Method for realizing arbitrary adjustable Poisson's ratio of amorphous system based on machine learning

Abstract

A method for realizing any adjustable Poisson's ratio of an amorphous system based on machine learning comprises the following steps: constructing a two-dimensional amorphous network model which comprises nodes and connections between the nodes; connection in the two-dimensional amorphous network model is optimized and regulated through a machine learning method, a global optimization network is generated, an optimized data set is fed back to an actual network, and continuous regulation of the Poisson ratio v of the actual two-dimensional amorphous network system from-0.9 to 0.9 is achieved. An intelligent optimization machine learning algorithm is utilized, and-0.9 lt;-0.9 lt is comprehensively and effectively realized in various two-dimensional amorphous systems; vlt; the adjustment of 0.9 is close to-1lt of a two-dimensional isotropic system; vlt; 1 is a theoretical limit. In addition, the invention also finds that the system v is determined by a few low-frequency intrinsic modes. By implementing machine learning design, the material with a series of positive and negative v values is manufactured, and the gradient material based on the Poisson's ratio is further realized. The method contributes to the actual progress of the design of the auxetic material.

Description

Technical Field

[0001] The invention relates to a method for realizing an arbitrarily adjustable Poisson's ratio of an amorphous system based on machine learning. Background Art

[0002] Poisson's ratio v describes the elastic deformation properties of solid materials. In particular, materials that exhibit negative Poisson's ratios, or so-called auxetic behavior, are known for their superior energy absorption and resistance to fracture. Therefore, the ability to continuously adjust v from positive to negative is of great interest. When solid materials are subjected to external tension or compression, they deform longitudinally and transversely along the direction of the load. Poisson's ratio ν quantifies the relationship between transverse and longitudinal deformations. A positive ν indicates that the material expands in one direction and contracts in the other (see Figure 1A In contrast, a negative ν indicates expansion or contraction in two directions simultaneously, known as auxetic behavior (see Figure 1A (right image of a in Figure 1). Although most solid materials have positive ν, auxetic materials with negative ν exhibit superior properties in terms of energy absorption and fracture resistance. As a result, these materials are widely used in various fields, including seat belts, bulletproof vests, shoe soles, shock absorbers, and packaging materials. In addition to mechanical applications, auxetic materials can also be used in smart sensors, fatigue resistance, filtration devices, flexible electronics, and ceramic sintering. Therefore, materials that can adjust the Poisson's ratio from positive to negative and achieve auxetic behavior are crucial for diverse industrial applications.

[0003] Although auxetic behavior can be achieved through certain specific structures, universal tuning methods for general amorphous networks remain elusive because complex emergent phenomena in large amorphous systems make it difficult to predict their collective behavior. Classical design methods for creating auxetic materials are mainly based on periodically arranged special structures such as concave polygonal structures, rotated polygonal structures, and chiral structures. However, these periodic structures usually exhibit anisotropic behavior, and their response depends on the direction, limiting their practicality under uncontrollable arbitrary loads. In contrast, amorphous systems are isotropic and show consistent responses in all directions, making them more robust and versatile. Recent studies have shown that negative Poisson's ratios may be achieved in certain amorphous systems, such as packing-derived systems, thus extending the design of auxetic materials to isotropic amorphous systems. However, these findings are limited to specific amorphous systems, and universal tuning methods for arbitrary amorphous systems are still lacking. In addition, the universal and fundamental physical mechanism that determines the Poisson's ratio ν in amorphous systems remains unknown.

[0004] It should be noted that the information disclosed in the above background technology section is only used for understanding the background of the present application, and therefore may include information that does not constitute prior art known to ordinary technicians in the field. Summary of the invention

[0005] The main purpose of the present invention is to overcome the defects of the above-mentioned background technology and provide a method for realizing arbitrarily adjustable Poisson's ratio of an amorphous system based on machine learning.

[0006] To achieve the above object, the present invention adopts the following technical solutions:

[0007] A method for realizing an arbitrarily adjustable Poisson's ratio of an amorphous system based on machine learning, comprising: constructing a two-dimensional amorphous network model, the model including nodes and connections between nodes; optimizing and regulating the connections in the two-dimensional amorphous network model by a machine learning method to generate a global optimized network, and feeding the optimized data set back to the actual network to realize continuous adjustment of the Poisson's ratio of the actual two-dimensional amorphous network system from -0.9 to 0.9.

[0008] Furthermore, the machine learning method uses a simulated annealing (SA) algorithm to control the Poisson's ratio, comprising the following steps:

[0009] Select an initial network structure and set the initial temperature, initial solution and cooling plan of the simulated annealing algorithm;

[0010] Execute the outer loop, and start the simulated annealing algorithm according to the set initial temperature and initial solution, gradually reducing the temperature from the initial temperature to the final temperature, and each iteration corresponds to an annealing temperature;

[0011] In each iteration of the outer loop, an inner loop is performed, the current state is set as the initial solution, and the current network structure and the corresponding Poisson's ratio are recorded;

[0012] For the inner loop, in each iteration, a certain number of connections are removed or added to the network system;

[0013] For the inner loop, in each iteration, a random number from a uniform distribution is generated and compared with the exponential function calculated by the Metropolis criterion; preferably, the random number is a random number in the range [0,1];

[0014] If the random number is less than the result of the exponential function, accept the new solution and update the network structure and the corresponding Poisson's ratio; otherwise, accept the new solution with a dynamic acceptance probability according to the Metropolis criterion and update the network structure and the corresponding Poisson's ratio;

[0015] If the preset termination condition is reached, such as the maximum number of iterations or the preset final temperature, the algorithm terminates and outputs the final network structure.

[0016] Furthermore, the dynamic acceptance probability P is:

[0017]

[0018] Among them, π t and π t+1 represents the network structure at iteration steps t and t+1, ν(π t ) and ν(π t+1 ) represents the Poisson’s ratio of the amorphous system under the corresponding network structure, T t represents the temperature of the simulated annealing algorithm at iteration step t.

[0019] Furthermore, for expansion systems of different sizes, the Poisson's ratio of the network after Poisson's ratio regulation converges to 0.9; the expansion structure of the system has the following characteristics: (1) it is amorphous and isotropic in all directions; (2) it is composed entirely of convex polygons; and (3) it does not have any chiral characteristics.

[0020] Furthermore, the network of the amorphous system is a twisted triangular lattice network, and the construction of the two-dimensional amorphous network model includes: constructing a perfect triangular lattice network; Acting on each node within the boundary, a deformation is introduced, where the displacement vector of the αth node is expressed as where θ α Randomly distributed in the range of [0,2π], represents the displacement direction, r represents the degree of distortion, and r is less than half of the lattice constant; after deformation, the connection between the nodes is replaced by a new relaxed spring.

[0021] Furthermore, the network of the amorphous system is a Delaunay triangulated random network, and constructing a two-dimensional amorphous network model includes: randomly distributing a group of nodes in a rectangular area, creating relaxed spring connections through Delaunay triangulation, and setting the average coordination number of the network to 6.

[0022] Furthermore, the network of the amorphous system is a packing-derived network, and the construction of the two-dimensional amorphous network model includes: constructing a two-dimensional bidisperse filling system, which is composed of two kinds of particles of the same number and a radius ratio of 1:1.4; replacing each particle with the same sphere, and the contact between adjacent particles is replaced by the same relaxation spring.

[0023] Further, the method includes: determining the eigenmodes of the two-dimensional amorphous network model at the frequency corresponding to the peak value of the positive Poisson's ratio and the eigenmodes at the frequency corresponding to the peak value of the negative Poisson's ratio; calculating the projection probabilities or weights of the eigenmodes of these two frequencies respectively according to the actual deformation field under the external load; adjusting the connections in the two-dimensional amorphous network model so that the weight of the eigenmode corresponding to the frequency of the peak value of the negative Poisson's ratio dominates to adjust the Poisson's ratio to a negative value, or so that the weight of the eigenmode corresponding to the frequency of the peak value of the positive Poisson's ratio dominates to adjust the Poisson's ratio to a positive value.

