Dynamic crack growth calculation method and system based on finite element-peridynamics

Abstract

The present invention belongs to the field of numerical simulation technology and provides a finite element-peridynamics-based dynamic crack propagation calculation method and system. The present invention first calculates the deformation and stress conditions of the finite element unit to determine whether the finite element unit has reached the failure condition. The calculation area is divided into a finite element grid and peridynamic material points. Then, cracks initiate between the peridynamic material points. As the crack propagates, the finite element units that have reached the failure condition are gradually replaced by peridynamic material points. A dynamic conversion method for the finite element grid and peridynamic material points is constructed, avoiding the problem of requiring a preset finite element and peridynamic calculation range. Finally, the position and stress of the finite element unit nodes and material points are calculated, realizing dynamic modeling and efficient calculation of the crack propagation process.

Description

Technical Field

[0001] The present invention belongs to the technical field of numerical simulation, and in particular relates to a dynamic crack propagation calculation method and system based on finite element-peridynamics. Background Art

[0002] The basic solution idea of ​​the Finite Element Method (FEM) is to divide the computational domain into a finite number of non-overlapping units, then analyze each unit, and finally integrate and solve the overall displacement and force. The advantage of the FEM is its high computational efficiency, but because this method is based on the continuity assumption, it is difficult to simulate crack propagation problems. Peridynamics is a method that builds a model based on the idea of ​​non-local action and describes the physical and mechanical behavior of materials by solving spatial integral equations. It avoids the singularity of traditional methods based on the continuity assumption and solving spatial differential equations when facing discontinuous problems, and has significant advantages in aspects such as crack propagation. However, compared with the FEM, the computational efficiency of peridynamics is low and it is difficult to apply to large-scale engineering calculations. Therefore, by coupling the FEM with peridynamics, it is possible to improve the computational efficiency and make up for the shortcomings of the FEM in crack propagation simulation.

[0003] The inventors found that although some scholars have proposed the finite element-peridynamics coupling method, the existing method requires a preset finite element and peridynamic calculation range, that is, the finite element and peridynamic models are fixed, which is impossible to achieve for the problem of difficult to predict the crack propagation path and range. Summary of the Invention

[0004] In order to solve the above problems, the present invention proposes a dynamic crack propagation calculation method and system based on finite element-peridynamics. The present invention divides the calculation area into finite element grids and peridynamic material points, and constructs a dynamic conversion method between finite element grids and peridynamic material points, thereby realizing dynamic modeling and efficient calculation of the crack propagation process.

[0005] In order to achieve the above object, the present invention is implemented through the following technical solutions:

[0006] In a first aspect, the present invention provides a method for calculating dynamic crack growth based on finite element-peridynamics, comprising:

[0007] Perform finite element meshing on the calculation area to obtain multiple finite element units;

[0008] Calculate the deformation and stress of the finite element unit and determine whether the finite element unit has reached the failure condition;

[0009] Finite element elements that have reached failure conditions are replaced by a limited number of peridynamic material points. Cracks initiate between peridynamic material points, and as the cracks propagate, the finite element elements that have reached failure conditions are gradually replaced by peridynamic material points.

[0010] The finite element-peridynamics coupling calculation method is used to calculate the positions and forces of finite element unit nodes and material points.

[0011] Furthermore, the finite element method is used to calculate the deformation and force of the finite element unit, the interpolation function is used to solve the internal stress of the finite element unit node and the finite element unit, and the macro strength criterion is used to determine whether the finite element unit reaches the failure condition.

[0012] Furthermore, the macro strength criterion adopts the maximum tensile stress theory, the maximum tensile strain theory or the Mohr-Coulomb yield criterion.

[0013] Furthermore, the mass, material parameters, velocity, and displacement of the peridynamic material points are inherited from the finite element.

[0014] Furthermore, the finite element unit includes a triangular unit, a quadrilateral unit, a tetrahedral unit or a hexahedral unit.

[0015] Furthermore, the finite element elements replaced by a finite number of peridynamic material points are set as dead elements, the stiffness of the dead elements is multiplied by a preset coefficient, the element load associated with the dead elements is set to 0, and the mass and damping of the dead elements are set to 0.

[0016] Furthermore, displacement and stress cloud maps are drawn to visually display the entire process of crack initiation, expansion and penetration.

