Proxy material model explicit topological optimization method for boundary unit reconstruction

Abstract

A proxy material model explicit topological optimization method for boundary unit reconstruction comprises the steps of defining a design domain and components of a proxy material model, dividing grid units in the design domain into boundary inner units, boundary outer units and boundary units through boundary description, calculating the density of all divided units, and calculating the density of all the divided units; the topological structure is optimized through finite element analysis and volume calculation, after the density of all the units is calculated, the density of the boundary units is updated and replaced, then finite element analysis and volume calculation are carried out, the finite element analysis accuracy can be improved, and then the topological optimization precision is improved. According to the grid refinement based on the boundary units, only the boundary units are subjected to grid reconstruction, and compared with global grid refinement, the number of reconstruction units can be greatly reduced. And new nodes formed by reconstruction are only used for calculating the boundary element density and do not participate in finite element analysis, so that the influence on the overall calculation efficiency is small.

Description

Technical Field

[0001] The invention relates to a topology optimization method of a proxy material model, in particular to an explicit topology optimization method of a proxy material model with boundary unit reconstruction. Background Art

[0002] Surrogate model, also known as metamodel or response surface model, replaces a complex and computationally expensive model by creating an approximate model with low computational cost. Surrogate model is widely used in the field of material science and engineering design. Surrogate material model can obtain calculation results close to theory with very low computational cost. The topology optimization methods of surrogate material model can be divided into two categories according to the different topological descriptions of the structure: topology optimization based on density description and topology optimization based on boundary description. Topology optimization methods based on density description, such as solid isotropic material with penalization (SIMP), are to make pseudo-density continuous and penalize the design variables to discrete solutions of 0 / 1 by penalization. SIMP method based on density description has the advantages of stable convergence process and easy handling of complex structural design problems such as multi-class variables, multi-constraints and multi-disciplinary optimization. It has been successfully integrated into many commercial CAE software such as Abaqus, Ansys and Hyperworks. However, it is difficult to identify and control geometric features (such as structural size, interface and closed holes) of structural topology optimization based on density description.

[0003] There are topology optimization methods based on boundary description, such as level set method (LSM) and moving morphable component (MMC) "GUO X, ZHANG W, ZHONG W. Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework [J]. Computer Methods in Applied Mechanics and Engineering, 2014, 81: 81009". The level set method uses the zero contour of the higher-dimensional level set function to implicitly characterize the boundary of the topological structure, and the MMC method uses an explicit boundary to describe each component in the structure. Although the topology optimization method based on density description is relatively mature, the topology optimization method based on boundary description has a clearer structural boundary, which can further reduce the human intervention in post-processing. Therefore, it has become a research hotspot in topology optimization technology and an important development direction of topology optimization technology in the future. Topology optimization based on boundary description realizes the evolution of topological structure by driving boundary evolution. The boundary divides the grid cells in the design domain into three categories: cells inside the boundary, cells outside the boundary, and cut cells, and obtains the density of the cells based on node information. In the MMC method, cutting cells will produce intermediate density. Due to the choice of cutting cell processing strategy by the boundary description-based topology optimization method, cells of different areas may have the same cell density, which will also affect the calculation accuracy. Summary of the invention

[0004] The technical problem to be solved by the present invention is to overcome the problem that inaccurate boundary unit density affects model accuracy in existing topology optimization methods based on boundary description, and to provide an explicit topology optimization method for proxy material models with boundary unit reconstruction.

[0005] The technical solution adopted by the present invention to solve the above technical problems is: an explicit topology optimization method of a proxy material model with boundary unit reconstruction, defining the design domain and components of the proxy material model, dividing the grid units in the design domain into boundary inner units, boundary outer units and boundary units through boundary description, calculating the density of all divided units, and optimizing the topological structure through finite element analysis and volume calculation. After calculating the density of all units, the density of the boundary units is updated and replaced, and then the finite element analysis and volume calculation are performed. The boundary unit density updating method includes the following steps:

[0006] (1) Identify boundary elements based on the calculation results of the proxy material model;

[0007] (2) Reconstruct the boundary cells, layout the same number of nodes with the same spacing along the edges of the boundary cells in both directions, and generate corresponding nodes inside the boundary cells to achieve grid subdivision and reconstruction of the boundary cells;

[0008] (3) Calculate the topological function values ​​of the newly added nodes based on the description functions of all components and identify the nodes inside the boundary in the boundary cells;

[0009] (4) Calculate the number of nodes located inside the boundary of the boundary cell, and use the ratio of the number of nodes located inside the boundary of the boundary cell to the total number of nodes of the boundary cell as the new cell density of the boundary cell, thereby completing the update and replacement of the boundary cell density.

