CN112862972A - Surface structure grid generation method - Google Patents
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Abstract
The invention provides a surface structure grid generation method, which comprises the following steps: performing extensible local shot parameterization on the three-dimensional model, optimizing anti-overturning energy in the process of mesh parameterization, and indirectly optimizing distortion energy by minimizing a group of simpler proxy energy; the mesh subdivision is carried out in the result of the parameterized two-dimensional plane domain, the complex region is decomposed, the structured mesh is generated on the sub-region plane domain, the mesh is optimized on the plane domain, the angle of the mesh is adjusted based on the fairing algorithm of the angle, the optimization is carried out, the obtained two-dimensional structure mesh is inversely mapped, the two-dimensional structure mesh is inversely mapped to the three-dimensional space, the three-dimensional model surface structure mesh of the paster body is obtained, the structured mesh generated by the method reduces the difficulty of generating the structured mesh in the three-dimensional space, the structured mesh generated in the parameterized domain is easy to realize, and the mesh generated by the method has good quality and high precision.
Description
Technical Field
The invention relates to computer graphics and finite element mesh subdivision, in particular to a surface structure mesh generation method.
Background
Mapping is an important tool in computer graphics and geometry processing, and is one of the most studied subjects in computer graphics, and mesh parameterization is one of the most basic and widespread purposes, and in practice, many applications, such as texture mapping, re-meshing, shape transformation and feature attribute transfer, rely on the calculation of low-distortion parameterization. This problem has been extensively studied and, as there are a number of algorithms, linear methods provide an efficient way of parameterizing computations, which when grid boundaries are fixed, result in a high degree of grid distortion, so that the injective parameterization can be ensured. With the advent of powerful processors, nonlinear optimization has become feasible, allowing one to perform free boundary calculations, allowing high quality shots and shots to be calculated, current nonlinear methods typically require long time calculations and do not scale well to large data sets.
Surface parameterization of three-dimensional models is an important component in computer graphics, such as filtering, compression, recognition, texture mapping and deformation. It involves computing a bijective mapping between a piecewise linear triangulated surface and a suitable parametric domain. Normally, parameterization will result in some metric distortion, since only the developable surface is flattened onto a plane will there be no distortion. The goal of the parameterization is therefore to find a bijective mapping that preserves as much as possible some of the original geometric properties, e.g. equiangular mapping preserving area, conformal mapping preserving angle, equiangular mapping preserving length, or some combination of these mappings. Each individual triangle can be easily parameterized without distortion, but they will no longer fit on the plane.
Mesh generation is a fundamental and critical problem in the modeling and processing of geometric data, and in most computer-aided engineering tasks, involves numerical simulation, the use of finite elements or finite volumes to solve partial differential equations in geometric regions or objects, the first step being to discretize geometric figures or geometries using polygonal or polyhedral meshes. The quality of the mesh may seriously affect the accuracy, efficiency and stability of numerical calculation, and the generated mesh may be classified into a structured mesh and an unstructured mesh. Structured grids are those with vertices in a constant degree and regular arrangement of cells. An unstructured grid, in contrast, allows its elements to be arranged in irregular connectivity.
Grid generation is an important step of numerical calculation, and the quality of generated grids influences the calculation result. With the rapid development of scientific technology, models needing to be solved are more and more complex, and automatic generation of high-quality grids is an important problem which needs to be solved urgently at present. With the rapid development of computer technology and computer aided design technology, the number of three-dimensional models in the fields of engineering drawing, video games, scientific research, biomedicine and the like is increasing explosively. The three-dimensional model has the advantages that the three-dimensional model is closer to the real world, stronger visual impact can be given to people, the impact degree is far higher than that of a two-dimensional picture, and the multi-view visual angle of three-dimensional data enables the model to be more vivid. Due to the wide application prospect of the three-dimensional model, many researchers in computer graphics and computer vision are interested in the field, and therefore good development opportunities are provided for the three-dimensional model.
The grid quality is an important part in a finite element method, although the grid generation is widely applied in finite element analysis, the grid generation quality cannot be guaranteed, some serious distortion or deformation of grid units can be generated in the grid dividing and grid encrypting processes, the performance and the precision of a finite element solution can be seriously influenced by the quality problem of the grid, the solving time can be shortened by the high-quality grid division, and the solving precision can be improved, so that the grid quality improvement method is frequently used in the grid generation. There are three grid improvement methods: node insertion and deletion optimization, topology optimization and geometric optimization. Geometric optimization is also called fairing because it improves mesh quality by repositioning mesh vertices while preserving mesh topology, and therefore fairing plays an important role in mesh optimization.
