A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity

Dai, Honghua; Yue, Xiaokui; Yuan, Jianping; Atluri, Satya N.

doi:10.1016/j.jcp.2014.03.063

Cited by 43 publications

(38 citation statements)

References 25 publications

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“…Normally, the numerical approximation of the Jacobian matrix is calculated in each iteration step via numerical difference techniques. The explicitly derived Jacobian 204 HONGHUA DAI, XIAOKUI YUE AND SATYA N. ATLURI matrix may significantly accelerate the computing rate of the algebraic solver [Dai et al 2014a]. The effect of using the explicitly derived Jacobian matrix rather than the numerically calculated one on the computational efficiency has been intensively analyzed in [Dai et al 2014b] in a two-degree-of-freedom airfoil problem.…”

Section: Solutions Of the Von Kármán Plate Equations Bymentioning

confidence: 99%

Solutions of the von Kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix

Dai

Yue

Atluri

2014

J. Mech. Mater. Struct.

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Large deflections of a simply supported von Kármán plate with imperfect initial deflections, under a combination of in-plane loads and lateral pressure, are analyzed by a semianalytical global Galerkin method. While many may argue that the dominance of the finite element method in the marketplace may make any other attempts to solve nonlinear plate problems to be redundant and obsolete, semi-and precise analytical methods, when possible, simply serve as benchmark solutions if nothing else. Also, since parametric variations are simpler to access through such analytical methods, they are more useful in studying the physics of the phenomena. In the present method, the Galerkin scheme is first applied to transform the governing nonlinear partial differential equations of the von Kármán plate into a system of general nonlinear algebraic equations (NAEs) in an explicit form. The Jacobian matrix, the tangent stiffness matrix of the system of NAEs, is explicitly derived, which speeds up the Newton-Raphson iterative method if it is used. The present global Galerkin method is compared with the incremental Galerkin method, the perturbation method, the finite element method and the finite difference method in solving the von Kármán plate equations to compare their relative accuracies and efficiencies. Buckling behavior and jump phenomenon of the plate are detected and analyzed. Besides the classical NewtonRaphson method, an entirely novel series of scalar homotopy methods, which do not need to invert the Jacobian matrix (the tangent stiffness matrix), even in an elastostatic problem, and which are insensitive to the guesses of the initial solution, are introduced. Furthermore, we provide a comprehensive review of the newly developed scalar homotopy methods, and incorporate them into a uniform framework, which renders a clear and concise understanding of the scalar homotopy methods. In addition, the performance of various scalar homotopy methods is evaluated through solving the Galerkin-resulting NAEs. The present scalar homotopy methods are advantageous when the system of NAEs is very large in size, when the inversion of the Jacobian may be avoided altogether, when the Jacobian is nearly singular, and the sensitivity to the initially guessed solution as in the Newton-Raphson method needs to be avoided, and when the system of NAEs is either over-or under-determined.

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Section: Solutions Of the Von Kármán Plate Equations Bymentioning

confidence: 99%

Solutions of the von Kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix

Dai

Yue

Atluri

2014

J. Mech. Mater. Struct.

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“…Hence, the MSTDC method is regarded as an extension of the TDC method. The TDC method has been shown to be a simple but powerful tool to solve non-linear oscillatory problems [1,7,10]. For solving the NAEs, the Jacobian matrix B of the MSTDC algebraic system is required, and is derived by differentiating F i with respect to a 0 ; a k and b k .…”

Section: Formulation Of the Mstdc In Solving The Duffing Equationmentioning

confidence: 99%

“…Some representatives are the perturbation methods [4], the differential transformation method [5], the harmonic balance (HB) method [6,7], the high dimensional harmonic balance (HDHB) method [8,9], and the time domain collocation (TDC) method [1]. In Dai et al [1], the TDC method was initially developed as a simple alternative method to solve non-linear oscillatory problems, wherein it was successfully applied to investigate the complex responses of the Duffing oscillator.…”

Section: Introductionmentioning

confidence: 99%

“…Dai et al [10] extended the TDC method via increasing the number of collocation points to successfully eliminate the fake solutions which may arise from the HDHB solution procedure. Furthermore, Dai et al [7] conducted a comparative study of the classical HB method, the TDC method, and the HDHB method in the framework of solving a two dimensional airfoil in subsonic flow, wherein they clearly explained the aliasing mechanism of the HDHB method.…”

Section: Introductionmentioning

confidence: 99%

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A multiple scale time domain collocation method for solving non-linear dynamical system

Dai

Yue

Liu

2014

International Journal of Non-Linear Mechanics

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“…The assumption of linear aerodynamics is adequate to treat subsonic and supersonic flow regimes . In transonic conditions, which are relevant for passengers' transport aircraft jets, the flow is dominated by nonlinear effects and exhibits complex interactions between shock waves and boundary layer (Alder, 2015;Dai et al, 2014a;Dai et al, 2014b).…”

Section: Introductionmentioning

confidence: 99%

Computational fluid dynamics-based transonic flutter suppression with control delay

Zhou

Ronch

et al. 2016

Journal of Fluids and Structures

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This work investigates the effects of control input time delay on closed-loop transonic computational aeroelastic analysis. Control input time delays are becoming critical as the demand for high frequency control actions is increasing. The flow in transonic conditions exhibits strong nonlinearities which require accurate physical modelling techniques, in turn resulting in large dimensional systems that are computationally costly to solve. A unified framework is demonstrated for the robust and efficient generation of reduced order models. Once generated, the reduced order model is employed for the flutter boundary search, and excellent agreement with the large order coupled model is demonstrated. The aero-servo-elastic reduced order model is then exploited to design a feedback control law, which is implemented in the fully coupled computational fluid/structural dynamics solver. As expected, the controller effectiveness is found to degrade for increasing time delay, up to a critical value where the controller fails to suppress flutter. It is shown that a controller for a time-delay system may be designed using the same aero-servo-elastic reduced order model, incurring in no extra costs or complications. The new controller is found to achieve excellent flutter suppression characteristics. The aero-servo-elastic reduced order model may also be used to identify, for a given feedback controller, the critical value of control input time delay at which the closed-loop aero-servo-elastic system loses its stability. The test cases are for a two-dimensional pitch-plunge aerofoil section and the AGARD 445.6 wing modified with a trailing-edge control surface.

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A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity

Cited by 43 publications

References 25 publications

Solutions of the von Kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix

Solutions of the von Kármán plate equations by a Galerkin method, without inverting the tangent stiffness matrix

A multiple scale time domain collocation method for solving non-linear dynamical system

Computational fluid dynamics-based transonic flutter suppression with control delay

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