Quadrilateral Mesh Generation Research Articles

Template mapping method for plane quadrilateral mesh generation with controllable density transitions

Template mapping method for plane quadrilateral mesh generation with controllable density transitions

Quadrilateral surface mesh generation with improved quality by combination of triangles

AbstractQuadrilateral mesh is preferred in numerical simulation compared with triangular mesh, but high‐quality and robust quadrilateral mesh generation for complex geometries remains a challenging problem. In this study, an enhanced indirect method by the combination of triangles is proposed for the generation of high‐quality quadrilateral mesh. It takes a triangular mesh as the input, and except that the number of elements needs to be even, the triangular mesh can be generated by the commonly used AFT method or Delaunay method. To ensure the quality of the combined quads, a local‐global triangular remeshing procedure is conducted first, and size gradation and vertex valence are focused in this step. Then, the well‐known Blossom algorithm is used to find the perfect matching triangles for combination. Finally, in order to reduce the number of unnecessary singularities, a connectivity optimization procedure is introduced to replace the irregular elements in local regions with regular elements and thus improve the overall quality of the combined quads. Numerical experiments on meshes in both 2D and 3D cases are presented to demonstrate the effectiveness of the proposed method.

Feature-preserving quadrilateral mesh Boolean operation with cross-field guided layout blending

Compared to triangular meshes, high-quality quadrilateral meshes offer significant advantages in the field of simulation. However, generating high-quality quadrilateral meshes has always been a challenging task. By synthesizing high-quality quadrilateral meshes based on existing ones through Boolean operations such as mesh intersection, union, and difference, the automation level of quadrilateral mesh modeling can be improved. This significantly reduces modeling time. We propose a feature-preserving quadrilateral mesh Boolean operation method that can generate high-quality all-quadrilateral meshes through Boolean operations while preserving the geometric features and shape of the original mesh. Our method, guided by cross-field techniques, aligns mesh faces with geometric features of the model and maximally preserves the original mesh's geometric shape and layout. Compared to traditional quadrilateral mesh generation methods, our approach demonstrates higher efficiency, offering a substantial improvement to the pipeline of mesh-based modeling tools.

Boundary constrained quadrilateral mesh generation based on domain decomposition and templates

Compared to triangular or mixed meshes, pure quadrilateral meshes generally offer higher accuracy and computational efficiency in numerical simulations. Recently, the main obstacles in the generation of quadrilateral meshes have revolved around the identification of mesh singularities and the subsequent decomposition of the domain. In this study, a novel quadrilateral mesh generation method based on domain decomposition and templates is presented. The basic idea is to decompose the domain by separating the mesh singularities rather than identifying their precise locations. Surfaces composed of a set of surface patches are meshed independently. For each surface patch, the problem domain is divided into several simple polygonal subregions, including quadrangles, triangles, and pentagons, using constrained Delaunay triangulation (CDT) and recursive bisections. Subsequently, mesh templates with a limited number of singularities are employed to rapidly generate high-quality quadrilateral meshes over those subregions. By constraining the boundary discretization of each surface patch, mesh conformability is automatically achieved. An integer linear programming equation is presented to ensure that the total number of boundary nodes on each surface is an even value. Several experimental results are presented and discussed, demonstrating the reliability and efficiency of the proposed method.

Quadrilateral Mesh Generation Method Based on Convolutional Neural Network

The frame field distributed inside the model region characterizes the singular structure features inside the model. These singular structures can be used to decompose the model region into multiple quadrilateral structures, thereby generating a block-structured quadrilateral mesh. For the generation of block-structured quadrilateral mesh for two-dimensional geometric models, a convolutional neural network model is proposed to identify the singular structure inside the model contained in the frame field. By training the network model with a large number of model region decomposition data obtained in advance, the model can identify the vectors of the frame field in the region located in the segmentation field. Then, the segmentation streamline is constructed from the annotation. Based on this, the geometric region is decomposed into several small regions, regions which are then discretized with quadrilateral mesh elements. Finally, through two geometric models, it is verified that the convolutional neural network model proposed in this study can effectively identify the singular structure inside the model to realize the model region decomposition and block-structured mesh generation.

