Optimization Method in 2D Thermal Cloaking Problems
Gennady Alekseev Dmitry Tereshko





Optimization Method in 2D Thermal Cloaking Problems

G. V. Alekseev^1,2 and  D. A. Tereshko^1

^1Institute of Applied Mathematics FEB RAS, Vladivostok, Russia

^2School of Natural Sciences, Far Eastern Federal University, Vladivostok, Russia






Inverse problems associated with designing cylindrical 
thermal cloaking shells are studied.
Using the optimization method these inverse problems
are reduced to corresponding control problems in which 
the diagonal components of diagonal in polar coordinates 
tensor of thermal conductivities play the role of 
passive controls. A numerical algorithm based on the 
particle swarm optimization is proposed and the results 
of numerical experiments are discussed.
Rigorous optimization analysis shows that high cloaking efficiency of the shell 
can be achieved either using a highly anisotropic single-layer shell 
or using a multilayer shell composed of only three isotropic natural materials with optimally selected thermal conductivities.





§ INTRODUCTION

In recent years much attention has been given to creation of tools of material objects masking from
detection with the help of electromagnetic or acoustic location. Beginning with pioneering papers
<cit.> the large number of publications was devoted to developing different methods of solving
the cloaking problems. Transformation optics (TO) is the most popular method of designing cloaking
devices (hereafter, cloaks). The methodology of cloaking based on this method obtained the name
of direct design because it is based in fact on solving direct problems of electromagnetic scattering.

The first works in this field focused on the electromagnetic cloaking, i.e.
cloaking objects from detection by electromagnetic location. Then the main results of the electromagnetic cloaking
theory were expanded to an acoustic cloaking <cit.> and to
cloaking magnetic, electric, thermal and other static fields <cit.>. 
It should be noted that the invisibility devices (hereafter, cloaks) designed on the
basis of direct strategies possess serious drawbacks. The main one is the difficulty of their
technical realization. For example, the design of the TO-based cloaks involves extreme values of
constitutive parameters and spatially varying distributions of the permittivity and permeability
tensors which are very difficult to implement <cit.>.

That is why the another cloak design strategy began develop recently. It obtained the name of
inverse design as it is related with solving inverse electromagnetic (acoustic or static) problems
(see <cit.>). The optimization method forms the core of the inverse design methodology.
Just this enables to overcome some substantial limitations of previous cloaking solutions. 
A growing
number of papers is devoted to applying the inverse design methodology in various cloaking
problems. Among them we mention the fist papers  <cit.> where numerical optimization algorithms are
applied for finding the unknown material parameters of TO-based cloak and 
papers <cit.> associated with the use of topology optimization and discrete material 
optimization for solving problems of designing cloaks, concentrators and other thermal functional devices.
Papers <cit.> are devoted to theoretical analysis of cloaking problems using the optimization approach.

Optimization method is applied and in this paper for solving inverse problems for the 2D stationary  model of heat transfer. 
These problems arise when developing the technologies of designing thermal cloaking devices  having the form of the cylindrical shell. It is assumed that the desired cloaking shell consists of finite number of  layers every of which is filled with homogeneous anisotropic medium. Radial and azimuthal thermal conductivities k_r and k_θ of the medium filling every layer play the role of controls. As a result our cloaking problems are reduced to solving the corresponding finite-dimensional extremum problems. We propose a numerical algorithm of solving these finite-dimensional extremum problems which is based on using the particle swam optimization (PSO) (see <cit.>) 
according to the scheme proposed in <cit.> and discuss some results of numerical experiments.



§ STATEMENT OF DIRECT CONDUCTIVITY PROBLEM. PROPERTIES OF EXACT SOLUTION
 
We begin with statement of the general direct conductivity problem, considered in a rectangle D={ x≡ (x,y):|x|<x_0, |y|<y_0} with 
specified numbers x_0>0 and y>0 (see Figure 1a). We will assume that an external temperature field
T^e is created by two vertical plates x = ± x_0
which are kept at different values T_1 and T_2, while the upper and lower plates y=± y_0 are thermally insulated. We assume further that
there is a material shell (Ω,κ) inside D. Here, Ω is a
circular layer a < | x| < b}, b<max{x_0, y_0} and κ is the thermal conductivity tensor of the inhomogeneous anisotropic medium filling the domain Ω. It is also assumed that the
interior Ω_i: | x|<a and exterior Ω_e: | x|>b of Ω are
filled with homogeneous media having constant thermal conductivity k_b > 0  (see Figure 1b).

