3.2.1 Overall approach

Investigations on the leading edge airfoil and morphing mechanism provided at the start of the project concerned the design of the laminate composite wing skin characteristics to be chosen. The design parameters [Reference Koreanschi, Sugar Gabor, Acotto, Brianchon, Portier, Botez, Mamou and Mebarki29] considered for the skin are laminate plies thickness, material and plies orientation to obtain an appropriate stress distribution along the skin. The other parameter to determine is the minimum possible value of the actuator force that should be applied on the morphing mechanism to match as closely as possible the desired leading-edge target airfoil geometry.

Initially, the mechanism geometry and the original wing airfoil were schematised and imported in the MSC PATRAN/NASTRAN FEM solver, in which the structure was reconstructed and meshed. A first guess for the laminate composite characteristics (thickness, material and plies orientation) and the pressure distribution acting on the optimised profile and boundary conditions has been implemented. At the very beginning, a wing skin characterised by relatively high stiffness was considered and analysed. Successive analyses were then performed to obtain a more deformable configuration under the same applied actuation loads. For all the cases, symmetric and balanced composite laminates were preferred, as suggested by manufacturability indications [Reference Gay, Hoa and Tsai30]. For each stacking, analyses were conducted by progressively increasing the actuator force, starting from 100 N. The results were evaluated in terms of profile displacement and stress distribution on the skin. A total of seven iterations were carried out, progressively refining the laminate configuration toward higher deformability under the same loading conditions. As indicated in Table 1, the final configuration is labeled Stacking 7, denoting the outcome of these seven iterative analyses.

Table 1.

Omega stringer section dimensions

Table 1 Long description

The diagram illustrates the dimensions of an omega stringer section. It includes a cross-sectional view with labeled measurements: H is 5 millimeters, t is 1.5 millimeters, W is 5 millimeters, and W1 is 2.5 millimeters. The diagram also shows the relationship between these dimensions, indicating the height, thickness, and width of the stringer section.

Based on the initial estimated laminate configuration, different FEM analyses were conducted by imposing several values of applied forces on the mechanism. To ensure a cost-effective procedure, both linear and non-linear FEM analyses were conducted. For each analysis, a cost function was estimated using the control points chosen on the morphed and target airfoil geometry to assess the closeness of the two airfoils. Equation 1 gives a formulation of the cost function J, calculated as a least square estimation, in which $n$ is the number of control points on an airfoil section, $po{s_i}$ is the coordinate position of the considered control point in the morphed configuration and $po{s_{i,target}}$ is the coordinate position relative to the target airfoil. The symbols appearing in Equations (1–3) are described in the text for clarity.

(1) \begin{align} J = {\rm{\;}}\mathop \sum \limits_{i = 1}^n \sqrt {{{\left( {po{s_i} - po{s_{i,target}}} \right)}^2}} {\rm{\;}} \end{align}

The general formulation of the fitness function can be written in three-dimensional coordinates, as in Equation (2):

(2) \begin{align} {\rm{J\;}} = \mathop \sum \limits_{i = 1}^n \sqrt {{{\left( {{x_i} - {x_{i,Target}}} \right)}^2} + {{\left( {{y_i} - {y_{i,Target}}} \right)}^2} + {{\left( {{z_i} - {z_{i,Target}}} \right)}^2}} \end{align}

where:

$i$ = each node belonging to a section of the airfoil;

$n$ = the number of control points for each section (156);

${x_i}\;,\;{y_i}\;,\;{z_i}\;$ = the x, y and z coordinates of the control point ‘ $i$ ’ belonging to the morphed airfoil; and

${x_{i,Target}}$ , ${y_{i,Target}}$ , $\;{z_{i,Target}}$ = the x, y and z coordinates of the control point ‘ $i$ ’ belonging to the target airfoil shape.

By neglecting the out-of-section plane displacements, the cost function can be described by Equation (3), which is used in the optimisation process.

