A method for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method

Abstract

This invention relates to a method for analyzing large deformations of saturated soil and structures based on a semi-implicit material point method, comprising: decoupling pore pressure and motion variables using an incremental step method; introducing a penalty function for an enhanced frictional contact algorithm; introducing an artificial compressibility parameter into the pressure Poisson equation; employing the B-bar technique to reduce volume lock-in effects; using an affine particle cell method; modifying the calculation formula for axisymmetric problems, considering volume integrals and gradient evaluation in cylindrical coordinates; and conducting verification. The beneficial effects of this invention are: by employing an incremental step method, artificial fluid compressibility, and the B-bar technique, this invention avoids numerical instability caused by coupling problems in traditional methods, ensures the convergence of the simulation process, and avoids non-physical stress oscillations.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology in geotechnical engineering, and more specifically, to a method for analyzing large deformations of saturated soil and structures based on a semi-implicit material point method. Background Technology

[0002] In geotechnical engineering, large deformation problems in saturated soils (such as pile foundation installation and slope instability) are highly nonlinear and complex. Traditional finite element methods (FEM) are prone to computational failure due to mesh distortion when dealing with large deformations. The material point method (MPM), as a meshless method, can effectively handle large deformation problems, but explicit MPM suffers from time step limitations and pore pressure oscillations when dealing with water-mechanical coupling problems. Existing semi-implicit MPM methods alleviate these problems to some extent, but still suffer from insufficient stability and low computational efficiency when dealing with soil-structure interactions, axisymmetric problems, and low-permeability materials.

[0003] Although existing techniques disclose the use of polynomial pressure projection stabilization in FEM analysis to alleviate checkerboard stress patterns caused by volume locking, some non-physical oscillations still exist. Summary of the Invention

[0004] The purpose of this invention is to address the shortcomings of existing technologies by proposing a method for analyzing large deformations of saturated soil and structures based on a semi-implicit material point method.

[0005] Firstly, a method for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method is provided, including:

[0006] Step 1: Decouple pore pressure and motion variables using the incremental step method;

[0007] Step 2: Introduce the penalty function of the enhanced frictional contact algorithm, adjust the velocity correction of the contact node, and use a lumped mass matrix to simplify the calculation of contact force;

[0008] Step 3: Introduce an artificial compressibility parameter into the pressure Poisson equation to alleviate pore pressure oscillations in low-permeability materials, and optimize the selection of the time step by adjusting the artificial compressibility parameter;

[0009] Step 4: Use B-bar technique to reduce volume lock-in effect by modifying the B matrix to calculate particle strain increment and nodal forces in the solid phase;

[0010] Step 5: Using the Affine Particle Cell Method (APIC) scheme, add a local affine description to each particle;

[0011] Step 6: Modify the calculation formula for axisymmetric problems, consider volume integrals and gradient evaluation in cylindrical coordinates, and expand the application scope of MPM in axisymmetric problems.

[0012] Step 7: By analyzing one-dimensional consolidation problems, Cryer sphere problems, and strip foundation problems, verify the effectiveness of semi-implicit MPM in handling soils with different permeability, the accuracy of coupled axisymmetric formulas, and the stability and accuracy of semi-implicit MPM in handling soil-structure interaction problems.

[0013] Preferably, step 1 is divided into a prediction step and a correction step; in the prediction step, the pore pressure gradient term is ignored when solving for the intermediate velocity, and in the correction step, the intermediate velocity is explicitly corrected by implicitly solving the pressure Poisson equation.

[0014] Preferably, in step 2, the penalty function is used to simulate the contact between an impermeable rigid structure and a deformable saturated medium.

[0015] Preferably, in step 6, the modification of the calculation formula includes: modifying the calculation formulas for particle volume, strain, pressure Poisson equation, and nodal internal forces.

[0016] Preferably, in step 6, the application scope of MPM in axisymmetric problems is expanded by introducing additional shape functions and gradient formulas.

