An efficient numerical analysis method based on regular space grid

Abstract

The present invention discloses an efficient numerical analysis method based on a regular space grid, which comprises the following steps: step 1, meshing a structural model, dividing it into a plurality of grid units, the grid unit where the boundary of the structural model is located is called a boundary unit; step 2, multi-level subdividing the boundary unit into a plurality of subdomains, the subdomain where the boundary of the structural model is located is called a boundary subdomain; reconstructing the boundary within the boundary subdomain to obtain a reconstructed boundary; step 3, generating the equilibrium equation of the structural model based on the reconstructed boundary, and solving to obtain the field variables of the structural model; post-processing the field variables to obtain the analysis result of the structural model. The method described in the present invention can better balance the calculation efficiency and calculation accuracy, and can obtain more accurate results in a faster way compared to the existing analysis method.

Description

Technical Field

[0001] The present invention relates to the technical field of structural numerical analysis, and in particular to an efficient numerical analysis method based on a regular space grid. Background Art

[0002] In the numerical analysis of engineering structures, the finite element method is the most popular numerical analysis tool. However, the process of generating high-quality body-fitting meshes using the finite element method is extremely complicated and time-consuming, especially for geometrically complex structural models.

[0003] When using the numerical analysis method based on regular space grid, it becomes difficult to accurately integrate on the boundary cells, and the processing of boundary conditions and post-processing of results become complicated because the boundary cells of the space grid do not fit the physical boundaries of the structural model. In order to accurately integrate on the boundary cells, the boundary cells are usually subdivided by quadtree / octree, so that the subdivided subdomains are as close to the physical boundaries of the structural model as possible, and the subdomains of the boundary cells are integrated using Gaussian integrals to complete the integration of the boundary cells at the cell level. However, when there are fewer Gaussian integral points, the error of this integration method of the boundary cells is large, especially for some boundary cells with only a few areas located inside the structural model, which cannot meet the requirements of calculation accuracy. In order to improve the calculation accuracy of the boundary cells, the number of Gaussian points must be increased, but the increase in Gaussian points will increase the amount of calculation at the cell level, greatly increasing the amount of calculation for numerical analysis. In the processing of boundary conditions of regular space grids, since there is no grid that fits the physical boundaries of the structural model, we can only indirectly apply the boundary conditions to the nodes of the non-fitting grid, so complex processing is required. In the post-processing of the existing regular space grid method, if the gradient of the field variable is to be accurately calculated, it is necessary to ensure that the integration of the structural model at the unit level is very accurate. However, the amount of calculation in three-dimensional problems is huge and cannot be achieved in engineering applications.

[0004] Therefore, how to reduce the amount of calculation while ensuring the accuracy of the results on the basis of the existing numerical analysis method based on regular space grids is a technical problem that needs to be solved urgently by technical personnel in this field. Summary of the invention

[0005] In view of this, the purpose of the present invention is to provide an efficient numerical analysis method based on regular space grids to address the deficiencies of the prior art, solve the problems of the existing regular space grid method in that it is difficult to accurately integrate the boundary cells, and the boundary condition processing and result post-processing are complicated. The boundary cells are integrated at the cell level by performing multi-level subdivision and reconstructing the boundary of the structural model, the boundary conditions of the structural model are processed by the penalty function method, and the post-processing is performed by constructing triangle / tetrahedron units at the boundary position of the structural model. The calculation efficiency and calculation accuracy can be better balanced, and more accurate results can be obtained in a faster way than the existing analysis methods. The present invention includes the following steps:

[0006] Step 1: Mesh the structural model into a plurality of mesh units, wherein the mesh unit where the boundary of the structural model is located is called a boundary unit;

[0007] Step 2: subdividing the boundary unit into multiple subdomains at multiple levels, and calling the subdomain where the boundary of the structural model is located as the boundary subdomain; reconstructing the boundary in the boundary subdomain to obtain a reconstructed boundary;

[0008] Step three: generating the equilibrium equation of the structural model based on the reconstructed boundary, and solving it to obtain the field variables of the structural model; post-processing the field variables to obtain the analysis results of the structural model.

[0009] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, the step 1 includes dividing the grid unit into internal units, boundary units and external units according to the positional relationship between the structural model and the grid unit; the specific steps of determining the positional relationship between the structural model and the grid unit are:

[0010] Randomly scatter some sampling points in the grid unit, and use the ray method to draw a ray in any direction with the sampling point as the starting point. If the number of intersections between the ray and the boundary of the structural model is an odd number, the sampling point is located inside the structural model and is an internal point. If the number of intersections is an even number, the sampling point is located outside the structural model and is an external point.

[0011] If all the sampling points of the grid cell are internal points, the grid cell is an internal cell; if all the sampling points of the grid cell are external points, the grid cell is an external cell; if all the sampling points of the grid cell have both internal points and external points, the grid cell is a boundary cell.

[0012] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, the multi-level subdivision in step 2 includes:

[0013] The boundary unit is subdivided multiple times using a quadtree / octree, and the boundary unit is subdivided into multiple subdomains. Multiple sampling points are randomly scattered in the subdomains. If all the sampling points in the subdomain include internal points and external points, the subdomain is a boundary subdomain.

[0014] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, the step of boundary reconstruction in step 2 is:

[0015] Selecting a sideline on the boundary subdomain, the corner point at one end of the sideline is an internal point, and the corner point at the other end of the sideline is an external point;

[0016] The signed distance function value from the corner point as an internal point to the boundary of the structural model is recorded as a positive distance, and the signed distance function value from the corner point as an external point to the boundary of the structural model is recorded as a negative distance. According to the signed distance function values ​​from the two corner points to the boundary of the structural model, the signed distance function from each point on the edge line to the boundary of the structural model is linearly interpolated, and a point on the edge line whose signed distance function value to the boundary is 0 is selected as a new boundary point, and two adjacent new boundary points are connected by a straight line to obtain the reconstructed boundary;

[0017] The coordinates of the new boundary point are (x 0 ,y 0 ), where x 0 ,y 0 The calculation formula is:

[0018]

[0019] Among them, (x 0 ,y 0 ) represents the coordinates of the new boundary point, (x 1 ,y 1 ) and (x 2 ,y 2 ) represent the coordinates of corner point 1 and corner point 2 on the edge line, d 1 and d 2 They are the signed distance function values ​​from corner point 1 and corner point 2 to the boundary respectively.

[0020] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, solving the equilibrium equation in step 3 includes:

[0021] 3.1 completing the integration at the unit level based on the structural model reconstructed at the boundary;

[0022] 3.2 Using penalty function method to process the boundary conditions of the structural model;

[0023] 3.3 Solve the equilibrium equation based on the integration results at the unit level and the boundary condition processing results.

