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Non-uniform rational B-spline (NURBS) surfaces are a generalisation of the B-spline surface patches; the key departure from NURPS is the weighting of the control points which makes NURBS surfaces rational. Let u and v define a bivariate patch (u, v _∈

shape determined by control points, Pi,j, weights wi,j and the NURBS basis functions Ni,m andNj,n: S(u, v) = Pm i=0 Pn

j=0Ni,p(u)Nj,q(v)wi,jPi,j

Pm

i=0 Pn

j=0Ni,p(u)Nj,q(v)wi,j

. (5.1)

For a more detailed review of NURBS curves and surfaces, please refer to Piegl and Tiller (1996).

The continuity of this type of surface depends only on the basis functions and not on the control points. NURBS have all the advantages of B-splines, while extending liberty of modelling. Rational cubic surfaces allow the construction of conic sections such as spheres and cylinders as well as free-form shapes. In contrast to the NURPS surface previously described, NURBS are simply rational polynomial patches of a fixed degree.

The construction of NURBS surfaces requires a quadrilateral mesh of m_×n points;

very few objects can be constructed appropriately with a single rectangular patch. A combination of patches to describe complex shapes can, however, lead to complications. These complications are often in the form of continuity at points where patches meet. For complex surfaces, deforming a NURBS patch can result in a number of problems:

• the patches can tear apart at the seams leaving a discontinuous geometry, or • the continuity could be made to force the curvature to zero creating flat spots on

the geometry in these regions.

In order to address these problems, high degree patches could replace the patches at these problem points. However, this adds more control points to the set of parameters, increasing the complexity of the model.

Another disadvantage of NURBS surfaces is that, if a local shape deformation is required, detail is added to the mesh of control points by adding more if necessary, but, unless the local patch can be trimmed and located to a specific position, extra control points are added where they are not necessary, thus increasing the parameter set without good cause. This is illustrated in Figure 5.4. If trimming of the surfaces is avoided, allowing the patch to span a section of the NURBS surface from inlet to filter, and if a small local

Figure 5.4: _{NURBS surface representation with a mesh of control points over a de-}

formation patch

Figure 5.5: _{An example of the inner wall surface of a real artery}

area within this patch is to be deformed, a complex deformation may require a much finer mesh of control points than those shown in Figure 5.4. This would add control points where no deformation is necessary and increase the complexity of the entire model, perhaps making this method unsuitable for a local geometry manipulation.

For a three-dimensional airbox study, these surfaces provide a good representation of the global shape and a surface can be defined with a coarse net of control points. The positions of these points can easily be modified to alter the shape of the airbox wall. The position of each control point requires only three design variables. If the modification of more than one control point at a time is required, then a compact set of design variables can still be retained for an efficient global optimization process for the first stage in the two stage process. Local manipulation, however, may increase the parameter count to beyond the practicable limit for use in optimization.

Figure 5.6: _{NURBS surface representation of artery (left) with the net of control}

points defining the lower right hand side of the bifurcation (right)

Figure 5.5 shows a surface fitted through a cloud of points collected via a scan of a patient artery cast. It is clear that to represent a realistic parametric carotid artery bifurcation model, a combination of NURBS surface patches is necessary to capture sufficient detail. For such a complex geometry, the general shape can be captured and an example of a CAD fit to the real artery shape can be seen on the left of Figure 5.6. The right of Figure 5.6 shows the net of control points controlling the NURBS patch defining the lower right hand side of the carotid artery bifurcation. To capture the global shape of the surface, a large number of patches is required and a fine net of control points to capture the complexity of the shape. The patch is not as straightforward as the deformation patch shown for the airbox. It is because of this that a global surface fit can be achieved but, even with the tangency conditions set at the boundary of the patch, when a control point near the boundary is displaced, resulting surface deformations become problematic with seam discontinuities.

Figure 5.7 illustrates the displacement of a control point near the patch boundary, its resulting surface deformation and the seam discontinuity at the join with its adjacent surface patch. It is because of this problem that this technique could not be considered for any local surface deformations.