Disclosure of Invention
The invention solves the technical problem, overcomes the defects of the prior art, and provides a surface structure grid generation method for converting the problem of generating a surface structure grid in a three-dimensional space into the problem of generating the surface structure grid in a two-dimensional space by a parameterization method because the generation of a structured grid in the three-dimensional space is very difficult at present. The grid quality and speed of generating the surface structure grid in the three-dimensional space are improved. The difficulty of generating the surface structure grid by the three-dimensional model is reduced.
The invention adopts a technical scheme that the method specifically comprises the following steps:
step S1: inputting three-dimensional model data, acquiring triangular mesh vertex coordinate information, surface patch quantity information and edge quantity of the three-dimensional model, performing extensible local shot parameterization operation on the input three-dimensional model, wherein the distortion measurement of target minimum energy is continuous symmetric Dirichlet energy, parameterizing the three-dimensional model to a two-dimensional plane domain, and obtaining a parameterized result in the two-dimensional plane domain;
step S2: generating a structured quadrilateral mesh in the two-dimensional plane domain according to the parameterization result obtained in the step S1, performing region decomposition on the complex region by using a region decomposition method when the number of boundaries forming the region is more than 4, decomposing the complex region into a plurality of regular quadrilateral sub-regions, and generating the structured quadrilateral mesh in the sub-regions by using a mapping method;
step S3: performing grid fairing processing on the two-dimensional structural grid in the step S2, comparing all adjacent angles incident to each grid node by using a grid fairing method for changing angles, adjusting the angles, enabling the angles to be equal in quadrilateral grids, enabling the angles adjacent to each pair to be equal or to be in a certain proportion, and obtaining an optimized structured grid in a two-dimensional plane domain;
step S4: and (4) inversely mapping the two-dimensional structured grid obtained in the step (S3) back to the three-dimensional space, expressing the coordinates of the vertex of the two-dimensional structured grid into a linear equation of the coordinate of the triangle where the two-dimensional structured grid is located, and calculating the gravity center weight of the vertex to obtain the coordinates in the space so as to obtain the surface structure grid in the three-dimensional space.
In step S1, the input triangle mesh M is (V, F), V is a vertex, and F is a set of patches
Representing a parameterization, mapping the input triangular mesh into a planar domain, using a continuously piecewise affine function, mapping M into the planar domain, trianglesF belongs to F, the triangle is affine mapped, the affine mapping function is realized by phifRepresents; the triangle F ∈ F is mapped to the planar triangle, and the vector is expressed by pf:
to represent; flattening the input grid parameterization into a plane domain, introducing certain geometric distortion, wherein the geometric distortion is quantified through distortion measurement, the change of area angles before and after mapping is reflected by the distortion measurement, and the distortion measurement selects continuous symmetrical Dirichlet energy, so that deformation energy is minimized, the parameterization quality is high, and the parameterization has expandability; the minimized deformation energy is expressed as follows:
wherein
As a distortion measure, AfIs the area of element f, measures distortion
For translations to be invariant, the metric distortion can be transformed by every affine transformation phifIs formulated with a 2x2 jacobian matrix, the affine transformation is expressed as: phi is af(ii) a The jacobian matrix of the affine transformation is represented as:
wherein Jf(x) Is a linear function of x; the parameterization is an efficient and extensible algorithm to calculate high-quality equidistant parameterization, a triangular mesh is input and comprises vertex and surface information, the input triangular mesh is mapped into a plane domain in the parameterization process, a continuous piecewise affine function is adopted, different affine mappings are kept consistent on a common edge, the input mesh is flattened into a two-dimensional space, and certain geometric distortion is caused. Distortion is quantified by a distortion metric that reflects the planes before and after mappingThe product angle changes. The distortion measure is only from the shape of a triangle before and after mapping, the translation and the rotation are kept unchanged, the distortion measure which minimizes the nonlinear energy and is as rigid as possible is realized through alternate local global optimization, the distortion measure which is as rigid as possible is not ideal for the element processing effect of the overturning, and the distortion measure selected by the energy with the minimized target energy is continuous symmetric Dirichlet energy.