[0024] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, the specific steps of step 3.1 are:

[0025] Acquire basic information of the internal unit, the external unit and the boundary unit, wherein the basic information includes node coordinates, material properties and unit type;

[0026] According to the basic information, using Gaussian integral to calculate the integral of the internal unit at the unit level, to obtain the integral of the internal unit;

[0027] multiplying the integral of the internal unit by a predetermined coefficient to obtain the integral of the external unit;

[0028] In the boundary cell, the reconstructed boundary divides the boundary cell into an internal area and an external area, and Gaussian points are scattered in the internal area. Gaussian integrals are used to calculate the integral on the Gaussian points to obtain the integral of the internal area, thereby obtaining the integral of the boundary cell at the cell level.

[0029] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, step 3.2 comprises the following steps:

[0030] 3.2.1 Obtain boundary conditions and determine an application unit for applying the boundary conditions;

[0031] 3.2.2 Using the penalty function method, the boundary condition is integrated at the unit level within the applied unit to obtain the additional term generated by the boundary condition.

[0032] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, in step 3.2.2, if the boundary condition is applied to a point, the integral of the point at the unit level is calculated in the unit where the point is located to obtain the additional term generated by the boundary condition; if the boundary condition is applied to a line, the integral at the unit level is performed along the line in the unit where the line is located to obtain the additional term generated by the boundary condition; if the boundary condition is applied to a surface, the integral at the unit level is performed on the surface in the unit where the surface is located to obtain the additional term generated by the boundary condition;

[0033] Wherein, when the boundary condition is zero, the integral of the boundary condition at the unit level includes the additional matrix term; when the boundary condition is non-zero, the integral of the boundary condition at the unit level includes the additional matrix term and the load vector term.

[0034] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, step 3.3 solves the equilibrium equation, which is specifically:

[0035] Numbering all nodes of the grid unit, and obtaining degree of freedom numbers of the nodes of the grid unit according to the node numbers of the grid unit;

[0036] According to the degree of freedom number, the integration results and boundary condition processing results at the unit level are assembled to obtain the overall matrix and vector of the structural model;

[0037] The overall matrix and the vector are used to solve the equilibrium equation to obtain the field variables of the structural model.

[0038] In certain embodiments, in the efficient numerical analysis method based on a regular space grid, the calculation results of the equilibrium equation in step 3 are post-processed to obtain the analysis results of the structural model, which are specifically:

[0039] The field variable value of the point to be calculated in the structural model is calculated according to the field variable result, and the specific process is as follows:

[0040] For any of the points to be determined, the grid unit where the point to be determined is located is determined according to the coordinates of the point to be determined, and the field variable value of the grid unit node is obtained from the calculation result of the equilibrium equation; the field variable value of the point to be determined is determined from the field variable value of the grid node using the unit shape function;

[0041] The gradient formula is used to calculate the field variable gradient of the point to be calculated in the structural model according to the field variable result. The specific process is as follows:

[0042] When the point to be determined is located inside the structural model, the grid unit where the point to be determined is located is determined according to the coordinates of the point to be determined, and the field variable value of the grid unit node is obtained from the calculation result of the equilibrium equation; the field variable gradient value of the point to be determined is determined from the field variable value of the grid unit node using the gradient formula;

[0043] When the point to be determined is located at the boundary of the structural model, if the structural model is a two-dimensional model, an enclosing circle is constructed with the point to be determined as the center; the circumference of the enclosing circle is discretized into line segments, and the line segments whose two end points are both located inside the structural model are selected as internal line segments; the coordinate information of the point to be determined and the end points of the internal line segments are obtained, and the grid units corresponding to the point to be determined and the end points on the structural model are determined according to the coordinate information; the field variable values ​​of each node of the above-mentioned grid unit are obtained according to the calculation results of the equilibrium equation; interpolation is performed according to the field variable values ​​of each node to obtain the field variable values ​​of the point to be determined and the end points; the point to be determined and the two end points of the internal line segment are connected to form a triangular unit located inside the structural model, and the field variable gradient value of the triangular unit is obtained by using the gradient formula according to the field variable values ​​of the three points in the triangular unit; the field variable gradient values ​​of all the triangular units are averaged to obtain the field variable gradient value of the point to be determined;

[0044] When the point to be determined is located at the boundary of the structural model, if the structural model is a three-dimensional model, a bounding sphere is constructed with the point to be determined as the center of the sphere; the surface of the bounding sphere is discretized into triangular patches, and the three vertices of the triangular patches are selected to be located inside the structural model; the coordinate information of the point to be determined and the vertices of the internal triangular patches is obtained; the mesh units of the point to be determined and the vertices of the internal triangular patches on the structural model are determined according to the coordinate information; the field variable values ​​of each node of the above-mentioned mesh units are obtained according to the calculation results of the equilibrium equation; interpolation is performed according to the field variable values ​​of the nodes to obtain the field variable values ​​of the point to be determined and the vertices; the internal triangular patches and the point to be determined are connected to form a tetrahedral unit located inside the structural model; according to the field variable values ​​of each vertex of the tetrahedral unit, the field variable gradient value of the tetrahedral unit is obtained by using a gradient formula; the field variable gradient values ​​of all the tetrahedral units are averaged to obtain the field variable gradient value of the point to be determined.

[0045] The beneficial effects of the present invention are:

[0046] The present invention discloses an efficient numerical analysis method based on regular space grids, which solves the problems of difficulty in accurate integration on boundary cells, complex boundary condition processing and result post-processing in existing regular space grid methods. The boundary cells are subdivided into multiple levels and the boundaries of the structural model are reconstructed to complete the integration of the boundary cells at the cell level. The boundary conditions of the structural model are processed by a penalty function method. Post-processing is performed by constructing triangular or tetrahedral cells at the boundary positions of the structural model. The method can better balance the calculation efficiency and calculation accuracy. Compared with the existing analysis methods, more accurate results can be obtained in a faster way.

[0047] The present invention discloses an efficient numerical analysis method based on regular space grid, with further optimized space, accurate integration at unit level, processing of boundary conditions and post-processing of results are very important steps in numerical analysis methods based on regular space grid. Among them, when there are few Gaussian integral points, the unit level integration is directly performed using the rectangular / cuboid subdomain after the boundary unit is subdivided, and the error is large, especially for some boundary units with only a few areas located inside the structural model, which cannot meet the requirements of calculation accuracy. Increasing the number of Gaussian integral points will greatly increase the amount of calculation, which is unacceptable in engineering practice. Based on the boundary subdomain after subdivision, the boundary of the structural model is reconstructed by replacing the curve with a straight line, and different integration strategies are adopted for units (internal units, external units, boundary units) at different positions of the structural model, when the calculation accuracy is the same, the calculation efficiency is improved; when the number of Gaussian integral points is the same, the calculation accuracy is improved. In the penalty function method for processing boundary conditions, the present application only adds additional items to the calculation results of standard units (regular grid units), and the overall calculation scheme is very similar to the standard finite element calculation, so it is relatively easy to implement programming. In the post-processing of the results, the present application obtains better and more accurate post-processing results at a lower cost by constructing triangle / tetrahedron units at the boundary positions of the structural model. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 A schematic diagram of a flow chart of an efficient numerical analysis method based on a regular space grid according to the present invention;

