Saturated porous media dynamic analysis method, device, storage medium and electronic equipment

Abstract

The present invention discloses a method, device, storage medium and electronic device for large deformation dynamic analysis of saturated porous media, belonging to the field of computer numerical simulation technology. The method comprises: obtaining geometric parameters and boundary conditions of the dynamic analysis problem to be solved in the saturated porous medium; establishing a two-dimensional numerical analysis model based on the geometric parameters and boundary conditions; bringing the two-dimensional numerical analysis model into the framework of the mathematical programming particle finite element method to solve and obtain the dynamic analysis conclusion of the saturated porous medium; wherein, the dynamic analysis demonstration examples in the saturated porous medium include debris flow, dam destruction and particle column collapse. The device, storage medium and electronic device can be used to implement the method. It is of great significance for the accurate and rapid solution of large deformation dynamic problems in saturated porous media.

Description

Technical Field

[0001] The present invention relates to the technical field of computer numerical simulation, and in particular to a saturated porous medium dynamic analysis method, device, storage medium and electronic equipment. Background Art

[0002] Landslides, debris flows, and other engineering disasters caused by the dynamic failure of saturated soils can threaten transportation routes and the safety of people and property along these routes, thereby impacting local economic development and the safety of people. Accurately and rapidly assessing the potential for dynamic failure in these saturated soils is crucial for preventing these engineering disasters. While the traditional finite element method (FEM) plays a crucial role in geotechnical engineering analysis and design, the standard Lagrangian finite element method (FEM) remains limited in its ability to address geotechnical engineering problems involving large deformations and free-form surface evolution, such as landslides and debris flows. Summary of the Invention

[0003] In view of this, the present invention provides a saturated porous media dynamic analysis method, device, storage medium and electronic equipment, which are of great significance for the accurate and rapid solution of saturated porous media dynamic problems, and are therefore more suitable for practical use.

[0004] In order to achieve the first objective above, the technical solution of the saturated porous media dynamic analysis method provided by the present invention is as follows:

[0005] The saturated porous medium dynamic analysis method provided by the present invention comprises the following steps:

[0006] Obtain the geometric parameters and boundary conditions of the dynamic analysis problem to be solved in saturated porous media;

[0007] Establishing a two-dimensional numerical analysis model according to the geometric parameters and boundary conditions;

[0008] Bringing the two-dimensional numerical analysis model into the framework of particle finite element method to solve and obtain the dynamic analysis results of the saturated porous medium;

[0009] The example problems of dynamic analysis in saturated porous media include one or more of debris flow, dam destruction and particle column collapse.

[0010] The saturated porous media dynamic analysis method provided by the present invention can also be further implemented by adopting the following technical measures.

[0011] Preferably, the saturated porous medium dynamic analysis method further comprises the following steps:

[0012] Based on the UP model for dynamic analysis of saturated porous media, the governing equations of saturated porous media are given. The standard θ-method is introduced into the governing equations to perform time discretization on the effective stress σ', velocity v, and pressure p in the UP model, and a minimum-maximum problem equivalent to the time-discrete governing equations for dynamic analysis in saturated porous media is derived.

[0013] Based on the hybrid triangular element, the minimum-maximum problem of the time-discrete control equation for dynamic analysis in saturated porous media obtained in step 1 is transformed into a standard second-order cone programming problem.

[0014] Preferably, the solid-state static elastic Hellinger-Reisner functional is expressed by means of a minimum-maximum optimization method based on the Hellinger-Reisner variational principle of static elasticity. The minimum-maximum optimization method is to minimize the displacement in the objective function and maximize the stress in the objective function as follows:

[0015] For statically elastic solids, the Hellinger-Reissner functional is expressed as:

[0016]

[0017] Where σ is the Cauchy stress, u is the displacement, b is the body force, and t is the surface force. is the elastic compliance matrix, Ω and Γ are the integration domain and boundary, respectively. The operator S for the plane strain case takes the following form:

[0018]

[0019] The minimum-maximum problem form of this functional is:

[0020]

[0021] Among them, based on the Hellinger-Reissner variational principle in elastic mechanics, the generalized HR variational principle has been developed to analyze elastic-plastic problems, elastic-viscoplastic problems and quasi-static porous elastic-plastic problems.

[0022] Preferably, based on the UP model of saturated porous media dynamic analysis, the control equation of saturated porous media is given, and relevant data are introduced into the control equation to derive the minimum-maximum problem equivalent to the time-discrete control equation of dynamic analysis in saturated porous media:

[0023] Using the up model for dynamic analysis of saturated porous media, in the two-dimensional case, the governing equations for a saturated porous medium with a region Ω and a boundary Γ are as follows:

[0024] Momentum balance equation for a mixture

[0025]

[0026] Darcy's Law

[0027]

[0028] Pore ​​fluid mass balance equation

[0029]

[0030] Strain decomposition and stress-strain relationship

[0031] ε=ε e +ε p (7)

[0032] here

[0033]

[0034] In the above formula: ρ is the density of the mixture; u is the solid skeleton displacement; ρ f is the fluid density; σ' is the effective stress acting on the solid skeleton; p is the pore water pressure when the tensile pore water pressure is positive; b is the mixture body force; b f is the fluid body force; γ f is the fluid density; k is the Darcy permeability coefficient; w is the velocity of the pore fluid relative to the solid skeleton; ε is the strain vector, defined as ε=Su, given by ε e (elastic strain) and ε p (plastic strain) composition; λ is the plasticity multiplier, λ ≥ 0; F is the yield function, F ≤ 0; G is the plastic potential; C is the elastic compliance matrix; σ is the total Cauchy stress of the mixture; m = [1; 1; 1; 0],

