The problem of determining the thermal conductivity coefficient that depends on temperature is studied. The consideration is based on the Dirichlet boundary value problem for the two-dimensional unsteadystate heat equation. The inverse coefficient problem is reduced to a variation problem. The mean-root-square deviations of the temperature field from the experimental data is used as the objective functional. An algorithm for the numerical solution of the inverse coefficient problem is proposed. It is based on the modern approach of Fast Automatic Differentiation technique, which made it possible to solve a number of difficult optimal control problems for dynamic systems. An expression for the gradient of the objective functional is obtained for the discrete optimal control problem. The examples of solving the inverse coefficient problem confirm the efficiency of the proposed algorithm.
Introduction
convective and radiative heat transfer must be taken into account. The thermal conductivity coefficients in this
case typically depend on the temperature. To estimate these coefficients, various models of the medium are
used. As a result, one has to deal with a complex nonlinear model that describes the heat propagation in the
composite material
In [Zubov, 2016] the inverse coefficient problems are studied for the one-dimensional unsteady-state heat equation. There was considered the case of continuous thermal conductivity coefficient. In [Albu et al., 2017] the case of a discontinuous thermal conductivity coefficient is investigated.
In this paper we consider the problem studied in [Zubov, 2016] for the two-dimensional unsteady-state heat equation. The inverse coefficient problem is reduced to a variation problem. The mean-root-square deviation of the temperature field is used as the objective functional. An algorithm for the numerical solution of the inverse coefficient problem is proposed. It is based on the using of Fast Automatic Differentiation technique. 1
Formulation of the Problem A layer of material of length L and width R is considered. The temperature of this layer at the initial time is given. It is also known how the temperature on the boundary of this layer changes in time. The distribution of the temperature field at each instant of time is described by the following initial boundary value (mixed) problem:
C T (x; y; 0)
Here (x; y) are the Cartesian coordinates of the point in the layer; t is the time; T (x; y; t) is the temperature of the material at the point with the coordinates (x; y) at the time t; and C are the density and the heat capacity of the material, respectively; K(T ) is the thermal conductivity coefficient; w0(x; y) is the given temperature of the layer at the initial time; w1(y; t) and w2(y; t) are the given temperatures for x = 0 and for x = L respectively; w3(x; t) and w4(x; t) are the given temperatures for y = 0 and for y = R respectively.
If the dependence of the thermal conductivity coefficient K(T ) on the temperature T is known, then we can solve the mixed problem (1)–(6) to find the temperature distribution T (x; y; t) in Q [0; Θ]. Below, problem (1)–(6) will be called the direct problem.
If the dependence of the thermal conductivity coefficient on the temperature is not known, it is of interest to determine this dependence. A possible statement of this problem (it is classified as a model’s parameters identification problem) is as follows: find the dependence K(T ) on T under which the temperature field T (x; y; t) obtained by solving problem (1)–(6) is close to the field Y (x; y; t) obtained experimentally. The quantity
L R ∫ ∫ ∫ 0 0 0 Φ(K(T )) = [T (x; y; t)
Y (x; y; t)]2 (x; y; t)dx dy dt may be used as the measure of difference between these functions. Here (x; y; t) 0 is a given weighting function.
Thus, the optimal control problem is to find the optimal control K(T ) and the corresponding optimal solution T (x; y; t) of problem (1)–(6) that minimizes functional (7). 2
Finding the Gradient of Functional
The optimal control problem formulated above was solved numerically. The objective functional was minimized
using the gradient method. It is well known that it is very important for the gradient methods to determine
accurate values of the gradients. For this reason, in this paper we used the efficient Fast Automatic Differentiation
technique
One of the main elements of the proposed numerical method is the solution of the mixed problem (1)–(6). To [0; Θ] is decomposed by the grid lines fxngn=0, fyigi=0, and {tj }J
N I j=0 into this end, the domain [0; L] [0; R] parallelepipeds. At each node (xn; yi; tj ) characterized by the indices (n; i; j), all the functions are determined by their values at the point (xn; yi; tj ) (e.g., T (xn; yi; tj ) = Tnji). In each parallelepiped, the thermal balance must be preserved. As a result, using two-layer implicit scheme with weights we obtain the following finite difference scheme that approximates the mixed problem (1)–(6):
Sni
j A(T j+1) + (1 ) j A(T j ); The system of nonlinear algebraic equations (8)–(13) was solved iteratively using the method: s Tnji+1 =
Sni j
C
A ( s−1 )
T j+1 + (1 Sni ) j C
A (T j )+ Tni; j
0 where Tnji+1 = Tni:
j so that the nodes at the points {(Tem; km) This approach was used in the work to solve the mixed problem (1)–(6), and the function T (x; y; t) (more precisely, its approximation Tnji) was found.
