ISSN 1028-3358, Doklady Physics, 2017, Vol. 62, No. 10, pp. 465–469. © Pleiades Publishing, Ltd., 2017. Original Russian Text © G.V. Alekseev, V.A. Levin, D.A. Tereshko, 2017, published in Doklady Akademii Nauk, 2017, Vol. 476, No. 5, pp. 512–517.
MECHANICS
The Optimization Method in Design Problems of Spherical Layered Thermal Shells G. V. Alekseeva,b*, Academician V. A. Levinb,c, and D. A. Tereshkoa,b Received May 11, 2017
Abstract—The inverse problem of designing multilayered spherical shells, intended for thermal cloaking a spherical body or concentrating heat in it, has been analyzed. The stationary heat conduction equation for an anisotropic medium is applied as an original mathematical model. The optimization method is used to reduce this inverse problem to an extreme problem, where the role of controls is played by the thermal conductivities of shell layers. A numerical algorithm for solving the problem is proposed, and the results of computational experiments are discussed. DOI: 10.1134/S1028335817100044
In recent years, significant attention has been paid to a new line of research in heat and mass transfer, which is related to the design of special functional devices (cloaking shells, heat concentrators or converters) for proper control of heat flows [1–4]. With some simplifying suggestions, the problems of designing functional thermal devices have been considered in some studies (see, e.g., [2, 4] and references therein), in which a method for solving them based on analysis of the exact solutions (obtained by the Fourier method) of the corresponding direct heat transfer problem was proposed. However, this method is applicable only when rigid simplifying assumptions, providing for the construction of an approximate or exact solution of the direct heat transfer problem in an explicit form, are valid. The purpose of this study was to develop and analyze an efficient numerical algorithm for solving axisymmetric problems of designing thermal cloaking shells and spherical heat concentrators, based on fairly general assumptions concerning the initial data. When studying the aforementioned problems, we will proceed from the fact that, according to their statement, they belong to inverse ones. Therefore, it is quite natural to solve them using the optimization method. This approach, starting with Tikhonov’s fundamental studies [5], is widely used in analysis of the inverse heat and mass transfer problem (see, e.g., [6–8]). As a a Institute
of Applied Mathematics, Far East Branch, Russian Academy of Sciences, Vladivostok, Russia b Far East State University, Vladivostok, Russia c Institute of Automation and Control Processes, Far East Branch, Russian Academy of Sciences, Vladivostok, Russia *e-mail:
[email protected]
procedure of numerical optimization, we will apply the method of particle swarm optimization according to the scheme proposed (and briefly described) in [9] for solving the 2D problem of thermal cloaking of a cylindrical body. Recently, particle swarm optimization, which was introduced in 1995 [10], has been used to solve applied inverse and extreme problems for various models in mechanics and physics [11]. An important field of application is related to the optimization of the shape of aircraft (see, e.g., [12]). Particle swarm optimization was used to reconstruct the thermal conductivity and shape of inclusions in a solid in [13] and to determine the temperature-dependent heat capacity of a body in [14]. Let us start with the statement of the general heat dissipation problem, considered in a three-dimensional cylindrical domain D = {x ≡ (x, y, z): –z0 < z < z0,
x 2 + y 2 < c 2} , with specified numbers z 0 > 0 and c > 0 (see Fig. 1). Let an external thermal field T e be formed by two horizontal bases z = ± z 0 heated to different temperatures, T1 and T2 , and the lateral surface Γ of cylinder D be thermally insulated. We assume that there is a material shell (Ω, κ) inside D. Here, Ω is a spherical layer (a < | x| < b , b < c ) and κ is the thermal conductivity tensor of the inhomogeneous anisotropic medium filling the domain Ω . We assume also that the interior Ω i : | x| < a and exterior Ω e : | x| > b of Ω are filled with homogeneous media having (generally different) constant thermal conductivities: k i > 0 and ke > 0. In this case, the direct heat dissipation problem implies determination of a triad of functions: Ti in Ω i , T in Ω , and Te in Ω e , which satisfy the equations