[0017] Secondly, a system for analyzing large deformations of saturated soil and structures based on a semi-implicit material point method is provided, for performing any of the methods described in the first aspect, including:

[0018] The decoupling module is used to decouple pore pressure and motion variables using an incremental step method;

[0019] The adjustment module is used to introduce the penalty function of the enhanced friction contact algorithm, adjust the velocity correction of the contact node, and simplify the calculation of contact force by using a lumped mass matrix;

[0020] The mitigation module is used to introduce an artificial compressibility parameter into the pressure Poisson equation to mitigate pore pressure oscillations in low-permeability materials and optimize the selection of the time step by adjusting the artificial compressibility parameter.

[0021] The first modification module is used to reduce the volume lock-in effect by employing B-bar technology and calculating the particle strain increment and nodal internal forces of the solid phase by modifying the B matrix.

[0022] Add a module to add a local affine description for each particle using the affine particle cell method.

[0023] The second modification module is used to modify the calculation formula for axisymmetric problems, taking into account volume integrals and gradient evaluation in cylindrical coordinates, and expanding the application scope of MPM in axisymmetric problems.

[0024] The analysis and verification module is used to verify the effectiveness of the semi-implicit MPM in handling soils with different permeability, the accuracy of the coupled axisymmetric formula, and the stability and accuracy of the semi-implicit MPM in handling soil-structure interaction problems by analyzing one-dimensional consolidation problems, Cryer sphere problems, and strip foundation problems.

[0025] Thirdly, a computer storage medium is provided, wherein a computer program is stored therein; when the computer program is run on a computer, the computer causes the computer to perform any of the methods described in the first aspect.

[0026] Fourthly, an electronic device is provided, comprising:

[0027] Memory, used to store computer programs;

[0028] A processor for executing the computer program to implement the method as described in any of the first aspects.

[0029] The beneficial effects of this invention are:

[0030] 1. This invention employs incremental step method, artificial fluid compressibility, and B-bar technique. The incremental step method effectively decouples pore pressure and motion variables, avoiding numerical instability caused by coupling problems in traditional methods. The introduction of artificial fluid compressibility parameters alleviates pore pressure oscillations in low-permeability materials, ensuring the convergence of the simulation process. The B-bar technique further enhances the stability of the simulation by reducing volume lock-in effects and avoids non-physical stress oscillations.

[0031] 2. This invention employs the APIC scheme and a lumped mass matrix. The APIC scheme enhances the momentum conservation property by adding a local affine description to each particle, reducing energy dissipation and numerical noise, while improving the accuracy of information transfer between particles and the background mesh. The use of the lumped mass matrix simplifies the calculation of contact forces, reduces computational complexity, and significantly improves the computational efficiency of large-scale numerical simulations.

[0032] 3. This invention reduces checkerboard stress patterns and improves the accuracy of simulation results by employing an enhanced frictional contact algorithm and B-bar technology. The enhanced frictional contact algorithm, through the introduction of a penalty function and adjustment of the velocity correction at contact nodes, reduces spurious gaps and numerical fluctuations in the soil-structure contact region, ensuring accurate calculation of contact forces.

[0033] 4. This invention is applicable to a variety of geotechnical engineering problems, such as pile foundation installation and soil-structure interaction. Attached Figure Description

[0034] Figure 1 A schematic diagram of a single-point MPM description for modeling two-phase porous media provided by the present invention;

[0035] Figure 2 The implementation process of the coupled semi-implicit MPM algorithm provided by this invention;

[0036] Figure 3 The geometry and boundary conditions for one-dimensional consolidation testing provided by this invention;

[0037] Figure 4 Pore ​​pressure dissipation provided for this invention: (a) Comparison of MPM simulation and Terzaghi solution; (b) Evolution at point A under different hydraulic conductivity;

[0038] Figure 5 The Cryer sphere problem provided for this invention includes: (a) geometry and boundary conditions; and (b) pore pressure distribution at different times when ν = 0.3.

[0039] Figure 6 This invention provides the evolution of pore pressure at the center of a sphere under different Poisson's ratios.

