Analysis method for contact interaction between ellipsoidal particles

Abstract

The invention discloses a method for analyzing contact interaction between ellipsoidal particles. The method comprises the following steps: constructing a general ellipsoidal particle model, generating ellipsoidal particles according to the general ellipsoidal particle model, and determining the mass and the initial motion state of the ellipsoidal particles; determining ellipsoidal particles in contact with each other; obtaining initial semi-analytic geometric iteration points of the ellipsoidal particles, and determining nearest points among the ellipsoidal particles; and determining the contact force between the ellipsoidal particles based on the nearest points between the ellipsoidal particles, and obtaining the spatial position and the motion state of the ellipsoidal particles after the contact interaction according to the mass and the initial motion state of the ellipsoidal particles and the contact force between the ellipsoidal particles. According to the method, the contact interaction between the ellipsoidal particles can be analyzed more accurately, more efficiently and more stably.

Description

Technical Field

[0001] The invention relates to the field of simulation of geotechnical engineering, engineering geology and granular medium mechanics, and in particular to an analysis method for contact interaction between ellipsoidal particles. Background Art

[0002] The ellipsoid is a common geometric shape of natural particles or artificial materials. Its parametric equation based on the spherical coordinate system was proposed very early. In the simulation field of geotechnical engineering, engineering geology and granular medium mechanics, by changing the ratio of the semi-axis lengths in the three directions of x, y and z, ellipsoidal particles of various sizes can be obtained, which can be used to describe or approximate granular media such as gravel and pebbles with relatively smooth surfaces and relatively regular shapes in geotechnical engineering and engineering geology.

[0003] Numerical simulation is a modern technology to reproduce and predict the motion of ellipsoidal granular media. The contact interaction between ellipsoidal particles plays a controlling role in the motion of the entire particle system. The correctness of the calculation or processing of the contact interaction between ellipsoidal particles determines the reliability and availability of the entire particle system.

[0004] At present, the prior art provides an algebraic optimization method based on the parameter equations of two ellipsoidal particles. This method can reflect the interaction between the two ellipsoidal particles to a certain extent, but has the following shortcomings: (1) This method involves four optimization (or undetermined) parameters, and it is difficult to obtain an analytical solution; (2) When the shapes and sizes of the two ellipsoidal particles are quite different, this method has the problems of poor calculation accuracy and insufficient stability; (3) This method requires the selection of appropriate optimization initial values ​​and depends to a certain extent on the shape and size of the two ellipsoidal particles, so it has limitations in practical application.

[0005] In summary, there is an urgent need in this field for a more accurate, efficient and stable method for analyzing the interactions between ellipsoidal particles. Summary of the invention

[0006] In view of the above-mentioned deficiencies in the prior art, the present invention provides a method for analyzing contact interactions between ellipsoidal particles.

[0007] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is: A method for analyzing contact interactions between ellipsoidal particles comprises the following steps: S1. constructing a general ellipsoidal particle model, generating ellipsoidal particles according to the general ellipsoidal particle model, and determining the mass and initial motion state of the ellipsoidal particles; S2, determining the ellipsoidal particles that are in contact with each other; S3, obtaining initial semi-analytical geometric iteration points of the ellipsoidal particles based on the ellipsoidal particles in contact with each other, and determining the closest points between the ellipsoidal particles according to the initial semi-analytical geometric iteration points of the ellipsoidal particles; S4. Determine the contact force between the ellipsoidal particles based on the closest point between the ellipsoidal particles, and obtain the spatial position and motion state of the ellipsoidal particles after contact interaction according to the mass of the ellipsoidal particles, the initial motion state and the contact force between the ellipsoidal particles.

[0008] Further, in step S1, constructing a general ellipsoidal particle model includes the following steps: A1. Construct a standard ellipsoidal particle model with the center at the origin of the coordinate system in a spherical coordinate system; A2. Based on the standard ellipsoidal particle model with the center of the sphere at the origin of the coordinate system in the spherical coordinate system, the rotation, scaling and translation characteristics of the standard ellipsoidal particle are used to construct a general ellipsoidal particle model.

[0009] Furthermore, in step A1, a standard ellipsoidal particle model with the center at the origin of the coordinate system is constructed, which is expressed as: , ,

[0010] in: is any coordinate on the surface of a standard ellipsoidal particle, is the diagonal matrix of standard ellipsoidal particles, is the column vector of standard ellipsoidal particles, is the first semi-axis of a standard ellipsoidal particle, is the second semi-axis of the standard ellipsoidal particle, is the third semi-axis of the standard ellipsoidal particle, is the cosine function, is the first angle parameter of the standard ellipsoidal particle, is the second angle parameter of the standard ellipsoidal particle, is a sine function.

