Differential Quadrature Method Research Articles

The free vibration analysis of ring-stiffened FG conical sandwich shells with auxetic and non-auxetic honeycomb cores

In this paper, the natural frequencies of a ring-stiffened conical sandwich shell with a functionally graded (FG) honeycomb core are calculated. The core can be either a non-auxetic honeycomb (NAH) or an auxetic honeycomb (AH). The honeycomb core and face layers are fabricated from metal-ceramic functionally graded material (FGM) in which the volume fraction of the ceramic increases from zero at the inner surface of the shell to one at its outer surface based on either power-law function (P-FGM), sigmoid function (S-FGM), or exponential function (E-FGM). The sandwich shell is modeled based on Murakami’s zig-zag theory and the intermediate ring support is modeled as a rigid structure. Hamilton’s principle is employed to derive the governing equations, compatibility conditions, and boundary conditions. A semi-analytical solution is presented which includes an exact solution in the circumferential direction followed by an approximate numerical solution via the differential quadrature method (DQM) in the meridional direction. It is concluded that for each vibrational mode, optimal values can be found for the inclined angle, core-to-shell thickness ratio, and location of the ring support which bring about the highest natural frequency. The presented work is the first theoretical work associated with the free vibration analysis of a ring-stiffened FG conical sandwich shell with a honeycomb core which provides a reduction in the mass and improvement in the thermo-mechanical characteristics, simultaneously. Such properties make such a structure widely used in the aerospace industry. Considering the importance of the dynamic characteristics of aerospace structures, the results of this research can be used by these industries in the analysis and optimal design of the main body of airplanes and missiles.

In-plane vibration analysis of elastically restrained FGM skew plates using variational differential quadrature method

This work presents an accurate in-plane vibration analysis of functionally graded material (FGM) skew plates with elastically restrained boundaries using the variational differential quadrature method (VDQM). The weak form of the governing equations is derived by integrating two-dimensional elasticity theory with Hamilton's principle. The differential and integral operators are directly converted into matrix forms, thereby removing the necessity for higher-order derivative approximations in the displacement field. Transformation matrices are then developed for both interior and boundary nodes to link the governing equations with the boundary conditions, leading to the formulation of the vibration eigenvalue equations for FGM skew plates. Various factors, including aspect ratios, skew angles, boundary restraints, and gradient indices, are considered to investigate the in-plane vibration mode characteristics of FGM skew plates. Detailed solution procedures are provided, and numerical examples using the proposed solutions indicate that the VDQM exhibits superior numerical convergence and stability compared to other existing methods. The study also investigates the influence of highly skewed plates (75°), where stress singularities arise at the corners. This aspect is crucial for in-plane vibration analysis and has garnered limited attention in the existing literature. Furthermore, the dynamic analysis of FGM skew plates reveals a coupling between normal and tangential vibration modes.

Advanced dynamic thermal vibration of thick magnetostrictive FGM plates-cylindrical shells in unsteady yawed supersonic flow

The thermal vibration of magnetostrictive functionally graded material (FGM) thick plates-cylindrical shells in advanced dynamic unsteady yawed supersonic flow with nonlinear third-order piston theory is studied by using numerical method. The effects of yawed angle, controlled gain, and third-order shear deformation theory (TSDT) on the thermal stress and center displacement are analyzed. The nonlinear coefficient of TSDT and advanced shear correction provide the main effects on the time response values of displacements and stresses. Also the controlled gain provides the additional effects on the transient response values to control the displacements and stresses into smaller values. The novelties of the present article are the advanced shear coefficient included with TSDT displacement, velocity feedback control law, and nonlinear third-order piston theory used to obtain the thermal vibration modeling equation and the simulation results for FGM plates-shells in nonlinear unsteady yawed supersonic flow. The FGM structures are usually used for high temperature area to provide better resistance in the deflection and stress. The generalized differential quadrature (GDQ) methods used in the numerical calculation for getting the simulation results for FGM plates-shells in nonlinear unsteady yawed supersonic flow.

