## 3 The Multilevel Approach
### First Level
The first and last columns are related to the leading and trailing edge boundary conditions and can be used to define the spanwise deflection law. When the parameterization technique is used to introduce the morphing deformations, the matrix can be partitioned into three sub-matrices: the first and last portions are related to morphing the leading and trailing edge, respectively, while the middle one is related to the wing-box shape. In this case of morphing, the extra coefficients acting on the leading and trailing edge represent the optimization variables, while the other extra coefficients can be evaluated by a least-squares fitting that can meet the constraints related to the 3D wing-box volume. The mathematical formulation of this implicit constraint is similar to the parametric identification described in [34], but it is extended to the three-dimensional domain. Two different shapes of a morphing trailing edge installed on the GRA wing have been generated while considering a chordwise rear spar position equal to 80% and two different vertical trailing edge position laws reported in Figure 5. The corresponding shapes are shown in Figure 6 where the wing representation was divided into three sub-elements, which were naturally defined by two kinks. Figure 5: Vertical trailing-edge position parameter. This first level of optimization aims at defining the best wing shape changes that satisfy specific mission requirements. When the CST is used within the aero-structural optimization loop, the aerodynamic optimality can concern the efficiency with respect to reducing the aircraft's fuel consumption, or it can concern an improvement of an arbitrary performance index \(I_{aero}\) over a wide range of different flight conditions. The optimal morphing shape is found according to the structural constraints that limit the structural response of the skin. A Constant Cross section Length (CCL) strategy is adopted to maintain a zero value for the axial strain and to limit the bending stress. The structural model of the skin is not based on the finite element method, but it is embedded into the CST formulation and exploits the capability to analytically describe the shape changes due to the morphing. Due to these constraints, in addition to an implicit constraint on the wing-box volume, aerodynamic analyses are performed only on feasible morphing shapes. A typical aerodynamic shape optimization problem, including morphing skin structural limitations, can be written as follows,  
\[\min|\max~{}I_{aero} \tag{7}\]  
such that  
\[\mathbf{T}_{m\cdot(n-r)}^{T}\mathbf{T}_{m\cdot(n-r)}\mathbf{a}^{T} =\mathbf{T}_{m\cdot(n-r)}^{T}\mathbf{f}\] \[\Delta\psi_{wing-box}(\eta) \equiv(\psi_{fs}(\eta),\,\psi_{rs}(\eta))\] \[\mid\Delta A\mid \leq k_{a}\cdot\,A_{u}\] \[\mid\Delta L_{skin} \mid =0.\] Figure 4:0  
According to the optimization algorithm described in [34], the first term represents the implicit wing-box constraint. It acts on the three-dimensional wing-box region, which is delimited by the front and rear spar positions, and it is described by \(l+1\) pairwise distinct coordinates coming from the CAD model of the undeformed wing. \(\mathbf{T}_{m\cdot(n-r)}\) represents the \((l+1)\times m\cdot(n-r)\)_reduced CST-Vandermonde matrix_, where \(r\) is the number of perturbed coefficients, and \(\mathbf{f}\) is the corresponding \((l+1)\) vector. The unique solution of this linear system is equivalent to solving, in the least squares sense, the over-determined system that allows us to compute the \(m\cdot(n-r)\) extra coefficients of the matrix \(\mathbf{A}_{(\mathbf{m},\mathbf{n})}\), reorganized in the vector \(\mathbf{a}\). The \(l+1\) pairwise distinct real points are given and placed along the wing-box region, which must be kept undeformed. The \(r\) extra coefficients for each row of the matrix \(\mathbf{A}_{(\mathbf{m},\mathbf{n})}\) are the optimization variables that affect only the domain outside the wing-box region.