Optimization Algorithm for Radiative-Conductive Heat Transfer Model with Boundary Conditions of Cauchy Type ? ??

Abstract

To solve the initial boundary-value problem for a quasi-static approximate model of radiative heat transfer, an optimization algorithm is proposed. The analysis of the optimal control problem is carried out, the optimality system is obtained. It is shown that the sequence of solutions of extremal problems converges to the solution of a problem with boundary conditions of Cauchy type for temperature. The results of theoretical analysis are illustrated by numerical examples.

Keywords

• Equation of radiative heat transfer
• Di usion approxima- tion
• Optimal control problem
• Cauchy type conditions

-

') = 0; ' + a(' j j 3) = 0; x 2

; 0 < t < T ; ( 1 ) ( 2 ) ( 3 )

Here, is the normalized temperature, ' is the normalized intensity of radiation averaged over all directions. The positive parameters a, b, a, and , describing medium properties, are determined by a standard way [18]. The function r(x; t); x 2 ; t 2 (0; T ) is given, and the unknown function u(x; t); x 2 ; t 2 (0; T ) is a control. By @n we denote the derivative in direction of the outward normal n.

Extremal problem is to nd a triple f ; ' ; u g such that

J ( ; u) =

T 1 Z Z 2 0 ( b)2d dt +

T

Z Z 2 0 u2d dt ! inf ( 4 ) on solutions of the problem ( 1 )-( 3 ). The function b(x; t); x 2 ; t 2 (0; T ) and the regularization parameter > 0 are given.

The optimal control problem ( 1 ){( 4 ) if r := a( b + qb), where qb is a given function on = (0; T ), is for small values of an approximation of the boundary-value problem for equation ( 1 ) for which the boundary conditions for the intensity of radiation ' are unknown. Instead them the boundary temperature and ow are set, j = qb: ( 5 )

Mathematical modeling of heat exchange accounting for the radiation e ects is used in various applications, e.g., electron microscopic diagnosis [22, 24], glass manufacturing [13, 14], and laser thermotherapy [20]. A detailed theoretical and numerical analysis of various formulations of boundary and inverse problems, as well as control problems for the equations of radiation heat transfer within the P1{approximation for the radiation transfer equation, is presented in [7{ 11, 16, 18, 19, 23]. Interesting boundary value problems associated with radiative heat transfer are studied in [2{6]. The nonlocal solvability of nonstationary and steady-state boundary-value problems for the equations of complex heat transfer without boundary conditions on the radiation intensity and with the conditions ( 5 ) for temperature is proved in [12].

The main results of the work are to obtain a priori estimates for the solution of the problem ( 1 ), ( 2 ), on the basis of which the solvability of the optimal control problem ( 1 ){( 4 ) is proved and an optimality system is derived. It is shown that the sequence f ; ' ; u g of solutions to the extremal problem ( 1 ){( 4 ) for ! +0 converges to the solution of the initial-boundary value problem ( 1 ), ( 5 ) with conditions of Cauchy type for temperature. An algorithm for solving the control problem is presented. 2

Formalization of the Control Problem In what follows, we assume that R3 is a bounded strictly Lipschitz domain whose boundary consists of a nite number of smooth pieces. By Ls, 1 s 1 we denote the Lebesgue space, and by Hs the Sobolev space W2s. Let H = L2( ); V = H1( ). By V 0 we denote the dual space of V , and by Ls(0; T ; X) the Lebesgue space of functions from Ls, de ned on (0; T ), with values in the space X. The space H is identi ed with the space H0 so that V H = H0 V 0. We denote by k k the standard norm in H, and by (f; v) the value of the functional f 2 V 0 at the element v 2 V , which coincides with the inner product in H if f 2 H.