Snap-through of functionally graded graphene origami-enabled auxetic metamaterial doubly curved nonlinear shells

The lightweight design of thin-walled curved structures as spherical shells are frequently implemented in architecture, aerospace, mechanical, automotive, nuclear, and defense structures. Under the transverse loads, shells may largely deform and hence snap from one equilibrium position to the other. Thus, this work aims to develop a mathematical model and computational solution to investigate the nonlinear bending and snap-through behavior of doubly curved auxetic metamaterial shell subjected to transversal loading, for the first time. The shell is composed of several layers through the shell thickness, each layer is manufactured from a copper (Cu) matrix reinforced with a specified weight fraction of graphene origami auxetic metamaterial (GOAM). The mechanical properties of the GOAM shell panel are presented and described by functions of the GOAM volume fraction and folding degree. Three types of GOAM-distributions are considered, which are U-type, X-type, and O-type. The theoretical framework of Kirchhoff–Love hypotheses for thin shells and von Karman type nonlinearity are used to derive the governing equations. The differential quadrature method (DQM) is implemented to discretize the space domain and convert the nonlinear partial differential equation to nonlinear algebraic equations in terms of displacement field. An efficient incremental iterative procedure is developed to solve the nonlinear equations and predict the snap-through behavior. A model conversion and validation with isotropic spherical shell is considered. Several numerical results are conducted considering the effects of changing GOAM content, distribution pattern, folding degree, and shell curvature and thickness. For panels exhibiting snap-through behavior, increasing the shell curvature leads to higher snap-through limiting load; however, the instability gap is enlarged.

Advanced Dynamic Vibration Control Algorithms of Materials Terfenol-D Si3N4 and SUS304 Plates/Cylindrical Shells with Velocity Feedback Control Law

A numerical, generalized differential quadrature (GDQ) method is presented on applied heat vibration for a thick-thickness magnetostrictive functionally graded material (FGM) plate coupled with a cylindrical shell. A nonlinear c1 term in the z axis direction of a third-order shear deformation theory (TSDT) displacement model is applied into an advanced shear factor and equation of motions, respectively. The equilibrium partial differential equation used for the thick-thickness magnetostrictive FGM layer plate coupled with the cylindrical shell under thermal and magnetostrictive loads can be implemented into the dynamic GDQ discrete equations. Parametric effects including nonlinear term coefficient of TSDT displacement field, advanced nonlinear varied shear coefficient, environment temperature, index of FGM power law and control gain on displacement, and stress of the thick magnetostrictive FGM plate coupled with cylindrical shell are studied. The vibrations of displacement and stress can be controlled by the control gain algorithms in velocity feedback control law.

Quadrature Solution for Fractional Benjamin–Bona–Mahony–Burger Equations

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics.

Sound Insulation Characteristic Analysis of Corrugated Core Sandwich Panel by Using Differential Quadrature Finite Element Method

This paper proposes a universal vibroacoustic coupling theoretical analysis model for investigating the sound insulation characteristic of corrugated core sandwich panel (CCSP) under different boundary conditions. The vibroacoustic coupling model consists of CCSP (structural field) and two three-dimensional acoustic cavities which simulate the incident sound field and accept the sound field, respectively. Based on first-order shear deformation plate and shell theories and acoustic theory, the dynamic model of CCSP and three-dimensional acoustic cavities are established by employing the differential quadrature finite element method in conjunction with the energy principle. The coupling of CCSP and three-dimensional acoustic cavities is realized by the velocity continuity condition at the acoustic–structure interfaces. The validation of the vibroacoustic coupling model is verified by comparing the results calculated by the established model with the corresponding results calculated by ABAQUS. The influence of model parameters on the vibroacoustic characteristic of vibroacoustic coupling model is investigated systematically for determining the critical structure parameters which have influence on the vibroacoustic behaviors. The effect of critical structure parameters on the sound insulation characteristic of CCSP is analyzed systematically for the optimization of the sound insulation performance of CCSP and provides theoretical basis and technique guidance.

AN INVESTIGATION ON NONLINEAR OPTION PRICING BEHAVIOURS THROUGH A NEW FRÉCHET DERIVATIVE-BASED QUADRATURE APPROACH

Abstract Complicated option pricing models attract much attention in financial industries, as they produce relatively better accurate values by taking into account more realistic assumptions such as market liquidity, uncertain volatility and so forth. We propose a new hybrid method to accurately explore the behaviour of the nonlinear pricing model in illiquid markets, which is important in financial risk management. Our method is based on the Newton iteration technique and the Fréchet derivative to linearize the model. The linearized equation is then discretized by a differential quadrature method in space and a quadratic trapezoid rule in time. It is observed through computations that the accurate solutions for the model emerge using very few grid points and time elements, compared with the finite difference method in the literature. Furthermore, this method also helps to avoid consideration of the convergence issues of the Newton approach applied to the nonlinear algebraic system containing many unknowns at each time step if an implicit method is used in time discretization. It is important to note that the Fréchet derivative supports to enhance the convergence order of the proposed iterative scheme.

