On the Optimal Shape for the Heat Insulation Energy Problem Qinfeng - - PowerPoint PPT Presentation

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On the Optimal Shape for the Heat Insulation Energy Problem Qinfeng - - PowerPoint PPT Presentation

Mar 24, 2024 133 likes •383 views

Introduction Existence of Optimal Shapes Stability Conclusion On the Optimal Shape for the Heat Insulation Energy Problem Qinfeng Li (Joint work with Hengrong Du and Prof. Changyou Wang) Department of Mathematics Purdue University PDE

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Introduction Existence of Optimal Shapes Stability Conclusion

On the Optimal Shape for the Heat Insulation Energy Problem

Qinfeng Li (Joint work with Hengrong Du and Prof. Changyou Wang)

Department of Mathematics Purdue University

PDE Seminar, 2017

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Introduction Existence of Optimal Shapes Stability Conclusion Background :Optimization Problems in Thermal Insulation The Asymptotic Problem

Motivation

Σǫ := {σ + tν(σ)|σ ∈ ∂Ω , 0 ≤ t ≤ ǫh(σ)} is the layer. Ω thermally conducting body, u temperature function, f ∈ L2(Ω) heat source, and h insulation material function of total amount m > 0, that is, h ∈ Hm, where Hm = {h ≥ 0 :

  • ∂Ω

h(σ)dσ = m}.

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Introduction Existence of Optimal Shapes Stability Conclusion Background :Optimization Problems in Thermal Insulation The Asymptotic Problem

Thermal Insulation Problem

We want to design the optimal shape of Ω and find the

  • ptimal distribution of insulation material surrounding Ω.

We assume the layer is very thin compared to the size of Ω, hence the study of the thermal insulation problem is to mathematically consider the limit of Fǫ as ǫ → 0, where Fǫ(u, h, Ω) = 1 2

  • Ω

|∇u|2dx + ǫ 2

  • Σǫ

|∇u|2dx −

  • Ω
  • fudx. (1)

The limit is in the context of Γ-convergence.

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Introduction Existence of Optimal Shapes Stability Conclusion Background :Optimization Problems in Thermal Insulation The Asymptotic Problem

The Asymptotic Problem

(Acerbi-Buttazzo 86’) Fǫ Γ-converge to F(u, h, Ω) := 1 2

  • Ω

|∇u|2dx + 1 2

  • ∂Ω

u2 h dσ −

  • Ω

fudx. To find the best distribution of insulation material of total amount m > 0 around Ω, is to find the solution of minimization problem min

h∈Hm

min

u∈H1(Ω) F(u, h, Ω)

(2) To find the optimal shape of thermal insulation body with prescribed volume V0 > 0, is to find the solution of inf{ min

h∈Hm

min

u∈H1(Ω) F(u, h, Ω) : |Ω| = V0}

(3)

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Introduction Existence of Optimal Shapes Stability Conclusion Background :Optimization Problems in Thermal Insulation The Asymptotic Problem

(2) is equivalent to inf{J(u, Ω) : u ∈ H1(Ω)}, where J(u, Ω) := 1 2

  • Ω

|∇u|2dx + 1 2m

  • ∂Ω

|u|dHn−1 2 −

  • Ω

fudx (4) To find the optimal shape of thermal insulation body with prescribed volume V0 > 0, is to find the solution of inf

  • J(u, Ω) : u ∈ H1(Ω), |Ω| = V0
  • (5)

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Introduction Existence of Optimal Shapes Stability Conclusion Background :Optimization Problems in Thermal Insulation The Asymptotic Problem

(Bucur-Buttazzo-Nitsch 16’) Fix any domain Ω, J(u, Ω) admits a unique minimizer uΩ ∈ H1(Ω). (Bucur-Buttazzo-Nitsch 16’) If Ω = BR and f ≡ 1, then uΩ = R2−|x|2

2n

+

m n2wnRn−2 .

(Bucur-Buttazzo-Nitsch 16’) BR is stationary shape when f ≡ 1.

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Introduction Existence of Optimal Shapes Stability Conclusion Background :Optimization Problems in Thermal Insulation The Asymptotic Problem

Questions : (Bucur-Buttazzo-Nitsch 17’) : Can the infimum of (5) be attained at a pair (u, Ω)? If so, is BR an optimal shape for some R > 0?