[0035] The density of the mixture is ρ = n f ρ f +(1-n f )ρ s , where ρ s and n f are the density of the solid and the porosity of the mixture, respectively. Substitute the relative velocity w in the above equation (5) into equation (6):

[0036]

[0037] The boundary conditions are:

[0038]

[0039] in and are the prescribed displacements, traction forces, pore water pressures, and fluid fluxes. N is the outward vector normal to the corresponding boundary surface;

[0040] The standard θ-method is introduced to perform time discretization of effective stress σ', velocity v and pressure p:

[0041] σ′=θ1σ′ n+1 +(1-θ1)σ′ n (13)

[0042]

[0043] p=θ3p n+1 +(1-θ3)p n (15)

[0044] The subscripts n and n+1 denote the known and unknown states, respectively, and Δt is the time increment. Then, the momentum balance equation (4) of the mixture and Darcy's law can be rearranged to replace equation (8):

[0045]

[0046] in

[0047]

[0048] In the equation According to the known velocity field, The traction boundary condition (10) and the fluid flux boundary condition (12) are expressed as:

[0049]

[0050] The time-discrete governing equation for incremental dynamic analysis of saturated porous media is equivalent to the minimum-maximum problem:

[0051]

[0052] The validity of the min-max problem is demonstrated by deriving the associated Karush-Kuhn-Tucher conditions:

[0053] For the associated plastic flow, the Lagrangian format of problem (22) is constructed as follows:

[0054]

[0055] The KKT condition of formula (22) can be expressed as:

[0056]

[0057] Complementary conditions

[0058] ΔλF(σ′ n+1 )=0 (28)

[0059] Original feasibility

[0060] F(σ′ n+1 )≤0 (29)

[0061] Dual feasibility

[0062] Δλ≥0 (30)

[0063] Equation (24) consists of the discretized linear moment balance equation of the mixture, Equation (16) and the boundary conditions in (20); Equations (25), (28)-(30) are the incremental forms of the constitutive equations; Equation (26) represents the governing equations in (17) and the boundary conditions related to the fluid flux; Equation (27) is the inertial force γ n+1 The expression,

[0064] For the non-associative flow rule, for the two-dimensional case, the yield function F and plastic potential G of the non-associative Mohr-Coulomb model are:

[0065]

[0066] in, is the effective friction angle, c′ is the effective cohesion, and ψ is the dilatancy angle, using the approximate form of F

[0067]

[0068] Among them, Treated as a constant and passed

[0069]

[0070] The subscript 0 in equation (34) represents the state solved from the previous analysis step.

[0071] Preferably, the minimum-maximum problem of the time-discrete governing equation for dynamic analysis in saturated porous media obtained in step 1 is transformed into a standard second-order cone programming problem based on the hybrid triangular element:

[0072] The mixed triangle element is used for discretization, and the field variables are approximately:

[0073]

[0074] in, and is a vector composed of stress, displacement, dynamic force and pressure at the unit node. For the sake of simplicity, the intermediate variable and is the vector of intermediate variables at element nodes. N σ ,N u ,Nγ ,N p and N χ is the matrix of the corresponding shape function;

[0075] Step 32, replace formulas (35)-(39) into the optimization problem (22) to obtain its discretized form:

[0076]

[0077] Limited by

[0078]

[0079] Among them, n σ is the total number of Gaussian integration points, where:

[0080]

[0081] In Eq. (40), the underlined terms represent displacement boundary conditions, and the newly introduced variable r n+1 is the reaction force at the element node with the prescribed displacement, E u is an index matrix consisting of 1s and 0s, indicating nodes with / without the prescribed displacements. Verify its correctness and derive the associated Lagrangian with respect to the field variables Partial derivative of:

[0082]

[0083] in, is the prescribed displacement The discrete form of . This relation is obviously the displacement boundary condition in equation (9). For the min-max problem, the minimization part can be solved analytically, leading to the maximization problem,

[0084]

[0085] Limited by

[0086]

[0087] The discretized optimization problem (43) can be transformed into the standard SOCP problem form

[0088]

[0089] Limited by

[0090] ax=b

[0091]

[0092] where x=(x1,x2,...,x n )T consists of field variables, a, b and c are matrices and vectors of factors, is the tensor product of second-order cones such that Second-order cones can be of the following types:

[0093] Standard quadratic cone:

[0094]

[0095] Rotating the quadratic cone:

[0096]

[0097] The standard SOCP (44) is the minimization of a linear objective function subject to linear constraints and / or second-order cones, which reformulates the discretized maximization problem (43) as the following SOCP problem:

[0098]

[0099] Limited by

[0100]

[0101]

[0102] Where ρ, H, and d are determined according to the Mohr-Coulomb yield criterion (33):

[0103] ρ=(ρ1,ρ2,ρ3),σ=(σ′ xx ,σ′ yy ,σ′ zz ,σ′ xy ) (48)

[0104]

[0105] The program (47) follows the standard mathematical procedure using second-order cone programming.

[0106] Preferably, within a given time interval, the PFEM solution comprises the following steps:

[0107] Update the mesh node positions according to the incremental displacement obtained from the analysis;

[0108] Based on the updated grid nodes, identify the boundary of the calculation area;

[0109] Triangulate the identified area to generate a new mesh;

[0110] Mapping historical field variables from the old grid to the new grid;

[0111] Solve the equation on the new mesh.

