Differential Quadrature Method Research Articles

Particle Swarm Optimization for Solving Nonlinear Fisher’s Equation

The term "optimization" describes a way of increasing the favourable features of a function in a mathematical form while reducing the negative ones. Most of the problems encountered in the real world may be formulated as optimization problems. Particle swarm optimization is a tool which has been used extensively to handle problems across different fields. The purpose of this study is to develop a numerical method for solving nonlinear reaction-diffusion Fisher’s equations, whose numerical solutions exhibit the solitons form and have a wide range of applications. This study includes differential quadrature method using exponential basis function and particle swarm optimization. In the work so far documented in the literature, a random number was used to approximate the involved parameter in the basis functions. This value is now determined using the implemented optimization technique. The numerical solutions obtained for the equation is comparable to the data available in the literature. The research presented through figures and tables, yields favourable results.

Multi-Channel Signals in Dynamic Force-Clamp Mode of Microcantilever Sensors for Detecting Cellular Peripheral Brush.

The development of numerous diseases, such as renal cyst, cancer, and viral infection, is closely associated with the pathological changes and defects in the cellular peripheral brush. Therefore, it is necessary to develop a potential new method to detect lesions of cellular peripheral brush. Here, a piecewise linear viscoelastic constitutive model of cell is established considering the joint contribution of the peripheral brush and intra-cellular structure. By combining the Laplace transformation and its inverse transformation, and the differential method in the temporal domain and differential quadrature method (DQM) in the spatial domain, the signal interpretation models for quasi-static and dynamic signals of microcantilever are solved. The influence mechanisms of the peripheral brush on the viscoelastic properties of cells and quasi-static/dynamic signals of microcantilever are clarified. The results not only reveal that the peripheral brush has significant effects on the complex modulus of the cell and multi-channel signals of the microcantilever, but also suggest that an alternative mapping method by collecting multi-channel signals including quasi-static and higher frequency signals with more brush indexes could be potentially used to identify cancerous cells.

Multiscale dynamic behavior of imperfect hybrid matrix/fiber nanocomposite nested conical shells with elastic interlayer

This investigation delves into the free vibration characteristics of coupled nested conical shells (CNCSs) made of porous composite materials. These two conical shells are connected by a mid-layer of elastic springs. The composite materials used in the shells consist of epoxy, nanofillers, and fibers. Two types of nanofillers are considered: Graphene Nanoplatelets (GNPs) and Carbon Nanotubes (CNTs), while E-glass fiber is used as the fiber. The nanofillers are distributed in four different patterns within the shell section. Porosity is uniformly distributed along the shell section and characterized by a coefficient. The rule of mixtures is employed to ascertain the equivalent material properties of the hybrid materials, while the Chamis approach is utilized for three-phase materials. First-order shear deformation theory (FSDT) and Donnell's theory are utilized for modeling the conical shells. The governing equations of motion are established through Hamilton's principle are solved using the generalized differential quadrature method (GDQM). Seven different boundary conditions (BCs) are considered to encompass the full range of BCs for CNCSs and four type of BCs for single truncated conical shell (STCS). The solution's accuracy is verified, and the effects of various parameters on the natural frequency parameter (NFP) of the shell are investigated, such as BCs, circumferential wave number (n), nanofillers pattern, semi-vertex angle, nanofillers angle, and mid-layer stiffness. Initially, a comprehensive investigation into the vibration behavior of a STCS is presented, followed by an analysis of the NFP of the CNCSs. The results demonstrate that the stiffness of the elastic mid-layer significantly influences the NFP of the system. The orientation of the nanofillers in the shell can increase or decrease the NFP. Additionally, the relationship between mode number and n depends on the type of BCs of the shells.