[0024] Further, a Poisson's ratio-based functionally graded material without obvious structural or compositional gradients is adjusted, and networks with different Poisson's ratios are integrated in a single system.

[0025] A computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the method described above is implemented.

[0026] The present invention has the following beneficial effects:

[0027] The present invention provides a method for realizing arbitrarily adjustable Poisson's ratio in an amorphous system based on machine learning. For the constructed two-dimensional amorphous network model, the connections in the two-dimensional amorphous network model are optimized and regulated by machine learning methods to generate a globally optimized network, and the optimized data set is fed back into the actual network, so as to realize the continuous adjustment of the Poisson's ratio of the actual two-dimensional amorphous network system from -0.9 to 0.9. The inventor has discovered the general adjustment principle and clarified the basic physical mechanism of ν in the amorphous system in this paper. By using the intelligent optimization machine learning algorithm, the present invention realizes the continuous adjustment of the Poisson's ratio from -0.9 to 0.9, approaching the theoretical limit -1 < ν < 1 in the 2D case. The present invention determines the physical origin of ν, that is, one or two low-frequency vibration modes are excited, and reveals the basic physical principle of Poisson's ratio under general conditions. Using machine learning, the present invention can further design functionally graded materials based on Poisson's ratio and without obvious structural gradients, and can realize them by 3D printing.

[0028] As described above, by using the intelligent optimization machine learning algorithm, the present invention comprehensively and effectively realizes the adjustment of -0.9 < ν < 0.9 in various two-dimensional amorphous systems, approaching the theoretical limit of -1 < ν < 1 of the two-dimensional isotropic system. In addition, the present invention also finds that the system ν is determined by a few low-frequency eigenmodes. Through the implementation of machine learning design, the present invention manufactures materials with a series of positive and negative ν values, and further realizes gradient materials based on Poisson's ratio. The present invention contributes to the practical progress of auxetic material design.

[0029] Other beneficial effects in the embodiments of the present invention will be further described below. Description of the Drawings

[0030] Figure 1A to Figure 1B Tuning of ν from positive to negative values ​​in three types of amorphous networks is shown.

[0031] Figure 2 The differences between the auxetic structure of the embodiment of the present invention and the existing designs are shown.

[0032] FIG. 3A to FIG. 3B The relationship between the actual Poisson's ratio and the eigenmodes in an amorphous system is shown.

[0033] Figure 4 The experimental implementation of 3D printing according to an embodiment of the present invention is shown.

[0034] Figure 5 It is shown that an embodiment of the present invention realizes a ν gradient material.

[0035] Figure 6 Auxetic random networks of embodiments of the present invention with sizes ranging from 10×10 to 32×32 are shown.

[0036] Figure 7 It shows that the random networks of the embodiments of the present invention are adjusted to a more positive Poisson's ratio, with sizes ranging from 10×10 to 32×32.

[0037] Figure 8 The statistics of the change of each connection length under stretching along the x and y directions for the twisted triangular lattice and the Packing-derived network according to the embodiment of the present invention are shown.

[0038] Fig. 9 The modal analysis of a random network according to an embodiment of the present invention is shown.

[0039] Fig.10 The modal analysis of the Packing-derived network according to the embodiment of the present invention is shown.

[0040] Fig.11 The simulated deformation of the fiber network to be 3D printed in the tensile expansion performance experiment calculated by the finite element method according to the embodiment of the present invention is shown.

[0041] Fig.12 It is shown that embodiments of the present invention simulate both non-uniform and uniform deformations. DETAILED DESCRIPTION

[0042] The following is a detailed description of the embodiments of the present invention. It should be emphasized that the following description is only exemplary and is not intended to limit the scope and application of the present invention.

[0043] It should be understood that the terms "length", "width", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", etc., indicating the orientation or position relationship, are based on the orientation or position relationship shown in the accompanying drawings, and are only for the convenience of describing the embodiments of the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation on the present invention.

[0044] In addition, the terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of the embodiments of the present invention, the meaning of "plurality" is two or more, unless otherwise clearly and specifically defined.

[0045] An embodiment of the present invention provides a method for realizing an arbitrarily adjustable Poisson's ratio of an amorphous system based on machine learning, comprising: constructing a two-dimensional amorphous network model, the model including nodes and connections between nodes; optimizing and regulating the connections in the two-dimensional amorphous network model by a machine learning method to generate a global optimization network, and feeding the optimized data set back to the actual network to realize continuous adjustment of the Poisson's ratio of the actual two-dimensional amorphous network system from -0.9 to 0.9.

[0046] In a preferred embodiment, the machine learning method uses a simulated annealing (SA) algorithm to control the Poisson's ratio, including the following steps: selecting an initial network structure, and setting the initial temperature, initial solution, and cooling plan of the simulated annealing algorithm; executing an external loop, starting to execute the simulated annealing algorithm according to the set initial temperature and initial solution, gradually reducing from the initial temperature to the final temperature, and each iteration corresponds to an annealing temperature; in each iteration of the external loop, performing an internal loop, setting the current state as the initial solution, and recording the current network structure and the corresponding Poisson's ratio value; for the internal loop, in each iteration, a certain number of connections are removed or added to the network system; For the inner loop, in each iteration, a random number from a uniform distribution range is generated, such as a random number in the range of [0,1], and compared with the exponential function calculated by the Metropolis criterion; if the random number is less than the result of the exponential function, the new solution is accepted and the network structure and the corresponding Poisson's ratio are updated; otherwise, the new solution is accepted with a dynamic acceptance probability according to the Metropolis criterion and the network structure and the corresponding Poisson's ratio are updated; if the preset termination condition is met, such as the maximum number of iterations or the preset final temperature, the algorithm terminates; the final network structure and the corresponding Poisson's ratio are output, which is the global optimal solution.

[0047] In a preferred embodiment, the dynamic acceptance probability P is:

[0048]

[0049] Among them, π t and π t+1 represents the network structure at iteration steps t and t+1, ν(π t ) and ν(π t+1 ) represents the Poisson’s ratio of the amorphous system under the corresponding network structure, T t represents the temperature of the simulated annealing algorithm at iteration step t.

[0050] In some embodiments, for expansion systems of different sizes, the Poisson's ratio of the network after Poisson's ratio regulation converges to 0.9; the expansion structure of the system has the following characteristics: (1) it is amorphous and isotropic in all directions; (2) it is composed entirely of convex polygons; and (3) it does not have any chiral characteristics.

[0051] In some embodiments, the network of the amorphous system is a twisted triangular lattice network, and the construction of a two-dimensional amorphous network model includes: constructing a perfect triangular lattice network; Acting on each node within the boundary, a deformation is introduced, where the displacement vector of the αth node is expressed as where θ α Randomly distributed in the range of [0,2π], represents the displacement direction, r represents the degree of distortion, and r is less than half of the lattice constant; after deformation, the connection between the nodes is replaced by a new relaxed spring.

[0052] In some embodiments, the network of the amorphous system is a Delaunay triangulated random network, and constructing a two-dimensional amorphous network model includes: randomly distributing a group of nodes in a rectangular area, creating relaxed spring connections through Delaunay triangulation, and setting the average coordination number of the network to 6.

[0053] In some embodiments, the network of the amorphous system is a packing-derived network, and constructing a two-dimensional amorphous network model includes: constructing a two-dimensional bidisperse filling system, which consists of two kinds of particles of the same number with a radius ratio of 1:1.4; replacing each particle with the same sphere, and replacing the contact between adjacent particles with the same relaxation spring.

