A meshless large deformation simulation method of displacement-pore pressure non-uniform support domain interpolation

Abstract

A meshless large deformation simulation method using displacement-pore pressure non-uniform support domain interpolation is disclosed, belonging to the field of numerical simulation technology for civil engineering and hydraulic engineering. The steps are as follows: S1, read in the node coordinates, background mesh information, material information, and load information of the meshless model; S2, generate Gaussian points within the background mesh, calculate the displacement support domain and pore pressure support domain of the Gaussian points, and search for meshless nodes within each support domain; S3, calculate the displacement shape function and pore pressure shape function of the Gaussian points based on the nodes within the displacement and pore pressure support domains, and determine the displacement and pore pressure at each Gaussian point; S4, calculate the matrices required in the meshless governing equations; S5, conduct meshless large deformation analysis based on the shape function matrix and strain matrix. This invention improves simulation accuracy by calculating displacement and pore pressure shape functions through nodes within different support domains, and is of great significance for simulating large deformation of soil weakening and liquefaction in practical engineering.

Description

Technical Field

[0001] This invention belongs to the field of numerical simulation technology in civil engineering and hydraulic engineering, and relates to a meshless large deformation simulation method using displacement-pore pressure non-uniform support domain interpolation. Background Technology

[0002] Soil, as a typical saturated porous medium, may undergo weakening-liquefaction under cyclic shear loading, leading to a decrease in strength and potentially large deformation failure. For example, the 2001 Bhuj earthquake in India caused weakening-liquefaction of the overburden soil in the Chang and Shivlakha dams, resulting in overall dam subsidence and large lateral deformation pointing upstream. The 2008 Wenchuan earthquake triggered widespread liquefaction of foundation soil, causing loss of bearing capacity and leading to the collapse of numerous buildings. The 2023 torrential rains in Shanxi Province caused a large-scale increase in excess pore pressure, resulting in landslides. It is evident that large deformation failure of saturated soil can cause significant property damage. Therefore, accurately locating high porosity areas in saturated soil and reproducing the development process of large deformation failure is of great importance for practical engineering. However, simulating such large deformation is complex and has become a key challenge in current numerical simulation.

[0003] Currently, large deformation simulation of soil mainly employs two approaches: element-based large deformation methods and node-based large deformation analysis methods. The Finite Element Method (FEM), as a typical element-based analysis method, is feature-rich and mature, with wide applications in engineering. However, its solution accuracy is limited by the mesh, and in large deformation analysis, mesh distortion or frequent mesh re-distribution may occur, severely impacting its application in practical engineering large deformation analysis. In contrast, node-based large deformation methods (such as meshless methods, material point methods, and particle finite element methods) do not require mesh topology relationships, giving them a natural advantage in large deformation analysis. Among them, the meshless method based on a global weak form has a solution scheme consistent with the standard finite element scheme and exhibits good stability, and has been successfully applied to large deformation analysis in practical engineering projects such as slope stability and foundation liquefaction.

[0004] In the large deformation analysis of meshless saturated soil, each node requires solving for two degrees of freedom: displacement and pore pressure. The orders used for these two degrees of freedom in discretizing the governing equations are different: the displacement degree of freedom uses shape functions, while the pore pressure degree of freedom uses the first-order partial derivatives of the shape functions. Numerical results show that pore pressure is more sensitive to nodal conditions than displacement. If both displacement and pore pressure are calculated using shape functions from meshless points within the same support domain, it may lead to a decrease in accuracy. For example, Chinese invention patent CN202310964065.7 discloses a method for analyzing large deformations of soil liquefaction based on meshless RBF mapping technology. This method uses simultaneous interpolation of displacement and pore pressure degrees of freedom to reflect the accumulation-dissipation process of pore pressure. However, it exhibits numerical instability in simulations of saturated soils with low permeability coefficients. Chinese invention patent CN202510248540.X discloses a method for analyzing large deformations of saturated soils and structures based on a semi-implicit material point method. This method attempts to alleviate numerical instability in the analysis of large deformations of saturated soils by introducing artificial fluid compressibility and B-bar technology, but its simplified pore pressure calculation model fails to accurately reflect the dissipation process of pore pressure. In practical engineering, it is often necessary to both accurately reflect the dissipation process of pore pressure and ensure stable numerical calculations.

