The optimal control problems for SP1 and Rosseland approximations of evolution radiative heat transfer are considered. The problems are solved by weak form technique and Lagrange method. Numerical experiments for borosilicate glass are done. Numerical convergence of optimal control problem for SP1 approximation to Rosseland is studied.
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 powers of an integral operator applied to a certain start function. The terms can be calculated 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 coupled with the radiative transfer equation can be found in [4]–[8]. In particular, efficient 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 problems 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.
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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 solving problems using a hierarchy of approximate models.
The numerical analysis of optimal control problem of complex heat transfer for SP1 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 numerical convergence of optimal control problem in SP1 approximation to Rosseland. 2
Let, . The normalized evolution diffusion model describing radiative, conductive, and convective heat transfer in a bounden region has the following form: here, denotes the normalized temperature, sion, and radiation, respectively. The constant and and denote the coefficient of diffuare defined as follows: here, denotes the heat conductivity, the density, the heat capacity, the Stephan-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:
(
For problem (
The main subject on the analysis in this paper is the following initial boundary control problem:
w.r.t.
subject to system (
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 (
Denote the Lagrange function , . with . Here we only consider the case . It is worth to note that is typically called a Lagrange multiplier or adjoint variable.
For the minimization of (
The adjoint equation is defined as follows
By denoting
, the gradient of may be expressed as
(
For solving problem (
2. 3. REPEAT (a) Choose (b) FOR Find (c) (d) FOR (e) (f) REPEAT (i) FOR (ii) FOR Find (iii) (g) UNTIL (h)
DO
from (
DO
from (
using (
, 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.
Consider experiment results (Fig. 1-2). It is worth noting that the proposed algorithm 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.
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]
(
. Note the week form for (
denotes several test function in Sobolev space
Note the week form for (
For problem (
w.r.t.
subject to system (
For the minimization of (
In this section, we consider a numerical convergence of the optimal control
problem in SP1 approximation to solution in Rosseland when
In the following, we denote as the optimal solution of (
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 (
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 – black curve; gray curve.
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 temperature 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 approximation 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.
The research was supported by the Ministry of Education and Science of the Russian Federation (Project 14.Y26.31.0003).