An optimization procedure for the shape design of morphing aircraft is presented. The process is coupled with a knowledge-based framework combining parametric geometry representation, multidisciplinary modelling, and genetic algorithm. The parameterization method exploits the implicit properties of the Bernstein polynomial least squares fitting to allow both local and global shape control. The framework is able to introduce morphing shape changes in a feasible way, taking into account the presence of structural parts, such as the wing-box, the physical behaviour of the morphing skins, and the effects that these modifications have on the aerodynamic performances. It inherits CAD capabilities of generating 3D deformed morphing shapes and it is able to automatically produce aerodynamic and structural models linked to the same parametric geometry. Dedicated crossover and mutation strategies are used to allow the parametric framework to be efficiently incorporated into the genetic algorithm. This procedure is applied to the shape design of Reference Aircraft (RA) and to the assessment of the potential benefits that morphing devices can bring in terms of aircraft performances. It is adopted for the design of a variable camber morphing wing to investigate the effect of conformal leading and trailing edge control surfaces. Results concerning four different morphing configurations are reported.

The very challenging targets of new environmental requirements for transport aircraft force the researchers to look for more advanced aircraft configurations, based on more efficient aerodynamics and structures together with more sophisticated flight control systems. Focusing on European transport, it appears as clear that the pressure will increase for large scheduled European carriers to reequip their short haul fleets with more fuel-efficient types, in order to remain competitive with low-cost rivals and to ensure they will not be unduly penalized when European emissions trading comes into full force.

The Bréguet range equation [

While current aircraft are already equipped with systems able to introduce in-flight geometrical variations such as wing area change, variable camber, and retractable landing gear, the morphing of next generation still has challenges and leads to the design of morphing wings based on conformable control surfaces [

The design of morphing wing devices must combine two opposite requirements, often named kinematic and structural requirements, respectively: a flexible structure so to minimize the energy necessary to adapt its shape as expected and at the same time an enough rigid structure able to maintain the new shape under the aerodynamic loads when the morphing mechanism is not actuated. The approach proposed by the authors [

The work presented in this paper focuses on the first level of a wider morphing design framework having the following capabilities: wing shape optimization able to combine aerodynamic performances with optimal deformation of the skins (first level); optimal design of compliant mechanism able to produce, once actuated, the optimal shape coming out from the first level (second level); integration of the morphing devices into a high-fidelity model representing the complete aircraft for final aeroelastic assessment to evaluate the reliability of the designed morphing solutions [

Initially, the first level was implemented as a simple 2D shape optimization linked to a viscous and subsonic 2D aerodynamic solver, while the second one represented a general code for the synthesis of compliant mechanisms [

The paper describes the novel progress beyond this point about the first level optimization which is now based on a comprehensive knowledge-based engineering (KBE) framework able to define the optimal morphing wing shape in terms of mission profile performances, directly in the three-dimensional space. It is based on coupling a parametric geometry representation, able to predict the structural response of morphing skin, with Computer-Aided Engineering (CAE) capabilities, Object-Oriented Programming (OOP), genetic algorithm, and aerodynamic analyses. The results obtained applying it to a Reference Aircraft (RA), coming from the FP7 EU NOVEMOR project (Novel Air Vehicle Configurations: from Fluttering Wings to Morphing Flight) and adopted as a benchmark to evaluate the optimal benefits that can bring in terms of global performances, are reported to validate the proposed procedure and to evaluate the impact of continuous chordwise and spanwise camber variation in terms of lift to drag ratio and aerodynamic load distribution.

The shape design procedure presented in this work consists of a shape optimization, based on genetic algorithm and coupled with the parametric framework introduced in the following section, able to design the optimal morphing shapes from the aerodynamic point of view, under skin structural requirements. While genetic algorithms have been already coupled with 2D shape optimization of airfoil [

The set of parameterized cross section shapes is the starting point of the whole process shown in Figure

Scheme of the genetic algorithm based on the 3D parametric shape representation.

This parametric shape representation is suitable to be incorporated into genetic algorithm and to be parallelized [

Each individual making up the population is divided in as many parts as the wing sections are and each part is composed of the same set of optimization variables. The generation of each new population is produced by in-house crossover and mutation dedicated strategies, while a code parallelization allows decreasing the computational cost. Two possible code parallelization modes can be adopted. The first one computes in parallel the fitness functions in terms of global aerodynamic performance. The second one divides the wing problem into a number of subproblems, equal to the number of wing sections, that are solved simultaneously.