[0011] Further, in step A2, based on the standard ellipsoidal particle model with the center at the coordinate origin in the spherical coordinate system, the rotation characteristics, scaling characteristics and translation characteristics of the standard ellipsoidal particle are used to construct a general ellipsoidal particle model, which is expressed as: ,

[0012] in: are the coordinates of any point on the surface of a general ellipsoidal particle, is a matrix of shape No. Line Column elements, , =1,2,3, are the centroid coordinates of a general ellipsoidal particle, is the shape matrix of a general ellipsoidal particle, is the scaling matrix for general ellipsoidal particles, is the rotation matrix of a general ellipsoidal particle.

[0013] Further, in step S2, the ellipsoidal particles that are in contact with each other are determined. The specific process is: obtain the first semi-axes and centroid coordinates of the two ellipsoidal particles, calculate the centroid distance of the two ellipsoidal particles according to the centroid coordinates of the two ellipsoidal particles, and determine whether the centroid distance of the two ellipsoidal particles is greater than or equal to the sum of the first semi-axes of the two ellipsoidal particles and less than a set threshold; if so, they are determined to be ellipsoidal particles that are in contact with each other, otherwise, they are determined to be ellipsoidal particles that are not in contact with each other.

[0014] Further, in step S3, initial semi-analytical geometric iteration points of the ellipsoidal particles are obtained based on the ellipsoidal particles in contact with each other, including the following steps: B1. determining the centroid of the first ellipsoidal particle and the centroid of the second ellipsoidal particle based on the ellipsoidal particles in contact with each other; B2. Taking the centroid of the first ellipsoidal particle and the centroid of the second ellipsoidal particle as fixed points respectively, the nearest point on the ellipsoidal particle is obtained by using the nearest point solution method from the fixed point to the ellipsoidal particle, and the initial semi-analytical geometric iteration point of the ellipsoidal particle is obtained by selecting it.

[0015] Further, in step B2, the nearest point on the ellipsoidal particle is obtained by using a method for solving the nearest point from a fixed point to an ellipsoidal particle. The specific process is: select a second angle parameter to generate a first type of ellipse on the ellipsoidal particle, obtain the nearest point from the initial fixed point to the first type of ellipse, and update the nearest point from the initial fixed point to the first type of ellipse as a fixed point; select a first angle parameter to generate a second type of ellipse on the ellipsoidal particle, obtain the nearest point from the updated fixed point to the second type of ellipse, and continue to update the nearest point from the updated fixed point to the second type of ellipse as a fixed point, repeat the above process until the angle between the vector formed by the updated fixed point and the initial fixed point and the unit external normal vector is less than the set threshold, and the corresponding updated fixed point is determined as the nearest point on the ellipsoidal particle.

[0016] Furthermore, a second angle parameter is selected to generate a first type of ellipse on the ellipsoidal particle, which is expressed as:

[0017] in: is the expression symbol for the first kind ellipse, The parameters to be determined are The coordinates of any point on the ellipse, is the first semi-axis of a standard ellipsoidal particle, is the second semi-axis of the standard ellipsoidal particle, is the third semi-axis of the standard ellipsoidal particle, is the cosine function, is the second angle parameter of the standard ellipsoidal particle, is a sine function, is a matrix of shape No. Line Column elements, , =1,2,3, , are the centroid coordinates of a general ellipsoidal particle, is the transpose symbol; The first angle parameter is selected to generate a second type of ellipse on the ellipsoidal particle, expressed as:

[0018] in: is the expression symbol for the second kind ellipse, is the first angle parameter of a standard ellipsoidal particle.

[0019] Furthermore, in step S4, the contact force between the ellipsoidal particles is determined based on the nearest point between the ellipsoidal particles. The specific process is: based on the nearest point between the ellipsoidal particles, the contact point between the ellipsoidal particles is constructed using the median method, the contact direction between the ellipsoidal particles is constructed based on the nearest point and the contact point between the ellipsoidal particles, and the contact force between the ellipsoidal particles is determined based on the contact point and the contact direction between the ellipsoidal particles.