[0049] Figure 2 A schematic diagram of generating a regular space grid in a two-dimensional model by using an efficient numerical analysis method based on a regular space grid according to the present invention;

[0050] Figure 3 It is a schematic diagram of a boundary unit subdivision method in a two-dimensional model in an efficient numerical analysis method based on a regular space grid according to the present invention;

[0051] Figure 4 A schematic diagram of boundary reconstruction in a two-dimensional model of an efficient numerical analysis method based on a regular space grid according to the present invention;

[0052] Figure 5 A schematic diagram of a case of a high-efficiency numerical analysis method based on a regular space grid in boundary element integration according to the present invention;

[0053] Figure 6 It is a result diagram of boundary unit integration in an efficient numerical analysis method based on regular space grids described in the present invention and an existing regular space grid method;

[0054] Figure 7A schematic diagram of a case of a high-efficiency numerical analysis method based on a regular space grid in the result post-processing of the present invention;

[0055] Figure 8 To use the existing regular space grid method to Figure 7 The stress contour result obtained by post-processing a quarter of a two-dimensional ring is shown;

[0056] Fig. 9 The invention provides an efficient numerical analysis method based on regular space grids. Figure 7 The stress contour diagram results obtained by post-processing a quarter of a two-dimensional ring are shown. DETAILED DESCRIPTION

[0057] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention.

[0058] like Figures 1 to 4 As shown, an efficient numerical analysis method based on a regular space grid comprises the following steps:

[0059] Step 1: Mesh the structural model into a plurality of mesh units, wherein the mesh unit where the boundary of the structural model is located is called a boundary unit;

[0060] 1.1 Determine the physical boundary information of the structural model to be analyzed based on the size information of the structural model, and generate a bounding box that can completely surround the structural model based on the physical boundary information of the structural model. If the structural model is a two-dimensional problem, the bounding box is a rectangle; if the structural model is a three-dimensional problem, the bounding box is a cuboid.

[0061] 1.2 According to the geometric features of the structural model and the size information of the bounding box, a suitable grid size is selected, and the bounding box is gridded to divide the bounding box into a plurality of grid units, wherein the grid units are regular rectangular / cuboid units.

[0062] 1.3 According to the positional relationship between the structural model and the grid unit, the grid unit is divided into an internal unit, a boundary unit and an external unit; the specific steps of determining the positional relationship between the structural model and the grid unit are:

[0063] Some sampling points are randomly scattered in the grid unit, and a ray in any direction is drawn with the sampling point as the starting point using the ray method. If the number of intersections between the ray and the boundary is an odd number, the sampling point is located inside the structural model and is an internal point. If the number of intersections is an even number, the sampling point is located outside the structural model and is an external point. There are multiple sampling points, and the multiple sampling points are evenly scattered in the grid unit.

[0064] If all the sampling points of the grid cell are internal points, the grid cell is an internal cell; if all the sampling points of the grid cell are external points, the grid cell is an external cell; if all the sampling points of the grid cell have both internal points and external points, the grid cell is a boundary cell.

[0065] like Figure 2 As shown in the figure, the gray area surrounded by dotted lines is the two-dimensional structural model to be analyzed, and the outermost square wireframe is the bounding box of the structural model. Efficient grid division is performed on the bounding box to generate a regular square space grid. Figure 2 The boundary information of the two-dimensional model shown in the figure, that is, the endpoint coordinates of the boundary curve segment and the connection relationship between the endpoints, and obtain the node coordinate information of the spatial grid unit; randomly scatter some sampling points in the spatial grid, and use the ray method to judge the internal and external position relationship of all sampling points of the spatial grid relative to the structural model (with Figure 2 Taking points A and B in the figure as an example): Starting from point A of the spatial grid, a ray is sent horizontally to the right. The ray passes through the boundary of the two-dimensional model once, which is an odd number of times, and the sampling point A is judged to be inside the two-dimensional model; starting from point B, a ray is sent horizontally to the right. The ray passes through the boundary of the two-dimensional model twice, which is an even number of times, and the sampling point B is judged to be outside the two-dimensional model. According to the internal and external position relationship of the sampling point of the grid unit relative to the structural model, the internal and external position relationship of all grid units relative to the structural model is judged (based on Figure 2 For example, take cells 1, 2, and 3 in the figure as examples): all sampling points of grid cell 1 are internal points, so cell 1 is determined to be an internal cell; some sampling points of grid cell 2 are inside the structural model and some are outside the structural model, so grid cell 2 is determined to be a boundary cell; all sampling points of grid cell 3 are outside the two-dimensional model, so grid cell 3 is an external cell.

[0066] Step 2: subdivide the boundary unit into multiple subdomains at multiple levels, and the subdomain where the boundary of the structural model is located is called the boundary subdomain; reconstruct the boundary in the boundary subdomain to obtain a reconstructed boundary.

[0067] 2.1 The boundary unit is subdivided multiple times using a quadtree / octree, and the boundary unit is subdivided into multiple subdomains. Multiple sampling points are randomly scattered in the subdomains. If all the sampling points in the subdomain include internal points and external points, the subdomain is a boundary subdomain.

[0068] The number of subdivisions n can be an integer of 1, 2, 3, 4... When the result of calculating the equilibrium equation when the number of subdivisions is n is basically the same as the result of calculating the equilibrium equation when the number of subdivisions is n+1, the value of the number of subdivisions is determined to be n.

