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CN118094937B - An efficient numerical analysis method based on regular space grid - Google Patents

An efficient numerical analysis method based on regular space grid Download PDF

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CN118094937B
CN118094937B CN202410287623.5A CN202410287623A CN118094937B CN 118094937 B CN118094937 B CN 118094937B CN 202410287623 A CN202410287623 A CN 202410287623A CN 118094937 B CN118094937 B CN 118094937B
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CN118094937A (en
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任林娟
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Beijing Youjie Future Technology Co ltd
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Abstract

本发明公开一种基于规则空间网格的高效数值分析方法，其包括以下步骤：步骤一、对结构模型进行网格划分，将其划分成多个网格单元，结构模型边界所在的网格单元称为边界单元；步骤二、将边界单元多级细分成多个子域，所述结构模型边界所在子域称为边界子域；对边界子域内的边界进行重建得到重建边界；步骤三、在重建边界的基础上生成结构模型的平衡方程，并求解得到结构模型的场变量；对场变量进行后处理，得到结构模型的分析结果。本发明所述的方法，能够较好地平衡计算效率和计算精度，相对于现有分析方法，能够以更快的方法获得更精确的结果。

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

Efficient numerical analysis method based on regular space grid
Technical Field
The invention relates to the technical field of structural numerical analysis, in particular to a high-efficiency numerical analysis method based on a regular space grid.
Background
In engineering structure numerical analysis, a finite element method is the most popular numerical analysis tool, but the finite element method is utilized to generate high-quality body-attached grids, and particularly for geometrically complex structural models, the process is extremely complex and time-consuming.
When a numerical analysis method based on a regular space grid is adopted, since the boundary unit of the space grid is not attached to the physical boundary of the structural model, accurate integration on the boundary unit becomes difficult, and processing of boundary conditions and post-processing of results become complicated. In order to accurately integrate on the boundary unit, quadtree/octree subdivision is usually performed on the boundary unit, so that the sub-field obtained by subdivision is close to the physical boundary of the structural model as much as possible, and the sub-field of the boundary unit is integrated by using Gaussian integration, so that the integration of the boundary unit on the unit level is completed. However, when the gaussian integration points are few, the error of this integration method of the boundary unit is large, and especially for some boundary units with only a few areas located inside the structural model, the requirement of calculation accuracy cannot be met. In order to improve the calculation accuracy of the boundary cell, the number of gaussian points must be increased, but the increase of gaussian points causes an increase in the calculation amount at the cell level, which greatly increases the calculation amount of numerical analysis. In the processing of boundary conditions of a regular space grid, since there is no grid attached to the physical boundary of a structural model, we can only apply boundary conditions indirectly to nodes of a grid which is not attached to a body, and thus a complicated 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, the integration of the structural model at the unit level needs to be ensured to be very accurate, but the calculation amount in the three-dimensional problem is huge, and the calculation amount cannot be realized in engineering application.
Therefore, how to reduce the calculation amount and ensure the accuracy of the result on the basis of the existing numerical analysis method based on the regular space grid is a technical problem to be solved by the technicians in the field.
Disclosure of Invention
In view of the above, the present invention aims to overcome the shortcomings of the prior art, and provides a high-efficiency numerical analysis method based on a regular space grid, which solves the problems of difficult accurate integration on boundary units, complex boundary condition processing and post-processing of results in the conventional regular space grid method, performs multi-level subdivision on boundary units, completes the integration of boundary units on a unit level by reconstructing the boundary of a structural model, processes the boundary conditions of the structural model by a penalty function method, performs post-processing by constructing triangle/tetrahedron units at the boundary positions of the structural model, and can better balance the calculation efficiency and calculation accuracy, and obtain more accurate results in a faster method than the conventional analysis method. The invention comprises the following steps:
step one, carrying out grid division on a structural model, dividing the structural model into a plurality of grid cells, wherein the grid cells where the boundaries of the structural model are positioned are called boundary cells;
Step two, subdividing the boundary unit into a plurality of subfields in a multistage manner, and enabling the subfields where the boundary of the structural model is positioned to be called boundary subfields;
and thirdly, generating a balance equation of the structural model on the basis of the reconstruction boundary, solving to obtain a field variable of the structural model, and performing post-processing on the field variable to obtain an analysis result of the structural model.
In some embodiments, the first step of the high-efficiency numerical analysis method based on a regular space grid includes dividing the grid unit into an internal unit, the boundary unit and an external unit according to the positional relationship between the structural model and the grid unit, and the specific step of determining the positional relationship between the structural model and the grid unit is as follows:
Randomly scattering some sampling points in the grid unit, drawing a ray in any direction by using a ray method by taking the sampling points as starting points, wherein if the intersection points of the ray and the boundary of the structural model are odd numbers, the sampling points are positioned in the structural model and are internal points, and if the number of the intersection points is even numbers, the sampling points are positioned outside the structural model and are external points;
The grid unit is an internal unit if all the sampling points of the grid unit are internal points, the grid unit is an external unit if all the sampling points of the grid unit are external points, and the grid unit is a boundary unit if all the sampling points of the grid unit are both internal points and external points.
In some embodiments, in the method for efficient numerical analysis based on a regular spatial grid, the multi-level subdivision in the step two includes:
and subdividing the boundary unit for a plurality of times by using a quadtree/octree, subdividing the boundary unit into a plurality of subfields, randomly scattering a plurality of sampling points in the subfields, and if all the sampling points in the subfields comprise an internal point and an external point, determining the subfields as boundary subfields.
In some embodiments, in the method for high-efficiency numerical analysis based on a regular spatial grid, the step of reconstructing the boundary in the step two is:
Selecting an edge line on the boundary subdomain, wherein a corner point at one end of the edge line is an inner point, and a corner point at the other end of the edge line is an outer point;
Marking the distance between the corner point as the inner point and the boundary of the structural model as positive distance, marking the distance between the corner point as the outer point and the boundary of the structural model as negative distance, performing linear interpolation on the symbol distance function between each point on the boundary line and the boundary of the structural model according to the symbol distance function between the two corner points and the boundary of the structural model, selecting the point with the symbol distance function value of 0 on the boundary line as a new boundary point, and connecting the adjacent 2 new boundary points in a straight line to obtain the reconstruction boundary;
The new boundary point coordinate is (x 0,y0), wherein the calculation formula of x 0,y0 is:
Wherein, (x 0,y0) represents the coordinates of a new boundary point, (x 1,y1) and (x 2,y2) represent the coordinates of the corner point 1 and the corner point 2 on the side line, and d 1 and d 2 are the sign distance function values from the corner point 1 and the corner point 2 to the boundary.
