Complex particle three-dimensional morphology digital twinning method based on spherical harmonic function and principal component analysis

Abstract

The invention discloses a complex particle three-dimensional morphology digital twinning method based on a spherical harmonic function and principal component analysis, and the method comprises the steps: (1) employing a three-dimensional scanner to scan a real particle morphology so as to obtain a triangular mesh data file of a particle surface morphology, and employing a self-adaptive curved surface optimization algorithm to optimize a scanned particle triangular mesh; (2) performing shape scaling on the particles to obtain spheroids by using triangular surface grid data after particle scanning optimization, and performing 30-order spherical harmonic function expansion on the spheroids to realize parameterization and digital reconstruction on the particles with complex morphology; and (3) performing principal component analysis on the shape parameters and the spherical harmonic function parameters of all the scanned particles to obtain characteristic values and characteristic vectors of a parameter matrix, constructing random morphology parameters based on the characteristic values and the characteristic vectors, and constructing a brand-new random particle surface morphology through the reverse process of the step (2). According to the method, basic guarantee is provided for quantitative analysis of the three-dimensional morphology of the surface of the complex granular material and performance analysis of the granular composite material.

Description

Technical Field

[0001] This invention belongs to the field of digital modeling technology for civil engineering materials, specifically relating to a method for generating three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis. Background Technology

[0002] Particulate materials are widely found in chemical, powder, pharmaceutical, and civil engineering fields. As the smallest unit of particulate materials, the morphology of particles significantly influences their physical, chemical, and mechanical properties. In civil engineering, particulate materials are prevalent in building materials such as cement concrete, asphalt mixtures, soil and rock, and roadbeds. Their particle morphology not only affects their transport properties during construction but also influences the spatial arrangement and distribution of particles in the microstructure after compaction, thus further impacting the mechanical properties of the material. Therefore, accurately characterizing and constructing the complex morphological features of particles is fundamental to their quantitative analysis and exploration in particulate materials such as concrete, asphalt mixtures, and roadbeds.

[0003] Currently, the morphology of individual particles is often described using simple shape indicators such as aspect ratio and roundness. While these indicators are convenient, they neglect the microscopic characteristics of the particles. In the microscopic simulation analysis of particulate materials, particles are often simplified to spheres, ellipsoids, and polyhedra. Although simplifying particles can improve computational efficiency, the lack of microscopic features leads to significant errors in particle arrangement and microscopic stress transmission, thus greatly compromising the reliability of the simulation results. Therefore, quantifying particle morphology features and constructing high-precision particle morphologies is the cornerstone for effective quantitative analysis and microscopic simulation of complex particles. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a method for generating complex particle morphology. Based on real scanned particles, the method quantifies particle morphology features through shape scaling and spherical harmonic function parameterization. Subsequently, by performing principal component analysis on the parameters of the scanned particles, the method generates new particle morphologies of the same type, providing a fundamental guarantee for the accurate description of particle morphology and multi-scale analysis of particulate materials.

[0005] To achieve the above objectives, this invention provides a method for generating three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis. The method includes the following steps:

[0006] (1) The complex particle surface morphology is scanned by high-precision three-dimensional scanning to obtain the triangular mesh data file of the real particle surface morphology; the mesh data file is in STL format and contains the vertex coordinates and connection relationships of the particle triangular mesh; the scanned particle triangular mesh is further optimized by an adaptive surface optimization algorithm to improve mesh quality and reduce mesh density.

[0007] (2) Based on the vertex data of the real particle surface, obtain the three principal inertial axes of the particle; with the particle centroid as the origin and the principal inertial axes as the coordinate axes, rotate and translate the particle so that it is arranged in the direction of the principal axes; perform shape scaling on the particle to make it into a spherical particle; perform 30th order spherical harmonic function expansion on the spherical particle; thus, the parameterization and reconstruction of the particle morphology are realized through shape parameters and spherical harmonic function parameters.

