CN109035410B

CN109035410B - Discretization-based multiple curved surface building mesh division method

Info

Publication number: CN109035410B
Application number: CN201810797185.1A
Authority: CN
Inventors: 高博青; 李铁瑞
Original assignee: Zhejiang University ZJU
Current assignee: Zhejiang University ZJU
Priority date: 2018-07-19
Filing date: 2018-07-19
Publication date: 2020-08-07
Anticipated expiration: 2038-07-19
Also published as: CN109035410A

Abstract

The invention discloses a discretization-based multi-free-form surface building mesh division method, and belongs to the field of building mesh division. In order to realize the multi-free-form surface building mesh division with cutting and holes, firstly, the multi-free-form surface is respectively dispersed and sewn to form a discretization surface consisting of a large number of small triangular surface patches. Secondly, an improved error diffusion algorithm is adopted, and initial point distribution is carried out on the discretization curved surface according to a certain density. And then, carrying out primary homogenization on the point cloud by adopting a particle dynamic relaxation algorithm of the space distance, and carrying out secondary homogenization by applying a k-means algorithm of the curved surface distance. Then, a triangular mesh is calculated for the uniform point cloud. And finally, carrying out topology optimization and fairing optimization on the grid. The algorithm of the invention can effectively process multiple free-form surfaces with cutting and holes and obtain a uniform smooth triangular mesh.

Description

Discretization-based multiple curved surface building mesh division method

Technical Field

The invention belongs to the field of building mesh division, and particularly relates to a discretization-based multi-curved-surface building mesh division method.

Background

With the development of computer aided design technology, especially modeling technology, the creativity of architects can be better exerted, and the architectural form of the free-form surface is favored by people due to the powerful visual effect. However, it is difficult to mesh a complex curved surface, and there is no effective method.

For the free-form surface, the expression can not be realized by adopting an explicit equation or an implicit equation, and the NURBS technology is often adopted for modeling. For particularly complex surfaces, it is difficult or even impossible to express the surface with a single NURBS surface, and multiple surfaces are used for modeling. Since NURBS surfaces are represented by higher-order non-linear parametric equations, some classical algorithms are difficult to apply to NURBS surfaces, and more difficult to tailor surfaces, especially multi-surfaces.

Early scholars proposed methods such as wavefront propulsion, Delaunay, mapping, etc. These methods have their limitations in free-form surface applications. The existing method for dealing with the free-form surface comprises the following steps: the problem of curved surface Delaunay is solved by using an empty ellipse criterion instead of the traditional empty circle criterion; applying a Riemann metric to improve mapping distortion; and the equal-area curved surface unfolding method is adopted to improve the mapping relation from the plane to the curved surface. For multiple curved surfaces, the existing method is to divide meshes on each curved surface respectively and then adjust the meshes. However, the curved surface junction of the method may have an obvious bending phenomenon, and because the curves belong to a plurality of curved surfaces, the curve is difficult to optimize and adjust together, and the design requirement is difficult to achieve.

In another case, an architect first makes a physical model, and then digitalizes the physical model by using a 3-D scanning technology to obtain discretized surface data, so that the algorithm of the invention can directly perform mesh division without performing a complicated surface fitting reconstruction step.

Disclosure of Invention

The invention adopts the discretization idea to solve the problem of multiple curved surface building mesh division with cutting and holes. The basic idea is as follows: firstly, the multiple curved surfaces are respectively dispersed and then sewn, and are integrated into a discretized curved surface consisting of a large number of triangular surface patches, so that negative effects such as cutting boundaries, holes, multi-curved-surface splicing and the like can be eliminated. Secondly, an improved error diffusion algorithm is adopted to distribute initial points, and although the point distribution result of the algorithm is not uniform enough, the density distribution is basically met, and a feasible initial value is provided for a subsequent homogenization algorithm. And then, carrying out primary homogenization on the point cloud by adopting a particle dynamic relaxation algorithm based on the spatial distance, and carrying out secondary homogenization on the point cloud by applying a k-means algorithm based on the curved surface distance to obtain a uniform dot matrix on the curved surface. Then, a discrete surface Voronoi diagram is solved in a mode of solving the shortest path of the surface distance by adopting a Dijkstra algorithm, and a surface Delaunay diagram, namely a triangular mesh, is obtained. The algorithm can ensure that the obtained topological graph has the optimal quality of the triangle. Meanwhile, topology optimization is carried out, and partial topology connection can be modified interactively. And finally, optimizing the triangular mesh by adopting a spring mass point relaxation algorithm to obtain a uniform and smooth final mesh.

