Optimization Iterative Procedure for Radiative-Conductive Heat Transfer Model

Abstract

A boundary optimal control problem for radiative-conductive heat transfer model in a layered medium is considered. The problem consists in minimization of a given cost functional by controlling the boundary temperature. The solvability of this control problem is proved, and optimality conditions are derived. An iteration algorithm is proposed, and numerical experiments are performed.

Keywords

• optimal control
• radiative heat transfer
• conductive heat transfer

Introduction

maximization of the energy outflow from the model domain by controlling reflection properties of the boundary. In [11], the problem of constructing the desired temperature and(or) intensity of radiation in part of the model domain is solved.

In this paper, an optimal control problem of minimization of a given cost functional by controlling the boundary temperature is studied. Particularly, it can be interpreted as a problem of obtaining a desired temperature in whole layer. Similar problems appear in many engineering applications and draw attention of many researchers. In the current work, the solvability of this problem is proved, an optimality system is derived, and the numerical algorithm is implemented. 2

Formulation of the optimal control problem

Let us consider the boundary-value problem for radiative-conductive heat transfer model in a layered medium [12]: − θ′′(x) + ασ(|θ(x)|θ3(x) − ϕ(x)) = 0,

− ϕ(x)′′ + α(ϕ(x) − |θ(x)|θ3(x)) = 0, x ∈ ( 0, 1 ), ( 1 ) θ(0) = u1, θ( 1 ) = u2,

B1ϕ := ϕ(0) − β1ϕ′(0) = u14, B2ϕ := ϕ( 1 ) + β2ϕ′( 1 ) = u24. ( 2 ) Here, θ is the normalized temperature, and ϕ the normalized intensity of radiation averaged over all direction. The given positive constants α, σ, β1, β2 describe properties of the medium and boundaries. Specifically, α = 3τ02(1 − ω), σ = 1/3Nc, where ω is the albedo of single scattering, Nc the conduction-to-radiation parameter, and τ0 the optical depth of the layer. The coefficients βi = 2 (2 − εi) , i = 1, 2

3τ0εi describe the reflection properties of the boundaries. Here, ε1 and ε2 are the emissivity coefficients for the boundary surfaces.

We will consider a vector u = (u1, u2) ∈ R2 as the boundary control. The optimal control problem is to find functions θ, ϕ and vector u ∈ Uad ⊂ R2 that satisfy ( 1 ), ( 2 ) and minimize a cost functional: 1 Jμ(θ, ϕ, u) = J (θ, ϕ) + μ|u|2 → inf .

2 ( 3 ) Here, μ ≥ 0, |u|2 = u21 + u22, and Uad is a nonempty set of admissible controls. Particularly, the functional J can describe the mean square deviation between the temperature θ and a desired temperature θd ∈ L2( 0, 1 ), that is

1 J = ||θ − θd||2L2( 0,1 ).

2 Taking into account that the temperature (and the control) is normalized, we can assume that Uad = [0, 1]×[0, 1]. But this condition is not necessary from a mathematical point of view.

Formalization of the optimal control problem

Let H = L2( 0, 1 ), and W = W22( 0, 1 ) be a Sobolev space. By Y = W × W , we denote the state space of the controlled system, and V = H × H × R × R × R × R the space of constraints.

Let us determine an operator F : Y × R2 → V , F (θ, ϕ, u) = {−θ′′ + ασ(|θ|θ3 − ϕ), −ϕ′′ + α(ϕ − |θ|θ3),

θ(0) − u1, θ( 1 ) − u2, B1ϕ − u14, B2ϕ − u24}.

Then the problem ( 1 )-( 3 ) can be written as follows:

1 Jμ(θ, ϕ, u) = J (θ, ϕ) + μ|u|2 → inf, F (θ, ϕ, u) = 0, u ∈ Uad.

2 ( 4 ) Theorem 1. Let (i) Uad is a closed convex set; Uad is a bounded set, if μ = 0. (ii) J : Y → R is weakly lower semicontinuous.

Then there exists a solution of problem ( 4 ).

Proof. Notice that for a solution of the problem ( 1 ), ( 2 ) the following estimates hold [12]:

m ≤ θ ≤ M, |m|m3 ≤ ϕ ≤ |M |M 3, where m = min{u1, u2}, M = max{u1, u2}.Therefore,

||θ||W + ||ϕ||W ≤ C, where C depends only on m, M , α, and σ.

Let {θk, ϕk, uk} be a minimizing sequence of the problem ( 4 ),

uk ∈ Uad, F (θk, ϕk, uk) = 0, Jμ(θk, ϕk, uk) → inf Jμ.

It is obvious that the sequence {uk} ⊂ R2 is bounded if μ > 0 and the condition (i) guarantees the boundedness if μ = 0. Therefore, by ( 5 ), sequences {θk}, {ϕk} are bounded in W . Thus, we can assume that

uk → ub in R2, θk → θb, ϕk → ϕb weakly in W, and in addition ub ∈ Uad. The convergence allows to pass to limit in ( 6 ), i.e. a triple {θb, ϕb, ub} is admissible for the problem ( 4 ) and, by the condition (ii), it is a solution. 4

Optimality conditions

To derive the optimality system, we apply the principle of Lagrange for smooth convex extremal problems [13]. This principle requires only the convexity of the functional Jμ with respect to control. Let ub ∈ Uad be the optimal control, and yb = {θb, ϕb} be the ( 5 ) ( 6 ) optimal state. We suppose that (iii) J : Y → R is Frechet differentiable in {θb, ϕb}.

Let us prove that Im Fy′ (y, u) = V . Here, Fy′ (yb, ub) : Y → V is a derivative of the b b constraint operator with respect to state.