[0008] (3) Based on all the scanned particle parameters, construct a parameter matrix; perform principal component analysis on the parameter matrix to obtain the eigenvalues ​​and eigenvectors of the particle parameters and construct the parameter vector of random particles based on this, including shape parameters and shoe function parameters; based on the constructed parameters, generate new three-dimensional random particles by reverse construction (constructing spherical particles from spherical harmonic function particles and then scaling their shapes).

[0009] As a further improvement of the present invention, step (1) includes: scanning the complex surface morphology of the particles using a high-resolution three-dimensional scanner to obtain the particle surface mesh data, with each particle having no less than 50,000 vertices; secondly, since there are singular meshes, the original mesh needs to be further optimized by an adaptive surface optimization algorithm to improve the mesh quality and reduce the mesh density.

[0010] As a further improvement of the present invention, step (2) includes the following sub-steps:

[0011] (2.1) Based on the triangular mesh data of the real particle surface, obtain the three principal inertial axes of the particle; with the particle's centroid as the origin and the principal inertial axes as the coordinate axes, rotate and translate the particle so that it is aligned with the principal axis directions;

[0012] .

[0013] It is the centroid of the particle, and is the weighted average of the areas of all triangular faces; where , , ;in It is the first The area of ​​each triangular facet; It is the first The coordinates of the centroids of the three triangular faces; the inertia matrix of the triangular faces is:

[0014]

[0015] in , , , , , , , , The coordinates of the triangular facets relative to the centroid are given. The inertia matrix has three eigenvectors. , , That is, the three principal axes of the particle.

[0016] (2.2) Shape scaling of the vertices on the surface of the scanned particles transforms the particles into spherical particles:

[0017]

[0018] In the formula The coordinates of the vertices of the scanned particles; These are the vertex coordinates of the spherical particles after shape scaling. These are the three principal axis eigenvalues; The diameter of the particle-equivalent ellipsoid;

[0019] (2.3) Convert the surface vertices of the spherical particles in the Cartesian coordinate system to spherical coordinate system:

[0020]

[0021] In the formula, To scan the centroid of the particles; The polar coordinates of the vertices on the surface of the spherical particle;

[0022] Establish a polar coordinate system with the center of mass of the spherical particle as the origin, connect each point on the particle surface to the origin, and determine the polar angle of that point. azimuth and polar radius ;

[0023] (2.4) Expand the vertex of the spherical particle in spherical coordinates using spherical harmonic functions of order n=30, i.e.:

[0024]

[0025] In the formula The coefficients of the spherical harmonic function, for Spend A basis of spherical harmonic functions of order 1, defined as follows: ; The imaginary unit; The Legendre function is defined as follows: ;

[0026] To represent particle morphology at different scales, the polar radius function can be rewritten as a sum of spherical harmonic functions of different orders:

[0027]

[0028] In the formula, For all The sum of spherical harmonic functions of order 1. The maximum series chosen for the expansion of the spherical harmonic function. The larger the value, the higher the particle fit. A value of 15 is typically sufficient to accurately describe particle morphology.

[0029] Because the number of points on the particle surface is much greater than the coefficient of the spherical harmonic function. Number of Therefore, the spherical harmonic coefficients in equation (1) are solved by standard least squares estimation using the known polar radius. ;

[0030] (2.5) Using the coefficients determined in step (2) The polar coordinates of the vertices of spherical particles can be reconstructed using spherical harmonic functions:

[0031]

[0032] in, , , , , and They are respectively and The number of mesh nodes in the direction; by transforming the polar coordinates of the spherical object to Cartesian coordinates and scaling it back to the original particle shape, the original particle surface morphology can be reconstructed.

[0033]

[0034] in and These represent the coordinates of the reconstructed sphere and the reconstructed original particle vertices, respectively; different orders of the maximum spherical harmonic function are selected. It is possible to construct particle morphologies with different levels of fineness.

[0035] As a further improvement of the present invention, step (3) includes the following sub-steps:

[0036] (3.1) After performing shape scaling and spherical harmonic function expansion on each of the scanned particles, its morphological parameter vector can be obtained. It is expressed as follows:

[0037] Let be the particle morphology parameter vector, with dimension . The morphology of the j-th particle can be completely reconstructed through it.

