Method for characterizing spatial variability of mechanical properties of a curved composite material structure

Abstract

The application discloses a method for characterizing spatial variability of mechanical properties of a curved composite structure, and belongs to the field of mechanical property characterization of composite materials. Firstly, a flow structure is divided into three-dimensional discrete units, and coordinates of center points of the units are obtained; the coordinates of the center points are converted into two-dimensional plane coordinates through dimension reduction mapping means. Then, mechanical properties of part of the discrete units are obtained, and in combination with the two-dimensional plane coordinates corresponding to the center points of the units, a suitable kernel function is selected by using a Gaussian random field to obtain the mechanical properties at two-dimensional center points of other discrete units. Finally, the mechanical property characterization results of the two-dimensional plane are mapped onto the curved structure to realize the characterization of the spatial variability of the mechanical properties of the curved composite structure. In the application, the distance measurement in the covariance kernel function is replaced by the geodesic distance on the curved surface, so that the problem of not satisfying the manifold characteristics caused by directly establishing the spatial mapping based on the Euclidean distance is avoided.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of mechanical property characterization of composite materials, and particularly relates to a method for characterizing spatial variability of mechanical properties of composite structures by combining a planar mapping method of curved surface structure manifold with a Gaussian random field. BACKGROUND

[0002] Curved surface-containing composite structures are widely used in many fields such as aerospace, however, the mechanical properties of composite materials are affected by the randomness of microstructure and the non-uniformity of fiber weaving, especially in curved surface structures, the mechanical properties of materials at different positions of the structure have obvious randomness and difference, and the complex geometric characteristics of the curved surface bring challenges to accurately characterize the spatial variability of the above-mentioned mechanical properties of materials.

[0003] Currently, in the modeling and analysis of curved surface-containing composite structures, the same mechanical properties of materials are usually used at different positions of the structure, and there is little research on the spatial variability of the mechanical properties of materials. Even if the difference in mechanical properties between different positions of the structure is considered, the different properties of different positions are usually directly assigned based on the test results of destructive sampling, and there is a lack of effective characterization of the correlation between the mechanical properties of different positions. At the same time, existing research has shown that the spatial variability of material properties has a significant impact on the modal and vibration mode characteristics, stress localization and structural damage evolution behavior of the structure. The spatial variability of the modulus of a specific material system can be up to 20% or more, causing a difference in structural strength of up to 15% or more. The lack of spatial variability characterization methods for curved surface-containing composite structures seriously restricts the accurate analysis of the response of curved surface-containing composite structures and the reliable evaluation of the bearing capacity of the structure.

[0004] Therefore, it is of great value to establish a method for characterizing the spatial variability of the mechanical properties of composite materials for curved surface structures, taking into full account the influence of the difference and correlation between the mechanical properties of different positions of the structure on the mechanical behavior of the structure, and to accurately design the composite structure and reliably evaluate the state of the structure during service. SUMMARY

[0005] The purpose of the present application is to solve the above-mentioned problems existing in the background art, and to provide a method for characterizing the spatial variability of the mechanical properties of composite structures by combining a planar mapping method of curved surface structure manifold with a Gaussian random field.

[0006] To achieve the above-mentioned purpose, the technical solutions adopted by the present application are as follows:

[0007] A method for characterizing the spatial variability of the mechanical properties of curved surface-containing composite structures, the method being

[0008] Step one: two-dimensional plane mapping of curved structure manifold; this step avoids the problem of large error caused by directly using Euclidean distance to characterize complex curved surfaces in traditional methods, and better maintains the positional relationship on the structure.

[0009] Step two: mechanical property characterization based on plane Gaussian random field; this step improves the traditional method of destructive sampling to assign different properties to different positions, and can better characterize the mechanical property correlation between different positions.

[0010] Step three: mechanical property spatial variability characterization of curved surface structure based on mapping correspondence.

