Discrete Singular Convolution Research Articles

Free vibration analysis of laminated cylindrical panels using discrete singular convolution

Based on three-dimensional theory of elasticity, semi-analytical solutions for free vibration of cross-ply laminated cylindrical panels are derived applying the state space approach (SSA) and discrete singular convolution (DSC) algorithm. One pair of opposite edges are assumed to be with simply supported and the other pair of edges arbitrary boundary conditions. The thickness direction of panels is chosen as the transfer direction in SSA, and the DSC is employed to discretize the axial direction of panel. Hence, the original partial differential equations are transformed into a state equation consisting of first-order ordinary differential equations. The application of DSC can deal with various boundary conditions, which cannot be solved in the conventional SSA. Accuracy and convergence for laminated cross-ply panels are validated through numerical examples.

Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel

The present study deals with buckling analysis of simply supported conical panels based on the Donnell's shell theory. Different material properties have been considered such as isotropic, composite laminated and functionally graded (FG). The governing differential equation for buckling of laminated conical panel is derived. These equations are discrete using method of discrete singular convolution (DSC). Shannon's delta kernel is used for trial functions. To check the presented DSC method and computer program, the critical buckling loads for isotropic and composite conical panels are calculated which compare very well with earlier available results. The effect of some geometric parameters and material parameters on critical buckling of panels is also investigated. It is noticed that the present DSC methodology can predict accurately the buckling loads of conical panels.

DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix

A simple mechanical model for buckling behavior of boron nitride nanotube (BNNT) surrounded by an elastic matrix is presented. A nonlocal-continuum model is proposed for BNNT using the Euler–Bernoulli beam theory on an elastic matrix. The elastic matrix surrounded of the BNNT is modeled via linear spring model using the Winkler and Pasternak elastic foundation models. The equation is obtained by variational approach for buckling and has been solved by two different approaches. Separation of variables and method of discrete singular convolution are used for computations. The influences of some geometric parameters of BNNT on buckling behavior are investigated in detail. The effect of mode numbers and nonlocal parameter on buckling behavior of BNNT has also been investigated. Finally, some parametric results are presented for BNNT buckling. It is noticed that the present DSC approach can predict accurately the buckling loads for nano-scaled structures.

Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique

AbstractIn the present manuscript, free vibration response of circular cylindrical shells with functionally graded material (FGM) is investigated. The method of discrete singular convolution (DSC) is used for numerical solution of the related governing equation of motion of FGM cylindrical shell. The constitutive relations are based on the Love’s first approximation shell theory. The material properties are graded in the thickness direction according to a volume fraction power law indexes. Frequency values are calculated for different types of boundary conditions, material and geometric parameters. In general, close agreement between the obtained results and those of other researchers has been found.

Static and dynamic response of sector-shaped graphene sheets

ABSTRACTNonlocal elasticity theory is presented for the free vibration and bending analysis of nano-scaled graphene sheets having a sector shape. An eight-node curvilinear domain is used for transformation of the governing equation of motion of sector graphene from physical region to computational region in conjunction with the Kirchhoff plate theory. The discrete singular convolution method is employed for numerical solutions of resulting nonlocal governing differential equations and related boundary conditions. Then, the effects of nonlocal parameters, mode numbers, sector angles, and radius ratios on the static and vibration results of nano-scaled sector-shaped graphene sheets are discussed.

Dynamic Stability Analysis of Beams with Shear Deformation and Rotary Inertia Subjected to Periodic Axial Forces by Using Discrete Singular Convolution Method

AbstractThis paper presents dynamic stability analysis of beams with shear deformation and rotary inertia subjected to periodic axial forces by employing a discrete singular convolution (DSC) algorithm with regularized Shannon kernel. The shear effect has been taken into account based on Engesser’s and Haringx’s theories, respectively. A modified algorithm is proposed to solve the governing equations of beam motion. The influence of rotary inertia, shear deformation, and damping on dynamic instability regions has been investigated. The obtained numerical results are compared with those of the existing method. Numerical results indicate that the modified algorithm is a reliable approach for the solutions of this kind of problem in this investigation. The differences between dynamic instability regions of beams based on Engesser’s and Haringx’s theories are also presented.