[0017] In a second aspect, the present invention further provides a dynamic crack growth calculation system based on finite element-peridynamics, comprising:

[0018] The dynamic crack growth calculation system based on finite element and peridynamics includes:

[0019] The meshing unit is configured to: perform finite element meshing on the calculation area to obtain a plurality of finite element units;

[0020] The judgment unit is configured to: calculate the deformation and stress of the finite element unit and judge whether the finite element unit meets the failure condition;

[0021] A material point replacement unit is configured to replace finite element elements that have reached failure conditions with a finite number of peridynamic material points. Cracks initiate between the peridynamic material points, and as the crack propagates, the finite element elements that have reached failure conditions are gradually replaced by the peridynamic material points.

[0022] The calculation unit is configured to calculate the positions and forces of the finite element unit nodes and material points by using a finite element-peridynamics coupling calculation method.

[0023] In a third aspect, the present invention further provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the dynamic crack propagation calculation method based on finite element-peridynamics described in the first aspect.

[0024] In a fourth aspect, the present invention also provides an electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the program, the steps of the dynamic crack propagation calculation method based on finite element-peri-field dynamics described in the first aspect are implemented.

[0025] Compared with the prior art, the present invention has the following beneficial effects:

[0026] 1. The present invention first divides the computational domain into a finite element grid and peridynamic material points. Cracks then initiate between the peridynamic material points. As the crack propagates, finite element elements that reach failure conditions are gradually replaced by peridynamic material points. This method constructs a dynamic conversion method between the finite element grid and peridynamic material points. This avoids the problem of requiring pre-set finite element and peridynamic calculation ranges, and achieves dynamic modeling and efficient calculation of the crack propagation process.

[0027] 2. The present invention avoids the problem that traditional static methods require pre-setting of finite element and peridynamic calculation areas, and are not suitable for simulating difficult-to-predict crack propagation paths and material damage areas, thus achieving a wider range of applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] The drawings constituting a part of the specification of this embodiment are used to provide a further understanding of this embodiment. The schematic embodiments and descriptions of this embodiment are used to explain this embodiment and do not constitute an improper limitation on this embodiment.

[0029] Figure 1 Schematic diagram of the initial finite element network model of Example 1 of the present invention;

[0030] Figure 2 This is a schematic diagram of a model using material points to replace a destruction unit according to Example 1 of the present invention;

[0031] Figure 3 Schematic diagram of dynamic modeling of a crack propagation process unit-material point in Example 1 of the present invention;

[0032] Figure 4 Schematic diagram of the calculation process of Example 1 of the present invention. DETAILED DESCRIPTION

[0033] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0034] It should be noted that the following detailed descriptions are exemplary and are intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which the present application belongs.

[0035] Example 1:

[0036] This paper provides a dynamic crack growth calculation method based on finite element-peridynamics, including:

[0037] Perform finite element meshing on the calculation area to obtain multiple finite element units;

[0038] Calculate the deformation and stress of the finite element unit and determine whether the finite element unit has reached the failure condition;

[0039] Finite element elements that have reached failure conditions are replaced by a limited number of peridynamic material points. Cracks initiate between peridynamic material points, and as the cracks propagate, the finite element elements that have reached failure conditions are gradually replaced by peridynamic material points.

[0040] The finite element-peridynamics coupling calculation method is used to calculate the positions and forces of finite element unit nodes and material points.

[0041] Specifically, the calculation area is divided into finite element grids, and the deformation and force of the unit are calculated using the finite element method. The unit nodes and their internal stresses are solved by interpolation functions, and the macro-strength criterion is used to determine whether the unit has reached the failure condition. For the units that have reached the failure condition, a limited number of peridynamic material points are used to replace them. The physical and mechanical information such as the mass, material parameters, velocity, and displacement of the peridynamic material points are inherited from the finite element unit, and then the finite element-peridynamic coupling model is used to continue solving. The cracks start between the peridynamic material points, and as the cracks propagate, the finite element units that have reached the failure condition are gradually replaced by peridynamic material points. Finally, the finite element-peridynamic coupling method is used to achieve dynamic modeling and efficient calculation of the crack propagation process. By dividing the calculation area into finite element grids and peridynamic material points, and constructing a dynamic conversion method between finite element grids and peridynamic material points, dynamic modeling and efficient calculation of the crack propagation process are achieved. It can be understood that the calculation area is the area on the material to be calculated, that is, the solution object can be concrete, steel, and other materials that can crack or the corresponding models of the materials.

