A Multi-Resolution PD-FEM Structural Fracture Analysis Method Based on Companion Virtual Particle Coupling

Abstract

The present invention discloses a multi-resolution PD-FEM structural fracture analysis method based on coupled companion virtual particles, comprising: S1, establishing a fracture simulation analysis model and initializing it; S2, discretizing the fracture simulation analysis model by regions using FEM grids and PD particles to obtain the material parameters and initial field variable information of the grid elements and particles; S3, dividing a transition region in the FEM region to obtain the companion virtual particles of each grid element in the transition region and their field variables and related parameter information; S4, performing a neighboring particle search to determine the adjacent particles within the near fields of the companion virtual particles and PD particles, and establishing a coupled "bond" connection between the two types of particles; S5, transmitting interface boundary information and calculating the coupled force through the coupled "bond"; S6, transmitting the coupled force to the PD particles and unit nodes and then updating the field variable information based on the PD and FEM motion equations; S7, determining whether to return to step S5 or stop the calculation.

Description

Technical Field

[0001] The present invention belongs to the field of engineering structure damage and fracture analysis, and particularly relates to a multi-resolution PD-FEM structural fracture analysis method based on coupled companion virtual particles. Background Art

[0002] Structural fracture analysis plays a crucial role in fields such as civil engineering, hydraulic engineering, aerospace, etc. When traditional continuum mechanics deals with discontinuous problems, due to the non-existence of displacement derivatives, stress singularities occur at the crack tip. Peridynamics (PD) uses non-local integral equations, successfully avoiding this problem and being able to naturally simulate crack initiation and propagation, thus being widely applied. However, as a non-local method, PD has inherent defects such as large computational amount and low computational efficiency.

[0003] To balance the accuracy of crack simulation and computational efficiency, combining PD with the Finite Element Method (FEM) has become an effective approach. Currently, there are mainly two types of PD-FEM (Peridynamics-Finite Element) coupling schemes: the shared node coupling scheme and the interface element coupling scheme. The shared node coupling scheme transfers interaction information through nodes or particles in the coupling region, which is applicable to different mesh types, but requires the mesh size in the coupling region to match the size of PD particles, restricting its flexibility. The interface element coupling scheme uses dynamic virtual PD particles within the element to transfer interaction information, without the requirement of size matching, but a large number of dynamic virtual particles need to be arranged within the interface element, increasing the computational complexity and having limited applicable element types. Aiming at the deficiencies of the existing technology, the present invention proposes a multi-resolution PD-FEM structural fracture analysis method based on coupled companion virtual particles. Summary of the Invention

[0004] Aiming at the deficiencies of the existing technology, the present invention proposes a multi-resolution PD-FEM structural fracture analysis method based on coupled companion virtual particles. This method discretizes the solid continuous region using FEM and the vulnerable region using PD, taking into account both the solution of discontinuous problems and computational efficiency. At the same time, by proposing a coupled companion virtual particle scheme, problems such as strict mesh size requirements and complex virtual particle arrangement in the existing technology are overcome, and the flexibility of the method is improved on the premise of ensuring accuracy and efficiency.

[0005] To achieve the above object, the present invention provides the following solution:

[0006] A multi-resolution PD-FEM structural fracture analysis method based on coupled companion virtual particles, comprising the following steps:

[0007] S1. Establish and initialize a fracture simulation analysis model in the field of engineering structure damage and fracture analysis;

[0008] S2. Discretize the initialized fracture simulation analysis model by regions using FEM meshes and PD particles to obtain the material parameters and initial field variable information of the mesh elements and particles;

[0009] S3. Divide a transition region in the FEM region to obtain the companion virtual particles of each mesh element in the transition region and the corresponding field variable and related parameter information;

[0010] S4. Conduct a neighboring particle search to determine the adjacent particles within the near field of the companion virtual particles and PD particles, and establish a coupled "bond" connection between the two types of particles;

[0011] S5. Transmit the interface boundary information and calculate the coupled forces F PD-FEM and F FEM-PD ;

[0012] S6. Transmit the coupled forces F PD-FEM and F FEM-PD to the PD particles and element nodes, then update the field variable information based on the PD and FEM motion equations, and calculate the PD damage field based on the field variables;

[0013] S7. Determine whether the simulation time t reaches the total simulation duration t max . If not, return to S5 to perform the calculation for the next time step. If so, stop the calculation.

