The Research On The Boundary Face Method Based On Parametric Surface

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. The geometric errors introduced by coarse mesh can lead to accuracy problems. However, In the BFM both boundary integration and variable approximation are performed in the 2-D parametric space of the boundary surface of solid represented by the boundary representation data (B-rep). For boundary integration, the geometric data at Gaussian integration points, such as the physical 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.In the beginning of this paper, we deduced self-regular boundary integral equation of the potential problems governed by Laplace's equation simply based on the Green formula. Then the schemes for the boundary integration and variable approximation based on the surface element in parametric space are discussed. Each surface element is defined by a set of nodes given in 2-D parametric space of the surface which is different from the surface element defined in 3-D physical space as used in the BEM. In order to deal with thin and slender structures effectively, an adaptive integration scheme for nearly singular integrals and singular integrals has been developed based on 2-D surface element subdivision. The process of subdivision is performed directly in parametric space making its implementation simple and effective. We have improved the implementation of the advancing front method (AFM) according its general principle. And an adaptive method for generating surface elements has also been developed. Surface elements are generated along with an adaptive quad-tree procedure used to control the element size.The BFM has real potential to seamlessly interact with CAD software, integrating easily geometric design and engineering analysis into a completely unified framework, because its implementation can be directly based on a CAD model through its B-rep data. We have developed an interface between BFM and UG-NX with C++ code. Numerical examples involving complicated geometry have demonstrated that the integration of BFM and UG-NX is successful, which may provide an important step toward automatic simulation. The BFM has been verified through a number of numerical examples with different geometries and boundary condition type. It is demonstrated that our method has high convergence rate and yields very accurate numerical results. Compared with the conventional BEM, the BFM can not only provide more accurate results than the BEM, but also is less sensitive to the coarseness of the mesh. What's more, our method can deal with the structures with features in small size or thin structures to simulate 3-D potential problems efficiently.