Discontinuous Galerkin methods for solving Helmholtz isotropic wave equations for seismic applications

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Discontinuous Galerkin methods for solving Helmholtz

isotropic wave equations for seismic applications

Marie Bonnasse-Gahot, Stephane Lanteri, Julien Diaz, Henri Calandra

To cite this version:

Marie Bonnasse-Gahot, Stephane Lanteri, Julien Diaz, Henri Calandra. Discontinuous Galerkin

meth-ods for solving Helmholtz isotropic wave equations for seismic applications. HOSCAR - 3rd

Brazil-French workshop on High performance cOmputing and SCientific dAta management dRiven by highly

demanding applications (INRIA-CNPq), Sep 2013, Bordeaux, France. 2013. �hal-00929971�

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❉✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣

❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥s ❢♦r

s❡✐s♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s

❆❞✈✐s♦rs✿ ❙té♣❤❛♥❡ ▲❛♥t❡r✐✱ ■◆❘■❆✱ ◆❛❝❤♦s

❏✉❧✐❡♥ ❉✐❛③✱ ■◆❘■❆✱ ▼❛❣✐q✉❡ ✸❉

❍❡♥r✐ ❈❛❧❛♥❞r❛✱ ❚❖❚❆▲

▼❛r✐❡ ❇♦♥♥❛ss❡✲●❛❤♦t

■◆❘■❆✱ ▼❛❣✐q✉❡ ✸❉

❙❡♣t❡♠❜❡r ✷✱ ✷✵✶✸

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▼♦t✐✈❛t✐♦♥ ✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s ❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

▼♦t✐✈❛t✐♦♥

❊①❛♠♣❧❡s ♦❢ t❤❡ s❡✐s♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s

✶✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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▼♦t✐✈❛t✐♦♥

❊①❛♠♣❧❡s ♦❢ t❤❡ s❡✐s♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s

✶✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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▼♦t✐✈❛t✐♦♥

■♠❛❣✐♥❣ ♠❡t❤♦❞ ✿ t❤❡ ❢✉❧❧ ✇❛✈❡ ✐♥✈❡rs✐♦♥

◗✉❛♥t✐t❛t✐✈❡

❤✐❣❤ r❡s♦❧✉t✐♦♥

✐♠❛❣❡s ♦❢ t❤❡ s✉❜s✉r❢❛❝❡ ♣❤②s✐❝❛❧

♣❛r❛♠❡t❡rs

❋♦r✇❛r❞ ♣r♦❜❧❡♠ ♦❢ t❤❡ ✐♥✈❡rs✐♦♥ ♣r♦❝❡ss

❊❧❛st✐❝ ✇❛✈❡s ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❤❛r♠♦♥✐❝ ❞♦♠❛✐♥ ✿

❍❡❧♠❤♦❧t③

❡q✉❛t✐♦♥

✷✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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▼♦t✐✈❛t✐♦♥

■♠❛❣✐♥❣ ♠❡t❤♦❞ ✿ t❤❡ ❢✉❧❧ ✇❛✈❡ ✐♥✈❡rs✐♦♥

◗✉❛♥t✐t❛t✐✈❡

❤✐❣❤ r❡s♦❧✉t✐♦♥

✐♠❛❣❡s ♦❢ t❤❡ s✉❜s✉r❢❛❝❡ ♣❤②s✐❝❛❧

♣❛r❛♠❡t❡rs

❋♦r✇❛r❞ ♣r♦❜❧❡♠ ♦❢ t❤❡ ✐♥✈❡rs✐♦♥ ♣r♦❝❡ss

❊❧❛st✐❝ ✇❛✈❡s ♣r♦♣❛❣❛t✐♦♥ ✐♥ ❤❛r♠♦♥✐❝ ❞♦♠❛✐♥ ✿

❍❡❧♠❤♦❧t③

❡q✉❛t✐♦♥

✷✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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▼♦t✐✈❛t✐♦♥

❙❡✐s♠✐❝ ✐♠❛❣✐♥❣ ✐♥ ❤❡t❡r♦❣❡♥❡♦✉s ❝♦♠♣❧❡① ♠❡❞✐❛

❈♦♠♣❧❡① t♦♣♦❣r❛♣❤②

❍✐❣❤ ❤❡t❡r♦❣❡♥❡✐t✐❡s

❉● ♠❡t❤♦❞

❯s❡ ♦❢ tr✐❛♥❣✉❧❛r ✉♥str✉❝t✉r❡❞ ♠❡s❤❡s

❋❧❡①✐❜❧❡ ❝❤♦✐❝❡ ♦❢ ✐♥t❡r♣♦❧❛t✐♦♥ ♦r❞❡rs ✭♣ − ❛❞❛♣t❛t✐✈✐t②✮

✸✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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▼♦t✐✈❛t✐♦♥

