Keywords

2.1 Opening Remarks

The emergence of diffusion metamaterials [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] marks a transformative era in materials science, with theoretical advancements [17,18,19] in heat and mass transfer being actualized into dynamic, practical innovations [20,21,22]. This chapter navigates through the intricate terrain of simulation methodologies that are pivotal to this evolution, especially when contending with complex geometries. It dissects the role of finite-element simulations in visualizing physical field distributions, the strategic use of particle swarm optimization for crafting optimal designs [23, 24], the ingenuity behind topology optimization [25, 26] for emulating inhomogeneous and anisotropic properties, and the profound impact of machine learning algorithms [27] in refining design parameters and enhancing material functionality. Together, these simulation strategies, supported by intelligent algorithms and finite element analysis, are not merely reshaping metamaterials; they are setting the stage for a future where the versatility and ingenuity of these materials [28,29,30] could fundamentally alter the domain of thermal management and metamaterial engineering [31,32,33,34,35,36].

2.2 Finite-Element Simulation

Finite-element simulation [37] stands as a cornerstone in the design and analysis of diffusion metamaterials, providing a robust computational framework to predict and visualize the behavior of complex systems. At its core, this technique subdivides a large problem into smaller, simpler parts that are easier to understand, analyze, and solve—these are known as finite elements. What sets finite-element simulation apart is its adaptability and exactitude. It initiates with the crafting of a geometric mesh that echoes the actual structure of the material or system under study. This mesh comprises elements, each endowed with attributes that mirror the actual properties of the material, encompassing thermal conductivity, density, and specific heat capacity. The simulation then invokes the fundamental laws of physics, like the principles of energy and momentum conservation, expressed through differential equations. These equations are methodically spread across the mesh, effectively turning a continuous space into a solvable numerical puzzle of algebraic equations. This pivotal step permits the computation of critical field variables—temperature, pressure, stress—at specific junctures, thereby shedding light on the material’s behavior in diverse scenarios. Finite-element simulations shine particularly when applied to materials with unconventional geometries and boundary conditions—scenarios where analytical solutions are elusive. They provide granular analysis of variations and distributions within the metamaterial, thereby aiding the fine-tuning of thermal attributes and forecasting performance under real-world pressures. When enhanced by advanced algorithms, finite element analysis transcends mere prediction; it becomes an oracle that guides the iterative journey of design, leading to the conception of metamaterials with bespoke diffusion traits. This approach is a catalyst in the evolution of metamaterials, clearing a path for groundbreaking advancements in thermal regulation, wave control, and more. As our exploration deepens into the science of diffusion metamaterials, the role of finite element simulation becomes ever more crucial. It acts as a bridge from the theoretical to the tangible, helping to craft materials that rise to meet the multifaceted challenges of today’s technological landscape.

2.3 Particle Swarm Optimization

Particle swarm optimization is a computational method inspired by the social behaviors observed in flocks of birds or schools of fish. This technique, conceptualized by Kennedy, a social psychologist, and Eberhart, an electrical engineer, treats the search for solutions as a process of collective intelligence, much like a swarm’s natural movement toward the most promising paths [38]. The particle swarm optimization journey begins with assigning a flock of candidate solutions throughout the available solution space, with each ’bird’ representing a possible answer to the problem at hand, evaluated by a designated fitness function that measures the quality of the solution. The method then involves calculating the fitness for each candidate’s current location and pinpointing the best solutions found by any individual and the collective swarm up to that point. As the optimization unfolds, the swarm’s individuals adjust their trajectories based on the best solutions identified, leveraging the wisdom of the group to navigate towards the optimal outcome. This iterative process of evaluation and adjustment continues until a candidate solution satisfies a predetermined threshold for success, signaling the end of the search. Particle swarm optimization proves particularly adept as an inverse design tool for complex problems that defy traditional analytical approaches. It has recently been harnessed for the reverse engineering of diffusion metamaterials, allowing for the deduction of geometric or material parameters based on specific desired properties [24]. This makes particle swarm optimization a powerful ally in the development of metamaterials, where the target performance dictates the design parameters.