[0054] The embodiment of the present invention further provides the following method, including: determining the eigenmode of the two-dimensional amorphous network model at the frequency corresponding to the positive Poisson's ratio peak and the eigenmode at the frequency corresponding to the negative Poisson's ratio peak; calculating the projection probability or weight of the eigenmodes of these two frequencies according to the actual deformation field under the action of external load; adjusting the connection in the two-dimensional amorphous network model so that the weight of the eigenmode of the frequency corresponding to the negative Poisson's ratio peak becomes dominant to adjust the Poisson's ratio to a negative value, or so that the weight of the eigenmode of the frequency corresponding to the positive Poisson's ratio peak becomes dominant to adjust the Poisson's ratio to a positive value.

[0055] In some embodiments, the method of the present invention is used to adjust a Poisson's ratio-based functional gradient material without obvious structural or composition gradient, and a network with different Poisson's ratios is integrated in a single system.

[0056] The specific embodiments of the present invention are further described below.

[0057] like Figure 1A and Figure 1B As shown, ν is adjusted from positive to negative values ​​in three types of amorphous networks. Where: a, schematic diagram showing positive and negative ν behavior. The arrow indicates uniaxial stretching, the original state does not fit the arrow, and the deformed state fits the arrow tightly, indicating that the network is stretched to a specific position by an external force. When ν>0, it stretches longitudinally and shrinks transversely; when ν<0, it stretches in both directions. bd, three types of amorphous networks are constructed: twisted triangular lattice, Delaunay triangular random network, and stacking-derived network. e, schematic diagram showing local minima and global minima. fh, adjustment to positive or negative Poisson's ratio from the same initial state, for the three types of amorphous networks shown on the left. Note that all network structures in the entire regulation process contain only convex polygons. i, statistics of the change in length of a single connection under stretching. Each data point represents a connection: its x and y coordinates are the length changes under x and y stretching, respectively. The middle figure is the original system, and the top and bottom figures are the systems adjusted after ν increase and ν decrease, respectively. Clearly, systems with positive and negative ν exhibit different data distributions: the former are mainly in quadrants II and IV, while the latter are mainly in quadrants I and III. j, The renormalized variance of data points similar to i is plotted against ν, and a good collapse is observed for all systems. Note that the left branch of the data curve passes through σ at ν = 0. 2 Renormalize, the right branch passes through σ at ν = 1 / 3 2 Renormalize.

[0058] In the embodiments of the present invention, the above two problems are solved through numerical and experimental methods. By using intelligent optimization machine learning algorithms, the present invention comprehensively and effectively realizes the regulation of -0.9 < v < 0.9 in various two-dimensional amorphous systems, approaching the theoretical limit of -1 < v < 1 for two-dimensional isotropic systems. In addition, the present invention also discovers that the system ν is determined by a few low-frequency eigenmodes. Through the implementation of machine learning design, the present invention manufactures materials with a series of positive and negative ν values, and further realizes gradient materials based on Poisson's ratio. The present invention contributes to the practical progress of auxetic material design and the fundamental understanding of Poisson's ratio.

[0059] Regulating Poisson's ratio using intelligent optimization

[0060] In the amorphous system of the present invention, the present invention utilizes a network model composed of N b connections and N nodes, where the mass of each node is m, and each connection is an ideal massless spring with a spring constant of k. The coordination number z = 2N b / N represents the average number of connections per node. The present invention constructs three different types of amorphous networks and verifies the universality of the results. These network types include: (1) the distorted triangular lattice ( Figure 1A in b), (2) the Delaunay triangulated random network ( Figure 1B in c), and (3) the Packing-derived network ( Figure 1A in d). In each type, the present invention tests networks of multiple configurations to ensure the robustness of the conclusions. In all these systems, Poisson's ratio is regulated by selectively removing specific connections, which is a practical and commonly used technique in previous studies.

[0061] Using machine learning, the present invention proposes a general and effective method for regulating ν. Although in previous studies, ν could be adjusted to negative values in the packing-derived network, there is still a lack of a general adjustment method for arbitrary amorphous networks. In addition, the previous connection pruning process was based on the importance of a single connection to ν, which usually led to locally optimized results rather than globally optimized results (see Figure 1A in e). Therefore, the adjustment range or efficiency is quite limited. The present invention realizes a regulation result close to global optimization through a machine learning algorithm developed in reinforcement learning. The algorithm of the present invention has a feedback mechanism that can modify the optimization strategy and allows the algorithm to accept bad results to "jump out" of local minima. Eventually, the system will approach the global minimum, thus better capturing the system. After generating a large number of globally optimized networks, the data set is fed back into the actual network to obtain the best bond cutting procedure for ν adjustment.

[0062] By adopting the simulated annealing algorithm, the present invention achieves a Poisson ratio adjustment range of -0.9 < v < 0.9, which is close to the theoretical limit of -1 < v < 1 for a two-dimensional isotropic system. To ensure direction independence, the present invention separately determines the Poisson ratios ν x and ν y of the system under tensile stretching along the x and y directions. The average value of these values ν = (ν x + ν y ) / 2 represents the overall ν of the system. It is worth noting that due to the isotropic nature of the system of the present invention, ν x and ν y exhibit similar behaviors. As shown by f, g, h in Figure 1A , the machine learning algorithm of the present invention has successfully achieved -0.9 < v < 0.9 in all three types of amorphous networks, thus demonstrating its excellent adjustment range and applicability.

[0063] To visualize the changes occurring in each connection during the regulation process, the inventors analyzed the length changes of each individual connection in the network system under tensile stretching along the x and y directions. The results are statistically shown in Figure 1B as i: The middle figure depicts the initial configuration, while the upper and lower figures respectively show the situations after the adjustment of increasing ν and decreasing ν. In these figures, each point represents a connection, and its x and y coordinates respectively represent the changes in its length in the network under tensile stretching along the x and y directions. Obviously, the original data points in the middle figure are widely scattered in quadrants II and IV, which is due to the deformation results with opposite signs caused by the positive Poisson ratio. However, after increasing or decreasing ν, the data points converge towards the center, indicating that both types of adjustments will reduce the degree of connection length change during the stretching process, and only a small number of connections will exhibit large deformations. Interestingly, adjusting ν to a negative value causes the points to redistribute to quadrants I and III, which is in sharp contrast to the positive ν situation described above. Given that quadrants I and III require x and y deformations with the same sign, this behavior is in good agreement with the auxetic phenomenon, where both directions experience expansion or contraction simultaneously.

[0064] Therefore, Figure 1B in i clearly illustrates that under load, different distribution patterns of connection length changes correspond to different ν values. To quantify these distribution patterns, the variance of each distribution is calculated, denoted as and σ 2 is renormalized to 1 at ν = 0 (the regulation process of decreasing ν) or ν = 1 / 3 (the regulation process of increasing ν) respectively. This normalization allows for systematic comparison across different systems. Surprisingly, when the curves of these normalized σ 2 values are plotted against ν, the data from different amorphous systems show perfect coincidence on the curve, as shown in Figure 1BThis behavior indicates that the statistical variance of the change in the length of a single connection under load, σ 2 There is a close correlation between σ and ν. In addition, the machine learning method of the present invention effectively converts σ 2 Redistribution to quadrants I and III, thus achieving a negative ν system.