❙❡✐s♠✐❝ ✐♠❛❣✐♥❣ ✐♥ ❤❡t❡r♦❣❡♥❡♦✉s ❝♦♠♣❧❡① ♠❡❞✐❛

❈♦♠♣❧❡① t♦♣♦❣r❛♣❤②

❍✐❣❤ ❤❡t❡r♦❣❡♥❡✐t✐❡s

❉● ♠❡t❤♦❞

❯s❡ ♦❢ tr✐❛♥❣✉❧❛r ✉♥str✉❝t✉r❡❞ ♠❡s❤❡s

❋❧❡①✐❜❧❡ ❝❤♦✐❝❡ ♦❢ ✐♥t❡r♣♦❧❛t✐♦♥ ♦r❞❡rs ✭♣ − ❛❞❛♣t❛t✐✈✐t②✮

✸✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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▼♦t✐✈❛t✐♦♥

❉r❛✇❜❛❝❦ ♦❢ ❉● ♠❡t❤♦❞

■♠♣♦rt❛♥t ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st

▼❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤❡ t❤❡s✐s

❉❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ❤②❜r✐❞✐③❛❜❧❡ ❉● ✭❍❉●✮ ♠❡t❤♦❞

❉❡✈❡❧♦♣♠❡♥t ♦❢ ❛ r❡❢❡r❡♥❝❡ ♠❡t❤♦❞✱ ❛ ❝❧❛ss✐❝❛❧ ❉●

♠❡t❤♦❞

✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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▼♦t✐✈❛t✐♦♥

❉r❛✇❜❛❝❦ ♦❢ ❉● ♠❡t❤♦❞

■♠♣♦rt❛♥t ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st

▼❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤❡ t❤❡s✐s

❉❡✈❡❧♦♣♠❡♥t ♦❢ ❛♥ ❤②❜r✐❞✐③❛❜❧❡ ❉● ✭❍❉●✮ ♠❡t❤♦❞

❉❡✈❡❧♦♣♠❡♥t ♦❢ ❛ r❡❢❡r❡♥❝❡ ♠❡t❤♦❞✱ ❛ ❝❧❛ss✐❝❛❧ ❉●

♠❡t❤♦❞

✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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❚❛❜❧❡ ♦❢ ❝♦♥t❡♥ts

✶

✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

✷

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s

❈❡♥t❡r❡❞ ✢✉① ❉● s❝❤❡♠❡

❯♣✇✐♥❞ ✢✉① ❉● s❝❤❡♠❡

✸

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

P❧❛♥❡ ✇❛✈❡ ✐♥ ❛♥ ❤♦♠♦❣❡♥❡♦✉s ♠❡❞✐✉♠

❈✐r❝✉❧❛r ❞✐✛r❛❝t✐♦♥

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s

❘❡s✉❧ts ✇✐t❤ t❤❡ ♣

− ❛❞❛♣t❛t✐✈✐t②

✹

❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

✺✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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❈♦♥t❡♥ts

✶

✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

✷

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s

✸

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

✹

❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

✻✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

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✷❉ ❍❡❧♠❤♦❧t③ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

❋✐rst ♦r❞❡r ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❍❡❧♠♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

① = (①, ②) ∈ Ω ⊂ R

✷

_✱

_✐ω

ρ

(①)

✈

(①) = ∇·

σ

(①) +

❢

_s

(①)

✐ω

σ

(①) =

❈

(①) ε(

✈

(①))

❋r❡❡ s✉r❢❛❝❡ ❝♦♥❞✐t✐♦♥ ✿

σ

♥ = ✵ ♦♥ Γ

_❧

❆❜s♦r❜✐♥❣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✿

σ

♥ =

✈

_♣

(

✈

· ♥)♥ +

✈

s

(

✈

· t)t ♦♥ Γ

❛

✈

✿ ✈❡❧♦❝✐t② ✈❡❝t♦r

σ

✿ str❡ss t❡♥s♦r

ε

✿ str❛✐♥ t❡♥s♦r

✼✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(14)

✷❉ ❍❡❧♠❤♦❧t③ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

❋✐rst ♦r❞❡r ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❍❡❧♠♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

① = (①, ②) ∈ Ω ⊂ R

✷

_✱

_✐ω

ρ

(①)

✈

(①) = ∇·

σ

(①) +

❢

_s

(①)

✐ω

σ

(①) =

❈

(①) ε(

✈

(①))

❋r❡❡ s✉r❢❛❝❡ ❝♦♥❞✐t✐♦♥ ✿

σ

♥ = ✵ ♦♥ Γ

_❧

❆❜s♦r❜✐♥❣ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ✿

σ

♥ =

✈

_♣

(

✈

· ♥)♥ +

✈

_s

(

✈

· t)t ♦♥ Γ

_❛

ρ

✿ ♠❛ss ❞❡♥s✐t②

❈

✿ t❡♥s♦r ♦❢ ❡❧❛st✐❝✐t②

❝♦❡✣❝✐❡♥ts

❢

s

✿ s♦✉r❝❡ t❡r♠✱ ❢

s

∈

▲

✷

(Ω)

✈

♣

✿ P✲✇❛✈❡ ✈❡❧♦❝✐t②

✈

s

✿ ❙✲✇❛✈❡ ✈❡❧♦❝✐t②

✼✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(15)

✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

❋✐rst ♦r❞❡r ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❍❡❧♠♦❧t③ ✐s♦tr♦♣✐❝ ✇❛✈❡ ❡q✉❛t✐♦♥s











