Particle type earth and rockfill dam large deformation calculation method based on node integration

Abstract

The invention discloses a particle earth and rockfill dam large deformation calculation method based on node integration, and relates to the technical field of geotechnical engineering large deformation. The method comprises the following steps: establishing a generalized shape function and a strain displacement matrix of a proportion boundary finite element SBFEM under a Laplacian equation; under an updated Lagrange framework, state variables at Gaussian integral points of the triangular SBFEM unit are projected to nodes through a strain smoothing technology, a momentum equation is converted into a balance equation based on node integration through a virtual work principle and a Green divergence theorem, and the dynamic equation is solved through a central difference method; and when the grid distortion degree reaches a significant level, discarding the current grid and performing re-grid division. All state variables such as stress, displacement, speed and the like are concentrated on nodes of abandoned grids, so that the subsequent grid generation and calculation process can be seamlessly continued. According to the invention, continuous simulation of the whole large deformation process based on the proportional boundary finite element method is realized.

Description

Technical Field

[0001] This invention relates to the field of large deformation technology in geotechnical engineering, and in particular to a particle-based method for calculating large deformation of earth-rock dams based on nodal integrals. Background Technology

[0002] In the field of earth-rock dam engineering, large deformation disasters such as dam landslides, dam slope instability, and earthquake-induced liquefaction-induced dam failures occur frequently. These problems are directly related to the stability, seepage prevention, and safety of downstream areas of the dam structure. Accurate numerical characterization of the mechanical state during its evolution is a core engineering requirement that urgently needs to be addressed. Although traditional small deformation theories (such as the classical finite element method) are widely used in scenarios such as static analysis of conventional earth-rock dams, they have significant limitations under large strain: the severe mesh distortion accompanying large deformations can lead to a sharp decrease in numerical accuracy, or even cause the solution to terminate, making it impossible to accurately capture the complex mechanical behavior of earth-rock dam construction materials (such as riprap and clay core materials).

[0003] To address these challenges, various numerical methods have been developed in the prior art, but all have significant drawbacks. One category is mesh-based methods, such as the arbitrary Lagrange-Euler method, which balances material tracking and mesh stability by dynamically adjusting node positions, and the coupled Euler-Lagrange method, which simulates fluids and solids using fixed Euler meshes and Lagrangian motions respectively. However, both methods rely on mesh topology, requiring complex adaptive remapping in large deformation scenarios where the earth-rock dam body is coupled with the foundation. Another category is particle-based methods, such as the smoothed particle hydrodynamic method, which discretizes the continuum into particles and approximates field variables using kernel functions. However, it suffers from numerical diffusion problems, making it difficult to accurately reflect the stress transfer characteristics within the earth-rock dam body. While the Galerkin meshless method eliminates mesh dependence, it requires complex shape function construction, resulting in low computational efficiency and failing to meet the requirement for accurate stress-strain transfer during large deformation processes in earth-rock dams. In addition, although hybrid methods such as the material point method and the particle finite element method combine the advantages of particle tracking and mesh solving, the state variable mapping of the material point method is prone to errors and is difficult to adapt to the nonlinear constitutive relationship of earth-rock dam construction materials; the stepwise mesh regeneration process of the particle finite element method is cumbersome and it is difficult to balance the accuracy and efficiency of large deformation analysis of earth-rock dams.

