SURFACE MODELING FOR CAD / CAM BASED ON NURBS

In the digital design process, surface modeling is required to be as accurate as possible for the effective support of production as well as for numerical performance analysis. This article reviews the geometric modeling techniques, based on non-uniform rational B-spline (NURBS). The NURBS surface can be readily translated into ma ny CAD/CAM packages (Computer Aided Design/Computer Aided Manufacturing), which is more convenient for visualization performance and finite element methods.


INTRODUCTION
Surface modeling is the key to integration of design, analysis, manufacturing, and other calculation [3].There are two principle categories for surface fitting techniques.The first one works with reverse engineering in two steps; first the given data points are rearranged into the rectangular mesh, then the surface is constructed by some matrices inversion and Bspline algorithm procedures.The first problem is that the data points are often scattered.The second problem comes from the matrices inversion.In fact, the matrices inversion gets the ill conditioned problem, and it must also

OVERVIEW OF THE B-SPLINE FITTING ALGORITHM
Surface fitting basis B-spline surface data D(u,w) is given by: , TAÏ P CHÍ PHAÙ T TRIEÅ N KH&CN, TAÄ P 14, SOÁ K4 -2011 Trang 45 Let x i and y j be knots value and B i,j is the vertex point of B-spline for data fitting, and Ni,k(u) and Mj,l(w) are the basis functions.
Eq. 1 shows that shape of NUB surface will be regenerated if we change the location of vertex point and knot value, as in Fig. 1.In GA process, we changed the shape of surface automatically in this way.Therefore, the final surface can get the good solution time by time.
In the matrix form of NUB surface point of Eq. ( 2), moving in space with 2 degrees of freedom, u and w, is given by:  For curve fitting, an optimization method with GA has been presented by Yoshimoto [4].
However, the matrices inversion is not a solution for irregular given data points.Instead, moving the location of the vertex point using GA is efficient way to improve the surface quality.

NEW APPROACH TO SURFACE FITTING FOR THE GIVEN INTERIOR
DATA POINT BY USING GA

Vertices Encoding for Initial Population
For an initial population, usually the base surface is created by vertices.The vertices can be generated with the same x and z coordinate value of the given data points and deviation δ in y direction (see in Fig. 3).The variable design has a deviation with the deviation size δ as given by Eq. ( 6).
[ ] [ ] Therefore, the surface quality improves a good result after each time.Deformation of a NUB surface can be achieved by moving the vertices that define it (see Fig. 5)., In that case, the location of vertices will be optimized to fit the given data points using GA technique.

1. Ship Hull Surface
The Fig. 6 illustrates the given data points of surface.In this case, the surface shows total error values converged at 20,000th generation and the best fitting after 20,000 generations (see Fig. 7&8).The normalized error value after 20,000 generations is computed in Table 1.

Complicated Surface
The complicated mesh (Fig. 9) consists of 91 given data points.The initial result and the best result at 20,000th generation were illustrated from Fig. 10 and 11.Finally, a value of normalized error converged at the 20,000th generation (see Table 2).

MÔ HÌNH HÓA BỀ MẶT TRONG CAD/CAM DỰA TRÊN THUẬT TOÁN NURBS
Lê Tất Hiển (1) , Nguyễn Xuân Hùng (2) , Võ Trọng Cang (1) (1) Trường ðại học Bách Khoa, ðHQG-HCM avoid round off error magnifications in backsubstitution calculation and large storage capacity.The second surface generation technique is to approximate the given data points by B-spline algorithm.Many conventional methods have been proposed [8,1].The main problem is the parameterization of B-spline surface.It needs to be estimated from an initial unknown surface.The approximation of surface fitting using genetic algorithm is developed by Birmingham [5], Le et al [6].However, this method gets stuck in case of complicated surfaces.The main contribution of this research is to use non uniform B-spline (NUB) surface fitting through a GA technique.This approach has more advantages with regard to surface representation, and fairness of the interior surface.

Fig 1 . 1 . 1 .
Fig 1. Rectangular vertex points Where [D] is an r x s x 3 matrix containing the 3D coordinates of surface data points, [C] is an r x s x n x m matrix of the products of the basis functions, and [B] is an n x m x 3 matrix of the 3D coordinates of the required vertices.If [C] is not square, the solution is given by:

Fig 2 .
Fig 2. The difficulties of fitted surface in matrices inversion method

Fig 3 . 47 Fig 4 .
Fig 3.The generation of vertices from the given data points 3.2.Crossover Process The idea here is to exchange a single knot value in two parents to form two new individuals.Fig. 4 shows that the individual 1 received a new sub-string from individual 2 to generate a new individual in a next population.

Fig 5 . 3 . 4 .
Fig 5.The effect of moving one of the vertex points of the surface