[0040] Figure 7 A schematic diagram of the geometry and boundary conditions of the strip foundation model provided by this invention;

[0041] Figure 8 The evolution of (a) contact force and (b) pore pressure at point A during the penetration process of the strip foundation provided by the present invention;

[0042] Figure 9 Pore ​​pressure at the end of penetration provided for this invention: (a) Comparison of MPM proposed in this invention and (b) FEM results from Monforte et al. (2023);

[0043] Figure 10 The pore pressure contour lines at the end of penetration provided for the MPM results of this invention: (ab) displacement and deviatoric strain, (cd) effective vertical stress and mean stress;

[0044] Figure 11 Load-displacement curves for large penetration analysis provided by this invention;

[0045] Figure 12 The simulation results of the strip foundation at the end of the large penetration provided by the present invention. Detailed Implementation

[0046] The present invention will be further described below with reference to embodiments. The description of the embodiments below is only for the purpose of helping to understand the present invention. It should be noted that those skilled in the art can make several modifications to the present invention without departing from the principle of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

[0047] Example 1:

[0048] As one example, such as Figure 1 and Figure 2 As shown, this semi-implicit material point method for analyzing large deformations of saturated soil and structures uses a single layer of material points to represent the solid and liquid phases.

[0049] This numerical simulation method for large deformation of saturated soil based on the semi-implicit material point method includes the following steps:

[0050] Step 1, Incremental Fractional Step Method: The incremental fractional step method is used to decouple pore pressure and motion variables, and is divided into a prediction step and a correction step.

[0051] In the prediction step, the pore pressure gradient term is ignored when solving for the intermediate velocity; in the correction step, the intermediate velocity is explicitly corrected by implicitly solving the pressure Poisson equation, as follows:

[0052] (1) According to the mixture theory, saturated soil is homogenized into a continuous medium composed of solid and liquid phases. The partial density of each component... (representing the solid and liquid phases respectively) is determined by their volume fraction and inherent density ρ. α express:

[0053]

[0054] Where n is porosity, ρ m It is the total density of the mixture.

[0055] The Terzaghi effective stress in a saturated porous medium is expressed as:

[0056] σ=σ'-pI (2)

[0057] Where σ is the total stress tensor (positive for tension), σ' is the effective stress tensor, p is the pore pressure (positive for compression), and I is the unit tensor. The partial stresses in the solid and liquid phases are defined as follows:

[0058] σ s =σ'-(1-n)pI, σ l =-npI (3)

[0059] Assuming both the solid particles and the liquid are incompressible, the mass balance equations for the solid and liquid phases can be written as follows:

[0060]

[0061]

[0062] Among them, v α It is the velocity vector of phase α. This represents the time derivative with respect to the α phase. In a single-point two-phase MPM, the motion of the liquid phase's time derivative relative to the solid phase is described as follows:

[0063]

[0064] Combining equations (4a), (4b) and (5), while neglecting the porosity gradient (i.e. The mass balance equation for the mixture can be expressed as:

[0065]

[0066] The motion of a saturated medium is governed by the conservation of momentum in both the solid and liquid phases involved.

[0067]

[0068] Where b is the volume force vector, f d This resistance is due to the interaction between solid and liquid. This invention considers laminar flow, where the resistance satisfies Darcy's law:

[0069]

[0070] Where g is the acceleration due to gravity (g = 9.81 m / s²) 2 ), where k is the hydraulic conductivity.

[0071] The sum of equations (7a) and (7b) gives the momentum balance equation for the mixture:

[0072]

[0073] (2) By discretizing the velocity variable using the forward Euler scheme, the time-discrete form of the momentum equation can be written as:

[0074]

[0075] Where the superscript k indicates time t k The discrete approximation at point Δt is the time increment.

[0076] (3) The incremental step algorithm is used to decouple the pore pressure and motion variables in the solid-liquid coupled system, which facilitates equal-order interpolation of the displacement and pressure fields. According to the incremental step program, the momentum equation is modified by introducing an intermediate velocity. The process is divided into a prediction step and a correction step. The drag term in equation (10b) is formulated based on the intermediate velocity, which makes the time step independent of the permeability. The prediction step is expressed as:

[0077]

[0078] The calibration steps are represented as follows:

[0079]

[0080] Substituting equation (12) into equation (6), the pressure Poisson equation can be derived from the mass balance equation of the mixture, as follows:

[0081]

[0082] (4) This invention employs the Generalized Interpolated Material Point (GIMP) method and B-spline-MPM to spatially discretize the momentum balance equation (Equation 11) in the prediction step as follows:

[0083]

[0084] Among them, v α,i , and Let M represent the nodal velocity, internal force, and external force at node i in phase α, respectively. The nodal matrices related to mass and drag are represented by M. α and Q:

[0085]

[0086] Among them, S iq This is the shape function evaluated at matter point p; the subscripts i and p represent the node and matter point, respectively; N p V is the total number of material points associated with node i. p This represents the volume of each point mass. It's important to note that although the drag term is implicitly calculated based on intermediate velocities, the above momentum equation can be modified by replacing M with a lumped matrix. α The expression Q is converted into an explicit node expression. Explicit node expressions simplify the calculation process and improve computational efficiency while maintaining prediction accuracy.

