CN116484444A - Algorithm for generating quadrilateral grid based on function interpolation or fitting - Google Patents
Algorithm for generating quadrilateral grid based on function interpolation or fitting Download PDFInfo
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- CN116484444A CN116484444A CN202310002705.6A CN202310002705A CN116484444A CN 116484444 A CN116484444 A CN 116484444A CN 202310002705 A CN202310002705 A CN 202310002705A CN 116484444 A CN116484444 A CN 116484444A
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- 238000000034 method Methods 0.000 claims abstract description 10
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- 238000012545 processing Methods 0.000 claims description 7
- 238000012892 rational function Methods 0.000 claims description 7
- 230000010354 integration Effects 0.000 claims description 3
- 238000012800 visualization Methods 0.000 claims description 2
- 238000011960 computer-aided design Methods 0.000 description 1
- 238000013461 design Methods 0.000 description 1
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- 238000004088 simulation Methods 0.000 description 1
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Abstract
The invention discloses an algorithm for generating a quadrilateral grid based on function interpolation or fitting, which uses grid characteristics as function initial values, obtains complex functions through the function interpolation or fitting algorithm and integrates the complex functions on the grid to generate the quadrilateral grid. The algorithm expresses grid singular point information by using a sub-pure function, and further expresses global characteristics of the grid. The method is based on the algorithm for generating the quadrilateral mesh based on function interpolation or fitting, the quadrilateral mesh is rapidly generated according to the characteristic information of the triangular mesh, the mesh is represented in a simple and easy-to-express mathematical form, and the method is a heuristic method for generating the comprehensive function interpolation or fitting and the mesh.
Description
Technical Field
The invention belongs to the field of quadrilateral mesh generation, and particularly relates to an algorithm for generating a quadrilateral mesh based on function interpolation or fitting.
Background
The quadrilateral mesh generation has more good characteristics than the traditional triangular mesh in the fields of computer aided design, digital simulation and the like, such as tensor product structure suitable for spline fitting, better maintenance of local geometric characteristics and the like. The theoretical basis of a quadrilateral mesh is very wide, wherein important singular points are directly related to the topological properties of a mesh surface, but the problem of positioning the singular points of the mesh is a critical problem in the generation of a structured quadrilateral mesh, and the number and the positions of the singular points in the structured mesh can roughly determine the topological properties of the mesh. At present, the main means for solving the problem of singular point positioning is a method based on a crossed field, but the method is difficult to control the position of the singular point.
Disclosure of Invention
The invention aims to provide an algorithm for generating a quadrilateral mesh based on function interpolation or fitting, which solves the problem that the singular point position control is difficult in the technology.
In order to achieve the above object, the present invention provides an algorithm for generating a quadrilateral mesh based on function interpolation or fitting, comprising the steps of:
s1, constructing an initial triangular grid, firstly parameterizing the initial grid onto a plane, and corresponding to an expanded complex plane, wherein required information is the characteristics of the parameterized grid;
s2, sampling the parameterized grid, recording sampling information and processing sampling data;
s3, interpolating the sampled data, fitting and calculating a rational function approaching to the original data according to the original numerical value and the function value, and calculating a zero point and a pole of the original function;
s4, integrating the function obtained in the step S3 on a parameterized grid, wherein under the condition that singular points do not coincide, the integral result can automatically induce singular points with 5 singular degrees and 3 singular degrees at zero points and poles respectively;
s5, through step S4, each point on the original parameterized grid is provided with an integral value, and the integral values are used as new parameterized coordinates of each point to generate a quadrilateral grid which maintains the characteristics of the original grid.
Preferably, when the sampled data is processed in the step S2, each sampling point is required to correspond to an original numerical value, and the fourth power of the unit normal vector at the point is used as a function value and recorded together with a complex value.
Preferably, the step S2 of sampling adopts a method of locally representing the whole, and features of the original grid are revealed after the sampling result is separated from the original model for re-visualization.
