基于栅格法的三维六面体网格自适应生成算法及优化技术研究
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摘要
有限元法是计算科学和工程问题的通用数值分析方法,是计算机辅助设计的重要组成部分,其基本思想是离散化和分片插值,即用网格单元来描述所分析物体的空间。采用有限元法可以模拟各种复杂的材料结构、荷载关系和边界条件,因此,该方法广泛地应用于金属成形、机械、建筑、岩土工程、流体力学、生物医学工程、快速成形与制造、计算机图形学等领域。对于三维问题,通常采用四面体、六面体或二者混合网格。三维六面体网格由于在计算精度、生成网格数量、单元抗畸变程度和再划分次数等方面比四面体网格具有明显的优势,近几年来得到了广泛的应用。但由于六面体网格自适应生成技术自身的复杂性,仍然有很多问题未能得到解决。研究任意三维空间内六面体网格的自动生成算法,建立可靠高效的三维模型离散软件平台,具有重要的理论意义和工程应用价值。
在各种常用的三维六面体网格自动生成方法中,基于栅格法具有高度的自动化,易于网格局部加密,比较适合六面体网格的自动生成。所以,本文以基于栅格法作为六面体网格自动生成的基本算法,根据基于栅格法的研究现状、存在问题以及发展方向,开展三维六面体网格自适应生成算法的深入研究,以增强基于栅格法的稳定性、可靠性和强壮性。本文研究了基于实体模型STL文件的几何特征识别关键技术；建立了一套较完善的加密模板；提出了基于实体模型几何特征建立加密信息场的基本判据；研究了基于栅格法的边界拟合技术,提出了优先点和相对位置关系相结合的边界拟合方法；深入研究了网格单元的质量优化技术,包括拓扑优化技术和节点平滑技术；建立了相对完善的局部加密技术和自主控制的多次加密技术。
本文采用由CAD造型软件生成的STL文件来反映三维实体模型的表面几何信息,建立了具有拓扑关系的STL数据文件,避免了冗余数据的重复出现,提高了计算效率；建立了基于STL三角面片曲率变化的几何特征识别的关键技术,给出了特征边、特征点、子表面、边界边的识别方法以及特征边和边界边的凹凸属性和曲直属性的判断方法,并给出了特征点属性的识别方法；为了实现加密区域和非加密区域网格单元的协调过渡,本文在Ito模板的基础上建立了一套较为完善的六面体网格加密模板,改进了两个拐角模板(相邻面和相邻边加密模板),提高了子单元的质量和尺寸均匀性,增加了孤立点加密模板,针对实体模型上固有的孤立点和局部加密中选择的孤立点进行加密。拐角模板和孤立点模板的加入避免了加密信息场的蔓延,实现了网格加密区域的可控制性。根据实体模型的表面曲率特征、局部厚度特征和小特征等,建立了基于实体模型表面几何特征的自适应加密判据。为了同时考虑到实体模型的曲率变化和STL三角面片密度分布的影响,提出了相对曲率的概念,根据相对曲率建立了加密源点信息场和加密单元信息场,并通过实例证明了相对曲率判据可以获得更合理的网格密度分布和更好的加密效果；根据实体模型小厚度区域网格单元的属性特点和分布规律,提出了新的基于实体模型局部厚度的网格加密判据,并提出了部分骑边单元的补充加密判据,实现了加密区域的准确判断和合理识别。
深入研究了基于栅格法六面体网格自动生成的边界拟合技术,尤其是相对位置关系法的基本算法和关键技术,把优先点引入到相对位置关系法中,提出了一种优先点与相对位置关系相结合的边界拟合方法。总结归纳并给出了边界拟合的几何要求和拓扑连接要求；建立了八种网格表面自由四边形面片类型和五种补充面片类型,并给出了相应的拟合规则；提出了优先点的识别方法,包括边界拟合前第一、二级优先点和部分第三级优先点的识别方法以及边界拟合过程中优先点的逐级更新方法；建立了优先点与相对位置关系结合方法的基本算法和实现步骤；提出了一种基于几何拓扑关系和优先点的特征点拟合方法；提出了针对少数不合理拟合节点的修正方法和特殊子表面的处理方法,实现了特征边的有效拟合；针对凹曲边的拟合问题,提出了一种对不合理退化单元的表面节点进行拟合属性调整的方法,将同一条凹曲边上退化面片的指向调整为一致,有效解决了凹曲边的拟合与优化问题。实例证明本文提出的优先点与相对位置关系相结合的边界拟合方法以及凹曲边问题的处理方法能够实现网格模型表面与所分析实体模型表面的精确拟合,而且能够使网格边界单元和拟合节点的几何条件与拓扑连接关系符合边界拟合的基本要求,从而显著提高了所生成网格的精度。
