Whereas Free-Form Deformation show great flexibility and easiness of handling, they suffer from some limitations, since (i) the control points cannot be chosen freely, being the nodes of a lattice
(ii) it is not possible to perform a boundary control and (iii) the process is not interpolatory. In
particular, using a rectangular lattice to describe deformations of irregular or complex shapes makes the choice of control points crucially important; for example, control points located far from the boundary to be optimized have less influence. Choosing the subset of active points is therefore a critical issue, highly problem-dependent.
In order to overcome these limitations, an alternative method can be used for shape parametriza- tion – still based on a set of control points, but which is in addition interpolatory: the Radial Basis Function (RBF) technique. Originally used in neural networks and later applied to the solution of PDEs in what is known as meshless methods, RBFs are now a widely used method for scattered interpolation and reconstruction of surfaces and volumes. In particular, they provide a general and flexible way of interpolating data in multi-dimensional spaces, even for unstructured data where it is often impossible to apply polynomial or spline interpolation [155]. For a general introduction on RBF method see for instance the monographs by Buhmann [50] or Wendland [320].
We first describe the RBF interpolation procedure (which is employed in the following chapters also for different goals than shape parametrization, see e.g. Sect. A.3.2), focusing afterwards on the construction of parametric maps based on this strategy. Without loss of generality, we restrict ourselves to the two-dimensional case, where we denote by X = {x1, . . . xk} ⊆R2a set of k > 2
non-collinear points (usually called centers, with xi= (xi1, xi2)) and by Y = {y1, . . . yk} ⊆R the
Radial Basis Function Φ(r)
Linear r
Spline type (Rλ) |r|λ, λ odd
Thin-Plate Spline (TPSλ) |r|λlog |r|, λ even
Multiquadric (MQ) (1 + r2)1/2 Inverse Multiquadric (IMQ) (1 + r2)−1/2
Inverse Quadratic (IQ) (1 + r2)−2
Gaussian (GS) exp(−r2)
Table 2.1: Several options for radial basis functions
Then, the RBF interpolant τ :R2→R has the form
τ (x) = π(x) +
k
X
i=1
wi Φ(kx − xik). (2.62)
In (2.62), the function Φ is a fixed basis function, which is radial with respect to the Euclidean distance kxk, π(·) :R2→R is a bivariate polynomial function of (low) degree p, {w
i}ki=1, wi∈R
is a set of coefficients corresponding to the centers xi. Common choices for RBFs are listed in
Tab. 2.1. Moreover, in many cases it is appropriate to scale the basis function with a so-called shape parameter , so that the basis function is replaced by Φ(r) = Φ(r).
The coefficients {wi}ki=1 and the polynomial π(·) are determined so that τ interpolates the data
y1, . . . , yk (interpolation constraints)
τ (xi) = yi, 1 ≤ i ≤ k (2.63)
and satisfies the additional requirements (side constraints)
k X i=1 wiϕ(xi) = 0 ∀ϕ ∈ Π q 2, q ≤ p, (2.64)
where Πq2 is the space of all polynomials of degree up to q ≥ 1 in 2 unknowns. By imposing the side constraints, we require that each polynomial up to degree p is interpolated exactly; we discuss in Sect. 4.4 the geometrical meaning of this requirement, providing a general analysis of RBF schemes and some conditions on the well-posedness of RBF parametric mappings. For the sake of simplicity, we consider π(x) to be a polynomial function of degree p = 1, so that
π(x) = γ0+ γ1x1+ γ2x2;
moreover, we denote by w = (w1, . . . , wk)T ∈Rk and γ = (γ0, γ1, γ2)T ∈R3 the vectors whose
components are the coefficients w1, . . . , wk for the RBFs and for the polynomial, respectively.
Constraints (2.63)-(2.64) thus lead to the following linear system for the coefficients vector [w γ]T ∈Rk+3: M P PT 0 w γ = y 0 (2.65)
where y = (y1, . . . , yk)T ∈Rk, M ∈ Rk×k is the interpolation matrix of components
(M)ij = Φ(kxi− xjk), 1 ≤ i, j ≤ k
P = 1 x11 x12 .. . ... ... 1 xk1 xk2 .
