CN116597109B - Complex three-dimensional curved surface co-grid generation method - Google Patents
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Abstract
The invention discloses a complex three-dimensional curved surface co-grid generation method, which comprises the following steps: s1, giving a regular curved surface, projecting on an XOY plane to obtain a rectangular area, determining the subdivision length of the transverse direction and the longitudinal direction, and performing grid subdivision of the XOY plane; s2, mapping the center point coordinates of each grid obtained through subdivision from a Z plane to a W plane; s3, determining a mapping scheme from the space curved edge quadrangle to the plane triangle unit, and generating a coplanar triangle grid. The generated projection grid of the complex curved surface grid on the two-dimensional plane has the corner-preserving characteristic, the plane triangle grid subdivision problem of the curved surface structure can be realized, and the conversion from the quadrilateral grid to the triangle grid is completed.
Description
Technical Field
The invention relates to grid generation, in particular to a complex three-dimensional curved surface common-mode grid generation method.
Background
With the rapid development of computer software and hardware technology, the industry commonly uses advanced CAD and CAE technology to construct a target electromagnetic geometric model and a grid model to complete the simulation of products so as to predict and verify the electromagnetic performance of the products, thereby not only improving the flexibility of product design, but also remarkably reducing the research and development period and economic cost of the products. CAD, however, typically uses a continuous boundary representation (boundary representation, B-Rep) to represent the solid model, and CAE typically uses a discrete mesh model as the data input; meanwhile, since a high-quality grid model is critical to an electromagnetic simulation link, how to accurately and effectively construct a high-quality discrete grid model with certain physical and geometric meanings becomes a critical problem in an electromagnetic calculation process.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, and provides a complex three-dimensional curved surface co-grid generation method, wherein the projection grid of the generated complex curved surface grid on a two-dimensional plane has a corner-preserving characteristic, so that the plane triangle grid subdivision problem of a curved surface structure can be realized, and the conversion from a quadrilateral grid to a triangle grid is completed.
The aim of the invention is realized by the following technical scheme: the complex three-dimensional curved surface common-type grid generation method comprises the following steps:
s1, giving a regular curved surface, projecting on an XOY plane to obtain a rectangular area, determining the subdivision length of the transverse direction and the longitudinal direction, and performing grid subdivision of the XOY plane;
s2, mapping the center point coordinates of each grid obtained through subdivision from a Z plane to a W plane;
s3, determining a mapping scheme from a space curved edge quadrangle to a plane triangle unit, and generating a coplanar triangle grid
The beneficial effects of the invention are as follows: the method has the advantages that the conformal grids have conformality, the included angle attribute of the geometric grids before and after deformation is reserved, if the grid lines before deformation are mutually orthogonal, the grid lines after deformation also have orthogonality, the orthogonality can facilitate discretization processing of an electromagnetic calculation numerical algorithm (such as a moment method and a finite element method) on the grids, the numerical calculation process is simplified, the orthogonality of the grid lines can greatly reduce the scale of a system matrix, reduce calculation time consumption and resources, and further improve calculation accuracy.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a schematic diagram of an XOY planar rectangular grid;
FIG. 3 is a schematic diagram of the mapping principle from the Z plane to the W plane;
FIG. 4 is a schematic illustration of an XOY planar co-grid;
FIG. 5 is a schematic diagram of a mapping relationship between a quadrilateral mesh and a triangular mesh;
FIG. 6 is a schematic diagram of a co-grid obtained by the method one in the example;
FIG. 7 is a schematic diagram of a co-grid obtained by the second method in the embodiment.
Detailed Description
The technical solution of the present invention will be described in further detail with reference to the accompanying drawings, but the scope of the present invention is not limited to the following description.