Free vibrational mode shapes and frequencies of sandwich folded plates standing on folded foundation and structured by auxetic honeycomb core and FG-GPLRC coating layers

In this article, for the first time, the free vibration and modal shapes of sandwich folded plate with honeycomb core are studied. The current article investigates the free vibrational mode shapes and frequencies of sandwich folded plate made of auxetic honeycomb core and graphene platelet-reinforced functionally graded composite (FG-GPLRC) coating layers which standing on the normal-shear folded supports. The structures are supported by normal-shear folded supports. The negative Poisson’s ratio exhibited by honeycomb materials results in distinct physical and mechanical properties, profoundly impacting the behavior of structures constructed from this material. While structures with basic shapes like rectangular, circular, and spherical configurations find limited applications across various industries, the versatility of folded structures makes them highly suitable for a wide range of industrial applications, given their more complex geometries. Dynamic equations are formulated within the framework of the first-order shear deformation theory (FSDT) for each flat section of the folded plate. The continuous equality conditions for displacements, rotations, and stresses at the junction edges of the two flat sections are rigorously satisfied. To effectively solve these equations, a robust numerical technique known as the generalized differential quadrature method (GDQM) is employed specifically tailored for the 2D analysis of folded plates. Furthermore, the study investigates the influence of various parameters, including material properties and geometry, on the frequencies and mode shapes of the folded structures, shedding light on the intricate interplay between these factors in shaping the dynamic response of the structures.

Nonlinear Analysis of the Multi-Layered Nanoplates

This text investigates the bending/buckling behavior of multi-layer asymmetric/symmetric annular and circular graphene plates through the application of the nonlocal strain gradient model. Additionally, the static analysis of multi-sector nanoplates is addressed. By considering the van der Waals interactions among the layers, the higher-order shear deformation theory (HSDT), and the nonlocal strain gradient theory, the equilibrium equations are formulated in terms of generalized displacements and rotations. The mathematical nonlinear equations are solved utilizing either the semi-analytical polynomial method (SAPM) and the differential quadrature method (DQM). Also, the available references are used to validate the results. Investigations are conducted to examine the effect of small-scale factors, the van der Waals interaction value among the layers, boundary conditions, and geometric factors.

Size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams with elastic boundary conditions and thermal effects based on two‐phase local/nonlocal elastic model

AbstractWe present predictive models of the size‐dependent buckling loads of non‐uniform Bernoulli–Euler beams under thermal effects based on the two‐phase local/nonlocal elastic model. The beam ends are assumed to be constrained by elastic springs with translational and rotational stiffness to simulate general boundary conditions. In contrast to most literature in this field, both the bending and thermal deformations of the beams are simultaneously considered to be two‐phase local/nonlocal of two phases, that is, the thermal effect is taken as equivalent to a size‐dependent thermal load. By using the fully equivalent differential form of the local/nonlocal equation with a set of constitutive boundary conditions, the problem is solved numerically with the aid of the generalized differential quadrature method (GDQM). Through conducting validation study, several parametric studies are given for examining the effects of the slope of beams’ thickness variation, nonlocal parameter, and elastically supported conditions on the buckling loads of non‐uniform beams. The results show that constrained stiffness has a drastic influence on the critical bucking loads of the beams. Furthermore, the consideration of the two‐phase thermal load will further reduce the actual buckling load of the beams.

A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations.

Analysis of Vibration Characteristics for Rotating Braided Fiber-Reinforced Composite Annular Plates with Perforations.

In the current study, a comprehensive numerical model for analyzing the vibrational characteristics of braided fiber-reinforced composite (BFRC) rotating annular plate with perforations under diverse boundary constraints was introduced. This model employs the differential quadrature finite element method (DQFEM), which was developed based on the first-order shear deformation theory (FSDT) and coordinate transformation approach. The BFRC material, specifically a two-dimensional biaxial orthogonal fabric, was utilized to fabricate the annular plate with two distinct types of holes: circular and sector-shaped. The model's convergence, accuracy, numerical stability, and reliability were confirmed through comparative assessments utilizing data from the literature, from ABAQUS software, and from experimental findings. The analysis focuses on studying the influences of structural properties, material parameters, and boundary restraints on the frequencies of vibration for BFRC rotating annular plates with holes. This theoretical model helps provide scientific basis and technical guidance for the stability and lightweight design of rotating annular plates, such as rotor structures in aircraft engines.