[0038] (3.2) Combine the morphology parameter vectors of all scanned particles to obtain a morphology parameter matrix containing all particles. :

[0039]

[0040] Topographic parameter matrix Dimensions ,in This represents the total number of particles scanned.

[0041] (3.3) By averaging the morphology parameter matrix, we can obtain the average vector of particle morphology parameters:

[0042]

[0043] Let be the average topographic parameter vector, with dimension . ;

[0044] (3.3) Calculate the topographic parameter matrix variance matrix :

[0045]

[0046] In the formula, It is a unit vector with dimension . ;

[0047] (3.4) Perform principal component analysis to obtain its eigenvalues. and the corresponding feature vector And sorted in descending order of eigenvalues. In the Projection vectors of eigenvectors for:

[0048]

[0049] for The vector, after standardization, can be written as Its dimensions are ;

[0050] (3.5) Based on the eigenvalues ​​and eigenvectors, the random particle coefficient vector can be constructed using the following formula. :

[0051]

[0052] In the formula, From vector Any value chosen from the options. Based on This allows for the generation of new particle surface morphologies; different maximum principal order can be selected. This means that new particles with different levels of fineness can be generated.

[0053] The beneficial effects of the present invention are: through the above scheme, parametric analysis and reconstruction of complex particle morphologies, including flat and non-star-shaped particles, can be performed; and random generation of different fine particle morphologies of the same type can be achieved. Attached Figure Description

[0054] Figure 1 This is a flowchart of the random generation of complex particle three-dimensional morphology provided in Embodiment 1 of the present invention;

[0055] Figure 2 The schematic diagram of the particle morphology surface mesh and mesh optimization obtained by the three-dimensional scanner in Embodiment 1 of the present invention.

[0056] Figure 3 This is a schematic diagram of particle shape scaling and particle morphology reconstruction after expansion of different spherical harmonic functions in Embodiment 1 of the present invention.

[0057] Figure 3 This invention relates to a method for generating three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis, which provides reconstructed images of particles under different spherical harmonic function series.

[0058] Figure 4 This invention relates to a method for generating three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis, which randomly generates particles at different principal component orders.

[0059] Figure 5 This is a schematic diagram of the morphology of a complex particle three-dimensional morphology generation method based on spherical harmonic functions and principal component analysis according to the present invention.

[0060] Figure 6 This is an optimized schematic diagram of a method for generating three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis according to the present invention. Detailed Implementation

[0061] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for illustrative and explanatory purposes only and are not intended to limit the scope of the invention.

[0062] like Figures 1 to 4 As shown, the main steps of a method for generating three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis, according to the present invention, are as follows:

[0063] (1) such as Figure 2As shown in Figure a, a high-precision 3D scanner is used to scan the complex particle surface morphology to obtain a triangular mesh data file of the real particle surface morphology; the mesh data file is in STL format and contains the vertex coordinates of the particle triangular mesh and their connection relationships (each particle has no less than 50,000 vertices), such as... Figure 2 As shown in b; subsequently, the obtained particle surface mesh is optimized using an adaptive surface optimization algorithm to correct singular triangular meshes, improve mesh quality, and reduce mesh density, such as... Figure 2 As shown in c;

[0064] (2) Based on the optimized triangular mesh data of the particle surface, obtain the three principal inertial axes of the particle; with the particle's centroid as the origin and the principal inertial axes as the coordinate axes, rotate and translate the particle so that it is aligned with the principal axis directions. Figure 3 As shown in a; subsequently, the shape is scaled to spherical particles before a 30th-order spherical harmonic function expansion is performed, allowing flat particles and non-star-shaped particles to be accurately described, avoiding the shape singularity problem caused by directly performing spherical harmonic function expansion on the particle morphology (such as...). Figure 3 (as shown in b); specifically:

[0065] (2.1) Based on the optimized triangular mesh data of the particle surface, obtain the three principal inertial axes of the particle; with the particle's centroid as the origin and the principal inertial axes as the coordinate axes, rotate and translate the particle so that it is aligned with the principal axis directions. Figure 4 a) The centroid is calculated as follows:

[0066]

[0067] In the formula Let be the centroid of the particle, and be the weighted average of the areas of all triangular faces; , , ;in It is the first The area of ​​each triangular facet; It is the first The coordinates of the centroids of the three triangular faces; the inertia matrix of the triangular faces is:

[0068]

[0069] in , , , , , ; , , The coordinates of the triangular facets relative to the centroid are given. The inertia matrix has three eigenvectors. , , That is, the three principal axes of the particle.

