Differential Quadrature Research Articles

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.

Double cross magnetic wall decoupling for quadrature transceiver RF array coils using common-mode differential-mode resonators

In contrast to linearly polarized RF coil arrays, quadrature transceiver coil arrays are capable of improving signal-to-noise ratio (SNR), spatial resolution, and parallel imaging performance. Owing to a reduced excitation power, a low specific absorption rate can also be obtained using quadrature RF coils. However, due to the complex nature of their structure and their electromagnetic properties, it is challenging to achieve sufficient electromagnetic decoupling while designing multichannel quadrature RF coil arrays, particularly in ultra-high fields. In this work, we proposed a double-cross magnetic wall decoupling for quadrature transceiver RF arrays and implemented the decoupling method on common-mode differential mode quadrature (CMDM) quadrature transceiver arrays at an ultrahigh field of 7 T. The proposed magnetic decoupling wall, comprised of two intrinsically decoupled loops, is used to reduce the mutual coupling between all the multi-mode currents present in the quadrature CMDM array. The decoupling network has no physical connection with the CMDMs’ resonators, which provides less design constraint over size-adjustable RF arrays. To validate the feasibility of the proposed cross-magnetic decoupling wall, systematic studies on the decoupling performance based on the impedance of two intrinsic loops are numerically performed. A pair of quadrature transceiver CMDMs is constructed along with the proposed decoupling network, and their scattering matrix is characterized using a network analyzer. The measured results indicate that all the current modes from coupling are simultaneously suppressed using the proposed cross-magnetic wall. Moreover, field distribution and local specific absorption rate (SAR) are numerically obtained for a well-decoupled 8-channel quadrature knee-coil array.

Static and dynamic instability of functionally graded graphene origami-enabled auxetic metamaterial beams with variable thickness in fluid

Functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterial (GOEAM) structures have shown great potential for various engineering applications due to their exceptional mechanical and physical properties such as high strength-to-weight ratio, tuneable stiffness and strength, and negative Poisson's ratio (NPR). This paper aims to investigate the buckling and dynamic instability behaviours of FG-GOEAM beams with variable thickness immersed in a fluid, with a particular focus on the influence of NPR. The material properties such as Young's modulus and Poisson's ratio of the GOEAM are determined by using a genetic programming (GP)-based micromechanics model. Within the framework of the first-order shear deformation theory and by employing Hamilton's principle and modelling the fluid effect as added mass, the governing equations of motion are established and are then discretised by means of differential quadrature (DQ) method to obtain a linear system of Mathieu-Hill equations from which the principal instability regions of the FG-GOEAM beam are determined by Bolotin's method. A comprehensive parametric study is conducted to reveal the effects of GOri's folding degree, distribution, weight fraction, as well as fluid density and beam dimensions on the static and dynamic instability behaviours of the FG-GOEAM beam immersed in fluid. Numerical results show that the FG-GOEAM beam with NPR considerably outperforms its pristine metallic counterpart in terms of resistance against static buckling and dynamic stability.

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.

Stability analysis for internal flow-induced vertical cantilevered pipe subject to multiple lumped masses

This study investigates the effects of both single and multiple lumped masses attached to a vertical cantilevered pipe conveying internal flow. By introducing delta and Heaviside functions, the governing equation of the vibration of a pipe with lumped masses is newly established based on Hamilton’s principle. The gravitational force of the lumped mass should be considered in the theoretical model but has been essentially ignored in previous research. The harmonic differential quadrature (HDQ) method is applied to solve the governing vibration equation. The newly developed model is validated through a comparison with published data. Numerical results show that the gravity force of the lumped mass plays an important role, and ignoring this force leads to inaccurate results in the investigation of lumped mass effects. The instability of the pipe is analyzed using this new model with various lumped mass weights, positions, and fluid mass ratios. The numerical results showed that lumped masses significantly affect the critical flow velocity, vibration frequency, and modal shapes. The conclusions are expected to supply valuable guidance to reduce pipe vibration using lumped masses in engineering applications.