By U we denote the space L2( ) with the norm kuk =

Z u2d dt 1=2 : We will also use the space W = fy 2 L2(0; T ; V ) : y0 2 L2(0; T; V 0)g, where y0 = dy=dt:

We will assume that (i) a; b; ; a; = Const > 0; (ii) b; qb 2 U; r = a( b + qb) 2 L5( ) 0 2 L5( ):

Let us de ne the operators A : V ! V 0, B : U ! V 0, using the following equalities, which are valid for any y; z 2 V , w 2 L2( ): (Ay; z) = (ry; rz) + yzd ; (Bw; z) =

wzd : Z

Z The bilinear form (Ay; z) de nes the inner product in the space V , and the corresponding norm kzkV = p(Az; z) is equivalent to the standard norm in V . Therefore, the continuous inverse operator is de ned A 1 : V 0 7! V: Note that for any v 2 V , w 2 L2( ), g 2 V 0 the following inequalities hold: kvk2

C0kvk2V ; kvkV 0

C0kvkV ; kBwkV 0 kwk ; kA 1gkV kgkV 0 : Here, the constant C0 > 0 depends only on the domain :

In what follows, we use the following notation [h]s := jhjssign h, s > 0, h 2 R for the monotone power function.

De nition. The pair 2 W; ' 2 L2(0; T ; V ) is called a weak solution of the problem ( 1 )-( 3 ) if Problem (OC). Find the triple f ; '; ug 2 W L2(0; T ; V )

U such that

J ( ; u)

kuk2 ! inf; F ( ; '; u) = 0:

Solvability of the Problem (OC) Let us rst prove the unique solvability of the problem ( 1 )-( 3 ).

Lemma 1. Let conditions (i), (ii), u 2 U hold. Then there is a unique weak solution to the problem ( 1 ){( 3 ) and, moreover,

= [ ]5=2 2 L1(0; T ; H) \ L2(0; T ; V ); [ ]4 2 L2(0; T ; H): Proof. Let us express ' from the last equation of ( 6 ) and substitute it into the rst one. As a result, we obtain the following Cauchy problem for an equation with operator coe cients:

L = ( 7 ) ( 8 ) ( 9 ) 0 + aA + L[ ]4 = Br + f; Let us obtain a priori estimates for the solution of the problem ( 7 ), on the basis of which the solvability of this problem is derived in a standard way. Let [ ; ] = (( I + aA 1) ; ), 2 V 0; 2 V: Note that the expression [[ ]] = p[ ; ] de nes the norm in H equivalent to the standard one.

Multiplying, in the sense of the inner product of H, the equation in ( 7 ) by ( I + aA 1) , we obtain 1 d 5 2 dt [[ ]]2 + a (A ; ) + a ak k2 + b ak kL5( ) = [Br; ] + [f; ]: The equality ( 8 ) implies the estimate k kL1(0;T ;H) + k kL2(0;T ;V ) + k kL5(Q) C1; where C1 depends only on a; b; ; a, kf kL2(0;T ;H); k 0k; krkL2( ): Further, let = [ ]5=2: Note that ( 0; [ ]4) = 15 ddt k k2;

(A ; [ ]4) = 1265 kr k2 + k k2L2( ): Multiplying, in the sense of the inner product of H, the equation in ( 7 ) by [ ]4 = [ ]8=5, we obtain The equality ( 10 ) implies the estimate k kL1(0;T ;H) + k kL2(0;T ;V ) + k[ ]4kL2(0;T ;H) C2; ( 11 ) where C2 depends only on a; b; ; a, kf kL2(0;T ;H); k 0kL5( ); krkL5( ): Further, let us estimate k 0kL2(0;T ;V 0) taking into account 0 = Br + f aA L[ ]4: Due to the conditions on the initial data, it is true that Br; f 2 L2(0; T ; V 0): Since 2 L2(0; T ; V ), then A 2 L2(0; T ; V 0): Let = L[ ]4: Then + aA 1 = Multiplying, in the sense of the inner product of H, the last equality by , we obtain k k2 + a(A 1 ; ) = b a([ ]4; ) (k k2 + Therefore, k kV 0 = (A 1 ; )

2 ( 11 ), we obtain

4ab2 k[ ]4k2 and, by virtue of the estimates ( 9 ), k 0kL2(0;T ;V 0) kBr + f kL2(0;T ;V 0) + aC1 + p abC2: ( 12 ) The estimates ( 9 ){( 12 ) are su cient to prove the solvability of the problem.

