Theory for Invisible Thermal Sensors: Optimization Scheme

Abstract

Metamaterial-based devices have been extensively explored for their intriguing functions, such as cloaking, concentrating, rotating, and sensing. However, they are usually achieved by employing metamaterials with extreme parameters, critically restricting engineering preparation. In this chapter, we propose an optimization model with particle swarm algorithms to simplify parametric designs to realize bilayer thermal sensors composed of bulk isotropic materials (circular structure). For this purpose, the fitness function is defined to evaluate the difference between the actual and expected temperatures. By choosing suitable materials for different regions and treating the sensor, inner shell, and outer shell radii as design variables, we finally minimize the fitness function via particle swarm optimization. The designed scheme is easy to implement in applications and shows excellent performances in detective accuracy and thermal invisibility, which are confirmed by finite-element simulations and laboratory experiments. The optimization model can also be flexibly extended to a square case. This method can calculate numerical solutions for difficult analytical theories (circular structure) and optimal solutions for problems without analytical theories (such as square structure), providing new inspiration for simplifying the design of metamaterials in various communities.

1 Opening Remarks

Novel meta-devices [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] have been researched continuously over the decades in various fields since the pioneering theoretical proposals of transformation theory [1,2,3,4]. Recently, many fruitful strategies have been proposed for offering new avenue of devising thermal meta-devices such as neutral inclusion [6], bilayer schemes [10], illusion thermotics [13], regionalization transformation [14], and many-particle thermal invisibility [21]. However, most experimental devices are prepared by employing metamaterials with unconventional thermal conductivities (i.e., anisotropic, graded or singular), which remain to be overcomed for engineering applications. New schemes deserve exploring for purpose of simplifying engineering preparation and developing novel functional meta-devices.

Optimization method has been conjectured as an effective tool for the design of metamaterials in the macro [25,26,27,28,29,30,31,32,33] or micro [34,35,36] scale, which is applied comprehensively in recent years. A gradient-based numerical optimization algorithm was used to control the heat flow in the printed circuit board [27]. Also, non-gradient-based black-box algorithms, including evolutionary algorithms, have been used in the reverse structural design of thermal metamaterials [31]. Based on topology optimization [29, 30, 32, 33], thermal meta-devices are reversely designed for specific objective functions, providing excellent performance. Topology optimization usually involves a change in structural topology, resulting in complex structural parameters. This feature inspires us to explore simpler structural meta-devices with equally high performance by employing optimization algorithms.

Here, we propose an optimization model with particle swarm algorithms [37] (PSA) for designing bilayer thermal sensors composed of bulk isotropic materials. For example, we design a circular bilayer thermal sensor with detective accuracy and thermal invisibility. For this purpose, two objective functions are constructed simultaneously, one for detecting the temperature distribution of the region occupied by the sensor with accuracy and the other for undisturbing the temperature distribution of the original background. When choosing suitable material analogies for different regions, we treat the radius of the sensor, inner shell, and outer shell as design variables. By adopting PSA, the characteristics of the prescribed optimized structure are precisely and efficiently found. The designed scheme not only simplifies practical fabrication but shows almost perfect performance, as both simulation and experimental results exhibit. The optimization model can also be flexibly extended to a square case.

Fig. 12.1

Adapted from Ref. [38]

a Schematic diagram for bilayer thermal sensors. b Discretized finer mesh for optimization.

2 Theoretical Foundation

The scheme for a bilayer thermal sensor is shown in Fig. 12.1a. A sensor (with radius of $R_1$, Material (1) coated with a bilayer shell (inner shell with radius of $R_2$, Material (2); outer shell with radius of $R_3$, Material (3) is put in the center of background (Material 4) for detection of the temperature distribution of region occupied by it. Hot source and cold source are, respectively, set at the right-most and left-most boundaries. The up-most and down-most boundaries are thermally insulated. Temperature distributions follow the Laplace equation with passive heat conduction at steady state,

$$\begin{aligned} \boldsymbol{\nabla }\cdot \left( -\kappa \left( \boldsymbol{x}\right) \cdot \boldsymbol{\nabla }T\right) =0, \end{aligned}$$

