A first case of free shape representation is given by the Free-Form Deformation (FFD) technique, introduced first on computer graphics by Sederberg and Parry in the late 80s [297]. This technique consists of embedding the shape to be deformed inside a control volume and then of modifying – by acting on a lattice of control points – the metrics of this space and thus the shape embedded into it, rather than modifying the shape directly. A modification of the control points position thus results in a deformation inside the control volume and, automatically, of the computational FE mesh. A simple physical analogy for FFD is to consider a parallelepiped of clear matter, flexible (at the limit, like a jelly) material in which we embed an object we wish to deform. The object is also flexible, so that it can be deformed along with the external parallelepiped.
Based on tensor product of splines, FFD inherits from boundary parametrization techniques the possibility to handle with global deformations by acting on a set of control points [10, 40], but provides an easier tool – since any explicit parametrization is required – which can be applied to virtually any geometrical model.
A simple version of the FFD construction is defined as follows, as a mapping from Rd toRd
through a d-variate tensor product Bernstein polynomial. For the sake of simplicity, we restrict ourselves to the two-dimensional case (d = 2), extension to three dimensions is straightforward.
Given a fixed rectangular domain D containing the reference domain Ω ⊂ D we wish to deform, we introduce an affine map
Ψ : D → ˆD ≡ (0, 1)2, x = Ψ(x),ˆ x ∈ D,
with Ψ(·) a monotonic function, in order to define FFD – in a simpler way – in the coordinates ˆ
x = (ˆx1, ˆx2) of the spline parameter space (0, 1)2. We thus select an ordered lattice of control
points Pl,m∈ (0, 1)2 (the unperturbed configuration), so that
Pl,m= l/L m/M , l = 0, . . . , L, m = 0, . . . , M.
A perturbation of the control points positions is specified by a set of (L + 1)(M + 1) parameter vectors µl,m∈R2 so that the perturbed configuration of the control points results in
Pol,m(µl,m) = Pl,m+ µl,m, (2.55)
giving in total 2(L + 1)(M + 1) possible degrees of freedom. Very often, only small subsets of these are selected as design variables if we want to perform a sensible geometrical model order reduction; moreover, several rows or columns of control points can be fixed to obtain desired levels of continuity or to “anchor” certain parts of the domain. In general, we indicate the effectively free scalar-valued parameters chosen as design variables (or actual degrees of freedom) as µ1, . . . , µp,
each corresponding to the displacement of a control point in either the ˆx1 or the ˆx2 direction, i.e.
From now on we assume that the shape parametrization involves only those µl,m corresponding to the actual degrees of freedom, omitting the parameters that have been fixed. in this way, we denote by µ = (µ1. . . , µp), even if also other control points – which do not correspond to
effective design variables – obviously go under displacement.
We thus construct a parametric domain map ˆT (·; µ) : ˆD → ˆDo(µ) by which the updated geometry
is computed as follows: ˆ T (ˆx; µ) = L X l=0 M X m=0 bL,Ml,m (ˆx)Pol,m(µl,m) = L X l=0 M X m=0 bL,Ml,m (ˆx)(Pl,m+ µl,m), (2.56) where bL,Ml,m (ˆx) = bLl(ˆx1)bMm(ˆx2) = L l M m (1 − ˆx1)L−lˆxl1(1 − ˆx2)M −mxˆm2 (2.57)
are (Bézier) tensor products of the univariate Bernstein basis polynomials
bLl(ˆx1) = L l ˆ xl1(1 − ˆx1)L−l, bMm(ˆx2) = M m ˆ xm2(1 − ˆx2)M −m.
defined on the unit square ˆD with local variables (ˆx1, ˆx2) ∈ (0, 1)2. The shape and continuity
of a deformation within the volume is related to the degree L, M of the Bernstein polynomials. Finally, the FFD mapping T (·, µ) is obtained as the composition
T (·; µ) : D → Do(µ), T (x; µ) = Ψ−1◦ ˆT ◦ Ψ(x; µ); (2.58)
in particular, the parametrized domain Ωo(µ) is obtained as Ωo(µ) = Ψ−1◦ ˆT ◦ Ψ(Ω; µ); see
Fig. 2.2 for a representation of the mapping construction. An example of FFD mapping to represent deformations of a two-dimensional airfoil of the NACA family is shown in Fig. 2.3. In this case, the design parameters are given by the p = 8 vertical displacements of the control points located in the interior part of the domain.