Let 1;2 be solutions of the problem ( 7 ), = 1 2: Then Multiplying, in the sense of the inner product of H, the last equation by ( I + aA 1) , we obtain 1 d 2 dt [[ ]]2 + a (A ; ) + a ak k2 + b a([ 1]4 The last term on the left-hand side is non-negative and therefore, integrating the resulting equality over time, we derive = 1 2 = 0; which means the uniqueness of the solution. The lemma is proved.

Theorem 1. Let conditions (i), (ii) hold. Then there is a solution of the problem (OC): Proof. Let j = inf J on the set u 2 U , F ( ; '; u) = 0: We choose a minimizing sequence um 2 U; m 2 W; 'm 2 L2(0; T ; V ), J ( m; um) ! j ; m0 + aA m + b a([ m]4 'm) = Br; m(0) = 0; A'm + a('m [ m]4) = Bum: ( 13 ) The boundedness of the sequence um in the space U implies, by Lemma 1, the estimates k mkL2(0;T ;V )

C; k mkL1(0;T ;L5( ))

C; k m0kL2(0;T ;V 0)

C; T Z Z

C:

Here, C > 0 denotes the largest of the constants limiting the corresponding norms and not depending on m. Passing, if necessary, to subsequences, we conclude that there is a triple fub; b; 'bg 2 U W L2(0; T ; V );

um ! ub weakly in U; m ! b weakly in L2(0; T ; V ); strongly in L2(Q);

'm ! 'b weakly in L2(0; T ; V ): Moreover, b 2 L8(Q) \ L1(0; T ; L5( )):

Convergence results allow us to pass to the limit in ( 13 ). In this case, the passage to the limit in the nonlinear terms follows from the following inequality, which is valid for 2 C1(Q):

T Z 0

Therefore,

j([ m]4 [b]4; )jdt

5=3 4=3 5=3 4=3 2 max j j k mkL5( )k mkL8( ) + kbkL5( )kbkL8( ) k m

Q bkL2(Q): b0 + aAb + b a([b]4 ') = Br; b b(0) = 0;

A' + a('b b [b]4) = Bu; b and wherein j J (b; ub) limJ ( m; um) = j . Thus, the triple fb; 'b; ubg is a solution of the problem (OC): 4

Optimality Conditions To obtain an optimality system, it is su cient to use the Lagrange principle for smooth-convex extremal problems [15, 17]. Let us check the validity of the key condition that the image of the derivative of the constraint operator Fy0 (y; u), where y = f ; 'g 2 W L2(0; T ; V ), coincides with space L2(0; T ; V 0) L2(0; T ; V 0) H: It is this condition that guarantees the nondegeneracy of the optimality conditions. Recall that Lemma 2. Let the conditions (i),(ii) hold. If yb 2 W L2(0; T ; V ); ub 2 U is a solution of the problem (OC), then the following equality is valid: ImFy0 (yb; ub) = L2(0; T ; V 0)

L2(0; T ; V 0)

H: Proof. It is enough to check that the problem 0 + aA + b a(4jbj3 ) = f1; is solvable for all f1;2 2 L2(0; T ; V 0); 0 2 H: Let us express from the last equation and substitute it into the rst one. As a result, we get the following problem: 0 + aA + 4L(jbj3 ) = f1 + b a( A + aI) 1f2; (0) = 0: ( 14 ) The unique solvability of the linear problem ( 14 ) is proved similarly to Lemma 1. According to Lemma 2, the Lagrangian of the problem (OC) has the form L( ; '; u; p1; p2; q) = J ( ; u) + ( 0 + aA + b a([ ]4 ')

