on the numerical condition of bernstein bezier subdivision processes the linear map m that takes the bernstein coefficients of a polynomial p t on a given interval a b into those on any subinterval a b is specified by a stochastic matrix which depends only on the degree n of p t and the size and location of a t relative to a b we show that in the 11 110j norm the condition number of m has the simple form k m 2f max u7 vo where u5 m a b a and v b m b a are the barycentric coordinates of the subinterval midpoint m 2 a b and f denotes the zoom factor b a b a of the subdivision map this suggests a practical rule of thumb in assessing how far bezier curves and surfaces may be subdivided without exceeding prescribed worst case bounds on the typical errors in their control points the exponential growth of koo m with n also argues forcefully against the use of high degree forms in computer aided geometric design applications

solution of poisson and laplace equations by quadrilateral quadrature element the quadrature element method qem is a new concept in numerical methods which was introduced only recently a quadrilateral quadrature element is developed in the paper and applied to the solution of two dimensional potential problems governed by poisson or laplace equations the results of three examples are in good agreement with the available solutions it is shown that the present quadrilateral quadrature element is very efficient and effective most significantly it can be employed to solve differential equations of physical problems with irregular geometry and complex boundary conditions

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

free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule the generalized differential quadrature rule proposed recently by these authors is applied here to the free vibration analyses of solid circular plates with radially varying thickness and elastic restraints the thickness can vary radially in any given continuous form and it varies exponentially and linearly in this work two regularity conditions corresponding to the plate center are expressed in explicit formulae since they have been inexactly or even wrongly expressed in the literature such errors are pointed out in the application of the differential quadrature techniques and in the expression of the regularity conditions at the plate center numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

exploiting parity in converting to and from bernstein polynomials and orthogonal polynomials abstract the coefficients of a polynomial in the bernstein basis can be converted to the coefficients of a legendre or chebyshev series by a simple matrix vector multiply at a cost of o 2 n 1 2 operations where n is the degree of the polynomial in this note we show that by exploiting parity with respect to the center of the interval x 0 1 is possible to halve the cost in d dimensions with a tensor product basis the savings are a factor of two independent of d

the weighted dual functionals for the univariate bernstein basis abstract we find an explicit formula for the weighted dual functions of the bernstein polynomials with respect to the jacobi weight function using the usual inner product in the hilbert space l 2 0 1 we define the weighted dual functionals of the bernstein polynomials which are used to find the coefficients in the least squares approximation

a comparison of implicit time integration methods for nonlinear relaxation and diffusion several time integration methods for nonlinear systems are compared all of the time discretizations are based on the method but differ in their treatment of the implicit nonlinear terms one method converges the implicit nonlinear terms to a small tolerance and is often referred to as nonlinearly consistent nc newton s method or its approximation newton krylov is used to converge the nonlinearities the other methods considered are linearized and comparisons are made for a relaxation problem and a radiation diffusion problem the linearized one step method that uses the full jacobian is shown to have similar accuracy as nc methods the lagged linearization method and an extension that is second order accurate are also studied a trunction error analysis complements the numerical results for the relaxation problem it is shown that each of the second order accurate linearized methods may be more accurate than an nc method depending on the degree of nonlinearity in the problem for the radiation diffusion problem in general the nc method is most accurate and allows a larger time step however the linearized methods perfom surprisingly well

direct estimation of density functionals using a polynomial basis a number of fundamental quantities in statistical signal processing and information theory can be expressed as integral functions of two probability density functions such quantities are called density functionals as they map density functions onto the real line for example information divergence functions measure the dissimilarity between two probability density functions and are useful in a number of applications typically estimating these quantities requires complete knowledge of the underlying distribution followed by multidimensional integration existing methods make parametric assumptions about the data distribution or use nonparametric density estimation followed by high dimensional integration in this paper we propose a new alternative we introduce the concept of data driven basis functions functions of distributions whose value we can estimate given only samples from the underlying distributions without requiring distribution fitting or direct integration we derive a new data driven complete basis that is similar to the deterministic bernstein polynomial basis and develop two methods for performing basis expansions of functionals of two distributions we also show that the new basis set allows us to approximate functions of distributions as closely as desired finally we evaluate the methodology by developing data driven estimators for the kullback leibler divergences and the hellinger distance and by constructing empirical estimates of tight bounds on the bayes error rate