[0010] In the step (1), boundary cells are identified based on the calculation results of the proxy material model, and intermediate density cells with density between material-free cells and solid cells in the calculation results are identified as boundary cells.

[0011] The grid cells in the design domain are numbered in sequence, and after the boundary cells are identified, the boundary cell coordinates are determined according to the sequence numbers of the boundary cells.

[0012] An xy coordinate system is established with a corner of the design domain as the origin, and the design domain is located in the first quadrant of the coordinate system; each grid unit divided in the design domain is a 4-node bilinear unit, and each grid unit is surrounded by 4 nodes distributed at the 4 corners; the numbering order of the grid units and the nodes starts from a column close to the y-axis, and each column is numbered sequentially from farthest from the x-axis to close to the x-axis.

[0013] The node close to the origin of each grid unit is taken as the reference node, and its reference node number Nj is calculated according to the sequence number of the identified boundary unit. The formula is:

[0014] Among them, Ei is the unit number, β is a very small positive value, nely is the number of grid units in the y direction, |·| integer Indicates taking the absolute value of an integer;

[0015] Determine the boundary element reference node coordinates Nj(x e ,y e ), the formula is:

[0016]

[0017] Where EW is the grid unit length and EH is the grid unit width.

[0018] In the step (2), the number of newly added nodes is determined according to the required reconstruction accuracy of the boundary unit. The more newly added nodes are, the higher the reconstruction accuracy is.

[0019] In the step (2), nodes are set in the boundary unit, and the boundary unit is divided into m grids in both the x-direction and the y-direction. The boundary unit can be reconstructed into:

[0020]

[0021] Among them, x N and N are the horizontal and vertical coordinates of all nodes of the reconstructed boundary unit.

[0022] In the step (3), the topological function value of the boundary unit node is mapped to the node using the Heaviside function, and the node with a value of 1 is identified as the node located inside the boundary of the boundary unit.

[0023] When the boundary only falls on a node at one corner of the boundary element, resulting in a density calculation value of 0, the new density value of the boundary element is set to a minimum value.

[0024] During the finite element analysis and volume calculation, the nodes added during the boundary element reconstruction process do not participate in the calculation.

[0025] The beneficial effects of the present invention are as follows: the boundary units are re-meshed and reconstructed to form a local encrypted network at the boundary, which is beneficial to the ratio of the number of entity nodes in the boundary unit to the total number of nodes as the new density of the boundary unit, thereby obtaining more accurate unit density information, which can provide the accuracy of finite element analysis, and thus improve the accuracy of topological optimization. Mesh refinement based on boundary units only reconstructs the mesh of boundary units, which can greatly reduce the number of reconstructed units compared to global mesh refinement. Moreover, the new nodes formed by reconstruction are only used to calculate the density of boundary units, do not participate in finite element analysis (do not calculate the unit stiffness matrix and displacement), and do not participate in sensitivity analysis, so the impact on the overall calculation efficiency is small. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a schematic diagram of unit grid cutting.

[0027] Figure 2 It is a schematic diagram of the boundary unit grid reconstruction model of the present invention.

[0028] Figure 3 It is a schematic diagram of the boundary unit density update and replacement process of the present invention.

[0029] Figure 4 It is a schematic diagram of grid unit division in the design domain.

[0030] Figure 5 It is a schematic diagram of the problem example.

[0031] Figure 6 This is a schematic diagram comparing the optimization results of the Michell truss.

[0032] Figure 7 It is a schematic diagram comparing the short beam optimization results. DETAILED DESCRIPTION

[0033] The technical solution of the present invention is described clearly and completely below in conjunction with the accompanying drawings and specific implementation methods. The specific contents listed in the following embodiments are not limited to the technical features necessary for the technical problems to be solved by the technical solutions recorded in the claims. At the same time, the enumerated embodiments are only part of the present invention, not all embodiments.