Br; p1)dt T Z 0 [ ]4) T Z 0 + ( A' + a('

Bu; p2)dt + (q; (0) 0): Here, p = fp1; p2g 2 L2(0; T ; V ) L2(0; T ; V ) Is the conjugate state, q 2 H is the Lagrange multiplier for the initial condition. If fb; 'b; ubg is a solution of the problem (OC), then by virtue of the Lagrange principle [15, Ch. 2, Theorem 1.5] the variational equalities hold 8 2 L2(0; T ; V ); v 2 U b); ) + ( 0 + aA + 4b ajbj3 ; p1) a(4jbj3 ; p2) dt+(q; (0)) = 0; T Z 0 (B(b T Z T Z ( (u; v) b (( A + a ; p2) b a( ; p1)) dt = 0; (Bv; p2)) dt = 0: 0 0 Thus, from the obtained conditions, we derive the following result. Theorem 2. Let the conditions (i),(ii) hold. If fb; 'b; ubg is a solution of the problem (OC), then there is a unique pair fp1; p2g 2 W W such that p01 + aAp1 + 4jbj3 a(bp1 p2) = B( b b); p1(T ) = 0; and wherein u = p2j : b 5

Approximation of a Problem Conditions of Cauchy Type Ap2 + a(p2

bp1) = 0 ( 15 ) with Boundary Let us consider an initial-boundary value problem for the equations of complex heat transfer, in which there are no boundary conditions on the radiation intensity: a + b a([ ]4

= b; = qb on ;

Theorem 3. Let the conditions (i),(ii) hold and there is a solution ; ' 2 L2(0; T ; H2( )) of the problem ( 16 ), ( 17 ). If f ; ' ; u g is a solution of the problem (OC) for > 0, then as ! +0 ! weakly in L2(0; T ; V ); strongly in L2(Q); '

! ' weakly in L2(0; T ; V ): Proof. Let ; ' 2 L2(0; T ; H2( )) be a solution of the problem ( 16 ), ( 17 ), u = (@n' + ') 2 U: Then where r := a( b + qb): Therefore, taking into account that j = b, J ( ; u ) = Therefore, one can choose a sequence ! +0 such that u ! u

weakly in U; ! weakly in L2(0; T ; V ); strongly in L2(Q); ' ! '

weakly in L2(0; T ; V ): The obtained results on convergence allow, as in Theorem 1, to pass to the limit as ! +0 in the equations for ; ' ; u and then Wherein j = b: From the rst equation in ( 18 ), taking into account that r = a( b + qb), we derive a = b; = qb a:e: in From the second equation in ( 18 ) it follows that ' + a(' [ ]4) = 0 almost everywhere in Q: Thus, the pair ; ' is a solution of the problem ( 16 ), ( 17 ). Since the solution to this problem is unique [12], then = ; ' = ': 6

Numerical Algorithm and Examples Let us present an algorithm for solving the control problem. Let

Je (u) = J ( (u); u); where (u) is the component of solution to the problem ( 1 ){( 2 ) corresponding to the control u 2 U . According to ( 15 ), the gradient of the functional Je (u) is de ned as follows: Je0 (u) = u p2. Here, p2 is the corresponding component of the conjugate state of the system ( 15 ), where b := (u).

The proposed algorithm for solving the problem (OC) is as follows: Algorithm 1 Gradient descent algorithm 1: Choosing the value of the gradient step ", 2: Choosing the number of iterations N , 3: Choosing an initial approximation for the control u0 2 U , 4: for k 0; 1; 2; : : : ; N do : 5: For a given uk, calculate the state yk = f k; 'kg, a solution of the problem ( 1 )-( 3 ). 6: We calculate the value of the quality functional J ( k; uk). 7: From equations ( 15 ), we calculate the conjugate state pk = fp1k; p2kg, where 8: b := k; ub := uk.

