A method for constructing random pore model of internal perturbations in polycrystalline materials based on parallel progression

Abstract

The present invention relates to a method for constructing a random pore model of perturbations in a polycrystalline body based on parallel progression, which comprises the following steps: step 1: obtaining a three-dimensional Thiessen polygon representative volume element model of a crystal material; step 2: obtaining unit information and all node information of a grain set based on the polycrystalline representative volume element model; step 3: determining the internal node information of a grain set based on parallel progression; step 4: determining a pore set that needs to be removed; step 5: renumbering the unit information and internal node information of the model after the pores are removed; and step 6: establishing a multi-pore model between crystals. The present invention uses Thiessen polygons to establish a grain set, and based on a random method of perturbations in a polycrystalline body based on parallel progression, the established pore structure model is more similar to the structure of an actual crystal material; the present invention can control the morphology of the pores by controlling the weights of the pores in different coordinate directions, and design pore structures of different sizes and structures, making the operation more flexible.

Description

Technical Field

[0001] The present application relates to the field of macro-microscopic modeling of porous materials, and in particular to a method for constructing a random pore model of polycrystalline internal disturbances based on parallel progression. Background Art

[0002] In an ideal smelting process, the crystalline material is a homogeneous material with no pores between the grains. However, in the actual metallurgical smelting process or even the recrystallization process of metal materials, it is difficult to ensure that there are no pores between the grains of the material. In order to simulate the deformation evolution law of porous materials during deformation through the finite element simulation platform. Analyze the key role of pores in the material deformation process, such as stress / strain concentration caused by pores, pore compaction during deformation, etc. It is necessary to express the distribution state structure of pores in porous materials more accurately, especially different pore morphologies, establish a representative volume element model of multi-pores with different morphological distributions, and then apply different deformation conditions or loading schemes to simulate the complex deformation and stress-strain evolution law caused by pores with different morphologies in porous materials.

[0003] Regarding the establishment of porous material models, most finite element models currently use homogeneous materials. On this basis, they use spatial construction capabilities to cut out holes with regular shapes (mostly round or square) in the homogeneous matrix, without distinguishing the distribution relationship of the grains, and it is even more difficult to construct relevant pores on the crystal scale. In the finite element simulation of crystal scale models, the most common method is to use the representative volume element method to establish a finite element model. And the modeling method of the representative volume element model is based on the Thiessen polygon structure model established by the Voronoi diagram, which represents the collection in the model as grains. However, the connections between the grains of the polycrystalline representative volume element model established by this method are bound together without any gaps, and share nodes, and it is impossible to remove any collection, so this method cannot directly establish pores.

[0004] Based on the Thiessen polygon structure representative volume element model established based on the Voronoi diagram, some units are removed again to construct a crystal scale model with a pore structure. The construction of this model is more in line with the actual porous material. The new model not only has different grain structures, but also has pore structures with different morphologies. In the simulation process of different deformation behaviors or loading conditions, this model can realize the evolution behavior of pores with different morphologies. Summary of the invention

[0005] In order to overcome the shortcomings of the prior art, the present invention uses Thiessen polygons to establish a grain set, and based on a parallel progressive multi-crystal internal perturbation random method, completes the construction of the pore structure model. Compared with the actual crystal material, the error is smaller, pore structures of different sizes and structures are designed, and the operation is more flexible.

[0006] To achieve the above purpose, the solution adopted by the present invention is:

[0007] A method for constructing a random pore model of polycrystalline internal disturbance based on parallel progression comprises the following steps:

[0008] Step 1: Obtain a three-dimensional Thiessen polygon representative volume element model of the crystal material;

[0009] Establish a three-dimensional Thiessen polygon representative volume element model of the crystal material, divide it into multiple grain sets, and obtain a polycrystalline representative volume element model;

[0010] Step 2: According to the polycrystalline representative volume element model, obtain the unit information and all node information of the grain set;

[0011] Since the polycrystalline representative volume element model in step 1 includes a large number of grain sets, and each grain set contains multiple units; according to the structural characteristics of the grain set, the unit number of each grain set is extracted and exported and stored in the form of key-value pairs. The specific expression form is as follows:

[0012] M={[Gset[n]:E1,…E i , …], …}

[0013] Where: M represents the polycrystalline body model; Gset represents the grain set; n represents the sequence number of the grain set; E represents the unit of the grain set; i represents the unit number;