In this case, the direct conduction problem consists of determination of a triplet of functions, namely T_i in Ω_i, T in Ω, and T_e in Ω_e, which satisfy the equations

    k_b  Δ T_i = 0   Ω_i,


    div  (κ  grad  T) = 0  Ω,


    k_b  Δ T_e = 0   Ω_e,

obey the following boundary conditions on the boundary ∂ D of D

    T_e|_x= -x_0 = T_1,   T_e|_x=x_0 = T_2,   ∂T_e/∂ y|_y=± y_0 = 0

and the matching conditions on internal Γ_i and external Γ_e components of the boundary Γ of the shell Ω having the form

    T_i = T,    k_b ∂ T_i/∂ n = (κ∇ T) · nΓ_i,    
    T_e = T,    k_b ∂ T_e/∂ n = (κ∇ T) · nΓ_e.




As already mentioned, our goal is to analyze the cloaking problems for the model (<ref>)–(<ref>).
Let us remind that the general inverse cloaking problem in the exact formulation
consists in finding thermal conductivity κ providing the
following cloaking conditions <cit.>:

    T_e= T^e   Ω_e,   ∇ T_i= 0   Ω_i.

These conditions imply that temperature field must be equal to the external field T^e  in Ω_e and 
the gradient ∇ T_i must be equal to zero in Ω_i.
Besides this general problem we will also study the external cloaking problem. It consists in finding 
conductivity κ providing the fulfillment of the first condition in (<ref>).


Preliminarily we will study some properties of solutions to direct problem (<ref>)–(<ref>).
We will assume firstly that the tensor κ is diagonal in polar coordinates r, θ and its diagonal components (radial and polar conductivities) k_r and k_θ are the positive constants. 
This condition is widely used when analysing two-dimensional problems of designing annular functional devices. 
Moreover, in the particular case when the boundary functions T_1 and T_2 are constants, k_b= const and 
the so called admissibility condition k_r k_θ=k_b^2 is satisfied, problem (<ref>)–(<ref>) has an exact 
solution which can be found by Fourier method. This solution (T_i,T,T_e) is described by 

    T_i (r, θ) = T_0/x_0 K(s)   rcosθ 
    +T_1 +T_2/2  in Ω_i,   
    T (r, θ) =  T_0/x_0( r/b)^s-1 rcosθ 
    +T_1+T_2/2  in Ω,


    T_e (r, θ) =  T_0/x_0 r cosθ +T_1+T_2/2  Ω_e

where

    s=√(k_θ/k_r),     K(s) =(a/b)^s-1, 
      T_0=T_2-T_1/2.

The parameter s=√(k_θ/k_r) characterizes the degree of anisotropy of the respective homogeneous shell 
(Ω, k_r, k_θ). In the particular case when s=1 (provided that k_r=k_θ = k_b so that the entire medium filling the domain D is homogeneous and isotropic as shown in Figure 1a), formula (<ref>) is transformed into the unified formula

    T^e (x)=  T_0/x_0 r cosθ +T_1+T_2/2 = T_0 x/x_0 +T_1+T_2/2,

which describes the external applied field T^e (x). Just this field T^e is used to detect objects located in domain D. 

Assuming that s≥ 1 let us define function M(s)=1- K(s) where K(s) is given in (<ref>). Since a<b we derive from (<ref>) that

    0≤ K (s)≤ 1,    
    0≤ M(s)≤ 1,    M(s) +  K(s) = 1  s ≥ 1.

Moreover, we have

    K (s)→ 1,   M (s)→ 0   as  s→ 1,   K (s)→ 0,   M (s)→ 1   as  s→∞.