[0014] Preferably, before establishing the fracture simulation analysis model in S1 in the field of engineering structure damage and fracture analysis, it includes: determining the overall size of the computational domain, the size of the PD region, the size of the FEM region, and the total simulation duration.

[0015] Preferably, in S1, the method for establishing and initializing the fracture simulation analysis model in the field of engineering structure damage and fracture analysis includes: setting boundary conditions, where the boundary conditions include the force boundary conditions, displacement boundary conditions, and contact boundary conditions of the FEM region and the PD region.

[0016] Preferably, in S2, the method for discretizing the initialized fracture simulation analysis model by regions using FEM meshes and PD particles to obtain the material parameters and initial field variable information of the mesh elements and particles includes:

[0017] Determine the sizes of the FEM meshes and PD particles, and discretize the continuous FEM region, the damaged PD region, and the PD boundary in the fracture simulation analysis model into elements, solid PD particles, and virtual PD particles respectively;

[0018] Mass, density, elastic modulus, Poisson's ratio, displacement, and velocity field variable values are assigned to the unit, the physical PD particles, and the virtual PD particles.

[0019] Preferably, in step S3, the method for dividing a transition region in the FEM region and obtaining the companion virtual particles of each grid cell in the transition region and the corresponding field variable and related parameter information includes:

[0020] The FEM region within a preset range from the PD-FEM interface is divided into a transition region, where the calculation method of the preset range is:

[0021] L PD-FEM = 2.0 × m PD l e

[0022] where m PD is the nonlocal factor and l e is the element characteristic length;

[0023] A unique companion virtual particle located at the centroid is matched to each element in the transition region to obtain the relationship between the element and the companion virtual particle, where the relationship between the element and the companion virtual particle is:

[0024]

[0025] where the subscripts a and b represent the element number and the companion virtual particle number, respectively;

[0026] Based on the relationship between the element and the companion virtual particle, the element field variable information is assigned to the companion virtual particle, and the calculation method is:

[0027]

[0028] where N I is the shape function of node I, X b and Pos I ini represent the position tensors of the companion virtual particle and the node in the initial configuration, respectively, and u b , v b , V b and ρ b represent the displacement, velocity, volume, and density of the companion virtual particle, respectively.

[0029] Preferably, in step S4, the method for performing a neighboring particle search to determine the adjacent particles within the near fields of the companion virtual particles and the PD particles and establishing a coupled "bond" connection between the two types of particles includes:

[0030] Determine the near fields of the companion virtual particles and the PD particles, and the calculation method is:

[0031]

[0032] Among them, the subscript i represents the PD particle number, and b represents the companion virtual particle number. and are the near-field radii of the companion virtual particle b and the PD particle i respectively, and dx PD is the PD particle size, and ndim represents the spatial dimension;

[0033] Couple the companion virtual particle b and the PD particle i with a neighboring particle relationship by a "coupling bond", which is expressed as:

[0034] ξ′ = X b -X i .

[0035] Among them, ξ′ represents the "coupling bond" composed of the PD particle i and the virtual particle b.

[0036] Preferably, in the S5, the interface boundary information and the calculation of the coupling force F PD-FEM and F FEM-PD are transmitted through the coupling "bond" and include:

[0037] Based on the velocity and displacement information of the companion virtual particle and the PD particle, determine the relative deformation amount of the coupling "bond", and the calculation method is:

[0038] Y = ξ′ + η

[0039] Among them, Y is the "bond" deformation tensor, and η is the relative displacement tensor;

[0040] Calculate the coupling force according to the relative deformation amount of the coupling "bond" and the local external force compensation technology, and the calculation method is:

[0041]

[0042] Among them, and represent the near-field ranges of the PD particle i and the virtual particle b respectively, t i (Y) and t b (Y) represent the force scalar states of the PD particle i and the virtual particle b respectively, M(Y) is the unit direction vector state of the "bond" after deformation, and V i is the volume of the PD particle i.

