Research Of The B-spline Boundary Face Method And The Isogeometric Analysis In The Boundary Integral Equation

B-spline is firstly from the research of approximation theory in the subject of math. Because of well mathematical properties and strong performance of free form of curve and surface, B-spline has been widely used and has played an very important role in the Computer Aided Geometric Design(CAGD). A kind of B-spline expressed by a rational expression after B-spline is proposed, called non-uniform rational B-spline(NURBS), overcomes the shortcoming of B-spline that it cannot exactly describe quadratic curve and surface, that make NURBS becomes the sole mathematical method in International Standardization Organization(ISO) to define the shape of industry products. Actually, the flexible capability of modeling of B-spline or NURBS is depend on the their well mathematical properties. Their properties, such as partition of unity, diminishing variation and strong convex, and so on, are the key factors to determine the exactness and stability of numerical approximation algorithm. Here we use B-spline or NURBS as approximation method to combine with Boundary Integral Equation(BIE). In the frame of Boundary Face Method(BFM), we proposed a B-spilne BFM in which B-spline is used to apprpximate the boundary field variable. Furthermore, we use B-spline or NURBS to modeling and approximation of boundary field variable and proposed an Isogeometric Method based on BIE.When to use bivariate B-spline function to approximate the boundary field variable, a lot of singular and nearly singular integrals require to be dealed with on some boundary surface, such as a sphere surface, where singular point(The point is degenerated from the edge of a regular parametic plane when it affined to the Cartesion Coordinates) exists due to B-spline tensor product restriction. This phenomena will not only increase the compution cost but also deteriorate the numerical result. To slove the problem, a local form of B-spline (Called as local B-spline) and the expressions of bivariate local B-spline and NURBS and their derivation, will be introduced. It will efficiently avoid the computation of singular and nearly singular integrals around the singular point of parametric surface. Therefore, it will distinctly improve the computational efficiency and exactness comparing with the global B-spline. The solution of3-D potential and linear elasticity will testify these properties in the paper. To more exhibit the superiority of B-spline, the MLS method that posess high interpolation exactness will be introduced for comparison in the same BFM frame. The final numerical results show that B-spline has higher exactness and convegence rate.To actually implement the seamless connection between CAD and CAE(i.e., exact geometric imfomation is used to analysis and interaction between CAD and CAE only needs one time when to perform adaptive analysis), the isogeometic analysis that first proposed in the realm of Finite Elment Method(FEM) is creatively combined with BIE and the B-spline isogeometic analysis based on the BIE is proposed. This method inherits the dimensional reduction of boundary element method. To solve3-D problems, not only the geometric model can be expressed by the B-rep representation(i.e., solid model can be bounded by the B-spline or NURBS surfaces), but also the approximation of boundary field variable only requires bivariate B-spline in the CAE. Comparing with triviriate B-spline is required for geometic modeling and variable approximation in the FEM, the implementation of isogeometic analysis and mesh generation are more simple in the BIE. For the implementation, we write program for solution of3-D potential and linear elasticity problem. Numerical results will testify the exactness and stability of the algorithm. Note that we will use the bivariate local B-spline to approximate the boundary field variable in the frame of isogeometric analysis, numerical results show that local B-spline has higher computational exactness and efficiency.In order to solve more complicated engineering problem in the frame of BFM and isogeometric analysis, the weld structure widely used in the engineering is considered. To perform the CAE analysis for the true weld structure, we require building the weld mold that quitely like the weld structure in the reality. But this module to simulate the true weld structure is not contained in the CAD software. Considering the above, the author develop a module of modeling by C++language on the UG platform of second development, in which B-spline is used to simulate the true weld. The module can be performed with menu response in the UG. It also supply the model condition for the subsequent CAE analysis.