Discrete Singular Convolution Method Research Articles

Vibration analysis of a sandwich Timoshenko beam reinforced by GOAM/CNT with various boundary conditions using VIM

In the present study, a sandwich beam reinforced by carbon nanotubes (CNTs)/graphene origami auxetic metamaterials (GOAM) and porous cores is studied. Various distributions of CNTs in facesheets and porosity in the core are stated in terms of thickness. The equations of motion for a sandwich beam are derived and solved under various boundary conditions using variational iteration method (VIM), in which, it is a powerful technique with higher convergence with the computational method that gives analytical response to a linear problem. The effect of the types of CNT distributions, porosity coefficient, types of porous cores, porosity parameter, weight fraction and folding degree of GOAM, different geometric dimensions and different B.C.’s on the natural frequency is investigated. The obtained results are compared with the other literature, that there is a good agreement between them. On the other hands, the maximum error percentage of the present work (VIM) with Civalek and Kiracioglu [66] (Discrete singular convolution method) for the first five frequencies is equal to 0.3% according to Table 2, while in Table 3, the maximum error percentage of the present work (VIM) with Yas and Samadi [67] (differential quadrature method) for the first frequency with various boundary conditions and nanocomposite material is equal to 0.22%. In this regard, the present study can be a suitable platform to find the natural frequency and prevent the phenomenon of resonance and increase the life of the structure. Moreover, it is seen that with enhancing of weight fraction of GOAM, frequency increases because increasing the stiffness of structures. The importance of this research is to extract the formulation of sandwich beam structure for GOAM under different boundary conditions and compare the results between them by repeating the changes.

Mathematical Investigation of the Infection Dynamics of COVID-19 Using the Fractional Differential Quadrature Method

In recent times, the global community has been faced with the unprecedented challenge of the coronavirus disease (COVID-19) pandemic, which has had a profound and enduring impact on both global health and the global economy. The utilization of mathematical modeling has become an essential instrument in the characterization and understanding of the dynamics associated with infectious illnesses. In this study, the utilization of the differential quadrature method (DQM) was employed in order to anticipate the characterization of the dynamics of COVID-19 through a fractional mathematical model. Uniform and non-uniform polynomial differential quadrature methods (PDQMs) and a discrete singular convolution method (DSCDQM) were employed in the examination of the dynamics of COVID-19 in vulnerable, exposed, deceased, asymptomatic, and recovered persons. An analysis was conducted to compare the methodologies used in this study, as well as the modified Euler method, in order to highlight the superior efficiency of the DQM approach in terms of code-execution times. The results demonstrated that the fractional order significantly influenced the outcomes. As the fractional order tended towards unity, the anticipated numbers of vulnerable, exposed, deceased, asymptomatic, and recovered individuals increased. During the initial week of the inquiry, there was a substantial rise in the number of individuals who contracted COVID-19, which was primarily attributed to the disease’s high transmission rate. As a result, there was an increase in the number of individuals who recovered, in tandem with the rise in the number of infected individuals. These results highlight the importance of the fractional order in influencing the dynamics of COVID-19. The utilization of the DQM approach, characterized by its proficient code-execution durations, provided significant insights into the dynamics of COVID-19 among diverse population cohorts and enhanced our comprehension of the evolution of the pandemic. The proposed method was efficient in dealing with ordinary differential equations (ODEs), partial differential equations (PDEs), and fractional differential equations (FDEs), in either linear or nonlinear forms. In addition, the stability of the DQM and its validity were verified during the present study. Moreover, the error analysis showed that DQM has better error percentages in many applications than other relevant techniques.

Buckling analysis of multi-span non-uniform beams with functionally graded graphene-reinforced foams

This paper conducts the buckling analysis of multi-span functionally graded (FG) beams reinforced by three-dimensional graphene foams (3D-GFs) restrained by elastic supports. The graphene foams are considered as graded variation in the thickness direction according to four different porosity distributions. The width and height of the non-uniform beams vary in the longitudinal direction in accordance with the power-law principle, and the multi-span beams consist of two types of optional beams. The governing equations of buckling behavior are derived based on the Timoshenko beam theory and the principle of virtual displacements, and solved by the discrete singular convolution (DSC) method. The purpose of this paper is to obtain the critical buckling load of FG 3D-GFs reinforced beams under the consideration of variable cross-section and multi-span beams with 2–4 spans. The effect of porosity coefficients, slenderness ratio, and spring constants on the buckling characteristic also are studied. The results suggest that the taper ratio of beams and the configuration of multi-span beams lead to remarkable changes in the critical buckling load of beams.