[0043] Preferably, in the S6, after transmitting the coupling forces F PD-FEM and F FEM-PD to the PD particles and the element nodes, update the field variable information based on the PD and FEM motion equations. The method for calculating the PD damage field based on the field variables includes: transmitting the coupling forces F PD-FEM and FFEM-PD Perform the transformation and solve the motion equation according to the boundary conditions. The calculation method is as follows:

[0044]

[0045] Among them, f(X i , Y, t) represents the force density transformed by the coupling force F FEM-PD , represents the component force of the coupling force F assigned to node I PD-FEM , ρ i represents the density of PD particle i, represents the acceleration of PD particle i, J represents the numbers of other nodes in the same element as node I, represents the acceleration of node J, L u (X i , t) represents the resultant force density of PD particle i exerted by other PD particles in the neighborhood, b(Xi, t) is the external force density acting on the PD particle, M IJ is the mass matrix of the FEM system, which is independent of time; is the external force of node I, is the internal force of node I.

[0046] Preferably, in S7, it is judged whether the simulation time t reaches the total simulation duration t max . If not, return to S5 to perform the calculation of the next time step. If so, the method for stopping the calculation includes:

[0047] The explicit time integration scheme with the central difference format is adopted for solving the PD and FEM motion equations. The method for selecting a reasonable time step is as follows:

[0048]

[0049] Among them, Δt PD-FEM , Δt PD and Δt FEM respectively represent the time step increments of PD-FEM coupling calculation, PD calculation, and FEM calculation. The constant C CFL needs to satisfy 0 ≤ C CFL ≤ 1, δ is the near-field radius, E is the elastic modulus, v is the Poisson's ratio, ρ is the density, and c′ represents the dilatational wave speed in the solid material.

[0050] Compared with the prior art, the beneficial effects of the present invention are:

[0051] Based on the PD-FEM coupling method, the present invention discretizes the solid structure by regions. In terms of the interface treatment of PD-FEM, a coupled scheme of companion virtual particles is proposed. This scheme presets a coupled transition region in the FEM region and generates a unique companion virtual particle at the centroid position of the elements in the transition region, greatly simplifying the complexity of virtual particle arrangement. At the same time, the coupling scheme introduces the local external force compensation (LEFC) technology to calculate the coupling force between companion virtual particles and PD particles with different sizes, which not only overcomes the requirement of element-particle size matching but also ensures the momentum conservation of the system. Comparing the simulation results with those of a single method, it is found that the two results are in good agreement, and the method of the present invention takes less time. The present invention can quickly and effectively perform structural fracture analysis in engineering problems, featuring high precision, high efficiency, and high flexibility. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] In order to more clearly illustrate the technical solutions of the present invention, the drawings required for use in the embodiments are briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present invention, and those of ordinary skill in the art can also obtain other drawings based on these drawings without creative efforts.

[0053] Figure 1 It is a schematic flow chart of the multi-resolution PD-FEM structural fracture analysis method based on the coupled companion virtual particles in the embodiments of the present invention;

[0054] Figure 2 It is a schematic diagram of the geometric model and PD-FEM modeling in the embodiments of the present invention;

[0055] Figure 3 It is a schematic diagram of the coupled scheme of companion virtual particles in the embodiments of the present invention;

[0056] Figure 4 It is a schematic diagram of the calculation of the coupling force in the embodiments of the present invention;

[0057] Figure 5 It is a schematic diagram of the crack opening displacement at t = 16.7 μs without allowing damage in the embodiments of the present invention;

[0058] Figure 6 It is a schematic diagram of the crack length change during the crack propagation process and the final damage situation when λ = 5.0 in the embodiments of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0059] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0060] To make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0061] Embodiment 1

[0062] Figure 1 This is a flow chart of the multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling in the embodiments of the present invention. A rapid tensile example of a cracked square plate will be described in detail below.

[0063] The embodiment is mainly to evaluate the calculation accuracy and efficiency of the PD-FEM multi-resolution coupling scheme based on companion virtual particles. Before establishing the fracture simulation model in the field of engineering structure damage and fracture analysis in step 1, according to the problem to be solved, the overall size of the calculation domain, the size of the PD region, the size of the FEM region, and the total simulation duration are determined. The plate is the most commonly used structure in ships and ocean structures. Under the action of wave loads and internal loads, small cracks will occur in the flat plate. When encountering harsh environments, the cracks will rapidly expand, causing the structure to fracture, thus seriously threatening the safety of the hull structure. As Figure 2 (a) shows, the side length of the square plate is L = 50 mm, and the horizontal initial crack length at the center of the square plate is a = 10 mm. As Figure 2 (b) shows, the area with a height of H = 20 mm in the middle of the square plate is selected as the PD region, and the rest is the FEM region. During the simulation, the time step increment is set to Δt PD-FEM = 1.3367×10 -8 s, and the total simulation time is 16.7 μs.

