Fully mechanized caving mining coal caving rule prediction method and system based on graph neural network

Abstract

The invention discloses a fully-mechanized caving mining coal caving rule prediction method and system based on a graph neural network, and relates to the field of mining engineering top coal caving mining, and the method comprises the steps: carrying out a numerical simulation test of horizontal segment fully-mechanized caving mining through discrete element software, and building a graph structure data set based on a test result; using the graph structure data set to train a graph neural network, and when the loss function converges, obtaining a trained graph neural network; nodes of the graph are positions and physical information of particles in discrete elements, edges of the graph are contact states of the particles, and global features of the graph are segment height, coal seam thickness, coal seam inclination angle and time step of the horizontal segment fully mechanized caving face; particle node position information and a coal rock interface after coal caving are obtained through the trained graph neural network, and a released body form is obtained through inversion of released particle coordinates. According to the method, the top coal caving body and the coal rock interface can be accurately predicted, and a quick and efficient research means is provided for researching the complex coal caving rule of horizontal section fully mechanized caving mining.

Description

Technical Field

[0001] The present invention relates to the technical field of top coal caving mining in mining engineering, and more specifically, to a method and system for predicting the coal caving law of fully mechanized top coal caving mining based on a graph neural network. Background Art

[0002] The fully mechanized top coal caving mining technology is an effective method for mining thick coal seams. Among them, the horizontal sectional fully mechanized top coal caving mining technology is one of the main methods for mining thick coal seams. Studying the top coal caving law of the top coal caving face is of great significance for guiding the actual operation process of top coal caving mining, optimizing the coal caving parameters, and improving the top coal recovery rate of the working face.

[0003] In the horizontal sectional fully mechanized top coal caving working face, due to the different occurrence conditions of coal seams, the differences in coal seam thickness, coal seam dip angle, and sectional height will cause changes in the coal caving results of horizontal sectional fully mechanized top coal caving mining, which are specifically manifested as differences in the development forms of the top coal caving body and the coal-rock interface. At present, the research methods for the top coal caving body and the coal-rock interface of the top coal caving face are limited, mainly through numerical simulation tests, physical tests, on-site measurements, or theoretical analysis. The specific situations are as follows: (1) Using numerical simulation tests can simulate the coal caving process under various coal seam thicknesses and coal seam dip angles and can accurately count the forms of the top coal caving body and the coal-rock interface. However, due to the large thickness of the top coal in horizontal sectional fully mechanized top coal caving mining, the numerical simulation model will require a large number of particles to carry out, resulting in a long calculation time for numerical simulation tests. If we want to explore the development laws of the top coal caving body and the coal-rock interface forms under different sectional heights, coal seam thicknesses, and coal seam dip angles, it will take a lot of calculation time; (2) When using physical tests to explore the coal caving law, due to the overly cumbersome process of building the physical test bench and the difficulty of the inversion process of the caving body, the physical tests under different coal seam thicknesses and coal seam dip angles have poor repeatability, are time-consuming and laborious, and the data processing process is cumbersome; (3) In the actual horizontal sectional fully mechanized top coal caving working face, it is basically unrealistic to conduct on-site measurements of the caving body and the coal-rock interface. Although there are test equipment such as top coal migration trackers that can track and monitor the coal-rock interface on-site, the procedures for arranging the test instruments are cumbersome, and at the same time, affected by the complex construction environment, the test results often fail to meet the research needs; (4) When using theoretical analysis to study the coal caving law, it is very difficult to consider the influence of complex boundary conditions and the random flow of bulk materials in the establishment of the theoretical equation, resulting in poor generalization of the theoretical analysis results and it is very difficult to be well applied under different working face parameters (sectional height, coal seam thickness, coal seam dip angle).

[0004] To make up for the deficiencies of the above research methods, prediction models based on deep learning algorithms, as an efficient, fast, and accurate technical means, have been widely used in various industries. Compared with numerical simulation tests, the algorithm model can predict and output any unknown input data based on training data, with a faster calculation speed and more convenient results. Compared with physical tests and on-site tests, the algorithm model has a lower implementation cost and can minimize the consumption of human and material resources to the greatest extent. Compared with traditional theoretical methods, the algorithm model can fully absorb and integrate multi-source data and has good applicability to working conditions with different geological conditions and boundary conditions.