[0014] Each unit of a grain is composed of nodes. The node information corresponding to the unit is extracted. At the same time, according to the unit information of the grain set extracted by the above formula, all the node information contained in the grain set is determined. The specific expression form is as follows:

[0015]

[0016] Where: N represents the node; j represents the node number;

[0017] Step 3: Determine the internal node information of the grain set based on parallel progression;

[0018] Compare all node numbers contained in one grain set with all node numbers of the remaining grain sets. If the node numbers are the same, it is considered that the node coexists in two grain sets. Then the node is a common node between the grain sets and also a node on the surface of the grain set. The specific expression is as follows:

[0019] B=Gset[n]∩Gset[n+1]

[0020] Where: B represents the grain boundary set;

[0021] In this selection among a large number of values, two adjacent grain sets are first selected, and the node information of one grain is compared with all the node information in the remaining grain sets; if the node also exists in the second grain set, its node is stored in the interface list, and the node is deleted from the two selected grain sets to ensure uniqueness, so that the node is not compared repeatedly in the comparison with the adjacent grains. The specific expression is as follows:

[0022] NGset[n]=Gset[n]-B

[0023] Where: NGset represents the set of grains excluding interface nodes;

[0024] In this way, the comparison, screening and classification of node information are carried out simultaneously, and the interface nodes and the internal nodes of the set are quickly screened out from a large number of values, saving a lot of comparison operations; the internal node information of the grain set is determined by removing the nodes on the surface of the set from the grain set established in step 2 through a parallel progressive method, and stored in a list form;

[0025] Step 4: Determine the set of pores that need to be removed;

[0026] First, the overall model is divided into multiple standard regions with a volume of a×b×c, so that each standard region contains at least 4 complete grain sets to avoid the selected pore set becoming pores in the grain or the entire grain set; the position of the pore in the standard region is determined according to the central disturbance parameter r;

[0027] Step 5: Renumber the unit information and internal node information of the model after removing the pores;

[0028] Remove the unit information of the pore set and renumber the units of the model: according to the pore set information selected in step 4, delete the units in the pore set; this will cause the unit numbers in the model to be discontinuous, so the remaining units are renumbered from small to large, but the grain set information and the node information contained in the unit are not changed;

[0029] Remove the internal node information of the pore set and renumber the nodes of the model: according to the pore set information selected in step 4 and the internal node information of the set determined in step 3, delete the nodes in the pore set; this will cause the node numbers to be discontinuous, so the remaining nodes are renumbered in order from small to large, but the unit information containing the node and the position information contained in the node are not changed;

[0030] Step 6: Establish a multi-pore model between crystals;

[0031] According to the unit information and internal node information re-edited in step 5, the multi-pore model between crystals is constructed.

[0032] Preferably, the three-dimensional Thiessen polygons in step 1 represent a volume element model, which can construct an equiaxed grain model and a fibrous grain model by imitating actual crystal materials.

[0033] Preferably, in step 3, the intersection node of the two grain sets is deleted from the two selected grain sets, so that this node is no longer repeatedly compared with the adjacent grains; the comparison, screening and classification of the node information are performed simultaneously, and the interface nodes and the internal nodes of the grain sets are quickly and progressively screened out in parallel.

[0034] Preferably, the pore set is selected in step 4 to separate the overall model, and the random pore positions are determined using the perturbation parameter r to avoid direct random selection leading to pore connection and the generation of large pores; the pore positions are determined based on the perturbation parameter r of the center within the standard area as follows;

[0035] Then, the area where the pores are located is selected from all standard areas, and then the position of the pores is determined in the standard area according to the central disturbance parameter r, which is expressed as:

[0036]

[0037] Where: X1, Y1 and Z1 represent the horizontal, vertical and vertical coordinates of the pore location respectively; X0, Y0 and Z0 represent the horizontal, vertical and vertical coordinates of the minimum point of the standard area respectively; r represents the disturbance parameter, which is a random number between 0 and 1 and is automatically generated by the computer; a, b and c represent the length, width and height of the cube of the standard area;

[0038] Complete the random selection of pore sets from the entire set. The pores selected by this perturbation random method avoid the direct random selection that leads to the connection of pores and the generation of larger pores, which is more similar to the actual pore distribution;

[0039] In order to make the randomly generated coordinate positions in the model space present a regular distribution state, the pore set to be removed is regularly selected according to the weights k, g and h of the pores in the horizontal, vertical and vertical coordinate directions; at this time, the overall model is divided into standard areas, and the pore shape satisfies the following relationship:

[0040]

[0041] Where: X m , Y m and Z m Represents the horizontal, vertical and vertical coordinates of the center position of the standard area; k, g and h represent the weights of the horizontal, vertical and vertical coordinate directions respectively.