[0064] Step 1, establish an engineering structure fracture simulation analysis model and initialize it. Specifically, set the boundary conditions according to the requirements of the problem to be solved. The boundary conditions include the force boundary conditions, displacement boundary conditions, and contact boundary conditions of the FEM region and the PD region. As Figure 2 (a) shows, the boundaries of the upper and lower sides of the square plate selected as the FEM region are subjected to a tensile load with a rate of v y = 20 m / s. The schematic diagram of PD-FEM modeling is shown in Figure 2 (b).

[0065] Step 2: Determine the FEM mesh, PD particle size, and material type. Discretize the fracture simulation analysis model regionally based on the elements and particles, and assign material parameters and initial field variable information to the elements and particles. Specifically, determine the mesh and particle sizes, and discretize the continuous FEM region, the possible damaged PD region, and the PD boundary in the fracture simulation analysis model into elements, solid PD particles, and virtual PD particles respectively. Assign mass, density, elastic modulus, Poisson's ratio, displacement, and velocity field variable values to the elements, solid PD particles, and virtual PD particles. According to the problem to be solved, the PD particle size is fixed at dx PD = 0.1 mm, while the mesh element sizes dx FEM are 0.1 mm, 0.2 mm, and 0.5 mm respectively, and the corresponding element-particle size ratios are λ = 1.0, λ = 2.0, and λ = 5.0. Table 1 lists the PD and PD-FEM discretization schemes for simulating this problem. The relevant material parameters include: density ρ s = 8000 kg / m 3 , elastic modulus E = 192 GPa, and Poisson's ratio ν = 0.333.

[0066] Table 1 Layout of the scheme for tensile simulation of a cracked square plate

[0067]

[0068] Step 3: Divide the transition region in the FEM region and obtain the companion virtual particles of each mesh element in the transition region and their field variable and related parameter information. According to Figure 3 , specifically, divide the FEM region within a preset range from the PD-FEM interface into the transition region, where the calculation method of the preset range is:

[0069] L PD-FEM = 2.0 × m PD l e

[0070] where m PD is the non-local factor and l e is the element characteristic length.

[0071] Match a unique companion virtual particle located at the centroid of each element in the transition region. The relationship between the element and the companion virtual particle is:

[0072]

[0073] where the subscripts a and b represent the element number and the companion virtual particle number respectively.

[0074] Assign the element field variable information to the companion virtual particle. The calculation method is:

[0075]

[0076] Among them, N I is the shape function of node I, X b and Pos I ini represent the position tensors of the companion virtual particle and the node in the initial configuration respectively, and u b , v b , V b and ρ b represent the displacement, velocity, volume and density of the companion virtual particle respectively.

[0077] Step 4: Conduct a neighboring particle search to determine the neighboring particles within the near fields of the companion virtual particles and PD particles, and establish a coupled "bond" connection between the two types of particles. According to Figure 3 , specifically, determine the near fields of the companion virtual particles and PD particles (including solid PD particles and virtual PD particles), and the calculation method is as follows:

[0078]

[0079] Among them, the subscript i represents the PD particle number, and b represents the companion virtual particle number, and are the near-field radii of the companion virtual particle b and the PD particle i respectively, dx PD is the PD particle size, and ndim represents the spatial dimension.

[0080] Couple and "bond" connect the companion virtual particle b and the PD particle i with the neighboring particle relationship, which is expressed as:

[0081] ξ′ = X b - X i .

[0082] Among them, ξ′ represents the "coupled bond" formed by the PD particle i and the virtual particle b.

[0083] Step 5: Transmit the interface boundary information and calculate the coupled force F PD-FEM and F FEM-PD through the coupled "bond". According to Figure 4 , based on the velocity and displacement information of the companion virtual particle and the PD particle, determine the relative deformation of the coupled "bond", and the calculation method is as follows:

[0084] Y = ξ′ + η

[0085] Among them, Y is the "bond" deformation tensor, and η is the relative displacement tensor.

[0086] Calculate the coupled force based on the relative deformation of the coupled "bond" and the local external force compensation (LEFC) technology, and the method is:

[0087]

[0088] Among them, and respectively represent the near-field ranges of PD particle i and virtual particle b, and t i ( Y ) and t b (Y) respectively represent the force scalar states of PD particle i and virtual particle b, and M(Y) is the unit direction vector state of the "bond" after deformation, and V i is the volume of PD particle i.