[0065] like Figure 2 As shown, the classical auxetic structure is fundamentally different from the auxetic structure discovered by machine learning of the present invention. ac, three classical auxetic structures: concave structure, rotational structure, and chiral structure. dl, three rows correspond to three different amorphous systems. In each row, the plots in the middle column show the original structure, and the plots in the left and right columns show the structures with negative and more positive Poisson's ratios discovered by machine learning. Obviously, in the left column of plots of the auxetic structure, the concave and chiral structures that are usually present in the classical auxetic structure are not present. Also note that the negative ν of the left column of plots is very similar in structure to the positive ν on the right, and z is almost the same.

[0066] Next, use Figure 2 The fundamental differences between the auxetic structures of the present invention and existing designs are illustrated. In the first row, three classical auxetic designs are shown: re-entrant, rotational, and chiral. Rows 2 to 4 show the present invention's amorphous networks derived from twisted triangular lattices (row 2), Delaunay triangulated random networks (row 3), and Packing-derived (row 4). In the middle of each row, the original network with ν of about 0.3 is shown, while the networks with extremely negative and positive Poisson's ratios generated by removing connections are shown in the left and right columns, respectively. It is worth noting that the auxetic structures in the left column are very different from the classical designs in the first row: (1) the present structure is amorphous and isotropic in all directions, while the classical designs show direction-dependent responses; (2) the present structure is composed entirely of convex polygons, while the previous re-entrant and rotational structures rely on concave polygons to achieve auxetic behavior; (3) the present structure does not have any chiral features. Therefore, the auxetic structures generated by the present machine learning method form a unique class that is different from previous designs. It is noteworthy that the auxetic behavior of the isotropic structures of the present invention is independent of direction, and the structures consisting only of convex polygons make them more resistant to compression under external loads, making them strong and more suitable for practical applications. In addition, the present invention also regulates the Poisson's ratio for different system scales (from 12×12 nodes to 32×32 nodes). The results consistently demonstrate the tunability of the Poisson's ratio, such as Figure 6 and Figure 7As shown. Clearly, regardless of the system size, for auxetic networks, the Poisson ratio always converges to -0.9, and for networks with a more positive Poisson ratio, its value always converges to 0.9. This result emphasizes the consistent tunability of the machine learning algorithm of the present invention across different system dimensions and scales.

[0067] Interestingly, structures with extremely negative and extremely positive values of ν look very similar. Both exhibit isotropic structures with similar connectivity (i.e., similar z), as Figure 2 shown in the left and right column diagrams of d-l in. However, when subjected to an external load, these visually similar structures exhibit completely opposite Poisson ratios. This high structural similarity between structures with different positive and negative ν has never been reported before, further demonstrating the uniqueness and novelty of the machine learning-generated structures. By combining these visually similar structures, functionally graded materials without any structural gradients can be created, as shown below.

[0068] The Poisson ratio originates from a few eigenmodes

[0069] The present invention uses modal analysis to elucidate the nature of ν. Generally, the present invention finds that only a few low-frequency vibration eigenmodes (usually one or two) can determine the Poisson ratio of an amorphous system. For a two-dimensional amorphous network with N nodes, the present invention constructs a 2N×2N dynamical matrix and calculates its 2N eigenvalues and eigenmodes, corresponding to the vibration frequencies and normal modes respectively. Over a wide range of -0.9 < v < 0.9 and in all systems of the present invention, it is observed that ν is generally determined by one or two vibration normal modes.

[0070] To characterize the Poisson ratio of the eigenmodes, an effective Poisson ratio ν′ is defined for each mode. As Figure 3A shown in a and b in, when the vibration vector of the eigenmode is regarded as the displacement vector on the nodes, the original structure undergoes a "deformation", thus generating the effective Poisson ratio ν′ of this mode. The mode may exhibit a positive or negative ν′ (see Figure 3A a, b in). The present invention proves that the actual ν is determined by the superposition of several important mode ν′.

[0071] To identify the important eigenmodes among them, the present invention projects the actual deformation of the network under load onto all eigenmodes and analyzes the projection probability or weight. Since the eigenmodes are orthogonal to each other and form an orthonormal basis vector set consisting of 2N modes, the present invention can project any actual deformation field onto this set of bases:

[0072]

[0073] Here |δr> is the actual deformation field under the action of an external load, which is a 2N×1-dimensional vector normalized to unit amplitude. |ωi > is the frequency ω i The i-th eigenmode of C ωi =<ω i |δr> is |δr> in the modal |ω i >The projection factor on |C ωi | 2 It has the physical meaning of projection probability: it gives a certain mode |ω i >The weight or importance of the actual deformation field |δr>. In order to obtain direction-independent results, external loads are applied to the network in the x and y directions respectively, and the average value is obtained as the final projection probability.

[0074] Therefore, the importance or weight of each mode in the actual deformation can be obtained by Figure 3B Draw in c|C ωi | 2 With ω i It is noteworthy that two distinct peaks can be observed in the low-frequency range, indicating the importance of two modes, while the contribution of other modes is negligible (note that the weight is zero at high frequencies and only the low-frequency range is significant). Interestingly, the two modes generally exhibit opposite effective Poisson's ratios ν′: one exhibits a highly positive ν′, while the other exhibits a highly negative ν′ (in Figure 3B In the c, they are represented by ω + and ω - ). By considering their weight-based superposition, the actual Poisson's ratio ν can be determined as This relationship is achieved through Figure 3B The linear relationship described by d in is verified.

[0075] As shown in Figure 3, the effective Poisson's ratio ν of the eigenmode ′ Determines the actual ν of the system. a, the left figure shows the frequency ω + The mode of the network, the box represents the original area of ​​the network. The right figure shows the configuration after adding the vibration vector: Obviously, the network expands in the y direction and contracts in the x direction, which is a typical feature of a positive Poisson's ratio. b, frequency ω - The eigenmode at shows expansion in both the x and y directions, which is a typical auxetic behavior. Shows the weight or importance of the different eigenmodes in the actual tensile deformation. Only the low frequency range is important, the high frequencies are all zero. Clearly there are two peaks in the inset: one at ω + Where, and ν ′ is positive, and the other peak is at ω - Where, and ν ′ is a negative value. ′ Based on the weighted superposition, Determine the actual ν. d, from the eigenmode shows a good linear relationship with the actual ν, and good coincidence of the data curves of various systems is observed. Obviously, one or two eigenmodes can universally determine the actual Poisson's ratio of various amorphous systems. e, from top to bottom, as ν is adjusted to negative values, ω - The weight of is increasing to dominate, while ω + The weight of ν decreases to a negligible value. When two modes have similar weights, their ν ′ The two peaks first approach each other, then overlap, and finally separate. f, from top to bottom, as ν is adjusted to a more positive value, ω + The weight of is increasing to dominate, while ω - The weight of is decreasing to be negligible, and the two peaks are separating.

[0076] Therefore, the origin of ν essentially comes from the ν′ of the eigenmodes. Certain low-frequency modes have positive or negative ν′. Applying an external load will significantly excite one or two such modes. The superposition of these ν′ values ​​determines the actual ν of the system. It is worth noting that this mechanism is universally applicable to all studied amorphous systems, including twisted triangular lattices, random networks, and Packing-derived networks. Figure 3B The good overlap of different types of network data in d demonstrates this universality.

[0077] The basic origin mechanism of ν from the intrinsic mode ν′ provides the present invention with a clearer understanding of the Poisson's ratio regulation process. Figure 3B The two important modes ω are shown in Figure e + and ω - How does the competition between determine the actual value of ν. When ν is adjusted from the original value of 0.3 to a negative value, ω - The weight of increases and eventually becomes dominant, while ω + The weight of is reduced and becomes negligible. This causes ν to effectively decrease from a positive value to a negative value. In addition, ω + and ω - Initially approaching each other, then overlapping, and finally separating, as Figure 3B The process from top to bottom is shown. It is noteworthy that when the two modes overlap, they exhibit the same height, resulting in the cancellation of positive and negative ν′. Therefore, a ν value close to zero is obtained (see Figure 3B In contrast, when ν is adjusted from the original value of 0.3 to a more correct value, Figure 3B The opposite trend is observed in f. The two peaks continue to separate, ω + The height increases, while ω- The height of ω decreases and eventually becomes negligible. + The increase in weight contributes to the overall increase in the actual value of ν.