GDQ computation for thermal vibration of thick FGM plates by using third-order shear deformation theory

The third-order shear deformation theory (TSDT) effects on the thick functionally graded material (FGM) plates under thermal vibration are investigated by using the generalized differential quadrature (GDQ) method. The nonlinear coefficient term of displacement field of TSDT is used to derive the equations of motion for thermal vibration of thick FGM plates. The simpler form stiffness of FGM plates under linear temperature rise is considered in the thermo-elastic stress–strain relations. The dynamic equilibrium differential equations of FGM plates in terms of partial derivatives of displacements and shear rotations subjected to partial derivatives of thermal loads, mechanical loads and inertia terms are derived. Two parametric: environment temperature and FGM power law index effects on the thermal stress and center deflection of FGM thick plates are investigated.

Buckling analysis of nanocomposite beams based on refined beam theory by considering the Agglomeration effect of CNTs

The present work deals with buckling load behavior of nanocomposite beams by considering the agglomeration effect of single-walled carbon nanotubes (CNTs) and different patterns CNTs in polymeric matrix. The material properties of nanocomposite beams are estimated using the Eshelby–Mori– Tanaka approach based on an equivalent fiber. The equations of motion are derived based on refined beam strain gradient theory, employing Hamilton’s principle and using Differential Quadrature Finite Element Method (DQFEM) derived from the differential quadrature method (DQM). The results are compared with analytical results in the literature. It is remarked that mechanical properties and therefore critical buckling loads of nanocomposite beam are seriously affected by CNTs agglomeration.

Free vibration analysis of laminated doubly-curved shells with arbitrary material orientation distribution employing higher order theories and differential quadrature method

In the present work, the dynamic behaviour of laminated anisotropic doubly-curved shells characterized by a generalized distribution of the material orientation angle is investigated employing higher order theories. The structural problem is developed following the Equivalent Single Layer (ESL) methodology, setting up a unified approach for the assessment of the displacement field variable with higher order theories. Accordingly, a generalized three-dimensional distribution of the material orientation angle is associated to each layer of the stacking sequence, accounting for an in-plane bivariate power distribution and out-of-plane symmetric and unsymmetric profiles described with both polynomial and non-polynomial analytical expressions. The fundamental equations are derived from the Hamiltonian Principle, and they are numerically tackled employing the Generalized Differential Quadrature (GDQ) method directly in the strong form. Moreover, a generalized three-dimensional set of linear elastic springs is implemented for the assessment of con-conventional boundary conditions. Furthermore, a generalized isogeometric mapping of the physical domain accounts for arbitrarily-shaped structures. The model is validated successfully with respect to refined three-dimensional classical models, and it is, then, applied systematically to check for the sensitivity of the mechanical response to the structural curvature, external constraints, and material orientation angle distributions.

A study on the crack presence effect on dynamical behaviour of bi-directional compositionally imperfect material graded micro beams

This work uses the Quasi-3D (Q3D) shear and normal deformation beam concept in combination with the length scale capturing theory through the “Modified Couple Stress Theory” (MCST) and the novel computational approach namely, Differential Quadrature Finite Element Method (DQFEM) to investigate free vibratory conducts of an imperfect BD-FG microbeams with a crack. To obtain the relevant governing equations, the Lagrange's rule is utilized. “The differential quadrature” (DQ) and “Gauss-Lobatto quadrature” procedures for weighting coefficient matrices are used to easily construct the mass and stiffness matrices. The Gauss-Lobatto node scheme is used to determine the griding knots. In order to show the efficiency and exactness of the described numerical resolution method, the attained outcomes are authenticated with the similar results found in the literature. The effects of the crack position, crack depth, beam thickness by material length scale aspect, the volume fraction of material imperfection, edge supports state types, length through thickness factor, and the two-way material grading indexes on the nondimensional natural frequencies of 2D porous FG microbeams are inspected. This work will provide a numerical basis for the design of FG microstructures in the field of micromechanics. These findings can be used to construct porous FG microstructures in engineering.

Radial Basis Function-Based Differential Quadrature Approach to Study Reaction–Diffusion of Ca2+ in T Lymphocyte

T lymphocytes have a primary role in both health and disease. Extracellular and intracellular signals determine whether a T-cell activates different cells, divides, or begins apoptosis. The reaction–diffusion process of Ca2+ ions is critical for the initiation, sustenance, and termination of the immunological function of T cell. A nonlinear spatio-temporal dynamics of Ca2+ in T cells is modeled incorporating parameters Sarco/endoplasmic reticulum Ca2+-ATPase (SERCA) pump, Ryanodine receptor, source amplitude, and buffers. A numerical meshless approach using multiquadric radial basis functions (MQRBF), differential quadrature, and Runge–Kutta method is developed for the solution. The results obtained here give better insights of calcium dynamics in T cells.

