A metamaterial self-similar nested structure and a construction method and application thereof

Abstract

The application discloses a metamaterial self-similar nested structure and a construction method and application thereof, relates to the technical field of self-similar metamaterial structure design, and aims to solve the technical problem of the deficiency of the prior art.The construction method of the metamaterial self-similar nested structure comprises the following steps: taking a simple cubic unit structure, a face-centered cubic unit structure and a simple cubic-face-centered cubic mixed unit structure as a basic unit, nesting n self-similar units with coincident geometric centers in the basic unit, and connecting the inner and outer units in a body-centered cubic unit connection mode to obtain a new metamaterial self-similar nested structure; wherein the edge length of the i-th self-similar unit from the inside to the outside is i / n of the edge length of the outermost basic unit, and n represents the order of the unit.

Description

Technical Field

[0001] This application relates to the field of self-similar metamaterial structure design technology, specifically to a metamaterial self-similar nested structure, its construction method, and its application. Background Technology

[0002] In existing technologies, the gradient intensity distribution method constructs an intensity progression relationship through a global array, which can extend the energy absorption stage but requires complex control. The multi-platform method achieves layered collapse through the design of multi-level intensity regions within the unit cell, offering greater structural design flexibility. Improved designs based on FCC lattices significantly enhance energy absorption capacity by introducing structures such as cross beams to increase the number of plastic hinges. Current designs generally suffer from complex correlations between geometric configuration and nonlinear mechanical response, and difficulties in precisely controlling the stress triggering threshold of multi-platforms. There is an urgent need to develop novel self-similar metamaterial design methods to achieve controllable multi-level energy absorption. Summary of the Invention

[0003] The main purpose of this application is to provide a metamaterial self-similar nested structure, its construction method and application, in order to solve the technical problems of the shortcomings of the prior art.

[0004] To achieve the above objectives, the technical solutions adopted in the embodiments of this application are as follows: In a first aspect, embodiments of this application provide a method for constructing a metamaterial self-similar nested structure, comprising the following steps: Using simple cubic cell structure, face-centered cubic cell structure and simple cubic-face-centered cubic hybrid cell structure as basic cells, new metamaterial self-similar nested structures are obtained by nesting n self-similar cells with coincident geometric centers in the basic cells and connecting the inner and outer cells in the same way as body-centered cubic cells. In this context, the side length of the i-th self-similar cell from the inside out is i / n of the outermost basic cell, where n represents the order of the cell.

[0005] As some optional embodiments of this application, the side length of the simple cubic cell embedded in the first-order cell of the simple cubic cell structure is 1 / 2 of the basic cell, and the side lengths of the simple cubic cells embedded in the second-order cell are 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside to the outside.

[0006] As some optional embodiments of this application, the side length of the face-centered cubic cell embedded in the first-order cell of the face-centered cubic cell structure is 1 / 2 of the basic cell, and the side lengths of the face-centered cubic cells embedded in the second-order cell are 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside to the outside.

[0007] As some optional embodiments of this application, the side length of the simple cubic-face-centered cubic hybrid cell embedded in the first-order cell of the simple cubic-face-centered cubic hybrid cell structure is 1 / 2 of the basic cell, and the side lengths of the simple cubic-face-centered cubic hybrid cell embedded in the second-order cell are 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside to the outside.

[0008] Secondly, this application also provides a metamaterial self-similar nested structure, which is constructed by the method described above.

[0009] As some optional embodiments of this application, the cell array is a 3×3×3 structure.

[0010] As some optional embodiments of this application, the cell is filled with 27 zero-order simple cubic cells, 27 zero-order face-centered cubic cells, or 27 zero-order simple cubic-face-centered cubic hybrid cells.

[0011] As some optional embodiments of this application, the cell filling consists of 9 zero-order simple cubic cells, 9 first-order simple cubic cells, or 9 second-order simple cubic cells, filled from top to bottom.

[0012] As some optional embodiments of this application, the cell is filled from top to bottom with 9 zero-order simple cubic-face-centered cubic mixed cells, or 9 first-order simple cubic-face-centered cubic mixed cells, or 9 second-order simple cubic-face-centered cubic mixed cells.

[0013] As some optional embodiments of this application, the cell is filled from top to bottom with 9 second-order face-centered cubic cells, 9 second-order simple cubic cells, or 9 second-order simple cubic-face-centered cubic hybrid cells. As can be seen from the above, the construction method and the self-similar nested structure of metamaterials provided in this application simplify the correlation mechanism between geometric configuration and mechanical response by adopting a self-similar nested structure and a fixed ratio side length design. It has the advantages of simplifying the correlation mechanism between geometric configuration and mechanical response and achieving precise control of stress triggering threshold on multiple platforms. Attached Figure Description

[0014] Figure 1 This is a flowchart illustrating the construction method of the metamaterial self-similar nested structure involved in the embodiments of this application; Figure 2 This is a schematic diagram of the simple cubic (SC) cell structure involved in the embodiments of this application; Figure 3 This is a schematic diagram of the face-centered cubic (FCC) cell structure involved in the embodiments of this application; Figure 4 This is a schematic diagram of a simple cubic-face-centered cubic (SF) hybrid cell structure involved in the embodiments of this application; Figure 5 This is a schematic diagram of a pure zero-order simple cubic (SC) cell-filled array structure involved in the embodiments of this application; Figure 6 This is a schematic diagram of a pure zero-order face-centered cubic (FCC) cell-filled array structure involved in the embodiments of this application; Figure 7 This is a schematic diagram of a pure zero-order simple cubic-face-centered cubic (SF) cell-filled array structure involved in the embodiments of this application; Figure 8 This is a schematic diagram of a 0-2 order simple cubic (SC) cell-filled array structure involved in the embodiments of this application; Figure 9 This is a schematic diagram of a 0-2 order simple cubic (SC) cell-filled array structure involved in the embodiments of this application; Figure 10 This is a schematic diagram of a 0-2 order simple cubic (SC) cell-filled array structure involved in the embodiments of this application; Figure 11 This is a schematic diagram of a specific embodiment of the hybrid-filled 3×3×3 cell array involved in this application. Detailed Implementation

[0015] The technical solutions of this application will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are merely some embodiments of this application, and not all embodiments. The components of this application described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of this application provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely to illustrate selected embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without inventive effort are within the scope of protection of this application.

[0016] It should be noted that similar reference numerals and letters in the following figures indicate similar items; therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. Furthermore, in the description of this application, terms such as "first," "second," etc., are used only to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0017] In the field of metamaterial design, there is a strong coupling relationship between geometric configuration and nonlinear mechanical response, which makes it difficult to accurately control the stress triggering threshold of multiple platforms. This significantly limits the controllability of the energy absorption stage. Specifically, small changes in structural geometric parameters can easily trigger nonlinear abrupt changes in mechanical response, resulting in a lack of predictability in the design optimization process, which in turn affects the stable realization of multi-level energy absorption functions.

[0018] For example, in the application of collision energy-absorbing structures in automotive safety systems, metamaterials are used as the front energy-absorbing zone. The complexity of their cell geometry causes uncontrollable fluctuations in the stress triggering threshold under dynamic loads. Actual testing has shown that inner cells trigger prematurely during low-speed collisions while outer cells fail to participate effectively, or the triggering sequence is disordered during high-speed collisions, resulting in inconsistent collapse processes. This reduces energy absorption efficiency and weakens the overall structural stability. If this problem is not resolved, the energy absorption process will not proceed as expected in stages, increasing the risk of localized stress concentration, raising the likelihood of structural failure, and ultimately leading to a decrease in the reliability of the safety system.

[0019] To achieve controllable multi-stage energy absorption, such as Figure 1 As shown, the method for constructing a self-similar nested structure of metamaterials includes the following steps: Using simple cubic cell structure, face-centered cubic cell structure and simple cubic-face-centered cubic hybrid cell structure as basic cells, new metamaterial self-similar nested structures are obtained by nesting n self-similar cells with coincident geometric centers in the basic cells and connecting the inner and outer cells in the same way as body-centered cubic cells. In this context, the side length of the i-th self-similar cell from the inside out is i / n of the outermost basic cell, where n represents the order of the cell.