[0005] Therefore, it is necessary to propose a displacement-pore pressure non-uniform support domain interpolation method and provide reasonable ratio suggestions for the corresponding support domain radii. Summary of the Invention

[0006] To address the shortcomings of meshless large deformation methods in simulating saturated soil, this invention provides a meshless large deformation simulation method using displacement-pore pressure non-uniform support domain interpolation. This method addresses the issue of different orders for the two degrees of freedom (displacement and pore pressure) when discretizing the meshless governing equations. It improves simulation accuracy by calculating the displacement and pore pressure shape functions through nodes within different support domains, and provides suggested ranges for the ratio of the displacement support domain radius to the pore pressure support domain radius. This method is of significant importance for simulating large deformations caused by soil weakening and liquefaction in practical engineering.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A meshless large deformation simulation method using displacement-pore pressure non-uniform support domain interpolation, the meshless large deformation simulation method comprising the following steps: S1, reads in the node coordinates, background mesh information, material information, and load information of the meshless model. Specifically: S1.1, Read in the number of nodes in the meshless model, and then read in the orientation coordinates of the nodes in sequence; S1.2, Read in the background grid size L back The background mesh size is the average node spacing in S1.1. 2 times, calculated according to formula (1): (1) In the formula, n To determine the number of unmesh nodes within the solution domain; i , j These represent two distinct nodes; , express i Nodes x direction and y Direction coordinates; , express j Nodes x direction and y Direction coordinates.

[0008] S1.3, based on the background grid size in S1.2 L back Generate a square background mesh that covers the entire solution domain.

[0009] S1.4, then read in the material and load information of the meshless model.

[0010] S2 generates Gaussian points within the background grid based on the background grid information in S1. (See [link]). Figure 2 The displacement support domain and pore pressure support domain of the Gaussian point are calculated separately, and the meshless nodes within each support domain are searched. Specifically: S2.1, both the displacement support domain and the pore pressure support domain of the Gaussian point are circular, with their centers at the coordinates of the Gaussian point ( xx k , yy k The radius is determined by the average spacing between the unmesh nodes in S1. dis ave Displacement support domain coefficient alfs dis Pore ​​pressure support domain coefficient alfs pore Calculation determined; calculated according to formulas (2) and (3): (2) (3) In the formula, Indicates the radius of the displacement support domain; Indicates the radius of the pore pressure support domain; The displacement support domain coefficient alfs dis and pore pressure support domain coefficient alfs pore All are constants. alfsdis The recommended range is 2-3. alfs pore The recommended range is 1.2 to 2.4. alfs pore / alfs dis The recommended value range is 0.6 to 0.8; S2.2, based on the displacement support domain radius in S2.1 and pore pressure support domain radius Search for unmeshable nodes within the displacement support domain and pore pressure support domain of each Gaussian point. Unmeshable nodes that satisfy formula (4) are identified as nodes within the displacement support domain, and unmeshable nodes that satisfy formula (5) are identified as nodes within the pore pressure support domain. Formulas (4) and (5) are as follows: (4) (5) In the formula, ( x i , y i () represents the coordinates of a gridless node; S3, based on the Gaussian point and the nodes within the displacement support domain and pore pressure support domain in S2, calculate the displacement shape function and pore pressure shape function of the Gaussian point respectively, and then determine the displacement and pore pressure at each Gaussian point respectively. Specifically: S3.1, using radial basis functions based on MQ, calculate the Gaussian point displacement shape function and the Gaussian point hole compression shape function; S3.2, using the Gaussian point displacement shape function and Gaussian point pore pressure shape function from S3.1, the displacement vector and pore pressure at each Gaussian point are expressed as follows: (6) (7) In the formula, Represents the displacement vector of the Gaussian point; Indicates the pore pressure at the Gaussian point; m dis and m pore These represent the number of meshless nodes in the Gaussian point displacement support domain and the number of meshless nodes in the Gaussian point pore pressure support domain, respectively. u e This represents a displacement vector that has no mesh nodes within the displacement support domain. p e This represents the pore pressure vector without mesh nodes within the pore pressure support domain. and These are vectors representing the Gaussian point displacement shape function and the Gaussian point hole compression shape function, respectively.