A symplectic method for structure-preserving modelling of damped acoustic waves

In this paper, a symplectic method for structure-preserving modelling of the damped acoustic wave equation is introduced. The equation is traditionally solved using non-symplectic schemes. However, these schemes corrupt some intrinsic properties of the equation such as the conservation of both precision and the damping property in long-term calculations. In the method presented, an explicit second-order symplectic scheme is used for the time discretization, whereas physical space is discretized by the discrete singular convolution differentiator. The performance of the proposed scheme has been tested and verified using numerical simulations of the attenuating scalar seismic-wave equation. Scalar seismic wave-field modelling experiments on a heterogeneous medium with both damping and high-parameter contrasts demonstrate the superior performance of the approach presented for suppression of numerical dispersion. Long-term computational experiments display the remarkable capability of the approach presented for long-time simulations of damped acoustic wave equations. Promising numerical results suggest that the approach is suitable for high-precision and long-time numerical simulations of wave equations with damping terms, as it has a structure-preserving property for the damping term.

Thermal buckling of temperature dependent FG-CNT reinforced composite conical shells

In this research, linear thermal buckling of a composite conical shell made from a polymeric matrix and reinforced with carbon nanotube fibres is investigated. Distribution of reinforcements across the shell thickness is assumed to be uniform or functionally graded. Thermomechanical properties of the constituents are temperature dependent. Under the assumption of first order shear deformation shell theory, Donnell kinematic assumptions and von Kármán type of geometrical nonlinearity, the complete set of equilibrium equations and boundary conditions of the shell are obtained. A linear membrane analysis is carried out to obtain the pre-buckling thermal stresses of the shell. Adjacent equilibrium criterion is implemented to establish the stability equations associated with the buckling state. The resulting equations are discreted by means of trigonometric expansion through the circumferential direction and discrete singular convolution method through the shell length. The established eigenvalue problem is solved iteratively to obtain the critical buckling temperature and critical mode number. Parametric studies are presented to explore the influences of semi-vertex angle, volume fraction of CNTs, distribution pattern of CNTs and boundary conditions. It is shown that, conical shells with intermediate carbon nanotube volume fraction do not have, necessarily, intermediate critical buckling temperature.

Erratum for “Numerical Analysis of Free Longitudinal Vibration of Nonuniform Rods: Discrete Singular Convolution Approach” by Mohammad Shokrollahi and Amir Zayeri Baghlani Nejad

C-F −2.5 4.884 2.96 × 10−02 3.455 8.28 × 10−02 2.904 1.17 × 10−01 1.674 2.59 × 10−01 −0.5 3.233 −4.78 × 10−03 2.109 3.06 × 10−02 1.684 4.98 × 10−02 0.812 −5.66 × 10−03 0.5 2.350 −1.65 × 10−02 1.432 −5.79 × 10−02 1.100 1.42 × 10−02 0.470 1.39 × 10−03 2.5 0.864 2.99 × 10−03 0.424 2.87 × 10−02 0.295 5.03 × 10−03 0.101 4.05 × 10−02 C-C −2.5 5.286 5.77 × 10−02 4.486 9.52 × 10−02 4.316 1.02 × 10−01 4.353 8.18 × 10−02 −0.5 3.622 5.47 × 10−03 3.309 4.21 × 10−02 3.384 3.95 × 10−02 4.015 1.88 × 10−03 0.5 2.817 −3.78 × 10−03 2.906 −2.94 × 10−03 3.154 2.59 × 10−03 4.115 −2.03 × 10−03 2.5 2.030 3.89 × 10−02 3.074 5.56 × 10−02 3.598 3.99 × 10−02 5.012 2.38 × 10−03

A Fourier error analysis for radial basis functions and the Discrete Singular Convolution on an infinite uniform grid, Part 1: Error theorem and diffusion in Fourier space