[0042] like Figure 1-3 As shown, this embodiment provides a finite element-peridynamic modeling method for dynamically simulating the crack propagation process, specifically including:

[0043] like Figure 1 As shown in the figure, the initial finite element modeling is as follows: the calculation area is divided into finite element grids, the deformation and force of the unit are calculated using the finite element method, the unit nodes and their internal stresses are solved by the interpolation function, and whether the unit reaches the failure condition is judged according to the macro strength criterion.

[0044] Finite element models can use a variety of unit types according to the research object. For example, for two-dimensional models, triangular units or quadrilateral units can be selected, and for three-dimensional models, tetrahedral units or hexahedral units can be selected.

[0045] There are many types of macro-strength criteria that can be selected according to the research object, such as maximum tensile stress theory, maximum tensile strain theory, Mohr-Coulomb yield criterion, etc.

[0046] like Figure 2 As shown in the figure, finite element unit-peridynamic material point replacement modeling: for the units that reach the failure condition, a finite number of peridynamic material points are used to replace them. The physical and mechanical information such as the mass, material parameters, velocity, and displacement of the peridynamic material points are inherited from the finite element unit, and then the finite element-peridynamic coupling model is used to continue the solution.

[0047] Units that reach the failure condition can be replaced by a finite number of peridynamic material points, but the number of peridynamic material points must be above a certain critical value, that is, the peridynamic model must be able to meet basic computational requirements.

[0048] When there are no elements that meet the failure condition in the calculation area, the finite element method is continued to be used for solution. However, once elements that meet the failure condition appear in the calculation area and are replaced by peridynamic material points, the finite element-peridynamic coupling solution is adopted.

[0049] like Figure 3 As shown in the figure, the finite element unit-peridynamic material point dynamic modeling is as follows: the crack initiates between the peridynamic material points, and as the crack propagates, the finite element units that reach the failure condition are gradually replaced by the peridynamic material points, thereby realizing the construction of the finite element-peridynamic dynamic model during the crack propagation process.

[0050] The finite element unit needs to be judged for strength at each time step, and the replacement of the finite element unit by the peridynamic material point during the dynamic modeling process is irreversible, that is, only the finite element units that reach the failure condition are allowed to be replaced by the peridynamic material point, and the conversion of the peridynamic material point into the peridynamic unit is not allowed.

[0051] like Figure 4 As shown, this embodiment also provides a finite element-peridynamic calculation method for dynamically simulating the crack propagation process, including:

[0052] S1. Divide the calculation area into a finite number of finite element grids and divide the calculation time into a finite number of time steps.

[0053] The calculation area mentioned above refers to the complete calculation area excluding the initial damage, which usually meets the requirements of finite element calculation.

[0054] The finite element mesh refers to a mesh model used for finite element method calculations. For a two-dimensional model, triangular elements or quadrilateral elements can usually be selected, and for a three-dimensional model, tetrahedral elements or hexahedral elements can usually be selected.

[0055] The time step refers to dividing the actual simulation time into multiple small time steps evenly according to the loading process, that is, T = n × Δt, where T refers to the total duration; n refers to the number of time steps; and Δt refers to the time step.

[0056] S2. Based on the initially divided finite element mesh, the finite element method is used to solve the problem and construct the finite element stiffness matrix. The displacement and force of the node are solved according to the boundary conditions, and the shape function is used for interpolation to solve the stress state of the unit node and its internal stress. If the unit node or its internal stress meets the macro strength criterion in a certain time step, the unit is judged to be a damaged unit.

[0057] The finite element method solution refers to the basic finite element equation [K][U]=[P], where [K] is the stiffness matrix; [U] is the node displacement; and [P] is the equivalent node load. By constructing the unit stiffness matrix and forming the overall stiffness matrix, the forced boundary conditions are introduced to solve the finite element equation to obtain the node displacement and load, and the shape function is used for interpolation to solve the stress state of the unit node and any point inside it.

[0058] The macro strength criterion is a theoretical basis for determining whether the stress state of a unit node or any internal point reaches the yield condition or the failure condition. Usually, the maximum tensile stress theory, the maximum tensile strain theory, the Mohr-Coulomb yield criterion and other judgment criteria can be selected.

[0059] The failure unit refers to the unit node or its internal stress meeting the macro strength criterion. For example, according to the maximum tensile stress criterion, when the internal stress of the unit meets σ>σ c , where σ is the element internal stress, σ c If it is the allowable stress of the material when stretched, it is judged that the unit has a tendency to be destroyed and is about to enter the destruction stage.

