Fractal geometry-based three-dimensional structure unit cell configuration multi-physical field performance prediction method and regulation and control method

Abstract

The invention provides a fractal geometry-based three-dimensional structure unit cell configuration multi-physics field performance prediction method and a fractal geometry-based three-dimensional structure unit cell configuration multi-physics field regulation and control method, and relates to the field of electric digital data processing. Comprising the steps that according to a specific three-dimensional structure unit cell configuration, the fractal dimension of the whole configuration is calculated, and the duty ratio and the thickness of each layer of the three-dimensional structure unit cell configuration are determined; constructing a mapping relation model of fractal dimensions and electromagnetic, mechanical and optical properties; and predicting the electromagnetic, mechanical or optical performance of the unit cell configuration of the three-dimensional structure according to the determined fractal dimension in combination with the mapping relation model. According to the method, a precise design tool can be provided for wave-absorbing materials, high-strength structural materials, optical regulation and control materials and the like.

Description

Technical Field

[0001] This invention relates to the field of electronic digital data processing, and proposes a theoretical modeling method for the directional control of electromagnetic, mechanical, and optical properties by quantitatively characterizing the geometric complexity of microstructure unit cells using fractal dimension. Based on the multi-scale geometric feature characterization of fractal geometry, and combined with equivalent medium theory and multi-physics coupling models, this method constructs a unified design paradigm for "mathematically defined materials," providing theoretical support for the precise design of advanced functional materials. Background Technology

[0002] Traditional functional material design relies on empirical trial-and-error methods, optimizing performance by adjusting material composition or macroscopic structure, but struggles to achieve precise control across multiple physics dimensions. Fractal geometry, as a mathematical tool for describing complex irregular structures, has been used for the geometric characterization of material microstructures, such as quantitatively describing the geometric complexity of honeycomb and stepped cone structures using box-counting dimension calculations. However, current technologies only establish a single mapping between electromagnetic fractal dimension and wave impedance, while fractal control models for mechanical strength and optical bandgap remain lacking. In the electromagnetic domain, although a corrected relationship between fractal dimension and wave impedance has been established, the fractal control mechanisms for mechanical properties (such as strength and toughness) and optical properties (such as reflectivity and bandgap) still lack theoretical support. Furthermore, a unified framework for fractal-performance mapping across multiple physics fields has not yet been established, hindering the transition of material design from "empirical trial-and-error" to "mathematical definitions." Therefore, there is an urgent need to develop multiphysics performance prediction methods based on fractal dimensions. Summary of the Invention

[0003] The purpose of this invention is to establish a method for predicting and controlling the multiphysics performance of three-dimensional structural unit cell configurations with fractal dimension as the core design parameter.

[0004] The technical solution of this invention is: a multiphysics performance prediction method for three-dimensional structural unit cell configurations based on fractal geometry, comprising: Based on the specific three-dimensional structural unit cell configuration, calculate the fractal dimension of the overall configuration, and determine the duty cycle and thickness of each layer of the three-dimensional structural unit cell configuration. Construct a mapping model between fractal dimension and electromagnetic, mechanical, and optical properties; predict the electromagnetic, mechanical, or optical properties of three-dimensional structural unit cell configurations based on the determined fractal dimension and the mapping model.

[0005] Preferably, the mapping model between fractal dimension and electromagnetic properties is constructed based on the equivalent medium theory: The experiment measured the basic electromagnetic parameters of the loss material used in the three-dimensional structural unit cell configuration. It was assumed that the loss material was an isotropic medium, and the equivalent electromagnetic parameters of each layer of the configuration were calculated according to the duty cycle. The basic electromagnetic parameters included negative permittivity and negative permeability. By using the basic electromagnetic parameters, fractal dimension, and duty cycle of the loss material, a structural resonance effect factor is constructed, and then the impedance gradient correction coefficient is corrected. A mapping model between fractal dimension and electromagnetic performance is established based on the corrected impedance gradient correction coefficient, the thickness of each layer, and the equivalent electromagnetic parameters. This mapping model is used to calculate the interface wave impedance and surface reflectivity of the three-dimensional structural unit cell configuration.

