Spectrum-based metamaterial structure diversity metric method

Abstract

The application discloses a kind of based on spectrum's metamaterial structure diversity measurement method, it is related to the technical field of metamaterial design optimization, this method includes: obtaining the discrete density field of first microstructure and the discrete density field of second microstructure, discrete density field is used to characterize the material distribution of microstructure;The discrete density field of first microstructure and the discrete density field of second microstructure are respectively Fourier transformed, obtain first frequency domain result and second frequency domain result;From first frequency domain result, extract first amplitude spectrum, from second frequency domain result, extract second amplitude spectrum;Based on first amplitude spectrum and second amplitude spectrum, the diversity distance between first microstructure and second microstructure is calculated as the index for measuring the structural difference between microstructure.The method of the application can eliminate the diversity illusion caused by the periodic metamaterial translational invariance, and truly reflect the essential topological structure difference of microstructure.

Description

Technical Field

[0001] This invention relates to the field of metamaterial design optimization technology, and in particular to a method for measuring the structural diversity of metamaterials based on the spectrum. Background Technology

[0002] Metamaterials are a new class of materials that achieve extraordinary physical properties through the artificial design of microstructures rather than relying on chemical compositions, and they hold significant application potential in cutting-edge fields such as electromagnetics, acoustics, thermodynamics, and mechanics. Among these, topology optimization, with its extremely high degree of design freedom, has become an important method for designing metamaterial microstructures.

[0003] However, traditional topology optimization methods typically converge to a single mathematically optimal solution, which is insufficient to meet the demands of diverse alternative solutions in practical engineering applications, addressing manufacturing errors, operating condition fluctuations, and adaptability to multiple scenarios. Therefore, developing topology optimization methods capable of generating a batch of microstructures with similar performance but significantly different topological structures is of significant engineering importance. In diversity-driven topology optimization methods, accurately and efficiently quantifying the structural diversity among microstructures is the core foundation of the entire technology system.

[0004] Currently, the geometric similarity and diversity of metamaterial microstructures are mainly characterized by spatial domain distance metrics. A typical approach is to use Euclidean distance based on density fields, which involves summing the squares of the material density differences between two microstructures at all spatial locations and then taking the square root to measure their structural differences. However, while spatial domain distance metrics are effective for similarity assessment of conventional single structures, they have insurmountable limitations for periodic metamaterials composed of periodically arranged microstructures. Specifically, the macroscopic properties of periodic metamaterials are determined by the periodic arrangement of their microstructures. Microstructures with different spatial translational patterns can be periodically extended to form metamaterials with identical structural properties; that is, periodic metamaterials possess translational invariance. However, spatial domain-based distance metrics treat microstructures as images fixed at specific locations in space, directly comparing them point by point without considering the inherent physical property of translation invariance. This means that microstructures that produce different spatial domain morphologies due to spatial translation but essentially constitute the same metamaterial will be judged as different individuals with significant structural differences under this metric, thus creating a serious illusion of diversity. As a result, multiple alternative solutions output by the optimization design are essentially the same metamaterial, causing metamaterial optimization design selection failure and wasted R&D costs. Summary of the Invention

[0005] To address some or all of the technical problems existing in the prior art, this invention provides a spectrum-based method for measuring the diversity of metamaterial structures, which can eliminate the illusion of diversity caused by the translational invariance of periodic metamaterials and truly reflect the essential topological differences of microstructures.

[0006] The technical solution of the present invention is as follows: A spectrum-based method for measuring the structural diversity of metamaterials is provided, including: The discrete density fields of the first microstructure and the second microstructure are obtained, and the discrete density fields are used to characterize the material distribution of the microstructure. Fourier transforms are performed on the discrete density fields of the first microstructure and the second microstructure respectively to obtain the first frequency domain result and the second frequency domain result; Extract the first amplitude spectrum from the first frequency domain result, and extract the second amplitude spectrum from the second frequency domain result; Based on the first amplitude spectrum and the second amplitude spectrum, the diversity distance between the first microstructure and the second microstructure is calculated as an indicator to measure the structural differences between the microstructures.