In some embodiments, in the method for efficient numerical analysis based on a regular spatial grid, the solving the equilibrium equation in the third step includes:
3.1, integrating on a unit level on the basis of the structural model of boundary reconstruction;
3.2 processing boundary conditions of the structural model by using a penalty function method;
And 3.3, solving a balance equation based on the integration result and the boundary condition processing result on the unit level.
In some embodiments, the specific steps of step 3.1 in the high-efficiency numerical analysis method based on the regular space grid are as follows:
Acquiring basic information of the internal unit, the external unit and the boundary unit, wherein the basic information comprises node coordinates, material properties and unit types;
calculating the integral of the internal unit on the unit level by utilizing Gaussian integral according to the basic information to obtain the integral of the internal unit;
multiplying the integral of the internal unit by a predetermined coefficient to obtain an integral of the external unit;
and in the boundary unit, dividing the boundary unit into an inner area and an outer area by the reconstruction boundary, scattering Gaussian points in the inner area, and calculating the integral on the Gaussian points by Gaussian integral to obtain the integral of the inner area, so as to obtain the integral of the boundary unit on the unit level.
In some embodiments, the step 3.2 of the method for high-efficiency numerical analysis based on a regular space grid includes the steps of:
3.2.1 acquiring boundary conditions, and determining an application unit for applying the boundary conditions;
3.2.2 integrating the boundary conditions at the unit level within the application unit using a penalty function method results in additional terms resulting from the boundary conditions.
In some embodiments, in the method for high-efficiency numerical analysis based on a regular space grid, if the boundary condition is applied to a point in the step 3.2.2, calculating the integral of the point on the unit level in the unit where the point is located, to obtain an additional term generated by the boundary condition; if the boundary condition is applied to the line, integrating at the unit level along the line in the unit where the line is located to obtain an additional term generated by the boundary condition; if the boundary condition is applied to the surface, integrating the surface on the unit level in the unit where the surface is located to obtain an additional item generated by the boundary condition;
Wherein when the boundary condition is zero, the integral of the boundary condition on the unit level comprises an additional matrix item, and when the boundary condition is non-zero, the integral of the boundary condition on the unit level comprises the additional matrix item and a load vector item.
In some embodiments, in the efficient numerical analysis method based on a regular space grid, step 3.3 solves a balance equation, which specifically includes:
numbering all the 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;
Assembling an integration result and a boundary condition processing result on a unit level according to the degree of freedom number to obtain an overall matrix and a vector of the structural model;
And solving a balance equation by utilizing the overall matrix and the vector to obtain a field variable of the structural model.
In some embodiments, in the high-efficiency numerical analysis method based on a regular space grid, the step three is to perform post-processing on the calculation result of the equilibrium equation to obtain the analysis result of the structural model, which specifically includes:
According to the field variable result, calculating a field variable value of a point to be solved in the structural model, wherein the field variable value comprises the following specific processes:
For any point to be solved, determining a grid cell where the point to be solved is located according to the point to be solved coordinates, and acquiring a field variable value of a grid cell node from a calculation result of the balance equation;
calculating the field variable gradient of the point to be solved in the structural model according to the field variable result by utilizing a gradient formula, wherein the specific process is as follows:
when the point to be solved is positioned in the structural model, determining a grid cell in which the point to be solved is positioned according to the coordinate of the point to be solved, and acquiring a field variable value of a grid cell node from a calculation result of the balance equation;
When the point to be solved is positioned at the boundary of the structural model, if the structural model is a two-dimensional model, constructing an enclosing circle by taking the point to be solved as a circle center, dispersing the circumference of the enclosing circle into line segments, selecting the line segments with two end points positioned in the structural model as internal line segments, acquiring coordinate information of the point to be solved and the end points of the internal line segments, determining grid units corresponding to the point to be solved and the end points on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of all the nodes to obtain the field variable values of the point to be solved and the end points, connecting the point to be solved and the 2 end points of the internal line segments into triangular units positioned in the structural model, obtaining field variable gradient values of the triangular units according to the field variable values of three points in the triangular units, and obtaining average field variable gradient values of all the triangular units according to a gradient formula;
when the point to be solved is located at the boundary of the structural model, if the structural model is a three-dimensional model, constructing a bounding sphere by taking the point to be solved as a sphere center, dispersing the surface of the bounding sphere into triangular patches, selecting internal triangular patches with three vertexes of the triangular patches located in the structural model, acquiring coordinate information of the point to be solved and the vertexes of the internal triangular patches, determining grid units of the point to be solved and the vertexes of the internal triangular patches on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of the nodes to obtain the field variable values of the point to be solved and the vertexes, connecting the internal triangular patches with the point to be solved into tetrahedron units located in the structural model, obtaining the field variable gradient values of the tetrahedron units according to the field variable values of all vertexes of the tetrahedron units by utilizing gradient formulas, and obtaining the field variable gradient values of all tetrahedron units to be the field variable gradient values of the tetrahedron units to be average variable values of the point to be solved.
The beneficial effects of the invention are as follows:
The invention discloses a high-efficiency numerical analysis method based on a regular space grid, which solves the problems of difficult accurate integration on boundary units, complex boundary condition processing and result post-processing in the conventional regular space grid method, integrates the boundary units on a unit level by carrying out multistage subdivision on the boundary units and reconstructing the boundary of a structural model, processes the boundary conditions of the structural model by a penalty function method, carries out post-processing by constructing triangular or tetrahedral units at the boundary positions of the structural model, can better balance the calculation efficiency and calculation precision, and can obtain more accurate results by a faster method compared with the conventional analysis method.
The application discloses a high-efficiency numerical analysis method based on a regular space grid, which has a further optimized space, and the accurate integration on the unit level, the processing of boundary conditions and the post-processing of results are very important steps in the numerical analysis method based on the regular space grid. When the number of Gaussian integral points is small, rectangular/cuboid sub-domains subdivided by the boundary units are directly utilized to integrate the unit layers, errors are large, and particularly for some boundary units with only a small area positioned in the structural model, the requirement of calculation accuracy cannot be met, the number of the Gaussian integral points is increased, the calculation amount is greatly increased, and the method is not acceptable in engineering practice. The application reconstructs the boundary of the structural model by using a straight-substitution curve mode based on the subdivided boundary subdomain, adopts different integration strategies relative to units (an internal unit, an external unit and a boundary unit) at different positions of the structural model, improves the calculation efficiency when the calculation accuracy is the same, and improves the calculation accuracy when the number of Gaussian integration points is the same. In the application, in the process of the boundary condition by the penalty function method, the extra item is only added on the calculation result of the standard unit (regular grid unit), and the whole calculation scheme is very similar to the standard finite element calculation, so that the programming implementation is easier. In the post-processing of the result, the application constructs triangle/tetrahedron units at the boundary position of the structural model, thereby obtaining better and more accurate post-processing result with lower cost.