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dition 2k r k θ = k 02 + k 0k r is satisfied, problem (1)–(3) has an exact solution independent of ϕ:
T = T2 z
T0 z0 T T (r , θ) = 0 z0
Ti (r , θ) =
∇T ⋅ n = 0
Ω
Ωe
Ωi
y
k i Δ Ti = 0 in Ω i , div(κ grad T ) = 0 in Ω, k e Δ Te = 0 in Ω e , and the boundary conditions
(1)
T | z = z 0 = T2 ,
∂ Ti = (κ∇ T ) ⋅ n| Γ i , ∂n (3) ∂T Te = T | Γ e , k e e = (κ∇ T ) ⋅ n| Γ e . ∂n In spite of the different geometries (cylindrical and spherical) of the above-introduced domains D and Ω ⊂ D , their common important property is that they are both axisymmetric. This property will be essentially used below, along with the following assumptions: (i) tensor κ is diagonal in spherical coordinates r , θ, ϕ, and its diagonal components (radial, polar, and azimuthal thermal conductivities), k r , k θ , and kϕ, are independent of ϕ and satisfy the assumptions ki
k θ ∈ L∞ (Ω),
k r > k r0 = const > 0,
k θ > k θ0 = cons t > 0,
k ϕ = k θ.
Ωi,
in
Ω,
(5)
T0 T + T2 r cos θ + 1 2 z0 T + T2 = T0 z + 1 in D, z0 2
T e (z) =
∂T = 0. (2) ∂n Γ Also fulfilled are the matching conditions at boundaries Γ i and Γ e of layer Ω , which have the form [4]
a nd
in
The parameter l = k θ / k r characterizing the degree of anisotropy of the shell (Ω, k r , k θ ) was introduced in (5). In the particular case where l = 1 (provided that k r = k θ = k 0 , so that the entire medium filling the domain D is homogeneous and isotropic), formula (5) is transformed into the unified formula
Fig. 1. Spherical shell Ω placed in cylinder D.
kr
T1 + T2 2 q −1 T + T2 r cos θ + 1 2 r cos θ +
T0 T + T2 in Ω e , r cos θ + 1 2 z0 k T − T1 . q ≡ − 1 + 1 + 8l , l = θ , T0 = 2 kr 2 2
T = T1
Ti = T | Γ i ,
q −1
Te (r , θ) =
x
T | z =− z0 = T1,
() (br ) a b
(4)
The condition k ϕ = k θ in (4) is widely used when analyzing three-dimensional problems of designing spherical thermal functional devices. Moreover, in the particular case where the parameters k r , k θ , T1, and T2 are constant (and k i = k e = k 0 = const ) and the con-
(6)
which describes the external applied field T e (z), used to detect objects located in domain D. It follows from (5) and (6) that
|∇T | = e
T0 , z0
| ∇ Ti | =
T0 _ z0
(7) q −1 a in Ω i , _(l ) ≡ . b Then we introduce the quantity }(l ) = 1 − _(l ) , which was referred to as the cloaking efficiency of shell (Ω, k r , k θ ) at l > 1 in [4]. Obviously, 0 ≤ }(l ) ≤ 1 at l ≥ 1; and q(l ) → 1, _ → 1, } → 0 at l → 1; and q(l ) → ∞ , _ → 0 , and } → 1 at l → ∞ . Thus, the cloaking efficiency of the homogeneous anisotropic shell (Ω, kr, k θ ) considered here, described by formulas (5), increases with an increase in the anisotropy parameter l = k θ / k r . In the limit at l → ∞ , we obtain an exact cloaking shell with maximum cloaking efficiency } = 1. However, in the limit at l → 1, we have a thermal shell (Ω, k r , k θ ) with } = 0. Let us now consider the case l < 1. Since q(l ) < 1 and a < b, _(l ) > 1. This means that the shell (Ω, k r , k θ ) acts as a heat concentrator at l < 1. The quantity _ = (b/ a)1−q , which has the meaning of the concentration efficiency of the shell (Ω, k r , k θ ) at l < 1 [4], increases with an increase in the ratio b/a and a decrease in the parameter l = k θ / k r ; note that _(l ) → b/ a at l → 0 . Let us now complicate the structure of the shell under consideration, assuming that (Ω, k r , k θ ) is a layDOKLADY PHYSICS
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ered shell consisting of M concentric spherical layers Ω j , j = 1, 2, … , M . Each of these layers is filled with a homogeneous and (generally) anisotropic medium, described by constant thermal conductivities k rj > 0 and k θj > 0 , j = 1, 2, … , M . The parameters k r , k θ , and kϕ of this shell are given by the formulas M
k r (x) =
∑k j =1
M
rj χ j (x),
k θ (x) =
k ϕ(x) = k θ(x).