[0087] The pressure Poisson equation in equation (13) can be discretized in space as follows:

[0088]

[0089] The velocity is updated via an explicit correction procedure, as shown in equation (12), which is spatially discretized as follows:

[0090]

[0091] Where, φ l,p =n p and φ s,p =1-n p These represent the volume fractions of the solid and liquid phases, respectively, and a lumped mass matrix was used in the correction step.

[0092] Step 2, Enhanced Friction Contact Algorithm: A penalty function is introduced to reduce spurious gaps and numerical fluctuations in the soil-structure contact region by adjusting the velocity correction of the contact nodes. A lumped mass matrix is ​​used to simplify the calculation of contact forces and improve computational efficiency.

[0093] The contact constraints on the interface nodes are determined using the impenetrable detection condition as follows:

[0094] (v rigid,i -v α,i )·n r,i >0 (20)

[0095] Where, n r,i It is the outward normal unit vector of the rigid body at node i, determined by its mass gradient, v rigid,i Represents the velocity vector of a rigid body.

[0096] When interface nodes are detected to be in contact, the nodal velocities and accelerations of the deformable body during the prediction and correction steps should be adjusted to avoid mutual penetration. (Node intermediate velocity) Correction and the corresponding contact force applied to phase α The calculation is as follows:

[0097]

[0098] Where ω is the unit tangential vector of the contact surface, and μ is the interfacial friction coefficient. c It is the Coulomb friction coefficient. The first term on the right-hand side of equation (24) represents the normal velocity component constrained by the impenetrability condition and the velocity continuity assumption, while the second term represents the tangential velocity correction controlled by the Coulomb friction criterion. Therefore, for the liquid phase, the tangential correction associated with the friction term should be discarded (i.e., μ = 0). For simplification, a lumped mass matrix is ​​used to allow explicit and nodal corrections to nodal velocities.

[0099] Node velocity correction in the calibration step and contact force The calculation is as follows:

[0100]

[0101] In summary, the nodal contact force applied to the saturated medium is obtained. for:

[0102]

[0103] Step 3, Artificial Fluid Compressibility: An artificial compressibility parameter is introduced into the pressure Poisson equation to effectively mitigate pore pressure oscillations in low-permeability materials. Furthermore, by adjusting the artificial compressibility parameter and optimizing the time step selection, numerical stability is improved.

[0104] Step 3 is as follows:

[0105]

[0106] Where ε is a sufficiently small parameter used to control the degree of artificial compressibility, expressed as:

[0107]

[0108] The degree of artificial compressibility can be adjusted by the parameter β0, with the optimal solution being Δt≈Δt. cr Critical time step Δt cr Represented as:

[0109]

[0110] Among them, L min It is the smallest unit size, and K and G are the bulk modulus and shear modulus of the solid, respectively.

[0111] Step 4, B-bar technique: The B-bar technique is used to reduce the volume lock-in effect. By modifying the B matrix, the particle strain increment and the nodal internal forces of the solid phase are calculated, thereby reducing non-physical stress oscillations.

[0112] The B-bar matrix for plane strain configuration is given by employing reduced integrals of the volume components and full integrals of the partial components:

[0113]

[0114] in, and The gradient of the shape function evaluated at material point p, in the x and y directions, respectively. and It is the gradient of the shape function evaluated at the center of the cell.

[0115] Step 5, Affine Particle Cell Method (APIC) scheme: By adding a local affine description to each particle, the momentum conservation property is enhanced, and energy dissipation and numerical noise are reduced.

