CN-122021104-A - Node integration-based particle earth-rock dam large deformation calculation method

CN122021104ACN 122021104 ACN122021104 ACN 122021104ACN-122021104-A

Abstract

The invention discloses a node integration-based particle earth-rock dam large deformation calculation method, and relates to the technical field of geotechnical engineering large deformation. The method comprises the steps of establishing a generalized shape function and a strain displacement matrix of a proportional boundary finite element SBFEM under a Laplace equation, projecting state variables at Gaussian integral points of a triangle SBFEM unit to nodes under an updated Laplace frame through a strain smoothing technology, converting a momentum equation into a balance equation based on node integral through a virtual work principle and a Green divergence theorem, solving a power equation by utilizing a central difference method, and discarding a current grid and carrying out re-grid division when the grid distortion degree reaches a significant level. Since all state variables such as stress, displacement, speed and the like are concentrated on the nodes of the discarded grid, the subsequent grid generation and calculation process can be continued seamlessly. The invention realizes the continuous simulation of the whole large deformation process based on the proportional boundary finite element method.

Inventors

GAN LEI
SONG XINWEI
ZHANG WENQIANG
LIU JUN
YE WENBIN
WANG PEIQING
LIU XIAOLI
CHEN LIANG
ZHANG ZHEN
SUN YIQING

Assignees

西藏农牧大学
大连理工大学
河海大学

Dates

Publication Date: 20260512
Application Date: 20251217

Claims (9)

1. The method for calculating the large deformation of the particle earth-rock dam based on node integration is characterized by comprising the following steps of: Establishing a proportional boundary finite element discrete model, constructing a geometric model and a local coordinate system of a triangle unit, and deriving a generalized shape function and a unit strain displacement matrix based on a Laplace equation; projecting the strain displacement matrix to a node through a strain smoothing technology, constructing a non-overlapping smoothing unit taking the node as a center, and calculating the node smoothing strain; The principle of virtual work and the principle of green's divergence are combined, and a momentum conservation differential equation is discretized into a node integral balance equation comprising a node mass matrix, an external force vector and an internal force vector; Performing time dispersion on the balance equation by adopting an explicit center difference method, and iteratively updating speed, displacement and stress state variables according to CFL (computational fluid dynamics) stability time step; the mesh quality is evaluated after each time step, and when the mesh distortion exceeds a threshold, a high quality triangle mesh is regenerated based on the state variables stored by the nodes until all calculations are completed.
2. The method for calculating the large deformation of the particle-class earth-rock dam based on node integration according to claim 1, wherein the method is characterized in that the method comprises the following steps of The triangle proportion boundary finite element unit is adopted to carry out discretization to establish a proportion boundary finite element coordinate system The expression for interpolating each cell using a shape function is: Wherein the method comprises the steps of For a similar center coordinate, As a linear cell shape function.
3. The method for calculating the large deformation of the particle-class earth-rock dam based on node integration according to claim 2, wherein the relation of coordinate transformation under the Cartesian coordinate system is as follows: Wherein, the Jacobian matrix for coordinate transformation with determinant as 。
4. The method for calculating the large deformation of the particle earth-rock dam based on node integration according to claim 3, wherein the corresponding unit control equation is derived by using a laplace equation, specifically: Wherein the method comprises the steps of For any scalar field variable, In the case of a differential operator, , And Is a coefficient matrix of the cell.
5. The method for calculating the large deformation of the particle-class earth-rock dam based on node integration according to claim 4, wherein the step of introducing the variable is as follows And internal flux Composition variable The Laplace equation is converted into a first-order ordinary differential equation, specifically: Wherein, the Representing intermediate variables With respect to coordinates Is used for the first order partial derivative of (a), For the coefficient matrix of the structural domain, the superscript-1 represents the inverse sign of the matrix; For a pair of Performing eigenvalue decomposition to obtain corresponding eigenvalues and eigenvectors: Wherein, the And Is a eigenvector matrix and an eigenvalue matrix corresponding to the finite field.
6. The method for computing large deformation of a particle-based earth-rock dam based on node integration according to claim 5, wherein the radial analytic function is obtained by eigenvalue decomposition The method comprises the following steps of: The obtained proportional boundary finite element generalized shape function is as follows: For any point coordinates, the coordinates are also expressed as a form function interpolation form, and the formula is as follows: Wherein, the Representing node coordinates on the boundary, using the chain law, the corresponding coordinate transformation from the reference coordinate system to the physical coordinate system is: in particular, for a triangle scale boundary finite element, the transformation matrix and corresponding strain displacement matrix of the triangle scale boundary finite element are: Wherein, the Is that Is used for the inverse matrix of (a), Representing the strain displacement matrix of the ith node of the corresponding triangle cell.
7. The method for calculating the large deformation of the particle-class earth-rock dam based on node integration according to claim 1, wherein the method is characterized in that the strain displacement matrix of all units around the node is concentrated on the node corresponding to the smooth strain unit, and specifically comprises the following steps: Wherein, the Is a node Is used for the smooth strain of (c), Representing nodes Is used for the purpose of determining the coordinates of (a), Representation and node The number of all nodes to be connected, Representing smooth strain cells The strain displacement matrix of (2) is given by: Wherein, the Representing smooth strain cells Is defined by the area of the (c), Indicating the boundary of the two-dimensional space, Representing the projected length of the normal unit vector in either the x or y direction, Is a node In the corresponding smoothing unit, the first The individual nodes are at The strain displacement coefficient of the direction is set, Is a node In the corresponding smoothing unit, the first The individual nodes are at Strain displacement coefficient of direction.
8. The method for calculating the large deformation of the particle earth-rock dam based on node integration according to claim 1, wherein the formula of the continuous medium momentum equation is as follows: Wherein, the Indicating the density of the material and, Indicating the acceleration of the material, Is the second-order cauchy stress tensor, Is a physical load; The momentum equation is converted into the following by adopting the virtual work principle and the green's theorem of divergence: after introducing strain smooth node integration, the method is converted into Wherein the method comprises the steps of Is the load of the external force, Is the force of the internal node and, Is the quality of the node and, Or (b) Is the node stress.
9. The method for calculating the large deformation of the particle-class earth-rock dam based on node integration according to claim 8, wherein the time increment is defined as: Thus, the speed and displacement of the next time step are correspondingly obtained as: consider a time stabilization step size of: Wherein the method comprises the steps of Is the stability factor of the light-emitting diode, For the length of the unit feature, Is the current wave speed.

