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

Abstract

The invention discloses a saturated porous medium large deformation dynamic analysis method and device, a storage medium and electronic equipment, and belongs to the technical field of computer numerical simulation. The method comprises the following steps: acquiring geometric parameters and boundary conditions of a dynamic analysis problem in a to-be-solved membership saturated porous medium; establishing a two-dimensional numerical analysis model according to the geometric parameters and the boundary conditions; the two-dimensional numerical analysis model is substituted into a framework of a mathematical programming particle finite element method to be solved, and a saturated porous medium dynamic analysis conclusion is obtained; wherein the dynamic analysis demonstration examples in the saturated porous medium comprise debris flow, dam damage, particle column collapse and the like. The device, the storage medium and the electronic equipment can be used for realizing the method. The method is of great significance to accurate and rapid solving of the large deformation power problem of the saturated porous medium.

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] Engineering disasters caused by dynamic damage of saturated soils such as landslides and debris flows will threaten the safety of transportation routes and the personal and property of people along the routes, thus affecting the economic development and life safety of the people in the area. How to accurately and quickly measure whether these saturated soils will be damaged dynamically is extremely important for preventing these engineering disasters. The traditional finite element method (FEM) occupies an important position in geotechnical engineering analysis and design, but the standard Lagrangian finite element still has limitations in dealing with geotechnical engineering problems such as landslides and debris flows with large deformations and free surface evolution. Summary of the invention

[0003] In view of this, 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.

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

[0007] According to the geometric parameters and boundary conditions, a two-dimensional numerical analysis model is established;

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

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

[0010] The saturated porous medium 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 of dynamic analysis of saturated porous media, the control equation of saturated porous media is given. The standard θ-method is introduced into the control equation to time discretize the effective stress σ', velocity v and pressure p in the UP model, and the minimum-maximum problem equivalent to the time-discrete control equation of 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, based on the Hellinger-Reisner variational principle of static elasticity, the solid static elastic Hellinger-Reisner functional is expressed by means of a minimum-maximum optimization method, wherein 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] In the formula, 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 min-max problem form of this functional is:

[0020]

[0021] Among them, based on the Hellinger-Reissner variational principle of elasticity mechanics, a 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 the dynamic analysis of saturated porous media, the control equation of the saturated porous media is given, and the relevant data are introduced into the control equation to derive the minimum-maximum problem equivalent to the time-discrete control equation of the dynamic analysis in the 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; 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 time discretize the 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 equations for incremental dynamics analysis of saturated porous media are 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:

[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 of

[0064] For the non-associated flow rule, for the two-dimensional case, the yield function F and plastic potential G of the non-associated 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, is considered as a constant and passed

[0069]

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

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

[0072] The mixed triangle unit 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 simplicity, the intermediate variable and is the vector of intermediate variables at the element nodes. N σ ,Nu ,N γ ,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 equation (40), the underlined term represents the displacement boundary condition, and the newly introduced variable r n+1 is the reaction force at the node of the element with the prescribed displacement, E u is an index matrix consisting of 1s and 0s, indicating nodes with / without the specified displacements, verifying its correctness and deriving the associated Lagrangian with respect to the field variables Partial derivative of:

[0082]

[0083] in, is the specified displacement The relationship is clearly 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,...,xn ) 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] Rotate 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] Equation (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 displacements obtained by analysis;

[0108] Based on the updated grid nodes, identify the boundary of the computational region;

[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 grid.

[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 is a hybrid of the particle method and Lagrangian FEM, which regards the 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 boundaries 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 geometric parameters and boundary conditions of the dynamic analysis problem to be solved in the saturated porous medium;

[0117] A model building module, used to build 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 the first 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, and 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 technical solution of the electronic device provided by the present invention is as follows:

[0123] The electronic device provided by the present invention comprises 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 saturated porous media dynamic analysis method based on the generalized Hellinger-Reissner variational principle and particle finite element method (PFEM), which has the following beneficial effects compared with the existing technology:

[0125] (1) Based on the UP model of dynamic analysis of saturated porous media, the present invention derives the minimum-maximum problem of the time-discrete control equation equivalent to the dynamic analysis in saturated porous media, and establishes the 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 converted into a standard second-order cone programming (SOCP) problem that can be solved using the interior point method.

[0127] (3) Compared with traditional finite element algorithms, SOCP has advantages. 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 formulas obtained in SCOP are incorporated into the PFEM for the dynamic problem of saturated porous media with large deformation and free surface evolution. This allows the problem to be coupled to the framework of PFEM and to gain the advantages of the particle method in dealing with large deformation problems, 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 of ordinary skill in the art by reading the detailed description of the preferred embodiments below. The accompanying drawings are only for the purpose of illustrating the preferred embodiments and are not to be considered as limiting the present invention. Moreover, the same reference symbols are used throughout the accompanying drawings to represent the same components. In the accompanying 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 the prescribed displacement, traction, pore water pressure and fluid flux respectively;

[0132] Figure 3 is an example of a hybrid 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 sedimentation 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 1 s (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 It is the evolution of displacement at point A and pore water pressure at point B obtained by simulation using different time steps and time integration parameters;

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

[0139] Fig.10 It is a schematic diagram of the signal flow relationship between various functional modules in the saturated porous medium dynamic analysis device provided by an embodiment of the present invention;

[0140] Fig.11 A 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 is of great significance for the accurate and rapid solution of saturated porous medium dynamic problems, and is therefore more suitable for practical use.