[0006] Preferably, the formula for calculating the structural resonance effect factor is as follows:

[0007] In the formula, Let k be the duty cycle of the material in the k-th layer of the configuration. The fractal dimension of the three-dimensional structure's unit cell configuration. The baseline fractal dimension is 2.

[0008] Preferably, the mapping model between fractal dimension and electromagnetic properties is as follows: Interface impedance

[0009] reflectivity

[0010] In the formula, the impedance gradient correction coefficient of the k-th layer is... ; It is the equivalent dielectric constant; It is the equivalent permeability; The interface wave impedance of the surface of the configuration. This is the structural resonance effect factor.

[0011] Preferably, the mapping model between fractal dimension and mechanical properties is established based on fractal fracture mechanics theory: elastic modulus Fracture strength ; Energy absorption capacity

[0012] in , , For material constants, The value range is 1-10. The value range is 0-1. The value range is 0-1. , , The elastic modulus, fracture strength, and energy absorption capacity of the materials used in the respective configurations are given, where D is the fractal dimension.

[0013] Preferably, the mapping model between fractal dimension and optical performance is established based on fractal photonic crystal theory: Photonic bandgap center frequency Optical reflectivity ; Refractive index , where ω, , These are optical characteristic coefficients, all taking values ​​in the range of (0~1). , , The center frequency, reflectivity, and refractive index of the materials used in the respective configurations are given, and D is the fractal dimension.

[0014] A multiphysics method for controlling the configuration of a three-dimensional structural unit cell based on fractal geometry, comprising: Based on the multiphysics performance prediction method for three-dimensional single-cell configurations, a mapping model between fractal dimension and electromagnetic, mechanical, and optical properties is constructed. Based on specific application requirements, the above mapping relationship model can be used to achieve targeted control of macroscopic physical properties by adjusting the fractal dimension.

[0015] Preferably, the electromagnetic performance is directionally controlled based on the principle that the smaller the fractal dimension, the better the electromagnetic loss performance.

[0016] Preferably, the mechanical properties are directionally controlled according to the principle that the larger the fractal dimension, the stronger the energy dissipation capability of the configuration, and the greater the compressive strength.

[0017] Preferably, the optical performance is directionally controlled based on the power function relationship between the fractal dimension and the bandgap frequency of the configuration.

[0018] The advantages of this invention compared to the prior art are as follows: This invention innovatively proposes a new paradigm of "mathematically defined materials" with fractal dimension as the core design parameter, breaking through the limitations of traditional empirical design. Mathematically defined materials refer to the precise prediction and reverse control design of physical properties through mathematical equations of geometric parameters such as fractal dimension and duty cycle. It establishes a quantitative mapping relationship between fractal dimension and electromagnetic, mechanical, and optical properties, enabling the prediction and targeted control of material properties. Specifically, it uses the box-counting dimension method of fractal geometry to characterize the geometric complexity of microstructure unit cells, combines equivalent medium theory to construct a macroscopic equivalent model of electromagnetic properties, introduces fractal fracture mechanics theory to establish a mechanical property prediction model, and derives optical property control equations based on fractal photonic crystal theory, forming a unified theoretical modeling system for multiple physics fields. This method can realize forward design and reverse optimization from microstructure geometric parameters to macroscopic physical properties, providing precise design tools for microwave absorbing materials, high-strength structural materials, and optical control materials. Attached Figure Description

[0019] Figure 1: Flowchart of mathematically defined material theory modeling, showing the entire process from geometric characterization to multiphysics performance prediction; Figure 2: Schematic diagram of the configuration of electromagnetic fractals (honeycomb structure); Figure 3: Schematic diagram of the structure of an optical fractal (fractal photonic crystal). Detailed Implementation