[0112] Preferably, the two-dimensional numerical analysis model is brought into the framework of particle finite element method for solution, specifically:

[0113] The particle finite element method (PFEM) for dynamic analysis of saturated porous media is a hybrid of the particle method and Lagrangian FEM, which treats mesh nodes as free particles but uses Lagrangian FEM to solve the governing equations. The basic steps of PFEM are to use the Alpha-shape method to identify the boundary of the computational domain and then perform mesh generation for Lagrangian finite element analysis.

[0114] In order to achieve the above second purpose, the technical solution of the saturated porous medium dynamic analysis device provided by the present invention is as follows:

[0115] The saturated porous medium dynamic analysis device provided by the present invention comprises:

[0116] A data acquisition module is used to obtain the geometric parameters and boundary conditions of the dynamic analysis problem in the saturated porous medium to be solved;

[0117] A model building module, used for building a two-dimensional numerical analysis model according to the geometric parameters and boundary conditions;

[0118] A solution module is used to bring the two-dimensional numerical analysis model into the framework of particle finite element method to solve and obtain the dynamic analysis conclusion of the saturated porous medium;

[0119] The dynamic analysis problems in the saturated porous medium include one or more of debris flow, dam destruction and particle column collapse.

[0120] In order to achieve the third objective above, the technical solution of the computer-readable storage medium provided by the present invention is as follows:

[0121] The computer-readable storage medium provided by the present invention stores a saturated porous medium dynamic analysis program. When the saturated porous medium dynamic analysis program is executed by a processor, the steps of the saturated porous medium dynamic analysis method provided by the present invention are implemented.

[0122] In order to achieve the fourth objective, the present invention provides an electronic device with the following technical solutions:

[0123] The electronic device provided by the present invention includes a memory and a processor, wherein the memory stores a saturated porous medium dynamic analysis program, and when the saturated porous medium dynamic analysis program is executed by the processor, the steps of the saturated porous medium dynamic analysis method provided by the present invention are implemented.

[0124] The present invention provides a dynamic analysis method for saturated porous media based on the generalized Hellinger-Reissner variational principle and particle finite element method (PFEM), which has the following advantages over existing technologies:

[0125] (1) Based on the UP model for dynamic analysis of saturated porous media, the present invention derives the minimum-maximum problem equivalent to the time-discrete control equation for dynamic analysis in saturated porous media, and establishes a finite element formula for analyzing the dynamics of saturated porous media using the Hellinger-Reissner variational principle.

[0126] (2) The generalized Hellinger-Reissner variational principle for dynamic analysis in saturated porous media is discretized using mixed finite elements and then transformed into a standard second-order cone programming (SOCP) problem that can be solved using the interior point method.

[0127] (3) It has advantages over traditional finite element algorithms. For example, compared with the traditional Newton-Raphson iterative format finite element algorithm, the finite element algorithm in SOCP has advantages in the convergence of analytical solutions. In addition, it directly handles singular points in the widely used Mohr-Coulomb model and Bingham model, as well as directly extends from single-sided plasticity to multi-sided plasticity.

[0128] (4) The finite element formula obtained in SCOP is incorporated into the PFEM for the dynamic problem of saturated porous media with large deformation and free surface evolution, so that the problem is coupled to the framework of PFEM and the advantages of the particle method in dealing with large deformation problems are obtained, providing an efficient and accurate method for solving similar problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0129] Various other advantages and benefits will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiment below. The accompanying drawings are for illustration purposes only and are not to be considered as limiting the present invention. The same reference symbols are used throughout the drawings to represent the same components. In the drawings:

[0130] Figure 1 is a flowchart of the steps of the saturated porous medium analysis method provided by an embodiment of the present invention;

[0131] Figure 2 is the area of ​​saturated medium and its boundary demarcation. u , Γ t , Γ p and Γ q They are subjected to specified displacement, traction, pore water pressure and fluid flux respectively;

[0132] Figure 3 is an example of a mixed triangle element;

[0133] Figure 4 It is the geometric model of the embankment. In the simulation, the lateral boundaries are set as free rollers and the bottom is fixed;

[0134] Figure 5 It is the final sediment profile and displacement distribution after slope failure;

[0135] Figure 6 It is the geometrical dimensions, boundary conditions and loading mode of the computational model of one-dimensional dynamic consolidation problem;

[0136] Figure 7 When k = 10 -2 m / s, Δt=5×10 -3 Schematic diagram of the change of displacement at point A and pore water pressure at point B within 1s (where Figure 7 a is a schematic diagram of the displacement change of point A within 1s, Figure 7 b is a schematic diagram of the change of pore water pressure at point B within 1 s);

[0137] Figure 8 is the evolution of the displacement at point A and the pore water pressure at point B obtained by simulation using different time steps and time integration parameters;

[0138] Figure 9 This is the comparison of the numerical solution and analytical solution of the top displacement of point A over time within 0.5s, k = 10 -2 m / s;

[0139] Figure 10 Schematic diagram of the signal flow relationship between the functional modules in the saturated porous medium dynamic analysis device provided by an embodiment of the present invention;

[0140] Figure 11 Schematic diagram of the structure of a saturated porous medium dynamic analysis device in the hardware operating environment provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0141] In order to solve the problems existing in the prior art, the present invention provides a saturated porous medium dynamic analysis method, device, storage medium and electronic equipment, which are of great significance for the accurate and rapid solution of saturated porous medium dynamic problems, and are therefore more suitable for practical use.