The temperature interval [a; b] (the interval of interest) is partitioned by the points Te0 = a, Te1; Te2; : : : ; TeM = b into M parts (they can be equal or of different lengths). Each point Tem (m = 0; : : : ; M ) is assigned a number km = K(Tem). The function K(T ) to be found is approximated by a continuous piecewise linear function with }M K(T ) = km−1 +
Tem−1) for Tem−1
T
Tem; (m = 1; : : : ; M ): The objective functional (7) was approximated by a function F (k0; k1; : : : ; kM ) of the finite number of variables as Φ(K(T ))
J I−1 N∑−1 ( F = ∑ ∑ (Tnji
j 2 Yni)
jnihxnhiy j ): km Tem m=0 km−1 (T Tem−1 According to [Evtushenko, 1998], approximation (8)-(13) of the mixed problem (1)-(6) is reduced to the j j (K(Tnj+1;i) + K(Tnji)) (Tnj+1;i Tni + ani
Tni j ) a˜ni j (
K(Tnji) + K(Tn−1;i) j ) ( j
Tni K(Tnji) + K(Tn;i+1) j ) ( j
Tn;i+1
Tni j ) ˜bjni (K(Tnji) + K(Tnj;i−1) ) ( j
Tni The Fast Automatic Differentiation technique makes it possible to formally write relations determining the j 1, i = 0; I 1, j = 1; J ): adjoint problem of (8)-(13) for the conjugate variables pi , (n = 1; N
j pni pjn+i1 + fani j
j Xni
j a˜ni
j j Yni + bni
j Uni ˜j bni
j Vnig pjn+i1 + an−1;i j
j Yni pj+1 n−1;i j a˜n+1;i fcjn−i1 c˜jn−+11;i
Xni pjn++11;i + bn;i−1 j j
Vni pjn+;i1−1
j j Xni
j Xni c˜jn−i1
Yni + djn−i1 j
j Uni
d˜jn−i1 j j pn+1;i + dn;i−1
j j Vni pn;i−1 ˜j bn;i+1
j
Vnig d˜jn−;i1+1
Unji pjn+;i1+1 + pni + cj−1 j
n−1;i j Uni
j pn;i+1 +
Ynji pjn−1;i j Xni
j Yni
j Uni
j
Vni
= = = = =
K′(Tnji) (Tn+1;i j
j K′(Tnji) (Tn;i+1 j
j
Tn−1;i) + K(Tnji) + K(Tn−1;i); j j
j K(Tn+1;i);
j
K(Tn;i+1); function are calculated by the formula: canonical form ∑J ∑I ∑N ∑J ∑I−1 N∑−1
g=0 r=0 l=0
j pni
g @K(Tlr) ; g @K(Tlr) that appear in (14) are calculated by the formulas m = 0; M : (14) j Tni Tem+1
Tem Tem ; if Tem
j Tni
Tem+1; otherwise. = j Tni
Tem+1 0;
Tem men ; if Tem
j Tni
Tem+1; otherwise.
The value of the gradient of the objective function F (k0; k1; : : : ; kM ) calculated by formula (14) is exact for the chosen approximation of the optimal control problem.
The function F (k0; k1; : : : ; kM ) was minimized numerically using the gradient method. 3
Numerical Results To illustrate the efficiency of the proposed algorithm the following variation problem was considered: x2 + 7xy
K(T ) y2; x2; + 1;
(0 (t 0; x 0 2; y 0
y 1);
1); (t 0; 0 x 2): Here Q = f(0 x 2) (0 y 1)g.
Note that the inverse problem with above-indicated data has an analytical solution; indeed, the function Y (x; y; t) = x2 + 7xy y2 + exp( t) sin ( x) sin (2 y) is the solution of the mixed problem (1)-(6) with the
In more details, the process of finding the thermal conductivity coefficient is as follows. If the experimental temperature field is determined by the solution of the direct problem with the given K(T ) = 174 2 , i.e., Ynji = Tnji, then the objective functional changes from 1:53445 10−2 to 3:47387 10−30, and the maximum of the gradient’s absolute value decreases from 9:40155 10−1 to 1:71804 10−15, while the coefficient of thermal conductivity 4 coincides with K(T ) = 17 2 accurate to the machine precision.
Acknowledgements This work was supported by the Russian Foundation for Basic Research (project no. 17-07-00493 a), by the Program “Leading Scientific Schools”, no. NSh-8860.2016.1 and by the Program I.33 of the Presidium of RAS “Mathematical models and tools to study the economic and physical processes using high-performance computing”.