[0011] S4, based on the Gaussian point displacement shape function and Gaussian point hole compression shape function in S3, calculates the matrix required in the meshless governing equations, specifically: Based on the shape functions of Gaussian point displacements and their partial derivatives, the displacement-related shape function matrix N... dis and strain matrix B dis Represented as: (8) (9) Based on the Gaussian point pore compression shape function and its partial derivative, the pore compression-related shape function matrix N pore and strain matrix B pore Represented as: (10) (11) S5, based on the shape function matrix and strain matrix in S4, performs meshless large deformation analysis, specifically: S5.1, based on the shape function matrix and strain matrix in S4, discretizes the meshless control equations, including the equilibrium equations and the continuity equations; S5.2, solve for the displacement and pore pressure of each meshless node, calculate the stress and strain at each Gauss point, and update the stress and strain at the Gauss point through field variable mapping technology, see formula (12), thereby realizing the spatial redistribution of the stress state of saturated soil during large deformation.

[0012] (12) In the formula, Represents the field variable mapping function vector, m dis This indicates the number of meshless nodes in the support domain for Gaussian point displacement; This represents the stress and strain at the Gaussian point before mapping. This represents the stress and strain at the Gaussian point after mapping.

[0013] S5.3, loop through S2 to S5.2 until the solution is completed and the calculation results are output, thus completing the meshless large deformation simulation process of displacement-pore pressure non-uniform support domain interpolation.

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) This invention improves the accuracy reduction problem caused by displacement-pore pressure same-order interpolation, and can better reflect the high pore pressure ratio region of saturated soil, and thus more reasonably capture potential liquefaction zones; (2) The displacement-pore pressure non-uniform support domain shape function calculation process adopted in this invention is similar to that of the uniform support domain shape function, and will not significantly increase the programming complexity and computational burden; (3) This invention is applicable to any Lagrange-Eulerian (ALE) frame and has high robustness in large deformation analysis; In summary, this invention constructs a non-uniform displacement-pore pressure support domain interpolation and calculates the corresponding shape function by employing different displacement and pore pressure support domain coefficients, thereby achieving the discretization of the governing equations for meshless saturated soil. This invention improves upon the accuracy reduction problem caused by displacement-pore pressure interpolation of the same order in large deformation simulations using meshless methods, and more effectively captures the high pore pressure ratio region of saturated soil. Attached Figure Description

[0015] Figure 1 This is a schematic diagram of the main process of the method of the present invention; Figure 2 This is a schematic diagram showing the nodes without mesh and the background mesh. Figure 3 Schematic diagram of a meshless displacement-pore pressure consistent support domain; Figure 4 This is a schematic diagram of a non-uniform support domain for displacement-pore pressure without a mesh. Figure 5 Geometric model of the Mandel problem; Figure 6 For the Mandel problem, a meshless node model is used. Figure 7 for t =0.05s under different operating conditions ( alfs pore / alfs dis Comparison of excess pore pressure distribution in soil (=0.5, 0.8, 1.2); Figure 8 for t Different operating conditions at 0.5s ( alfs pore / alfs dis Comparison of excess pore pressure distribution in soil (=0.5, 0.8, 1.2); Figure 9 for t Results for each working condition at 0.5s ( alfs pore / alfs dis = [0.3-1.3] The average relative deviation from the theoretical solution at intervals of 0.1; Figure 10 This is a typical breakwater geometric model; Figure 11 This is a meshless node model of a typical breakwater. Figure 12 This is the time history curve of seismic wave acceleration; Figure 13 This represents a typical post-earthquake high porosity region of a breakwater (non-uniform support domain interpolation). Figure 14 This represents a typical post-earthquake high porosity region of a breakwater (consistent support domain interpolation). Figure 15 This is a typical post-earthquake deformation diagram of a breakwater without grid nodes (non-uniform support domain interpolation). Figure 16 This is a typical post-earthquake deformation diagram of a breakwater without grid nodes (consistently supports domain interpolation). In the diagram: L represents the side length of the geometric model of the Mandel problem; q represents the magnitude of the uniformly distributed load. Detailed Implementation

[0016] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but the scope of protection of the present invention is not limited thereto.

