A simulation method for solving continuum structural mechanics problems using multi-scale discrete solid elements

Abstract

The present invention discloses a simulation method for solving continuum structural mechanics problems using multi-scale discrete entity elements. The multi-scale discrete entity element model includes a large ball region, a small ball region and a transition region. The spherical elements are connected by a spring system, and the spring system includes an axial spring and tangential springs in two directions. Based on the theory of continuum mechanics and the principle of energy equivalence, the analytical relationship between the microscopic model parameters (spring stiffness) and the macroscopic elastic constants (elastic modulus and Poisson's ratio) is derived. The present invention can effectively reduce the number of spherical elements and springs, improve calculation efficiency, and enable the discrete entity element method to provide more possibilities, which is suitable for complex mechanical simulations such as large deformation, damage fracture and collapse of structures or components.

Description

Technical Field

[0001] The invention relates to a multi-scale discrete entity unit method mechanical simulation method, which is used to solve strong nonlinear mechanical problems such as large deformation, damage fracture and collapse of structures or components, and belongs to the technical field of discrete element analysis. Background Art

[0002] The discrete element method is a discontinuous numerical calculation method proposed by American scholar Cundall in 1971. [1-2] , was originally used to analyze the movement of rock slopes. ([1] Cundall PA, Strack OD L. A discrete numerical model for granular assemblies [J]. geotechnique, 1979, 29 (1): 47-65. [2] Cundall PA, Hart R D. Numerical modelling of discontinua [J]. Engineering computations, 1992, 9 (2): 101-113.) The basic principle of the discrete element method is to divide the research object into discrete units. During the process of structural deformation and movement, the units can be separated, that is, a unit can contact with its neighboring units or be separated. [3-4] . ([3] Xing Jibo, Wang Yongjia. Improvement of discrete element method and its application in granular media research [J]. Journal of Geotechnical Engineering, 1990, 12(5): 51-57. [4] Wang Yongjia, Xing Jibo. Discrete element method and its application in rock and soil mechanics. Northeast Polytechnic University Press [M]. 1991.) The traditional discrete element method mainly studies granular bodies and is not applicable to continuum structures. In recent years, some scholars have begun to try to use the discrete element method for numerical calculations of steel structure collapse, stability, and impact damage problems. Ye Jihong Research Group [5-7] A discrete element model of the bar structure is proposed, the stiffness coefficient of the spring in the model is derived, and a discrete element method plastic hinge model suitable for the bar structure, a fiber model considering the plastic development of the section, a fracture criterion and a unit collision model are established. ([5] Qi Nian, Ye Jihong. Geometric nonlinear large deformation analysis of bar structures based on discrete element method [J]. Journal of Southeast University (Natural Science Edition), 2013(5):917-922. [6] Qi Nian. Research on DEM / FEM coupling calculation method and its application in the simulation of lattice shell collapse failure [D]. Nanjing: Southeast University, 2016. [7] Qin Yanan. Numerical simulation of static and dynamic response of simple structures based on discrete element method [D]. Nanjing: Southeast University, 2015.).

[0003] The inventor's research group is Feng Ruoqiang[8-10] A discrete solid element method was proposed, and the setting rules of contact springs between spherical elements were improved. Springs were used to connect the edges and diagonals of the spherical cube arrangement model. The springs between spherical elements included normal springs and tangent springs in two directions, which could accurately reflect the mechanical properties of the continuum structure while considering the Poisson's ratio effect of the material. ([8]Zhu B, Feng R, Wang Mechanics,2019:1-17.

[10] Zhu B,Feng R.Investigation of a boundary simulation of continuity using the discrete solid element method[J].Advances in Mechanical Engineering,2019,11(1):1687814018822397.)

[0004] In summary, the existing discrete solid element method can perform mechanical analysis on simple continuum structures, but there are still problems such as large number of spherical elements and low computational efficiency in solving complex structures with complex forces. Therefore, there is an urgent need for multi-scale discrete element calculation methods to efficiently simulate the mechanical problems of complex structures with complex forces. Summary of the invention

[0005] Purpose of the invention: The traditional discrete element method cannot simulate continuous media well. The existing discrete solid element method can perform mechanical analysis on simple continuum structures, but there are still problems such as a large number of spherical elements and low calculation efficiency in solving complex structures with complex forces. The present invention provides a multi-scale discrete solid element method. Based on the energy equivalence principle, a multi-scale discrete solid element model is constructed, and the analytical relationship between the microscopic model parameters (spring stiffness) and the macroscopic elastic constants (elastic modulus and Poisson's ratio) is derived, thereby solving the problems of a large number of spherical elements and low calculation efficiency in the calculation of complex structures with complex forces using the traditional discrete solid element method.