[0089] The interface velocity and displacement boundary conditions are indirectly applied to the PD particles through the relative deformation amount of the coupled "bond"; the coupled force caused by the relative deformation of the coupled "bond" is calculated using the LEFC technique, which is applicable to the working conditions where the companion virtual particle and the PD particle are adjacent particles to each other and only one particle is an adjacent particle of the other particle, ensuring interaction symmetry and momentum conservation.

[0090] Step 6: After transferring the coupled force to the PD particles and the element nodes, update the field variable information based on the PD and FEM motion equations, and calculate the PD damage field based on the field variables. Specifically, transform the coupled forces F PD-FEM and F FEM-PD , and solve the motion equation according to the boundary conditions. The calculation method is as follows:

[0091]

[0092] Among them, f(X i , Y, t) represents the force density transformed from the coupled force F FEM-PD , represents the component force of the coupled force F PD-FEM assigned to node I, ρ i represents the density of PD particle i, represents the acceleration of PD particle i, J represents the numbers of other nodes in the same element as node I, represents the acceleration of node J, and L u (X i , t) represents the resultant force density of PD particle i exerted by other PD particles in its neighborhood, and b(Xi, t) is the external force density acting on the PD particle. M IJ is the FEM system mass matrix, which is independent of time; is the external force of node I, is the internal force of node I.

[0093] According to the statistics of the number of broken "bonds" on each PD particle, the concept of PD local damage is defined. The calculation method is as follows:

[0094]

[0095] Among them, D i represents local damage, V j is the volume of PD particle j, μ ij is a discontinuous function, characterizing the fracture state of the "bond" between PD particles i and j, and the calculation method is as follows:

[0096]

[0097] Among them, s 0 is the critical elongation rate, which can be defined as Among them, G 0 is the energy release rate, K is the bulk modulus, δ is the near-field radius; s ij is the elongation rate of the "bond" between PD particles i and j, and the calculation method is as follows:

[0098]

[0099] Among them, ξ represents the relative position vector of PD particles i and j in the initial configuration; η is the relative displacement vector between PD particles i and j. The local damage D i has a value range of 0 ≤ D i ≤ 1. When the cumulative fracture of the "bond" forms a surface, a macroscopic crack is formed. In the present invention, D i = 0.38 is used as the crack formation criterion.

[0100] Step 7, determine whether the simulation time t reaches the total simulation duration t max , if not, then return to step S5 to perform the calculation of the next time step, if so, then stop the calculation. Specifically, for the solution of the PD and FEM motion equations, an explicit time integration scheme with a central difference format is adopted, and the method for selecting a reasonable time step is as follows:

[0101]

[0102] Among them, Δt PD-FEM , Δt PD and Δt FEM respectively represent the time step increments of PD-FEM coupling calculation, PD calculation, and FEM calculation. The constant C CFL needs to satisfy 0 ≤ C CFL ≤ 1, δ is the near-field radius, E is the elastic modulus, v is the Poisson's ratio, ρ is the density, and c' represents the dilatational wave speed in the solid material.

[0103] Determine whether the simulation time t reaches the total simulation duration t max = 16.7 μs;

[0104] Not reached, return to step S5 to perform the calculation of the next time step;

[0105] Reach and end the calculation.

[0106] In this example simulation, the crack opening displacement at t = 16.7 μs without allowing damage is first obtained. As Figure 5 shown, the crack opening displacement curves obtained using the PD-FEM method are basically consistent with the results of using the PD method alone, and these results are not affected by the mesh-particle size ratio. At the same time, different from the elliptical crack opening displacement in classical continuum mechanics, the crack tip displacement obtained based on the PD method is sharp, showing the physical significance of the PD method in describing the crack opening shape. The crack opening shapes of PD-FEM and PD are consistent, indicating that the PD-FEM coupling method successfully inherits the advantages of the PD method. When damage is allowed to occur, the square plate shows a phenomenon of crack propagation under tensile load. Figure 6 (a) shows the crack length variation process under different discretization schemes. The figure shows that as the velocity load is applied, the opening displacement of the initial crack gradually increases, and at the same time, the damage of the PD particles also accumulates continuously. Using the local damage value D = 0.38 of the PD particles as the cracking criterion, the crack initiation time for all discretization schemes is about 6.55 μs. Subsequently, the crack continues to expand until a typical Mode I crack as shown in Figure 6 (b) is formed at t = 16.7 μs. The crack length variation curves obtained by the PD-FEM method under different element-particle size ratios are basically consistent and in good agreement with the results of the PD method alone. At the same time, according to the obtained curves, the crack propagation speed is calculated to be about 1610 m / s, which is lower than the upper limit of the Mode I crack propagation speed of 2800 m / s (Rayleigh wave speed), indicating that the simulation results are reasonable.

