Multi-resolution near-field dynamics efficient simulation method for material deformation and fracture analysis

Abstract

The invention discloses a multi-resolution near-field dynamics efficient simulation method for material deformation and fracture analysis. The method comprises the following steps: establishing a near-field dynamics numerical model; discretization of different spatial resolutions is carried out on the near-field dynamics numerical model, and physical information is given to each substance point; determining the interaction between different substance points; the interaction between the substance points with different resolutions is corrected, and effective interaction between the substance points is ensured; applying boundary conditions to the near-field dynamic numerical model; judging the positions of the material points in the near-field range, and correcting the volumes of the material points; selecting a proper time integration strategy according to the characteristics of the solved problem; counting and updating the dynamic information of the material points, and describing the deformation condition of the material; according to the fracture criterion, the damage condition of the material points in the computational domain is updated, and the fracture process of the material is described. According to the method, the numerical simulation efficiency is improved on the premise that the precision is guaranteed, and the requirement for computer hardware in the calculation process is reduced.

Description

Technical Field

[0001] The present invention belongs to the field of computational solid mechanics, and in particular relates to a multi-resolution peridynamics efficient simulation method for material deformation and fracture analysis. Background Art

[0002] The fracture failure problem of solid materials has been a difficult problem faced by the academic and engineering communities since the 20th century, and is also a research focus in related fields. When dealing with the fracture failure problem of materials, the numerical methods based on the traditional continuum theory will inevitably encounter the contradiction between the continuity assumption and the actual discontinuity. Peridynamics (PD) uses the method of non-local internal force integration to effectively overcome the singularity problem of the crack tip in the traditional method, and thus shows significant advantages in solving the discontinuity problem.

[0003] Due to the large spatial scale of actual engineering problems, numerical simulation methods often face problems of computational efficiency and accuracy. Compared with traditional local methods, the non-local characteristics of peridynamics determine its low computational efficiency. Most studies related to peridynamics use a uniform discretization method, which is convenient for programming and control of computational errors. Peridynamics itself does not impose any restrictions on the discretization method, so some non-uniform discretization peridynamic models have been proposed to improve computational efficiency and accuracy. Non-uniform discretization poses certain challenges when dealing with interactions between material points of different resolutions, and it is easy to generate false forces, linear momentum and angular momentum imbalance, causing false wave reflection problems, which will ultimately affect the accuracy of the calculation results. Developing appropriate multi-resolution peridynamic simulation technology, accurately describing the interactions between material points of different resolutions, improving computational efficiency, and ensuring computational accuracy are of great significance for the promotion and use of peridynamic methods. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention proposes a multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis, which can effectively simulate the deformation and failure problems of materials with better accuracy while ensuring efficiency.

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

[0006] The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis includes the following steps:

[0007] S1: Establish a peridynamic numerical model based on the material deformation and fracture analysis process;

[0008] S2: discretizing the peridynamic numerical model with different spatial resolutions and assigning physical information to each material point;

[0009] S3: Determine the interaction between material points of different resolutions;

[0010] S4: Correct the interaction between material points of different resolutions to obtain effective interaction between material points;

[0011] S5: applying boundary conditions to the peridynamic numerical model;

[0012] S6: Determine the position of the material point in the near field and correct the volume of the material point;

[0013] S7: According to the characteristics of material deformation and fracture analysis, a time integration strategy that meets the preset requirements is selected;

[0014] S8: Based on the effective interaction between the material points, the boundary conditions, the corrected material point volume and the time integration strategy, the dynamic information of the material points is counted and updated to obtain the deformation of the material;

[0015] S9: Based on the deformation of the material and in accordance with the fracture criterion, the material point damage is updated to obtain the fracture process of the material.

[0016] Preferably, in S1, the method for establishing a peridynamic numerical model according to the material deformation and fracture analysis process includes:

[0017]

[0018] Among them, ρ i is the density; H Xi is the material point X i Near field range; is the material point X i The acceleration at time t; T[X i ,t]<ξ> means that the material point X at time t i The "bond" vector ξ is mapped to the force vector state in N / m 6 ; T [X j ,t]<-ξ> means that the material point X at time t j The "bond" vector ξ is mapped to the force vector state in N / m 6 ; V j is the material point X j Volume; is the force acting on particle X at time t i physical density.