Feature Preserving Parameterization for Quadrilateral Mesh Generation Based on Ricci Flow and Cross Field

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.

Reinforcement learning for automatic quadrilateral mesh generation: A soft actor–critic approach

This paper proposes, implements, and evaluates a reinforcement learning (RL)-based computational framework for automatic mesh generation. Mesh generation plays a fundamental role in numerical simulations in the area of computer aided design and engineering (CAD/E). It is identified as one of the critical issues in the NASA CFD Vision 2030 Study. Existing mesh generation methods suffer from high computational complexity, low mesh quality in complex geometries, and speed limitations. These methods and tools, including commercial software packages, are typically semiautomatic and they need inputs or help from human experts. By formulating the mesh generation as a Markov decision process (MDP) problem, we are able to use a state-of-the-art reinforcement learning (RL) algorithm called “soft actor-critic” to automatically learn from trials the policy of actions for mesh generation. The implementation of this RL algorithm for mesh generation allows us to build a fully automatic mesh generation system without human intervention and any extra clean-up operations, which fills the gap in the existing mesh generation tools. In the experiments to compare with two representative commercial software packages, our system demonstrates promising performance with respect to scalability, generalizability, and effectiveness.

Indirect all-quadrilateral meshing based on bipartite topological labeling

Quadrilateral meshes offer certain advantages compared to triangular ones, such as reduced number of elements and alignment with problem-specific directions. We present a pipeline for the generation of quadrilateral meshes on complex geometries. It is based on two key components: robust surface meshing and efficient indirect conversion of a triangular mesh to an all-quad one. The input is a valid geometric surface mesh, i.e., a triangulation that accurately represents the geometry of the model. A right-angled triangular surface mesh is initially created by continuously modifying the input mesh while always preserving its topological validity. The main advantages of our local mesh modification-based approach are to (i) allow the generation of meshes that are globally aligned with a given direction field and (ii) to reliably handle non-manifold feature edges (in multi-volume models) and small features. The final quadrilateral mesh is obtained by merging pairs of triangles into quadrilaterals. We develop a novel bipartite labeling scheme in order to identify and correct inconsistent pairings. The procedure is based on local operations and is much more efficient than previous global strategies for tri-to-quad conversion. The whole pipeline is tested on a large number of models with diverse characteristics.

Quadrilateral mesh generation III : Optimizing singularity configuration based on Abel–Jacobi theory

This work proposes a rigorous and practical algorithm for quad-mesh generation based the Abel–Jacobi theory of algebraic curves. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quad-mesh, in terms of the deck-transformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a T-mesh; second, the edge lengths of the T-mesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quad-mesh.In the first stage, the algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the Abel–Jacobi map for a given divisor; optimize the divisor to satisfy the Abel–Jacobi condition by integer programming; compute a flat Riemannian metric with cone singularities at the divisor by Ricci flow; isometrically immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct a motorcycle graph to generate a T-Mesh.In the second stage, the deck transformation constraints are formulated as a linear equation system of the edge lengths of the T-mesh. The solution provides a flat metric with integral deck transformations, which leads to the final quad-mesh.The proposed method is rigorous and practical. The T-mesh and quad-mesh results can be applied for constructing Splines directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results on surfaces with complicated topologies and geometries.

Combinatorial Construction of Seamless Parameter Domains

AbstractThe problem of seamless parametrization of surfaces is of interest in the context of structured quadrilateral mesh generation and spline‐based surface approximation. It has been tackled by a variety of approaches, commonly relying on continuous numerical optimization to ultimately obtain suitable parameter domains. We present a general combinatorial seamless parameter domain construction, free from the potential numerical issues inherent to continuous optimization techniques in practice. The domains are constructed as abstract polygonal complexes which can be embedded in a discrete planar grid space, as unions of unit squares. We ensure that the domain structure matches any prescribed parametrization singularities (cones) and satisfies seamlessness conditions. Surfaces of arbitrary genus are supported. Once a domain suitable for a given surface is constructed, a seamless and locally injective parametrization over this domain can be obtained using existing planar disk mapping techniques, making recourse to Tutte's classical embedding theorem.