[0070] (2.2) As Figure 4 As shown in b, scaling the shape of the vertices on the particle surface transforms the particle into a spherical particle:

[0071]

[0072] In the formula The coordinates of the vertices of the scanned particles; These are the vertex coordinates of the spherical particles after shape scaling. These are the three principal axis eigenvalues; The diameter of the particle-equivalent ellipsoid;

[0073] (2.3) Convert the surface vertices of the spherical particles in the Cartesian coordinate system to spherical coordinate system:

[0074]

[0075] In the formula, To scan the centroid of the particles; The polar coordinates of the vertices on the surface of the spherical particle;

[0076] Establish a spherical coordinate system with the center of mass of the spherical particle as the origin, connect each point on the particle surface to the origin, and determine the polar angle of that point. azimuth and polar radius ;

[0077] (2.4) Expand the vertex of the spherical particle in spherical coordinates using spherical harmonic functions of order n=30, i.e.:

[0078]

[0079] In the formula The coefficients of the spherical harmonic function, for Spend A basis of spherical harmonic functions of order 1, defined as follows: ; The imaginary unit; The Legendre function is defined as follows: ;

[0080] To represent particle morphology at different scales, the polar radius function can be rewritten as a sum of spherical harmonic functions of different orders:

[0081]

[0082] In the formula, For all The sum of spherical harmonic functions of order 1. The maximum series chosen for the expansion of the spherical harmonic function. The larger the value, the higher the particle fit. A value of 15 is typically sufficient to accurately describe particle morphology.

[0083] Because the number of points on the particle surface is much greater than the coefficient of the spherical harmonic function. Number of Therefore, the spherical harmonic coefficients in equation (1) are solved by standard least squares estimation using the known polar radius. ;

[0084] (2.5) Using the coefficients determined in step (2) It is possible to reconstruct spherical particles (such as spherical harmonic functions) Figure 4 c), whose vertex polar coordinates are:

[0085]

[0086] in, , , , , and They are respectively and The number of mesh nodes in the direction; transforming the polar coordinates of the spherical object to Cartesian coordinates and scaling it back to the original particle shape allows for the reconstruction of the original particle surface morphology (e.g., Figure 4 d):

[0087]

[0088] in and These represent the coordinates of the reconstructed sphere and the reconstructed original particle vertices, respectively; different orders of the maximum spherical harmonic function are selected. It is possible to construct particle morphologies with different levels of fineness, such as Figure 5 As shown, the higher the order of the spherical harmonic function, the more detailed the particle morphology is reconstructed.

[0089] (3) Based on all scanned particle parameters, construct a parameter matrix; perform principal component analysis on the parameter matrix to obtain the eigenvalues ​​and eigenvectors of the particle parameters, and based on this, construct the parameter vector of random particles, including shape parameters and shoe function parameters; based on this constructed parameter, generate new three-dimensional random particles through reverse construction (constructing spherical particles from spherical harmonic function particles and then scaling their shapes). The specific steps are as follows:

[0090] (3.1) After performing shape scaling and spherical harmonic function expansion on each of the scanned particles, its morphological parameter vector can be obtained. It is expressed as follows:

[0091] Let be the particle morphology parameter vector, with dimension . The morphology of the j-th particle can be completely reconstructed through it.

[0092] (3.2) Combine the morphology parameter vectors of all scanned particles to obtain a morphology parameter matrix containing all particles. :

[0093]

[0094] Topographic parameter matrix Dimensions ,in This represents the total number of particles scanned.