[0019] Preferably, in S2, the method of discretizing the peridynamic numerical model with different spatial resolutions and assigning physical information to each material point comprises:

[0020] According to the material deformation and fracture analysis process, different regions are set for spatial multi-resolution discretization, and the material point discrete spacing dx and near-field radius δ of different regions are set;

[0021] Assign physical information to each material point in the computational domain, including density, volume, displacement, velocity, and acceleration.

[0022] Preferably, in S3, the method for determining the interaction between material points with different resolutions comprises:

[0023] T = T ( Y )= T ( Y )<ξ>,

[0024] in, M ( Y )<ξ> represents the deformation direction vector state, which is defined as:

[0025]

[0026] in, Y is the deformation vector state, y is the deformed scalar state, t is the force scalar state.

[0027] Preferably, in S4, the method of correcting the interaction between material points with different resolutions to obtain effective interaction between material points includes:

[0028] According to the material deformation and fracture analysis process, the calculation domain is divided into regions;

[0029] Based on the divided areas, the interaction between material points with different near-field ranges and different resolutions is improved according to the local external force compensation technology, the average near-field radius technology or the maximum near-field radius technology to obtain effective interaction between the material points.

[0030] Preferably, in S6, the method of determining the position of the material point in the near field range and correcting the volume of the material point includes:

[0031] According to the coordinate information of the spatially discrete material points and the near-field radius of different resolutions, the neighboring material points of each material point are determined;

[0032] According to the search results, for the neighborhood material points whose partial volumes are within the near field of the material points, the volume correction technology is introduced to correct the volumes of the neighborhood material points.

[0033] Preferably, in S8, the method of collecting and updating the dynamic information of the material points to obtain the deformation of the material includes:

[0034] Based on the boundary conditions, the discretized peridynamic numerical model is solved to obtain the acceleration of the material point;

[0035] Based on the acceleration of the material point, the velocity and displacement of the material point are solved to complete the description of the material deformation.

[0036] Preferably, in S9, based on the deformation of the material and according to the fracture criterion, the material point damage is updated to obtain the fracture process of the material, which method includes:

[0037] According to the displacement of the material point, the bond elongation is calculated and the bond elongation is obtained by comparing it with the initial bond length:

[0038]

[0039] where ξ represents the “key” vector, e represents the “bond” vector ξ stretched scalar state, x <ξ> represents the position scalar state of the “key” vector ξ, X <ξ> represents the position vector state of the "key" vector ξ, Y <ξ> represents the deformation vector state of the “key” vector ξ;

[0040] Compare the bond elongation with the critical elongation of the bond to determine whether bond breaking occurs:

[0041]

[0042] Among them, μ ij (X j ,X i ,t) is the material point X i With X j The discontinuity function for forming a bond, s ij is the material point X i With X j The elongation of bond formation, s0 is the critical elongation;

[0043] All broken bonds of each material point are counted to obtain the damage value of the material point, completing the description of the fracture process of the material:

[0044]

[0045] Where D = 0 means that the material point X i There is no damage at the material point X. D = 1 means that the material point X i Total destruction.

[0046] Compared with the prior art, the present invention has the following beneficial effects:

[0047] The present invention adopts the peridynamic method, discretizes the material into material points of different resolutions, sets the material parameters of the material points, determines the near-field range of material points of different resolutions, calculates the volume of the material points, gives boundary conditions according to the problem to be solved, considers the interaction between material points of different resolutions, performs time integration, and realizes efficient simulation of material deformation and fracture, which not only ensures the accuracy of the simulation, but also reduces the number of material points and improves the calculation efficiency, which is of great significance for promoting the application of perifield dynamics to the simulation of practical problems. In the PD local refinement modeling and analysis of the Kalthoff-Winkler impact experiment, it is found that the current PD model can effectively describe the crack propagation and structural destruction process. The introduced non-local contact model can effectively handle the interaction between different material points. The present invention can effectively simulate the deformation and destruction problems of materials, and has better accuracy while ensuring efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] In order to more clearly illustrate the technical solution of the present invention, the following briefly introduces the drawings required for use in the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0049] Figure 1 It is a schematic diagram of the geometric model of the experiment of the embodiment of the present invention;