[0020] The schematic diagram of the simple cubic (SC) cell structure is shown below. Figure 2 As shown, the basic simple cubic cell is as follows: Figure 2 As shown in a, the corresponding first-order cells and second-order cells are as follows: Figure 2 b and Figure 2 As shown in c; the side length of the simple cubic cell embedded in the first-order cell is 1 / 2 of the basic cell, and the side lengths of the simple cubic cells embedded in the second-order cell are 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside to the outside.

[0021] The schematic diagram of the face-centered cubic (FCC) cell structure is shown below. Figure 3 As shown, the basic face-centered cubic cell is as follows: Figure 3 As shown in a, the corresponding first-order cells and second-order cells are as follows: Figure 3 b and Figure 3 As shown in c; the side length of the face-centered cubic cell embedded in the first-order cell is 1 / 2 of the basic cell, and the side lengths of the face-centered cubic cells embedded in the second-order cell are 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside to the outside.

[0022] The schematic diagram of the simple cubic-face-centered cubic (SF) hybrid cell structure is shown below. Figure 4 As shown, the basic simple cubic-face-centered cubic hybrid cell is as follows: Figure 4 As shown in a, the corresponding first-order cells and second-order cells are as follows: Figure 4 b and Figure 4As shown in c; the side length of the simple cubic-face-centered cubic hybrid cell embedded in the first-order cell is 1 / 2 of the basic cell, and the side lengths of the simple cubic-face-centered cubic hybrid cell embedded in the second-order cell are 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside to the outside.

[0023] like Figure 5 As shown, after obtaining the corresponding n-order cells, the cells can be filled into a 3×3×3 cell array to form a corresponding cell array. The 3×3×3 cell array is filled with zero-order simple cubic cells, forming a pure zero-order simple cubic cell array. For example... Figure 6 As shown, the 3×3×3 cell array is filled with zero-order face-centered cubic cells, forming a pure zero-order face-centered cubic cell array. Figure 7 As shown, the 3×3×3 cell array is filled with zero-order simple cubic-face-centered cubic hybrid cells, forming a pure zero-order simple cubic-face-centered cubic hybrid cell array. Figure 8 As shown, the 3×3×3 cell array is filled with zero-order, first-order, and second-order simple cubic cells from top to bottom in a gradient manner, forming a 0-2 gradient simple cubic cell array. Figure 9 As shown, the 3×3×3 cell array is filled with zero-order, first-order, and second-order face-centered cubic cells from top to bottom in a gradient manner, forming a 0-2 gradient face-centered cubic cell array. Figure 10 As shown, the 3×3×3 cell array is filled from top to bottom with zero-order, first-order, and second-order simple cubic-face-centered cubic mixed cells, forming a 0-2 gradient simple cubic-face-centered cubic mixed cell array. Figure 11 As shown, the 3×3×3 cell array is filled from top to bottom with second-order face-centered cubic cells, second-order simple cubic cells, and second-order simple cubic-face-centered cubic hybrid cells, forming a hybrid cell array.

[0024] It should be noted that the above embodiments are only preferred 3×3×3 array cell filling methods of this application, which can make the coupling effect of the metamaterial self-similar nested structure more uniform. When using this application, other distribution methods can also be used for filling according to different application requirements.

[0025] For ease of understanding, the following explains some key terms in this embodiment: Metamaterial self-similar nested structures refer to structures composed of repeating units (cells) that exhibit geometric self-similarity at different scales, where smaller cells are nested inside larger cells. This structural design aims to achieve specific physical properties through multi-level hierarchies.

[0026] The construction method refers to a series of steps or processes followed to create the aforementioned self-similar nested metamaterial structures. This method defines the formation path of the structure from basic units to the final complex form.

[0027] The basic cell is the most fundamental unit for constructing self-similar nested structures in metamaterials. This basic cell can be a simple cubic cell structure, a face-centered cubic cell structure, or a hybrid cell structure combining features of both simple cubic and face-centered cubic structures. The choice of the basic cell provides the initial geometric framework for the subsequent nesting process.

[0028] Self-similar cells are geometrically similar to the basic cell, but scaled down. These cells are key elements in nested structures, repeating the geometric features of the basic cell at different scales.

[0029] Geometric center coincidence means that all nested self-similar cells and basic cells share the same precise geometric center point. This concentric arrangement ensures the symmetry of the structure in all directions and the uniform transmission of forces.

[0030] Body-centered cubic (BCC) cell connectivity refers to a specific connectivity topology inspired by the connectivity patterns of BCC lattices. In this connectivity, internal structural elements are typically connected to the center or specific nodes of surrounding or nested elements, thereby establishing robust structural connections and load transfer paths between cells of different scales.

[0031] The order (n) of a cell represents the depth or complexity of a self-similar nested structure. A higher order means more nested levels and a higher degree of refinement of the structure at different scales.

[0032] The rule that the side length of the i-th self-similar cell from the innermost to the outermost layer is i / n times the side length of its outermost basic cell defines the scaling relationship of nested cells. For a given order n, the side length of the i-th self-similar cell from the innermost layer (i=1) to the outermost layer (i=n) is i / n times the side length of its outermost basic cell. This proportional relationship ensures a systematic progression in the cell size of each layer in the nested structure.

[0033] The main features of the above construction method will be explained in more detail below: This method first involves the selection of the basic cell. This basic cell can be a simple cubic cell structure, a face-centered cubic cell structure, or a hybrid simple cubic-face-centered cubic cell structure. For example, a simple cubic cell structure can be chosen as the basis, characterized by its simplicity and ease of construction. Alternatively, a face-centered cubic cell structure can be chosen, which has higher packing density and different connectivity properties. A hybrid cell structure can also be selected to combine the advantages of different cell types, providing richer geometric properties.

[0034] Furthermore, n self-similar cells with concentric geometric centers are nested within the selected basic cell. This means that a series of self-similar cells of gradually decreasing size are placed concentrically inside the basic cell. For example, if two self-similar cells need to be nested (i.e., n=2), a smaller self-similar cell is placed at the geometric center of the basic cell, and the basic cell itself is considered the outermost self-similar cell in the nesting sequence. This concentric nesting arrangement ensures that the structure maintains geometric continuity and symmetry at different scales.

[0035] Furthermore, the inner and outer cells are connected using body-centered cubic cells. This connection method ensures mechanical coupling between cells in each layer of the nested structure. Specifically, the connecting member can extend from the center of a cell to a specific connection point, such as a corner or face center, of its adjacent or nested cells. This connection pattern helps to effectively transfer forces from one scale to another when the structure is subjected to external loads, thereby achieving a coordinated response of the overall structure.

[0036] Therefore, the side length of the i-th self-similar cell from the inside out is set to i / n of the side length of the outermost basic cell, where n represents the order of the cell. This side length relationship is the core of constructing the self-similarity property. For example, if the order n of the cell is set to 2, then the side length of the innermost self-similar cell (i=1) will be 1 / 2 of the side length of the basic cell. The side length of the second-layer self-similar cell (i=2) will be 2 / 2 of the side length of the basic cell, that is, the same size as the basic cell. This precise size scaling allows the structure to exhibit a controllable geometric gradient at different scales.

[0037] The following example will provide a more detailed explanation of the above technical solution: Suppose we need to construct a metamaterial structure to solve the problems in existing technologies that exist in achieving controllable multi-level energy absorption, such as complex regulation, complex correlation between geometric configuration and nonlinear mechanical response, and difficulty in accurately controlling the stress triggering threshold of multiple platforms.