[0069] Figure 3 This is a schematic diagram of a boundary unit subdivision method in a two-dimensional model in an efficient numerical analysis method based on a regular space grid of the present application, wherein, taking a boundary unit in the two-dimensional model as an example, a quadtree / octree is used to perform multi-level subdivision of the boundary unit. Figure 3 The k in represents the order of quadtree partitioning. In this schematic diagram, the maximum value of k is 3. The specific subdivision steps are as follows: in the first-order quadtree partitioning (i.e., k=1), the boundary unit is subdivided into 4 subdomains, and the corner point coordinate information of the 4 subdomains is obtained according to the node coordinate information of the boundary unit. Some sampling points are randomly scattered in the 4 subdomains, and the inside and outside properties of all sampling points are judged. The inside and outside properties of the 4 subdomains are judged according to the inside and outside properties of the sampling points. It can be seen that 2 of the 4 subdomains are cut by the boundary of the two-dimensional model; in the second-order quadtree partitioning (i.e., k=2), only the 2 subdomains cut by the boundary of the two-dimensional model obtained by the first-order quadtree partitioning are subdivided, and each subdomain is subdivided into 4 small subdomains. The corner point information of the small subdomain after the second-order subdivision is obtained according to the corner point information of the subdomain when k=1, and some sampling points are randomly scattered in the small subdomain, and the sampling points are randomly scattered. The inside and outside of the sampling points are judged, and the inside and outside of the small subdomains obtained by the second-order subdivision are judged according to the inside and outside of the sampling points. It can be seen that after the second-order subdivision, the boundary unit has a total of 10 subdomains, of which 5 subdomains are cut by the boundary of the two-dimensional model; in the third-order quadtree partition, only the 5 subdomains cut by the boundary of the two-dimensional model are subdivided, and the corner point information of the smaller subdomains after the third-order subdivision is obtained from the corner point information of the 5 subdomains. Similarly, some sampling points are randomly scattered in the smaller subdomains, and the inside and outside of the sampling points are judged. The inside and outside of the smaller subdomains obtained by the third-order subdivision are judged according to the inside and outside of the sampling points. It can be seen that after the third-order subdivision, the boundary unit has a total of 25 subdomains, of which 8 subdomains are cut by the boundary of the two-dimensional model. At this time, k has reached the maximum value of 3, and no further subdivision is performed, completing the multi-level subdivision of the boundary unit. The result of solving the equilibrium equation when k = 3 is basically the same as the result when k = 4, so k = 3 is taken, because further subdivision will only increase the workload and will not have a great impact on the calculation results.

[0070] 2.2 Select a sideline on the boundary subdomain, the corner point at one end of the sideline is an internal point, and the corner point at the other end of the sideline is an external point.

[0071] The signed distance function value from the corner point as an internal point to the boundary of the structural model is recorded as a positive distance, and the signed distance function value from the corner point as an external point to the boundary of the structural model is recorded as a negative distance. According to the signed distance function values ​​from these two corner points to the boundary of the structural model, the signed distance function from each point on the edge line to the boundary of the structural model is linearly interpolated to find the point on the edge line whose signed distance function value from the boundary of the structural model is 0 as the new boundary point, and the coordinates of the new boundary point are recorded; two adjacent new boundary points are connected by a straight line to obtain the reconstructed boundary.

[0072] The coordinates of the new boundary point are (x 0 ,y 0 ), where x 0 ,y 0 The calculation formula is:

[0073]

[0074] Among them, (x 0 ,y 0 ) represents the coordinates of the new boundary point, (x 1 ,y 1 ) and (x 2 ,y 2 ) represent the coordinates of the edge corner point 1 and corner point 2, respectively, d 1 and d 2 They are the signed distance function values ​​from corner point 1 and corner point 2 to the boundary respectively.

[0075] like Figure 4 As shown in the figure, taking the boundary reconstruction in a boundary unit as an example, the internal area of ​​the boundary unit located inside the two-dimensional model is represented by gray. The specific reconstruction steps are: obtaining the basic information of the boundary of the two-dimensional model, that is, the endpoint coordinates of the curve segment model in the boundary subdomain and the topological relationship between the endpoints; based on Figure 3The subdivided boundary unit shown in the figure obtains the corner point coordinate information and topological relationship of the boundary subdomain cut by the two-dimensional model boundary in the boundary unit; according to the basic information of the boundary of the two-dimensional model and the corner point information of the subdomain, the internal and external position relationship of the corner point of the subdomain relative to the two-dimensional model is identified; the edge line on the boundary subdomain is selected, and the two end points of the edge line (also the corner points of the boundary subdomain) are one internal point and the other external point; the signed distance function values ​​from the two corner points to the two-dimensional boundary are calculated respectively, and the signed distance function values ​​from the corner points of the internal points to the boundary of the two-dimensional model are recorded as positive distances, and the signed distance function values ​​from the corner points of the external points to the boundary of the two-dimensional model are recorded as negative distances. According to the signed distance function values ​​from the two corner points to the boundary of the two-dimensional model, the signed distance function from each point on the edge line where the two corner points are located to the boundary of the two-dimensional model is linearly interpolated, and the point on the edge line of the subdomain with a signed distance function value of 0 to the boundary of the two-dimensional model is found as the new boundary point, with Figure 4 Take subdomain 1 in as an example: boundary subdomain 1 contains 4 corner points, of which a and d are internal points of the two-dimensional model, b is exactly on the boundary of the two-dimensional model, and c is an external point of the two-dimensional model. Then the two corner points (endpoints) of the cd edge line in subdomain 1 have different internal and external properties. Calculate the signed distance function values ​​from point c and point d to the boundary of the two-dimensional model, where c takes a negative distance and d takes a positive distance. According to the signed distance function values ​​from point c and point d to the boundary of the two-dimensional model, perform linear interpolation on the signed distance function from the points on the cd edge to the boundary of the two-dimensional model, and find the point whose signed distance function value to the boundary of the two-dimensional model is 0, that is, point e as the new boundary point. Repeat the above steps to find all points on the boundary subdomain edge line in the boundary unit whose signed distance function value to the boundary of the two-dimensional model is 0. These points are used as new boundary points of the two-dimensional model. Connect the new boundary points with straight lines to complete. Figure 4 Boundary reconstruction within the mid-boundary cell.

[0076] Step three: generating the equilibrium equation of the structural model based on the reconstructed boundary, and solving it to obtain the field variables of the structural model; post-processing the field variables to obtain the analysis results of the structural model.

[0077] 3.1 completing the integration at the unit level based on the structural model reconstructed at the boundary;

[0078] 3.1.1 Obtain basic information of the internal unit, the external unit and the boundary unit, wherein the basic information includes node coordinates, material properties and unit type;

[0079] 3.1.2 Based on the basic information, the integral of the internal unit at the unit level is calculated using Gaussian integral to obtain the integral of the internal unit, that is, the integral of all internal units at the unit level;

[0080] The integral of the internal unit is multiplied by a predetermined coefficient to obtain the integral of the external unit, that is, the integral of all external units at the unit level; the predetermined coefficient is 10 -6 .

[0081] In the boundary cell, the reconstructed boundary divides the boundary cell into an internal area and an external area, and Gaussian points are scattered in the internal area. Gaussian integral is used to calculate the integral on the Gaussian points to obtain the integral of the internal area, thereby obtaining the integral of the boundary cell at the cell level; in the external area, Gaussian integral is not performed.

[0082] 3.2 Using penalty function method to process the boundary conditions of the structural model;

[0083] 3.2.1 Obtaining boundary conditions and determining an application unit for applying the boundary conditions; the boundary conditions include a location and a value for applying the boundary conditions;

[0084] 3.2.2 Using the penalty function method, the boundary condition is integrated at the unit level within the applied unit to obtain the additional term generated by the boundary condition.