[0142] To further illustrate the technical means and effectiveness of the present invention in achieving its intended objectives, the following, in conjunction with the accompanying drawings and preferred embodiments, describes in detail the specific implementation, structure, features, and effectiveness of the saturated porous media dynamic analysis method, apparatus, storage medium, and electronic device proposed by the present invention. In the following description, different references to "one embodiment" or "embodiment" do not necessarily refer to the same embodiment. Furthermore, features, structures, or characteristics of one or more embodiments may be combined in any suitable manner.

[0143] The term "and / or" in this article is merely a description of the association relationship between associated objects, indicating that three relationships may exist. For example, A and / or B, specifically understood as: A and B may be included at the same time, A may exist alone, or B may exist alone, and any of the above three situations may exist.

[0144] The present invention proposes a dynamic analysis method for saturated porous media based on the generalized Hellinger-Reissner variational principle and particle finite element method (PFEM). The method is established on a two-dimensional saturated porous media model and meshes are divided; the Hellinger-Reissner variational principle is expressed in its functional form with the help of the minimum-maximum optimization method; the minimum-maximum problem of the time-discrete control equation of dynamic analysis in saturated porous media is constructed with the help of the up model of saturated porous media dynamic analysis and the standard θ-method; then, based on the hybrid triangular unit, the dynamic analysis problem of saturated porous media can be constructed as a standard second-order cone programming problem based on the FEM framework, which is further introduced into the PFEM framework to establish a second-order cone programming particle finite element method suitable for classical continuum; then, according to the geometric parameters and boundary conditions required by the problem, a mathematical optimization solver is used to solve it, and the water-mechanical coupling result of the saturated soil dynamic problem with large deformation and free surface evolution can be obtained. The specific steps are as follows:

[0145] Step 1: Based on the UP model for dynamic analysis of saturated porous media (the UP form of the Boit equation), the governing equations of saturated porous media are given, and the standard θ-method is introduced to time-discretize the effective stress σ', velocity v, and pressure p in the UP model. Finally, a minimum-maximum problem equivalent to the time-discrete governing equations for dynamic analysis in saturated porous media is derived, and the Lagrangian of this minimum-maximum problem is constructed. The feasibility of the problem is proved by the Karush-Kuhn-Tucher (KKT) condition.

[0146] Step 11, the UP model for dynamic analysis of saturated porous media is used. This model ignores the derivative of the relative velocity of the fluid with respect to the solid and is widely used in low-frequency loading situations. In the two-dimensional case, the governing equations for a saturated porous medium with a region Ω and a boundary Γ are as follows (see the boundary for details). Figure 2):

[0147] Momentum balance equation for a mixture

[0148]

[0149] Darcy's Law

[0150]

[0151] Pore ​​fluid mass balance equation

[0152]

[0153] Strain decomposition and stress-strain relationship

[0154] ε=ε e +ε p (4)

[0155] here

[0156]

[0157] In the above formula: ρ is the density of the mixture; u is the solid skeleton displacement; ρ f is the fluid density; σ' is the effective stress acting on the solid skeleton; p is the pore water pressure when the tensile pore water pressure is positive; b is the mixture body force; b f is the fluid body force; γ f is the fluid density; k is the Darcy permeability coefficient; w is the velocity of the pore fluid relative to the solid skeleton; ε is the strain vector, defined as ε=Su, given by ε e (elastic strain) and ε p (plastic strain); λ is the plasticity multiplier, λ≥0; F is the yield function, F≤0; G is the plastic potential; is the elastic compliance matrix; σ is the total Cauchy stress of the mixture; m = [1; 1; 1; 0].

[0158] The density of the mixture is ρ = n f ρ f +(1-n f )ρ s , where ρ s and n f are the density of the solid and the porosity of the mixture, respectively. Substitute the relative velocity w in the above equation (2) into equation (3):

[0159]

[0160] The boundary conditions are (see Figure 2 ):

[0161]

[0162] in and are the prescribed displacements, traction forces, pore water pressures, and fluid fluxes. N is the outward vector normal to the corresponding boundary surface.

[0163] Step 12, introduce the standard θ-method to perform time discretization on the effective stress σ', velocity v and pressure p:

[0164] σ′=θ1σ′ n+1 +(1-θ1)σ′ n (10)

[0165]

[0166] p=θ3p n+1 +(1-θ3)p n (12)

[0167] The subscripts n and n+1 represent the known and unknown states, respectively, and Δt is the time increment. Then, the momentum balance equation (1) of the mixture and Darcy's law can be rearranged to replace equation (5):

[0168]

[0169] in

[0170]

[0171] Note that in the equation According to the known velocity field, The traction boundary condition (7) and the fluid flux boundary condition (9) are expressed as:

[0172]

[0173] In step 13, we can now formulate the min-max problem equivalent to the time-discrete governing equations for the incremental dynamics analysis of saturated porous media:

[0174]

[0175] The validity of this min-max problem can be proved by showing the relevant Karush-Kuhn-Tucher (KKT) conditions. To do this, we first construct the Lagrangian of problem (19), which is:

[0176]

[0177] The KKT condition of formula (20) is:

[0178]

[0179] Complementary conditions

[0180] ΔλF(σ′ n+1 )=0 (26)

[0181] Original feasibility

[0182] F(σ′ n+1 )≤0 (27)

[0183] Dual feasibility

[0184] Δλ≥0 (28)

[0185] It is clear that Equation (21) consists of the discretized linear equilibrium equations of the mixture, Equation (16) and the boundary conditions in (20); Equations (25), (28)-(30) are the incremental forms of the constitutive equations; Equation (26) represents the governing equations in (17) and the boundary conditions related to the fluid flux; Equation (27) is the inertial force γ n+1 expression.