[0005] After the algorithm model based on deep learning is established, there is no need for complex code to model different working conditions and can be directly used by most practitioners, greatly improving the application value. However, through research on existing literature and patents, it is found that there are few studies on the prediction of coal caving laws in horizontal sublevel caving mining based on graph neural networks. Therefore, proposing a prediction scheme for coal caving laws in sublevel caving mining based on graph neural networks is of great significance for revealing the coal caving laws in horizontal sublevel caving mining, optimizing the coal caving process, and improving the resource recovery rate, and it is a technical problem that needs to be solved urgently by those skilled in the art. Summary of the Invention

[0006] In view of this, the present invention provides a method and system for predicting coal caving laws in sublevel caving mining based on graph neural networks, which solves the problems existing in the background technology.

[0007] To achieve the above object, the present invention provides the following technical solutions: A method for predicting coal caving laws in sublevel caving mining based on graph neural networks, comprising the following steps: Conduct a numerical simulation test of horizontal sublevel caving mining through discrete element software, and establish a graph structure data set based on the test results; Use the graph structure data set to train the graph neural network. When the loss function converges, obtain the trained graph neural network; wherein, the nodes of the graph are the positions and physical information of the particles in the discrete element, the edges are the contact states of the particles, and the global features are the sublevel height, coal seam thickness, coal seam dip angle, and time step of the horizontal sublevel caving working face; Use the trained graph neural network to obtain the position information of the particle nodes and the coal-rock interface after coal caving, and obtain the shape of the caved body by inverting the coordinates of the caved particles.

[0008] Optionally, conducting a numerical simulation test of horizontal sublevel caving mining through discrete element software is specifically: In the PFC software, establish a numerical simulation basic model of horizontal sublevel caving mining, conduct coal caving tests under different sublevel heights, working face lengths, coal seam dip angles, and coal caving processes, and count various information of the initial model, includingx Axis coordinates, y axis coordinates, particle radius, and category.

[0009] Optionally, the graph structure dataset consists of the results of numerical simulation tests conducted under different sectional heights, coal seam thicknesses, and coal seam dip angles, specifically including: Node features: , where is the id of the i th particle, is the i th particle's x coordinate, is the y coordinate of the i th particle, is the type of the i th particle, is the particle size; Edge features: , where is the edge index matrix composed of contact particle ids, is the edge feature matrix composed of contact force features; Global attributes of the graph u include sectional height, coal seam thickness, coal seam dip angle, and time step.

[0010] Optionally, during the training of the graph neural network, the particle nodes and contact information are iteratively updated through L-layer message passing. The operations for each layer are as follows: Message generation: For each edge , generate the message ; where is the feature of node i at the l th layer, is the feature of edge at the l th layer, is the message function, is the feature of node j at the l th layer, is the global attribute u at the l th layer; Message aggregation: Based on , aggregate the messages of the contact nodes for each node; where is the aggregation operator, representing attention weighting; is the message of edge at the l th layer, is the feature of node i at the lMessage aggregation result of the layer; Node update: According to Update the node i 's features; where is the update function, is the node i at the l +1 layer's features.

[0011] Optionally, the message function selects a multi-layer perceptron MLP or a linear transformation.

[0012] Optionally, the update function selects a gated recurrent unit GRU.

[0013] Optionally, the method for obtaining the loss function of the graph neural network is specifically: The main loss function uses the mean squared error at the position, and the expression is: In the formula: is the mean squared error loss, N is the number of samples, is the particle i 's predicted value of the position, is the particle i 's true value of the position; The expression of the caving body shape error is: In the formula: is the sectional height; h is the distance from the center of the coal discharge opening to the origin of the coal discharge opening, , D is the width of the coal discharge opening, is the maximum migration angle of the granular material; is the shape correction coefficient on the floor side, is the shape correction coefficient on the roof side; Let the left side of the equation be the caving body equation, expressed as ; where is the particle node position information, is the coal discharge opening coordinates, S is the shape correction coefficient, then the position constraint of the particle nodes inside or discharged from the caving body is written as: ; For each coal-rock interface equation In the formula: is the function of the distance from the point on the equation to the coal discharge opening, is the k th coal-rock interface equation for coal caving, For the k -1 theoretical equation of the discharge body for coal caving; It is the loss at the coal-rock interface; The expression of the total error is: Among them, is the regularization coefficient of the loss at the coal-rock interface.