[0142] In order to further explain the technical means and effects adopted by the present invention to achieve the predetermined invention purpose, the following is a detailed description of the saturated porous medium dynamic analysis method, device, storage medium and electronic device proposed by the present invention, its specific implementation, structure, characteristics and effects in combination with the accompanying drawings and preferred embodiments. In the following description, different "one embodiment" or "embodiment" does not necessarily refer to the same embodiment. In addition, the features, structures, or characteristics of one or more embodiments may be combined in any suitable form.

[0143] The term "and / or" in this article is merely a description of the association relationship of associated objects, indicating that three relationships may exist, for example, A and / or B. Specifically, it is understood that: 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 be met.

[0144] The present invention proposes a saturated porous media dynamic analysis method based on the generalized Hellinger-Reissner variational principle and particle finite element (PFEM), wherein 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; further, based on the mixed triangular unit, the saturated porous media dynamic analysis problem can construct a standard second-order cone programming problem based on the FEM framework, which is further introduced into the PFEM framework, thereby establishing a second-order cone programming particle finite element method suitable for classical continua; further, 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 of dynamic analysis of saturated porous media (up form of Boit equation), the control equation of saturated porous media is 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, the minimum-maximum problem equivalent to the time-discrete control equation of dynamic analysis in saturated porous media is derived, and the Lagrangian of this minimum-maximum problem is constructed. The feasibility of this 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; 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 time discretize 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 denote 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] Step 13, now we can 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, 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] Obviously, Eq. (21) consists of the discretized linear equilibrium equations of the mixture, Eq. (16) and the boundary conditions in (20); Eqs. (25), (28)-(30) are the incremental forms of the constitutive equations; Eq. (26) represents the governing equations in (17) and the boundary conditions related to the fluid flux; Eq. (27) is the inertial force γ n+1 expression.

[0186] Note that the above is for associated plastic flow. Considering the non-associated flow rules, specifically, for the two-dimensional case, the yield function F and plastic potential G of the non-associated 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, is considered as a constant and passed

[0191]

[0192] The subscript 0 in equation (32) represents 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 invention using mixed triangular elements. The discretized optimization problem is then transformed into a standard second-order cone programming problem, which can be solved using existing optimization engines.

[0195] Step 21, discretization using Figure 3 The hybrid triangular element 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 simplicity, the intermediate variable and is the vector of intermediate variables at the 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 formulas (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 node of the element with the prescribed displacement. E u is an index matrix consisting of 1s and 0s, indicating nodes with / without the prescribed displacement. To verify its correctness, the associated Lagrangian with respect to the field variables is derived The partial derivative of

[0206]

[0207] in, is the specified displacement The discrete form of , and this relationship 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] Rotate 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, 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] Equation (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 in the saturated porous medium that needs to be solved (such as debris flow, dam rupture and particle column collapse, etc.), establish a two-dimensional numerical analysis model and bring it into the framework of particle finite element method (PFEM) for solution.

[0230] In this step, the particle finite element method (PFEM) is a hybrid of the particle method and Lagrangian FEM, which treats the mesh nodes as free particles but solves 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 includes: (a) updating the mesh node positions according to the incremental displacements obtained from 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 parts of the computational domain as well as the separation and reconnection of individual isolated particles, which can occur in fluid dynamics problems (such as wave breaking and water splashing) and geotechnical problems (such as granular flows and landslides). After separation, the motion of the isolated particle 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 operated. 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 acceptable limits. 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 continuation 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 weight of water γ f=10kN / m 3 , the weight of unsaturated soil is γ unsa =18.5kN / m 3 , the density of saturated soil in the slope is γ sa1 =20kN / m 3 , the weight of saturated soil in 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 = 0.01s. The final sedimentation profile and displacement distribution after the 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 saturated soil embankment dynamic analysis method proposed in the present invention.

[0233] Example 2

[0234] This example uses the method proposed in this invention 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, a 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 is used instead of θ 1,2,3 ) are all 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=5×10-3s, and the displacement of the upper surface (such as point A (x=1m, y=10m)) and the pressure at the bottom (such as point B (x=1m, y=0m)) are monitored. Selected parameters: Density of the mixture ρ=2000kg / m 3 , 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-integrated 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 displacements are 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 study 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 with the time integration 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 the surface load, and its analytical solution is available. The problem is considered as a plane strain problem with surface load f2(t). The setting 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 Fig. 9 ).

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

[0236] like Fig.10 As 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 optional user interface 1003 may also 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 Fig.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 less components than those shown in the figure, or combine certain components, or arrange the components differently.