[0186] Note that the above is for associated plastic flow. Considering the non-associative flow rules, specifically, for the two-dimensional case, the yield function F and plastic potential G of the non-associative Mohr-Coulomb model are given by:

[0187]

[0188] in is the effective friction angle, c′ is the effective cohesion, and ψ is the dilatancy angle. Using the approximate form of F

[0189]

[0190] Among them, Treated as a constant and passed

[0191]

[0192] The subscript 0 in equation (32) indicates the state solved from the last analysis step.

[0193] Step 2: Based on the hybrid triangular element, the minimum-maximum problem of the time-discrete control equation for dynamic analysis in saturated porous media obtained in step 1 is transformed into a standard second-order cone programming problem.

[0194] The standard finite element discretization of the optimization problem (19) is performed in the present invention using mixed triangular elements. The discretized optimization problem is then transformed into a standard second-order cone programming problem that can be solved using existing optimization engines.

[0195] Step 21, discretization using Figure 3 The hybrid triangular element is shown. In detail, the field variables are approximated as:

[0196]

[0197] in, and is a vector composed of stress, displacement, inertia force and pressure at the unit node. For the sake of simplicity, the intermediate variable and is the vector of intermediate variables at element nodes. N σ ,N u ,N γ ,N p and N χ is the matrix of the corresponding shape functions.

[0198] Step 22, substituting equations (33)-(37) into the optimization problem (19) leads to its discretized form:

[0199]

[0200] Limited by

[0201]

[0202] where n σ is the total number of Gaussian integration points, where:

[0203]

[0204]

[0205] In Equation (38), the underlined term represents the displacement boundary condition, and the newly introduced variable r n+1 is the reaction force at the element node with the prescribed displacement. E u is an index matrix consisting of 1 and 0, indicating nodes with / without the prescribed displacement. To verify its correctness, the associated Lagrangian with respect to the field variables is derived Partial derivative of

[0206]

[0207] in, is the prescribed displacement The discrete form of , and this relation is obviously the displacement boundary condition in equation (6). For the min-max problem, the minimization part can be solved analytically, leading to the maximization problem,

[0208]

[0209] Limited by

[0210]

[0211] Step 23, the discretized optimization problem (41) is reformulated in this section as a standard second-order cone programming (SOCP) problem. A standard SOCP problem is of the form

[0212]

[0213] Limited by

[0214] ax=b

[0215]

[0216] where x=(x1,x2,...,x n ) T consists of field variables, a, b and c are matrices and vectors of factors, is the tensor product of second-order cones such that Second-order cones can be of the following types:

[0217] Standard quadratic cone:

[0218]

[0219] Rotating the quadratic cone:

[0220]

[0221] As shown above, the standard SOCP (42) is the minimization of a linear objective function subject to linear constraints and / or second-order cones. The discretized maximization problem (41) can be reformulated as the following SOCP problem

[0222]

[0223] Limited by

[0224]

[0225] In the above, ρ, H, and d are determined according to the Mohr-Coulomb yield criterion (31):

[0226] ρ=(ρ1,ρ2,ρ3),σ=(σ′ xx ,σ′ yy ,σ′ zz ,σ′ xy ) (46)

[0227]

[0228] The program (45) follows the standard mathematical procedure using second-order cone programming.

[0229] Step 3: Obtain the geometric parameters and boundary conditions of the dynamic analysis problem to be solved in saturated porous media (such as debris flow, dam rupture, and particle column collapse), establish a two-dimensional numerical analysis model, and bring it into the particle finite element method (PFEM) framework for solution.

[0230] In this step, the particle finite element method (PFEM) is a hybrid of particle methods and Lagrangian FEM, treating mesh nodes as free particles but solving the governing equations using Lagrangian FEM. The basic steps of PFEM are to identify the boundaries of the computational domain using the Alpha-shape method, followed by mesh generation for Lagrangian finite element analysis. At a given time interval, the PFEM solution involves: (a) updating the mesh node positions based on the incremental displacements obtained in the previous analysis step; (b) identifying the boundaries of the computational domain using the Alpha-shape technique based on the updated mesh nodes; (c) triangulating the identified region to generate a new mesh; (d) mapping the historical field variables from the old mesh to the new mesh; and (e) solving the equations on the new mesh using Lagrangian FEM. Due to step (b), PFEM has the ability to model the separation and reconnection of portions of the computational domain and of individual isolated particles, which can occur in fluid dynamics problems (such as wave breaking and water splash) and geotechnical problems (such as granular flows and landslides). After separation, the motion of the isolated particles is treated as free fall and can be solved analytically, while the motion of the subdomain is solved by finite elements. When they approach the main domain, the separated particles and subdomains will reconnect to the main domain when step (b) is performed. Note that boundary detection using the Alpha-shape method may artificially induce volume changes (or changes in total mass). By using appropriate α values ​​and mesh refinement, the volume can be kept within an acceptable range. It is worth noting that although the developed formulation for the dynamic analysis of saturated porous media is based on small deformation theory, the material configuration is updated after each incremental analysis. The idea of ​​using a series of incremental analyses based on infinitesimal strain theory, with updated geometry, for large deformation problems has been widely used in continuous limit analysis.