[0050] Figure 2 A schematic diagram of multi-resolution modeling of peridynamics in an embodiment of the present invention;

[0051] Figure 3 Schematic diagram of the symmetric interaction between material points in the peridynamics of the present invention; wherein (a) is a schematic diagram of the local external force compensation (LEFC) technology; (b) is a schematic diagram of the average near-field radius (AHR) technology; (c) is a schematic diagram of the maximum near-field radius (MHR) technology;

[0052] Figure 4 It is a schematic diagram of the peridynamic volume correction technology in the present invention;

[0053] Figure 5 Schematic diagram of crack initiation and propagation in the Kalthoff-Winkler impact test of the present invention; wherein (a) is a schematic diagram of the LEFC technology; (b) is a schematic diagram of the AHR technology; (c) is a schematic diagram of the MHR technology;

[0054] Figure 6 The material point distribution and damage cloud diagram at the interface between the refinement and non-refinement regions at t=72μs in the present invention; wherein (a) is a schematic diagram of the LEFC technology; (b) is a schematic diagram of the AHR technology; (c) is a schematic diagram of the MHR technology;

[0055] Figure 7 Schematic diagram of the process flow of the multi-resolution peridynamics efficient simulation method for material deformation and fracture analysis in an embodiment of the present invention. DETAILED DESCRIPTION

[0056] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0057] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0058] Embodiment 1

[0059] like Figure 7 As shown, the present invention provides a multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis, including:

[0060] S1. According to the material deformation and fracture analysis process, establish a peridynamic numerical model and set material parameters;

[0061] S2. According to practical problems and computational efficiency, the peridynamic numerical model is discretized with different spatial resolutions to give physical information to each material point.

[0062] S3, determine the interaction between different material points;

[0063] S4. Correct the interaction between material points of different resolutions to ensure effective interaction between material points;

[0064] S5. Apply boundary conditions to the peridynamic numerical model;

[0065] S6. Determine the position of the material point in the near field range and correct the volume of the material point;

[0066] S7. Select appropriate time integration strategy according to the characteristics of material deformation and fracture analysis process;

[0067] S8, counting and updating the dynamic information of the material points to describe the deformation of the material;

[0068] S9, updating the damage of material points according to the fracture criterion to describe the fracture process of the material;

[0069] S10. Determine whether to return to step S7 or terminate the calculation.

[0070] In this embodiment, step S1 of establishing a peridynamic numerical model and setting material parameters includes:

[0071] According to the problem to be solved, the computational domain is set and the corresponding control equations are established using the peridynamic model:

[0072]

[0073] Among them, ρ i is the density; H Xi is the material point X i Near field range; is the material point X i The acceleration at time t; T[X i ,t]<ξ> means that the material point X at time t i The "bond" vector ξ is mapped to the force vector state in N / m 6 ; T [X j ,t]<-ξ> means that the material point X at time t j The "bond" vector ξ is mapped to the force vector state in N / m 6 ; V j is the material point X j Volume; is the force acting on particle X at time t i physical density.

[0074] Initialize the elastic modulus, Poisson's ratio, and fracture energy release rate of the material.

[0075] In this embodiment, step S2 discretizes the simulation model with different spatial resolutions according to actual problems and computational efficiency, and assigns physical information to each material point, including:

[0076] According to the problem to be solved, different regions are set for spatial multi-resolution discretization, and the material point discrete spacing dx and the near-field radius δ of different regions are set;

[0077] Assign physical information such as density, volume, displacement, velocity and acceleration to each material point in the computational domain.

[0078] In this embodiment, S3 determines the interaction between different material points including:

[0079] Establish a peridynamic numerical model of the corresponding problem and construct the governing equations in peridynamic form;

[0080] Derive the point force equations of the material under study to describe the interaction between different material points:

[0081] T=T(Y)=tM(Y)<ξ>

[0082] in, M ( Y )<ξ> represents the deformation direction vector state, which is defined as:

[0083]

[0084] in, Y is the deformation vector state, y is the deformed scalar state, t is the force scalar state.

