Generalized Differential Quadrature Research Articles

Natural Frequency Response of FG-CNT Coupled Curved Beams in Thermal Conditions

This study investigates the sensitivity of dynamic properties in coupled curved beams reinforced with carbon nanotubes (CNTs) to thermal variations. Temperature-dependent (TD) mechanical properties are considered for poly methyl methacrylate (PMMA) to be strengthened with single-walled CNTs (SWCNTs), employing the basic rule of mixture to define the equivalent mechanical properties of nanocomposites. The governing equations of motion are derived using a first-order shear deformation theory (FSDT) and Hamilton’s principle, accounting for elastic interfaces modeled using elastic springs. A meshfree solution method based on a generalized differential quadrature (GDQ) approach is employed to discretize the eigenvalue problem and to obtain the frequency response of the structure. The proposed numerical procedure’s accuracy is verified against predictions in the literature for homogeneous structural cases under a fixed environmental temperature. The systematic investigation assesses the impact of various geometric and material properties, including curvature, boundary conditions, interfacial stiffness, and CNT distribution patterns, on the vibrational behavior.

Efficient analysis on buckling of FG-CNT reinforced composite joined conical–cylindrical laminated shells based on GDQ method under multiple loading conditions

This work presents an efficient analysis of buckling for functionally graded carbon nanotube reinforced composite (FG-CNTRC) joined conical–cylindrical laminated shells with lateral pressure and axial compressive load considering thermal effect. The governing equilibrium equation of the joined conical–cylindrical shell (CO-CYL) is determined using the first-order shear deformation theory (FSDT), Donnell kinematic assumptions, von-Kármán geometrical nonlinearity, and considering thermal effect. A semi-analytical solution, generalized differential quadrature (GDQ) method, and artificial springs are applied during the solution procedure. The comparison studies have demonstrated a remarkable 97% reduction in running time compared to the finite element method (FEM).

Artificial neural networks coupled with numerical approach for the stability prediction of non-uniform functionally graded microscale cylindrical structures

Functionally graded materials (FGMs) have garnered significant attention in research due to their potential for creating innovative materials with diverse applications. This study focuses on investigating the behavior of steel-concrete functionally graded structures under axial loading, specifically exploring nonlinear buckling. The research considers cylindrical structures with simply-supported boundary conditions and analyzes four different configurations with varying thickness. The governing equations are derived using the modified couple stress theory, incorporating assumptions of nonlinear von-Kármán theory coupled with classical beam theory. Numerical solutions are obtained using the general differential quadrature (GDQ) method. A notable aspect of this work is the use of Artificial Neural Networks (ANN) to predict nonlinear buckling forces, greatly enhancing the accuracy and efficiency of the predictions. This approach offers a novel method for analyzing concrete structures. The results are validated by comparing them with data from other published works, using dimensionless parameters for consistent comparisons. The findings provide a comprehensive analysis of various parameters, including wall thickness function and porosity, on the buckling stability of the cylindrical structure. Both numerical and ANN approaches are employed to report the effects. The study concludes that the uniform thickness cylinder exhibits the highest resistance to axial loading, resulting in buckling.

Aero-Hygro-Thermoelastic size-dependent analysis of NCMF-reinforced GNPs sector microplates located between piezoelectric patches in supersonic flow considering surface stress effects

Lately, metal foams have been widely used in various engineering structures due to their exceptional mechanical properties, including their lightweight. However, scientists have proposed adding nano-fillers like graphene nanoplatelets (GNPs) to overcome the issue of low stiffness caused by increased porosity. Hence, aerodynamic stability and vibrational behaviors of nanocomposite metal foams (NCMF)-reinforced GNPs sector microplate with piezoelectric/metallic face sheets are provided in this work. The microstructure is sited on Kerr elastic substrate and also is in a hygro-thermo-electrical media, subjected to supersonic flow based on first-order piston’s theory (FPT). Gaussian random field (GRF), Halpin-Tsai, and extended rule of mixture (ERM) in conjunction with Gurtin-Morduch, modified strain gradient theories, and Hamilton’s principle are used to derive the governing equations and linked boundary conditions. Generalized differential quadrature (GDQ) scheme is chosen to attain the results. By validating the results’ precision, the effects of different parameters on the natural frequencies (NFs) and critical aerodynamic pressures (CAPs) are observed. It is perceived that adding GNPs enhances both NFs and CAPs significantly, and increasing porosity up to seventy percent, leads the results to drop. Various industries use simpler models to evaluate the microstructure, therefore the present model can pave the way for future uses and push the boundaries of knowledge in these industries based on the results it produces.