The specific technical scheme of the invention is as follows:

the multiple curved surface building mesh division method based on discretization comprises the following steps:

1) respectively discretizing and sewing the multiple curved surfaces to form a discretized representation consisting of a large number of small triangular patches;

2) performing initial point distribution on the discretization curved surface according to a certain density by adopting an improved error diffusion algorithm;

3) carrying out primary homogenization on the point cloud by adopting a particle dynamic relaxation algorithm based on the spatial distance;

4) carrying out homogenization again by adopting a k-means algorithm based on the curved surface distance;

5) solving a Voronoi diagram based on the curved surface distance from the uniform point cloud, and obtaining a triangular mesh;

6) carrying out topology optimization on the triangular mesh;

7) and performing fairing optimization on the triangular mesh by adopting a spring mass point method.

In the above scheme, each step can be specifically realized by adopting the following mode:

the improved error diffusion algorithm comprises the following steps:

2.1) calculating the base density ρ 'of each triangular patch according to the area of the patch'_T

Wherein n is the total number of points to be arranged on the curved surface, A_TThe area of the triangular patch is shown, and A is the total area of the discrete curved surface;

2.2) correcting the basic density according to the given density distribution function and the curvature function to obtain the density rho of each triangular patch_T

ρ_T＝ρ′_Tf_ρ(T)f_k(T)

Wherein f is_ρ(T) is a density correction function, defined by the user; f. of_k(T) calculating according to the position of the triangular patch as a curvature correction function;

2.3) transferring the density of the triangular patch to the node

Where ρ is_NFor node density, N ∈ T indicates that node N belongs to threeAn angle T;

2.4) selecting a point as a starting node and adding the point into a queue;

2.5) taking out a queue head node, if the density value of the node is accumulated to a threshold value t, carrying out random judgment once, arranging a point at the node with probability p, and subtracting 1 from the density of the node; the threshold value t adopts a variation value and is gradually increased along with the increase of the number of access points;

2.6) evenly diffusing the density of the nodes (if the points are arranged in the step 2.5, the density is negative) to surrounding nodes which are not accessed, adding the nodes into a queue, and marking the node as accessed;

2.7) if the queue is empty, ending and going to step 3); otherwise, go to step 2.5).

The particle power relaxation algorithm based on the spatial distance comprises the following specific steps:

regarding each node as a particle with mass and electric quantity, wherein repulsion exists between two particles with the distance less than the critical distance, and the particles continuously move under the action of the repulsion; another constraint of the algorithm is: the node can not leave the curved surface, after each time step is finished, the node is moved to a triangular surface nearest to the discrete curved surface, after a certain step number is calculated in an iterative mode, the node is stabilized to be close to the balance position, and uniform point cloud is obtained.

The k-means algorithm based on the curved surface distance comprises the following specific steps:

for the n points obtained in step 3), alternately performing the following two steps until convergence:

step 4.1: taking n points as a center, creating a curved surface Voronoi diagram;

step 4.2: moving the n points to the center of the respective Voronoi diagram regions;

after a certain number of steps of iteration, more uniform point cloud is obtained.

The step 5) is as follows:

for the n homogenized points, calculating the curved surface distance from the n homogenized points to each small triangular patch, wherein the small triangular patch belongs to the point with the closest distance; after the point to which each small triangular patch belongs is determined, n areas are formed;

and after a Voronoi diagram of the lattice is obtained, solving a dual diagram to obtain the triangular mesh.

In the step 6), the method for performing topology optimization on the triangular mesh comprises the following steps: traversing each rod piece, forming a quadrangle by two surfaces on two sides of the rod piece, and if the number of the internal node connecting rods is closer to 6 and the number of the boundary node connecting rods is closer to 4 in four vertexes of the quadrangle by exchanging diagonals of the quadrangle, carrying out exchange.