[0231] Example 1

[0232] This example uses the proposed method to calculate the failure of the embankment. The dimensions of the geometric model are as follows: Figure 4 As shown in , the model uses 2277 units for discretization. Figure 5 As shown in Figure 1, the model consists of four parts: water, saturated foundation, unsaturated soil, and saturated slope, which are marked with different colors. For simplicity, the unsaturated soil is considered as dry soil. The specific gravity of water γ f =10kN / m 3, the specific gravity of unsaturated soil is γ unsa =18.5kN / m 3 , the specific gravity of saturated soil on the slope is γ sa1 =20kN / m 3 , the weight of saturated soil of foundation is γ sa2 =22kN / m 3 The elastic modulus of saturated soil and unsaturated soil in the slope is E1 = 25 MPa, and the internal friction angle is Cohesion c1 = 4 kPa, expansion angle ψ1 = 0°, Poisson's ratio ν1 = 0.3. For foundation soil, elastic modulus E2 = 50 MPa, internal friction angle The cohesion c2 = 10 kPa, the expansion angle is ψ2 = 0°, and the Poisson's ratio ν2 = 0.3. The cohesion angle and friction angle of water are set to zero, and the Poisson's ratio is set to 0.499, which is approximately incompressible. In the gravity loading step, the entire model is set to stable by adopting large cohesion and internal friction angles. Then, a dynamic analysis is performed by reducing the strength to a certain specified value, thereby inducing embankment failure. The time step used in the dynamic analysis is Δt = 0.01 s. The final sediment profile and displacement distribution after embankment failure are shown in Figure 2. Figure 5 As shown. Figure 5 It can be seen that the final sedimentary profile obtained using the method proposed in the present invention is consistent with the SPH model, and the shear zone is also consistent with the Bishop sliding surface, indicating the correctness of the dynamic analysis method for embankments composed of saturated soil proposed in the present invention.

[0233] Example 2

[0234] This example uses the method proposed in this paper to simulate a one-dimensional dynamic consolidation problem. The geometric dimensions and boundary conditions of the calculation model are as follows: Figure 6 As shown in Figure 1, the saturated soil column is subjected to two loading types, f1(t) and f2(t). Since the standard θ method is used for time discretization, if all three constants, θ1, θ2 and θ3 (for simplicity, θ 1,2,3 If θ is greater than or equal to 0.5, the time integration scheme is implicit and unconditionally stable. Otherwise, the time integration scheme is explicit. 1,2,3 = 0.5, the integration format is the midpoint rule, and when θ 1,2,3 = 1, the integration format is the backward Euler format. In order to illustrate the characteristics of the time integration scheme, a vertical load f1(t) is applied to the surface of the soil column. Three sets of time integration parameters (such as θ 1,2,3 = 0.495, 0.5, 1.0). The time step is Δt = 5 × 10-3 s, and the displacement of the upper surface (such as point A (x = 1 m, y = 10 m)) and the pressure at the bottom (such as point B (x = 1 m, y = 0 m)) are monitored. Selected parameters: density of the mixture ρ = 2000 kg / m3 , Young's modulus E = 10 4 kPa, Poisson's ratio υ=0.2, porosity n f =0.35, Darcy permeability k=10 -2 m / s. Figure 7 As shown, when all time integration parameters (e.g., θ 1,2,3 = 0.5 and θ 1,2,3 = 1.0) Using values ​​of 0.5 and 1.0 ensures a stable solution for displacement and pore water pressure. 1,2,3 = 0.495, the pore water pressure starts to oscillate severely from t = 0.2s, and unstable displacement is observed from t = 0.75s. Compared with the displacement response at point A, the pressure response at point B is more likely to be unstable. In order to investigate the convergence of the numerical solution with respect to the time step, two other time steps (e.g., Δt = 2.5 × 10 -3 s and 1×10 -2 s) respectively use the time integral parameter θ 1,2,3 = 0.5 and 1.0 to simulate the problem. The simulation results are compared with the simulation results using Δt = 5×10-3s. Figure 8 As shown, satisfactory agreement is obtained in all cases. In the following, all simulations representing the anti-Euler scheme are performed with the time integration parameter θ 1,2,3 = 1.0. The proposed formula further verifies the dynamic response of the infinite half-space under surface loads, and its analytical solution is available. The problem is considered as a plane strain problem with surface load f2(t). The material parameters of the problem are: solid density ρ s =2000kg / m 3 , fluid density ρ f =1000kg / m 3 , porosity n f =0.33, the Lame constant μ of the solid skeleton s =5.583MPa,λ s =8.375MPa. Comparing the formula proposed in this paper with the analytical solution, the dynamic response of the top surface displacement is very consistent with the analytical solution (see Figure 9 ).

[0235] Reference Figure 10 , Figure 10 This is a schematic diagram of the structure of a saturated porous medium dynamic analysis device in the hardware operating environment involved in an embodiment of the present invention.

[0236] like Figure 10As shown, the saturated porous media dynamic analysis device may include: a processor 1001, such as a central processing unit (CPU), a communication bus 1002, a user interface 1003, a network interface 1004, and a memory 1005. Among them, the communication bus 1002 is used to realize the connection and communication between these components. The user interface 1003 may include a display screen (Display), an input unit such as a keyboard (Keyboard), and the user interface 1003 may optionally include a standard wired interface and a wireless interface. The network interface 1004 may optionally include a standard wired interface and a wireless interface (such as a wireless fidelity (WIreless-FIdelity, WI-FI) interface). The memory 1005 may be a high-speed random access memory (Random Access Memory, RAM) memory, or a stable non-volatile memory (Non-Volatile Memory, NVM), such as a disk memory. The memory 1005 may also be a storage device independent of the aforementioned processor 1001.