[0078] Experimental Implementation

[0079] like Figure 4 As shown, it is an experimental realization of 3D printing. Where: a, experimental setup for ν measurement. The top surface of the sample is subjected to a compressive load driven by two stepper motors. The compressive strain magnitude and speed are precisely controlled by the motor. Poisson's ratio is measured based on the change in area of ​​the sample during compression, which is recorded by a camera in front of the sample. bj, three rows show the deformation of three amorphous systems. Each photo was taken under a compressive strain of ε = -0.1, and the outer frame represents the original boundary without loading. Obviously, the left column of figures presents negative ν, the middle column of figures presents close to zero ν, and the right column of figures presents positive ν.

[0080] like Figure 5 As shown, a ν gradient material is realized. Among them: ac, a gradient material with positive and negative ν is demonstrated by a schematic diagram, and then realized by simulation and experiment. Under compression, the upper part expands and the lower part contracts, realizing positive and negative ν in one system. df, a gradient material with negative-positive-negative (NPN) ν, is demonstrated by a schematic diagram, and then realized by simulation and experiment. gi is a gradient material with positive-negative-positive (PNP) ν, which is demonstrated by a schematic diagram, and then realized by simulation and experiment. Please note that regions with different ν have similar amorphous structures, so the present invention obtains a ν gradient material without obvious structural gradient.

[0081] The present invention has successfully achieved continuous regulation of ν from extremely positive to extremely negative values ​​by using a machine learning optimization algorithm in a general amorphous network. This discovery has great practical potential, and the present invention has now experimentally realized it. 3D printed amorphous elastomer networks with negative, positive and zero ν. For experimental convenience, Figure 2 Compared to the simulated structures in the paper, the 3D-printed structures were smaller in size. But the machine learning methods used for both the experiments and simulations were exactly the same.

[0082] In the experiments, a compressive strain of ε = -0.1 was applied to each network. Then, the corresponding deformation field and Poisson's ratio were measured using optical imaging. For practical reasons, only compressive strain was applied in the experiments to avoid connection breakage and experimental failure caused by tensile strain. At the same time, the tensile strain was numerically simulated and the Poisson's ratio results consistent with the compressive strain were obtained. In order to verify the robustness of the design, all three types of amorphous networks were tested, such as Figure 4As shown in the three rows of b-j. It is worth noting that for each network type, negative, positive, and nearly zero ν values were successfully achieved. The left figure shows contraction or negative ν in the horizontal response, the right figure shows expansion or positive ν, and the middle figure shows little change or zero ν. The outer frame represents the initial position of the system. The exact deformation process can be observed in Movie-1 to Movie-9. The range of Poisson's ratio achieved in the experiment is -0.6 < v < 0.5, which is narrower than the simulation range of -0.9 < v < 0.9. However, from a practical perspective, this range is still quite impressive. A key reason for this difference is that only pure spring interactions were assumed in the simulation, while in reality, there are inevitably bending effects in the connections of the elastomer network. Despite this inherent bending complexity, all amorphous systems exhibit a significant regulation range, thus highlighting the robustness and universality of the machine learning design of the present invention. Another reason is the static friction at the top and bottom boundaries of the sample, which limits the horizontal movement and results in a reduced adjustment range.

[0083] Through this effective regulation method, networks with different ν values can be integrated into a single system to achieve a functionally graded material of Poisson's ratio. Functionally graded materials are versatile and powerful systems that exhibit different properties at different positions and have shown a wide range of applications in the fields of biological and engineering materials. Generally, these materials rely on different structures or components or a combination of both. However, this also poses challenges in terms of compatibility and integration. In contrast, the machine learning design principle of the present invention enables the present invention to achieve very different ν values with very similar amorphous structures, as Figure 5 shown. This enables the present invention to easily combine networks with different ν values into a single system and achieve a gradient material without an obvious structural or compositional gradient. Figure 5 a of shows the combination of positive and negative ν values to create a simple ν-gradient system that is Figure 5 achieved through numerical simulation and experiment in b-c of: the upper and lower parts clearly exhibit positive and negative ν values, respectively. In addition, more complex systems can also be achieved, such as negative-positive-negative (NPN) and positive-negative-positive (PNP) ν materials, as Figure 5 shown in the second and third rows of (d-f and g-i, respectively) (refer to Videos 10 to 12). Therefore, the research of the present invention provides a basic framework for achieving ν-gradient materials without an obvious gradient in structure or composition, which has potential applications in intelligent mechanical materials.

[0084] In summary, the present invention solves two important problems related to the study of Poisson's ratio of amorphous systems. First, the present invention successfully achieves effective regulation of the global optimal ν using intelligent optimization machine learning technology, and realizes this regulation experimentally. Secondly, the present invention theoretically explains ν through the concept of eigenmodes, provides a basic explanation for the Poisson's ratio in amorphous networks, and provides a research paradigm of modal analysis for understanding the Poisson's ratio. On a practical level, the machine learning algorithm of the present invention enables the present invention to generate a series of amorphous networks with a continuous range of Poisson's ratios (ranging from -0.9 to +0.9). These networks can be converted into elastic materials with potential industrial applications through 3D printing. The Poisson's ratio of actual materials ranges from -0.6 to +0.5. Importantly, the algorithm of the present invention is able to continuously regulate the Poisson's ratio starting from the initial structure of the network, thereby enabling the design of gradient and adjustable materials that can respond intelligently to external stimuli. For example, these smart materials can be designed to be sensitive to electrical, magnetic or temperature control signals. In general, the research of the present invention enhances the basic understanding of Poisson's ratio and provides a systematic framework for designing practical systems with a wide range of adjustable Poisson's ratios.

[0085] Generation of amorphous networks

[0086] The triangular lattice distorts the network. The network is initially a perfect triangular lattice, such as Figure 1A As shown in the left figure of b. In order to introduce deformation, the displacement vector Acts on each node within the boundary, as shown in the right figure. The displacement vector of the αth node is expressed as where θ α Randomly distributed in the range [0,2π], represents the displacement direction, and r represents the degree of distortion. It is worth noting that r should be less than half of the lattice constant to avoid crossover of different connections. After deformation, the connection between nodes is replaced by a new relaxed spring.

[0087] Randomly scattered Delaunay triangulated network. Figure 1A Figure c illustrates the process of building a random network. On the left, a set of nodes (represented as balls) are randomly distributed within a rectangular region. On the right, the creation of connections (relaxed springs) through Delaunay triangulation while ensuring that the average coordination number of the amorphous network ( <z>) is set to 6.

[0088] Packing-derived network. Figure 1A As shown in the left figure of d, the network originates from a typical two-dimensional bidisperse packing system. The system consists of two kinds of particles with the same number and a radius ratio of 1:1.4. In the next step, the particles are replaced by identical spheres and the contacts between adjacent particles are replaced by identical relaxed springs, as shown in Figure 1A As shown in the right picture of middle d.