[0020] The following is in conjunction with the appendix Figure 1-3 This invention provides a detailed description of the present invention. The present invention provides a method for predicting the multiphysics performance of a three-dimensional structural unit cell configuration based on fractal geometry, comprising: (1) Based on the specific three-dimensional structural unit cell configuration, calculate the fractal dimension of the overall configuration and determine the duty cycle and thickness of each layer of the three-dimensional structural unit cell configuration; The three-dimensional structural unit cell configurations include, but are not limited to, honeycomb configurations, stepped cone configurations, stacked wood configurations, box configurations, and combined configurations; the honeycomb configuration is a multi-layered honeycomb gradient arrangement, stacked from largest to smallest; the stepped cone configuration is a multi-layered cube gradient arrangement, stacked from largest to smallest; the stacked wood configuration is a multi-layered staggered arrangement of cuboids; the box configuration is a multi-layered cube gradient arrangement, stacked from largest to smallest; and the combined configuration is a multi-layered arrangement of cuboids and boxes.

[0021] The fractal dimension is currently generally calculated using the box-counting method, which involves filling the geometric object with a small sphere of radius r. This invention provides a preferred embodiment for calculating the fractal dimension in the following manner: Construct a geometric model of a three-dimensional structural unit cell configuration, and use cubic meshes of different scales to contain the three-dimensional space occupied by the geometric model; Calculate the number of cubic meshes containing objects, Nr; right and In least squares regression analysis, the slope of the fitted line is the fractal dimension. Where r is the side length of the cube mesh.

[0022] With honeycomb structure (electromagnetic fractal) Figure 2 Taking the stepped cone structure (mechanical fractal) as an example, the box-counting dimension method is used to calculate the fractal dimension: First, a cubic mesh with side lengths of 0.2 mm, 0.05 mm, and 0.002 mm is used to cover the three-dimensional configuration space. The number of meshes N containing the configuration at each scale is counted. Least square regression analysis is performed on ln N and ln(1 / r), and the slope of the fitted curve is the fractal dimension D. For example, the fractal dimension of a certain cross-section of a honeycomb structure is 2.9695, and the fractal dimension of a typical cross-section of a stepped cone structure is 2.9898.

[0023] (2) Construct a mapping model between fractal dimension and electromagnetic, mechanical, and optical properties; 2.1) Mapping model between fractal dimension and electromagnetic properties The experiment measured the basic electromagnetic parameters of the loss material used in the three-dimensional structural unit cell configuration. It was assumed that the loss material was an isotropic medium, and the equivalent electromagnetic parameters of each layer of the configuration were calculated according to the duty cycle. The basic electromagnetic parameters included negative permittivity and negative permeability. By utilizing the fundamental electromagnetic parameters, fractal dimension, and duty cycle of the loss material, a structural resonance effect factor is constructed, which in turn corrects the impedance gradient correction coefficient. The formula for calculating the structural resonance effect factor is as follows:

[0024] In the formula, Let k be the duty cycle of the material in the k-th layer of the configuration. The fractal dimension of the three-dimensional structure's unit cell configuration. The baseline fractal dimension is 2.

[0025] A mapping model between fractal dimension and electromagnetic performance is established based on the corrected impedance gradient correction coefficient, the thickness of each layer, and the equivalent electromagnetic parameters. This mapping model is used to calculate the interface wave impedance and surface reflectivity of the three-dimensional structural unit cell configuration.

[0026] Interface impedance

[0027] reflectivity

[0028] In the formula, the impedance gradient correction coefficient of the k-th layer is... ; It is the equivalent dielectric constant; It is the equivalent permeability; The interface wave impedance of the surface of the configuration. is the structural resonance effect factor, and k represents the kth layer in the three-dimensional structural unit cell configuration.

[0029] Equivalent dielectric constant With equivalent permeability ; Let k be the duty cycle of the k-th layer. The dielectric constant of the material, The dielectric constant of air, The magnetic permeability of the material, The magnetic permeability of air 2.2) Mapping model between fractal dimension and mechanical properties elastic modulus Fracture strength ; Energy absorption capacity

[0030] in , , For material constants, The value range is 1-10. The value range is 0-1. The value range is 0-1. , , The elastic modulus, fracture strength, and energy absorption capacity of the materials used in each configuration.

