solution of the poisson equation by differential quadrature the method of differential quadrature is demonstrated by solving the two dimensional poisson equation the results for three test problems are compared with the exact analytical solutions and the numerical solutions obtained by others for the galerkin the control volume and the five point finite difference methods the method of differential quadrature leads to more accurate results for comparable levels of computational effort

an eight node curvilinear differential quadrature formulation for reissner mindlin plates a first endeavour to exploit the differential quadrature dq method as a simple accurate and efficient numerical technique for the bending of quadrilateral reissner mindlin plates with curvilinear boundaries is made the curvilinear quadrilateral is mapped onto a square domain by using the geometric coordinate transformation this geometric mapping technique is employed to transform the governing differential equations and boundary conditions of the problem from the physical domain onto the computational domain the dq procedures are then applied to discretise the transformed set of differential equations and boundary conditions into a set of linear algebraic equations the numerical solutions to the problem are obtained by solving this set of linear algebraic equations examples illustrating the accuracy and convergence of the dq method for reissner mindlin plates with curvilinear boundaries are presented the great development potential of the dq method as an alternative to other approximate techniques for solving problems related to engineering sciences and structural mechanics is exploited in the present study

a dqem for transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions in this paper a differential quadrature element method dqem for free transverse vibration analysis of multiple cracked non uniform timoshenko beams with general boundary conditions is proposed governing equations the compatibility conditions at the damaged cross sections and implementation of the external boundary conditions are derived and formulated by the differential quadrature analogue the accuracy convergence and versatility of the proposed method are confirmed by the exact solution of the uniform beam which has been presented by other authors and 2d finite element method fem numerical results for non uniform beam after the validation of the presented method the effect of quantity depth and location of the cracks on the frequency values of vibrations are investigated the achieved results show that the existence of the crack leads to a decrease in the frequencies of the vibrations through decrease in the stiffness of the beam meanwhile the compatibility conditions at the damaged section is considered as a discontinuity in slope and vertical displacement where the effect of the discontinuity in the slope is more considerable as many authors have neglected the discontinuity in the vertical displacement as it will be shown consideration of the discontinuity in the vertical displacement causes more decrease in the frequencies

separable projection integrals for higher order correlators of the cosmic microwave sky we present a case study describing efforts to optimise and modernise modal the simulation and analysis pipeline used by the planck satellite experiment for constraining general non gaussian models of the early universe via the bispectrum or three point correlator of the cosmic microwave background radiation we focus on one particular element of the code the projection of bispectra from the end of inflation to the spherical shell at decoupling which defines the cmb we observe today this code involves a three dimensional inner product between two functions one of which requires an integral on a non rectangular domain containing a sparse grid we show that by employing separable methods this calculation can be reduced to a one dimensional summation plus two integrations reducing the overall dimensionality from four to three the introduction of separable functions also solves the issue of the non rectangular sparse grid this separable method can become unstable in certain scenarios and so the slower non separable integral must be calculated instead we present a discussion of the optimisation of both approaches we demonstrate significant speed ups of 100 arising from a combination of algorithmic improvements and architecture aware optimisations targeted at improving thread and vectorisation behaviour the resulting mpi openmp hybrid code is capable of executing on clusters containing processors and or coprocessors with strong scaling efficiency of 98 6 on up to 16 nodes we find that a single coprocessor outperforms two processor sockets by a factor of 1 3 and that running the same code across a combination of both microarchitectures improves performance per node by a factor of 3 38 by making bispectrum calculations competitive with those for the power spectrum or two point correlator we are now able to consider joint analysis for cosmological science exploitation of new data

three dimensional elasticity solutions to some orthotropic plate problems abstract this paper presents a three dimensional analysis of rectangular orthotropic plates by employing the differential quadrature dq method the derivation of the governing equations from the governing equations and the stress strain relationship of the three dimensional elasticity model is detailed the constrained conditions of the orthotropic plate edges are given the governing equations and boundary conditions are first normalized and discretized according to the dq procedure example problems pertaining to the bending buckling and free vibration of orthotropic plates with generic boundary conditions are selected to illustrate the simplicity and applicability of the dq method the convergence characteristics of the dq method are first obtained based on numerical studies after which the dq solutions are compared where possible with exact solutions it is found that the dq method yields accurate results for the plate problems under investigation