[0237] Those skilled in the art will understand that Figure 10 The structure shown in the figure does not constitute a limitation on the saturated porous medium dynamic analysis device, and may include more or fewer components than shown in the figure, or combine certain components, or arrange the components differently.

[0238] like Figure 10 As shown, the memory 1005 as a storage medium may include an operating system, a data storage module, a network communication module, a user interface module, and a saturated porous media dynamic analysis program.

[0239] exist Figure 10 In the saturated porous medium dynamic analysis device shown, the network interface 1004 is mainly used for data communication with the network server; the user interface 1003 is mainly used for data interaction with the user; the processor 1001 and the memory 1005 in the saturated porous medium dynamic analysis device of the present invention can be set in the saturated porous medium dynamic analysis device, and the saturated porous medium dynamic analysis device calls the saturated porous medium dynamic analysis program stored in the memory 1005 through the processor 1001, and executes the saturated porous medium dynamic analysis method provided by the embodiment of the present invention.

[0240] Although the preferred embodiments of the present invention have been described, those skilled in the art may make additional changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0241] Obviously, those skilled in the art may make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if such changes and modifications fall within the scope of the claims and their equivalents, the present invention is intended to include such changes and modifications.

Claims

1. A method for dynamic analysis of saturated porous media, characterized in that: The following steps are involved: Obtain the geometric parameters and boundary conditions of the dynamic analysis problem to be solved in saturated porous media; Establishing a two-dimensional numerical analysis model according to the geometric parameters and boundary conditions; The two-dimensional numerical analysis model is brought into the framework of particle finite element method to obtain the dynamic analysis conclusion of the saturated porous medium; The example problems of dynamic analysis in saturated porous media include one or more of debris flow, dam failure and particle column collapse; The following steps are also included: Based on the UP model for dynamic analysis of saturated porous media, the governing equations of saturated porous media are given. The standard θ-method is introduced into the governing equations to perform time discretization on the effective stress σ', velocity v, and pressure p in the UP model, and a minimum-maximum problem equivalent to the time-discrete governing equations for dynamic analysis in saturated porous media is derived. The governing equations for dynamic analysis of saturated porous media after time discretization are spatially discretized using hybrid triangular elements. Based on the Hellinger-Reissner variational criterion for static elasticity, a generalized Hellinger-Reissner variational criterion (HR variational criterion) for dynamic analysis of saturated porous media is established, which is further transformed into a standard second-order cone programming problem. Based on the UP model for dynamic analysis of saturated porous media, the governing equations of saturated porous media are given. Relevant data are introduced into the governing equations, and the minimum-maximum problem equivalent to the time-discrete governing equations for dynamic analysis in saturated porous media is derived: Using the up model for dynamic analysis of saturated porous media, in the two-dimensional case, the governing equations for a saturated porous medium with a region Ω and a boundary Γ are as follows: Momentum balance equation for a mixture Where T represents the matrix transpose, Darcy's Law Pore ​​fluid mass balance equation Where T represents the matrix transpose, Strain decomposition and stress-strain relationship e=e e +e p (7) here In the above formula: ρ is the density of the mixture; u is the solid skeleton displacement; ρ f is the fluid density; σ' is the effective stress acting on the solid skeleton; p is the pore water pressure when the tensile pore water pressure is positive; b is the mixture body force; b f is the fluid body force; γ f is the fluid density; k is the Darcy permeability coefficient; w is the velocity of the pore fluid relative to the solid skeleton; ε is the strain, defined as ε = Su, which is obtained from the elastic strain ε e and plastic strain ε p composition; λ is the plasticity multiplier, λ≥0; F is the yield function, F≤0; G is the plastic potential; is the elastic compliance matrix; σ is the total Cauchy stress of the mixture; m = [1; 1; 1; 0], The density of the mixture is ρ = n f ρ f +(1-n f )ρ s , where ρ s and n f are the density of the solid and the porosity of the mixture, respectively. Substituting the velocity w in the above equation (5) into equation (6): The boundary conditions are: in and are the prescribed displacements, traction forces, pore water pressures, and fluid fluxes, and N is the outward vector normal to the corresponding boundary surface; The standard θ-method is introduced to perform time discretization of effective stress σ', velocity v and pressure p: σ′=θ1σ′ n+1 +(1-θ1)σ′ n (13) Subscripts n and n+1 represent known and unknown states, respectively. Δt is the time increment, Δu is the displacement increment, and θ1, θ2, and θ3 are parameters. Then, the momentum balance equation (4) of the mixture and Darcy's law are replaced by equation (8) and rearranged as follows: in In the equation According to the known velocity field, The traction boundary condition (10) and the fluid flux boundary condition (12) are expressed as: The time-discrete governing equations for incremental dynamics analysis of saturated porous media are equivalent to the minimum-maximum problem:

2. The saturated porous media dynamic analysis method according to claim 1, characterized in that: A generalized Hellinger-Reisner variational criterion for the dynamic analysis of saturated porous media is established. With the help of mathematical programming methods, it is transformed into a maximum-minimization problem, where the optimization problem is to minimize the displacement in the objective function and maximize the stress in the objective function. Specifically, For statically elastic solids, the Hellinger-Reissner functional is expressed as: Where σ is the Cauchy stress, u is the displacement, b is the body force, and t is the surface force. is the elastic compliance matrix, Ω and Γ are the integration domain and boundary conditions, respectively, T denotes the matrix transpose, and the operator S for the plane strain case takes the following form: The min-max optimization class for this problem is: Among them, based on the Hellinger-Reissner variational principle in elastic mechanics, the generalized HR variational principle has been developed to analyze elastic-plastic problems, elastic-viscoplastic problems and quasi-static porous elastic-plastic problems.