Drawings
FIG. 1 is a flow chart of a method for efficient numerical analysis based on a regular spatial grid according to the present invention;
FIG. 2 is a schematic diagram of a rule space grid generated in a two-dimensional model by a high-efficiency numerical analysis method based on the rule space grid according to the present invention;
FIG. 3 is a schematic diagram of a boundary element subdivision method in a two-dimensional model in a high-efficiency numerical analysis method based on a regular space grid according to the present invention;
FIG. 4 is a schematic diagram of boundary reconstruction in a two-dimensional model according to the method for high-efficiency numerical analysis based on a regular space grid of the present invention;
FIG. 5 is a schematic diagram of a case of a high-efficiency numerical analysis method based on a regular spatial grid in boundary unit integration according to the present invention;
FIG. 6 is a graph showing the results of boundary unit integration in a rule space grid-based high-efficiency numerical analysis method and an existing rule space grid method according to the present invention;
FIG. 7 is a schematic diagram of a case of a rule space grid-based high-efficiency numerical analysis method in post-processing of results according to the present invention;
FIG. 8 is a graph of stress cloud obtained by post-processing the quarter two-dimensional torus of FIG. 7 using a conventional regular spatial grid method;
fig. 9 is a stress cloud graph result obtained by post-processing the quarter two-dimensional ring shown in fig. 7 by the high-efficiency numerical analysis method based on the regular space grid.
Detailed Description
The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is apparent that the described embodiments are only some of the embodiments of the present invention.
As shown in fig. 1 to 4, a method for efficient numerical analysis based on a regular spatial grid includes the steps of:
step one, carrying out grid division on a structural model, dividing the structural model into a plurality of grid cells, wherein the grid cells where the boundaries of the structural model are positioned are called boundary cells;
1.1 determining physical boundary information of a structural model according to size information of the structural model to be analyzed, and generating a bounding box capable of completely bounding the structural model according to the physical boundary information of the structural model. If the structural model is a two-dimensional problem, the bounding box is rectangular, and if the structural model is a three-dimensional problem, the bounding box is rectangular.
1.2 Selecting proper grid size according to the geometric characteristics of the structural model and the size information of the bounding box, carrying out grid division on the bounding box, dividing the bounding box into a plurality of grid units, wherein the grid units are regular rectangular/cuboid units.
1.3 Dividing the grid cells into an internal cell, a boundary cell and an external cell according to the position relation between the structural model and the grid cells, wherein the specific steps of judging the position relation between the structural model and the grid cells are as follows:
Randomly scattering some sampling points in the grid unit, drawing a ray in any direction by using a ray method by taking the sampling points as starting points, wherein if the intersection points of the ray and the boundary are odd numbers, the sampling points are positioned in the structural model and are internal points, and if the number of the intersection points is even numbers, the sampling points are positioned outside the structural model and are external points, the number of the sampling points is a plurality, and the sampling points are uniformly scattered in the grid unit.
The grid unit is an internal unit if all the sampling points of the grid unit are internal points, the grid unit is an external unit if all the sampling points of the grid unit are external points, and the grid unit is a boundary unit if all the sampling points of the grid unit are both internal points and external points.
As shown in fig. 2, a gray area surrounded by a dotted line in the drawing is a two-dimensional structural model to be analyzed, a square wire frame at the outermost periphery is a bounding box of the structural model, and efficient grid division is performed on the bounding box, so that a regular square space grid is generated. The method comprises the steps of obtaining boundary information of a two-dimensional model shown in fig. 2, namely the connection relation between the endpoint coordinates of a boundary curve segment and the endpoints, obtaining node coordinate information of a space grid unit, randomly scattering some sampling points in the space grid, judging the internal and external position relation of all the sampling points of the space grid relative to a structural model by using a ray method (taking A, B points in fig. 2 as an example), namely emitting a ray from a point A of the space grid horizontally to the right, wherein the ray passes through the boundary of the two-dimensional model 1 time for odd times, judging that the sampling point A is inside the two-dimensional model, emitting a ray from a point B horizontally to the right, and the ray passes through the boundary of the two-dimensional model 2 times for even times, and judging that the sampling point B is outside the two-dimensional model. According to the internal and external positional relationship of the sampling points of the grid cells relative to the structural model, the internal and external positional relationship of all the grid cells relative to the structural model is judged (taking the cell 1, the cell 2 and the cell 3 in fig. 2 as examples), wherein all the sampling points of the grid cell 1 are internal points, so the cell 1 is judged to be an internal cell, some of all the sampling points of the grid cell 2 are inside the structural model, some of the sampling points are outside the structural model, so the grid cell 2 is judged to be a boundary cell, and all the sampling points of the grid cell 3 are positioned outside the two-dimensional model, so the grid cell 3 is an external cell.
And secondly, subdividing the boundary unit into a plurality of subfields in a multistage manner, wherein the subfields where the boundary of the structural model is positioned are called boundary subfields, and reconstructing the boundary in the boundary subfields to obtain a reconstructed boundary.
2.1 Subdividing the boundary unit for a plurality of times by using a quadtree/octree, subdividing the boundary unit into a plurality of subfields, randomly scattering a plurality of sampling points in the subfields, and if all the sampling points in the subfields comprise an internal point and an external point, the subfields are boundary subfields.
The number of subdivisions n may be an integer of 1,2, 3,4, the result of calculating the balance equation with the subdivision number of n is basically the same as the result of calculating the balance equation with the subdivision number of n+1, and the value of the subdivision number is determined to be n.