Torsion Analysis of Functionally Graded Iso/Orthotropic Sections Using Differential Quadrature and pb2-Rayleigh Ritz Methods

Torsion Analysis of Functionally Graded Iso/Orthotropic Sections Using Differential Quadrature and pb2-Rayleigh Ritz Methods

A novel unified variational finite difference (VFD) solution method for optimal control problems

This article is aimed to propose a simple yet efficient unified numerical strategy for solving both linear and non-linear optimal control problems. To do so, the general form of quadratic performance index function and nonlinear state equations are considered first. Then, the idea of variational differential quadrature method is used to convert the integral/differential equations to the equivalent algebraic form. Since time is the only independent variable is this research, the finite difference method with an equally-spaced discretization scheme would be a more appropriate technique rather than the differential quadrature approach. So, the implemented numerical solution is called now as the variational finite difference method. The method of Lagrange multipliers is then utilized for minimization purpose and, as a result, the final set of nonlinear algebraic equations are obtained. Finally, the quadratic and triadic forms of non-linearity are considered and an explicit formulation is represented for the residual and Jacobian of the Newton-based iterative solution procedure. To demonstrate the accuracy and efficiency of the proposed approach in the quadratic optimal control area, several benchmark problems involving linear time-invariant, linear time-variant and nonlinear examples are successfully solved and the results are confirmed with those existed in the literature.

Two-dimensional analysis of electrical behavior in composite semiconductor shell structures with magnetic field tuning

To precisely investigate the electric properties and reveal the intrinsic interaction mechanisms between multiple fields in the composite magneto-electric-semiconductor shell structure, two-dimensional analyses were conducted within the framework of the third-order shear deformation theory and the coupled field theory. The governing equations for the composite piezoelectric semiconductor (PS) shell structures were derived using the principle of virtual work. Field distributions of the composite PS shell structures were obtained by expanding basic field quantities into Fourier series along the width direction and using the differential quadrature method. Prior to analysis, the convergence and correctness of the adopted method were evaluated. Several numerical examples were presented to demonstrate how magnetic manipulation can tune electrical behavior. It was found that inhomogeneous shear strain induced by applied magnetic fields is the key factor affecting potential variation in the composite PS shells. The appearance of potential barriers and wells in the composite PS shell structure with an opposite c axis is explained thoroughly. Additionally, the effect of magnetic field non-uniformity on the composite PS shell structure was investigated using an equivalent body force model, offering valuable insights for designing thin-film devices.

Semi-analytical layerwise solutions for FG MEE complex shell structures

This paper presents a unique analytical layerwise solution for functionally graded magneto-electric-elastic shell with complex geometry. The middle surface of the shell is modeled by a parametric equation and the Lamé parameter and radii of curvature is modeled by Differential Geometry. The mechanical displacements, electrical and magnetic potentials are written in term of a simple cosine layerwise based on a unified formulation. The highly coupled differential equations are discretized by the Chebyshev-Gauss-Lobatto grid points and solved numerically via the Differential Quadrature Method (DQM). Lagrange interpolation polynomials are employed as the basis functions. The highly coupled differential equations are solved for shells subjected to different loads and boundary conditions. Since extremely few results on this topic is available in the literature, benchmarks complex shell problems and their solutions are introduced in this paper for the very first time.

On the measurement of the nonlinear dynamics of sandwich sector plate surrounded by the auxetic concrete foundation: Introducing a machine learning algorithm for nonlinear problems

The measurement of the nonlinear dynamics of sandwich rotary sector plates is crucial for measurement engineers as it enables the precise analysis and understanding of the behavior of advanced composite structures under dynamic conditions. These measurements help in identifying and mitigating potential issues related to vibration, stability, and fatigue, which are critical in industries like aerospace, automotive, and civil engineering. Accurate data on nonlinear dynamics aids in the optimization of design, enhancing performance, durability, and safety. Furthermore, it supports the development of predictive maintenance strategies, reducing downtime and operational costs, and contributing to the advancement of engineering materials and techniques. So, in the current work, for the first time, nonlinear dynamics of sandwich rotary sector plate via a mathematical modeling and machine learning algorithm is presented. First, due to a lack of information on the nonlinear dynamics of sandwich rotary sector plates, a dataset for training, testing, and validating the machine learning algorithm is presented. For this purpose, using Hamilton’s principle, Von-Karman nonlinearity, and first-order shear deformation theory (FSDT), the nonlinear governing equations are obtained. Also, due to increasing stiffness and stability, an auxetic concrete foundation covers the sandwich structure. After that using the two-dimensional generalized differential quadrature method (GDQM) and Newmark’s time integration method (NTIM), the nonlinear equations are solved. This research provides valuable insights for engineers in designing advanced composite structures with improved dynamic properties. The results also contribute to the broader understanding of nonlinear dynamic interactions in complex material systems, paving the way for innovative applications in various engineering fields.