[0034] The explicit topology optimization method of the proxy material model reconstructed by boundary cells of the present invention is based on the mobile deformable component method (MMC), which defines the design domain of the proxy material model and the components in the design domain, and each component is explicitly described by an explicit topological description function, and the proxy material model is optimized by operations such as movement, deformation, and covering of the component and topological optimization. In the mobile deformable component method, the grid cells in the design domain are divided into three categories: boundary inner cells, boundary outer cells, and boundary cells by describing the boundary of the component, and the density of all the divided cells is calculated based on the node information, and the topological structure is optimized by finite element analysis and volume calculation.

[0035] The proxy material model mainly defines the unit material properties based on the node material properties. For a single material, a 4-node bilinear unit is used, that is, the nodes located at the four corners form a grid unit, and the following unit density function can be defined:

[0036]

[0037] Where H = H(x) is the Heaviside function, i=1,…,4 represents the topological value of the unit node, q represents the penalty factor (usually q=2). Then the unit equivalent stiffness can be expressed as:

[0038]

[0039] The above formula is the proxy material model, where E is the elastic modulus of the solid element. In order to ensure the stability of the numerical implementation, H(x) is usually regularized as follows during finite element analysis:

[0040]

[0041] Where ε is a regularization parameter that controls the width of the smooth transition of the Heaviside function, and α is a small positive number that simulates the stiffness of the hole structure in the proxy material model to avoid singular phenomena in the overall stiffness matrix of the structure.

[0042] In the usual mobile deformable component method, the proxy material model obtains the density of the unit from the node information. When the boundary cuts the unit, the density of the same unit may be calculated at different boundary positions. Figure 1 The figure shows the schematic diagram of unit mesh cutting. Among them, Figure (a) is a unit without material, (b) is a unit with solid material, (c) and (d) are schematic diagrams of cutting units cut in the oblique and horizontal directions respectively. There are three cutting methods in each direction: ①, ②, and ③. For ease of understanding, the regularized form of the Heaviside function is not considered. Figure 1 (c) and Figure 1 The calculated density and actual density of the cut unit shown in (d) are shown in the following table. c1, c2, c3 in the table are the actual densities of the units in the three different cutting methods ①, ②, ③ in Figures (c) and (d).

[0043]

[0044] It can be seen that the calculated density is only one case of the actual density. The actual density has a large variation range under the same calculated density. If the calculated density is used for finite element analysis, a large analysis error may occur. In addition, the accuracy of the cutting unit also directly affects the calculation of the actual volume. The solid volume is often used as a constraint, and its result needs to be brought into the solver to update the design variables. Therefore, the actual volume of the cutting unit will also have a certain impact on the variable update, which may affect the final structural form.

[0045] In order to solve the problem of inaccurate calculation of boundary unit density, the density of the boundary unit is updated and replaced after calculating the density of all units, and then the finite element analysis and volume calculation are performed. The core idea of ​​this method is to reconstruct the boundary unit (i.e., the cutting unit) by secondary mesh division to form a local encrypted mesh of the boundary, such as Figure 2 As shown. Then the unit density of the boundary unit is recalculated based on the new unit node information. The two key points of reconstructing the proxy material model of the boundary unit are the identification of the boundary unit and the reconstruction unit density calculation strategy. The identification of the boundary unit can be achieved by screening the calculation results of the proxy material model, that is, screening out the intermediate density units. The reconstruction unit density calculation strategy still uses the calculation strategy based on node information, considering that the calculation amount of the secondary calculation using the proxy material model is large and the implementation is more complicated. The present invention uses a relatively simple statistical method, which uses the ratio of the number of physical nodes to the total number of nodes as the new density of the unit. The process of updating and replacing the boundary unit density is as follows: Figure 3 As shown in the figure, the process can be embedded into the existing MMC method program. After the original program calculates all unit densities using the proxy material model, the following process is implemented:

[0046] (1) Identify boundary elements. Identify boundary elements based on the calculation results of the proxy material model; the density of the material-free element in the calculation results is α 2 , the density of the solid unit is 1, α is a small positive number, which is much smaller than 1, and the density is between α 2 The elements with density between 1 and 1, that is, the elements with density between the material-free element and the solid element, are identified as boundary elements.