vibration analysis of non circular curved panels by the differential quadrature method free vibration characteristics of cantilever non circular curved panels are analyzed by using the differential quadrature method dqm in this paper the equations of motion of a curved panel are based on the love s hypothesis and are expressed in an orthogonal curvilinear co ordinate system by applying the differential quadrature formulation and the proposed modified relationships for specified boundary conditions the free vibration equations of motion of the curved panel are transformed to a set of algebraic equations natural frequencies of a cantilever flat plate and a circular curved panel are obtained for verifying the applicability of the present approach good convergent trend and accuracy are observed effects of shallowness thickness and aspect ratios on the natural frequencies of a cantilever curved panel are also investigated furthermore natural frequencies of parabolic curved panels are obtained in all cases studied the efficiency and convenience of the dqm are illustrated

stability and accuracy of differential quadrature method in solving dynamic problems in this paper the differential quadrature method is used to solve dynamic problems governed by second order ordinary differential equations in time the legendre radau chebyshev chebyshev gauss lobatto and uniformly spaced sampling grid points are considered besides two approaches using the conventional and modified differential quadrature rules to impose the initial conditions are also investigated the stability and accuracy properties are studied by evaluating the spectral radii and truncation errors of the resultant numerical amplification matrices it is found that higher order accurate solutions can be obtained at the end of a time step if the gauss and radau sampling grid points are used however the conventional approach to impose the initial conditions in general only gives conditionally stable time step integration algorithms unconditionally stable algorithms can be obtained if the modified differential quadrature rule is used unfortunately the commonly used chebyshev gauss lobatto sampling grid points would not generate unconditionally stable algorithms

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

evaluation of the derivative of a polynomial in bernstein form in geometric modelling and computer aided design a polynomial is usually represented in bernstein form forward error analysis for the computation of the derivative of a polynomial in bernstein form is performed the corresponding running error is also analyzed we have compared our results with those corresponding to a polynomial represented in power basis and using legendre and chebyshev forms since the conversions between the bernstein and the power bases and between bernstein and chebyshev polynomials were already studied in the literature for the sake of completeness the conversion between bernstein and chebyshev polynomials of the first kind is analyzed in an appendix

a test based on conversion to the bernstein polynomial basis for an interval to be free of zeros applicable to polynomials in chebyshev form and to transcendental functions approximated by chebyshev series abstract polynomials of high degree are much less vulnerable to roundoff error when expressed as truncated chebyshev series rather than the usual power series form recent articles have developed subdivision methods in which all real roots on the canonical chebyshev interval x 1 1 are found by subdividing the interval and finding the roots of separate chebyshev series of moderate degree on each subdomain this strategy can be applied either to polynomials or to transcendental functions if the latter are analytic on the search interval and thus have rapidly convergent chebyshev polynomial approximations the last step is to compute the eigenvalues of the chebyshev frobenius companion matrix for each local polynomial here we propose a simple strategy for flagging some subdomains as zero free so that eigensolving can be omitted the test requires conversion of the polynomial from chebyshev form to a bernstein polynomial basis the interval is zero free if all coefficients in the bernstein basis are of the same sign we give the conversion matrices for various small and moderate n and quote rababah s formulas for general n we show how to exploit parity so as to halve the cost of the conversion for large n to about n 2 floating point operations we show that the conversion matrices have condition numbers that are approximately 5 8 2 n