网格的质量直接影响数值分析的精度和效率,为了提高网格的质量,本文对三维六面体网格的质量优化技术进行了深入研究。建立了六面体单元最小雅可比矩阵行列式值的网格质量评价准则,将六面体网格的质量进行量化,使网格质量的检测更加简便和精确。研究了从内向外栅格法所生成的六面体网格边界单元之间的拓扑连接关系,分析了边界网格几何形状和拓扑连接问题出现的原因和拓扑优化的必要性。提出了插入新单元技术、单元退化技术和插入与退化相结合的技术,显著改善了表面网格边界单元之间的拓扑连接关系。在现有拓扑优化模式的基础上,提出了五种新的插入模式、四种新的退化模式和三种新的插入和退化相结合的混合模式,并用图表的形式对采用的十一种插入模式、七种退化模式和六种混合模式进行了归纳总结,给出了每个模式的几何特点、类型以及应用时单元数量的变化情况。建立了各种模式尤其是混合模式的协调应用规则以及相容性问题的处理方法。研究了拉普拉斯平滑技术,针对传统的拉普拉斯平滑算法往往引起实体模型体积变化的问题,提出了相应的改进方法。采用基于曲率拉普拉斯平滑算法对边界拟合点进行平滑,控制了边界拟合点相对于实体模型边界的偏离；提出了一种基于映射的拉普拉斯平滑算法对一般表面节点进行平滑,解决了传统的拉普拉斯平滑算法容易出现的几何形状失真和体积误差问题；提出了一种基于节点和面积相结合的拉普拉斯平滑方法对内部节点进行平滑,以保证内部节点和表面节点平滑的步调一致,从而提高了表面单元的质量。
对网格局部加密技术进行了深入研究,包括针对节点、单元、单元面、单元边、网格边界边、网格表面和局部区域等的加密。总结了各种局部加密技术应用的原因和必要性；对于节点加密技术,给出了孤立点的判断依据和加密方法；对于网格边界边和网格表面的局部加密,需要首先识别出网格的边界边和网格表面,为此,本文给出了识别网格边界边和网格表面的基本方法和详细步骤；提出了多次加密技术,对实体网格的加密次数进行自主控制,使最终网格能够更精确地描述实体模型的几何特征和物理特征,进一步提高了有限元分析的精度。
在对上述基于栅格法六面体网格自动生成算法和关键技术的研究基础上,对本课题组前期开发的三维六面体网格自动生成软件AUTOMESH-3D进行了改进和完善,进一步提高了其可靠性、稳定性和通用性,为科学与工程问题的研究提供了建立三维六面体网格模型的通用平台。介绍了该软件的模块结构及功能特点,并通过大量的六面体网格自动生成实例验证了所开发软件的精度和可靠性。
Finite element method is a general-purpose method used for numerical analysis of computational science and engineering problems. It is the important part of computer-aided design. The basic idea of finite element method is discretization and slice interpolation. That is, using mesh elements to describe the space of analyzed object. Finite element method can simulate various types of complex material structures, load relationships and boundary conditions. It is widely used in metal forming, mechanics, architecture, geotechnical engineering, solid and fluid mechanics, biomedical engineering, rapid prototyping and manufacture, computer graphics and other fields. For three-dimensional issues, tetrahedra, hexahedra and a combination of them are usually used. Three-dimensional hexahedral element mesh expresses much more advantages than tetrahedral element mesh in computational accuracy, the element count, mesh distortion and re-meshing number, etc., thus receiving significant attentions. However, due to the complexity of adaptive hexahedral mesh generation itself, there remain a lot of problems yet to be further resolved. Therefore, it has important theoretical significance and engineering application values to study automatic hexahedral element meshing algorithms in arbitrary space and establish reliable and efficient discretion software platform for three-dimensional models.