∑k
θ j χ j (x),
j =1
(8)
Here, χ j (x) is a characteristic function of layer Ω j , which is equal to 1 inside of Ω j and 0 outside of Ω j . We should emphasize that, in the case of a layered shell, the inverse problems considered here (thermal cloaking and heat concentration) become finitedimensional, because they are reduced to determination of unknown coefficients k rj and k θj (entering formula (8)), which form a 2M-dimensional vector k ≡ (k r1, k θ1, … , k rM , k θM ) . In turn, using the optimization method, one can reduce these problems to the minimization of a certain quality functional J. To find the explicit form of functional J for each problem, we denote the solution to the direct problem, corresponding to parameters (8) in Ω and coefficients k i in Ω i and k e in Ω e , as T (k) ≡ T (k r1, k θ1, … , k rM , k θM ) . It is assumed below that the vector k ≡ (k r1, k θ1, ..., k rM , k θM ) belongs to the bounded set
U = { k = (k r1, k θ1, … , k rM , k θM ): m j ≤ k rj , k θj ≤ M j , j = 1, 2, … , M }
(9)
for specified positive constants m j , M j , j = 1, 2, … , M . Assuming that
J 1(k) = J 2(k) =
|| T (k) − Td || L2(Q) || Td || L2(Q)
|| ∇ T (k)|| L2(Ω ) i
|| ∇ T e || L2(Ω )
,
, (10)
Td ∈ L (Q), 2
i
where Td is a field specified in the subdomain Q ⊂ Ω i ∪ Ω e , we will consider the following two problems:
J 1(k) ≡ J 1(k r1, k θ1, … , k rM , k θM ) → min, J 2(k) → max, k ∈ U .
(11)
Choosing a particular subset Q in Ω i ∪ Ω e and the field Td specified in Q , we can investigate various inverse problems. In particular, when Q ⊆ Ω e and Td = T e , J 1(k) has the meaning of the relative mean squared integral perturbation of external field T e over the domain Q. Therefore, the parameter k opt = (k ropt 1 , DOKLADY PHYSICS
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opt opt k θopt 1 , ..., k rM , k θ M ), at which its minimum is obtained, corresponds to the approximate solution of the thermal cloaking problem, whereas the solution kopt to the maximization problem J 2(k) → max corresponds to the design of an approximate heat concentrator. To solve each problem in (11), we will use an algorithm based on the particle swarm optimization [11]. Within this method, the desired parameters determining the value of minimized functional J are presented in the form of the coordinates of the radius vector k = (k r1, k θ1, … , k rM , k θM ) of some abstract particle. A particle swarm is considered to be any finite set of particles k 1, … , k N . Within the particle swarm optimization, one sets the initial swarm position k 0j , j = 1, 2,… , N , and the iterative displacement procedure for all particles k j , which is described by the formulas (see [9]) i +1
i +1
kj = kj +vj , i +1
i
v j = w v j + c1r1(p j − k j ) + c2r2(p g − k j ). i
i
i
i
i
(12)
Here, p j (or p g ) is the personal (or global) best position of the jth particle; v j is the displacement vector; w, c1, and c2 are constant parameters; and r1 and r2 are random variables, which are uniformly distributed over the interval (0, 1). Their choice was considered in more detail in [11]. The subscript j ∈ {1, 2, … , N } in (12) denotes the particle number, and the superscript i ∈ {0,1, … , L} indicates the iteration number. The most time-consuming part of the abovedescribed algorithm is the calculation of the values J (k ij ) of the minimized J(k) functional for the position of particle k ij at different i and j values. This procedure includes two stages. At the first stage, the solution T (k ij ) to direct problem (1)–(3) is calculated. To this end, we used the FreeFEM++ software package (www.freefem.org), designed for numerical solution of two- and three-dimensional boundary-value problems by the finite-element method. All calculations were performed in dimensionless variables; the domain D and shell Ω were determined for the values z 0 = 3, a = 1, b = 1 . 