✐ω

✈

①

=

✶

_ρ

∂

_∂

σ

_①

①①

+

∂

_∂

σ

_③

①③

✐ω

✈

③

=

✶

ρ

∂

σ

_①③

∂①

+

∂

σ

_③③

∂③

✐ω

σ

_①①

=

(

λ

+

✷

µ

)

∂

✈

①

∂

①

+

λ

∂

✈

_③

∂

③

✐ω

σ

_③③

=

λ

∂

✈

①

∂①

+ (

λ

+

✷

µ

)

∂

✈

_③

∂③

✐ω

σ

_①③

=

µ

∂

✈

①

∂

③

+

∂

✈

_③

∂

①

λ

❛♥❞

µ

▲❛♠é✬s ❝♦♥st❛♥ts ❛♥❞

✈

_♣

=

s

λ

+

✷

µ

ρ

❛♥❞

✈

s

=

r

µ

ρ

✽✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(16)

✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

❱❡❝t♦r✐❛❧ ❢♦r♠

✐ω

◗

+

❆

_①

∂

◗

∂

①

+

❆

③

∂

◗

∂

③

=

✵

✇❤❡r❡

◗

= (

✈

_①

,

✈

_③

,

σ

_①①

,

σ

_③③

,

σ

_①③

)

❚

❛♥❞ ✿

❆

①

= −







✵

✶

ρ

✵ ✵

✵

✵ ✵ ✵

✶

ρ

λ

+

✷

µ

✵ ✵ ✵ ✵

λ

✵ ✵ ✵ ✵

✵

µ

✵ ✵ ✵







,

❆

_③

= −







✵

✵ ✵

✶

ρ

✵

✶

ρ

✵

λ

✵ ✵ ✵

✵

λ

+

✷

µ

✵ ✵ ✵

µ

✵

✵ ✵ ✵







✾✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(17)

▼♦t✐✈❛t✐♦♥ ✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s ❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s ❈❡♥t❡r❡❞ ✢✉① ❉● s❝❤❡♠❡ ❯♣✇✐♥❞ ✢✉① ❉● s❝❤❡♠❡

❈♦♥t❡♥ts

✶

✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

✷

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s

❈❡♥t❡r❡❞ ✢✉① ❉● s❝❤❡♠❡

❯♣✇✐♥❞ ✢✉① ❉● s❝❤❡♠❡

✸

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

✹

❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

✶✵✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(18)

❉● ♠❡t❤♦❞s ✐♥ t✐♠❡ ❞♦♠❛✐♥ ❢♦r s❡✐s♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s

▼✳ ❉✉♠❜s❡r ❛♥❞ ▼✳ ❑äs❡r✱ ❆♥ ❛r❜✐tr❛r② ❤✐❣❤✲♦r❞❡r

❞✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ❢♦r ❡❧❛st✐❝ ✇❛✈❡s ♦♥

✉♥str✉❝t✉r❡❞ ♠❡s❤❡s ✲ ■■ ❀ ❚❤❡ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ✐s♦tr♦♣✐❝

❝❛s❡✱ ✷✵✵✻ ✭

✉♣✇✐♥❞ s❝❤❡♠❡

✮

❙✳ ❉❡❧❝♦✉rt❡✱ ▲✳❋❡③♦✉✐ ❛♥❞ ◆✳ ●❧✐♥s❦②✲❖❧✐✈✐❡r✱ ❆ ❤✐❣❤ ♦r❞❡r

❞✐s❝♦♥t✐♥✉♦✉s ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ❢♦r t❤❡ s❡✐s♠✐❝ ✇❛✈❡

♣r♦♣❛❣❛t✐♦♥✱ ✷✵✵✾ ✭

❝❡♥t❡r❡❞ s❝❤❡♠❡

✮

✶✶✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(19)

◆♦t❛t✐♦♥s ❛♥❞ ❞❡✜♥✐t✐♦♥s

◆♦t❛t✐♦♥s

Γ

_❧

❢r❡❡ s✉r❢❛❝❡ ❜♦✉♥❞❛r②

Γ

_❛

t❤❡ ❛❜s♦r❜✐♥❣ ❜♦✉♥❞❛r②

T

_❤

♠❡s❤ ♦❢ Ω ❝♦♠♣♦s❡❞ ♦❢ tr✐❛♥❣❧❡s ❑

F

_❤

s❡t ♦❢ ❛❧❧ ❢❛❝❡s ❋ ♦❢ T

_❤

♥ t❤❡ ♥♦r♠❛❧ ♦✉t✇❛r❞ ✈❡❝t♦r ♦❢ ❛♥ ❡❧❡♠❡♥t ❑

✶✷✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(20)

◆♦t❛t✐♦♥s ❛♥❞ ❞❡✜♥✐t✐♦♥s

❉❡✜♥✐t✐♦♥s

❏✉♠♣ [[·]] ♦❢ ❛ ✈❡❝t♦r ✉ ❢♦r ❋ ✿

[[✉]] = ✉

+

_·♥

+

_+✉

−

· ♥

−

=

✉

+

·♥

+

−✉

−

· ♥

+

❏✉♠♣ ♦❢ ❛ t❡♥s♦r σ ❢♦r ❋ ✿

[[σ]] = σ

+

♥

+

+ σ

−

♥

−

= σ

+

♥

+

− σ

−

♥

+

K

+

K

−

n

+

n

−

✶✸✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(21)