[0042] Compared with the prior art, the present invention has the following beneficial effects:

[0043] (1) The technical solution proposed in the present invention uses Thiessen polygons to establish a grain set, obtains internal information of the grain set, and establishes a polycrystalline pore structure model based on a parallel progressive polycrystalline internal perturbation random method. The model structure established by this method is more accurate and has a smaller error compared with the structure of the actual crystal material;

[0044] (2) The present invention can control the shape and structure of the pore structure model by controlling the proportion of the weight of the pores in different coordinate directions, which is convenient for designing pore structure models of different sizes and structures, making it more flexible for operators to use and more convenient and free in actual application. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 This is a flow chart of a method for constructing a random pore model of a polycrystalline internal disturbance based on parallel progressiveness according to an embodiment of the present invention;

[0046] Figure 2 A flowchart of a procedure for establishing pores of different shapes in a modeling method for a microscopic model of porous materials at a crystal scale according to an embodiment of the present invention;

[0047] Figure 3 A schematic diagram of tetrahedral units and their nodes in a collection in a modeling method for a microscopic model of porous materials at a crystal scale according to an embodiment of the present invention;

[0048] Figure 4 A polycrystalline representative volume element model diagram in a modeling method for a microscopic model of a porous material at a crystal scale according to an embodiment of the present invention;

[0049] Figure 5 A pore model diagram of 5% pores uniformly distributed on the same axis in the modeling method of a microscopic model of a porous material at a crystal scale in an embodiment of the present invention;

[0050] Figure 6 This is a pore model diagram of 1:1:3 long strips with 5% uniformly distributed pores in the modeling method of a microscopic model of a porous material at a crystal scale in an embodiment of the present invention;

[0051] Figure 7 This is a cloud diagram of pore morphology and matrix strain during compression deformation in a modeling method for a microscopic model of porous materials at a crystal scale in an embodiment of the present invention. DETAILED DESCRIPTION

[0052] Hereinafter, embodiments of the present invention will be described with reference to the drawings.

[0053] The present invention introduces a method for constructing a porous structure model. Figure 1The flowchart of the method for constructing a random pore model based on a parallel progressive polycrystalline internal perturbation in an embodiment of the present invention is shown; in this case, a grain set is established using Thiessen polygons, and the pore structure model is constructed based on a parallel progressive polycrystalline internal perturbation random method. The error is small compared with the actual crystal material, and pore structures of different sizes and structures can be designed, making the operation more flexible. Figure 2 It is a flowchart of a procedure for establishing pores of different shapes in a modeling method for a microscopic model of porous materials at a crystal scale according to an embodiment of the present invention;

[0054] The embodiment of the present invention provides a method for constructing a random pore model of a polycrystalline internal disturbance based on parallel progression. In order to prove the applicability of the present invention, the method is applied to an example, which specifically includes the following steps:

[0055] S1: Obtain a three-dimensional Thiessen polygon representative volume element model of the crystal material;

[0056] A three-dimensional Thiessen polygon representative volume element model of a crystal material or a crystal multi-grain material is established, and it is divided into multiple grain sets to obtain a polycrystalline representative volume element model. According to the actual crystal material, an equiaxed grain model and a fibrous grain model can be constructed; Figure 4 The figure shows a polycrystalline representative volume element model diagram in the modeling method for a microscopic model of a porous material at a crystal scale according to an embodiment of the present invention. The inp file of a three-dimensional Thiessen polygon representative volume element model of a polycrystalline material is established by using the "-T" and "-M" modules of the Neper software and imported into the Abaqus finite element software.

