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In this paper we introduce a new method called the Dirac Assisted Tree (DAT) method, which can handle 1D heterogeneous Helmholtz equations with arbitrarily large variable wave numbers. DAT breaks an original global problem into many parallel tree-structured small local problems, which are linked together to form a global solution by solving small linking problems. To solve the local problems in DAT, we propose a compact finite difference method (FDM) with arbitrarily high accuracy order and low numerical dispersion for piecewise smooth coefficients and variable wave numbers. This compact FDM is particularly appealing for DAT, because the local problems and their fluxes in DAT can be computed with high accuracy. DAT with such compact FDMs can solve heterogeneous Helmholtz equations with arbitrarily large variable wave numbers accurately by solving small linear systems — 4 × 4 matrices in the extreme case — with tridiagonal coefficient matrices in a parallel fashion. Several numerical examples are provided to illustrate the effectiveness of DAT using the Mth order compact FDMs with M = 6 , 8 for numerically solving heterogeneous Helmholtz equations with variable wave numbers. We shall also discuss how to solve some special 2D Helmholtz equations using DAT.

This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.

This paper presents a conjugate gradient (CG) method for accurate and robust downward continuation of potential field data. Utilizing the Block-Toeplitz Toeplitz-Block (BTTB) structure, the storage requirement and the computational complexity can be significantly reduced. Unlike the wavenumber domain regularization methods based on fast Fourier transform, the BTTB-based conjugate gradient method induces little artifacts near the boundary. The application of a re-weighted regularization in a space domain significantly improves the stability of the CG scheme for noisy data. The synthetic data with different levels of added noise and real field data are used to validate the effectiveness of the proposed scheme, and the computed results are compared with those based on recently proposed wavenumber domain methods and the Taylor series method. The simulation results verify that the proposed scheme is superior to the existing methods considered in this study in terms of accuracy and robustness. The proposed scheme is a powerful computational tool capable of applications for large scale data with modest computational cost.

This paper presents new finite difference schemes for solving the Helmholtz equation in the polar and spherical coordinates. The most important result presented in this study is that the developed difference schemes are pollution free, and their convergence orders are independent of the wave number k. Let h denote the step size, it will be demonstrated that when solving the Helmholtz equation at large wave numbers and considering kh is fixed, the errors of the proposed new schemes decrease as h decreases even when k is increasing and kh > 1.

The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

We investigate the effects of parameter uncertainties on the dynamical response of an aeroelastic model representing an oscillating airfoil in pitch and plunge with linear aerodynamics and cubic structural nonlinearities. An approach based on the stochastic normal form is proposed to determine the effects due to the variations in the flow speed and the structural stiffness terms on the stability of the aeroelastic system near the Hopf bifurcation point. This approach allows us to study analytically the bifurcation scenario and to predict the amplitude and frequency of the limit cycle oscillation (LCO). The results show that the amplitude of LCO corresponding to the supercritical Hopf bifurcation increases with the intensity of the noise perturbing the pitch and plunge cubic terms, but there is almost no effect on the LCO frequency. Uncertainties in the flow speed produce a shift in the bifurcation point, and unstable subcritical behavior may occur for values of parameters for which the corresponding deterministic model is stable. The stochastic normal form confirms and extends previously known numerical results regarding the effect of parameter variations, and offers an effective way to perform sensitivity analysis of the system's response.

A stochastic collocation method is proposed to investigate the secondary bifurcation of a two-dimensional aeroelastic system with structural nonlinearity represented by cubic restoring forces, and uncertainties expressed by random parameters in the cubic stiffness coefficient and in the initial pitch angle. The accuracy of the stochastic collocation method is improved by incorporating higher order schemes, such as piecewise cubic interpolation and piecewise cubic spline interpolation, instead of a piecewise linear interpolation formula. For an aeroelastic problem with the uncertainty expressed by a time dependent combination of five random variables, an efficient collocation method is developed using a sparse grid approach with a dimension adaptive strategy. Numerical simulations are carried out to demonstrate the effectiveness of the proposed method for long term computation and discontinuous problems.

In this paper we present an expert system to perform steady-state response predictions. We consider an aeroelastic model simulating a two degree-of-freedom airfoil oscillating in pitch and plunge with a freeplay nonlinearity in the pitch degree-of-freedom. In the proposed data-driven methodology, a freeplay is first confirmed, and then the locations of the switching points are determined. A state-space formulation is constructed to model the piece-wise linear system. The parameters of the system are estimated using the Kalman filter and the expectation maximization algorithm. The attractive feature of the present approach is its ability to accurately predict the steady-state behavior of the nonlinear aeroelastic system with freeplay, using only a limited amount of transient input data. To demonstrate the effectiveness of the proposed methodology, we present applications to freeplay aeroelastic data arising from wind tunnel experiments and numerical simulations.