[0006] Technical solution: In order to achieve the above-mentioned invention object, the multi-scale discrete entity unit method mechanical simulation method proposed in the present invention comprises the following steps:

[0007] Step 1: According to the simulation requirements, the structure or component is discretized into a multi-scale discrete element model, the spherical elements of the model are regularly arranged, and the deformation of the material is completely stored and represented by the contact springs between the spherical elements;

[0008] The multi-scale discrete solid element model includes a large ball region, a small ball region and a transition region. The ball elements are connected by a spring system, which includes an axial spring and tangential springs in two directions.

[0009] Step 2: At t=0, t≥0, the external force, internal force, displacement, velocity and acceleration of each particle unit are assigned initial values.

[0010] Step 3, for all particle units: calculate the external force and damping force at time t, apply Newton's second law to solve the motion control equation, and obtain the position and velocity of the particle unit at time t+Δt.

[0011] Step 4: Calculate the displacement increment ΔU and the trial contact force increment Thus, the contact force F of t+Δt is obtained, and the elastic-plastic state of the current particle unit is determined according to the yield criterion.

[0012] Step 5: According to the elastic-plastic contact constitutive equation, the contact force increment ΔF of the particle unit within the Δt time step is calculated, and the contact force between the particle units at the current moment is updated.

[0013] Step 6: According to the equilibrium equation, calculate the internal force, damping force and external force of each particle unit to prepare for the calculation at time t+2Δt. Repeat this cycle until the model reaches a stable state.

[0014] Specifically, it includes the following steps:

[0015] (1) Preliminary estimation of the complex stress areas of the simulation object

[0016] According to the concept of mechanics and experience, the complex force area of ​​the simulation object is estimated, the radii of the large ball and small ball elements are determined, the complex force area is divided into the small ball area, the simpler force area is divided into the large ball area, and the transition area is between the small ball area and the large ball area.

[0017] (2) Discretize the simulation object into a multi-scale discrete element model

[0018] According to the large ball area, small ball area and transition area determined in step 1, a multi-scale discrete element model of the simulation object is established. The spherical elements in the large ball area, small ball area and transition area are arranged according to the rules of their respective areas. The deformation of the material is completely stored and represented by the contact spring of the unit. The model parameters and external load information are input to determine the boundary conditions.

[0019] The spring stiffness in the large ball area can be expressed as:

[0020] Normal spring rate:

[0021] Tangential spring rate:

[0022] in, is the normal spring stiffness of the prism spring in the large sphere area, is the normal spring stiffness of the angular spring in the large spherical area, They are the tangential spring stiffness in two directions of the prism spring in the large ball area, are the tangential spring stiffness in two directions of the diagonal spring in the large ball area, R is the radius of the large ball, E is the elastic modulus, and μ is the Poisson's ratio of the material.

[0023] The spring stiffness of the ball area can be expressed as:

[0024] Normal spring rate:

[0025] Tangential spring rate:

[0026] in, is the normal spring stiffness of the spherical area edge spring, is the normal spring stiffness of the angular spring in the ball area, are the tangential spring stiffness of the spherical area edge spring in two directions, are the tangential spring stiffness in two directions of the angular spring in the ball area, r is the radius of the ball, E is the elastic modulus, and μ is the Poisson's ratio of the material.

[0027] The spring stiffness in the transition region can be expressed as:

[0028] Normal spring rate:

[0029] Tangential spring rate:

[0030] in, They are the axial stiffness of the edge springs in the part connecting the transition area with the large ball area and the small ball area; They are the axial stiffness of the angular spring at the connecting part between the transition area and the large ball area and the small ball area; are the axial stiffness of the edge spring and the face angle spring in the transition region, respectively; They are the tangential stiffness in two directions of the prism springs connecting the transition area with the large ball area and the small ball area; They are the tangential stiffness in two directions of the angular spring of the connecting part between the transition area and the large ball area and the small ball area; They are the tangential stiffness in two directions of the edge spring and the diagonal spring in the transition area respectively; R is the radius of the ball, r is the radius of the ball, E is the elastic modulus, and μ is the Poisson's ratio of the material.

