=Paper=
{{Paper
|id=Vol-1623/papermp14
|storemode=property
|title=Convergence of Solutions of an Optimal Control Problem for SP1 and Rosseland Approximations
|pdfUrl=https://ceur-ws.org/Vol-1623/papermp14.pdf
|volume=Vol-1623
|authors=Andrei Sushchenko, Tatiana Park, René Pinnau, Oliver Tse
|dblpUrl=https://dblp.org/rec/conf/door/SushchenkoPPT16
}}
==Convergence of Solutions of an Optimal Control Problem for SP1 and Rosseland Approximations==
https://ceur-ws.org/Vol-1623/papermp14.pdf
Convergence of Solutions of an Optimal Control Problem
for SP1 and Rosseland Approximations
Andrei Sushchenko1,2,*, Tatiana Park1, René Pinnau3, Oliver Tse3
1
Far Eastern Federal University, Sukhanova st. 8, Vladivostok 690950, Russia
2
Institute of Applied Mathematics FEB RAS, Radio st. 5, Vladivostok 690041, Russia
3
Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany
sushchenko.aa@dvfu.ru
Keywords: optimal control, SP1 approximation, Rosseland approximation,
radiative heat transfer.
Abstract. The optimal control problems for SP 1 and Rosseland approximations
of evolution radiative heat transfer are considered. The problems are solved by
weak form technique and Lagrange method. Numerical experiments for borosil-
icate glass are done. Numerical convergence of optimal control problem for SP 1
approximation to Rosseland is studied.
1 Introduction
The interest in studying problems for complex heat transfer models [1], where the
radiative, convective, and conductive contributions are simultaneously taken into
account, is motivated by their importance for many engineering applications. The
common feature of such processes is the radiative heat transfer dominating at high
temperatures. The radiative heat transfer equation is a first order integro-differential
equation governing the radiation intensity. The radiation traveling along a path is
attenuated as a result of absorption and scattering. Solutions to the radiative transfer
equation can be represented in the form of the Neumann series whose terms are pow-
ers of an integral operator applied to a certain start function. The terms can be calcu-
lated using a Monte Carlo method (see e.g. [2], [3]), which may be interpreted as
tracking the history of energy bundles from emission to absorption at a surface or
within a participating medium.
Numerical and theoretical analysis of one-dimensional heat transfer models cou-
pled with the radiative transfer equation can be found in [4]–[8]. In particular, effi-
cient numerical algorithms are proposed in [6], [7]. In [5]–[8] the unconditional
unique solvability of one-dimensional steady-state complex heat transfer problems is
proved. Papers [9]–[12] state conditional unique solvability of three dimensions prob-
lems for complex heat transfer models. In [13] unconditional unique solvability of
boundary-value problem for P1 approximation of 3D complex heat transfer model is
proved.
Copyright © 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
Convergence of Solutions of an Optimal Control Problem for SP1 285
A considerable number of works of optimal control problems for complex heat
transfer models is devoted to the problems of controlling evolutionary systems (see
e.g. [14]–[17]). In [18], [19] problems of optimal boundary control for a steady-state
complex heat transfer model were considered. On the basis of new a priori estimates
of solutions of the control systems, the solvability of the optimal control problems
was proved.
In paper [20] optimal control problems in radiative transfer are solved by means of
the space mapping technique. Authors constructed fast numerical algorithm for solv-
ing problems using a hierarchy of approximate models.
The numerical analysis of optimal control problem of complex heat transfer for SP 1
approximation using weak form technique and Lagrange method was considered in
[21]. Using a similar approach we study the optimal control problem for a simplified
model given by the Rosseland approximation. Moreover, we further consider the nu-
merical convergence of optimal control problem in SP1 approximation to Rosseland.
2 Rosseland Approximation
Let, . The normalized evolution
diffusion model describing radiative, conductive, and convective heat transfer in a
bounden region has the following form:
(1)
here, denotes the normalized temperature, and denote the coefficient of diffu-
sion, and radiation, respectively. The constant and are defined as follows:
here, denotes the heat conductivity, the density, the heat capacity, the Steph-
an-Boltzmann constant, the refractive index, the maximum temperature in the
unnormalized model, and the absorption coefficient.
Assume that the function satisfy to the following condition on the boundary:
(2)
and the initial condition:
(3)
Here, denotes temperature of sources on the boundary and describes the
reflective properties of the boundary.
For problem (1)-(3) we consider the cost functional of tracking type
(4)
286 A. Sushchenko,T. Park, R. Pinnau, and O. Tse
where solves (1)–(3). Here, is a specified desired temperature profile. Further-
more, the positive constant allows one to adjust the weight of the penalty term.