[0107] To further evaluate the computational efficiency of the coupling method, Table 2 presents the computational costs required for the PD calculation model and the PD-FEM calculation model of the square plate rapid tension problem. The results show that the computational efficiency of the PD model is low, and the total simulation time reaches 2207.25 s. However, when using the PD-FEM model and only setting the element-particle size ratio to λ = 1.0, the total simulation time required will be reduced to 63.2% of the PD model; as the mesh-particle size ratio increases to λ = 5.0 and the number of meshes further decreases (fixing the PD particle size), the computational time of the PD-FEM model drops to 37.4% of the PD model. These results indicate that the PD-FEM coupling method not only inherits the ability of the PD method to handle discontinuous problems but also fully exploits the advantages of the FEM in terms of computational efficiency.

[0108] Table 2 Total computational time for the simulation of the cracked square plate problem

[0109]

[0110] Through the present invention, it is possible to realize the efficient simulation of the fracture problem of structures by using the multi-resolution PD-FEM coupling method. In the multi-resolution PD-FEM coupling modeling and analysis of the rapid tensile problem of a cracked square plate, it is found that the PD-FEM coupling method based on the PGP scheme proposed by the present invention not only successfully simulates complex structural fracture problems, but also proves its ability to accurately capture the details of structural fractures without the condition of precise mesh-particle matching, and the simulation results are consistent. By comparing the calculation time of PD-FEM under different conditions with the calculation time of using PD alone, it is proved that the PD-FEM coupling method not only ensures the calculation accuracy, but also improves the calculation efficiency.

[0111] The embodiments described above are only descriptions of the preferred embodiments of the present invention, and do not limit the scope of the present invention. Without departing from the design spirit of the present invention, various deformations and improvements made by those of ordinary skill in the art to the technical solutions of the present invention shall fall within the protection scope determined by the claims of the present invention.

Claims

1. A multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling, characterized in that: The following steps are involved: S1. Establish and initialize the fracture simulation analysis model in the field of engineering structure damage and fracture analysis; S2, using FEM grid and PD particles to discretize the initialized fracture simulation analysis model into regions, and obtain the material parameters and initial field variable information of the grid unit and particles; S3, dividing the transition region in the FEM region, obtaining the companion virtual particles and corresponding field variables and related parameter information of each grid unit in the transition region; S4. Conduct neighboring particle search to determine the neighboring particles within the near field of the companion virtual particles and PD particles, and establish a coupling "bond" connection between the two types of particles; S5. Transmit interface boundary information and calculate coupling force F through coupling "key" connection PD-FEM and F FEM-PD ; S6, the coupling force F PD-FEM and F FEM-PD After being transmitted to the PD particles and unit nodes, the field variable information is updated based on the PD and FEM motion equations, and the PD damage field is calculated based on the field variables; S7, determine whether the simulation time t reaches the total simulation time t max If not, return to S5 to perform the calculation of the next time step. If yes, stop the calculation; In S3, the method of dividing the transition region in the FEM region and obtaining the companion virtual particles and corresponding field variables and related parameter information of each grid unit in the transition region includes: The FEM area within a preset range from the PD-FEM interface is divided into a transition area, wherein the preset range is calculated by: L PD-FEM =2.0×m PD L e Among them, m PD is a non-local factor, l e is the unit characteristic length; Each unit in the transition region is matched with a unique companion virtual particle located at the centroid to obtain a relationship between the unit and the companion virtual particle, wherein the relationship between the unit and the companion virtual particle is: Wherein, subscripts a and b represent the unit number and companion virtual particle number, respectively; Based on the relationship between the unit and the companion virtual particle, the unit field variable information is assigned to the companion virtual particle, and the calculation method is: Among them, N I is the shape function of node I, X b and Pos I ini denote the position tensors of the companion virtual particles and nodes in the initial configuration, u b 、v b 、V b and ρ b They represent the displacement, velocity, volume and density of the companion virtual particle respectively.