Fig. 3 is a schematic diagram of a boundary unit subdivision method in a two-dimensional model in a high-efficiency numerical analysis method based on a regular space grid according to the present application, wherein, taking one boundary unit in the two-dimensional model as an example, a quadtree/octree is used to perform multistage subdivision on the boundary unit. In the schematic diagram, k represents the order of quadtree division, and k is maximally taken to be 3, wherein the specific subdivision steps are that when the quadtree is divided into a first order (namely k=1), a boundary unit is subdivided into 4 subfields, angular point coordinate information of the 4 subfields is obtained according to node coordinate information of the boundary unit, sampling points are scattered randomly in the 4 subfields, the inner and outer properties of all the sampling points are judged, the inner and outer properties of the 4 subfields are judged according to the inner and outer properties of the sampling points, and it can be seen that 2 subfields in the 4 subfields are cut by the boundary of the two-dimensional model; in the second order quadtree division (i.e., k=2), only 2 sub-fields cut by the boundary of the two-dimensional model obtained by the first order quadtree division are subdivided into 4 small sub-fields, the corner information of the small sub-fields after the second order subdivision is obtained according to the corner information of the sub-fields when k=1, some sampling points are scattered randomly in the small sub-fields, and the inner and outer properties of the sampling points are judged, the inner and outer properties of the small sub-fields obtained by the second order subdivision are judged according to the inner and outer properties of the sampling points, it can be seen that after the second order subdivision, there are 10 sub-fields in total, 5 sub-fields are cut by the boundary of the two-dimensional model, in the third order quadtree division, only the 5 sub-fields cut by the boundary of the two-dimensional model are subdivided, the corner information of the 5 sub-fields are obtained, some sampling points are scattered randomly in the small sub-fields after the third order subdivision, and the inner and outer properties of the sampling points are also scattered randomly in the small sub-fields, and the inner and outer properties of the small sub-fields obtained by the third order subdivision are judged according to the inner and outer properties of the sampling points, after the third-order subdivision, the boundary unit has 25 subfields, wherein 8 subfields are cut by the boundary of the two-dimensional model, at the moment, k reaches the maximum value of 3, no further subdivision is performed, and the multi-level subdivision of the boundary unit is completed. The result of solving the equilibrium equation at k=3 is basically the same as the result at k=4, so taking k=3, since continuing subdivision only increases the workload, and does not have a great influence on the calculation result.
2.2 Selecting the boundary line on the boundary subdomain, wherein the corner point at one end of the boundary line is an inner point, and the corner point at the other end of the boundary line is an outer point.
The method comprises the steps of marking the function value of the sign distance from the corner point of an internal point to the boundary of the structural model as a positive distance, marking the function value of the sign distance from the corner point of an external point to the boundary of the structural model as a negative distance, carrying out linear interpolation on the function of the sign distance from each point on the boundary line to the boundary of the structural model according to the function value of the sign distance from the two corner points to the boundary of the structural model, finding out the point with the function value of the sign distance from the boundary of the structural model to 0 on the boundary line as a new boundary point, recording the coordinates of the new boundary point, and connecting the adjacent 2 new boundary points in a straight line to obtain the reconstruction boundary.
The new boundary point coordinate is (x 0,y0), wherein the calculation formula of x 0,y0 is:
Wherein, (x 0,y0) represents the coordinates of a new boundary point, (x 1,y1) and (x 2,y2) represent the coordinates of the corner point 1 and the corner point 2 of the side line, and d 1 and d 2 are the sign distance function values from the corner point 1 and the corner point 2 to the boundary.
Taking boundary reconstruction in a boundary unit as an example, and gray representing an internal region in the boundary unit positioned in the two-dimensional model as shown in fig. 4, wherein the specific reconstruction steps include obtaining basic information of the boundary of the two-dimensional model, namely, the endpoint coordinates and the topological relation between endpoints of the curve segment model in the boundary subdomain; based on the subdivided boundary unit shown in figure 3, acquiring the angular point coordinate information and the topological relation of boundary subdomains cut by the boundary of the two-dimensional model in the boundary unit, identifying the internal and external position relation of angular points of the subdomains relative to the boundary of the two-dimensional model according to the acquired basic information of the boundary of the two-dimensional model and the angular point information of the subdomains, selecting an edge line on the boundary subdomains, wherein one of two endpoints (also the angular points of the boundary subdomains) of the edge line is an internal point and the other is an external point, respectively calculating the symbol distance function value from the two angular points to the boundary of the two-dimensional model, marking the symbol distance function value from the angular point of the internal point to the boundary of the two-dimensional model as a positive distance, marking the symbol distance function value from the angular point of the external point to the boundary of the two-dimensional model as a negative distance, linearly interpolating the symbol distance function from the angular point on the boundary of the two-dimensional model to the boundary of the two-dimensional model according to the symbol distance function value from the two angular points of the two angular points, finding the point on the boundary of the subdomain the boundary of the subdomains as a new boundary point, taking 1 in figure 4 as an example, wherein a and d is the boundary 1 in the two-dimensional model, c is the point of the boundary point of the two-dimensional model and c is the boundary point between the two-dimensional point and the boundary 1, and c is a negative distance, d is a positive distance, linear interpolation is carried out on the sign distance function from the point on the cd side to the two-dimensional model boundary according to the sign distance function value from the point c and the point d to the two-dimensional model boundary, and a point with the sign distance function value of 0 to the two-dimensional model boundary is found, namely, a point e is used as a new boundary point. Repeating the steps to find all points with the symbol distance function value of 0 to the boundary of the two-dimensional model on the boundary sub-field boundary line in the boundary unit, wherein the points are used as new boundary points of the two-dimensional model, and the new boundary points are connected by straight lines to finish the boundary reconstruction in the boundary unit in fig. 4.
And thirdly, generating a balance equation of the structural model on the basis of the reconstruction boundary, solving to obtain a field variable of the structural model, and performing post-processing on the field variable to obtain an analysis result of the structural model.
3.1, Integrating on a unit level on the basis of the structural model of boundary reconstruction;
3.1.1 obtaining basic information of the internal unit, the external unit and the boundary unit, wherein the basic information comprises node coordinates, material properties and unit types;
3.1.2 calculating the integral of the internal unit on the unit level by using Gaussian integral according to the basic information to obtain the integral of the internal unit, namely obtaining the integral of all the internal units on the unit level;
Multiplying the integral of the internal unit by a preset coefficient to obtain the integral of the external unit, namely obtaining the integral of all the external units on the unit level, wherein the preset coefficient is 10 -6.
And dividing the boundary unit into an inner area and an outer area by the reconstruction boundary in the boundary unit, scattering Gaussian points in the inner area, calculating the integral on the Gaussian points by Gaussian integral to obtain the integral of the inner area, and obtaining the integral of the boundary unit on a unit level, wherein the outer area does not carry out Gaussian integral.