An experimental and numerical effort on the vibration behavior of additively manufactured recycled polyethylene terephthalate glycol components

AbstractThe importance of recycling engineering components and thus obtaining low‐cost production solutions has become prominent in today's world. In this study, the mechanical and dynamic behaviors of three‐dimensional‐printed recycled polyethylene terephthalate glycol (RePET‐G) beams were investigated numerically and experimentally for the first time in the literature. Initially, the governing equations of the beams were determined according to the Bernoulli–Euler beam theory, and these equations were numerically solved using the differential quadrature method and ANSYS program. Subsequently, to validate the accuracy of the numerical models, the obtained natural frequencies were compared with experimental results. It was observed that the numerical results showed good agreement with the experimental results. Finally, the effects of beam length, infill rate, and building direction on the natural frequencies of RePET‐G beams were investigated. The outcomes showed that as the beam length changed, natural frequencies were significantly affected. Increasing the infill rate, especially for beams with vertical building direction, from 20% to 100% led to a slight decrease in the natural frequency values of the structure. Moreover, it was found that for beams with an infill rate of 100%, the natural frequency values obtained in the horizontal building direction were higher than those obtained in the vertical building direction.Highlights Printable recycled filaments have great potential for vibration applications. Sample length affects the first natural frequency value of RePETG parts. Differential quadrature and ANSYS methods can be utilized for the vibration. For the 3D‐printed samples, rising infill rate causes a natural frequency drop.

Nonlocal buckling analysis of five-layer laminated nanocomposites on kerr foundation: A refined zigzag theory approach

In this paper, the buckling analysis of a five-layer laminated nanocomposite resting on a Kerr foundation is presented. In order to describe the non-continuous behavior of composite plate through its thickness, the displacement field is determined using the Refined Zigzag Theory (RZT). Additionally, the constitutive relations of piezo electromagnetic isotropic materials, orthotropic composites, and Functionally Graded Porous Materials (FGPMs) are presented. With respect to Bi- and Uniaxial loading in the nanoplate, the Hamilton’s principle is utilized to derive the equation of motion of this nanoplate. To study the small-scale effect in nanoplates, both Nonlocal Eringen Theory and Nonlocal Strain Gradient Theory (NSGT) are employed to account for nonlocal effects. Finally, the coupled equations of motion are solved using the Differential Quadrature Method (DQM). This paper introduces the newly used Kerr foundation and its effect on the buckling analysis. It also investigates the influence of plate dimensions, piezo electromagnetic terms, boundary conditions, and loading on the dimensionless critical buckling load.

Thermal buckling analysis of reinforced composite conical shells in acidic environments: Numerical and experimental investigation on the effects of nanoparticles

