摘要
以边界积分方程为基础的边界面法继承了边界元法的所有优点,例如可以使求解问题域降低一维,也可以求解无限域问题和裂纹扩展等奇异性问题。由于该法的边界积分和场变量插值都是在以边界表征的实体边界曲面的参数空间里进行,积分点的坐标,法向量以及雅克比等几何数据都可以由曲面本身计算获得,而传统边界元法是通过网格单元插值近似得到,边界面法避免了几何误差,因此计算精度比传统边界元法高。
在边界面法中,前处理网格生成对计算结果有着非常重要的影响,而且网格是定义在曲面的参数空间中,因此有必要对适合边界面法的网格生成算法进行研究。工程实际应用中,四边形网格在计算精度和计算效率方面都要优于三角形网格,本文对四边形网格生成算法—铺砖法及程序实现进行了详细研究,同时对铺砖法进行改进,把铺砖法推广到任意曲面的四边形网格生成,最后把生成的网格用于边界面法分析。
本文研究主要内容如下:
(1)根据边界面法的特点和铺砖法算法的一般流程对参数曲面四边形网格生成进行了多方面研究。重点研究运用黎曼度量在参数空间生成四边形网格,并且在原有算法的基础上进行了改进,文中引入点的迁移算子对在周期曲面上生成四边形网格进行了研究。
(2)利用VC++与UG二次开发技术,建立边界面法前处理四边形网格模块。通过VC++进行程序设计实现了四边形网格生成程序,并对程序数据管理和网格生成算法关键问题的程序实现作了详细介绍。程序中通过调用UG/Open API中函数获取CAD模型边界表征信息和显示最终生成的网格。
(3)利用生成的网格数据调用边界面法分析模块进行三维位势问题和线弹性问题分析。四边形网格是在CAD模型曲面参数空间生成的,网格数据保留了CAD几何模型的数据,利用网格数据调用边界面法分析程序进行计算,成功实现了CAD/CAE一体化,最后通过与精确解和有限元商业软件计算结果进行对比,证明算法的正确性和边界面法的优越性。
The boundary face method (BFM), which is based on the boundary integral equation (BIE), inherits all advantages of the boundary element method (BEM) such as a lower computational scale of one order and abilities to solve problems on infinite domain and problems with singularities including crack propagation. In the BFM, the integration and variable approximation are both performed in the parametric space of the boundary surface. The geometric data such as coordinates, outward normals and Jacobians on integral points can be calculated directly from the parametric surface.In the conventional BEM, however, the geometric data are calculated through element interpolation,thus no geometric errors are introduced. The BFM is usually more accurate than the conventional BEM.
In the implementation of the BFM, the grid generation is of great importance to the computation. The mesh in the BFM, however, is defined in the parametric space of the boundary surface. The mesh generation algorithm that is suitable for the BFM should be studied. In engineering applications, quadrilateral mesh has many advantages over the triangular mesh on both accuracy and efficiency. This paper studies the paving method, which is one of the quadrilateral mesh generation method, and its program application. The paving method is improved and extended to generate quadrilateral mesh on arbitrary surface. The improved method is implemented in the BFM to solve 3D potential problems and 3D elasticity problems.
Contents of this paper are listed as follows:
(1) According to the feature of the BFM and the general procedure of the paving method, a full study on the generation of the quadrilateral mesh on parametric surface has been done. The mesh generation in parametric space applying Riemann metric is emphasized. The original paving method has been improved. Moreover, mesh generation on closed surface has been studied.
(2) The pre-process module for quadrilateral mesh in the BFM has been developed by using both language of the visual C++ and the secondary developing technology of the Uni-Graphics. Using the language of the visual C++, the program for quadrilateral mesh generation has been developed. Furthermore, the data management and critical problems in the mesh generation algorithm in the program has been discussed in details. By calling the functions that are available in the UG/Open API, data of the boundary represented solid model have been obtained and the final meshes have been displayed on the model.
(3) BFM analyses have been performed with the mesh generated by the developed program. The quadrilateral mesh is generated in the parametric space of the boundary surface of the CAD solid models. The topology data of the original CAD geometric model are preserved. Thus the BFM has integrated CAD and CAE successfully. The BFM has been verified by numerical examples in which comparison between the numerical solution and the analytical solution was made. Furthermore, comparison study between the BFM and the finite element method (FEM) has been made to illustrate the advantages of the BFM.
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