[0014] A coal caving law prediction system for fully mechanized top coal caving mining based on a graph neural network, which executes a coal caving law prediction method for fully mechanized top coal caving mining based on a graph neural network described in any one of the above, includes a data acquisition module, a training module, and a prediction module connected in sequence; The data acquisition module is used to perform numerical simulation experiments on horizontal sectional fully mechanized top coal caving mining through discrete element software, and establish a graph structure data set based on the experimental results; The training module is used to train the graph neural network through the graph structure data set. When the loss function converges, a trained graph neural network is obtained; among them, the nodes of the graph are the positions and physical information of the particles in the discrete element, the edges are the contact states of the particles, and the global features are the sectional height, coal seam thickness, coal seam dip angle, and time step of the horizontal sectional fully mechanized top coal caving face; The prediction module is used to obtain the position information of the particle nodes and the coal-rock interface after coal caving through the trained graph neural network, and obtain the shape of the discharge body by inverting the coordinates of the discharged particles.

[0015] It can be seen from the above technical solutions that compared with the prior art, the present invention discloses a coal caving law prediction method and system for fully mechanized top coal caving mining based on a graph neural network, which has the following beneficial effects: (1) To solve the problems of long calculation time and difficulty in repeated experiments in the numerical simulation experiment method, the present invention, through the constructed graph neural network model, can obtain the particle distribution characteristics after coal caving only by inputting parameters such as the position information, contact information, sectional height, coal seam thickness, and coal seam dip angle of the starting model, and then obtain the coal-rock interface and the shape of the discharge body, and obtain the experimental results more quickly and easily; (2) To solve the problems of cumbersome processes and data being easily affected by the environment in traditional physical experiments and on-site measurements, the present invention is based on a graph neural network, and can achieve the high efficiency, convenience, and safety of the experimental research process; (3) To solve the problem of poor applicability of theoretical research, the present invention combines the numerical simulation experiment results, physical experiment results, and numerical correction of on-site measurements, realizes the fusion of multi-source data, and enhances the applicability of the final output results; (4) To solve the problem that traditional graph neural network algorithms are difficult or unable to be applied in the horizontal sublevel caving mining, the present invention proposes to introduce global features (sublevel height, coal seam thickness, coal seam dip angle) and position loss and coal-rock interface loss to improve the accuracy of the algorithm in the experimental results of horizontal sublevel caving mining, and reduce the error caused by the traditional graph neural network's inability to fully fuse and absorb the special boundary conditions and working condition parameters in the study of horizontal sublevel caving mining. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only the embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on the provided drawings.

[0017] Figure 1 It is a flowchart of the sublevel caving law prediction method based on graph neural network provided by the present invention; Figure 2 It is a schematic diagram of the graph neural network provided by the present invention for predicting the particle position after the caving is completed; Figure 3 It is the process of obtaining the shape of the top coal caving body through the output node information provided by the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0018] The following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0019] The embodiments of the present invention disclose a sublevel caving law prediction method based on graph neural network, as Figure 1 shown, including the following steps: Conduct a numerical simulation experiment of horizontal sublevel caving mining through discrete element software, and establish a graph structure data set based on the experimental results; Use the graph structure data set to train the graph neural network. When the loss function converges, obtain the trained graph neural network; wherein, the nodes of the graph are the positions and physical information of the particles in the discrete element, the edges are the contact states of each particle (including contact particles and contact forces), and the global features are the sublevel height, coal seam thickness, coal seam dip angle, and time step of the horizontal sublevel caving working face; The position information of particle nodes and the coal-rock interface after the end of coal caving are obtained by using the trained graph neural network, and the shape of the caved body is obtained by inverting the coordinates of the caved particles (the positions of the particles in the initial model or the model before coal caving).

