Three-dimensional node splitting method based on graph theory

Abstract

The invention discloses a three-dimensional node splitting method based on a graph theory, and the method comprises the following steps: S1, starting from a current checking step, detecting each node in sequence from small to large according to node serial numbers, and determining whether the current node has a surface reaching a damage condition or not; if yes, entering the step S2; if not, the detection of the current node is completed, the step S1 is repeated, and the next node is detected in sequence; s2, searching closed non-repeated separation paths connected with the current node, and when there are multiple closed non-repeated separation paths, screening out an optimal separation path according to the most flat separation surface; s3, performing node splitting, and performing attribution judgment on original new nodes, edges and surfaces by taking a separation surface with the minimum directed distance from a unit as a reference; s4, repeating the steps S1 to S3 until the detection of the last node is completed; and S5, repeating the steps S1 to S4 until the current simulation work is completed. According to the method, node splitting can be carried out more accurately, and technical support is provided for simulating rock dynamic fracture.

Description

Technical Field

[0001] The 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 process of oil and gas reservoir development, studying the dynamic fracture of rocks during drilling and blasting is one of the effective methods to design new drilling tools and improve rock breaking efficiency. However, due to the opacity of rocks and the high complexity and rapidity of the fracture process during deep excavation, it is difficult for indoor experiments to reflect the fracture state and internal characteristics of the rock mass. Numerical simulation has become one of the main methods for studying engineering problems in many engineering fields due to its fast and convenient characteristics.

[0003] At present, the methods for simulating dynamic fracture of rocks can be mainly divided into three categories, namely, methods based on continuum, methods based on discontinuum, and methods based on coupling of continuum and discontinuum. Among them, the methods based on continuum mainly include finite element method (FEM), extended finite element method (XFEM), etc. This method divides the complex geometric area into a large number of grid units, and the units are connected by nodes. It is often used for mechanical problems of continuous media such as elastic-plastic deformation. The methods based on discontinuum include discrete element method (DEM), discontinuous deformation analysis method (DDA), etc. Taking the discrete element method as an example, the constitutive physical model of contact is established by analyzing the contact between blocks of discrete units. This method allows nonlinear large deformation characteristics such as large displacement and block separation. The traditional finite element method cannot simulate the fracture of rock and the contact behavior between blocks. However, the discrete element method is inefficient and takes a long time to calculate when realizing large-scale model calculations. The coupling law couples the finite-discrete element method (FDEM), which can effectively simulate the mechanical response of the complete rock and the dynamic fracture process of the rock, and is often widely used in rock fracture and damage analysis. FDEM uses an explicit solution scheme, occupies 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 damage problems.

[0004] However, most FDEM currently use the interface crack propagation method to simulate the initiation and propagation of rock cracks. This method achieves crack propagation by globally inserting zero-thickness joint elements between finite elements in advance, but its disadvantage is that the constitutive model of the joint element has an initial elastic stage, that is, before the material breaks, the bonding element will also bear the elastic deformation of part of the material, resulting in artificial reduction of the material stiffness, unrealistic deformation of the material and reduced calculation 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 solution of the present invention is as follows:

[0007] A three-dimensional node splitting method based on graph theory comprises 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 there is a surface that reaches the destruction condition at the current node;

[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-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;

[0012] S3: Perform node splitting according to 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 work is completed.

[0015] Preferably, in step S1, when detecting a node, the status of all connection surfaces connected to the current node are 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 is 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 has occurred.

[0021] Preferably, in step S2, the closed non-repetitive separation path needs to 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 taking the separation surface with the smallest directed distance from the distance unit as the reference to determine the affiliation 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 impact of the initial grid on the crack propagation morphology, and effectively simulate the initiation and expansion 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 uses a depth-first algorithm based on graph theory to search for the closed non-repetitive separation path, 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 drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. 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 creative labor.

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

[0030] Figure 2 It is a 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 a flow chart of path finding of a 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 judgment condition for element boundary / surface failure;

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

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

[0035] Figure 7 A schematic diagram of the determination principle of the original node and the new node belonging to a specific embodiment;

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

[0037] Fig. 9 is a schematic diagram of n1 node splitting in three dimensions in another specific embodiment;

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

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

[0040] Fig.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] Fig.13 A schematic diagram of the flattest separation surface in three dimensions in another specific embodiment;

[0042] Fig.14 is a result diagram of splitting of the edge node n1 of the model in three dimensions in another specific embodiment;

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

[0044] The present invention is further described below in conjunction with the accompanying drawings and embodiments. 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 generally understood by those of ordinary skill in the art to which this application belongs. The words "including" or "comprising" and the like used in the disclosure of the present invention mean 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 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 so, go to step S2; if not, the current node is checked, and step S1 is repeated to check 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 not damaged; if the connection surface has been damaged, the connection surface is marked as damaged; if there is no connection surface that reaches the damage condition, the current node does not have 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 has 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 units 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 needs to meet the following conditions: at least one connection surface in the path has a state that reaches 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, selecting 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 according to 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 work is completed.

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

[0058] Since 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 checking 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, such as Figure 5 The 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 (2) As shown in 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 replication, and generating a new unit boundary e13, as shown in Figure 5 (3) as shown.

