Automatic Quadrilateral Mesh Research Articles

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.

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.

An algorithm for decomposition of sub-domains and quadrilateral mesh generation with line constraints

The technique for quadrilateral mesh generation on arbitrary 2-D domain has been quite mature; however, the existing methods and software can not directly deal with mesh generation with internal line constraints. In flood analysis and other analysis, the analysis models often contain a large number of constraint lines. In mesh generation, the sides of elements besides constraint lines must be attached to constraint lines, so that boundary conditions can be defined on these constraint lines. In this paper, a general method for automatic quadrilateral mesh generation with internal line constraints is presented. The mesh generation process is mainly divided into two stages, i.e. the automatic decomposition of sub-domains and quadrilateral mesh generation in each sub-domain. The sub-domains enclosed by constraint lines and boundaries are determined automatically with the proposal algorithm in this paper. Then nodes are generated on each edge of sub-domains based on mesh size requirements. By solving an integer linear programming problem, the number of nodes on each edge is adjusted automatically, so that the total number of boundary nodes in each sub-domain is even, satisfying the necessary condition for generating all-quadrilateral mesh. For free constraint lines in each sub-domain, the method of regarding constraint lines as holes with zero area is proposed, so that constraint lines can be treated as internal boundaries. The merging method of constraint lines and outer boundary is given, which can deal with all possible distribution of constraint lines and has generality. The examples of mesh generation with a large number of constraint lines are given to demonstrate the reliability of proposed method.

A robust combinatorial approach to reduce singularities in quadrilateral meshes

There are many automatic quadrilateral mesh generators that can produce a high-quality mesh with low distortion. However, they typically generate a large number of singularities that could be detrimental to downstream applications. This paper introduces Minimum Singularity Templates (MST) to reduce the number of singularities in a pure quad mesh. These templates are easy to encode with high-level grammar rules for complete automation or interactive control. The MST exploits two important properties of quadrilateral meshes: (1) every submesh has even number of quad edges on its boundary, and (2) every submesh with 3, 4 or 5 topological convex corners on its boundary can be transformed into a patch that has at most two interior singularities. The MST (1) does not change the boundary edges of the patch, (2) avoids corner picking on a patch and solving NP-hard interval matching algorithm to select divisions, (3) is extremely fast with time complexity of O(1) in template creation, and (4) has low memory footprint and is robust. To illustrate the concepts, we consider quadrilateral meshes generated using Abaqus, Gmsh, and Cubit, and reduce the singularities within these meshes.Besides improving the topological quality of an existing quad mesh, we describe two additional algorithms that exploit the above MST operations to (1) generate an ab-initio high-quality quadrilateral mesh in a complex domain, and (2) convert a quad-dominant mesh to a pure quadrilateral mesh.

A new quadtree-based approach for automatic quadrilateral mesh generation

In this paper, we present a new method for adaptive all-quadrilateral mesh generation for two-dimensional domains, including domains modeled by constraints with complex geometry or with varying scales. The method subdivides the domain's bounding box using a new extended quadtree scheme. In this subdivision process, the quadtree node corners are moved onto the geometrical constraints using local deformation criteria during the tree refinement steps. We define new subdivision patterns as part of our extended quadtree to add flexibility in the adaptation and guarantee that geometrical constraints are entirely modeled by tree edges. During the process, we ensure grid alignment with constraint accuracy and element quality at every scale. Our proposal converts the tree structure into a mesh with only quadrilateral elements. Results showed that our method generates elements of reasonable quality even for complex geometries and varying scales. The small number of parameters controlling the process is intuitive and makes our method efficient and user friendly.

Automatic quadrilateral mesh generation and quality improvement techniques for an improved combination method

This paper developed an automatic quadrilateral meshing system to resolve the numerical simulation problems in the fields of geological mechanics, hydraulics, hydrology, and water resources. An improved combination method was proposed to automatically convert Delaunay triangular meshes into quadrilateral meshes by combining adjacent pairs of triangles through searching advancing-front edges. In order to implement the full conversion, seven combination patterns were established to select appropriate pairs of adjacent triangles. To handle the residual triangles, each nearest pair of residual triangles were moved to each other along a proper path and recombined into a quadrilateral. The movement of residual triangles was implemented through the destruction and reconstruction of neighboring quadrilaterals. A transformation template was established to convert the only residual triangle in each domain into quadrilaterals. To ensure the geometric topology of the resulting mesh, nine topological optimization modes were proposed to improve the topological connections of the degenerated quadrilaterals especially those on the boundaries of the mesh. An area-weighted Laplacian method was used to smooth the interior nodes, and an objective function method was employed to adjust the positions of special nodes. Finally, practical examples of two treatment strategies for river and overland flows were provided to demonstrate the accuracy and reliability of the meshing and optimization algorithms proposed in this paper.