[0020] According to Figure 1 the process shown, in this embodiment, based on the results of numerical simulation tests, using physical tests, on-site measured data, and reasonable physical condition constraints, the graph neural network established can predict the coal caving results under different sectional heights, coal seam thicknesses, and coal seam dips, and accurately obtain the top coal caved body and the coal-rock interface. This scheme has high computational efficiency, can make up for the disadvantages of long time consumption and cumbersome process in numerical simulation tests, physical tests, and on-site tests, improves the accuracy of the model prediction results by fusing multi-scale physical fusion constraints, and makes up for the disadvantage of poor applicability of theoretical methods; the model has good generalization and applicability, and provides a fast and efficient research method for studying the complex coal caving law of horizontal slicing fully-mechanized top coal caving mining.

[0021] Furthermore, numerical simulation tests of horizontal slicing fully-mechanized top coal caving mining are carried out through discrete element software, specifically as follows: In the PFC software, a numerical simulation basic model of horizontal slicing fully-mechanized top coal caving mining is established, and coal caving tests are carried out under different sectional heights, working face lengths, coal seam dips, coal caving processes, etc., and various information of the initial model is statistically analyzed, including x x-axis coordinates, y y-axis coordinates, particle radius, and category. As shown in Table 1, taking the two-dimensional numerical simulation test as an example, if it is a three-dimensional test, the z z-axis coordinate also needs to be added.

[0022] Table 1 Node information of the graph structure of horizontal slicing fully-mechanized top coal caving mining Based on the recorded data, a graph structure data set is established, including the edge messages shown in Table 2 and the global messages shown in Table 3.

[0023] Table 2 Edge messages of the graph structure of horizontal slicing fully-mechanized top coal caving mining Table 3 Global messages of the graph structure of horizontal slicing fully-mechanized top coal caving mining Furthermore, the graph structure data set is composed of the results of numerical simulation tests carried out under different combinations of sectional heights, coal seam thicknesses, and coal seam dips, specifically including: Node features: , where is the id of the i th particle, is thei The x coordinates of a particle, i where is the y - coordinate of the i th particle, is the type of the th particle (0 for coal, 1 for gangue), and is the particle size; Edge features: where u is the edge index matrix composed of contact particle IDs,

[0024] and is the edge feature matrix composed of contact force features;

[0025] Global attributes of the graph include the sectional height, coal seam thickness, coal seam dip angle, and time step. Therefore, the model input is i.e., the particle position information, contact information, and global information (sectional height, coal seam thickness, coal seam dip angle) input to the initial model. Furthermore, during the training of the graph neural network, the particle nodes and contact information are iteratively updated through L - layer message passing. The operations of each layer are as follows: Message generation: For each edge i from i to l ), generate a message ; where is the feature of node l at the th layer, [[ID=S0]] is the feature of edge j at the l th layer; is the message function, which can be a multi - layer perceptron MLP, linear transformation, etc.; u is the feature of node l at the th layer, is the feature of the global attribute at the th layer; Message aggregation: Based on l ), aggregate the messages of its contact nodes (particles) for each node (particle information); where is the aggregation operator, representing attention weighting; i is the message of edge l at the th layer, Update Node i characteristics; among which, For the update function, select the gated recurrent unit GRU; For nodes i In the l +1 layer of features.

[0026] After the above process is iterated through L layers, the particle position prediction is output: Output all particle position information after coal placement is completed , by obtaining the particle position information , after distinguishing the characteristics of coal and gangue, the coal-rock interface results can be obtained, such as Figure 2 Afterwards, the ID of the released particle can be used to invert the released body morphology in the original model, as shown in Figure 3 shown.