[0038] First, a simple cubic cell structure is chosen as the basic cell. This basic cell has well-defined geometric boundaries and connection points. Next, the order n of the self-similar cells to be nested is determined to be 2. This means that a nested structure containing two levels of self-similarity will be constructed.

[0039] According to this method, two self-similar cells with coincident geometric centers are nested within the aforementioned simple cubic basic cell. Specifically, the side length of the first self-similar cell (i=1) from the inside out is set to 1 / 2 of the side length of the basic cell. The side length of the second self-similar cell (i=2) is set to 2 / 2 of the side length of the basic cell, i.e., the basic cell itself. Therefore, inside the basic cell, there exists a concentric simple cubic cell with half its size.

[0040] Subsequently, body-centered cubic cells are used to connect the internal self-similar cells to the external basic cells. For example, connecting rods can be designed to extend from the center of the basic cell to each corner of the internal self-similar cell, and from the center of the internal self-similar cell to its own corner, thus forming a mutually supporting internal skeleton.

[0041] Thus, a metamaterial with a two-level self-similar nested structure was obtained. When this structure is subjected to an external impact load, the smaller internal self-similar cells, due to their relatively low critical deformation load, will deform first, absorbing some of the impact energy and forming a first energy absorption plateau. As the load continues to increase, when the internal cells reach their deformation limit, the load is transferred to the external basic cells through body-centered cubic connections. At this point, the external basic cells begin to deform, absorbing additional energy and forming a second energy absorption plateau. This geometry-driven, stepwise deformation mechanism enables the metamaterial structure to achieve controllable multi-level energy absorption.

[0042] Compared to existing gradient intensity distribution methods that require complex control to construct intensity progression relationships using a global array, this method achieves the endogenous geometric construction of the energy absorption platform by nesting n geometrically similar cells within a basic cell and following a size scaling rule that "the side length of the i-th self-similar cell from the inside out is i / n of the outermost basic cell." This design approach allows the triggering threshold for multi-level energy absorption to be determined by the geometric parameters of the structure itself, eliminating the need for complex external control and thus simplifying the design and manufacturing process.

[0043] Furthermore, addressing the challenges of complex correlations between geometric configurations and nonlinear mechanical responses, and difficulties in precisely controlling stress triggering thresholds across multiple platforms in existing multi-platform methods, this approach offers a systematic solution by introducing self-similar nested structures and body-centered cubic cell connections. Self-similarity ensures predictable correlations in mechanical responses at different scales, while the body-centered cubic connection guarantees effective load transfer between cell levels. Through precise control of geometric scales, this method achieves intrinsic regulation of stress triggering thresholds across multiple platforms, effectively reducing design complexity and enhancing the controllability of the structure in multi-level energy absorption. This construction method provides an effective pathway for developing novel self-similar metamaterials, enabling controllable multi-level energy absorption.

[0044] In some of the above embodiments, a method for constructing a metamaterial self-similar nested structure is proposed by nesting n geometrically coincident self-similar cells within a basic cell, connecting the inner and outer cells using a body-centered cubic cell connection method. The side length of the i-th self-similar cell from the inside out is i / n of the outermost basic cell. However, in practical applications, the specific nesting ratio (i / n) for cells of different orders (n) needs to be precisely defined to ensure that the constructed metamaterial exhibits the expected physical properties and structural stability. The lack of specific dimensional constraints on specific orders and cell types may lead to design uncertainties, affecting the precise control of metamaterial properties and reproducible fabrication.

[0045] In this regard, this application further proposes that the side length of the simple cubic cell embedded in the first-order cell of the simple cubic cell structure is 1 / 2 of that of the basic cell, and the side lengths of the simple cubic cells embedded in the second-order cell are 1 / 3 and 2 / 3 of that of the basic cell, respectively, from the inside to the outside.

[0046] In this structure, the side length of the simple cubic cell embedded within the first-order cell is half that of the basic cell. This feature clarifies the dimensional relationship between the embedded simple cubic cell and the basic cell when the order n is 1, i.e., in a first-order nested structure. Specifically, the side length of the embedded cell is set to half the side length of the basic cell. This precise proportional setting ensures that the structure has clear geometric features in the case of a single-layer nesting. This proportion can be accurately modeled using computer-aided design (CAD) software and realized using additive manufacturing (such as 3D printing) technology, or precisely replicated at the microscale using microfabrication techniques such as photolithography or etching. Furthermore, the side lengths of the simple cubic cells embedded within the second-order cell are 1 / 3 and 2 / 3 of the basic cell from the inside out, respectively. This feature further clarifies the dimensional relationship between the two embedded simple cubic cells and the basic cell when the order n is 2, i.e., in a second-order nested structure. From the innermost layer to the outermost layer, the side length of the first embedded cell is set to one-third of the side length of the base cell, while the side length of the second embedded cell is set to two-thirds of the side length of the base cell. This hierarchical and precise definition of size proportions allows for more complex hierarchy and finer geometric control in the second-order nested structure. In terms of implementation, in addition to the aforementioned CAD modeling and additive manufacturing, it can also be constructed through multi-step assembly or layered manufacturing processes. For example, smaller embedded cells can be manufactured first, then larger embedded cells can be manufactured or assembled around them, and finally the entire structure can be placed within the base cell.

[0047] This application's solution addresses the generalization problem of size proportions in basic construction methods by specifically quantifying the side lengths of embedded cells in simple cubic cell structures at different orders. Specifically, when constructing a first-order nested simple cubic structure, the side length of the embedded cell is precisely set to 1 / 2 of the side length of the base cell, creating a clear size hierarchy between the inner and outer cells. When constructing a second-order nested simple cubic structure, the side lengths of the two embedded cells are further defined as 1 / 3 and 2 / 3 of the base cell's side length, respectively. This precise size definition, combined with the general rule in basic construction methods that "the side length of the i-th self-similar cell from the inside out is i / n of the outermost base cell," concretizes the abstract proportional relationship. This approach not only ensures the coincidence of the geometric centers of the self-similar nested structures and the effective implementation of the body-centered cubic connection method, but also gives the size relationship of each nested layer clear physical meaning and operability. This precise dimensional control is crucial for the performance regulation of metamaterials. For example, by adjusting these proportions, the effective density, elastic modulus, acoustic bandgap, or electromagnetic response frequency of a material can be precisely designed, thereby achieving fine-grained control over the macroscopic properties of metamaterials.

[0048] The following is a concrete example. Assume there exists a simple cubic basic cell with a side length of L. In one specific implementation, when constructing a first-order metamaterial self-similar nested structure, a simple cubic cell with a side length of L / 2 is nested inside the simple cubic basic cell with side length L. The geometric centers of these two simple cubic cells coincide and are connected through a body-centered cubic cell connection. For example, connecting rods can be designed to extend from the center of the nested cell to the eight vertices of the basic cell, or from the vertices of the nested cell to the center of the basic cell, forming a stable structure. In another specific implementation, when constructing a second-order metamaterial self-similar nested structure, two simple cubic cells are nested inside the simple cubic basic cell with side length L. The innermost simple cubic cell has a side length of L / 3, and the outermost simple cubic cell has a side length of 2L / 3. The geometric centers of these two nested cells and the basic cell all coincide. Similarly, these cells are connected by body-centered cubic cell connections. For example, a connection structure can be designed so that the innermost cell and the middle cell, as well as the middle cell and the basic cell, maintain a body-centered cubic geometric relationship, ensuring the stability and functionality of the entire multi-layer nested structure.

[0049] Through the above technical solutions, this application provides specific and operable dimensional parameters for constructing self-similar nested structures of metamaterials, solving the design uncertainty problem caused by the generalization of dimensional ratios in basic construction methods. By specifying the specific side length ratios of simple cubic cell structures in first-order and second-order nesting, the geometric design of metamaterials becomes more precise and controllable. This precise dimensional definition facilitates refined management of the internal structural hierarchy of metamaterials, thereby enabling more accurate prediction and control of their macroscopic physical properties. For example, it allows for optimization of the material's mechanical response, acoustic filtering characteristics, or electromagnetic wave modulation capabilities. Furthermore, this explicit dimensional specification greatly improves the repeatability and reliability of the metamaterial manufacturing process, reduces the complexity of design and manufacturing, and enables the constructed metamaterials to stably exhibit the expected functional properties.

