Variational Analysis and Aerospace Engineering

In the framework of structural optimization via a level-set method, we develop an approach to handle the directional molding constraint for cast parts. A novel molding condition is formulated and a penalization method is used to enforce the constraint. A first advantage of our new approach is that it does not require to start from a feasible initialization, but it guarantees the convergence to a castable shape. A second advantage is that our approach can incorporate thickness constraints too. We do not address the optimization of the casting system, which is considered a priori defined. We show several 3d examples of compliance minimization in linearized elasticity under molding and minimal or maximal thickness constraints. We also compare our results with formulations already existing in the literature.

Given a linear continuous-time infinite-dimensional plant on a Hilbert space and disturbances of known waveform but unknown amplitude and phase, we show that there exists a stabilizing direct model reference adaptive control law with persistent disturbance rejection and robustness properties. The plant is described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. For this paper, the plant will be weakly minimum phase, i.e., there will be a finite number of unstable zeros with real part equal to zero. All other zeros will be exponentially stable.

The central result will show that all errors will converge to a prescribed neighborhood of zero in an infinite-dimensional Hilbert space even though the plant is not truly minimum phase. The result will not require the use of the standard Barbalat’s lemma which requires certain signals to be uniformly continuous. This result is used to determine conditions under which a linear infinite-dimensional system can be directly adaptively controlled to follow a reference model. In particular we examine conditions for a set of ideal trajectories to exist for the tracking problem. Our principal result will be that the direct adaptive controller can be compensated with a zero filter for the unstable zeros which will produce the desired robust adaptive control results even though the plant is only weakly minimum phase. Our results are applied to adaptive control of general linear infinite-dimensional systems described by self-adjoint operators with compact resolvent.

Aeroelasticity of PrandtlPlane configurations is a yet unexplored field. The overconstrained structural system and the mutual aerodynamic interference of the wings enhance the complexity of the aeroelastic response. In this work the aeroelastic behavior of several models based on wing system of 250-seat PrandtlPlane design is studied. When an aluminum version of the structure is considered, flutter is associated with a coalescence of the first two elastic modes, the first being characterized by a classic upward bending of both wings, and the second one being associated with an out-of-phase bending of the two wings and tilting of the lateral joint. Analyses show that energy is injected in the structure mainly at the tip of the front wing, close to the aileron. Effects of freeplay of mobile surfaces are evaluated, showing how, in some cases, an increase in the flutter speed is observed. When flutter analyses are repeated considering the configuration free to pitch and plunge, flutter speed does increase due to a particular interaction between rigid-body pitching and elastic modes. Several of the above findings are demonstrated on more detailed structural models considering also the local stiffness distribution, and taking also into consideration compressibility effects. When composite materials are employed, flutter issues are completely overcome.

About two decades ago years researchers began to apply a new approach, using evolutionary algorithms or metaheuristics, to solve continuous optimal control problems. The evolutionary algorithms use the principle of “survival of the fittest” applied to a population of individuals representing candidate solutions for the optimal trajectories. Metaheuristics optimize by iteratively acting to improve candidate solutions, often using stochastic methods. Because of certain compromises that are usually necessary when transcribing the problem for solution by these methods it has been thought that they were not capable of yielding accurate solutions. However that is a misconception as is demonstrated by examples in this work.

The evolution of the response of an elastic-plastic bar from the initial unstressed state to rupture is studied with a one-dimensional model based on incremental energy minimization. The model can reproduce both brittle and ductile fracture, as well as an intermediate fracture mode, called ductile–brittle, in which, due to an extreme localization of the plastic deformation, the bar suddenly breaks after a more or less protracted plastic regime. Numerical simulations obtained from the model’s implementation are compared with the results of tensile tests on bars made of steel and of non-reinforced concrete. With an accurate choice of the analytical shapes of the plastic strain energy, not only the overall behavior, but also many details of the experimental response can be captured.

An analytical formulation for the induced drag minimization of generic single-wing non-planar systems, biwings, and closed systems is presented. The method is based on a variational approach, which leads to the Euler–Lagrange integral equations in the unknown circulation distributions. The relationship between quasi-closed C-wings, biwings, and closed systems is discussed and several induced drag theorems/properties are introduced. It is shown that under optimal conditions these systems present the same minimum induced drag and the circulation can be obtained from a fundamental one by just adding a constant. The shape of the optimal aerodynamic load on the Box Wing is showed to change with the distance between the wings; differently that what assumed in previous works, it is not the superposition of a constant and an elliptical function.

Three surrogate models, or reduced-order models (ROMs), are constructed using the proper orthogonal decomposition (POD) applied in the parameter space. We use a reduced snapshot set adopting full and fractional factorial planes together with quadtree distribution for the initial positioning of the snapshots. To compute the POD coefficients, response surface methodology is employed. In the first application a ROM is constructed in order to analyze the subsonic flow past a two-dimensional airfoil. The second example regards a transonic two-dimensional flow in a five-dimensional shape parameter space. In the last case a surrogate model for database generation considering a three-dimensional aircraft configuration is constructed. In all the three cases a posteriori error estimates were performed and the surrogate models showed good agreement with the CFD reference solution.