[0095] (3.3) By averaging the morphology parameter matrix, we can obtain the average vector of particle morphology parameters:

[0096]

[0097] Let be the average topographic parameter vector, with dimension . ;

[0098] (3.3) Calculate the topographic parameter matrix variance matrix :

[0099]

[0100] In the formula, It is a unit vector with dimension . ;

[0101] (3.4) Perform principal component analysis to obtain its eigenvalues. and the corresponding feature vector And sorted in descending order of eigenvalues. In the Projection vectors of eigenvectors for:

[0102]

[0103] for The vector, after standardization, can be written as Its dimensions are ;

[0104] (3.5) Based on the eigenvalues ​​and eigenvectors, the random particle coefficient vector can be constructed using the following formula. :

[0105]

[0106] In the formula, From vector Any value chosen from the options. Based on This means that new particle surface morphology can be generated; Figure 6 As can be seen from the particles of different orders, the higher the order of the principal components, the more refined the new particles can be generated.

[0107] This invention provides a method for generating the three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis, which has the following advantages:

[0108] The endpoints and any values ​​of the ranges disclosed herein are not limited to the precise ranges or values, and these ranges or values ​​should be understood to include values ​​close to these ranges or values. For numerical ranges, the endpoint values ​​of the various ranges, the endpoint values ​​of the various ranges and individual point values, and individual point values ​​can be combined with each other to obtain one or more new numerical ranges, which should be considered as specifically disclosed herein.

[0109] The preferred embodiments of the present invention have been described in detail above; however, the present invention is not limited thereto. Within the scope of the inventive concept, various simple modifications can be made to the technical solutions of the present invention, including combinations of various technical features in any other suitable manner. These simple modifications and combinations should also be considered as the content disclosed in the present invention and are all within the protection scope of the present invention.

Claims

1. A digital twin method for the three-dimensional morphology of complex particles based on spherical harmonic functions and principal component analysis, characterized in that, The method includes the following steps: (1) The surface morphology of the particles is scanned by a high-precision three-dimensional scanner to obtain a triangular mesh data file of the real particle surface morphology; the mesh data file is in STL format and contains the vertex coordinates and connection relationships of the particle triangular mesh; an adaptive surface optimization algorithm is used to optimize the scanned particle triangular mesh to improve mesh quality and reduce mesh density. (2) Based on the optimized particle morphology triangular mesh, calculate the three principal inertial axes of the particles; with the particle centroid as the origin and the principal inertial axes as the coordinate axes, rotate and translate the particles to align them in the direction of the principal axes; perform shape scaling on the particles to make them into spherical particles; perform 30th order spherical harmonic function expansion on the spherical particles; realize the parameterization and digital reconstruction of the particle morphology through shape parameters and spherical harmonic function parameters; (3) Based on the shape parameters and spherical harmonic function parameters of the scanned particles, construct the parameter matrix of all scanned particles; perform principal component analysis on the parameter matrix to obtain the eigenvalues ​​and eigenvectors of the particle parameters and based on this, construct the parameter vector of random particles, which includes shape parameters and spherical harmonic function parameters; based on the constructed parameters, construct spherical particles through the reverse process of step (2), i.e., spherical harmonic function particles, and then perform shape scaling to randomly generate new three-dimensional particles with similar morphological features to the scanned particles.

2. The method for digital twinning of complex particle three-dimensional morphology based on spherical harmonic functions and principal component analysis according to claim 1, characterized in that, In step (1), a high-resolution 3D scanner is used to scan the 3D irregular particles to obtain the triangular mesh data of the particle surface. The data format is STL file, which contains the vertex coordinates of the particle triangular mesh and their connection relationship. Each particle has no less than 50,000 points. An adaptive meshing algorithm is used to correct singular meshes, reducing mesh density while improving mesh quality.