[0050] In some embodiments described above, this application proposes a method for constructing a metamaterial self-similar nested structure. This method involves nesting self-similar cells within a basic cell and connecting the inner and outer cells using a body-centered cubic (FCC) cell connection. However, in actual construction, for specific types of basic cells, such as FCC structures, accurately determining the specific side length ratios of the nested self-similar cells of different orders (e.g., first-order, second-order) is crucial for achieving the desired structural properties and functional optimization. Without specific dimensional constraints, the metamaterial design may lack precision, making it difficult to effectively control its macroscopic properties.

[0051] In this regard, this application further proposes that the side length of the face-centered cubic cell embedded in the first-order cell of the face-centered cubic cell structure is 1 / 2 of that of the basic cell, and the side lengths of the face-centered cubic cells embedded in the second-order cell are 1 / 3 and 2 / 3 of that of the basic cell, respectively, from the inside to the outside.

[0052] Face-centered cubic (FCC) cell structure is a common crystal structure type, characterized by a structural unit at each of the eight vertices and six face centers of a cubic unit cell. In metamaterial design, this structure serves as a fundamental geometric configuration, providing specific symmetries and connectivity that influence the mechanical, acoustic, or electromagnetic properties of the metamaterial. This structure can be realized through various manufacturing techniques, such as 3D printing using photopolymer resins, metal alloys, or ceramic materials, forming units with FCC arrangement characteristics through layer-by-layer stacking or selective laser sintering. Alternatively, microfluidic technology or template-assisted growth can be used to construct microstructures with FCC symmetry. The side length of the FCC cells embedded within the first-order cell is half that of the basic cell; this feature clarifies the specific size ratio of the embedded cells when first-order nesting occurs within the basic FCC cell. Setting the side length of the embedded cell to half that of the basic cell ensures a clear geometric relationship between the inner and outer cells. This is crucial for precisely controlling the hierarchical structure and functional gradient of metamaterials, directly affecting the material's effective modulus, band gap characteristics, and energy absorption capacity. During the design phase, computer-aided design (CAD) software can precisely set the dimensions of the embedded cell, ensuring its side length is strictly equal to half the side length of the basic cell. During manufacturing, such as using high-precision 3D printing technology, precise control of printing parameters (e.g., layer thickness, nozzle diameter) and material shrinkage rate is necessary to guarantee the dimensional accuracy of the final structure. The face-centered cubic cells embedded within the second-order cell have side lengths of 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside out. This feature further refines the dimensional setting of the face-centered cubic cell structure during second-order nesting. By introducing two embedded cells of different sizes, with side lengths of 1 / 3 and 2 / 3 of the basic cell, and specifying their arrangement from the inside out, more complex hierarchical structures can be constructed, achieving more precise functional control. This multi-level nesting can introduce multi-scale effects, such as acoustic or electromagnetic responses across different frequency ranges, or mechanical behavior at different strain levels. In numerical simulations and structural optimization, the precise dimensions of these two second-order embedded cells can be determined through parametric design based on the desired physical performance targets. In actual manufacturing, high-precision 3D printing technologies such as micro-stereolithography or electron beam lithography can be used to ensure the realization of multiple nested structures of different sizes within a limited space.

[0053] This application concretizes the construction method of the aforementioned self-similar nested structure of metamaterials, especially for face-centered cubic (FCC) cell structures, providing precise dimensional constraints. When using a FCC cell structure as the basic cell, if a first-order nested structure is to be constructed, the side length of the embedded FCC cell is precisely set to 1 / 2 of the side length of the basic cell. This explicit proportional relationship ensures the geometrical coordination between the inner and outer cells, thereby forming a metamaterial with specific mechanical or physical responses on a macroscopic scale. Furthermore, if a second-order nested structure is to be constructed, the side lengths of the two FCC cells embedded sequentially from the inside out are set to 1 / 3 and 2 / 3 of the side length of the basic cell, respectively. This multi-level, precise dimensional nesting allows the metamaterial to form structural hierarchies with different characteristic scales within it. These hierarchies are connected through body-centered cubic (BCC) cells, ensuring the integrity and stability of the structure. This layered and precisely dimensionally controlled nesting method enables the metamaterial to exhibit multi-scale coupling effects, such as different acoustic bandgap characteristics at different frequencies or unique deformation mechanisms under different stresses. In this way, the proposed solution concretizes the abstract concept of self-similar nesting into operable geometric parameters, thereby making the design and performance prediction of metamaterials more accurate and controllable.

[0054] The following is a concrete example to illustrate this. Assume there exists a face-centered cubic (FCC) cell structure as the basic cell, with a side length of 100 units. If a first-order nested metamaterial self-similar structure needs to be constructed, then according to the above scheme, the embedded FCC cells will be designed with a side length of 50 units (half of 100 units). This 50-unit embedded cell will be geometrically centered within the 100-unit basic cell. If a second-order nested metamaterial self-similar structure needs to be constructed, then two FCC cells will be nested within the 100-unit basic cell. The innermost cell will have a side length of approximately 33.33 units (one-third of 100 units), while the outermost cell will have a side length of approximately 66.67 units (two-thirds of 100 units). These two embedded cells are also geometrically aligned within the basic cell, with the 33.33-unit cell completely surrounded by the 66.67-unit cell. These precisely proportioned cells can be manufactured using high-precision additive manufacturing techniques, such as selective laser sintering or photopolymerization 3D printing, using polymers or metals. After manufacturing, the cells at different levels are structurally connected via body-centered cubic cell connections, for example, by designing connecting rods or beams to ensure structural integrity.

[0055] Through the above technical solution, this application provides a precise dimensional design basis for the self-similar nested construction of face-centered cubic (FCC) metamaterials. By specifying the side length ratio of the first-order and second-order embedded cells—that is, the first-order cell is 1 / 2 of the basic cell, and the second-order cells, from the inside out, are 1 / 3 and 2 / 3 of the basic cell, respectively—the design inaccuracies caused by unclear dimensional parameters in the basic scheme are solved. This precise geometric constraint makes the structural design of metamaterials highly controllable and predictable, thereby enabling fine-tuning of the macroscopic properties of the material. For example, the effective modulus, acoustic bandgap position, or electromagnetic response frequency of the metamaterial can be precisely designed according to specific application requirements, avoiding performance deviations caused by dimensional uncertainties. Compared with schemes that only provide general nesting rules, this scheme significantly improves the designability and functional realization efficiency of metamaterial structures by providing specific dimensional ratios, enabling the constructed metamaterials to exhibit the expected physical properties more stably and reliably.

[0056] In some embodiments described above in this application, a method for constructing a metamaterial self-similar nested structure is proposed. This method involves nesting n geometrically coincident self-similar cells within a basic cell and connecting the inner and outer cells using a body-centered cubic cell connection method. However, for certain types of cell structures, such as simple cubic-face-centered cubic hybrid cell structures, the specific side length ratios of self-similar cells of different orders are not yet clear. This may lead to difficulties in precisely controlling the geometric properties and functional performance of the structure during actual construction.

[0057] To address this, this application further proposes a method for constructing the aforementioned metamaterial self-similar nested structure, wherein the side length of the simple cubic-face-centered cubic hybrid cell embedded within the first-order cell of the simple cubic-face-centered cubic hybrid cell structure is half that of the base cell. This feature clarifies the proportional relationship between the side length of the embedded simple cubic-face-centered cubic hybrid cell and the side length of the outermost base cell when the nesting order n is 1, i.e., in the first-order nested structure. Specifically, the side length of the embedded cell is set to half that of the base cell. This setting ensures a clear and uniform size ratio between the embedded cell and the base cell in a single-layer nested structure, contributing to the formation of a stable geometric configuration. For example, precise mold manufacturing or 3D printing technology can be used to precisely construct the embedded cell, half the size of the base cell, inside the base cell. Another implementation method is to use parametric design software to automatically calculate and generate the geometric model of the embedded cell after defining the size of the base cell, ensuring that its side length is half that of the base cell.