analysis of a monte carlo method for nonlinear radiative transfer it has recently been proved that solutions of nonlinear radiative transfer problems satisfy a maximum principle andreev kozmanov and rachilov u s s r comput math math phys 23 104 1983 mercier siam j math anal in press in this article it is shown that monte carlo solutions of such problems obtained using the method of fleck and cummings j comput phys 8 313 1971 must satisfy this maximum principle for sufficiently small time steps but can violate it for sufficiently large time steps analyses of the frequency dependent and grey cases are given and a numerical solution violating the maximum principle is discussed

nonlinear variants of the tr bdf2 method for thermal radiative diffusion we apply the trapezoidal bdf2 tr bdf2 temporal discretization scheme to nonlinear grey radiative diffusion this is a scheme that is not well known within the radiation transport community but we show that it offers many desirable characteristics relative to other second order schemes several nonlinear variants of the tr bdf2 scheme are defined and computationally compared with the crank nicholson scheme it is found for our test problems that the most accurate tr bdf2 schemes are those that are fully iterated to nonlinear convergence but the most efficient tr bdf2 scheme is one based upon a single newton iteration it is also shown that neglecting the contributions to the jacobian matrix from the cross sections which is often done due to a lack of smooth interpolations for tabular cross section data has a significant impact upon efficiency

second order discretization in space and time for radiation hydrodynamics second order accurate discretizations for radiation hydrodynamics are currently an area of interest in the high energy density laboratory physics and astrophysics communities second order methods used to solve the hydrodynamics equations and second order methods used to solve the radiation transport equation often differ fundamentally making it difficult to combine them in a second order manner here we present an implicitexplicit imex method for solving the 1d equations of radiation hydrodynamics that is second order accurate in space and time our radiation hydrodynamics model consists of the 1d euler equations coupled with a gray radiation s2 approximation our rh method combines the muscl hancock method for solving the euler equations with the tr bdf2 time integration scheme and the linear discontinuous galerkin finite element spatial discretization scheme for the s2 radiation equations the muscl hancock method is commonly used for hydrodynamic calculations and the linear discontinuous galerkin scheme is the standard for the sn equations of radiative transfer while somewhat similar these schemes vary fundamentally with respect to the treatment of spatial slopes we address the challenges inherent to coupling these different numerical methods and demonstrate how these challenges can be overcome using the method of manufactured solutions we show that the method is second order accurate in space and time for both the equilibrium diffusion and streaming limit and we show that the method is capable of computing radiative shock solutions accurately by comparing our results with semi analytic solutions

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 balanced approximations for time integration of multiple time scale systems the effect of various numerical approximations used to solve linear and nonlinear problems with multiple time scales is studied in the framework of modified equation analysis mea first mea is used to study the effect of linearization and splitting in a simple nonlinear ordinary differential equation ode and in a linear partial differential equation pde several time discretizations of the ode and pde are considered and the resulting truncation terms are compared analytically and numerically it is demonstrated quantitatively that both linearization and splitting can result in accuracy degradation when a computational time step larger than any of the competing fast time scales is employed many of the issues uncovered on the simple problems are shown to persist in more realistic applications specifically several differencing schemes using linearization and or time splitting are applied to problems in nonequilibrium radiation diffusion magnetohydrodynamics and shallow water flow and their solutions are compared to those using balanced time integration methods

preconditioning strategies for fully implicit radiation diffusion with material energy transfer in this paper we present a comparison of four preconditioning strategies for jacobian systems arising in the fully implicit solution of radiation diffusion coupled with material energy transfer the four preconditioning methods are block jacobi schur complement and operator splitting approaches that split the preconditioner solve into two steps one splitting method includes the coupling of the radiation and material fields that appears in the matrix diagonal in the first solve and the other method puts this coupling into the second solve all preconditioning approaches use multigrid methods to invert blocks of the matrix formed from the diffusion operator the schur complement approach is clearly seen to be the most effective for a large range of weightings between the diffusion and energy coupling terms in addition tabulated opacity studies were conducted where again the schur preconditioner performed well last a parallel scaling study was done showing algorithmic scalability of the schur preconditioner