As shown in fig. 1, a complex three-dimensional curved surface co-model grid generating method includes the following steps:
s1, giving a regular curved surface, projecting on an XOY plane to obtain a rectangular area, determining the subdivision length of the transverse direction and the longitudinal direction, and performing grid subdivision of the XOY plane;
s101, displaying a known regular curved surface and a boundary as follows: z=f (x, y) x 1 ≤x≤x 2 y 1 ≤y≤y 2 The method comprises the steps of carrying out a first treatment on the surface of the Projection of the curved surface on the XOY plane of the world coordinate system XYOZ is a length l=x 2 -x 1 And a width w=y 2 -y 1 Then there is a mapping of the x and y coordinates in this region to the Z-plane, i.e. z=x+jy, whereWherein the Z plane is the XOY plane; the X axis is the real axis of the Z plane, and the Y axis is the imaginary axis of the Z plane; this is a familiar coordinate establishment for complex analysis;
s102, knowing that the maximum length of the subdivision on the XOY plane is Deltal, the transverse subdivision number nL=MAX (10, ceil (L/Deltal)), and the longitudinal subdivision number nW=MAX (10, ceil (W/Deltal)), wherein MAX is a function of two numbers to ensure that the minimum grid number of the transverse and vertical axes is not lower than 10, and ceil is an upward rounding function; obtaining a transverse subdivision length deltax=l/(nl+1) and a longitudinal subdivision length deltay=w/(nw+1);
s103, establishing x by transverse subdivision length Deltax 1 ≤x≤x 2 (nl+1) vertices and nL interval ranges within the range including the end points, each vertex constituting a longitudinal grid line, the j-th longitudinal grid line being expressed as:
establishing y from longitudinal split length deltay 1 ≤y≤y 2 Included within the scope are endpointsThe (nW+1) vertices and nW interval ranges therein, each vertex constituting a longitudinal grid line, the ith longitudinal grid line being expressed as:
forming rectangular grid units of (nL+1) x (nW+1) vertexes and nL x nW rectangular interval ranges by interlacing (nL+1) longitudinal grid lines with (nW+1) transverse grid lines; as shown in fig. 2;
in the figure, set G ij The rectangular area is enclosed by the ith longitudinal grid line, the (i+1) th longitudinal grid line, the (j) th transverse grid line and the (j+1) th transverse grid line; p (P) ij (x ij ,y ij ) Is G ij Is defined by the two-dimensional coordinates of the center point of (a), namely:
s2, mapping the center point coordinates of each grid obtained through subdivision from a Z plane to a W plane;
as shown in fig. 3, the Z-plane to W-plane co-model mapping function is known as g, which is expressed as w=g (Z) =ζ+jη;
then there are ζ=re (g (x+jy)) and η=im (g (x+jy)) where Re is a real function and Im is an imaginary function; the mapped W-surface curved surface mesh is shown in fig. 4:
g for any grid ij Center point coordinate P ij (x ij ,y ij ) C mapping from XOY plane to ζO η plane ij (ξ ij ,η ij ) The following steps are:
ξ ij ＝Re(g(x ij +Jy ij ) Sum eta) ij ＝Im(g(x ij +Jy ij ))；
The mapping method adopts any one of the following steps:
mapping method one: first using x ij And y ij Calculating z ij ＝f(x ij ,y ij ) Reuse of x ij And y ij Calculation of xi ij And eta ij
Then there is obtained a three-dimensional map (x ij ,y ij ,z ij )→(ξ ij ,η ij ,f(x ij ,y ij ))
And a mapping method II: first using x ij And y ij Calculation of xi ij And eta ij Reuse of z ij ＝f(ξ ij ,η ij ) Calculating z ij
Then there is obtained a three-dimensional map (x ij ,y ij ,z ij )→(ξ ij ,η ij ,f(ξ ij ,η ij ))。
S3, determining a mapping scheme from the space curved edge quadrangle to the plane triangle unit, and generating a coplanar triangle grid.