(12.1)

where $\kappa \left( \boldsymbol{x}\right)$ is thermal conductivity, denoted by

$$\begin{aligned} \kappa \left( \boldsymbol{x}\right) ={\left\{ \begin{array}{ll} \kappa _{1}~~\textrm{for}~\boldsymbol{x}~\mathrm{in~region~of~Material~1},\\ \kappa _{2}~~\textrm{for}~\boldsymbol{x}~\mathrm{in~region~of~Material~2},\\ \kappa _{3}~~\textrm{for}~\boldsymbol{x}~\mathrm{in~region~of~Material~3},\\ \kappa _{4}~~\textrm{for}~\boldsymbol{x}~\mathrm{in~region~of~Material~4}. \end{array}\right. } \end{aligned}$$

(12.2)

Considering expanding Eq. (12.1) in cylindrical coordinates and symmetry of boundary conditions, the general solution of the temperature distribution in four regions can be expressed as

$$\begin{aligned} T_1&= A_0+Ar\cos \theta ,\end{aligned}$$

(12.3)

$$\begin{aligned} T_2&= A_0+Br\cos \theta +Cr^{-1}\cos \theta ,\end{aligned}$$

(12.4)

$$\begin{aligned} T_3&= A_0+Dr\cos \theta +Er^{-1}\cos \theta ,\end{aligned}$$

(12.5)

$$\begin{aligned} T_4&= A_0+Fr\cos \theta +Gr^{-1}\cos \theta , \end{aligned}$$

(12.6)

where $A_0$ is the temperature at $\theta =\pm \pi /2$, and $F=|\boldsymbol{\nabla }T_0|$ represents the modulus of an external linear thermal field $\boldsymbol{\nabla }T_0$.

Boundary continuity conditions of temperature and normal heat flow should be satisfied,

$$\begin{aligned} {\left\{ \begin{array}{ll}T_1\left( R_1\right) =T_2\left( R_1\right) ,\\ T_2\left( R_2\right) =T_3\left( R_2\right) ,\\ T_3\left( R_3\right) = T_4\left( R_3\right) ,\\ \left( -\kappa _1\frac{\partial T_1}{\partial r}\right) _{R_1}=\left( -\kappa _{2}\frac{\partial T_2}{\partial r}\right) _{R_1},\\ \left( -\kappa _{2}\frac{\partial T_2}{\partial r}\right) _{R_2}=\left( -\kappa _3\frac{\partial T_3}{\partial r}\right) _{R_2},\\ \left( -\kappa _{3}\frac{\partial T_3}{\partial r}\right) _{R_3}=\left( -\kappa _4\frac{\partial T_4}{\partial r}\right) _{R_3}.\end{array}\right. } \end{aligned}$$

(12.7)

When thermal sensor works, there is no thermal disturbances in sensor and background regions, which means we have following two equations,

$$\begin{aligned} {\left\{ \begin{array}{ll} A=F,\\ G=0. \end{array}\right. } \end{aligned}$$

(12.8)

We simplify the form of equations that are composed of Eqs. (12.7) and (12.8)

$$\begin{aligned} {\left\{ \begin{array}{ll} FR_1=BR_1+CR_1^{-1},\\ BR_2+CR_2^{-1}=DR_2+ER_2^{-1},\\ DR_3+ER_3^{-1}=FR_3,\\ -\kappa _{1}F=-\kappa _{2}\left( B-CR_1^{-2}\right) ,\\ -\kappa _{2}\left( B-CR_2^{-2}\right) =-\kappa _{3}\left( D-ER_2^{-2}\right) ,\\ -\kappa _{3}\left( D-ER_3^{-2}\right) =-\kappa _{4}F. \end{array}\right. } \end{aligned}$$

(12.9)

For given $R_1$, $\kappa _{1}$, $\kappa _{2}$, $\kappa _{3}$ and $\kappa _{4}$, we do have six unknown coefficients (B, C, D, E, $R_2$ and $R_3$) determined by six equations (Eq. (12.9)) uniquely. However, due to the nonlinear coupling of multiple unknowns, it is difficult to find an explicit analytical expression for any one radius. When the thermal conductivity of the shell (circular or elliptic structure) is taken as unknown coefficients, analytic expressions can be obtained, which have great limitations. In this way, the physical image of the influence of geometric size on the performance of thermal sensors cannot be given intuitively from the analytical theory. There is a mapping relationship between the radius of the circles (say, $R_1$, $R_2$ and $R_3$) and the performance of the thermal sensor. We turn it into an optimization problem and reversely design the geometry size according to the performance.