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

static analysis of frame structures by the differential quadrature element method in this paper a new version of differential quadrature method dqm is proposed and then extended to analyse frame structures the new method called the differential quadrature element method dqem retains all advantages of the earlier version of the differential quadrature method and overcomes some critical shortcomings existing in the original dqm the proposed method is however different from the quadrature element method qem proposed earlier by striz et al the methodology is explained in details herein via a differential quadrature beam element and some numerical examples are given to show the efficiency of the present method 1997 by john wiley sons ltd

bezier curves in the modification of boundary integral equations bie for potential boundary values problems the paper presents a modification of the classical boundary integral equation method biem for two dimensional potential boundary values problems the proposed modification consists in describing the boundary geometry by means of bezier curves as a result of this analytical modification of the biem a new parametric integral equation system pies was obtained the kernels of these equations include the geometry of the boundary this new pies is no longer defined on the boundary as in the case of the biem but on the straight line for any given domain the solution of the new pies does not require a boundary discretization since it can be reduced merely to an approximation of boundary functions to solve this pies a pseudospectral method has been proposed and the results obtained were compared with exact solutions

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

asymptotic analysis of discrete schemes for non equilibrium radiation diffusion motivated by providing well behaved fully discrete schemes in practice this paper extends the asymptotic analysis on time integration methods for non equilibrium radiation diffusion in 2 to space discretizations therein studies were carried out on a two temperature model with larsen s flux limited diffusion operator both the implicitly balanced ib and linearly implicit li methods were shown asymptotic preserving in this paper we focus on asymptotic analysis for space discrete schemes in dimensions one and two first in construction of the schemes in contrast to traditional first order approximations asymmetric second order accurate spatial approximations are devised for flux limiters on boundary and discrete schemes with second order accuracy on global spatial domain are acquired consequently then by employing formal asymptotic analysis the first order asymptotic preserving property for these schemes and furthermore for the fully discrete schemes is shown finally with the help of manufactured solutions numerical tests are performed which demonstrate quantitatively the fully discrete schemes with ib time evolution indeed have the accuracy and asymptotic convergence as theory predicts hence are well qualified for both non equilibrium and equilibrium radiation diffusion provide ap fully discrete schemes for non equilibrium radiation diffusion propose second order accurate schemes by asymmetric approach for boundary flux limiter show first order ap property of spatially and fully discrete schemes with ib evolution devise subtle artificial solutions verify accuracy and ap property quantitatively ideas can be generalized to 3 dimensional problems and higher order implicit schemes

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 simple matrix form for degree reduction of bezier curves using chebyshev bernstein basis transformations we use the matrices of transformations between chebyshev and bernstein basis and the matrices of degree elevation and reduction of chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of bezier curves with respect to the weighted l2 norm for the interval 0 1 using the weight function w x 1 4x 4x2 the error of the degree reduction scheme is given and the degree reduction with continuity conditions is also considered

an efficient differential quadrature methodology for free vibration analysis of arbitrary straight sided quadrilateral thin plates in this paper a new differential quadrature dq methodology is employed to study free vibration of irregular quadrilateral straight sided thin plates a four nodded super element is used to map the irregular physical domain into a square domain in the computational domain second order transformation schemes with relative ease and less computation are employed to transform the fourth order governing equation of thin plates between the two domains the only degree of freedom within the domain is the displacement whereas along the boundaries the displacement as well as the second order derivative of the displacement with respect to associated normal co ordinate variable in computational domain are the two degrees of freedom implementing the method the formulation for the dq method for the free vibration analysis of plates of straight sided shapes was presented together with the implementation procedure for the different boundary conditions to demonstrate the accuracy convergency and stability of the new methodology detail studies are made on isotropic plates at acute angles with different geometries boundary and loading conditions including dq free edge boundary condition implementations accurate results even with fewer degrees of freedom than for those of comparable numerical algorithms were achieved