[0004] The Scaled Boundary Finite Element Method (SBFEM), as a semi-analytical method, only requires discretizing element boundaries and applying analytical solutions radially. It performs exceptionally well in wave propagation and stress singularity problems and has recently been extended to plate and shell mechanics, fluid-structure interaction, and other fields. However, when SBFEM is extended to medium-finite strain scenarios, it still relies on the initial mesh topology and characterizes deformation stiffness by combining the geometric stiffness matrix and the material stiffness matrix. Under extreme large deformation scenarios such as overall sliding and local collapse of earth-rock dams, mesh distortion leads to unreliable computational results, making it difficult to meet the core requirements of extreme large deformation analysis in earth-rock dam engineering. This deficiency significantly limits its application in earth-rock dam engineering. Therefore, there is an urgent need for a numerical simulation method that can integrate the analytical advantages of SBFEM with the flexibility of the particle method to solve the problems of mesh distortion, computational interruption, and the imbalance between accuracy and efficiency under large deformation in earth-rock dams. Summary of the Invention

[0005] To address the aforementioned technical problems, this application discloses a particle-based method for calculating large deformations of earth-rock dams based on nodal integrals, comprising:

[0006] A scaled boundary finite element discrete model is established, and a geometric model and local coordinate system of triangular elements are constructed. The generalized shape function and element strain-displacement matrix are derived based on the Laplace equation.

[0007] The strain-displacement matrix is ​​projected onto the nodes using strain smoothing technology to construct non-overlapping smoothed elements centered on the nodes, and the smoothed strain of the nodes is calculated.

[0008] Combining the principle of virtual work and Green's divergence theorem, the momentum conservation differential equation is discretized into a nodal integral equilibrium equation containing nodal mass matrices, external force vectors, and internal force vectors.

[0009] The equilibrium equations are discretized in time using the explicit central difference method, and the velocity, displacement, and stress state variables are updated iteratively according to the CFL steady-state time step.

[0010] The mesh quality is evaluated after each time step. When the mesh distortion exceeds the threshold, a high-quality triangular mesh is regenerated based on the state variables stored in the nodes until all calculations are completed.

[0011] Preferably, the coordinate system is in Cartesian coordinates. The following discretization is performed using triangular scaled boundary finite element elements, and a scaled boundary finite element coordinate system is established. The expression for interpolating each element using shape functions is as follows:

[0012]

[0013]

[0014] in For the coordinates of the similarity center, It is a linear unit shape function.

[0015] Preferably, the coordinate transformation relationship in the Cartesian coordinate system is as follows:

[0016]

[0017]

[0018]

[0019] in, The Jacobian matrix for coordinate transformation has the following determinant: .

[0020] Preferably, the derivation of the corresponding unit governing equations through the Laplace equation specifically involves:

[0021]

[0022]

[0023]

[0024]

[0025]

[0026]

[0027] in Let be any scalar field variable. For differential operators, , and The coefficient matrix is ​​the unit.

[0028] Preferably, the step involves introducing variables... and internal flux Composition of variables The Laplace equation is transformed into a first-order ordinary differential equation, specifically:

[0029]

[0030]

[0031]

[0032] in, intermediate variables Regarding coordinates The first-order partial derivative, Let be the coefficient matrix of the structural domain, with the superscript -1 indicating the inverse of the matrix;

[0033] right Eigenvalue decomposition yields the corresponding eigenvalues ​​and eigenvectors:

[0034]

[0035] in, and These are the eigenvector matrix and eigenvalue matrix corresponding to a finite field.

[0036] Preferably, the radial analytic function obtained through eigenvalue decomposition... Transform into:

[0037]

[0038] The generalized shape function of the proportional boundary finite element is obtained as follows:

[0039]

[0040]

[0041] For any point coordinate, which is also in the form of shape function interpolation, the formula is:

[0042]

[0043]

[0044] in, For the global node coordinates on the boundary, using the chain rule, the corresponding coordinate transformation from the reference coordinate system to the physical coordinate system is as follows:

[0045]

[0046]

[0047] Specifically, for the triangular scaled boundary finite element method, its isoparametric transformation matrix and corresponding strain-displacement matrix are:

[0048]

[0049]

[0050] in, for The inverse matrix, This represents the strain-displacement matrix of the i-th node of the corresponding triangular element.