On an infinite grid with uniform spacing h, the cardinal basis Cj(x; h) for many spectral methods consists of translates of a “master cardinal function”, Cj(x; h) = C(x/h − j). The cardinal basis satisfies the usual Lagrange cardinal condition, Cj(mh) = δjm where δjm is the Kronecker delta function. All such “shift-invariant subspace” master cardinal functions are of “localized-sinc” form in the sense that C(X) = sinc(X)s(X) for a localizer function s which is smooth and analytic on the entire real axis and the Whittaker cardinal function is sinc(X) ≡ sin (πX)/(πX). The localized-sinc approximation to a general f(x) is flocalized−sinc(x;h)≡∑j=−∞∞f(jh)s([x−jh]/h)sinc([x−jh]/h). In contrast to most radial basis function applications, matrix factorization is unnecessary. We prove a general theorem for the Fourier transform of the interpolation error for localized-sinc bases. For exponentially-convergent radial basis functions (RBFs) (Gaussians, inverse multiquadrics, etc.) and the basis functions of the Discrete Singular Convolution (DSC), the localizer function is known exactly or approximately. This allows us to perform additional error analysis for these bases. We show that the error is similar to that for sinc bases except that the localizer acts like a diffusion in Fourier space, smoothing the sinc error.

Discrete Singular Convolution Method for Dynamic Stability Analysis of Beams under Periodic Axial Forces

AbstractIn the present study, the discrete singular convolution (DSC) method is used to analyze the dynamic stability of beams subjected to different boundary conditions under periodic axial forces. A unified DSC algorithm is established to solve the governing equations of beam motion under different boundary conditions. The regularized Shannon’s delta kernel is selected as singular convolution to illustrate the present algorithm. The matched interface and boundary method is applied to the treatment of elastic restraint boundary conditions. The effects of a constant term in the periodic axial force and damping on dynamic instability regions are also investigated. The obtained numerical results are compared with those of Bolotin’s method. Numerical results indicate that the DSC method is a reliable method for dynamic stability analysis of beams.

MODAL ANALYSIS OF TAPERED BEAM-COLUMN EMBEDDED IN WINKLER ELASTIC FOUNDATION

Modal analysis of tapered piles embedded in elastic foundations is investigated. The pile is modeled via Bernoulli-Euler beam theory and discrete singular convolution is used for modeling. Some parametric results have been presented for tapered pile in elastic foundation

Fast Wideband Analysis of Reverberation Chambers Using Hybrid Discrete Singular Convolution-Method of Moments and Adaptive Frequency Sampling

This paper reports a fast and wideband analysis of reverberation chambers (RCs) by combining hybrid discrete singular convolution (DSC)-method of moments (MoM) with an adaptive frequency sampling (AFS). Utilizing RCs’ geometrical characteristics, the hybrid DSC-MoM significantly reduces the CPU time requirement. However, due to the strong resonance nature of RCs, wideband analysis is still time consuming. This is because a large number of samples are required through a uniform sampling. In order to accelerate its wideband analysis, the AFS is used to adaptively select samples so that the resonant frequencies are accurately resolved using a small number of samples. Simulation results show that combining the hybrid DSC-MoM and AFS substantially accelerates the wideband analysis of RCs.

A simple and accurate numeric solution procedure for nonlinear buckling model of drill string with frictional effect

Abstract The nonlinear buckling analysis of drill string in a rigid well is presented in this paper. Considering effects of friction and boundary constraints, this problem could be taken as a model of a rod laterally constrained in a rigid cylinder (horizontal, oblique and vertical rigid cylinder could be included). After introducing a new variable, the resulting coupled nonlinear integral–differential equations are successfully solved by employing the extended system shooting method. Examples with various friction coefficients and combinations of boundary conditions are proposed. It is found that the axial frictional force plays a more significant part on buckling load for horizontal well than vertical one. Compared to experimental data or results obtained by using the discrete singular convolution algorithm (DSC) and the finite element method (FEM), the accuracy of the formulations and solution procedures, is verified. What’s more, the nonlinear buckling behaviors of two instances of vertical scientific wells are analyzed. The present results are useful for practical design applications related to calculation of buckling loads and selection of bottom hole assembly (BHA) elements.