◆♦t❛t✐♦♥s ❛♥❞ ❞❡✜♥✐t✐♦♥s

❉❡✜♥✐t✐♦♥s

❆✈❡r❛❣❡ {·} ♦❢ ❛ ✈❛r✐❛❜❧❡ ✉✱ ❢♦r ❋ ✿

{✉} =

✉

+

✉

−

✷

K

+

K

−

n

+

n

−

✶✸✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(22)

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥

▲♦❝❛❧ ❉● ❢♦r♠✉❧❛t✐♦♥

✐ω

◗

+

❆

_①

∂

◗

∂

①

+

❆

③

∂

◗

∂

③

=

✵

✶✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(23)

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥

▲♦❝❛❧ ❉● ❢♦r♠✉❧❛t✐♦♥

✐ω

◗

ϕ

+

❆

_①

∂

◗

∂

①

ϕ

+

❆

③

∂

◗

∂

③

ϕ

=

✵

✶✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(24)

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥

▲♦❝❛❧ ❉● ❢♦r♠✉❧❛t✐♦♥

Z

❑

✐ω

◗

❑

_ϕ

₋

Z

❑

❆

①

❑

∂

◗

❑

∂

①

ϕ

−

Z

❑

❆

③

❑

∂

◗

❑

∂

③

ϕ

=

✵

✶✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(25)

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥

▲♦❝❛❧ ❉● ❢♦r♠✉❧❛t✐♦♥

Z

❑

✐ω

◗

❑

_ϕ

₋

Z

❑

❆

①

❑

_◗

❑

∂ϕ

∂

①

−

Z

❑

❆

③

❑

∂ϕ

∂

③

+

X

❋ ∈F(❑)

Z

❋

❉

♥

◗

ϕ

=

✵

❉

♥

=

♥

①

❆

①

+

♥

③

❆

③

= −







✵

♥

①

ρ

✵

♥

③

ρ

✵

♥

③

ρ

♥

①

ρ

♥

①

(

λ

+

✷

µ

)

♥

③

λ

✵

♥

①

λ

♥

③

(

λ

+

✷

µ

)

✵

♥

③

µ

♥

_①

µ

✵







✶✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(26)

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥

●❧♦❜❛❧ ❉● ❢♦r♠✉❧❛t✐♦♥

X

❑∈T❤

Z

❑

✐ω

◗

ϕ

−

X

❑∈T❤

Z

❑

❆

①

◗

∂ϕ

∂

①

−

X

❑∈T❤

Z

❑

❆

③

◗

∂ϕ

∂

③

+

X

❋ ∈F❤

Z

❋

[[

❉

♥

◗

ϕ]] =

✵

✶✺✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(27)

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥

●❧♦❜❛❧ ❉● ❢♦r♠✉❧❛t✐♦♥

X

❑∈T❤

Z

❑

✐ω

◗

ϕ

−

X

❑∈T❤

Z

❑

❆

①

◗

∂ϕ

∂

①

−

X

❑∈T❤

Z

❑

❆

③

◗

∂ϕ

∂

③

+

X

❋ ∈F

❤

Z

❋

[[

❉

♥

◗

ϕ]] =

✵

[[

❉

_♥

◗

ϕ]] ≃ (

❉

_♥

◗

) [[ϕ]]

✶✺✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(28)

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥

●❧♦❜❛❧ ❉● ❢♦r♠✉❧❛t✐♦♥

X

❑∈T❤

Z

❑

✐ω

◗

ϕ

−

X

❑∈T❤

Z

❑

❆

①

◗

∂ϕ

∂

①

−

X

❑∈T❤

Z

❑

❆

③

◗

∂ϕ

∂

③

+

X

❋ ∈F❤

Z

❋

(

❉

♥

◗

) [[ϕ]] =

✵

✶✺✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(29)

❈❡♥t❡r❡❞ ✢✉① ❉● s❝❤❡♠❡

❈❡♥t❡r❡❞ ✢✉① ♦♥ ❛ ❢❛❝❡ ❋

(

❉

_♥

◗

) |

_❋

= {

❉

_♥

◗

} =

✶

✷

❉

♥

❑

◗

❑

+

❉

♥

❑

′

◗

❑

′

❈❡♥t❡r❡❞ ✢✉① ❉● s❝❤❡♠❡

Z

❑

✐ω

◗

❑

_ϕ

₋

Z

❑

❆

①

❑

_◗

❑

∂ϕ

∂

①

−

Z

❑

❆

③

❑

_◗

❑

∂ϕ

∂

③

+

X

❋

Z

❋

✶

✷

❉

♥

❑

◗

❑

+

❉

♥

❑

′

◗

❑

′

ϕ

=

✵

✶✻✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(30)

❯♣✇✐♥❞ ✢✉① ❉● s❝❤❡♠❡

❉❡✜♥✐t✐♦♥

❉

♥

+

=

❘

♥

Γ

+

(

❘

♥

)