The mesh division method can effectively process complex conditions such as cutting, punching, violent change, multi-surface splicing and the like, obtain uniform and smooth triangular meshes, and finally form meshes with good uniformity.

Drawings

FIG. 1 shows an embodiment of a local calculation for a ship-shaped curved surface, wherein (a) the original curved surface, (b) the discrete result, (c) the initial layout, (d) the homogenized dot matrix, (e) the triangular mesh, (f) the optimized final mesh;

FIG. 2 is a comparison example of a ship-shaped curved surface local algorithm, in which (a) two curved surfaces are respectively meshed and (b) two curved surfaces are respectively optimized;

FIG. 3 shows an example of a crescent-shaped surface cut out, in which (a) the original surface and (b) the algorithm mesh.

Detailed Description

The invention will be further elucidated and described with reference to the drawings and the detailed description. The technical features of the embodiments of the present invention can be combined correspondingly without mutual conflict.

Step 1: discretization

For a parametric surface, such as a NURBS surface, if the surface is complete, the parametric domain is adopted for uniform point distribution, and a grid is formed and mapped to the surface. For the cutting or hole-carrying condition, uniform points are formed firstly, then the points outside the boundary are abandoned, and then triangulation is carried out on the plane and the plane is mapped to the curved surface. For the case of multiple curved surfaces, each curved surface is discretized and then stitched at the junction of the two curved surfaces. The specific idea is to set a distance threshold, merge points with similar distances between the boundaries of the two curved surfaces, and delete the overlapped edges.

Step 2: initial stationing

The invention employs an improved error diffusion algorithm. The specific algorithm steps are as follows:

step 2.1: calculating the base density rho 'of each triangular patch according to the area of the patch'_T

Wherein n is the total number of points to be arranged on the curved surface, A_TThe area of the triangular patch, and A is the total area of the discrete curved surface.

Step 2.2: correcting the basic density according to the given density distribution function and curvature function to obtain the density rho of each triangular patch_T

ρ_T＝ρ′_Tf_ρ(T)f_k(T)

Wherein f is_ρ(T) is a density correction function, defined by the user; f. of_kAnd (T) is a curvature correction function and is calculated according to the position of the triangular patch.

Step 2.3: transferring the density of the triangular patch to the node

Where ρ is_NFor node density, N ∈ T indicates that node N belongs to triangle T;

step 2.4: and selecting a point as a starting node and adding the point into the queue.

Step 2.5: and taking out the head node of the queue. If the density value of the node has accumulated to the threshold value t, a random decision is made to place a point at the node with probability p and subtract 1 from the density of the node. The threshold value t is a variable value and gradually increases as the number of access points increases.

Step 2.6: the density of the nodes (negative if points are placed in step 5) is spread evenly to surrounding nodes that are not accessed and added to the queue, marking the node as accessed.

Step 2.7: if the queue is empty, ending and turning to the step 3; otherwise, go to step 2.5.

And step 3: point cloud homogenization based on spatial distance

The invention adopts a particle power relaxation algorithm, and the basic idea is as follows: each node is regarded as a particle with certain mass and electric quantity, and repulsion exists between two particles with close distance. Under the action of this repulsion force, the particles will continue to move. Another constraint is that the node cannot leave the surface. And after each time step is finished, moving the node to the triangular face nearest to the discrete curved surface. After a certain number of steps of iterative computation, the node will stabilize near the equilibrium position. And obtaining more uniform point cloud.

The particle power relaxation algorithm adopted by the invention comprises the following specific steps:

consider each node as having a mass m_iQ quantity of electricity_iOf any two particles V_i，V_jThere is a repulsive force f_i,j：

Where k is a coefficient for controlling the strength of the repulsive force, and e is an index for controlling the effect of the distance, generally taken as-2, r_i,jIs node V_iAnd V_jThe distance of (c). Irrespective of gravitational forces in this context, distances greater than a critical distance r_cAfter that, no interaction force is considered to be present.

The resultant force of each point is as follows:

wherein i_i，jIs node V_jRelative to node V_iThe unit direction vector of (2).