[0051] Preferably, the step of concentrating the strain displacement matrices of all elements around the node onto the node corresponding to the smooth strain element specifically involves:

[0052]

[0053] in, For nodes Smooth strain, For nodes coordinates For nodes The number of all connected nodes, Smooth strain element The strain-displacement matrix is ​​given by the formula:

[0054]

[0055]

[0056] in, Smooth strain element area, For its boundary, The projection length of the normal unit vector onto the x or y direction. For nodes In the corresponding smoothing unit, the first Each node Strain-displacement coefficient in the direction, For nodes In the corresponding smoothing unit, the first Each node Strain displacement coefficient in the direction.

[0057] Preferably, the momentum equation for the continuous medium is expressed as:

[0058]

[0059] in, For material density, For material acceleration, It is the second-order Cauchy stress tensor. For physical loads;

[0060] Using the principle of virtual work and Green's divergence theorem, the momentum equation is transformed into:

[0061]

[0062] After introducing strain-smooth nodal integrals, it is further transformed into

[0063]

[0064]

[0065] in It is an external load. It is an internal nodal force. It's about node quality. or For nodal stress.

[0066] Preferably, the time increment of the time-discrete momentum equation using the central difference method is defined as:

[0067]

[0068] Therefore, the velocity and displacement at the next time step are obtained as follows:

[0069]

[0070]

[0071]

[0072] Considering the time-stable step size, it is:

[0073]

[0074] in It is the stability coefficient. The feature length of the unit. This represents the current wave speed.

[0075] Compared with the prior art, the technical solution of this application has the following technical effects:

[0076] This invention effectively solves the core pain points of numerical methods in simulating large deformation of earth-rock dams by combining the semi-analytical advantages of the proportional boundary finite element method with the flexibility of the particle method. It relies on the initial mesh and is prone to interruption due to distortion. By using nodes to store complete state variables and a mesh regeneration mechanism, it can achieve continuous calculation of extreme large deformation processes without complex mesh re-deformation operations. This ensures the stability and continuity of numerical simulation in scenarios such as dam landslides and dam failures, and provides uninterrupted mechanical data support for engineering risk assessment.

[0077] This invention is based on a generalized shape function derived from the Laplace equation, which can accurately characterize the geometric and mechanical properties of triangular elements. Combined with strain smoothing technology, the element strain is projected to the nodes, which greatly reduces local integration errors. At the same time, the combination of the updated Lagrange framework and the explicit central difference method is used to accurately capture the nonlinear mechanical behavior of earth-rock dam construction materials. It is adapted to the large deformation response law of complex materials such as rockfill and clay core wall material, making the calculation results more consistent with engineering practice.

[0078] The nodal integral equilibrium equations constructed in this invention discretize the momentum conservation equations using the principle of virtual work and Green's divergence theorem, directly performing mechanical calculations with nodes as the core. This simplifies the complex unit integration and matrix assembly process of traditional methods. This efficient computational architecture not only shortens the numerical simulation cycle but also adapts to the computational needs of large-scale earth-rock dam models, meeting the practical requirements for rapid comparison and optimization of multiple schemes in engineering design, and significantly improving design efficiency.

[0079] This invention accurately outputs key mechanical parameters such as nodal displacement, stress, and strain during the large deformation process of earth-rock dams, clearly presenting the evolution path of crack initiation, propagation, and overall dam instability. These data provide precise numerical basis for the anti-sliding stability design, reinforcement of weak points, and disaster prevention of earth-rock dams, helping engineers to predict risk points in advance, optimize engineering plans, and reduce safety hazards and economic losses caused by large deformation disasters.

[0080] The above description is only an overview of the technical solution of this application. In order to better understand the technical means of this application and implement it in accordance with the contents of the specification, and to make the above and other objects, features and advantages of this application more obvious and understandable, the preferred embodiments of this application are described in detail below with reference to the accompanying drawings.

[0081] The above and other objects, advantages and features of this application will become more apparent to those skilled in the art from the following detailed description of specific embodiments in conjunction with the accompanying drawings. Attached Figure Description

[0082] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In all drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn to scale.