2. The multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling according to claim 1 is characterized in that: Before establishing the fracture simulation analysis model in the field of engineering structure damage and fracture analysis in S1, it includes: determining the overall size of the calculation domain, the PD area size, the FEM area size and the total simulation time.

3. The multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling according to claim 1 is characterized in that: In S1, the method of establishing and initializing a fracture simulation analysis model in the field of engineering structure damage and fracture analysis includes: setting boundary conditions, wherein the boundary conditions include force boundary conditions, displacement boundary conditions and contact boundary conditions in the FEM area and the PD area.

4. The multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling according to claim 3 is characterized in that: In S2, the method of using FEM grids and PD particles to discretize the initialized fracture simulation analysis model by regions, and obtaining material parameters of grid units and particles and initial field variable information includes: Determine the size of the FEM grid and the PD particles, and discretize the continuous FEM area and the damaged PD area and PD boundary in the fracture simulation analysis model into units, real PD particles and virtual PD particles respectively; The units, the real PD particles and the virtual PD particles are assigned mass, density, elastic modulus, Poisson's ratio, displacement and velocity field variable values.

5. The multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling according to claim 1, characterized in that: In S4, the method of conducting a neighboring particle search, determining neighboring particles within the near field of the companion virtual particles and the PD particles, and establishing a coupling "key" connection between the two types of particles includes: The near-field range of the companion virtual particle and the PD particle is determined by: Wherein, the subscript i represents the PD particle number, b represents the companion virtual particle number, and are the near field radii of the companion virtual particle b and PD particle i, dx PD is the PD particle size, ndim represents the spatial dimension; The companion virtual particle b with the neighboring particle relationship is coupled with the PD particle i by a "bond", which is expressed as: ξ′=X b -X i Among them, ξ′ represents the "coupling bond" composed of PD particle i and virtual particle b.

6. The multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling according to claim 5 is characterized in that: In S5, the interface boundary information is transmitted and the coupling force F is calculated through the coupling "key". PD-FEM and F FEM-PD include: Based on the velocity and displacement information of the companion virtual particle and the PD particle, the relative deformation of the coupling "key" is determined, and the calculation method is: Y =ξ′+η in, Y is the "key" deformation tensor, η is the relative displacement tensor; The coupling force is calculated based on the relative deformation of the coupling "key" and the local external force compensation technology. The calculation method is: in, and denote the near-field ranges of PD particle i and virtual particle b respectively, t i ( Y )and t b ( Y ) represent the force scalar states of PD particle i and virtual particle b, respectively. M ( Y ) is the unit direction vector state of the "key" after deformation, V i is the volume of PD particle i.

7. The multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling according to claim 6 is characterized in that: In S6, the coupling force F PD-FEM and F FEM-PD After being transmitted to the PD particles and unit nodes, the field variable information is updated based on the PD and FEM motion equations. The method for calculating the PD damage field based on the field variables includes: PD-FEM and F FEM-PD Transform and solve the equation of motion according to the boundary conditions. The calculation method is: Among them, f(X i , Y ,t) represents the coupling force F FEM-PD The force density of the transformation, represents the coupling force F assigned to node I PD-FEM The component of force, ρ i represents the density of PD particle i, represents the acceleration of PD particle i, J represents the number of other nodes in the same unit as node I, and ü J represents the acceleration of node J, L u (X i ,t) represents the combined force density of PD particle i on other PD particles in the neighborhood. is the external force density acting on the PD particles, M IJ is the mass matrix of the FEM system, which is independent of time; is the external force at node I, is the internal force at node I.

8. The multi-resolution PD-FEM structural fracture analysis method based on companion virtual particle coupling according to claim 1 is characterized in that: In S7, it is determined whether the simulation time t reaches the total simulation time t max If not, then return to S5 to perform the calculation of the next time step. If so, the method of stopping the calculation includes: The explicit time integration scheme of the central difference format is used to solve the PD and FEM motion equations. The method for selecting a reasonable time step is: Among them, Δt PD-FEM , Δt PD and Δt FEM They represent the time step increments for PD-FEM coupling calculation, PD calculation and FEM calculation respectively, and the constant C CFL Must satisfy 0≤C CFL ≤1, δ is the near-field radius, E is the elastic modulus, v is the Poisson's ratio, ρ is the density, and c' represents the expansion wave velocity in the solid material.

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