[0060] S3. For units that have reached the failure condition, a limited number of peridynamic material points are used to replace them. That is, the unit is set as a dead unit and no longer participates in the subsequent calculation process. The physical and mechanical information of the unit is assigned to the corresponding peridynamic material point, and then the finite element-peridynamic coupling model is used to carry out the subsequent calculation process.

[0061] The dead element is a finite element dead element principle, which is obtained by multiplying the element stiffness by a very small coefficient, usually about 1.0×10 -6 , and set the unit load associated with the dead unit to 0, and set all parameters that affect the calculation, such as unit mass and damping, to 0.

[0062] The peridynamics mentioned above refers to replacing the destruction unit with a certain number of peridynamic material points and considering the interaction relationship between material points within a certain range, that is, H = {x j ∈R:||x i -x j ||≤δ}, where x i and x j are the coordinates of the material points numbered i and j, respectively; R is the calculation region; δ is the neighborhood range, thus constructing the basic peridynamic equation based on the idea of ​​nonlocal action:

[0063]

[0064] Where ρ is the material density; is the material point acceleration; is the velocity of the material point; u is the displacement of the material point; f is the interaction force between material points; b is the body density of the material point; t is time.

[0065] As an optional implementation, peridynamics can be expressed using the finite element format, that is, a non-local peridynamic stiffness matrix and boundary conditions in the finite element format are constructed, and the material point displacements and forces are obtained by direct solution.

[0066] As an optional implementation, peridynamics can be solved explicitly or implicitly, that is, based on Newton's second law, the integral equation of non-local action is constructed to solve the acceleration, velocity and displacement of the material point, and then solve the force on the material point.

[0067] The finite element-peridynamics coupling model refers to a finite element-peridynamics coupling calculation method constructed by using any of the above-mentioned peridynamics solution methods to achieve finite element and peridynamics coupling simulation.

[0068] S4. Based on the peridynamic calculation results, determine whether the material point has reached the damage condition. Once the damage value of the material point exceeds the critical value, it is considered that the crack has started.

[0069] The damage condition refers to the situation where the bond between the material points in the peridynamics exceeds a certain critical value, that is, s>s0, where s is the elongation of the bond between the material points and s0 is the critical elongation of the bond between the material points. The bond between the material points breaks. At this time, a scalar function μ is introduced to characterize it, that is,

[0070]

[0071] Among them, μ = 1 means that the bonds between material points are intact, and there is interaction force between material points; μ = 0 means that the bonds between material points are broken, and there is no interaction between material points.

[0072] The critical value is the ratio of the number of broken bonds at the material point to the initial state, and the local damage of the material point is defined as follows:

[0073]

[0074] in, is the local damage value of the material point; V H is the volume of other material points in the neighborhood of the material point. It can be seen that the local damage of the material point is a value ranging from 0 to 1. When , all the bonds interacting with the material point are broken; when When all the bonds interacting with the material points are intact. Local damage represents the formation process of cracks inside the material. It can usually be considered that when When it is greater than a certain value, for example When the crack surface is formed.

[0075] S5. In subsequent time steps, the finite element-peridynamic coupling calculation is used to determine in real time whether the finite element unit has reached the failure condition. The units that have reached the failure condition are gradually replaced with peridynamic material points.

[0076] The stepwise replacement means that in subsequent time steps, steps 2, 3, and 4 are used to sequentially calculate whether any units in the finite element model have reached the failure condition, and the units that have reached the failure condition are gradually replaced with peridynamic material points, thereby realizing the coupled finite element-peridynamic modeling and calculation.

[0077] S6. In subsequent time steps, the material point damage is determined in real time through finite element-peridynamics coupling calculations, and the crack propagation process is described.

[0078] The crack propagation process described above refers to the stress concentration phenomenon at the crack tip. In the subsequent calculation process, the stress state of the unit near the crack tip can be obtained. Through step five, the units that reach the failure condition are gradually replaced with peridynamic material points, and the crack propagation process is described by peridynamic material points, thereby achieving effective simulation of the crack propagation process.

[0079] S7. Through the above steps, a finite element-peridynamic calculation method for dynamic simulation of crack propagation process was constructed. By outputting and saving the calculation results and using a computer to draw cloud maps of displacement, stress, etc., the entire process of crack initiation, propagation and penetration is intuitively displayed.