The shape design of morphing aircraft, able to smoothly change their external geometry, requires interfacing the aerodynamics that can be optimized for different flight conditions of the mission profile and the structure of the skin that constitutes the outermost component of the aircraft. CAE systems represent parametric tools, including Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD), able to assist the engineering in the shape design, but the ability to introduce complex changes is limited and the only rules that can be embedded are restricted to simple geometrical statements. KBE systems are designed to allow implicit rules, physics-based solvers, and evolutionary algorithms to be linked to CAD system. A specific knowledge-based framework, able to provide advanced implicit parametric capabilities and more direct shape control than common CAE systems, is here presented. It is a fully integrated multiphysics environment based on a parametric geometry representation of the aircraft that incorporates the ability to predict the structural response of the skin to the shape changes, provides couplings with aerodynamic solvers, and uses genetic algorithm to tackle the optimization problem related to the definition of the best morphing shapes.

The knowledge of the morphing skin structural behaviour is included into a so-called skin structural (KBSS) feedback that consists of an inference engine that uses rules to deduce new shapes consistent with the structural response of the morphing skin. This knowledge-based technology is strictly related to an Object-Oriented Programming-based geometry parameterization, that is, the Class Shape Transformation (CST), which is integrated with a Computer-Aided Engineering (CAE) tool. The integration with CAD allows enriching the parametric capabilities of the CST formulation with the parametric features of the CAD systems, for a complete and accurate 3D shape definition. When this knowledge-based system is coupled with genetic algorithm, as described in Section

The architecture of the framework is shown in Figure

The parametric framework.

The structural, aerodynamic, and CAD modelling capabilities are implicitly connected and inherit their properties from a main class able to combine in three dimensions the set of the most important parameterized cross sections of the aircraft model. A specific code directly connected to the CST method is also available for the automatic and high-fidelity 2D aerodynamic analyses of each cross section.

The KBE framework revolves around PHORMA (Parametric sHapes for aerOdynamic and stRuctural Modelling of Aircraft) that is an object oriented code composed by a suite of modules conceived to exchange and handle different shapes, corresponding to the set of the most important cross sections of the aircraft model, in order to generate the 3D geometry. The shapes are parameterized and combined in the 3D space through CAD-based interpolation techniques so that local shapes changes can be spread out. Resulting geometry is shared by the aerodynamic, structural, and CAD models that inherit the parametric capabilities from the so-called class/shape function transformation (CST) technique, originally proposed by Kulfan [

The general mathematical expression representing the airfoil geometry, already presented in [

CSTv3 geometric parameters for airfoil sections.

The first term of (

The

The first and last terms of the vector

The perturbation of one of the extra coefficients proportionally changes the value that the CST function assumes at the chordwise station of the peaks of every

PHORMA is implemented by the Object-Oriented Programming so that different cross-sectional shapes of a complete aircraft can be identified in order to reproduce the corresponding reference shape and to have a 3D mathematical model, based on CST parameterization, suitable to introduce morphing shape changes. In this way the estimation of the morphing skin structural behaviour, as well as the generation of the aerodynamic or finite element models, of the aircraft inherits all methods coming from its parametric representation.

In order to have a parametric model suitable to introduce morphing shape modifications, an efficient CST identification of preexisting CAD models is available. By means of a completely automatic procedure, the shapes corresponding to the set of the most important cross sections of the aircraft model are parameterized and corresponding CST models are generated. The upper and lower extra coefficients (

If

Taking into account that

The approximation can be improved extracting the boundary condition terms from the definition of the

The parametric framework introduced in this section is a dedicated knowledge-based engineering (KBE) environment able to force the shape design to move inside a domain where the global shape is driven to change in a feasible way. First of all, a morphing wing shape is feasible if it is able to take into account the structural behaviour of the skins. It depends on how the actuation is introduced and how the morphing skin responds to the actuation input together with the external loads. A Knowledge-Based Skin Structural (KBSS) feedback embedded into the shape parameterization scheme allows taking into account the skin structural response before the morphing mechanism has been defined.

One of the most important obstacles in the wing morphing is due to two main aspects:

The presence of main bearing structures.

The structural contribution of the skin.

Indeed, even if almost all the proposed approaches for morphing wings are based on a different structural configuration of the ribs, the structural contribution of the skin still remains. Moreover the skin is directly connected to the structural box which represents an obstacle for a full deformation and restricts outside deformable regions. In the approach here presented the knowledge-based parameterization is able to take into account all of these aspects in an implicit way. It represents a tool that assists the engineers in determining the best morphing shape to limit the deformation energy of the skin and, at the same time, the actuator power necessary to control the shape change. This approach is general and can be easily applied to skin made of different type of material.