A Robust Combinatorial Approach to Reduce Singularities in Quadrilateral Meshes

There are many automatic quadrilateral mesh generators that can produce high quality mesh with low distortion. However, they typically generate a large number of singularities that could be detrimental to downstream applications. This paper introduces Minimum Singularity Templates (MST) to reduce the number of singularities in an existing pure quad mesh. These templates are easy to encode with high-level grammar rules for complete automation, or interactive control. The MST exploits two important properties of quadrilateral meshes: (1) every submesh has even number of quad edges on its boundary, and (2) every submesh with 3, 4 or 5 topological convex corners on its boundary has at most two interior singularities. The MST (1) does not change the boundary edges of the patch, (2) avoids corner picking on a patch and solving NP hard internal matching algorithm to select divisions, (3) is extremely fast with time complexity of O(1) in template creation, and (4) has low memory footprint and is robust. To illustrate the concepts, we consider quadrilateral meshes generated using Abaqus, Gmsh, and Cubit, and reduce the singularities within these meshes.

An Improved Paving Method of Automatic Quadrilateral Mesh Generation

This article describes an improved paving method of automatic quadrilateral mesh generation. Paving, which was first proposed by Blacker and Stephenson [1], is a kind of direct method for generating a quadrilateral mesh and has been widely used since it was presented. This article aims to improve some weaknesses of the traditional paving method by generating high-quality quadrilateral grids without employment of a background mesh. Through efficient intersection resolution and other optimization measures, the improved paving method can generate well-aligned rows of quadrilateral elements almost parallel to the boundary of the domain, automatically and quickly.

NURBS based scheme for automatic quadrilateral mesh generation for FE and BEM analysis

In this paper, a Non Uniform Rational B-Spline (NURBS) based approach is proposed to generate structured quadrilateral meshes over three dimensional surfaces. The surfaces to be meshed are defined by NURBS. The surfaces can also have a hole of arbitrary shape. Initially an adaptive subdivision scheme applied to split the surface in parametric domain to match the boundary nodes, and then mesh generation process reduces to the formation of nodes and elements in parametric domain. In order to do that, special care is taken to maintain the continuity of the splitted surfaces. The primary objective of this paper is to recognize the neighbouring surfaces and split automatically to match the boundary points. None of the presently available CAD software has this type of feature. The proposed scheme is applied initially with simply connected surface, and then it is applied for more complicated structures. Finally the proposed method is applied to a part of a ship shaped body consisting with planar and curved surfaces to demonstrate the effectiveness of the proposed algorithm.

Automated quadrilateral mesh generation for digital image structures

With the development of advanced imaging technology, digital images are widely used. This paper proposes an automatic quadrilateral mesh generation algorithm for multi-colour imaged structures. It takes an original arbitrary digital image as an input for automatic quadrilateral mesh generation, this includes removing the noise, extracting and smoothing the boundary geometries between different colours, and automatic all-quad mesh generation with the above boundaries as constraints. An application example is provided to demonstrate the usefulness and effectiveness of the proposed approach.

Structured mesh generation by kriging with local refinement with a new elliptic scheme

Accurate results in finite element analysis are strongly related to mesh quality. In this paper, an automatic quadrilateral mesh generation methodology using kriging interpolation is described, a quality mesh study is conducted, and the development of a new local refinement scheme, called the elliptic scheme, is presented. The new elliptic refinement scheme is evaluated using four standard structural cases, and it is shown that it compares very well with octree-based refinement schemes and other local refinement methods.

A development of data structure and mesh generation algorithm for whole ship analysis modeling system

In the whole ship structure and vibration analysis, the FEA (finite element analysis) model of whole ship structure is required in the early design stage before the 3D CAD model is defined. Because ship structure has a complex curved surface, and many associated structural members, the whole ship analysis modeling job has become a time consuming job. For the effective support of the whole ship analysis modeling, a method to generate the analysis model using initial design information within the ship design process, hull form offset data and compartment data, is developed. To easily handle initial design information and FE model information, a flexible data structure is proposed. An automatic quadrilateral mesh generation algorithm using initial design information to satisfy the constraints imposed by the ship structure is also proposed. With the proposed data structure and mesh generation algorithm, whole ship analysis modeling job for various ship types can be effectively supported and these results are presented.

Finite element analysis of thixoforming process with arbitrary shape die

Numerical simulations for industrial net shape forming application by the finite element method are based on the integral form of the mechanical equations governing the physical process. However, current commercial packages cannot analyze the die filling and solidification for materials in the semi-solid state because they lack material models for liquid and solid materials and do not address segregation of the liquid and solid phases. Thus, this work presents an overview of the development of a new finite element program entitled SEMI-FORM S/W to predict the solid-phase deformation of semi-solid materials (SSM), filling defects and liquid segregation. The automatic mesh generation scheme by the modified looping algorithm using quadrilateral elements is proposed in this present study by a new loop splitting method and a local finite element control. An automatic quadrilateral mesh generator with automatic mesh refinement is developed in order to meet special demands found in the thixoforming process analysis with arbitrary shaped dies by the node renumbering and modifications of the nodal position. The remeshing due to B-spline form is carried out by applying the remeshing criterion. The SEMI-FORM S/W developed in this present study would be recognized as efficient and economical tool for designing and optimizing thixoformed products with arbitrary shaped dies.