Also, it follows from (<ref>), (<ref>) that external and internal fields T_e and T_i corresponding to the homogeneous shell (Ω,k_r,k_θ) satisfy conditions

    T_e= T^e,   |∇ T_e|= |∇ T^e|= |T_0|/x_0  Ω_e, 
    |∇ T_i|= |T_0|/x_0 K (s) Ω_i.


Using the above mentioned properties of functions M(s) and K(s) and taking into account terminology of <cit.> we will refer below to 
M(s) and K(s) as the cloaking efficiency and visibility measure of the respective cloak (Ω,k_r,k_θ). 
We emphasize that cloaking efficiency of the homogeneous anisotropic shell (Ω,k_r,k_θ) increases while the visibility measure decreases with an increase 
of the anisotropy parameter s=√(k_θ/ k_r).

In the limit as s →∞ we obtain an exact cloaking shell with maximum cloaking efficiency M = 1 and minimum visibility measure K=0. 
On contrary, in the limit as s→ 1, we have  a shell (Ω, k_r, k_θ) with minimal cloaking efficiency M(1)= 0 but maximum visibility measure K(1)=1. 



§ CONTROL PROBLEMS FOR LAYERED SHELL


Now we consider more complicated scenario when (Ω, k_r, k_θ) is a layered shell consisting of M concentric circular layers
Ω_j ={ r_j-1<r=| x|<r_j}, j = 1,2,…,M, where r_0 = a, r_M = b. Each of these layers is filled with a homogeneous and (generally) anisotropic medium, described by constant conductivities 
k_rj > 0 and k_θ j > 0, j = 1,2,…,M. 
The parameters k_r and k_θ of this layered shell are given by

    k_r ( x) = ∑_j=1^M k_rjχ_j ( x),   
    k_θ ( x) = ∑_j=1^M k_θ jχ_j ( x).

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, 
while k_rj and k_θ j are some constant unknown coefficients. It should be noted that the case when k_rj=k_θ j corresponds to M-layered isotropic cloak.

We remind that our purpose is the numerical analysis of inverse problems for 2D model of heat transfer (<ref>)–(<ref>) 
arising when developing technologies of designing thermal cloaking devices. In the case of a layered shell with parameters (<ref>) these problems consist of finding unknown coefficients k_rj and k_θ j, j=1,2,...,M in 
(<ref>) forming a 2M-dimensional vector k=(k_r1,k_θ 1,...,k_rM,k_θ M) from the cloaking conditions. 
We will refer to k as conductivity vector because it consists of conductivities k_rj, k_θ j
of separate sublayers Ω_j, j=1,2,...,M comprising the M-layered shell (Ω, k).
In the particular case of the M-layered isotropic shell when  k_rj=k_θ j=k_j>0
conductivities of separate sublayers form a M-dimensional vector 
k=(k_1,,...,k_M).
 
By optimization method our inverse problems are reduced to extremum problems of minimization of certain cost functionals which adequately correspond to inverse problems of designing devices for approximate cloaking <cit.>. 
 In order to formulate the control problems under study, we denote by 
T [ k] ≡ T [k_r 1, k_θ 1, … , k_r M, k_θ M] the solution to the direct problem (<ref>)–(<ref>) corresponding to parameters 
(k_r j, k_θ j) in Ω_j, j=1,2,…,M, and to conductivity k_b in Ω_i and Ω_e. 
We will assume below that the vector k = (k_r 1, k_θ 1, … , k_r M, k_θ M) belongs to the bounded set

    K= { k≡ (k_r 1, k_θ 1, … , k_r M, k_θ M):   
    m_r  ≤ k_rj≤ M_r,   m_θ≤  k_θ j≤ M_θ,   j=1,…,M }

to which we will refer to as a control set.  Here given positive constants  m_r, m_θ and M_r,  M_θ 
are lower and upper bounds of the control set K. Let us define two cost functionals

    J_e ( k) = T[ k]-T^e_L^2(Ω_e)/T^e_L^2(Ω_e), 
       J_i ( k) = ∇ T [ k]_L^2(Ω_i)/∇ T^e_L^2(Ω_i),
 
where, in particular, 

    T^e^2_L^2(Ω_e) = ∫_Ω_e |T^e|^2 d  x,   T[ k]-T^e^2_L^2(Ω_e) = ∫_Ω_e |T[ s]-T^e|^2 d  x,

and formulate following two control problems:

    J_e ( k)   →inf,    k∈ K,
 

    J( k)= J_e ( k)+J_i ( k)/2→inf,    k∈ K.
 
