞

❈♦♥s❡r✈❛t✐♦♥ ❧❛✇

∂tU + ∂xf(U) = 0 ∂xf(U) →        ❋✐♥✐t❡ ❞✐✛❡r❡♥❝❡s ❋✐♥✐t❡ ✈♦❧✉♠❡s ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ❙P❍

• ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❡r♠ ∂xf(U) ✐♥ ❙P❍ ❄

❈♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t ♣r♦♣❡rt✐❡s ✐s ❛♣♣r♦❛❝❤❡❞ ❜② ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✺ ✴ ✷✺

❙P❍ ❜❛❝❦❣r♦✉♥❞

❈♦♥s❡r✈❛t✐♦♥ ❧❛✇

∂tU + ∂xf(U) = 0 ∂xf(U) →        ❋✐♥✐t❡ ❞✐✛❡r❡♥❝❡s ❋✐♥✐t❡ ✈♦❧✉♠❡s ❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ❙P❍

• ❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t❡r♠ ∂xf(U) ✐♥ ❙P❍ ❄

❈♦♥✈♦❧✉t✐♦♥ ♣r♦❞✉❝t ♣r♦♣❡rt✐❡s

f ∗ δ =

• R f(y)δ(x − y)dy = f

∇f ∗ g = f ∗ ∇g

δ ✐s ❛♣♣r♦❛❝❤❡❞ ❜② ❛ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ W

∇fi ≈ ∇f ∗ W = f ∗ ∇W =

j

wjfj∇Wij

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✺ ✴ ✷✺

❙P❍ ❜❛❝❦❣r♦✉♥❞

❇❡♥❡✜ts

• ▼❡s❤❧❡ss ✭♣❛rt✐❝❧❡s✮
• ▲❛❣r❛♥❣✐❛♥ ♠❡t❤♦❞

❲❡❛❝❦♥❡ss❡s

• ▲♦✇ s♣❛t✐❛❧ ♦r❞❡r
• ◆♦ ❝♦♥✈❡r❣❡♥❝❡ t❤❡♦r❡♠

✧❈♦♥✈❡r❣❡♥❝❡✧ ❝♦♥❞✐t✐♦♥s

• h

∆x → ∞

❛♥❞ h → 0

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✻ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✼ ✴ ✷✺

P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t t❤❡♦r②

❙♦♠❡ ♣❛r❛♠❡t❡rs

• ❙♣❛❝✐♥❣ ♣❛r❛♠❡t❡r ǫ ∈ [0, 1]
• ❘❛❞✐✉s r❛t✐♦ α ∈]0, 1]
• ❙✇✐t❝❤ ♣❛r❛♠❡t❡r γ ∈ [0, 1]
• Rm

∆xm

❘❡✜♥❡♠❡♥t Pr♦❝❡ss

• ǫ∆xm

Rm αRm

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✽ ✴ ✷✺

P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t t❤❡♦r②

❙✇✐t❝❤ ♣❛r❛♠❡t❡r ♣r♦♣❡rt✐❡s

• γ = 1 → ♣❛rt✐❝❧❡ ✐s t✉r♥❡❞ ♦♥ ✭❛❝t✐✈❡ ♣❛rt✐❝❧❡✮
• γ = 0 → ♣❛rt✐❝❧❡ ✐s t✉r♥❡❞ ♦✛ ✭♣❛ss✐✈❡ ♣❛rt✐❝❧❡✮

1 γ

❇✉✛❡r ③♦♥❡ ❇✉✛❡r ③♦♥❡

γm γf

❘❡✜♥❡❞ ❛r❡❛

▼♦❞✐✜❡❞ ❙P❍ ♦♣❡r❛t♦r

• ∇fi =

j

fjωj∇Wijγj

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✾ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✵ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✶ ✴ ✷✺

❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss

d xi dt = v0i, ✭✶✮ dωi dt = ωi

• j

ωj( v0j − v0i)∇Wij, ✭✷✮ dωiρi dt = −ωi

• j

ωj2ρE( vE − v0( xij))∇Wij, ✭✸✮ dωiρi vi dt = −ωi

• j

ωj2

• ρE

vE ⊗ ( vE − v0( xij)) + PE ¯ ¯ I

• ∇Wij + ωiρi

g, ✭✹✮

✸ ♣♦ss✐❜✐❧✐t✐❡s ✿ v0i =   

• vi

✭▲❛❣r❛♥❣✐❛♥✮ ✭❊✉❧❡r✐❛♥✮

• vi +

δvi ✭❆▲❊✮

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✷ ✴ ✷✺

❆▲❊ ♣r♦❝❡ss ✿ ❜❡♥❡✜ts ♦♥ ❛ ❞❛♠❜r❡❛❦ t❡st ❝❛s❡

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✸ ✴ ✷✺

❲❤② ✉s✐♥❣ s✉❝❤ ❛♥ ❛♣♣r♦❛❝❤ ❄

Pr❡ss✉r❡ ✐♥st❛❜✐❧✐t✐❡s ❛t t❤❡ ✜♥❡✴❝♦❛rs❡ ✐♥t❡r❢❛❝❡

• P❛rt✐❝❧❡s ♠♦✈❡ ✐♥ ❛ ▲❛❣r❛♥❣✐❛♥ ✇❛② ✭❙tr❡❛♠❧✐♥❡s✮

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✹ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✺ ✴ ✷✺