−

✶

❉

♥

−

=

❘

♥

Γ

−

(

❘

♥

)

−

✶

✇❤❡r❡

Γ

−

= −







✈

♣

✵ ✵ ✵ ✵

✵

✈

s

✵ ✵ ✵

✵

✵ ✵ ✵ ✵

✵

✵ ✵ ✵ ✵

✵

✵ ✵ ✵ ✵







,

Γ

+

₌







✵ ✵ ✵ ✵

✵

✵ ✵ ✵ ✵

✵

✵ ✵ ✵ ✵

✵

✵ ✵ ✵

✈

s

✵

✵ ✵ ✵ ✵

✈

♣







✶✼✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(31)

❯♣✇✐♥❞ ✢✉① ❉● s❝❤❡♠❡

❉❡✜♥✐t✐♦♥

❉

♥

+

=

❘

♥

Γ

+

(

❘

♥

)

−

✶

❉

♥

−

=

❘

♥

Γ

−

(

❘

♥

)

−

✶

❛♥❞

❘

♥

=







♥

①

✈

♣

−♥

③

✈

s

✵

♥

③

✈

s

−♥

①

✈

♣

♥

③

✈

♣

♥

①

✈

s

✵

−♥

①

✈

s

−♥

③

✈

♣

λ

+

✷♥

✷

_①

µ

−✷♥

_③

♥

_①

µ

♥

✷

_③

−✷♥

_③

♥

_①

µ

λ

+

✷♥

_①

✷

µ

λ

+

✷♥

_③

✷

µ

✷♥

_③

♥

_①

µ

♥

✷

_①

✷♥

_③

♥

_①

µ

λ

+

✷♥

_③

✷

µ

✷♥

③

♥

①

µ

(♥

✷

①

−

♥

✷

③

)

−♥

①

♥

③

µ

(♥

✷

①

−

♥

✷

③

)

✷♥

③

♥

①

µ







✶✼✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(32)

❯♣✇✐♥❞ ✢✉① ❉● s❝❤❡♠❡

❯♣✇✐♥❞ ✢✉① ♦♥ ❛ ❢❛❝❡ ❋

(

❉

_♥

◗

) |

_❋

=

❉

_♥

❑

+

◗

❑

+

❉

_♥

❑

′

−

◗

❑

′

❯♣✇✐♥❞ ✢✉① ❉● s❝❤❡♠❡

Z

❑

✐ω

◗

❑

_ϕ

₋

Z

❑

❆

①

❑

_◗

❑

∂ϕ

∂

①

−

Z

❑

❆

③

❑

_◗

❑

∂ϕ

∂

③

+

X

❋

Z

❋

❉

♥

❑

+

◗

❑

₊

❉

♥

❑

′

−

◗

❑

′

ϕ

=

✵

✶✽✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(33)

▼♦t✐✈❛t✐♦♥ ✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s ❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s P❧❛♥❡ ✇❛✈❡ ✐♥ ❛♥ ❤♦♠♦❣❡♥❡♦✉s ♠❡❞✐✉♠ ❈✐r❝✉❧❛r ❞✐✛r❛❝t✐♦♥

❈♦♥t❡♥ts

✶

✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

✷

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s

✸

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

P❧❛♥❡ ✇❛✈❡ ✐♥ ❛♥ ❤♦♠♦❣❡♥❡♦✉s ♠❡❞✐✉♠

❈✐r❝✉❧❛r ❞✐✛r❛❝t✐♦♥

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s

❘❡s✉❧ts ✇✐t❤ t❤❡ ♣

− ❛❞❛♣t❛t✐✈✐t②

✹

❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

✶✾✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(34)

P❧❛♥❡ ✇❛✈❡

4000

❈♦♠♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥ Ω

s❡tt✐♥❣

P❤②s✐❝❛❧ ♣❛r❛♠❡t❡rs ✿

ρ

=

✷.✶✵

✸

❦❣.♠

−✸

λ

=

✶, ✻.✶✵

✶

✵P❛

µ

=

✽.✶✵

✾

P❛

✈

♣

=

✹.✶✵

✸

♠.s

−✶

✈

s

=

✷.✶✵

✸

♠.s

−✶

❇♦✉♥❞❛r② ✿

❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ♦♥ ∂Ω

s✉❝❤ ❛s ✿

✉ = ∇❡

✐(❦ ❝♦s θ①+❦ s✐♥ θ②)

✇❤❡r❡ ❦ =

_✈

ω

♣

♦r ❦ =

ω

✈

s

✷✵✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(35)

P❧❛♥❡ ✇❛✈❡ ❢♦r ❛ ❢r❡q✉❡♥❝② ❢ = ✷ ❍③✱ ❝♦♠♣♦♥❡♥t ❱

①

❊①❛❝t s♦❧✉t✐♦♥

P

✷

❝❡♥t❡r❡❞ ✢✉① ❉● ❢♦r♠✉✲

❧❛t✐♦♥

✷✶✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(36)