[0007] Furthermore, in some embodiments, the Fourier transform is a two-dimensional discrete Fourier transform.

[0008] Furthermore, in some implementations, the discrete density field of the microstructure is expressed as a Fourier transform as follows: ; in, Represents the spatial index of a discrete density field. Represents the row index of a discrete density field. The column index represents the discrete density field. This represents the total number of rows in the discrete density field. This represents the total number of columns in the discrete density field. Describes the first in a discrete density field Line 1 Column elements, Represents the spatial index in the frequency domain. Represents the row index in the frequency domain. Column index representing the frequency domain, The range of values ​​and The range of values ​​is the same. The range of values ​​and The range of values ​​is the same. To represent a complex unit, This represents an exponential function with base e. Representation matrix The Middle Line 1 Column elements, matrix This represents the matrix obtained after performing a two-dimensional discrete Fourier transform on the discrete density field of the microstructure.

[0009] Furthermore, in some implementations, the amplitude spectrum is extracted from the frequency domain results using the following formula: ; in, Represents the amplitude spectrum. Representation matrix The length of the module.

[0010] Furthermore, in some implementations, the diversity distance between microstructures is calculated using the following formula: ; in, Indicates the first The microstructure and the first The diversity distance between individual microstructures and Represents a microstructure index. Represents the frequency weighting matrix The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 The elements of the column.

[0011] Furthermore, in some implementations, the frequency weighting matrix Determined in the following ways: ; in, Represents the frequency weighting matrix The Middle Line 1 Column elements, This represents the corrected spatial index. , , This indicates the preset penalty intensity control parameter.

[0012] The main advantages of the technical solution of this invention are as follows: The spectrum-based metamaterial structure diversity measurement method of this invention transforms the evaluation of microstructure diversity from the spatial domain to the frequency domain and extracts the amplitude spectrum with translation invariance as the feature descriptor of the microstructure. This fundamentally eliminates the illusion of diversity caused by the translation invariance of periodic metamaterials. It ensures that microstructures that differ in shape only due to spatial translation but essentially constitute the same metamaterial are accurately identified as equivalent. This avoids outputting essentially duplicated false alternatives in diversity-driven metamaterial topology optimization design, effectively preventing selection failure and waste of R&D costs, and providing a reliable basic measurement guarantee for obtaining real and diverse metamaterial configuration designs. Attached Figure Description

[0013] The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and constitute a part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a schematic diagram of an existing periodic metamaterial structure and its three corresponding microstructures. Figure 2 A flowchart illustrating a method for measuring the structural diversity of metamaterials based on the spectrum, provided in an embodiment of the present invention; Figure 3 A schematic diagram of a periodic metamaterial and its three corresponding microstructures and the amplitude spectrum corresponding to the three microstructures provided in Example 1 of the present invention; Figure 4 for Figure 3 A schematic diagram showing the Euclidean distance distribution between the three microstructures is shown. Figure 5 for Figure 3 The diagram shows the diversity distance distribution among the three microstructures. Figure 6 A schematic diagram of a set of microstructures with fundamentally different topologies provided in Example 2 of the present invention; Figure 7 for Figure 6 The diagram shows the diversity distance distribution among a group of microstructures. Detailed Implementation

[0014] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0015] The technical solutions provided by the embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0016] For periodic metamaterials composed of periodically arranged microstructures, different microstructures may constitute the same metamaterial. For example... Figure 1 The three different microstructures shown, despite their significant morphological differences, can form identical metamaterials through periodic arrangement, demonstrating the translational invariance of periodic metamaterials. However, existing spatial domain-based distance metrics treat microstructures as images fixed at specific spatial locations, directly comparing them point-by-point without considering this inherent physical property of translational invariance. This leads to microstructures with different spatial domain morphologies due to spatial translation, but essentially constituting the same metamaterial, being classified as distinct individuals with significant structural differences under this metric. This creates a severe illusion of diversity, resulting in multiple alternative solutions in the optimization design output that are essentially the same metamaterial, causing metamaterial optimization design selection failures and wasted R&D costs.