It follows from previous analysis (see also <cit.>) that problem (<ref>) is aimed to finding an approximate (optimal) solution of the external cloaking problem while problem  (<ref>) is aimed to finding an approximate solution of the general cloaking problem. 


Two particular cases of problems (<ref>) and (<ref>) are of particular interest.
The first case is a single-layer shell with constant parameters k_r^*, k_θ^*
satisfying the admissibility condition k_r^* k_θ^* =k_b^2. As was mentioned before,  in this case there exists 
the exact solution (T_i,T,T_e) of problem  (<ref>)–(<ref>) which is described by 
(<ref>), (<ref>) and satisfies the conditions (<ref>). Using (<ref>)
one easily shows that for any admissible pair (k_r^*,k_θ^*) the following relations hold:

    J_e(k_r^*,k_θ^*)=0,     J_i(k_r^*,k_θ^*)=  K(s^*),   
    J(k_r^*,k_θ^*)= 0.5   K(s^*),    s^*=√(k_θ^* / k_r^*).

Here K(s^*) is the visibility measure  of the cloak (Ω,k_r^*,k_θ^*)
connected with the cloaking efficiency by the relation
K(s^*)=1- M(s^*). From here and from (<ref>) follows that value J_i(k_r^*,k_θ^*)
of the functional J_i as well as value J(k_r^*,k_θ^*) of the functional J 
characterizes the cloaking  efficiency of the corresponding shell (Ω,k_r^*,k_θ^*).
The smaller value J(k_r^*,k_θ^*) corresponds to smaller value of visibility measure 
which in turn corresponds to higher cloaking efficiency of the 
shell (Ω,k_r^*,k_θ^*) and vice versa. 
We emphasize that the value J( k) of the functional J on any element 
k≡ (k_r 1, k_θ 1, … , k_r M, k_θ M)
can be considered as a visibility measure of the respective cloak (Ω, k^opt)
which characterize a cloaking efficiency and in the general case for 
M-layered anisotropic (or isotropic) shell (Ω, k).
Therefore our goal will consist of finding a conductivity vector (an optimal solution of (<ref>))
k^opt∈ K for which the functional J takes a minimum value J^opt=J( k^opt)
on the set K and therefore the cloak 
(Ω, k^opt) possess a maximum cloaking efficiency.




To solve problems (<ref>), (<ref>) we use an algorithm based on the particle swarm optimization <cit.>.
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_r 1, k_θ 1, … , k_r M, k_θ M)
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_0^j, j=1,2,…,N, and the iterative displacement procedure 
k_j^i+1= k_j^i+ v_j^i+1 for all particles k_j, which is described by the formula (see <cit.>)

    k_j^i+1=w  v_j^i + c_1 d_1 ( p_j^i- k_j^i)
     + c_2 d_2 ( p_g- k_j^i).

After each displacement we calculate the value J ( k_j^i+1) of the functional J for the new position 
k_j^i+1, compare it to the current minimum value and, if necessary, update the personal and
global best positions p_j and p_g.
In the end of this iteration process, all particles have to come at the global minimum point.

Here, v_j is the displacement vector; w, c_1 and c_2 are constant parameters; and d_1 and d_2 are
random variables, which are uniformly distributed over the interval (0,1). Their choice was considered in
more detail in <cit.>. The subscript j ∈{1 , 2, … , N } in (<ref>) denotes the particle number, and the superscript i ∈{0 , 1, … , L } indicates the iteration number.