A variational quantum algorithm-based numerical method for solving potential and Stokes flows

This paper presents a numerical method based on the variational quantum algorithm to solve potential and Stokes flow problems. In this method, the governing equations for potential and Stokes flows can be respectively written in the form of Laplace's equation and Stokes equations using velocity potential, stream function and vorticity formulations. Then the finite difference method and the generalized differential quadrature (GDQ) method are applied to discretize the governing equations. For the prescribed boundary conditions, the corresponding linear systems of equations can be obtained. These linear systems are solved by using the variational quantum linear solver (VQLS), which resolves the potential and Stokes flow problems equivalently. To the best of authors' knowledge, this is the first study that incorporates the GDQ method which is inherently a high-order discretization method with the VQLS algorithm. Since the GDQ method can utilize much fewer grid points than the finite difference method to approximate derivatives with a higher order of accuracy, the size of the input matrix for the VQLS algorithm can be smaller. In this way, the computational cost may be saved. The performance of the present method is comprehensively assessed by two representative examples, namely, the potential flow around a circular cylinder and Stokes flow in a lid-driven cavity. Numerical results validate the applicability and accuracy of the present VQLS-based method. Furthermore, its time complexity is evaluated by the heuristic scaling, which demonstrates that the present method scales efficiently in the number of qubits n and the precision ε. This work brings quantum computing to the field of computational fluid dynamics. By virtue of quantum advantage over classical methods, promising advances in solving large-scale fluid mechanics problems of engineering interest may be prompted.

Higher order theories for the modal analysis of anisotropic doubly-curved shells with a three-dimensional variation of the material properties

The manuscript investigates the dynamic properties of doubly-curved shell structures laminated with innovative materials employing the Generalized Differential Quadrature (GDQ) method. A higher order generalized expression of the displacement field variable is assumed following the Equivalent Single Layer (ESL) approach. The geometrical description of the structures follows the differential geometry basics, whereas arbitrarily-shaped domains are distorted by means of generalized isogeometric blending functions. A three-dimensional generally anisotropic elastic constitutive behaviour is assumed within each layer of laminates, and a smooth variation of the material properties is implemented. In particular, a two-dimensional distribution alongside the physical domain is considered, whereas a through-the-thickness dispersion of the material properties is associated to each layer, taking into account both polynomial and non-polynomial analytical expressions. The fundamental equations are derived from the Hamiltonian principle, and they are solved directly in strong form via the GDQ method taking into account a non-uniform discrete computational grid. After some preliminary validating examples, some parametric investigations are performed systematically, showing the influence of the variation of the material properties on the modal response of the structures characterized by different lamination schemes, curvatures and boundary conditions. The proposed model allows for the dynamic analysis of the structures of different curvatures characterized by very complicated dispersions of the material properties following a simple and accurate methodology.

Self-weight and thermal bifurcation analysis of functionally graded vertical annular plates

The present paper thoroughly investigates the influence of structural weight on the thermal buckling of standing circular and annular plates. The structure is composed of symmetrically functionally graded materials (FGM) in terms of its thickness. The first-order theory of shear deformation along with nonlinear strain–displacement equations has been employed to derive the governing equations of the FG plate. First, the asymmetric pre-buckling path of the structure under the influence of weight and heat is calculated by a semi-analytical method. The plate's stability relations are extracted using the adjacent equilibrium criterion and virtual work principle. Then, the stability equations are analyzed by the 2D generalized differential quadrature (GDQ) method. Parametric results are presented based on the analysis of the FGM power-law index, geometric ratios, and edge constraints on the buckling of the vertical plate. The study reveals the critical power-law index and thickness at which the structure becomes unstable, shedding light on the previously neglected calculation of weight-induced buckling in vertical structures. Also, the effect of weight on the buckling mode shapes of heavy plates and disturbing its symmetry is demonstrated. Additionally, this study presents the first investigation into the effect of weight on the thermal stability of annular FG plates.