Figure 2.2: Schematic diagram of FFD technique: unperturbed control points Pl,m, perturbed
control points Po
l,m(µl,m), map ˆx = Ψ(x), ˆT (ˆx; µ) and resulting FFD map T (x; µ) = (Ψ−1◦ ˆT ◦
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
Figure 2.3: Example of shape deformation of a NACA0012 airfoil obtained through FFD mapping: reference configuration and control points (left), deformed configuration and control points (right).
Bernstein basis polynomials are used mainly because they fulfill the partition of unity
L X l=1 bLl(ˆx1) = M X m=1 bMm(ˆx2) = 1 (2.59) and positivity bL,Ml,m (ˆx) ≥ 0 ∀ˆx ∈ (0, 1)2
properties. Using the first property, it can be shown that FFD are a particular case of the transformations defined as perturbation of the identity, i.e. T (x; 0) = x. A more detailed analysis of this class of mappings will be addressed in sect. 4.4. One benefit of the simple FFD defined above is that the parametric transformations of the reference domain are simple polynomial functions in spatial coordinates and parameter, and they can be evaluated numerically through the stable De Casteljau algorithm [95], while their derivatives can be expressed in terms of tensor products of lower order Bernstein polynomials by means of the following formula [95]:
∇bL,Ml,m (ˆx) = " L(bL−1l−1(ˆx1) − bL−1l (ˆx1)bMm(ˆx2)) M (bM −1m−1(ˆx2) − bM −1m (ˆx2)bLl(ˆx1)) #T . (2.60)
This gradient formula is particularly useful for evaluating the parametrized tensors (2.18)–(2.20) appearing in the parametrized formulation (1.1)-(1.2) of our PDE problems, which depend on the Jacobian (2.19) of the map. Exploiting the partition of unity property (2.59), thanks to the formula (2.60), in the FFD case we obtain
JT(x; µ) = JΨ−1(x) " I + L X l=0 M X m=0 ∇bL,Ml,m (Ψ(x))µl,m # JΨ(x). (2.61)
Thus, not only the FFD map T (·; µ) can be stably evaluated, but also its Jacobian JT(·; µ) as well
as the parametrized tensors (2.18)–(2.20). Moreover, the determinant of the Jacobian provides us with control over the volume change that a body experiences under FFD: by computing | det(JT)|,
it can be shown that the largest and smallest polynomial coefficients provide an upper and a lower bound, respectively, on the volume change FFD. A condition on the well-posedness of the FFD map, ensuring also that the connectivity of the lattice of control points is preserved, will be briefly discussed in Sect. 4.4.
Concerning generality and quality of shapes, FFD techniques lead to low-dimensional parametriza- tions without loss of accuracy, since perturbations on parameters yield smooth shape deformations even if control points are not related to the shape boundary. In particular, parametric curves and surfaces remain parametric under FFD, geometrical singularities can be taken into account (since the initial shape including its singularities is deformed) and the smoothness of the deformation is controlled thanks to the Bernstein polynomials appearing in the map.
Regarding efficiency, the number of design variables depends on the user’s choice: we can keep fixed a subset of control points or only allow them to move in one direction; this allows the user to keep the number of FFD parameters to a desired low level (in our case P < 10 is typical). In particular, the number and position of control points chosen have a deep impact on FFD flexibility: it is crucial to maximize the influence of the control points by placing them close to the more sensitive regions of the configuration. Adaptive procedures for the selection of the control points based on sensitivity analysis and correlations are also available (see e.g Sect. 4.8).
FFD have been largely employed for parametrization and optimal design of aerodynamic surfaces such as wings, for instance by the group of Désidéri [88, 82, 11] and by Samareh [287]. A nice study of shape optimization of fluid domains by means of FFD techniques for some examples of engineering interest can be found in a paper by Lehnhäuser and Schäfer [189].
In the RB context, FFD has previously been proposed as a parametrization technique for inverse airfoils design in potential flows by Lassila and Rozza [185] and for thermal flows control by the author with Lassila and Rozza [281]. In this Thesis, we aim at enhancing the computational performance in the shape optimization process for viscous flows, coupling FFD and RB methods for Stokes/Navier-Stokes flows, developing what started in these two previous works.