§ SIMULATION RESULTS

In this section we discuss the results which were obtained using the PSO algorithm 
for designing anisotropic single-layer and isotropic multilayer cloaking shells.
The most time-consuming part of the above-described algorithm is the calculation of the values
J ( k_j^i) of the functional J  for the particle position k_j^i+1
at different i and j values. This procedure includes two stages. At the first stage, the solution
T[ k_j^i] to direct problem (<ref>)–(<ref>) 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.  After determining T[ k_j^i], at the second stage we calculate the mean squared integral norms 
entering the definition of functionals J_e or J_i in (<ref>) using the formulas of numerical integration.

Numerical experiments were performed for values N=25, L=50, w=0.4, c_1=1, c_2=1.5 of 
PSO parameters and at the following given data:

    x_0 = y_0 = 3 ,   a = 1  ,   b = 2 ,   T_1 = 100 ^∘,    T_2 = 0 ^∘,   k_b= k_0 ≡ 1 ·.
 
The role of the external field was played by the field T^e determined in (<ref>) at T_1 = 100 C^∘,
T_2 = 0 C^∘. It is characterized by a constant gradient ∇ T^e with a modulus |∇ T^e| = |T_0| / x_0 and with straight isolines T^e= const, oriented perpendicular to the x axis (see Figure 2). 




Firstly, we solve the design problem for a single-layer anisotropic shell 
(Ω,k_r,k_θ) assuming that the background material in Ω_i and Ω_e
is glass (k_b=k_0) while lower and upper bounds in (<ref>) are determined by

    m_r=0.05   k_0,   M_r=1.0   k_0,    m_θ=5    k_0,    M_θ=15    k_0.

Numerical solution of control problem (<ref>), (<ref>) 
with the help of PSO algorithm gives after 50 iterations the following results:

    k_r^opt=0.18    k_0,    k_θ^opt=5.54    k_0,    
    J_e^opt =2.27 × 10^-6,   J_i^opt =4.31 × 10^-2,  
    J^opt =2.15 × 10^-2.

Isolines for corresponding temperature field T^opt = T[k_r^opt, k_θ^opt] are shown in Figure 3a.
The isolines are straight as if there is no shell in domain D, and it creates the illusion of the absence of the shell Ω for an external observer. 

As the second test we solve (under the mentioned conditions (<ref>), (<ref>)  to the data) the problem (<ref>), (<ref>) using the developed PSO algorithm. After 50 iterations we obtain the following results:

    k_r^opt=7.42 × 10^-2  k_0,   k_θ^opt=13.46     k_0,   
    J_e^opt =7.92 × 10^-5,   J_i^opt =1.75 × 10^-4,  
    J^opt =1.27 × 10^-4.






The isolines of calculated temperature field T^opt = T[k_r^opt, k_θ^opt] are plotted in Figure 3b.
Again we note that the isolines are close to isolines in Ω_e in Figure 2 and Figure 3a.
It should be noted however that this high cloaking efficiency is achieved in both cases due to the high anisotropy
of respective cloak (Ω,k_r^opt,k_θ^opt). It is characterized by anisotropy coefficient 
s^opt=√(k_θ^opt/ k_r^opt) which is equal to 5.5 for the solution 
of problem (<ref>), (<ref>) and  is equal to 13.5 for the solution of problem (<ref>), (<ref>).

As mentioned above the technical realization of highly anisotropic cloaking shells is associated with great difficulties because appropriate materials are not found in nature and should be created using metamaterials approach.
One of the ways of overcoming these difficulties consist of using 
layered shell consisting of M concentric circular layers
Ω_j ={ r_j-1<r=| x|<r_j}, j = 1,2,…,M, where r_0 = a, r_M = b. Each of these layers is filled with a homogeneous and isotropic medium, described by constant conductivity 
k_j, j = 1,2,…,M.
Set k=(k_1, k_2, …, k_M) and define the control set K by 

    K={ k=(k_1, k_2, …, k_M):    k_min≤ k_j ≤  k_max,   j=1,…,M }.