[0116] Furthermore, the APIC scheme is adopted to improve the accuracy of information transfer between particles and the background mesh, and to enhance the stability of the calculation, as follows:

[0117] To balance energy conservation and numerical stability, this invention employs the APIC scheme. In APIC, the velocity transfer operator enhances momentum conservation by adding a local affine description to each particle. (Nodal mass m) i The nodal momentum is obtained by combining the particle mass and local affine velocity. The results obtained by mapping to grid nodes are as follows:

[0118]

[0119] The local affine velocity field of a point mass consists of translational and linear components, specifically expressed as follows:

[0120]

[0121] Among them, C α,p V represents the linear transformation of α at point p. α,p This represents the translational velocity, and x is the spatial position vector.

[0122] Affine matrix C α,p The calculation is as follows:

[0123] C α,p =B α,p (D p ) -1 (33)

[0124]

[0125]

[0126] Among them, B α,p Denotes the affine state containing angular momentum, D p This is the inertial-like matrix derived by maintaining affine motion.

[0127] Step 6, Axisymmetric Formula: For axisymmetric problems, the calculation formulas for particle volume, strain, pressure Poisson equations, and nodal forces are modified, considering volume integrals and gradient evaluation in cylindrical coordinates. By introducing additional shape functions and gradient formulas, the application scope of MPM in axisymmetric problems is expanded.

[0128] The axisymmetric formula is characterized by the rz plane (where r and z represent the radial and axial directions, respectively). In the cylindrical coordinate system, the circumferential direction θ is taken as 1 radian. The particle volume within the 1-radian wedge region is calculated as follows:

[0129]

[0130] Where, χ pIt is the particle characteristic function in GIMP. In standard linear MPM or B-spline-MPM, the Dirac delta function is usually chosen. p and Let p represent the area and radial position of particle p.

[0131] In axisymmetric MPMs, an additional shape function T is required. ip Let T represent the cyclic variable. ip =S ip / r p Shape function S ip and gradient Calculated using the plane formula.

[0132] The components of the strain increment Δε in axisymmetric coordinates are calculated using the following formula:

[0133]

[0134] Among them, (v s,i ) r and (v) s,i ) z These are the radial and axial components of the nodal velocity, respectively. and These are the gradients of the shape function in the r and z directions, respectively. Compared to the plane strain condition, an additional circumferential strain increment (Δε) is considered. s,p ) θθ .

[0135] For the pressure Poisson equation, the right-hand side of formula (16) representing volume change should be adjusted by considering circumferential deformation, as shown in the following expression:

[0136]

[0137] Internal forces in the solid and liquid phases, with the addition of circumferential stress (σ). α,p ) θθ Represented as:

[0138]

[0139] The corresponding terms in the correction steps should also be modified by evaluating the contribution of the circumferential pore pressure increment as follows:

[0140]

[0141] Using the B-bar technique, the expression under axisymmetric conditions is adjusted as follows:

[0142]

[0143] Here, variables with the subscript p represent values ​​evaluated at particle positions, while variables with the subscript c represent values ​​evaluated at cell centers.

[0144] Step 7, Application and Results Output: By analyzing one-dimensional consolidation problems, Cryer sphere problems, and strip foundation problems, the effectiveness of semi-implicit MPM in handling soils with different permeability, the accuracy of coupled axisymmetric formulas, and the stability and accuracy of semi-implicit MPM in handling soil-structure interaction problems are verified.

[0145] Example 2:

[0146] Based on Example 1, Example 2 of this application uses 2D plane strain MPM analysis to simulate the classic one-dimensional consolidation problem, verifying the effectiveness of semi-implicit MPM in treating different permeable soils, as detailed below:

[0147] A classic one-dimensional consolidation problem is simulated using 2D plane strain MPM analysis. For example... Figure 3 As shown, a constant vertical stress of 10 kPa is applied to the top of a saturated soil column, 1.0 m high and 0.02 m wide. Drainage is allowed only at the top surface, rolling boundaries are applied on both sides, and the bottom is fully constrained. The computational domain is discretized using linear quadrilateral elements of size 0.02 m × 0.02 m, each containing 2 × 2 material points. The solid phase is treated as linearly elastic, and the model parameters are set as follows: Young's modulus E = 1 × 10⁻⁶. 7 Pa, Poisson's ratio ν = 0.3, soil particle density ρ s =2650kg / m 3 pore fluid density ρ l =1000kg / m 3 Initial porosity n0 = 0.3, hydraulic conductivity k = 1 × 10⁻⁶ -3 m / s. Assume initial pore pressure and effective stress are zero, and neglect gravity. Take a time step of Δt = 1.0 × 10⁻⁶ m / s. -4 s.