An Explicit Modeling for the Pseudoelastic Response of Porous SMA Thick-Walled Cylinders under Internal and External Pressure

Shape-memory alloy (SMA) structures, either dense or porous, are widely used because of their functional characteristics arising from shape-memory effect and pseudoelasticity. Hence, the precise modeling of these materials is essential due to design issues. In this study, the solution algorithm based on an explicit phenomenological constitutive model is investigated to analyze pressurized cylinders. The proposed method is extracted following return mapping scheme to find the evolution of phase transformation in these structures. The formulation is validated by comparing uniaxial tension–compression curves during loading, unloading and thermal actuation against published literature. By the aid of this model, assuming elastic and phase transformation strains, a comprehensive study is presented to inspect stress, strain and displacement fields along the thickness of internally and externally pressurized thick-walled cylinders. To this end, the governing nonlinear equation in elastic–inelastic zones is discretized by the efficient method of generalized differential quadrature (GDQ) and the resultant relations are solved numerically. The influences of internal and external pressure on stress distribution along the cylinder wall during loading–unloading for different temperatures including phase transformation and saturation states are discussed. Also, the closed-form solution for critical pressure at the onset of the phase transformation is studied.

Generalized Kelvin-Voigt viscoelastic modeling and numerical study of Free-Damped vibrations in MR elastomer reinforced with graphene platelets

This study focuses on the modeling and numerical analysis of a novel composite plate, which consists of a laminated magnetorheological elastomer (MRE) reinforced with graphene platelets (GPLs). The investigation begins with the determination of the mechanical properties of the MRE, utilizing a modified generalized Kelvin-Voigt viscoelastic model. Through nonlinear regression analysis and the nonlinear least squares technique, the dependencies of the storage and loss modulus of the magnetorheological (MR) matrix are evaluated, considering factors such as the magnetic field, iron particles, and excitation frequency. The proposed model is validated by comparing the obtained results with existing experimental data from the literature, employing root mean square error and correlation coefficients as metrics of consistency. Next, the homogenization process is applied to the composite media, which involves integrating the MR elastomer matrix and GPL reinforcements using the Halpin-Tsai micromechanical approach. This procedure enables the extraction of effective material properties governing the behavior of the composite structure. The theoretical framework, encompassing third-order plate theory, linear elasticity, and viscoelasticity, is then employed to derive the dynamic equations of the composite plate, employing Hamilton's principle as a guiding principle. To solve the dynamic problem and obtain the complex frequencies characterizing the system, the generalized differential quadrature (GDQ) method is implemented. This numerical technique offers a robust and accurate solution approach, providing comprehensive insights into the vibrational behavior of the composite plate. The study conducts a thorough investigation, exploring the performance achieved by incorporating GPL reinforcements within the MRE matrix. Specifically, the effects of different volume fractions of iron particles and varying magnetic fields are comprehensively examined and analyzed. Through this comprehensive exploration, a profound understanding of the behavior of the composite plate and its response to external stimuli is attained. The findings highlight the potential advantages of integrating GPL reinforcements and offer valuable insights for optimizing the design and performance of such composite structures in practical applications. The research contributes to the advancement of knowledge in the field of composite materials and opens up new possibilities for innovative and high-performance structural designs.

Nonlinear large-amplitude vibration analysis of annular sector plates made of FGMs subjected to cooling shock

A numerical approach is developed in the present article to study the geometrically nonlinear vibrations of annular sector plates made of functionally graded materials (FGMs) due to being exposed to cooling shock. The first-order shear deformation plate theory (FSDT) is used in the modeling of plate, and the governing equations of motion are derived based on Hamilton's principle considering von Kármán nonlinear kinematic relations. It is also considered that the properties of FGMs are dependent on temperature and distribution of materials. According to the uncoupled thermoelasticity theory, the temperature distribution is obtained via 1D Fourier-type transient heat conduction equation. To numerically solve the problem, the generalized differential quadrature (GDQ) method and Newmark-beta integration scheme are utilized. Considering two different types of thermal loading as cooling shock, the effects of various parameters including thermal load rapidity time, geometry, magnitude of thermal load and material properties on the large-amplitude vibrations of annular sector plates are investigated.