[0083] Based on the description of the figures and their corresponding technical content in the document, the titles of the figures are as follows:

[0084] Figure 1 This is a flowchart illustrating a particle-based method for calculating large deformations of earth-rock dams based on nodal integration, according to the present invention.

[0085] Figure 2 This is a simplified schematic diagram of the proportional boundary finite element coordinate transformation according to an embodiment of the present invention;

[0086] Figure 3 This is a simplified schematic diagram of a smooth strain unit according to an embodiment of the present invention;

[0087] Figure 4 This is a comparison chart of the free vibration displacement time history of the cantilever beam calculated by the present invention and other methods. Detailed Implementation

[0088] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. In the following description, specific details such as specific configurations and components are provided merely to help fully understand the embodiments of this application. Therefore, those skilled in the art should understand that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. In addition, for clarity and brevity, descriptions of known functions and structures are omitted in the embodiments.

[0089] It should be understood that the phrase "an embodiment" or "this embodiment" throughout the specification means that a specific feature, structure, or characteristic related to the embodiment is included in at least one embodiment of this application. Therefore, "an embodiment" or "this embodiment" appearing throughout the specification does not necessarily refer to the same embodiment. Furthermore, these specific features, structures, or characteristics can be combined in any suitable manner in one or more embodiments.

[0090] Furthermore, reference numerals and / or letters may be repeated in different examples within this application. Such repetition is for the purpose of simplification and clarity and does not in itself indicate a relationship between the various embodiments and / or settings discussed.

[0091] In this article, the term "and / or" is merely a description of the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can mean: A exists alone, B exists alone, and A and B exist simultaneously. The term " / and" in this article describes another type of relationship between related objects, indicating that two relationships can exist. For example, A / and B can mean: A exists alone, and A and B exist alone. In addition, the character " / " in this article generally indicates that the related objects before and after it are in an "or" relationship.

[0092] In this article, the term "at least one" is merely a description of the relationship between related objects, indicating that there can be three relationships. For example, "at least one of A and B" can mean: A exists alone, A and B exist simultaneously, or B exists alone.

[0093] It should also be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion.

[0094] Example 1

[0095] This embodiment mainly describes a particle-based method for calculating large deformations of earth-rock dams based on nodal integration, such as... Figure 1 As shown, it specifically includes:

[0096] S1. Establish a scaled boundary finite element discrete model. In the Cartesian coordinate system, construct a geometric model of the geotechnical engineering problem using triangular elements as the calculation units, and establish a scaled boundary local coordinate system. Based on the Laplace equation, derive the generalized shape function of the scaled boundary finite element. This shape function is combined with the line element shape function, eigenvector matrix, and eigenvalue matrix to construct the corresponding element strain-displacement matrix.

[0097] S2. Construct nodal integral equilibrium equations based on strain smoothing. Within the updated Lagrange framework, for the Gaussian integration points of the triangular SBFEM element, project the strain-displacement matrix obtained in S1 onto the nodes using strain smoothing technology. Using each node as the center, connect the centroid and edge midpoints of adjacent elements to construct non-overlapping smoothed elements; calculate the smoothed strain of the nodes based on the area and boundary normal vector of the smoothed elements.

[0098] S3. Combining the principle of virtual work and Green's divergence theorem, the differential equation of momentum conservation is discretized into an equilibrium equation based on nodal integrals. This equation includes the nodal mass matrix, the external force vector covering gravity and boundary traction, and the nodal internal force vector calculated by the smooth strain in S2.

[0099] S4. Time Discretization and State Variable Update: The dynamic equilibrium equations obtained in S3 are discretized in time using the explicit central difference method. The total calculation time is divided into several time steps according to the CFL steady-state time step. The state variable iteration is completed for each time step through the logic of velocity update, displacement update and stress update.

