Aeroelastic tailoring considering uncertainties in material properties

The robustness of aeroelastic design optimization with respect to uncertainties in material and structural properties is studied both numerically and experimentally. The model consists of thin orthotropic composite wings virtually without fuselage. Three different configurations with consistent geometry but varying orientation of the main stiffness axis of the material are investigated. The onset of aeroelastic instability, flutter, is predicted using finite element analysis and the doublet-lattice method for the unsteady aerodynamic forces. The numerical results are experimentally verified in a low-speed wind tunnel. The optimization problem is stated as to increase the critical air speed, above that of the bare wing by massbalancing. It is seen that the design goals are not met in the experiments due to uncertainties in the structural performance of the wings. The uncertainty in structural performance is quantified through numerous dynamic material tests. Once accounting for the uncertainties through a suggested reformulation of the optimization problem, the design goals are met also in practice. The investigation indicates that robust and reliable aeroelastic design optimization is achievable, but careful formulation of the optimization problem is essential.     

Sensitivity analysis of bridge flutter with respect to mechanical parameters of the deck

A methodology for carrying out an analytical sensitivity analysis of the flutter phenomenon in long-span bridges is developed. Decks of bridges are generally bluff bodies and therefore the aeroelastic forces under wind action have to be experimentally evaluated in wind tunnels. Such forces depend on the response frequency of the bridge, which is not known until the problem is solved. Consequently, the calculus of the critical wind speed that initiates the flutter instability comprises a complex nonlinear eigenvalue problem. During the design phase, the sensitivity analysis gives very interesting information about the gradient of the flutter speed with respect to the key chosen design variables, moments of inertia of the bridge deck. The presented method is applied to the Great Belt and Vasco da Gama Bridges, as well as to the old Tacoma Bridge.     

太阳能无人机机翼颤振动力学建模与分析

太阳能无人机作为一种大展弦比轻质飞行器,其机翼的气动弹性效应显著,其中颤振问题尤为关键。此类飞机具有大尺寸和低刚度特点,通过风洞试验研究机翼颤振问题,成本高而且难度大,难以实现,因此仿真计算是分析此类飞机颤振问题的主要手段。针对国内某翼展为40米的太阳能无人机大展弦比机翼,首先对机翼有限元模型进行工程化处理,在此基础上开展结构动力学分析和颤振计算,重点计算了机翼上不同吊舱布置下的颤振速度。经过仿真计算,得到该太阳能无人机机翼颤振速度为26m/s, 满足设计要求,进一步分析表明,可以通过增加发动机连杆的长度、在发动机上增加配重以及改变吊舱在机翼上的展向站位等手段来提高此无人机的颤振速度。     

Finite prism method based topology optimization of beam cross section for buckling load maximization

The use of the finite element method (FEM) for buckling topology optimization of a beam cross section requires large numerical cost due to the discretization in the length direction of the beam. This investigation employs the finite prism method (FPM) as a tool for linear buckling analysis, reducing degrees of freedom of three-dimensional nodes of FEM to those of two-dimensional nodes with the help of harmonic basis functions in the length direction. The optimization problem is defined as the maximization problem of the lowest eigenvalue, for which a bound variable is introduced and set as the design objective to treat mode switching phenomena of multiple eigenvalues. The use of the bound formulation also helps the proposed optimization to treat beams having local plate buckling modes as the fundamental modes as well as beams having global buckling modes. The axial stress is calculated according to the distribution of material modulus which is interpolated using the SIMP approach. Optimization problems finding cross-section layouts from rectangular, L-shaped and generally-shaped design domains are solved for various beam lengths to ascertain the effectiveness of the proposed method.     

Multiple eigenvalues in structural optimization problems

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.Dedicated to the memory of Ernest F. MasurGuest professor during the period 16 November to 11 December, 1992 and 15 November to 12 December, 1993.     

Reliability based optimization in aeroelastic stability problems using polynomial chaos based metamodels

In this work, reliability based design optimization (RBDO) of two aeroelastic stability problems is addressed: (i) divergence, which arises in static aeroelasticity, and (ii) flutter, which arises in dynamic aeroelasticity. A set of design variables is considered as random variables, and the mean mass is minimized for a given set of constraints — including the probability of failure by divergence or flutter. The optimization process requires repeated evaluation of reliability, which is a major contributor to the total computational cost. To reduce this cost, a polynomial chaos expansion (PCE)-based metamodel is created over a grid in the parameter space. These precomputed PCEs are then interpolated for reliability calculation at intermediate points in the parameter space, as demanded by the optimization algorithm. Two new modifications are made to this method in this work. First, the Gauss quadrature rule is used — instead of statistical simulation — to estimate the chaos coefficients for higher computational speed. Second, to increase this computational gain further, a non-uniform grid is chosen instead of a uniform one, based on relative importance of the design parameters. This relative importance is found from a global sensitivity analysis. This new modified method is applied on a rectangular unswept cantilever wing model. For both optimization problems, it is observed that the proposed method yields accurate results with a considerable computational cost reduction, when compared to simulation based methods. The effect of grid spacing is also explored to achieve the best computational efficiency.     

Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.     

Reliability-based shape optimization of structures undergoing fluid–structure interaction phenomena

Fluid–structure interaction phenomena are often roughly approximated when the stochastic nature of a system is considered in the design optimization process, leading to potentially significant epistemic uncertainty. In this paper, after reviewing the state-of-the-art methods in robust and reliability-based design optimization of problems undergoing fluid–structure interaction phenomena, a computational framework is presented that integrates a high-fidelity aeroelastic model into reliability-based design optimization. The design optimization problem is formulated pursuant to the reliability index and performance measure approaches. The system reliability is evaluated by a first-order reliability analysis method. The steady-state aeroelastic problem is described by a three-field formulation and solved by a staggered procedure, coupling a potentially detailed structural finite element model and a finite volume discretization of the Euler flow. The design and imperfection sensitivities are computed by evaluating the analytically derived direct and adjoint coupled aeroelastic sensitivity equations. The computational framework is verified by the optimization of three-dimensional wing structures. The lift-to-drag ratio is maximized, subject to stress constraints, by varying shape, thickness, and material properties. Uncertainties in structural parameters, including design parameters, operating conditions, and modeling uncertainties are considered. The results demonstrate the need for reliability-based optimization methods, for the design of structures undergoing fluid–structure interaction phenomena, and the applicability of the proposed framework to realistic design problems. Comparing the optimization results for different levels of uncertainty shows the importance of accounting for uncertainties in a quantitative manner.     

Reliability-based design optimization of aeroelastic structures

Aeroelastic phenomena are most often either ignored or roughly approximated when uncertainties are considered in the design optimization process of structures subject to aerodynamic loading, affecting the quality of the optimization results. Therefore, a design methodology is proposed that combines reliability-based design optimization and high-fidelity aeroelastic simulations for the analysis and design of aeroelastic structures. To account for uncertainties in design and operating conditions, a first-order reliability method (FORM) is employed to approximate the system reliability. To limit model uncertainties while accounting for the effects of given uncertainties, a high-fidelity nonlinear aeroelastic simulation method is used. The structure is modelled by a finite element method, and the aerodynamic loads are predicted by a finite volume discretization of a nonlinear Euler flow. The usefulness of the employed reliability analysis in both describing the effects of uncertainties on a particular design and as a design tool in the optimization process is illustrated. Though computationally more expensive than a deterministic optimum, due to the necessity of solving additional optimization problems for reliability analysis within each step of the broader design optimization procedure, a reliability-based optimum is shown to be an improved design. Conventional deterministic aeroelastic tailoring, which exploits the aeroelastic nature of the structure to enhance performance, is shown to often produce designs that are sensitive to variations in system or operational parameters.     

Continuation and direct solution of the flutter equation

A new formulation of the flutter equation allowing efficient solutions both by a continuation and a direct method is herewith presented.The continuation method differentiates the flutter equation with respect to the speed giving rise to a system of differential equations whose solution permits an easy and efficient tracking of the aeroelastic modes, frequencies and approximate dampings.If only the flutter point, i.e. its mode, speed and frequency, is required, a direct solution of the nonlinear algebraic system of the flutter equation is performed.Some examples of practical applications are briefly presented and discussed.     

Coupling of a nonlinear finite element structural method with a Navier-Stokes solver

A new three-dimensional viscous aeroelastic solver is developed in the present work. A well validated full Navier-Stokes code is coupled with a nonlinear finite element plate model. Implicit coupling between the computational fluid dynamics and structural solvers is achieved using a subiteration approach. Computations of several benchmark static and dynamic plate problems are used to validate the finite element portion of the code. This coupled aeroelastic scheme is then applied to the problem of three-dimensional panel flutter. Inviscid and viscous supersonic results match previous computations using the same aerodynamic method coupled with a finite difference structural solver. For the case of subsonic flow, multiple solutions consisting of static, upward and downward deflections of the panel are discussed. The particular solution obtained is shown to be sensitive to the cavity pressure specified underneath the panel.     

Optimum design of composite plate wings for aeroelastic characteristics using lamination parameters

The present paper treats the flutter and divergence characteristics of composite plate wings with various sweep angles. First, the effect of laminate configuration on the flutter and divergence characteristics is investigated for composite plate wings. To examine the effect of laminate configuration, the flutter and divergence characteristics are represented on the lamination parameter plane. Next, a minimum weight design of composite plate wings subjected to the constraints on the flutter and divergence speeds is conducted by using a genetic algorithm in which lamination parameters are used as design variables. The effectiveness of aeroelastic tailoring is demonstrated through the optimization results.     

基于优化设计的印制板组件精细化动力学建模方法

以模态测试结果作为修正的目标值,提出了采用ANSYS优化设计对螺栓连接型印制板组件进行精细化动力学建模的方法.首先使用ANSYS概率设计模块计算出印制板组件特征值对结构材料参数的灵敏度.然后基于灵敏度分析结果与螺栓连接处的建模误差,使用ANSYS优化设计模块对印制板组件进行参数修正,最后通过比较不同边界条件下组件的固有特性验证了修正后参数的有效性.研究表明:基于灵敏度分析和建模误差的优化设计方法可以有效提高螺栓连接型印制板组件动力学模型的建模精度.     