Vibration of smart sandwich plate with an auxetic core and dual-FG nanocomposite layers integrated with piezoceramic actuators

In the current research, a thorough investigation of the free vibrational characteristics of a novel smart sandwich plate has been performed utilizing two different analytical and numerical solution techniques. The proposed structure is a sandwich plate with an auxetic core and super stiff dual-FG nanocomposite layers integrated with piezoceramic actuators. The nanocomposite layers of the structure are made of FG metal-ceramic matrix to which carbon nanotubes (CNTs) are added through various FG distribution patterns. Besides, two thin piezoceramic actuators are attached to the nanocomposite layers, which causes the structure to be controllable by applying an electrical voltage. Using such super stiff dual-FG nanocomposite layers together with piezoceramic actuators and ultra-light auxetic core with negative Poisson’s ratio results in the light and smart sandwich plate with high structural stiffness, which its natural frequency responses will be analyzed in this study. For this purpose, the governing equations are derived according to the first-order shear deformation theory (FSDT) and then solved using two different strategies, including numerical and analytical solution methods. The responses obtained by these solution methods by the name of generalized differential quadrature (GDQ) and Navier are compared and verified with each other and other reputed references in the literature. Accordingly, detailed parametric studies in the framework of tabulated and graphical results are performed to reveal the impacts of all involved parameters on the natural frequency responses of the proposed smart sandwich plate.

Free vibration analysis of pre-stretched hyperelastic micromorphic continua with arbitrary shapes

A novel numerical strategy is developed in this article to study the free vibrations of hyperelastic micromorphic continua under bending load. In the proposed approach, the variational differential quadrature (VDQ) method and the idea of position transformation are used. The 3D micromorphic hyperelasticity is first formulated via vector-matrix relations which can be readily utilized in the coding process of numerical methods. The present numerical approach is able to address problems with irregular domains. To this end, the domain of elements is transformed into a regular one by the technique of mapping of position field based on the finite element shape functions. Being locking-free, simple implementation, computational efficiency and fast convergence rate are other features of the present variational approach. Three numerical examples including rectangular, sector and circular plates under bending load are considered whose free vibration behavior is analyzed. It is shown that the method is capable of predicting the relation between natural frequencies of micromorphic hyperelastic continua with load factor in an efficient way. The effects of internal length scale, scale-transition parameter and mode transition on the results are investigated.

Buckling analysis and design optimization of trapezoidal composite plates under hygrothermal environments

The main goal of the present work is to study the buckling analysis and optimization of trapezoidal composite plates under hygrothermal conditions. The stability equations are first derived based on the first-order shear deformation theory (FSDT) using the energy method. Then, the generalized differential quadrature (GDQ) method is implemented to calculate the critical buckling temperature and moisture content. The effects of various parameters such as temperature, humidity, lay-up, boundary conditions, and geometry are also examined on the stability of trapezoidal composite plates. The numerical results indicate the significant effects of high-temperature and high-humidity environments on the buckling response of the structure. Finally, genetic algorithm (GA) is used to optimize the arrangement of layers for maximum buckling load. The optimization results are obtained to indicate the effecs of geometric parameters, boundary conditions, and hygrothermal loads on the optimal design of trapezoidal composite plates.

Numerical Simulation and Dynamics of Burgers’ Equation Using the Modified Cubic B-Spline Differential Quadrature Method

In the present work, a numerical approach using the Crank–Nicolson scheme along with the modified cubic B-spline differential quadrature (CN-MCDQ) method is proposed to find the numerical approximations to Burgers’ equation. After applying the well-known Crank–Nicolson technique, Burgers’ equation is solved in this study by using the differential quadrature approach to approximate the derivatives that lead to a system of equations to be solved. When compared to other methods for obtaining numerical solutions, the proposed method is shown to be efficient and easy to implement while still providing accurate results. The obtained results are in agreement with the earlier available approaches and are even better in comparison in terms of less domain partition. Three test problems were used to evaluate the methodology, and the results are tabulated and graphically shown below.

