A three-dimensional node splitting method based on graph theory

Abstract

The present invention discloses a three-dimensional node splitting method based on graph theory, comprising the following steps: S1: starting from the current inspection step, each node is detected in order from small to large according to the node sequence number to confirm whether the current node has a surface that reaches the failure condition; if so, proceed to step S2; if not, the current node detection is completed, step S1 is repeated, and the next node is detected in sequence; S2: a closed non-repeating separation path connected to the current node is searched, and when there are multiple closed non-repeating separation paths, the optimal separation path is screened out according to the flattest separation surface; S3: node splitting is performed, and the separation surface with the smallest directed distance from the distance unit is used as a reference to determine the ownership of the original and new nodes, edges, and faces; S4: steps S1-S3 are repeated until the last node is detected; S5: steps S1-S4 are repeated until the current simulation is completed. The present invention can perform node splitting more accurately and provide technical support for simulating dynamic rock fracture.

Description

Technical Field

[0001] The present invention relates to the technical field of engineering numerical simulation, and in particular to a three-dimensional node splitting method based on graph theory. Background Art

[0002] In the development of oil and gas reservoirs, studying the dynamic fracture of rock during drilling and blasting is an effective method for designing new drilling tools and improving rock-breaking efficiency. However, due to the opacity of rock and the complexity and rapidity of the fracture process during deep excavation, laboratory experiments are unable to accurately reflect the fracture state and internal characteristics of the rock mass. Numerical simulation, due to its rapidity and ease of use, has become a primary method for studying engineering problems in many engineering fields.

[0003] Currently, methods for simulating dynamic rock fracture can be divided into three main categories: continuum-based methods, discontinuum-based methods, and coupled continuum-discontinuum-based methods. Continuum-based methods primarily include the finite element method (FEM) and the extended finite element method (XFEM). These methods divide complex geometric regions into a large number of mesh elements connected by nodes and are commonly used for mechanical problems involving continuous media, such as elastic-plastic deformation. Discontinuum-based methods include the discrete element method (DEM) and discontinuum deformation analysis (DDA). For example, the DEM establishes a constitutive physical model of contact by analyzing the interblock contact of discrete elements. This method allows for large nonlinear deformation characteristics such as large displacements and block separation. Traditional finite element methods are unable to simulate rock fracture and interblock contact behavior. Furthermore, DEM methods are inefficient and time-consuming for large-scale model calculations. Coupled methods, however, combine finite and discrete element methods (FDEM), effectively simulating the mechanical response of intact rock and its dynamic fracture process, and are widely used in rock fracture and damage analysis. FDEM uses an explicit solution scheme, takes up less memory during the calculation process, and the program is relatively easy to implement GPGPU parallel processing. It is one of the most promising simulation methods for simulating large-scale engineering failure problems.

[0004] However, most current FDEM methods use the interfacial crack propagation method to simulate the initiation and propagation of rock cracks. This method achieves crack propagation by pre-emptively inserting zero-thickness joints between finite elements. However, its drawback is that the constitutive model of the joint elements has an initial elastic phase. That is, even before material rupture occurs, the bonded elements also experience some elastic deformation of the material, resulting in an artificial reduction in the material's stiffness, unrealistic deformation, and reduced computational efficiency. Summary of the Invention

[0005] In view of the above problems, the present invention aims to provide a three-dimensional node splitting method based on graph theory.

[0006] The technical solutions of the present invention are as follows:

[0007] A three-dimensional node splitting method based on graph theory includes the following steps:

[0008] S1: Starting from the current check step, each node is checked in order from small to large according to the node number to confirm whether the current node has a surface that reaches the destruction condition;

[0009] If it exists, go to step S2;

[0010] If it does not exist, the current node is detected and step S1 is repeated to detect the next node in turn;

[0011] S2: Find a closed non-repeating separation path connected to the current node. When there are multiple closed non-repeating separation paths, select the optimal separation path based on the flattest separation surface.