[0057] S2: According to the polycrystalline representative volume element model, the unit information and all node information of the grain set are obtained;

[0058] Since the polycrystalline representative volume element model in S1 includes a large number of grain sets, and a grain set contains multiple units; according to the structural characteristics of the grain set, the unit number of each grain set is extracted and exported and stored in the form of key-value pairs. The specific expression is as follows:

[0059] M={[Gset[n]:E1,…E i , …], …}

[0060] Where: M represents the polycrystalline body model; Gset represents the grain set; n represents the sequence number of the grain set; E represents the unit of the grain set; i represents the number of the internal information of the unit;

[0061] Each cell of a grain is composed of nodes. Figure 3The figure shows a schematic diagram of tetrahedral units and their nodes in a set in a modeling method for a microscopic model of porous materials at a crystal scale according to an embodiment of the present invention; the representative volume element model is a C3D4 unit, each unit is composed of 4 nodes, and a list storage of nodes corresponding to each unit is established.

[0062] The node information corresponding to the unit is extracted, and at the same time, all the node information contained in the grain set is determined according to the unit information of the grain set extracted by the above formula. The specific expression form is as follows:

[0063]

[0064] Where: N represents the node; j represents the node number;

[0065] S3: Determine the internal node information of the grain set based on parallel progression;

[0066] Compare all node numbers contained in one grain set with all node numbers of other remaining grain sets. If the node numbers are the same, it is considered that the node exists in two grain sets. Then the node is a common node between the grain sets and also a node on the surface of the grain set. The specific expression is as follows:

[0067] B=Gset[n]∩Gset[n+1]

[0068] Where: B represents the grain boundary set;

[0069] In this selection among a large number of values, two adjacent grain sets are first selected, and the node information of one grain is compared with all the node information in the remaining grain sets; if the node also exists in the second grain set, for the convenience of expression, such another grain set is defined as the second grain; its node is stored in the interface list, and at the same time, this node is deleted from the two selected grain sets to ensure uniqueness, so that this node is no longer compared repeatedly in the comparison with the adjacent grains. The specific expression form is as follows:

[0070] NGset[n]=Gset[n]-B

[0071] Where: NGset represents the set of grains excluding interface nodes;

[0072] In this way, the comparison, screening and classification of node information are carried out simultaneously, and the interface nodes and the internal nodes of the set are quickly screened out from a large number of values, saving a lot of comparison operations; through the parallel progressive method, the S2 is used to establish the grain set, and the nodes on the surface of the set are removed to determine the internal node information of the grain set, which is stored in the form of a list;

[0073] S4: determine the set of pores that need to be removed;

[0074] First, the overall model is divided into multiple standard regions with a volume of a×b×c, so that each standard region contains at least 4 complete grain sets; the pore set is selected to divide the overall model, and the random pore position is determined using the perturbation parameter r to avoid direct random selection that causes the pores to be connected and lead to the formation of large holes; the position of the pore is determined in the standard region based on the perturbation parameter r of the center as shown below;

[0075] Then, the area where the pores are located is selected from all standard areas, and then the position of the pores is determined in the standard area according to the central disturbance parameter r, which is expressed as:

[0076]

[0077] Where: X1, Y1 and Z1 represent the horizontal, vertical and vertical coordinates of the pore location respectively; X0, Y0 and Z0 represent the horizontal, vertical and vertical coordinates of the minimum point of the standard area respectively; r represents the disturbance parameter, which is a random number between 0 and 1 and is automatically generated by the computer; a, b and c represent the length, width and height of the cube of the standard area;

[0078] Complete the random selection of pore sets from the entire set. The pores selected by this perturbation random method avoid the direct random selection that leads to the connection of pores and the generation of larger pores, which is more similar to the actual pore distribution;

[0079] In order to make the randomly generated coordinate positions in the model space present a regular distribution state, the pore set to be removed is regularly selected according to the weights k, g and h of the pores in the horizontal, vertical and vertical coordinate directions; at this time, the overall model is divided into standard areas, and the pore shape satisfies the following relationship:

[0080]

[0081] Where: X m , Y m and Z m Represents the horizontal, vertical and vertical coordinates of the center position of the standard area; k, g and h represent the weights of the horizontal, vertical and vertical coordinate directions respectively.

