摘要
随着三维建模技术和数值方法的发展，针对三维实体的有限元技术在工程领域得到了越来越广泛的应用。实体网格的生成是三维实体有限元分析的重要前提，六面体网格由于其在计算精度、单元数量等方面明显优于四面体网格，正成为有限元分析中的主流网格。目前，任意三维实体的六面体网格全自动生成问题仍然没有获得真正意义上的解决。本文针对六面体网格自动生成中的实体模型表达、实体分解、基于扫掠的六面体网格生成及六面体网格质量优化等关键技术，进行了深入细致的研究，开发了三维实体网格生成软件。主要研究成果如下：
(1)提出了基于局部刚性和整体线弹性反向变形有限元法的曲面参数化算法。克服了组合曲面不易于参数化的缺点，利用从局部到整体的思想，通过分析作用在局部三角形上的仿射变换雅可比矩阵的奇异值，寻求局部三角形到参数域的刚性变换，然后采用线弹性有限元反向变形方法整体求解参数化结果。
(2)提出了基于扫掠体识别的三维实体模型分解方法。首先将模型的所有边分类，采用宽度优先搜索算法提取出模型的所有组合面；然后选择一张复杂度最高的组合面作为源面，利用深度优先搜索算法确定源面对应的约束面和连接面；最后针对不同的约束面类型构造裁剪面，将实体模型分解为多个扫掠体的组合。分解后的实体可直接采用映射法或扫掠法生成六面体网格。
(3)提出了基于内在特征和能量约束的曲面网格投影方法。通过网格参数化方法将源网格和目标基准网格展开到平面上；根据平面网格伸缩内在特征、平面网格变形能及边界位置约束建立二次能量方程，并采用最小二乘法最小化该方程，从而将平面上的源网格变形到目标基准网格上；最终映射到目标曲面上得到目标网格。实现了多张曲面之间的网格投影，并且保证了目标网格和源网格同胚。
(4)提出了从双八面体局部变换到整体线弹性有限元法的六面体网格优化方法。采用基于双八面体的局部变换将每个六面体单元进行局部规则化，得到网格质量较高的局部优化网格；比较局部优化前后的六面体单元，利用线弹性有限元法求得网格节点残余内力；将表面网格组成的几何特征作为约束条件，进而利用不平衡的网格节点残余内力整体求解得到六面体网格优化结果。优化后六面体网格保留了原有网格模型的几何特征，显著提高了六面体网格的质量。
在对上述方法深入研究的基础上，借助Open CASCADE几何造型平台，自主开发了三维实体网格生成软件。该软件具有标准的几何和有限元数据输入输出接口、六面体网格自动生成以及网格质量检查等功能。基于软插件技术，实现了三维实体网格生成软件的功能扩展，方便用户进行功能扩展。通过对航空零件的六面体网格自动生成，表明该软件网格生成效率较高，六面体网格质量较好，满足了三维实体有限元数值模拟的要求。
With the development of three-dimensional modeling technology and numerical method, thefinite element technology for three-dimensional entities has been used more widly in engineering. Thesolid mesh generation is an important premise of three-dimensional finite element analysis, and thehexahedral mesh is becoming the main element in finite element analysis for its superiority incalculation accuracy, element number and other aspect compared with the tetrahedral mesh. Now,automatic hexahedral mesh generation for arbitrary three-dimensional volumes is still not trulyresolved. In this thesis, the key techniques in automatic hexahedral mesh generation, such as solidmodel expression, volume decomposition, swept volume hexahedral mesh generation, hexahedralmesh quality optimization, etc are deeply studied, and a mesh generation software forthree-dimensional volumes has been developed. The research achievements are as follows:
(1) A local rigid and global linear-elastic reverse deformation finite element method basedsurfaces parameterization method has been proposed, whcih overcomes the shortcomings that thecomposite surfaces are difficult to parameterize. Base on the local to global methodology, the localrigid transformation can be taken by analyzing the Jacobian of affine transformation of each triangleand adjusting the corresponding singular values, and then the final parameterization result can beobtained by linear-elastic finite element method globally.
(2) A three-dimensional volume decomposition method based on swept volume recognition ispresented. Firstly, all boundary edges are classified and all composite surfaces are extracted bywidth-first searching algorithm; Then, the constraint faces and linking faces corresponding to thesource face are determined by depth-first searching algorithm. Finally, the cutting surfaces can begenerated for different type of constraint faces, and the entire geometric models can be decomposedinto the combination of multiple swpet volumes. The divided meshable pieces can generatehexahedral mesh by applying mapping or sweeping method directly.
(3) A surface mesh projection algorithm based on intrinsic characteristics and energyconstraints is presented. The source surface mesh and the target base mesh are flattened by meshparameterization method; A quadratic energy equation based on scaled intrinsic characteristics ofplanar mesh, mesh deformation energy and boundary constraints is established, then a least-squaremethod is used to minimize this energy function to morph the planar source mesh onto the planartarget base mesh; The target surface mesh is then obtained by mapping the plane source mesh onto the target base mesh. Mesh projection between multi surfaces can be implemented by using this method,which ensures that the target mesh is homeomorphism to source mesh.
(4) A double octahedron based local transformation and global linear-elastic finite elementmethod hexahedral mesh quality optimization method is presented. A double octahedron based localtransformation method is used to regularize a hexahedron locally, and regular hexahedral elements areobtained for every single mesh. Locally by analyzing the displacement of hexahedral element nodesbetween pre and post regularization, and the residual internal force can be calculated by linear elasticfinite element method. Globally by treating the surface features as constraints, the hexahedral meshquality optimization result can be obtained by loading the unbalanced residual internal force into itscorresponding node. After the optimization, the hexahedral mesh could preserve the geometricalfeatures of original model, and the hexahedral mesh quality is highly improved.
On the basis of deep study of these methods, and with the Open CASCADE geometric modelingplatform, a multi-functional three-dimensional solid mesh generation software is developed, includingstandard geometric and finite element data exchange interface, automatic hexahedral mesh generationand hexahedral mesh quality measurement and so on. By utilizing soft plug-in technology, thefunction extended interface is implemented to provide convenience for user extensions. The practicalhexahedral mesh generation examples of aeronautical blade and other complex parts show that thedeveloped software has higher mesh generation efficiency and can generate good quality hexahedralmesh, which meets the requirements of three-dimensional finite element numerical simulation.
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