[0148] Figure 4 (a) shows the different dimensionless times T (i.e., T = c). v t / H 2 , where c v Here, H is the consolidation coefficient, and H is the column height. The spatial distribution of pore pressure along the column depth is shown. It can be seen that the numerical results of the coupled MPM agree well with Terzaghi's analytical solution. By fixing the time step Δt = 1.0 × 10⁻⁶, the results show the pore pressure distribution. -4 s, simulated a series of consolidation tests with different hydraulic conductivity. Figure 4(b) Simulated pore pressure and corresponding analytical solutions were plotted for a selected material point (i.e., point A at h = 0.5 m). The numerical results are close to the theoretical solutions, although at lower permeability (k = 1.0 × 10⁻⁶ m), the results are relatively close. -6 There may be slight deviations in the time step (m / s), which can be addressed by reducing the time step. When considering the consolidation of low-permeability soils (such as clay), a smaller time step can be used.

[0149] It should be noted that the parts in this embodiment that are the same as or similar to those in Embodiment 1 can be referred to each other, and will not be repeated in this application.

[0150] Example 3:

[0151] Based on Example 1, Example 3 of this application uses an axisymmetric MPM to simulate the Cryer's sphere problem, verifying the accuracy of the coupled axisymmetric formula. Specifically:

[0152] The semi-implicit MPM and simulation results were compared with the theoretical solution to verify the coupled axisymmetric formula. Figure 5 This paper demonstrates how to construct an axisymmetric two-dimensional (2D) model in cylindrical coordinates (r, z, θ), where θ = 1 radian and radius R = 0.5 m. Free drainage is allowed from the outer surface, and impermeable boundaries are assigned along the vertical axis of the sphere. An isotropic pressure P0 = 10 kPa is applied to the surface of the sphere. The background mesh is constructed from 0.05 m × 0.05 m cells, each containing 4 material points. To improve the accuracy of the applied external pressure, material points near the outer surface are uniformly repositioned, allowing for the calculation of particle traction forces based on constant curvature. Initial pore pressure and stress are assumed to be zero. The material properties of the isotropic elastic sphere are: E = 1 × 10⁻⁶. 7 Pa, ρ s =2143kg / m 3 , ρ l =1000kg / m 3 n0 = 0.3, k = 1.0 × 10 -3 m / s. Considering various Poisson ratios, the time increment is 2.0 × 10⁻⁶ m / s. -4 s.

[0153] Figure 6 This shows the change in pore pressure at the center of the sphere with respect to dimensionless time T (i.e., T = c) under different Poisson's ratios. v t / R 2 The changes were observed. The MPM results showed good agreement with the Cryer analysis results. Figure 5 (b) shows the pore pressure distribution within the sphere for ν = 0.3 at T = 0.05, 0.15, and 0.20 (i.e., t = 0.0091 s, 0.0182 s, and 0.0273 s).

[0154] It should be noted that the parts in this embodiment that are the same as or similar to those in Embodiment 1 can be referred to each other, and will not be repeated in this application.

[0155] Example 4:

[0156] Based on Example 1, Example 4 of this application verifies the stability and accuracy of the semi-implicit MPM in handling soil-structure interaction problems through small deformation and large deformation analysis.

[0157] Plane strain strip foundation analysis was performed using the coupled MPM proposed in this invention. Figure 7 This is a schematic diagram of the geometry and boundary conditions of a strip foundation model. The foundation width B = 1m. The width and height of the computational domain are both set to W = H = 4B for small deformation analysis, and extended to W = 12BB and H = 10B for large deformation. The sides are sliding supports, and the bottom is a fixed support. Drainage is allowed on the top surface of the foundation, and the soil on both sides of the foundation is subjected to a vertical uniformly distributed load q = 10kPa. The rigid foundation is vertically pushed into saturated MCC soil at a specified constant speed. Gravity is ignored, the initial pore pressure is set to zero, and the initial effective vertical and horizontal stresses are set to 10kPa. The slope of the unloading-reloading line is κ = 0.01, the slope of the normal compression line is λ = 0.1, the slope of the critical state line is M = 1, Poisson's ratio is ν = 0.3, and the initial preconsolidation pressure is p. co =15kPa, soil particle density ρ s =2650kg / m 3 Fluid density ρ l =1000kg / m 3 The initial porosity n0 = 0.45, and the permeability k = 1.0 × 10⁻⁶. -10 m / s. The B-bar GIMP scheme and artificial compressibility (β0 = 3) were adopted to achieve better modeling stability.