(3) \begin{align} {\rm{J\;}} = \mathop \sum \limits_{i = 1}^n \sqrt {{{\left( {{x_i} - {x_{i,Target}}} \right)}^2} + {{\left( {{z_i} - {z_{i,Target}}} \right)}^2}} \end{align}

Some authors propose different fitness function formulations, assigning empirical weight values to the control points, thereby giving more importance to those related to critical values of air pressure acting on the airfoil [Reference Wang and Yang16]. This is done to avoid the acceptance of eventual low values of least square error, which would correspond to unacceptable deformations. However, in this study, it was decided to create a more detailed visualisation of the airfoil matching quality by simply splitting the cost function evaluation into the upper and lower airfoil camber, as described in Equations (4–6).

(4) \begin{align} {J_{up}} = \mathop \sum \limits_{i = 1}^{n = 81} \sqrt {{{\left( {{x_i} - {x_{i,Target}}} \right)}^2} + {{\left( {{z_i} - {z_{i,Target}}} \right)}^2}} \end{align}

(5) \begin{align} {J_{low}} = \mathop \sum \limits_{i = 82}^{n = 156} \sqrt {{{\left( {{x_i} - {x_{i,Target}}} \right)}^2} + {{\left( {{z_i} - {z_{i,Target}}} \right)}^2}} \end{align}

(6) \begin{align} J = {J_{up}} + {J_{low}}\end{align}

Figure 4.

Flowchart of the skin optimisation procedure introduced in this research.

Figure 4 Long description

The flowchart illustrates the skin optimization procedure. It starts from a relatively high stiffness skin configuration. The process involves analyzing geometry, material properties, pressure, and force using finite element method analysis. If the deformation delta is less than the target, the skin stacking and thickness are changed. If not, the minimum cost function is estimated. The stress on the wing skin is then calculated. If the stress is too high or too low, the process loops back to change the skin stacking and thickness. If the stress is acceptable, the profile deformation matches the target, and the stress is optimized.

The control points are the nodes obtained by meshing the original section airfoils. The cost function has been evaluated in relation to three different sections of the leading edge: the free edge $({J_1}$ ), the mechanism section $({J_2}$ ) and the midsection between the mechanisms $({J_3}$ ) (Fig. 4). In this paper, the average of these three cost functions is considered as the most relevant parameter with which to assess the quality of the morphing airfoil deformation (Equation (6)). Moreover, the cost function is calculated for both the upper and lower camber airfoils, as described in Equations (7) and (8).

(7) \begin{align} {J_{avg}} = \frac{{{J_1} + {J_2} + {J_{3{\rm{\;}}}}}}{3} \end{align}

(8) \begin{align} {J_{avg\_up}} = \frac{{{J_{1\_up}} + {J_{2\_up}} + {J_{3\_up{\rm{\;}}}}}}{3} \end{align}

(9) \begin{align} {J_{avg\_low}} = \frac{{{J_{1\_low}} + {J_{2\_low}} + {J_{3\_low{\rm{\;}}}}}}{3} \end{align}

Another relevant parameter with which to evaluate the smoothness of the airfoil along the wingspan is the mean absolute deviation δ of the three sections considered around the average cost function value, as expressed in Equation (10). This parameter is also evaluated for both upper and lower camber airfoil (Equation (11) and (12), respectively).

(10) \begin{align} \delta = \frac{{\mathop \sum \nolimits_{i = 1}^3 \left| {{\rm{\;}}{J_i} - {J_{avg}}} \right|}}{3} \end{align}

(11) \begin{align} {\delta _{up}} = \frac{{\mathop \sum \nolimits_{i = 1}^3 \left| {{\rm{\;}}{J_{i\_up}} - {J_{avg}}} \right|}}{3} \end{align}

(12) \begin{align} {\delta _{low}} = \frac{{\mathop \sum \nolimits_{i = 1}^3 \left| {{\rm{\;}}{J_{i\_low}} - {J_{avg}}} \right|}}{3} \end{align}