[0085] In step 3.2.2, if the boundary condition is applied to a point, the integral of the point at the unit level is calculated in the unit where the point is located, and the additional term generated by the boundary condition is obtained; if the boundary condition is applied to a line, the integral is performed at the unit level along the line in the unit where the line is located, and the additional term generated by the boundary condition is obtained; if the boundary condition is applied to a surface, the integral is performed at the unit level on the surface in the unit where the surface is located, and the additional term generated by the boundary condition is obtained. Wherein, when the boundary condition is zero, the integral of the boundary condition at the unit level includes the additional matrix term; when the boundary condition is non-zero, the integral of the boundary condition at the unit level includes the additional matrix term and the load vector term.

[0086] 3.3 Solve the equilibrium equation based on the integration results at the unit level and the boundary condition processing results.

[0087] 3.3.1 numbering all nodes of the grid unit, and obtaining the degree of freedom number of the nodes of the grid unit according to the node number of the grid unit;

[0088] 3.3.2 According to the degree of freedom number, assemble the integral results and boundary condition processing results at the unit level to obtain the overall matrix and vector of the structural model;

[0089] 3.3.3 Using the overall matrix and the vector, the equilibrium equation is solved to obtain the field variables of the structural model.

[0090] 3.4 The calculation results of the equilibrium equation are post-processed to obtain the analysis results of the structural model, which are specifically:

[0091] 3.4.1 Calculate the field variable value of the unknown point in the structural model according to the field variable results, and the specific process is: for any unknown point, determine the grid unit where the unknown point is located according to the coordinates of the unknown point, and obtain the field variable value of the grid unit node from the calculation result of the equilibrium equation; use the unit shape function to determine the field variable value of the unknown point from the field variable value of the grid node.

[0092] 3.4.2 Using the gradient formula, calculate the field variable gradient of the point to be determined in the structural model based on the field variable results.

[0093] When the point to be determined is located inside the structural model, the grid unit where the point to be determined is determined according to the coordinates of the point to be determined, and the field variable values ​​of the grid unit nodes are obtained from the calculation results of the equilibrium equation; and the field variable gradient value of the point to be determined is determined from the field variable values ​​of the grid unit nodes using the gradient formula.

[0094] When the point to be determined is located at the boundary of the structural model, if the structural model is a two-dimensional model, an enclosing circle is constructed with the point to be determined as the center; the circumference of the enclosing circle is discretized into multiple line segments, and the line segment whose two endpoints are both located inside the structural model is selected as the internal line segment; the coordinate information of the point to be determined and the endpoints of the internal line segment is obtained, and the grid units corresponding to the point to be determined and the endpoints of the internal line segment on the structural model are determined according to the coordinate information; the field variable values ​​of each node of the above-mentioned grid unit are obtained according to the calculation results of the equilibrium equation; interpolation is performed according to the field variable values ​​of each node to obtain the field variable values ​​of the point to be determined and the endpoints of the internal line segment; the point to be determined and the two endpoints of the internal line segment are connected to form a triangular unit located inside the structural model, and the field variable gradient value of the triangular unit is obtained by using the gradient formula according to the field variable values ​​of the three points in the triangular unit; the field variable gradient values ​​of all the triangular units are averaged as the field variable gradient value of the point to be determined.

[0095] When the point to be determined is located at the boundary of the structural model, if the structural model is a three-dimensional model, a bounding sphere is constructed with the point to be determined as the center of the sphere; the surface of the bounding sphere is discretized into triangular patches, and the three vertices of the triangular patches are selected to be located inside the structural model; the coordinate information of the point to be determined and the vertices of the internal triangular patches is obtained; the mesh units of the point to be determined and the vertices of the internal triangular patches on the structural model are determined according to the coordinate information; the field variable values ​​of each node of the above-mentioned mesh units are obtained according to the calculation results of the equilibrium equation; interpolation is performed according to the field variable values ​​of the nodes to obtain the field variable values ​​of the point to be determined and the vertices; the internal triangular patches and the point to be determined are connected to form a tetrahedral unit located inside the structural model; according to the field variable values ​​of each vertex of the tetrahedral unit, the field variable gradient value of the tetrahedral unit is obtained by using a gradient formula; the field variable gradient values ​​of all the tetrahedral units are averaged to obtain the field variable gradient value of the point to be determined.

[0096] Example 1: Using the efficient numerical analysis method based on regular space grids described in the present invention, mechanical problems in engineering practice are studied. In the numerical analysis process, relevant formulas of mechanics are applied in some steps, as follows:

[0097] Step 1: Mesh the structural model into a plurality of mesh units, wherein the mesh unit where the boundary of the structural model is located is called a boundary unit;

[0098] 1.1 Determine the physical boundary information of the structural model to be analyzed based on the size information of the structural model, and generate a bounding box that can completely surround the structural model based on the physical boundary information of the structural model. If the structural model is a two-dimensional problem, the bounding box is a rectangle; if the structural model is a three-dimensional problem, the bounding box is a cuboid.

[0099] 1.2 According to the geometric features of the structural model and the size information of the bounding box, a suitable grid size is selected, and the bounding box is gridded to divide the bounding box into a plurality of grid units, wherein the grid units are regular rectangular / cuboid units.

[0100] 1.3 According to the positional relationship between the structural model and the grid unit, the grid unit is divided into an internal unit, a boundary unit and an external unit; the specific steps of determining the positional relationship between the structural model and the grid unit are:

[0101] Randomly scatter some sampling points in the grid unit, and use the ray method to draw a ray in any direction with the sampling point as the starting point. If the number of intersections between the ray and the boundary is an odd number, the sampling point is located inside the structural model and is an internal point. If the number of intersections is an even number, the sampling point is located outside the structural model and is an external point.

[0102] If all the sampling points of the grid cell are internal points, the grid cell is an internal cell; if all the sampling points of the grid cell are external points, the grid cell is an external cell; if all the sampling points of the grid cell have both internal points and external points, the grid cell is a boundary cell.

[0103] Step 2: subdivide the boundary unit into multiple subdomains at multiple levels, and call the subdomain where the boundary of the structural model is located a boundary subdomain; reconstruct the boundary in the boundary subdomain to obtain a reconstructed boundary.

[0104] 2.1 The boundary unit is subdivided multiple times using a quadtree / octree, and the boundary unit is subdivided into multiple subdomains. Multiple sampling points are randomly scattered in the subdomains. If all the sampling points in the subdomain include internal points and external points, the subdomain is a boundary subdomain.

[0105] 2.2 Select a sideline on the boundary subdomain, the corner point at one end of the sideline is an internal point, and the corner point at the other end of the sideline is an external point.

[0106] The signed distance function value from the corner point as an internal point to the boundary of the structural model is recorded as a positive distance, and the signed distance function value from the corner point as an external point to the boundary of the structural model is recorded as a negative distance. According to the signed distance function values ​​from these two corner points to the boundary of the structural model, the signed distance function from each point on the edge line to the boundary of the structural model is linearly interpolated to find the point on the edge line whose signed distance function value from the boundary of the structural model is 0 as the new boundary point, and the coordinates of the new boundary point are recorded; two adjacent new boundary points are connected by a straight line to obtain the reconstructed boundary.

