CN109408973B - Distance potential function based two-dimensional deformable convex polygon block discrete unit method - Google Patents
Distance potential function based two-dimensional deformable convex polygon block discrete unit method Download PDFInfo
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Abstract
The invention discloses a two-dimensional deformable convex polygon block discrete unit method based on a distance potential function, which comprises the following steps: establishing a deformable discrete unit system, and determining the time step of the system; calculating the contact force of each contact unit and the target unit at the current time step, and converting the contact force into an equivalent node force vector of the load; establishing a power control equation in a system increment form by using the equivalent node force vector of the load to obtain the displacement of the finite element; updating coordinate information of each finite element according to the displacement of the finite element; and repeating the calculation until all time steps are calculated. The method realizes the deformation of the discrete element system, realizes the contact detection and contact force calculation of the units with different sizes and forms, reduces the number of actual division units, improves the calculation efficiency, enables the discrete element model to more accurately reflect the stress and strain conditions in the block body, and can be used for simulating more engineering actual problems.
Description
Technical Field
The invention relates to a two-dimensional deformable convex polygon block discrete unit method based on a distance potential function, and belongs to the technical field of deformable discrete elements.
Background
The discrete element unit method is a numerical simulation method specially used for solving the problem of discontinuous media, and can accurately capture the discontinuous deformation characteristics of separation, slippage damage, overturning rotation and the like of a block system. While the deformable discrete elements may be compressed, separated, or slid. The discrete element can simulate the large deformation characteristic of the rock mass more truly, so that the discrete element method is greatly developed in a plurality of fields such as theoretical research, engineering application and the like.
The finite discrete element method is currently proposed by professor a. munji za, uk, by dividing the study object into tetrahedral bulk elements of uniform size and establishing a potential function definition based on the centroid of the elements to calculate the contact force between the elements.
MUNJIZA teaches a deformable discrete element based on a potential function method, and combines a discrete element method and a finite element method to solve the problem of the deformable discrete element. Munjiza solves finite elements using an explicit solution, avoiding an iterative process for solving a finite element nonlinear equation set. There are still some problems: the tetrahedral units with uniform sizes are applied, on one hand, the model is not consistent with the actual situation, and on the other hand, when the tetrahedral units are applied in actual application, the number of the divided block units can be greatly increased by the uniform unit size and the simplest unit form, and the calculation efficiency is reduced. Discrete cell methods based on distance potential functions solve these problems, but do not take into account the deformability of the discrete elements and are therefore not fully in accordance with engineering practice.
Disclosure of Invention
The invention aims to solve the technical problem that any convex polygonal block discrete unit method based on a distance potential function is not deformable in the prior art, provides a two-dimensional deformable convex polygonal block discrete unit method based on the distance potential function, and overcomes the defects of the discrete unit method so that numerical simulation is more practical.
The invention adopts the following technical scheme for solving the technical problems:
the invention provides a two-dimensional deformable convex polygon block discrete unit method based on a distance potential function, which comprises the following steps of:
the method comprises the following steps: establishing a deformable discrete unit system, wherein the system comprises a plurality of discrete units and finite units formed by meshing the discrete units;
step two: determining a time step delta t of the system;
step three: carrying out contact detection on the grid cells on one layer of the periphery of the discrete cells;
step four: converting the resultant force of the contact forces acting on the contact unit and the target unit obtained by calculation in the third step into an equivalent node force vector of the current moment load of the grid unit by using a shape function
Step five: establishing a power control equation in a system increment form according to the equivalent node force vector of the load obtained by calculation in the step four, and solving to obtain the displacement of the t + delta t finite element at the next moment;
step six: updating the coordinate information of the vertex of each grid unit according to the displacement of the limited unit in the step five, and finishing the calculation at the current moment;
step six: and repeating the calculation from the second step to the sixth step until all time steps are calculated.
Further, in step two, the time step Δ t must satisfy:
Δt＝min(ΔtD,Δts)，
Δts≤L/C，
wherein, Δ tDCalculating a time step for a discrete unit; ξ is the damping ratio of a discrete unit and
m is the mass of the discrete unit block, c is the damping coefficient, k is the stiffness coefficient, Δ tsIs the time step of the finite element, L is the minimum side length of all finite elements, and C is 10000.
