The mechanical background of the bivariate spline space of degree 2 and smoothness 1 on rectangular partition is presented constructively. Making use of mechanical analysis method, by acting couples along the interior edges with suitable evaluations, the deflection surface is divided into piecewise form, therefore, the relation between a class of bivariate splines on rectangular partition and the pure bending of thin plate is established. In addition, the interpretation of smoothing cofactor and conformality condition from the mechanical point of view is given. Furthermore, by introducing twisting moments, the mechanical background of any spline belong to the above space is set up.
Isopaxametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and conformality. A 17onode quadrilateral element has been developed using the bivaxiate quaxtic spline interpolation basis and the triangular area coordinates, which can exactly model the quartic displacement fields. Some appropriate examples are employed to illustrate that the element possesses high precision and is insensitive to mesh distortions.
One major goal of mesh parameterization is to minimize the conformal distortion. Measured boundary parameteri-zations focus on lowering the distortion by setting the boundary free with the help of distance from a center vertex to all the boundary vertices. Hence these parameterizations strongly depend on the determination of the center vertex. In this paper,we introduce two methods to determine the center vertex automatically. Both of them can be used as necessary supplements to the existing measured boundary methods to minimize the common artifacts as a result of the obscure choice of the center vertex. In addition,we propose a simple and fast measured boundary parameterization method based on the Poisson's equation. Our new approach generates less conformal distortion than the fixed boundary methods. It also generates more regular domain boundaries than other measured boundary methods. Moreover,it offers a good tradeoff between computation costs and conformal distortion compared with the fast and robust angle based flattening (ABF++).