[0107] The coordinates of the new boundary point are (x 0 ,y 0 ), where x 0 ,y 0 The calculation formula is:

[0108]

[0109] Among them, (x 0 ,y 0 ) represents the coordinates of the new boundary point, (x 1 ,y 1 ) and (x 2 ,y 2 ) represent the coordinates of the edge corner point 1 and corner point 2, respectively, d 1 and d 2 They are the signed distance function values ​​from corner point 1 and corner point 2 to the boundary respectively.

[0110] Step three: generating the equilibrium equation of the structural model based on the reconstructed boundary, and solving it to obtain the field variables of the structural model; post-processing the field variables to obtain the analysis results of the structural model.

[0111] 3.1.1 Obtain basic information of the internal unit, the external unit and the boundary unit, wherein the basic information includes node coordinates, material properties and unit type;

[0112] 3.1.2 Calculating the integral of the grid unit at the unit level using Gaussian integral according to the basic information;

[0113] The integration at the element level in mechanical problems is the calculation of the element stiffness matrix:

[0114]

[0115] Among them, K e represents the unit stiffness matrix, Ω e represents the unit area, B is the strain-displacement matrix, D is the elastic matrix, and h is the characteristic function, which takes different values ​​according to the location of the integration point. When the integration point is located inside the structural model, h takes 1, and when the integration point is located outside the structural model, h takes a very small number, which is 10 -6 .

[0116] 3.2 The penalty function method is used to process the boundary conditions of the structural model. The penalty function method processes the boundary conditions of the structural model. Taking the mechanical problem as an example, the additional term generated by the penalty function method for the unit where the boundary condition is located is:

[0117]

[0118]

[0119] in, and They represent the additional stiffness matrix and additional load vector generated by the unit where the boundary condition is located. represents the area where the displacement boundary condition is applied, and β represents the penalty coefficient of the penalty function method, which is generally 10 6 , N is the shape function matrix, is the displacement vector given by the displacement boundary condition.

[0120] 3.3 Solve the equilibrium equation based on the integration results at the unit level and the boundary condition processing results.

[0121] 3.3.1 numbering all nodes of the grid unit, and obtaining the degree of freedom number of the nodes of the grid unit according to the node number of the grid unit;

[0122] 3.3.2 According to the degree of freedom number, assemble the integral results and boundary condition processing results at the unit level to obtain the overall matrix and vector of the structural model;

[0123] 3.3.3 Using the overall matrix and the vector, solve the equilibrium equation to obtain the field variables of the structural model. Taking the mechanical problem as an example, the equilibrium equation of the system is:

[0124] KU=F

[0125] Among them, K represents the overall stiffness matrix of the system, U represents the displacement vector of the system to be calculated, and F represents the load vector of the system. The specific calculation formula is:

[0126]

[0127]

[0128] Among them, n ele represents the number of units in the system. The first term in F is the contribution of the surface force t. It represents the boundary of the surface force. The second term in F is the contribution of the body force b. The indicative function h takes different values ​​according to the different positions of the integration point. The specific value is the same as that of the integration at the unit level. When the integration point is located inside the structural model, h takes 1. When the integration point is located outside the structural model, h takes 10. -6 The summation symbol in the above formula represents the assembly process.

[0129] 3.4 The calculation results of the equilibrium equation are post-processed to obtain the analysis results of the structural model, which are specifically:

[0130] 3.4.1 Calculate the field variable value of the point to be determined in the structural model based on the field variable results.

[0131] For any of the points to be determined, the grid unit where the point to be determined is located is determined according to the coordinates of the point to be determined, and the field variable values ​​of the grid unit nodes are obtained from the calculation results of the equilibrium equation; the field variable values ​​of the point to be determined are determined from the field variable values ​​of the grid nodes using the unit shape function.

[0132] 3.4.2 Using the gradient formula, calculate the field variable gradient of the point to be determined in the structural model based on the field variable results.

[0133] When the point to be determined is located inside the structural model, the grid unit where the point to be determined is determined according to the coordinates of the point to be determined, and the field variable values ​​of the grid unit nodes are obtained from the calculation results of the equilibrium equation; and the field variable gradient value of the point to be determined is determined from the field variable values ​​of the grid unit nodes using the gradient formula.

[0134] When the point to be determined is located at the boundary of the structural model, if the structural model is a two-dimensional model, an enclosing circle is constructed with the point to be determined as the center; the circumference of the enclosing circle is discretized into multiple line segments, and the line segment whose two endpoints are both located inside the structural model is selected as the internal line segment; the coordinate information of the point to be determined and the endpoints of the internal line segment is obtained, and the grid units corresponding to the point to be determined and the endpoints of the internal line segment on the structural model are determined according to the coordinate information; the field variable values ​​of each node of the above-mentioned grid unit are obtained according to the calculation results of the equilibrium equation; interpolation is performed according to the field variable values ​​of each node to obtain the field variable values ​​of the point to be determined and the endpoints of the internal line segment; the point to be determined and the two endpoints of the internal line segment are connected to form a triangular unit located inside the structural model, and the field variable gradient value of the triangular unit is obtained by using the gradient formula according to the field variable values ​​of the three points in the triangular unit; the field variable gradient values ​​of all the triangular units are averaged as the field variable gradient value of the point to be determined.

[0135] When the point to be determined is located at the boundary of the structural model, if the structural model is a three-dimensional model, a bounding sphere is constructed with the point to be determined as the center of the sphere; the surface of the bounding sphere is discretized into triangular patches, and the three vertices of the triangular patches are selected to be located inside the structural model; the coordinate information of the point to be determined and the vertices of the internal triangular patches is obtained; the mesh units of the point to be determined and the vertices of the internal triangular patches on the structural model are determined according to the coordinate information; the field variable values ​​of each node of the above-mentioned mesh units are obtained according to the calculation results of the equilibrium equation; interpolation is performed according to the field variable values ​​of the nodes to obtain the field variable values ​​of the point to be determined and the vertices; the internal triangular patches and the point to be determined are connected to form a tetrahedral unit located inside the structural model; according to the field variable values ​​of each vertex of the tetrahedral unit, the field variable gradient value of the tetrahedral unit is obtained by using a gradient formula; the field variable gradient values ​​of all the tetrahedral units are averaged to obtain the field variable gradient value of the point to be determined.

[0136] In addition, when using the method described in the present invention to study thermal or other problems, formulas in related fields are used, such as thermal equilibrium equations.

