Continuum shape sensitivity analysis for aeroelastic gust using an arbitrary lagrangian-eulerian reference frame

Continuum Sensitivity Analysis (CSA), a method to determine response derivatives with respect to design variables, is derived here for the first time in an arbitrary Lagrangian-Eulerian (ALE) reference frame. CSA differentiates nonlinear governing system of equations to arrive at a linear system of partial differential continuum sensitivity equations (CSEs), here, for fluid-structure interaction (FSI). The CSEs and associated sensitivity boundary conditions are derived here for the first time for FSI, using the boundary velocity formulation, carefully distinguishing design velocity from flow velocity and ALE mesh velocity. Whereas boundary conditions must be differentiated using the material (total) derivative, it is sometimes advantageous to derive the CSEs using local (partial) derivatives. The benefit is that geometric sensitivity, known as design velocity, may not be required in the domain. It is shown here that this advantage is realized when the ALE frame undergoes only the rigid body motion associated with the structure to which it is attached. It is further shown that the advantage is not realized when the ALE mesh deforms due to the flexible motion of the fluid-structure interface. The equations for the transient gust response of a two-dimensional airfoil in compressible flow, flexibly attached to a rigid body mass, are presented as a model problem to illustrate a detailed derivation.     

Rapid heating vibrations of FGM annular sector plates

In this paper, thermally induced vibration of annular sector plate made of functionally graded materials is analyzed. All of the thermomechanical properties of the FGM media are considered to be temperature dependent. Based on the uncoupled linear thermoelasticity theory, the one-dimensional transient Fourier type of heat conduction equation is established. The top and bottom surfaces of the plate are under various types of rapid heating boundary conditions. Due to the temperature dependency of the material properties, heat conduction equation becomes nonlinear. Therefore, a numerical method should be adopted. First, the generalized differential quadrature method (GDQM) is implemented to discretize the heat conduction equation across the plate thickness. Next, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson time marching technique. The obtained thermal force and thermal moment resultants at each time step from temperature profile are applied to the equations of motion. The equations of motion, based on the first-order shear deformation theory (FSDT), are derived with the aid of the Hamilton principle. Using the GDQM, two-dimensional domain of the sector plate and suitable boundary conditions are divided into a number of nodal points and differential equations are turned into a system of ordinary differential equations. To obtain the unknown displacement vector at any time, a direct integration method based on the Newmark time marching scheme is utilized. Comparison investigations are performed to validate the formulation and solution method of the present research. Various examples are demonstrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the plate, temperature dependency, sector opening angle, values of the radius, in-plane boundary conditions, and type of rapid heating boundary conditions on thermally induced response of the FGM plate under thermal shock.

     

On closed form solutions for the differential matrix Riccati equation problem

A new formulation of the differential matrix Riccati equation is presented and a closed analytical solution is obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space. The formulation generalizes the results of [1] on algebraic equations to differential matrix Riccati equations. To illustrate the usefulness of the method, a closed analytical solution of the differential matrix Riccati equation is obtained inR^{2 times 2}.     

A moving Kriging interpolation-based element-free Galerkin method for structural dynamic analysis

In this paper, a meshfree method based on the moving Kriging interpolation is further developed for free and forced vibration analyses of two-dimensional solids. The shape function and its derivatives are essentially established through the moving Kriging interpolation technique. Following this technique, by possessing the Kronecker delta property the method evidently makes it in a simple form and efficient in imposing the essential boundary conditions. The governing elastodynamic equations are transformed into a standard weak formulation. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard implicit Newmark time integration scheme. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in details. As a consequence, it is found that the method is very efficient and accurate for dynamic analysis compared with those of other conventional methods.     

On the choice of an appropriate imbedding algorithm for the solution of unstable linear boundary value problems

The method of Extended (or Generalized) Invariant Imbedding solves unstable linear boundary value problems in a stable manner. The method requires the integration of one of four different matrix Riccati differential equations (four different algorithms). In this contribution the questions of existence, stability, boundedness and error propagation of these Riccati equations are discussed in order to help the user to choose the appropriate algorithm.     

Mixed finite-difference scheme for analysis of simply supported thick plates

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.     

A PDE breach to the SDRE

The state‐dependent Riccati equation (SDRE) is a nonlinear optimal controller derived from applying optimality conditions on a Hamiltonian equation. A co‐state vector is involved in the derivation process. This has been commonly considered a function of time only, despite the existence of states in the co‐state vector. This has resulted in a series of nonlinear coupled ordinary differential equations (ODEs) with a final boundary condition, known as the SDRE. In this work, for the first time, the co‐state vector is regarded as a function of time and states that results in a partial differential equation (PDE) instead of an ODE. The new governing equation is named partial differential state‐dependent Riccati equation (PDSDRE), and the PDE provides a tensor for gain over domains of time and states. Since the generated PDE is highly nonlinear, the solution to the PDSDRE is proposed based on the method of lines (MOL), which is an extension to the finite difference method (FDM). The proposed approach is implemented on both scalar and second order systems and is compared with an SDRE technique to validate the results and show the advantages of proposed structure.     

