Reconfigurable Three-Dimensional Thermal Dome

Yuhong Zhou; Fubao Yang; Liujun Xu; Pengfei Zhuang; Dong Wang; Xiaoping Ouyang; Ying Li; Jiping Huang

doi:10.1016/j.eng.2024.07.021

Engineering ›› 2025, Vol. 46 ›› Issue (3) :250 -258. DOI: 10.1016/j.eng.2024.07.021

Research Metamaterials—Article

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Reconfigurable Three-Dimensional Thermal Dome

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Abstract

Thermal metamaterial represents a groundbreaking approach to control heat conduction, and, as a crucial component, thermal invisibility is of utmost importance for heat management. Despite the flourishing development of thermal invisibility schemes, they still face two limitations in practical applications. First, objects are typically completely enclosed in traditional cloaks, making them difficult to use and unsuitable for objects with heat sources. Second, although some theoretical proposals have been put forth to change the thermal conductivity of materials to achieve dynamic invisibility, their designs are complex and rigid, making them unsuitable for large-scale use in real three-dimensional (3D) spaces. Here, we propose a concept of a thermal dome to achieve 3D invisibility. Our scheme includes an open functional area, greatly enhancing its usability and applicability. It features a reconfigurable structure, constructed with simple isotropic natural materials, making it suitable for dynamic requirements. The performance of our reconfigurable thermal dome has been confirmed through simulations and experiments, consistent with the theory. The introduction of this concept can greatly advance the development of thermal invisibility technology from theory to engineering and provide inspiration for other physical domains, such as direct current electric fields and magnetic fields.

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Thermal domes / Reconfigurable metamaterials / Three-dimensional invisibility

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Yuhong Zhou, Fubao Yang, Liujun Xu, Pengfei Zhuang, Dong Wang, Xiaoping Ouyang, Ying Li, Jiping Huang. Reconfigurable Three-Dimensional Thermal Dome. Engineering, 2025, 46(3): 250-258 DOI:10.1016/j.eng.2024.07.021

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1. Introduction

The urgent need to render objects invisible to infrared detection has driven substantial research into thermal cloaking [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Traditionally, thermal invisibility cloaks were designed by first enveloping the target with insulating materials and then guiding heat flow around the cloaked area to achieve invisibility. Approaches such as transformation thermotics [1], [2], [19], [20], [21], [22], [23], scattering cancellation [24], [25], [26], [27], [28], and topology optimization [29], [30], [31], [32], [33] have been developed following this methodology.

However, the thermal invisibility devices designed based on these principles face significant engineering challenges, unlike other metamaterials with vast engineering potential [34], [35], [36], [37]. These devices are difficult to manufacture and install, have limited reusability, and their fully enclosed designs cannot accommodate internal heat sources. A continuous rise in internal temperature can lead to catastrophic outcomes [38]. Remarkably, scenarios requiring the concealment of heat-generating objects are common but have been largely overlooked in previous studies. To address these limitations, researchers have started exploring non-traditional cloak designs [39], [40] that do not require full enclosure, allowing interaction with the external environment. Yet, these studies primarily focus on two-dimensional structures and heavily rely on negative thermal conductivity, which poses significant challenges in practical applications. Moreover, these new “external cloaks” fail to solve the critical issue of dissipating internal heat sources, essential for many practical applications. Thus, further research is needed to address these issues and potentially open new avenues in the field of thermal cloaking.

Reconfigurable capabilities greatly enhance a device’s ability to adapt to dynamic requirements, thereby increasing its applicability and acceptance across various engineering sectors. Existing thermal cloaking devices, however, are usually tailored to specific backgrounds, making environmental changes necessitate a redesigned cloak, which is inconvenient and economically inefficient. Researchers have proposed various methods, such as nonlinearity [41], chameleon-like behaviors [42], [43], convection [44], [45], [46], and height manipulation [47], [48], to regulate the thermal conductivity of materials and meet dynamic stealth requirements. However, implementing these techniques in three-dimensional (3D) spaces presents significant hurdles, making them impractical for many real-world applications.