When local perturbations are introduced in a wing shape, one of the main feasible constraints is represented by the wing regions that must be kept undeformed. Considering a conventional wing equipped with active camber, this region corresponds to the structural box. Introducing this equality constraint in an explicit way, for example, during an optimization process, is an expensive approach in terms of computational effort. A different approach is to exploit the

The CST formulation offers the possibility to capture a second structural feedback related to the behavior of the portion of morphing skin outside the undeformed regions. The estimation of the skin structural deformation here proposed is directly inherited from the analytical CST formulation and it is included in the

The structural behavior of the skin has a key role in the shape changes that the skin can assume during the deformation. It can be obstacle or a help to achieve the desired shape. This link is preserved by the CST formulation which embeds, in the same parameterization technique, the ability to introduce shape changes and, at the same time, to evaluate the corresponding stress distribution along the skin. The first ability has been previously described, while the second one is based on geometrical consideration, already described in [

Once a parametric model is available and morphing shape changes are introduced, corresponding CAD, FEM, or CFD models can be linked to the same shape geometry. While the framework described in this section allows exporting the most common CAD formats, such as IGES and STEP, as well as CATIA files, parametric meshing strategies, able to follow the shape changes based on the CST formulation, are implemented to automatically generate structural and aerodynamic models. It interacts with different classes and methods, as shown in Figure

PHORMA is equipped with a CAD interface to automatically interpolate in spanwise direction the set of cross sections describing the wing. Corresponding parametric shapes are automatically transformed into spline-based curves by means of a user-defined number of points extracted from the CST formulation and combined to generate a multisection NURBS-based surface by sweeping them along a user-defined spine. In order to accurately describe the original geometry, one or more guide curves can be defined and slope and curvature boundary conditions can be introduced at the ends of the surface. Comparisons between original wing geometries and identified shapes are performed to identify the correct combination Bernstein polynomial order/number of sections in spanwise direction able to guarantee an approximation error tolerance.

The generation of aerodynamic models is strictly connected to the full parametric description of the geometry. Both 2D and 3D models can be produced by different in-house procedure depending on the desired model quality.

In the case of 2D aerodynamic analyses, a specific code able to automatically produce a 2D structured mesh suitable to perform Navier-Stokes computations is directly linked to the CST-based code. The generation of structured mesh around a parameterized airfoil is based on a script for Ansys ICEMCFD [

The implemented procedure for 2D CFD analyses embedded into the CST class.

Parametric meshing around undeformed airfoil and corresponding morphing leading and trailing edge shapes.

The whole process allows adding a set of high-fidelity polar curves to the set of the most important parameterized cross sections used by PHORMA to describe the complete wing model. This could be used to make local corrections to a 3D low-fidelity simulation with the computational cost of a reduced number of 2D flow solutions equal to the number of cross sections [

In the case of 3D aerodynamic analyses, the cross sections are combined in order to spread the specific geometry details along the wing via particular interpolation surfaces able to accurately reproduce the correct wing thickness distribution in spanwise direction. The implemented procedure allows imposing different transition laws among different 2D airfoils, as well as modifying their properties such as Angle of Attack, dihedral or tow angles. Different kind of low-fidelity and high-fidelity aerodynamic models can be generated exploiting the parametric methods that PHORMA inherits from the CST formulation.

Another way to perform 3D simulations is to directly generate full volume meshes for medium/high-fidelity analyses. PHORMA is linked in batch mode to the software

The hybrid mesh generation has been applied to the Reference Aircraft. The prismatic layer around the fuselage and around a wing section placed near the root equipped with a morphing leading edge is shown in Figure

3D hybrid mesh for RANS computation around complete aircraft w/o engine (a) and around the morphing leading edge (b).

The skin structural requirements, introduced in Section

When a perturbation is introduced in the CST geometry by means of a subset of coefficients

This approach is very efficient for the following main reasons:

It allows introducing local perturbations by means of a global geometry representation method.

It preserves the

It reduces the number of external variables, from

It makes implicit constraints along the regions which must be kept undeformed.

Comparison between a generic airfoil (black), its parameterization (blue), and corresponding shapes obtained introducing the same perturbation with (red) and without (green) solving the reduced system.

The approach here presented is extended to three dimensions by the CAD interpolation capabilities described in Section

Local perturbation and fixed wing-box parametric representation in three dimensions.

This method allows implementing implicit wing-box constraints into the shape optimization. Every time, the

Wing-box constraints are very common in the aircraft shape design and several approaches that allow achieving this goal can be found in the literature. Different parameterization methods could be used to improve the approximation in the undeformed region [

Combining the estimation of the structural behavior of the morphing skin and the introduction of feasible perturbation described in Sections

Algorithm which implements (

The simplest example to demonstrate how the

Parametric CAD model of a morphing winglet.