[0017] See Figure 1 A meshless large deformation simulation method using displacement-pore pressure non-uniform support domain interpolation includes the following steps: S1, read in the node position coordinates, background mesh information, material information, and load information of the meshless model; S2 generates Gaussian points within the background grid based on the background grid information in S1. (See [link]). Figure 2 The displacement support domain and pore pressure support domain of the Gaussian point are calculated respectively, and the meshless nodes in each support domain are searched. S3 is based on the Gaussian points in S2 and the meshless nodes within the displacement support domain and pore pressure support domain. Figure 3 This diagram illustrates the displacement-pore pressure consistency support domain interpolation. Figure 4 This example illustrates non-uniform support domain interpolation of displacement and pore pressure. In this instance, uniform support domain interpolation of displacement and pore pressure is used to calculate the displacement shape function and pore pressure shape function of Gaussian point, respectively. S4, based on the Gaussian point displacement shape function and Gaussian point hole compression shape function in S3, calculates the matrix required for the meshless governing equations; S5, based on the matrix calculated in S4, performs meshless large deformation analysis using displacement-pore pressure non-uniform support domain interpolation.

[0018] Example 1: The classic Mandel problem; This case study employs the classic Mandel problem, aiming to discuss the impact on simulation accuracy by comparing simulation results with theoretical solutions, and thus provide suggestions on the range of values ​​for this ratio.

[0019] S1, reads in the node coordinates, background mesh information, material information, and load information of the meshless model; specifically: S1.1, refer to Figure 5The geometric model of the Mandel problem is given, where L represents the side length of the geometric model (L=2m) and q represents the magnitude of the uniformly distributed load. The location information of the meshless nodes in the Mandel problem is then read in. Figure 6 A meshless node model of the Mandel problem; S1.2, Read in the background grid size L back The background mesh size is the average node spacing in S1.1. The average node spacing in this scheme is twice that of the given value, calculated according to formula (1). The depth is 0.2m, and the background grid size is 0.4m; (1) In the formula, n To determine the number of unmesh nodes within the solution domain; i , j These represent two distinct nodes; , express i Nodes x direction and y Direction coordinates; , express j Nodes x direction and y Direction coordinates.

[0020] S1.3, based on the background grid size in S1.2 L back Generate a square background mesh that covers the entire solution domain.

[0021] S1.4, then read in the material and load information of the meshless model. The saturated soil adopts a linear elastic model, and the elastic modulus is... E =100MPa, Poisson's ratio v =0.25, porosity n =0.363, soil particle bulk modulus is k s =1×10 6 GPa, fluid bulk modulus is k f =2.2GPa, permeability coefficient k =1×10 -4 m / s, apply a vertical uniformly distributed load to the top of the model. q =1kPa, and then the soil consolidates along the horizontal direction. The simulation time step is 0.001s, with a total of 500 steps and a total calculation time of 0.5s.