[0012] S3: Perform node splitting based on the path finding result of step S2, and take the separation surface with the smallest directed distance from the unit as the reference to determine the ownership of the original and new nodes, edges and faces;

[0013] S4: Repeat steps S1-S3 until the last node is detected;

[0014] S5: Repeat steps S1-S4 until the current simulation is completed.

[0015] Preferably, in step S1, when detecting a node, the status of all connection surfaces connected to the current node is detected:

[0016] If the connection surface is not damaged, the connection surface is marked as undamaged;

[0017] If the connection surface has been damaged, the connection surface will be marked as damaged;

[0018] If there is no connection surface that reaches the destruction condition, the current node does not meet the splitting condition;

[0019] If there is a connection surface that reaches the destruction condition, the connection surface is marked as reaching the destruction condition, and the current node meets the splitting condition.

[0020] Preferably, the Mohr-Coulomb criterion is used as a basis for judging whether damage occurs.

[0021] Preferably, in step S2, the closed non-repetitive separation path must meet the following conditions: at least one connection surface in the path is in a state of reaching a destruction condition, and the states of the remaining connection surfaces are all destroyed.

[0022] Preferably, in step S2, the closed non-repetitive separation path is found based on a depth-first algorithm based on graph theory.

[0023] Preferably, in step S2, selecting the optimal separation path according to the flattest separation surface specifically includes the following sub-steps:

[0024] The sum of the normal angles of adjacent separation surfaces in each path is calculated; the sum of the absolute values ​​of the angles in each path is calculated, and the path with the minimum value is taken as the optimal separation path.

[0025] Preferably, in step S3, when the separation surface with the smallest directed distance from the distance unit is used as a reference to determine the ownership of the original and new nodes, edges and faces, those in the negative direction of the normal of the separation surface belong to the original nodes, edges and faces, and those in the positive direction of the separation surface belong to the new nodes, edges and faces.

[0026] The beneficial effects of the present invention are:

[0027] The present invention can fully consider the impact of stiffness differences caused by inserting joint units in advance, can update the topological information of related units, and provide conditions for adaptive insertion of joint units. When there are multiple closed non-repetitive separation paths, the present invention selects the optimal separation path based on the flattest separation surface, can avoid the influence of the initial grid on the crack propagation morphology, and effectively simulate the initiation and propagation of cracks inside the rock. In addition, when searching for a path, due to the complex spatial topological structure in three-dimensional space, the present invention searches for the closed non-repetitive separation path based on the depth-first algorithm of graph theory, and can easily retrieve multiple non-repetitive closed separation paths. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0029] Figure 1 Schematic diagram of the process of the three-dimensional node splitting method based on graph theory of the present invention;

[0030] Figure 2 Schematic diagram of the process of node detection of the three-dimensional node splitting method based on graph theory of the present invention;

[0031] Figure 3 A schematic diagram of the process of path finding for the three-dimensional node splitting method based on graph theory of the present invention;

[0032] Figure 4 Schematic diagram of the Mohr-Coulomb criterion, which is the criterion for determining element boundary / surface failure;

[0033] Figure 5Schematic diagram of a unilateral destruction process in a two-dimensional case in a specific embodiment;

[0034] Figure 6 Schematic diagram of a two-dimensional multilateral destruction process in a specific embodiment;

[0035] Figure 7 A diagram showing the principle of determining the ownership of an original node and a new node in a specific embodiment;

[0036] Figure 8 A schematic diagram of a decomposition of a model surface in a three-dimensional case in another specific embodiment;

[0037] Figure 9 Schematic diagram of splitting the n1 node in three dimensions in another specific embodiment;

[0038] Figure 10 A schematic diagram of depth-first search in another specific embodiment;

[0039] Figure 11 A schematic diagram of finding a closed, non-repetitive, separating path using a depth-first algorithm based on graph theory in another specific embodiment;

[0040] Figure 12 A schematic diagram of a path search for an internal node n2 of a model in a three-dimensional case in another specific embodiment;

[0041] Figure 13 A schematic diagram of the flattest separation surface in three dimensions in another specific embodiment;