A trust region filter-SQP method is used for wing multi-fidelity aerostructural optimization. Filter method eliminates the need for a merit function, and subsequently a penalty parameter. Besides, it can easily be modified to be used for multi-fidelity optimization. A low fidelity aerostructural analysis tool is presented, that computes the drag, weight, and structural deformation of lifting surfaces as well as their sensitivities with respect to the design variables using analytical methods. That tool is used for a mono-fidelity wing aerostructural optimization using a trust region filter-SQP method. In addition to that, a multi-fidelity aerostructural optimization has been performed, using a higher fidelity CFD code to calibrate the results of the lower fidelity model. In that case, the lower fidelity tool is used to compute the objective function, constraints, and their derivatives to construct the quadratic programming subproblem. The high fidelity model is used to compute the objective function and the constraints used to generate the filter. The results of the high fidelity analysis are also used to calibrate the results of the lower fidelity tool during the optimization. This method is applied to optimize the wing of an A320 like aircraft for minimum fuel burn. The results showed about 9 % reduction in the aircraft mission fuel burn.

In this paper, an extended version of Zolesi et al. (Proceedings of the 42nd ICES (AIAA 2012-3601), San Diego, CA, 2012), we describe a tensegrity ring of innovative conception for deployable space antennas. Large deployable space structures are mission-critical technologies for which deployment failure cannot be an option. The difficulty to fully reproduce and test on ground the deployment of large systems dictates the need for extremely reliable architectural concepts. In 2010, ESA promoted a study focused on the pre-development of breakthrough architectural concepts offering superior reliability. This study, which was performed as an initiative of ESA Small Medium Enterprises Office by Kayser Italia at its premises in Livorno (Italy), with Università di Roma TorVergata (Rome, Italy) as sub-contractor and consultancy from KTH (Stockholm, Sweden), led to the identification of an innovative large deployable structure of tensegrity type, which achieves the required reliability because of a drastic reduction in the number of articulated joints in comparison with non-tensegrity architectures. The identified target application was in the field of large space antenna reflectors. The project focused on the overall architecture of a deployable system and the related design implications. With a view toward verifying experimentally the performance of the deployable structure, a reduced-scale breadboard model was designed and manufactured. A gravity off-loading system was designed and implemented, so as to check deployment functionality in a 1-g environment. Finally, a test campaign was conducted, to validate the main design assumptions as well as to ensure the concept’s suitability for the selected target application. The test activities demonstrated satisfactory stiffness, deployment repeatability, and geometric precision in the fully deployed configuration. The test data were also used to validate a finite element model, which predicts a good static and dynamic behavior of the full-scale deployable structure.

This contribution gathers some of the ingredients presented at Erice during the third workshop on “Variational Analysis and Aerospace Engineering.” It is a collection of several previous publications on how to set up an uncertainty quantification (UQ) cascade with ingredients of growing computational complexity for both forward and reverse uncertainty propagation. It uses data analysis ingredients in a context of existing deterministic simulation platforms. It starts with a complexity-based splitting of the independent variables and the definition of a parametric optimization problem. Geometric characterization of global sensitivity spaces through their dimensions and relative positions through principal angles between vector spaces bring a first set of information on the impact of uncertainties of the functioning parameters on the optimal solution. Joining the multi-point descent direction and probability density function quantiles of the optimization parameters permits to define the notion of directional extreme scenarios (DES) without sampling of large dimension design spaces. One goes beyond DES with ensemble Kalman filters (EnKF) after the multi-point optimization algorithm is cast into an ensemble simulation environment. This formulation accounts for the variability in large dimension. The UQ cascade continues with the joint application of the EnKF and DES leading to the concept of ensemble directional extreme scenarios which provides a more exhaustive description of the possible extreme scenarios. The different ingredients developed for this cascade also permit to quantify the impact of state uncertainties on the design and provide confidence bounds for the optimal solution. This is typical of inverse designs where the target should be assumed uncertain. Our proposal uses the previous DES strategy applied this time to the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using available adjoint-based gradients of the functional. Eventually, we go beyond inverse design and apply the method to general optimization problems. The ingredients of the paper have been applied to constrained aerodynamic performance analysis problems.

In this paper the aerodynamic efficiency of wind turbines with horizontal axis is discussed and the so-called box blade concept, inspired by the Prandtl’s ‘Best Wing System’ is analysed; this wind turbine configuration is proved to be efficient than a conventional blade. Moreover, other non-planar blades, such as the winglet and C extension are analysed via vortex theory with the numerical method of Ribner and Foster for the optimum circulation and the recent model of Okulov and Sørensen for the performance evaluation; a generalization of the above mentioned models is also presented in this work. Finally, the box blades are verified by means of a commercial CFD software.