In this dissertation, grid-based method was used as the basic algorithm for automatic creation of hexahedral element meshes. According to the research situations, existent problems and development trends of grid-based method, we carried out the deep research on adaptive generation algorithm of three-dimensional hexahedral mesh so as to enhance the stability, reliability and robustness of the method. The key technique of geometric feature identification based on the STL files of solid models was studied. A relative perfect set of refinement templates was built. The main criteria for establishing refinement fields were proposed on the basis of the geometric characteristics of the solid model. The characteristic edge (C-edge) match technique was studied deeply. A new method combined priority nodes with relative position relationships was proposed for C-edge match. This dissertation also made an intensive study of quality improvement techniques, including two aspects:topological optimization and node position smoothing. In addition, a relative complete technique of local refinement was put forward, and the refinement level could be self-controlled arbitrarily.
The STL files generated using CAD modeling software were used to reflect the surface geometry information of three-dimensional solid models. We established the STL files containing topological connections to avoid the repeated appearance of redundant data and further improve the computing efficiency. The essential technique of geometric feature identification was set up based on the curvature change of adjacent STL triangle facets. The identification methods of C-edges, characteristic points (C-points), sub-surfaces and boundary lines were given. The judgement methods of concave-convex and curve-straight attributes of C-edges and boundary lines, as well as the determination methods of the attributes of C-points, were also proposed. In order to achieve the conformal transition of the elements between refined regions and non-refined regions, this dissertation proposed a relatively perfect set of refinement templates on the basis of the research of Ito. The templates for two-face refinement and two-edge refinement were modified for the quality improvement and size uniformity of subdivided elements. The isolated node refinement template was added to refine the isolated nodes existing on the surface of the solid model or selected for local refinement. The insertion of these three templates avoided the expansion of refinement fields and implemented the effective control of refinement mesh areas.
Four adaptive refinement criteria were proposed according to the geometric features of model surface, such as curvature characteristics, local thickness and small features, etc. In order to take both of the curvature change of the solid model and the density distribution of STL triangle facets into consideration, the definition of relative curvature was proposed. And on the basis of that, the refinement source point fields and element fields were constructed. Cases verified that the relative curvature criterion could allow more reasonable distribution of mesh density and obtain better refinement results. A new criterion based on local thickness was proposed according to the status properties and distribution patterns of the mesh elements in small thickness regions of the solid model. Aiming at some of the straddling elements, a supplementary refinement criterion was proposed in order to realize the accurate judgement and reasonable identification of refinement domains.
Boundary matching is an important problem in automatic generation of three-dimensional finite element meshes. In this dissertation, the research and investigation of the C-edge matching technique of grid-based hexahedral element meshing method were performed. We made a deep study on the basic algorithm and key technique of relative position relationship method. A C-edge matching algorithm which combined priority nodes with relative position relationships was proposed by introducing priority nodes into the relative position relationship method. The geometric and topologic conditions for C-edge match were concluded and summarized. Eight types as well as five complementary types of free facet configurations were established, and their corresponding matching rules were also provided. The identification method for each level of priority nodes was presented, containing the identification of the first two levels and some of the third levels of priority nodes before C-edge match and the level-updated identification of priority nodes during C-edge match. The basic algorithms and implementation strategies of the combined method of priority nodes and relative position relationships were introduced in detail. An effective method for C-point match was also proposed based on geometric topology and priority nodes. A correction method was proposed to handle the irrational phenomenon where a small number of nodes were fitted to undesired C-edges. A method for treating the special sub-surface that intersected itself on a boundary line was proposed to ensure the effective match of special boundary line. For the C-edge matching problem on concave curves, this dissertation proposed a method to adjust the matching properties of the surface nodes associated with unreasonable degenerate elements. This method could unify the orientations of the degenerate facets fixed on a common concave curve in order that they pointed to the same sub-surface. Several examples demonstrated that the combined method proposed in this dissertation and the treatment methods for corresponding problems especially for concave curve matching problems could achieve accurate match between the surface of the solid model and the surface of the mesh. More than that, the geometric and topologic characteristics of the boundary elements and matched nodes could fully conform to the basic conditions of C-edge match, and the meshing accuracy was improved significantly.