[0100] S5. Set the mesh distortion judgment threshold and evaluate the mesh quality after each time step in S4. When the mesh distortion exceeds the threshold, discard the current poor quality mesh and regenerate a high-quality triangular mesh using the complete state variables stored in the nodes, including displacement, velocity, stress, etc., as the data source.

[0101] Furthermore, such as Figure 2As shown, the structure in S1 is discretized using triangular scaled boundary finite element elements, and a scaled boundary finite element coordinate system is established. The expression for interpolating each element using shape functions is as follows:

[0102]

[0103]

[0104] in For the coordinates of the similarity center, It is a linear unit shape function.

[0105] The coordinate transformation relationship in S1 under the Cartesian coordinate system is as follows:

[0106]

[0107]

[0108]

[0109] in, The Jacobian matrix for coordinate transformation has the following determinant: .

[0110] Furthermore, the corresponding element governing equations in S1 are derived from the Laplace equation:

[0111]

[0112]

[0113]

[0114]

[0115]

[0116]

[0117] in Let be any scalar field variable. For differential operators, , and The coefficient matrix is ​​the unit.

[0118] Introducing variables and internal flux Composition of variables The equation can be transformed into a first-order ordinary differential equation as follows:

[0119]

[0120]

[0121]

[0122] in, Indicate intermediate variables Regarding coordinates The first-order partial derivative, Let be the coefficient matrix of the structural domain, and the superscript -1 indicates the inverse of the matrix.

[0123] right Eigenvalue decomposition yields the corresponding eigenvalues ​​and eigenvectors:

[0124]

[0125] in, and These are the eigenvector matrix and eigenvalue matrix corresponding to a finite field.

[0126] Radial analytic function through eigenvalue decomposition Transform into:

[0127]

[0128] The generalized shape function of the proportional boundary finite element is obtained as follows:

[0129]

[0130]

[0131] Furthermore, after obtaining the proportional boundary finite element shape function corresponding to the Laplace equation in S1, the coordinates of any point are also in the form of shape function interpolation, as follows:

[0132]

[0133]

[0134] in, This represents the coordinates of the nodes on the boundary. Using the chain rule, the corresponding coordinate transformation from the reference coordinate system to the physical coordinate system is:

[0135]

[0136]

[0137] Specifically, for the triangular scaled boundary finite element method, its isoparametric transformation matrix and corresponding strain-displacement matrix are:

[0138]

[0139]

[0140] in, for The inverse matrix, This represents the strain-displacement matrix of the i-th node of the corresponding triangular element.

[0141] Furthermore, such as Figure 3 As shown, in S2, the strain displacement matrix of all elements around the node is concentrated onto the node corresponding to the smooth strain element, as follows:

[0142]

[0143] in, Represents a node coordinates Represents nodes The number of all connected nodes, Represents smooth strain element The strain-displacement matrix is ​​expressed as:

[0144]

[0145]

[0146] in, Represents smooth strain element area, Indicate its boundary, It represents the projection length of the normal unit vector onto the x or y direction.

[0147] Furthermore, the momentum equation for the continuous medium in S3 is expressed as:

[0148]

[0149] in, Indicates the density of the material. Indicates the acceleration of the material. It is the second-order Cauchy stress tensor. For physical loads.

[0150] Using the principle of virtual work and Green's divergence theorem, the momentum equation is transformed into:

[0151]

[0152] After introducing strain-smooth nodal integrals, it is further transformed into

[0153]

[0154]

[0155] in It is an external load. It is an internal nodal force. It's about node quality. or For nodal stress.

[0156] Furthermore, in S4, the time increment of the discrete momentum equation using the central difference method is defined as:

[0157]

[0158] Therefore, the velocity and displacement at the next time step are obtained as follows:

[0159]

[0160]

[0161]

[0162] Considering the time-stable step size, it is:

[0163]

[0164] in It is the stability coefficient. Indicates the characteristic length of the unit. This indicates the current wave speed.

