We propose a new method to generate surface quadrilateral mesh by calculating a globally defined parameterization with feature constraints. In the field of quadrilateral generation with features, the cross field methods are well-known because of their superior performance in feature preservation. The methods based on metrics are popular due to their sound theoretical basis, especially the Ricci flow algorithm. The cross field methods’ major part, the Poisson equation, is challenging to solve in three dimensions directly. When it comes to cases with a large number of elements, the computational costs are expensive while the methods based on metrics are on the contrary. In addition, an appropriate initial value plays a positive role in the solution of the Poisson equation, and this initial value can be obtained from the Ricci flow algorithm. So we combine the methods based on metric with the cross field methods. We use the discrete dynamic Ricci flow algorithm to generate an initial value for the Poisson equation, which speeds up the solution of the equation and ensures the convergence of the computation. Numerical experiments show that our method is effective in generating a quadrilateral mesh for models with features, and the quality of the quadrilateral mesh is reliable.

The finite element method has become one of the most effective numerical analysis methods in computer graphics and industrial fields. Geometric models need to be described in the computer first so as to further modeling, analysis and simulation. The main task of the finite element method is to construct a discrete finite element mesh that accurately describes the geometric shape of the 3D model. The quality of the finite element mesh will affect the results of numerical calculation directly, so constructing a high-quality discrete finite element mesh is an important part. There are many methods to represent surfaces in the computer field, such as spline function, implicit surface, point cloud and polygon mesh. The most commonly used meshes are triangle meshes. Although triangle meshes are simple and flexible, the quadrilateral mesh elements can naturally align with the feature edges and the geometric features, such as the principal curvature direction and feature lines on surfaces. In addition, quadrilateral meshes perform well in computational accuracy and efficiency. Therefore, it has become more and more necessary to study quadrilateral mesh generation algorithms. At the same time, with the rapid development of CAE, the demands for modeling industrial models are rapidly rising, which requires higher feature preservation. A robust feature-preserving quadrilateral mesh generation algorithm is expected gradually.

Singularities are significant in quadrilateral mesh generation. In short, in quadrilateral meshes, if the number of faces around an inner vertex is not equal to 4 or the number of faces around a boundary vertex is not equal to 2, such vertices are called singular points. The number and position of singular points are crucial in quadrilateral mesh generation. But the existence of singular points is almost inevitable for complex models, especially where geometric characteristics are involved. Singularities are also related to the models’ genus and boundary number. This shows that the quadrilateral mesh generation still faces many challenges at present. Because of the important role quadrilateral meshes play in industrial software, researches on quadrilateral mesh generation are very popular at home and abroad. Many methods have been proposed for generating quadrilateral meshes, mainly including the methods based on metric [

The Q-Morph method is stable and widely used, but its efficiency depends on the quality of the triangular mesh input. In order to generate a full quadrilateral mesh, a few elements of poor quality may be generated. The block method first computes an initial skeleton map based on the triangular mesh input. This map can roughly divide the mesh into several topological quadrilateral blocks. There are also methods [

Cross field based method has been widely studied recently. The methods based on field theory have been proposed successively to characterize the geometric features with field distribution. Each approach should choose a way to represent a cross first, for example, N-RoSy representation [

Metric based method is also a popular method. This method uses the metric of mesh to establish a one-to-one correspondence between the points on the surface and the points on the parameter domain. The essence is to calculate an immersion of manifold on the parameter domain. Quadrilateral mesh is generated on the parameter domain first, and then reflected back to the physical space of surface [

However, methods based on metric usually perform badly on feature preserving. The exact singular points information is also needed to ensure the smallest area deformation of the obtained quadrilateral meshes. Cross field methods’ outstanding performance at feature preserving inspires us to combine the metric method with the cross field method, using the Ricci flow to reduce computation cost by mesh parameterization and provide a reliable, stable and convergent initial value for the Poisson equation. Because of its orthogonality, the cross field is consistent with the structured quadrilateral mesh. Singularities are also naturally included in the global cross field information. Compared with other methods that require additional representation, this is advantageous for subsequent calculation and representation. A large number of previous studies on cross field methods have made it easier to obtain the cross field which fits the characteristics and boundary constraints. We use the energy optimization method to obtain the required cross field. Since the study of the Ricci flow method, there have been a complete theoretical system and calculation methods about how to obtain the preliminary metric. Once the Ricci flow method is used to design metric, its convergence is guaranteed by Hamilton and Chow’s theorems.

The work is arranged as follows:

Ricci flow was proposed by Hamilton originally to solve low dimensional topology problems, such as poincaré conjecture and Thurston’s conjecture. Its specific form is

Hamilton first proved that Ricci flow could quickly converge to a constant curvature measure on a 3-manifold which has an initial measure with good curvature conditions. Observing its form, it is not difficult to see that Ricci flow continuously updates surfaces’ measure according to its Gaussian curvature. In fact, it is a process of heat flow diffusion and finally stabilized in a normal state, that is, a constant curvature.