The main subject on the analysis in this paper is the following initial boundary
control problem:
w.r.t.
(5)
subject to system (1)–(3)
This optimal control problem is considered as a constrained minimization problem
and the adjoint variables are used for the construction of numerical algorithms.
For solving (5) and finding the optimal pair authors use the method of La-
grange multipliers. First of all, denote the week form for (1)–(3) as follows
(6)
where denotes a test function in the Sobolev space , .
Denote the Lagrange function
(7)
with . Here we only consider the case . It is worth to note that is typical-
ly called a Lagrange multiplier or adjoint variable.
For the minimization of (7), we solve the first-order optimality system
(8)
The state equation for (8) is defined as follows
(9)
Here and in the following, denote several test functions.
The adjoint equation is defined as follows
(10)
By denoting , the gradient of may be expressed as
Convergence of Solutions of an Optimal Control Problem for SP1 287
(11)
For solving problem (5) we use an iterative algorithm. On the first stage, the state
equation (9) is solved by a time-discrete approximation and semi-implicit scheme. On
the second stage, the adjoint equation (10) is solved by time-discrete approximation,
starting from the terminal time ( ). On the third stage, a new control variable is
computed by using (11) and the Armijo rule. Afterwards, this algorithm is repeated
until the relative gradient of is less than a small parameter . The construction of the
algorithm is presented below.
1. Choose
2.
3. REPEAT
(a) Choose
(b) FOR DO
Find from (9)
(c)
(d) FOR DO
Find from (10)
(e)
(f) REPEAT
(i) FOR DO
using (11)
(ii) FOR DO
Find from (9)
(iii)
(g) UNTIL
(h)
4. UNTIL
In the algorithm, we used the following notation for variables:
, where denotes the control function on step # . The set
denotes a uniform grid on the interval . It is worth noting that step 3(f-g)
describes the Armijo rule.
Further, we consider a glass cooling process which was already considered for Pn
in [22]. A numerical experiment for borosilicate glass is investigated in the domain
and .
Area parameters are represented in Table 1.
288 A. Sushchenko,T. Park, R. Pinnau, and O. Tse
Table 1. Area Parameters
Consider experiment results (Fig. 1-2). It is worth noting that the proposed algo-
rithm for this example converges after 4 iterations. Convergence of the cost functional
to the minimum value of 89.59 is given in graphical form in Fig. 1. Temperature
profile of glass, limited to goal temperature , is shown in Fig. 2.
.
Convergence of Solutions of an Optimal Control Problem for SP1 289
3 SP1 Approximation
Furthermore, we consider one more diffusion model of complex heat transfer in
SP1 approximation. The process of propagation of heat transfer is investigated in the
same medium with same boundary and initial conditions. It is known that the
Rosseland approximation is a simplification of the SP1 approximation [1], [23].
Rosseland approximation is valid when the medium is optically thick.
The normalized evolution diffusion model describing radiative, conductive, and
convective heat transfer in a bounded region has the following form [24]
(12)
(13)
(14)
(15)
(16)
here, . Note the week form for (12), (14), (16) as follows
here denotes several test function in Sobolev space , .
Note the week form for (13), (15) as follows
here p_1 denotes several test function in Sobolev space H^1 (Q), analogically.
For problem (12)–(16) authors consider the same cost functional (4) and solve ana-
logical initial-boundary control problem
290 A. Sushchenko,T. Park, R. Pinnau, and O. Tse
w.r.t.
(17)
subject to system (12)–(16)
We then construct the Lagrange function
(18)
in a similar fashion, where and denote the weak form of (12),
(14), (16) and (13), (15) respectively.
For the minimization of (18), we solve the corresponding optimality system
(19)
We designed a similar algorithm for solving (19). The analysis of numerical experi-
ments for SP1 approximation for borosilicate glass is considered in [21].
4 Studying a convergence of solutions
In this section, we consider a numerical convergence of the optimal control prob-
lem in SP1 approximation to solution in Rosseland when
(22)
In the following, we denote as the optimal solution of (17) for SP 1 approxi-
mation with the cost functional and denotes the optimal solution of (5) for
Rosseland approximation with cost functional .
The numerical experiment for borosilicate glass is done. An exemplary case study
is a thin bar (5x5 cm) that has been cooled during 300 sec. On the Fig. 3, we present
the values of the cost functional which depends on for (17) and for (5). One
clearly observes that the functional values of (17) converge to the one of (5).
Thus, we have shown, numerically, that solutions of the optimal control problem
for the SP1 approximation converges to the solution of optimal control problem for
the Rosseland approximation for any initial temperature and medium when
Convergence of Solutions of an Optimal Control Problem for SP1 291
Fig. 3. Functions: – black curve; gray curve.