[0017] refer to Figure 2 To address the aforementioned technical problems, this invention provides a method for measuring the structural diversity of metamaterials based on the spectrum. This method includes the following steps: Step 1: Obtain the discrete density field of the first microstructure and the discrete density field of the second microstructure. The discrete density field is used to characterize the material distribution of the microstructure. Step 2: Perform Fourier transforms on the discrete density fields of the first microstructure and the second microstructure respectively to obtain the first frequency domain result and the second frequency domain result. Step 3: Extract the first amplitude spectrum from the first frequency domain result and extract the second amplitude spectrum from the second frequency domain result; Step 4: Based on the first amplitude spectrum and the second amplitude spectrum, calculate the diversity distance between the first microstructure and the second microstructure as an indicator to measure the structural differences between the microstructures.

[0018] In this embodiment of the invention, the material distribution of the microstructure is characterized as a discrete density field. Specifically, a microstructure is discretized into a spatial grid containing M rows and N columns of grid cells, with each grid cell corresponding to a material density value. The discrete density field of this microstructure can be represented by an M×N matrix. Representation, matrix The Middle Line 1 Column elements The value is taken as the first value of the microstructure. Line 1 The material density values ​​of the column mesh cells. The specific values ​​of M and N are set according to actual requirements.

[0019] In this embodiment of the invention, a Fourier transform is performed on the discrete density field of the microstructure to obtain the corresponding frequency domain result. The amplitude spectrum is extracted from the frequency domain result and used as a translation-invariant feature descriptor for the microstructure in the frequency domain. Based on the amplitude spectrum, the diversity distance between microstructures is calculated. According to the translation theorem of the Fourier transform, a translation of a signal in the spatial domain only leads to a linear change in the phase spectrum of its frequency domain result, while the amplitude spectrum remains absolutely unchanged. For periodic metamaterials, spatial translation only changes the morphology of the microstructure within a unit cell, without altering the macroscopic metamaterial structure formed after periodic extension. Therefore, in this embodiment of the invention, diversity assessment is transformed from the translation-dependent spatial domain to the translation-invariant frequency domain. By utilizing the translation properties of the Fourier transform, the illusion of diversity caused by the translation invariance of periodic metamaterials can be fundamentally eliminated, thereby achieving accurate quantification of the differences in microstructure topology.

[0020] The spectrum-based metamaterial structure diversity measurement method provided in this invention transforms the evaluation of microstructure diversity from the spatial domain to the frequency domain and extracts the amplitude spectrum with translation invariance as the feature descriptor of the microstructure. This fundamentally eliminates the illusion of diversity caused by the translation invariance of periodic metamaterials. It ensures that microstructures that differ in form only due to spatial translation but essentially constitute the same metamaterial are accurately identified as equivalent. This avoids outputting essentially duplicated false alternatives in diversity-driven metamaterial topology optimization design, effectively preventing selection failures and wasted R&D costs, and providing a reliable basic measurement guarantee for obtaining real and diverse metamaterial configuration designs.

[0021] Furthermore, in this embodiment of the invention, the Fourier transform is a two-dimensional discrete Fourier transform. Specifically, the corresponding frequency domain result is obtained by performing a two-dimensional discrete Fourier transform on the discrete density field of the microstructure.

[0022] In this embodiment of the invention, the Fast Fourier Transform is used to implement the Discrete Fourier Transform.

[0023] In this embodiment of the invention, the two-dimensional discrete Fourier transform of the discrete density field of the microstructure is specifically expressed as follows: ; in, Represents the spatial index of a discrete density field. Represents the row index of a discrete density field. The column index represents the discrete density field. This represents the total number of rows in the discrete density field. This represents the total number of columns in the discrete density field. Describes the first in a discrete density field Line 1 Column elements, Represents the spatial index in the frequency domain. Represents the row index in the frequency domain. Column index representing the frequency domain, The range of values ​​and The range of values ​​is the same. The range of values ​​and The range of values ​​is the same. To represent a complex unit, This represents an exponential function with base e. Representation matrix The Middle Line 1 Column elements, matrix This represents the matrix obtained after performing a two-dimensional discrete Fourier transform on the discrete density field of the microstructure.