Provided that the interpolation matrixM is nonsingular, we can invert (2.65) and obtain
w = (I − M−1PMPPT)M−1y, γ =MPPTM−1y,
whereMP= (PTMP)−1, so that the RBF interpolant (2.62) is uniquely identified.
For shape parametrization in d dimensions – as well as for mesh deformation – we need d interpolating functions, one for each coordinate direction. In the two-dimensional case, we shall consider the centers {P1, . . . Pk} ⊆R2 as the unperturbed configuration of the control points in
the reference domain Ω, being Pi= (Pi1, Pi2) for any i = 1, . . . , k, and {Po1, . . . P
o
k} ⊆R
2 as the
perturbed configuration of the control points in the original domain Ωo.
By introducing a set of (now vector-valued) coefficients {wi}ki=1, wi = (wi1, wi2)T ∈R2, and
denoting by π(x) = c +Ax a polynomial function of degree 1, with c ∈ R2 andA ∈ R2×2, the RBF map (2.62) can be rewritten in a compact form as
τ (x) = c +Ax + WTs(x), (2.66)
where s(x) = (Φ(kx − P1k), . . . , Φ(kx − Pkk))T ∈Rk andW = [w1, . . . , wk]T ∈Rk×2. Regarding
shape parametrizations, the polynomial of degree 1 represents the affine part of a deformation (rotation and/or scaling), while the termWTs(x) depending on the control points adds a nonaffine
contribution. The RBF map (2.66) is thus a function of 2k + 6 coefficients in the two-dimensional case, which are determined by looking for a transformation such that: (i) each control point
Pi in the unperturbed configuration is mapped onto the corresponding control point Poi in the
perturbed configuration, and that (ii) each affine transformation is recovered exactly (i.e. the RBF interpolation is invariant with respect to rigid motions). This is equivalent to impose both the interpolation and the side constraints (2.63)-(2.64), which now read:
τ (Pi) = Poi, i = 1, . . . , k, (2.67) k X i=1 wi= 0, k X i=1 Pi1wi= k X i=1 Pi2wi= 0. (2.68)
We point out that when dealing with mesh deformation or coupling between two different phases, interpolation and side constraints (2.67)-(2.68) guarantee the conservation of virtual work, total load and total momentum when the displacements are transferred from one phase to another [155, 217].
In order to fit the RBF technique in our parametrized framework, let us express the deformed positions Po
i of the control points as
Poi(µi) = Pi+ µi, i = 1, . . . , k,
where µi= (µi1, µi2) is the displacement of the i-th control points. As in the FFD case, only
small subsets of p ≤ 2k displacements are selected as design variables – accordingly to some problem-dependent criteria – if we want to perform a sensible geometrical reduction. We indicate the actual degrees of freedom as µ = (µ1, . . . , µp), each corresponding to the displacement of a
control point in either the x1 or the x2 direction, i.e. to one of the components of a vector µi.
The parametric map T (·; µ) : Ω → Ωo(µ) is thus given by
Figure 2.4: Schematic diagram of the RBF technique: on the left, the reference (or initial) configuration Ω and the unperturbed control points Xi, i = 1, . . . , k; on the right, the deformed
(or target) configuration Ωo(µ) and the displaced control points Yi, i = 1, . . . , k.
where the coefficients c(µ), A(µ), W(µ) satisfy the constraints (2.67)-(2.68) and now depend on µ ∈ Rp. As before, by denoting M ∈ Rk×k the interpolation matrix whose components
(M)ij = Mij are given by
Mij = Φ(kPi− Pjk), 1 ≤ i, j ≤ k (2.70)
andP ∈ Rk×3the matrix defined by imposing the side constraints: P = 1 P11 P12 .. . ... ... 1 Pk1 Pk2 .
