Differential Quadrature Rule Research Articles

A differential quadrature-based approach à la Picard for systems of partial differential equations associated with fuzzy differential equations

Departing from a numerical method designed to solve ordinary differential equations, in this manuscript we extend such approach to solve problems involving fuzzy partial differential equations. The method proposed in this work is a non-recursive technique that combines differential quadrature rules and a Picard-like scheme in order to obtain general solutions of systems of partial differential equations derived from a general fuzzy partial differential model. The property of stability and a bound for the Hausdorff distance are established under suitable conditions. Several numerical examples are provided in order to show the effectiveness of the proposed technique.

Free vibration analysis of soft-core sandwich beams by the novel weak form quadrature element method

The free vibration of soft-core sandwich beams is analyzed by using the weak form quadrature element method. An N-node novel weak form quadrature sandwich beam element is established based on the extended high-order sandwich panel theory and the differential quadrature rule. Gauss quadrature is used to obtain the element stiffness and consistent mass matrices numerically. Detailed formulations are given. Convergence study shows the fast rate of convergence. The validity and the capability of the novel beam element are examined through comparison with analytic results based on the same theory and two-dimensional finite element data. It is shown that one proposed beam element can yield very accurate frequencies with a relatively small number of nodal points. New data of soft-core sandwich beams with boundary conditions other than the simply supported one, not available in literature, are provided. These data may be referential with which other researchers can compare their results during developing new numerical methods.

Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method

Based on the classical beam theory (CBT) and differential quadrature (DQ) rule, an N-node novel weak form quadrature functionally graded (FG) beam element is established. Both Young’s modulus and mass density of the beam materials vary exponentially through the thickness. The element node points can be different from the integration points. Either Gauss–Lobatto–Legendre (GLL) quadrature or Gauss quadrature can be used to obtain the element stiffness matrix and mass matrix. Detailed formulations are given. Convergence study is performed. For verification, results are compared with available solutions in literature. It is shown that the proposed thin beam element can yield very accurate frequencies with relatively small number of nodal points. New accurate results are presented for functionally graded beams with nine different boundary conditions. The tabulated results will be a reference with which other researchers can compare their results during developing new numerical method.

Nonlinear elastic/plastic buckling analysis of thick/thin skew plates under uniaxial and biaxial loading

In this study, elastic/plastic buckling analysis of thick skew plates subjected to uniaxial compression or biaxial compression/tension loading using the generalized differential quadrature method is presented for the first time. The governing differential equations are derived based on the incremental and deformation theories of plasticity and first-order shear deformation theory. The elastic/plastic behavior of the plates is described by the Ramberg–Osgood model. Generalized differential quadrature discretization rules in association with an exact coordinate transformation are simultaneously used to transform and discretize the equilibrium equations and the related boundary conditions. The results are compared with the previously published data to verify the established methodology and procedures. The effect of skew angle and thickness ratio on the convergence and accuracy of the method are studied. Moreover, the effects of aspect, loading and thickness ratios, skew angle, incremental, and deformation theories and different types of boundary conditions on the buckling coefficients are presented in detail. The results show that the difference between the incremental and deformation theories becomes greater with increasing thickness ratio and constraints at boundary conditions. Furthermore, the skew angle also has an important effect on differences between those theories.

A new least-squares DQ-based method for the buckling of structures with elastic supports

This note introduces a new computational scheme based on a numerical Picard-like method, which combines successive approximations with differential quadrature rules in order to improve the computational efficiency. Despite of its recursive nature, this scheme can generate the approximate solution in terms of operational matrices and vectors of known quantities. This allows to define explicitly an error functional which can be minimized in order to easily compute the critical load. The performance of this new scheme is examined by means of two application examples. A comparison with the available solutions is carried out, by showing that the method behaves satisfactorily.

A novel variational numerical method for analyzing the free vibration of composite conical shells

This paper proposes an efficient numerical method in the context of variational formulation and on the basis of Rayleigh–Ritz technique to address the free vibration problem of laminated composite conical shells. To this end, the energy functional of Hamilton’s principle is written in a quadratic form using matrix relations first. Displacements are then approximated via a linear combination of base functions, by which the number of final unknowns reduces. After that, the strain tensor is discretized by means of matrix differential quadrature (DQ) operators. In the next step, using Taylor series and DQ rules, a matrix integral operator is constructed which is embedded into the stiffness matrix so as to discretize the quadratic representation of energy functional. Finally, the reduced form of mass and stiffness matrices are readily obtained from the aforementioned discretized functional. To obtain the natural frequencies of conical shell, hybrid harmonic-beam base functions are employed as modal displacement functions. The accuracy of the present numerical method is examined by comparing its results with those from the published literature. It is revealed that the method is capable of accurately solving the problem with a little computational effort and ease of implementation.

Static analysis of higher order sandwich beams by weak form quadrature element method

Based on the extended high order sandwich panel theory (EHSAPT) and differential quadrature rule, an N-node novel weak form quadrature sandwich beam element is established. Gauss Lobatto Legendre (GLL) points are utilized as element nodes and GLL quadrature is used to obtain the element stiffness matrix and work equivalent load. Detailed formulations are given. Convergence study is performed. Numerical results are presented for sandwich beams with different boundary conditions subjected to distributed loadings. Different materials for the face sheets and core of the beam structure are considered. It is shown that the proposed beam element can yield very accurate displacements and stresses as compared to theoretical solutions.