This study focuses on the Rosenau–Burgers equation u t + u x x x x t − α u x x + f ( u ) x = 0 with a periodic initial boundary condition. It is proved that with smooth initial value the global solution uniquely exists. Furthermore, for α > 0 , the global solution converges time asymptotically to the average of the initial value in an exponential form, and the convergence rate is optimal; while for α = 0 , the unique solution oscillates around the initial average all the time. Finally, the numerical simulations are reported to confirm the theoretical results.

This paper focuses on the phase transitions of a 2$\times$2 system of mixed type for viscosity-capillarity with periodic initial-boundary condition in a viscoelastic material. By the Liapunov functional method, we prove the existence, uniqueness, regularity and uniform boundedness of the solution. The results are correct even for large initial data.

We present some new results on the asymptotic behavior of the periodic solution to a 2$\times$2 mixed-type system of viscosity-capillarity in a viscoelastic material. We prove that the solution converges to a certain stationary solution as time approaches to infinity, in particular, when the viscosity is large enough or the mean of the initial datum is in the hyperbolic regions, the solution converges exponentially to the trivial stationary solution with it any large initial datum. The location of the initial datum and the amplitude of viscosity play a key role for the phase transitions. Furthermore, we obtain the convergence rate to the stationary solutions. Finally, we carry out numerical simulations to confirm the theoretical predictions.

Background Real Time Cell Analysis (RTCA) technology is used to monitor cellular changes continuously over the entire exposure period. Combining with different testing concentrations, the profiles have potential in probing the mode of action (MOA) of the testing substances. Results In this paper, we present machine learning approaches for MOA assessment. Computational tools based on artificial neural network (ANN) and support vector machine (SVM) are developed to analyze the time-concentration response curves (TCRCs) of human cell lines responding to tested chemicals. The techniques are capable of learning data from given TCRCs with known MOA information and then making MOA classification for the unknown toxicity. A novel data processing step based on wavelet transform is introduced to extract important features from the original TCRC data. From the dose response curves, time interval leading to higher classification success rate can be selected as input to enhance the performance of the machine learning algorithm. This is particularly helpful when handling cases with limited and imbalanced data. The validation of the proposed method is demonstrated by the supervised learning algorithm applied to the exposure data of HepG2 cell line to 63 chemicals with 11 concentrations in each test case. Classification success rate in the range of 85 to 95 % are obtained using SVM for MOA classification with two clusters to cases up to four clusters. Conclusions Wavelet transform is capable of capturing important features of TCRCs for MOA classification. The proposed SVM scheme incorporated with wavelet transform has a great potential for large scale MOA classification and high-through output chemical screening. Electronic supplementary material The online version of this article (doi:10.1186/s13040-016-0098-0) contains supplementary material, which is available to authorized users.

The purpose of this paper is to study the zero-dispersion limit of the water wave interaction equations which arise in modelling surface waves in the present of both gravity and capillary modes. This topic is also of interest in plasma physics. For the smooth solution, the limiting equation is given by the compressible Euler equation with a nonlocal pressure caused by the long wave. For weak solution, when the coupling coefficient λ is small order of ε, λ=o(ε), the wave map equation is derived and the scattering sound wave is shown to satisfy a linear wave equation.

Aging aircraft and combat aircraft that carry heavy external stores potentially face problems arising from nonlinearities in structure. An expert data mining system is proposed that is capable of predicting the asymptotic behavior of an aeroelastic system with structural nonlinearities represented by polynomial restoring forces or freeplay models. The input is represented only by a limited set of transient data. The output provides a long-term nonlinear aeroelastic response, and the prediction is made when certain rule-based reasoning conditions are satisfied. An attractive feature of this new approach is that no information about the system parameters is needed. In the prediction module, we propose two methods, based on nonlinear time series models and the unscented Kalman filter. To our knowledge, these approaches have not been reported so far for predicting the long-term nonlinear aeroelastic responses. Compared with the classical extended Kalman filter, the unscented filter does not require differentiability and can be applied to nonlinear aeroelastic models with freeplay and hysteresis. The performances of the expert data mining system are demonstrated for simulated data and wind-tunnel experimental aeroelastic data resulting from a two-degree-of-freedom airfoil oscillating in pitch and plunge.

A new approach is proposed, based on the use of artificial neural networks, for predicting nonlinear aeroelastic oscillations. Our objective is to reconstruct the asymptotic state of the nonlinear behavior of an aeroelastic model when only a limited segment of the transient data is known. An original neural network architecture is proposed and is used to predict the nonlinear motions of an aeroelastic system modeling a self-excited two-degree-of-freedom airfoil oscillating in pitch and plunge. When a segment of the transient state of the given signal is used for training, the neural network is capable of correctly predicting the corresponding limit-cycle oscillations, damped oscillations, or unstable divergent oscillations. The network training set consists of numerically generated data or data obtained from a wind-tunnel experiment. A neural network used in conjunction with a wavelet decomposition is presented, which proves to be capable of extracting the values of the damping coefficients and frequencies from the predicted signal. Neural networks, thus, prove to be useful tools in nonlinear aeroelastic analysis.