[0042] Figure 14 2 is a diagram showing the result of splitting the edge node n1 of the model in a three-dimensional case in another specific embodiment;

[0043] Figure 15 This is a result diagram of the splitting of the internal node n2 of the model in three dimensions in another specific embodiment. DETAILED DESCRIPTION

[0044] The present invention is further described below with reference to the accompanying drawings and examples. It should be noted that, in the absence of conflict, the embodiments in this application and the technical features in the embodiments can be combined with each other. It should be noted that, unless otherwise specified, all technical and scientific terms used in this application have the same meanings as those commonly understood by those of ordinary skill in the art to which this application belongs. The use of similar words such as "include" or "comprising" in the present invention means that the elements or objects appearing before the word cover the elements or objects listed after the word and their equivalents, without excluding other elements or objects.

[0045] like Figure 1-3 As shown, the present invention provides a three-dimensional node splitting method based on graph theory, comprising the following steps:

[0046] S1: Starting from the current inspection step, each node is inspected in order from small to large according to the node number to confirm whether the current node has a surface that reaches the destruction condition; if so, proceed to step S2; if not, the current node is inspected and step S1 is repeated to inspect the next node in turn.

[0047] In a specific embodiment, when detecting a node, the status of all connection surfaces connected to the current node is detected: if the connection surface has not been damaged, the connection surface is marked as undamaged; if the connection surface has been damaged, the connection surface is marked as damaged; if no connection surface reaches the damage condition, the current node does not meet the splitting condition; if there is a connection surface that reaches the damage condition, the connection surface is marked as reaching the damage condition, and the current node meets the splitting condition.

[0048] In a specific embodiment, the Mohr-Coulomb criterion is used as a basis for judging whether damage has occurred. Figure 4 As shown, according to the normal stress σ on the common boundary of adjacent elements n and shear stress τ s When the tensile or shear failure conditions are met, the element boundaries / surfaces fail.

[0049] S2: Find a closed non-repetitive separation path connected to the current node. When there are multiple closed non-repetitive separation paths, select the optimal separation path based on the flattest separation surface.

[0050] In a specific embodiment, the closed non-repetitive separation path must meet the following conditions: at least one connection surface in the path is in a state of reaching a destruction condition, and the states of the remaining connection surfaces are all destroyed.

[0051] In a specific embodiment, the closed non-repetitive separation path is found using a depth-first algorithm based on graph theory.

[0052] In a specific embodiment, screening out the optimal separation path based on the flattest separation surface specifically includes the following sub-steps: calculating the sum of the normal angles of adjacent separation surfaces in each path; finding the sum of the absolute values ​​of the angles in each path, and taking the path with the minimum value as the optimal separation path.

[0053] S3: Perform node splitting based on the path search result of step S2, and take the separation surface with the smallest directed distance from the unit as the basis to determine the ownership of the original and new nodes, edges and faces.

[0054] In a specific embodiment, when the separation surface with the smallest directed distance from the distance unit is used as a reference to determine the ownership of the original and new nodes, edges and faces, those in the negative direction of the normal of the separation surface belong to the original nodes, edges and faces, and those in the positive direction of the separation surface belong to the new nodes, edges and faces.

[0055] S4: Repeat steps S1-S3 until the last node is detected.

[0056] S5: Repeat steps S1-S4 until the current simulation is completed.

[0057] It should be noted that when using the present invention, the points, edges or faces at the model boundary are all in a separable state by default.

[0058] Because the three-dimensional situation is more complicated than the two-dimensional situation, in a specific embodiment, a two-dimensional model is first used to illustrate the present invention. Figure 5 As shown in (1), in one check step, all nodes of the model are traversed and judged from small to large sequence numbers. If the node currently being checked has an edge that reaches the destruction condition and there are two or more separable boundaries, then the node can be split. In this embodiment, the unit edge of the model boundary is in a separable state by default, as shown in Figure 5 Edges e1, e2, e3, e4, e5, e6 in (1).