On the supersonic vibrations of the advanced composite quasi-3D hyperbolic refined high-order sector annular/circular structure in thermal environment

Functionally graded materials (FGMs) have been used in many high-tech engineering fields because they could improve material properties. Some well-known applications of these materials are over different geometrical shapes like shells and plates. Multi-directional FGMs have the main role in the improvement of multi-functional materials by introducing modern materials. A detailed study of sector annular/circular disks with different thickness directions has been carried out. In practice, the material properties need to be categorized in other directions as well. Sector annular/circular disks made of multi-dimensional FGM (MD-FGM) are vulnerable to high-temperature gradients and enormous deflections. Using the discrete singular convolution method (DSCM), we investigated asymmetric deformations of MD-FG sector annular/circular disk tolerating thermo-aerodynamics loads. What makes this research unique is using thickness classification for the material properties of the disk while radial and circumferential directions follow an exponential trend. In practical conditions, this assumption would result in a more accurate simulation. Having developed equations based on quasi-3D new refined theory (Q3D-NRT) considering the effect of thickness stretching, we solved the equations based on our conditions. With the aid of COMSOL multi-physics finite element simulation, the results are verified and new recommendations for improving the stability of presented MD-FG sector disks are reported. This research might be helpful for scientists working in the field of thermoelastic theory of disks. Since it is very important to have small and uniform thermal stresses in every direction of the materials, the study’s results could be used as a trustworthy database to determine a material property function.

Dynamic characteristics of non-uniform multi-span functionally graded 3D graphene foams reinforced beams with elastic restraints

This paper investigates the dynamic characteristics of non-uniform multi-span functionally graded (FG) 3D graphene foams (3D-GFs) reinforced beams with elastic boundary conditions. It is assumed that the foams are distributed according to four porosity distributions along the thickness direction and the beams sections varies in height and width along the length with the power-law distribution. Based on Timoshenko beam theory and Hamilton’s principle, the equations of motion for multi-span beams are derived and the discrete singular convolution (DSC) method is adopted to solve them numerically. The present analysis is rigorously validated through direct comparisons against the results by those available in open literatures. A comprehensive parametric study is conducted to study the effects of porosity distribution, slenderness ratio, taper ratio and different combination types of multi-span beams with non-uniform sections. This study provides a new conception of multi-span beam composition, providing designers with choices of beam forms under vibration response.

On exact wave propagation analysis in a beach volleyball ball

Wave propagation and vibration state in the sports balls have important effects on the stability in movement and accuracy of hits. In some sports, similar to football and volleyball, the balls are subjected to continuous strong hits from players. In the current study, the wave propagation in a beach volleyball ball under the effect of temperature change, different material characteristics, different thicknesses, and the total mass is investigated. In this regard, the ball is modeled using a spherical shell structure and a higher-order displacement field is considered. The classical theory of elasticity is employed along with Hamilton’s principle for the spherical shell. The outcome equations with different boundary conditions are solved by engaging the discrete singular convolution method (DSCM). In addition, the finite element method is used to obtain the modal shape and natural frequency. The formulations and solution method are utilized in different steps of convergence examination, validation, and parametric study to acquire a comprehensive insight into the behavior of beach volleyball balls under the effect of the dynamic loading conditions. In specific, we are more interested in the phase velocity of the ball. The results are presented for different temperature changes and geometrical and material properties.

On the dynamics of the axisymmetric circular system via an exact continuum elasticity theory and discrete singular convolution method

The vibration for the circular sector plates made of reinforced composite with carbon nanotubes based on a 3D-poroflexibility theory is presented in this research. A discrete singular convolution integration method (DSC-IM) is used. By using the renowned rule of mixtures, the effective mechanical properties are related to reinforced composites. The present results, formulations, and solution procedures are verified. The novelty of this work considers the effects of 3D- elasticity theory’s initial radial and environmental stresses and various boundary conditions via DSC-IM. Consequently, the outcomes demonstrate that in-plane boundary conditions, CNT’s volume fraction, Biot’s coefficient, and radial and circumferential initial stresses greatly impact the vibration frequency of the circular sector plates. Also, the effect of initial circumferential stress is more prominent than radial ones in determining the vibration frequency. Also, in higher open-angle, the impact of volume fraction related to CNTs on the vibrational frequency of the system is more apparent.