When the shape changes are introduced by means of the shape optimization of Section

According to the constant cross section length (CCL) concept [

Once the wing-box region is detected, a subset of CST parameters can be defined, as described in Section

Both crossover and mutation subroutines execute the

The procedure described in the previous sections has been applied for the optimization of the so-called Reference Aircraft (RA). The morphing wing configuration and some information about the wing planform, such as the total wing span and the aileron span position, are shown in Figure

Reference Aircraft with morphing wing configuration and corresponding wing planform.

Different sections were chosen in order to identify the parametric shape of the Reference Aircraft and to build up the mathematical model suitable to be used to introduce the shape changes into the leading and trailing edge morphing regions. The whole process, described in Section

Parametric identification applied to the Reference Aircraft model.

In the following subsections, some results for morphing leading and trailing edge at different flight conditions are summarized. Since different design requirements have been provided for each morphing device, different shape optimization problems have been defined. They represent four aerodynamic shape optimizations including structural constraints on the morphing skin. According to the approach described in Section

At the beginning, a 2D shape optimization based on RANS computations has been applied to the wing section placed at 9 m along the wing span, in the reference system of Figure

The parametric airfoil shape is described by Bernstein Polynomials Order

The vertical deflection which results from the shape optimization is equal to

Morphing leading edge in high speed conditions: comparison between reference and deformed shape 2D results in terms of

The complete procedure described in Section

The genetic algorithm started with an initial population of

The shape optimization process provided a spanwise distribution of optimal leading edge deflections

The results are reported in Figure

Comparison between the

Comparison between reference and morphing shape 3D results in terms of

The optimization algorithm acted to decrease the reference lift coefficient from

A second 3D optimization process was run using an objective function that tries to introduce the maximum droop deflection along the wing span, instead of an aerodynamic performance estimation. This may be suitable for a low speed condition, corresponding to the landing condition of the RA. A set of

The genetic algorithm started with an initial population of

Parametric CAD model of the

Curvature difference functions along the wing span for

Different types of

The shape optimization procedure was applied to the wing of the Reference Aircraft to improve the performances by means of a morphing trailing edge optimized for the cruise condition. The fitness function is related to the aerodynamic efficiency

The population size and the grid size of the aerodynamic model are more or less the same used for the study described in Section

Parametric CAD model of the

The aerodynamic results are reported in Figure

Comparison between the

The optimization algorithm was able to improve the lift over drag ratio of

Morphing seems a potentially promising technology allowing matching the new stringent requirements in terms of environmental impact of next generation aircraft. Unfortunately the results available in the literature concerning the application of morphing concepts still combine lights and shadows in terms or real benefits and open issues like certification, fatigue, and so on. For sure the lack of design procedures specifically dedicated to the optimal design of morphing mechanisms appears clearly. This paper introduced in a general way the work carried out at Politecnico di Milano to set up a complete framework for the optimal design of morphing mechanisms based on compliant mechanisms. In particular the paper focused on the first level of the proposed multilevel design procedure. Indeed, the so-called knowledge-based shape optimization procedure has been introduced and described that is able to combine aerodynamic with structural performances, mainly related to the behavior of the skin during the shape variation. The proposed procedure has been evaluated in the design of morphing wing based on conformable leading and trailing edge surfaces able to adapt the wing camber during the mission profile. An external optimization loop works on the most important design variables that affect the camber morphing and it is dedicated to the aerodynamic performance evaluation. A nested loop, based on a particular development of CST method, guarantees that the outer loop works only on feasible shapes able to satisfy wing-box volume constraints and morphing skin requirements.

Some examples are reported, concerning the so-called Reference Aircraft, a regional-type aircraft developed inside NOVEMOR project and used as a test bench for a final assessment on morphing applications. The results obtained for the leading edge device at the subsonic regime and the trailing edge device at the transonic flight regime have been selected and the mission analyses have been completed to quantify possible benefits in terms of fuel saving. During these studies, only the aerodynamic behaviour and the structural behaviour of the morphing skin were considered. Neither the impact on the maximum takeoff weight of the morphing mechanisms nor the effect of aeroelasticity on the behavior of morphing devices has been considered. A family of optimal shapes have been produced and verified using both medium and high-fidelity tools showing promising results as well proving the versatility of the proposed approach. During the second level of the same procedure, all the optimal shapes can be adopted as multiple targets for the optimal design of the morphing mechanism, according to the multiobjective design strategy.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Special thanks go to Alexandre Antunes, Felipe Odaguil, and Grace Lima from EMBRAER for the CFD and mission analysis performed on Reference Aircraft. The research leading to these results has been partially funded by the European Union’s Seventh Framework Programme [FP7/2007–2013] under Grant Agreement no. 285395.