[0020] The present invention has the following beneficial effects: (1) The present invention transforms the problem of the closest point between two separated ellipsoidal surfaces in an ellipsoidal particle into the problem of the closest point between a series of points and two types of ellipses on the ellipsoidal surface. The initial semi-analytical geometric iteration points of the ellipsoidal particle are obtained to further determine the closest point between the ellipsoidal particles. This process fully utilizes the analytical and geometric properties of the ellipsoidal surface. Each iteration involves only one undetermined parameter and can be directly solved using an analytical formula, which can effectively overcome the shortcomings of the existing methods. (2) The present invention determines the mass and initial motion state of the ellipsoidal particles, then determines the ellipsoidal particles that are in contact with each other, obtains the initial semi-analytical geometric iteration points of the ellipsoidal particles based on the ellipsoidal particles that are in contact with each other, and determines the closest point between the ellipsoidal particles based on the initial semi-analytical geometric iteration points of the ellipsoidal particles, and finally determines the contact force between the ellipsoidal particles based on the closest point between the ellipsoidal particles, and obtains the spatial position and motion state of the ellipsoidal particles after the contact interaction based on the mass, initial motion state and contact force between the ellipsoidal particles. Through the above process, the contact interaction between the ellipsoidal particles can be analyzed more accurately, efficiently and stably; (3) The present invention generates ellipsoidal particles according to a general ellipsoidal particle model by constructing the general ellipsoidal particle model. There is no need to select an appropriate optimized initial value. The generated ellipsoidal particles do not depend on a specific shape and size, and therefore have greater practicality. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] Figure 1 It is a schematic diagram of the process of analyzing the contact interaction between ellipsoidal particles; Figure 2 Schematic diagram of standard ellipsoidal particles; Figure 3 Schematic diagram of two general ellipsoidal particles; Figure 4 Schematic diagram of the iterative process for finding the closest point between the centroid C1 and the ellipsoidal particle E2; Figure 5 2D schematic diagram for determining the initial semi-analytical geometry iteration points; Figure 6 Schematic diagram of the closest point between two general ellipsoidal particles and their connecting line. DETAILED DESCRIPTION

[0022] The specific implementation modes of the present invention are described below so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of the specific implementation modes. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the attached claims, these changes are obvious, and all inventions and creations utilizing the concept of the present invention are protected.

[0023] like Figure 1 As shown, a method for analyzing contact interactions between ellipsoidal particles includes steps S1-S4, which are specifically as follows: S1. Construct a general ellipsoidal particle model, generate ellipsoidal particles according to the general ellipsoidal particle model, and determine the mass and initial motion state of the ellipsoidal particles.

[0024] In an optional embodiment of the present invention, the present invention constructs a general ellipsoidal particle model, comprising the following steps: A1. Construct a standard ellipsoidal particle model with the center at the origin of the coordinate system in the spherical coordinate system, expressed as: , ,

[0025] in: is the coordinate of any point on the surface of a standard ellipsoidal particle, is the diagonal matrix of standard ellipsoidal particles, is the column vector of standard ellipsoidal particles, is the first semi-axis of a standard ellipsoidal particle, is the second semi-axis of the standard ellipsoidal particle, is the third semi-axis of the standard ellipsoidal particle, is the cosine function, is the first angle parameter of the standard ellipsoidal particle, is the second angle parameter of the standard ellipsoidal particle, is a sine function.

[0026] A2. Based on the standard ellipsoidal particle model with the center of the sphere at the origin of the coordinate system in the spherical coordinate system, the rotation, scaling and translation characteristics of the standard ellipsoidal particle are used to construct a general ellipsoidal particle model, which is expressed as: ,

[0027] in: are the coordinates of any point on the surface of a general ellipsoidal particle, is a matrix of shape No. Line Column elements, , =1,2,3, are the centroid coordinates of a general ellipsoidal particle, is the shape matrix of a general ellipsoidal particle, is the scaling matrix for general ellipsoidal particles, is the rotation matrix of a general ellipsoidal particle.

[0028] The calculation formula of the scaling matrix of general ellipsoidal particles is expressed as:

[0029] in: , , They are x , y , z The scaling factors in the axis directions. If the three are the same, it means equal scaling.

[0030] The calculation formula of the rotation matrix of a general ellipsoidal particle is expressed as:

[0031] in: , , Respectively relative to x , y , z Euler angles for axis rotation.

[0032] The present invention generates ellipsoidal particles according to a general ellipsoidal particle model. Specifically, different ellipsoidal particles have different centroid coordinates and shape matrices. The present invention can generate two ellipsoidal particles according to the general ellipsoidal particle model by determining two sets of different centroid coordinates, three semi-axes, three scaling factors and three Euler angles.