[0047] (2) Boundary unit reconstruction. Nodes with the same number and equal spacing are arranged along the edges of the boundary unit in both directions, and corresponding nodes are generated inside the boundary unit to achieve grid subdivision and reconstruction of the boundary unit. The reconstruction accuracy is determined by the number of newly added nodes. The more newly added nodes, the higher the reconstruction accuracy. The number of newly added nodes is determined according to the required boundary unit reconstruction accuracy.

[0048] (3) Identification of nodes within the boundary. The topological function value of the newly added node is calculated based on the description functions of all components, and the Heaviside function is used to identify the nodes within the boundary.

[0049] (4) Recalculate the boundary cell density. Calculate the number of nodes located inside the boundary in the boundary cell, and use the ratio of the number of nodes located inside the boundary in the boundary cell to the total number of nodes in the boundary cell as the new cell density of the boundary cell, thus completing the update and replacement of the boundary cell density.

[0050] In the specific process, after the boundary unit is identified, it is necessary to locate the boundary unit, that is, determine the boundary unit coordinates. Then calculate the coordinates of the internal nodes of the boundary unit after reconstruction and the topological function values ​​of the reconstructed boundary unit nodes. The boundary unit coordinates can be determined according to the boundary unit sequence number. Figure 4 As shown, the grid units divided in the design domain are 4-node bilinear units. The square on the right side of the figure is a legend of the grid unit, which has four nodes: 1th, 2th, 3th, and 4th. The unit length is EW and the unit width is EH. Figure 4 The left side of the figure shows an example of 3×3 unit grid division. In the figure, an xy coordinate system is established with a corner of the design domain as the origin, and the design domain is located in the first quadrant of the coordinate system. The total number of x-direction units nelx = 3, and the total number of y-direction units nely = 3. The grid units and nodes are numbered in sequence, starting from the column close to the Y axis, and numbering in each column from the farthest from the X axis to the closest to the X axis. After one column is numbered, the numbering continues from the next column. The numbering of each grid unit and node is shown in the figure.

[0051] The node of each grid unit close to the origin (i.e. the second node in the figure) is taken as the reference node, and its reference node number Nj is calculated according to the sequence number of the identified boundary unit. The formula is:

[0052]

[0053] Where Ei is the unit number and β is a very small positive value to ensure that the formula is also applicable to the bottom row of units and to avoid errors in the calculation of the first row of units. In general, β = 1 × 10 -3 nely is the number of grid cells in the y direction, |·| integer Indicates taking the absolute value of an integer.

[0054] Determine the boundary element reference node coordinates Nj(x e ,y e ), the formula is:

[0055]

[0056] Taking the reconstruction accuracy as m (m is an integer) as an example, in the reconstruction process, nodes are arranged on the edges and inside of the boundary cells in both directions, and the boundary cells are divided into m grids in both the x and y directions. The meshgrid function can be used to reconstruct the boundary cells as follows:

[0057]

[0058] In the formula, x N and N are the horizontal and vertical coordinates of all nodes of the reconstructed boundary unit.

[0059] Calculate the topological function values ​​of all components at the boundary unit nodes after reconstruction, and then map the topological function values ​​to the nodes through the Heaviside function (the regularized form of the Heaviside function is not required at this time), and identify the nodes with a value of 1 as the nodes located inside the boundary of the boundary unit. Count the number of nodes with a value of 1 at the boundary unit nodes after reconstruction, and the ratio of this value to the total number of nodes is recorded as the density of the boundary unit. For example Figure 2 As shown, Figure (a) is a schematic diagram of unit reconstruction, and the reconstructed unit has a total of 11×11 nodes. Figure (b) is a schematic diagram of the identified nodes within the boundary, where the arc line is the boundary, the black dots are the identified nodes located inside the boundary, there are 24 boundary stacked nodes (solid nodes), and the density of the boundary unit is calculated to be 24 / 121. Compared with the unit calculation density of 1 / 4 of the original proxy material model of the MMC method, the accuracy is significantly improved. It should be noted that some situations may cause the new density calculation value to be 0, such as when the boundary only falls on the node at one corner of the boundary unit (for example, it falls on the first node of the unit 1th). At this time, in order to avoid matrix singularity, the new density is assigned a minimum value. Finally, the density of the boundary unit is replaced.