This paper investigates the dynamic response of a two-dimensional aeroelastic system with structural non-linearity represented by hysteresis. The formulations of the point transformation method developed in Part 1 of this study for the aeroelastic system with a freeplay model is extended for a hysteresis model. These formulations can be applied not only to predict the amplitude and frequency of limit cycle oscillations, but also to detect complex aeroelastic responses such as periodic motion with harmonics, period doubling, chaotic motion and the coexistence of stable limit cycles. It is shown that the point transformation technique is the most suitable to analyze the aeroelastic response of systems containing piecewise continuous restoring forces.

A point transformation technique is developed to investigate the non-linear behavior of a two-dimensional aeroelastic system with freeplay models. Two formulations of the point transformation method are presented, which can be applied to accurately predict the frequency and amplitude of limit cycle oscillations. Moreover, it is demonstrated that the developed formulations are capable of detecting complex aeroelastic responses such as periodic motions with harmonics, period doubling, chaotic motions and the coexistence of stable limit cycles. Applications of the point transformation method to several test examples are presented. It is concluded that the formulations developed in this paper are efficient and effective.

In this study, a frequency relation for limit cycle oscillations of a two-degree-of-freedom aeroelastic system with structural non-linearities represented by cubic restoring spring forces is derived. The centre manifold theory is applied to reduce the original system of nine-dimensional first order ordinary differential equations to a governing system in two dimensions near the bifurcation point. The principle of normal form is used to simplify the non-linear terms of the lower dimensional system. Using the frequency relation and the amplitude–frequency relationships derived from a previous study, limit cycle oscillations (LCOs) for self-excited systems can be predicted analytically. The mathematical technique proposed here has been applied to investigate LCO near a Hopf-bifurcation for an aeroelastic system with cubic restoring forces. Not only that an excellent agreement is obtained compared to the numerical results from solving the original system of eight non-linear differential equations by Runge–Kutta time integration scheme, but we also demonstrate that the use of a mathematical approach leads to a better understanding of non-linear aeroelasticity.

We present a fully implicit and discrete energy conserving finite difference scheme for the solution of an initial-boundary value problem of the nonlinear Klein-Gordon equation. A theoretical analysis is performed, and it has been demonstrated that the numerical scheme is particularly attractive when long time solutions are sought.

New formulations for the algebraic multigrid (AMG) method are presented. A new interpolation operator is developed, in which the weighting could be negative. Numerical experiments demonstrate that the use of negative interpolation weights is necessary in some applications. New approaches to construct the restriction operator and the coarse-grid equations are discussed. Two new AMG methods are proposed. Theoretical study and convergence analysis of the AMG methods are presented. The main contributions of this paper are to improve the convergence rate and to extend the range of applications of an AMG method. Numerical experiments are reported for matrix computations that resulted from partial differential equations, signal processing, and queueing network problems. The success of the proposed new AMG algorithms is clearly demonstrated by applications to non-diagonally dominant matrix problems for which the standard AMG method fails to converge.

In calculations of transonic flows it is necessary to limit the domain of computation to a size that is manageable by computers. At the boundary of the computational domain, boundary conditions are required to ensure a unique solution. Since wave solutions exist in the unsteady transonic flow field, incorrect boundary conditions may result in spurious reflections from the computational boundary. This may introduce errors into the solution. To prevent the spurious reflections, absorbing boundary conditions are often used on the computational boundary. In this paper we describe a method to derive absorbing boudary conditions for transonic calculations. We demonstrate both theoretically and numerically that the use of the absorbing boundary conditions will reduce the spurious reflections in the calculation.

As an effect of continual advances in high performance computing, numerical simulations are widely used to study complex aeroelastic systems. Uncertainty quantification (UQ) methods are an important part of the verification and validation process of the computational simulation results, because small variations in the system parameters can have a large influence on the aeroelastic response. In this paper, we use an approach based on the stochastic normal form to analyze the impact of parameters uncertainty on the Hopf bifurcation and on the amplitude and frequency of the limit cycle oscillations (LCO) of a nonlinear airfoil structure. We start from the following deterministic model describing a two-degree-of-freedom airfoil oscillating in pitch and plunge

This paper is concerned with the eigenvalues of Strum-Liouville problems with periodic and semi-periodic boundary conditions by a shooting algorithm. The proposed technique is based on the application of the Floquet theory. Convergence analysis and a general guideline to provide the starting values for the computed eigenvalues are presented. Some numerical results are also reported.