The equation

Fy′ (yb, ub)hhi = z, h = {h1, h2} ∈ Y, z = {z1, z2, z3, z4, z5, z6} ∈ V is equivalent to the boundary-value problem −h′1′ + ασ(4|θb|3h1 − h2) = z1, −h′2′ + α(h2 − 4|θb|3h1) = z2, x ∈ ( 0, 1 ), h1(0) = z3, h1( 1 ) = z4, B1h2 = 4θb3(0)z3 + z5, B2h2 = 4θb3( 1 )z4 + z6. Lemma 1. The boundary-value problem ( 8 ), ( 9 ) is the unique solvable for all z1, z2 ∈ H, zk ∈ R, k = 2, 6 Proof. Due to Fredholm property of the problem ( 8 ), ( 9 ), to prove the lemma, it is sufficient to show that the homogeneous problem has only the zero solution. Set z = 0 in ( 8 ), ( 9 ). Let rε(s) = (s/|s|, |s| ≥ ε,

s/ε, |s| < ε.

Multiplying the first equation in ( 8 ) by rε(h1), the second by σrε(h2), then integrating the result over ( 0, 1 ) and adding, we obtain ( 7 ) ( 8 ) ( 9 ) (h′1, rε′(h1)h′1)σ(h′2, rε′(h2)h′2) +

h2(0)rε(h2(0)) + σ β2 σ β1 h2( 1 )rε(h2( 1 )) + ασ(4|θb|3h1 − h2, rε(h1) − rε(h2)) = 0.

Notice that rε′(s) ≥ 0, s ∈ R. Dropping the first two terms and passing to limit as ε → +0, we obtain

σ(β1−1|h2(0)| + β2−1|h2( 1 )|) + ασ(4|θb|3h1 − h2, signh1 − signh2) ≤ 0.

Therefore, h2(0) = h2( 1 ) = 0, and Thus, h1 + σh2 = 0, and hence

(h1 + σh2)′′ = 0, x ∈ ( 0, 1 ); (h1 + σh2)|x=0;1 = 0.

−h′1′ + α(1 + 4σ|θb|3)h1 = 0, x ∈ (0; 1); h1(0) = h1( 1 ) = 0.

As a result h1 = 0, and consequently h2 = 0. This proves the lemma. Since, the derivative of the constraint operator with respect to state is epimorphism, then we can apply the Lagrange principle [14, Cor.2, Th. 1.5].

Let L = Jμ + (−θ′′ + ασ(|θ|θ3 − ϕ), p1) + (−ϕ′′ + α(ϕ − |θ|θ3), p2)

+ (θ(0) − u1)q1 + (θ( 1 ) − u2)q2 + (B1ϕ − u41)q3 + (B2ϕ − u42)q4.

Here, p1,2 ∈ H, and qk ∈ R, k = 1, 4 are Lagrange multipliers.

Equating to zero derivatives of Lagrange function L with respect to θ and ϕ, we obtain: −p′1′ + 4α|θb|3(σp1 − p2) = −Jθ′ (θb, ϕb), −p′2′ + α(p2 − σp1) = −J ϕ′(θb, ϕb), p1(0) = p1( 1 ) = 0, B1p2 = B2p2 = 0, q = {q1, q2} =

p′1(0) + 4β−1p2(0)ub13, −p′1( 1 ) + 4β2−1p2( 1 )ub23 .

From condition we obtain

(L′u, ub − v)R2 ≤ 0 ∀v ∈ Uad, (μub − q, ub − v)R2 ≤ 0 ∀v ∈ Uad. ( 10 ) ( 11 ) ( 12 ) ( 13 ) Thus, we obtain the following optimality conditions of the first order. Theorem 2. Let {θb, ϕ} be an optimal state, u an optimal control, and condition (iii) b b holds. Then there exists a unique adjoint state p = {p1, p2} ∈ Y satisfying ( 10 ),( 11 ), and the variational inequality ( 13 ) holds. 5

Numerical algorithm

The algorithm is based on solving the optimality system consisting from boundaryvalue problem ( 1 ), ( 2 ), where θ = θb, ϕ = ϕ and conditions ( 10 )-( 13 ). The system is b solved by an iterative procedure based on method of the gradient projection of the original extremal problem:

uk+1 = PUad uk − λ(μuk − qk) , k = 0, 1, 2, ...

Here, u0 is a given initial approximation, λ is an iterative parameter, PUad the projection operator to Uad. To find qk, at first, the problem ( 1 ),( 2 ) for u = uk is solved. Further, we solve the adjoint system ( 10 ),( 11 ), and then we find qk from ( 12 ), where u = uk. b

In conclusion, let us consider the numerical experiment. We took the following parameters of the model (see [15], Problem 2): ω = 0.9, τ0 = 3, ε1 = 0.7, ε2 = 0.6, and Nc = 0.05. To determine the cost functional, we set θd = 0.8 − 0.4x and μ = 0.01. In figure 1, the optimal temperature is shown (solid curve). The small value of μ practically means neglecting the second term in the cost functional. In this case, the optimal temperature approximates the given function θd (dashed line). Further, we consider the same model data as in the first experiment with the exception of Nc = 0.00001 (see [16], Problem 2). This corresponds to the case of a high temperature. To determine the cost functional, we set θd = 0.8−0.2x and μ = 0.01. In figure 2, the obtained optimal temperature (solid curve) and given function θd (dashed line) are shown.

We did not study theoretically the rate of the convergence. Nevertheless, it was sufficient 10 iterations for convergence of the iterative procedure. The numerical experiments demonstrate the efficiency of the proposed algorithm.

0.8 0.75 0.7 Fig. 2. The optimal normalized temperature θ (solid curve) and given temperature θd = 0.8 − 0.2x (dashed line) for Nc = 0.00001.

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