[0027] Furthermore, the loss function of the graph neural network is obtained as follows: The main loss function uses the mean square error of position, which is expressed as: Where: is the mean square error loss, N is the number of samples, For particles i The predicted value of the location, For particles i The true value of the position; The expression of the release body shape error is: Where: is the segment height; h is the distance from the center of the coal caving port to the origin of the coal caving port, , D is the width of the coal opening, is the maximum migration angle of the bulk; is the bottom plate side shape correction coefficient, is the top plate side shape correction factor; Let the left side of the equation be the discharge body equation, expressed as ;in, is the particle node location information, is the coordinate of the coal caving port, S is the shape correction coefficient, then the position constraint of the released particle node or the released particle node is written as: ; For each coal-rock interface equation Where: is a function of the distance from the point on the equation to the coal discharge port, is the coal-rock interface equation for the k th coal discharge, is the theoretical equation of the discharged body for the k -1 coal discharge; is the loss of the coal-rock interface; The expression of the total error is: where, is the regularization coefficient of the loss of the coal-rock interface.

[0028] Corresponding to the Figure 1 method described above, the embodiment of the present invention also provides a prediction system for the coal discharge law of fully-mechanized caving mining based on a graph neural network, which is used for the specific implementation of the method in Figure 1 . The prediction system for the coal discharge law of fully-mechanized caving mining based on a graph neural network provided by the embodiment of the present invention can be applied to a computer terminal or various mobile devices, and specifically includes a data acquisition module, a training module, and a prediction module that are connected in sequence; The data acquisition module is used to perform numerical simulation experiments on the horizontal sectional fully-mechanized caving mining through discrete element software, and establish a graph structure data set based on the experimental results; The training module is used to train the graph neural network through the graph structure data set. When the loss function converges, a trained graph neural network is obtained; wherein, the nodes of the graph are the positions and physical information of the particles in the discrete element, the edges are the contact states of the particles, and the global features are the sectional height, coal seam thickness, coal seam dip angle, and time step of the horizontal sectional fully-mechanized caving working face; The prediction module is used to obtain the position information of the particle nodes and the coal-rock interface after the coal discharge through the trained graph neural network, and obtain the shape of the discharged body by inverting the coordinates of the discharged particles.

[0029] In summary, this embodiment innovatively proposes a coal caving law prediction model based on a graph neural network to study the coal caving law of horizontal slicing fully-mechanized top coal caving mining. Among them, the inputs of the model include node information (ID information, particle position information, physical information, category information), edge messages (contact information and contact force between particles), and global attributes (slicing height, coal seam thickness, coal seam dip angle, time step), and the output includes the particle position information after coal caving, that is, the shape of the coal-rock interface. It should be particularly noted that the shape of the coal-rock interface at each moment during the continuous coal caving process can be output, and the shape of the top coal caving body can be obtained by inverting the caving body. In addition, the loss function of the graph neural network proposed in this embodiment is innovative. Compared with the node or edge prediction of other graph neural networks, this embodiment makes the model results more accurate by introducing global features (slicing height, coal seam thickness, coal seam dip angle), position loss, coal-rock interface loss, and physical constraints on the shape of the caving body.

[0030] The various embodiments in this specification are described in a progressive manner. The key point of each embodiment is to illustrate the differences from other embodiments. The same or similar parts among the various embodiments can be referred to each other. For the system disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant parts can be referred to the description in the method part.

[0031] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be obvious to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but will be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for predicting the coal caving law in fully-mechanized caving mining based on graph neural network, characterized in that, Including the following steps: Conduct numerical simulation experiments on horizontal sublevel caving mining through discrete element software, and establish a graph-structured dataset based on the experimental results; Use the graph-structured dataset to train a graph neural network. When the loss function converges, obtain the trained graph neural network. Among them, the nodes of the graph are the positions and physical information of the particles in the discrete element, the edges are the contact states of each particle, and the global features are the sublevel height, coal seam thickness, coal seam dip angle, and time step of the horizontal sublevel caving working face; Use the trained graph neural network to obtain the position information of the particle nodes and the coal-rock interface after coal caving is completed, and obtain the shape of the caved body by inverting the coordinates of the caved particles.

2. The coal caving law prediction method based on graph neural network according to claim 1, wherein Conduct numerical simulation experiments on horizontal sublevel caving mining through discrete element software, specifically: In the PFC software, a numerical simulation basic model for horizontal slicing top coal caving mining is established. Coal caving tests are carried out under different slicing heights, working face lengths, coal seam dips, and coal caving technologies, and various information of the initial model is statistically analyzed, including x axial coordinates, y axial coordinates, particle radius and category.