3.2 Processing boundary conditions of the structural model by using a penalty function method;
3.2.1 obtaining boundary conditions, and determining an application unit for applying the boundary conditions, wherein the boundary conditions comprise the application position and the value of the boundary conditions;
3.2.2 integrating the boundary conditions at the unit level within the application unit using a penalty function method results in additional terms resulting from the boundary conditions.
If the boundary condition is applied to a point, calculating the integral of the point on the unit level in the unit where the point is located in the step 3.2.2 to obtain an additional item generated by the boundary condition, if the boundary condition is applied to a line, performing the integral on the unit level along the line in the unit where the line is located to obtain an additional item generated by the boundary condition, and if the boundary condition is applied to a plane, performing the integral on the unit level on the plane in the unit where the plane is located to obtain an additional item generated by the boundary condition. Wherein when the boundary condition is zero, the integral of the boundary condition on the unit level comprises an additional matrix item, and when the boundary condition is non-zero, the integral of the boundary condition on the unit level comprises the additional matrix item and a load vector item.
And 3.3, solving a balance equation based on the integration result and the boundary condition processing result on the unit level.
3.3.1 Numbering all the 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;
3.3.2, according to the degree of freedom number, assembling the integration result and the boundary condition processing result on the unit level to obtain an overall matrix and a vector of the structural model;
And 3.3.3, solving a balance equation by using the overall matrix and the vector to obtain a field variable of the structural model.
3.4, Post-processing the calculation result of the balance equation to obtain an analysis result of the structural model, wherein the analysis result specifically comprises the following steps:
3.4.1 calculating the field variable value of the point to be solved in the structural model according to the field variable result, wherein the specific process is that for any point to be solved, the grid unit where the point to be solved is positioned is determined according to the point coordinate to be solved, the field variable value of the grid unit node is obtained from the calculation result of the balance equation, and the field variable value of the point to be solved is determined by the field variable value of the grid node by utilizing a unit shape function.
3.4.2 Calculating the field variable gradient of the point to be solved in the structural model according to the field variable result by using a gradient formula.
When the point to be solved is positioned in the structural model, determining a grid cell where the point to be solved is positioned according to the coordinate of the point to be solved, acquiring a field variable value of a grid cell node from a calculation result of the balance equation, and determining a field variable gradient value of the point to be solved according to the field variable value of the grid cell node by utilizing a gradient formula.
When the point to be solved is located at the boundary of the structural model, if the structural model is a two-dimensional model, constructing an enclosing circle by taking the point to be solved as the center of a circle, dispersing the circumference of the enclosing circle into a plurality of line segments, selecting the line segments with two endpoints located in the structural model as internal line segments, acquiring coordinate information of the point to be solved and the endpoints of the internal line segments, determining grid units corresponding to the points to be solved and the endpoints of the internal line segments on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of all the nodes to obtain field variable values of the point to be solved and the endpoints of the internal line segments, connecting the point to be solved and the 2 endpoints of the internal line segments into a triangle unit located in the structural model, obtaining field variable gradient values of the triangle unit according to the field variable values of three points in the triangle unit by utilizing a gradient formula, and obtaining the field variable gradient values of all the triangle unit as the field variable values of the point to be solved.
When the point to be solved is located at the boundary of the structural model, if the structural model is a three-dimensional model, constructing a bounding sphere by taking the point to be solved as a sphere center, dispersing the surface of the bounding sphere into triangular patches, selecting internal triangular patches with three vertexes of the triangular patches located in the structural model, acquiring coordinate information of the point to be solved and the vertexes of the internal triangular patches, determining grid units of the point to be solved and the vertexes of the internal triangular patches on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of the nodes to obtain the field variable values of the point to be solved and the vertexes, connecting the internal triangular patches with the point to be solved into tetrahedron units located in the structural model, obtaining the field variable gradient values of the tetrahedron units according to the field variable values of all vertexes of the tetrahedron units by utilizing gradient formulas, and obtaining the field variable gradient values of all tetrahedron units to be the field variable gradient values of the tetrahedron units to be average variable values of the point to be solved.
Embodiment 1, using the high-efficiency numerical analysis method based on the regular space grid of the present invention, researches on mechanical problems in engineering practice, wherein in the numerical analysis process, relevant formulas of mechanics are applied in part of the steps, specifically as follows:
step one, carrying out grid division on a structural model, dividing the structural model into a plurality of grid cells, wherein the grid cells where the boundaries of the structural model are positioned are called boundary cells;
1.1 determining physical boundary information of a structural model according to size information of the structural model to be analyzed, and generating a bounding box capable of completely bounding the structural model according to the physical boundary information of the structural model. If the structural model is a two-dimensional problem, the bounding box is rectangular, and if the structural model is a three-dimensional problem, the bounding box is rectangular.
1.2 Selecting proper grid size according to the geometric characteristics of the structural model and the size information of the bounding box, carrying out grid division on the bounding box, dividing the bounding box into a plurality of grid units, wherein the grid units are regular rectangular/cuboid units.
1.3 Dividing the grid cells into an internal cell, a boundary cell and an external cell according to the position relation between the structural model and the grid cells, wherein the specific steps of judging the position relation between the structural model and the grid cells are as follows:
Randomly scattering some sampling points in the grid unit, drawing a ray in any direction by using a ray method by taking the sampling points as starting points, wherein if the intersection points of the ray and the boundary are odd numbers, the sampling points are positioned in the structural model and are internal points, and if the number of the intersection points is even numbers, the sampling points are positioned outside the structural model and are external points;
The grid unit is an internal unit if all the sampling points of the grid unit are internal points, the grid unit is an external unit if all the sampling points of the grid unit are external points, and the grid unit is a boundary unit if all the sampling points of the grid unit are both internal points and external points.
And secondly, subdividing the boundary unit into a plurality of subfields in a multistage manner, namely, designating the subfields where the boundary of the structural model is positioned as boundary subfields, and reconstructing the boundary in the boundary subfields to obtain a reconstructed boundary.
2.1 Subdividing the boundary unit for a plurality of times by using a quadtree/octree, subdividing the boundary unit into a plurality of subfields, randomly scattering a plurality of sampling points in the subfields, and if all the sampling points in the subfields comprise an internal point and an external point, the subfields are boundary subfields.