Dynamics and pull-in instability of electrostatically actuated microbeams conveying fluid

The purpose of this paper is to develop a theoretical model for predicting the dynamics and pull-in mechanism of electrostatically actuated microbeams containing internal fluid flow. By considering the effects of nonuniform profile of the flow velocity, material length scale parameter of the microbeam and nonlinear electrostatical force, the equation of motion of the microbeam has been presented. The lateral displacement of the microbeam consists of two parts: a static (steady) displacement and a perturbation displacement about the static. Based on the general differential quadrature rule, the static deflection of the microbeam is calculated numerically. The obtained static deflection is then used to solve the equation governing the perturbed displacement. The natural frequency and flow-induced instability of the microbeam are analyzed for both clamped–clamped and cantilevered boundary conditions. Results show that the internal fluid flow could dramatically affect the static deflection of the microbeam and hence the pull-in voltage. The electric voltage, on the other hand, would significantly influence the dynamics of the microbeam.

Novel weak form quadrature element method with expanded Chebyshev nodes

Abstract Based on the principle of minimum potential energy and the differential quadrature rule, novel weak form quadrature element method is proposed. Different from the existing ones, expanded Chebyshev grid points are used as the element nodes. A simple but general way is proposed to compute the strains at the integration points explicitly by using the differential quadrature rule. For illustration and verification, quadrature bar and beam elements are established. Several examples are given. Numerical results indicate that the proposed quadrature element method allows a longer time step as compared to elements with other nodes and is an accurate and efficient method for structural analysis.

Free Vibration Analysis of a Rotating Non-Uniform Blade with Multiple Open Cracks Using DQEM

In this paper, differential quadrature element method (DQEM) is used to analyze the free transverse vibration of a multiple cracked non-uniform Timoshenko blade rotating with a constant angular velocity. The blade’s deflection is assumed to be small so that the nonlinear terms can be neglected. Governing equations, compatibility conditions at the damaged cross-sections and boundary conditions are derived and formulated by the differential quadrature rules. First, the precision of the proposed method is confirmed by the exact solutions available in the literature. Then, the effect of angular velocity value as well as the location, depth and numbers of the cracks are investigated on the natural frequencies of the blade. The advantages of the proposed method are applicable for blades with any arbitrary variable section and less time-consuming for cases with several cracks.

Soft Core Plane State Structures Under Static Loads UsingGDQFEM and Cell Method

The aim of this work is to study the static behavior of 2D soft core plane state structures, which are very common in many engineering fields. Deflections and interlaminar stresses caused by forces can have serious consequences for strength and safety of these structures. Therefore, an accurate identification of the variables in hand is of considerable importance for their technical design. In particular, it will be shown how a soft core structure can be tailored as a function of the stress profile of the two constituent materials. It is obvious that, since a complex plane structure is under consideration, there is no analytical solution which can be used to solve the physical system problem. Only numerical procedures can be considered to solve a two-material system subjected to generic static loads. In this study two advanced numerical technique will be used: the Generalized Differential Quadrature Finite Element Method (GDQFEM) procedure and the Cell Method (CM). The first is a numerical technique based on the classic Generalized Differential Quadrature (GDQ) rule [1,2,3] and it operates differently from the well-known Finite Element Method (FEM). In fact, compatibility conditions are enforced between the element boundaries, such as in the CM contact problems [4,5]. In order to capture the discontinuity between two boundaries, the compatibility conditions have to be applied at each disconnection, when the GDQFEM is used. New results are presented to illustrate comparisons between GDQFEM, CM and FEM.

Free vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses using DQEM

A B S T R A C T In this paper, a differential quadrature element method (DQEM) is developed for free transverse vibration analysis of a non-uniform cantilever Timoshenko beam with multiple concentrated masses. Governing equations, compatibility and boundary conditions are formulated according to the differential quadrature rules. The compatibility conditions at the position of each concentrated mass are assumed as the continuity in the vertical displacement, rotation and bending moment and discontinuity in the transverse force due to acceleration of the concentrated mass. The effects of number, magnitude and position of the masses on the value of the natural frequencies are investigated. The accuracy, convergence and efficiency of the proposed method are confirmed by comparing the obtained numerical results with the analytical solutions of other researchers. The two main advantages of the proposed method in comparison with the exact solutions available in the literature are: 1) it is less time-consuming and subsequently moreefficient; 2) it is able to analyze the free vibration of the beams whose section varies as an arbitrary function which is difficult or sometimes impossible to solve with analytical methods. }}

Static analysis of functionally graded doubly-curved shells and panels of revolution

The Generalized Differential Quadrature (GDQ) Method is applied to study four parameter functionally graded and laminated composite shells and panels of revolution. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT), in particular on the Toorani-Lakis Theory. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element method. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of functionally graded shell structures made of a mixture of ceramics and metal.

Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method

The Generalized Differential Quadrature (GDQ) Method is applied to study laminated composite shells and panels of revolution. The mechanical model is based on the so called First-order Shear Deformation Theory (FSDT) deduced from the three-dimensional theory, in order to analyze the above moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin theory. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The results are obtained taking the two co-ordinates into account, without using the Fourier expansion methodology, as done in semi-analytical methods. After the solution of the fundamental system of equations in terms of displacements and rotations, the generalized strains and stress resultants can be evaluated by applying the Differential Quadrature rule to the generalized displacements themselves. The transverse shear and normal stress profiles through the laminate thickness are reconstructed a posteriori by simply using local three-dimensional equilibrium equations. No preliminary recovery or regularization procedure on the extensional and flexural strain fields is needed when the Differential Quadrature technique is used. By using GDQ procedure through the thickness, the reconstruction procedure needs only to be corrected to properly account for the boundary equilibrium conditions. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with 3D finite element methods. Stresses of several composite shell panels are evaluated. Very good agreement is observed without using mixed formulations and higher order kinematical models. Various examples of stress profiles for different shell elements are presented to illustrate the validity and the accuracy of GDQ method.

Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery

The Generalized Differential Quadrature (GDQ) Method is applied to study laminated composite degenerate shell panels such as rectangular and annular plates. The theoretical treatment is maintained general in order to expose in a unique way the procedure adopted to obtain the stress profiles through the thickness of plates without specifying the equations for rectangular and annular plates. By simply imposing some geometrical relations the equations governing the problem of plates under consideration, that are degenerate shells, are inferred from the theory of shells of revolution. The mechanical model is based on the so called First-order Shear Deformation Theory (FSDT) deduced from the three-dimensional theory in order to analyse the above moderately thick structural elements. The solution is given in terms of generalized displacement components of points lying on the middle surface of the plate. After the solution of the fundamental system of equations in terms of displacements and rotations, the generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the laminate thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. No preliminary recovery or regularization procedure on the extensional and flexural strain fields is needed when the Differential Quadrature technique is used. By using GDQ procedure through the thickness, the reconstruction procedure needs only to be corrected to properly account for the boundary equilibrium conditions. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element methods. Stresses of several composite plates are evaluated. Very good agreement is observed without using mixed formulations and higher order kinematical models. Various examples of stress profiles for rectangular and annular plate elements are presented to illustrate the validity and the accuracy of GDQ method.

Solving 2D-wave problems by the iterative differential quadrature method

In this paper, the numerical stability of an iterative method based on differential quadrature (DQ) rules when applied to solve a two-dimensional (2D) wave problem is discussed. The physical model of a vibrating membrane, with different initial conditions, is considered. The stability analysis is performed by the matrix method generalized for a 2D space-time domain. This method was presented few years ago by the same author as an analytical support to check the stability of the iterative differential quadrature method in 1D space-time domains. The stability analysis confirms here the conditionally stable nature of the method. The accuracy of the solution is discussed too.

Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin and Toorani–Lakis theory. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the general formulation and the Classical Reissner–Mindlin and Classical Toorani–Lakis theory are presented. New results are presented in order to investigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the free vibrations of anisotropic shells of revolution with a free-form meridian.

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner–Mindlin and the Toorani–Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.

A differential quadrature analysis of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field. Fundamental equations of plane electromagnetic, temperature and elastic fields are formulated. To the best of the authors’ knowledge, this is the first attempt to apply the DQM to magneto-thermo-elasticity and the first attempt to analyze a finite two-dimensional magneto-thermo-elastic problem. The fundamental equations and the inhomogeneous time-dependent boundary conditions are discretized in spatial and temporal domain by differential quadrature (DQ) rules. The unknowns satisfying the governing equations, the boundary and initial conditions simultaneously are computed in the entire domain by means of DQM with high efficiency and accuracy using dramatically less grid points in both spatial and time domain. Solutions of magnetic field, eddy current, temperature change and dynamic solutions of stresses and deformations are illustrated graphically.

A DIFFERENTIAL QUADRATURE FINITE ELEMENT METHOD

This paper studies the differential quadrature finite element method (DQFEM) systematically, as a combination of differential quadrature method (DQM) and standard finite element method (FEM), and formulates one- to three-dimensional (1-D to 3-D) element matrices of DQFEM. It is shown that the mass matrices of C 0 finite element in DQFEM are diagonal, which can reduce the computational cost for dynamic problems. The Lagrange polynomials are used as the trial functions for both C 0 and C 1 differential quadrature finite elements (DQFE) with regular and/or irregular shapes, this unifies the selection of trial functions of FEM. The DQFE matrices are simply computed by algebraic operations of the given weighting coefficient matrices of the differential quadrature (DQ) rules and Gauss-Lobatto quadrature rules, which greatly simplifies the constructions of higher order finite elements. The inter-element compatibility requirements for problems with C 1 continuity are implemented through modifying the nodal parameters using DQ rules. The reformulated DQ rules for curvilinear quadrilateral domain and its implementation are also presented due to the requirements of application. Numerical comparison studies of 2-D and 3-D static and dynamic problems demonstrate the high accuracy and rapid convergence of the DQFEM.