=Paper=
{{Paper
|id=Vol-1623/papermp5
|storemode=property
|title=Optimization Iterative Procedure for Radiative-Conductive Heat Transfer Model
|pdfUrl=https://ceur-ws.org/Vol-1623/papermp5.pdf
|volume=Vol-1623
|authors=Alexander Chebotarev, Andrey Kovtanyuk, Veronika Pestretsova
|dblpUrl=https://dblp.org/rec/conf/door/ChebotarevKP16
}}
==Optimization Iterative Procedure for Radiative-Conductive Heat Transfer Model==
https://ceur-ws.org/Vol-1623/papermp5.pdf
Optimization Iterative Procedure
for Radiative-Conductive Heat Transfer Model
Alexander Chebotarev1,2⋆ , Andrey Kovtanyuk1,2, and Veronika Pestretsova1,2
1
Far Eastern Federal University,
Sukhanova st. 8, 690950 Vladivostok, Russia,
2
Institute for Applied Mathematics,
Radio st. 7, 690041 Vladivostok, Russia
{cheb@iam.dvo.ru,kovtanyuk.ae@dvfu.ru,nika02061994@mail.ru}
Abstract. A boundary optimal control problem for radiative-conductive heat
transfer model in a layered medium is considered. The problem consists in min-
imization 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
1 Introduction
Radiative-conductive heat transfer in a scattering and absorbing medium bounded
by two reflecting and radiating planes is examined. The study of the coupled heat
transfer [1] where the radiative and conductive contributions are simultaneously taken
into account is important for many engineering applications. So, in [2–5], the radiative-
conductive heat transfer model is studied in context of glass manufacturing. Modeling of
thermal processes in the presence of radiation effects in nanofluids is performed in [6,7].
Notice that nanofluids have numerous applications in engineering and biomedicine (e.g.,
design of cooling systems, cancer therapy, etc.).
Usually, the process of radiative-conductive heat transfer is described by a nonlinear
system of two differential equations: an equation of the radiative heat transfer and an
equation of the conductive heat exchange. In general, the problem is characterized by
anisotropic scattering and by specularly and diffusely reflecting boundaries.
A considerable number of works devoted to optimal control problems for radiative-
conductive heat transfer models consider evolutionary systems (see e.g. [3–5, 8, 9]).
Optimal control problems for steady-state radiative-conductive heat transfer model
were studied weaker. Here, we can mention the works [10, 11] where optimal bound-
ary control problems are studied. In [10], the optimal problem is formulated as the
⋆
The research was supported by the Russian Foundation for Basic Research (Project No.
14-01-00037)
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
Optimization Iterative Procedure 179
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 ,
B1 ϕ := ϕ(0) − β1 ϕ′ (0) = u41 , B2 ϕ := ϕ(1) + β2 ϕ′ (1) = u42 . (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
2 (2 − εi )
βi = , i = 1, 2
3τ0 εi
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 . (3)
2
Here, µ ≥ 0, |u|2 = u21 + u22 , and Uad is a nonempty set of admissible controls. Particu-
larly, 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.
180 A. Chebotarev, A. Kovtanyuk, V. Pestretsova
3 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 , B1 ϕ − u41 , B2 ϕ − u42 }.
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 . (4)
2
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 , (5)
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 . (6)
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 → u b ϕk → ϕ
b in R2 , θk → θ, b weakly in W,
and in addition ub ∈ Uad . The convergence allows to pass to limit in (6), i.e. a triple
b ϕ,
{θ, b u
b} 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 = {θ, b ϕ}
b be the
Optimization Iterative Procedure 181
optimal state. We suppose that
b ϕ}.
(iii) J : Y → R is Frechet differentiable in {θ, b
′
Let us prove that Im Fy (b b) = V . Here, Fy′ (b
y, u y, u
b) : Y → V is a derivative of the
constraint operator with respect to state.
The equation
Fy′ (b
y, u
b)hhi = z, h = {h1 , h2 } ∈ Y, z = {z1 , z2 , z3 , z4 , z5 , z6 } ∈ V (7)
is equivalent to the boundary-value problem
b 3 h1 − h2 ) = z1 , −h′′ + α(h2 − 4|θ|
−h′′1 + ασ(4|θ| b 3 h1 ) = z2 , x ∈ (0, 1), (8)
2
h1 (0) = z3 , h1 (1) = z4 , B1 h2 = 4θb3 (0)z3 + z5 , B2 h2 = 4θb3 (1)z4 + z6 . (9)
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 (
s/|s|, |s| ≥ ε,
rε (s) =
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
σ
(h′1 , rε′ (h1 )h′1 )σ(h′2 , rε′ (h2 )h′2 ) + h2 (0)rε (h2 (0))
β1
σ b 3 h1 − h2 , rε (h1 ) − rε (h2 )) = 0.
+ h2 (1)rε (h2 (1)) + ασ(4|θ|
β2
Notice that rε′ (s) ≥ 0, s ∈ R. Dropping the first two terms and passing to limit as
ε → +0, we obtain
b 3 h1 − h2 , signh1 − signh2 ) ≤ 0.
σ(β1−1 |h2 (0)| + β2−1 |h2 (1)|) + ασ(4|θ|
Therefore, h2 (0) = h2 (1) = 0, and
(h1 + σh2 )′′ = 0, x ∈ (0, 1); (h1 + σh2 )|x=0;1 = 0.
Thus, h1 + σh2 = 0, and hence
b 3 )h1 = 0, x ∈ (0; 1); h1 (0) = h1 (1) = 0.
−h′′1 + α(1 + 4σ|θ|
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].