2.2 Selecting the boundary line on the boundary subdomain, wherein the corner point at one end of the boundary line is an inner point, and the corner point at the other end of the boundary line is an outer point.
The method comprises the steps of marking the function value of the sign distance from the corner point of an internal point to the boundary of the structural model as a positive distance, marking the function value of the sign distance from the corner point of an external point to the boundary of the structural model as a negative distance, carrying out linear interpolation on the function of the sign distance from each point on the boundary line to the boundary of the structural model according to the function value of the sign distance from the two corner points to the boundary of the structural model, finding out the point with the function value of the sign distance from the boundary of the structural model to 0 on the boundary line as a new boundary point, recording the coordinates of the new boundary point, and connecting the adjacent 2 new boundary points in a straight line to obtain the reconstruction boundary.
The new boundary point coordinate is (x 0,y0), wherein the calculation formula of x 0,y0 is:
Wherein, (x 0,y0) represents the coordinates of a new boundary point, (x 1,y1) and (x 2,y2) represent the coordinates of the corner point 1 and the corner point 2 of the side line, and d 1 and d 2 are the sign distance function values from the corner point 1 and the corner point 2 to the boundary.
And thirdly, generating a balance equation of the structural model on the basis of the reconstruction boundary, solving to obtain a field variable of the structural model, and performing post-processing on the field variable to obtain an analysis result of the structural model.
3.1.1 Obtaining basic information of the internal unit, the external unit and the boundary unit, wherein the basic information comprises node coordinates, material properties and unit types;
3.1.2 calculating the integration of the grid cells on a cell level by using Gaussian integration according to the basic information;
the integral on the unit level in the mechanical problem is the calculation of the unit stiffness array:
Wherein K e represents a cell stiffness matrix, Ω e represents a cell region, B is a strain-displacement matrix, D is an elastic matrix, h is an indication function, which is different according to the difference of the positions of integration points, h is1 when the integration points are located inside the structural model, and h is a small number, 10 -6, when the integration points are located outside the structural model.
3.2 Processing the boundary condition of the structural model by using a penalty function method, wherein the penalty function method is used for processing the boundary condition of the structural model, and taking a mechanical problem as an example, the penalty function method generates additional terms for units where the boundary condition is located as follows:
Wherein, AndRepresenting the additional stiffness matrix and the additional load vector generated by the cell where the boundary condition is located respectively,Representing the region where the displacement boundary condition is imposed, β represents the penalty coefficient of the penalty function method, typically taking 10 6, N as the form function matrix,A displacement vector given for a displacement boundary condition.
And 3.3, solving a balance equation based on the integration result and the boundary condition processing result on the unit level.
3.3.1 Numbering all the 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;
3.3.2, according to the degree of freedom number, assembling the integration result and the boundary condition processing result on the unit level to obtain an overall matrix and a vector of the structural model;
and 3.3.3, solving a balance equation by using the overall matrix and the vector to obtain a field variable of the structural model. Taking the mechanical problem as an example, the equilibrium equation of the system is:
KU=F
wherein, K represents the overall rigidity matrix of the system, U represents the displacement vector to be solved by the system, F represents the load vector of the system, and the specific calculation formula is as follows:
Where n ele represents the number of units of the system, the first term in F is the contribution of the face force t, The second term in F represents the boundary of the action of the face force, which is the contribution generated by the physical force b, the indication function h takes different values according to different positions of the integration point, the specific value is the same as that of integration on a unit level, when the integration point is positioned in the structural model, h takes 1, and when the integration point is positioned outside the structural model, h takes 10 -6. The summation symbols in the above equation represent the assembly process.
3.4, Post-processing the calculation result of the balance equation to obtain an analysis result of the structural model, wherein the analysis result specifically comprises the following steps:
and 3.4.1, calculating the field variable value of the point to be solved in the structural model according to the field variable result.
And determining the grid cell of the point to be solved according to the coordinates of the point to be solved, acquiring the field variable value of the grid cell node from the calculation result of the balance equation, and determining the field variable value of the point to be solved by using a unit shape function from the field variable value of the grid node.
3.4.2 Calculating the field variable gradient of the point to be solved in the structural model according to the field variable result by using a gradient formula.
When the point to be solved is positioned in the structural model, determining a grid cell where the point to be solved is positioned according to the coordinate of the point to be solved, acquiring a field variable value of a grid cell node from a calculation result of the balance equation, and determining a field variable gradient value of the point to be solved according to the field variable value of the grid cell node by utilizing a gradient formula.
When the point to be solved is located at the boundary of the structural model, if the structural model is a two-dimensional model, constructing an enclosing circle by taking the point to be solved as the center of a circle, dispersing the circumference of the enclosing circle into a plurality of line segments, selecting the line segments with two endpoints located in the structural model as internal line segments, acquiring coordinate information of the point to be solved and the endpoints of the internal line segments, determining grid units corresponding to the points to be solved and the endpoints of the internal line segments on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of all the nodes to obtain field variable values of the point to be solved and the endpoints of the internal line segments, connecting the point to be solved and the 2 endpoints of the internal line segments into a triangle unit located in the structural model, obtaining field variable gradient values of the triangle unit according to the field variable values of three points in the triangle unit by utilizing a gradient formula, and obtaining the field variable gradient values of all the triangle unit as the field variable values of the point to be solved.
When the point to be solved is located at the boundary of the structural model, if the structural model is a three-dimensional model, constructing a bounding sphere by taking the point to be solved as a sphere center, dispersing the surface of the bounding sphere into triangular patches, selecting internal triangular patches with three vertexes of the triangular patches located in the structural model, acquiring coordinate information of the point to be solved and the vertexes of the internal triangular patches, determining grid units of the point to be solved and the vertexes of the internal triangular patches on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of the nodes to obtain the field variable values of the point to be solved and the vertexes, connecting the internal triangular patches with the point to be solved into tetrahedron units located in the structural model, obtaining the field variable gradient values of the tetrahedron units according to the field variable values of all vertexes of the tetrahedron units by utilizing gradient formulas, and obtaining the field variable gradient values of all tetrahedron units to be the field variable gradient values of the tetrahedron units to be average variable values of the point to be solved.
In addition, related art formulas are used in studying thermal or other problems using the methods described herein. Such as using thermal equilibrium equations, etc.