This paper introduces a groundbreaking solution to these challenges: the thermal dome (Fig. 1). Designed with practical applications in mind, the thermal dome features an open hidden area, facilitating easy installation and reuse. Remarkably, this device achieves thermal invisibility for heat-generating objects—a pioneering achievement in the field. Inspired by Lego structures [49], [50], we combined an open architecture with a multi-layered design [51], giving the thermal dome a reconfigurable nature. Users can intuitively assemble it to meet specific requirements and adapt to varying environments, much like assembling Lego blocks. This flexibility and adaptability set the thermal dome apart from conventional thermal cloaks, highlighting its exceptional engineering significance. Through solving differential equations, we designed a semi-ellipsoidal thermal dome and validated its functionality in a hemispherical form using common bulk materials. The introduction of this thermal dome concept marks a paradigm shift in thermal invisibility devices, propelling them towards practical application and inspiring further exploration of their feasibility in real-world scenarios.

2. Design principles of thermal domes

We consider 3D heat transfer without convection and radiation. In a homogeneous medium with a constant thermal conductivity of κ_b, heat flows uniformly from the high-temperature surface to the low-temperature surface. However, introducing a target object disturbs the heat flow due to the object’s different thermal conductivity (κ_o). To eliminate this disturbance, a thermal dome can be placed over the target, cloaking it as an object with the same thermal conductivity as the background, thus achieving stealth. To achieve this functionality, the shape and material of the thermal dome must be carefully designed. While the thermal dome can take any shape, choosing a shape with poor symmetry can result in irregular faces, complicating the design process. The semi-ellipsoidal shape has excellent symmetry and can form various shapes by adjusting the lengths of its three axes to meet the designer’s needs, making it an ideal choice for the thermal dome. As shown in Fig. 2(a), the semi-axis of the core (dome) is specified as l_c_i (l_d_i) along the x_i axis, where i = 1, 2, and 3 represents the three dimensions. The heat transfer equation in the ellipsoidal coordinate system is written as Ref. [52]

(1)

\frac{\partial}{\partial ρ_{1}} [g (ρ_{1}) \frac{\partial T}{\partial ρ_{1}}] + \frac{g (ρ_{1})}{ρ_{1} + l_{i}^{2}} \frac{\partial T}{\partial ρ_{1}} = 0

where

g (ρ_{1}) = \prod_{i} {(ρ_{1} + l_{i}^{2})}^{1 / 2}

, ρ₁ stands for the radius in spherical coordinates. T represents temperature, and

l_{i}

represents the semi-axis of the ellipsoid. We implement an external thermal field along the x_i axis as depicted in Fig. 2(b). The semi-ellipsoid is visualized as a halved complete ellipsoid, with the solution procedure identical to that of the complete ellipsoid except for an additional boundary condition: The entire surface temperature at the base of the dome must be equalized to ensure the background temperature field remains undisturbed. Additionally, to solve the differential equations, we need to impose further boundary conditions, namely, the equality of temperatures and normal heat fluxes at the interfaces between different regions. By applying the generalized solution to these boundary conditions, we obtain the design requirement for the thermal dome:

(2)

κ_{b} = \frac{L_{c i} κ_{c} + (1 - L_{c i}) κ_{d} + (1 - L_{d i}) (κ_{c} - κ_{d}) f}{L_{c i} κ_{c} + (1 - L_{c i}) κ_{d} - L_{d i} (κ_{c} - κ_{d}) f} κ_{d}

where

f = g (ρ_{c}) / g (ρ_{d}) = \prod_{i} l_{c i} / l_{d i}

denotes the volume fraction, L_c_i and L_d_i are the shape factors of the core and the dome along the x_i axis, respectively. κ_d and κ_c represent the thermal conductivities of the thermal dome and the core, respectively. The inner and outer boundaries of the thermal dome can be denoted as ρ_c and ρ_d. The detailed solution steps are outlined in Section S1 in Appendix A.