[0031] 2.3) Mapping model between fractal dimension and optical performance Photonic bandgap center frequency Optical reflectivity ; Refractive index , where ω, , These are optical characteristic coefficients, all taking values ​​in the range of 0-1. , , The center frequency, reflectivity, and refractive index of the materials used in the respective configurations are determined.

[0032] The electromagnetic, mechanical, or optical properties of a three-dimensional structural unit cell configuration can be predicted based on the determined fractal dimension and the mapping relationship model.

[0033] The present invention further provides a multi-physics field control method for three-dimensional structural unit cell configuration based on fractal geometry. Specifically, the mapping relationship model between fractal dimension and electromagnetic, mechanical and optical properties is first constructed according to the above steps (1) and (2). Then, step (3) is executed. Based on specific application requirements, the above mapping relationship model is used to achieve directional control of macroscopic physical properties by adjusting the fractal dimension.

[0034] Research has found that electromagnetic performance is better with a smaller fractal dimension.

[0035] Research has found that in the field of mechanics, the larger the fractal dimension, the higher the topological complexity of the microstructure unit cell, and the weaker the stress concentration effect when the material is under load. The fractal dimension increases the tortuosity of the crack propagation path, thereby improving the energy dissipation capacity of the material. Simulation experiments show that for every 0.1 increase in fractal dimension, the compressive strength of the stepped cone structure increases by about 7%.

[0036] Research has found that in the field of optics, the fractal dimension of a microstructure unit cell has a power function relationship with the bandgap frequency. By designing the number of fractal layers and the duty cycle, optical reflection / transmission in the visible to infrared bands can be controlled.

[0037] Example 1: Electromagnetic fractal Based on the equivalent transmission line theory, a mapping model between fractal dimension and electromagnetic parameters is established: when the duty cycle dielectric constant of the medium magnetic permeability When, the equivalent dielectric constant equivalent permeability Through fractal dimension Calculate the resonance effect factor Impedance gradient correction factor Then, the electromagnetic performance prediction is completed using the following formula: calculate the interface wave impedance. Corresponding reflectivity .

[0038] Furthermore, this invention can be used to address radar stealth requirements in a specific electromagnetic band by adjusting the fractal dimension of the configuration to achieve effective electromagnetic loss in that band.

[0039] Example 2: Mechanical Fractal Based on fractal fracture mechanics theory, a quantitative relationship between fractal dimension and mechanical properties is established: a benchmark strength is set. Material constants When fractal dimension At that time, fracture strength Reference elastic modulus ,index At that time, elastic modulus Fractal dimension increases the tortuosity of crack propagation paths, thereby improving the energy dissipation capacity of materials. Simulation experiments show that for every 0.1 increase in fractal dimension, the compressive strength of the stepped cone structure increases by approximately 7%.

[0040] To meet the structural mechanical load-bearing requirements, a lightweight design that satisfies mechanical strength is achieved by adjusting the fractal dimension of the configuration.

[0041] Example 3: Optical Fractal Based on fractal photonic crystals (e.g.) Figure 3 Based on the self-similarity property of fractal dimension, a mapping model between fractal dimension and optical properties is constructed: reference frequency. Optical coefficients Fractal dimension Corresponding photonic bandgap center frequency (Near-infrared band); reference reflectivity attenuation coefficient At that time, optical reflectivity . , n = 1.91 + 0.45 2.8875 .

[0042] Fractal dimension enables precise control of band gap position and width by regulating the spatial filling density and periodic distribution of microstructure unit cells. The error between theoretical calculations and simulation results of fractal dendritic photonic crystals is less than 5%.

[0043] For the visible light stealth requirement of a specific frequency band, the present invention can adjust the bandgap center frequency and reflectivity by controlling the fractal dimension.

[0044] The parts of this invention not described in detail are common knowledge to those skilled in the art.

Claims

1. A method for predicting the multiphysics performance of three-dimensional structural unit cell configurations based on fractal geometry, characterized in that... include: Based on the specific three-dimensional structural unit cell configuration, calculate the fractal dimension of the overall configuration, and determine the duty cycle and thickness of each layer of the three-dimensional structural unit cell configuration. Construct a mapping model between fractal dimension and electromagnetic, mechanical, and optical properties; predict the electromagnetic, mechanical, or optical properties of three-dimensional structural unit cell configurations based on the determined fractal dimension and the mapping model.