Further, in step four, a contact force algorithm of a discrete unit method based on a distance potential function is adopted when the contact force of each contact unit and the target unit at the current moment is calculated.
Further, in step five, a motion control equation in a system increment form is established, and the displacement of the finite element at the next time t + Δ t is obtained by solving, specifically including:
step 5.1: establishing a reference coordinate system, and selecting the configuration before deformation as a reference configuration;
step 5.2: adopting a generalized Newmark method to carry out time domain dispersion and predicting the mechanical quantity at the current moment;
step 5.3: establishing a power control equation in the form of system increments
Solving to obtain the acceleration increment from the current time t to the next time t + delta t
Where M is the mass matrix of the finite elements, D is the damping matrix of the finite elements, K is the stiffness matrix of the finite elements,
is the acceleration increment of a finite element,
the speed increment of the finite element, the delta u is the displacement increment of the finite element, and the equivalent node force vector R of the load of the grid element at the current moment;
step 5.4: performing time domain dispersion by a generalized Newmark method, and calculating the displacement of each finite element at the next moment t + delta t, wherein the displacement of each grid element at the next moment t + delta t is obtained through calculation because the grid elements are consistent with the node coordinates of the finite elements;
further, in step six, the coordinate formula of the vertex of each mesh unit is updated as follows:
x(t+Δt)＝x(t)+(r(t+Δt))x
y(t+Δt)＝y(t)+(r(t+Δt))y
wherein x (t + Δ t), y (t + Δ t) are the coordinates x (t) of the vertex of the grid unit at the current time step, y (t) is the coordinates (r (t + Δ t))x、(r(t+Δt))yThe components of the displacement of the grid cell in the x, y directions, respectively.
The invention achieves the technical effects that: the method realizes the deformation of the discrete element system, so that the discrete element model can more accurately reflect the stress and strain conditions in the block body, and can be used for simulating more practical engineering problems; the contact detection and contact force calculation problems of different size and form units are realized, the number of actually divided units is reduced, and the calculation efficiency is improved.
Drawings
FIG. 1 is a schematic diagram of a method of an embodiment of the present invention in which two discrete units are further subdivided into finite elements;
FIG. 2 is a schematic view of the contact overlap of a contact unit and a target unit according to an embodiment of the method of the present invention;
3-6 are schematic diagrams illustrating the process of landslide destruction of rock slopes at different calculation times;
FIG. 3 illustrates a stabilized condition of the rock slopes of the exemplary embodiment;
fig. 4-6 show the movement of the slope under the influence of external conditions along the sliding surface in the embodiment.
Detailed Description
The technical scheme of the invention is further explained in detail by combining the attached drawings:
the invention provides a two-dimensional deformable convex polygon block discrete unit method based on a distance potential function, which comprises the following steps of:
step one, establishing a deformable discrete unit system, as shown in fig. 1, wherein the system comprises a plurality of discrete units and a finite unit formed by meshing the discrete units; each grid after the grid is divided by the discrete unit is defined as a grid unit, the node coordinate of the grid unit is consistent with the node coordinate of the finite unit, wherein the parameters of the discrete unit comprise the node coordinate, the mass, the damping ratio and the rigidity of the discrete unit, and the parameters of the finite unit comprise the node coordinate, the mass matrix, the damping matrix and the rigidity matrix of the finite unit;
step two, timing step length, calculating time step length delta t and satisfying the following requirements:
Δt＝min(ΔtD,Δts)
Δts≤L/C
wherein, Δ tDCalculating a time step for the discrete element; ξ is the damping ratio of the system,
m is the mass of the block unit, c is the damping coefficient, k is the stiffness coefficient, Δ tsThe time step length of the finite element is shown, L is the minimum side length of all finite elements, and the value of C is 10000.