Space curved edge quadrilateral gridTo two plane triangle units T 2k And T 2k+1 The map of (2) is shown in FIG. 5:
for space curved edge quadrilateral gridsThe four vertexes are respectively: />Since the four vertices are not necessarily in a plane, they cannot be characterized by a planar quadrilateral mesh, which is diagonally cut into two planar triangular units T 2k And T 2k+1 According to the figure, there is T 2k The unit comprises three vertexes:
T 2k+1 the unit comprises three vertexes:
assume thatTwo-dimensional traversal of vertex number is performed in the order of i and then j, i is nW+1, and starting from 0, there is +.>Point number->
Then there is a segmented point connection T 2k And T 2k+1 The point connections of (a) are respectively:and
the specific process comprises the following steps:
step one: generating an XOY plane rectangular grid list { G ij I.e. [0,1,2, …, nW-1 ]]，j∈[0,1,2,…,nL-1]The ith horizontal and jth vertical grid is denoted as G ij By the following constitutionAnd the four points are formed. P (P) ij For the mesh vertex list { P ij Elements of } represent the ith, jth, longitudinal vertex in the transverse direction, where i e [0,1,2, …, nW)]，j∈[0,1,2,…,nL]。
Step two: initializing triangle mesh list { T ] k [0,1,2, …, nT-1 ]]Nt=2 (nl+1) (nw+1) represents the number of triangle units; initializing k=0;
step three: traversing a planar rectangular grid list { G ] ij [ MEANS FOR SOLVING PROBLEMS ] to obtain G ij G is mapped by using a common-mode mapping method ij Mapping into
Step four: mapping method from space curved edge quadrangle to plane triangle unitSplit into two triangular units T 2k And T 2k+1 Wherein T is 2k Point connection number->T 2k+1 Point connection number of (2)
Step five: k++, cycling from step three to step five until { G } is complete ij Traversal of }.
Step six: export outSum { T ] k And the list is the generated common plane triangle mesh.
In the examples of the present application, nl=40, nw=40,
wherein,
the first method is adopted, and the obtained common grid is shown in fig. 6:
let nl=50, nw=50, f (x, y) =2.5 SINC (x) SINC (y) x 1 ＝0.1,x 2 ＝3.3,y 1 ＝30,y 2 ＝150
g(z)＝ln(z)
Adopting a second method, the obtained common grid is shown in figure 7;
since the grids of fig. 6 and 7 are planar rectangular grids generated by the co-model grid technique of the present patent, the planar grid is also a planar rectangular grid, which is the projection of the finally generated three-dimensional grid onto the XOY plane.
The method has the advantages that the conformal grids have conformality, the included angle attribute of the geometric grids before and after deformation is reserved, if the grid lines before deformation are mutually orthogonal, the grid lines after deformation also have orthogonality, the orthogonality can facilitate discretization processing of an electromagnetic calculation numerical algorithm (such as a moment method and a finite element method) on the grids, the numerical calculation process is simplified, the orthogonality of the grid lines can greatly reduce the scale of a system matrix, reduce calculation time consumption and resources, and further improve calculation accuracy.
Finally, it should be noted that: the above description is only a preferred embodiment of the present invention, and the present invention is not limited thereto, but it is to be understood that the present invention is described in detail with reference to the above embodiments, and modifications, for example, variations in the names of the methods, etc. may be made to the methods described in the above embodiments by those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (1)
1. A complex three-dimensional curved surface co-grid generation method is characterized in that: the method comprises the following steps:
s1, giving a regular curved surface, projecting on an XOY plane to obtain a rectangular area, determining the subdivision length of the transverse direction and the longitudinal direction, and performing grid subdivision of the XOY plane;
the step S1 includes:
s101, displaying a known regular curved surface and a boundary as follows: z=f (x, y) x 1 ≤x≤x 2 y 1 ≤y≤y 2 The method comprises the steps of carrying out a first treatment on the surface of the Projection of the curved surface on the XOY plane of the world coordinate system XYOZ is a length l=x 2 -x 1 And a width w=y 2 -y 1 Then there is a mapping of the x and y coordinates in this region to the Z-plane, i.e. z=x+jy, whereWherein the Z plane is the XOY plane;