[0059] When the edge e11 connected to the n1 node reaches the destruction condition, the n1 node cannot be split because there is only one separable edge connected to the n1 node. At this time, the next node is tested according to the sequence number until the node n5. Figure 5 As shown in (2), Figure 5 (2) It can be seen that there are three separable boundaries of node n5, so the node can be split. At this time, node n5 is split, generating a new node n8, destroying the edge e11 copy, and generating a new unit boundary e13, as shown in Figure 5 (3) shown.

[0060] After the node splitting and boundary copying are completed, the number assignment needs to be determined. It is stipulated that the elements in the positive direction of the separation surface normal use the new nodes and edges, and the elements in the negative direction use the original nodes and numbers. Node n1 points to node n5, and the separation surface normal is as follows Figure 5 (3) As shown in Figure 3, the c4 element is in the positive direction of the normal to the separation surface (the normal direction of the separation surface can be determined according to the right-hand screw rule), and the c5 element is in the negative direction. It is determined that the new node n8 and the new boundary e13 generated after the splitting of the n5 node are assigned to the c4 element, while the original node n5 and the original boundary e11 are still located in the c5 element.

[0061] like Figure 6As shown in (1), when it is detected that the edge e10 connected to node n1 reaches the destruction condition, the separable edges connected to node n1 are e10, e11, and e13, and there is an edge e10 that reaches the destruction condition, then node n1 meets the splitting condition. When node n1 splits, the edge e10 is copied to generate a new node n9 and a new edge e14, and the second node splitting process is completed. Figure 6 (3) Similarly, after the splitting of n1 is completed, the global search continues and it is detected that the n4 node can be split, generating a new node n10. At this time, unit c4 is peeled off from the model.

[0062] In actual engineering simulation problems, the complexity and uncertainty of grid model division easily lead to confusion in the assignment of original nodes and new nodes after node splitting. Figure 7 As shown in (1), the edges e10 and e13 connected to node n1 are two unit edges that have reached the failure condition. According to the above process, the separable edges are determined to be e6-e13-e10-e2, and the pairs n1, n3, and n6 will be split in sequence. After the splitting of n1 is completed, the units c3, c4, and c5 are all in the positive direction of the normals of the two separation surfaces, while the c6 unit is in the negative direction of the normal, but is located in the positive direction of the normal of the boundary e10. When the splitting reaches node n6, if the judgment is still based on the positive direction of the normal of the separation surface, the splitting of the n6 node will be disordered.

[0063] In mathematical applications, the directed distance is defined as the orthogonal distance from a given point to a set of boundaries of a metric space. If the point is inside the space, it is a positive value, otherwise it is a negative value if it is outside the space. This concept is used in the present invention to determine the above situation, and the breaking edge of the "minimum directed distance" is used for determination. Figure 7 As shown in (2), the distances from the centroid of unit c6 (marked as a red dot) to the failure edges e13 and e10 are calculated as −L2 and +L1 by directed distance calculation. Since |L1| > |L2|, so edge e13 is used as the judgment edge of unit c6. At this time, units c5, c4, and c3 are in the positive direction of the normal of the split boundary. After splitting, new nodes and new edges are applied, and the original positions are retained as the original nodes and original edges. After node n1 is split, a new node n9 is generated. The edges e13 and e10 are copied and destroyed to generate new edges e15 and e16. The edge e10 connects units c2 and c3, and e13 connects units c5 and c6. First, node n1 is split to generate a new node n9, and then the edge e10 is separated in sequence to generate a new edge e15. According to the minimum directed distance, e15 is assigned to unit c3, and e10 belongs to unit c2. Then the edge e13 is separated. Similarly, e16 belongs to unit c5, and e13 belongs to unit c6. The splitting result is as follows Figure 7 (3) shown.

[0064] In another specific embodiment, the present invention is used to perform node splitting on a three-dimensional model. Factors such as model size, unit shape, and spatial distribution in three-dimensional space will greatly increase the complexity of node splitting. The present invention is now described using the C3D8 unit as an example. Figure 8 As shown in (1), the three-dimensional model of this embodiment is a cube composed of 8 C3D8 units. Since the unit faces at the model boundary are separable by default, the unit faces outside the model can be peeled off. At this time, the model consists of three inner planes, and the unit faces on the three inner planes are numbered in sequence, as shown in the following example: Figure 8 (3) shown.