This study investigates the effects of acid penetration and temperature on the buckling behavior of conical composite shells, to enhance structural integrity and longevity in corrosive environments. The study explores the impact of acid exposure on thermal properties and examines the efficacy of incorporating nano-silica and nano-clay in preventing buckling. Additionally, it analyzes the influence of nanoparticles on the thermal, moisture, and mechanical properties of the composite material. Experimental assessments are conducted to measure material properties during exposure to a sulfuric acid solution, providing a comprehensive understanding of the material's behavior under extreme conditions. However, due to the complexity of investigating the combined effects of temperature, acid, and nanoparticles on composite shell buckling, a combined numerical and experimental approach is adopted to predict the critical buckling load. To this end, equations of conical shells under hygrothermal loading are derived, and the critical buckling load is determined through pre-buckling analysis. The Generalized Differential Quadrature (GDQ) method is employed to solve the hygrothermal buckling of the composite shell using experimentally obtained material properties. Comparative results are presented for different nanoparticles, shell geometries, and exposure times in acidic environments. The experiments reveal that adding nanoparticles enhances mechanical properties and reduces thermal and moisture expansion coefficients. Conversely, the acidic conditions deteriorate these properties. Numerical analysis demonstrates that incorporating nanoparticles significantly increases the critical buckling temperature, with nano-silica and nano-clay particles resulting in an 11.5 % and 34.2 % increase, respectively. However, acidic environments decrease the critical buckling temperature, with reductions of 32 % for unreinforced, 29 % for nano-silica reinforced, and 46 % for nano-clay reinforced composites after three months of exposure.

Thermal dynamics of nanoparticle aggregation in MHD dissipative nanofluid flow within a wavy channel: Entropy generation minimization

This paper examines how nanoparticle aggregation and a consistent magnetic field influence the peristaltic movement of a dissipative nanofluid, which is caused by the sinusoidal deformation of the boundary. The viscosity of TiO2/H2O nanofluids is accurately determined by the Krieger-Dougherty model with nanoparticle aggregation, while thermal conductivity (TC) is estimated through the Bruggeman model. The set of governing equations are modeled in a fixed frame by utilizing the conservation laws of energy, mass and momentum. Galilean transformation is utilized to transform the system of equations into a wave frame, which is then converted into a dimensionless form. The assumption of a small Reynolds number and long wavelength serve to further simplify the set of equations, which are subsequently addressed through the implementation of the differential quadrature method (DQM), a highly effective numerical technique. Quantities of interest, namely velocity, pressure gradient, temperature, trapping phenomena, heat transfer, and volumetric entropy generation are analyzed across a range of physical parameters, including the solid volume fraction (Φ=0.01−0.04), Eckert number (Ec=0.0−0.1), Hartman number (Mh=0.2−2.2), Grashof number (Gr=1.0−3.0) and temperature ratio parameter (θd=0.5−2.5). A comparative analysis is conducted between the scenario involving aggregation and the one without aggregation. It is observed that nanoparticle aggregation significantly alters these quantities.

Nonlocal dynamic response of FGP sandwich microbeam with 2D PSH network incorporating adatoms-surface interactions energy under magnetic field

The present work models and studies the resonance frequency change due to adsorption phenomena in a simply-supported adatom-microbeam system under an axial magnetic environment. The microbeam structure is a functionally graded (FG) sandwich that includes a rectangular configuration and a ceramic perforated core with periodic square holes network and FG porous material faces. To evaluate the adsorption-induced energies produced by van der Waals interaction (vdW), the Morse and Lennard-Jones (6–12) potentials are included in conjunction with nonlocal material elasticity to adopt the small-scale impact. Moreover, the explicit formulas for the inertia moments and shear force are developed by the Timoshenko beam equations, considering the residual stress influence as an additional axial load. The Navier-type solution and the differential quadratic method (DQM) are implemented to compute the solution of the governing equations. It is established that the resonance frequency change for the O/Si (100) and H/Si (100) is dependent on the geometrical, perforation, and porosity parameters, adsorption density, mode number, and the magnetic field intensity as well. Therefore, these numerical results have been discussed in detail to properly examine dynamic vibration behavior and a suitable mass detection design of M/NEMS structures.