3. The prediction method of the coal caving law in fully-mechanized caving mining based on graph neural network according to claim 1, wherein Graph-structured dataset Composed of the results of numerical simulation tests carried out under different combinations of sectional heights, coal seam thicknesses, and coal seam dips, specifically including: Node features: , where is the id of the i th particle, is the i th particle's x coordinates, is the y coordinate of the i th particle, is the type of the i th particle, is the particle size; Edge feature: , where is the edge index matrix composed of contact particle IDs, is the edge feature matrix composed of contact force features; Global attributes of the figure u Including sectional height, coal seam thickness, coal seam dip angle, and time step.

4. A method for predicting the coal caving law in fully-mechanized caving mining based on a graph neural network according to claim 1, characterized in that, When training the graph neural network, iteratively update the particle nodes and contact information through L-layer message passing. The operations of each layer are as follows: Message generation: For each edge , generate a message ; where is the feature of node i at the l th layer, is the feature of edge at the l th layer, is the message function, is the feature of node j at the l th layer, is the feature of the global attribute u at the l th layer; Message aggregation: Based on , aggregate the messages of each node's contact nodes; among them, is the aggregation operator, representing attention weighting; is the edge at the l -th layer's message, is the message aggregation result of node i at the l -th layer; Node update: According to Update the node i Features; among them, Is an update function, Is the feature of node i At the l +1 layer.

5. A method for predicting the coal caving law in fully-mechanized caving mining based on a graph neural network according to claim 4, characterized in that, Message function Select the multi-layer perceptron MLP or linear transformation.

6. The prediction method for the coal caving law in fully-mechanized caving mining based on graph neural network according to claim 4, wherein Update function Select the gated recurrent unit (GRU).

7. A method for predicting the coal caving law in fully-mechanized caving mining based on graph neural network according to claim 1, characterized in that The specific method for obtaining the loss function of the graph neural network is as follows: The main loss function uses the mean squared error of position, and the expression is: In the formula: is the mean squared error loss, N is the number of samples, is the predicted value of the position of the particle i ; is the true value of the position of the particle i . The expression for the error of the caved body shape is: In the formula: is the sectional height; h is the distance from the center of the coal discharge opening to the origin of the coal discharge opening, , D is the width of the coal discharge opening, is the maximum migration angle of the bulk material; is the shape correction coefficient on the floor side, is the shape correction coefficient on the roof side; Let the left side of the equation be the equation of the discharging body, expressed as ; where is the particle node position information, is the coordinate of the coal discharging port, S is the shape correction coefficient, then the position constraint of the particle nodes inside or discharged from the discharging body is written as: ; For the coal-rock interface equation each time In the formula: is a function of the distance from the point on the equation to the coal discharge port, is the coal-rock interface equation for the k th coal discharge, is the theoretical equation of the discharged body for the k -1 coal discharge; is the loss of the coal-rock interface The expression for the total error is: Among them, is the regularization coefficient for the loss at the coal-rock interface.

8. A coal caving law prediction system based on graph neural network, characterized in that, Implement a method for predicting the coal caving law in sublevel caving mining based on a graph neural network according to any one of claims 1-7, including a data acquisition module, a training module, and a prediction module connected in sequence; The data acquisition module is used to conduct numerical simulation experiments on horizontal sublevel caving mining through discrete element software, and establish a graph-structured dataset based on the experimental results; The training module is used to train a graph neural network through the graph-structured dataset. When the loss function converges, obtain the trained graph neural network. Among them, the nodes of the graph are the positions and physical information of the particles in the discrete element, the edges are the contact states of each particle, and the global features are the sublevel height, coal seam thickness, coal seam dip angle, and time step of the horizontal sublevel caving working face; The prediction module is used to obtain the position information of the particle nodes and the coal-rock interface after coal caving is completed through the trained graph neural network, and obtain the shape of the caved body by inverting the coordinates of the caved particles.

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