Buckling analysis of non-uniform Timoshenko columns under localised fire

The buckling analysis of non-uniform steel columns with various boundary conditions under localised fire is investigated based on the Timoshenko beam theory (TBT). The temperature-dependent (TD) material properties are taken into account. First, based on the differential quadrature (DQ) method, the temperature field in the column is determined by solving the heat transfer equation considering the convection and radiation boundary conditions. Then, a segmented model is presented to solve the stability problem for the column with the variable flexural rigidity resulting from the non-uniform temperature distribution and cross-section. The transfer-matrix method is employed to obtain the critical buckling load and corresponding buckling mode. Finally, an example is conducted to study the temperature field and buckling response of tapered circular hollow steel columns under localised fire. The feasibility of the present method is verified by comparing the present results with those predicted by the finite element (FE) method and those reported in the literature. Numerical results show the influences of the localised fire and shear deformation on the buckling response of the column.

Vibrational characteristic simulations regarding connecting two different semi-spheroidal shells and a full-spheroidal shell with a conical shell categorized in underwater structures

Recently, the related structures of marine, underwater, and ocean structures, covering, ice-breaking devices, riser pipes, ballast water gears, gliders, and underwear noise migrations, can be counted by Connected Shells (CSs). Accordingly, for the first time, the investigation related to vibration outputs associated with connecting two different semi-spheroidal shells and a full-spheroidal shell with a conical shell, namely Connected Spheroidal-Conical Shell Assembly (CSCSA) organized in three assembly groups is examined here. In detail, General Shell Hypothesis (GSH) and the First Shear Deformation Theory (FSDT) are connected to find the main formulations of the CSCSAs' shell elements. Addedly, the Motion Governing Differential Equations (MGDEs) associated with each shell element related to the CSCSAs are identified as implementing Hamilton's principle. Addedly, the MGDEs associated with the CSCSA components are discretized, realizing the Generalized Differential Quadrature Method (GDQM). Thereafter, the Natural Frequency Factors (NFFs) of the CSCSAs are determined by implementing the eigenvalue determination. Also, the NFFs measured by the suggested context are linked to the NFFs determined using classical FEM-constructed commercial software to validate the provided approach since previous research has not been conducted on these structures in the literature (except one type related to the CSCSAs). More importantly, the largest observed inaccuracy is less than 1%. Consistent with this, the responses of this investigation may be operated as the benchmark. Finally, the impacts of the geometrical parameters connected to the CSCSAs on the NFFs of the structure are identified via the design and solution of original cases.

Nonlinear dynamics of viscoelastic fluid-conveying pipe installed within uniform external cross flow by pipe clamps

Nonlinear dynamic response of a viscoelastic fluid-conveying pipe excited by uniform external cross flow is investigated, which is installed by pipe clamps. A pipe clamp theoretical model is proposed for the actual pipe clamp instead of the commonly used simply supported or clamped boundary conditions. Based on the model, equivalent support stiffness and torsional stiffness are obtained from structure and material parameters of the pipe clamp. The vortex-induced lift force and the mean drag force are described by van der Pol oscillator model. The nonlinear equations of motion with the boundary conditions of different constraint stiffness are discretized by Differential Quadrature Method (DQM) and then solved by Runge-Kutta algorithm. The results are verified by Galerkin method for the pinned-pinned pipe, and by ANSYS for the pipe with two arbitrary constraint stiffnesses. The dependence of critical external fluid velocity upon constraint stiffness and pipe length is investigated. Moreover, the effects of support stiffness, torsional stiffness, internal fluid velocity, and viscoelastic coefficient on the amplitude response of the pipe are studied in detail. The influence of constraint stiffness on characteristics of lock-in region is elucidated. Consequently, pipe clamp spacing, material and structure parameters of the rubber mat can be designed to ensure the requirement of economy and safety, which are helpful to guide the layout and design of pipe clamps.

Application of GDQ method to large amplitude response of FGM joined spherical-conical shells under rapid surface heating

Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) joined spherical-conical shells are analyzed in the current research. Thermo-mechanical properties of the shells are assumed to be temperature and position dependent. The system of joined spherical-conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and Crank-Nicolson methods. This equation is non-linear since the thermo-mechanical properties of the shells are temperature dependent. The total functional of the shells is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well-validated with the available data for the case of single FGM spherical shell. Parametric studies are conducted to examine the influences of conical and spherical shell geometries, material composition, temperature dependence, in-plane and out-of-plane mechanical boundary conditions, various configuration of conical/spherical shell system, and thermal boundary conditions. It is highlighted that thermally induced vibrations indeed exists.