As previously mentioned, besides the thermal conductivity condition, another requisite for the semi-ellipsoidal structure to not impact the background temperature field is to have the entire surface on which its base is located at an equal temperature. Furthermore, if the thermal conductivity of the thermal dome needs to be independent of the core region, the core region can be insulated, whereby κ_c can be considered as equal to 0 W∙m⁻¹∙K⁻¹, making the designed thermal dome applicable to arbitrary objects. Specifically, for a hemispherical thermal dome, as shown in Fig. 2(c), we can establish its thickness (d) in relation to its geometric size and thermal conductivity:

(3)

d = [{(\frac{2 r + 1}{2 r - 2})}^{\frac{1}{3}} - 1] l_{c}

where r = κ_d/κ_b represents the ratio of the thermal conductivity of the thermal dome to that of the background. Importantly, at this stage, we have introduced an insulating layer beneath the hemispherical thermal dome to ensure its functionality across various objects.

The semi-axis l_c of the core is determined by the specific usage scenario, while the thickness of the dome is closely related to the choice of material, as seen from Eq. (3). The thickness variation of a single-layer thermal dome as a function of l_c and r is depicted in Fig. 2(d). The ratio of r has a significant impact on the thickness d, with d becoming infinite when the device uses the same material as the background (i.e., r = 1). Conversely, when the thermal conductivity of the thermal dome κ_d is much larger than that of the background κ_b (i.e., r is a large number), the thickness of the thermal dome becomes very small. This is because the thermal dome acts as a compensation for thermal conductivity, and if its material has a high thermal conductivity, only a small amount of material is needed to compensate. Therefore, selecting an appropriate material to manufacture the thermal dome based on the specific scenario is crucial. For instance, if the background is made of cement and a thin thermal dome is preferred, copper would be a better choice. According to Eq. (3), the thickness of the layer decreases to 0.16 mm when l_c is set to 10 cm in this case.

The aforementioned approach can be readily expanded to accommodate core–shell structures with n-layered shells. We can employ computational software to determine the parameters for the thermal domes in each layer. An expedited method for designing a multilayer thermal dome involves leveraging effective medium theory, where the design process can be conducted iteratively, layer by layer. Further details are provided in Section S2 in Appendix A. Additionally, two recent articles on machine learning could also be referenced to inspire the design of multilayer thermal domes [53].

3. Function verification and simulation results

We utilized the commercial software COMSOL Multiphysics to perform finite-element simulations and validate our theoretical design. We conducted steady-state simulations with the heat transfer module, and the transient results will be elaborated in Section S3 in Appendix A. For simplicity, we used the hemispherical thermal dome for verification. The background is dimensioned at 30 cm × 30 cm × 15 cm and features a thermal conductivity κ_b of 10 W∙m⁻¹∙K⁻¹. A hemispherical object represented by a core with radius l_o = 9 cm and thermal conductivity κ_o = 500 W∙m⁻¹∙K⁻¹ is introduced. The temperature and isotherm distributions in the background demonstrate the perturbation of heat flow. Once heat transfer reaches equilibrium, we analyze the efficacy of our thermal dome through the examination of the temperature distribution in three distinct groups.

As depicted in Fig. 3(a), the background devoid of the object exhibits a uniform temperature distribution and straight isotherms. Upon placing of the thermal dome over the object (Fig. 3(b)), with a shell having a diameter l_a = 9 cm, thickness d_a = 1 cm, and thermal conductivity κ_a = 0.023 W∙m⁻¹∙K⁻¹ representing the adiabatic layer, and another shell having diameter l₁ = 10 cm, thickness d₁ = 1 cm, and thermal conductivity κ₁ = 55 W∙m⁻¹∙K⁻¹ representing the single-layer thermal dome, the temperature distribution and isotherms in the background precisely match those of the reference group. Here, the subscript “1” is used to denote the layer number of the thermal dome. For a single-layer thermal dome, there is only layer 1. It is worth noting that when counting the number of thermal dome layers here, we only consider the layers designed using Eq. (3), excluding the adiabatic layers. In contrast, the presence of the object distorts the temperature distribution in the background, causing the isotherms to bend and reflect the disturbance, as shown in Fig. 3(c).

To accurately contrast their differences, we exported data from a cross-section at z = 7.5 cm (−15 cm < x < 15 cm, Fig. 3(d)). We utilized a dimensionless temperature T* = 100(T₀ − T)/T₀ and a dimensionless position x* = 2x/L, where, T₀ and L denote the temperature of the reference and the length of the background, respectively. The temperature without the thermal dome (depicted by an orange line) diverges from the temperature of the reference across the entire space. Conversely, the blue line (with the thermal dome) aligns perfectly with T₀ in the background, signifying the successful achievement of the thermal cloaking effect by the thermal dome.

