基于参数曲面三维势问题的边界面法

A boundary face method for 3D potential problems based on parametric surface

提出了边界元法(BEM)的一种新的实现方法——边界面法(BFM)。在传统的边界元法中,单元不仅用来进行边界积分和函数插值,而且用来近似几何体。当离散网格较稀疏时,会引起较大几何误差,因而影响计算精度。本文基于参数曲面,将几何实体的边界曲面离散为参数空间里的曲面单元,边界积分和场变量的插值都是在曲面参数空间里进行。积分点的几何数据,如坐标、雅可比、外法向量都是直接由曲面算得,而不是通过单元插值近似,从而避免了几何误差。另外,该方法的实现是直接基于边界表征的CAD模型,做到了与CAD软件的无缝连接。三维位势问题的数值实例表明,该方法不仅比传统边界元法具有更高的精度,而且使用非常方便,容易做到自动分析。

This work presents a new implementation of the boundary element method（BEM）,here called the boundary face method（BFM）.The conventional BEM uses the standard elements for boundary integration and approximation of the geometry,and thus introduces errors of geometry.In this paper,the boundary faces of the geometry are discretized by patches in parametric space.Both boundary integration and variable approximation are also performed in the parametric space.The geometric data at Gaussian integration points,such as the coordinates,the Jacobians and the out normals are calculated directly from the faces rather than from elements,and thus no geometric error will be introduced.The BFM has real potential to seamlessly interact with CAD software,because its implementation can be directly based on a CAD model through its Brep data.Numerical examples for 3D potential problems demonstrate that the new method provides not only more accurate results than the conventional BEM,but also a new way toward automatic simulation,as simulations can be greatly simplified with our method.