[0137] In order to better illustrate the integration of the boundary unit by the integration method proposed in this application, when the calculation accuracy is the same, fewer Gaussian points are required, and the efficiency is higher; when the number of Gaussian points is the same, the absolute value of the relative error of the integration method proposed in this application is smaller, and the calculation result is more accurate. Figure 5 The numerical analysis is performed using a square boundary element with a side length of 1. Figure 5In the figure, gray represents the internal area of ​​the structural model. The gray area is a part of a circle with (-0.4, 0) as the center and 1 as the radius. 2 +y 2 As the objective function, the number of Gaussian points required by the two integration methods for different orders of quadtree partitioning, the corresponding Gaussian integration results and the absolute value of the error of the Gaussian integration results relative to the accurate integration results are obtained. Among them, the accurate integration result in the gray area of ​​the boundary unit is 0.1076. Table 1 shows the results of the existing regular space grid integration method, and Table 2 shows the results of the integration method proposed in this application. According to the data in Tables 1 and 2, we get Figure 6 The curves of the absolute value of the relative error of the two methods changing with the number of Gaussian points, where the dotted line represents the result of the existing integration method, and the solid line represents the result of the integration method proposed in this application. Comparing the two curves, it can be seen that when the absolute value of the relative error is the same, the integration method proposed in this application requires fewer Gaussian points, the amount of calculation of the boundary unit integration is smaller, and the efficiency is higher; when the number of Gaussian points is the same, the absolute value of the relative error of the integration method proposed in this application is smaller, and the accuracy of the calculation result is higher.

[0138] Table 1 Integration methods in the prior art Figure 5 Integration results for the case shown

[0139] Quadtree Order 1 2 3 4 5 Gaussian Points 27 90 180 378 792 Gaussian integral result 0.1250 0.1094 0.1068 0.1080 0.1078 Absolute value of relative error 0.1618 0.0170 0.0079 0.0041 0.0018

[0140] Table 2 The integration method proposed in this application Figure 5 Integration results for the case shown

[0141] Quadtree Order 1 2 3 4 5 Gaussian Points 13 61 125 271 578 Gaussian integral result 0.0896 0.1031 0.1065 0.1073 0.1075 Absolute value of relative error 0.1675 0.0420 0.0102 0.0025 0.0006

[0142] Embodiment 2: Figure 7 This is a schematic diagram of an example of an efficient numerical analysis method based on a regular space grid in the result post-processing of this application. Figure 7 It is a quarter-two-dimensional ring under internal pressure, with an inner diameter of 15 mm, an outer diameter of 20 mm, and an internal pressure of 1 MPa. The true stress at any position of the ring can be calculated from the analytical expression of the ring, where the Mises stress at the inner boundary of the ring takes the maximum value, which is 4.1625 MPa, and the Mises stress at the outer boundary of the ring takes the minimum value, which is 2.5714 MPa.

[0143] Figure 8 and Fig. 9 The existing regular space grid method and the method in this application are respectively used to Figure 7 The stress contour diagram results obtained by post-processing a quarter of a two-dimensional ring are shown in the figure, where: Figure 8 The stress range in is [2.5086,4.8496]MPa, Fig. 9 The stress range in is [2.5748,4.1445]MPa. Figure 8 It can be seen that some abnormal stress points appear in the post-processing results of the existing method, resulting in abnormal stress concentration. These abnormal stress points appear because only a small area of ​​the boundary units where these points are located is located inside the ring. The integral on these boundary units must be very accurate to ensure the correctness of the stress results. However, for the existing method, accurate integration within these boundary units requires a lot of Gaussian points. The computational cost brought by Gaussian points is huge and unacceptable in engineering practice. When we adopt the method proposed in this application, Fig. 9 The stress cloud diagram shown in is very smooth, and the maximum error with the theoretical solution does not exceed 0.5%. The method of the present application improves the accuracy of the integral of boundary cells with only a small area inside the structural model when the number of Gaussian points is the same. At the same time, the method of constructing triangle / tetrahedron cells in post-processing improves the accuracy of the post-processing results at a lower cost. Compared with the existing regular space grid method, the method proposed in the present application can more accurately describe the real stress situation of the structure and is more suitable for the numerical analysis of engineering structures.

[0144] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Other modifications or equivalent substitutions made to the technical solution of the present invention by ordinary technicians in the field should be included in the scope of the claims of the present invention as long as they do not depart from the spirit and scope of the technical solution of the present invention.

Claims

1. An efficient numerical analysis method based on regular space grids, characterized in that: The following steps are involved: Step 1: Mesh the structural model into a plurality of mesh units, wherein the mesh unit where the boundary of the structural model is located is called a boundary unit; Step 2: subdividing the boundary unit into multiple subdomains at multiple levels, wherein the subdomain where the boundary of the structural model is located is called the boundary subdomain; reconstructing the boundary within the boundary subdomain to obtain a reconstructed boundary; Step 3: generating the equilibrium equation of the structural model based on the reconstructed boundary, and solving it to obtain the field variables of the structural model; Post-processing the field variables to obtain analysis results of the structural model; In step 3, the calculation results of the equilibrium equation are post-processed to obtain the analysis results of the structural model, which are specifically: Calculating the field variable value of the point to be determined in the structural model according to the field variable result, the specific process is: for any of the points to be determined, determining the grid unit where the point to be determined is located according to the coordinates of the point to be determined, and obtaining the field variable value of the grid unit node from the calculation result of the equilibrium equation; Using the unit shape function, the field variable value of the point to be determined is determined from the field variable value of the grid unit node; The solution of the equilibrium equation includes: 3.1 Based on the structural model reconstructed by the boundary, the integration at the unit level is completed. In the mechanical problem, the integration at the unit level is the calculation of the unit stiffness matrix: Among them, K e represents the unit stiffness matrix, Ω e represents the unit area, B is the strain-displacement matrix, D is the elastic matrix, and h is the characteristic function, which takes different values ​​according to the location of the integration point. When the integration point is located inside the structural model, h is 1, and when the integration point is located outside the structural model, h is 10 -6 ; 3.2 Using penalty function method to process the boundary conditions of the structural model; 3.3 Solve the equilibrium equation based on the integration results at the unit level and the boundary condition processing results; Step 3.3 solves the equilibrium equation, which is: Numbering all nodes of the grid unit, and obtaining degree of freedom numbers of the nodes of the grid unit according to the node numbers of the grid unit; According to the degree of freedom number, the integration results and boundary condition processing results at the unit level are assembled to obtain the overall matrix and vector of the structural model; Using the overall matrix and the vector, solving the equilibrium equation to obtain the field variables of the structural model; In mechanical problems, the equilibrium equation of the system is: KU=F Among them, K represents the overall stiffness matrix of the system, U represents the displacement vector of the system to be calculated, and F represents the load vector of the system. The specific calculation formula is: in, and They represent the additional stiffness matrix and additional load vector generated by the unit where the boundary condition is located, N is the shape function matrix, and n ele represents the number of units in the system. The first term in F is the contribution of the surface force t. represents the boundary where the surface force acts, the second term in F is the contribution from the body force b, and the summation symbol in the above formula represents the assembly process.