If the displacement obtained by the results from FEM linear analyses does not reach the required displacement, another composite configuration with lower stiffness must be considered and the procedure re-iterated. The most suitable case among those analysed is the one corresponding to the minimum cost function value. This case would then be investigated in terms of stress distribution acting on the skin through non-linear FEM analyses. To assess whether the stress level in the composite skin was acceptable, a maximum-stress approach was adopted. Stress values exceeding approximately 50% of the material’s ultimate strength were considered too high, indicating an overstressed configuration, while values below about 25% of the ultimate strength were regarded as too low, suggesting that the laminate stiffness was excessive and underutilised. These thresholds were defined as engineering guidelines rather than fixed limits, since the acceptable range can vary depending on the specific composite material, the applied load conditions, and the desired safety margin. In case of a too-low or excessive stress value, a further change in the laminate composite characteristics must be made, and the procedure repeated from its beginning. At the end of this algorithm, the final obtained airfoil is the one closest to the target (using the initial mechanism design). At the same time, an optimised composite laminate skin is found in terms of displacements and the force exerted by the actuator on the mechanism. This workflow is shown in Fig. 5.

Figure 5.

Geometrical schematisation (left) and PATRAN 3D model (right) of the morphing airfoil.

Figure 5 Long description

The image contains two elements: a geometrical schematisation on the left and a PATRAN 3D model on the right. The geometrical schematisation shows a detailed diagram of the morphing airfoil mechanism with various nodes and coordinates marked. The PATRAN 3D model presents a three-dimensional view of the airfoil, highlighting its structure and design. The two images together illustrate the design and modeling process of the morphing airfoil.

3.2.2 Structural modeling of the leading edge

Starting from the original airfoil and morphing mechanism, a 3D model of 800mm of the wing was developed, approximated as a rectangular wing. While some authors have analysed only small wing sections, each containing only one DN mechanism, in this study, the presence of two consecutive mechanisms in the same section is considered to assess their influence on the stresses and displacements in the midsection [Reference Yuzhu, Wenjie, Jin, Yonghong, Donglai, Zhuo and Dianbiao8]. For this reason, along the 3D model of the 800mm wing length, two equally spaced mechanisms were placed, with, as a first attempt, a total of six mechanisms in a single wingspan, according to Li et al. [Reference Yuzhu, Wenjie, Jin, Yonghong, Donglai, Zhuo and Dianbiao8]. The 3D wing model was imported into PATRAN 2020 to perform the FEM analyses (Figs 6 and 7).

Table 2.

Properties of fiber/epoxy plies – adapted from D. Gay [Reference Gay, Hoa and Tsai30]

Table 2 Long description

A table listing properties of fiber/epoxy plies for carbon material. The table includes specific mass, longitudinal tensile strength, longitudinal compressive strength, transverse tensile strength, transverse compressive strength, longitudinal elastic modulus, transverse elastic modulus, shear modulus, and Poisson ratio. The table has 10 rows and 2 columns. Row 1: Specific mass, 1,530 kilograms per cubic meter. Row 2: Longitudinal tensile strength, 1,270 megapascals. Row 3: Longitudinal compressive strength, 1,130 megapascals. Row 4: Transverse tensile strength, 42 megapascals. Row 5: Transverse compressive strength, 141 megapascals. Row 6: Longitudinal elastic modulus, 134,000 megapascals. Row 7: Transverse elastic modulus, 7,000 megapascals. Row 8: Shear modulus, 4,200 megapascals. Row 9: Poisson ratio, 0.25.

Figure 6.

Surface model of a DNLE UAS-S45 airfoil leading edge (left), and its modelling in Patran (right).

Figure 6 Long description

The left image shows a surface model of a DNLE UAS-S45 airfoil leading edge, displaying a detailed 3D representation of the airfoil's surface. The right image illustrates the modeling of the same airfoil leading edge in Patran software. The model is divided into sections labeled as Mechanism 1, Mechanism 2, and Midsection, each separated by 200 millimeters. The free edge is also labeled, indicating the end of the leading edge. The model includes various nodes and control points, with annotations showing relative coordinates and other technical details.

Figure 7.

Oriented stringers with no offset (left) and with offset (right).