[0022] S2, based on the background mesh information in S1, generates 2×2 Gaussian points within each background mesh, then calculates the displacement support domain and pore pressure support domain for each Gaussian point, and searches for the number of mesh-free nodes within the entire displacement support domain and pore pressure support domain; specifically: S2.1, both the displacement support domain and the pore pressure support domain of the Gaussian point are circular, with their centers at the coordinates of the Gaussian point ( xx k , yy k The radius is determined by the average spacing between the unmesh nodes in S1. dis ave Displacement support domain coefficient alfs dis Pore ​​pressure support domain coefficient alfs pore The calculation is determined according to formulas (2) and (3): (2) (3) In the formula, Indicates the radius of the displacement support domain; Indicates the radius of the pore pressure support domain; In this simulation, the radius of the displacement support domain is three times the average node spacing, i.e. alfs dis =3, the pore pressure support domain range is set to 0.5 to 1.5 times the displacement support domain. alfs pore / alfs dis =[0.5, 1.5], with a spacing of 0.1, for a total of 11 groups; S2.2, based on S2.1 and Search for unmeshable nodes within the displacement support domain and pore pressure support domain of each Gaussian point. Unmeshable nodes that satisfy formula (4) are identified as nodes within the displacement support domain, and unmeshable nodes that satisfy formula (5) are identified as nodes within the pore pressure support domain. Formulas (4) and (5) are as follows: (4) (5) In the formula, ( x i , y i () represents the coordinates of a gridless node; S3, based on the Gaussian point and the nodes within the displacement support domain and pore pressure support domain in S2, calculate the displacement shape function and pore pressure shape function of the Gaussian point respectively, and then determine the displacement and pore pressure at each Gaussian point respectively. Specifically: S3.1, using the radial basis function based on MQ, calculate the Gaussian point displacement shape function and the Gaussian point hole compression shape function, and calculate the constants to be determined according to the following two equations. a i , b j The shape functions and their partial derivatives can be solved: (13) (14) In the formula, , These are the radial basis matrix and the linear additional basis matrix, respectively; k Denotes an additional linear basis, with values ​​of k =3; m represents the radial basis dimension, which is the number of meshless nodes within a Gaussian point displacement support domain and a pore pressure support domain of equal size.

[0023] S3.2, using the shape function from S3.1, the displacement vector and pore pressure at each Gaussian point are expressed as follows: (6) (7) In the formula, Represents the displacement vector of the Gaussian point; Indicates the pore pressure at the Gaussian point; m dis and m pore These represent the number of meshless nodes in the Gaussian point displacement support domain and the number of meshless nodes in the Gaussian point pore pressure support domain, respectively. u e This represents a displacement vector that has no mesh nodes within the displacement support domain. p e This represents the pore pressure vector without mesh nodes within the pore pressure support domain. and These represent the Gaussian point displacement shape function and the Gaussian point hole compression shape function vectors, respectively.

[0024] S4, based on the Gaussian point displacement shape function and Gaussian point hole compression shape function in S3, calculates the matrix required in the meshless governing equations; specifically: Based on the shape functions of Gaussian point displacements and their partial derivatives, the displacement-related shape function matrix N... dis and strain matrix B dis Represented as: (8) (9) Based on the Gaussian point pore compression shape function and its partial derivative, the pore compression-related shape function matrix N pore and strain matrix Bpore Represented as: (10) (11) S5, based on the shape function matrix and strain matrix in S4, performs meshless large deformation analysis; specifically: S5.1, based on the shape function matrix and strain matrix in S4, discretizes the meshless control equations, including the equilibrium equations and the continuity equations, as shown in the following two equations; (15) (16) In the formula, , , , , These represent the acceleration, velocity, displacement, rate of change of pore pressure, and pore pressure without mesh nodes, respectively. , These represent the displacement-related load vector and the pore pressure-related load vector, respectively. M , C , K , Q , S , H These represent the mass matrix, damping matrix, stiffness matrix, fluid-structure interaction matrix, seepage matrix, and compression matrix, respectively.

[0025] S5.2, solve for the displacement and pore pressure of each meshless node, calculate the stress and strain at each Gauss point, and update the stress and strain at the Gauss point through field variable mapping technology, see formula (12), thereby realizing the spatial redistribution of the stress state of saturated soil during large deformation.

[0026] (12) In the formula, Represents the field variable mapping function vector, m dis This indicates the number of meshless nodes in the support domain for Gaussian point displacement; This represents the stress and strain at the Gaussian point before mapping. This represents the stress and strain at the Gaussian point after mapping.

[0027] S5.3, loop through S2 to S5.2 until the solution is completed and the calculation results are output, thus completing the meshless large deformation simulation process of displacement-pore pressure non-uniform support domain interpolation.