[0165] In this embodiment, the generalized shape function and strain-displacement matrix of the Scaled Boundary Finite Element Method (SBFEM) are derived, specifically for the Laplace equation. Under the updated Lagrangian framework, strain smoothing techniques are used to project the state variables at the Gaussian integration points of the triangular SBFEM elements to the nodes, facilitating the solution of the equilibrium equations. When the mesh distortion reaches a significant level, the current mesh is discarded and remeshed. Since all state variables, including stress, displacement, and velocity, are concentrated on the nodes of the discarded mesh, subsequent mesh generation and calculation processes can continue seamlessly, thus achieving continuous simulation of the entire large deformation process.

[0166] Based on Example 1, this example details the implementation and verification of the particle-based earth-rock dam large deformation calculation method based on nodal integrals in a dynamic deformation scenario. By demonstrating consistency with traditional numerical methods within a small to medium deformation range, a cantilever beam free vibration case is selected as the verification under dynamic deformation conditions. This serves as a classic benchmark for verifying the dynamic response capability of the method in geotechnical engineering dynamic analysis. Specifically:

[0167] Given an undamped cantilever beam model with the following geometric parameters: beam length 2m, height 0.2m, one end completely fixed, the other end free; material parameters: elastic modulus 80MPa, Poisson's ratio 0.25, density 1850kg / m³; load conditions: a sudden gravitational acceleration of 10m / s² is applied vertically downwards, causing the beam to vibrate around its static equilibrium position; the deflection at the free end is defined as the vertical displacement relative to the initial stress-free state. Figure 4 As shown, the deflection time history curve of this invention highly overlaps with the results obtained from the updated Lagrange finite element method and the smoothed particle finite element method. All three exhibit a consistent periodic vibration trend, and the vibration amplitude and phase show no significant deviation. To quantitatively verify the accuracy, the root mean square error (RMSE) of the deflection and velocity time histories was calculated: compared with the updated Lagrange finite element method, the RMSE of the deflection is 0.0109 m; compared with the smoothed particle finite element method, the RMSE of the deflection is reduced to 0.0088 m, and the errors are both controlled at a low level.

[0168] To further verify the simulation capability of this invention in the large deformation flow problem of granular materials in geotechnical engineering, the case of aluminum particle column collapse was selected as the core benchmark for further verifying the large deformation numerical method for particle flow, shear band evolution, and sliding distance, as shown in Table 1 below:

[0169] Table 1 Comparison of the results of this method, experiments, and SPH calculations for smooth particle dynamics.

[0170] The model geometry and boundary conditions are as follows: the particle column is a two-dimensional rectangle with a width of 200 mm and a height of 100 mm, placed on a rigid bottom surface (constraining vertical displacement). A normal constraint is applied to the left boundary, and a temporary normal constraint is applied to the initial right boundary to maintain the initial shape. The material parameters are determined based on the experimental aluminum particle mixing characteristics: volumetric weight 20.4 kN / m³, elastic modulus 70 GPa, Poisson's ratio 0.3, cohesion 0 Pa, internal friction angle 30°, and dilatation angle 5°. As shown in the table, at time t = 0.335 s, the error compared to the experimental results is 1.2%, and the relative error compared to the smooth particle dynamics simulation results is 4.2%, further demonstrating the effectiveness and accuracy of the method of this invention.

[0171] The above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. For those skilled in the art, the present invention can have various modifications and variations. Any changes, modifications, substitutions, integrations, and parameter changes made to these embodiments within the spirit and principles of the present invention, without departing from the principles and spirit of the present invention, through conventional substitutions or to achieve the same function, fall within the scope of protection of the present invention.