[0089] Simulated annealing algorithm

[0090] The simulated annealing (SA) algorithm mimics the gradual cooling process observed in metals. At each iteration, corresponding to the annealing temperature, the algorithm generates new potential solutions by modifying the current state. New solutions are then accepted or rejected based on the Metropolis criterion, and this process continues until convergence. The SA algorithm is known for its ability to avoid local optima and approach global optimal solutions. At each iteration, a certain number of connections are removed or added to the network system, causing the system's Poisson's ratio to change from ν(π t ) becomes ν(π t+1 ), where π t and π t+1 Represents the network structure at step t and t+1. The algorithm consists of an outer loop and an inner loop. The annealing process is applied to the outer loop, and the system starts from the initial temperature T0 and cools to the next step T at a cooling rate α. t+1 =αT t t. When the system reaches the final temperature T f The annealing process is terminated when . The inner loop involves the implementation of the Metropolis principle. At each temperature T t Next, iterate L k times in order to find the best solution by regulating the connection of the system. Here, L k In the inner loop, the acceptance probability P is introduced, which is similar to the transition matrix used in the Markov Chain Monte Carlo (MCMC) algorithm to facilitate the system to escape from the local optimum and approach the global optimum.

[0091] Next, we will use the case of decreasing ν to illustrate the function of the SA algorithm. After removing specific connections, the network structure changes from π t becomes π t+1 The acceptance probability P can be expressed by the following expression, which depends on ν(π t ) and ν(π t+1 ):

[0092]

[0093] Therefore, if ν decreases after one iteration step, the operation is accepted without question (P = 1). On the contrary, if ν increases, the algorithm considers accepting the bond-breaking strategy based on the Metropolis principle. It generates a random number ∈ in the range [0,1] and compares it with the exponential function P. If ∈ ≤ P, the strategy is accepted. Otherwise, the algorithm proceeds to the next iteration.

[0094] Visualizing the solution space as a landscape, the optimal connection pruning strategy can be viewed as a ball falling into a pit. The goal of finding the global optimal strategy is to guide the ball to find the deepest pit in the landscape. However, without a reliable algorithm, the ball may get stuck in a local minimum, such as Figure 1A To solve this problem, the SA algorithm implements a dynamic acceptance probability, which is determined by the temperature T at time t. t and the Poisson's ratio difference between two consecutive states, Δν=ν(π t+1 )-ν(π t ). By adopting this dynamic acceptance probability and the Metropolis criterion, the ball in the solution space can be perturbed, which helps the ball move out of the local minimum and continue to explore the solution space until it eventually reaches the deepest pit in the world. Once the ball reaches the deepest pit, it is difficult to move out and it has to stay. Similarly, for the increase of ν, the acceptance probability P of the worse solution becomes exp(-(ν(π t+1 )-ν(π t )) / T t )If ν(π t+1 ) <v(π t ).

[0095] Regulating the Poisson's ratio of networks of different scales

[0096] In order to verify the universality of the machine learning algorithm of the present invention in adjusting the Poisson's ratio in amorphous networks of different sizes, Figure 6 and Figure 7 Results for auxetic and Poisson’s ratio-corrected networks of varying sizes are presented in Figure 2. Depending on the size of the system, the Poisson’s ratio consistently converges to -0.9 for the auxetic and to 0.9 for the Poisson’s ratio-corrected networks. This finding highlights the tunability of the proposed machine learning algorithm across different system dimensions and scales. Furthermore, it is important to highlight that all optimized auxetic networks consisted of only convex polygons. This is in stark contrast to conventional design approaches that traditionally rely on reentrant structures in amorphous networks.

[0097] Figure 6 Auxetic random networks with sizes ranging from 10×10 to 32×32 are shown.

[0098] Figure 7 Random networks adjusted to a more positive Poisson's ratio are shown, with sizes ranging from 10×10 to 32×32.

[0099] Distribution of connection length changes in triangular lattice twisted networks and Packing-derived networks

[0100] For the twisted triangular lattice and Packing-derived network, the statistics of the change in length of each link under stretching in the x and y directions are as follows Figure 8 As shown in Figure 2. The middle columns b and e show the original network, and the left and right columns show the distribution after adjusting for decreasing and increasing ν, respectively. Clearly, for the original distribution in the middle, the points are widely distributed in quadrants II and IV. Adjusting to negative ν will redistribute the points to quadrants I and III, as shown in Figure 2. Figure 8 As shown in a and d in Figure 8 This is in contrast to the positive ν case plotted in c and f in Figure 1. Since the x and y coordinates have the same sign in quadrants I and III, this behavior is very consistent with auxetic behavior where both the x and y deformations expand or contract simultaneously.

[0101] like Figure 8 Figure 2. Statistics of the change in connection length under stretching for the ac twisted triangular lattice and the df Packing-derived network. Each data point represents a connection: its x and y coordinates are the change in length of the network under stretching along the x and y directions, respectively. The middle column is the original system, and the left and right columns are the systems after ν is reduced and ν is increased, respectively. Obviously, the systems with positive ν and negative ν show different data distributions: the former are mainly in quadrants II and IV, while the latter are mainly in quadrants I and III.

[0102] Derivation of Dynamical Matrix

[0103] For a two-dimensional central force network, define a 2N-dimensional vector |F> to represent the force on each node and define an N b The stress along each connection is represented by a dimensional vector |T>. The force vector is related to the stress vector through the equilibrium matrix Q:

[0104] |F>=Q|T>#(E1)

[0105] Similarly, the 2N-dimensional vector |δr> representing the displacement of each node is coupled to another N-dimensional vector representing the extension of the connection through the compatibility matrix C. b dimensional vector |E>:

[0106] |E>=C|δr>#(E2)

[0107] According to the generalized Hooke's law, the energy change of the network under infinitesimal deformation can be expressed as:

[0108]

[0109] where k is N b ×N b dimensional diagonal matrix, representing the spring constant, D = CkC T =Q T kQ is the dynamical matrix that contains all the information of the network.

[0110] Modal Analysis of Random and Packing-derived Networks

[0111] As Figure 3A and Figure 3B As shown in the figure, the Poisson's ratio ν of the amorphous network is given by |C ω+ | 2 and |C ω- | 2 The superposition of Figure 3A and Figure 3B In addition to the twisted triangular lattice shown in , modal analysis of random networks and Packing-derived networks is also shown here, such as Fig. 9 and Fig.10 shown. Fig. 9 The ac and Fig.10 The ac of these two types of networks shows two obvious peaks at ν = 0.26 and ν = 0.33 and their corresponding eigenvector fields. Figure 3A and Figure 3B similar, Fig. 9 A and Fig.10 The network in a represents a positive Poisson ratio mode, while Fig. 9 b and Fig.10 The network in b represents the negative ν mode. Fig. 9 In a vertical stretch or Fig.10 When the network is stretched horizontally in a, positive ν behavior can be observed, resulting in contraction in the orthogonal direction. Similarly, when the network is subjected to Fig. 9 The stretching of b or Fig.10 When the compression in b occurs, tensile expansion behavior occurs. Fig. 9 of de and Fig.10 In the de, the evolution of the different modal contributions can be tracked when both the random and packing-derived systems are tuned to negative and positive ν. As expected, the more auxetic behavior is attributed to |C ω- | 2 On the other hand, the corrected ν comes from |C ω+ | 2 When ν reaches Fig. 9 -0.70 in d and Fig.10 When the d is -0.72, only |C ω- | 2 The peak of the auxetic mode represented by , is present, while all the modal peaks associated with positive ν disappear. Similarly, when ν reaches Fig. 9 The e of 0.85 and Fig.10 When the e is 0.88, there is only one ω+ | 2 The peaks of the modal modes are not found, while all the peaks with auxetic modes disappear. The results confirm that modal analysis is generally applicable to all amorphous networks tested.