In step S2, in the structured quadrilateral mesh, the fully regular mesh topology is too strict, and some singular points need to be introduced to flexibly reduce distortion. When the number of the boundaries of the formed area is more than 4, the area decomposition is carried out on the complex area, a divide-and-conquer strategy is adopted to carry out the area decomposition method, the complex area is divided into solvable sub-areas, the complex area is decomposed into a plurality of regular quadrilateral sub-areas, a large and complex geometric area is divided into small and simple solvable sub-areas, and through the area decomposition method, the advantages of the mapping method can be exerted through the mapping method, and the problem that the mapping method is difficult to generate a structural grid in the complex area is solved.
In step S3, performing mesh optimization on the two-dimensional planar domain structured quadrilateral mesh in step S2, calculating degrees of adjacent angles by using an angle-based fairing algorithm, calculating a difference between the degrees of the adjacent angles to determine a rotation angle, for each mesh node, comparing all adjacent angles incident to the node, adjusting the angles, making the angles equal in the quadrilateral mesh, making each pair of adjacent angles equal or in a certain proportion, and finally calculating new coordinates of the node, thereby optimizing the two-dimensional structured mesh to obtain the two-dimensional structured mesh; after the fairing algorithm, the size and the shape quality of the grid are good, and the method can prevent the grid unit from being inverted and ensure that the size of the grid unit is as uniform as possible;
the angle-based fairing algorithm is as follows:
(1) for adjacent vertex Nj-1，Nj，Nj+1The calculation formula of two adjacent angles is as follows:
wherein v isj-1,vj,vj+1Is a shared vertex NjVector of (a)1And alpha2Is the angle determined by these three vectors;
(2) calculating the difference between two adjacent angles and determining the vector vjBy the angle of rotation, βj＝(α2-α1)/2，βjIs by a vector vjThe angle caused by the movement;
(3) will vector vjRotation vjAngle, NiThe new coordinates of (a) are:
x′＝x0+(x-x0)cosβj-(y-y0)sinβj
y′＝y0+(x-x0)sinβj+(y-y0)cosβj
wherein (x)0,y0) Is node NjIs node N, (x, y)iIs node N, (x ', y') is the old coordinate ofiNew coordinates of (2);
(4) by traversing all the neighbouring nodes, the same node NiWith a number k of new positions, k being the number of adjacent nodes, node N is calculated by taking all adjacent nodes and calculating the average of (x', yiThe coordinate formula is:
in step S4, the process of inverse mapping the structural mesh in the two-dimensional planar domain to the three-dimensional space is: the two-dimensional coordinates are inversely mapped back to a three-dimensional space, the mapping relation from two-dimensional points to three-dimensional points is realized, concrete coordinates in the three-dimensional space corresponding to the vertexes of the generated two-dimensional structure grid are found by a gravity center weighting method, the vertex set of the two-dimensional grid is input in the inverse mapping process, the inversely mapped three-dimensional grid is output, the vertex set of the two-dimensional grid is traversed, triangles where the vertexes are located are found, the area ratios of the triangles are respectively calculated, the area ratios are used as inverse mapping edge weights, an inverse mapping function is expressed by a linear equation, and the coordinates after vertex inverse mapping are calculated, so that the surface structure grid in the three-dimensional space.
The invention has the beneficial effects that a surface structure grid generation method is designed, a parameterization result which can generate a structured grid in a plane domain can be obtained through parameterization, the parameterization effect has good expansibility, the structured grid is generated in a sub-region through region decomposition in the plane domain, grid density and density of the grid in the plane domain are controlled, and the structured grid is mapped back to a three-dimensional space through inverse mapping to obtain the structured grid of a jointed three-dimensional model surface; compared with the prior art, the mesh generation method has the advantages that compared with the mesh generation method which is directly carried out on the surface of the three-dimensional model, mesh generation operation carried out on the parameterized region is simpler, calculation complexity is lower, and the mesh generation method is easier to realize.
Drawings
FIG. 1 is a schematic diagram of an embodiment of a surface structure mesh generation process;
FIG. 2 is a schematic diagram of the two-dimensional flat domain structured grid domain decomposition method;
fig. 3 is a schematic input/output diagram of the method.
Detailed Description
The invention will be further described with reference to the accompanying drawings.
As shown in fig. 1, the specific implementation steps of the present invention are as follows:
in step S1, the input triangular mesh M is (V, F), V is a vertex, and F is a set of patches. For parameterisation
To indicate. Mapping M into the flat domain with a continuously piecewise affine mapping. Triangle F belongs to F, is affine mapped and utilizes phi to countfAnd (4) showing. The different affine mappings remain consistent on the common edge. By passing
Representing the mapped coordinates of the vertices.