3. The method for digital twinning of complex particle three-dimensional morphology based on spherical harmonic functions and principal component analysis according to claim 2, characterized in that, In step (2), the particles are arranged along the principal axis of inertia; then, the shape is scaled to spherical particles before spherical harmonic function expansion is performed, so that flat particles and non-star-shaped particles can also be accurately reconstructed, avoiding the shape singularity problem caused by directly expanding the particle morphology with spherical harmonic function; specifically: (2.1) Based on the optimized surface triangular mesh data of real particles, obtain the three principal inertial axes of the particles; with the particle's centroid as the origin and the principal inertial axes as the coordinate axes, rotate and translate the particles so that they are aligned in the direction of the principal axes; In the formula It is the centroid of the particle, and is the weighted average of the areas of all triangular faces; where , , ; It is the first The area of ​​each triangular facet; It is the first The coordinates of the centroids of the three triangular faces; the inertia matrix of the triangular faces is: In the formula , , , , , ; , , The coordinates of the triangular face relative to the centroid; the inertia matrix has three eigenvectors. , , That is, the three principal axes of the particle Cartesian coordinate system; (2.2) Shape scaling based on the vertices of the rotated scanned particle surface transforms the particles into spherical particles: In the formula The coordinates of the vertices of the triangular mesh of the scanned particle after rotation along the main axis; These are the vertex coordinates of the spherical particles after shape scaling. These are the eigenvalues ​​corresponding to the three principal axes; Let be the diameter of the particle's equivalent ellipsoid; the above equation can be further rewritten as: ,in, This refers to the particle length index. is a flatness index; both are shape parameters describing the particle profile. (2.3) Convert the surface vertices of the spherical particles in the Cartesian coordinate system to spherical coordinate system: In the formula, The center of mass of the spherical particles; The polar coordinates of the vertices of the mesh on the surface of the spherical particles; (2.4) Polar radius of vertex in spherical coordinates for spherical particles Perform a spherical harmonic expansion of order n=30, i.e.: In the formula The coefficients of the spherical harmonic function, for Spend A basis of spherical harmonic functions of order 1, defined as follows: ; The imaginary unit; The Legendre function is defined as follows: ; The maximum order chosen for the expansion of the spherical harmonic function; the larger the order, the more refined the reconstructed particle morphology. (2.5) Using the coefficients determined in step (2) Reconstruct the polar coordinates of the vertices of the spherical particles using spherical harmonic functions: ,in, , , , , and They are respectively and The number of mesh nodes in the direction; by transforming the polar coordinates of the sphere-like vertices to Cartesian coordinates and scaling them back to the original particle shape, it is possible to reconstruct the surface morphology of the original particles under different orders of spherical harmonic functions: ,in and These represent the vertex coordinates of the reconstructed sphere and the reconstructed particle, respectively; different orders of the maximum spherical harmonic function are selected. Reconstruct the surface morphology of the scanned particles at different levels of fineness.

4. The method for digital twinning of complex particle three-dimensional morphology based on spherical harmonic functions and principal component analysis according to claim 3, characterized in that, In step (3), new morphology parameters are constructed by performing principal component analysis on the morphology parameters of the scanned particles, thereby achieving the construction of particles with the same morphological characteristics; in addition, as the order of the principal components increases, the refinement of the generated particle morphology also increases, specifically: (3.1) Based on the step (2) for the first By scaling the shape of the scanned particle and expanding it using spherical harmonic functions, the morphology parameters of the scanned particle are obtained and written as a vector. The format is as follows: in, Dimensions The three-dimensional morphology of the scanned particle can be completely reconstructed using this parameter vector. (3.2) Combine the morphology parameter vectors of all scanned particles to obtain a particle morphology matrix containing all particle morphology parameters. : morphological parameter matrix Dimensions ,in This represents the total number of particles scanned. (3.3) Calculate the morphology matrix of the morphological particles variance matrix : In the formula, It is a unit vector with dimension . ; Let be the average vector of particle morphology parameters, with dimension . ; (3.4) Perform principal component analysis to obtain its eigenvalues. and the corresponding feature vector And sorted in descending order of eigenvalues; In the Projection vectors of eigenvectors for: , for The vector, after standardization, can be written as Its dimensions are ; (3.5) Based on the eigenvalues ​​and eigenvectors, the random particle coefficient vector can be constructed using the following formula. : In the formula, From vector Any value chosen from the options; based on This allows for the generation of new particle surface morphologies; different maximum principal component orders can be selected. This means that new particles with different levels of fineness can be generated.

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