Conclusions
We studied the optimal control problems of complex heat transfer with diffusion
approximations. For a special cost functional that allows one to find the optimal tem-
perature of sources on the boundary and get target temperature in the medium, we
designed iterative algorithms. For some examples we showed a convergence of the
optimal costs in the SP1 approximations to optimal cost in the Rosseland approxima-
tion in the limit. Since the simplified Rosseland model is more applicable for
computing because it needs less computation for solving, such results could be used
in glass production.
Acknowledgment
The research was supported by the Ministry of Education and Science of the Rus-
sian Federation (Project 14.Y26.31.0003).
References
1. Modest, M.F.: Radiative Heat Transfer. Academic Pres (2003)
2. Kovtanyuk, A.E., Nefedev, K.V., Prokhorov, I.V.: Advanced computing method for solv-
ing of the polarized-radiation transfer equation. LNCS 6083, 268-276 (2010)
3. Prokhorov, I.V.: The Cauchy problem for the radiative transfer equation with generalized
conjugation conditions. Comput. Math. Math. Phys. 53 (5), 588–600 (2013)
4. Andre, S., Degiovanni, A.: A theoretical study of the transient coupled conduction and ra-
diation heat transfer in glass: phonic diffusivity measurements by the flash technique: Int.
J. Heat Mass Transfer 38 (18), 3401-3412 (1995)
292 A. Sushchenko,T. Park, R. Pinnau, and O. Tse
5. Kelley, C.T.: Existence and uniqueness of solutions of nonlinear systems of conductive-
radiative heat transfer equations. Transport Theory Statist. Phys. 25 (2), 249-260 (1996)
6. Banoczi, J.M., Kelley, C.T.: A fast multilevel algorithm for the solution of nonlinear sys-
tems of conductive-radiative heat transfer equations. SIAM J. Sci. Comput. 19 (1), 266-
279 (1998)
7. Kovtanyuk, A.E., Chebotarev, A.Yu.: An iterative method for solving a complex heat
transfer problem. Applied Math. Comput. 219, 9356-9362 (2013)
8. Kovtanyuk, A.E., Chebotarev, A.Yu., Botkin, N.D., Hoffmann, K.-H.: Solvability of P1
approximation of a conductive-radiative heat transfer problem. Applied Math. And
Comput. 249, 247-252 (2014)
9. Amosov, A.A.: On the solvability of a problem of radiative heat transfer. Soviet Physics
Doklady 24 (1), 261-262 (1979)
10. Amosov, A.A.: The limiting relation between two problems on radiative heat transfer. So-
viet Physics Doklady 24 (11), 439-441 (1979)
11. Kovtanyuk, A.E., Chebotarev, A.Yu., Botkin, N.D., Hoffmann, K.-H.:The unique solvabil-
ity of a complex 3D heat transfer problem. J. Math. Anal. Appl. 409, 808-815 (2014)
12. Kovtanyuk, A.E., Chebotarev, A.Yu.: Steady-state problem of complex heat transfer.
Comput. Math. and Math. Phys. 54 (4), 719-726 (2014)
13. Kovtanyuk, A.E., Chebotarev, A.Yu., Botkin, N.D., Hoffmann, K.-H.: Unique solvability
of a steady-state complex heat transfer model. Communications in Nonlinear Science and
Numerical Simulation 20, 776-784 (2015)
14. Pinnau, R.: Analysis of optimal boundary control for radiative heat transfer modeled by the
SPN system. Comm. Math. Sci., 5(4) (2007) 951-969.
15. 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)
16. Tse, O., Pinnau, R.: Optimal control of a simplified natural convection-radiation model.
Comm. Math. Sci. 11(3), 679-707 (2013)
17. 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)
18. 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. of Math.
Analysis and Applications 412, 520-528 (2014)
19. 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)
20. Pinnau, R., Thommes, G.: Model hierarchies and optimal control of radiative transfer.
Progress in industrial mathematics at ECMI 2008 15, 419-424 (2010)
21. Sushchenko, A.: Optimal control problem of complex heat transfer for SP1 approximation.
Key Engineering Materials 685, 99-103 (2016)
22. Frank, M., Klar, A., and Pinnau, R.: Optimal control of glass cooling using simplified P-n
theory. Transport theory and statistical physics 39 (2-4), 282-311 (2010)
23. Tse, O., Pinnau, R., Siedow, N.: Identification of temperature dependent parameters in a
simplified radiative heat transfer. Australian Journal of Basic and Applied Sciences 5(1),
7-14 (2011)
24. Grenkin, G. V., Chebotarev, A. Yu.: A nonstationary problem of complex heat transfer.
Computational Mathematics and Mathematical Physics 54 (11), 1737-1747 (2014)