In order to better explain the integration of the boundary unit by the integration method, the application has the advantages of fewer Gaussian points and higher efficiency when the calculation accuracy is the same, and smaller absolute value of relative error and higher accuracy of calculation result when the Gaussian points are the same. The numerical analysis method and the numerical analysis method in the prior art are respectively used for carrying out numerical analysis on the square boundary unit with the side length of 1 in fig. 5, gray represents the internal area of the structural model, wherein the gray area takes (-0.4, 0) as the center of a circle, 1 is a part of a circle with a radius, f (x, y) =x 2+y2 as an objective function, and the number of gauss points required by the two integration methods when the quadtree is divided into different orders, and the absolute value of the corresponding gauss integration result and the error of the gauss integration result relative to the accurate integration result are obtained, wherein the accurate integration result in the gray area of the boundary unit is 0.1076, table 1 is the result of the conventional regular space grid integration method, and table 2 is the result of the integration method proposed by the application. From the data in tables 1 and 2, a curve of the absolute value of the relative error of the two methods in fig. 6 as a function of the gaussian point number is obtained, wherein the dashed line represents the result of the existing integration method and the solid line represents the result of the integration method proposed by the present application. Comparing the two curves, the application has the advantages that when the absolute values of the relative errors are the same, the required Gaussian points of the integration method provided by the application are fewer, the calculation amount of boundary unit integration is smaller, the efficiency is higher, and when the Gaussian points are the same, the absolute values of the relative errors of the integration method provided by the application are smaller, and the accuracy of the calculation result is higher.
Table 1 integration results of the prior art integration method in the case shown in fig. 5
Quadtree order 1 2 3 4 5
Gauss points 27 90 180 378 792
Gaussian integration results 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
Table 2 integration results of the integration method proposed by the present application in the case shown in fig. 5
Quadtree order 1 2 3 4 5
Gauss points 13 61 125 271 578
Gaussian integration results 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
In the second embodiment, fig. 7 is a schematic diagram of a case of the high-efficiency numerical analysis method based on the regular space grid in post-processing of results, and fig. 7 is a quarter two-dimensional ring subjected to internal pressure, wherein the internal diameter of the ring is 15mm, the external diameter of the ring is 20mm, and the internal pressure is 1MPa. The true stress of any position of the circular ring can be calculated by the analytic expression of the circular ring, wherein the maximum value of Mises stress at the inner boundary of the circular ring is 4.1625MPa, and the minimum value of Mises stress at the outer boundary of the circular ring is 2.5714MPa.
FIGS. 8 and 9 are stress cloud results from post-processing the quarter two-dimensional ring of FIG. 7 using the prior art regular space grid method and the method of the present application, wherein the stress ranges in FIG. 8 are [2.5086,4.8496] MPa and in FIG. 9 are [2.5748,4.1445] MPa, respectively. As can be seen from fig. 8, in the post-processing results of the prior art method, some points with abnormal stress occur, which cause abnormal stress concentration, because the boundary units where the points are located have only few areas located inside the circular ring, and the integration on the boundary units must be very accurate to ensure the accuracy of the stress results, but in the prior art method, the accurate integration in the boundary units requires very many gaussian points, and the calculation cost caused by the gaussian points is huge, which is unacceptable in engineering practice. When we use the method proposed by the present application, the stress cloud shown in fig. 9 is very smooth, and the maximum error from the theoretical solution is not more than 0.5%. The method of the application improves the integration precision of the boundary units when the number of the Gaussian points is the same for the boundary units with only a few areas positioned in the structure model, and simultaneously adopts a method for constructing triangle/tetrahedron units during post-processing, thereby improving the precision of post-processing results with lower cost. Compared with the existing regular space grid method, the method provided by the application can describe the real stress situation of the structure more accurately and is more suitable for numerical analysis of engineering structures.
Finally, it is noted that the above-mentioned embodiments are merely for illustrating the technical solution of the present invention, and that other modifications and equivalents thereof by those skilled in the art should be included in the scope of the claims of the present invention without departing from the spirit and scope of the technical solution of the present invention.

Claims (8)

1. The high-efficiency numerical analysis method based on the regular space grid is characterized by comprising the following steps of:
step one, carrying out grid division on a structural model, dividing the structural model into a plurality of grid cells, wherein the grid cells where the boundary of the structural model is positioned are called boundary cells;
step two, subdividing the boundary unit into a plurality of subfields in a multistage manner, wherein the subfields where the boundary of the structural model is positioned are called boundary subfields;
Thirdly, generating a balance equation of the structural model on the basis of the reconstruction boundary, and solving to obtain a field variable of the structural model;
In the third step, the calculation result of the balance equation is post-processed to obtain the analysis result of the structural model, which specifically comprises the following steps:
The method comprises the steps of calculating the field variable value of a point to be solved in the structural model according to the field variable result, determining a grid cell where the point to be solved is located according to the coordinates of any point to be solved, and obtaining the field variable value of a grid cell node from the calculation result of the balance equation;
Wherein solving the equilibrium equation comprises:
3.1, finishing integral on the unit level on the basis of the structural model of boundary reconstruction, wherein in the mechanical problem, the integral on the unit level is the calculation of the unit stiffness array:
Wherein K e represents a unit stiffness matrix, omega e represents a unit area, B is a strain-displacement matrix, D is an elastic matrix, h is an indication function, which is different according to different values of the position of an integration point, when the integration point is positioned in the structural model, h is 1, and when the integration point is positioned outside the structural model, h is 10 -6;
3.2 processing boundary conditions of the structural model by using a penalty function method;
3.3, solving a balance equation based on the integral result and the boundary condition processing result on the unit level;
step 3.3, solving a balance equation, which specifically comprises the following steps:
numbering all the 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;
Assembling an integration result and a boundary condition processing result on a unit level according to the degree of freedom number to obtain an overall matrix and a vector of the structural model;
Solving a balance equation by utilizing the overall matrix and the vector to obtain a field variable of the structural model;
In the mechanical problem, the equilibrium equation of the system is:
KU=F
wherein, K represents the overall rigidity matrix of the system, U represents the displacement vector to be solved by the system, F represents the load vector of the system, and the specific calculation formula is as follows:
Wherein, AndRespectively representing an additional stiffness matrix and an additional load vector generated by a unit where the boundary condition is located, wherein N is a shape function matrix, N ele represents the number of units of the system, the first term in F is the contribution generated by the face force t,Representing the boundary of the action of the face forces, the second term in F is the contribution of the physical force b, and the summation symbols in the above formula represent the assembly process.