[0112] like Fig. 9 The figure shows the modal analysis of a random network. a. Frequency ω + The eigenmode at ω indicates that the network expands in the y direction and contracts in the x direction, which is a typical feature of a positive Poisson’s ratio. b, frequency ω - The eigenmode at shows expansion in both the x and y directions, which is a typical auxetic behavior. Shows the weight or importance of the different eigenmodes in the actual tensile deformation. Clearly there are two peaks: one at ω + Where, and ν ′ is positive; the other peak is at ω - Where, and ν ′ d, from top to bottom, as ν is adjusted to a negative value, ω - The weight of is increasing to dominate, while ω + The weight of is gradually reduced to be negligible. When two modes have similar weights, their ν ′ The two peaks first approach each other, then overlap, and finally separate. e, from top to bottom, as ν is adjusted to more positive values, ω + The weight of is increasing to dominate, while ω - The weight of is decreasing to be negligible, and the two peaks are separating.

[0113] like Fig.10 The figure shows the modal analysis of the Packing-derived network. a. Frequency ω + The eigenmode at ω indicates that the network expands in the x direction and contracts in the y direction, which is a typical feature of a positive Poisson’s ratio. b, frequency ω - The eigenmode at shows that both the x and y directions contract simultaneously, which is also a typical tensile behavior. Shows the weight or importance of different eigenmodes in the actual tensile deformation. Similarly, there are two peaks: one at ω + , and ν ′ is positive; the other peak is at ω - , and ν ′ d, from top to bottom, as ν is adjusted to a negative value, ω - The weight of continuously increases to a dominant level, while that of ω + continually decreases to a negligible level. When the two modes have similar weights, their ν ′ cancel each other out, and the system exhibits a ν close to zero (see the middle part of Fig.10 ). The positions of the two peaks first approach each other, then overlap, and finally separate. e, from top to bottom, as ν is adjusted to a more positive value, the weight of ω + continually increases to a dominant level, while that of ω - continually decreases to a negligible level, and the positions of the two peaks keep separating.

[0114] The relationship between ν and ν′

[0115] The actual displacement field is a linear superposition of several main modes. It is further demonstrated here that the actual ν is also a linear superposition of the ν′ of these modes. Consider a network stretched horizontally with 2N nodes. The node displacement field is represented by a 2N - dimensional vector |u x >, where the elements represent the vertical or horizontal displacements of each node. Similarly, if the network is stretched vertically, the displacement field of the network is represented as |u y >. By solving the eigenvalue problem of the dynamic matrix, a set of normal coordinates of the system, i.e., normalized eigenvectors, is obtained. These eigenvectors form a 2N - dimensional space, and each eigenvector is a basis vector of this space. The actual node displacement field can be linearly mapped to this space:

[0116]

[0117] where |i> is the i - th eigenvector of the dynamic matrix; x i and y i are the projections of the two displacement fields on the coordinate component |i>, respectively. In the research of the present invention, it is assumed that the number of nodes on the top and bottom boundaries of the network is almost equal. Define two 2N - dimensional vectors |X> and |Y> composed of scalars 0, 1 / L, - 1 / L:

[0118]

[0119]

[0120] where L is the number of nodes on the boundary, and s1 + s2 + s3 = d1 + d2 + d3 = 2N - 4L. If the displacements of the boundary nodes in |u x > or |u y > are concentrated into <X| and <Y| corresponding to a line segment of length 2L, then <X|u x >, <X|u y >, <Y|u x >, and <Y|u y > to obtain the average displacement of the boundary nodes in a particular direction. Note that |X> and |Y> can also be linearly mapped to the eigenvector space:

[0121]

[0122] where p i and q i is the corresponding projection. Note<X|Y> =0, that is Where <...> means taking the average value. It can be observed that p i and q i satisfies the Cauchy-Lorentz distribution, so <p i >= i >=0. If two quantities such as n and m are independent of each other, then <nm> ≈ <n> <m>Considering that different eigenvectors are orthogonal to each other, the projections with different subscripts are almost independent of each other. Based on this relationship, we can write:

[0123]

[0124] Likewise,

[0125]

[0126] If the network is stretched horizontally, Poisson's ratio can be obtained:

[0127]

[0128] If the network is stretched vertically, the corresponding Poisson's ratio is:

[0129]

[0130] It is worth noting that the projections p of the three trivial zero modes representing translation and rotation i and q i are all 0, so these modes contribute zero to the network Poisson's ratio. ν can be further simplified to x The expression is:

[0131]

[0132] Likewise, y can be simplified to:

[0133]

[0134] V x and ν y Taking the arithmetic mean, we can finally get the average Poisson's ratio of the network as:

[0135]

[0136] The physical meaning of the formula is clear: the Poisson's ratio of the system is determined by the projection of the displacement field on the eigenvector space and the average amplitude of the boundary nodes in each eigenvector.

[0137] Function x i ,y i ,p i , and q i depends on the frequency ω. In the previous derivation, four average approximations were used: <x i p i >≈ <x i > <p i >, <x i q i >≈ <x i > <q i >, <y i p i >≈ <y i > <p i >, and <y i q i >≈ <y i > i >. In general, there is a small correction on the right side of the approximation equation. If the two functions are uncorrelated, this correction is considered to be zero. However, if there is a correlation between the two functions, the correction on the right side cannot be ignored, which means that the approximation is no longer valid.

[0138] Equation G11 can be simplified by considering only the contributions of the local peaks of P(ω) in the ten lowest-frequency normal modes. These modes have dominant contributions (∑ i P(ω i )| peak >0.8), and the above four approximations strictly maintain the local peak. For other modes, such as Figure 3A and Figure 3B As shown, the correction caused by the approximation distorts the scatter plot of ν′ versus ν, which is expected to be a straight line.

[0139] According to the functional correlation theorem, if the rank of the Jacobian matrix exist If s = m, then the functions are independent at this point. For example, consider the approximation <x i p i >≈ <x i > <p i > and focus on x i (ω) and p i (ω) is the pre-inertial relationship between these two functions. The Jacobian matrix of these two functions is a two-dimensional vector. Therefore, at the local peak frequency ω peak There are where all elements of the row vector form a maximum linearly independent group. The rank of a vector is determined by the number of elements in its maximum linearly independent group. Therefore, the rank of the Jacobian matrix is ​​equal to its dimension, indicating that the two functions are peak is strictly independent at the local peak. Similarly, the other three approximations can be shown to be strictly correct at the local peak. Finally, equation G11 can be simplified to

[0140] Poisson's ratio is the same for compression and tension loads

[0141] The deformation of the auxetic random network under compressive and tensile loads is described in the simulations as Fig.11 ​In fact, the Poisson's ratio remains the same under both loading conditions, enabling the selection of the best loading method for experimental measurement. In the experimental measurement of the present invention, compression loading was selected as the preferred method for practical considerations, as tensile loading may destroy the bond and cause experimental failure.

[0142] like Fig.11 As shown, the simulated deformation of the fiber network to be 3D printed in the tensile performance experiment calculated by the finite element method, where: a. under compression load and b. under tension load.

[0143] Discussion on the influence of non-uniform bow deformation on ν

[0144] To investigate the effect of static friction causing non-uniform bow deformation, simulations were performed for both non-uniform and uniform deformation. Fig.12 The results are illustrated in Figure 1, showing an ideal uniform deformation on the left and a non-uniform bow deformation due to friction on the right. The simulations show that the non-uniform case due to static friction makes the Poisson’s ratio less adjustable than the uniform case.

[0145] like Fig.12 Figure 2 shows uniform deformation and inhomogeneous bow deformation, where: a and b show the shape deformation of the auxetic amorphous network under uniform (a) and inhomogeneous (b) compressive loads. The outer contours in both figures are the final shapes of the auxetic behavior under compression. In cd, the arrows indicate the simulated displacement field corresponding to each node in the network.

[0146] An embodiment of the present invention further provides a storage medium for storing a computer program, which at least performs the above method when executed.

[0147] An embodiment of the present invention further provides a control device, comprising a processor and a storage medium for storing a computer program; wherein the processor is configured to execute at least the method described above when executing the computer program.