[0238] like Fig.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 Fig.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 other 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 can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A method for dynamic analysis of saturated porous media, characterized in that: The following steps are involved: Obtaining geometric parameters and boundary conditions of the dynamic analysis problem to be solved in saturated porous media; According to the geometric parameters and boundary conditions, a two-dimensional numerical analysis model is established; The two-dimensional numerical analysis model is brought into the framework of particle finite element method to solve and obtain the dynamic analysis conclusion of the saturated porous medium; The example problems of dynamic analysis in saturated porous media include the first one or more of debris flow, dam destruction and particle column collapse.

2. The saturated porous media dynamic analysis method according to claim 1, characterized in that: The following steps are also included: Based on the UP model of dynamic analysis of saturated porous media, the control equation of saturated porous media is given. The standard θ-method is introduced into the control equation to time discretize the effective stress σ', velocity v and pressure p in the UP model, and the minimum-maximum problem equivalent to the time-discrete control equation of dynamic analysis in saturated porous media is derived; The hybrid triangular unit is used to spatially discretize the control equations of dynamic analysis of saturated porous media after time discretization in step one. Based on the Hellinger-Reissner variational criterion of static elasticity, the 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.

3. The saturated porous media dynamic analysis method according to claim 2, characterized in that: A generalized Hellinger-Reisner variational criterion for dynamic analysis of saturated porous media is established. With the help of mathematical programming methods, it is transformed into a maximum-minimization problem, in which 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: In the formula, σ is the Cauchy stress, u is the displacement, b is the body force, and t is the surface force. is the elastic flexibility matrix, Ω and Γ are the integration domain and boundary conditions, respectively, 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 of elasticity mechanics, a generalized HR variational principle has been developed to analyze elastic-plastic problems, elastic-viscoplastic problems and quasi-static porous elastic-plastic problems.

4. The saturated porous media dynamic analysis method according to claim 2, characterized in that: Based on the up model of dynamic analysis of saturated porous media, the control equation of saturated porous media is given. Relevant data are introduced into the control equation, and the minimum-maximum problem equivalent to the time-discrete control equation of dynamic analysis in saturated porous media is derived as follows: 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 Darcy's Law Pore ​​fluid mass balance equation 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; 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 strain, defined as ε = Su, from 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], 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 superficial velocity w in the above equation (5) into equation (5): The boundary conditions are: in and are the specified 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 time discretize the effective stress σ', velocity v and pressure p: σ′=θ1σ′ n+1 +(1-θ1)σ′ n (13) p=θ3p n+1 +(1-θ3)p n (15) Subscripts n and n+1 denote 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): 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 obtained as the equivalent min-max problem: The validity of the min-max problem is demonstrated by the associated Karush-Kuhn-Tucher condition: For the associated plastic flow, the Lagrangian expression of problem (22) is constructed as: 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 control equations in (17) and the boundary conditions related to the fluid flux; equation (27) is the inertial force γ n+1 The expression of For the non-associated flow rule, for the two-dimensional case, the yield function F and plastic potential G of the non-associated 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, is considered as a constant and passed The subscript 0 in equation (34) represents the state solved from the previous analysis step.

5. The saturated porous media dynamic analysis method according to claim 2, characterized in that: 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, 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 simplicity, an intermediate variable is introduced 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; Step 32, replace formulas (35)-(39) into the optimization problem (22) to obtain its discretized form: Limited by Among them, n σ is the total number of Gaussian integration points, where: In equation (40), 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 specified displacement, E u is an index matrix consisting of 1s and 0s, indicating nodes with / without the specified displacements. Verify its correctness and derive the associated Lagrangian with respect to the field variables Partial derivative of: in, is the specified 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 to produce 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 can be of the following types: Standard quadratic cone: Rotate 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): ρ=(ρ1,ρ2,ρ3),σ=(σ′ xx ,s′ yy ,s′ zz ,s′ xy ) (48) The planning problem (47) conforms to the standard second-order cone programming form.

6. The saturated porous media dynamic analysis method according to claim 1, characterized in that: At a given time interval, the PFEM solution consists of the following steps: Update the mesh node positions according to the incremental displacements obtained by analysis; Based on the updated grid nodes, identify the boundary of the computational region; 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 grid.

7. 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 for solution, specifically: The particle finite element method (PFEM) is a hybrid of the particle method and the Lagrangian FEM, which regards the mesh nodes as free particles but uses the Lagrangian FEM to solve the control equations. The basic steps of PFEM are to use the Alpha-shape method to identify the boundary of the computational domain, and then use the mesh generated by Delaunay triangulation to perform Lagrangian finite element analysis.

8. A saturated porous medium dynamic analysis device, characterized in that: include: A data acquisition module is used to obtain geometric parameters and boundary conditions of the dynamic analysis problem to be solved in the saturated porous medium; 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 result of the saturated porous medium; The example problems of dynamic analysis in saturated porous media include the first one or more of debris flow, dam destruction and particle column collapse.

9. A computer-readable storage medium, characterized in that: The computer-readable storage medium 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 described in any one of claims 1-7 are implemented.

10. An electronic device, characterized in that: It 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 described in any one of claims 1 to 7 are implemented.

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