[0058] Furthermore, the side lengths of the simple cubic-face-centered cubic hybrid cells embedded within the second-order cell are 1 / 3 and 2 / 3 of the basic cell's side length, respectively, from the inside out. This feature further refines the proportional relationship between the side lengths of the two simple cubic-face-centered cubic hybrid cells from the innermost to the outermost layer and the side length of the basic cell when the nesting order n is 2, i.e., in a second-order nested structure. Specifically, the side length of the innermost cell is set to 1 / 3 of the basic cell's side length, while the side length of the second innermost cell (i.e., the second cell from the inside out) is set to 2 / 3 of the basic cell's side length. This hierarchical sizing setting creates a progressive sizing gradient between cells in a multi-layered nested structure, thus providing a richer structural hierarchy while maintaining self-similarity. For example, in a computer-aided design (CAD) environment, the size of the basic cell can be defined, and then the geometric model of the two layers of embedded cells can be automatically generated based on the 1 / 3 and 2 / 3 ratio. Alternatively, during micro-nano fabrication, the dimensions of different cell levels can be precisely controlled through multi-step photolithography or electron beam etching to achieve these specific side length ratios.

[0059] This application's solution, in the aforementioned method for constructing self-similar nested metamaterial structures, explicitly defines the specific side length ratios of first-order and second-order nested cells for simple cubic-face-centered cubic hybrid cell structures. The basic construction method provides a general framework for building metamaterials based on specific cells, through self-similar nesting with coincident geometric centers and body-centered cubic connections. Building upon this, this application further provides precise dimensional definitions for simple cubic-face-centered cubic hybrid cell structures. Specifically, for first-order nesting, the side length of the embedded cell is set to 1 / 2 of the base cell; for second-order nesting, the side lengths of the two embedded cells from the inside out are set to 1 / 3 and 2 / 3 of the base cell, respectively. This precise proportional setting overcomes the dimensional uncertainties that may arise from the general i / n rule when constructing self-similar nested simple cubic-face-centered cubic hybrid cell structures, ensuring a precise and controllable dimensional gradient between cell levels. In this way, the constructed metamaterial structure not only maintains self-similarity but also has a more refined and ordered internal geometry, thus enabling more effective control over the material's macroscopic physical properties. The body-centered cubic connection method further ensures the structural integrity and mechanical transfer efficiency between these precisely sized nested cells.

[0060] The following is a concrete example. Suppose we need to construct a first- or second-order self-similar nested metamaterial based on a simple cubic-face-centered cubic hybrid cell structure. As a specific implementation, when the basic cell is a simple cubic-face-centered cubic hybrid cell structure, if a first-order nested structure is required, a simple cubic-face-centered cubic hybrid cell with its geometric center coincidentally is precisely embedded inside the basic cell. The side length of this embedded cell is set to half the side length of the basic cell. For example, if the side length of the basic cell is 10 units, the side length of the embedded cell will be 5 units. If a second-order nested structure is required, two simple cubic-face-centered cubic hybrid cells with their geometric centers coincidentally are embedded inside the basic cell. The innermost simple cubic-face-centered cubic hybrid cell has a side length set to 1 / 3 of the basic cell's side length (e.g., approximately 3.33 units), while the next innermost (the second from the inside) simple cubic-face-centered cubic hybrid cell has a side length set to 2 / 3 of the basic cell's side length (e.g., approximately 6.67 units). These embedded cells are connected to the outer cells or basic cells via body-centered cubic cell connections, for example, by connecting rods or beams between the cell center points, to form a complete metamaterial structure with specific geometric properties.

[0061] Through the above technical solution, this application clarifies the specific side length ratios of simple cubic-face-centered cubic hybrid cell structures at different nesting orders, solving the problem of accurately controlling their geometric dimensions and hierarchical structure when constructing such complex metamaterial structures. This precise dimensional definition enables the constructed metamaterial structure to possess highly predictable geometric properties and structural stability. Especially in first-order and second-order nesting, by setting explicit 1 / 2, 1 / 3, and 2 / 3 side length ratios between embedded cells and basic cells, it is possible to ensure the formation of an ordered and uniform size gradient between cells at each level, thereby optimizing the mechanical, acoustic, or electromagnetic response of the material. This allows designers to more precisely control the function of metamaterials to achieve specific performance goals, such as energy absorption, vibration isolation, or waveguide properties within a specific frequency range.

[0062] In other embodiments, this application proposes a metamaterial self-similar nested structure, which is constructed using the method described above. This metamaterial self-similar nested structure includes a basic cell and n self-similar cells nested within the basic cell. The basic cell is a simple cubic cell structure, a face-centered cubic cell structure, or a simple cubic-face-centered cubic hybrid cell structure. All self-similar cells coincide with the geometric center of the basic cell, and the inner and outer cells are interconnected through body-centered cubic cell connections. The side length of the i-th self-similar cell from the inside out is i / n of the outermost basic cell, where n represents the order of the cell.

[0063] The core innovation of this embodiment lies in combining the connection of self-similar nested structures with body-centered cubic cells in a manner where the geometric centers coincide, and following the size scaling rule that the side length of the i-th self-similar cell from the inside out is i / n of the outermost basic cell, thereby achieving the endogenous construction of a multi-level energy absorption platform. This design allows the stress triggering threshold of the multi-platform to be directly determined by the structural geometric parameters, without the need for complex external adjustments, effectively solving the problems of complex correlation between geometric configuration and nonlinear mechanical response and difficulty in accurately controlling the stress triggering threshold of the multi-platform in existing technologies.

[0064] Specifically, when the metamaterial structure is subjected to external loads, due to the small size of its internal self-similar cells and low critical deformation load, it first undergoes plastic deformation, absorbing impact energy and forming the first energy absorption plateau. As the load continues to increase, the deformation is transmitted to the external cells through body-centered cubic connections, and the external cells deform sequentially, forming subsequent energy absorption plateaus. This step-by-step collapse mechanism ensures the controllability and efficiency of the energy absorption process.

[0065] For example, in a structure with order n=2, the innermost self-similar cell has a side length half that of the basic cell. When subjected to impact, this inner cell deforms first. Once the inner cell reaches its deformation limit, the load is transferred to the outer basic cell, causing it to deform as well, thus achieving two-stage energy absorption. By adjusting the order n and the size ratio, the number of energy absorption platforms and the trigger threshold can be precisely controlled to meet the needs of different application scenarios. Compared to the gradient intensity distribution method in existing technologies, which requires complex control to construct an intensity progression relationship through a global array, this structure achieves intrinsic control of the energy absorption platform through precise control of the geometric scale, significantly improving the controllability of multi-stage energy absorption.

[0066] In some of the embodiments described above in this application, a method for constructing a metamaterial self-similar nested structure and the metamaterial self-similar nested structure obtained by the method are proposed. However, in its practical application, a single metamaterial self-similar nested structure may have limitations in size, making it difficult to directly apply to scenarios that require a large volume or specific macroscopic performance. Furthermore, the stability and predictability of its overall performance may not be sufficient to meet engineering requirements.

[0067] In this regard, this application further proposes that the above-mentioned metamaterial self-similar nested structure can be configured as a cell array, which is a 3×3×3 structure.