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

M-conformal domain

Definition We say Ω is M-conformal domain if there is a bi-Lipschitz map Φ with constant M such that Φ(B) = Ω and Φ(∂B) = ∂Ω. Remark If Ω is a M-conformal domain, then P(Ω) ≤ Mn−1Hn−1(∂B) = C(M, n). Example Ω ⊂ BR0, convex and |Ω| = V0 > 0, M = M(n, V0, R0). Ω is uniformly star-shaped, M depends on the star-shaped constant.

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

If Ω is M-conformal and u ∈ H1(Ω), then Trace inequality ||u||L2(∂Ω,dσ) ≤ C(M, n)||u||H1(Ω), (6) Poincaré inequality

  • Ω

u2 ≤ C(M, n)

  • Ω

|∇u|2dx +

  • ∂Ω

|u|dHn−1 2 , (7) Extension property ||Eu||H1(Rn) ≤ C(n, M)||u||H1(Ω). (8)

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

Lower semicontinuity

(7) and (8) allows us to carry out direct method, in the sense that if (ui, Ωi) is minimizing sequence realizing (4), then it follows that ui as extension funtions are bounded in H1(Rn) and thus have H1 weak limit u. It is clear we can find M-conformal domain Ω as L1 limit of Ωi, then

  • Ω

|∇u|2 ≤ lim inf

i→∞

  • Ωi

|∇ui|2 (9) follows by showing ∇uiχΩi ⇀ uχΩ weakly in L2(Rn). (6) together with weak convergence argument gives

  • ∂Ω

|u| ≤ lim inf

i→∞

  • ∂Ωi

|u|. (10)

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

Existence of optimal shape in energy problem

Theorem Given f ∈ L2(Rn) and m, M are fixed positive constants, then the infimum of J(u, Ω) := 1 2

  • Ω

|∇u|2dx + 1 2m

  • ∂Ω

|u|dHn−1 2 −

  • Ω

fu (11) can be attained over all u ∈ H1(Ω) and M-conformal domain Ω in BR with |Ω| = V0 > 0.

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

Flow Map

Definition Let η be a smooth vector field, and Ft(x) := F(t, x) solve the ODE

  • d

dt F(t, x) = η(F(t, x))

F0(x) = x. (12) We call Ft is the flow map generated by η. JFt = 1 + t div η + t2 2

  • ∇(div η) · η + (div η)2

+ O(t3) (∇xFt(x))−1 = I−t∇xη(x)+t2 2

  • (∇xη(x))2 − ∇2

xη(x) · η(x)

  • +O(t3)

where entij

  • ∇2η · η
  • = ηi

jkηk.

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

Always assume

  • Ω div η = 0. u = uΩ is the unique minimizer of

infu∈H1(Ω) J(u, Ω) Definition We say Ω is a stationary solution to (4) if for any smooth variation vector field η, d dt J(uΩ ◦ F −1

t

, Ft(Ω)) = 0, where Ft is the flow map generated by η.

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

Definition We say Ω is stable under smooth variation vector field η if d2 dt2 J(uΩ ◦ F −1

t

, Ft(Ω)) ≥ 0, where Ft is the flow map generated by η. Definition We say Ω is a stable shape if Ω is stable under any smooth variation vector fields.

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Introduction Existence of Optimal Shapes Stability Conclusion Direct Approach First Variation and Stationary Condition

First Variation and Stationary Condition

Let I(t) = J(ut, Ωt) = I1(t) + I2(t), where ut := u ◦ F −1

t

, Ωt := Ft(Ω), I1(t) := 1 2

  • Ωt

|∇ut|2 −

  • Ωt

u and I2(t) = 1 2m

  • ∂Ωt

ut 2 . I′(0) = 1 2

  • ∂Ω
  • |∂τu|2 − |∂νu|2 − 2u +

2 m

  • ∂Ω

u

∂Ω

uH

  • (η·ν)

(13) Therefore, we obtain the following stationary condition : |∂τu|2 − |∂νu|2 − 2u + 2 m

  • ∂Ω

u

∂Ω

uH ≡ const. (14)

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Introduction Existence of Optimal Shapes Stability Conclusion Second Variation and Stability