[0065] Now let’s start with node n1 to explain the node splitting algorithm process in three-dimensional case. Figure 9 As shown, the faces connected to node n1 are four external faces f13, f14, f15, and f16, and one internal face f11. As mentioned above, in this invention, element faces at the edge of the model are automatically marked as "destroyed faces." The node can be split if at least one of the faces connected to the split node meets the failure condition. That is, when f11 meets the failure condition, node n1 can split.

[0066] After determining the splitting node, it is necessary to find a closed non-repetitive separation path. The surface f11 reaches the destruction condition, and the edges connecting this surface to the splitting node n1 are e4 and e10. Starting from e4 or e10, different closed non-repetitive separation paths can be found. In this embodiment, the depth-first algorithm based on graph theory is used to find the closed non-repetitive separation path. The core of the depth-first algorithm is "recursion". Starting from the target source, it traverses and explores the unvisited adjacent edges, checks whether the current edge has been visited, and if not, marks it as "visited" to record the visited edges and prevent repeated searches for the same edges. Secondly, check whether the face connected to the current edge returns to the target source. If the path length of the face at the current position to the target source is greater than 1, it is added to the path and returned, and the mark status of the visited edge is removed. Figure 10 (1). In this embodiment, the edge is the detection object and the adjacent surface is the path, then Figure 9 The situation is abstracted as a DFS search graph, such as Figure 10 As shown in (2), the points in the figure are the edges of the model, and the edges in the figure are the faces of the model.

[0067] Since the edges e4 and e10 in the failure surface f11 connected to the node n1 are both connected to the node n1, taking the edge e4 as an example, two non-repeated closed paths 1 and 2 can be found. The results are as follows: Figure 11As shown. Edge e4 is the common boundary of faces f14, f11, and f15. Face f14 is detected from edge e4. Edge e7 connected to node n1 is detected in face f14. Face f13 is detected from edge e7. Face f13 has edge e10 connected to node n1. At this time, edge e10 is detected back to face f11 to form a closed path of f11-e4-f14-e7-f13-e10-f11. Similarly, different paths can be formed starting from different edges and different directions, but in essence, they all start from face f11 and finally return to face f11. For example, Figure 11 Path 2 in .

[0068] When searching for closed and non-repeated separation paths for nodes inside the model, it can also be abstracted as a DFS search graph. Taking node n2 as an example, the result is as follows: Figure 12 When all adjacent faces connected to node n2 reach the destruction condition, the path search graph shown in the figure can be abstracted. The core idea can be summarized as "edges find faces, faces find edges."

[0069] The rock fracture process is complex and rapid. According to the maximum energy release rate theory, the smoother the fracture surface from the crack tip, the faster the energy release rate. If the crack spreads to the surrounding area, the rate of crack expansion will gradually decrease. When multiple closed non-repeating separation paths are found, the present invention selects the optimal separation path based on the smoothest separation surface to meet the rock fracture criteria. Figure 13 As shown in (1), there are three path faces f11, f15, and f16 of path 2. The normal vectors of their respective faces can be determined by the node number, such as the normal vector of face f11 is n11, the normal vector of face f15 is n15, and the normal vector of face f16 is n16. At this time, the angles θ1, θ2, and θ3 between the normal vectors n11, n15, and n16 can be calculated based on the inverse operation of vector dot multiplication. The calculation formulas are:

[0070]

[0071] It is easy to see from the diagram that the three angles are all 90°, and the sum of the absolute values ​​of the angles is 270°. Similarly, the flatness of path 1 is also 270°, so the flatness of the two paths is the same, and either path can be selected for splitting.