Voxel-based quadrilateral mesh generation from point cloud

In recent years point cloud has been used more and more widely in CAD and computer graphics communities due to the availability of fast and accurate laser scan devices. Many works focus on reconstructing a triangle mesh from a point cloud but few consider the quadrangulate of a point cloud directly. In this paper, a novel method which is able to achieve quad mesh models directly from a point cloud is proposed. The main idea is to use voxel as the intermediate medium between point cloud and grid to generate voxel model, then by mapping vertices of a voxel model outer surface to points to construct quad meshes. Furthermore, quad meshes are refinement through point remapping and meshes regularization to obtain high quality meshes. The experimental results show that our method can efficiently and flexible generate quad mesh based on point cloud directly with better quality and different scales. Especially, this method is much more suitable to tasks expecting less computation time and less manual interaction in rapid prototyping and reverse engineering.

Quadrilateral mesh generation II: Meromorphic quartic differentials and Abel–Jacobi condition

This work discovers the equivalence relation between quadrilateral meshes and meromorphic quartic differentials. Each quad-mesh induces a conformal structure of the surface, and a meromorphic quartic differential, where the configuration of singular vertices corresponds to the configurations of the poles and zeros (divisor) of the meromorphic differential. Due to Riemann surface theory, the configuration of singularities of a quad-mesh satisfies the Abel–Jacobi condition. Inversely, if a divisor satisfies the Abel–Jacobi condition, then there exists a meromorphic quartic differential whose divisor equals the given one. Furthermore, if the meromorphic quartic differential is with finite trajectories, then it also induces a quad-mesh, the poles and zeros of the meromorphic differential correspond to the singular vertices of the quad-mesh.Besides the theoretic proofs, the computational algorithm for verification of Abel–Jacobi condition is also explained in detail. Furthermore, constructive algorithm of meromorphic quartic differential on genus zero surfaces is proposed, which is based on the global algebraic representation of meromorphic differentials.Our experimental results demonstrate the efficiency and efficacy of the algorithm. This opens up a novel direction for quad-mesh generation using algebraic geometric approach.

An automatic approach to constrained quadrilateral mesh generation

Purpose Arbitrary constraints might be included into the problem domain in many engineering applications, which represent specific features such as multi-domain interfaces, cracks with small yield stresses, stiffeners attached on the plate for reinforcement and so on. To imprint these constraints into the final mesh, additional techniques need to be developed to treat these constraints properly. Design/methodology/approach This paper proposes an automatic approach to generate quadrilateral meshes for the geometric models with complex feature constraints. Firstly, the region is decomposed into sub-regions by the constraints, and then the quadrilateral mesh is generated in each sub-region that satisfies the constraints. A method that deals with constraint lines and points is presented. A distribution function is proposed to represent the distribution of mesh size over the region by using the Laplace equation. The density lines and points can be specified inside the region and reasonable mesh size distribution can be obtained by solving the Laplace equation. Findings An automatic method to define sub-regions is presented, and the user interaction can be avoided. An algorithm for constructing loops from constraint lines is proposed, which can deal with the randomly distributed constraint lines in a general way. A method is developed to deal with constraint points and quality elements can be generated around constraint points. A function defining the distribution of mesh size is put forward. The examples of constrained quadrilateral mesh generation in actual engineering analysis are presented to show the performance of the approach. Originality/value An automatic approach to constrained quadrilateral mesh generation is presented in this paper. It can generate required quality meshes for special applications with complex internal feature constraints.

Mesh generation techniques for numerical integration of arbitrary function over polygonal domain by finite element method

In this paper, a new approach is introduced for approximating two dimensional surface integral proble ms numerically over a convex, non convex reg ions such integrals are typically occur in boundary element method (BEM), the approach given here is domain discretized method by influence of quadrilateral mesh generation technique, then apply Gauss Legendre quadrature rule to generate Gaussian points over convex, nonconvex region. The performances of this method are illustrated with numerical examp les.