5, and c = 3. The role of the external field was played by the field T e in (6) at T1 = 0 , T2 = 100 , which is characterized by a constant gradient ∇ T e with a modulus | ∇ T e | = T0 / z 0 and straight temperature isolines T = const , oriented perpendicular to the z axis. In view of the axial symmetry of the direct boundary-value problem, its solution in spherical coordinates is independent of the angle φ. Therefore, an approximate solution was sought in the cross section D2 of the three-dimensional domain D by the y = 0 plane. All calculations were performed in spherical coordinates r , θ , using two-dimensional secondorder splines; the calculation domain was divided into
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N 3
z 3 96.0
96.0 92.0
92.0
88.0
88.0
2
84.0
2
84.0 80.0
80.0 76.0
76.0
72.0
1
72.0 68.0
64.0
56.0
52.0
0
0
48.0 44.0
48.0
32.0 28.0
−1
36.0 32.0 28.0 24.0
20.0 16.0
20.0
−2
12.0
16.0 12.0
8.0
8.0
4.0
−3
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40.0
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−2
52.0
40.0
36.0
−1
64.0
60.0
60.0
56.0
68.0
1
4.0
2
3 x
Fig. 2. Temperature isolines T ( x, 0, z) in the cross section by the y = 0 plane in the presence of a body Ω i .
25000 triangles. After determining T (k ij ), we calculated the mean squared integral norm, entering the definition of functionals J 1 or J 2 in (10), using numerical integration over the cross section of Ω e or Ω i by the y = 0 plane. In our computational experiments for problem (11) at i = 1, we assumed that N = 25 ; L = 50 ; w = 0.5; c1 = 1; c2 = 1 . 5; m j = 0 . 1; M j = 10; and j = 1, 2, … , M . First, we will report the results of the numerical solution of the direct problem corresponding to the following scenario: a body (Ω i , k i ) is placed in the domain D at ki = 0 . 2, and it is assumed that k e = 1 in D Ω i . The introduction of (Ω i , k i ) into D leads to a perturbation of the external field T e , which manifests itself, in particular, in the bending of temperature isolines in the domain D (see Fig. 2). Specifically, this bending indicates the presence of body (Ω i , k i ) in D . To cloak the body (Ω i , k i ), we will surround it by an Mlayered shell (Ω, k) , where k = (k r1, k θ1, … , k rM , k θM ), and determine the desired parameters k rj and k θj by solving the minimization problem in (11). Using the particle swarm optimization, we obtained the optimal values k rj and k θj for different values of number M, in particular, k r = 6 . 372 , k θ = 1 . 466, and J 1opt = 5.21 × 10–4 for a single-layer shell (M = 1) and k r1 = 0 . 988, k θ1 = 1 . 882 , k r 2 = 1 . 073, k θ2 = 1 . 836 , k r 3 = 1.162, k θ3 = 1.419, k r 4 = 0 . 961, k θ4 = 1 . 753 , k r 5 = 0 . 925, k θ5 =
−3
−2
−1
0
1
2
3 x
Fig. 3. Temperature isolines T ( x, 0, z) for a five-layer shell.
1.224, and J 1opt = 5 . 76 × 10 −5 for M = 5. For all the values M = 1, 2, …, the external thermal field Te corre-
sponding to the optimal values k rjopt and k θopt j turned out to be similar to the external field T e (for M = 5, the isolines of the calculated field T in the cross section of D2 are shown in Fig. 3); specifically this fact forms an illusion of the absence of the perturbing object in D.