Accuracy and convergence of a finite element algorithm for laminar boundary layer flow

The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equation appears extensible to the non-linear equations characteristic of laminar boundary layer flow.     

Riccati discrete time transfer matrix method for elastic beam undergoing large overall motion

An efficient method for dynamics simulation for elastic beam with large overall spatial motion and nonlinear deformation, namely, the Riccati discrete time transfer matrix method (Riccati-DT-TMM), is proposed in this investigation. With finite segments, continuous deformation field of a beam can be decomposed into many rigid bodies connected by rotational springs. Discrete time transfer matrices of rigid bodies and rotational springs are used to analyze the dynamic characteristic of the beam, and the Riccati transform is used to improve the numerical stability of discrete time transfer matrix method of multibody system dynamics. A predictor-corrector method is used to improve the numerical accuracy of the Riccati-DT-TMM. Using the Riccati-DT-TMM in dynamics analysis, the global dynamics equations of the system are not needed and the computation time required increases linearly with the system’s number of degrees of freedom. Three numerical examples are given to validate the method for the dynamic simulation of a geometric nonlinear beam undergoing large overall motion.     

A new energy harvester using a cross-ply cylindrical membrane shell integrated with PVDF layers

In this paper, we proposed a smart cylindrical membrane shell panel (SCMSP) model for vibration-based energy harvester. The SCMSP is made of an orthotropic elastic core covered by outer PVDF layers with transverse polarization vector. Electrodynamics governing equations of motion are derived by applying extended Hamilton’s principle. The governing equations are based on Donnell’s linear thin shell theory. The SCMSP displacement fields are expanded by means of double Fourier series satisfying immovable edges with free rotation boundary conditions and coupled system of linear partial differential equations are obtained. The discretized linear ordinary differential equations of motion are obtained using Galerkin method. The output power is taken as an indicating criterion for the generator. A parametric study for MEMS applications is conducted to predict the power generated due to radial harmonic ambient vibration. Optimal resistance value is also obtained for the particular electrode distribution that gives maximum output power. A low vibration amplitude (5?Pa), and a low-frequency (471.79?Hz) vibration source is targeted for the resonance operation, in which the output power of 0.4111?μW and peak-to-peak voltage of 0.2952?V are predicted.     

Solving parabolic PDE's by a decomposition method

The paper deals with the application of the hybrid method of decomposition for the solution of two-dimensional parabolic partial differential equations with constant coefficients. The convergence and accuracy of the CSDT method are analysed, deriving the form of the relationship between ‘eigenvalues’ and the time step. Theoretical problems of this method of decomposition are mentioned. Also, the expression for computing the coefficients for various types of boundary problems has been derived. Practical applications of the method are presented. The cubic spline function interpolation is used for the continuous signal reproduction in the hybrid system. Some results obtained are presented at the end.     

A note on the stability of CSDT methods for solving parabolic equations

In this note we study the stability aspects of CSDT methods for solving parabolic partial differential equations. We define two types of stability and discuss the stability of various CSDT methods.     

Computationally efficient, numerically exact design space derivatives via the complex Taylor's series expansion method

Within numerical design optimization, discrete sensitivity analysis is often used to estimate the derivative of an objective function with respect to the design variables. Discrete sensitivity analysis estimates these derivatives by taking advantage of additional derivative information available in an implicit computational fluid dynamics (CFD) solver of the discretized governing partial differential equations. The key benefits of steady-state discrete sensitivity analysis are its computational efficiency and numerical accuracy. More recently, the complex Taylor's series expansion (CTSE) method has been used to generate these design space derivatives to machine accuracy, by analyzing a complex perturbation of the objective function. For fortran codes, this method is quite easy to implement, for both implicit and explicit codes; unfortunately, the CTSE method can be quite time consuming, because it requires a complex solution of the governing partial differential equations. In this paper, the authors demonstrate that the direct formulation of discrete sensitivity analysis and the CTSE method solve the same iterative sensitivity equation, which sheds light on the most efficient use of the CTSE method. Finally, these methods are demonstrated via application to numerical simulations of one-dimensional and two-dimensional open-channel flows.     