[0031] (3) At t = 0, the velocity and acceleration of the spherical element are initialized, and the internal force of the spring system is initialized;

[0032] (4) For all spherical elements, apply Newton's second law based on the external force and damping force at time t to obtain the position and velocity of the spherical element at time t+Δt;

[0033] Take any spherical element i, assume that n external forces act on spherical element i, and there are m spherical elements adjacent to it. The external force acting on spherical element i is The sum of the forces exerted by adjacent spheres on it is The damping force of spherical element i is F i c According to Newton's second law, the motion control equation is shown in formula (1).

[0034]

[0035] Among them, m i is the mass of spherical element i; is the acceleration of spherical element i; is the jth external force on spherical element i; is the jth contact internal force on spherical element i; F i c is the damping force of spherical element i.

[0036] On the basis of formula (1), the central finite difference algorithm is used to determine the acceleration of the center of the sphere, and then the displacement of the center of the sphere. The displacement of sphere i at time t+△t is calculated as shown in formula (2).

[0037]

[0038] Among them, u i (t+Δt) is the displacement of spherical element i at time t+Δt; u i (t) is the displacement of spherical element i at time t; For spherical element i The acceleration of time.

[0039] (5) The relationship between the increment of the spring normal and tangential contact force and the increment of the normal and tangential displacement within the calculation time step Δt is shown in formula (3). The coordinate transformation matrix is ​​used to convert the increment of the spring normal and tangential contact force into the increment of the contact force in the global coordinate, as shown in formula (4). The contact force between the spherical elements is calculated in the form of increments, and the relationship between the spring contact force in the global coordinate at time t+Δt and time t is shown in formula (5).

[0040]

[0041]

[0042]

[0043] (6) In the solid element method calculation model, a spherical element is connected to several spherical elements through a spring. The sum of the spring contact force and the external force is the unbalanced force of the spherical element at time t+Δt. This unbalanced force will be used to solve the motion calculation of the spherical element in the next calculation step t+2Δt, and then obtain the acceleration, velocity and displacement of the spherical element.

[0044] Beneficial effects:

[0045] (1) Based on the theory of continuum mechanics and the principle of energy equivalence, the present invention obtains the analytical relationship between the microscopic model parameters (spring stiffness) and the macroscopic elastic constants (elastic modulus and Poisson's ratio). This effectively reduces the number of spherical elements and springs, improves the computational efficiency, and provides more possibilities for the discrete solid element method, which is suitable for complex mechanical simulations such as large deformation, damage, fracture, and collapse of structures or components.

[0046] (2) In the multi-scale discrete solid element method, there is no need to assemble the stiffness matrix and perform iterative solution, and the calculation process is simple and clear. The more complex the structure, the more units there are, and the corresponding amount of calculation is also greater, but it is just a cyclic repetition of calculations, which will not bring substantial difficulties to the calculation.

[0047] (3) Compared with traditional methods, the discrete solid element method will show greater advantages in the simulation of large structures and complex mechanical behaviors. The constructed three-dimensional discrete solid element model can reflect the Poisson's ratio effect and realize the connection of spherical elements of different scales. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 is a flow chart of the present invention;

[0049] Figure 2 It is a schematic diagram of the physical model of multi-scale discrete entity element method;

[0050] 1-large ball area surface spring, 2-large ball area ridge spring, 3-transition area ridge spring, 4-small ball area surface spring, 5-small ball area ridge spring, 6-transition area ridge spring, 7-transition area surface spring;

[0051] A-large ball area, B-transition area, C-small ball area;

[0052] a-end fixed constraint, b-end large ball fixed constraint;

[0053] Figure 3 It is a schematic diagram of the partitioning of the physical model of the multi-scale discrete entity element method;

[0054] Figure 4 It is a discrete entity metamodel;

[0055] Figure 5 It is a multi-scale discrete entity metamodel;

[0056] Figure 6 It is the deformation diagram of discrete entity metamodel;

[0057] Figure 7 It is the deformation diagram of multi-scale discrete entity metamodel;

[0058] Figure 8 It is a deformation comparison diagram of discrete solid element (DSEM), multi-scale discrete solid element (MDSEM) and finite element model (FEM). DETAILED DESCRIPTION

[0059] The technical solution of the present invention will be specifically described below in conjunction with embodiments:

[0060] Reference Figure 1 As shown, the simulation method of the present invention using multi-scale discrete entity elements to solve the problem of continuum structural mechanics is as follows:

[0061] (1) Preliminary estimate of the complex force area of ​​the simulation object. According to mechanical concepts and experience, estimate the complex force area of ​​the simulation object, determine the radius of the large ball and small ball elements, divide the complex force area into the small ball area, and divide the relatively simple force area into the large ball area. The area between the small ball area and the large ball area is the transition area.