We We recalculate the control uk+1 = uk "( uk p2)

The parameter " is chosen empirically so that the value of "( uk p2) is a signi cant correction for uk+1. The number of iterations N is chosen su cient to satisfy the condition J ( k; uk) J ( k+1; uk+1) < , where > 0 determines the accuracy of the calculations.

The example considered below illustrates the performance of the proposed algorithm for small, which is important, values of the regularization parameter 10 12: Note that for the numerical solution of a direct problem with a given control, the simple iteration method was used to linearize the problem and solve it by the nite element method. Solving a conjugate system that is linear at a given temperature is straightforward. For numerical simulation, we used the solver FEniCS [1, 21].

Let us compare the work of the proposed algorithm with the results of the article [12]. The problem is considered in the domain ( L; L), where = fx = (x1; x2) : 0 < x1;2 < dg and for large L reduces to a two-dimensional problem for the computational domain . The following values of the problem parameters were chosen: d = 1(m), a = 0:92 10 4 (m2=s), b = 0:19 (m=s), = 0:0333 (m) a = 1 (m 1). The parameters correspond to air at normal atmospheric pressure and temperature 400 C. The function b, qb for the boundary condition ( 5 ) are set as follows: b = bj , qb = @n bj , where b = (x1 0:5)2 0:5x2 + 0:75.

An approximate solution to the problem ( 16 ), ( 17 ) with Cauchy data, presented in [12] (see Fig. 1), was obtained by solving a fourth-order parabolic problem for temperature.

0.4 0.6 0.8 1

The solution stabilized after 120 seconds, but the calculations at each time step were quite expensive [12]. Fig. 2 shows the steady-state temperature eld obtained by the method proposed in the current article.

The presented example illustrates that the proposed algorithm successfully nds a numerical solution to the problem ( 16 ), ( 17 ) with the boundary conditions of the Cauchy type.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.9 0.84 0.2 0.4 0.55 0.65 0.7 0.74 0.79 0.79 0.84 0.9 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

References

1. Alnaes , M.S. , Blechta , J. , Hake , J. , Johansson , A. , Kehlet , B. , Logg , A. , Richardson , C. , Ring , J. , Rognes , M.E. , Wells , G.N. : The FEniCS project version 1.5. Archive of Numerical Software 3 ( 100 ), 9 { 23 ( 2015 )

2. Amosov , A.A. : Stationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency . J. Math. Sci . 164 ( 3 ), 309 { 344 ( 2010 )

3. Amosov , A.A. : Unique solvability of a nonstationary problem of radiativeconductive heat exchange in a system of semitransparent bodies . Russian J. of Math. Phys . 23 ( 3 ), 309 { 334 ( 2016 )

4. Amosov , A.A. : Unique solvability of stationary radiative-conductive heat transfer problem in a system of semitransparent bodies . J. Math. Sci . 224 ( 5 ), 618 { 646 ( 2017 )

5. Amosov , A.A. : Asymptotic behavior of a solution to the radiative transfer equation in a multilayered medium with di use re ection and refraction conditions . J. Math. Sci. 244 , 541 { 575 ( 2020 )

6. Amosov , A.A. , Krymov , N.E. : On a nonstandard boundary value problem arising in homogenization of complex heat transfer problems . J. Math. Sci. 244 , 357 { 377 ( 2020 )

7. Chebotarev , A.Yu. , Kovtanyuk , A.E. , Grenkin , G.V. , Botkin , N.D. , Ho mann, K.-H.: Nondegeneracy of optimality conditions in control problems for a radiative-conductive heat transfer model . Appl. Math. Comput . 289 , 371 { 380 ( 2016 )

8. Chebotarev , A.Yu. , Grenkin , G.V. , Kovtanyuk , A.E. : Inhomogeneous steady-state problem of complex heat transfer . ESAIM Math. Model. Numer. Anal . 51 ( 6 ), 2511 { 2519 ( 2017 )