[0060] The following example analysis is carried out. The numerical example program based on the MMC framework draws on the existing technology, and the units are dimensionless. The short beam problem and the Michell truss problem are selected as the research objects, such as Figure 5 As shown, Figure (a) is a short beam problem, Figure (b) is a Michell truss problem, and F is the load.

[0061] Taking volume as constraint and structural flexibility as objective function, the topology optimization problem of minimizing structural flexibility is considered. The specific optimization formula is as follows:

[0062] Find D=((D 1 ) T ,…,(D i ) T ,…,(D n ) T ) T ,u(x)∈Ω s (D)

[0063]

[0064] St

[0065]

[0066] Where D represents the given structural design domain, Ω s (D) represents the area occupied by the entire solid material of the structure, u(x) represents the displacement field function of the structure in the design domain, which is only related to the coordinate x, and U ad ={v(x)v(x)∈H 1 (D),v(x)=0onΓ U} is the admissible set of displacement trial functions that can be selected, and v(x) represents the trial displacement function in finite element analysis. Denotes the Dirichlet frontier Γ U The displacement boundary conditions are given in advance according to the specific problem, and t represents the Newman boundary Γ t The boundary conditions of the force are given by the external load acting on the structure. E(x)=(E ijkl (x)) represents the fourth-order elastic tensor distributed in the design domain D, Where II represents the fourth-order unit tensor, δ represents the second-order unit tensor, Represents tensor product. H(x) represents the Heaviside function. When x>0, H=1, and when x<0, H=0. Since the Heaviside function is not a continuous function, in order to ensure the differentiability of sensitivity and the numerical stability of iterative optimization, the regularized form of the Heaviside function H is often used. ε (x).

[0067] Only the sensitivity analysis of the optimization problem with the objective function of structural flexibility is discussed. The sensitivity of the objective function to any design variable a (component geometric parameter) is:

[0068]

[0069] Where K is the overall stiffness matrix of the structure, k s is the topological function value The element stiffness matrix corresponding to the element with i = 1, ..., 4 and E = 1. In formula (13), NE is the total number of elements in the design domain.

[0070] The sensitivity of the constraint function, i.e., the volume constraint function, to the design variables is:

[0071]

[0072] For the Michell truss problem, the design domain length and height are 2×1, the two end points of the bottom edge are fixed, a vertical downward load F=1 is applied to the midpoint of the bottom edge, the volume constraint is set to 0.4, and three grid densities are set: 80×40, 100×50, and 120×60. The short beam optimization results based on the proxy material model and boundary element reconstruction proxy material model are as follows: Figure 6 It can be seen that the optimization results of the proposed boundary unit reconstruction proxy material model are similar to those of the original proxy material model. The reason why the topological optimization structures of the two models are similar is that the case structure is relatively simple, which also verifies the effectiveness of the proposed method.

[0073] Regarding the short beam problem, the design domain length and height are 2×1, the left side is fixed, a vertical downward load F=1 is applied to the midpoint of the right side, the volume constraint is set to 0.4, and three grid densities are set: 80×40, 100×50, and 120×60. The short beam optimization results based on the proxy material model and boundary unit reconstruction proxy material model are as follows: Figure 7 It can be seen that the boundary unit reconstruction proxy material model can obtain a better topological structure under different grid densities and can further reduce the grid dependence.

[0074] The boundary unit reconstruction proxy material topology optimization method of the present invention is proposed to obtain more accurate unit density information to improve the accuracy of finite element analysis, thereby improving the accuracy of topology optimization. Compared with the global mesh refinement, which greatly increases the amount of finite element calculations, the mesh refinement based on the boundary unit only reconstructs the boundary unit, and the number of reconstructed units is greatly reduced. And the new nodes formed by reconstruction are only used to calculate the density of the boundary unit, do not participate in the finite element analysis (do not calculate the unit stiffness matrix and displacement), do not participate in sensitivity analysis, and therefore have little impact on the overall calculation efficiency. The boundary unit reconstruction proxy material model can also further decouple the optimization results and the background mesh.