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

vibration analysis of edge cracked beam on elastic foundation with axial loading using the differential quadrature method abstract the eigenvalue problems of clamped free and hinged hinged bernoulli euler beams on elastic foundation with a single edge crack axial loading and excitation force were numerically formulated using the differential quadrature method dqm appropriate boundary conditions accompanied the dqm to transform the partial differential equation of a bernoulli euler beam with a single edge crack into a discrete eigenvalue problem the dqm results for the natural frequencies of cracked beams agree well with other literature values the sampling point number effect the location of the crack effect and the depth of the crack effect on the accuracy variation of calculated natural frequencies are presented by using two elements in this work the effects of axial loading foundation stiffness opening crack and closing crack are also studied

modified jacobi bernstein basis transformation and its application to multi degree reduction of bezier curves this paper reports new modified jacobi polynomials mjps we derive the basis transformation between mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the new polynomials together with the geometrical insight of bernstein polynomials the mjps with indexes corresponding to the number of endpoints constraints are the natural basis functions for least square approximation of bezier curves using mjps leads us to deal with the constrained jacobi polynomials and the unconstrained jacobi polynomials as orthogonal polynomials the mjps are automatically satisfying the homogeneous boundary conditions thereby the main advantage of using mjps in multi degree reduction of bezier curves on computer aided geometric design cagd is that the constraints in cagd are also satisfied and that decreases the steps of multi degree reduction algorithm several numerical results for the multi degree reduction of bezier curves on cagd are given we establish and prove the basis transformation between the modified jacobi polynomials mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the mjps with the geometrical insight of the bernstein form the mjps with indexes corresponding to the number of endpoint constraints are the natural basis functions for least square approximation of bezier curves several numerical results are considered

weighted dual functions for bernstein basis satisfying boundary constraints in this paper we consider the issue of dual functions for the bernstein basis which satisfy boundary conditions the jacobi weight function with the usual inner product in the hilbert space are used some examples of the transformation matrices are given some figures for the weighted dual functions of the bernstein basis with respect to the jacobi weight function satisfying boundary conditions are plotted we discuss special cases of the jacobi weight function as the legendre weight function and the chebyshev weight functions of the first second and third kinds

differential quadrature domain decomposition method for a class of parabolic equations in this paper the differential quadrature method dqm and the domain decomposition method ddm are combined to form the differential quadrature domain decomposition method dqddm in which the boundary reduction technique brm is adopted the dqddm is applied to a class of parabolic equations which have discontinuity in the coefficients of the equation or weak discontinuity in the initial value condition two numerical examples belonging to this class are computed it is found that the application of this method to the above mentioned problems is seen to lead to accurate results with relatively small computational effort

gauss lobatto to bernstein polynomials transformation the aim of this paper is to transform a polynomial expressed as a weighted sum of discrete orthogonal polynomials on gauss lobatto nodes into bernstein form and vice versa explicit formulas and recursion expressions are derived moreover an efficient algorithm for the transformation from gauss lobatto to bernstein is proposed finally in order to show the robustness of the proposed algorithm experimental results are reported

the generalized differential quadrature rule for initial value differential equations abstract the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to solve initial value differential equations of the 2nd to 4th order differential quadrature expressions are derived based on the gdqr for these equations the hermite interpolation functions are used as trial functions to obtain the explicit weighting coefficients for an easy and efficient implementation of the gdqr the numerical solutions for example problems demonstrate that the gdqr has high efficiency and accuracy a detailed discussion on the present method is presented by comparing with other existing methods the present method can be extended to other types of differential equation systems