3 Optimization Problem Description

In principle, a thermal sensor should have the ability of reproducing temperature distributions in sensor and background regions, which are the same as those in corresponding regions of original background [39]. The heat field to be studied is discretized for numerical optimization by using finer mesh in COMSOL Multiphysics, as shown in Fig. 12.1b. Considering optimization problem, two objective functions for bilayer thermal sensors with accuracy and invisibility are, respectively, defined as

$$\begin{aligned} \boldsymbol{\Psi }_{s}=\frac{1}{N_s}\sum _{i=1}^{N_s} \left| T\left( i\right) -T_{ref}\left( i\right) \right| , \end{aligned}$$

(12.10)

$$\begin{aligned} \boldsymbol{\Psi }_{b}=\frac{1}{N_b}\sum _{i=1}^{N_b} \left| T\left( i\right) -T_{ref}\left( i\right) \right| , \end{aligned}$$

(12.11)

where i, T, $T_{ref}$, $N_s$, and $N_b$ represent sequence number of nodes, temperature distribution controlled by bilayer thermal sensor, temperature distribution in pure background, number of nodes in sensor and background regions after discretization, respectively. Then we add Eqs. (12.10) and (12.11) to represent the fitness function,

$$\begin{aligned} \boldsymbol{\Psi }=\boldsymbol{\Psi }_{s}+\boldsymbol{\Psi }_{b}. \end{aligned}$$

(12.12)

As a swarm intelligence optimization algorithm, PSA (Fig. 12.2) has a very high convergence rate, adopted extensively for inverse problems. PSA gets the optimal solution through the coordination of particles in the solution space, and particles constantly follow the current optimal particle. To solve the optimal problem mentioned above, we first initialize N particles $\boldsymbol{R}_j^0$ $j=1,2,...,N$, in the feasible solution space $\boldsymbol{K}$, given as

$$\begin{aligned} \boldsymbol{K}=\left\{ \boldsymbol{R}=\left( R_1,~R_2,~R_3\right) : R_{min}\le R_i<R_j\le R_{max},~i<j;~i,~j\in \left\{ 1,2,3\right\} \right\} . \end{aligned}$$

(12.13)

The characteristics of each particle are represented by position, velocity and fitness function in $\boldsymbol{K}$ space. Particles move constantly in the solution space, updating the position and velocity of individuals by tracking individual and group extremum points. Here, individual and group extremum points are the positions with the minimum fitness function among all the positions experienced by the individual and group particles. Each time positions of particles are updated, the fitness function of which are calculated. During each iteration, the updating formula of particle velocity and position are

$$\begin{aligned} \boldsymbol{V}_j^{i+1}=w\boldsymbol{V}_j^i+c_1d_1\left( \boldsymbol{P}_j^i-\boldsymbol{R}_j^i\right) +c_2d_2\left( \boldsymbol{P}_g-\boldsymbol{R}_j^i\right) , \end{aligned}$$

(12.14)

$$\begin{aligned} \boldsymbol{R}_j^{i+1}=\boldsymbol{R}_j^i+\boldsymbol{V}_j^{i+1}, \end{aligned}$$

(12.15)

where i is iteration number, j is sequence number of each particle, $\boldsymbol{P}_j^i$ is individual extremum point of j-th particle at the i-th iteration, and $\boldsymbol{P}_g$ is group extremum point. w is inertia weight usually taken as a linear decreasing function, denoted by

$$\begin{aligned} w=w_s-\left( w_s-w_e\right) \frac{i}{I_{max}}, \end{aligned}$$

(12.16)

where $w_s$, $w_e$, and $I_{max}$ are respectively initial inertia weight (for global search), final inertia weight (for local search), and total number of iteration. $c_1$ and $c_2$ are empirical constant. $d_1$ and $d_2$ are random numbers between 0 and 1. Relevant parameters of the optimization model are shown in Table 12.1. When finishing total iterations, termination conditions are met (or say minimum fitness function converged). Therefore, we get the optimal solution for design variables. Also, we have finished the design of bilayer thermal sensors.

On the same footing, we extend bilayer thermal sensors to a square case, where there is no strict analytical theory of bilayer thermal sensors. For a given set of four different bulk isotropic materials, the sides of three squares ($L_1$, $L_2$ and $L_3$, from inside to outside) are selected as design variables. Using PSA, we can obtain the geometrical size of the bilayer thermal sensor with the best performance of accuracy and invisibility.

Fig. 12.2

Adapted from Ref. [38]

a Illustration of the algorithm of PSA. b Schematic diagram of a particle swarm in search of an optimal solution. c Contour map of (b).