[0011] Further, the step one is specifically:

[0012] Step one: discrete the three-dimensional curved surface manifold structure into n units , each unit center point is represented by a three-dimensional coordinate point (1≤i≤n, the upper indices 1, 2 and 3 represent the horizontal coordinate, vertical coordinate and vertical coordinate respectively), that is, three-dimensional coordinate data is obtained from the manifold space;

[0013] Step two: adopt neighborhood search algorithm (such as nearest neighbor or neighborhood) to construct the local adjacency graph of each point, calculate the local tangent space of each point to describe the local geometric features of the manifold; connect all local adjacency graphs to build a global manifold graph structure, and the distance between points is represented by the adjacency distance l;

[0014] Step three: based on graph embedding algorithm (such as Laplacian Eigenmaps, Diffusion Maps or customized topology preserving algorithm), two-dimensional expansion of three-dimensional manifold, under the premise of maintaining the adjacency distance l between all points and points unchanged to the largest extent, it is mapped to two-dimensional plane to obtain two-dimensional coordinate points (1≤i≤n), which is one-to-one corresponding to three-dimensional points (1≤i≤n).

[0015] Further, the step two is specifically:

[0016] Step two: obtain the material mechanical property data of m (m<n) structure typical positions , the typical position corresponds to two-dimensional coordinate point , the selection of typical position should be representative and cover the whole structure as much as possible;

[0017] Step two two: combine the data points with the mechanical property data of the typical positions, select the appropriate kernel function form (such as linear kernel function, exponential kernel function, square exponential kernel function, periodic kernel function) according to whether the data has periodicity, monotonicity and other characteristics, and establish a Gaussian random field model

[0018]

[0019] Wherein, GP is a Gaussian random field, f(x) represents the mechanical property of x at two-dimensional coordinates, is the mean function, generally 0, is the kernel function, used to represent the correlation between the points and two points, according to the actual situation, different forms are selected, in the formula, E is the expectation.

[0020] Further, the step three is specifically:

[0021] Step three one: based on the selected kernel function, the covariance matrix between different typical positions is calculated by using the coordinate vector of the typical positions:

[0022]

[0023] Step three two: according to the solving process of the Gaussian random field model, the mechanical property of the atypical position is solved by using the calculated covariance matrix, and the joint Gaussian distribution is established:

[0024]

[0025] In the formula, N is the Gaussian distribution, is the two-dimensional coordinate vector of the atypical position, is the mechanical property corresponding to , , , through conditioning, the result of is obtained:

[0026]

[0027]

[0028]

[0029] Wherein, the mechanical property of the atypical position can be considered as is the confidence degree (variance) of the corresponding atypical position , is the noise variance of the training point data (generally, the sensor noise variance is 0.1, which can also be optimized by maximum likelihood estimation), I is the unit matrix;

[0030] Through the generated Gaussian random field, the distribution of the mechanical properties in space can be described.

[0031] Step three: through the correspondence between the two-dimensional points and the three-dimensional discrete units, the generated Gaussian random field data is mapped to specific discrete units, so that the mechanical property distribution of the whole structure is obtained.

[0032] The beneficial effects of the present application relative to the prior art are: 1. The present application replaces the distance metric in the covariance kernel function with the geodesic distance on the curved surface, avoiding the problem of not meeting the manifold characteristics caused by directly establishing the space mapping based on the Euclidean distance. 2. The present application uses the statistical method of Gaussian random field to represent the overall performance based on partial position performance, which can effectively represent the mechanical property correlation between different positions. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 The flowchart of the present application.

[0034] Figure 2 The specific flowchart of the example. DETAILED DESCRIPTION

[0035] The technical solutions of the present application will be further described below in conjunction with the drawings and examples, but are not limited thereto, any modification or equivalent replacement to the technical solutions of the present application without departing from the spirit and scope of the present application shall be covered in the protection scope of the present application.