In general, after parameterizing the input mesh, the input mesh is flattened, which introduces some geometric distortion, which is quantified by a distortion metric
The change in the combination of the area and angle of the triangle between the front and rear sides and the latter side between the reaction maps is shown. The distortion metric depends only on the shape of the triangle before and after the mapping, and remains unchanged for rotation and translation.
It is assumed that each triangle F e F maps to a planar triangle, all with its own arbitrary local equidistant mapping, with pf：
To indicate. The parameterized mapping of triangle f can be written as
Wherein the affine mapping in the planar domain uses phif：
To indicate.
Measuring distortion
For translations to be invariant, it can be transformed by every affinefIs formulated as a 2x2 jacobian matrix. Called:
wherein Jf(x) Is a linear function of x.
The energy minimized is:
wherein
As a distortion measure, AfIs the area of element f. The metric distortion is an isometric metric distortion if the metric distortion is minimal for rotation.
A metric distortion that is as stiff as possible is defined as:
minimizing (1) the associated nonlinear energy with a distortion metric that is as rigid as possible is achieved with an alternating local global algorithm. The local global optimization algorithm has very good characteristics, namely when an originally understood point is initialized, a great step is carried out in the process of minimization, the characteristic enables the local global method to obtain a very good result after a plurality of iterations, the local global method can rapidly progress in the initial iteration, and the local global optimization algorithm is also very important in realizing the interactive behavior.
Minimize the use of (1)
Is a continuous symmetric Dirichlet energy, minimizing the energy of the distortion metric components ensures that there is no triangle degradation or flipping, and provides a higher overall quality parameterization. The parameterization method in the invention is called as a local global expansion optimization algorithm, and minimizes energy except distortion measurement which is as rigid as possible.
Symmetric Dirichlet re-weighted local global algorithm, input: mesh M, vertex set V, patch element set F, output: mapping coordinates, minimizing energy
The optimization process starts with 1 iteration number to the maximum iteration number, and for each Jacobian Jf(xk-1) Calculating the most recent rotation
By using
Updating weights for each F ∈ F
Solving for
Thereby obtaining pkCalculating dk，dk：＝pk-xk-1Find alphamaxA bisector search is performed to find the step size α, from α ═ min {1, 0.8 αmaxStart, interval [0, α ]max]。
Jf(xk-1) Is a Jacobian determinant in which the number of the target,
is the most recent rotation of the rotor that is,
is the weight of the update, alpha is the update step,
is a distortion measure, AfIs the area of element f.
The local global algorithm describes the parameterized problem as an optimization problem with both local and global elements, looking for local variations that minimize the deformation of each triangular mesh.
Assuming that the triangle number of the 3D triangle mesh is T1 to T, the area of the 3D triangle is atAssuming that each 3D triangle uses triangles in one plane, each has its own local equal-length parameterization
Our goal is to find a single parameterization for the entire mesh, e.g. a piecewise linear mapping from 3D mesh to 2D planes, to numbersEach of the vertices of n is assigned a 2D coordinate u. For triangular t, 2D coordinates may be used
And (4) showing. Under this setting, xtAnd utThe mapping between the two is related by a 2x2 jacobian matrix. This matrix at triangle t is denoted Jt(u) to indicate its dependence on u. He represents the linear part of the affine mapping, from xtTriangle to u of descriptiontThe depicted triangle.
Energy u defining the parametric coordinates, and an auxiliary linear transformation T, and L ═ L1，..，LTTherein of
This energy can be described in x, u coordinate form.
To solve the minimized ARAP mapping, a local global algorithm is adopted, and iteration is carried out in the two stages, wherein the first stage is a local stage, and the optimal iteration of each triangle is calculated. And in the second part of global stage, solving the optimal u as a sparse linear system. Each of which ensures a reduction in energy that will eventually converge. Since the matrix of the global phase is invariant over iterations, it only needs to be decomposed once, and reused in each iteration. The local phase can be calculated by singular value decomposition, and for ARAP, the singular value decomposition can be directly carried out by using a logarithm covariance matrix.
As shown in fig. 2, the method can not only take advantage of the mapping method, but also solve the problem that the mapping method is difficult to generate the structural grid in the complex area by performing the determination by the boundary information, performing the area decomposition if the complex area is a complex area, and performing the area decomposition if the number of the boundaries constituting the area is more than 4 by using the area decomposition method, and performing the area decomposition on the complex area to decompose the complex area into a plurality of regular quadrangular sub-areas, and generating the structural quadrangular grid in the sub-areas by using the mapping method.