And step three, performing contact detection on the grid cells on one layer of the periphery of the discrete unit to determine a target unit and a contact unit, wherein the grid cells which are contacted with one layer of the periphery of the discrete unit are defined as the target unit, and the grid cells which are contacted with the target unit and are positioned on one layer of the periphery of the discrete unit are defined as the contact unit. Calculating the normal contact force and the tangential contact force of the current time step acting on the target unit and the contact unit according to the definition of the arbitrary convex polygonal block discrete unit method based on the distance potential function as shown in FIG. 2;
step four, converting the resultant force of the contact forces acting on the contact unit and the target unit calculated in the step three into an equivalent node force vector of the current moment load of the grid unit by using a shape function
Wherein,
is the equivalent node force vector of the current moment load of the grid cell,
and
load vectors of grid cell physical force and surface force at the current moment respectively, N is a shape function of grid cell nodes, V0Is the volume of the grid cell, A0tIs the surface area of the grid cell at the current time t, A0Is the surface area of the grid cell;
step five, solving a power control equation in a system increment form by the equivalent node force vector of the load obtained by calculation in the step four to obtain the displacement of the finite element;
step 5.1: establishing a reference coordinate system, and selecting the configuration before deformation as a reference configuration;
step 5.2: adopting a generalized Newmark method to carry out time domain dispersion and predicting the mechanical quantity at the current moment;
step 5.3: establishing a power control equation in the form of system increments
Solving to obtain the acceleration increment from the current time t to the next time t + delta t
Where M is the mass matrix of the finite elements, D is the damping matrix of the finite elements, K is the stiffness matrix of the finite elements,
is the acceleration increment of a finite element,
is the velocity increment of the finite element, and Δ u is the displacement increment of the finite element;
step 5.4: performing time domain dispersion by a generalized Newmark method, and calculating the displacement of each finite element at the next moment t + delta t, wherein the displacement of each grid element at the next moment t + delta t is obtained through calculation because the grid elements are consistent with the node coordinates of the finite elements;
in other embodiments, other values of the variables of the finite elements, such as velocity or acceleration at time t + Δ t for each grid element, may be solved for by the power control equations in the form of system increments proposed by the present invention.
Step six, updating the geometrical information such as the coordinates of the vertexes of each finite element according to the displacement of the finite element in the step five, and finishing the calculation of the current time step;
the coordinate formula for updating the vertices of each mesh cell is:
x(t+Δt)＝x(t)+(r(t+Δt))x
y(t+Δt)＝y(t)+(r(t+Δt))y
wherein x (t + Δ t), y (t + Δ t) are the coordinates x (t) of the vertex of the grid unit at the current time step, y (t) is the coordinates (r (t + Δ t))x、 (r(t+Δt))yThe components of the displacement of the grid cell in the x, y directions, respectively.
And step seven, repeating the step two to the step six to calculate the next time step until all the time steps are calculated.
Example (b):
due to the existence of a weak interlayer in a rock slope, the rock slope may cause landslide geological disasters under the influence of external conditions such as earthquake, rainfall and the like. By adopting the method provided by the invention, a discrete unit model is established for the rock slope, and as shown in fig. 3, 130 limited units are divided into a discrete unit landslide body and a slope body, so that 130 grid units are also provided.
Fig. 3 shows the rock slope in a stable state.
Fig. 4 to 6 show the movement of the slope under the influence of external conditions along the sliding surface to produce a slide. When the landslide body discrete unit starts to slide under the action of gravity, the landslide body discrete unit is in contact with the slope body, contact forces are generated between the landslide body discrete unit and between the landslide body discrete unit and the slope body discrete unit due to contact, the contact forces are converted to the limited unit through the equivalent node vector of the load, the displacement of the limited unit is changed, the node coordinates of the grid unit are updated, the node coordinates of the discrete unit are updated, contact detection is carried out again, new contact is generated, new contact forces are generated, the displacement of the limited unit is changed again, and the process is circulated until the movement is finished. The distance potential function two-dimensional deformable block discrete unit method provided by the invention is used for simulating the landslide process of the rock slope, can clearly describe the process of the rock slope damaged along the sliding surface under the influence of adverse load, can well analyze whether the rock slope is safe under the load, and can visually display the landslide damage process and the form, the volume, the scale and the like of an accumulation body formed by the landslide body if the landslide damage process is generated.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.