[0068] In this context, a cell array refers to the combination of multiple independent metamaterial self-similar nested structural units according to a predetermined spatial arrangement rule, forming a larger-scale overall structure. This array configuration can be achieved by periodically repeating a lattice-like structure formed by a single metamaterial self-similar nested structural unit in three-dimensional space, or by stacking and connecting multiple metamaterial self-similar nested structural units according to a preset geometric rule to form a composite structure. The 3×3×3 structure specifically defines the scale and dimensions of the cell array, indicating that three metamaterial self-similar nested structural units are arranged in each of three mutually perpendicular spatial directions. This specific size defines the overall shape and macroscopic scale of the cell array, providing a foundation for the macroscopic properties of metamaterials and ensuring sufficient repeating units in all three dimensions, thereby enabling the expression of anisotropic or isotropic properties of metamaterials.

[0069] This application's solution arranges individual metamaterial self-similar nested structural units into a cell array according to a 3×3×3 rule, allowing the unique properties of each nested cell to accumulate and manifest on a macroscopic scale. This regular array configuration gives the overall structure of the metamaterial a clear periodicity and symmetry, enabling specific manipulation of physical fields such as electromagnetic waves, sound waves, or mechanical waves. For example, when a wave interacts with the periodically arranged cell array during propagation, its propagation characteristics are affected by the array structure, resulting in extraordinary physical phenomena such as negative refraction and band gaps. This 3×3×3 cubic array structure ensures sufficient repeating units in all directions, making the macroscopic response of the metamaterial more uniform and predictable, overcoming the size limitations and functional singularities that may exist for individual complex cells in practical applications. This array design means that the overall performance of the metamaterial is no longer simply the superposition of individual cells, but rather, through the interaction between cells and the overall periodic arrangement, new and more complex macroscopic properties are generated.

[0070] The following is a concrete example to illustrate this. First, a single metamaterial self-similar nested structure can be constructed using the method described above. For example, a simple cubic cell structure with a side length of L can be used as the base cell, and a self-similar simple cubic cell with a side length of L / 2 can be nested inside it. The inner and outer cells are then connected using the same method as body-centered cubic cells, resulting in a first-order metamaterial self-similar nested structure. Then, these 27 identical first-order metamaterial self-similar nested structure units are arranged and connected tightly in three-dimensional space according to a rule of 3 rows, 3 columns, and 3 layers. For example, these units can be fixed together by methods such as bonding, welding, sintering, or integral molding to form a cubic array with a side length of approximately 3L.

[0071] By organizing the self-similar nested structures of individual metamaterials into a 3×3×3 cell array, this application effectively extends the unique properties of individual complex cells to the macroscopic scale, thereby overcoming the size limitations and functional singularities that may exist in practical applications of individual cells. This regular array configuration makes the overall performance of metamaterials more stable and predictable, facilitating large-scale manufacturing and application, and enabling better macroscopic control of specific physical fields (such as electromagnetic fields and acoustic fields), thus providing a solid foundation for the engineering applications of metamaterials.

[0072] In some embodiments described above in this application, a metamaterial self-similar nested structure is proposed, with a 3×3×3 cell array. However, during its implementation, if the specific filling method of each cell in the 3×3×3 array is not clearly defined, the macroscopic performance of the overall structure may be difficult to predict and control, thereby limiting its performance in specific applications.

[0073] In this regard, this application further proposes that the cell is filled with 27 zero-order simple cubic cells, 27 zero-order face-centered cubic cells, or 27 zero-order simple cubic-face-centered cubic hybrid cells.

[0074] Specifically, "cell filling" refers to the internal structure of the 27 independent cells that make up the 3×3×3 array. These cells are not empty cavities, but are occupied by pre-defined structural units. This filling method directly determines the local characteristics of each cell, and thus affects the macroscopic behavior of the entire metamaterial.

[0075] "27 zero-order simple cubic cells" means that each position in this 3×3×3 array is composed of a zero-order simple cubic cell. The zero-order simple cubic cell is the most basic form of a simple cubic structure and does not contain any self-similar nesting levels. It can be a solid cube or a cubic lattice structure composed of rods or panels, without further self-similar fractal nesting inside.

[0076] "27 zero-order face-centered cubic cells" means that each position in this 3×3×3 array is composed of a zero-order face-centered cubic cell. The zero-order face-centered cubic cell is the most basic form of face-centered cubic structure and does not contain self-similar nesting levels. It can be a solid face-centered cubic unit or a face-centered cubic lattice structure composed of rods or panels, without further self-similar fractal nesting inside.

[0077] "27 zero-order simple cubic-face-centered cubic hybrid cells" means that each position in this 3×3×3 array is composed of a zero-order simple cubic-face-centered cubic hybrid cell. This hybrid cell is the most basic structural unit combining the characteristics of simple cubic and face-centered cubic structures, and does not contain self-similar nested levels. For example, it might be within a cubic frame, with some nodes following a simple cubic arrangement while others follow a face-centered cubic arrangement, forming a composite lattice structure.

[0078] The scheme in this application achieves uniformity and homogeneity of structural units at the macroscopic array level by filling each cell in the 3×3×3 cell array of the aforementioned metamaterial self-similar nested structure with 27 zero-order simple cubic cells, or 27 zero-order face-centered cubic cells, or 27 zero-order simple cubic-face-centered cubic hybrid cells. This means that among the 27 basic units constituting the entire 3×3×3 array, each unit has the same basic structural type, and these basic structures themselves do not undergo deeper self-similar nesting. This design ensures the structural consistency of the metamaterial at the local scale, making the mechanical, acoustic, or electromagnetic responses of the overall structure more stable and predictable. By selecting different types of zero-order cells for filling, the intrinsic properties of the metamaterial can be flexibly adjusted to meet different functional requirements while maintaining structural homogeneity. This explicit filling method, combined with the 3×3×3 array structure, jointly constructs a metamaterial system with controllable macroscopic properties.

[0079] The following is a concrete example. One implementation involves constructing a 3×3×3 metamaterial cell array. In this array, each cell is filled with a zero-order simple cubic cell. This zero-order simple cubic cell can be a solid cube made using additive manufacturing techniques (such as 3D printing), with a side length equal to 1 / 1 the size of any single cell in the array. Alternatively, the zero-order simple cubic cell can be a lattice structure formed by connecting thin rods at the vertices and center edges of the cube. The material and geometry of these rods are predetermined, and they do not contain any nested structures. In another implementation, each cell in the 3×3×3 array can be filled with a zero-order face-centered cubic cell. This cell can be a face-centered cubic lattice made of a specific material, with nodes located at the corners and center of the cube, and without self-similar nesting.

[0080] The aforementioned technical solution clarifies the specific filling method of each cell in the 3×3×3 cell array of metamaterials, resolving the uncertainty in macroscopic structural performance. This unified and zero-order cell filling method enables the metamaterial to exhibit high structural uniformity and predictability overall, thus allowing for precise design and control of its macroscopic physical properties. This not only simplifies the design and analysis process of metamaterials but also provides a solid foundation for realizing metamaterials with specific functions (such as high strength, lightweight, and specific acoustic or electromagnetic responses), greatly enhancing the practicality and application potential of metamaterials.

[0081] In some embodiments described above in this application, a metamaterial self-similar nested structure is proposed, with a 3×3×3 cell array. However, in practical applications, to achieve specific functional gradients or optimize material properties, it may be necessary to fill the cells in the array more finely and with specific patterns to meet the differentiated requirements of different regions for mechanical, acoustic, or electromagnetic properties.

[0082] In this regard, this application further proposes that the cell is filled from top to bottom with 9 zero-order simple cubic cells, 9 first-order simple cubic cells, or 9 second-order simple cubic cells.