on the performance of three iterative methods for solution of gdq algebraic equations abstract in this paper the three conventional iterative methods namely j oe or jacobi gauss seidel g s and sor methods are extended to solve the gdq algebraic equations the performances of these three iterative methods for the gdq and second order finite difference fd algebraic equations are comparatively studied by their application to solve the two dimensional poisson equation it was found that sor method gives the fastest convergence rate for both gdq and fd algebraic equations however the effect of relaxation factor for gdq algebraic equations is reduced the high efficiency and accuracy of gdq over second order fd scheme are also shown in the paper furthermore the performance of j oe and sor methods was validated by their application to a complicated problem that is numerical simulation of natural convection in the annulus between two concentric cylinders it was found that for this complicated case jacobi and gauss seidel methods cannot get converged solution and the efficiency of sor method is reduced to be around three times faster than the j oe method

static analysis of mindlin plates the differential quadrature element method dqem abstract the differential quadrature method dqm is a new numerical method for fast solving linear and nonlinear differential equations based on global basis functions in the present study a discrete approximate method based on the dqm the differential quadrature element method dqem is developed for the axisymmetric static analysis of moderately thick circular and annular plates described by the mindlin shear deformable theory the basic idea of the dqem is 1 to divide the whole variable domain into several sub domains elements 2 to form discretized element governing equations by applying the dqm to each element and 3 to assemble all the discretized element governing equations into the overall characteristic equations with the consideration of displacement and stress compatibility conditions between adjacent elements the annular and circular mindlin plate elements of differential quadrature dq are established the convergence characteristics of the proposed method are carefully investigated from the view points of element refinement and element grid refinement and some general regulations and suggestions on element division and element grid selection are provided a number of numerical examples are calculated and the results are compared with the corresponding exact solutions which exhibits high accuracy simplicity and applicability of the present method

vibration characteristics and transient response of shear deformable functionally graded plates in thermal environments free and forced vibration analyses for initially stressed functionally graded plates in thermal environment are presented material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents theoretical formulations are based on reddy s higher order shear deformation plate theory and include the thermal effects due to uniform temperature variation the plate is assumed to be clamped on two opposite edges with the remaining two others either free simply supported or clamped one dimensional differential quadrature technique galerkin approach and the modal superposition method are used to determine the transient response of the plate subjected to lateral dynamic loads comprehensive numerical results for silicon nitride stainless steel rectangular plates are presented in dimensionless tabular and graphical forms the roles played by the constituent volume fraction index temperature rise shape and duration of dynamic loads initial membrane stresses as well as the character of boundary conditions are studied the results reveal that when thermal effects are considered functionally graded plates with material properties intermediate to those of isotropic ones do not necessarily have intermediate natural frequencies and dynamic responses

application of differential quadrature to transport processes the methodology and numerical solution of problems concerning transport processes via the method of differential quadrature are presented application of the method is demonstrated by solving a simple one dimensional time dependent transient diffusion process involving an irreversible reaction without any flux across the end boundary in addition the same technique is used for the first time to the authors knowledge to solve a steady state problem for this purpose a convection diffusion problem involving an irreversible reaction is considered the demonstration is carried out in two ways 1 using the bellman et al technique which employs approximation formulas for higher order partial derivatives derived by iterating the linear quadrature approximation for the first order partial derivative and 2 using individual quadratures to approximate the partial derivatives of first as well as higher orders as suggested by mingle both approaches give the same results however the latter saves an appreciable amount of iterative computing effort despite the fact that it requires separate weighting coefficients for each individual quadrature since the technique of differential quadrature can produce solutions with sufficient accuracy even when using as few as three discrete points both the programming task and computational effort are alleviated considerably for these reasons the differential quadrature approach appears to be very practical in solving a variety of problems related to transport phenomena

