Research On Modeling Minimal Surfaces Based On Weighted B-spline

The problem of the minimal surface is to find a surface on bounded domains whose area is minimal, it originated from the experiments of Plateau in 18th century. It is an important topic in the theory of the differential geometry and PDE. The average curvature of the minimal surfaces is 0, so they have many good natures which make they have been extensively employed in many areas such as architecture, aviation, ship manufacture, art, biology, crystallogeny and so on. Although the theory of the minimal surface in the differential geometry is very rich, the researches of modeling the minimal surface in CAD/CAGD are few, and the NURBS has become a standard in CAD/CAGD, so the approximation to the solution of the minimal surface equation bounded by B-spline curves is meaningful. In this paper we have some elements works on minimal surface from the point of CAGD.The equation of the minimal surface is a nonlinear partial differential equation(PDE), its complexity makes it have no explicit expression of the solution, The finite element method is an effective way to solving this kind of problems. In this paper, we research the minimal surfaces on arbitrary boundary which are approximated by weighted extended B-spline surfaces. In theory, uniform B-spline dosen't have the the nature of interpolating of the endpoints, it cann't be zero on the boundary automatically, so using it as finite element is not feasible because of the boundary conditions. However, we can simply multiply tensor uniform B-spline by a fixed weighted function which vanished on the boundary, more precisely, multiply the distance function to the boundary which makes the weighted extended B-spline to be zero on the boundary to meet the needs of the boundary conditions. It yields high order accurate approximation.