[0158] (1) Small deformation analysis

[0159] The small deformation analysis of the foundation penetration involves vertically displacing the rigid strip foundation to the maximum penetration depth of 0.05m at a velocity of 0.05m / s, with a time step Δt = 2 × 10⁻⁶. -4 s, the interfacial friction coefficient μ=1.

[0160] Figure 8 Showing the base bottom ( Figure 7 The evolution of contact force and pore pressure at point A during penetration shows high agreement with the results obtained by Monforte et al. (2023) using nodal smoothing finite element methods with variable element types and sizes. The pore pressure contour lines at the end of penetration in the MPM results are shown below. Figure 9As shown, the results, along with those from Monforte et al. (2023), demonstrate good agreement. The displacement field, deviatoric strain, and effective stress field at the end of the simulation are shown below. Figure 10 As shown, the MPM results are also very similar to the FEM results given by Monforte et al. (2023). It is worth noting that although the polynomial pressure projection stabilization technique was used in the FEM analysis to alleviate the checkerboard stress mode caused by volume locking, some non-physical oscillations still exist, while these oscillations are avoided in the semi-implicit MPM analysis proposed in this invention.

[0161] (2) Large Deformation Analysis

[0162] The model setup was the same as for the small deformation analysis, except for the extended computational domain. Furthermore, to improve computational efficiency, the penetration velocity was doubled to 0.10 m / s, ultimately extending the penetration depth to 2 m. Parametric analysis was performed to investigate the influence of the interfacial friction coefficient μ on the contact force. Figure 10 Load-displacement curves obtained from coupled B-bar GIMP simulations are presented, showing different interfacial friction coefficients. While the small deformation analysis stabilizes to a finite limit, the large deformation drag increases steadily with increasing penetration depth at an almost constant rate and is insensitive to μ.

[0163] To mitigate contact oscillations and spurious gaps commonly observed during large deformations due to premature contact, an enhanced contact algorithm was employed in the analysis. Figure 11 The contact force curves in the data verify the effectiveness of the enhanced contact algorithm.

[0164] Figure 12 Numerical results for the rigid foundation at the end of penetration are presented, showing a smooth and continuous distribution of displacement and deviatoric strain contour lines. The pore pressure and effective vertical stress field exhibit similar patterns to those found in small deformation analyses, further validating the stability and accuracy of the proposed method in handling water-mechanical coupled soil-structure interaction problems involving large deformations.

[0165] It should be noted that the parts in this embodiment that are the same as or similar to those in Embodiment 1 can be referred to each other, and will not be repeated in this application.

[0166] Example 5:

[0167] Based on Example 1, Example 5 of this application provides a system for analyzing large deformations of saturated soil and structures based on a semi-implicit material point method, including:

[0168] The decoupling module is used to decouple pore pressure and motion variables using an incremental step method;

[0169] The adjustment module is used to introduce the penalty function of the enhanced friction contact algorithm, adjust the velocity correction of the contact node, and simplify the calculation of contact force by using a lumped mass matrix;

[0170] The mitigation module is used to introduce an artificial compressibility parameter into the pressure Poisson equation to mitigate pore pressure oscillations in low-permeability materials and optimize the selection of the time step by adjusting the artificial compressibility parameter.

[0171] The first modification module is used to reduce the volume lock-in effect by employing B-bar technology and calculating the particle strain increment and nodal internal forces of the solid phase by modifying the B matrix.

[0172] Add a module to add a local affine description for each particle using the affine particle cell method.

[0173] The second modification module is used to modify the calculation formula for axisymmetric problems, taking into account volume integrals and gradient evaluation in cylindrical coordinates, and expanding the application scope of MPM in axisymmetric problems.