Hygrothermally-Induced Vibration Analysis of Porous FGM Rectangular Mindlin Plates Resting on Elastic Foundation

The nonlinear vibration response of rectangular plates made of functionally graded porous materials (FGPs) induced by hygrothermal loading is investigated in this article using a numerical approach. The effect of elastic foundation on the vibrations is taken into account according to the Winkler–Pasternak model. Hygroscopic stresses produced due to nonlinear rise in moisture concentration are also considered. The temperature-dependent material properties of plate are computed based on the modified Voigt’s rule of mixture and Touloukian experiments for even and uneven distribution patterns of porosity. Within the framework of the first-order shear deformation plate theory and von-Kármán nonlinearity, Hamilton’s principle is utilized in order to derive the equations of motions. To achieve the temporal evolution of maximum lateral deflection of hygrothermally-induced plates, the generalized differential quadrature (GDQ) and Newmark integration methods are employed. Selected numerical results are presented to study the influences of temperature distribution, porosity volume fraction, moisture concentration, geometrical parameters, elastic foundation parameters and FG index on the geometrically nonlinear vibrations of FG porous plates with various boundary conditions.

Modeling of a Magnetorheological Elastomer-Based Nanocomposite Higher-Order Beam: Tunable Dynamic and Damping Characteristics

This study involves the comprehensive modeling of material properties and a numerical investigation of the free-damped vibrations exhibited by an innovative composite beam. The beam under consideration is a laminated magnetorheological (MR) elastomer, strategically reinforced with graphene platelets (GPLs). Initially, the mechanical properties of the MR matrix are meticulously calculated using a modified generalized Kelvin–Voigt viscoelastic model. To ensure the accuracy and credibility of the proposed model, nonlinear regression analysis based on the nonlinear least squares technique is employed. This modeling approach enables the tunability of the MR matrix’s storage and loss modulus, which depend on factors such as the magnetic field, iron particles, and excitation frequency. The proposed model is verified by comparing the results with experimental data from the literature, using analyses of root mean square error (RMSE) and correlation coefficients. Subsequently, the composite media, comprising the magnetorheological elastomer (MRE) matrix and GPL reinforcements, undergoes homogenization through the application of the Halpin–Tsai micromechanical procedure. Theoretical formulations, encompassing third-order beam theory, strains, linear elasticity, and viscoelasticity, are then utilized to derive the motion equations governing the dynamic behavior of the composite beam. Hamilton’s principle guides the derivation of these equations. To obtain the complex frequencies characterizing the system, the generalized differential quadrature (GDQ) method is implemented. In this study, particular emphasis is placed on investigating the performance achieved by adding GPL reinforcements to the MRE with different iron particle volume fractions under various magnetic fields.

On the mechanical properties and vibrational characteristics of the nanoparticle-reinforced composite cylindrical panels in the acidic environment: Numerical and experimental investigation

The purpose of this study is to determine the impact of nano-clay and nano-silica particles on the vibrational characteristics of composite cylindrical panels in corrosive environments. In this context, due to the presence of diluted sulfuric acid as the corrosive solution, the coupling effects of material degradation and hygroscopic loading on the natural frequencies are taken into consideration. In order to achieve this, it is first explored through experiment how adding nanoparticles affects the coefficient of hygroscopic expansion of the glass/epoxy specimens. Using this coefficient, the in-plane loading caused by corrosive solution is then applied in the equations. The mechanical properties and hygroscopic expansion coefficients of the composite samples exposed to the corrosive environment are first determined by certain experimental testing. The equations of motion are derived using the first-order shear deformation theory (FSDT), and the vibration problem is solved using the generalized differential quadrature (GDQ) technique. Investigation is conducted into the effects of various nanoparticles, immersion time, geometrical elements, and boundary conditions on the natural frequencies of the cylindrical panels. It is found that adding nanoparticles can significantly improve the mechanical properties, the corrosion resistance, and the natural frequencies of composites in which the nano-clay is more effective. For example, adding nano-clay and nano-silica particles, respectively, results in 32.32% and 26.29% increase in elasticity modulus of reinforced glass/epoxy composite. In addition, 12.71% and 47.32% improvements are observed in the natural frequency of the nano-clay reinforced composite panel before and after 90 days of immersion in sulfuric acid. In comparison, the nano-silica particles are responsible for 7.80% and 23.03% increase in natural frequency before and after immersion time.