182 A. Chebotarev, A. Kovtanyuk, V. Pestretsova
Let
L = Jµ + (−θ′′ + ασ(|θ|θ3 − ϕ), p1 ) + (−ϕ′′ + α(ϕ − |θ|θ3 ), p2 )
+ (θ(0) − u1 )q1 + (θ(1) − u2 )q2 + (B1 ϕ − u41 )q3 + (B2 ϕ − 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:
b 3 (σp1 − p2 ) = −J ′ (θ,
−p′′1 + 4α|θ| b ϕ), b ϕ),
b −p′′2 + α(p2 − σp1 ) = −Jϕ′ (θ, b (10)
θ
p1 (0) = p1 (1) = 0, B1 p2 = B2 p2 = 0, (11)
u31 , −p′1 (1) + 4β2−1 p2 (1)b
q = {q1 , q2 } = p′1 (0) + 4β −1 p2 (0)b u32 . (12)
From condition
(L′u , u
b − v)R2 ≤ 0 ∀v ∈ Uad ,
we obtain
(µb
u − q, u
b − v)R2 ≤ 0 ∀v ∈ Uad . (13)
Thus, we obtain the following optimality conditions of the first order.
Theorem 2. Let {θ, b ϕ}
b be an optimal state, ub an optimal control, and condition (iii)
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 boundary-
b ϕ=ϕ
value problem (1), (2), where θ = θ, b and conditions (10)-(13). The system is
solved by an iterative procedure based on method of the gradient projection of the
original extremal problem:
uk+1 = PUad uk − λ(µuk − q k ) , k = 0, 1, 2, ...
Here, u0 is a given initial approximation, λ is an iterative parameter, PUad the pro-
jection 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 q k from (12), where
b = uk .
u
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
Optimization Iterative Procedure 183
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 ex-
periments demonstrate the efficiency of the proposed algorithm.
0.8
0.75
0.7
Normalized temperature (θ)
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Points of layer (x)
Fig. 1. The optimal normalized temperature θ (solid curve) and given temperature θd =
0.8 − 0.4x (dashed line) for Nc = 0.05.
0.9
0.85
0.8
0.75
Normalized temperature (θ)
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Points of layer (x)
Fig. 2. The optimal normalized temperature θ (solid curve) and given temperature θd =
0.8 − 0.2x (dashed line) for Nc = 0.00001.
184 A. Chebotarev, A. Kovtanyuk, V. Pestretsova
References
1. Modest, M.F.: Radiative Heat Transfer. McGraw-Hill, New York (1993)
2. Klar, A., Siedow, N.: Boundary layers and domain decomposition for radiative heat transfer
and diffusion equations: applications to glass manufacturing process. Eur. J. Appl. Math.
9(4), 351–372 (1998)
3. Thömes, G., Pinnau, R., Seaı̈d, M., Götz, T., Klar, A.: Numerical methods and optimal
control for glass cooling processes. Trans. Theory Stat. Phys. 31(4-6), 513–529 (2002)
4. M. Frank, M., Klar, A., Pinnau, R.: Optimal control of glass cooling using simplified PN
theory. Trans. Theory Stat. Phys. 39(2-4), 282–311 (2010)
5. Clever, D., Lang, J.: Optimal control of radiative heat transfer in glass cooling with re-
strictions on the temperature gradient. Optimal Control Appl. and Meth. 33(2), 157–175
(2012)
6. Mabood, F., Shateyi, S., Rashidi, M.M., Momoniat, E., Freidoonimehr, N.: MHD stag-
nation point flow heat and mass transfer of nanofluids in porous medium with radiation,
viscous dissipation and chemical reaction. Advanced Powder Technology 27, 742–749 (2016)
7. Rashidi, M.M., Ganesh, N.V., Hakeem, A.K.A., Ganga, B.: Buoyancy effect on MHD flow
of nanofluid over a stretching sheet in the presence of thermal radiation. J. Molecular Liquids
198, 234–238 (2014)
8. Grenkin, G.V., Chebotarev, A.Yu., Kovtanyuk, A.E., Botkin, N.D., Hoffmann, K.-H.:
Boundary optimal control problem of complex heat transfer model. J. Math. Anal. Appl.
433, 1243–1260 (2016)
9. Sushchenko, A.: Optimal control problem of complex heat transfer for SP1 approximation.
Key Eng. Mater. 685, 99–103 (2016)
10. Kovtanyuk, A.E., Chebotarev, A.Yu., Botkin, N.D., Hoffmann, K.-H.: Theoretical analysis
of an optimal control problem of conductive-convective-radiative heat transfer. J. Math.
Anal. Appl. 412, 520–528 (2014)
11. Kovtanyuk, A.E., Chebotarev, A.Yu., Botkin, N.D., Hoffmann, K.-H.: Optimal boundary
control of a steady-state heat transfer model accounting for radiative effects. J. Math. Anal.
Appl. 439, 678–689 (2016)
12. Kovtanyuk, A.E., Chebotarev, A.Yu., Botkin, N.D., Hoffmann, K.-H.: Solvability of P1
approximation of a conductive-radiative heat transfer problem. Appl. Math. Comp. 249,
247–252 (2014)
13. Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. North-Holland, Amsterdam
(1979)
14. Fursikov, A.V.: Optimal Control of Distributed Systems. Theory and Applications. Amer.
Math. Soc. (2000)
15. Siewert, C.E., Thomas, J.R.: A computational method for solving a class of coupled
conductive-radiative heat-transfer problems. J. Quant. Spectrosc. Radiat. Transfer 45, 273–
281 (1991)
16. Siewert, C.E.: An improved iterative method for solving a class of coupled conductive-
radiative heat-transfer problems. J. Quant. Spectrosc. Radiat. Transfer 54, 599–605 (1995)