[0062] (2) Discretize the simulation object into a multi-scale discrete element model, establish a multi-scale discrete element model of the simulation object, arrange the spherical elements in the large sphere area, small sphere area and transition area according to the rules of their respective areas, and store and represent the deformation of the material completely by the contact spring of the unit. Input the model parameters and external load information, determine the boundary conditions, and refer to Figure 2 , 3 shown.

[0063] (3) At t = 0, the velocity and acceleration of the spherical element are initialized, and the internal force of the spring system is initialized;

[0064] (4) For all spherical elements, apply Newton's second law based on the external force and damping force at time t to obtain the position and velocity of the spherical element at time t+Δt;

[0065] (5) Calculate the normal and tangential displacement increments within the time step Δt, use the coordinate transformation matrix to convert the normal and tangential contact force increments of the spring into contact force increments in the global coordinates, and calculate the contact force between the spherical elements.

[0066] (6) In the solid element method calculation model, a spherical element is connected to several spherical elements through a spring. The sum of the spring contact force and the external force is the unbalanced force of the spherical element at time t+Δt. This unbalanced force will be used to solve the motion calculation of the spherical element in the next calculation step t+2Δt, and then obtain the acceleration, velocity and displacement of the spherical element.

[0067] Example 1

[0068] The following example verifies the superiority of multi-scale discrete solid elements in simulating large deformation of components.

[0069] The cantilever beam is 160 mm long, with a cross-sectional size of 30 mm × 30 mm and an elastic modulus of 2.06 × 10 5 MPa, Poisson's ratio is 0.2, density is 7850kg / m 3 , a shear force of 1000MPa is applied to the end of the beam.

[0070] The radius of the spherical element in the discrete solid element model is 2.5 mm, the number of spherical elements is 1617, and the number of connecting springs is 12092×3. Figure 4 As shown. The multi-scale discrete entity element model has a large ball area of ​​80 mm long, a large ball diameter of 10 mm, a small ball area of ​​70 mm long, a small ball diameter of 5 mm, a transition area of ​​10 mm long, 879 ball elements, and 6332×3 connecting springs. Figure 5 This embodiment examines the displacement change of the cantilever beam end in the Y direction under the action of the bending moment M.

[0071] Figure 6 and Figure 7 The deformation of the discrete solid element model and the multi-scale discrete solid element model under shear force. The comparison of the calculation results of discrete solid element, multi-scale discrete solid element and finite element is shown in Table 1.

[0072] Figure 8This is a deformation comparison diagram of discrete solid element, multi-scale discrete solid element and finite element model. Compared with the discrete solid element model, the number of spherical elements and connecting springs in the multi-scale discrete solid element model is reduced by about 50%, and the calculation efficiency is improved by 50%, but the calculation accuracy is reduced by about 2%, indicating that the calculation efficiency of the multi-scale discrete solid element is significantly improved while meeting the accuracy.

[0073] Table 1 Calculation results of Y-direction displacement of each point on the cantilever beam shear force (y=0, z=0) line

[0074]