Free vibration of layered magneto-electro-elastic beams by SS-DSC approach

Abstract Based on three-dimensional elasticity theory, semi-analytical solutions for free vibration of arbitrary layered magneto-electro-elastic beams are derived applying the state space approach (SSA) and discrete singular convolution (DSC) algorithm. The thickness direction of beams is chosen as the transfer direction in SSA, and the DSC is employed to discretize the length direction. Hence, the original partial differential equations are transformed into a state equation consisting of first-order ordinary differential equations. The application of DSC can implement various boundary conditions, which cannot be solved in the conventional SSA. Numerical examples are presented to study the method in details.

Dynamic Instability Analysis of a Rotating Ship Shaft under a Periodic Axial Force by Discrete Singular Convolution

Dynamic instability of a rotating ship shaft subjected to a periodic axial force is studied by using discrete singular convolution (DSC) with regularized Shannon’s delta kernel. The excitation frequency is related to the spinning speed and the number of blades on the propeller. Effects of number of blades, constant term in the periodic force, and damping on dynamic instability regions are investigated. The results have shown that the increase of number of blades and damping could improve the dynamic stability of rotating shaft with damping. The increase of constant term in the periodic force leads to dynamic instability regions shifting to lower frequencies, making the shaft more sensitive to periodic force. Those dynamic instability regions obtained by DSC method have been compared with those by Floquet’s method to verify the application of DSC method to dynamic instability analysis of rotating ship shaft.

Free vibration of simply supported and multilayered magneto-electro-elastic plates

Based on three-dimensional elasticity theory, semi-analytical solutions for free vibration of simply supported and multilayered magneto-electro-elastic plates have been derived applying a newly developed hybrid analysis, which combines the state space approach (SSA) and the discrete singular convolution (DSC) algorithm. The thickness direction of plate is selected as the transfer direction in SSA, and the DSC is applied to discretize the in-plane domains. Hence, the original partial differential equations are translated into a state equation consisting of first-order ordinary differential equations. The application of DSC makes it possible to treat various boundary conditions, and shows excellent performance for high frequency vibration. The accuracy and convergence of SS-DSC is validate through numerical examples.

Free vibration analysis of circular thin plates with stepped thickness by the DSC element method

Novel formulation is presented by using the discrete singular convolution (DSC) for free vibration analysis of circular thin plates with uniform and stepped thickness. Different from the commonly used ones in literature, regularity conditions are not needed at the circular plate center point to avoid singularity. DSC circular and annular thin plate elements are established. For the DSC circular plate element with radius of R1, the stiffness equation is first formulated in region [−R1, R1] with even number of nodes and then reduced to region [0, R1] by using either symmetric or anti-symmetric conditions. The proposed DSC circular and annular plate elements are used for obtaining frequencies of uniform/stepped circular thin plates or annular thin plates with different boundary conditions. Comparison of the present DSC results to existing analytic and numerical solutions verifies the proposed formulations. The present research extends the DSC method to free vibration of circular thin plates with stepped thicknesses.

Static analysis of laminated conical shells by Discrete Singular Convolution (DSC) approach

Static analyses of laminated conical and cylindrical shell are presented. The governing equation is derived using Love’s first approximation thin shell theory for conical geometry. Then, the method of Discrete Singular Convolution (DSC) is applied to the solution of the resulting governing equation and boundary conditions for bending. Numerical results for stress and deflections of laminated conical and cylindrical shells are presented for different geometric and material parameters. The numerical results show that the present method is quite easy to implement, accurate and efficient for the problems considered.

Thermo-electro-mechanical impedance based structural health monitoring of plates

This study presents a modeling of a plate structure in impedance-based structural health monitoring (ISHM) method using piezoelectric wafer active sensor (PWAS) considering temperature effects. For modeling of the plate, in plane and transverse vibrations of the plate has been considered. Rayleigh–Ritz method was used to discretize the system using three characteristic functions: one of them is plate characteristic function, the other one is Chebyshev pseudo spectral characteristic function and Discrete Singular Convolution (DSC) characteristic functions. After discretization, impedance of PWAS is determined using piezoelectric constitutive equations. In addition to the temperature effect, parameters uncertainly including Young’s modulus and density were considered in the modeling, because these uncertainties are affected on the impedance of PWAS. So, the Monte Carlo simulation is applied to consider parameters uncertainly on PWAS impedance.