[0028] Summarize and compare the simulation results with the theoretical solutions. Figure 7 and Figure 8 Three typical different working conditions are given respectively. alfs pore / alfs dis =0.5,0.8,1.2) at typical times ( t =0.05s and t Comparison of the distribution of normalized excess pore water pressure in soil samples with pore water pressure at 0.5 s (s = 0.5 s). Figure 9 Given different working conditions, t Simulation results for each working condition at 0.5s ( alfs pore / alfs dis = [0.3-1.3] Interval 0.1) Average relative deviation from the theoretical solution.

[0029] The results show that when alfs pore / alfs dis When the value is between 0.6 and 0.8, the simulation results agree relatively well with the theoretical solution; while when... alfs pore / alfs dis When the relative deviation exceeds 0.8 or is less than 0.6, the relative deviation increases significantly. This is because pore pressure is more sensitive to interpolation than displacement; as the support domain ratio increases, local pore pressure may be averaged out. Conversely, when the support domain ratio is too small, some Gaussian points lack sufficient nodes within the support domain, affecting the accuracy of the shape function solution. This leads to the conclusion... alfs pore / alfs dis The recommended value range for the ratio is 0.6 to 0.8.

[0030] Example 2: The problem of liquefaction deformation of a breakwater at a nuclear power plant; S1, reads in the node coordinates, background mesh information, material information, and load information of the meshless model; specifically: S1.1, Figure 10 This is a typical breakwater geometric model. The upper part of the breakwater consists of a 15.2m thick riprap layer, while the lower part contains an 18.5m thick layer of liquefiable sand and an 8.5m thick intermediate zone. The water level is 6m below the top of the riprap layer, and the slope extends 35m outwards on both sides of the toe. To simulate the large deformation phenomena of the sand and top riprap layers, meshless large deformation simulation is used in the riprap and sand layers, while the intermediate zone is still simulated using the finite element method. The meshless node model is calculated as follows: Figure 11 As shown; S1.2, Read in the background grid size L back The background mesh size is the average node spacing in S1.1. The average node spacing in this scheme is twice that of the given value, calculated according to formula (1). The size is 2m, and the background grid size is 4m; S1.3, based on the background grid size in S1.2 L back Generate a square background mesh that covers the entire solution domain.

[0031] S1.4, a generalized elastoplastic model was used to simulate the mechanical properties of the soil in each zone. The soil parameters for each zone are shown in Tables 1, 2, and 3. The buoyant unit weight and permeability coefficient of the underwater soil were used. k =5×10 -5 m / s, dry unit weight for soil above water. Seismic motion was input using the wave method, see [link / reference]. Figure 12 .

[0032] Table 1. Generalized elastoplastic parameters of soil in the rockfill area

[0033] Table 2. Generalized elastoplastic parameters of soil in sandy areas

[0034] Table 3. Generalized elastoplastic parameters of soil in the intermediate differentiation zone

[0035] S2, based on the background mesh information in S1, generates 2×2 Gaussian points within each background mesh, then calculates the displacement support domain and pore pressure support domain for each Gaussian point, and searches for the number of mesh-free nodes within the entire displacement support domain and pore pressure support domain; specifically: S2.1, both the displacement support domain and the pore pressure support domain of the Gaussian point are circular, with their centers at the coordinates of the Gaussian point ( xx k , yy k The radius is determined by the average spacing between the unmesh nodes in S1. dis ave Displacement support domain coefficient alfs dis Pore ​​pressure support domain coefficient alfs pore The calculations are determined according to formulas (2) and (3). In this simulation, non-uniform support domain interpolation is used respectively. alfs pore / alfs dis The liquefaction deformation and high porosity region were calculated using two methods: =0.8 and consistent support domain interpolation. S2.2, based on S2.1 and Search for unmeshable nodes within the displacement support domain and pore pressure support domain of each Gaussian point. Unmeshable nodes that satisfy formula (4) are identified as nodes within the displacement support domain, and unmeshable nodes that satisfy formula (5) are identified as nodes within the pore pressure support domain.