2. The method for predicting the multiphysics performance of three-dimensional structural unit cell configurations according to claim 1, characterized in that: The mapping model between fractal dimension and electromagnetic properties is constructed based on the equivalent medium theory: The experiment measured the basic electromagnetic parameters of the loss material used in the three-dimensional structural unit cell configuration. It was assumed that the loss material was an isotropic medium, and the equivalent electromagnetic parameters of each layer of the configuration were calculated according to the duty cycle. The basic electromagnetic parameters included negative permittivity and negative permeability. By using the basic electromagnetic parameters, fractal dimension, and duty cycle of the loss material, a structural resonance effect factor is constructed, and then the impedance gradient correction coefficient is corrected. A mapping model between fractal dimension and electromagnetic performance is established based on the corrected impedance gradient correction coefficient, the thickness of each layer, and the equivalent electromagnetic parameters. This mapping model is used to calculate the interface wave impedance and surface reflectivity of the three-dimensional structural unit cell configuration.

3. The method for predicting the multiphysics performance of three-dimensional single-cell configurations according to claim 2, characterized in that: The formula for calculating the structural resonance effect factor is as follows: In the formula, Let k be the duty cycle of the material in the k-th layer of the configuration. The fractal dimension of the three-dimensional structure's unit cell configuration. The baseline fractal dimension is 2.

4. The method for predicting the multiphysics performance of three-dimensional single-cell configurations according to claim 1, characterized in that: The mapping model between fractal dimension and electromagnetic properties is as follows: Interface impedance reflectivity In the formula, the impedance gradient correction coefficient of the k-th layer is... ; It is the equivalent dielectric constant; It is the equivalent permeability; The interface wave impedance of the surface of the configuration. This is the structural resonance effect factor.

5. The method for predicting the multiphysics performance of three-dimensional structural unit cell configurations according to claim 1, characterized in that: The mapping model between fractal dimension and mechanical properties is established based on fractal fracture mechanics theory: elastic modulus Fracture strength ; Energy absorption capacity in , , For material constants, The value range is 1-10. The value range is 0-1. The value range is 0-1. , , The elastic modulus, fracture strength, and energy absorption capacity of the materials used in the respective configurations are given, where D is the fractal dimension.

6. The method for predicting the multiphysics performance of a three-dimensional structural unit cell configuration according to claim 1, characterized in that, The model for the mapping relationship between fractal dimension and optical performance is established based on the theory of fractal photonic crystals: Photonic bandgap center frequency Optical reflectivity ; Refractive index , where ω, , These are optical characteristic coefficients, all taking values ​​in the range of 0-1. , , The center frequency, reflectivity, and refractive index of the materials used in the respective configurations are given, and D is the fractal dimension.

7. A multiphysics field control method for three-dimensional structural unit cell configurations based on fractal geometry, characterized in that... include: The method for predicting the multiphysics performance of three-dimensional single-cell configurations according to any one of claims 1-6 constructs a mapping model between fractal dimension and electromagnetic, mechanical, and optical properties; Based on specific application requirements, the above mapping relationship model can be used to achieve targeted control of macroscopic physical properties by adjusting the fractal dimension.

8. The method for multiphysics field manipulation of three-dimensional structure unit cell configuration according to claim 7, characterized in that: Electromagnetic performance is directionally controlled based on the principle that the smaller the fractal dimension, the better the electromagnetic loss performance.

9. The method for multiphysics field manipulation of three-dimensional single-cell configuration according to claim 7, characterized in that: Mechanical properties are directionally controlled based on the principle that the larger the fractal dimension, the stronger the energy dissipation capacity of the configuration, and the greater the compressive strength.

10. The method for multiphysics field manipulation of three-dimensional structure unit cell configuration according to claim 7, characterized in that: Optical performance can be directionally controlled based on the power function relationship between fractal dimension and bandgap frequency of the configuration.

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