Preferably, the number of the sampling points in the step S3 is not greater than 50 under the limitation of the interpolation algorithm.
Preferably, the minimum unit processing of the sampled data in the step S2 is in the form of [ (zx, zy), (f (zx_normal), f (zy_normal)) ] where (zx, zy) is a complex function value corresponding to a sampling point and (f (zx_normal), f (zy_normal)) is a complex function value corresponding to a fourth power of a unit normal vector at the sampling point.
Preferably, the rational function fitted in the step S3 is a sub-pure function.
Preferably, according to the step S4, the calculated sub-pure function is integrated on the triangular mesh in a discrete form with the edge on the mesh as a minimum unit of integration.
Therefore, the algorithm for generating the quadrilateral mesh based on function interpolation or fitting is adopted, the mesh characteristics are used as function initial values, the function interpolation algorithm is used for obtaining a sub-pure function, the function is integrated on the mesh, the rational function is used for expressing the singular point information of the mesh, the global characteristics of the mesh are further expressed, and the quadrilateral mesh is rapidly generated according to the characteristic information of the triangular mesh, so that the problem of difficult control of the singular point position in the prior art is solved.
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
Drawings
FIG. 1 is a flow chart of an implementation of an algorithm for generating a quadrilateral mesh based on function interpolation or fitting in accordance with the present invention;
FIG. 2 is a schematic illustration of parameterized grid sampling for an algorithm for generating a quadrilateral grid based on function interpolation or fitting in accordance with the present invention;
FIG. 3 is a sample data of an algorithm for generating a quadrilateral mesh based on function interpolation or fitting in accordance with the present invention;
fig. 4 is a graph of the effect of integrating and visualizing a quadrilateral mesh based on interpolation results of an algorithm for generating the quadrilateral mesh based on function interpolation or fitting according to the present invention.
Detailed Description
Examples
The following detailed description of the embodiments of the invention, provided in the accompanying drawings, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Referring to fig. 1-4, an algorithm for generating a quadrilateral mesh based on function interpolation or fitting includes the following steps:
s1, constructing an initial triangular grid, firstly parameterizing the initial grid onto a plane, and corresponding to an expanded complex plane, wherein required information is the characteristics of the parameterized grid;
s2, sampling the parameterized grid, recording sampling information and processing sampling data;
s3, interpolating the sampled data, fitting and calculating a rational function approaching to the original data according to the original numerical value and the function value, and calculating a zero point and a pole of the original function;
s4, integrating the function obtained in the step S3 on a parameterized grid, wherein under the condition that singular points do not coincide, the integral result can automatically induce singular points with 5 singular degrees and 3 singular degrees at zero points and poles respectively;
s5, through step S4, each point on the original parameterized grid is provided with an integral value, and the integral values are used as new parameterized coordinates of each point to generate a quadrilateral grid which maintains the characteristics of the original grid.
When processing the sampled data in step S2, each sampling point is required to correspond to an original numerical value, and the fourth power of the unit normal vector at the point is used as a function value to be recorded together with the complex value. When sampling is carried out, the characteristics of the parameterized grid are required to be reasonably and uniformly reflected as far as possible, and the characteristics of the original grid are displayed after the sampling result is separated from the original model and is visualized again by adopting a method of locally representing the whole. The minimum unit processing of the sampled data is in the form of [ (zx, zy), (f (zx_normal), f (zy_normal)) ] where (zx, zy) is a complex function value corresponding to the sampling point and (f (zx_normal), f (zy_normal)) is a complex function value corresponding to the fourth power of the unit normal vector at the sampling point.
Step S3 is limited by the interpolation algorithm, and the number of sampling points is not greater than 50, so that the sampling positions need to be reasonably allocated. The rational function fitted according to step S3 is a sub-pure function, and the complex function value (f (zx_normal), f (zy_normal)) is not exactly equal to that obtained after all (zx, zy) operations, but a result as close as possible is achieved, and when the difference between the operation result of all the sampling points and the original result is within a small error range, we consider that the fitted sub-pure function is acceptable.