A computational three-dimensional elasticity theory for bending and frequency analysis of the axisymmetric circular/annular plates via machine learning and discrete singular convolution integration methods

The present study deals with the frequency and bending behavior of annular/circular nanocomposite plates. The nanocomposite is made of three-phase materials of graphene oxide (nanoscale) and carbon fibers (microscale), and epoxy matrix (macro-scale) in multiple layers of different orientations. The governing equation extracted to investigate the multiscale hybrid laminated nanocomposite reinforced (MHLNR) is a three-dimensional elasticity theory. Furthermore, the whole structure is on the elastic substrate, which takes into account the horizontal forces in the calculations. The displacement compatibility and equilibrium condition are considered in the interlayers of the three, five, and seven layers of the composite structure. The bulk material properties at each layer are evaluated utilizing fiber micromechanics and Halpin–Tsai equations. The discrete singular convolution method is employed to transform the equations and solve equations. The obtained results are further verified using several references in this field. In addition, an adaptively tuned deep neural network model is designed based on the data of the current study to predict the results. Finally, a comprehensive parametric study is carried out to demonstrate the influences of different parameters on the internal stress field and frequency responses of the current sandwich structure.

Computer simulation as an aid to predict fundamental frequency of a sandwich system via discrete singular convolution method

Present article is the first attempt to analyze the vibration behavior of the composite sandwich cylindrical panels based on the integration of a statistical approaches with numerical solver of differential equations. The motion equations of the structure with functionally graded graphene-platelets reinforced composite (FG-GPLRC) face-sheets imbedded in linear and torsional spring-simulated elastic substrate and subjected to multi-orientational initial stresses are formulated based upon three-dimensional poroelasticity theory. Bi-directional discrete singular convolution method (DSCM) is implemented to find the natural frequency of the mentioned system at design-points. Then, the response surface method is utilized to acquire the closed form of the natural frequency and predict the vibration response of the system according to the reduced third-order polynomial functions. The modified format of Halpin-Tsai micromechanics is employed in order to approximate the elastic properties of the structure. The veracity of the solving procedure is verified by comparing its outcomes with those recorded by the high-quality literature. The results reveal that the span-angle of the panel plays the most important role in the free vibration response of the system compared with the other parameters whose effects are studied in this article. Additionally, it is concluded that reinforcing the face-layers of the sandwich panel by graphene-platelets (GPL) is the more effective way to increase the stiffness of the sandwich structure compared with the case of reinforcing the core section.

Discrete singular convolution method for modelling of waveguide interaction of beam-type structures with impedance boundaries

Discrete Singular Convolution (DSC) is an accurate local/global mathematical method for the vibration and acoustic analysis of individual continuous structures. In recent years, successful applications of the DSC method for handling connected structures have been performed. However, the studies are very limited and applicable only for common basic boundary conditions such as the pinned, clamped or free boundaries. This study extends the applicability of the DSC method to waveguide interactions of connected structures with the mechanical impedance boundaries. For this purpose, vibration analyses of structures composed of beams forming I-, L- and T-type structures are performed via the DSC. Taylor series expansion is utilized to implement the mechanical impedance boundary conditions. The structural models support the longitudinal and bending waveguides and related interactions. Natural frequencies and vibration displacement responses to a point force excitation are predicted via the proposed methodology. The results are validated by finite element and/or analytical solutions. It has been shown that the DSC method can be reliably applied for structures exhibiting waveguide interactions with impedance boundaries.

On the modelling of the vibration behaviors via discrete singular convolution method for a high-order sector annular system

On the modelling of the vibration behaviors via discrete singular convolution method for a high-order sector annular system

Sensitivity analysis of laminated composite plates with different orientations in low to high order modes

Laminated composite plates are utilized due to their high strength/weight ratios. Since such structures are light and have less stiffness, their natural frequencies are lower than usual isotropic structures. Therefore, their behaviours in higher order modes are of interest. On the contrary, sensitivity analysis of vibrating structures is generally performed for lower order modes. In this study, sensitivity analysis of symmetrical/anti-symmetrical/asymmetrical laminated composite plates is performed in low to high order modes. In this regard, angular orientation, Young’s modulus, shear modulus, Poisson’s ratio and density of the plate are selected as design parameters whereas natural frequencies are selected as design objective functions. Discrete singular convolution method is used as solver due to its high computational capability. Although some other conclusions are drawn, it can mainly be concluded that, the natural frequencies are very sensitive to small differences in Poisson ratio for orientation angles of considered plates.