The scenario of a reconfigured thermal dome is considered when the background thermal conductivity changes. In such a case, with κ_b changing from 10 to 23 W∙m⁻¹∙K⁻¹, the original thermal dome no longer satisfies the stealth requirement (Fig. 3(e)). One solution is to add another layer to the outer surface of the single-layer thermal dome (Fig. 3(f)). The Lego-like structure of the thermal dome facilitates easy layering or removal, a feat that is challenging for traditional thermal cloaks. Upon reconfiguration, the new thermal dome exhibits impressive performance in the new background. To verify the preceding discussion, we plot the dimensionless temperature T* (Fig. 3(g)). Additionally, the reconstructions shown above are of the thermal dome from the outer layers. In Section S4 in Appendix A, we demonstrate how to reconstruct the thermal dome from the inner layers. The choice of whether to reconstruct from the inside out or the outside in should be based on the specific application scenario.

Past designs of thermal invisibility cloaks failed to account for scenarios with heat sources within the concealed area. However, a significant number of real-world objects that need to be hidden do emit heat, rendering conventional thermal cloaks ineffective. The open structure of the thermal dome allows the hidden area to directly engage a cold source that absorbs the heat generated within the hidden area. This ensures that the invisibility function remains uncompromised, and the internal temperature does not continually elevate. We show the simulation validation of the thermal dome and the conventional thermal cloak in Figs. 3(h) and (i), respectively.

As time extends, the temperature within the thermal dome remains virtually stable, whereas the temperature within the traditional thermal cloak’s space continues to rise. For further analysis, we selected two points inside the hidden region and in the background area, and plotted their temperature values over time in Figs. 3(j) and (k). The comparison with the pure background reference group reveals the disadvantages of traditional thermal cloaks when heat sources are present in the hidden area: First, the internal temperature incessantly escalates over time; second, in the absence of entirely insulating materials, the elevated internal temperature impacts the temperature distribution in the background, leading to invisibility function failure. In contrast, the thermal dome not only preserves excellent invisibility but also maintains a stable internal temperature, akin to the cold source temperature. Thus, the thermal dome emerges as an effective solution for concealing objects with heat sources.

Throughout the preceding discussion, regardless of whether we consider a single-layer or multi-layer thermal dome, the temperature bias is vertical, directing heat flow from top to bottom. However, altering the direction of the temperature bias to horizontal does not compromise the perfect cloaking function of the thermal dome, as shown in Section S5 in Appendix A. When the temperature bias is arbitrary (Fig. S6 in Appendix A), the temperature field becomes non-uniform, and the cloaking function of the thermal dome is no longer perfect. Yet, the simulation results reveal that the thermal dome still offers a substantial amount of cloaking effect compared to the control group without the thermal dome. Under such conditions, the object remains undetectable by low-precision infrared cameras. Furthermore, other shapes of thermal domes will be discussed in Section S6 in Appendix A.

4. Experimental validation of the thermal dome

4.1. Experimental results

We conducted experimental validation of the hemispherical thermal dome, and the results are shown in Fig. 4. Due to constraints of the experimental setup, the heat source was placed below, and the cold source was positioned above. The temperature distribution within the background is inferred from its surface. Any disturbance in the background heat flow causes distortion in its surface isotherms; otherwise, the isotherms remain straight. Therefore, we used an infrared camera to assess the temperature distribution on the sample’s surface, validating the function of the thermal dome.

In the single-layer thermal dome experiment (Fig. 4(c)), layer 1 functioned as the thermal dome, while cement served as the background. Three samples were prepared: a reference group comprised solely of the background, a control group with the presence of objects but without a thermal dome, and a group with objects protected by the thermal dome. The experimental and simulation results for these three groups are presented in Fig. 4(d). Both the experimental and simulation outcomes demonstrate that the isotherms of the group without a thermal dome are distorted, whereas the isotherms of the group equipped with a thermal dome and the reference group remain straight. This indicates that the thermal dome shields the object, preventing its detection. For an intuitive comparison, data from the same line were analyzed, as shown in Fig. 4(e). Here, T* = 100(303 K − T)/303 K is a dimensionless temperature that represents deviation. Both the reference group and the group with a thermal dome exhibit a temperature close to 303 K, with deviations approximating 0 K. The group without a thermal dome shows a significantly larger temperature deviation.