1 2 3 4 5
Jmin 0.0010
0.0001
0
10
20
30
40
50 i
Fig. 4. Dependence of the minimum value Jmin of functional J1 on the iteration number i for different numbers of shell layers: M = 1, 2, 3, 4, 5. DOKLADY PHYSICS
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THE OPTIMIZATION METHOD IN DESIGN PROBLEMS Table 1. Calculated thermal conductivities k ropt and k θopt and the concentration efficiency _ the designed concentrator mj
Mj
kr
opt
kθ
=
J 2 (k ropt , k θopt )
l = kθ / k r
0.05
5
5
0.05
0.01
0.05 0.1 0.1 0.2 0.2
10 5 10 5 10
10 5 10 5 10
0.05 0.1 0.1 0.2 0.2
0.005 0.02 0.01 0.04 0.02
_
of
opt
1.55 1.59 1.53 1.57 1.50 1.54
Figure 4 shows the dependences of the minimal value Jmin of functional J 1 on the iteration number i for different numbers of shell layers: M = 1,2,3,4,5 . A further increase in the number of layers M does not lead to any significant enhancement of the cloaking effect. Note that different optimal controls kopt and related optimal temperature distributions are obtained for different numbers of cloaking shell layers. When a heat concentrator was designed by solving the problem of maximizing the functional J2 for any number of layers M , we obtained the values of coefficients k rj = M j , k θj = m j , j = 1, 2, … , M , where mj and Mj are the limitations on the set of controls introduced in (9). Thus, in all the cases under consideration, the solution to the problem of maximizing the functional J2 was in fact the same concentration shell, independent of the way of dividing a shell into layers. This effect can physically be explained as follows: to provide a maximum heat flow to the internal domain Ω i , the thermal conductivity should be maximum in the radial direction and minimum in the perpendicular direction. The values of the parameters k ropt and k θopt and the corresponding concentration efficiency
_ opt = J 2(k ropt , k θopt ) for some characteristic values of mj and Mj are listed in Table 1. A detailed analysis of
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the numerical experiments performed will be carried out by us in a separate paper. ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research (project no. 16-01-00365-a), the Russian Science Foundation (project no. 14-1100079), and the “Far East” Program (project no. 15I-4-036). REFERENCES 1. S. Guenneau, C. Amra, and D. Veynante, Opt. Express 20, 8207 (2012). 2. T. Han and Z.-M. Wu, Prog. Electromagn. Res. 143, 131 (2013). 3. T. Han and C.-W. Qiu, Optik 18, 044003 (2016). 4. G. V. Alekseev, Problem of Invisibility in Acoustics, Optics, and Heat Transfer (Dal’nauka, Vladivostok, 2016) [in Russian]. 5. A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1986) [in Russian]. 6. O. M. Alifanov, E. A. Artyukhin, and S. V. Rumyantsev, Extreme Solutions of Ill-Posed Problem and Their Application to Inverse Heat Exchange Problems (Nauka, Moscow, 1988) [in Russian]. 7. G. V. Alekseev, I. S. Vakhitov, and O. V. Soboleva, Comp. Math. Math. Phys. 52 (12), 1635 (2012). 8. G. V. Alekseev and V. A. Levin, Dokl. Phys. 61 (11), 546 (2016). 9. G. V. Alekseev, V. A. Levin, and D. A. Tereshko, Dokl. Phys. 62 (2), 71 (2017). 10. J. Kennedy and R. Eberhart, Proc. IEEE Int. Conf. Neural Networks IV, 1942 (1995). 11. R. Poli, J. Kennedy, and T. Blackwell, Swarm Intel. 1, 33 (2007). 12. K. Chiba, Y. Makino, and T. Takatoya, J. Comp. Sci. Tech. 2 (1), 268 (2008). 13. M. D. Ardakani and M. Khodadad, Inverse Probl. Sci. Eng. 17 (7), 855 (2009). 14. F.-B. Liu, Heat Mass Transfer 48, 99 (2012).
Translated by Yu. Sin’kov
