GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of highorder accuracy. To attain high accuracy in time discretization, the fourth-order AdamsBashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Hermite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation （nonlinear convection- diffusion problem） and KdV equation（single solitary and 2-solitary wave problems）, where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries （KdV） equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation （nonlinear convectiondiffusion problems） and the KdV equation （single solitary and 2-solitary wave problems）, where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.

CHEBYSHEV FINITE SPECTRAL METHOD FOR 2-D EXTENDED BOUSSINESQ EQUATIONS

In this article,an accurate Chebyshev finite spectral method for the 2-D extended Boussinesq equations is proposed.The method combines the advantages of both the finite difference and spectral methods.The Adams-Bashforth predictor and the fourth-order Adams-Moulton corrector are adopted for the numerical solution of the governing differential equations.An efficient wave absorption strategy is also proposed to effectively absorb waves at outgoing wave boundaries and reflected waves from the interior of the computational domain due to barriers and bottom slopes at the incident wave boundary to avoid contamination of the specified incident wave conditions.The proposed method is verified by a case where experimental data are available for comparison for both regular and irregular waves.The case is wave diffraction over a shoal reported by Vincent and Briggs.Numerical results agree very well with the corresponding experimental data.

EFFICIENT NUMERICAL METHOD FOR DYNAMIC ANALYSIS OF FLEXIBLE ROD HIT BY RIGID BALL

Impact dynamics of flexible solids is important in engineering practice. Obtaining its dynamic response is a challenging task and usually achieved by numerical methods. The objectives of the study are twofold. Firstly, the discrete singular convolution （DSC） is used for the first time to analyze the impact dynamics. Secondly, the efficiency of various numerical methods for dynamic analysis is explored via an example of a flexible rod hit by a rigid ball. Three numerical methods, including the conventional finite element （FE） method, the DSC algorithm, and the spectral finite element （SFE） method, and one proposed modeling strategy, the improved spectral finite element （ISFE） method, are involved. Numerical results are compared with the known analytical solutions to show their efficiency. It is demonstrated that the proposed ISFE modeling strategy with a proper length of con- ventional FE yields the most accurate contact stress among the four investigated models. It is also found that the DSC algorithm is an alternative method for collision problems.

Time-domain Spectral Finite Element Method for Wave Propagation Analysis in Structures with Breathing Cracks

Guided waves are generally considered as a powerful approach for crack detection in structures,which are commonly investigated using the finite element method(FEM).However,the traditional FEM has many disadvantages in solving wave propagation due to the strict requirement of mesh density.To tackle this issue,this paper proposes an efficient time-domain spectral finite element method(SFEM)to analyze wave propagation in cracked structures,in which the breathing crack is modeled by definiiig the spectral gap element.Moreover,novel orthogonal polynomials and Gauss-Lobatto-Legendre quadrature rules are adopted to construct the spectral element.Meanwhile,a separable hard contact is utilized to simulate the breathing behavior.Finally,a comparison of the numerical results between the FEM and the SFEM is conducted to demonstrate the high efficiency and accuracy of the proposed method.Based on the developed SFEM,the nonlinear features of waves and influence of the incident mode are also studied in detail,which provides a helpful guide for a physical understanding of the wave propagation behavior in structures with breathing cracks.

Applying the spectral stochastic finite element method in multiple-random field RC structures

This paper uses the spectral stochastic finite element method(SSFEM)for analyzing reinforced concrete(RC)beam/slab problems.In doing so,it presents a new framework to study how the correlation length of a random field(RF)with uncertain parameters will affect modeling uncertainties and reliability evaluations.It considers:1)different correlation lengths for uncertainty parameters,and 2)dead and live loads as well as the elasticity moduli of concrete and steel as a multi-dimensional RF in concrete structures.To show the SSFEM’s efficiency in the study of concrete structures and to evaluate the sensitivity of the correlation length effects in evaluating the reliability,two examples of RC beams and slabs have been investigated.According to the results,the RF correlation length is effective in modeling uncertainties and evaluating reliabilities;the longer the correlation length,the greater the dispersion range of the structure response and the higher the failure probability.

Effect of turbulence intensity on airfoil flow:numerical simulations and experimental measurements

The effect of the turbulence intensity of the oncoming stream on the aerodynamic characteristics of the NACA-0012 airfoil is investigated by a direct numerical simulation. The numerical results are found to be consistent with the experimental measurements. Based on the finite spectral QUICK scheme, the simulation gets the high accuracy results. Both the simulation and the experiment reveal that the airfoil stall does not exist for the low turbulence intensity, however, occurs when the turbulence intensity increases sufficiently. Besides, the turbulence intensity has a significant effect on both the airfoil boundary layer and the separated shear layer.