[0085] In this embodiment, S4 corrects the interaction between material points of different resolutions to ensure effective interaction between material points including:

[0086] According to the problem requirements, the computational domain is divided into regions;

[0087] The interaction between material points with different near-field ranges and resolutions is improved based on the local external force compensation technology, the average near-field radius technology or the maximum near-field radius technology to ensure effective interaction between material points:

[0088] (1) Using local external force compensation (LEFC) technology to maintain the material point X i and X j The near-field range of the material point X i The governing equations of remain unchanged, and the material point X is modified j The control equation is:

[0089]

[0090] Among them, L u (X i ,t) represents the material point X at time t j Acting on material point X i The internal force density of the system is introduced to ensure the conservation of momentum. i For material point X j Local external force:

[0091] F LEFC (X j ,X i ,t)=- t [X i ,t]<ξ> M ( Y )<ξ>V ij V i

[0092] in, t [X i ,t]<ξ> means that the material point X at time t i The “bond” vector ξ is mapped to the force scalar state, V ij Represents material point X j At material point X i The volume occupied by the near field, V i is the material point X i volume.

[0093] (2) Using the Average Horizon Radius (AHR) technique, first determine X i and X j The average near field radius and is the material point X i and X j If the distance between two material points exceeds the average near-field radius δ m , then there is no interaction between them; otherwise, using the uniformly discrete peridynamic formula, according to the average near-field radius δ m Calculate the interaction between material points, which can be expressed as:

[0094] F ij =-F ji =( t [X i ,t]<ξ>V ij (δ m )V i - t [X j ,t]<-ξ>V ji (δ m )V j ) M ( Y )<ξ>

[0095] Among them, F ij is the material point X j For material point X i The force, F ji is the material point X i For material point X j The force, V ij (δ m ) is the material point X calculated based on the average near-field radius j The effective volume, V ji Represents material point X i At material point X j The volume occupied by the near field.

[0096] (3) Using the Maximum Horizon Radius (MHR) technique, first determine the maximum near-field radius of the two material points. To ensure that both are within their respective near fields. The interaction between material points is:

[0097] F ij =-F ji =( t [X i ,t]<ξ>V ij (δ max )V i - t [X j ,t]<-ξ>V ji (δ max )V j ) M ( Y )<ξ>

[0098] In this embodiment, S5 applies boundary conditions to the peridynamic numerical model including:

[0099] According to the problem to be solved, select the corresponding type of boundary conditions, including: force boundary conditions, velocity boundary conditions;

[0100] According to the characteristics of peri-field dynamics, a virtual material point layer is set and the boundary conditions are processed and converted into a boundary condition form that can be applied to the virtual material point layer.

[0101] In this embodiment, S6 determines the position of the material point in the near field range, and correcting the volume of the material point includes:

[0102] According to the coordinate information of the spatially discrete material points and the near-field radius of different resolutions, the neighboring material points of each material point are determined;

[0103] According to the search results, for the neighboring material points whose partial volumes are within the near field of the material point, the volume correction technology is introduced to correct the volume of the neighboring material points;

[0104]

[0105] Among them, V ij Represents material point X j At material point X i The volume occupied by the near field.

[0106] In this embodiment, S7 selects a suitable time integration strategy according to the characteristics of the problem to be solved, including:

[0107] Determine the characteristics of the problem to be solved: dynamic problem, static problem or quasi-static problem;

[0108] The dynamic problem uses the central difference method of Velocity-Verlet format for time integration:

[0109] I. According to the initial conditions u0, The acceleration of the material point at the initial moment is obtained by using the control equation

[0110] II. Calculate in each time step according to the following steps

[0111] (1) Calculation The speed of time

[0112]

[0113] (2) Calculate t+Δt PD Displacement of time

[0114]

[0115] Among them, u t is the displacement at time t, PD is the abbreviation of Peridynamics in English, and has no special meaning. The subscript here refers to the time step of time integration using this method.

[0116] (3) According to t+Δt PD Displacement of time Calculate t+Δt PD Acceleration of time

[0117]

[0118] (4) Calculate t+Δt PD The speed of time

[0119]

[0120] The central difference method of the Velocity-Verlet format is conditionally stable, that is, the time step increment Δt PD Small enough to meet the convergence and accuracy requirements; but the time step increment Δt PD It should be large enough to meet the computational efficiency requirements. According to the Courant-Friedrichs-Lewy (CFL) condition, the time step increment should satisfy:

[0121]

[0122] Where 0≤C CFL≤1; c′ represents the expansion wave velocity in the solid material, δ is the near-field radius, E is the elastic modulus, ρ is the density, and ν is the Poisson's ratio.