Discrete singular convolution–polynomial chaos expansion method for free vibration analysis of non-uniform uncertain beams

This article enhances the discrete singular convolution method for free vibration analysis of non-uniform thin beams with variability in their geometrical and material properties such as thickness, specific volume (inverse of density) and Young’s modulus. The discrete singular convolution method solves the differential equation of motion of a structure with a high accuracy using a small number of discretisation points. The method uses polynomial chaos expansion to express these variabilities simulating uncertainty in a closed form. Non-uniformity is locally provided by changing the cross section and Young’s modulus of the beam along its length. In this context, firstly natural frequencies of deterministic uniform and non-uniform beams are predicted via the discrete singular convolution. These results are compared with finite element calculations and analytical solutions (if available) for the purpose of verification. Next, the uncertainty of the beam because of geometrical and material variabilities is modelled in a global manner by polynomial chaos expansion to predict probability distribution functions of the natural frequencies. Monte Carlo simulations are then performed for validation purpose. Results show that the proposed algorithm of the discrete singular convolution with polynomial chaos expansion is very accurate and also efficient, regarding computation cost, in handling non-uniform beams having material and geometrical variabilities. Therefore, it promises that it can be reliably applied to more complex structures having uncertain parameters.

Free vibration and buckling analyses of CNT reinforced laminated non-rectangular plates by discrete singular convolution method

This paper presents the free vibration and buckling analyses of functionally graded carbon nanotube-reinforced (FG-CNTR) laminated non-rectangular plates, i.e., quadrilateral and skew plates, using a four-nodded straight-sided transformation method. At first, the related equations of motion and buckling of quadrilateral plate have been given, and then, these equations are transformed from the irregular physical domain into a square computational domain using the geometric transformation formulation via discrete singular convolution (DSC). The discretization of these equations is obtained via two-different regularized kernel, i.e., regularized Shannon’s delta (RSD) and Lagrange-delta sequence (LDS) kernels in conjunctions with the discrete singular convolution numerical integration. Convergence and accuracy of the present DSC transformation are verified via existing literature results for different cases. Detailed numerical solutions are performed, and obtained parametric results are presented to show the effects of carbon nanotube (CNT) volume fraction, CNT distribution pattern, geometry of skew and quadrilateral plate, lamination layup, skew and corner angle, thickness-to-length ratio on the vibration, and buckling analyses of FG-CNTR-laminated composite non-rectangular plates with different boundary conditions. Some detailed results related to critical buckling and frequency of FG-CNTR non-rectangular plates have been reported which can serve as benchmark solutions for future investigations.

Buckling and free vibrations of CNT-reinforced cross-ply laminated composite plates

This article deals with the investigation of free vibration and buckling behaviors of carbon nanotube (CNT)-reinforced cross-ply laminated composite plates. The plate kinematics is assumed to follow a first-order shear deformation theory (FSDT). After the coupled equations of motion and buckling are derived, the method of discrete singular convolution (DSC) is used for the numerical solution of the problems. Both the regularized Shannon’s and Lagrange’s delta kernels are used for spatial discretizing of the resulting governing equations of buckling and vibration of CNT-reinforced laminated plates. Natural frequencies and critical buckling loads are obtained for different cases. Wherever possible, the present DSC results are verified by comparing them with the existing analytical results available in the literature. Then, a detailed parametric study is performed to examine the effects of boundary conditions, CNT distributions and volume fraction, aspect ratio, length-to-width ratio, and number of layers on the frequencies and buckling loads.

Buckling of carbon nanotube (CNT)-reinforced composite skew plates by the discrete singular convolution method

A geometric transformation method based on discrete singular convolution (DSC) is firstly applied to solve the buckling problem of a functionally graded carbon nanotube (FG-CNT)-reinforced composite skew plate. The straight-sided quadrilateral plate geometry is mapped into a square domain in the computational space using a four-node DSC transformation method. Hence, the related governing equations of plate buckling and boundary conditions of the problem are transformed from the physical domain into a square computational domain by using the geometric transformation-based singular convolution. The discretization process is achieved via the DSC method together with numerical differential and two different regularized kernels such as regularized Shannon’s delta and Lagrange-delta sequence kernels. The accuracy of the present DSC results is first verified, and then, a detailed parametric study is presented to show the impacts of CNT volume fraction, CNT distribution pattern, geometry of the skew plate and skew angle on the axial and biaxial buckling responses of FG-CNTR composite skew plates with different boundary conditions. Some new results related to critical buckling of an FG-CNT-reinforced composite skew plate are also presented, which can serve as benchmark solutions for future investigations.