9. Chebotarev , A.Y. , Grenkin , G.V. , Kovtanyuk , A.E. , Botkin , N.D. , Ho mann, K.-H.: Di usion approximation of the radiative-conductive heat transfer model with Fresnel matching conditions . Comm. Nonlin. Sci. Num . Simulat. 57 , 290 { 298 ( 2018 )

10. Chebotarev , A.Yu. , Grenkin , G.V. , Kovtanyuk , A.E. , Botkin , N.D. , Ho mann, K.-H.: Inverse problem with nite overdetermination for steadystate equations of radiative heat exchange . J. Math. Anal. Appl . 460 ( 2 ), 737 { 744 ( 2018 )

11. Chebotarev , A.Yu. , Pinnau , R.: An inverse problem for a quasi-static approximate model of radiative heat transfer . J. Math. Anal. Appl . 472 ( 1 ), 314 { 327 ( 2019 )

12. Chebotarev , A.Y. , Kovtanyuk , A.E. , Botkin , N.D. : Problem of radiation heat exchange with boundary conditions of the Cauchy type . Comm. Nonlin. Sci. Num . Simulat. 75 , 262 { 269 ( 2019 )

13. Clever , D. , Lang , J.: Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient . Optimal Control Appl. Meth . 33 ( 2 ), 157 { 175 ( 2012 )

14. Frank , M. , Klar , A. , Pinnau , R.: Optimal control of glass cooling using simpli ed PN theory . Trans. Theory Stat. Phys . 39 ( 2 {4), 282 { 311 ( 2010 )

15. Fursikov , A.V. : Optimal control of distributed systems . Theory and applications . American Math. Soc. , Providence, Rhode Island ( 2000 )

16. Grenkin , G.V. , Chebotarev , A.Yu. , Kovtanyuk , A.E. , Botkin , N.D. , Ho mann, K.-H.: Boundary optimal control problem of complex heat transfer model . J. Math. Anal. Appl . 433 ( 2 ), 1243 { 1260 ( 2016 )

17. Io e, A.D., Tikhomirov , V.M. : Theory of extremal problems . North-Holland, Amsterdam ( 1979 )

18. Kovtanyuk , A.E. , Chebotarev , A.Yu. , Botkin , N.D. , Ho mann, K.-H.: Unique solvability of a steady-state complex heat transfer model . Commun. Nonlinear Sci. Numer . Simul. 20 ( 3 ), 776 { 784 ( 2015 )

19. Kovtanyuk , A.E. , Chebotarev , A.Yu. , Botkin , N.D. , Ho mann, K.-H.: Optimal boundary control of a steady-state heat transfer model accounting for radiative e ects . J. Math. Anal. Appl . 439 ( 2 ), 678 { 689 ( 2016 )

20. Kovtanyuk , A.E. , Chebotarev , A.Yu. , Astrakhantseva , A.A. , Sushchenko , A.A. : Optimal control of endovenous laser ablation . Optics and Spectroscopy 128 ( 9 ), 1508 { 1516 ( 2020 )

21. Logg , A. , Wells , G.N. : DOLFIN: Automated Finite Element Computing. ACM Trans. Math. Software . 37 ( 2 ), 20 ( 2010 )

22. Maslovskaya , A. , Pavelchuk , A. : Simulation of heat conductivity and charging processes in polar dielectrics induced by electron beam exposure . IOP Conf. Series: Materials Sci. Eng . 81 ( 1 ), 012119 ( 2015 )

23. Pinnau , R. : Analysis of optimal boundary control for radiative heat transfer modeled by SP1-system . Commun. Math. Sci. 5 ( 4 ), 951 { 969 ( 2007 )

24. Sogr , A.A. , Maslovskaya , A.G. : Estimation of SEM imaging resolution of the domain structure of pyroprobe-heated ferroelectrics . Tech. Phys . 49 ( 11 ), 1505 { 1508 ( 2004 )