[0123] Dynamic relaxation technology is introduced to solve quasi-static problems. When using the central difference method in Velocity-Verlet format to solve the problem, artificial damping terms are introduced to absorb the kinetic energy of the system to ensure that a static or quasi-static solution is obtained:

[0124] The motion equation with artificial damping is expressed as:

[0125]

[0126] In the formula, C d For artificial damping; Represents the sum of the internal force density and the external force density. When solving using the central difference method in Velocity-Verlet format, the displacement and velocity expressions for the next time step are:

[0127]

[0128] When using dynamic relaxation techniques to deal with quasi-static problems, it is necessary to introduce an iterative convergence criterion to reasonably terminate the calculation. The displacement convergence criterion is a commonly used convergence criterion, which is based on the following principle: when the displacement increment of the material point is small enough, the system is considered to have reached an equilibrium state:

[0129]

[0130] In the formula, Δu n and u n represent the displacement increment and displacement of the nth iteration respectively; is the displacement convergence tolerance, usually set to 1×10 -5 .

[0131] In this embodiment, S8 counts and updates the dynamic information of the material points to describe the deformation of the material, including:

[0132] Based on the boundary conditions, the discretized control equations are solved to obtain the acceleration of the material point;

[0133] Solve the velocity and displacement of the material point to complete the description of the material deformation.

[0134] In this embodiment, S9 updates the damage of the material point according to the fracture criterion to describe the fracture process of the material, including:

[0135] According to the displacement of the material point, the bond elongation is calculated and the bond elongation is obtained by comparing it with the initial bond length:

[0136]

[0137] where ξ represents the “key” vector, e represents the “bond” vector ξ stretched scalar state, x <ξ> represents the position scalar state of the “key” vector ξ, X <ξ> represents the position vector state of the "key" vector ξ, Y <ξ> represents the deformation vector state of the "key" vector ξ.

[0138] Compare with the critical elongation of the bond to determine whether bond breaking occurs:

[0139]

[0140] Among them, μ ij (X j ,X i ,t) is the discontinuity function of the bond between material point Xi and Xj, s ij is the material point Xi and X j The elongation for forming a bond, s0 is the critical elongation.

[0141] All broken bonds of each material point are counted and divided by the total number of initial bonds in the neighborhood of the material point to obtain the damage value of the material point, completing the description of the fracture process of the material:

[0142]

[0143] Where D = 0 means that the material point X i There is no damage at the material point X. D = 1 means that the material point X i Total destruction.

[0144] In this embodiment, S10 determines whether to return to step S7 or terminate the calculation, including:

[0145] Determine whether the maximum calculation time step has been reached;

[0146] If not reached, return to step S7 to continue calculation;

[0147] Reached, terminate the calculation.

[0148] Embodiment 2

[0149] The following is a detailed description of the multi-resolution peridynamics efficient simulation technology for material deformation and fracture analysis of the present invention in conjunction with an example of target plate impact destruction.

[0150] Step 1: Implement the example to demonstrate the performance of the current peridynamic model in simulating structural dynamic fracture. According to the problem to be solved, establish a peridynamic numerical model and set material parameters. Specifically, establish a peridynamic numerical model and set material parameters for the Kalthoff-Winkler impact experiment. The experimental settings are as follows: Figure 1 As shown, a rigid impactor moves at a constant speed v p =30m / s hits a rectangular target with two symmetrical initial cracks. The dimensions of the target are length L = 200mm, height H = 100mm, initial crack length l is 50mm, and tip diameter s1 is 1.5mm. The impactor has a height H1 = 20mm and a diameter L1 = 50mm, which is equal to the distance between the two initial cracks. Both the target and the impactor are made of maraging steel X2NICoMo 1895, and its material parameters are: elastic modulus E = 190GPa, density ρ s =8000kg / m3, Poisson's ratio ν=0.3.

