5. Translational surfaces

<sup>L</sup>Int <sup>ξ</sup>i; <sup>η</sup><sup>j</sup> 

4.1. Creating a surface of interpolation

s are associated with η.

16 Topics in Splines and Applications

(right) are shown.

[5, 8, 9].

j 0 <sup>¼</sup> xij; <sup>S</sup>Int

The following steps describe creating a surface of interpolation:

Figure 15. We depict the mapping between physical and computational spaces.

klijð Þ¼ <sup>ξ</sup>; <sup>η</sup> <sup>C</sup><sup>k</sup>

In applications of practical interest, usually cubic piecewise continuous curves are preferred because they provide a global C<sup>2</sup> representation that is smooth enough, called a bicubic surface

1. An input control polygon, whose points are in R3, is provided. They correspond to data that is structured and ordered, which is usually a matrix-type array of points (see left side of Figure 16). For simplicity, points in the i-direction are associated with the ξ parameter while

Figure 16. The first two steps to interpolate structured data are depicted: input control polygon (left) and ξ-interpolants

<sup>i</sup>ð Þ <sup>ξ</sup> <sup>⊗</sup> <sup>D</sup><sup>l</sup> j

ð Þ η ; ð Þ ξ; η ∈ Ω (34)

These surfaces are again a two-parameter mapping, <sup>σ</sup><sup>T</sup> : <sup>R</sup><sup>2</sup> ! <sup>R</sup><sup>3</sup> , but their construction procedure is simpler than interpolation surfaces; see, for instance, [5, 8, 9]. The idea here is just literally translating a curve α along another curve β, which yields

$$
\underline{\sigma}^T(\xi, \eta) = \underline{\sigma}(\xi) + \underline{\beta}(\eta). \tag{35}
$$

This idea became very popular in CAGD systems long time ago. Those systems usually support a command which allows extruding a geometrical entity, for instance, a cylinder can be easily created by extruding a circle along a straight line. Figure 18 shows the above procedure applied to an aircraft wing where an NURBS airfoil profile was translated or extruded along a straight line accordingly. We interpolated an NACA 65 polyline with a NURBS curve as we mentioned in Section 3.4.

This procedure becomes very useful in the geometrical reconstruction of oil reservoirs (RS). Indeed, we reconstructed the geometry of RS in [12] by using B-spline surfaces. The technique exploits input mesh's simplicity to build a robust piecewise continuous geometrical representation using Bèzier bicubic patches. We manage the reservoir's topology with interpolation

surfaces, while translational surfaces allow extrapolating it toward its side burdens. After that, transfinite interpolation can be applied to generate decent hexahedral meshes. Figure 19 shows a sample translational surface that we obtain by extruding a curve that interpolates the reservoir's edge as shown. We render the surfaces in blue color with a white wireframe, while the RS is the color-contoured surface that represents the porosity, a scalar property. We tackle the RS itself after interpolating the control polygon that Figure 20 highlights in red color. The polygon is a 17 � 9 array of points representing the RS topology. The procedure works well for a variety of so-called open-to-the-public RS data sets that we reconstructed in [12]. It is also possible to utilize these NURBS curves and surfaces as interfaces for gluing nonmatching

Scalar and Parametric Spline Curves and Surfaces http://dx.doi.org/10.5772/intechopen.74929 19

In the context of applications in statistical analysis involving very high dimensional data sets, response surfaces are growing popularity. By running the simulations at a set of points (e.g., experimental design) and fitting response surfaces, i.e., splines, for instance, to the resulting input-output data that is characterized by sparsity, we can obtain fast surrogates for the objective function for optimization purposes [14, 15]. The appeal of the latter approach goes beyond reducing runtime. Since the method begins with experimental design, statistical analyses can be done to identify which input variables are the most important, and thus we can create "main effect plots" to visualize input-output relationships [14]. We must recognize interpolation methods in which the basis functions are fixed and those in which they have parameters that are tuned (e.g., kriging, which has a statistical interpretation that allows one to construct an estimate of the potential error in the interpolator). We refer the reader to [14, 15]

There are different ways to approximate a function of several variables: multivariate piecewise polynomials, splines, and tensor product methods, among others. All these approaches have advantages and drawbacks, but if the rank of the linear system to solve may become large, a natural choice is radial basis functions, which are also useful in lower dimensional problems [14, 16, 17]. This may be particularly true if the input data is scattered, which excludes tensor product methods at first glance. Duchon splines are a class of positive definite and compactly supported radial functions, which consist of univariate polynomial within their support. It can be proven that they are of minimal degree and unique up to a constant factor, for given smoothness and space dimension [18]. They are particularly suitable to compute interpolants

interfaces for the finite element method as we showed in [13].

6. Duchon splines

for further reading.

for very large scatter datasets [17].

Duchon splines, denoted herein as s, are defined by [17, 18]

<sup>s</sup>ð Þ¼ <sup>x</sup> <sup>X</sup> j

<sup>φ</sup>ð Þ¼ <sup>r</sup> <sup>r</sup><sup>2</sup>

<sup>r</sup><sup>j</sup> <sup>¼</sup> <sup>x</sup> � xj � � � �

ln r,

λ<sup>j</sup> � φ r<sup>j</sup>

� � <sup>þ</sup> pnð Þ<sup>x</sup> ; n <sup>¼</sup> <sup>2</sup>, <sup>3</sup>

(36)

Figure 18. An aircraft wing by translating an NACA profile accordingly.

Figure 19. A translational surface.

Figure 20. An interpolation surface.

surfaces, while translational surfaces allow extrapolating it toward its side burdens. After that, transfinite interpolation can be applied to generate decent hexahedral meshes. Figure 19 shows a sample translational surface that we obtain by extruding a curve that interpolates the reservoir's edge as shown. We render the surfaces in blue color with a white wireframe, while the RS is the color-contoured surface that represents the porosity, a scalar property. We tackle the RS itself after interpolating the control polygon that Figure 20 highlights in red color. The polygon is a 17 � 9 array of points representing the RS topology. The procedure works well for a variety of so-called open-to-the-public RS data sets that we reconstructed in [12]. It is also possible to utilize these NURBS curves and surfaces as interfaces for gluing nonmatching interfaces for the finite element method as we showed in [13].