[0036] The present application provides a curved surface composite material structure mechanical property spatial variability representation method combining the planar mapping method of curved surface structure manifold with Gaussian random field. Firstly, the manifold structure is divided into three-dimensional discrete units and the coordinates of the center points of each unit are obtained; the center point coordinates are converted into two-dimensional plane coordinates through dimension reduction mapping means. Then the mechanical properties of part of the discrete units are obtained, combined with the two-dimensional plane coordinates corresponding to the unit center points, the appropriate kernel function is selected by using Gaussian random field, and the mechanical properties at the two-dimensional center points of other discrete units are obtained. Finally, the performance representation results of the two-dimensional plane are mapped to the curved surface structure, realizing the representation of the mechanical property spatial variability of the curved surface composite material structure. The present application provides a complete method from the two-dimensional planar mapping of the curved surface structure manifold to the mechanical property representation based on the planar Gaussian random process model to the mechanical property spatial variability representation of the curved surface structure based on the mapping correspondence, which provides support for the mechanical property spatial variability representation of the curved surface composite material structure.

[0037] Example 1:

[0038] As Figure 1 shown, the present application includes the following stages:

[0039] Phase one, two-dimensional plane mapping of curved structure manifold;

[0040] Phase two, mechanical property characterization based on plane Gaussian random field;

[0041] Phase three, mechanical property spatial variability characterization of curved structure based on mapping correspondence;

[0042] Phase one includes the following steps:

[0043] Step 1: Discretize the three-dimensional curved manifold structure into n units , and represent each unit center point with a three-dimensional coordinate vector (1≤i≤n and i∈N), that is, obtain three-dimensional coordinate data from the manifold space; this example takes a curved shell structure as an example to illustrate the overall process framework as shown in Figure 2 . The curved shell structure is divided into 6×23 three-dimensional discrete units , and each unit center point represents each discrete unit, as shown in Figure 2 .

[0044] Step 2: Use neighborhood search algorithm (such as nearest neighbor or neighborhood) to construct the local adjacency graph of each point and calculate the local tangent space of each point to depict the local geometric features of the manifold. Connect all local adjacency graphs to build a global manifold graph structure, and the distance between points is represented by the adjacency distance l; in this example, the discrete points are , and the adjacency graph of each point is obtained using k-nearest neighbors, and then the adjacency graphs of each point are connected to build a global manifold graph structure.

[0045] Step 3: Based on graph embedding algorithm (such as Laplacian Eigenmaps, Diffusion Maps or customized topology preserving algorithm), perform two-dimensional expansion on the three-dimensional manifold, and map it to a two-dimensional plane under the premise of maintaining the adjacency distance l between all points unchanged to obtain two-dimensional points (1≤i≤n), which are one-to-one corresponding to three-dimensional points (1≤i≤n); in this example, Maps algorithm is used, the adjacency distance matrix of the discrete points is input, and the Euclidean distance matrix of the points after dimensionality reduction is output, so that the curved surface is reduced to a two-dimensional plane.

[0046] Phase two includes the following steps:

[0047] Step 4: Obtain the material mechanical property data of m (m<n) typical positions , and the typical positions correspond to two-dimensional coordinate points The selection of typical locations should be representative and cover the entire structure as much as possible; in this example, typical locations of the structure are selected. Its three-dimensional coordinates are ,in , corresponding to two-dimensional coordinate vector ,in Elastic modulus data corresponding to typical locations. .

[0048] Step 5: Combining the reduced-dimensional data points with the mechanical performance data at typical locations, select an appropriate kernel function (such as linear kernel function, exponential kernel function, quadratic exponential kernel function, periodic kernel function) based on the characteristics of the data, such as periodicity and monotonicity, to establish a Gaussian random field model.

[0049]

[0050] Where GP is a Gaussian random field, and f(x) represents the mechanical properties at point x in two-dimensional coordinates. It is a mean function, and is generally set to 0. The kernel function is used to represent the sum of x points and x. ’ The correlation between two points can be expressed in different forms depending on the actual situation. In the formula, E represents the desired outcome; in this example, the Young's modulus of elasticity of the material is selected as the characterization target, and a squared exponential kernel function is chosen:

[0051]

[0052] in, It is the signal variance. , Let be the coordinate vector of two points. It is a relevant length scale. It is the noise variance. It is the Kronecker delta function. , , Joint optimization can be achieved through maximum likelihood estimation.