In the following discussion, we use

We know that a low dimensional parameterization can be achieved by setting the target curvature to a constant in Ricci flow algorithm. For points that are not singular points, set their target curvature as

We could obtain the canonical homology group bases which form a cut graph

The existence of singular points is almost inevitable, especially in complex models. We explain the influence of singular points in the combined method and how to deal with singular points from the perspective of quadrilateral mesh directly. Suppose

For a quadrilateral mesh

In practice, we are more interested in the loop case, parallel transporting a tangent vector at

Because

Cross field is an important tool in various manifold operations, such as finite element subdivision, remeshing or texture mapping. A 4-symmetry direction field is defined on the tangent plane of each vertex of the surface, which is invariant by rotation of

Each face on a quadrilateral mesh is close to a square. As posted before, the internal points’ topology degree is four, which is adjacent to four faces, the topological degree of a singular point is not equal to four. Select any loop composed of faces on the quadrilateral mesh embracing a singular point, put a unit frame on the first face, the coordinate axis is parallel to the two opposite sides of the quadrilateral, and then move it to the next face in parallel. In this way, it moves parallel along the circuit until it returns to the original face. There is a rotation operation between the final frame and the initial frame, and the rotation angle is an integral multiple of

The next step is computing the potential function with singular information. The final parameterization should align with the given input field as well as possible, i.e., the parameterization

Suppose the input surface S is discretized as a triangular mesh.

1. Calculate initial parameterization with discrete dynamic surface Ricci flow algorithm and obtain a flat metric

2. Compute a cut graph

3. Isometrically immerse

4. Feature and boundary structure deformation. Given corresponding restrictions for feature edges and boundary edges in the Poisson equation. Such that each pair of dual boundary segments of

5. Trace the critical graph to construct a motorcycle graph, and then the motorcycle graph induces a quadrilateral partition of surface where T-junctions are allowed. From which the final quad mesh could be generated.

In the following, we explain each step in detail.

As mentioned above, Ricci flow is a process of metric changing under a conformal map. For each vertex

After introducing some necessary background knowledge, We can lead to the discrete Ricci energy. For a triangle

Then the discrete Ricci energy for the whole mesh is

In actual operation, the Hessian matrix is calculated with the cotangent edge weight, assume

Under the condition of regularization,

After this step, an initial metric

Suppose triangle mesh

We can isometrically immerse

Like most standard cross field methods, we compute the final position distribution of points by solving a Poisson equation with constraints. Poisson equation is a partial differential equation commonly used in mathematics, electrostatics, mechanical engineering and theoretical physics. It is a non-homogeneous Laplace equation. The meaning of the equation is that the liquid flow passing through any closed surface is equal to the total amount of liquid produced by the fluid source contained in the surface. According to this character, the equation to be solved can be written as

This energy should be minimized.

So the final equation we have to solve can be written as:

The solution defines a parameterization on which all the feature conditions are satisfied. It is worth noting that after the parameterization of Ricci flow algorithm, the solution space of the Poisson equation is

In order to generate a seamless quadrilateral mesh, we also need to set the boundary holonomy conditions. Since we have set the target curvature at singular points in the Ricci flow algorithm, the holonomy conditions have been satisfied near the singular points. In fact, we have almost obtained an initial metric that globally satisfies the holonomy conditions. So we only need to set the boundary conditions and adjust the metric slightly to achieve our goal.

Let

We can modify the metric of mesh to achieve the holonomy conditions, such that all pairs of edges are vertical or vertical. Setting the boundary conditions, then we could calculate the coordinates of the interior vertices of

A seamless parameterization globally defined gives a quartic differential on a surface [

In this section, we give some examples and results. All the experiments were conducted on a server with 2.10 GHz Intel(R) Xeon(R) Silver 4110 CPU, 128G RAM, GeForce RTX 2080Ti with 11G RAM and 64-bit Ubuntu operating system. The running time is reported in

Twist | 7264 | 5166 | 24.815 | 5.729 | 5.437 |

Ring | 4622 | 2465 | 17.781 | 4.450 | 3.891 |

Elbow | 7068 | 9236 | 26.166 | 22.314 | 3.226 |

Hemisphere | 6730 | 5884 | 27.157 | 23.809 | 6.021 |

In

Compared to other field methods, our method performs well in computational efficiency because the Ricci flow algorithm provides a pretty good initial value for Poisson equation and greatly reduces the calculation cost through parameterization. With this initial value, there is little possibility of divergence during the optimization process. The Ricci flow algorithm reduces the dimension of computation, making our method more friendly to large models. In addition, due to the conformal property of the Ricci flow algorithm, our algorithm performs well in mesh quality.

For the models with very high genus, the Ricci flow algorithm still can be calculated, but in this case, surfaces are embedded into the Poincaré disk. The metric near the disk boundary is very small and its tiny value will always cause decisive errors in the 2D field mapping calculation, even if the embedding is translated to the middle area of the Poincaré disk by M

In this work, a new quadrilateral mesh generation algorithm is proposed based on the methods of metric and cross field. After the process of the discrete Ricci flow algorithm, a preliminary metric satisfying singularity condition is obtained, which provides a stable, convergence initial value and a low dimensional solution space for the Poisson equation. With the constraints of specific features, we solve the Poisson equation to attain the final metric. The existence and uniqueness of the solution can be ensured by the prior result of the Ricci flow algorithm. Our experimental results demonstrate the efficiency and effectiveness of the algorithm. Further work should focus on the combination of the Ricci flow and cross field methods for hexahedral mesh generation.

This work is supported by

The authors declare that they have no conflicts of interest to report regarding the present study.