Figure 7 Long description

The image shows two diagrams comparing oriented stringers. The left diagram illustrates stringers with no offset, while the right diagram shows stringers with an offset. Both diagrams feature a series of interconnected nodes and lines representing the structure and alignment of the stringers. The diagrams highlight the differences in the positioning and orientation of the stringers due to the presence or absence of an offset.

The wing skin material has an important role in achieving the leading-edge morphing. For this kind of application, the skin material must have a high curvature at breaking stress, and, at the same time, it must provide sufficient spanwise stiffness to withstand the aerodynamic loads, and thus to reduce the number of morphing mechanisms required. In this perspective, the selected material should have a high anisotropy [Reference Contell Asins, Landersheim and Schwarzhaupt19]. The choice of the wing skin material has been subjected to several design constraints, including being composed of a composite laminate material with the possibility to choose between commercial E-glass/epoxy, carbon/epoxy, and Kevlar/epoxy balanced fabric composite material. Without considering thickness or stacking configuration, the skin has been modelled as a 2D thin laminate shell with the orthotropic material properly oriented along the surface. The selected stringer material is a composite material composed of three plies of unidirectional carbon/epoxy, where each ply has a thickness of 0.5mm, for a total thickness of 1.5mm. A constant omega cross-section has been chosen, with its dimensions and material properties as indicated in Tables 2 and 3, respectively.

The mechanism’s material is an aluminum alloy 7075 widely used for aeronautical applications and available off-the-shelf, with a rectangular cross-section profile and characteristics; its properties and geometric profiles are indicated in Tables 4 and 5, respectively.

Table 3.

Aluminum 7075 properties

Table 3 Long description

The table presents the properties of Aluminum 7075. It contains four rows and four columns. The columns are labeled as Elastic modulus, Shear modulus, Density, and Poisson Ratio. The first row shows the elastic modulus as seventy-one thousand seven hundred megapascals. The second row indicates the shear modulus as twenty-six thousand nine hundred megapascals. The third row lists the density as two thousand eight hundred kilograms per cubic meter. The fourth row specifies the Poisson ratio as zero point three three.

Table 4.

Mechanism rectangular section geometry

Table 4 Long description

A table titled Mechanism rectangular section geometry presents the dimensions of a rectangular section. The table has three columns labeled W, H, t1, and t2, and four rows including the header. The width (W) is 4 millimeters, the height (H) is 6 millimeters, and both t1 and t2 are 1 millimeter each. The table provides a clear and concise representation of the geometric parameters of the mechanism's rectangular section.

Table 5.

Properties of balanced fabric/epoxy composites – adapted from D. Gay [Reference Gay, Hoa and Tsai30]

Table 5 Long description

The table presents the properties of balanced fabric/epoxy composites, specifically E-glass. It includes specific mass of 1,900 kilograms per cubic meter, tensile strength of 400 megapascals in both x and y directions, compressive strength of 390 megapascals in both x and y directions, elastic modulus of 20,000 megapascals in both x and y directions, shear modulus of 2,850 megapascals, elongation at break ranging from 2.5 to 3.7 percent, and a Poisson coefficient of 0.13.

3.2.3 Meshing

Meshes of the structure were realised within a PATRAN workspace. The surface of the LE wing skin was meshed using a curved-based mesh-seed for the side of the airfoil with a maximum allowable error equal to h_max=0.01mm. The straight sides of the surface were meshed using a uniform mesh seed. A total of 124,956 nodes and 127,200 CQUAD4 elements with dimensions of 1x1 mm were realised to mesh the wing surface. The choice of CQUAD4 elements and of the shell material was also determined by the fact that they support nonlinear analyses.

The number of nodes of each stringer is equal to the number of nodes in a single line along the spanwise direction of the considered wing. In this way, by employing the equivalence command, it is possible to place a stringer directly on the wing skin. The next step is to set an appropriate offset to the centre of mass of the stringer cross-section at each node, to guarantee a value of the inertia matrix that is more appropriate for the stringers, simulating a perfect adhesion between the stringers and the wing skin. Therefore, each stringer cross-section was oriented to follow the curvature of the skin, as shown in Fig. 4.