❚❤❡ ❆P❘ ❝♦♥❝❡♣t

Pr♦❧♦♥❣❛t✐♦♥

φd = φm + RDh(φ)m.(xd − xm)

• ◗✉❛♥t✐t②

❇❡❢♦r❡ r❡✜♥❡♠❡♥t ❆❢t❡r r❡✜♥❡♠❡♥t ▼❛ss mm

D

• j

mj ❑✐♥❡t✐❝ ❡♥❡r❣②

1 2 mm

um. um

1 2 D

• j

mj uj. uj ▲✐♥❡❛r ♠♦♠❡♥t✉♠ mm um

D

• j

mj uj ❆♥❣✉❧❛r ♠♦♠❡♥t✉♠

• xm × mm

um

D

• j
• xj × mj

uj

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✻ ✴ ✷✺

❚❤❡ ❆P❘ ❝♦♥❝❡♣t

Pr♦❧♦♥❣❛t✐♦♥

φd = φm + RDh(φ)m.(xd − xm)

❲✐t❤♦✉t ♣r♦❧♦♥❣❛t✐♦♥ ❲✐t❤ ♣r♦❧♦♥❣❛t✐♦♥

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✻ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✼ ✴ ✷✺

❚❤❡ ✉t✐❧✐t② ♦❢ ✇❡❧❧ ❞✐str✐❜✉t❡❞ t❛s❦s ❜❡t✇❡❡♥ ♣r♦❝❡ss♦rs

• ◆❡✐❣❤❜♦rs ♥✉♠❜❡r
• P❛rt✐❝❧❡ r❛❞✐✉s

P♦♦r ❞✐str✐❜✉t✐♦♥ ♦❢ ♣❛rt✐❝❧❡s ❜❡t✇❡❡♥ ♣r♦❝❡ss♦rs

• P❛rt✐❝❧❡s ✐♥s✐❞❡✴♥❡❛r t❤❡ ❜✉✛❡r ③♦♥❡ ❤❛✈❡ 2 t♦ 4 ♠♦r❡ ♥❡✐❣❤❜♦rs t❤❛♥

♦t❤❡r ♣❛rt✐❝❧❡s

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✽ ✴ ✷✺

◆❡✇ ❛❧❣♦r✐t❤♠ ❡✣❝✐❡♥❝②

▲♦❛❞ ❜❛❧❛♥❝✐♥❣

τ = min λ(k)

max λ(k) ✇❤❡r❡ λ(k) ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✐♥t❡r❛❝t✐♦♥s ♦❢ ♣r♦❝❡ss♦r k

• ❋♦r♠❡r ❛❧❣♦r✐t❤♠ ✭❝✉tt✐♥❣ ✐♥t♦ ❛♥

❡q✉❛❧ ♥✉♠❜❡r ♦❢ ♣❛rt✐❝❧❡s✮

• ◆❡✇ ❛❧❣♦r✐t❤♠ ✭❝✉tt✐♥❣ ✐♥t♦ ❛♥

❡q✉❛❧ ♥✉♠❜❡r ♦❢ ✐♥t❡r❛❝t✐♦♥s✮

❆t ❧❡❛st ♦❢ 30% ❈P❯ ❣❛✐♥

❖♥ t❤✐s t❡st ❝❛s❡✱ τ = 0.8 ✇✐t❤ t❤❡ ♥❡✇ ❛❧❣♦r✐t❤♠ ✭❛❣❛✐♥st τ = 0.54 ❜❡❢♦r❡✮

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✶✾ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✵ ✴ ✷✺

P❧❛♥

✶ ❙P❍ ❜❛❝❦❣r♦✉♥❞ ✷ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t t❤❡♦r② ✸ P❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♠♣r♦✈❡♠❡♥ts ❆r❜✐tr❛r② ▲❛❣r❛♥❣✐❛♥ ❊✉❧❡r✐❛♥ ✭❆▲❊✮ ♣r♦❝❡ss ❚❤❡ ❆P❘ ❝♦♥❝❡♣t P❛r❛❧❧❡❧✐③❛t✐♦♥ ✹ ❙♦♠❡ r❡s✉❧ts ❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✶ ✴ ✷✺

❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✿ ♣r❡ss✉r❡ ✜❡❧❞ ❛♥❛❧②s✐s

1.22 m 1.17 m 0.55 m 0.45 m 0.16 m 0.67 m 0.68 m 0.6 m

❘❡✜♥❡❞ ❛r❡❛

• ❈♦❛rs❡
• ❆P❘
• ❋✐♥❡

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✷ ✴ ✷✺

❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✿ ♣r❡ss✉r❡ ✜❡❧❞ ❛♥❛❧②s✐s

1.22 m 1.17 m 0.55 m 0.45 m 0.16 m 0.67 m 0.68 m 0.6 m

❘❡✜♥❡❞ ❛r❡❛

• ❈♦❛rs❡
• ❆P❘
• ❋✐♥❡

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✷ ✴ ✷✺

❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✿ ❈♦♠♣❛r✐s♦♥ ♦❢ ♣r❡ss✉r❡ s❡♥s♦rs

• ❙❡♥s♦r P✶
• ❙❡♥s♦r P✶ ✭❩♦♦♠✮

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✸ ✴ ✷✺

❑❧❡❡❢s♠❛♥ ❞❛♠❜r❡❛❦ ✿ ❈♦♠♣❛r✐s♦♥ ♦❢ ♣r❡ss✉r❡ s❡♥s♦rs

• ❙❡♥s♦r P✸
• ❑✐♥❡t✐❝ ❊♥❡r❣②

❙✐♠✉❧❛t✐♦♥ ♣❡r❢♦r♠❡❞ ❈P❯ ❚✐♠❡ ✭s❡❝♦♥❞s✮ Pr♦✜t ❋✐♥❡ 21 000 ✲ ❆P❘ 9000 57.2%

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✸ ✴ ✷✺

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥ ✦

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✹ ✴ ✷✺

❘❡❢❡r❡♥❝❡s

❉✳❆✳ ❇❛r❝❛r♦❧♦✱ ●✳ ❖❣❡r✱ ❛♥❞ ❉✳ ▲❡ ❚♦✉③é✳ ❆❞❛♣t✐✈❡ ♣❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ❛♥❞ ❞❡r❡✜♥❡♠❡♥t ❛♣♣❧✐❡❞ t♦ ❙♠♦♦t❤❡❞ P❛rt✐❝❧❡ ❍②❞r♦❞②♥❛♠✐❝s ♠❡t❤♦❞✳ ❏♦✉r♥❛❧ ♦❢ ❝♦♠♣✉t❡r ♣❤②s✐❝s✱ ✷✼✸ ✿✻✹✵✕✻✺✼✱ ✷✵✶✹✳ ❏✳❋❡❧❞♠❛♥ ❛♥❞ ❏✳❇♦♥❡t✳ ❉②♥❛♠✐❝ r❡✜♥❡♠❡♥t ❛♥❞ ❜♦✉♥❞❛r② ❝♦♥t❛❝t ❢♦r❝❡s ✐♥ ❙P❍ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ✢✉✐❞ ✢♦✇ ♣r♦❜❧❡♠s✳ ■♥t✳ ❏✳ ◆✉♠❡r✳ ▼❡t❤✳ ❊♥❣✱ ✼✷ ✿✷✾✺✕✸✷✹✱ ✷✵✵✼✳ ❙✳ ❑✐ts✐♦♥❛s ❛♥❞ ❆✳ ❲❤✐t✇♦rt❤✳ ❉❡✈❡❧♦♣♠❡♥t ♦❢ ❙P❍ ✈❛r✐❛❜❧❡ r❡s♦❧✉t✐♦♥ ✉s✐♥❣ ❞②♥❛♠✐❝ ♣❛rt✐❝❧❡ ❝♦❛❧❡s❝✐♥❣ ❛♥❞ s♣❧✐tt✐♥❣✳ ▼♦♥t❤❧② ◆♦t✐❝❡s ♦❢ t❤❡ ❘♦②❛❧ ❆str♦♥♦♠✐❝❛❧ ❙♦❝✐❡t②✱ ✸✸✵ ✿✶✷✾✕✶✸✻✱ ✷✵✵✷✳ ❑✳▼✳❚✳ ❑❧❡❡❢s♠❛♥✱ ●✳ ❋❡❦❦❡♥✱ ❆✳❊✳P✳ ❱❡❧❞♠❛♥✱ ❇✳ ■✇❛♥♦✇s❦✐✱ ❛♥❞ ❇✳ ❇✉❝❤♥❡r✳ ❆ ✈♦❧✉♠❡✲♦❢✲✢✉✐❞ ❜❛s❡❞ s✐♠✉❧❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ✇❛✈❡ ✐♠♣❛❝t ♣r♦❜❧❡♠s✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s✱ ✷✵✻ ✿✸✻✸✕✸✾✸✱ ✷✵✵✺✳ ❨✳ ❘✳ ▲♦♣❡③✱ ❉✳ ❘♦♦s❡✱ ❛♥❞ ❈✳ ❘✳ ▼♦r❢❛✳ ❉②♥❛♠✐❝ ♣❛rt✐❝❧❡ r❡✜♥❡♠❡♥t ✐♥ ❙P❍ ✿ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❢r❡❡ s✉r❢❛❝❡ ✢♦✇ ❛♥❞ ♥♦♥✲❝♦❤❡s✐✈❡ s♦✐❧ s✐♠✉❧❛t✐♦♥s✳ ❈♦♠♣✉t ▼❡❝❤✱ ✺✶ ✿✼✸✶✕✼✹✶✱ ✷✵✶✸✳ ❙✳ ▼❛rr♦♥❡✱ ❆✳ ❈♦❧❛❣r♦ss✐✱ ❉✳ ▲❡ ❚♦✉③é✱ ❛♥❞ ●✳ ●r❛③✐❛♥✐✳ ❋❛st ❢r❡❡✲s✉r❢❛❝❡ ❞❡t❡❝t✐♦♥ ❛♥❞ ❧❡✈❡❧✲s❡t ❢✉♥❝t✐♦♥ ❞❡✜♥✐t✐♦♥ ✐♥ ❙P❍ s♦❧✈❡rs✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ P❤②s✐❝s✱ ✷✷✾ ✿✸✻✺✷✕✸✻✻✸✱ ✷✵✶✵✳