[0083] The top-down filling of cells refers to filling cells layer by layer or column by column along the vertical direction (e.g., typically the Z-axis) in a 3×3×3 cell array of a metamaterial self-similar nested structure. This filling method aims to achieve gradient changes or layered functions of material properties in the vertical direction. This filling process can be achieved through additive manufacturing techniques, such as 3D printing, by depositing different types of cell materials layer by layer during the construction process; or it can be achieved through modular assembly, by pre-preparing different types of cell modules and then precisely stacking and connecting them in a top-down order. The zero-order simple cubic cell, first-order simple cubic cell, or second-order simple cubic cell are unit types used to fill specific layers or columns in the 3×3×3 cell array. In this context, a zero-order simple cubic cell refers to the most basic simple cubic cell, containing no nested structures. A first-order simple cubic cell contains a layer of self-similar cells nested within a basic simple cubic cell, with a side length half that of the basic cell. A second-order simple cubic cell contains two layers of self-similar cells nested within a basic simple cubic cell, with side lengths of 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside out. By selecting simple cubic cells of different orders for filling, the physical properties of the material, such as density, stiffness, acoustic impedance, or electromagnetic response, can be precisely controlled in different regions. For example, higher-order cells typically have more complex internal structures, potentially leading to lower effective density or more unique wave propagation characteristics. During the design phase, the type and order of cells to be filled in each layer or column can be predetermined based on the desired performance distribution. During manufacturing, automated equipment can precisely place the pre-defined zero-order, first-order, or second-order simple cubic cells into designated positions.

[0084] This application's solution, based on the aforementioned self-similar nested structure of metamaterials, achieves vertical stratification or gradient distribution of structural properties by filling cells in a 3×3×3 cell array from top to bottom. Specifically, this solution uniformly fills each layer (or a specific layer) of the array with nine simple cubic cells of the same order. For example, it can be filled entirely with zero-order simple cubic cells, or entirely with first-order simple cubic cells, or entirely with second-order simple cubic cells. This filling strategy ensures that each layer has a relatively uniform internal structure and physical properties, while different layers exhibit different properties due to differences in the order of the filled cells. For example, zero-order simple cubic cells have a relatively simple structure and may have high stiffness and density; while first-order or second-order simple cubic cells, due to their internal self-similar nested structure, may exhibit more complex mechanical responses, lower effective density, or specific wave propagation characteristics. In this way, the overall performance distribution of metamaterials in the vertical direction can be precisely designed and controlled according to actual application requirements, thereby achieving functional diversity while maintaining the integrity of the overall structure.

[0085] The following is a concrete example. As a specific implementation, a 3×3×3 metamaterial self-similar nested structure can be constructed. In this structure, the bottom nine cells can be filled entirely with zero-order simple cubic cells to provide basic support or specific stiffness. The middle nine cells can be filled entirely with first-order simple cubic cells, where the internal nested structure begins to emerge, potentially for energy absorption or waveguide functions. The top nine cells can be filled entirely with second-order simple cubic cells, and their more complex nested structure can be used to achieve more precise acoustic or electromagnetic manipulation. This layered filling can be achieved through layer-by-layer 3D printing, where the printer automatically switches materials or printing parameters after completing one layer to construct the next layer of cells of different orders.

[0086] Through the aforementioned technical solution, precise control over the cell type and order in the vertical direction can be achieved within a 3×3×3 cell array of a self-similar nested metamaterial structure. This top-down layered filling method allows the metamaterial to exhibit differentiated physical properties at different heights; for example, it can form gradient structures where stiffness, density, or wave propagation characteristics vary along the vertical direction. This significantly enhances the functional versatility and design flexibility of metamaterials, enabling them to better adapt to application scenarios requiring specific spatial performance distributions, thereby optimizing the overall structural performance.

[0087] In some embodiments described above in this application, a metamaterial self-similar nested structure is proposed, wherein the cell array is a 3×3×3 structure. However, in practical applications, merely limiting the size of the cell array may not be sufficient to precisely control the macroscopic properties of the metamaterial, such as its mechanical, acoustic, or electromagnetic responses, because the specific filling method within the cell array has a decisive influence on the overall properties of the material.

[0088] In this regard, this application further proposes that the above-mentioned cell is filled from top to bottom with 9 zero-order simple cubic-face-centered cubic mixed cells, or 9 first-order simple cubic-face-centered cubic mixed cells, or 9 second-order simple cubic-face-centered cubic mixed cells.

[0089] "Top-down filling" refers to the layering or gradient arrangement of cells along a vertical direction (e.g., the Z-axis) in a 3×3×3 cell array of a metamaterial self-similar nested structure. This filling method can be understood as dividing the entire 3×3×3 array into several 3×3 planar layers and configuring the cells within these layers in a specific way. For example, this can be achieved through layer-by-layer additive manufacturing techniques or by prefabricating different types of cell layers and stacking them together.

[0090] A "zero-order simple cubic-face-centered cubic hybrid cell" refers to a basic simple cubic-face-centered cubic hybrid cell without any self-similar nesting. It represents the most basic unit of this hybrid cell structure and does not contain smaller self-similar structures. Such cells typically have relatively high density and stiffness, or specific wave propagation characteristics.

[0091] A "first-order simple cubic-face-centered cubic hybrid cell" refers to a structure in which a layer of self-similar cells is nested within a basic simple cubic-face-centered cubic hybrid cell. According to the construction method described above, this first-order cell contains a self-similar cell with a geometrically coinciding center, whose side length is 1 / n of the outermost basic cell (when n=1, it is 1 / 1, but usually n>1; for example, if it is first-order, the side length of the nested cell can be 1 / 2 of the basic cell). This nested structure gives the cell a more complex internal geometry, which may alter its effective material properties, such as reducing density, adjusting the acoustic bandgap, or electromagnetic response.

[0092] A "second-order simple cubic-face-centered cubic hybrid cell" refers to a structure in which two layers of self-similar cells are nested within a basic simple cubic-face-centered cubic hybrid cell. According to the construction method described above, this second-order cell contains two self-similar cells with coincident geometric centers, whose side lengths are 1 / n and 2 / n of the outermost basic cell, respectively (for example, if n=2, the side lengths of the nested cells can be 1 / 3 and 2 / 3 of the basic cell). This multi-layer nesting further increases the structural complexity of the cell, enabling more precise control over material properties, such as producing wider band gaps or more unique and extraordinary mechanical properties.

[0093] This application proposes a scheme that constructs metamaterials with gradient or layered properties by selectively filling a 3×3×3 cell array with simple cubic-face-centered cubic hybrid cells of a specific order along the vertical direction. Specifically, this scheme allows for the uniform use of zero-order, first-order, or second-order simple cubic-face-centered cubic hybrid cells to fill a specific layer of the array (e.g., a 3×3 plane). This filling strategy makes the overall structure of the metamaterial no longer homogeneous, but allows different microstructural complexities and corresponding macroscopic physical properties to be exhibited between different layers. For example, by combining zero-order cell layers with first-order or second-order cell layers, continuous or stepwise variations in the material's stiffness, density, acoustic impedance, or electromagnetic dielectric constant along the vertical direction can be achieved. This design fully utilizes the tunable properties exhibited by self-similar nested cells at different orders and macroscopically integrates them into a three-dimensional array, thereby enabling the entire metamaterial structure to be functionally optimized and customized for specific application requirements.

[0094] The following is a concrete example. A 3×3×3 metamaterial self-similar nested structure can be considered as a stack of three 3×3 planar layers. In one specific implementation, the topmost 3×3 planar layer can be entirely composed of nine zero-order simple cubic-face-centered cubic hybrid cells. The next middle 3×3 planar layer can be entirely composed of nine first-order simple cubic-face-centered cubic hybrid cells. The bottommost 3×3 planar layer can be entirely composed of nine second-order simple cubic-face-centered cubic hybrid cells. In this configuration, the internal structural complexity of the metamaterial gradually increases from top to bottom, potentially leading to a gradient change in its mechanical, acoustic, or electromagnetic properties along the vertical direction. For example, the zero-order cell layer might provide higher initial stiffness, while as the order increases, the first- and second-order cell layers might exhibit lighter mass, lower effective stiffness, or unique wave propagation characteristics.