[0151] Step 2: According to the actual problem and computational efficiency, the simulation model is discretized with different spatial resolutions to give physical information to each material point. Specifically, in the peridynamic model, according to Figure 2 , in the lower left corner B area, a material point with dx = 0.5 mm is used for local refinement modeling, and the material point size in other areas is dx = 1.0 mm. The near-field radius of the material points in both areas is 3 times the size of the material point, that is, δ = 3.0dx. Time step increment Δt PD =8×10 -8 s.

[0152] Step 3, determine the interaction between different material points. Specifically, construct the governing equations of the peridynamic form of the Kalthoff-Winkler impact experiment:

[0153]

[0154] Determine the point force equations for the material under study, describing the interactions between different material points:

[0155] T=T(Y)=tM(Y)<ξ>

[0156]

[0157] in, K is the bulk modulus, G is the shear modulus; the volume expansion scalar is

[0158]

[0159] Where e is the stretched scalar state:

[0160] e=yx,x=|X

[0161] Weighted volume factor q v for:

[0162]

[0163] Step 4: Correct the interaction between material points of different resolutions to ensure effective interaction between material points. Figure 3 , the three technologies are:

[0164] (1) Local External Force Compensation (LEFC) technology to maintain the material point X i and X j The near-field range of the material point X i The governing equations of remain unchanged, and the material point X is modified j The control equation is:

[0165]

[0166] In order to ensure the conservation of momentum of the system, the material point X is introduced. i For material point X j Local external force:

[0167] F LEFC (X j ,X i ,t)=-t[X i ,t]<ξ>M(Y)<ξ>V ij V i

[0168] (2) Average Horizon Radius (AHR) technology, first determine X i and X j The average near field radius If the distance between two material points exceeds the average near-field radius δ m , then there is no interaction between them; otherwise, using the uniformly discrete peridynamic formula, according to the average near-field radius δ m Calculate the interaction between material points, which can be expressed as:

[0169] F ij =-F ji =( t [X i ,t]<ξ>V ij (δ m )V i - t [Xj ,t]<-ξ>V ji (δ m )V j ) M ( Y )<ξ>

[0170] Among them, V ij (δ m ) is the material point X calculated based on the average near-field radius j effective volume.

[0171] (3) Maximum Horizon Radius (MHR) technique: First, determine the maximum near-field radius of two material points. To ensure that both are within their respective near fields. The interaction between material points is:

[0172] F ij =-F ji =( t [X i ,t]<ξ>V ij (δ max )V i - t [X j ,t]<-ξ>V ji (δ max )V j ) M ( Y )<ξ>

[0173] Step 5: Apply boundary conditions to the peridynamic model of the Kalthoff-Winkler impact experiment. Specifically, according to the characteristics of peridynamics, a virtual material point layer is set to process the boundary conditions. The material points in the area where the model impactor is located are assigned velocity boundaries, and their constant velocity is v p =30m / s.

[0174] Step 6: determine the position of the material point in the near field range and correct the volume of the material point. Specifically, determine the neighboring material points of each material point based on the coordinate information of the spatially discrete material points and the near field radius of different resolutions;

[0175] like Figure 4 According to the search results, for the neighboring material points whose partial volumes are within the near field of the material point, the volume correction technology is introduced to correct the volume of the neighboring material points;

[0176]

[0177] Among them, V ij Represents material point X j At material point Xi The volume occupied by the near field.

[0178] Step 7: Select an appropriate time integration strategy based on the characteristics of the problem being solved. Specifically, the central difference method of the Velocity-Verlet format is used for time integration:

[0179] I. According to the initial conditions u0, The acceleration of the material point at the initial moment is obtained by using the control equation

[0180] II. Calculate in each time step according to the following steps

[0181] (1) Calculation The speed of time

[0182]

[0183] (2) Calculate t+Δt PD Displacement of time

[0184]

[0185] (3) According to t+Δt PD Displacement of time Calculate t+Δt PD Acceleration of time

[0186]

[0187] (4) Calculate t+Δt PD The speed of time

[0188]

[0189] The central difference method of the Velocity-Verlet format is conditionally stable, that is, the time step increment Δt PD Small enough to meet the convergence and accuracy requirements; but the time step increment Δt PD It should be large enough to meet the computational efficiency requirements. According to the Courant-Friedrichs-Lewy (CFL) condition, the time step increment should satisfy:

[0190]

[0191] Where 0≤C CFL ≤1; c′ represents the expansion wave velocity in the solid material.