[0036] S3, based on the Gaussian point and the nodes within the displacement support domain and pore pressure support domain in S2, calculate the displacement shape function and pore pressure shape function of the Gaussian point respectively, and then determine the displacement and pore pressure at each Gaussian point respectively. Specifically: S3.1, using the radial basis function based on MQ, calculate the Gaussian point displacement shape function and the Gaussian point hole compression shape function, and calculate the constants to be determined according to the following two equations. a i , b j The shape function and its partial derivatives can be solved, as shown in equations (13) and (14). S3.2, using the shape function in S3.1, the displacement vector and pore pressure at each Gaussian point are shown in equations (6) and (7). S4, based on the Gaussian point displacement shape function and Gaussian point hole compression shape function in S3, calculates the matrix required in the meshless governing equations; displacement-related shape function matrix N. dis and strain matrix B dis See equations (8) and (9); based on the Gaussian point pore compression shape function and its partial derivative, the pore compression related shape function matrix N pore and strain matrix B pore See equations (10) and (11).

[0037] S5, based on the shape function matrix and strain matrix in S4, performs meshless large deformation analysis; specifically: S5.1, based on the shape function matrix and strain matrix in S4, discretize the meshless control equations, including the equilibrium equation and the continuity equation, as shown in equations (15) and (16). S5.2, solve for the displacement and pore pressure of each meshless node, calculate the stress and strain at each Gauss point, and update the stress and strain at the Gauss point through field variable mapping technology, see formula (12), thereby realizing the spatial redistribution of the stress state of saturated soil during large deformation.

[0038] S5.3, loop through S2 to S5.2 until the solution is completed and the calculation results are output, thus completing the meshless large deformation simulation process of displacement-pore pressure non-uniform support domain interpolation.

[0039] The pore pressure ratio distribution and nodal deformation diagrams after the earthquake were compiled using the two methods. Figure 13 and Figure 14The distribution of high porosity ratio areas after earthquakes is presented using two different methods. Since porosity is more sensitive to interpolation than displacement, using a smaller porosity support domain in non-uniform support domain interpolation helps to capture local high porosity ratio areas, thus better reflecting local soil liquefaction phenomena. Figure 15 and Figure 16 Deformation diagrams of the breakwater without mesh nodes after the earthquake are presented for both methods. It can be seen that as the soil weakens and liquefies, both methods simulate the deformation patterns of breakwater top subsidence and slope toe lateral displacement. Due to the more pronounced local high pore pressure ratio in non-uniform interpolation, the overall deformation of the breakwater is greater than that in uniform support domain interpolation.

[0040] The above embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A meshless large deformation simulation method using displacement-pore pressure non-uniform support domain interpolation, characterized in that, The meshless large deformation simulation method includes the following steps: S1, read in the node position coordinates, background mesh information, material information, and load information of the meshless model; S2. Based on the background mesh information in S1, Gaussian points are generated within the background mesh. The displacement support domain and pore pressure support domain of the Gaussian points are calculated respectively, and meshless nodes within each support domain are searched. S3. Based on the Gaussian point and the nodes in the displacement support domain and the pore pressure support domain in S2, calculate the displacement shape function and the pore pressure shape function of the Gaussian point respectively, and then determine the displacement and pore pressure at each Gaussian point respectively. S4, based on the Gaussian point displacement shape function and Gaussian point hole compression shape function in S3, calculate the matrix required in the meshless governing equations; S5, based on the shape function matrix and strain matrix in S4, conducts meshless large deformation analysis.

2. The meshless large deformation simulation method for displacement-pore pressure non-uniform support domain interpolation according to claim 1, characterized in that, Specifically, S1 refers to: S1.1, Read in the number of nodes in the meshless model, and then read in the orientation coordinates of the nodes in sequence; S1.2, Read in the background grid size L back The background mesh size is the average node spacing in S1.

1. 2 times, calculated according to formula (1): （1） In the formula, n To determine the number of unmesh nodes within the solution domain; i , j These represent two distinct nodes; , express i Nodes x direction and y Direction coordinates; , express j Nodes x direction and y Direction coordinates; S1.3, based on the background grid size in S1.2 L back Generate a square background mesh that covers the entire solution domain. S1.4, then read in the material and load information of the meshless model.