[0075] The above description of the specific implementation mode is only used to help understand the technical concept and core idea of ​​the present invention. Although the technical solution is described and illustrated using a specific preferred embodiment, it should not be understood as a limitation of the present invention itself. Those skilled in the art may make various changes in form and details without departing from the technical concept of the present invention. These easily conceived changes or substitutions should all be included in the protection scope of the present invention.

Claims

1. An explicit topology optimization method for a proxy material model with boundary unit reconstruction, defining the design domain and components of the proxy material model, dividing the grid cells in the design domain into boundary inner cells, boundary outer cells and boundary cells through boundary description, calculating the density of all divided cells, and optimizing the topology structure through finite element analysis and volume calculation, characterized in that: After calculating the density of all cells, the density of the boundary cells is updated and replaced, and then the finite element analysis and volume calculation are performed. The boundary cell density update method includes the following steps: (1) Identify boundary elements based on the calculation results of the proxy material model; (2) Reconstruct the boundary cells, layout the same number of nodes with the same spacing along the edges of the boundary cells in both directions, and generate corresponding nodes inside the boundary cells to achieve grid subdivision and reconstruction of the boundary cells; (3) Calculate the topological function values ​​of the newly added nodes based on the description functions of all components and identify the nodes inside the boundary in the boundary cells; (4) Calculate the number of nodes located inside the boundary of the boundary cell, and use the ratio of the number of nodes located inside the boundary of the boundary cell to the total number of nodes of the boundary cell as the new cell density of the boundary cell, thereby completing the update and replacement of the boundary cell density.

2. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 1, characterized in that: In the step (1), the boundary unit is identified based on the calculation results of the proxy material model, and the density of the material-free unit in the calculation results is α 2 , the density of the solid unit is 1, α is a positive number much smaller than 1, and the density is between α 2 The cells between 1 and 1 are identified as boundary cells.

3. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 1, characterized in that: The grid cells in the design domain are numbered in sequence, and after the boundary cells are identified, the boundary cell coordinates are determined according to the sequence numbers of the boundary cells.

4. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 3, characterized in that: An xy coordinate system is established with a corner of the design domain as the origin, and the design domain is located in the first quadrant of the coordinate system; each grid unit divided in the design domain is a 4-node bilinear unit, and each grid unit is surrounded by 4 nodes distributed at the 4 corners; the numbering order of the grid units and the nodes starts from a column close to the y-axis, and each column is numbered sequentially from farthest from the x-axis to close to the x-axis.

5. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 4, characterized in that: The node close to the origin of each grid unit is taken as the reference node, and its reference node number Nj is calculated according to the sequence number of the identified boundary unit. The formula is: Among them, Ei is the unit number, β is a very small positive value, nely is the number of grid units in the y direction, |·| integer Indicates taking the absolute value of an integer; Determine the boundary element reference node coordinates Nj(x e ,y e ), the formula is: Where EW is the grid unit length and EH is the grid unit width.

6. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 1, characterized in that: In the step (2), the number of newly added nodes is determined according to the required reconstruction accuracy of the boundary unit. The more newly added nodes are, the higher the reconstruction accuracy is.

7. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 5, characterized in that: In the step (2), nodes are set in the boundary unit, and the boundary unit is divided into m grids in both the x-direction and the y-direction. The boundary unit can be reconstructed into: Among them, x N and N are the horizontal and vertical coordinates of all nodes of the reconstructed boundary unit.

8. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 1, characterized in that: In the step (3), the topological function value of the boundary unit node is mapped to the node using the Heaviside function, and the node with a value of 1 is identified as the node located inside the boundary of the boundary unit.

9. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 1, characterized in that: When the boundary only falls on a node at one corner of the boundary element, resulting in a density calculation value of 0, the new density value of the boundary element is set to a minimum value.

10. The method for explicit topology optimization of a proxy material model with boundary element reconstruction according to claim 1, characterized in that: During the finite element analysis and volume calculation, the nodes added during the boundary element reconstruction process do not participate in the calculation.

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