Symmetric and asymmetric vibration and buckling of a thermal rotating annular-plate with Pasternak foundation

This paper studies the symmetric and asymmetric vibration and buckling modes of a rotating thermal annular plate resting on a Pasternak foundation. With considering the von Kármán geometric nonlinearity and the first-order shear deformation theory, the governing equations of the system are established via Hamilton's principle, and then solved by using the differential quadrature method. With both inner and outer boundaries constrained, the modes of the rotating system vary differently depending on the mode symmetry. The symmetric modes are mainly affected by the generated centrifugal force, while the asymmetric ones are affected by both centrifugal and Coriolis forces. The additional Coriolis, with mainly two directions, strengthens the forward travelling parts of the asymmetric modes, but makes the backward counterparts buckled more easily. Also, the thermal effect, as well as boundary elasticity and foundation coefficients, are also investigated from the viewpoint of mode symmetry. Results show that the temperature rise can shift the fundamental vibration mode from the symmetric one to the asymmetric one and backward travelling waves for rotation, which also adds to the complexity of mode variation with joint parameters. For the joint effect of the temperature rise and rotating, the annular plate tends to buckle more readily while the effects of the two factors are linearly superimposed.

Computational Methods for Stationary and Nonstationary Fokker–Planck–Kolmogorov Equations with Random Coefficients

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated.

Tunable Bending and Buckling Behaviors of Circular Plates Made of FG Graphene Origami-Enabled Auxetic Metamaterials

Based on the variational differential quadrature (VDQ) method, the bending and buckling characteristics of circular plates made of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs) are numerically studied in this paper. It is considered that the plate is composed of multiple GOEAM layers with graphene origami (GOri) content that changes in layer-wise patterns. The results from genetic programming-assisted micromechanical models are also employed in order to estimate the material properties. The plate is modeled according to the first-order shear deformation plate theory whose governing equations are obtained using an energy approach in the context of VDQ technique. The governing equations are given in a new vector-matrix form which can be easily utilized in coding process of numerical methods. By means of VDQ matrix differential and integral operators, the governing equations are discretized and solved to calculate the lateral deflection and critical buckling load of plates under various boundary conditions. Selected numerical results are presented to investigate the influences of boundary conditions, GOri content, folding degree and distribution pattern on the buckling and bending behaviors of FG-GOEAM plates.

Numerical fatigue damage analysis and mathematical modeling of articular cartilage under cyclic load via hyperelasticity theory

The paper presents the modeling of damage to articular cartilage under cyclic daily loads using a curved beam model and hyper-elasticity theory. Mainly, articular cartilage (AC) is a kind of very important biological tissue that has the fundamental role of withstanding applied mechanical loads and providing smooth movement of joints. The existence of mechanical loads, however, has a huge influence on the behavior and the entire healthiness of AC. These loads, over time, can cause injury through fatigue-type damage because of frequent stresses. The basic aim of this study is to mainly offer a detailed mathematical model measuring the damage caused in AC under the action of mechanical forces incorporating different variants like age, body mass index, metabolic activity, functionally graded, porosity and prestresses. The structural energies including potential energy according to the neoHookean model as well as kinetic energy and external work are achieved through the use of strain-displacement and stress-strain relations. Then, the nonlinear governing equations of the articular cartilage are derived using Hamilton's principle. Furthermore, the mathematical model has been implemented numerically through the differential quadrature method (DQM). A qualitative correspondence between the numerical predictions and experimental data has led us to conclude that this model has the potential to serve as a valuable tool for physicians and therapists. The results of this research indicate that among the factors affecting the increase of damage in cartilage, the most important factor is the body mass index, followed by a person's age, hormonal conditions, and cartilage thickness with a negative effect. The probability of damage for an athlete is about 33 percent higher than a normal person, and for a weightlifter (heavy sports) it is about 140 percent higher than a normal person.