The quality of the mesh directly impacts on the accuracy and efficiency of numerical analysis. In this dissertation, we deeply studied the quality improvement techniques of three-dimensional hexahedral element meshes. The quality metric was constructed based on the minimal determinant value of Jacobian matrix, which quantified the quality of hexahedral meshes and made the evaluation of mesh quality easier and more accurate. We studied the topological connections of boundary elements in the hexahedral mesh created by inside-out grid-based method, and analyzed the reason why geometric and topologic problems of boundary mesh were created and the necessity of topological optimization. New element inserting technique, old element collapsing technique and the mixed technique which combined the inserting and collapsing techniques were employed. These three techniques improved the geometric topology of surface mesh remarkably. On the base of the current topological optimization modes, five inserting modes, four collapsing modes and three mixed modes were newly proposed. We also summarized all of the eleven inserting modes, seven collapsing modes and six mixed modes in the pattern of tables to show their geometric characteristics, types and the change of element number. The coordinate application rules of topological optimization modes especially mixed modes and the treatment methods of conformal problems were given. For node position smoothing technique, the Laplacian smoothing algorithm was studied in detail. Aiming at the issue that the conventional Laplacian method could not conserve volumes of solid models, modified methods were proposed accordingly. The curvature-based Laplacian method was used to smooth the nodes fitted on C-edges to prevent the deviation of the mesh nodes from the boundary line of the solid model. In order to resolve the geometry distortion and volume error that usually appeared while using conventional Laplacian method, a projection-based Laplacian method was proposed for the smooth of the remaining boundary nodes. A node-and area-based Laplacian equation was proposed to adjust the position coordinates of interior nodes in order that the interior nodes and the remaining boundary nodes had the same smoothing amplitude so as to further improve the quality of the surface elements.
The research on local refinement technique was conducted. This technique could achieve the effective local refinement for individual or groups of nodes, elements, element-surfaces, element-edges, mesh-boundaries, mesh-faces and local regions, etc. This dissertation stated the reason why each category of local refinement was needed. For the technique of node refinement, this dissertation gave the judgement criterion and refinement method of the isolated nodes. For the technique of mesh-boundary or-face refinement, it was required first to identify the mesh-boundaries or-faces of the resulting mesh. For that reason, this dissertation proposed the basic algorithms and detailed procedures of mesh-boundary or-face identifications. Multi-level local refinement technique was also proposed. The levels of local refinement could be self-controlled effectively. The multi-level local refinement technique was able to perform arbitrary times of local refinement for certain geometric domains of the solid model, so that the final mesh could capture the geometric and physical features of the solid model more accurately and further increase the accuracy of finite element analysis.
On the basis of the research on grid-based hexahedral element mesh generation algorithms and key techniques described above, the hexahedral mesh automatic generation software AUTOMESH-3D previously developed by our groups was modified and improved. The reliability, stability and versatility of AUTOMESH-3D were further enhanced, which built a general-purpose platform to generate three-dimensional hexahedral element meshes for various science and engineering problems. In this dissertation, the module structure and function properties of this software were introduced. Several hexahedral element meshing examples for solid models with complex geometric shapes were provided to demonstrate the accuracy and reliability of the developed software.