[0033] S2. Determine the ellipsoidal particles that are in contact with each other.

[0034] In an optional embodiment of the present invention, the present invention determines the ellipsoidal particles that are in contact with each other. The specific process is: obtain the first semi-axis and centroid coordinates of the two ellipsoidal particles, calculate the centroid distance of the two ellipsoidal particles according to the centroid coordinates of the two ellipsoidal particles, and determine whether the centroid distance of the two ellipsoidal particles is greater than or equal to the sum of the first semi-axes of the two ellipsoidal particles and less than a set threshold; if so, they are determined to be ellipsoidal particles in contact with each other, otherwise, they are determined to be ellipsoidal particles that are not in contact with each other.

[0035] Specifically, the first semi-axes of the two ellipsoidal particles E1 and E2 are , ; Their centroids are C1 and C2 respectively. If the distance dc1c2 between the two centroids satisfies the following conditions: + + d0>dc1c2 ≥ +

[0036] in + + d0 is a set threshold value. In the present invention, d0 is 0.001, and the two ellipsoidal particles are initially in contact, which meets the use conditions of the present invention, and the subsequent steps of the present invention can be continued; otherwise, the spatial positions of the two ellipsoidal particles do not meet the use conditions of the present invention.

[0037] S3. Acquire initial semi-analytical geometric iteration points of the ellipsoidal particles based on the ellipsoidal particles in contact with each other, and determine the closest points between the ellipsoidal particles according to the initial semi-analytical geometric iteration points of the ellipsoidal particles.

[0038] In an optional embodiment of the present invention, the present invention obtains initial semi-analytical geometric iteration points of ellipsoidal particles based on mutually contacting ellipsoidal particles, comprising the following steps: B1. Based on the ellipsoidal particles in contact with each other, determine the centroid C1 of the first ellipsoidal particle and the centroid C2 of the second ellipsoidal particle.

[0039] B2. Taking the centroid of the first ellipsoidal particle and the centroid of the second ellipsoidal particle as fixed points respectively, the nearest point on the ellipsoidal particle is obtained by using the nearest point solution method from the fixed point to the ellipsoidal particle, and the initial semi-analytical geometric iteration point of the ellipsoidal particle is obtained by selecting it.

[0040] The present invention utilizes a method for solving the nearest point from a fixed point to an ellipsoidal particle to obtain the nearest point on the ellipsoidal particle. The specific process is: selecting a second angle parameter to generate a first type of ellipse on the ellipsoidal particle, obtaining the nearest point from the initial fixed point to the first type of ellipse, and updating the nearest point from the initial fixed point to the first type of ellipse as a fixed point; selecting a first angle parameter to generate a second type of ellipse on the ellipsoidal particle, obtaining the nearest point from the updated fixed point to the second type of ellipse, and continuing to update the nearest point from the updated fixed point to the second type of ellipse as a fixed point, repeating the above process until the angle between the vector formed by the updated fixed point and the initial fixed point and the unit external normal vector is less than a set threshold, and the corresponding updated fixed point is determined as the nearest point on the ellipsoidal particle.

[0041] The present invention selects the second angle parameter e = e 1( e 1∈[- π / 2, π / 2]) to generate a first-class ellipse on the ellipsoidal particle e 1, expressed as:

[0042] in: is the expression symbol for the first kind ellipse, The parameters to be determined are The coordinates of the point on the ellipse, is the first semi-axis of a standard ellipsoidal particle, is the second semi-axis of the standard ellipsoidal particle, is the third semi-axis of the standard ellipsoidal particle, is the cosine function, is the second angle parameter of the standard ellipsoidal particle, is a sine function, is a matrix of shape No. Line Column elements, , =1,2,3, , are the centroid coordinates of a general ellipsoidal particle, is the transpose symbol.

[0043] The present invention obtains the fixed point P to the ellipse e 1's closest point I1( a 1, e 1). The subscript 1 in I1 indicates the first iteration. a 1 and e 1 is the corresponding angle parameter. In the present invention, for the next iteration, the fixed point is set equal to I1.

[0044] The present invention selects the first angle parameter a = a 1, to generate a second type of ellipse on the ellipsoidal particle a 1, expressed as:

[0045] in: is the expression symbol for the second kind ellipse, is the first angle parameter of a standard ellipsoidal particle.