2.4 Topology Optimization

Topology optimization has emerged as a versatile algorithm for crafting thermal metamaterials, enabling the fine-tuning of natural material distributions to achieve exceptional thermal performances. This method, pioneered by Svanberg [39], has catalyzed the development of metadevices that leverage the optimized arrangement of their components to enhance their thermal properties. The power of topology optimization shines in its ability to fabricate transformation-theory based metamaterials, which traditionally grapple with the need for materials exhibiting inhomogeneous and anisotropic parameters—qualities rarely found in nature. Researchers have embraced this challenge by innovating a concept of topological functional cells, a strategy that permits the crafting of materials with the required complex properties [26]. Their methodology unfolds in a three-step process: First, they compute the desired thermal conductivity distribution across the metamaterial domain, guided by transformation theory. Next, they apply topology optimization to each functional cell to sculpt the desired thermal conductivity tensor. Finally, these optimized cells are assembled to form a metamaterial endowed with tailored inhomogeneous and anisotropic conductivities. The optimization of a single topological functional cell adheres to the following formulation:

$$\begin{aligned} \begin{aligned} &\textrm{min}~ C = \frac{1}{|V|}\sum _{e=1}^N \rho _e, \\ &s.t.: {\textbf{K}}(\rho _e) {\textbf{T}} = {\textbf{Q}}, \\ &G=f((\kappa _{lm}^\textrm{Output} - \kappa _{lm}^\textrm{Input})^2) = 0, \\ &0 \le \rho _e \le 1,~e = 1,~2...N,\\ &\kappa _{lm}^\textrm{Input} = \begin{pmatrix} \kappa _{11}^\textrm{Input} &{} \kappa _{12}^\textrm{Input} \\ \kappa _{21}^\textrm{Input} &{} \kappa _{22}^\textrm{Input} \end{pmatrix}(l,m=1,2),\\ &\kappa _{lm}^\textrm{Output} = \begin{pmatrix} \kappa _{11}^\textrm{Output} &{} \kappa _{12}^\textrm{Output} \\ \kappa _{21}^\textrm{Output} &{} \kappa _{22}^\textrm{Output} \end{pmatrix}(l,m=1,2). \end{aligned} \end{aligned}$$
(2.1)

In this formula, \(\rho _e\) signifies a variable dictating the material makeup, which ranges from \(\rho _e=0\) (material 1) to \(\rho _e=1\) (material 2). N represents the amount of \(\rho _e\), and |V| encapsulates the volume of the whole topological functional cell. \({\textbf{K}}(\rho _e)\), \({\textbf{T}}\), and \({\textbf{Q}}\) are globally heat conduction, temperature, and heat load matrices, respectively. \(\kappa _{lm}^\textrm{Input}\) and \(\kappa _{lm}^\textrm{Output}\) represent target and optimized thermal conductivity tensors, respectively. f is a function to evaluate the discrepancy between \(\kappa _{lm}^\textrm{Output}\) and \(\kappa _{lm}^\textrm{Input}\). The distribution of \(\rho _e\) is refined via the classical method of moving asymptotes, ensuring a precise and efficient optimization process.

2.5 Machine Learning

Intelligent materials that incorporate artificial intelligence algorithms into their design have garnered significant attention in diverse fields like optics, nanotechnology, and acoustics. Yet, in the realm of diffusion physics, the paucity of controllable degrees of freedom presents a formidable challenge to applying these innovative technologies to diffusion metamaterials. Thermal metamaterials provide a pertinent case in point. Current advancements have primarily been confined to inverse design strategies concerning geometrical or material parameters. For instance, researchers have adeptly utilized machine learning algorithms to optimize a four-layer thermal cloak [40]. They trained an artificial neural network to process the thermal conductivities \(k_1,k_2,k_3\), and \(k_4\) of four isotropic materials (four middle layers) and output two objective functions that assess the cloak’s efficacy. One function gauges the temperature uniformity within the cloaked area,