[0080] Example 2:

[0081] This embodiment proposes a dynamic crack growth calculation system based on finite element-peridynamics, including:

[0082] The meshing unit is configured to: perform finite element meshing on the calculation area to obtain a plurality of finite element units;

[0083] The judgment unit is configured to: calculate the deformation and stress of the finite element unit and judge whether the finite element unit has reached a failure condition;

[0084] A material point replacement unit is configured to replace finite element elements that have reached failure conditions with a finite number of peridynamic material points. Cracks initiate between the peridynamic material points, and as the crack propagates, the finite element elements that have reached failure conditions are gradually replaced by the peridynamic material points.

[0085] The calculation unit is configured to calculate the positions and forces of the finite element unit nodes and material points by using a finite element-peridynamics coupling calculation method.

[0086] The working method of the system is the same as the dynamic crack growth calculation method based on finite element-peridynamics in Example 1, and will not be repeated here.

[0087] Example 3:

[0088] This embodiment provides a computer-readable storage medium having a computer program stored thereon. When the program is executed by a processor, the steps of the dynamic crack growth calculation method based on finite element-peridynamics described in Example 1 are implemented.

[0089] Example 4:

[0090] This embodiment provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, the steps of the dynamic crack propagation calculation method based on finite element-peri-field dynamics described in Example 1 are implemented.

[0091] The above description is merely a preferred embodiment of this embodiment and is not intended to limit this embodiment. Those skilled in the art will readily appreciate that this embodiment may be modified and varied in various ways. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of this embodiment shall be within the scope of protection of this embodiment.

Claims

1. A dynamic crack growth calculation method based on finite element-peridynamics, characterized by: include: Perform finite element meshing on the calculation area to obtain multiple finite element units; Calculate the deformation and stress of the finite element unit and determine whether the finite element unit has reached the failure condition; Finite element elements that have reached failure conditions are replaced by a limited number of peridynamic material points. Cracks initiate between peridynamic material points, and as the cracks propagate, the finite element elements that have reached failure conditions are gradually replaced by peridynamic material points. The finite element elements replaced by a finite number of peridynamic material points are set as dead elements, the stiffness of the dead elements is multiplied by a preset coefficient, the element loads associated with the dead elements are set to 0, and the mass and damping of the dead elements are set to 0; The finite element-peridynamics coupling calculation method is used to calculate the positions and forces of finite element unit nodes and material points.

2. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: The finite element method is used to calculate the deformation and force of the finite element unit, the interpolation function is used to solve the internal stress of the finite element unit node and the finite element unit, and the macro strength criterion is used to determine whether the finite element unit reaches the failure condition.

3. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 2, characterized in that: The macro strength criterion adopts the maximum tensile stress theory, the maximum tensile strain theory or the Mohr-Coulomb yield criterion.

4. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: The mass, material parameters, velocity, and displacement of the peridynamic material points are inherited from the finite element.

5. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: The finite element unit includes a triangular unit, a quadrilateral unit, a tetrahedral unit or a hexahedral unit.

6. The dynamic crack growth calculation method based on finite element-peridynamics according to claim 1, characterized in that: Draw displacement and stress cloud maps to intuitively display the entire process of crack initiation, expansion and penetration.

7. A dynamic crack growth calculation system based on finite element-peridynamics, characterized by: include: The meshing unit is configured to: perform finite element meshing on the calculation area to obtain a plurality of finite element units; The judgment unit is configured to: calculate the deformation and stress of the finite element unit and judge whether the finite element unit has reached a failure condition; A material point replacement unit is configured to replace finite element elements that have reached failure conditions with a finite number of peridynamic material points. Cracks initiate between the peridynamic material points, and as the crack propagates, the finite element elements that have reached failure conditions are gradually replaced by the peridynamic material points. The finite element elements replaced by a finite number of peridynamic material points are set as dead elements, the stiffness of the dead elements is multiplied by a preset coefficient, the element loads associated with the dead elements are set to 0, and the mass and damping of the dead elements are set to 0; The calculation unit is configured to calculate the positions and forces of the finite element unit nodes and material points by using a finite element-peridynamics coupling calculation method.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps of the dynamic crack growth calculation method based on finite element-peridynamics as claimed in any one of claims 1 to 6 are implemented.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the dynamic crack growth calculation method based on finite element-peridynamics are implemented as described in any one of claims 1 to 6.

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