Interplay of photonic, electrical, and inertial loads on the stability of rotating sector perovskite sandwich plates with a GPL-based nanocomposite core

The bifurcation stability of sandwich sector plates, primarily constructed from lead halide perovskite skins known for their significant photostrictive and electrostrictive properties, is explored. These properties render them highly relevant for multiphysics applications. The influence of a photo-induced thermal environment on the behavior of these plates is also examined. A notable challenge, the inherent stiffness of these structures, is addressed by integrating a nanocomposite laminated core composed of a polymer matrix and graphene platelet (GPL) reinforcers. The GPLs are distributed throughout the core layers according to functionally graded models, significantly enhancing structural integrity. To effectively model the core environment, the Halpin-Tsai micromechanical rule is employed. The structural displacement field is modeled using the first-order shear deformation theory. Moreover, the von-Kármán geometrically nonlinear strain-displacement relations are applied. The constitutive relationships are governed by the theory of linear photo-thermo-electro-elasticity, providing a framework for the analysis of perovskite-based structures. The reorganization of bifurcation points from the pre-buckling route and the linearization of stability equations are performed using the adjacent-equilibrium criterion. The generalized differential quadrature (GDQ) method is utilized to solve the equilibrium equations of pre-buckling and the stability equations of buckling. This comprehensive investigation reveals the critical influence of photonic, electrical, and rotational stimuli on the stability characteristics of advanced perovskite-based sandwich sector plates, demonstrating potential advancements in multiphysics applications.

Numerical investigation of unsteady MHD immiscible Casson micropolar and Jeffery fluid in a horizontal channel with heat transfer using MCB-DQM approach

The dynamics of immiscible Casson micropolar and Jeffery fluid flow offer new possibilities for enhancing momentum and heat transfer efficiencies. Motivated by numerous applications of immiscible fluid flows in natural and industrial systems, the current work presents a numerical investigation of unsteady, incompressible, magnetohydrodynamic (MHD) flow involving two immiscible fluids, namely, Casson micropolar and Jeffery fluid, in a horizontal channel. The flow, driven by a uniform pressure gradient, assumes no-slip conditions at the channel walls and a continuous, nondeformable interface between the immiscible fluids. The well-established rapid converging modified cubic B-spline differential quadrature method (MCB-DQM) is applied to solve the partial differential equations governing the flow. The impact of crucial fluid parameters comprising the Casson micropolar fluid parameter, Jeffery fluid parameter, thermal, magnetic, and various other fluid parameters on temperature, microrotation, and velocity profile is illustrated graphically. Notably, the parameters of each fluid influence the profiles in both fluid zones. The flow dynamics are significantly affected by viscous dissipation, Lorentz force, fluid–wall interactions, and interfacial conditions. Key parameters such as time, Jeffery fluid, Casson micropolar, ion slip and Hall parameters, Reynolds number, Eckert number, the ratio of viscosities, the ratio of densities, and pressure gradient show significant impacts on the velocity, microrotation, and temperature profiles. However, reverse or mixed behavior is observed for other parameters. The novelty of current work lies in modeling the unsteady, immiscible, MHD flow of Casson micropolar and Jeffery fluid and proposing a numerical solution by deploying the MCB-DQM method. The findings have potential applications in various industrial and engineering processes, such as the design of oil extraction, optimizing processes in the food processing and cosmetics industry, and developing effective strategies for environment conservation.

Numerical simulation of Volterra PIDE with singular kernel via modified cubic exponential and uniform algebraic trigonometric tension B-spline DQM

In this paper, two different numerical techniques are employed to solve the Volterra partial integro-differential equation (PIDE) with a weakly singular kernel: Uniform Algebraic Trigonometric tension (UAT) B-spline and Exponential B-spline. These techniques are further modified under certain conditions. The presented techniques transform the discretized Volterra PIDE into a linear algebraic system of equations. In this process, the forward difference formula is utilized to address the time derivative, while the differential quadrature method (DQM) is used for the spatial order derivative. The fusion of modified cubic exponential and modified cubic UAT tension B-spline with DQM is considered. The effectiveness of the methods is assessed via various types of errors considered through three distinct examples. Additionally, the validity of these results is shown by comparison with previous findings in the same environment. The comparison demonstrates that the modified cubic UAT tension B-spline produces less inaccuracy than the other. This work provides robust results that advance research toward developing more advanced and computationally efficient numerical techniques.