We begin with analysis of cloaking efficiency of a multilayer cloaking shell proposed in <cit.>. 
It is composed  of alternate layers of two materials (wood and stainless steel), whose conductivities are defined by the following parameters: k_1=k_3=…=k_M-1=5 × 10^-2  k_0, 
k_2=k_4=…=k_M=20   k_0 and, besides, k_b=k_0. 
Since all cloak parameters are specified, it remains to us  to determine the cloaking efficiency 
of the corresponding cloaking shell (Ω,  k) by calculating the values J( k), 
J_i( k) and J_e( k).
These values together with k_M = 20   k_0 are presented for six different shells (M=2, 4, 6, 8, 10 and 12)  in Table 1. 
The isolines of corresponding temperature field for a six-layer shell (M=6) are shown in Figure 4a.
It is seen that J( k) takes the value 1.6 × 10^-2 for M=12. 
Besides, the isolines outside cloak are curved. 
These results indicate low cloaking efficiency of the above designed cloak (Ω, k).









In order to design a cloaking shell with higher cloaking efficiency 
we apply the optimization method.
To this end we solve the control problem  (<ref>), (<ref>) for the case when
k_min=5 × 10^-2  k_0, k_max=20   k_0 and k_b=k_0. 
Our optimization analysis  using PSO algorithm showed that optimal values k_i^opt of  all parameters k_i 
with odd indices i= 3,5,…,M-1 coincide with  k_min,  optimal values of parameters 
k_2^opt,…,k_M-2^opt with even indices coincide with k_max
while k_M^opt changes from 20   k_0 at M=2 to  8.15   k_0 at M=12.
Thus we have 

    k_1^opt=k_3^opt=…=k_M-1^opt= k_min,   
    k_2^opt=k_4^opt=…=k_M-2^opt=k_max

while optimal value k_M^opt of the last control k_M and corresponding values J^opt, J_i^opt and J_e^opt for different M=2, 4,..., 12 are presented in Table 2. 
Isolines of corresponding field T^opt for a six-layer shell (M=6) are plotted in Figure 4b.




Analysis of Table 2 shows that the optimal values J^opt, J_i^opt and J_e^opt of all cost functionals decrease with increasing the number of layers M. 
In particular, J^opt decreases from 1.1 × 10^-1 at M=2 to 2.4 × 10^-3 at M=12.
Temperature isolines outside the cloak in Figure 4a are straight as if there is no shell in the domain.
This indicates the relatively high cloaking efficiency of the shell (Ω, k^opt).

The high cloaking efficiency of the designed shell can be explained, in fact, by optimal choice of the last layer material.
It should be noted that the quality of the optimal solution, which is characterized by the optimal value J^opt of the cost functional J, depends very substantially on the control set K defined in (<ref>).
If, in particular, one extends the set K, 
then this inevitably results in a decrease in the value J^opt and consequently to increase in the efficiency of the respective cloaking shell.
To see this, we consider as second test the scenario when again k_min = 0.05   k_0, 
k_b = k_0 while k_max = 236   k_0 (corresponds to aluminium). 
Applying PSO algorithm again results in relations (<ref>) at k_min = 0.05   k_0 and k_max = 236   k_0
for all k_i^opt except k_M^opt and to value k_M^opt which takes different values depending on M.
Optimal value k_M^opt of the last control and corresponding values J^opt, J_i^opt and J_e^opt for different M=2, 4,..., 12 are presented in Table 3. 
We note, in particular, that k_10^opt=2.85   k_0,  J^opt=1.9 × 10^-6 for M=10 and 
k_12^opt=0.75, J^opt=1.1 × 10^-6 for M=12.
More detailed results of the functional J iterative minimization for the second test are presented in Figure 5  that shows dependence of the current value J_min (i) on the iteration number i for different numbers of shell layers M = 4, 6, 8, 10, 12. Here J_min (i) is the minimum value of the functional J at the i-th iteration.
In all considered cases the algorithm converges faster then in 20 iterations. 
It is clearly seen that values of functional J  in Table 3 are substantially less than in Table 2.