3. The saturated porous media dynamic analysis method according to claim 1, characterized in that: The validity of the minimum-maximum problem is proved by the related Karush-Kuhn-Tucher condition: for the associated plastic flow, the Lagrangian expression of the minimum-maximum problem (22) is constructed: The KKT condition of formula (22) is: Complementary slack conditions ΔλF(σ′ n+1 )=0 (28) Original feasibility F(σ′ n+1 )≤0 (29) Dual feasibility Δλ≥0 (30) Equation (24) consists of the equilibrium equations after the mixture is discretized, and the boundary conditions in Equations (16) and (20); Equations (25), (28)-(30) are the incremental forms of the constitutive equations; Equation (26) represents the governing equations in (17) and the boundary conditions related to the fluid flux; Equation (27) is the inertial force γ n+1 The expression, For the non-associative flow rule, for the two-dimensional case, the yield function F and plastic potential G of the non-associative Mohr-Coulomb model are: in, is the effective friction angle, c′ is the effective cohesion, and ψ is the dilatancy angle, using the approximate form of F Among them, Treated as a constant and passed The subscript 0 in equation (34) represents the state solved from the previous analysis step.

4. The saturated porous media dynamic analysis method according to claim 1, characterized in that: The obtained minimum-maximum problem of the time-discrete governing equation for dynamic analysis in saturated porous media is transformed into a standard second-order cone programming problem based on the hybrid triangular element, specifically: Using mixed triangle elements for discretization, the field variables can be approximated as: in, and is a vector composed of stress, displacement, dynamic force and pressure at the unit node. For the sake of simplicity, the intermediate variable and is the vector of intermediate variables at the node, N σ ,N u ,N γ ,N p and N χ is the matrix of the corresponding shape function; Converting formulas (35)-(39) into optimization problems and substituting them into formula (22) yields the discretized form: Limited by Among them, n σ is the total number of Gaussian integration points, where: In Eq. (40), the underlined terms represent displacement boundary conditions, and the newly introduced variable r n+1 is the reaction force at the element node with the specified displacement, E u is an index matrix consisting of 1s and 0s, indicating nodes with or without the specified displacement. Verify its correctness and derive the associated Lagrangian with respect to the field variables Partial derivative of: in, is the prescribed displacement The discrete form of this relation is obviously the displacement boundary condition in the equation, that is, Equation (9). For the minimum-maximum problem, the minimization part can be solved analytically, resulting in a maximization problem. Limited by The discretized optimization problem (43) can be transformed into the form of a standard SOCP problem Limited by ax=b where x=(x1, x2, ..., x n ) T consists of field variables, a, b and c are matrices and vectors of factors, is the tensor product of second-order cones such that Second-order cones are of the following types: Standard quadratic cone: Rotating the quadratic cone: The standard SOCP (44) is a minimization problem of a linear objective function subject to linear constraints and / or second-order cones, so we reformulate the discretized maximization problem (43) as the following standard SOCP problem: Limited by Where ρ, H, and d are determined according to the Mohr-Coulomb yield criterion (33): p=(p1,p2,p3),σ=(σ′ xx ,s′ yy ,s′ zz ,s′ xy ) (48) The planning problem (47) conforms to the standard second-order cone programming form.

5. The saturated porous media dynamic analysis method according to claim 1, characterized in that: At a given time interval, the particle finite element method solution includes the following steps: Update the mesh node positions according to the incremental displacement obtained from the analysis; Based on the updated grid nodes, identify the boundary of the calculation area; Triangulate the identified area to generate a new mesh; Mapping historical field variables from the old grid to the new grid; Solve the equation on the new mesh.

6. The saturated porous media dynamic analysis method according to claim 1, characterized in that: The two-dimensional numerical analysis model is brought into the framework of particle finite element method to solve it, specifically: The particle finite element method (PFEM) is a hybrid of the particle method and the Lagrangian FEM. It treats mesh nodes as free particles but uses the Lagrangian FEM to solve the governing equations. The basic steps of PFEM are to use the Alpha-shape method to identify the boundary of the computational domain and then perform Lagrangian finite element analysis using the mesh generated by Delaunay triangulation.

7. A saturated porous medium dynamic analysis device based on the saturated porous medium dynamic analysis method according to any one of claims 1 to 6, characterized in that: include: A data acquisition module is used to obtain the geometric parameters and boundary conditions of the dynamic analysis problem in the saturated porous medium to be solved; A model building module, used to build a two-dimensional numerical analysis model according to the geometric parameters and boundary conditions; A solution module, used for bringing the two-dimensional numerical analysis model into the framework of particle finite element method to solve and obtain the dynamic analysis results of the saturated porous medium; The example problems of dynamic analysis in saturated porous media include one or more of debris flow, dam destruction and particle column collapse.

8. A computer-readable storage medium, characterized in that The computer-readable storage medium stores a saturated porous medium dynamic analysis program, which, when executed by a processor, implements the steps of the saturated porous medium dynamic analysis method according to any one of claims 1 to 6.

9. An electronic device, characterized in that: The invention comprises a memory and a processor, wherein a saturated porous medium dynamic analysis program is stored in the memory, and when the saturated porous medium dynamic analysis program is executed by the processor, the steps of the saturated porous medium dynamic analysis method according to any one of claims 1 to 6 are realized.

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