the coefficients (c(µ))m= cm(µ), (A(µ))mn= Amn(µ) and (W(µ))im= wim(µ), for 1 ≤ m, n ≤
2, 1 ≤ i ≤ k appearing in the map (2.69) are obtained by solving the following linear systems: M P PT 0 W1(µ) γ1(µ) = Po 1(µ) 0 , M P PT 0 W2(µ) γ2(µ) = Po 2(µ) 0 (2.71) being Wm(µ) = (w1m, (µ) . . . , wkm(µ))T, Pom(µ) = (P1m(µ), . . . , Pkm(µ))T, m = 1, 2 and γ1(µ) = (c1(µ), A11(µ), A21(µ))T, γ2(µ) = (c2(µ), A12(µ), A22(µ))T.
A representation of the mapping construction is shown in Fig. 2.4. An example of RBF mapping used to model deformations of a two-dimensional airfoil of the NACA family is shown in Fig. 2.5. In this case, the design parameters are given by the p = 6 vertical displacements of the control points located along the boundary sides.
Control points are usually chosen close to the boundary of the shape to be deformed (see Fig. 2.4), even if in our case the parametrization is constructed independently from the computational mesh. For a small number of control points, as in our approach, linear systems (2.71) can be efficiently solved by a suitable direct method (we remark that matrix factorization is not depending on the parameters). When using a large number of control points – as for example in fluid-structure interaction coupled problems or, more generally, when dealing with mesh motion through RBF – the matrix appearing in (2.71) may be badly conditioned and some difficulties may arise. In these cases, suitable scaling or preconditioning strategies may help, as discussed for example in [50].
In general, RBF maps provide more local control of the shape due to the ability to fine tune the shape by adding locally more control points and by choosing the type and support radius of the shape functions. The asymptotic behavior of Φ(h) is different among the selected basis functions, and the choice on the various possibilities is performed according to shape regularity and to convergence properties of the numerical method used to compute the coefficients appearing in (2.62). For example, the linear RBFs exhibit better convergence properties, while the cubic RBFs guarantee an enhanced shape smoothness.
RBFs can be divided into two groups, functions with compact support and functions with global support. Whenever Φ(·) is an increasing function with global support, the influence of a center on an evaluation node increases with the distance of the two nodes. Thus, the global character of these functions tend to smooth out local effects, so that a large support radius yields a good approximation order. On the other hand, a full matrix system has to be solved10. Introduced by
Wendland [319] and Wu [322], compactly supported RBFs attempt to reduce the bandwidth of the otherwise full interpolation matrix (2.70). However, in this case the choice of the support radius of the RBF functions, which is an additional parameter to be tuned, may become a critical issue. Increasing the support radius tends to produce ill-conditioned interpolation systems and lead to the Runge phenomenon [44]. On the other side, using a support radius that is too small yields a large interpolation error, even though the system can be more easily solved. To conclude, a drawback of the many different types of RBFs proposed in literature is that often the final shape is quite sensitive to the choice of the shape function and its support radius parameter.
0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 −0.1 0 0.1 0.2 0.3
Figure 2.5: Example of shape deformation of a NACA0012 airfoil obtained through RBF mapping: reference configuration and control points (left), deformed configuration and control points (right).
Depending on the application at hand, we will discuss some possible choices of RBFs functions. In any case, the versatility of RBFs comes at a price of complexity in guaranteeing the quality of the remeshing interpolant.
Because of their excellent approximation properties, RBFs have been successfully applied to many different areas, such as computer graphics, mesh deformation [269, 69] and interpolation between nonmatching meshes in FSI computations [25, 70]. Another application of RBFs is in simulation based optimization, where they are used to build a surrogate model for less expensive black-box optimization; see e.g. [156] and references therein. In optimal design problems, RBFs have been used as shape parametrization for airfoil shape optimization in [217, 155], as well as for topological optimization of structure in [318, 317].
10Actually, this is not a big issue in our context, since k is rather small – k = O(10) if we desire a low-dimensional
shape parametrization – but may become somehow undesirable in applications of larger dimensions, such as mesh deformations or interpolation between non-matching meshes.