Some researchers have addressed this problem by creating stringers made of surface elements and then by creating a permanent glued contact between the stringers and the skin (e.g. Ref. [Reference Wang and Yang16]). The primary benefit of a permanently glued contact is that it can connect dissimilar meshes. The higher level of accuracy in the analysis of the interaction between stringers and wing skin incurs a cost in terms of model complexity. The benefits of the proposed approach are in the simplicity and flexibility of creating different stringers’ shapes and geometries, which allows for faster testing of various possible solutions. On the other hand, this approach can lead to non-convergence problems in the nonlinear analysis due to the possible high differences in stiffness between adjacent elements around the same node. However, this issue can be addressed by geometrical adjustments or by using the most suitable NASTRAN solver. In the analysis conducted using NASTRAN nonlinear solver ‘sol400’, non-convergence problems never occurred. It is important to note that NASTRAN nonlinear solvers are not allowed to conduct analyses on beam elements with non-zero offsets. Therefore, when switching from linear to nonlinear analysis, the beam elements are set up with a zero offset. The offsets between the beam elements and the reference line of the morphing mechanism are on the order of a few millimeters. Neglecting these offsets slightly reduces the structural stiffness, resulting in a marginally more compliant model. This simplification was required due to solver limitations in the nonlinear NASTRAN environment, which does not allow offset definitions under large-deformation conditions. Although this assumption introduces a minor deviation from the physical stiffness distribution, its influence on the global deformation behaviour is expected to be limited and not significant for the present analysis. Correction factors or refined modeling strategies to compensate for the zero-offset assumption could be adopted to further increase the fidelity of the model.

The mechanism has been meshed by means of BEAM elements with a uniform length of 0.5mm. The entire wing model is made of 125,386 nodes and 126,822 elements. In the original mechanism concept representation provided by the LARCASE team, the rotation of the arms around each hinge axis is allowed. To realise this kind of kinematic structure, each arm of the mechanism was properly meshed, and multi-point constraints (MPCs) between matching nodes corresponding to the hinges of the mechanism were properly realised: MPCs were set up by means of RBAR elements allowing just one degree of freedom rotation around the y-axis. The wing structure is shown in Fig. 8.

Figure 8.

Mesh of the morphing leading edge structure on PATRAN (stringers on the left; mechanism on the right).

Figure 8 Long description

The image presents a detailed mesh of a morphing leading edge structure, divided into two sections. On the left, the mesh displays stringers, which are structural elements designed to provide strength and stability. The right section illustrates a mechanism, likely responsible for the morphing functionality. The mesh is color-coded in teal and black, highlighting different components and their intricate details. The structure is oriented horizontally, with the stringers extending along the length of the leading edge and the mechanism positioned centrally. The image provides a clear view of the internal geometry and spatial arrangement of the components, essential for understanding the design and functionality of the morphing leading edge.

3.2.4 Loads and boundary conditions

It has been shown that a significant amount of the forces exerted by a morphing mechanism is used to deflect the leading-edge skin, and to overcome the aerodynamic pressure load acting on it [Reference Kintscher, Monner and Heintze31]. Therefore, the aerodynamic pressure has been accurately modeled and implemented into the model using the tabular data of pressure acting on the skin of the droop-nose configuration airfoil, optimised at α = 6°, provided by the LARCASE laboratory. As shown in Fig. 9, the pressure distribution was considered as just the one relative to the droop-nose section, which is up to a value of x/c = 0.15. Figure 9 in the right side shows the pressure distribution on the leading edge implemented on PATRAN.

Figure 9.

Example of pressure distribution on the optimised droop-nose airfoil (pressure distribution along the chord on the left; pressure on the surface of the leading edge on the right).

Figure 9 Long description

The left side of the image features a line graph depicting pressure distribution along the chord of an optimized droop-nose airfoil. The x-axis represents the chord length, labeled as x over c, ranging from 0 to 1.05. The y-axis represents pressure in pascals, ranging from negative 1600 to 1000. The graph shows two distinct lines, one with a steep decline and another with a gradual increase. The right side of the image displays a 3D surface plot illustrating pressure on the surface of the leading edge of the airfoil. The plot uses a color gradient to represent different pressure values, with a scale on the right ranging from negative 2.36 to negative 14.03. The plot includes a coordinate system with axes labeled x, y, and z. The surface plot shows varying pressure distributions across the leading edge, with notable changes in color indicating different pressure regions.