Fig. 12.3

Adapted from Ref. [38]

Finite-element simulation results of circular bilayer thermal sensors. The simulation box is 22 $\times$ 22 cm$^{2}$. White lines represent isotherms. a Pure background for reference. b and c First case for bare sensor with $R_1=3.42$ cm and bilayer thermal sensor with $R_1=3.42$ cm, $R_2=3.64$ cm, and $R_3=5.95$ cm. f and g Second case for bare sensor with $R_1=2.73$ cm and bilayer thermal sensor with $R_1=2.73$ cm, $R_2=4.11$ cm, and $R_3=6.85$ cm. d and e Temperature difference between (b), (c), and (a). h and i Temperature difference between (f), (g), and (a).

Table 12.1 Parameter setting of optimization model

4 Finite-Element Simulation

With each suitable set of selected materials, we obtain the optimal solution $\boldsymbol{R}$ of the design variables, representing the sensor, inner shell, and outer shell radii. For numerical demonstrations, we choose two different materials for the inner shell (Inconel alloy 625 and Stainless steel 436 with thermal conductivity of 9.8 and 30 W m$^{-1}$ K$^{-1}$) to perform finite-element simulations with COMSOL MULTIPHYSICS. Sensor, outer shell, and background are Magnesium alloy, Copper, and Aluminum with thermal conductivity of 72.7, 400, and 220 W m$^{-1}$ K$^{-1}$. Thus, we get two sets of design variables, parameterizing two cases of bilayer thermal sensors.

Before discussing the results of bilayer thermal sensors, we first show two reference schemes; one for pure background (Fig. 12.3a), the other for bare sensor (Fig. 12.3b and f). The presence of a bare sensor disturbs the thermal field of the pure background, making the thermal field in the sensor region distorted. Figure 12.3c and g show the simulation results of two cases of bilayer thermal sensors designed by PSA. External isotherms in two cases are both vertical. In two cases, the interval between internal isotherms (in the sensor region) is almost identical to the pure background. We plot the temperature-difference distributions between various schemes and pure background to accentuate the temperature difference. From Fig. 12.3d and h, we get a significant temperature deviation in sensor and background regions imposed by bare sensors. On the contrary, the temperature difference in sensor and background regions in Fig. 12.3e and i is almost zero. Furthermore, the temperature distributions along the y axis on Line 1 in Fig. 12.3 are exported to contrast the performance of these schemes quantitatively; see Fig. 12.4. Each bilayer thermal sensor maintains the same temperature distributions in the sensor and background regions as those in the pure background, demonstrating its excellent performance. However, a bare sensor not only measures the sensor region inaccurately but also distorts the thermal field in the background region, whose temperature difference (TD) is shown in the upper right inset of Fig. 12.4.

Moreover, we also perform finite-element simulations of square bilayer thermal sensors with different geometrical sizes (structure with single shell, structure with bilayer shell of random size, and structure with bilayer shell of optimized size). As expected, optimized size dramatically improves the performance of the thermal sensor, reproducing temperature distributions in sensor and background regions from the original thermal field; see Fig. 12.5.

Fig. 12.4

Adapted from Ref. [38]

Quantitative comparison of Fig. 12.3. a Temperature distributions ($T_0$, $T_1$, and $T_2$) on Line 1 in Fig. 12.3a, b, and c. Upper-right inset shows the temperature difference on Line 1 between $T_1$ ($T_2$) and $T_0$. b Temperature distributions ($T_0$, $T_3$, and $T_4$) on Line 1 in Fig. 12.3a, f, and g. Upper-right inset shows the temperature difference on Line 1 between $T_3$ ($T_4$) and $T_0$.

Fig. 12.5

Adapted from Ref. [38]

Simulation results of square bilayer thermal sensors. The simulation box is 17 $\times$ 17 cm$^{2}$. White lines represent isotherms. Sensor, inner shell, outer shell, and background are respectively expanded Magnesium alloy, Stainless steel, Copper, and Aluminum with conductivity of 72.7, 30, 400, and 220 W m$^{-1}$ K$^{-1}$. a Thermal sensor of single layer size with $L_1=2$ cm and $L_2=3.82$ cm. b Bilayer thermal sensor of random size with $L_1=3$ cm, $L_2=4$ cm and $L_3=5$ cm. c Bilayer thermal sensor of optimized size with $L_1=2$ cm, $L_2=3.82$ cm and $L_3=6.59$ cm. d–f Temperature difference between a–c and Fig. 12.3a.