[0192] Step 8: Count and update the dynamic information of the material points to describe the deformation of the material. Specifically, based on the boundary conditions, the discretized control equations are solved to obtain the acceleration of the material points; the velocity and displacement of the material points are solved to complete the description of the material deformation.

[0193] Step 9: Update the damage of the material point according to the fracture criterion to describe the fracture process of the material. Specifically, calculate the elongation of the bond according to the displacement of the material point, and compare it with the initial bond length to obtain the bond elongation:

[0194]

[0195] where ξ represents the “key” vector, e represents the “bond” vector ξ stretched scalar state, x <ξ> represents the position scalar state of the “key” vector ξ, X <ξ> represents the position vector state of the "key" vector ξ, Y <ξ> represents the deformation vector state of the "key" vector ξ.

[0196] Compare with the critical elongation of the bond to determine whether bond breaking occurs:

[0197]

[0198] Among them, μ ij (X j ,X i ,t) is the material point X i With X j The discontinuity function for forming a bond, s ij is the material point X i With X j The elongation for forming a bond, s0 is the critical elongation.

[0199] All broken bonds of each material point are counted and divided by the total number of initial bonds in the neighborhood of the material point to obtain the damage value of the material point, completing the description of the fracture process of the material:

[0200]

[0201] Where D = 0 means that the material point X i There is no damage at the material point X. D = 1 means that the material point X i Total destruction.

[0202] Optionally, S10 determines whether to return to step S7 or terminate the calculation, including:

[0203] Determine whether the maximum calculation time step t = 103.2μs has been reached;

[0204] If not reached, return to step S7 to continue calculation;

[0205] Reached, terminate the calculation.

[0206] This example simulates different time periods, such as Figure 5 , the crack propagation process obtained based on three local refinement techniques. With the impact of the impactor, the initial crack tip damage gradually accumulates and cracks at t = 28μs, which is very close to the experimental observation of 29μs. Subsequently, the crack continues to propagate under the action of shear and tension, and the path angle relative to the vertical direction is 64°. At the same time, it can be seen that whether it is LEFC, AHR or MHR technology, the path angle obtained is exactly the same, indicating that the crack propagation is not affected by the local area refinement treatment. Figure 6 At t = 72μs, the crack has extended to the refinement area. Comparison shows that the crack path angles obtained by LEFC and AHR technology are basically the same, and the cracks are symmetrically distributed on the left and right sides; while the left crack path angle obtained by MHR technology differs from other results by 0.7° and 1°, respectively, and the right crack path angle differs by 1.1° and 1°, respectively. The left and right crack path angles differ by 2.3°, which may be affected by the stress wave generated by crack bifurcation in the refinement area. Finally, at t = 103.2μs, the cracks on both sides reach the target plate boundary. The overall path angles of the left crack obtained by the three technologies are 69.2°, 69.1° and 69.9°, respectively, which are highly consistent with the experimental value of 70°. The differences in the left and right overall path angles of the cracks obtained by LEFC and AHR technology are 0.1° and 0.2°, respectively, showing good symmetry. The difference in the left and right overall path angles of the cracks obtained by MHR technology is 1.6°. Although there is a difference, it can be ignored from a macroscopic point of view. In summary, the three local refinement processing techniques are all suitable for local refinement simulation of structural dynamic fracture, among which LEFC and AHR techniques perform better.

[0207] The present invention can realize efficient simulation of material deformation and fracture using multi-resolution peridynamics. In the peridynamic multi-resolution modeling and analysis of the Kalthoff-Winkler impact experiment, it is found that the current peridynamic model can effectively describe the crack propagation and structural damage process. Comparing the crack results of three local refinement processing technologies, it is found that all technologies can reasonably predict the crack initiation time and path angle, among which LEFC and AHR technologies perform better.