A high-order generalised differential quadrature element method for simulating 2D and 3D incompressible flows on unstructured meshes

In this paper, a high-order generalised differential quadrature element method (GDQE) is proposed to simulate two-dimensional (2D) and three-dimensional (3D) incompressible flows on unstructured meshes. In this method, the computational domain is decomposed into unstructured elements. In each element, the high-order generalised differential quadrature (GDQ) discretisation is applied. Specifically, the GDQ method is utilised to approximate the partial derivatives of flow variables and fluxes with high-order accuracy inside each element. At the shared interfaces between different GDQ elements, the common flux is computed to account for the information exchange, which is achieved by the lattice Boltzmann flux solver (LBFS) in the present work. Since the solution in each GDQ element solely relies on information from itself and its direct neighbouring element, the developed method is authentically compact, and it is naturally suitable for parallel computing. Furthermore, by selecting the order of elemental GDQ discretisation, arbitrary accuracy orders can be achieved with ease. Representative incompressible flow problems, including 2D laminar flows as well as 3D turbulent simulations, are considered to evaluate the accuracy, efficiency, and robustness of the present method. Successful numerical simulations, especially for scale-resolving 3D turbulent flow problems, confirm that the present method is efficient and high-order accurate.

Nonlinear generalized piezothermoelasticity of spherical vessels made of functionally graded piezoelectric materials

The present study investigates the thermoelastic response of a heterogeneous piezoelectric sphere under thermal shock loading. Boundary conditions as well as loading are considered as symmetric; thus, the response of the sphere is expected to be symmetric. All of the properties of the thick-walled sphere, including mechanical, electrical and thermal properties, are considered dependent on the radial position, except for the relaxation time, which is considered a constant value along the radius. The governing equations of the sphere have been derived under heterogeneous anisotropic assumptions. The general form of the second law of thermodynamics, which is nonlinear in nature, and is called nonlinear energy equation is used. The number of the established equations is three, which includes the motion equation, energy equation and Maxwell electrostatic equation of Maxwell. These equations are obtained in terms of radial displacement, temperature difference and electric potential. The energy equation is derived based on Lord and Shulman theory with a single relaxation time. In the next step, by introducing dimensionless variables, the governing equations are provided in dimensionless presentation. Then these equations have been discretized using generalized differential quadrature method. Also, in order to follow the solution of the equations in time domain, Newmark method has been used. Since the system of equations is nonlinear, Picard algorithm is applied as a predictor-corrector mechanism to solve the nonlinear system of equations. Then numerical results are presented to investigate the propagation of mechanical, thermal and electric waves inside the heterogeneous sphere and also their reflection from the outer surface of the sphere. By examining the results, it can be seen that mechanical and thermal waves propagate with a limited speed, while the speed of electric wave propagation is infinite.

Free vibration and supersonic flutter analyses of a sandwich cylindrical shell with CNT-reinforced honeycomb core integrated with piezoelectric layers

In this research, the free vibration and supersonic flutter analyses of a cylindrical sandwich shell with a reinforced honeycomb core (RHC) integrated with piezoelectric face sheets are investigated. The core is a honeycomb structure enriched with fibers and carbon nanotubes (CNTs). The layers of the shell are modeled based on the Cooper-Naghdi shell theory and the continuity conditions between the layers are considered. The aerodynamic pressure of the supersonic flow is modeled using the linear piston theory. Hamilton’s principle is used to derive the governing equations. A semi-analytical solution is presented including an exact solution in the circumferential direction followed by a numerical solution in the longitudinal direction via the differential quadrature method (DQM). The effects of several parameters on the frequencies, the corresponding damping coefficients, and the critical aerodynamic pressure are investigated. It is concluded that improving the stiffness of the honeycomb core with the CNTs does not necessarily improve the aeroelastic stability of the shell. The presented results can be used in the design and analysis of aerospace structures.