A systematic method for the hybrid dynamic modeling of open kinematic chains confined in a closed environment

This work presents a systematic method for the dynamic modeling of multi-rigid links confined within a closed environment. The behavior of the system can be completely characterized by two different mathematical models: a set of highly coupled differential equations for modeling the confined multi-link system when it has no impact with surrounding walls; and a set of algebraic equations for expressing the collision of this open kinematic chain system with the confining surfaces. In order to avoid the Lagrangian formulation (which uses an excessive number of total and partial derivatives in deriving the governing equations of multi-rigid links), the motion equations of such a complex system are obtained according to the recursive Gibbs–Appell formulation. The main feature of this paper is the recursive approach, which is used to automatically derive the governing equations of motion. Moreover, in deriving the motion equations, the manipulators are not limited to planar motions only. In fact, for systematic modeling of the motion of a multi-rigid-link system in 3D space, two imaginary links are added to the \(n\)-real links of a manipulator in order to model the spatial rotations of the system. Finally, a 2D and a 3D case studies are simulated to demonstrate the effectiveness of the proposed approach.     

Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation

Simply supported and clamped thin elastic plates resting on a two-parameter foundation are analyzed in the paper. The governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order. One of them is Poisson’s equation whereas the other one is Helmholtz’s equation. The local boundary integral equation method is used with meshless approximation for both the Poisson and the Helmholtz equation. The moving least square method is employed as the meshless approximation. Independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations. The Winkler foundation model follows from the Pasternak model if the second parameter is equal to zero. Numerical results for a square plate with simply and/or clamped edges are presented to prove the efficiency of the proposed formulation.     

Modeling and control of flexible space stations

In this article, a preliminary formulation of large space structures and their stabilization is considered. The system consists of a (rigid) massive body and flexible configurations that consist of several beams, forming the space structure. The rigid body is located at the center of the space structure and may play the role of experimental modules. A complete dynamics of the system has been developed using Hamilton's principle. The equations that govern the motion of the complete system consist of six ordinary differential equations and several partial differential equations together with appropriate boundary conditions. The partial differential equations govern the vibration of flexible components. The ordinary differential equations describe the rotational and translational motion of the central body.The dynamics indicate very strong interaction among rigid-body translation, rigid-body rotation, and vibrations of flexible members through nonlinear couplings. Hence, any rotation of the rigid body induces vibration in the beams and vice versa. Also, any disturbance in the orbit induces vibration in the beams and wobbles in the body rotation and vice versa. This makes the system performance unsatisfactory for many practical applications. In this article, stabilization of the above-mentioned system subject to external disturbances is considered. The asymptotic stability of the perturbed system by application of velocity feedback controls is proved using Lyapunov's method.Numerical simulations are carried out in order to illustrate the impact of dynamic coupling or interaction among several members of the system and the effectiveness of the suggested feedback controls for stabilization. This study is expected to provide some insight into the complexity of modeling, analysis, and stabilization of actual space stations.     

An optimization problem in distributed parameter systems†

The optimal control problem for a furnace heating a one-dimensional slab with a quadratic performance index is analysed. This system is a typical distributed parameter system. The Hamiltonian is defined and the canonical equations are obtained. A Riccati type matrix partial differential equation is obtained from the canonical equations. An approximate method to solve these equations is derived and an example is presented to illustrate this method.     

Petrov-Galerkin finite element solution for the first passage probability and moments of first passage time of the randomly accelerated free particle

A numerical solution to an initial boundary value problem governing the probability of failure of a randomly accelerated free particle is obtained using a Petrov-Galerkin finite element method. This direct solution is the first successful one, and no others have been reported in the literature.A solution of the Pontriagin-Vitt equation for the time to first passage of the particle is obtained first: in this case an analytical solution is available and used to evaluate the numerical algorithm. Extensions to the solution of other stochastic differential equations, in particular those governing the probability of failure of the linear oscillator, and applications to structural dynamics are discussed.     

LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C₀-semigroup generation property of such an operator is proven and it is shown that the generated C₀-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.     

A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings

The dynamic stiffness matrix of a composite beam that exhibits both geometric and material coupling between bending and torsional motions is developed and subsequently used to investigate its free vibration characteristics. The formulation is based on Hamilton’s principle leading to the governing differential equations of motion in free vibration, which are solved in closed analytical form for harmonic oscillation. By applying the boundary conditions the frequency dependent dynamic stiffness matrix that relates the amplitudes of loads to those of responses is then derived. Finally the Wittrick-Williams algorithm is applied to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of an illustrative example. The results are discussed and some conclusions are drawn. The theory can be applied for modal analysis of high aspect ratio composite wings and can be further extended to aeroelastic studies.