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Figure 1.10. Example of a hollow cylinder described with a quadratic NURBS volume

In the rest of this book, we may omit the degree of the functions when unnecessary. In this case, the notations of the form will simply turn into Ri. In addition, we will sometimes use a matrix form to define the spline objects for ease of reading, with respect to the implementation. In this respect, the general expression for NURBS (thus also including B-spline) geometries with parameter ξ ∈ d ⊂ ℝ^d (with d being the parametric dimension of the spline entity, i.e. d = 1 for curves, d = 2 for surfaces and d = 3 for volumes) will be simply written as:

[1.23]

      where R and x denote the matrix of the NURBS basis functions and the vector collecting the locations of the associated control points, respectively. Considering, for example, control points in 3D, R and x, formally read as:

      [1.24]

Pi in [1.23] is obviously the subset of x gathering the coordinates (xi, yi, zi) of the ith control point.

      1.2.3. Design features and shape parameterization

Besides being attractive for generating geometries, B-splines and NURBS are interesting because they offer great flexibility in terms of shape control. This certainly explains why most of the CAD software is based on these spline technologies. In this work, which ultimately intends to perform the shape optimization of structures, we will deeply benefit from many of the appealing properties of B-splines and NURBS, especially their efficient refinement procedures and their flexible shape manipulation possibilities. Key aspects regarding these two features are given in the following.

      1.2.3.1. Refinement

      B-splines and NURBS can be refined while leaving the underlying geometry intact. There are different refinement mechanisms. The first possibility consists of inserting new knots into the knot-vectors. This process is called knot-insertion. More precisely, let us take the case of a p-degree B-spline curve, defined from knot-vector with n₁ control points Pi. Knot-insertion consists of adding k₁ new (interior) knots in the knot-vector. The new refined version of the B-spline curve takes the same degree p, but is built from the new knot-vector, which concatenates the initial knots with the inserted ones (in ascending order). The new knot-vector counts n₁ + k₁ + p + 1 knots. Ultimately, the only input that remains to be able to define the refined spline object is the vector collecting the control points. In other words, we need to compute the coordinates of the n₁ + k₁ control points denoted here Interestingly, it takes the form of a linear application:

      [1.25]

      where D^p is the refinement matrix (size l(k₁ + n₁) × ln₁, with l being the dimension of the physical space) and vectors and x gather the control point positions associated with and Pi, respectively (the same notation as in [1.23]). The refinement matrix is sparse and is obtained recursively (Cottrell et al. 2009), since it is based on formula [1.13]:

      [1.26]

      [1.27]

An example of refinement through knot-insertion, starting with a quadratic B-spline curve, is presented in Figure 1.11 (middle). We can observe the changes for the basis functions and for the control points, while leaving the initial geometry intact.

Figure 1.11. Refinement procedures applied to a B-spline curve

      The other refinement possibility is the degree elevation. As its name suggests, it consists of increasing the polynomial degree of the B-spline basis functions. Hence, it has to be noted that through the order-elevation, all values in the knot-vector are repeated to preserve the initial discontinuities of the derivatives of the functions (ensuring that the refined spline space includes the initial coarse spline space). As an illustration, let us take the example from Figure 1.11. The initial knot-vector Ξ is for quadratic B-spline functions while the refined knot-vector is for quartic functions. More precisely, Ξ and read:

      [1.28]

At the (interior) knot location, the regularity of the refined curve is still C¹:. Indeed, we have p − m = 4 − 3 = 1 (with m being the multiplicity of the interior knots). The computation of the new control points can be performed in three steps (Piegl and Tiller 1997):

      Steps 1 and 3 are performed using the knot-insertion procedure. The formula for the degree-elevation of a Bézier curve can be found for the example in Piegl and Tiller (1997). It is possible to elevate the degree from p to p + t in one step:

[1.29] Скачать книгу