P❧❛♥❡ ✇❛✈❡ ❢♦r ❛ ❢r❡q✉❡♥❝② ❢ = ✷ ❍③✱ ❝♦♠♣♦♥❡♥t ❱

①

❊①❛❝t s♦❧✉t✐♦♥

P

✷

✉♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉✲

❧❛t✐♦♥

✷✷✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(37)

P❧❛♥❡ ✇❛✈❡

▲

✷

❡rr♦r ❢♦r t❤❡ ❝❡♥t❡r❡❞ ✢✉①

✷✸✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(38)

P❧❛♥❡ ✇❛✈❡

▲

✷

✲❡rr♦r ❢♦r t❤❡ ✉♣✇✐♥❞ ✢✉①

✷✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(39)

❈✐r❝✉❧❛r ❞✐✛r❛❝t✐♦♥

Γ

l

Γ

a

Ω

a b

❈♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣✉t❛✲

t✐♦♥❛❧ ❞♦♠❛✐♥ Ω

P❤②s✐❝❛❧ ♣❛r❛♠❡t❡rs ✿

ρ

=

✷.✶✵

✸

❦❣.♠

−✸

λ

=

✶, ✻.✶✵

✶

✵P❛

µ

=

✽.✶✵

✾

P❛

✈

♣

=

✹.✶✵

✸

♠.s

−✶

✈

s

=

✷.✶✵

✸

♠.s

−✶

❇♦✉♥❞❛r② ✿

Γ

❧

✐s ❛ ❢r❡❡ s✉r❢❛❝❡ ✿

σ

♥ = ✵

Γ

❛

❛❜s♦r❜✐♥❣ ❜♦✉♥❞❛r② ✿

σ

♥ =

✈

_♣

(

✈

· ♥)♥ +

✈

s

(

✈

· t)t

❛ = ✷✵✵✵♠

❜ = ✽✵✵✵♠

T

_❤

❝♦♠♣♦s❡❞ ♦❢ ✾✻✺✸

❡❧❡♠❡♥ts

✷✺✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(40)

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s ✿ ❢ = ✷ ❍③

❊①❛❝t s♦❧✉t✐♦♥

P

✶

❝❡♥t❡r❡❞ s❝❤❡♠❡

✷✻✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(41)

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s ✿ ❢ = ✷ ❍③

❊①❛❝t s♦❧✉t✐♦♥

P

✶

✉♣✇✐♥❞ s❝❤❡♠❡

✷✼✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(42)

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s ✿ ❢ = ✷ ❍③

◆❜ ❞♦❢

❈❡♥t❡r❡❞

❯♣✇✐♥❞

✶✹✹✼✾✺

❱

①

▲

✷

✲❡rr♦r

✸✳✹✹❡✲✵✶ ✶✳✹✾❡✲✵✶

✭P

✶

✮

❋❛❝t✳✲r❡s✳ t✐♠❡ ✭s✮

✶✸

✶✻

▼❡♠♦r②

✽✵✵

✾✽✵

✷✽✾✺✾✵

❱

①

▲

✷

✲❡rr♦r

✹✳✹✶❡✲✵✷ ✹✳✻✼❡✲✵✷

✭P

✷

✮

❋❛❝t✳✲r❡s✳ t✐♠❡ ✭s✮

✹✵

✺✼

▼❡♠♦r②

✶✾✵✵

✷✽✵✵

◆✉♠❡r✐❝❛❧ st❛t✐st✐❝s

✷✽✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(43)

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s ✿ ❢ = ✹ ❍③

❊①❛❝t s♦❧✉t✐♦♥

P

✷

❝❡♥t❡r❡❞ s❝❤❡♠❡

✷✾✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(44)

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s ✿ ❢ = ✹ ❍③

❊①❛❝t s♦❧✉t✐♦♥

P

✷

✉♣✇✐♥❞ s❝❤❡♠❡

✸✵✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(45)

❘❡s✉❧ts ❢♦r ✈❛r✐♦✉s ❢r❡q✉❡♥❝✐❡s ✿ ❢ = ✹ ❍③

◆❜ ❞♦❢

❈❡♥t❡r❡❞

❯♣✇✐♥❞

✶✹✹✼✾✺

❱

①

▲

✷

✲❡rr♦r

✶✳✶✾

✵✳✺✵✻

❋❛❝t✳✲r❡s✳ t✐♠❡ ✭s✮

✶✸

✶✻

▼❡♠♦r②

✽✵✵

✾✼✵

✷✽✾✺✾✵

❱

①

▲

✷

✲❡rr♦r

✹✳✷✹❡✲✵✶ ✶✳✻✻❡✲✵✶

❋❛❝t✳✲r❡s✳ t✐♠❡ ✭s✮

✹✵

✺✼

▼❡♠♦r②

✶✾✵✵

✷✽✵✵

◆✉♠❡r✐❝❛❧ st❛t✐st✐❝s

✸✶✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(46)