[0095] The above technical solutions effectively address the challenge of precisely controlling the macroscopic properties of metamaterials solely through array size, thereby enabling refined design and optimization of metamaterial functional characteristics. This layered filling method allows metamaterials to exhibit anisotropic or gradient functional properties. For example, in applications such as impact protection, acoustic filtering, or electromagnetic stealth, precise control of cell types and orders at different levels allows for directional regulation of energy absorption, wave propagation direction, or electromagnetic response, greatly expanding the application range and performance of metamaterials.

[0096] In some embodiments described above in this application, a metamaterial self-similar nested structure is proposed. This structure is obtained through a specific construction method and has a 3×3×3 cell array. However, only the overall size and construction method of the cell array are defined, without specifying the specific filling method and type of each cell in the array. Especially when different orders and basic cell types are involved, this may lead to difficulty in precisely controlling the functionality or specific properties of the structure, and making it impossible to effectively achieve gradient changes in material properties or anisotropic design.

[0097] In this regard, this application further proposes that the cell is filled from top to bottom with 9 second-order face-centered cubic cells, 9 second-order simple cubic cells, or 9 second-order simple cubic-face-centered cubic hybrid cells.

[0098] The term "top-down filling" describes a specific spatial arrangement strategy: in a three-dimensional cell array, filling begins from the top layer and proceeds downwards layer by layer or by specific regions, thus differentiating the cell types or orders of different layers. This is crucial for constructing metamaterials with gradient functions or anisotropic properties. In practice, this vertically ordered structural arrangement can be achieved through layer-by-layer manufacturing techniques, such as layer-by-layer deposition or selective area printing in additive manufacturing.

[0099] The phrase "9 second-order face-centered cubic cells, 9 second-order simple cubic cells, or 9 second-order simple cubic-face-centered cubic hybrid cells" specifies the exact type, order, and number of cells used for filling. "Second-order cells" indicates that the cell has a two-level nested self-similar structure, exhibiting a higher level of complexity than zero-order or first-order cells and displaying more unique physical properties. Choosing "face-centered cubic cells," "simple cubic cells," or "simple cubic-face-centered cubic hybrid cells" defines the basic geometry and internal connectivity of these second-order cells. The quantity "9" indicates that these specific types of second-order cells are used to fill a complete layer (e.g., a 3×3 plane) in a 3×3×3 array. These second-order cells can be pre-prepared and then assembled into the array, or generated directly at designated locations during manufacturing.

[0100] The solution in this application configures the cell array in the aforementioned metamaterial self-similar nested structure as a 3×3×3 structure. Furthermore, it fills the cells from top to bottom with nine second-order face-centered cubic cells, nine second-order simple cubic cells, or nine second-order simple cubic-face-centered cubic hybrid cells, achieving a high degree of customization of the metamaterial's internal structure. Specifically, this filling strategy means that in a specific layer of the 3×3×3 array (e.g., the top layer), all nine cells are composed of a specific type of cell with a two-level nested structure. Because second-order cells have more complex internal geometry and connections than zero-order or first-order cells, they can exhibit unique mechanical, acoustic, or electromagnetic responses. When these cells with specific complexity are filled into the array in a top-down, ordered manner, the entire metamaterial structure can form a performance gradient or functional partition along the vertical direction. This precise filling and arrangement method makes the macroscopic properties of metamaterials no longer uniform, but can exhibit differentiated characteristics in different regions according to design requirements. This enables fine control over the overall functional response of the material, such as exhibiting enhanced stiffness or specific wave propagation characteristics in a specific direction.

[0101] The following is a concrete example to illustrate this. Consider a metamaterial self-similar nested structure constructed using the method described above, with a 3×3×3 cell array. One specific implementation example is as follows: The top layer of this 3×3×3 cell array, i.e., the 3×3 plane at the top of the array, can be completely filled with nine second-order face-centered cubic cells. This means that each cell in this layer is a face-centered cubic cell with a two-level nested structure, containing two self-similar face-centered cubic cells with coincident geometric centers, connected in a manner similar to body-centered cubic cells. For example, if the side length of the outermost basic cell is L, then the side lengths of the two nested self-similar cells can be L / 3 and 2L / 3, respectively. This filling method allows the metamaterial structure to form a functional layer at the top composed of second-order face-centered cubic cells with specific complexity, thereby endowing this region with specific physical properties.

[0102] Through the aforementioned technical solution, in a metamaterial self-similar nested structure with a 3×3×3 cell array, a top-down approach is adopted to fill it with nine second-order face-centered cubic cells, nine second-order simple cubic cells, or nine second-order simple cubic-face-centered cubic hybrid cells. This allows for precise control over the composition and internal complexity of the metamaterial structure in the vertical direction. This layered filling strategy enables the metamaterial to exhibit the expected gradient functional characteristics or anisotropic behavior, such as different stiffness, acoustic impedance, or electromagnetic response at different levels. Compared to schemes that only define the overall array size, this scheme, by introducing cell filling of specific orders and types, greatly enhances the ability to control the macroscopic properties of the metamaterial. This allows for the design of advanced metamaterial structures with specific functional distributions and responses, effectively solving the technical problem of achieving precise functional control in complex arrays.

[0103] The above description is merely an embodiment of this application and is not intended to limit the scope of protection of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A method for constructing a metamaterial self-similar nested structure, characterized in that, Includes the following steps: Using simple cubic cell structure, face-centered cubic cell structure and simple cubic-face-centered cubic hybrid cell structure as basic cells, new metamaterial self-similar nested structures are obtained by nesting n self-similar cells with coincident geometric centers in the basic cells and connecting the inner and outer cells in the same way as body-centered cubic cells. In this context, the side length of the i-th self-similar cell from the inside out is i / n of the outermost basic cell, where n represents the order of the cell.

2. The method for constructing a metamaterial self-similar nested structure according to claim 1, characterized in that, The simple cubic cell embedded in the first-order cell of the simple cubic cell structure has a side length of 1 / 2 that of the basic cell, and the simple cubic cell embedded in the second-order cell has side lengths of 1 / 3 and 2 / 3 that of the basic cell, respectively, from the inside to the outside.

3. The method for constructing a metamaterial self-similar nested structure according to claim 1, characterized in that, The side length of the face-centered cubic cell embedded in the first-order cell of the face-centered cubic cell structure is 1 / 2 of that of the basic cell. The side lengths of the face-centered cubic cells embedded in the second-order cell are 1 / 3 and 2 / 3 of that of the basic cell, respectively, from the inside to the outside.

4. The method for constructing a metamaterial self-similar nested structure according to claim 1, characterized in that, The simple cubic-face-centered cubic hybrid cell structure has a first-order cell whose side length is half that of the basic cell. The simple cubic-face-centered cubic hybrid cell embedded in the second-order cell has side lengths of 1 / 3 and 2 / 3 of the basic cell, respectively, from the inside to the outside.

5. A metamaterial self-similar nested structure, characterized in that, Obtained by means of any one of claims 1-7.

6. The metamaterial self-similar nested structure according to claim 5, characterized in that, Its cell array has a 3×3×3 structure.

7. The metamaterial self-similar nested structure according to claim 6, characterized in that, The cell is filled with 27 zero-order simple cubic cells, 27 zero-order face-centered cubic cells, or 27 zero-order simple cubic-face-centered cubic mixed cells.

8. The metamaterial self-similar nested structure according to claim 6, characterized in that, The cell filling consists of 9 zero-order simple cubic cells, 9 first-order simple cubic cells, or 9 second-order simple cubic cells, arranged from top to bottom.

9. The metamaterial self-similar nested structure according to claim 6, characterized in that, The cell is filled from top to bottom with 9 zero-order simple cubic-face-centered cubic mixed cells, or 9 first-order simple cubic-face-centered cubic mixed cells, or 9 second-order simple cubic-face-centered cubic mixed cells.

10. The metamaterial self-similar nested structure according to claim 6, characterized in that, The cell is filled from top to bottom with 9 second-order face-centered cubic cells, 9 second-order simple cubic cells, or 9 second-order simple cubic-face-centered cubic hybrid cells.

Images