Fig. 4(f) illustrates the functionality of the reconfigured thermal dome in a new background, where stainless steel (316L) was used as the background material, and the thermal dome consisted of layer 1 and layer 2. Experimental results along with corresponding simulation results for the group with the new thermal dome and the group with the original thermal dome are displayed in Fig. 4(g). Additionally, T and T* are plotted in Fig. 4(h) to visually compare the efficacy of the new multi-layer thermal dome with the original single-layer dome. From Figs. 4(g) and (h), we can conclude the following: Multi-layer thermal domes, with their Lego-like structures, can adapt to various environments by simply adjusting the number of layers—a feat challenging to achieve with traditional thermal cloaks.

In real-world applications, there are inevitable additional factors that affect the functionality of the thermal dome. For example, during the assembly of a multi-layer thermal dome, the thermal contact resistance (TCR) between different layers can impact its performance. In Section S7 in Appendix A, we discuss various factors influencing the TCR and simulate the temperature distribution of the thermal dome considering TCR. We also explore strategies to mitigate the influence of TCR on the functionality of the thermal dome. Additionally, convective and radiative heat transfer between the sample and the external environment cannot be disregarded in practical applications. In Section S8 in Appendix A, we address these factors and conclude that the thermal dome maintains its functionality when convective and radiative heat transfers do not significantly alter the original temperature distribution—that is, when the temperature field remains relatively uniform.

4.2. Experimental setups

4.2.1. Background materials

Cement: dimensions: 15 cm × 15 cm × 7.5 cm; thermal conductivity: κ_b = 1.28 W∙m⁻¹∙K⁻¹. The cement used in this study is a common construction material. Its thermal conductivity varies depending on moisture content and curing time. Uniform thermal conductivity was maintained across all sample sets by using cement from the same batch and initiating the curing process simultaneously.

316L steel: dimensions: 15 cm × 15 cm × 7.5 cm; thermal conductivity: κ_b = 16.2 W∙m⁻¹∙K⁻¹. This steel variant was shaped using computer numerical control (CNC) machining.

4.2.2. Layers

Layer 1: A 316L stainless steel layer created using 3D printing, with a diameter l₁ of 6 cm, thickness d₁ of 0.25 cm, and thermal conductivity κ₁ of 16.2 W∙m⁻¹∙K⁻¹.

Layer 2: A copper layer created through CNC machining, having a diameter l₂ of 6.25 cm, thickness d₂ of 0.12 cm, and thermal conductivity κ₂ of 385 W∙m⁻¹∙K⁻¹.

4.2.3. Target object and adiabatic layer

Constructed of styrofoam for simplicity, both the target object and the adiabatic layer possess the same thermal conductivity, κ_o = κ_a = 0.042 W∙m⁻¹∙K⁻¹, and a maximum radius l_a of 6 cm. Please be aware that in this experiment, we have used insulating material as the target for the sake of convenience. However, in practical applications, particularly in cases involving internal heat sources, it is neither necessary nor advisable to entirely encase the target with an insulating layer, as this would hinder the efficient discharge of heat flow through the base. It suffices for the insulating layer to isolate the target from the thermal dome.

4.2.4. Thermal sources

Hot source: This consists of a heating table set to a constant temperature of 323 K and a copper oil bath pan, which is placed directly on the heating table and filled with silicone oil.

Cold source: A copper pan filled with an ice-water mixture acts as a cold source, maintaining a temperature of 273 K.

4.2.5. Sample preparation

Three sets of samples were prepared: a reference group, a thermal dome group, and a group without a thermal dome. We began by fabricating three 15 cm × 15 cm × 7.5 cm cuboid molds using polyvinyl chloride foam boards, followed by a common batch of cement. The reference group involved simply pouring the cement into the mold, followed by mixing and drying. For the thermal dome group, we fixed the thermal dome to the mold bottom, poured cement, mixed it, and allowed it to dry. The group without a thermal dome involved a similar process, but with a resin shell manufactured via 3D printing that matched the thermal dome’s size. After a fortnight, the cement was fully dried, the molds and resin shell removed, and the resulting voids filled with styrofoam. For samples featuring a 316L steel background, we directly placed the thermal dome into the pre-formed background.