Claims

1. A particle-based method for calculating large deformation of earth-rock dams based on nodal integrals, characterized in that, include: A scaled boundary finite element discrete model is established, and a geometric model and local coordinate system of triangular elements are constructed. The generalized shape function and element strain-displacement matrix are derived based on the Laplace equation. The strain-displacement matrix is ​​projected onto the nodes using strain smoothing technology to construct non-overlapping smoothing elements centered on the nodes, and the smoothed strain of the nodes is calculated. Combining the principle of virtual work and Green's divergence theorem, the momentum conservation differential equation is discretized into a nodal integral equilibrium equation containing nodal mass matrices, external force vectors, and internal force vectors. The equilibrium equations are discretized in time using the explicit central difference method, and the velocity, displacement, and stress state variables are updated iteratively according to the CFL steady-state time step. The mesh quality is evaluated after each time step. When the mesh distortion exceeds the threshold, a high-quality triangular mesh is regenerated based on the state variables stored in the nodes until all calculations are completed.

2. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 1, characterized in that, In the Cartesian coordinate system The following discretization is performed using triangular scaled boundary finite element elements, and a scaled boundary finite element coordinate system is established. The expression for interpolating each element using shape functions is as follows: in For the coordinates of the similarity center, It is a linear unit shape function.

3. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 2, characterized in that, The coordinate transformation relationship in the Cartesian coordinate system is as follows: in, The Jacobian matrix for coordinate transformation has the following determinant: .

4. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 3, characterized in that, The corresponding element governing equations are derived through the Laplace equation, specifically as follows: in Let be any scalar field variable. For differential operators, , and The coefficient matrix is ​​the unit.

5. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 4, characterized in that, The above is achieved by introducing variables and internal flux Composition of variables The Laplace equation is transformed into a first-order ordinary differential equation, specifically: in, Indicate intermediate variables Regarding coordinates The first-order partial derivative, Let be the coefficient matrix of the structural domain, with the superscript -1 indicating the inverse of the matrix; right Eigenvalue decomposition yields the corresponding eigenvalues ​​and eigenvectors: in, and These are the eigenvector matrix and eigenvalue matrix corresponding to a finite field.

6. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 5, characterized in that, The radial analytical function obtained through eigenvalue decomposition Transform into: The generalized shape function of the proportional boundary finite element is obtained as follows: The coordinates of any point can also be expressed in shape function interpolation form, with the following formula: in, The coordinates of the nodes on the boundary, using the chain rule, correspond to the coordinate transformation from the reference coordinate system to the physical coordinate system as follows: Specifically, for the triangular scaled boundary finite element method, its isoparametric transformation matrix and corresponding strain-displacement matrix are: in, for The inverse matrix, This represents the strain-displacement matrix of the i-th node of the corresponding triangular element.

7. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 1, characterized in that, The step of concentrating the strain displacement matrices of all elements around the node onto the node corresponding to the smooth strain element specifically involves: in, For nodes Smooth strain, Represents a node coordinates Represents nodes The number of all connected nodes, Represents smooth strain element The strain-displacement matrix is ​​given by the formula: in, Represents smooth strain element area, Indicate its boundary, The projection length of the normal unit vector onto the x or y direction. For nodes In the corresponding smoothing unit, the first Each node Strain-displacement coefficient in the direction, For nodes In the corresponding smoothing unit, the first Each node Strain displacement coefficient in the direction.

8. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 1, characterized in that, The momentum equation for the continuous medium is as follows: in, Indicates the density of the material. Indicates the acceleration of the material. It is the second-order Cauchy stress tensor. For physical loads; Using the principle of virtual work and Green's divergence theorem, the momentum equation is transformed into: After introducing nodal integrals for strain smoothing, it is transformed into in It is an external load. It is an internal nodal force. It's about node quality. or For nodal stress.

9. The method for calculating large deformation of particle-based earth-rock dams based on nodal integrals according to claim 8, characterized in that, The time increment of the time-discrete momentum equation using the central difference method is defined as: Therefore, the velocity and displacement at the next time step are obtained as follows: Considering the time-stable step size, it is: in It is the stability coefficient. The feature length of the unit. This represents the current wave speed.

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