pet image reconstruction using information theoretic anatomical priors we describe a nonparametric framework for incorporating information from co registered anatomical images into positron emission tomographic pet image reconstruction through priors based on information theoretic similarity measures we compare and evaluate the use of mutual information mi and joint entropy je between feature vectors extracted from the anatomical and pet images as priors in pet reconstruction scale space theory provides a framework for the analysis of images at different levels of detail and we use this approach to define feature vectors that emphasize prominent boundaries in the anatomical and functional images and attach less importance to detail and noise that is less likely to be correlated in the two images through simulations that model the best case scenario of perfect agreement between the anatomical and functional images and a more realistic situation with a real magnetic resonance image and a pet phantom that has partial volumes and a smooth variation of intensities we evaluate the performance of mi and je based priors in comparison to a gaussian quadratic prior which does not use any anatomical information we also apply this method to clinical brain scan data using fallypride a tracer that binds to dopamine receptors and therefore localizes mainly in the striatum we present an efficient method of computing these priors and their derivatives based on fast fourier transforms that reduce the complexity of their convolution like expressions our results indicate that while sensitive to initialization and choice of hyperparameters information theoretic priors can reconstruct images with higher contrast and superior quantitation than quadratic priors

on optimal selection of interior points for applying discretized boundary conditions in dq vibration analysis of beams and plates the direct implementation of derivative boundary conditions at edge points was demonstrated to be simple flexible and computationally efficient in the vibration analysis of beams and plates by using the differential quadrature dq method in the approach the discretized governing equations at certain interior points need to be replaced by the discretized boundary conditions naturally one may have questions as to whether the replacement points can be selected arbitrarily and how to choose the optimal replacement points this paper tries to answer these questions and its focus is to investigate systematically the effect of the replacement location on the accuracy of the numerical solution from the error distribution analysis of derivative approximation it is found that the optimal position for the replacement is the interior point just adjacent to the boundary this conclusion is confirmed by the numerical experiments of free vibration analysis for beams and plates with various boundary conditions

comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates this paper presents a comprehensive comparison study between the discrete singular convolution dsc and the well known global method of generalized differential quadrature gdq for vibration analysis so as to enhance the understanding of the dsc algorithm the dsc method is implemented through lagrange s delta sequence kernel dsc lk which utilizes local lagrange polynomials to calculate weighting coefficients whereas the gdq requires global ones moreover it is shown that the treatments of boundary conditions and the use of grid systems are different in the two methods comparison study is carried out on 21 rectangular plates of different combinations of simply supported s clamped c and transversely supported with nonuniform elastic rotational restraint e edges and five rectangular plates of mixed supporting edges some of which with a range of aspect ratios and rotational spring coefficients all the results of the dsc lk agree very well with both those in the literature and newly computed gdq results furthermore it is observed that the dsc lk performs much better for plates vibrating at higher order eigenfrequencies unlike the gdq the dsc lk is numerically stable for problems which require a large number of grid points

two new approximate methods for analyzing free vibration of structural components two approximate methods which have not previously been used for structural dynamics problems are applied to the free vibration analysis of various structural components the first method is a new version of the complementary energy method it is shown to be considerably more accurate than the conventional rayleigh and rayleigh schmidt methods when applied to spatially one dimensional free vibration problems prismatic and tapered bars prismatic beams and axisymmetric motion of circular membranes the second method is the differential quadrature method introduced by bellman and his associates it is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates

a new differential quadrature methodology for beam analysis and the associated differential quadrature element method this paper presents a computationally efficient and an accurate new methodology in differential quadrature analysis of beam elements the methodology would overcome the difficulties in boundary conditions implementations of fourth order differential equations encountered in such problems the methodology benefited from defining the second order derivatives along the boundaries as independent degrees of freedom would enable the differential quadrature method to exactly satisfy some types of boundary conditions where by most other conventional algorithms have to be satisfied approximately the weighting coefficients employed are not exclusive and any accurate and efficient method such as the generalized differential quadrature method may be used to produce the methods weighting coefficients by solving some typical stability deflection and frequency analysis beam problems and by comparing the results with those of exact solutions and or those of other methodologies accuracy convergency and efficiency of the methodology is asserted in order to generalize its application to large scale beam structures and to the cases with the discontinuity in loading conditions and geometry a new one dimensional differential quadrature element method formulation is presented and implemented