[0046] The present invention obtains the updated fixed point I1 to the ellipse a 1's closest point I2( a 1, e 2). The subscript 2 in I2 indicates the second iteration. a 1 and e 2 is the corresponding angle parameter. In the present invention, the fixed point is set equal to I2 for the next iteration.

[0047] because , , so the present invention converts the expressions of the first and second ellipses into Then use the existing analytical method to solve the quartic equation. , the present invention obtains 4 complex solutions, and then uses the real solutions to calculate the parameters , and then use the expressions of the above two types of ellipses to find the corresponding candidate points Finally, the present invention can find the closest point by comparing the distances from the fixed point to all candidate points.

[0048] The present invention repeats the above process until the vector formed by the updated fixed point and the initial fixed point and the unit external normal vector n The angle between them is less than the set threshold (1.0×10 -4 ), I k is the closest point from the fixed point P obtained by the kth iteration to the ellipsoidal particle (the corresponding angle parameter is recorded as a k , e k ), and the corresponding updated fixed point is determined as the nearest point on the ellipsoidal particle. k Unit external normal vector at point n , expressed as: .

[0049] The present invention selects the nearest point on the ellipsoidal particle to obtain the initial semi-analytical geometry iteration point of the ellipsoidal particle. The specific process is as follows: the present invention calculates the nearest point A from the centroid C1 to the ellipsoidal particle E2 and the nearest point B from the centroid C2 to the ellipsoidal particle E1, and then determines whether ∠BAC1 is smaller than ∠ABC2; if so, point A is the initial semi-analytical geometry iteration point of the ellipsoidal particle; otherwise, point B is the initial semi-analytical geometry iteration point of the ellipsoidal particle.

[0050] The present invention determines the closest point between ellipsoidal particles according to the initial semi-analytical geometric iteration point of the ellipsoidal particle. The specific process is: find the closest point Q1 from the initial semi-analytical geometric iteration point of the ellipsoidal particle to the ellipsoidal particle E1; then find the closest point Q2 from point Q1 to the ellipsoidal particle E2; repeat the above process until the angle ∠Q k-1 Q k C1 and angle ∠Q k Q k-1 C2 is less than the set threshold (1.0×10 -4 degrees), at this time, click Q k-1 and Q k That is, the closest point between the ellipsoidal particles E1 and E2, point Q k-1 On the ellipsoidal particle E1, point Q k On the ellipsoidal particle E2.

[0051] S4. Determine the contact force between the ellipsoidal particles based on the closest point between the ellipsoidal particles, and obtain the spatial position and motion state of the ellipsoidal particles after contact interaction according to the mass of the ellipsoidal particles, the initial motion state and the contact force between the ellipsoidal particles.

[0052] In an optional embodiment of the present invention, the present invention determines the contact force between the ellipsoidal particles based on the nearest point between the ellipsoidal particles. The specific process is: based on the nearest point between the ellipsoidal particles, the contact point between the ellipsoidal particles is constructed using the median method, the contact direction between the ellipsoidal particles is constructed based on the nearest point and the contact point between the ellipsoidal particles, and the contact force between the ellipsoidal particles is determined based on the contact point and the contact direction between the ellipsoidal particles.

[0053] Specifically, the present invention takes the closest point Q k-1 and Q k The midpoint of is the contact point C; k-1 Point to Q k Vector is the contact normal vector ; through the contact point C and perpendicular to the normal vector n c The plane is defined as the contact tangent plane T; a point D is randomly generated on plane T, and the vector is the first contact tangent vector; define the vector t c = n c × s c is the second contact tangent vector. Three contact vectors n c , s c , t c After the contact point and contact direction are constructed, the contact force between the ellipsoidal particles is determined by using the contact spring method, the Lagrange multiplier method or the variational inequality method. F c , and then the contact force is applied to the two ellipsoidal particles respectively, and the acceleration is calculated according to Newton's second law of motion, and then the spatial position and motion state of the two ellipsoidal particles are updated.

[0054] Verification process: like Figure 2 As shown, the present invention provides a standard ellipsoidal particle with its centroid at the origin of the coordinate system in a spherical coordinate system. Its parameter equation is:

[0055] Among them: the three semi-axes are r u =1, r v =1 / 2, r w =1 / 3. Based on this, the present invention randomly gives two groups of different parameters: the first group, the centroid coordinates ( x ,y , z )=(0,0,0), three proportional scaling factors ( S x , S y , S z ) = (3.163513, 3.163513, 3.163513) and three Euler angles ( γ x , γ y , γ z )=(5.840123, 2.573058, 0.002145); The second group, centroid coordinates ( x , y , z )=(7.493145,0,0), three proportional scaling factors ( S x , S y , S z ) = (4.329334, 4.329334, 4.329334) and three Euler angles ( γ x , γ y , γ z )=(4.703020, 3.413651, 2.124548). The present invention establishes Figure 3 Two generally ellipsoidal particles E1 and E2 are shown.