[0082] S5: renumber the unit information and internal node information of the model after removing the pores;

[0083] Remove the unit information of the pore set and renumber the units of the model: according to the pore set information selected by S4, delete the units in the pore set; this will cause the unit numbers in the model to be discontinuous, so the remaining units are renumbered from small to large, but the grain set information and the node information contained in the unit are not changed;

[0084] Remove the internal node information of the pore set and renumber the nodes of the model: according to the pore set information selected by S4 and the internal node information of the set determined by S3, delete the nodes in the pore set; this will cause the node numbers to be discontinuous, so the remaining nodes are renumbered in order from small to large, but the unit information containing the node and the position information contained in the node are not changed;

[0085] S6: Establish a multi-pore model between crystals;

[0086] According to the unit information and internal node information re-edited by S5, the multi-pore model between crystals is constructed. Write it into a new inp file, and import the newly written inp file into Abaqus software.

[0087] like Figure 5 FIG. 1 is a pore model diagram of 5% equiaxially distributed pores in the modeling method of a microscopic model of a porous material at a crystal scale in an embodiment of the present invention, wherein the pores are substantially equiaxial and their shapes satisfy the Voronoi shape characteristics, and are not regular spheres or squares, which are more similar to the pores in actual crystal materials; Figure 6 Shown is a pore model diagram of 1:1:3 long strips with 5% uniformly distributed pores in the modeling method of the microscopic model of porous materials at the crystal scale in an embodiment of the present invention, wherein the pores exhibit long strip characteristics in the vertical direction, and the shape satisfies the Voronoi shape characteristics, rather than being a regular rectangle, and is more similar to the pores in actual crystal materials.

[0088] The model established by the construction method of the present invention has pores with different morphological states, and the pores are evenly distributed in the representative volume element model. By assigning material properties to the solid unit (i.e., a collection of grains) and simulating the deformation behavior of the representative volume element model, it is possible to realize the morphological change law of pores with different morphologies along with the overall model, and to realize the detailed expression of the influence of pores with different morphologies on the stress and strain of the matrix, such as Figure 7 The figure shows the pore morphology and matrix deformation results during the compression deformation process in the modeling method of the microscopic model of porous materials at the crystal scale in an embodiment of the present invention. This result is more realistic for simulating the deformation of pores in crystal materials.

[0089] In summary, the prediction results of this case prove to be very effective.

[0090] (1) The technical solution proposed in the embodiment of the present invention uses Thiessen polygons to establish a grain set, obtains internal information of the grain set, and establishes a polycrystalline pore structure model based on a parallel progressive polycrystalline internal perturbation random method. The model structure established by this method is more accurate and has a smaller error compared with the structure of the actual crystal material;

[0091] (2) The embodiment of the present invention can control the shape and structure of the pore structure model by controlling the proportion of the weight of the pores in different coordinate directions, which is convenient for designing pore structure models of different sizes and structures, making it more flexible for operators to use and more convenient and free in actual application effects.