2. The method of claim 1, wherein the first step comprises dividing the grid cells into an internal cell, a boundary cell and an external cell according to the positional relationship between the structural model and the grid cells, and the specific step of determining the positional relationship between the structural model and the grid cells comprises the following steps:
Randomly scattering some sampling points in the grid unit, drawing a ray in any direction by using a ray method by taking the sampling points as starting points, wherein if the intersection points of the ray and the boundary of the structural model are odd numbers, the sampling points are positioned in the structural model and are internal points, and if the number of the intersection points is even numbers, the sampling points are positioned outside the structural model and are external points;
The grid unit is an internal unit if all the sampling points of the grid unit are internal points, the grid unit is an external unit if all the sampling points of the grid unit are external points, and the grid unit is a boundary unit if all the sampling points of the grid unit are both internal points and external points.
3. The method of claim 1, wherein the multi-stage subdivision in the second step comprises:
and subdividing the boundary unit for a plurality of times by using a quadtree/octree, subdividing the boundary unit into a plurality of subfields, randomly scattering a plurality of sampling points in the subfields, and if all the sampling points in the subfields comprise an internal point and an external point, determining the subfields as boundary subfields.
4. The method for efficient numerical analysis based on regular spatial grid according to claim 2, wherein the step of boundary reconstruction in the step two is:
Selecting an edge line on the boundary subdomain, wherein a corner point at one end of the edge line is an inner point, and a corner point at the other end of the edge line is an outer point;
Marking the distance between the corner point as the inner point and the boundary of the structural model as positive distance, marking the distance between the corner point as the outer point and the boundary of the structural model as negative distance, performing linear interpolation on the symbol distance function between each point on the boundary line and the boundary of the structural model according to the symbol distance function between the two corner points and the boundary of the structural model, selecting the point with the symbol distance function value of 0 on the boundary line as a new boundary point, and connecting the adjacent 2 new boundary points in a straight line to obtain the reconstruction boundary;
The new boundary point coordinate is (x 0,y0), wherein the calculation formula of x 0,y0 is:
Wherein (x 1,y1) and (x 2,y2) represent coordinates of corner 1 and corner 2 on the edge, respectively, and d 1 and d 2 are sign distance function values of corner 1 and corner 2 to the boundary, respectively.
5. The method for high-efficiency numerical analysis based on regular spatial grid according to claim 2, wherein the specific steps of step 3.1 are as follows:
Acquiring basic information of the internal unit, the external unit and the boundary unit, wherein the basic information comprises node coordinates, material properties and unit types;
calculating the integral of the internal unit on the unit level by utilizing Gaussian integral according to the basic information to obtain the integral of the internal unit;
multiplying the integral of the internal unit by a predetermined coefficient to obtain an integral of the external unit;
and in the boundary unit, dividing the boundary unit into an inner area and an outer area by the reconstruction boundary, scattering Gaussian points in the inner area, and calculating the integral on the Gaussian points by Gaussian integral to obtain the integral of the inner area, so as to obtain the integral of the boundary unit on the unit level.
6. A method of efficient numerical analysis based on a regular spatial grid according to claim 1, characterized in that said step 3.2 comprises the steps of:
3.2.1 acquiring boundary conditions, and determining an application unit for applying the boundary conditions;
3.2.2 integrating said boundary conditions at a unit level within said application unit using a penalty function to obtain additional terms resulting from said boundary conditions;
In the mechanical problem, the penalty function method generates additional terms for the unit where the boundary condition is located as follows:
Wherein, Represents the area applying displacement boundary conditions, beta represents the penalty coefficient of the penalty function method, 10 6 is taken as a shape function matrix,A displacement vector given for a displacement boundary condition.
7. The method according to claim 6, wherein in the step 3.2.2, if the boundary condition is applied to a point, the integral of the point on the unit level is calculated in the unit where the point is located to obtain an additional term generated by the boundary condition, if the boundary condition is applied to a line, the integral on the unit level is performed along the line in the unit where the line is located to obtain an additional term generated by the boundary condition, and if the boundary condition is applied to a plane, the integral on the unit level is performed on the plane in the unit where the plane is located to obtain an additional term generated by the boundary condition;
wherein when the boundary condition is zero, the integral of the boundary condition on the unit level comprises an additional matrix item, and when the boundary condition is non-zero, the integral of the boundary condition on the unit level comprises the additional matrix item and a load vector item.
8. The method for efficient numerical analysis based on regular space grids according to claim 1, wherein a gradient formula is used to calculate a field variable gradient of a point to be solved in the structural model according to the field variable result, and the specific process is as follows:
when the point to be solved is positioned in the structural model, determining a grid cell in which the point to be solved is positioned according to the coordinate of the point to be solved, and acquiring a field variable value of a grid cell node from a calculation result of the balance equation;
When the point to be solved is positioned at the boundary of the structural model, if the structural model is a two-dimensional model, constructing a surrounding circle by taking the point to be solved as a circle center, dispersing the circumference of the surrounding circle into line segments, selecting the line segments with two endpoints positioned in the structural model as internal line segments, acquiring coordinate information of the point to be solved and the endpoints of the internal line segments, determining grid units corresponding to the point to be solved and the endpoints on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of the nodes to obtain the field variable values of the point to be solved and the endpoints, connecting the point to be solved and the 2 endpoints of the internal line segments into triangular units positioned in the structural model, obtaining field variable gradient values of the triangular units according to the field variable values of three points in the triangular units, and obtaining average field variable gradient values of all the triangular units to be the field variable gradient values of the point to be solved;
when the point to be solved is located at the boundary of the structural model, if the structural model is a three-dimensional model, constructing a bounding sphere by taking the point to be solved as a sphere center, dispersing the surface of the bounding sphere into triangular patches, selecting internal triangular patches with three vertexes of the triangular patches located in the structural model, acquiring coordinate information of the point to be solved and the vertexes of the internal triangular patches, determining grid units of the point to be solved and the vertexes of the internal triangular patches on the structural model according to the coordinate information, obtaining field variable values of all nodes of the grid units according to a calculation result of a balance equation, interpolating according to the field variable values of the nodes to obtain the field variable values of the point to be solved and the vertexes, connecting the internal triangular patches with the point to be solved into tetrahedron units located in the structural model, obtaining the field variable gradient values of the tetrahedron units according to the field variable values of all vertexes of the tetrahedron units by utilizing gradient formulas, and obtaining the field variable gradient values of all tetrahedron units to be the field variable gradient values of the tetrahedron units to be average variable values of the point to be solved.
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