2. The efficient numerical analysis method based on regular space grid according to claim 1, characterized in that: The step 1 includes dividing the grid unit into internal units, boundary units and external units according to the positional relationship between the structural model and the grid unit; the specific steps of determining the positional relationship between the structural model and the grid unit are: Randomly scatter some sampling points in the grid unit, and use the ray method to draw a ray in any direction with the sampling point as the starting point. If the number of intersections between the ray and the boundary of the structural model is an odd number, the sampling point is located inside the structural model and is an internal point. If the number of intersections is an even number, the sampling point is located outside the structural model and is an external point. If all the sampling points of the grid cell are internal points, the grid cell is an internal cell; if all the sampling points of the grid cell are external points, the grid cell is an external cell; if all the sampling points of the grid cell have both internal points and external points, the grid cell is a boundary cell.

3. The efficient numerical analysis method based on regular space grid according to claim 1, characterized in that: The multi-level segmentation in step 2 includes: The boundary unit is subdivided multiple times using a quadtree / octree, and the boundary unit is subdivided into multiple subdomains. Multiple sampling points are randomly scattered in the subdomains. If all the sampling points in the subdomain include internal points and external points, the subdomain is a boundary subdomain.

4. The efficient numerical analysis method based on regular space grid according to claim 2, characterized in that: The steps of boundary reconstruction in step 2 are: Selecting a sideline on the boundary subdomain, the corner point at one end of the sideline is an internal point, and the corner point at the other end of the sideline is an external point; The signed distance function value from the corner point as an internal point to the boundary of the structural model is recorded as a positive distance, and the signed distance function value from the corner point as an external point to the boundary of the structural model is recorded as a negative distance. According to the signed distance function values ​​from the two corner points to the boundary of the structural model, the signed distance function from each point on the edge line to the boundary of the structural model is linearly interpolated, and a point on the edge line whose signed distance function value to the boundary is 0 is selected as a new boundary point, and two adjacent new boundary points are connected by a straight line to obtain the reconstructed boundary; The coordinates of the new boundary point are (x0, y0), where the calculation formula of x0, y0 is: Among them, (x1, y1) and (x2, y2) represent the coordinates of corner point 1 and corner point 2 on the edge line respectively, and d1 and d2 represent the signed distance function values ​​from corner point 1 and corner point 2 to the boundary respectively.

5. The efficient numerical analysis method based on regular space grid according to claim 2, characterized in that: The specific steps of step 3.1 are: Acquire basic information of the internal unit, the external unit and the boundary unit, wherein the basic information includes node coordinates, material properties and unit type; According to the basic information, using Gaussian integral to calculate the integral of the internal unit at the unit level, to obtain the integral of the internal unit; multiplying the integral of the internal unit by a predetermined coefficient to obtain the integral of the external unit; In the boundary cell, the reconstructed boundary divides the boundary cell into an internal area and an external area, and Gaussian points are scattered in the internal area. Gaussian integrals are used to calculate the integral on the Gaussian points to obtain the integral of the internal area, thereby obtaining the integral of the boundary cell at the cell level.

6. The efficient numerical analysis method based on regular space grid according to claim 1, characterized in that: Step 3.2 The following steps are involved: 3.2.1 Obtain boundary conditions and determine an application unit for applying the boundary conditions; 3.2.2 using a penalty function method, integrating the boundary condition at the unit level within the applied unit to obtain an additional term generated by the boundary condition; In mechanical problems, the additional term generated by the penalty function method for the unit where the boundary condition is located is: in, represents the area where the displacement boundary condition is applied, and β represents the penalty coefficient of the penalty function method, which is 10 6 , N is the shape function matrix, is the displacement vector given by the displacement boundary condition.

7. The efficient numerical analysis method based on regular space grid according to claim 6, characterized in that: In step 3.2.2, if the boundary condition is applied to a point, the integral of the point at the unit level is calculated in the unit where the point is located, and the additional term generated by the boundary condition is obtained; if the boundary condition is applied to a line, the integral is performed at the unit level along the line in the unit where the line is located, and the additional term generated by the boundary condition is obtained; if the boundary condition is applied to a surface, the integral is performed at the unit level on the surface in the unit where the surface is located, and the additional term generated by the boundary condition is obtained; Wherein, when the boundary condition is zero, the integral of the boundary condition at the unit level includes the additional matrix term; when the boundary condition is non-zero, the integral of the boundary condition at the unit level includes the additional matrix term and the load vector term.

8. The efficient numerical analysis method based on regular space grid according to claim 1, characterized in that: The gradient formula is used to calculate the field variable gradient of the point to be calculated in the structural model according to the field variable result. The specific process is as follows: When the point to be determined is located inside the structural model, the grid unit where the point to be determined is located is determined according to the coordinates of the point to be determined, and the field variable value of the grid unit node is obtained from the calculation result of the equilibrium equation; Using the gradient formula, the field variable gradient value of the point to be determined is determined from the field variable value of the grid unit node; When the point to be found is located at the boundary of the structural model, if the structural model is a two-dimensional model, construct an enclosing circle with the point to be found as the center; discretize the circumference of the enclosing circle into line segments, and select the line segment whose two endpoints are both located inside the structural model as an internal line segment; Obtaining coordinate information of the point to be determined and the endpoint of the internal line segment, and determining the grid units corresponding to the point to be determined and the endpoint on the structural model according to the coordinate information; obtaining the field variable value of each node of the above grid unit according to the calculation result of the equilibrium equation; Interpolation is performed according to the field variable values ​​of the nodes to obtain the field variable values ​​of the point to be determined and the endpoints; the point to be determined and the two endpoints of the internal line segment are connected to form a triangular unit located inside the structural model, and the field variable gradient value of the triangular unit is obtained by using a gradient formula according to the field variable values ​​of the three points in the triangular unit; the field variable gradient values ​​of all the triangular units are averaged to obtain the field variable gradient value of the point to be determined; When the point to be determined is located at the boundary of the structural model, if the structural model is a three-dimensional model, a bounding sphere is constructed with the point to be determined as the center of the sphere; the surface of the bounding sphere is discretized into triangular patches, and the three vertices of the triangular patches are selected to be located inside the structural model; the coordinate information of the point to be determined and the vertices of the internal triangular patches is obtained; the grid unit of the point to be determined and the vertices of the internal triangular patches on the structural model is determined according to the coordinate information; and the field variable values ​​of each node of the above-mentioned grid unit are obtained according to the calculation results of the equilibrium equation; Interpolation is performed according to the field variable values ​​of the nodes to obtain the field variable values ​​of the point to be determined and the vertex; the internal triangular facets and the point to be determined are connected to form a tetrahedral unit located inside the structural model; according to the field variable values ​​of each vertex of the tetrahedral unit, the field variable gradient value of the tetrahedral unit is obtained using a gradient formula; the field variable gradient values ​​of all the tetrahedral units are averaged to obtain the field variable gradient value of the point to be determined.

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