Development of a multiblock procedure for automated generation of two-dimensional quadrilateral meshes of gear drives

This article describes a new multiblock procedure for automated generation of two-dimensional quadrilateral meshes of gear drives. The typical steps of the multiblock schemes have been investigated in depth to obtain a fast and simple way to mesh planar sections of gear teeth, allowing local mesh refinement and minimizing the appearance of distorted elements in the mesh.The proposed procedure is completed with two different mesh quality enhancement techniques. One of them is applied before the mesh is generated, and reduces the distortion of the mesh without increasing the computational time of the meshing process. The other one is applied once the mesh is generated, and reduces the distortion of the elements by means of a mesh smoothing method.The performance of the proposed procedure has been illustrated with several numerical examples, which demonstrate its ability to mesh different gear geometries under several meshing boundary conditions.

A high resolution PDE approach to quadrilateral mesh generation

We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute crosses. Instead, we compute a high order accurate guiding field using a continuous Galerkin (CG) or discontinuous Galerkin (DG) spectral element method to solve a Laplace equation for each of the field variables using the open source code Nektar++. The spectral method provides spectral convergence and sub-element resolution of the fields. The DG approximation allows meshing of corners that are not multiples of π/2 in a discretization consistent manner, when needed. The high order field can then be exploited to accurately find irregular nodes, and can be accurately integrated using a high order separatrix integration method to avoid features like limit cycles. The result is a mesh with naturally curved quadrilateral elements that do not need to be curved a posteriori to eliminate invalid elements. The mesh generation procedure is implemented in the open source mesh generation program NekMesh.

Quadrilateral mesh generation I : Metric based method

This work proposes a novel metric based algorithm for quadrilateral mesh generating. Each quad-mesh induces a Riemannian metric satisfying special conditions: the metric is a flat metric with cone singularities conformal to the original metric, the total curvature satisfies the Gauss–Bonnet condition, the holonomy group is a subgroup of the rotation group {eikπ∕2}, there is cross field obtained by parallel translation which is aligned with the boundaries, and its streamlines are finite geodesics. Inversely, such kind of metric induces a quad-mesh. Based on discrete Ricci flow and conformal structure deformation, one can obtain a metric satisfying all the conditions and obtain the desired quad-mesh.This method is rigorous, simple and automatic. Our experimental results demonstrate the efficiency and efficacy of the algorithm.

Rapid and Adaptive Quadrilateral Mesh Generation Algorithm from Dense Point Cloud

Rapid and Adaptive Quadrilateral Mesh Generation Algorithm from Dense Point Cloud

Generation of multi-axis swept mesh in a global way

Multi-axis sweeping is an important tool to generate hexahedral meshes for solid models which are composed of swept volumes with different sweep directions. However, traditional multi-axis sweeping algorithms either fail to handle complex grafting relationships between swept volumes or are easy to produce hexahedral elements with bad quality around the graft surfaces. To achieve a high-quality multi-axis swept mesh, this paper proposes a global approach to multi-axis swept mesh generation, which can robustly generate hexahedral meshes for solid models composed by swept volumes with different sweep directions. We first generate all surface meshes globally by applying an optimized structured quadrilateral mesh generation algorithm. After that, we generate a swept mesh for each swept volume. Finally, we determine an appropriate way to optimize the topology of the generated mesh so as to improve the mesh quality. The experimental results show the effectiveness and efficiency of the proposed method.

An automatic quadrilateral mesh generation algorithm applied to 2-D overland flow simulations

This paper presents an automatic quadrilateral mesh generation algorithm to provide 2-D meshes for simulating overland flow and transport problems in watershed systems. The conformal mapping method is studied deeply and specific treatment strategies are established according to the width of rivers and the area of other water storage zones. A conformity treatment method is proposed to deal with the incompatibility problems during discretizing the complex geometry with a plurality of sub-domains. This method ensures grid conformity through removing certain overlapped nodes and creating new appropriate nodes on common boundaries of the multi-domain mesh. Aiming at the undesired deviation of mesh boundaries from geometric contours, a relative position-percent interpolation method is proposed and a supplementary match method related to dead ends is presented. And, the double effects of projection and smooth for boundary nodes are achieved. An objective function method is proposed to locally optimize the degenerated quadrilaterals located on concave or large-curvature curves. In order to accommodate the need for overland simulations, the 1-D/2-D correspondence on river reaches, junctions with and without storage, ponds, lakes, dead ends and control structures is established. Finally, practical applications are provided to demonstrate the accuracy and reliability of the quadrilateral meshing algorithms proposed in this paper.