[0024] Furthermore, the Fourier transform has the following translation property: ; in, This represents the spatial index of the translated discrete density field. , , Indicates the amount of spatial translation. Indicates the amount of translation in the row direction. Indicates the amount of translation in the column direction. This represents the Fourier transform operator, specifically indicating the performance of a two-dimensional discrete Fourier transform on the signal within the brackets.

[0025] Based on the translation properties described above, when a density field undergoes arbitrary translation in the spatial domain, although its shape changes in the spatial domain, in the frequency domain, this change is merely manifested as the original spectrum multiplied by a pure phase rotation factor. Since the magnitude of this phase rotation factor is 1, the amplitude of the original spectrum and the amplitude of the translated spectrum are exactly the same. Therefore, in this embodiment of the invention, the corresponding amplitude spectrum is extracted from the frequency domain result, and the amplitude spectrum is used as a feature descriptor for the microstructure to have translation invariance in the frequency domain.

[0026] In this embodiment of the invention, the amplitude spectrum is extracted from the frequency domain results using the following formula: ; in, Represents the amplitude spectrum. Representation matrix The length of the module.

[0027] Furthermore, in this embodiment of the invention, to facilitate visualization and analysis, after extracting the amplitude spectrum from the frequency domain results, the amplitude spectrum can be further post-processed.

[0028] In this embodiment of the invention, post-processing of the amplitude spectrum includes: The amplitude spectrum is centered to move the low-frequency components to the center of the spectrum, resulting in a centered amplitude spectrum. Taking the logarithm of the centered amplitude spectrum yields the logarithmically scaled amplitude spectrum.

[0029] In this embodiment of the invention, the diversity distance between microstructures can be calculated based on the extracted original amplitude spectrum or the post-processed amplitude spectrum as an indicator to measure the structural differences between microstructures.

[0030] Furthermore, in this embodiment of the invention, the diversity distance between microstructures is calculated using the following formula: ; in, Indicates the first The microstructure and the first The diversity distance between individual microstructures and Represents a microstructure index. Represents the spatial index in the frequency domain. Represents the row index in the frequency domain. Column index representing the frequency domain, This represents the total number of rows in the discrete density field. This represents the total number of columns in the discrete density field. Represents the frequency weighting matrix The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 The elements of the column.

[0031] In this embodiment of the invention, considering that the contribution of each frequency component in the amplitude spectrum to the structural differences of the microstructure is not equal, with low-frequency components dominating the global load-bearing skeleton and overall mechanical behavior, while high-frequency components mainly characterize local details with minimal impact on mechanical performance, treating all frequency components equally could lead to the diversity distance calculation results being dominated by high-frequency noise. Therefore, to avoid interference from high-frequency noise in the diversity distance and to highlight physically meaningful structural differences, a frequency weighting matrix is ​​introduced into the aforementioned diversity distance calculation formula. Among them, the frequency weighting matrix Determined in the following ways: ; in, Represents the frequency weighting matrix The Middle Line 1 Column elements, This represents the corrected spatial index. , , This represents the preset penalty intensity control parameter, which controls the intensity of the penalty applied to the frequency component. The specific value of the penalty intensity control parameter is set according to actual needs; the smaller the value, the stronger the penalty applied to the frequency component. For example, set it to 20.