3. The meshless large deformation simulation method for displacement-pore pressure non-uniform support domain interpolation according to claim 2, characterized in that, Specifically, S2 is: S2.1, both the displacement support domain and the pore pressure support domain of the Gaussian point are circular, with their centers at the coordinates of the Gaussian point ( xx k , yy k The radius is determined by the average spacing between the unmesh nodes in S1. dis ave Displacement support domain coefficient alfs dis Pore ​​pressure support domain coefficient alfs pore Calculation determined; calculated according to formulas (2) and (3): （2） （3） In the formula, Indicates the radius of the displacement support domain; Represents the radius of the pore pressure support domain; where, the displacement support domain coefficient alfs dis and pore pressure support domain coefficient alfs pore All are constants; S2.2, based on the displacement support domain radius in S2.1 and pore pressure support domain radius Search for unmeshable nodes within the displacement support domain and pore pressure support domain of each Gaussian point. Unmeshable nodes that satisfy formula (4) are identified as nodes within the displacement support domain, and unmeshable nodes that satisfy formula (5) are identified as nodes within the pore pressure support domain. Formulas (4) and (5) are as follows: （4） （5） In the formula, ( x i , y i () represents the coordinates of a gridless node.

4. The meshless large deformation simulation method for displacement-pore pressure non-uniform support domain interpolation according to claim 3, characterized in that, In S2.1, the displacement support domain coefficient alfs dis The range is 2~3, and the pore pressure support domain coefficient is... alfs pore The range is 1.2 to 2.4; alfs pore / alfs dis The value range is 0.6 to 0.

8.

5. The meshless large deformation simulation method for displacement-pore pressure non-uniform support domain interpolation according to claim 3, characterized in that, Specifically, S3 refers to: S3.1, using radial basis functions based on MQ, calculate the Gaussian point displacement shape function and the Gaussian point hole compression shape function; S3.2, using the Gaussian point displacement shape function and Gaussian point pore pressure shape function from S3.1, the displacement vector and pore pressure at each Gaussian point are expressed as follows: （6） （7） In the formula, Represents the displacement vector of the Gaussian point; Indicates the pore pressure at the Gaussian point; m dis and m pore These represent the number of meshless nodes in the Gaussian point displacement support domain and the number of meshless nodes in the Gaussian point pore pressure support domain, respectively. u e This represents a displacement vector that has no mesh nodes within the displacement support domain. p e This represents the pore pressure vector without mesh nodes within the pore pressure support domain. and These are vectors representing the Gaussian point displacement shape function and the Gaussian point hole compression shape function, respectively.

6. The meshless large deformation simulation method for displacement-pore pressure non-uniform support domain interpolation according to claim 5, characterized in that, Specifically, S4 is: Based on the shape functions of Gaussian point displacements and their partial derivatives, the displacement-related shape function matrix N... dis and strain matrix B dis Represented as: （8） （9） Based on the Gaussian point pore compression shape function and its partial derivative, the pore compression-related shape function matrix N pore and strain matrix B pore Represented as: （10） （11）。 7. The meshless large deformation simulation method for displacement-pore pressure non-uniform support domain interpolation according to claim 6, characterized in that, Specifically, S5 is: S5.1, based on the shape function matrix and strain matrix in S4, discretizes the meshless control equations, including the equilibrium equations and the continuity equations; S5.2 solves the displacement and pore pressure of each meshless node, calculates the stress and strain at each Gaussian point, and updates the stress and strain at the Gaussian point through field variable mapping technology to realize the spatial redistribution of the stress state of saturated soil during large deformation. S5.3, loop through S2 to S5.2 until the solution is completed and the calculation results are output, thus completing the meshless large deformation simulation process of displacement-pore pressure non-uniform support domain interpolation.

8. The meshless large deformation simulation method for displacement-pore pressure non-uniform support domain interpolation according to claim 7, characterized in that, In S5.2, formula (12) is used to update the stress and strain at the Gaussian point; （12） In the formula, Represents the field variable mapping function vector, m dis This indicates the number of meshless nodes in the support domain for Gaussian point displacement; This represents the stress and strain at the Gaussian point before mapping. This represents the stress and strain at the Gaussian point after mapping.

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