[0056] At the same time, the masses and initial motion states of the two general ellipsoidal particles are determined as follows: the mass M1 of the ellipsoidal particle E1 is 1, the three linear velocities are (1.0, 0, 0), and the three angular velocities are all 0, that is, along x The mass of the ellipsoidal particle E2 is M2 = 2, the three linear velocities are (-1.0, 0, 0), and the three angular velocities are all 0, that is, along x The negative movement of the axis. That is, initially, the two ellipsoidal particles move towards each other.

[0057] After scaling, Figure 3 The maximum semi-axes of the two generally ellipsoidal particles E1 and E2 are shown as r u1 =3.163513, r u2=4.329334; their centroid coordinates are C1(0,0,0) and C2(7.493145,0,0). It is easy to get the distance between the two centroids r u1 + r u2 + d 0=7.493847> d c1c2 =7.493145> r u1 + r u2 =7.492847( d 0 is 0.001). Therefore, the two ellipsoidal particles are in contact with each other, which meets the use conditions of the present invention, and the subsequent steps of the present invention can be continued. Otherwise, the spatial positions of the two ellipsoidal particles do not meet the use conditions of the present invention.

[0058] The centroids of the two separated ellipsoidal particles E1 and E2 are C1 and C2 respectively.

[0059] The present invention obtains the closest point A from the centroid C1 to the ellipsoidal particle E2. Figure 4 As shown, the present invention provides a schematic diagram of an iterative process for finding the closest point from the centroid C1 to the ellipsoidal particle E2: 1) Take e = e 1=0( e 1∈[- π / 2, π / 2]), the first type of ellipse is generated on the ellipsoidal particle E2 e 1. Find the centroid C1 to the ellipse e 1's closest point I1( a 1, e 1) =I1(1.2, 0). The subscript 1 in I1 indicates the first iteration. a 1 and e 1 is the angle parameter.

[0060] 2) Take a = a 1=1.2, the second type of ellipse is generated on the ellipsoidal particle E2 a 1. Find the centroid C1 to the ellipse a 1's closest point I2( a 1, e 2) =I2(1.2, 0.47 π ). The subscript 2 in I2 indicates the second iteration. a 1 and e 2 is the angle parameter.

[0061] 3) Repeat the above iterative steps 1 and 2 until the vector and the unit external normal vector n The angle between θ Less than 1.0×10 -4 Here, the closest point I4 ( a 4, e 4)=I4(3.402,0.4701 π )=A, such as Figure 4 shown.

[0062] Secondly, the closest point B from the centroid C2 to the ellipsoidal particle E1 is obtained by the same steps. Finally, the present invention calculates that ∠BAC1=5.8°<∠ABC2=6.1°, so point A is the initial semi-analytical geometric iteration point of the ellipsoidal particle; otherwise, point B is the initial semi-analytical geometric iteration point of the ellipsoidal particle. Point A is on the ellipsoidal particle E2, and point B is on the ellipsoidal particle E1. Figure 5 shown.

[0063] The present invention then determines the closest point between two displaced ellipsoidal particles: 1) Find the closest point Q1 from point A to the ellipsoidal particle E1; 2) Find the closest point Q2 from point Q1 to the ellipsoidal particle E2; 3) Repeat the above iterative steps 1 and 2 until the angle ∠Q k-1 Q k C1 and angle ∠Q k Q k-1 C2 is less than the threshold (1.0×10 -4 Here, points Q3 and Q4 are the closest points between the two ellipsoidal particles E1 and E2, as shown in Figure 6 shown.

[0064] like Figure 6 As shown, the coordinates of the closest points between the two ellipsoidal particles E1 and E2 are Q3 (2.651773, 0.143960, 1.204024) and Q4 (2.651980, 0.144030, 1.203953), respectively. Figure 6 The green straight line in the middle represents the line from a certain iteration point on the ellipsoidal particle E2 to the nearest point on the ellipsoidal particle E1 during the iteration process; the blue straight line represents the line from a certain iteration point on the ellipsoidal particle E1 to the nearest point on the ellipsoidal particle E2 during the iteration process; the red straight line represents the line between the nearest points between the ellipsoidal particles E1 and E2 when the iteration ends, i.e., the line between points Q3 and Q4. These straight lines describe the iterative process of the semi-analytical geometric iterative algorithm proposed in the present invention.