Description

Node integration-based particle earth-rock dam large deformation calculation method Technical Field The invention relates to the technical field of large deformation of geotechnical engineering, in particular to a method for calculating large deformation of a particle earth-rock dam based on node integration. Background In the field of earth and rockfill dam engineering, large deformation disasters such as dam landslide, dam slope instability, seismic liquefaction induced dam break and the like frequently occur, the problems are directly related to stability, seepage resistance and downstream area safety of a dam structure, and the accurate numerical representation of the mechanical state in the evolution process is a core engineering requirement to be solved urgently. Although the traditional small deformation theory (such as a classical finite element method) is widely applied to the scenes of static analysis of a conventional earth-rock dam and the like, the method has obvious limitation under large strain that serious grid distortion accompanied by large deformation can lead to rapid reduction of numerical accuracy, even causes solution termination, and cannot accurately capture complex mechanical behaviors of earth-rock dam building materials (such as piled stones and clay core materials). To address the above challenges, a number of classes of numerical methods have been developed in the prior art, but all suffer from significant drawbacks. The method is a grid method, such as random Lagrange-Euler method, which is used for simulating fluid and solid by respectively using fixed Euler grids and Lagrange motions through dynamically adjusting node position balance material tracking and grid stability, but both methods depend on grid topology, and complex self-adaptive repainting is still needed under a large deformation scene of coupling an earth-rock dam body and a foundation. The other type is a particle method, such as a smooth particle fluid dynamic method, wherein a continuum is scattered into particles, the particles are approximated to a field variable through a kernel function, but the problem of numerical diffusion exists, the internal stress transfer characteristic of a dam body of the earth and rockfill dam is difficult to accurately reflect, while the Galerkin grid-free method is free of grid dependence, a complex shape function structure is needed, the calculation efficiency is low, and the requirement of accurate transfer of stress strain in the large deformation process of the earth and rockfill dam cannot be met. In addition, although the mixing method such as a material point method and a particle finite element method fuses the advantages of particle tracking and grid solving, the state variable mapping of the material point method is easy to generate errors, the nonlinear constitutive relation of the earth-rock dam building material is difficult to adapt, the gradual grid regeneration process of the particle finite element is complicated, and the accuracy and the efficiency of the earth-rock dam large deformation analysis are difficult to be considered. The proportional boundary finite element method (SBFEM) is used as a semi-analytical method, only discrete unit boundaries are needed, analytical solutions are adopted along the radial direction, the method is excellent in wave propagation and stress singularity problems, and the method is expanded to the fields of plate-shell mechanics, fluid-solid coupling and the like in recent years. However, when SBFEM is expanded to a medium-limited strain scene, the deformation rigidity is represented by combining a geometric rigidity matrix and a material rigidity matrix, and under the extremely large deformation scene such as integral sliding, local breaking and the like of the earth-rock dam body, the calculation reliability cannot be ensured due to grid distortion, the core requirement of the extremely large deformation analysis of the earth-rock dam engineering is difficult to adapt, and the application range of the method in the field of the earth-rock dam engineering is greatly limited by the defect. Therefore, there is a need for a numerical simulation method that can combine SBFEM analytic advantages and particle method flexibility and solve the problems of grid distortion, calculation interruption and precision efficiency imbalance under large deformation of earth-rock dams Disclosure of Invention Based on the technical problems, the application discloses a method for calculating the large deformation of a particle earth-rock dam based on node integration, which comprises the following steps: Establishing a proportional boundary finite element discrete model, constructing a geometric model and a local coordinate system of a triangle unit, and deriving a generalized shape function and a unit strain displacement matrix based on a Laplace equation; projecting the strain displacement matrix to a node through a strain smoothing technology, c