Claims

1. A simulation method for solving continuum structural mechanics problems using multi-scale discrete entity elements, characterized in that: Here are the steps: Step 1: According to the simulation requirements, the structure or component is discretized into a multi-scale discrete entity element model, the spherical elements of the model are regularly arranged, and the deformation of the material is completely stored and represented by the contact springs between the spherical elements; The multi-scale discrete entity element model includes a large ball area, a small ball area and a transition area, and the ball elements are connected by a spring system, and the spring system includes an axial spring and tangential springs in two directions; The radius of the big ball is R, the radius of the small ball is r, and the ratio of the big ball to the small ball is 2:1; The arrangement of the spheres in the large sphere area is a cube arrangement, the side length of the cube is 2R, the spheres are located at the corners of the cube, and the distance between the sphere centers is 2R and The spheres are connected by a spring system, and the spheres with a center spacing of 2R are connected by edge springs. The spherical elements are connected by face-angle springs; The arrangement of the spheres in the small ball area is a cube arrangement, the side length of the cube is 2r, the spheres are located at the corners of the cube, and the distance between the sphere centers is 2r and The spheres are connected by a spring system, and the spheres with a center spacing of 2r are connected by edge springs. The spherical elements are connected by face-angle springs; The transition area includes a large ball, a small ball and a spring system connecting the large ball and the small ball. The arrangement of the spherical elements on one side of the transition area is the same as that of the large ball area. The distance between the center of the large ball and the adjacent large ball is 2R, and the surface is connected to the large ball area. One side of the transition area has the same arrangement as the small ball area, the distance between the centers of the small balls and the adjacent small balls is 2r, and the surface is connected to the small ball area; The distance between the center of the big ball and the small ball is 2R and The spheres are connected by a spring system, and the spheres with a center distance of 2R between the large and small spheres are connected by an edge spring. The spherical elements are connected by face-angle springs; Step 2: at t=0, t≥0, assign initial values ​​to the external force, internal force, displacement, velocity and acceleration of each particle unit; Step 3: For all particle units: calculate the external force and damping force at time t, apply Newton's second law to solve the motion control equation, and obtain the position and velocity of the particle unit at time t+Δt; Step 4: Calculate the displacement increment ΔU and the trial contact force increment Thus, the contact force F of t+Δt is obtained, and the elastic-plastic state of the current particle element is determined according to the yield criterion; Step 5: According to the elastic-plastic contact constitutive equation, the contact force increment ΔF of the particle unit in the Δt time step is calculated, and the contact force between the particle units at the current moment is updated; Step 6: According to the equilibrium equation, calculate the internal force, damping force and external force of each particle unit to prepare for the calculation at time t+2Δt. Repeat this cycle until the model reaches a stable state.

2. The simulation method for solving continuum structural mechanics problems using multi-scale discrete entity elements according to claim 1, characterized in that: The analytical relationship between the microscopic model parameters and the macroscopic elastic constants of the large ball region model is: Normal spring rate: Tangential spring rate: in, is the normal spring stiffness of the prism spring in the large sphere area, is the normal spring stiffness of the large spherical area surface angle spring, They are the tangential spring stiffness in two directions of the prism spring in the large ball area, are the tangential spring stiffness in two directions of the diagonal spring in the large ball area, R is the radius of the large ball, E is the elastic modulus, and μ is the Poisson's ratio of the material; The microscopic model parameter is the spring stiffness, and the macroscopic elastic constants are the elastic modulus and Poisson's ratio.

3. The simulation method for solving continuum structural mechanics problems using multi-scale discrete entity elements according to claim 1, characterized in that: The analytical relationship between the microscopic model parameters of the ball region model and the macroscopic elastic constants is: Normal spring rate: Tangential spring rate: in, is the normal spring stiffness of the spherical area edge spring, is the normal spring stiffness of the angular spring in the ball area, are the tangential spring stiffness of the spherical area edge spring in two directions, are the tangential spring stiffness in two directions of the angular spring in the ball area, r is the radius of the ball, E is the elastic modulus, and μ is the Poisson's ratio of the material; The microscopic model parameter is the spring stiffness, and the macroscopic elastic constants are the elastic modulus and Poisson's ratio.

4. The simulation method for solving continuum structural mechanics problems using multi-scale discrete entity elements according to claim 1, characterized in that: The analytical relationship between the mesoscopic model parameters and the macroscopic elastic constants of the transition region model is: Normal spring rate: Tangential spring rate: in, They are the axial stiffness of the edge springs in the part connecting the transition area with the large ball area and the small ball area; They are the axial stiffness of the angular spring at the connecting part between the transition area and the large ball area and the small ball area; are the axial stiffness of the edge spring and the face angle spring in the transition region, respectively; They are the tangential stiffness in two directions of the prism springs connecting the transition area with the large ball area and the small ball area; They are the tangential stiffness in two directions of the angular spring of the connecting part between the transition area and the large ball area and the small ball area; They are the tangential stiffness in two directions of the edge spring and the diagonal spring in the transition area respectively; R is the radius of the ball, r is the radius of the ball, E is the elastic modulus, and μ is the Poisson's ratio of the material.

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