The actuator effect was considered by means of a force applied to the mechanism and with the same inclination indicated in Fig. 7. In each analysis, this load was modified to better realise the desired angular variation to the mechanism and to give first indications on the appropriateness of the considered wing skin thickness and stacking. As a first approximation, the leading edge can be analysed without considering the rest of the wing, as it is clamped to the wing box by means of the front spar. This approach is widely adopted in the literature. Analogously, the mechanism is rigidly connected to the front spar by employing a hinge that allows only rotation around the y-axis. The structure approximation is shown in Fig. 10.

Figure 10.

Boundary conditions’ approximation of a morphing leading edge.

Figure 10 Long description

The image shows a diagram of a morphing leading edge with fixed boundary conditions. The leading edge is represented by a series of black dots connected by lines, indicating the mechanism nodes. The fixed boundary is depicted on the right side of the diagram with diagonal lines. The leading edge is labeled with various parameters such as relative coordinates of the mechanism nodes, airfoil chord length, lift coefficient, drag coefficient, Young's modulus, actuator force, shear modulus, cost function, number of control points, coordinate position of a control point, ply thickness, and Cartesian coordinates.

3.3.1 Approach overview

Starting with the designed wing skin, the proposed mechanism’s geometry was investigated. The structure’s geometry was parameterised using ten parameters, and a MATLAB code was developed. Two assumptions were proposed: first, to treat the structure as one-degree kinematic geometry to find node positions after mechanism rotation; and second, to use a spline to interpolate displaced nodes from the undeformed airfoil geometry. Two MATLAB functions (genetic algorithm and particle swarm optimisation) optimised the code, determining the optimal parameters by minimising the airfoil’s cost function. The modified structure underwent non-linear FEM analyses under varying actuator forces. The solution with the lowest cost function was chosen, and actuator force, airfoil deformation, and stress distribution were analysed. Figure 11 presents this approach as a flowchart.

Figure 11.

Flowchart of the mechanism optimisation procedure adopted in this study.

Figure 11 Long description

Flowchart of mechanism optimization procedure with steps and decision points. The process starts from a designed skin and proceeds through structure parameterization and assumptions. It involves the estimation of the deformed airfoil and cost function J. The optimization uses genetic algorithms (GA) and particle swarm optimization (PSO) to find optimized parameters with minimum J value. New geometry, material properties, boundary conditions, pressure, and force are then analyzed using finite element method (FEM). The minimum J case from analysis is evaluated, and if the parameters alpha, J, and F are lower, the optimized structure is achieved. If not, assumptions and optimization options are changed, and the process repeats.

3.3.2 Parametrisation and determination of the deformed airfoil

A MATLAB code was developed to parameterise the mechanism’s initial profile using ten geometrical parameters, detailed in Fig. 12. These parameters include the x-coordinates of nodes (a_1x, a_2x, a_3x and n₁), angular orientations (α₁, α₂, α₃, θ₁ and θ₁), and an additional rotation (Δθ₁) of the main mechanism. Initially, the mechanism’s hinge and initial profile points remain fixed. Intersection points between the mechanism and the airfoil (MPC4, 5, 6, 7 in Fig. 12) can be found by varying the first nine parameters listed above.

Figure 12.

Parametrisation of the morphing mechanism.

Figure 12 Long description

The diagram illustrates a morphing mechanism with various labeled coordinates and components. The upper and lower chambers are marked, with control points labeled as MPC4, MPC5, MPC6, and MPC7. The coordinates a1x, a2x, a3x, and n1 are indicated, representing the relative positions of the mechanism nodes in millimeters. The angles alpha1, alpha2, alpha3, beta1, and beta2 are shown, along with the change in angle Delta theta1. The Cartesian coordinates x, y, and z are also depicted. The diagram includes annotations for the airfoil chord length c, the lift coefficient CL, the drag coefficient CD, Young's modulus E, the actuator force F, the shear modulus G, and the cost functions J, J1, J2, and J3. The ply thickness t is marked in millimeters.