[0208] The embodiments described above are only descriptions of the preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. An efficient multi-resolution peridynamic simulation method for material deformation and fracture analysis, characterized by: The following steps are involved: S1: Establish a peridynamic numerical model based on the material deformation and fracture analysis process; S2: discretizing the peridynamic numerical model with different spatial resolutions and assigning physical information to each material point; S3: Determine the interaction between material points of different resolutions; S4: Correct the interaction between material points of different resolutions to obtain effective interaction between material points; S5: applying boundary conditions to the peridynamic numerical model; S6: Determine the position of the material point in the near field and correct the volume of the material point; S7: According to the characteristics of material deformation and fracture analysis, a time integration strategy that meets the preset requirements is selected; S8: Based on the effective interaction between the material points, the boundary conditions, the corrected material point volume and the time integration strategy, the dynamic information of the material points is counted and updated to obtain the deformation of the material; S9: Based on the deformation of the material and in accordance with the fracture criterion, the material point damage is updated to obtain the fracture process of the material.

2. The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis according to claim 1 is characterized in that: In S1, according to the material deformation and fracture analysis process, the method for establishing a peridynamic numerical model includes: Among them, ρ i is the density; is the material point X i The near field range of u(X i ,t) is the material point X i The acceleration at time t; T[X i ,t]<ξ> means that the material point X at time t i The "bond" vector ξ is mapped to the force vector state in N / m 6 ; T [X j ,t]<-ξ> means that the material point X at time t j The "bond" vector ξ is mapped to the force vector state in N / m 6 ; V j is the material point X j Volume; is the force acting on particle X at time t i physical density.

3. The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis according to claim 1, characterized in that: In S2, the method of discretizing the peridynamic numerical model with different spatial resolutions and assigning physical information to each material point includes: According to the material deformation and fracture analysis process, different regions are set for spatial multi-resolution discretization, and the material point discrete spacing dx and near-field radius δ of different regions are set; Assign physical information to each material point in the computational domain, including density, volume, displacement, velocity, and acceleration.

4. The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis according to claim 2 is characterized in that: In S3, the method for determining the interaction between material points with different resolutions includes: T=T(Y)=tM(Y)<ξ>, Among them, M(Y)<ξ> represents the deformation direction vector state, which is defined as: in, Y is the deformation vector state, y is the deformed scalar state, t is the force scalar state.

5. The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis according to claim 1, characterized in that: In S4, the method for correcting the interaction between material points of different resolutions to obtain effective interaction between material points includes: According to the material deformation and fracture analysis process, the calculation domain is divided into regions; Based on the divided areas, the interaction between material points with different near-field ranges and different resolutions is improved according to the local external force compensation technology, the average near-field radius technology or the maximum near-field radius technology to obtain effective interaction between the material points.

6. The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis according to claim 1, characterized in that: In S6, the method of determining the position of the material point in the near field range and correcting the volume of the material point includes: According to the coordinate information of the spatially discrete material points and the near-field radius of different resolutions, the neighboring material points of each material point are determined; According to the search results, for the neighborhood material points whose partial volumes are within the near field of the material points, the volume correction technology is introduced to correct the volumes of the neighborhood material points.

7. The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis according to claim 1, characterized in that: In S8, the method of counting and updating the dynamic information of the material points to obtain the deformation of the material includes: Based on the boundary conditions, the discretized peridynamic numerical model is solved to obtain the acceleration of the material point; Based on the acceleration of the material point, the velocity and displacement of the material point are solved to complete the description of the material deformation.

8. The multi-resolution peridynamic efficient simulation method for material deformation and fracture analysis according to claim 1, characterized in that: In S9, based on the deformation of the material and according to the fracture criterion, the material point damage is updated to obtain the fracture process of the material, which includes: According to the displacement of the material point, the bond elongation is calculated and the bond elongation is obtained by comparing it with the initial bond length: Among them, ξ represents the "bond" vector, e represents the stretched scalar state of the "bond" vector ξ, x<ξ> represents the position scalar state of the "bond" vector ξ, X<ξ> represents the position vector state of the "bond" vector ξ, and Y<ξ> represents the deformation vector state of the "bond" vector ξ; Compare the bond elongation with the critical elongation of the bond to determine whether bond breaking occurs: Among them, μ ij (X j ,X i ,t) is the material point X i With X j The discontinuity function for forming a bond, s ij is the material point X i With X j The elongation of bond formation, s0 is the critical elongation; All broken bonds of each material point are counted to obtain the damage value of the material point, completing the description of the fracture process of the material: Where D = 0 means that the material point X i There is no damage at the material point X. D = 1 means that the material point X i Total destruction.

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