Three-dimensional elasticity solutions for doubly-curved composite shells by extended differential quadrature method

This paper presents the three-dimensional (3D) stress analysis of doubly-curved composite shells with general boundary conditions using the strong sampling surfaces (SaS) formulation. The SaS method is based on the choice of SaS parallel to the middle surface and located at Chebyshev polynomial nodes to introduce the displacements of these surfaces as unknown functions that leads to a non-conventional shell formulation, in which strain–displacement and stress–strain relationships are represented in terms of SaS displacements. This is due to the use of Lagrange polynomials in approximating displacements, strains and stresses in the thickness direction. The outer surfaces are not included into a set of SaS that makes it possible to uniformly minimize the error used by Lagrange interpolation. The strong SaS formulation, based on direct integration of elasticity equilibrium equations in the thickness direction by the extended differential quadrature (EDQ) method, can be applied efficiently for high-precision calculations of doubly-curved composite shells with clamped and free edges. This is because in the SaS/EDQ formulation, displacements, strains and stresses of SaS are interpolated in a rectangular domain specified in a curvilinear coordinate system using a Chebyshev-Gauss-Lobatto grid and Lagrange polynomials are also used as basis functions. The proposed approach deals with equilibrium equations in terms of SaS stresses, avoiding the integration of second order differential equations in terms of SaS displacements, that greatly simplifies the implementation of the EDQ method.

Differential quadrature lattice Boltzmann method for simulating fluid flows

In this study, a novel method for simulating incompressible flows using the lattice Boltzmann method is presented, aiming to improve robustness. This method is enhanced by the Qadyan numerical method. In the Qadyan method, Partial differential equations are solved using semi-discrete schemes combined with the differential quadrature method. To discretize the spatial derivatives of the lattice Boltzmann equation, the upwind differential quadrature method is employed, while the first-order forward difference scheme and/or fourth-order Runge-Kutta method handle the temporal term. The decision to exclude more complex methods stems from their requirement for refined computational grids. This work focuses on achievable results with similar types of uniform grids. The proposed scheme introduces and evaluates a novel method for solving both steady and unsteady problems. The present numerical method is validated by solving five benchmark problems, including heat diffusion on a slab, Stokes’ first problem, Taylor-Green vortex flow, flow around a flat plate, and flow in a lid-driven cavity. Reasonable agreements are obtained between the solutions obtained using this method and those from analytical/other numerical approaches. The Qadyan formulation delivers more accurate results compared to the finite-difference lattice Boltzmann method, (FDLBM), without requiring an increase in the number of grids. Notably, the novel Qadyan method achieves this increased accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids. This is possible due to the nature of the differential quadrature method, which utilizes more combinations of candidate stencils. The Qadyan method has a higher computational cost in comparison with the FDLBM.

RFOR-DQHFEM: Hybrid relaxed first-Order reliability and differential quadrature hierarchical finite element method for multi-physics reliability analysis of conical shells

In this current work, a hybrid reliability analysis and theoretical frequency technique are suggested for the reliability response of conical shells. Two levels of analyses are proposed as the main loop of the reliability method for finding the failure probability and the second level applied in the main loop for giving the performance function of frequency applied in conical shell structures with multi-physics vibration analysis. A dynamical adjusting procedure is proposed for computing the relaxed factor using the enough descent condition inside the reliability method. The superior convergence rate is considered for selecting the relaxed factor of the proposed first-order reliability method named RFORM. An elastic-electro-mechanical model based on the Higher-Order Shear Deformation Theory (HSDT) is extended for frequency analysis of conical shells. The innovative numerical procedure named Differential Quadrature Hierarchical Finite Element Method (DQHFEM) as a robust framework for giving the vibration behavior of studied mechanical structures is applied for solving motion equations. The developing DQHFEM and RFORM are applied for the laminated, nanocomposite, and piezoelectric conical shell structures with multi-source uncertainties. Increasing the volume percentage of nanoparticles from 0% to 10% significantly enhances the reliability index, with carbon nanoparticles showing a 132% increase, silica nanoparticles showing a 97% increase, and other nanoparticles showing an approximate 40% increase. Also, as moisture content increases from 0% to 30%, the reliability index for a thickness-to-large-radius ratio of 0.2 drops by about five times. Excessive moisture levels (above 20%) result in a negative reliability index, indicating a hazardous condition.

A local differential quadrature method for the generalized nonlinear Schrödinger (GNLS) equation

A local differential quadrature method based on Fourier series expansion numerically solves the generalized nonlinear Schrodinger equation. For time integration, a Runge-Kutta fourth-order method is used. Matrix stability analysis is used to examine the method's stability. Three test problems involving the motion of a single solitary wave, the interaction of two solitary waves, and a solution that blows up in finite time, respectively, demonstrate the accuracy and efficiency of the provided method. Finally, the numerical results obtained from the presented method are compared with the exact solution and those obtained in earlier works available in the literature.