In our third test we assume that k_min = 0.05   k_0 (wood), 
k_b = k_0 (glass) while k_max = 401   k_0 (copper). 
Numerical results for this test are presented in Table 4 (which is analogue of Tables 2 and 3).
As in the previous tests, the optimal values of all control parameters with odd indices coincide and they equal to the lower bound 
k_min=0.05   k_0 while all even controls except the last one coincide with upper bound k_max.
But it is clearly seen that in this case the optimal values J^opt of the functional J 
for all M are substantially less than in Table 2 and in Table 3. 
Thus for the new choice of bound k_max we obtained the shell with the highest cloaking efficiency.
This can be explained by the fact that for the third test the value (contrast) k_max/ k_min  is much larger than 
for the first and second tests (8020 versus 400 and 4720). 
Comparing the values of functional J in Tables 2, 3 and 4, we come to the conclusion 
that the cloaking efficiency of the shell increases with increasing number of layers M and the ratio k_max/ k_min.

Since the optimal values of all control parameters except the last one k_M^opt
are equal to the lower or upper bounds of the control set K, the designing a multilayer shell can be reduced to 
solving  one-parameter control problem for k_M. 
According to this reduced design,
conductivities of all odd layers are initially prescribed to k_min,
conductivities of all even layers except k_M are fixed to k_max 
and conductivity of the last layer is found by solving 
one-parameter control problem for k_M. 
As a result, we can use only three isotropic materials to construct a multilayer shell if 
the optimal conductivity k_M^opt corresponds to some natural material.
For example, the value k_10^opt=2.49   k_0 from Table 3 at M=10 is the conductivity of marble.
If the optimal value k_M^opt does not correspond to any natural material than
we can choose materials with conductivities k̃_M^opt  closed to optimal k_M^opt.
For example, instead of the value k_12^opt=8.04   k_0 from the last row of Table 2 we can use 
the close value k̃_12^opt=7.8    k_0 that corresponds to manganese.
Assuming k̃^opt=(k_1^opt,…,k_M-1^opt,k̃_M^opt) we obtain 
cloaking shell (Ω, k̃^opt) with J( k̃^opt)=2.8 × 10^-4
which corresponds to high cloaking efficiency.
Instead of the value k_12^opt=0.54   k_0 from the last row of Table 3 we can use 
the value k̃_12^opt=0.5    k_0 that corresponds to polyethylene.
In this case  J( k̃^opt)=2.49 × 10^-4.
If the choice of materials is limited but we want to increase cloaking efficiency 
then one can use the geometric parameters of the shell (e.g., the thickness of the last sublayer) as additional controls.



§ CONCLUSION

In this paper we studied inverse problems for 2D model of heat transfer (<ref>)–(<ref>)
associated with designing cylindrical layered thermal cloaking shells. Using the optimization method these inverse problems
were reduced to corresponding control problems in which the thermal conductivities of layers play the role of controls. 
For solving our finite-dimensional extremum problems 
we proposed numerical algorithm based on the particle swarm optimization and discussed simulation results. 
Optimization analysis showed that the optimal values of all control parameters with odd indices 
(conductivities of odd layers) coincide and equal to the lower bound 
of the control set while all even controls except the last one equal to the upper bound of the control set. 
As for the last control, its finding is reduced to solving a one-parameter control problem.
Thus the proposed method simplifies substantially the solution of the cloak design problem under study 
and provides a high cloaking efficiency and a simplicity of technical realization 
of cloaks designed with the help of our method. 
In fact, it is enough to use three different natural materials for obtaining high performance cloaks.
Two of them correspond to the initially specified natural materials with great contrast while 
the latter is found by solving a one-parameter control problem.
Finally, we note that the proposed method and results are not limited only to the temperature field. 
They also could be extended to dc electric, magnetic and other static fields and, besides, to design problems of concentrators, inverters and another functional devices.





§ ACKNOWLEDGEMENT

The first author was supported by the Russian Science Foundation (project No. 14-11-00079) and the Russian Foundation for Basic Research (project no. 16-01-00365-a),
the second author was supported by the Federal Agency for Scientific Organizations in the framework of the state task
(subject no. 0263-2018-0001).


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