In step S2, the finite element mesh is subdivided in the two-dimensional planar domain according to the obtained parameterization result to generate a structured mesh.
For the complex model, because the structured grid can not be directly generated, a region decomposition method is provided, if the structured grid can not be directly generated, the complex region is decomposed into a series of regular sub-regions, then the structured grid is generated in the relatively regular sub-regions by using a mapping method, and finally all the regions are combined to finally obtain the structured grid.
When the structured quadrilateral grids cannot be directly generated, the complex area is decomposed into quadrilateral sub-areas with simpler and more regular shapes by adding the auxiliary lines, then the structured quadrilateral grids are generated in each sub-area by adopting a mapping method, and finally the sub-areas are combined to generate the structured grids in the whole area.
When carrying out regional decomposition, go on the border information and traverse the region, if having traversed the inner border in the region, then this region is complicated region, consequently convert complicated region, be connected inner border and outer border, turn into simple region with many complicated regions, carry out the mapping method in the subregion, adopt transfinite interpolation method to carry out the grid to subregion and generate, complicated region can be turned into the calculation region through transfinite interpolation method, to the structured grid that generates, control grid unit density that this method can be better, the precision is higher.
In step S3, the angle-based mesh fairing algorithm steps are as follows: for each node, e.g. vertex Nj-1，Nj，Nj+1The calculation formula of two adjacent angles is as follows:
wherein v isj-1,vj,vj+1Is a shared vertex NjVector of (a)1And alpha2Is the angle determined by these three vectors.
Calculating the difference between two adjacent angles and determining the vector vjBy the angle of rotation, βj＝(α2-α1)/2，βjIs by a vector vjThe angle caused by the movement.
Vertex NjWill vector vjRotation vjAngle, NiThe new coordinates of (a) are:
x′＝x0+(x-x0)cosβj-(y-y0)sinβj
y′＝y0+(x-x0)sinβj+(y-y0)cosβj
by traversing all the neighbouring nodes, the same node NiWith a number k of new positions, k being the number of adjacent nodes, node N is calculated by taking all adjacent nodes and calculating the average of (x', yiThe coordinate formula is:
the diagonal nodes are used in addition to the nodes connected to the central node, ideally at 45 °, the above steps are repeated, the target angle is calculated iteratively, and the corresponding nodes are then moved. Wherein (x)0，y0) Is node NjIs node N, (x, y)iIs node N, (x ', y') is the old coordinate ofiThe new coordinates of (2).
In step S4, the structural mesh of the planar domain is inversely mapped to the three-dimensional space, the two-dimensional coordinates are mapped to the three-dimensional space, the mapping relationship from the two-dimensional points to the three-dimensional points is inversely mapped to the three-dimensional space by the centroid mapping method, the coordinates of the vertices of the two-dimensional mesh are expressed as the linear equation of the coordinates of the triangle where the two-dimensional mesh is located, and the coordinates in the space are calculated by calculating the centroid weight of the points, so as to obtain the surface structural mesh in the three-dimensional space. Assuming that the parameter domain has structural grid nodes, according to coordinate traversal, in which triangle of the parameter domain grid, the coordinates can be expressed as a linear equation set related to the coordinates of the triangle, wherein the gravity center weights corresponding to the vertices in the triangle are calculated, and finally, the gravity center weights can be expressed by a piecewise linear function, so that the coordinates of the points in the space are obtained, and the specific coordinates of the generated two-dimensional structural grid points in the three-dimensional space can be obtained in the same way. After the vertex coordinates are obtained, the connection relation is determined by the connection relation of the two-dimensional structural grid, so that the structural grid after inverse mapping is ensured, and the condition of non-coplanarity can be avoided.
As shown in fig. 3, a schematic diagram of output and output is described, and the generation problem of the three-dimensional model surface mesh is simplified into that of the structural mesh generated by the method of the present invention: the parameterization problem from the three-dimensional space to the two-dimensional plane domain and the generation problem of the structural grid in the two-dimensional plane domain reduce the calculation amount and the difficulty of generating the structural grid in the three-dimensional space, and the structural grid generated in the parameterization domain is easy to realize, and has good quality and high precision.