❘❡s✉❧ts ✇✐t❤ ♣ − ❛❞❛♣t❛t✐✈✐t②

❆r❡❛ ♦❢ t❤❡ tr✐❛♥❣❧❡ ■♥t❡r♣♦❧❛t✐♦♥ ♦r❞❡r ◆✉♠❜❡r ♦❢ tr✐❛♥❣❧❡s

❪✵ ❀✶✵✵✵✵❪

✵

✸

❪✶✵✵✵✵ ❀✶✺✵✵✵❪

✶

✶✼✹✺

❪✶✺✵✵✵ ❀✷✵✵✵✵❪

✷

✸✾✾✾

❪✷✵✵✵✵ ❀✷✺✵✵✵❪

✸

✷✻✺✽

❪✷✺✵✵✵ ❀✸✵✵✵✵❪

✹

✶✷✹✽

❉✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♦r❞❡rs

✸✷✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(47)

❘❡s✉❧ts ✇✐t❤ ♣ − ❛❞❛♣t❛t✐✈✐t② ✿ ❢ = ✷ ❍③

❊①❛❝t s♦❧✉t✐♦♥

✏♣✲❧♦❝❛❧✑ ❝❡♥t❡r❡❞ s❝❤❡♠❡

✸✸✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(48)

❘❡s✉❧ts ✇✐t❤ ♣ − ❛❞❛♣t❛t✐✈✐t② ✿ ❢ = ✷ ❍③

❊①❛❝t s♦❧✉t✐♦♥

✏♣✲❧♦❝❛❧✑ ✉♣✇✐♥❞ s❝❤❡♠❡

✸✹✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(49)

❘❡s✉❧ts ✇✐t❤ ♣ − ❛❞❛♣t❛t✐✈✐t② ✿ ❢ = ✹ ❍③

❊①❛❝t s♦❧✉t✐♦♥

✏♣✲❧♦❝❛❧✑ ❝❡♥t❡r❡❞ s❝❤❡♠❡

✸✺✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(50)

❘❡s✉❧ts ✇✐t❤ ♣ − ❛❞❛♣t❛t✐✈✐t② ✿ ❢ = ✹ ❍③

❊①❛❝t s♦❧✉t✐♦♥

✏♣✲❧♦❝❛❧✑ ✉♣✇✐♥❞ s❝❤❡♠❡

✸✻✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(51)

❘❡s✉❧ts ✇✐t❤ ♣ − ❛❞❛♣t❛t✐✈✐t②

❢ = ✷ ❍③

❢ = ✹ ❍③

❈❡♥t❡r❡❞

❯♣✇✐♥❞

❈❡♥t❡r❡❞

❯♣✇✐♥❞

❱

①

▲

✷

✲❡rr♦r

✺✳✵✷❡✲✵✷ ✻✳✵✵❡✲✵✷ ✷✳✽✸❡✲✵✶ ✷✳✼✻❡✲✵✶

❋❛❝t✳✲r❡s✳ t✐♠❡ ✭s✮

✺✼

✼✾

✺✼

✽✵

▼❡♠♦r②

✸✵✵✵

✸✽✵✵

✸✵✵✵

✸✽✵✵

◆✉♠❡r✐❝❛❧ st❛t✐st✐❝s ❢♦r ❜♦t❤ s❝❤❡♠❡s ❛s ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢r❡q✉❡♥❝②

✸✼✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(52)

❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ♣ − ❛❞❛♣t❛t✐✈✐t② ❛♥❞ ♣ − ❣❧♦❜❛❧ ❢♦r

❢ = ✷❍③

♣ − ❛❞❛♣t❛t✐✈✐t②

♣ − ❣❧♦❜❛❧

❈❡♥t❡r❡❞

❯♣✇✐♥❞

❈❡♥t❡r❡❞

❯♣✇✐♥❞

❱

①

▲

✷

✲❡rr♦r

✺✳✵✷❡✲✵✷ ✻✳✵✵❡✲✵✷ ✹✳✹✶❡✲✵✷ ✹✳✹✶❡✲✵✷

❋❛❝t✳✲r❡s✳ t✐♠❡ ✭s✮

✺✼

✼✾

✹✵

✺✼

▼❡♠♦r②

✸✵✵✵

✸✽✵✵

✶✾✵✵

✷✽✵✵

✸✽✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(53)

❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ ♣ − ❛❞❛♣t❛t✐✈✐t② ❛♥❞ ♣ − ❣❧♦❜❛❧ ❢♦r

❢ = ✹❍③

♣ − ❛❞❛♣t❛t✐✈✐t②

♣ − ❣❧♦❜❛❧

❈❡♥t❡r❡❞

❯♣✇✐♥❞

❈❡♥t❡r❡❞

❯♣✇✐♥❞

❱

①

▲

✷

✲❡rr♦r

✷✳✽✸❡✲✵✶ ✷✳✼✺❡✲✵✶ ✹✳✷✹❡✲✵✶ ✶✳✻✻❡✲✵✶

❋❛❝t✳✲r❡s✳ t✐♠❡ ✭s✮

✺✼

✽✵

✹✵

✺✼

▼❡♠♦r②

✸✵✵✵

✸✽✵✵

✶✾✵✵

✷✽✵✵

✸✾✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(54)

❈♦♥t❡♥ts

✶

✷❉ ❍❡❧♠❤♦❧t③ ✐s♦tr♦♣✐❝ ❡❧❛st✐❝ ❡q✉❛t✐♦♥s

✷

❉● ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥s

✸

◆✉♠❡r✐❝❛❧ r❡s✉❧ts

✹

❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

✹✵✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(55)

❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

❈♦♥❝❧✉s✐♦♥

❯♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ❣✐✈❡s ❜❡tt❡r r❡s✉❧ts ♦♥ ❝♦❛rs❡

♠❡s❤❡s ♦r ❢♦r ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s t❤❛♥ ❝❡♥t❡r❡❞ ✢✉① ❉●

❢♦r♠✉❧❛t✐♦♥

❲✐t❤ t❤❡ ✉♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ✇❡ ♦❜t❛✐♥ ♦♥❡

❝♦♥✈❡r❣❡♥❝❡ ♦r❞❡r ♠♦r❡ t❤❛♥ t❤❡ ❝❡♥t❡r❡❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥

P❡rs♣❡❝t✐✈❡s

❉❡✈❡❧♦♣ ✸❉ ✉♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ❢♦r ❍❡❧♠❤♦❧t③

❡q✉❛t✐♦♥s

❆❞❛♣t t❤❡ ♣r♦❣r❛♠ ❢♦r ♣❛r❛❧❧❡❧ ❝♦♠♣✉t✐♥❣

✹✶✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(56)

❈♦♥❝❧✉s✐♦♥✲P❡rs♣❡❝t✐✈❡s

❈♦♥❝❧✉s✐♦♥

❯♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ❣✐✈❡s ❜❡tt❡r r❡s✉❧ts ♦♥ ❝♦❛rs❡

♠❡s❤❡s ♦r ❢♦r ❤✐❣❤ ❢r❡q✉❡♥❝✐❡s t❤❛♥ ❝❡♥t❡r❡❞ ✢✉① ❉●

❢♦r♠✉❧❛t✐♦♥

❲✐t❤ t❤❡ ✉♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ✇❡ ♦❜t❛✐♥ ♦♥❡

❝♦♥✈❡r❣❡♥❝❡ ♦r❞❡r ♠♦r❡ t❤❛♥ t❤❡ ❝❡♥t❡r❡❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥

P❡rs♣❡❝t✐✈❡s

❉❡✈❡❧♦♣ ✸❉ ✉♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ❢♦r ❍❡❧♠❤♦❧t③

❡q✉❛t✐♦♥s

❆❞❛♣t t❤❡ ♣r♦❣r❛♠ ❢♦r ♣❛r❛❧❧❡❧ ❝♦♠♣✉t✐♥❣

✹✶✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(57)

P❡rs♣❡❝t✐✈❡s

❉r❛✇❜❛❝❦

▲✐♥❡❛r s②st❡♠ ✇✐t❤ ✺ ✉♥❦♥♦✇♥s t♦ st♦r❡

P❡rs♣❡❝t✐✈❡s

❯s❡ ♦❢ t❤❡ ✉♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ❛s ❛ r❡❢❡r❡♥❝❡ ♠❡t❤♦❞

❈♦♠♣❛r❡ ✉♣✇✐♥❞ ❉● ❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ❤②❜r✐❞✐③❛❜❧❡ ❉●

❢♦r♠✉❧❛t✐♦♥

❈♦♥str✉❝t✐♦♥ ♦❢ ❛ ❍❉● ❢♦r♠✉❧❛t✐♦♥

❉❡✈❡❧♦♣ ♦t❤❡r ❧✐♥❡❛r s♦❧✈❡rs ❢♦r s♣❛rs❡ ♠❛tr✐❝❡s ✭✐♥

❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ■◆❘■❆ t❡❛♠✱ ❍✐❡♣❛❝s✮

✹✷✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s

(58)

P❡rs♣❡❝t✐✈❡s

❉r❛✇❜❛❝❦

▲✐♥❡❛r s②st❡♠ ✇✐t❤ ✺ ✉♥❦♥♦✇♥s t♦ st♦r❡

P❡rs♣❡❝t✐✈❡s

❯s❡ ♦❢ t❤❡ ✉♣✇✐♥❞ ✢✉① ❉● ❢♦r♠✉❧❛t✐♦♥ ❛s ❛ r❡❢❡r❡♥❝❡ ♠❡t❤♦❞

❈♦♠♣❛r❡ ✉♣✇✐♥❞ ❉● ❢♦r♠✉❧❛t✐♦♥ ✇✐t❤ ❤②❜r✐❞✐③❛❜❧❡ ❉●

❢♦r♠✉❧❛t✐♦♥

❈♦♥str✉❝t✐♦♥ ♦❢ ❛ ❍❉● ❢♦r♠✉❧❛t✐♦♥

❉❡✈❡❧♦♣ ♦t❤❡r ❧✐♥❡❛r s♦❧✈❡rs ❢♦r s♣❛rs❡ ♠❛tr✐❝❡s ✭✐♥

❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ ■◆❘■❆ t❡❛♠✱ ❍✐❡♣❛❝s✮

✹✷✴✹✷ ▼✳ ❇♦♥♥❛ss❡✲●❛❤♦t ❉● ♠❡t❤♦❞s ❢♦r ❍❡❧♠❤♦❧t③ ✇❛✈❡ ❡q✉❛t✐♦♥s