non axisymmetric bending of thin annular plates due to circumferentially distributed moments abstract the non linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied a von karman plate model is adopted to formulate the equations of motion free free boundary conditions have been applied at both inner and outer edges the matrix formulation of the differential quadrature method is used to discretise and solve the governing equations linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome however the proposed non linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non axisymmetric cylindrical deformation this is the case when at least one of two parameters reaches a critical value these parameters are the non dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius furthermore it is noted that free free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk hence the principal axes of curvature in the deformed configuration do not have a preferred orientation therefore the present work describes a shell possessing infinite identical equilibria having different yet no favoured direction and hence links to previous researches on neutrally stable structures

three dimensional asymptotic nonlocal elasticity theory for the free vibration analysis of embedded single walled carbon nanotubes abstract within the framework of three dimensional 3d nonlocal elasticity theory the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported single walled carbon nanotubes swcnts non embedded or embedded in an elastic medium using the multiple time scale method eringen s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation the interactions between the swcnt and its surrounding medium are modeled as a two parameter pasternak foundation model after performing a series of mathematical processes including nondimensionalization asymptotic expansion and successive integration etc the authors obtain recurrent sets of motion equations for various order problems the nonlocal classical shell theory cst is derived as a first order approximation of the current 3d nonlocal elasticity problem and the equations of motion for higher order problems retain the same differential operators as those of the nonlocal cst although with different nonhomogeneous terms the current asymptotic solutions for the natural frequency parameters of non embedded or embedded swcnts and their corresponding through thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature

application of a new differential quadrature methodology for free vibration analysis of plates a new methodology is introduced in the differential quadrature dq analysis of plate problems the proposed approach is distinct from other dq methods by employing the multiple boundary conditions in a different manner for structural and plate problems the methodology employs the displacement within the domain as the only degree of freedom whereas along the boundaries the displacements as well as the second derivatives of the displacements with respect to the co ordinate variable normal to the boundary in the computational domain are considered as the degrees of freedom for the problem employing such a procedure would facilitate the boundary conditions to be implemented exactly and conveniently in order to demonstrate the capability of the new methodology all cases of free vibration analysis of rectangular isotropic plates in which the conventional dq methods have had some sort of difficulty to arrive at a converged or accurate solution are carried out excellent convergence behaviour and accuracy in comparison with exact results and or results obtained by other approximate methods were obtained the analogous dq formulation for a general rectangular plate is derived and for each individual boundary condition the general format for imposing the given conditions is devised it must be emphasized that the computational efforts of this new methodology are not more than for the conventional differential quadrature methods copyright 2002 john wiley sons ltd

solving initial value problems by differential quadrature method part 1 first order equations in this paper the differential quadrature method is used to solve first order initial value problems the initial condition is given at the beginning of a time interval the time derivative at a sampling grid point within the time interval can be expressed as a weighted linear sum of the given initial condition and the function values at the sampling grid points within the time interval the order of accuracy and the stability property of the quadrature solutions depend on the locations of the sampling grid points it is shown that the order of accuracy of the quadrature solutions at the end of a time interval can be improved to 2n 1 or 2n if the n sampling grid points are chosen carefully in fact the approximate solutions are equivalent to the generalized pade approximations the resultant algorithms are therefore unconditionally stable with controllable numerical dissipation the corresponding sampling grid points are found to be given by the roots of the modified shifted legendre polynomials from the numerical examples the accuracy of the quadrature solutions obtained by using the proposed sampling grid points is found to be better than those obtained by the commonly used uniformly spaced or chebyshev gauss lobatto sampling grid points copyright 2001 john wiley sons ltd

implementation of clamped and simply supported boundary conditions in the gdq free vibration analysis of beams and plates abstract in this paper a new methodology for implementing the clamped and simply supported boundary conditions is presented for the free vibration analysis of beams and plates using the generalized differential quadrature gdq method the proposed approach directly substitutes the boundary conditions into the governing equations and is referred to as sbcge approach the sbcge approach is presented to overcome the drawbacks of previous approaches in treating the boundary conditions a comparison of the sbcge approach with the method of modifying weighting coefficient matrices mwcm is made by their application to the vibration analysis of beams and plates with combinations of simply supported and clamped boundary conditions some details of the gdq method are also described in the paper

imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method one of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions there may be multiple boundary conditions involving higher order derivatives at the boundary points the boundary conditions can be imposed by modifying the weighting coefficient matrices directly however the existing method is not robust and is known to have many limitations in this paper a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations the given boundary conditions are imposed exactly furthermore it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established the other sampling grid points with no associated boundary conditions are not essential copyright 2002 john wiley sons ltd