Sensitivity analysis and optimization of thin laminated structures with a nonsmooth eigenvalue based criterion

This paper presents a discrete model for the design sensitivity analysis of thin laminated angle-ply composite structures using a plate shell element based on a Kirchhoff discrete theory for the bending effects. To overcome the nondifferentiability of multiple eigenvalues, which may occur during a structural optimization involving free vibrations or buckling design situations, a nonsmooth eigenvalue based criterion is implemented. Angle-ply design variables and vectorial distances from the laminated midle surface to the upper surface of each layer are considered as design variables. The design sensitivities and the directional derivatives are evaluated analytically. The efficiency and accuracy of the model developed is discussed with two illustrative cases which show the need to compute sensitivities of multiple eigenvalues as directional derivatives for laminated composite structures.     

Sensitivity analysis of vibro-acoustic systems in statistical energy analysis framework

The main objective of this paper is to present first and second-order sensitivity analysis of vibro-acoustic systems in the statistical energy analysis (SEA) frame work. Equations for computing these sensitivities for a general SEA model are obtained from two different approaches: (1) direct method and (2) adjoint method. The above equations are applied to a simple model of three plates, joined in the form of a ‘Z’, to minimize the total energy of one of the plates. It has been verified that these approaches lead to the same results and the difference between them is only with respect to the computational efficiency. The design sensitivity results calculated from the proposed analytical methods are compared with those obtained from the finite difference method, which show good agreement. The results of this paper can be useful to optimization of vibro-acoustic systems at the drawing board stage in the SEA framework.     

On structural optimization with aeroelasticity constraints

The optimal design of a cantilever wing in incompressible flow is considered. The wing is modelled as a full depth sandwich wing using finite element analysis. A doublet lattice panel method is used for computation of the unsteady aerodynamic loads. The weight of the wing is minimized using the thicknesses of the composite face sheets as design variables subject to constraints on flutter and divergence speed. Imperfection sensitivity of the final design is analysed and general aspects of imperfection sensitivity in optimization subject to aeroelasticity constraints are discussed in some detail.     

Multidisciplinary approach to aeroelastic studies of long-span bridges

This paper focuses on the design of long-span suspension bridges under aeroelastic constraints. Such challenging structures need to be protected against wind-induced instabilities as flutter. The authors envision that a set of scientific disciplines not currently used in bridge engineering may help along the design process and constitute a useful tool. First, the formulation of sensitivity analysis of flutter speed is described, indicating how this technique can be a guide for engineers making changes in the prototype; two examples of important bridges, as the Great Belt and Messina Strait, are used to demonstrate the capability of this approach. Then, the idea of producing computer animations to represent the aeroelastic deformation of bridges simulating virtual boundary layer wind tunnel testing is presented showing pictures of the Tacoma Narrows and Messina Bridges. Finally, the advantages of introducing distributed computing to make easier to implement the previously mentioned techniques are demonstrated. The authors have previously published papers related with sensitivity analysis in bridges or computer animations of the aeroelastic behaviour of suspension bridges. However, this is the first time that their comprehensive multidisciplinary approach is presented.     

A digital simulation method for flutter analysis

This paper presents a digital simulation method which yields the flutter speed, the flutter frequency, and the subcritical response of a system. The flutter system can be considered as a control system comprising a number of linear hysteresis elements, and the model of the digital simulation is obtained. The subcritical responses of all elements of the system are determined by the Runge-Kutta method, and all eigenvalues and eigenvectors are obtained by the QR method. Some numerical results are given. The flutter speed and frequency calculated by the present method and those by the V-g method agree quite well. Moreover, it yields the responses of all elements of the system unobtainable with other methods. These numerical results show that the digital simulation method for studying a flutter system is feasible, accurate, and convenient.     

Dynamic modeling of swept wing with PTFE method

A numerical approach for the solution of structural problems, based on the concept of parameter transfer finite elements, is here adopted to study the dynamic behaviour of composite wings.The purpose of this paper is to extend the methodology to a wing-like swept composite plate, which can be used, for example, for aeroelastic analysis, multidisciplinary optimization, structural control etc. The dynamics of composite swept wing is here analyzed and numerical results are compared with FEM solution. The effect of sweep angles and material anisotropicity on the aeroelastic behaviour of the wing is studied both from static divergency and dynamic flutter point of view. Several numerical results are presented.     

Computational-fluid-dynamics-based aeroelastic analysis and structural design optimization—a researcher’s perspective

The paper discusses current approaches for the use of computational fluid dynamics in aeroelastic analyses and structural design optimization applications. Current methods for computational fluid dynamics-based static maneuver load analysis and flutter analysis are reviewed, including related issues such as fluid-structure interface, and moving mesh. Present state-of-the-art examples from the literature are presented along with research studies by the author. The paper highlights the challenges of computational fluid dynamics-based aeroelastic methods and their incorporation into aircraft structural design.