[0148] An embodiment of the present invention further provides a processor, wherein the processor executes a computer program and at least executes the method described above.

[0149] The storage medium may be implemented by any type of volatile or non-volatile storage device, or a combination thereof. Among them, the non-volatile memory may be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), a magnetic random access memory (FRAM), a flash memory, a magnetic surface memory, an optical disc, or a compact disc read-only memory (CD-ROM); the magnetic surface memory may be a disk memory or a tape memory. The volatile memory may be a random access memory (RAM), which is used as an external cache. By way of example but not limitation, many forms of RAM are available, such as static random access memory (SRAM), synchronous static random access memory (SSRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous link dynamic random access memory (SLDRAM), direct memory bus random access memory (DRRAM). The storage media described in the embodiments of the present invention are intended to include, but are not limited to, these and any other suitable types of memory.

[0150] In the several embodiments provided by the present invention, it should be understood that the disclosed systems and methods can be implemented in other ways. The device embodiments described above are only schematic. For example, the division of the units is only a logical function division. There may be other division methods in actual implementation, such as: multiple units or components can be combined, or can be integrated into another system, or some features can be ignored or not executed. In addition, the coupling, direct coupling, or communication connection between the components shown or discussed can be through some interfaces, and the indirect coupling or communication connection of the devices or units can be electrical, mechanical or other forms.

[0151] The units described above as separate components may or may not be physically separated, and the components displayed as units may or may not be physical units, that is, they may be located in one place or distributed on multiple network units; some or all of the units may be selected according to actual needs to achieve the purpose of the present embodiment.

[0152] In addition, all functional units in the embodiments of the present invention may be integrated into one processing unit, or each unit may be separately used as a unit, or two or more units may be integrated into one unit; the above-mentioned integrated units may be implemented in the form of hardware or in the form of hardware plus software functional units.

[0153] A person skilled in the art can understand that: all or part of the steps of implementing the above method embodiment can be completed by hardware related to program instructions, and the aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it executes the steps of the above method embodiment; and the aforementioned storage medium includes: mobile storage devices, read-only memories (ROM, Read-Only Memory), random access memories (RAM, Random Access Memory), disks or optical disks, etc. Various media that can store program codes.

[0154] Alternatively, if the above-mentioned integrated unit of the present invention is implemented in the form of a software function module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the embodiment of the present invention can be essentially or partly reflected in the form of a software product that contributes to the prior art. The computer software product is stored in a storage medium and includes several instructions for a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the methods described in each embodiment of the present invention. The aforementioned storage medium includes: various media that can store program codes, such as mobile storage devices, ROM, RAM, magnetic disks or optical disks.

[0155] The methods disclosed in the several method embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new method embodiments.

[0156] The features disclosed in several product embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new product embodiments.

[0157] The features disclosed in several method or device embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new method embodiments or device embodiments.

[0158] The above contents are further detailed descriptions of the present invention in combination with specific preferred embodiments, and it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art of the present invention, several equivalent substitutions or obvious variations can be made without departing from the concept of the present invention, and the performance or use is the same, which should be regarded as belonging to the protection scope of the present invention.< / m> < / n> < / nm> ​< / z>

Claims

1. A method for realizing arbitrarily adjustable Poisson's ratio of an amorphous system based on machine learning, characterized in that: include: A two-dimensional amorphous network model is constructed, which includes nodes and connections between nodes. The connections in the two-dimensional amorphous network model are optimized and regulated by machine learning methods to generate a global optimized network. The optimized data set is fed back to the actual network to achieve continuous adjustment of the Poisson's ratio ν of the actual two-dimensional amorphous network system from -0.9 to 0.

9.

2. The method according to claim 1, characterized in that The machine learning method uses a simulated annealing (SA) algorithm to control the Poisson's ratio, and includes the following steps: Select an initial network structure and set the initial temperature, initial solution and cooling plan of the simulated annealing algorithm; Execute the outer loop, and start the simulated annealing algorithm according to the set initial temperature and initial solution, gradually reducing the temperature from the initial temperature to the final temperature, and each iteration corresponds to an annealing temperature; In each iteration of the outer loop, an inner loop is performed, the current state is set as the initial solution, and the current network structure and the corresponding Poisson's ratio are recorded; For the inner loop, in each iteration, a certain number of connections are removed or added to the network system; For the inner loop, in each iteration, a random number from a uniform distribution is generated and compared with the exponential function calculated by the Metropolis criterion; preferably, the random number is a random number in the range [0,1]; If the random number is less than the result of the exponential function, accept the new solution and update the network structure and the corresponding Poisson's ratio; otherwise, accept the new solution with a dynamic acceptance probability according to the Metropolis criterion and update the network structure and the corresponding Poisson's ratio; If the preset termination condition is reached, such as the maximum number of iterations or the preset final temperature, the algorithm terminates and outputs the final network structure.

3. The method according to claim 2, characterized in that The dynamic acceptance probability P is: Among them, π t and π t+1 represents the network structure at iteration steps t and t+1, ν(π t ) and ν(π t+1 ) represents the Poisson’s ratio of the amorphous system under the corresponding network structure, T t represents the temperature of the simulated annealing algorithm at iteration step t.

4. The method according to any one of claims 2 to 3, characterized in that: For expansion systems of different sizes, the Poisson's ratio of the network after Poisson's ratio regulation converges to 0.9; the expansion structure of the system has the following characteristics: (1) it is amorphous and isotropic in all directions; (2) it is composed entirely of convex polygons; and (3) it does not have any chiral characteristics.

5. The method according to any one of claims 1 to 4, characterized in that: The network of the amorphous system is a twisted triangular lattice network. The construction of a two-dimensional amorphous network model includes: constructing a perfect triangular lattice network; Acting on each node within the boundary, a deformation is introduced, where the displacement vector of the αth node is expressed as where θ α Randomly distributed in the range of [0,2π], represents the displacement direction, r represents the degree of distortion, and r is less than half of the lattice constant; after deformation, the connection between the nodes is replaced by a new relaxed spring.

6. The method according to any one of claims 1 to 4, characterized in that: The network of the amorphous system is a Delaunay triangulated random network. The construction of a two-dimensional amorphous network model includes: randomly distributing a group of nodes in a rectangular area, creating relaxed spring connections through Delaunay triangulation, and setting the average coordination number of the network to 6.

7. The method according to any one of claims 1 to 4, characterized in that The network of the amorphous system is a packing-derived network, and the construction of a two-dimensional amorphous network model includes: constructing a two-dimensional double-dispersed filling system, which is composed of two kinds of particles of the same number and a radius ratio of 1:1.4; replacing each particle with the same sphere, and replacing the contact between adjacent particles with the same relaxation spring.

8. The method according to any one of claims 1 to 7, characterized in that include: Determine the eigenmode of the two-dimensional amorphous network model at the frequency corresponding to the positive Poisson's ratio peak and the eigenmode at the frequency corresponding to the negative Poisson's ratio peak; calculate the projection probability or weight of the eigenmodes of these two frequencies according to the actual deformation field under the action of the external load; adjust the connection in the two-dimensional amorphous network model so that the weight of the eigenmode corresponding to the frequency of the negative Poisson's ratio peak becomes dominant to adjust the Poisson's ratio to a negative value, or so that the weight of the eigenmode corresponding to the frequency of the positive Poisson's ratio peak becomes dominant to adjust the Poisson's ratio to a positive value.

9. The method according to any one of claims 1 to 8, characterized in that Functionally graded materials based on Poisson's ratio are obtained without obvious structural or composition gradients, and networks with different Poisson's ratios are integrated into a single system.

10. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the method according to any one of claims 1 to 9 is implemented.

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