表面细分技术在求解边界积分方程中的应用

【作者】 王卫祥；

【导师】 文立华；

【作者基本信息】 西北工业大学 ， 工程力学， 2005， 硕士

【摘要】 表面细分技术是近二十年发展起来的计算机图形学中的一项新兴技术,它是按照计算机图形处理方法对复杂图形用网格控制点进行描述,可以根据描述精度要求,通过表面细分算法对网格进行细分,所描述的图形可以达到c~n连续,满足精确描述的目的。它能对图形进行自适应处理。目前,传统的边界元方法不能对复杂边界进行精确描述,也不能根据误差要求对网格进行自动加密,另一方面,CAD模型还不能直接用于数值模拟分析,必须借助网格模型划分器划分网格后才能进行分析,即使这样,有时CAD模型进入网格模型划分网格还是有一定的困难。本文探索了表面细分技术的算法原理,把表面细分技术应用到解决边界积分方程方法——边界元方法和边界谱方法中,能通过精确描述模型提高求解精度,而且可以实现网格自动加密,通过控制细分次数满足误差要求,同时在边界积分方程中提出了一种CAD和CAE一体化建模方法。数值算例显示,表面细分求解方法的精度有了较大提高,表明算法的有效性。 本文的工作主要集中在以下几个方面: (1) 介绍了表面细分技术的算法原理以及适合于本文求解的表面细分算法,介绍了边界的处理方式以及算法的收敛。 (2) 详细介绍了表面细分技术与边界元方法的结合过程,包括单元内的插值方法的细分、形函数的变化过程和坐标变换过程。 (3) 介绍了应用子波构建的边界谱方法的建立过程和系数计算具体表达,介绍了边界谱方法与表面细分技术的结合过程。 (4) 运用本文的方法对弹性扭转问题、多连域拉普拉斯方程问题、泊桑方程问题以及二维声辐射和声散射问题进行了数值仿真,并与一般方法所得结果进行了比较。更多还原

【Abstract】 Subdivision surface is a new technology that has been developing for about twenty years in Computer Graphics.it can depict complicated models with the controlling graphic grids using the Computer Graphic disposal method and fractionize the grids according to the precision requirements. The model can be depicted to C~n continuum with subdivision surface. So far. the traditional boundary element method can not depict the complicated model accurately or coarctate the meshes automatically; moreover, the CAD model cannot be used directly for numerical simulation. In this paper, a new paradigm for boundary element method is developed based on the use of subdivision surface to: i) accurately describe the boundary geometry to reduce the error from the disfigurement on depicting models when using the traditional boundary element method with which only the boundary curves or surfaces are discrete, ii) realize growing the mesh density automatically and iii) offer an approach to use the CAD models directly in CAE.The main research work and conclusions are listed as follow:(1) The arithmetic theory of subdivision surface is introduced as well as the subdivision surface method applied in this thesis; the disposal method of boundary and the convergence of subdivision surface are also presented.(2) How to apply the subdivision surface to boundary element method, including the interpolating method in single element, the changing of shape functions and the integral coordination, is depicted in detail.(3) Both of the constructing course based on the wavelet and the expressions of coefficients of the boundary spectral method are introduced; the combination of the boundary spectral method and subdivision surface is discussed too.(4) The elasticity torsion problem, problems with more than one surface of Laplace’s equation, Posson’s equation problem and the problem of two- dimension acoustic radiation and scattering are discussed with the method introduced in this paper; and finally, the results obtained with the method introduced in this paper are compared with those obtained with the traditional boundary element method.更多还原