Claims (6)
1. A two-dimensional deformable convex polygon block discrete unit method based on a distance potential function is characterized by comprising the following steps:
the method comprises the following steps: establishing a deformable discrete unit system, wherein the deformable discrete unit system comprises a plurality of discrete units and a finite unit formed by meshing the discrete units;
step two: determining a time step delta t of the system;
step three: carrying out contact detection on grid cells on one layer of the periphery of all discrete cells to determine a target cell and a contact cell, defining the grid cell which is in contact with one layer of the periphery of the discrete cells as the target cell, defining the grid cell which is in contact with the target cell and is on one layer of the periphery of the discrete cells as the contact cell, and calculating the normal contact force and the tangential contact force which are applied to the target cell and the contact cell at the current time step:
step four: converting the resultant force of the contact forces acting on the contact unit and the target unit obtained by calculation in the third step into an equivalent node force vector of the current moment load of the grid unit by using a shape function
The expression is as follows:
wherein,
is the equivalent node force vector of the current moment load of the grid cell,
and
load vectors of grid cell physical force and surface force at the current moment respectively, N is a shape function of grid cell nodes, V0Is the volume of the grid cell, A0tIs the surface area of the grid cell at the current time t, A0Is the surface area of the grid cell;
step five: establishing a power control equation in a system increment form according to the equivalent node force vector of the load obtained by calculation in the step four, and solving to obtain the displacement of the t + delta t finite element at the next moment;
step six: updating the coordinate information of the vertex of each grid unit according to the displacement of the limited unit in the step five, and finishing the calculation at the current moment;
step seven: and repeating the calculation from the second step to the sixth step until all time steps are calculated.
2. A two-dimensional deformable convex polygon block discrete unit method based on distance potential function according to claim 1, wherein the time step Δ t is satisfied:
Δt＝min(ΔtD,Δts)，
Δts≤L/C，
wherein, Δ tDCalculating a time step for a discrete unit; ξ is the damping ratio of a discrete unit and
m is the mass of the discrete unit block, c is the damping coefficient, k is the stiffness coefficient, Δ tsIs a limited sheetThe time step of the element, L is the minimum side length of all finite units, and C is 10000.
3. The two-dimensional deformable convex polygon bulk discrete unit method based on the distance potential function of claim 1, wherein in step three, a contact force algorithm of the discrete unit method based on the distance potential function is adopted when calculating the contact force between each contact unit and the target unit at the current moment.
4. The two-dimensional deformable convex polygon block discrete unit method based on the distance potential function according to claim 1, wherein in the fifth step, a motion control equation in a system increment form is established, and the displacement of the finite unit at the next time t + Δ t is obtained by solving, specifically comprising:
step 5.1: establishing a reference coordinate system, and selecting the configuration before deformation as a reference configuration;
step 5.2: adopting a generalized Newmark method to carry out time domain dispersion and predicting the mechanical quantity at the current moment;
step 5.3: establishing a power control equation in the form of system increments
Solving to obtain the acceleration increment from the current time t to the next time t + delta t
Where M is the mass matrix of the finite elements, D is the damping matrix of the finite elements, K is the stiffness matrix of the finite elements,
is the acceleration increment of a finite element,
is the velocity increment of the finite element, and Δ u is the displacement increment of the finite element;
step 5.4: and then carrying out time domain dispersion by a generalized Newmark method, and calculating the displacement of each finite element at the next moment t + delta t, wherein the displacement of each grid element at the next moment t + delta t is obtained by calculation because the grid elements are consistent with the node coordinates of the finite elements.
5. A two-dimensional deformable convex polygon block discrete unit method based on distance potential function as claimed in claim 1, wherein in the fifth step, the coordinate formula of the vertex of each finite unit is updated as follows:
x(t+Δt)＝x(t)+(r(t+Δt))x
y(t+Δt)＝y(t)+(r(t+Δt))y
wherein x (t + Δ t), y (t + Δ t) are coordinates of the vertex of the mesh unit at the current time step; x (t), y (t) are the coordinates of the vertices of the block unit at the last time step; (r (t + Deltat))x、(r(t+Δt))yThe components of the displacement of the grid cell in the x, y directions, respectively.
6. The method for the two-dimensional deformable convex polygonal block discrete unit based on the distance potential function of claim 1, wherein the parameters of the discrete unit comprise node coordinates, mass, damping ratio and rigidity of the discrete unit; the parameters of the finite element comprise node coordinates, a mass matrix, a damping matrix and a rigidity matrix of the finite element.
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