Fig. 12.6

Adapted from Ref. [38]

Laboratory experiments. a Experimental setup. b Composition of bilayer thermal sensor. c and e Real photos of reference and bilayer thermal sensor. d and f Measured results for (c) and (e). White lines represent isotherms. The sample size in (c) and (e) is the same as that for Fig. 12.3a and g.

5 Laboratory Experiment

To test the performance of the bilayer thermal sensor, we select one case (Fig. 12.3g) to experiment with a setup shown in Fig. 12.6a. For comparison, we prepare two samples, one for background with pure Aluminum (Fig. 12.6c); another for the bilayer scheme made of Magnesium alloy (AZ91D), Stainless steel (ASTM 436), Copper, and Aluminum (from inside to outside) (Fig. 12.6e). Both of them have the size of 22 $\times$ 22 $\times$ 0.5 cm$^{3}$, each with two tentacles 5 cm high on the left and right, respectively. Sensor ($R_1$), inner shell ($R_1$ and $R_2$), and outer shell ($R_2$ and $R_3$) have the circular boundaries with the radii of $R_1=2.73$ cm, $R_2=4.11$ cm, and $R_3=6.85$ cm. We process the structure of four materials by laser cutting and combine these parts using mechanics enchases craft, as shown in Fig. 12.6b. The left and right tentacles of samples are immersed in 283 and 313 K water baths, and an infrared camera FLIR E60 is used to measure the temperature distributions of two samples at the steady state (after 10 min). Figure 12.6d and f show the measured results of Fig. 12.6c and e, respectively. Though thermal contact resistance exists in the interface of different materials, our scheme exhibits excellent properties in both accuracy and thermal invisibility, which is well consistent with the simulation result of Fig. 12.3g.

6 Conclusion

In summary, we have proposed an optimization model with particle swarm algorithms for designing bilayer thermal sensors composed of bulk isotropic materials. For example, we design and fabricate a circular bilayer thermal sensor with high performance, as both simulation result and experimental result exhibit. Such a scheme removes the need for extreme parameters (anisotropic, graded, or singular), making engineering applications readily and efficiently. The optimization model can also be flexibly extended to a square case. Finally, an intelligent method of simplifying structures and materials can calculate numerical solutions for difficult analytical theories (such as circular structure) and optimal solutions for problems without analytical theories (such as square structure). This property provides an insight into the development of metamaterials in a wide range of communities.

7 Exercise and Solution

Exercise

• 1. Let $y=x^2$, we initialize two particles whose x coordinates are $x_1=1$ and $x_2=2$, respectively. Using particle swarm optimization, we can get the minimum value of y. Please write down the $x_1^{(2)}$ and $x_2^{(2)}$ coordinates of the two particles after the first two iterations.

  $$\begin{aligned} \boldsymbol{V}_j^{(i+1)}=w\boldsymbol{V}_j^{(i)}+\left( \boldsymbol{P}_j^{(i)}-\boldsymbol{R}_j^{(i)}\right) +\left( \boldsymbol{P}_g-\boldsymbol{R}_j^{(i)}\right) , \boldsymbol{R}_j^{(i+1)}=\boldsymbol{R}_j^{(i)}+\boldsymbol{V}_j^{(i+1)}, \end{aligned}$$

  (12.17)

  where $w=0.5$ and $\boldsymbol{V}_j^{(0)}=0$.

Solution

• 1. First iteration of particle j, we have

  $$\begin{aligned} \boldsymbol{V}_j^{(1)}=w\boldsymbol{V}_j^{(0)}+\left( \boldsymbol{P}_j^{(0)}-x_j^{(0)}\right) +\left( \boldsymbol{P}_g-x_j^{(0)}\right) , x_j^{(1)}=x_j^{(0)}+\boldsymbol{V}_j^{(1)}. \end{aligned}$$

  (12.18)

  Second iteration of particle j, we have

  $$\begin{aligned} \boldsymbol{V}_j^{(2)}=w\boldsymbol{V}_j^{(1)}+\left( \boldsymbol{P}_j^{(1)}-x_j^{(1)}\right) +\left( \boldsymbol{P}_g-x_j^{(1)}\right) , x_j^{(2)}=x_j^{(1)}+\boldsymbol{V}_j^{(2)}. \end{aligned}$$

  (12.19)

  After substituting the values, we get $x_1^{(2)}=1$ and $x_2^{(2)}=0.5$.