Discrete Singular Convolution Method for Acoustic Transmission Lines

Discrete singular convolution (DSC) algorithm is an accurate dynamic analysis method for single structures. However, vibration analysis of connected structures via the DSC method is limited to step beams and plates. This paper extends the applicability of the DSC method in handling acoustic transmission lines composed of several duct elements with different diameter ratios. Connections of elements are handled by setting up continuity equations at geometric discontinuities. Conformity of DSC equations at these connections is carried out using Taylor series expansion. In this study, natural frequencies, mode shapes and power transmission coefficients of acoustic transmission lines are performed to test the proposed DSC implementation. Power transmission coefficients are obtained through the two-load method using the data provided by the DSC approach. The results are compared with finite element method and analytical solutions (if applicable), and the analytical transfer matrix method is also used for validating the power transmission coefficients. This paper shows that the DSC method is accurate and reliable, and so applicable for acoustic transmission line analysis.

A DSC Regularized Dirac-Delta Method for Flexural Vibration of Elastically Supported FG Beams Subjected to a Moving Load

This research presents a numerical approach to address the moving load problem of functionally graded (FG) beams with rotational elastic edge constraints, in which the regularized Dirac-delta function is used to describe a time-dependent moving load source. The governing partial differential equations of the system, derived in accordance with the classical Euler–Bernoulli beam theory, are approximated by the discrete singular convolution (DSC) method. The resulting set of algebraic equations can then be solved by the Newmark-β integration scheme. Such a singular Dirac-delta formulation is also employed as the kernel function of the DSC method. In this work, the material properties of FG beams are assumed to be changed in the thickness direction. A convergence study is performed to validate the accuracy and reliability of the numerical results. In addition, the effects of moving load velocity and material distribution on the dynamic behavior of elastically restrained FG beams are also studied to serve as new benchmark solutions. By comparing with the available results in the existing literature, the present results show good agreement. More importantly, the major finding of this work indicates that the DSC regularized Dirac-delta approach is a good candidate for moving load problems, since the equally spaced grid system adopted in the DSC scheme can achieve a preferable representation of moving load sources.

Shear buckling analysis of functionally graded (FG) carbon nanotube reinforced skew plates with different boundary conditions

In this paper, a four-nodded straight-sided geometric element is proposed for shear buckling problem of functionally graded material (FGM) composite and carbon nanotube (CNT) reinforced composite skew plate. By using the transformation rule, quadrilateral plate field is mapped into a square domain in the computational space using discrete singular convolution (DSC) method. Related governing equations of skew plate buckling and boundary conditions of the problem are transformed from the physical domain into a square computational domain by using the geometric transformation based singular convolution. The discretization process is achieved via the DSC method together with numerical differential and two-different regularized kernel such as regularized Shannon's delta (RSD) and Lagrange delta sequence (LDS) kernels. The accuracy of the present DSC results is first verified via exiting results in literature. Then, some parametric studies have been presented to show the effects of CNT volume fraction, CNT distribution pattern, geometry of skew plate and skew angle on the shear buckling responses of FG-CNTR composite skew plates with different boundary conditions. Some new results related to critical buckling of FGM and CNT reinforced composite skew plate have been presented which can serve as benchmark solutions for future investigations.

Probabilistic stability analysis of functionally graded graphene reinforced porous beams

This paper presents the first attempt to study the probabilistic stability characteristics of functionally graded (FG) graphene platelets (GPLs) reinforced beams by taking into account the multidimensional probability distributions, such as stochastic porosity and GPL distribution patterns as well as random material properties. For this purpose, a non-inclusive Chebyshev metamodel (CMM), which is implemented on deterministic analysis using discrete singular convolution (DSC) method with excellent computational efficiency and accuracy, is proposed and used to obtain both deterministic and probabilistic results including probability density functions (PDFs), cumulative density functions (CDFs), means and standard deviations of the critical buckling load. The present analysis is rigorously validated through direct comparisons against the results obtained by a direct quasi-Monte Carlo simulation (QMCS) method and those available in open literature. The influences of material properties, porosity distribution, GPL dispersion pattern and boundary condition on probabilistic buckling behaviour of the FG-GPL beam are comprehensively investigated. The global sensitivity analysis is also conducted. The results suggest that the critical buckling load of the FG-GPL beam is most sensitive to porosity distribution, followed by porosity coefficient and GPL weight fraction.