• ✳ ❖❣❡r✱ ❙✳ ▼❛rr♦♥❡✱ ❛♥❞ ❉✳ ▲❡ ❚♦✉③é✳

❆ ❝♦♥s✐st❡♥t ❝♦♥t✐♥✉♦✉s ♣❛rt✐❝❧❡ r❡♦r❞❡r✐♥❣ ✐♥ ✇❡❛❦❧②✲❝♦♠♣r❡ss✐❜❧❡ ❙P❍ t❤r♦✉❣❤ ❛♥ ❆▲❊ ❢♦r♠❛❧✐s♠✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❙P❍❊❘■❈ ✇♦r❦s❤♦♣✱ ✷✵✶✺✳ ❘✳ ❱❛❝♦♥❞✐♦✱ ❇✳ ❉✳ ❘♦❣❡rs✱ P✳ ❑✳ ❙t❛♥s❜②✱ P✳ ▼✐❣♥♦s❛✱ ❛♥❞ ❏✳ ❋❡❧❞♠❛♥✳ ❉❡✈❡❧♦♣♠❡♥t ♦❢ ❙P❍ ✈❛r✐❛❜❧❡ r❡s♦❧✉t✐♦♥ ✉s✐♥❣ ❞②♥❛♠✐❝ ♣❛rt✐❝❧❡ ❝♦❛❧❡s❝✐♥❣ ❛♥❞ s♣❧✐tt✐♥❣✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✼t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❙P❍❊❘■❈ ✇♦r❦s❤♦♣✱ ♣❛❣❡ ✸✹✼✕✸✺✹✱ ✷✵✶✷✳ ❘✳ ❱❛❝♦♥❞✐♦✱ ❇✳ ❉✳ ❘♦❣❡rs✱ P✳ ❑✳ ❙t❛♥s❜②✱ P✳ ▼✐❣♥♦s❛✱ ❛♥❞ ❏✳ ❋❡❧❞♠❛♥✳ ❙♠♦♦t❤❡❞ P❛rt✐❝❧❡ ❍②❞r♦❞②♥❛♠✐❝s ✇✐t❤ ♣❛rt✐❝❧❡ s♣❧✐tt✐♥❣✱ ❛♣♣❧✐❡❞ t♦ s❡❧❢✲❣r❛✈✐t❛t✐♥❣ ❝♦❧❧❛♣s❡✳ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❙P❍❊❘■❈ ✇♦r❦s❤♦♣✱ ♣❛❣❡s ✸✹✼✕✸✺✹✱ ✷✵✶✷✳

▲❛✉r❡♥t ❈❤✐r♦♥ ❆❞❛♣t✐✈❡ P❛rt✐❝❧❡ ❘❡✜♥❡♠❡♥t ✭❆P❘✮ ❛♣♣❧✐❡❞ t♦ ❙P❍ ♠❡t❤♦❞ ✿ ❙t❛❜✐❧✐t②✱ ❛❝❝✉r❛❝② ❛♥❞ ❈P❯ ❛♥❛❧②s✐s ✶❡r ❥✉✐❧❧❡t ✷✵✶✺ ✷✺ ✴ ✷✺