在各种常用的三维六面体网格自动生成方法中,基于栅格法具有高度的自动化,易于网格局部加密,比较适合六面体网格的自动生成。所以,本文以基于栅格法作为六面体网格自动生成的基本算法,根据基于栅格法的研究现状、存在问题以及发展方向,开展三维六面体网格自适应生成算法的深入研究,以增强基于栅格法的稳定性、可靠性和强壮性。本文研究了基于实体模型STL文件的几何特征识别关键技术；建立了一套较完善的加密模板；提出了基于实体模型几何特征建立加密信息场的基本判据；研究了基于栅格法的边界拟合技术,提出了优先点和相对位置关系相结合的边界拟合方法；深入研究了网格单元的质量优化技术,包括拓扑优化技术和节点平滑技术；建立了相对完善的局部加密技术和自主控制的多次加密技术。
本文采用由CAD造型软件生成的STL文件来反映三维实体模型的表面几何信息,建立了具有拓扑关系的STL数据文件,避免了冗余数据的重复出现,提高了计算效率；建立了基于STL三角面片曲率变化的几何特征识别的关键技术,给出了特征边、特征点、子表面、边界边的识别方法以及特征边和边界边的凹凸属性和曲直属性的判断方法,并给出了特征点属性的识别方法；为了实现加密区域和非加密区域网格单元的协调过渡,本文在Ito模板的基础上建立了一套较为完善的六面体网格加密模板,改进了两个拐角模板(相邻面和相邻边加密模板),提高了子单元的质量和尺寸均匀性,增加了孤立点加密模板,针对实体模型上固有的孤立点和局部加密中选择的孤立点进行加密。拐角模板和孤立点模板的加入避免了加密信息场的蔓延,实现了网格加密区域的可控制性。根据实体模型的表面曲率特征、局部厚度特征和小特征等,建立了基于实体模型表面几何特征的自适应加密判据。为了同时考虑到实体模型的曲率变化和STL三角面片密度分布的影响,提出了相对曲率的概念,根据相对曲率建立了加密源点信息场和加密单元信息场,并通过实例证明了相对曲率判据可以获得更合理的网格密度分布和更好的加密效果；根据实体模型小厚度区域网格单元的属性特点和分布规律,提出了新的基于实体模型局部厚度的网格加密判据,并提出了部分骑边单元的补充加密判据,实现了加密区域的准确判断和合理识别。
深入研究了基于栅格法六面体网格自动生成的边界拟合技术,尤其是相对位置关系法的基本算法和关键技术,把优先点引入到相对位置关系法中,提出了一种优先点与相对位置关系相结合的边界拟合方法。总结归纳并给出了边界拟合的几何要求和拓扑连接要求；建立了八种网格表面自由四边形面片类型和五种补充面片类型,并给出了相应的拟合规则；提出了优先点的识别方法,包括边界拟合前第一、二级优先点和部分第三级优先点的识别方法以及边界拟合过程中优先点的逐级更新方法；建立了优先点与相对位置关系结合方法的基本算法和实现步骤；提出了一种基于几何拓扑关系和优先点的特征点拟合方法；提出了针对少数不合理拟合节点的修正方法和特殊子表面的处理方法,实现了特征边的有效拟合；针对凹曲边的拟合问题,提出了一种对不合理退化单元的表面节点进行拟合属性调整的方法,将同一条凹曲边上退化面片的指向调整为一致,有效解决了凹曲边的拟合与优化问题。实例证明本文提出的优先点与相对位置关系相结合的边界拟合方法以及凹曲边问题的处理方法能够实现网格模型表面与所分析实体模型表面的精确拟合,而且能够使网格边界单元和拟合节点的几何条件与拓扑连接关系符合边界拟合的基本要求,从而显著提高了所生成网格的精度。
网格的质量直接影响数值分析的精度和效率,为了提高网格的质量,本文对三维六面体网格的质量优化技术进行了深入研究。建立了六面体单元最小雅可比矩阵行列式值的网格质量评价准则,将六面体网格的质量进行量化,使网格质量的检测更加简便和精确。研究了从内向外栅格法所生成的六面体网格边界单元之间的拓扑连接关系,分析了边界网格几何形状和拓扑连接问题出现的原因和拓扑优化的必要性。提出了插入新单元技术、单元退化技术和插入与退化相结合的技术,显著改善了表面网格边界单元之间的拓扑连接关系。在现有拓扑优化模式的基础上,提出了五种新的插入模式、四种新的退化模式和三种新的插入和退化相结合的混合模式,并用图表的形式对采用的十一种插入模式、七种退化模式和六种混合模式进行了归纳总结,给出了每个模式的几何特点、类型以及应用时单元数量的变化情况。建立了各种模式尤其是混合模式的协调应用规则以及相容性问题的处理方法。研究了拉普拉斯平滑技术,针对传统的拉普拉斯平滑算法往往引起实体模型体积变化的问题,提出了相应的改进方法。采用基于曲率拉普拉斯平滑算法对边界拟合点进行平滑,控制了边界拟合点相对于实体模型边界的偏离；提出了一种基于映射的拉普拉斯平滑算法对一般表面节点进行平滑,解决了传统的拉普拉斯平滑算法容易出现的几何形状失真和体积误差问题；提出了一种基于节点和面积相结合的拉普拉斯平滑方法对内部节点进行平滑,以保证内部节点和表面节点平滑的步调一致,从而提高了表面单元的质量。
对网格局部加密技术进行了深入研究,包括针对节点、单元、单元面、单元边、网格边界边、网格表面和局部区域等的加密。总结了各种局部加密技术应用的原因和必要性；对于节点加密技术,给出了孤立点的判断依据和加密方法；对于网格边界边和网格表面的局部加密,需要首先识别出网格的边界边和网格表面,为此,本文给出了识别网格边界边和网格表面的基本方法和详细步骤；提出了多次加密技术,对实体网格的加密次数进行自主控制,使最终网格能够更精确地描述实体模型的几何特征和物理特征,进一步提高了有限元分析的精度。
在对上述基于栅格法六面体网格自动生成算法和关键技术的研究基础上,对本课题组前期开发的三维六面体网格自动生成软件AUTOMESH-3D进行了改进和完善,进一步提高了其可靠性、稳定性和通用性,为科学与工程问题的研究提供了建立三维六面体网格模型的通用平台。介绍了该软件的模块结构及功能特点,并通过大量的六面体网格自动生成实例验证了所开发软件的精度和可靠性。
Finite element method is a general-purpose method used for numerical analysis of computational science and engineering problems. It is the important part of computer-aided design. The basic idea of finite element method is discretization and slice interpolation. That is, using mesh elements to describe the space of analyzed object. Finite element method can simulate various types of complex material structures, load relationships and boundary conditions. It is widely used in metal forming, mechanics, architecture, geotechnical engineering, solid and fluid mechanics, biomedical engineering, rapid prototyping and manufacture, computer graphics and other fields. For three-dimensional issues, tetrahedra, hexahedra and a combination of them are usually used. Three-dimensional hexahedral element mesh expresses much more advantages than tetrahedral element mesh in computational accuracy, the element count, mesh distortion and re-meshing number, etc., thus receiving significant attentions. However, due to the complexity of adaptive hexahedral mesh generation itself, there remain a lot of problems yet to be further resolved. Therefore, it has important theoretical significance and engineering application values to study automatic hexahedral element meshing algorithms in arbitrary space and establish reliable and efficient discretion software platform for three-dimensional models.