Claims (4)
1. A method for generating a surface structure mesh, the method comprising the steps of:
step S1: inputting a three-dimensional model, acquiring vertex coordinate information, surface patch quantity information and the quantity of edges of a triangular mesh of the three-dimensional model, performing extensible local shot parameterization operation on the input three-dimensional model, wherein the distortion measure of target minimum energy is continuous symmetric Dirichlet energy, parameterizing the three-dimensional model to a two-dimensional plane domain, and obtaining a parameterized result in the two-dimensional plane domain;
step S2: generating a structured quadrilateral mesh in the two-dimensional plane domain according to the parameterized result obtained in the step S1, performing region decomposition on the complex region by using a region decomposition method when the number of boundaries forming the region is more than 4, decomposing the complex region into a plurality of regular quadrilateral sub-regions, and generating the structured quadrilateral mesh in the sub-regions by using a mapping method;
step S3: grid optimization is carried out on the structured quadrilateral grid in the step S2, degrees of adjacent angles are calculated through an angle-based fairing algorithm, a rotation angle is determined through calculation of a difference value of the degrees of the adjacent angles, and new coordinates of nodes are finally obtained through calculation, so that the two-dimensional structured grid is optimized, and the two-dimensional structured grid is obtained;
step S4: and (4) inversely mapping the two-dimensional structured grid obtained in the step (S3) back to the three-dimensional space, expressing the coordinates of the vertex of the two-dimensional structured grid into a linear equation of the coordinate of the triangle where the two-dimensional structured grid is located, and calculating the gravity center weight of the vertex to obtain the coordinates in the space so as to obtain the surface structure grid in the three-dimensional space.
2. The method of claim 1, wherein: in step S1, the triangle mesh M is input as (V, F), V is a vertex, and F is a set of patches
Representing a parameterization process, mapping an input triangular mesh into a plane domain, adopting a continuously segmented affine function to map M into the plane domain, enabling the triangle F to be in the same size as F, enabling the triangle to be in affine mapping, and enabling the affine mapping function to use phi to pass throughfRepresents; mapping the triangle F e F to the planar triangle
To represent; flattening the input grid parameterization into a plane domain, introducing certain geometric distortion, wherein the geometric distortion is quantified through distortion measurement, the change of area angles before and after mapping is reflected by the distortion measurement, and the distortion measurement selects continuous symmetrical Dirichlet energy, so that deformation energy is minimized, the parameterization quality is high, and the parameterization has expandability; minimum deformation energy meterShown below:
wherein
As a distortion measure, AfIs the area of element f, measures distortion
For translations to be invariant, the metric distortion can be transformed by every affine transformation phifIs formulated with a 2x2 jacobian matrix, the affine transformation is expressed as: phi is af(ii) a The jacobian matrix of the affine transformation is represented as:
wherein Jf(x) Is a linear function of x.
3. The method of claim 1, wherein: in step S3, the angle-based fairing algorithm is as follows:
(1) for adjacent vertex Nj-1，Nj，Nj+1The calculation formula of two adjacent angles is as follows:
wherein v isj-1，vj，vj+1Is a shared vertex NjVector of (a)1And alpha2Is the angle determined by these three vectors;
(2) calculating the difference between two adjacent angles and determining the vector vjBy the angle of rotation, βj＝(α2-α1)/2，βjIs by a vector vjThe angle caused by the movement;
(3) will vector vjRotation vjAngle, NiThe new coordinates of (a) are:
x′＝x0+(x-x0)cosβj-(y-y0)sinβj
y′＝y0+(x-x0)sinβj+(y-y0)cosβj
wherein (x)0，y0) Is node NjIs node N, (x, y)iIs node N, (x ', y') is the old coordinate ofiNew coordinates of (2);
(4) by traversing all the neighbouring nodes, the same node NiWith a number k of new positions, k being the number of adjacent nodes, node N is calculated by taking all adjacent nodes and calculating the average of (x', yiThe coordinate formula is:
4. the method of claim 1, wherein: in step S4, the process of inverse mapping the structural mesh in the two-dimensional planar domain to the three-dimensional space is: the two-dimensional coordinates are inversely mapped back to a three-dimensional space, the mapping relation from two-dimensional points to three-dimensional points is realized, concrete coordinates in the three-dimensional space corresponding to the vertexes of the generated two-dimensional structure grid are found by a gravity center weighting method, the vertex set of the two-dimensional grid is input in the inverse mapping process, the inversely mapped three-dimensional grid is output, the vertex set of the two-dimensional grid is traversed, triangles where the vertexes are located are found, the area ratios of the triangles are respectively calculated, the area ratios are used as inverse mapping edge weights, an inverse mapping function is expressed by a linear equation, and the coordinates after vertex inverse mapping are calculated, so that the surface structure grid in the three-dimensional space.
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