[0065] The present invention then determines the closest point between two displaced ellipsoidal particles: 1) Contact distance check. The distance between the closest points Q3 and Q4 can be known from their coordinates. ( is a given threshold value, and the present invention adopts 0.0001), indicating that the two ellipsoidal particles are in contact with each other, and the subsequent steps of the present invention can continue to be performed; otherwise, the use conditions of the present invention are not met.

[0066] 2) Construct the contact point and contact direction, and calculate the contact force. The midpoint of the closest points Q3 and Q4 is the contact point C = (Q3 + Q4) / 2 = (2.651876, 0.143995, 1.203988). The vector from point Q3 to point Q4 is the contact normal vector =(0.900861,0.307183, -0.306734); passes through contact point C and is perpendicular to the normal vector n c The plane is defined as the contact tangent plane T; a point D (6.537322, -8.713025, 3.744952) is randomly generated on plane T, and the vector is defined =(0.388544, -0.885702, 0.254096) is the first contact tangent vector; define the vector t c = n c × s c = (-0.193621, -0.348085, -0.917249) is the second contact tangent vector. Three contact vectors n c , s c , t c After constructing the contact point and contact direction, the present invention uses the variational inequality method to calculate the contact force F c =(121.345781, 34.871459, -26.814911).

[0067] 3) Update the spatial position and motion state of the two ellipsoidal particles. The contact force is applied to the two ellipsoidal particles respectively, and the acceleration is calculated according to Newton's second law of motion, and then their spatial positions and motion states are updated as follows: the centroid coordinates C1 of the ellipsoidal particle E2 are (-0.134255, 0.521679, 0.113520), the three linear velocities are (-0.134111, 0.145100, 0.138957), and the three angular velocities are (0.134111, 0.145100, 0.138957); the centroid coordinates C2 of the ellipsoidal particle E2 are (7.556910, -0.325688, -0.024580), the three linear velocities are (0.711444, -0.084590, -0.073331), and the three angular velocities are (0.063009, 0.076300, -0.057701).

[0068] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0069] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0070] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0071] The present invention uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as a limitation on the present invention.

[0072] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A method for analyzing contact interactions between ellipsoidal particles, characterized in that: The following steps are involved: S1. constructing a general ellipsoidal particle model, generating ellipsoidal particles according to the general ellipsoidal particle model, and determining the mass and initial motion state of the ellipsoidal particles; S2, determining the ellipsoidal particles that are in contact with each other; S3, obtaining initial semi-analytical geometric iteration points of the ellipsoidal particles based on the ellipsoidal particles in contact with each other, and determining the closest points between the ellipsoidal particles according to the initial semi-analytical geometric iteration points of the ellipsoidal particles; S4. Determine the contact force between the ellipsoidal particles based on the closest point between the ellipsoidal particles, and obtain the spatial position and motion state of the ellipsoidal particles after contact interaction according to the mass of the ellipsoidal particles, the initial motion state and the contact force between the ellipsoidal particles.

2. The method for analyzing contact interactions between ellipsoidal particles according to claim 1, characterized in that: In step S1, a general ellipsoidal particle model is constructed, including the following steps: A1. Construct a standard ellipsoidal particle model with the center at the origin of the coordinate system in a spherical coordinate system; A2. Based on the standard ellipsoidal particle model with the center of the sphere at the origin of the coordinate system in the spherical coordinate system, the rotation, scaling and translation characteristics of the standard ellipsoidal particle are used to construct a general ellipsoidal particle model.

3. The method for analyzing contact interactions between ellipsoidal particles according to claim 2, characterized in that: In step A1, a standard ellipsoidal particle model with the center at the origin of the coordinate system is constructed, which is expressed as: ， ， in: is the coordinate of any point on the surface of a standard ellipsoidal particle, is the diagonal matrix of standard ellipsoidal particles, is the column vector of standard ellipsoidal particles, is the first semi-axis of a standard ellipsoidal particle, is the second semi-axis of the standard ellipsoidal particle, is the third semi-axis of the standard ellipsoidal particle, is the cosine function, is the first angle parameter of the standard ellipsoidal particle, is the second angle parameter of the standard ellipsoidal particle, is a sine function.