Bending Vibration and Buckling Analysis Of Nonhomogeneous Nanotubes Using Nonlocal Elasticity Theory and Gdq Method

In this paper structural analysis of nonhomogeneous nanotubes has been carried out using nonlocal elasticity theory. Governing differential equations of nonhomogeneous nanotubes are derived. Nonlocal theory of elasticity has been employed to include the scale effect of the nanotubes. Nonlocal parameter, elastic modulus, density and diameter of the cross sections are assumed to be functions of spatial coordinates. General Differential Quadrature (GDQ) method has been employed to solve the governing differential equations of the nanotubes. Various boundary conditions have been applied to the nanotubes. Present results considering nonlocal theory are in good agreement with the results available in the literature. Effect of various geometrical and material parameters on the structural response of the nonhomogeneous nanotubes has been investigated. Present results of the nonhomogeneous nanotubes are useful in the design of the nanotubes.

Nonlinear vibration analysis of graphene platelets reinforced nanocomposite doubly curved concrete panels subjected to thermal shock: Development of an artificial intelligence technique

When the temperature gradient across a component quickly changes, a phenomenon known as thermal shock occurs that exposes the component to fast variable thermal stresses and large-scale strains. One of the goals of this abrupt loading is to increase the capacity of different systems to withstand thermal shock. The nonlinear thermally induced vibration of doubly curved graphene nanoplatelet reinforced nanocomposite panels subjected to abrupt thermal shock is being studied right now. The bottom portion of the curved panel system is held at a reference temperature while the top portion is thermally shocked. The Crank-Nicholson technique and generalized differential quadrature (GDQ) are used to build up and solve the one-dimensional transient heat equation. The double-curved panel’s thermomechanical characteristics are temperature dependent; hence this equation is nonlinear. Based on the law of decoupled thermoelasticity, Kant’s theory, and von Karman’s assumption of some kind of geometric nonlinearity, the overall function of the doubly curved panel is established. After modeling the current system mathematically and comparing the current approach’s findings to those of earlier research, we revalidated the findings in a Python environment by comparing them to those of the Extreme Gradient Boosting (XGBoost) machine learning algorithm. This approach is founded on the “boosting” idea, which combines additive training techniques with the identification of weak learners to produce strong learners. The findings demonstrate that certain structural geometrical and physical factors are significant contributors to the double-curved GPLRC panel’s nonlinear frequency under thermal shock.

Free vibration analysis of carbon-fiber plain woven reinforced composite conical-cylindrical shell under thermal environment with general boundary conditions

In this paper, the free vibration analysis of carbon fiber/epoxy resin and carbon/carbon plain woven conical-cylindrical shell under thermal environment with general boundary conditions using generalized differential quadrature (GDQ) is conducted. A two-step homogenization approach, in which 3D finite element method (3D FEM) based on 3D representative volume elements (3D RVE) are applied, is used to get the mechanical properties of the plain woven composites. According to first order shear deformation theory (FSDT) and Hamilton’s principle, considering a thermal environment, the governing equations of both the two shells are obtained. GDQ is applied during the solving procedure and general boundary conditions are considered through artificial springs. After several comparison studies focused on convergence, mechanical properties and thermal environment, the effects of boundary conditions, geometry, total fiber volume fraction and uniform, linear temperature field are studied comprehensively.