[0060] After the node splitting and boundary copying are completed, the numbering assignment needs to be determined. It is stipulated that the units in the positive direction of the separation surface normal use the new nodes and edges, and the units 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 unit is in the positive direction of the normal direction of the separation surface (the normal direction of the separation surface can be determined according to the right-hand screw rule), and the c5 unit is in the negative direction. It is determined that the new node n8 and the new boundary e13 generated after the n5 node is split are assigned to the c4 unit, while the original node n5 and the original boundary e11 are still located in the c5 unit.

[0061] like Figure 6As shown in (1), when it is detected that the edge e10 connected to the n1 node reaches the destruction condition, the separable edges connected to the node n1 are e10, e11, and e13, and there is an edge e10 that reaches the destruction condition, then the node n1 meets the splitting condition. The n1 node is split, and the e10 boundary is copied to generate a new node n9 and a new boundary e14. At this time, the second node splitting process is completed, as shown in Figure 6 (3) Similarly, after n1 is split, global search is continued and it is detected that 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 mesh model division can easily lead to confusion in the attribution 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 reach 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 are 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 the node n6, if it is still judged according to the positive direction of the normal of the separation surface, it will cause the n6 node splitting disorder.

[0063] In mathematical applications, the directed distance is defined as the orthogonal distance from a given point to a set of boundaries of the metric space. If the point is inside the space, it is a positive value, otherwise it is outside the space. The present invention uses this concept to make the above judgment, and takes the "minimum directed distance" breaking edge for judgment. Figure 7 (2) shows that the distances from the centroid of unit c6 (marked as a red dot) to the broken edges e13 and e10 are −L2 and +L1, respectively, as shown in the 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 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) as 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 faces 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, such as Figure 8 (3) as shown.

[0065] Now let's start with node n1 to explain the node splitting algorithm process in three dimensions. Fig. 9 As shown, the faces connected to the node n1 include four faces f13, f14, f15, and f16 on the outside and a face f11 inside the model. As mentioned above, in the present invention, the unit faces at the edge of the model are automatically marked as "destroyed faces". It is stipulated that if at least one of the faces connected to the splitting node reaches the destruction condition, the node can be split. That is, when f11 reaches the destruction condition, the n1 node can be split.

[0066] After determining the split node, it is necessary to find a closed, non-repetitive separation path. Face f11 reaches the destruction condition, and the edges connecting this face to the split node n1 are e4 and e10. Starting from e4 or e10, different closed, non-repetitive separation paths can be found. In this embodiment, a 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 unvisited adjacent edges, and checks whether the current edge has been visited. If not, it is marked as "visited" to record the visited edges and prevent repeated searches for the same edges. Secondly, it is checked whether the face connected to the current edge returns to the target source. If the path length from 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. Fig.10 (1). In this embodiment, the edge is taken as the detection object and the adjacent surface is taken as the path. Fig. 9 The situation is abstracted into a DFS search graph, such as Fig.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-repeating closed paths 1 and 2 can be found. The results are as follows: Fig.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 directions, but in essence, they all start from face f11 and finally return to face f11. For example, Fig.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: Fig.12 As shown. When all adjacent faces connected to node n2 reach the destruction condition, it can be abstracted into the path retrieval graph shown in the figure. Its 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-repetitive separation paths are found, the present invention selects the optimal separation path based on the smoothest separation surface to meet the rock fracture criteria. Fig.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 numbers, 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 not difficult to see from the figure 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 any one of the paths can be selected for splitting.

[0072] In addition, to illustrate other situations, different flatnesses are introduced for comparison, such as Fig.13 (2) is 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 in the figure is 0°, which is a closed plane path. Fig.13(1), whose flatness is the smallest, then this path is selected for splitting. After the splitting path is selected, taking path 1 as an example, node n1 is split to generate a new node n28, and face f11 is copied to generate a new face f37. The units in the positive direction of the normal of the splitting surface use the new nodes and the newly generated faces, and the units in the negative direction of the normal use the original nodes and faces. The splitting results are as follows: Fig.14 shown.

[0073] Proceed to the next node detection until all nodes in the current check step are detected. Fig.15 Shown Fig.12 The result of splitting node n2 along the path e15-f10-e14-f9-e17-f12-e16-f11-e15. According to the minimum directed distance, the new node n28 and new faces f37-f40 generated by the splitting are assigned to units c2, c3, c6 and c7, that is, node n2 is split from the inside of the model, causing the model to begin to separate. At this time, a series of tasks such as node n1 determination, path finding, node splitting, numbering update, etc. are completed, and the node is split. Repeating the detection of each node and the detection of each detection step can complete the node splitting of the entire three-dimensional model.

[0074] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present invention without departing from the content of the technical solution of the present invention still fall 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: The following steps are involved: 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 there is a surface that reaches the destruction condition at the current node; 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; 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; S3: Perform node splitting according to 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 work is completed.

2. The three-dimensional node splitting method based on graph theory according to claim 1 is characterized in that: In step S1, when detecting a node, the status of all connection surfaces connected to the current node is detected: If the connection surface is not damaged, the connection surface is marked as undamaged; If the connection surface has been damaged, the connection surface is 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.

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

4. The three-dimensional node splitting method based on graph theory according to claim 1 is 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 has reached a destruction condition, and the states of the remaining connection surfaces are all destroyed.

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

6. The three-dimensional node splitting method based on graph theory according to claim 1, characterized in that: In step S2, selecting the optimal separation path according to the flattest separation surface specifically includes the following sub-steps: 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.

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

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