4. The method for analyzing contact interactions between ellipsoidal particles according to claim 3, characterized in that: In step A2, based on the standard ellipsoidal particle model with the center at the origin of the coordinate system in the spherical coordinate system, the rotation characteristics, scaling characteristics and translation characteristics of the standard ellipsoidal particle are used to construct a general ellipsoidal particle model, which is expressed as: ， in: are the coordinates of any point on the surface of a general ellipsoidal particle, is a matrix of shape No. Line Column elements, , =1,2,3, are the centroid coordinates of a general ellipsoidal particle, is the shape matrix of a general ellipsoidal particle, is the scaling matrix for general ellipsoidal particles, is the rotation matrix of a general ellipsoidal particle.

5. The method for analyzing contact interactions between ellipsoidal particles according to claim 1, characterized in that: In step S2, the ellipsoidal particles that are in contact with each other are determined. The specific process is: obtain the first semi-axes and centroid coordinates of the two ellipsoidal particles, calculate the centroid distance of the two ellipsoidal particles according to the centroid coordinates of the two ellipsoidal particles, and determine whether the centroid distance of the two ellipsoidal particles is greater than or equal to the sum of the first semi-axes of the two ellipsoidal particles and less than a set threshold; if so, they are determined to be ellipsoidal particles that are in contact with each other, otherwise, they are determined to be ellipsoidal particles that are not in contact with each other.

6. The method for analyzing contact interactions between ellipsoidal particles according to claim 1, characterized in that: In step S3, initial semi-analytical geometric iteration points of the ellipsoidal particles are obtained based on the ellipsoidal particles in contact with each other, including the following steps: B1. determining the centroid of the first ellipsoidal particle and the centroid of the second ellipsoidal particle based on the ellipsoidal particles in contact with each other; B2. Taking the centroid of the first ellipsoidal particle and the centroid of the second ellipsoidal particle as fixed points respectively, the nearest point on the ellipsoidal particle is obtained by using the nearest point solution method from the fixed point to the ellipsoidal particle, and the initial semi-analytical geometric iteration point of the ellipsoidal particle is obtained by selecting it.

7. The method for analyzing contact interactions between ellipsoidal particles according to claim 6, characterized in that: In step B2, the nearest point on the ellipsoidal particle is obtained by using a method for solving the nearest point from a fixed point to an ellipsoidal particle. The specific process is: select a second angle parameter to generate a first type of ellipse on the ellipsoidal particle, obtain the nearest point from the initial fixed point to the first type of ellipse, and update the nearest point from the initial fixed point to the first type of ellipse as a fixed point; select a first angle parameter to generate a second type of ellipse on the ellipsoidal particle, obtain the nearest point from the updated fixed point to the second type of ellipse, and continue to update the nearest point from the updated fixed point to the second type of ellipse as a fixed point, repeat the above process until the angle between the vector formed by the updated fixed point and the initial fixed point and the unit external normal vector is less than a set threshold, and the corresponding updated fixed point is determined as the nearest point on the ellipsoidal particle.

8. The method for analyzing contact interactions between ellipsoidal particles according to claim 7, characterized in that: The second angle parameter is selected to generate a first-class ellipse on the ellipsoidal particle, expressed as: in: is the expression symbol for the first kind ellipse, The parameters to be determined are The coordinates of any point on the ellipse, is the first semi-axis of a standard ellipsoidal particle, is the second semi-axis of the standard ellipsoidal particle, is the third semi-axis of the standard ellipsoidal particle, is the cosine function, is the second angle parameter of the standard ellipsoidal particle, is a sine function, is a matrix of shape No. Line Column elements, , =1,2,3, , are the centroid coordinates of a general ellipsoidal particle, is the transpose symbol; The first angle parameter is selected to generate the second type of ellipse on the ellipsoidal particle, which is expressed as: in: is the expression symbol for the second kind ellipse, is the first angle parameter of a standard ellipsoidal particle.

9. The method for analyzing contact interactions between ellipsoidal particles according to claim 1, characterized in that: In step S4, the contact force between the ellipsoidal particles is determined based on the nearest point between the ellipsoidal particles. The specific process is: based on the nearest point between the ellipsoidal particles, the contact point between the ellipsoidal particles is constructed using the median method, the contact direction between the ellipsoidal particles is constructed based on the nearest point and the contact point between the ellipsoidal particles, and the contact force between the ellipsoidal particles is determined based on the contact point and the contact direction between the ellipsoidal particles.

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