Second Variation of Surface Energy

d2 dt2

  • ∂Ωt

ut = d2 dt2

  • u ◦ ψ
  • |g(t)|d˜

x = d dt

  • u ◦ ψ[ζH
  • |g(t)|] ◦ Ft ◦ ψd˜

x =

  • u ◦ ψ
  • ζ2H2 + ζ(−∆∂Ωtζ − |A|2ζ)

|g|

  • Ft ◦ ψ

+ u ◦ ψ ˙ ζH

  • |g| ◦ Ft ◦ ψd˜

x, where ˙ ζ := d

dt (ζ ◦ Ft ◦ ψ) can be calculated easily as

˙ ζ = d dt (η · ν) = ∇ηη, ν + η, ˙ ν = ∇ηη, ν, since η ⊥ ∂Ω

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Introduction Existence of Optimal Shapes Stability Conclusion Second Variation and Stability

Second Variation of Surface Energy

When Ω = BR, I′′

2(0) = 1

m

  • ∂BR

u

  • ∂BR

u

  • |∇∂Ωζ|2 + (H2 − |A|2)ζ2 + H ∇ηη, ν
  • =

m n3ωnRn−3

  • ∂BR

|∇∂BRζ|2 + (n − 1)(n − 2) R2 ζ2 + n − 1 R ∇ηη, ν Notice ∇ηη, ν = ζ div η − ζ2H, hence I′′

2(0) =

m n3wnRn−1

  • Sn−1 |∇Sn−1ζ|2 − (n − 1)ζ2
  • :A2

+ n − 1 nR

  • ∂BR

u(η · x) div η

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Introduction Existence of Optimal Shapes Stability Conclusion Second Variation and Stability

Second Variation of Bulk Energy

I′′

1(0) = d2

dt2

  • t=0

1 2

  • BR
  • ∇u
  • I − t∇η + t2

2

  • (∇η)2 − ∇2η · η
  • 2
  • 1 + t div η + t2

2 (∇(div η) · η + (div η)2

  • −
  • BR

u[∇(div η) · η + (div η)2] Integration by part yields I′′

1(0) = 1

n2

  • BR

|∇(x · η)|2 − 1 R

  • ∂BR

(x · η)2

  • :=A1

− R 2n2

  • ∂BR

(x · η) div η − 1 R

  • ∂BR

u(η · x) div η

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Introduction Existence of Optimal Shapes Stability Conclusion Second Variation and Stability

Therefore, I′′(0) = A1 + A2 − 1 n2 R 2 + m nwnRn−1

∂BR

(x · η) div η (15) A2 ≥ 0, since the first eigenvalue λ1 of ∆Sn−1 is n − 1. A1 ≥ 0, since the first non-trivial Stekloff eigenvalue σ2(BR) = 1/R. Recall σ2(Ω) := inf

h∈H1(Ω)\{0}

  • Ω |∇h|2
  • ∂Ω h2

:

  • ∂Ω

h = 0

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Introduction Existence of Optimal Shapes Stability Conclusion

Stablity

Theorem BR is a stable for any R > 0 under smooth normal variation vector field η with η = 0 or div η = 0 on ∂BR. In particular, BR is stable under pointwise divergence free vector fields, or those without moving the boundary.

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Introduction Existence of Optimal Shapes Stability Conclusion

Theorem BR is not a stable shape. Démonstration. Let η = xn|x|p(x1, . . . , xn) I′′(0) =R2p+4 n2

  • Rn(p2 + 6p + 8

2p + n + 4 − p + n + 3 2 ) − m(p + n + 1) nwn

  • ∂B1

x2

ndHn−1 + R2p+4

n2 nwnRn 2p + n + 4 :=R2p+4D(R, m, p) Fix R and m, we let p → ∞, then D(R, m, p) → −∞.

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Introduction Existence of Optimal Shapes Stability Conclusion

Future Problems

What can we say about the regularity of optimal shape? What are all the stationary shapes, the stable shapes and

  • ptimal shapes among M-conformal domains?

Do the optimal shapes exist over generic class of sets of finite perimeter? If so, are all the optimal shapes necessarily domains?

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Introduction Existence of Optimal Shapes Stability Conclusion

Thank You!

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