In this work the authors propose a new paradigm for the optimum design of variable angle tow (VAT) composites. They propose a generalisation of a multi-scale two-level (MS2L) optimisation strategy already employed to solve optimisation problems of anisotropic structures characterised by a constant stiffness distribution. In the framework of the MS2L methodology, the design problem is split into two sub-problems. At the first step of the strategy the goal is to determine the optimum distribution of the laminate stiffness properties over the structure, while the second step aims at retrieving the optimum fibres-path in each layer meeting all the requirements provided by the problem at hand. The MS2L strategy relies on: (a) the polar formalism for describing the behaviour of the VAT laminate, (b) the iso-geometric surfaces for describing the spatial variation of the stiffness properties and (c) a hybrid optimisation tool (genetic- and gradient-based algorithms) to perform the solution search. The effectiveness of the MS2L strategy is proven through a numerical example on the maximisation of the first buckling factor of a VAT plate subject to both mechanical and manufacturability constraints.

The conservation laws of continuum mechanic are naturally written in an Eulerian frame where the difference between a fluid and a solid is only in the expression of the stress tensors, usually with Newton’s hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. There are currently two favored approaches to Fluid Structured Interactions (FSI) both working with the equations for the solid in the initial domain; one uses an ALE formulation for the fluid and the other matches the fluid–structure interfaces using Lagrange multipliers and the immersed boundary method. By contrast the proposed formulation works in the frame of physically deformed solids and proposes a discretization where the structures have large displacements computed in the deformed domain together with the fluid in the same; in such a monolithic formulation velocities of solids and fluids are computed all at once in a single variational formulation by a semi-implicit in time and the finite element method. Besides the simplicity of the formulation the advantage is a single algorithm for a variety of problems including multi-fluids, free boundaries, and FSI. The idea is not new but the progress of mesh generators renders this approach feasible and even reasonably robust. In this article the method and its discretization are presented, stability is discussed showing in a loose fashion were are the difficulties and why one is able to show convergence of monolithic algorithms on fixed domains for fluids in compliant shell vessels restricted to small displacements. A numerical section discusses implementation issues and presents a few simple tests.

This work describes and applies the recently introduced, general-purpose perturbative guidance termed variable-time-domain neighboring optimal guidance, which is capable of driving an aerospace vehicle along a specified nominal, optimal path. This goal is achieved by minimizing the second differential of the objective function (related to the flight time) along the perturbed trajectory. This minimization principle leads to deriving all the corrective maneuvers, in the context of an iterative closed-loop guidance scheme. Original analytical developments, based on optimal control theory and adoption of a variable time domain, constitute the theoretical foundation for several original features. The real-time feedback guidance at hand is exempt from the main disadvantages of similar algorithms proposed in the past, such as the occurrence of singularities for the gain matrices. The variable-time-domain neighboring optimal guidance algorithm is applied to two typical aerospace maneuvers: (1) minimum-time climbing path of a Boeing 727 aircraft and (2) interception of fixed and moving targets. Perturbations arising from nonnominal propulsive thrust or atmospheric density and from errors in the initial conditions are included in the dynamical simulations. Extensive Monte Carlo tests are performed, and unequivocally prove the effectiveness and accuracy of the variable-time-domain neighboring optimal guidance algorithm.

This presentation gives an overview of the capacities of optimal shape design methods as actual engineering tools utilized for industrial applications, mostly for aerodynamic shape design. Numerical formulation and implementation are recalled and illustrations of applications are discussed.

The problem of the optimum propeller with straight blades was first solved by Goldstein; in this paper, a variational formulation is proposed in order to extend the solution to non-planar blades. First, we find a class of functions (the circulation along the blade axis) for which the thrust and the aerodynamic drag moment are well defined. In this class, the objective functional is proved to be strictly convex and then the global minimum exists and is unique. Then we determine the Euler equation in the case of a general blade and show that the numerical results are consistent with the Goldstein’s solution. Finally, some numerical results with the Ritz method are presented for optimum propeller blades.

The polar formalism is a mathematical method, based upon a complex variable transformation, proposed in 1979 by G. Verchery for representing plane tensors of any rank using invariants and angles. As such, it is particularly suited for representing anisotropic properties, in particular elasticity.

In this paper, we give a brief account of the fundamentals of the polar formalism, stressing in particular the role played by the polar invariants on the characterization of elastic symmetries, that leads to a new classification of them, based upon an algebraic criterion and that has allowed for the discovery of two special orthotropies.

Then, we focus on some special theoretical subjects: anisotropy of complex or rari-constant layers, some strange cases of interaction between geometry and anisotropy, the anisotropy of damaged layers initially isotropic.