[0032] The effectiveness of the spectrum-based metamaterial structure diversity measurement method provided by the embodiments of the present invention will be illustrated below with specific examples: refer to Figures 3-5 , Figure 3 This is a schematic diagram of a periodic metamaterial and its three corresponding microstructures, as well as the amplitude spectra corresponding to the three microstructures, provided in Example 1 of the present invention. Figure 4 for Figure 3 The diagram shows the Euclidean distance distribution between the three microstructures. Figure 5 for Figure 3 The diagram shows the diversity distance distribution among the three microstructures, where, Figure 3 amplitude spectrum , , All are post-processed amplitude spectra. Figure 4 middle This is the Euclidean distance. According to... Figures 3-5 It can be seen that although the three microstructures differ significantly in spatial domain morphology, their periodic arrangement constitutes an identical metamaterial. However, when the structural differences of the three microstructures are measured using Euclidean distance based on the density field, the Euclidean distances between them differ considerably, indicating a large structural difference and creating an illusion of diversity. In contrast, when the method provided in this embodiment measures the structural differences of the three microstructures, since their frequency domain amplitude spectra are almost identical, the calculated diversity distance between them is zero. This accurately reflects the essential similarity of the three microstructures and effectively eliminates the illusion of diversity caused by the translational invariance of periodic metamaterials.

[0033] refer to Figures 6-7 , Figure 6 This is a schematic diagram of a set of microstructures with fundamentally different topologies provided in Example 2 of the present invention. Figure 7 for Figure 6 This diagram illustrates the diverse distance distribution among a group of microstructures. According to... Figures 6-7 It can be seen that for a group of microstructures with essential differences in topology, when the structural differences of this group of microstructures are measured using the method provided in the embodiments of the present invention, the calculated diversity distances between each pair of microstructures are different. The diversity distances between microstructures with similar topologies are smaller, while the diversity distances between microstructures with large differences in topology are larger. This shows that the method provided in the embodiments of the present invention can not only eliminate the illusion of diversity caused by the translation invariance of periodic metamaterials, but also truly, effectively and precisely reflect the essential topological differences between microstructures.

[0034] It should be noted that, in this document, relational terms such as “first” and “second” are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms “comprising,” “including,” or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus.

[0035] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for measuring the structural diversity of metamaterials based on the spectrum, characterized in that, include: The discrete density fields of the first microstructure and the second microstructure are obtained, and the discrete density fields are used to characterize the material distribution of the microstructure. Fourier transforms are performed on the discrete density fields of the first microstructure and the second microstructure respectively to obtain the first frequency domain result and the second frequency domain result; Extract the first amplitude spectrum from the first frequency domain result, and extract the second amplitude spectrum from the second frequency domain result; Based on the first amplitude spectrum and the second amplitude spectrum, the diversity distance between the first microstructure and the second microstructure is calculated as an indicator to measure the structural differences between the microstructures; The diversity distance between microstructures can be calculated using the following formula: ； in, Indicates the first The microstructure and the first The diversity distance between individual microstructures and Represents a microstructure index. This represents the total number of rows in the discrete density field. This represents the total number of columns in the discrete density field. Represents the spatial index in the frequency domain. Represents the row index in the frequency domain. Column index representing the frequency domain, Represents the frequency weighting matrix The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 Column elements, Indicates the first Amplitude spectrum of microstructures The Middle Line 1 Column elements; Frequency weighting matrix Determined in the following ways: ； in, Represents the frequency weighting matrix The Middle Line 1 Column elements, This represents the corrected spatial index. , , This indicates the preset penalty intensity control parameter.

2. The method for measuring the diversity of metamaterial structures based on the spectrum according to claim 1, characterized in that, The Fourier transform is a two-dimensional discrete Fourier transform.

3. The method for measuring the diversity of metamaterial structures based on the spectrum according to claim 2, characterized in that, The discrete density field of the microstructure can be expressed as follows using a Fourier transform: ； in, Represents the spatial index of a discrete density field. Represents the row index of a discrete density field. The column index represents the discrete density field. Describes the first in a discrete density field Line 1 Column elements, The range of values ​​and The range of values ​​is the same. The range of values ​​and The range of values ​​is the same. To represent a complex unit, This represents an exponential function with base e. Representation matrix The Middle Line 1 Column elements, matrix This represents the matrix obtained after performing a two-dimensional discrete Fourier transform on the discrete density field of the microstructure.

4. The method for measuring the diversity of metamaterial structures based on the spectrum according to claim 3, characterized in that, The amplitude spectrum can be extracted from the frequency domain results using the following formula: ； in, Represents the amplitude spectrum. Representation matrix The length of the module.

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