[0053] Phase 3 includes the following steps:

[0054] Step 6: Based on the selected covariance function, calculate the covariance matrix between different typical locations; in this example, the covariance matrix of different typical sites is:

[0055]

[0056] Step 7: Using the calculated covariance matrix, solve for the mechanical properties at atypical locations according to the solution process of the Gaussian random field model, and establish a joint Gaussian distribution:

[0057]

[0058] In this example, is a two-dimensional coordinate vector of the atypical position, is the corresponding Young's modulus at the atypical position, , By conditioning, we can get the result:

[0059]

[0060]

[0061]

[0062] where, the elastic modulus at the atypical position can be considered as the confidence degree (variance) of the corresponding atypical position .

[0063] Through the generated Gaussian random field, the distribution of the elastic modulus in the two-dimensional plane can be characterized.

[0064] Step 8: Through the correspondence between the two-dimensional points and the three-dimensional discrete elements, the generated Gaussian random field data is mapped to the specific discrete elements, thereby obtaining the mechanical property distribution of the entire structure; in this example, the two-dimensional reduced dimension coordinate points , the three-dimensional original coordinate points , and the three-dimensional discrete elements have the same number, which can be directly mapped.​

Claims

1. A method for characterizing the spatial variability of mechanical properties of composite materials containing curved surfaces, characterized in that: The method is as follows Step 1: Two-dimensional planar mapping of the curved surface structure manifold; Step 1 specifically includes: Step 11: Discretize the three-dimensional curved surface manifold structure into n elements. Each unit center point is represented by a three-dimensional coordinate point. This means that 1≤i≤n, and the superscripts 1, 2, and 3 represent the x-coordinate, y-coordinate, and y-coordinate, respectively, which means obtaining three-dimensional coordinate data from the manifold space; Steps 1 and 2: Construct a local adjacency graph for each point using a neighborhood search algorithm, calculate the local tangent space for each point to characterize the local geometric features of the manifold; connect all local adjacency graphs to construct the global manifold graph structure, and the distance between points is represented by the adjacency distance l; Step 13: Based on the graph embedding algorithm, the 3D manifold is unfolded into 2D, and its 2D coordinates are obtained by mapping it to a 2D plane while keeping the adjacency distance l between all points unchanged to the greatest extent. , 1≤i≤n, and its relationship with three-dimensional points One-to-one correspondence; Step 2: Characterization of mechanical properties based on planar Gaussian random fields; Step 2 specifically includes: Step 21: Obtain m typical structural locations Material mechanical property data Typical locations correspond to two-dimensional coordinate points The selection of typical locations should be representative, covering as much of the entire structure as possible, where m <n; Step 22: Combining the reduced-dimensional data points with the mechanical performance data at typical locations, select an appropriate kernel function based on whether the data has periodicity and monotonicity, and establish a Gaussian random field model. Where GP represents a Gaussian random field. This represents the mechanical properties at point x in a two-dimensional coordinate system. It is a mean function. For kernel functions, used to represent Dot and The correlation between two points can be expressed in different forms depending on the specific circumstances. In the formula, E represents the expectation; the appropriate kernel function is one of the following: linear kernel function, exponential kernel function, quadratic exponential kernel function, or periodic kernel function. Step 3: Characterization of the spatial variability of the mechanical properties of curved structures based on mapping correspondence; Step 3 specifically includes: Step 31: Based on the selected kernel function, calculate the covariance matrix between different typical locations using typical location coordinate vectors: Step 32: Using the calculated covariance matrix, and following the solution process of the Gaussian random field model, solve for the mechanical properties at atypical locations and establish a joint Gaussian distribution: In the formula, N is a Gaussian distribution. It is a two-dimensional coordinate vector at an atypical location. yes The corresponding mechanical properties, , By conditionalization, we obtain Result: in, The mechanical properties at atypical locations can be considered. For corresponding atypical locations The degree of confidence, Let I be the noise variance of the training data, and let I be the identity matrix. The generated Gaussian random field can be used to describe the spatial distribution of mechanical properties; Step 33: By mapping the generated Gaussian random field data to specific discrete units through the correspondence between two-dimensional points and three-dimensional discrete units, the mechanical property distribution of the entire structure is obtained.

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