[0174] The analysis and verification module is used to verify the effectiveness of the semi-implicit MPM in handling soils with different permeability, the accuracy of the coupled axisymmetric formula, and the stability and accuracy of the semi-implicit MPM in handling soil-structure interaction problems by analyzing one-dimensional consolidation problems, Cryer sphere problems, and strip foundation problems.

[0175] It should be noted that the system provided in this embodiment is the system corresponding to the method provided in embodiment 1. Therefore, the parts in this embodiment that are the same as or similar to those in embodiment 1 can be referred to each other, and will not be described again in this application.

Claims

1. A method for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method, characterized in that, include: Step 1: Decouple pore pressure and motion variables using the incremental step method; Step 2: Introduce the penalty function of the enhanced frictional contact algorithm, adjust the velocity correction of the contact node, and use a lumped mass matrix to simplify the calculation of contact force; Step 3: Introduce an artificial compressibility parameter into the pressure Poisson equation to alleviate pore pressure oscillations in low-permeability materials, and optimize the selection of the time step by adjusting the artificial compressibility parameter; Step 4: Use B-bar technique to reduce volume lock-in effect by modifying the B matrix to calculate particle strain increment and nodal forces in the solid phase; Step 5: Using the affine particle cell method, add a local affine description to each particle; Step 6: Modify the calculation formula for axisymmetric problems, consider volume integrals and gradient evaluation in cylindrical coordinates, and expand the application scope of MPM in axisymmetric problems. Step 7: By analyzing one-dimensional consolidation problems, Cryer sphere problems, and strip foundation problems, verify the effectiveness of semi-implicit MPM in handling soils with different permeability, the accuracy of coupled axisymmetric formulas, and the stability and accuracy of semi-implicit MPM in handling soil-structure interaction problems.

2. The method for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method according to claim 1, characterized in that, Step 1 is divided into a prediction step and a correction step. In the prediction step, the pore pressure gradient term is ignored when solving for the intermediate velocity. In the correction step, the intermediate velocity is explicitly corrected by implicitly solving the pressure Poisson equation.

3. The method for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method according to claim 2, characterized in that, In step 2, the penalty function is used to simulate the contact between an impermeable rigid structure and a deformable saturated medium.

4. The method for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method according to claim 3, characterized in that, In step 6, the modification of the calculation formula includes: modifying the calculation formulas for particle volume, strain, pressure Poisson equation, and nodal internal forces.

5. The method for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method according to claim 3, characterized in that, In step 6, the application scope of MPM in axisymmetric problems is expanded by introducing additional shape functions and gradient formulas.

6. A system for analyzing large deformations of saturated soil and structures based on the semi-implicit material point method, characterized in that, For performing the method according to any one of claims 1 to 5, comprising: The decoupling module is used to decouple pore pressure and motion variables using an incremental step method; The adjustment module is used to introduce the penalty function of the enhanced friction contact algorithm, adjust the velocity correction of the contact node, and simplify the calculation of contact force by using a lumped mass matrix; The mitigation module is used to introduce an artificial compressibility parameter into the pressure Poisson equation to mitigate pore pressure oscillations in low-permeability materials and optimize the selection of the time step by adjusting the artificial compressibility parameter. The first modification module is used to reduce the volume lock-in effect by employing B-bar technology and calculating the particle strain increment and nodal internal forces of the solid phase by modifying the B matrix. Add a module to add a local affine description for each particle using the affine particle cell method. The second modification module is used to modify the calculation formula for axisymmetric problems, taking into account volume integrals and gradient evaluation in cylindrical coordinates, and expanding the application scope of MPM in axisymmetric problems. The analysis and verification module is used to verify the effectiveness of the semi-implicit MPM in handling soils with different permeability, the accuracy of the coupled axisymmetric formula, and the stability and accuracy of the semi-implicit MPM in handling soil-structure interaction problems by analyzing one-dimensional consolidation problems, Cryer sphere problems, and strip foundation problems.

7. A computer storage medium, characterized in that, The computer storage medium stores a computer program; when the computer program is run on the computer, it causes the computer to perform the method described in any one of claims 1 to 5.

8. An electronic device, characterized in that, include: Memory, used to store computer programs; A processor for executing the computer program to implement the method as described in any one of claims 1 to 5.

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