[0092] The embodiments described above are only descriptions of the preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A method for constructing a random pore model of polycrystalline internal disturbance based on parallel progression, characterized in that: It includes the following steps: Step 1: Obtain a three-dimensional Thiessen polygon representative volume element model of the crystal material; Establish a three-dimensional Thiessen polygon representative volume element model of the crystal material, divide it into multiple grain sets, and obtain a polycrystalline representative volume element model; Step 2: According to the polycrystalline representative volume element model, obtain the unit information and all node information of the grain set; Since the polycrystalline representative volume element model in step 1 includes a large number of grain sets, and each grain set contains multiple units; according to the structural characteristics of the grain set, the unit number of each grain set is extracted and exported and stored in the form of key-value pairs. The specific expression form is as follows: M={[Gset[n]:E1,…E i ,…],…} Where: M represents the polycrystalline body model; Gset represents the grain set; n represents the sequence number of the grain set; E represents the unit of the grain set; i represents the unit number; Each unit of a grain is composed of nodes. The node information corresponding to the unit is extracted. At the same time, according to the unit information of the grain set extracted by the above formula, all the node information contained in the grain set is determined. The specific expression form is as follows: Gset[n]=[N1,…N j ,…] Where: N represents the node; j represents the node number; Step 3: Determine the internal node information of the grain set based on parallel progression; Compare all node numbers contained in one grain set with all node numbers of the remaining grain sets. If the node numbers are the same, it is considered that the node coexists in two grain sets. Then the node is a common node between the grain sets and also a node on the surface of the grain set. The specific expression is as follows: B=Gset[n]∩Gset[n+1] Where: B represents the grain boundary set; In this selection among a large number of values, two adjacent grain sets are first selected, and the node information of one grain is compared with all the node information in the remaining grain sets; if the node also exists in the second grain set, its node is stored in the interface list, and the node is deleted from the two selected grain sets to ensure uniqueness, so that the node is no longer compared repeatedly in the comparison with the adjacent grains. The specific expression is as follows: NGset[n]=Gset[n]-B Where: NGset represents the set of grains excluding interface nodes; In this way, the comparison, screening and classification of node information are carried out simultaneously, and the interface nodes and the internal nodes of the set are quickly screened out from a large number of values, saving a lot of comparison operations; the internal node information of the grain set is determined by removing the nodes on the surface of the set from the grain set established in step 2 through a parallel progressive method, and stored in a list form; Step 4: Determine the set of pores that need to be removed; First, the overall model is divided into multiple standard regions with a volume of a×b×c, so that each standard region contains at least 4 complete grain sets, and avoids the selected pore set to become pores within the grain or the entire grain set; The location of the pores in the standard area is determined based on the disturbance parameter r at the center; Step 5: Renumber the unit information and internal node information of the model after removing the pores; Remove the unit information of the pore set and renumber the units of the model: according to the pore set information selected in step 4, delete the units in the pore set; this will cause the unit numbers in the model to be discontinuous, so the remaining units are renumbered from small to large, but the grain set information and the node information contained in the unit are not changed; Remove the internal node information of the pore set and renumber the nodes of the model: according to the pore set information selected in step 4 and the internal node information of the set determined in step 3, delete the nodes in the pore set; this will cause the node numbers to be discontinuous, so the remaining nodes are renumbered in order from small to large, but the unit information containing the node and the position information contained in the node are not changed; Step 6: Establish a multi-pore model between crystals; According to the unit information and internal node information re-edited in step 5, the multi-pore model between crystals is constructed.

2. The method for constructing a random pore model of a polycrystalline body with internal disturbance based on parallel progression according to claim 1, characterized in that: The three-dimensional Thiessen polygons in step 1 represent a volume element model, which can construct an equiaxed grain model and a fibrous grain model by imitating actual crystal materials.

3. The method for constructing a random pore model of a polycrystalline body with internal disturbance based on parallel progression according to claim 1, characterized in that: In step 3, the intersection nodes of the two grain sets are deleted from the two selected grain sets, so that this node is no longer repeatedly compared with the adjacent grains; the comparison, screening and classification of the node information are performed simultaneously, and the interface nodes and the internal nodes of the grain set are quickly screened out in parallel and progressively.

4. The method for constructing a random pore model of a polycrystalline body with internal disturbance based on parallel progression according to claim 1, characterized in that: The pore set selected in step 4 is separated into the whole model, and the random pore position is determined by using the perturbation parameter r to avoid the formation of large pores caused by direct random selection resulting in pore connection; the position of the pore is determined according to the perturbation parameter r of the center within the standard area as follows; Then, the area where the pores are located is selected from all standard areas, and then the position of the pores is determined in the standard area according to the central disturbance parameter r, which is expressed as: Where: X1, Y1 and Z1 represent the horizontal, vertical and vertical coordinates of the pore location respectively; X0, Y0 and Z0 represent the horizontal, vertical and vertical coordinates of the minimum point of the standard area respectively; r represents the disturbance parameter, which is a random number between 0 and 1 and is automatically generated by the computer; a, b and c represent the length, width and height of the cube of the standard area; Complete the random selection of pore sets from the entire set. The pores selected by this perturbation random method avoid the direct random selection that leads to the connection of pores and the generation of larger pores, which is more similar to the actual pore distribution; In order to make the randomly generated coordinate positions in the model space present a regular distribution state, the pore set to be removed is regularly selected according to the weights k, g and h of the pores in the horizontal, vertical and vertical coordinate directions; at this time, the overall model is divided into standard areas, and the pore shape satisfies the following relationship: Where: X m , Y m and Z m Represents the horizontal, vertical and vertical coordinates of the center position of the standard area; k, g and h represent the weights of the horizontal, vertical and vertical coordinate directions respectively.

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