According to step S4, the calculated sub-pure function is integrated in a discrete form on the triangular mesh with the edges on the mesh as the minimum unit of integration.
Therefore, the algorithm for generating the quadrilateral mesh based on function interpolation or fitting is adopted to solve the problem that the singular point position control is difficult in the prior art. The invention has reasonable design structure, can quickly generate quadrilateral grids according to the characteristic information of the triangular grids, and represents the grids in a simple and easy-to-express mathematical form, thereby being a heuristic method for interpolation or fitting of the comprehensive function and grid generation.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention and not for limiting it, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that: the technical scheme of the invention can be modified or replaced by the same, and the modified technical scheme cannot deviate from the spirit and scope of the technical scheme of the invention.
Claims (7)
1. An algorithm for generating a quadrilateral mesh based on function interpolation or fitting, comprising the steps of:
s1, constructing an initial triangular grid, firstly parameterizing the initial grid onto a plane, and corresponding to an expanded complex plane, wherein required information is the characteristics of the parameterized grid;
s2, sampling the parameterized grid, recording sampling information and processing sampling data;
s3, interpolating the sampled data, fitting and calculating a rational function approaching to the original data according to the original numerical value and the function value, and calculating a zero point and a pole of the original function;
s4, integrating the function obtained in the step S3 on a parameterized grid, wherein under the condition that singular points do not coincide, the integral result can automatically induce singular points with 5 singular degrees and 3 singular degrees at zero points and poles respectively;
s5, through step S4, each point on the original parameterized grid is provided with an integral value, and the integral values are used as new parameterized coordinates of each point to generate a quadrilateral grid which maintains the characteristics of the original grid.
2. An algorithm for generating a quadrilateral mesh based on function interpolation or fitting according to claim 1, characterized in that: when the step S2 processes the sampled data, each sampling point needs to correspond to an original numerical value, and the fourth power of the unit normal vector at the point is used as a function value and recorded together with a complex value.
3. An algorithm for generating a quadrilateral mesh based on function interpolation or fitting according to claim 1, characterized in that: and step S2, sampling adopts a method of locally representing the whole, and features of the original grid are displayed after the sampling result is separated from the original model for re-visualization.
4. An algorithm for generating a quadrilateral mesh based on function interpolation or fitting according to claim 2, characterized in that: the number of the sampling points is not greater than 50 under the limit of the interpolation algorithm in the step S3.
5. An algorithm for generating a quadrilateral mesh based on function interpolation or fitting according to claim 1, characterized in that: the minimum unit processing of the sampled data in the step S2 is in the form of [ (zx, zy), (f (zx_normal), f (zy_normal)) ] where (zx, zy) is a complex function value corresponding to a sampling point and (f (zx_normal), f (zy_normal)) is a complex function value corresponding to a fourth power of a unit normal vector at the sampling point.
6. An algorithm for generating a quadrilateral mesh based on function interpolation or fitting according to claim 2, characterized in that: the rational function fitted out in the step S3 is a sub-pure function.
7. An algorithm for generating a quadrilateral mesh based on function interpolation or fitting according to claim 1, characterized in that: according to the step S4, the calculated sub-pure function is integrated in a discrete form on the triangular mesh with the edges on the mesh as the minimum unit of integration.
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| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202310002705.6A CN116484444A (en) | 2023-01-03 | 2023-01-03 | Algorithm for generating quadrilateral grid based on function interpolation or fitting |
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| CN202310002705.6A CN116484444A (en) | 2023-01-03 | 2023-01-03 | Algorithm for generating quadrilateral grid based on function interpolation or fitting |
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- 2023-01-03 CN CN202310002705.6A patent/CN116484444A/en active Pending
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