s102, knowing that the maximum length of the subdivision on the XOY plane is Deltal, the transverse subdivision number nL=MAX (10, ceil (L/Deltal)), and the longitudinal subdivision number nW=MAX (10, ceil (W/Deltal)), wherein MAX is a function of two numbers to ensure that the minimum grid number of the transverse and vertical axes is not lower than 10, and ceil is an upward rounding function; obtaining a transverse subdivision length deltax=l/(nl+1) and a longitudinal subdivision length deltay=w/(nw+1);
s103, establishing x by transverse subdivision length Deltax 1 ≤x≤x 2 (nl+1) vertices and nL interval ranges within the range including the end points, each vertex constituting a longitudinal grid line, the j-th longitudinal grid line being expressed as:
establishing y from longitudinal split length deltay 1 ≤y≤y 2 (nW+1) vertices and nW interval ranges within the range including the endpoints, each vertex constituting a longitudinal gridline, the ith longitudinal gridline being expressed as:
forming rectangular grid units of (nL+1) x (nW+1) vertexes and nL x nW rectangular interval ranges by interlacing (nL+1) longitudinal grid lines with (nW+1) transverse grid lines;
set G ij The rectangular area is enclosed by the ith longitudinal grid line, the (i+1) th longitudinal grid line, the (j) th transverse grid line and the (j+1) th transverse grid line; p (P) ij (x ij ,y ij ) Is G ij Is defined by the two-dimensional coordinates of the center point of (a), namely:
s2, mapping the center point coordinates of each grid obtained through subdivision from a Z plane to a W plane;
the step S2 includes:
knowing the Z-plane to W-plane co-model mapping function as g, the mapping is expressed as w=g (Z) =ζ+jη;
then there are ζ=re (g (x+jy)) and η=im (g (x+jy)) where Re is a real function and Im is an imaginary function;
g for any grid ij Center point coordinate P ij (x ij ,y ij ) C mapping from XOY plane to ζO η plane ij (ξ ij ,η ij ) The following steps are:
ξ ij ＝Re(g(x ij +Jy ij ) Sum eta) ij ＝Im(g(x ij +Jy ij ))；
The mapping method adopts any one of the following steps:
mapping method one: first using x ij And y ij Calculating z ij ＝f(x ij ,y ij ) Reuse of x ij And y ij Calculation of xi ij And eta ij
Then there is obtained a three-dimensional map (x ij ,y ij ,z ij )→(ξ ij ,η ij ,f(x ij ,y ij ))
And a mapping method II: first using x ij And y ij Calculation of xi ij And eta ij Reuse of z ij ＝f(ξ ij ,η ij ) Calculating z ij
Then there is obtained a three-dimensional map (x ij ,y ij ,z ij )→(ξ ij ,η ij ,f(ξ ij ,η ij ))；
S3, determining a mapping scheme from the space curved edge quadrangle to the plane triangle unit, and generating a coplanar triangle grid;
the step S3 includes:
for space curved edge quadrilateral gridsThe four vertexes are respectively: />Since the four vertices are not necessarily in a plane, they cannot be characterized by a planar quadrilateral mesh, which is diagonally cut into two planar triangular units T 2k And T 2k+1 According to the figure, there is T 2k The unit comprises three vertexes:
T 2k+1 the unit comprises three vertexes:
assume thatTwo-dimensional traversal of vertex number is performed in the order of i and then j, i is nW+1, and starting from 0, there is +.>Point number->
Then there is a segmented point connection T 2k And T 2k+1 The point connections of (a) are respectively:and
the specific process comprises the following steps:
step one: generating an XOY plane rectangular grid list { G ij I.e. [0,1,2, …, nW-1 ]]，j∈[0,1,2,…,nL-1]The ith horizontal and jth vertical grid is denoted as G ij By the following constitutionThe four points are formed; p (P) ij For the mesh vertex list { P ij Elements of } represent the ith, jth, longitudinal vertex in the transverse direction, where i e [0,1,2, …, nW)]，j∈[0,1,2,…,nL]；
Step two: initializing triangle mesh list { T ] k [0,1,2, …, nT-1 ]]Nt=2 (nl+1) (nw+1) represents the number of triangle units; initializing k=0;
step three: traversing a planar rectangular grid list { G ] ij [ MEANS FOR SOLVING PROBLEMS ] to obtain G ij G is mapped by using a common-mode mapping method ij Mapping into
Step four: mapping method from space curved edge quadrangle to plane triangle unitSplit into two triangular units T 2k And T 2k+1 Wherein T is 2k Point connection number->T 2k+1 Point connection number of (2)
Step five: k++, cycling from step three to step five until { G } is complete ij Traversing;
step six: export outSum { T ] k And the list is the generated common plane triangle mesh.
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