General boundary conditions implementation for the static analysis of anisotropic doubly-curved shells resting on a Winkler foundation

In the present work an Equivalent Single Layer (ESL) formulation is proposed for the static analysis of doubly-curved anisotropic structures of arbitrary geometry and variable stiffness resting on a Winkler elastic foundation. In-plane and out-of-plane general distributions of linear elastic springs are provided for the implementation of general external constraints along the edges of the structure. The structure is geometrically described accounting for the principal curvatures of the shell object of analysis. A generalized set of blending functions based on Non-Uniform Rational Basis Spline (NURBS) curves is adopted so that arbitrary shaped structures can be modelled with the same approach. The fundamental governing equations are obtained in terms of displacement field unknowns, which has been effectively described accounting for a unified formulation based on the minimum potential energy principle. General anisotropic lamination schemes are considered, setting a general orientation of each lamina, as well as all possible material symmetries. The numerical implementation is performed by means of the Generalized Differential Quadrature (GDQ) method, thus allowing a strong formulation of the structural problem. A series of validation examples is performed on shells with zero, single and double curvatures in which the static structural response provided with the proposed formulation has been compared to that obtained from a refined three-dimensional finite element model, showing a great accordance between these different approaches. The research shows that the employment of higher order theories, together with the GDQ method, allows to obtain very accurate results with a reduced computational cost, compared to finite element simulations.

A failure strength analysis method for fiber reinforcement composite cylindrical shells under hydrostatic pressure based on generalized differential quadrature

In this paper, a numerical model is presented to study the failure strength of moderate thickness composite cylindrical shells under hydrostatic pressure. Considering the transverse shear effect on the mechanics of the moderate thickness cylindrical shell, the numerical model is proposed based on the First-order Shear Deformation Theory (FSDT). The Generalized Differential Quadrature (GDQ) method is applied to solve the 3D displacement and strain field of the static governing equations. Tsai-Wu tensor failure criteria is used to acquire the structural failure strength of the cylindrical shell. The calculated displacement and strain results are compared with the ones of FE analysis and show very good agreement of the distribution. The obtained failure strength based on proposed model is validated with an experimental test and the maximum error of the failure strength is 7.78%. The results and errors indicate that the proposed numerical model is valid and accurate enough for the failure strength analysis of the moderate thickness composite cylindrical shells under hydrostatic pressure.

Numerical modeling of vibration and damping of higher-order magnetorheological elastomer polar orthotropic composite sectorial/annular plates

This study contributes to the modeling and analysis of damping and vibration in the innovative composite sector and annular plates composed of a magnetorheological elastomer (MRE) reinforced with glass fibers. These plates are formed of laminate construction. For the purpose of this investigation, polar orthotropic laminas are used. The generalized Maxwell constitutive law is employed in order to describe the tunable viscoelastic properties of the MR elastomer, including storage and modal loss factor. The application of a magnetic field enables one to exert control over the storage and loss moduli of these materials. In addition, the effective mechanical properties of the MRE composite (MREC) are determined through a modified version of the Voigt micromechanical rule and the Halpin–Tsai rule. In order to define the displacement field of the plate, a hyperbolic higher-order shear deformation approach is used. By adopting Hamilton’s principle, one may get the equations describing the motion as well as the relevant boundary and continuity conditions. In order to solve this system of highly coupled partial differential equations (PDEs), the 2D generalized differential quadrature (GDQ) approach is exploited. A comparison was made between the developed model and those available in the literature to validate the accuracy and convergence quality of the solution technique and formulation. A complete investigation is carried out into the influence of the magnetic field on the free-damped response of the MREC sector and annular plates. In addition, the effects of the composite and geometric properties of the structure on the natural frequencies and modal loss factors of the structure have been examined.

A new numerical method for elastic-plastic plate analysis: Theoretical-experimental validation and its application in ratcheting phenomenon

In the present paper, two different numerical methods, generalized differential quadrature (GDQ) and dynamic relaxation method (DR) are employed to analyze elastic-plastic plates. These methods are initially evaluated for elastic beams and then they are utilized for elastic-plastic plates analysis. A new method, GDDR (Generalized Differential Dynamic Relaxation), is developed by the combination of two mentioned methods. The proposed method is validated by theoretical analysis for elastic problems, Abaqus simulation and experimental study for elastic-plastic plates. Reduction in the number of required discretizing nodes, lower computational time and decrease in the number of iterations (about 80%) are the advantages of introduced method compared with the conventional ones. Moreover, this method improves the accuracy of solutions, especially in terms of stresses, by increasing the differentiation accuracy of nodes for elastic-plastic analysis. Comparing the errors of proposed GDDR method with conventional DR method shows about 70% reduction in mean squared error. Moreover, this new method is able to solve problems by using only 3 nodes in each direction, while other methods need at least 8.