In this dissertation, grid-based method was used as the basic algorithm for automatic creation of hexahedral element meshes. According to the research situations, existent problems and development trends of grid-based method, we carried out the deep research on adaptive generation algorithm of three-dimensional hexahedral mesh so as to enhance the stability, reliability and robustness of the method. The key technique of geometric feature identification based on the STL files of solid models was studied. A relative perfect set of refinement templates was built. The main criteria for establishing refinement fields were proposed on the basis of the geometric characteristics of the solid model. The characteristic edge (C-edge) match technique was studied deeply. A new method combined priority nodes with relative position relationships was proposed for C-edge match. This dissertation also made an intensive study of quality improvement techniques, including two aspects:topological optimization and node position smoothing. In addition, a relative complete technique of local refinement was put forward, and the refinement level could be self-controlled arbitrarily.
The STL files generated using CAD modeling software were used to reflect the surface geometry information of three-dimensional solid models. We established the STL files containing topological connections to avoid the repeated appearance of redundant data and further improve the computing efficiency. The essential technique of geometric feature identification was set up based on the curvature change of adjacent STL triangle facets. The identification methods of C-edges, characteristic points (C-points), sub-surfaces and boundary lines were given. The judgement methods of concave-convex and curve-straight attributes of C-edges and boundary lines, as well as the determination methods of the attributes of C-points, were also proposed. In order to achieve the conformal transition of the elements between refined regions and non-refined regions, this dissertation proposed a relatively perfect set of refinement templates on the basis of the research of Ito. The templates for two-face refinement and two-edge refinement were modified for the quality improvement and size uniformity of subdivided elements. The isolated node refinement template was added to refine the isolated nodes existing on the surface of the solid model or selected for local refinement. The insertion of these three templates avoided the expansion of refinement fields and implemented the effective control of refinement mesh areas.
Four adaptive refinement criteria were proposed according to the geometric features of model surface, such as curvature characteristics, local thickness and small features, etc. In order to take both of the curvature change of the solid model and the density distribution of STL triangle facets into consideration, the definition of relative curvature was proposed. And on the basis of that, the refinement source point fields and element fields were constructed. Cases verified that the relative curvature criterion could allow more reasonable distribution of mesh density and obtain better refinement results. A new criterion based on local thickness was proposed according to the status properties and distribution patterns of the mesh elements in small thickness regions of the solid model. Aiming at some of the straddling elements, a supplementary refinement criterion was proposed in order to realize the accurate judgement and reasonable identification of refinement domains.