Under the action of the repulsive force, the particles will move continuously, and the acceleration, velocity and displacement of the particles are calculated as follows.

v_t+1,i＝v_t,i+a_t,iΔt；

Another constraint is that the node cannot leave the surface, and after each time step is over, the node is moved to the triangular patch nearest to the discrete surface. After a certain number of steps of iterative computation, the node will stabilize near the equilibrium position.

And 4, step 4: point cloud homogenization based on curved surface distance

The invention adopts a k-means algorithm popularized to a discrete curved surface, and the basic idea is that the following two steps are alternately carried out until convergence for n points which are initially arranged:

step 4.1: a curved surface Voronoi diagram is created with n points as centers.

Step 4.2: the n points are moved to the center of the respective Voronoi diagram regions.

And 5: point cloud triangulation network

The invention solves the triangular mesh by solving the curved surface Voronoi diagram firstly. The Voronoi diagram definition is generalized to discrete surfaces, i.e., each small triangular patch is closest in distance to the point to which it belongs. The discretized surface can be regarded as a topological graph with patches as vertexes and an edge between adjacent patches. When the dimension of the small triangular patch is smaller than a certain threshold (0.1) relative to the calculated triangular mesh, the shortest path between two points of the topological graph can be considered as the corresponding curved surface distance. The invention adopts Dijkstra algorithm to solve the shortest path of the single source. The specific method for solving the triangular mesh comprises the following steps:

and calculating the curved surface distance from the n homogenized points to each small triangular patch. The small triangular patch belongs to the closest point. After the point to which each small triangular patch belongs is determined, n regions are formed.

Step 6: topology optimization

For the triangular mesh, it is generally considered preferable that the internal nodes connect 6 rods and the boundary nodes connect 4 rods. The optimization method comprises the following specific steps:

traversing each rod piece, forming a quadrangle by two surfaces on two sides of the rod piece, and if the number of the internal node connecting rods is closer to 6 and the number of the boundary node connecting rods is closer to 4 in four vertexes of the quadrangle by exchanging diagonals of the quadrangle, carrying out exchange.

And 7: fairing optimization

And finally, carrying out fairing optimization on the obtained grid. The invention adopts a spring mass point method. The basic idea is to treat the rod as a spring with a certain stiffness and the node as a mass point. The optimization goal is that the rod pieces are equal in length. The node will move under the spring force and stabilize in an equilibrium position. And finally obtaining a smooth grid.

Examples

The selected calculation is shown in fig. 1 (a). The example is typical of complex multiple free-form surfaces. Due to the fact that the model is complex, when an architect builds a model, the mode that an upper curved surface and a lower curved surface are spliced is adopted, the upper curved surface has a cutting boundary, the upper curved surface is visible through a curved surface structure line, one end of the upper curved surface is sharply contracted, one edge of a parameter domain corresponds to one point of a space, and mapping deformation is huge.

The method comprises the following specific steps:

1) in this example, the total number n of points to be arranged on the curved surface is 700, and then the points are scattered into at least 70k small triangular patches according to the requirement of the algorithm of the present invention on the scattering scale. The upper and lower curved surfaces are discretized respectively and sewn at the boundary to obtain the discretized curved surface shown in fig. 1 (b). The discrete curved surface actually consists of 360k small triangular patches, and meets the calculation requirement.

2) On the discrete surface, an error diffusion algorithm is adopted. Taking the total number n of the points as 700; in this case, a uniform density distribution is considered, i.e.

Without taking into account curvature correction, i.e.

The probability factor p is 0.2; and (4) adopting a variable threshold value t, wherein the initial value is 1e-4, the increment is 1e-4 each time, and the maximum value is 0.4. The initial spotting result is shown in fig. 1 (c). As can be seen, even though the distribution is not uniform enough, a uniform density distribution is satisfied, providing a good initial value for the next step.

3) The point cloud is preliminarily homogenized by adopting a particle dynamic relaxation algorithm based on spatial distance, wherein the mass m is_iWith an electric quantity q_iAll take 1.0, the intensity coefficient k is artificially regulated and controlled according to the effect in the operation process, the index e of the distance control action takes-2, and the critical distance r_cTake 4 m. And (3) homogenizing the point cloud again by adopting a k-means algorithm based on the curved surface distance, wherein uniform density distribution is considered in the example, curvature correction is not considered, and uniform point cloud is obtained, as shown in fig. 1 (d). As can be seen from the figure, the point cloud distribution is relatively uniform.