$$\begin{aligned} \Delta T = |T_{x=r_1}-T_{x=-r_1}|, \end{aligned}$$
(2.2)

while the other measures the thermal neutrality against the ambient background,

$$\begin{aligned} M_V = \frac{\int _{\Omega }|T(x,y,z)-T_r(x,y,z)|d\Omega }{\int _{\Omega }d\Omega }, \end{aligned}$$
(2.3)

where \(\Omega \) represents the domain \(r > r_5\), and \(T_r\) is the temperature distribution of a uniform medium. After processing 10,000 design samples, the neural network achieved an inverse mapping between the output metrics {\(\Delta T\), \(M_V\)} and the input thermal conductivities. This mapping facilitated the calculation of optimal material conductivities for superior cloaking, resulting in a device engineered from copper and poly-dimethylsiloxane based on effective medium theory, which upheld a uniform temperature in the cloaked region without disturbing the surrounding thermal field. However, this machine-learning-assisted thermal cloak is inherently static, lacking the ability to adapt dynamically to changing conditions. In a leap forward, further integration of artificial intelligence and advanced hardware led to the conception of a new class of thermal metamaterial. This design is distinguished by its parameters that autonomously adjust to environmental changes [27]. A micro infrared camera monitors the temperature of a bilayer structure, and a computing system equipped with a trained artificial neural network dynamically modulates the core region’s thermal conductivity by altering the spinning angular velocity via a stepper motor. The network’s architecture, connecting input and output through four hidden layers, is defined as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \boldsymbol{H}^{(i+1)}=\textrm{ReLU} \left( \boldsymbol{W}^{(i)}\boldsymbol{T}^{(i)}+\boldsymbol{b}^{(i+1)}\right) ,~i~=~0\\ \boldsymbol{H}^{(i+1)}=\textrm{ReLU} \left( \boldsymbol{W}^{(i)}\boldsymbol{H}^{(i)}+\boldsymbol{b}^{(i+1)}\right) ,~0~<~i~<~4\\ \omega _1=\textrm{ReLU} \left( \boldsymbol{W}^{(i)}\boldsymbol{H}^{(i)}+\boldsymbol{b}^{(i+1)}\right) ,~i~=~4 \end{array}\right. } \end{aligned}$$
(2.4)

where \(\boldsymbol{H}^{(i)}\) represent the activations of the current layer, \(\textrm{ReLU}\left( \mathrm a\right) =\textrm{max}\left( 0,\mathrm a\right) \) is the rectified linear unit function, \(\boldsymbol{W}^{(i)}\) and \(\boldsymbol{b}^{(i)}\) are the weights and biases for neurons in the i-th layer. This sophisticated setup connects ambient temperature changes to thermal functionality, creating a responsive device that epitomizes the next wave in intelligent material design: self-adaptive metamaterials, which are poised to redefine the landscape of material science.

2.6 Outlook

Contemporary computational techniques, such as finite-element simulations, offer a direct method to model the physical field distributions of metamaterials under specified environmental conditions [41]. For problems lacking analytical solutions, particle swarm optimization can be utilized to find the optimal configurations. Notably, thermal sensors can achieve superior performance using conventional bulk materials by fine-tuning their dimensions [23, 24]. Topology optimization [25, 26] is instrumental in manipulating the placement of natural materials to simulate inhomogeneous and anisotropic thermal conductivities, enabling metamaterials to maintain high efficiency across various shapes. Machine learning algorithms have the potential to act as inverse design solvers, deducing the necessary geometric structures and material parameters based on desired physical attributes. Furthermore, machine learning can significantly enhance the adaptability and functionality of traditional static metamaterials. A recent landmark study introduced a deep learning-assisted active metamaterial [27] that functions as a configurable nonlinear thermal material. Such nonlinear materials pave the way for unidirectional heat transfer [42, 43], offering fresh perspectives for creating asymmetric thermal couplings used for the classical Su-Schrieffer-Heeger model. Consequently, these innovative active metamaterials may serve as platforms for exploring novel mechanisms in topological thermal transport and the development of advanced thermal metadevices.