application of generalized differential quadrature method to structural problems summary this paper presents the first endeavour to exploit a generalized differential quadrature method as an accurate efficient and simple numerical technique for structural analysis firstly drawbacks existing in the method of differential quadrature dq are evaluated and discussed then an improved and simpler generalized differential quadrature method gdq is introduced to overcome the existing drawback and to simplify the procedure for determining the weighting coefficients subsequently the generalized differential quadrature is systematically employed to solve problems in structural analysis numerical examples have shown the superb accuracy efficiency convenience and the great potential of this method numerical approximation methods for solving partial differential equations have been widely used in various engineering fields classical techniques such as finite element and finite difference methods are well developed and well known these methods can provide very accurate results by using a large number of grid points however in a large number of cases reasonably approximate solutions are desired at only a few specific points in the physical domain in order to get results even at or around a point of interest with acceptable accuracy conventional finite element and finite difference methods still require the use of a large number of grid points consequently the requirement for computer capacity is often unnecessarily large in such cases in seeking an alternate numerical method using fewer grid points to find results with acceptable accuracy the method of differential quadrature dq was introduced by bellman et a the method of dq is a global approximate method this method is based on the ideas that the derivative of a function with respect to a co ordinate direction can be expressed as a weighted linear sum of all the function values at all mesh points along that direction and that a continuous function can be approximated by a higher order polynomial in the overall domain the dq method differs from the finite element method fem in two aspects firstly the fem uses lower order polynomials to approximate a function on a local element level while the dq method approximates a function on the global area using higher order polynomials secondly the dq method directly approximates the derivatives of a function at a point while the fem approximates a function over a local element and the derivatives can then be derived from the approximate function in this aspect the dq method is more similar to the finite difference method fdm however the fdm is also a local approximation method based on lower order polynomial approximation in fact it can be shown that the fdm is just a special case of the dq method where it is applied locally on the range xi xi l owing to the higher order

thermally induced dynamic instability of laminated composite conical shells abstract thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method the laminated composite conical shells are subjected to static and periodic thermal loads the linear instability approach is adopted in the present study a set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs the formulation begins with three dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium after proper nondimensionalization asymptotic expansion and successive integration we obtain recursive sets of differential equations at various levels the method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible using the method of differential quadrature and bolotin s method and imposing the orthonormality and solvability conditions on the present asymptotic formulation we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner the principal instability regions of cross ply conical shells with simply supported simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory

differential quadrature and partial differential equations some numerical results abstract the differential quadrature method is used to solve partial differential equations arising in chemical reactor design the results are compared with the finite difference approach although only 9th order quadrature is used or only 9 grid points are calculated as compared to the calculation of 480 grid points in difference methods a reasonably good accuracy is obtained with differential quadrature

application of differential quadrature to static analysis of structural components the numerical technique of differential quadrature for the solution of linear and non linear partial differential equations first introduced by bellman and his associates is applied to the equations governing the deflection and buckling behaviour of one and two dimensional structural components separate transformations are used for higher order derivatives as suggested by mingle thus extending the method to treat fourth order equations and to include multiple boundary conditions in the respective co ordinate directions results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods the application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort

dynamic analysis of a spinning timoshenko beam by the differential quadrature method thedynamiccharacteristicsofaspinningtimoshenkobeamareinvestigatedbyusingthedifferentialquadrature method dqm the beam is subject to any combination of free simply supported clamped and elastically supported boundary conditions the weighting matrices in the differential quadrature formulation are modie ed to incorporate classical boundary conditions of the beam whereas boundary conditions of elastic supports are incorporated into the discretized equations numerical results of the spinning timoshenko beam obtained by the dqm are compared with the exact solutions orresults obtained by the e nite element method the results show the high accuracy and efe ciency of the differential quadrature method nomenclature a cross sectional area of beam c b damping matrix of elastic supports cxx cyy damping coefe cients e young s modulus g shear modulus g gyroscopic matrix i transverse moment of inertia jp polar mass moment of inertia k stiffness matrix k b stiffness matrix of elastic supports kxx kyy spring constants l