4.2.6. Experimental setup construction

To minimize convective heat transfer between the sample surface and the environment, we first wrapped two layers of foam around the sample. We then prepared a copper basin for the oil bath, which was placed on a heating platform set to 323 K, filled with silicone oil. This oil bath heating method ensured consistent heating of the bottom surface, eliminating inconsistencies caused by gaps between the sample and the heating platform. To serve as a cold source, another copper basin filled with an ice-water mixture was placed atop the sample, with the intervening gap filled with silicone grease to ensure effective contact. To avoid uneven temperatures caused by disparate ice distribution, we maintained the ice in a single block floating atop the water. The ambient temperature in the laboratory was kept at 298 K.

4.2.7. Data collection

Post-heating, we collected temperature data from the observation surface using an infrared camera at 10 s intervals. To mitigate environmental interference, we used a blackboard as a backdrop behind the sample. We also applied a transparent film on the sample surface and used a black foam backdrop to reduce the effect of material emissivity on temperature measurements. While these measures may not entirely eliminate emissivity effects, they ensure accurate temperature differences between samples under identical environmental conditions, providing reliable conclusions.

Actual photographs of the experimental setup can be found in Section S9 in Appendix A. Furthermore, in Section S10 in Appendix A, it is verified that convective heat exchange between the observation surface and the environment does not affect the analysis of the experimental results.

5. Conclusions and discussions

In this paper, we depart from traditional methods of designing thermal invisibility devices and propose a new approach that directly guides heat flow towards isothermal surfaces. Building on this foundation, combining classic scattering cancellation and effective thermal conductivity methods, we have designed an open-structured invisibility device called the thermal dome. Although the bilayer thermal cloak also utilizes scattering cancellation and appears similar in structure, significant differences exist.

First, due to different design philosophies, the thermal dome directly guides heat flow towards isothermal surfaces, resulting in an open structure, unlike the closed structure of the bilayer thermal cloak. This open structure provides several advantages, including simplicity in engineering applications and the ability to cloak self-heating objects. Furthermore, incorporating a multi-layer structure concept adds reconfigurability similar to building blocks. While the concept of multi-layer structures is not new, its potential to make thermal cloaks reconfigurable has not been fully explored due to the closed nature of previous designs, which made practical reconfiguration challenging.

Therefore, it is crucial not to equate the thermal dome with bilayer thermal cloaks solely based on their use of scattering cancellation. Clear distinctions must be made in terms of approach, structure, and application to fully appreciate the significance of this new concept in advancing thermal invisibility devices.

Notably, while we have set the base of the thermal dome to an isothermal boundary condition in our paper to achieve zero disturbance to the background, it is entirely feasible to replace this condition with a substantial heat reservoir in practical applications. Under such circumstances, the disturbance caused by heat flow directed towards the heat reservoir through the thermal dome can be considered negligible. For example, the vicinity of a large heat sink or the Earth can be approximated as a constant temperature heat source. Alternatively, if the goal is to prevent disturbance to the background temperature at a particular location caused by an object, the thermal dome can effectively guide the heat flow through the object and disperse it to less critical areas. This novel approach opens up a new research direction in the field of thermal invisibility. Further exploration of this concept could provide substantial theoretical support for the practical application of thermal cloaking devices.

Acknowledgments

This work was supported by the National Natural Science Foundation of China to Jiping Huang (12035004 and 12320101004), the Innovation Program of the Shanghai Municipal Education Commission to Jiping Huang (2023ZKZD06), the National Natural Science Foundation of China to Ying Li (92163123 and 52250191), the Zhejiang Provincial Natural Science Foundation of China to Ying Li (LZ24A050002), and the National Natural Science Foundation of China to Liujun Xu (12375040, 12088101, and U2330401).

Compliance with ethics guidelines

Yuhong Zhou, Fubao Yang, Liujun Xu, Pengfei Zhuang, Dong Wang, Xiaoping Ouyang, Ying Li, and Jiping Huang declare that they have no conflict of interest or financial conflicts to disclose.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.eng.2024.07.021.

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