Boundary matching is an important problem in automatic generation of three-dimensional finite element meshes. In this dissertation, the research and investigation of the C-edge matching technique of grid-based hexahedral element meshing method were performed. We made a deep study on the basic algorithm and key technique of relative position relationship method. A C-edge matching algorithm which combined priority nodes with relative position relationships was proposed by introducing priority nodes into the relative position relationship method. The geometric and topologic conditions for C-edge match were concluded and summarized. Eight types as well as five complementary types of free facet configurations were established, and their corresponding matching rules were also provided. The identification method for each level of priority nodes was presented, containing the identification of the first two levels and some of the third levels of priority nodes before C-edge match and the level-updated identification of priority nodes during C-edge match. The basic algorithms and implementation strategies of the combined method of priority nodes and relative position relationships were introduced in detail. An effective method for C-point match was also proposed based on geometric topology and priority nodes. A correction method was proposed to handle the irrational phenomenon where a small number of nodes were fitted to undesired C-edges. A method for treating the special sub-surface that intersected itself on a boundary line was proposed to ensure the effective match of special boundary line. For the C-edge matching problem on concave curves, this dissertation proposed a method to adjust the matching properties of the surface nodes associated with unreasonable degenerate elements. This method could unify the orientations of the degenerate facets fixed on a common concave curve in order that they pointed to the same sub-surface. Several examples demonstrated that the combined method proposed in this dissertation and the treatment methods for corresponding problems especially for concave curve matching problems could achieve accurate match between the surface of the solid model and the surface of the mesh. More than that, the geometric and topologic characteristics of the boundary elements and matched nodes could fully conform to the basic conditions of C-edge match, and the meshing accuracy was improved significantly.
The quality of the mesh directly impacts on the accuracy and efficiency of numerical analysis. In this dissertation, we deeply studied the quality improvement techniques of three-dimensional hexahedral element meshes. The quality metric was constructed based on the minimal determinant value of Jacobian matrix, which quantified the quality of hexahedral meshes and made the evaluation of mesh quality easier and more accurate. We studied the topological connections of boundary elements in the hexahedral mesh created by inside-out grid-based method, and analyzed the reason why geometric and topologic problems of boundary mesh were created and the necessity of topological optimization. New element inserting technique, old element collapsing technique and the mixed technique which combined the inserting and collapsing techniques were employed. These three techniques improved the geometric topology of surface mesh remarkably. On the base of the current topological optimization modes, five inserting modes, four collapsing modes and three mixed modes were newly proposed. We also summarized all of the eleven inserting modes, seven collapsing modes and six mixed modes in the pattern of tables to show their geometric characteristics, types and the change of element number. The coordinate application rules of topological optimization modes especially mixed modes and the treatment methods of conformal problems were given. For node position smoothing technique, the Laplacian smoothing algorithm was studied in detail. Aiming at the issue that the conventional Laplacian method could not conserve volumes of solid models, modified methods were proposed accordingly. The curvature-based Laplacian method was used to smooth the nodes fitted on C-edges to prevent the deviation of the mesh nodes from the boundary line of the solid model. In order to resolve the geometry distortion and volume error that usually appeared while using conventional Laplacian method, a projection-based Laplacian method was proposed for the smooth of the remaining boundary nodes. A node-and area-based Laplacian equation was proposed to adjust the position coordinates of interior nodes in order that the interior nodes and the remaining boundary nodes had the same smoothing amplitude so as to further improve the quality of the surface elements.
The research on local refinement technique was conducted. This technique could achieve the effective local refinement for individual or groups of nodes, elements, element-surfaces, element-edges, mesh-boundaries, mesh-faces and local regions, etc. This dissertation stated the reason why each category of local refinement was needed. For the technique of node refinement, this dissertation gave the judgement criterion and refinement method of the isolated nodes. For the technique of mesh-boundary or-face refinement, it was required first to identify the mesh-boundaries or-faces of the resulting mesh. For that reason, this dissertation proposed the basic algorithms and detailed procedures of mesh-boundary or-face identifications. Multi-level local refinement technique was also proposed. The levels of local refinement could be self-controlled effectively. The multi-level local refinement technique was able to perform arbitrary times of local refinement for certain geometric domains of the solid model, so that the final mesh could capture the geometric and physical features of the solid model more accurately and further increase the accuracy of finite element analysis.
On the basis of the research on grid-based hexahedral element mesh generation algorithms and key techniques described above, the hexahedral mesh automatic generation software AUTOMESH-3D previously developed by our groups was modified and improved. The reliability, stability and versatility of AUTOMESH-3D were further enhanced, which built a general-purpose platform to generate three-dimensional hexahedral element meshes for various science and engineering problems. In this dissertation, the module structure and function properties of this software were introduced. Several hexahedral element meshing examples for solid models with complex geometric shapes were provided to demonstrate the accuracy and reliability of the developed software.
引文
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