To determine the deformed airfoil after a rotation Δθ1, two assumptions were made:

1. 1. First assumption: The mechanism’s MPC points along the skin move as a one-degree kinematic geometry, resembling four-bar linkages. Two structural approximations were considered; each was assessed using a MATLAB code.

   • • In the initial approximation, the wing skin section was treated as three four-bar linkages, allowing precise node evaluation after Δθ₁ (Fig. 13);

     Figure 13.

     Movement of the nodes after a mechanism rotation in the first structure approximation.

     

     Figure 13 Long description

     The image presents a diagram illustrating the movement of nodes after a mechanism rotation in the first structure approximation. The left side of the diagram shows the initial positions of the nodes connected by lines of different colors, representing various structural components. The right side of the diagram depicts the new positions of the nodes after the mechanism rotation, with lines indicating the movement paths. The nodes are connected by lines of different colors, indicating the relative coordinates of the mechanism nodes in millimeters. The diagram includes labels for the nodes and their relative coordinates, as well as the airfoil chord length and other mechanical properties. The movement of the nodes is shown with arrows and lines, indicating the direction and extent of the rotation.

   • • The second approximation considered the upper wing skin as a single beam with connected rods and hinges (Fig. 14).

     Figure 14.

     Movement of the nodes after a mechanism rotation in the second structure approximation.

     

     Figure 14 Long description

     A flowchart illustrating the movement of nodes after a mechanism rotation in the second structure approximation. The flowchart consists of two main sections, each depicting a different stage of the mechanism's rotation. The left section shows the initial positions of the nodes, represented by black dots connected by lines of various colors. The right section illustrates the positions of the nodes after rotation, with additional elements such as circles and arrows indicating movement and interaction. Key components include nodes, lines representing connections, and colored paths showing the direction and extent of movement. The flowchart highlights the relative coordinates of the mechanism nodes, the airfoil chord length, and the forces acting on the structure.

2. 2. Second assumption: The leading edge of the deformed airfoil can be approximated by a spline passing through the displaced MPC points (Fig. 15). The airfoil was evaluated in MATLAB by means of a Catmull-Rom spline, given the position of interpolation points. The morphed airfoil was divided into 156 equally separated control points to allow the computation of the J cost function.

   Figure 15.

   Example of a deformed profile evaluation through a spline passing through displaced nodes.

   

   Figure 15 Long description

   The diagram illustrates a deformed profile evaluation using a spline that passes through displaced nodes. The spline is depicted as a curved line connecting several nodes, which are marked with red and black dots. The diagram includes various lines and arrows indicating the direction and extent of deformation. The nodes are connected by orange and blue lines, representing different sections of the mechanism. The spline is shown in black, highlighting the path through the displaced nodes. The overall structure suggests an analysis of mechanical deformation in a specific profile.

Using the MATLAB function built according to the procedure described above, it is possible to draw the morphed shape and to evaluate the cost function for each combination of geometrical parameters and to impose an additional rotation on the mechanism. This MATLAB function has been optimised, aiming at minimising the value of the cost function J and finding the optimal values of the ten geometrical parameters defined at the beginning of Section 3.3.2. The morphing structure approximated via the two options discussed above was optimised using a genetic algorithm and then a particle swarm optimisation method. A graphical representation of the iteration shapes is shown in Fig. 16.

Table 6.

Stacking final configuration and thickness

Figure 16.

Evaluation of different mechanism geometries and deformed profiles using the optimisation toolbox.

Figure 16 Long description

A scatter plot with hundreds of data points in blue and red colors representing different mechanism geometries and deformed profiles. The x-axis ranges from 0 to 60 degrees, and the y-axis ranges from 0 to 50 degrees. The data points form distinct clusters and patterns, with some points connected by dashed lines. The plot includes a legend indicating the color coding for different data sets. All values are approximated.