[0072] In addition, to illustrate other situations, different flatness levels are introduced for comparison, such as Figure 13 (2) As shown. The normal vectors of faces f9, f10, f11, and f12 are n9, n10, n11, and n12. It is not difficult to see that these vectors are parallel to each other and have the same direction. The flatness of the situation shown is 0°, which is a closed plane path. Figure 13(1), its flatness is the smallest, then this path is selected for splitting. After completing the splitting path optimization, taking path 1 as an example, node n1 is split to generate a new node n28, and surface f11 is copied to generate a new surface f37. The units in the positive direction of the splitting surface normal use the new nodes and newly generated surfaces, and the units in the negative direction of the normal use the original nodes and original surfaces. The splitting results are as follows Figure 14 shown.

[0073] Proceed to the next node detection work until all nodes in the current inspection step are detected. Figure 15 Shown Figure 12 The result of splitting node n2 along the path e15-f10-e14-f9-e17-f12-e16-f11-e15 is shown in Figure 1. Based on the minimum directed distance, the resulting new node n28 and new faces f37-f40 are assigned to units c2, c3, c6, and c7. This means that node n2 has split from within the model, causing the model to begin to separate. At this point, the entire process of determining node n1, finding the path, splitting the node, and updating the node number is complete, and the node is now split. Repeating the node testing and detection steps will complete the node splitting of the entire 3D model.

[0074] The above description is merely a preferred embodiment of the present invention and does not constitute any form of limitation to the present invention. Although the present invention has been disclosed as a preferred embodiment as above, it is not intended to limit the present invention. Any technician familiar with the present profession can make some changes or modifications to equivalent embodiments of equivalent changes using the technical contents disclosed above without departing from the scope of the technical solution of the present invention. However, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solution of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. A three-dimensional node splitting method based on graph theory, characterized in that: Used to simulate rock splitting, including the following steps: S1: Starting from the current check step, each node is checked in order from small to large according to the node number to confirm whether the current node has a surface that reaches the destruction condition; If it exists, go to step S2; If it does not exist, the current node is detected and step S1 is repeated to detect the next node in turn; When detecting a node, check the status of all connected surfaces connected to the current node: If the connection surface is not damaged, the connection surface is marked as undamaged; If the connection surface has been damaged, the connection surface will be marked as damaged; If there is no connection surface that reaches the destruction condition, the current node does not meet the splitting condition; If there is a connection surface that reaches the destruction condition, the connection surface is marked as reaching the destruction condition, and the current node meets the splitting condition; S2: Find a closed non-repeating separation path connected to the current node. When there are multiple closed non-repeating separation paths, select the optimal separation path based on the flattest separation surface. Screening the optimal separation path based on the flattest separation surface specifically includes the following sub-steps: Calculate the sum of the normal angles of adjacent separation surfaces in each path; calculate the sum of the absolute values ​​of the angles in each path, and take the path with the minimum value as the optimal separation path; S3: Perform node splitting based on the path finding result of step S2, and take the separation surface with the smallest directed distance from the unit as the reference to determine the ownership of the original and new nodes, edges and faces; S4: Repeat steps S1-S3 until the last node is detected; S5: Repeat steps S1-S4 until the current simulation is completed.

2. The three-dimensional node splitting method based on graph theory according to claim 1, characterized in that: The Mohr-Coulomb criterion is used as the basis for judging the occurrence of damage.

3. The three-dimensional node splitting method based on graph theory according to claim 1, characterized in that: In step S2, the closed non-repetitive separation path must meet the following conditions: at least one connection surface in the path is in a state of reaching a destruction condition, and the states of the remaining connection surfaces are all destroyed.

4. The three-dimensional node splitting method based on graph theory according to claim 3, characterized in that: In step S2, the closed non-repetitive separation path is found based on a depth-first algorithm of graph theory.

5. The three-dimensional node splitting method based on graph theory according to claim 1, characterized in that: In step S3, when the separation surface with the smallest directed distance from the unit is used as the basis for determining the ownership of the original and new nodes, edges and faces, those in the negative direction of the normal of the separation surface belong to the original nodes, edges and faces, and those in the positive direction of the separation surface belong to the new nodes, edges and faces.

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