4) And solving a Voronoi diagram based on the curved surface distance for the uniform point cloud, and solving a dual diagram to obtain a triangular mesh, as shown in FIG. 1 (e). As can be seen from the figure, the obtained triangle has good quality, is close to a regular triangle, has no long and narrow triangle, but has poor fluency.

5) And (f) performing topology and fairing optimization on the triangular mesh to obtain a final result, as shown in fig. 1 (f).

And the final grids are uniform and smooth, a small number of internal node connecting rod pieces are not 6, but the transition is smooth, and the building aesthetic requirement is met.

Comparative example

The multiple free-form surfaces in fig. 1(a) are taken as an example. By adopting a mapping method, the upper and lower boundaries of the upper curved surface are divided into 44 sections, and the side boundary is divided into 4 sections; the upper and lower boundaries of the lower curved surface are divided into 44 sections, which are aligned with the upper curved surface, and the side boundaries are divided into 16 sections; and connecting the corresponding points of the boundary in the parameter domain, and mapping to the space curved surface. The results are shown in FIG. 2 (a). It can be seen that the mapping method has a serious mapping deformation problem. The segmentation method can achieve an average rod length similar to the embodiment mesh, but the rod length variance is much larger than the embodiment. The results obtained by applying fairing optimization to the upper and lower curved surfaces respectively are shown in fig. 2(b), the negative influence of mapping deformation is relieved to a certain extent, but the grid streamline at the junction of the two curved surfaces is bent obviously, and the curve belongs to the two curved surfaces respectively, so that the adjustment is difficult. Another problem is that the lower lattice is too dense and the upper rods are too long.

Compared with the method of respectively optimizing the grids divided on a plurality of curved surfaces, the method for dividing the building grids based on the discrete multiple curved surfaces provided by the invention has the advantages that the obtained grids are uniform in size and smooth in transition, and a small number of nodes are inevitably connected with 5 or 7 rod pieces on the curved surfaces with severe changes. In general, the invention solves the problem of grid division of complex multiple free-form surfaces with cutting or holes, and meets the requirements of building aesthetics on grid uniformity and smooth transition.

The above-described embodiments are merely preferred embodiments of the present invention, which should not be construed as limiting the invention. Various changes and modifications may be made by one of ordinary skill in the pertinent art without departing from the spirit and scope of the present invention. Therefore, the technical scheme obtained by adopting the mode of equivalent replacement or equivalent transformation is within the protection scope of the invention.

Claims

1. A multi-curved-surface building mesh division method based on discretization is characterized by comprising the following steps:

1) respectively discretizing and sewing the multiple curved surfaces to form discretization representation consisting of triangular patches;

6) carrying out topology optimization on the triangular mesh;

7) performing fairing optimization on the triangular mesh by adopting a spring mass point method;

the improved error diffusion algorithm comprises the following steps:

ρ_T＝ρ′_Tf_ρ(T)f_k(T)

Wherein f is_ρ(T) is a density correction function; f. of_k(T) is a curvature correction function;

2.3) transferring the density of the triangular patch to the node

2.4) selecting a point as a starting node and adding the point into a queue;

2.6) uniformly diffusing the density of the nodes to surrounding nodes which are not accessed, adding the nodes into a queue, and marking the node as accessed;

2.7) if the queue is empty, ending and going to step 3); otherwise, go to step 2.5);

2. The discretization-based multiple curved building meshing method of claim 1, wherein the k-means algorithm based on the curved distance comprises the following specific steps:

3. The discretization-based multi-surface building meshing method of claim 1, wherein the step 5) is specifically as follows:

4. The discretization-based multi-surface building meshing method of claim 1, wherein in the step 6), the method for topology optimization of the triangular mesh comprises: traversing each rod piece, forming a quadrangle by two surfaces on two sides of the rod piece, and if the number of the internal node connecting rods is closer to 6 and the number of the boundary node connecting rods is closer to 4 in four vertexes of the quadrangle by exchanging diagonals of the quadrangle, carrying out exchange.