Forging process analysis with arbitrary shape die in semi-solid material with high solid fraction

Numerical simulations for industrial net shape forming application by the finite element method (FEM) are based on the integral form of the mechanical equations governing the physical process. However, current commercial packages cannot analyze the die filling and solidification for materials in the semi-solid state because they lack material models for liquid and solid materials and do not address segregation of the liquid and solid phases. Thus, this work presents an overview of the development of a new finite element program entitled SEMI-FORM S/W to predict the solid-phase deformation of semi-solid materials, filling defects and liquid segregation. The automatic mesh generation scheme by the modified looping algorithm using quadrilateral elements is proposed in this present study by a new loop splitting method and a local finite element control. An automatic quadrilateral mesh generator with automatic mesh refinement is developed in order to meet special demands found in the thixoforming process analysis with arbitrarily shaped dies by the node renumbering and modifications of the nodal position. Applying the remeshing criterion carries out the remeshing due to B-spline form. The SEMI-FORM S/W developed in this present study would be recognized as efficient and economical tool for designing and optimizing thixoformed products with arbitrarily shaped dies.

다양한 2차원 영역에서의 향상된 Paving법을 이용한 자동 사각 요소 생성

In part of mechanical design analysis, quadrilateral mesh is usually used because it provides less approximate errors than triangular mesh. Over the decades, Paving method has been considered as the most robust method among existing automatic quadrilateral element mesh generation methods. However, it also has some problems such as unpredictable node projection and relatively large element generation. In this study, the aforementioned problems are corrected by updating the Paving method. In so doing, a part of node projection process is modified by classifying nodes based on the interior angles. The closure check process is also modified by adding more nodes while generating elements. The result shows well shaped element distribution in the final mesh without any aforementioned problems.<br/>

The Construction of the Automatic Quadrilateral Mesh Generation System for the Hexahedral Mesh Generation : 1st Report, Proposal of the Algorithm

Recently the dual cycle elimination method was proposed by Muller-Hannemann for hexahedral mesh generation. This method starts with a surface quadrilateral mesh whose dual cycles have no self-intersections and, after the elimination of dual cycles, generates a hexahedral mesh while tracing back the reverse order of eliminations and supplementing hexahedrons inside the object step by step. This paper concerns with the automatic quadrilateral mesh generation system for the dual-cycle elimination method. We shall propose a simple method to construct such a surface mesh for a given polyhedron and nodes on its edges. Some kinds of templates play an important role on our method.

대변형 유한요소해석을 위한 요소망 자동 생성기법

An automatic quadrilateral mesh generator for large deformation finite element analysis such as metal forming simulation was developed. The NURBS interpolation method is used for modeling arbitrary 2-D free surface. This mesh generation technique is the modified paving algorithm, which is an advancing front technique with element-by-element resolving method for paving boundary intersection problem. The mesh density for higher analysis accuracy and less analysis time can be easily controlled with high-density points, maximum and minimum element size. A couple of application to large deformation finite element analysis is given as an example, which shows versatility and applicability of the proposed approach and the developed mesh generator for large deformation finite element analysis.

Remeshing for metal forming simulations—Part I: Two‐dimensional quadrilateral remeshing

AbstractIn this paper, a general framework of practical two‐dimensional quadrilateral remeshing, which includes the determination of remeshing time, automatic quadrilateral mesh generation, and data transfer process, will be formulated. In particular, the current work contains new algorithms of mesh density specification according to the distribution of effective strain‐rate gradients, mesh density smoothing by fast Fourier transform (FFT) and low‐pass filtering techniques, coarsening it by node placement scheme, and a modified Laplacian mesh smoothing technique. The efficiency of the developed remeshing scheme was tested through three practical two‐dimensional metal forming simulations. The results clearly indicate that the algorithms proposed in this study make it possible to simulate two‐dimensional metal forming problems efficiently and automatically. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Automatic quadrilateral mesh generation for FEM using dynamic bubble system

In this paper, a method for automatic quadrilateral (Quad) mesh generation using dynamic bubble system is presented. The proposed method can be separated into three separate computational processes: the first one generates nodes inside the analysis domain using physically-based system of bubbles, the second one for automatic generation of Quad finite elements using previously generated set of nodes, and the last process where the generated elements are smoothed using Laplacian smoothing operator. The proposed method is suitable and easily extendable for automatic meshing of a complex 3-D domains.

Efficient unstructured quadrilateral mesh generation

This work is devoted to the description of an algorithm for automatic quadrilateral mesh generation. The technique is based on a recursive decomposition of the domain into quadrilateral elements. This automatically generates meshes composed entirely by quadrilaterals over complex geometries (there is no need for a previous step where triangles are generated). A background mesh with the desired element sizes allows to obtain the preferred sizes anywhere in the domain. The final mesh can be viewed as the optimal one given the objective function is defined. The recursive algorithm induces an efficient data structure which optimizes the computer cost. Several examples are presented to show the efficiency of this algorithm. Copyright © 2000 John Wiley & Sons, Ltd.