on the degree elevation of bernstein polynomial representation the polynomials determined in the bernstein bezier basis enjoy considerable popularity in computer aided design cad applications the common situation in these applications is that polynomials given in the basis of degree n have to be represented in the basis of higher degree the corresponding transformation algorithms are called algorithms for degree elevation of bernstein polynomial representations these algorithms are only then of practical importance if they do not require the ill conditioned conversion between the bernstein and the power basis we discuss all the algorithms of this kind known in the literature and compare them to the new ones we establish some among the latter are better conditioned and not more expensive than the currently used ones all these algorithms can be applied componentwise to vector valued polynomial bezier representations of curves or surfaces

legendre bernstein basis transformations abstract the bernstein form of a polynomial offers valuable insight into its geometrical behavior and has thus won widespread acceptance as the basis for bezier curves and surfaces for least squares approximation problems on the other hand the use of orthogonal bases such as the legendre polynomials permits simple and efficient constructions for convergent sequences of approximants we derive the transformation matrices that map the bernstein and legendre forms of a degree n polynomial on 0 1 into each other and examine the stability of this linear map in the p 1 and norms the condition number of the legendre bernstein transformation matrix grows at a significantly slower rate with n than in the well studied power bernstein case and at a dramatically slower rate than for other common e g bernstein hermite or power hermite basis conversions the utility of legendre representations in approximation problems and their relatively stable transformation to bernstein bezier form argue for more widespread applications of legendre methods in cagd algorithms

differential quadrature for multi dimensional problems the technique of differential quadrature for the solution of partial differential equations introduced by bellman et al is extended and generalized to encompass partial differential equations involving multiple space variables approximation formulae for a variety of first and second order partial derivatives and typical weighting coefficients are presented application of these formulae is demonstrated on the solution of the convection diffusion equation for the two and three dimensional space dependent cases and for both the transient and steady state dispersion of inert neutrally buoyant pollutants from continuous sources into an unbounded atmosphere

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

application of chebyshev ii bernstein basis transformations to degree reduction of bezier curves a polynomial curve on 0 1 can be expressed in terms of bernstein polynomials and chebyshev polynomials of the second kind we derive the transformation matrices that map the bernstein and chebyshev coefficients into each other and examine the stability of this linear map in the p 1 and norms the condition number of the chebyshev bernstein transformation matrix grows at a significantly slower rate with n than in the power bernstein case and the rate is very close somewhat faster to the legendre bernstein case using the transformation matrices we present a method for the best multi degree reduction with respect to the t t 2 weighted square norm for the unconstrained case which is further developed to provide a good approximation to the best multi degree reduction with constraints of endpoints continuity of orders r s r s 0 this method has a quadratic complexity and may be ill conditioned when it is applied to the curves of high degree we estimate the posterior l 1 error bounds for degree reduction

free vibration analysis of isosceles triangular mindlin plates by the triangular differential quadrature method the triangular differential quadrature method is applied to the free flexural vibration of isosceles triangular mindlin plates the first six frequencies are sought and the convergence of triangular differential quadrature method in vibrational analysis of triangular mindlin plates is examined in comparison with available results good to excellent agreement is achieved for triangular plates with combinations of clamped and simply supported boundary conditions the implementation of various other boundary conditions in triangular differential quadrature analysis is also discussed and the sensitivity of solution to the implementation of corner conditions in vibrational analysis is highlighted

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

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

combining differential quadrature method with simulation technique to solve non linear differential equations nowadays most of the ordinary differential equations odes can be solved by modelica based approaches such as matlab simulink dymola and labview which use simulation technique st however these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain this limitation is one of the most important drawbacks of the st another method of solution differential quadrature method dqm leads to very accurate results using only a few grids on the domain on the other hand dqm is not flexible for the solution of non linear odes and it is not so easy to impose multiple conditions on the same location for these reasons the author aims to eliminate the mentioned disadvantages of the simulation technique st and dqm using favorable characteristics of each method in the other this work aims to show how the combining method cm works simply by solving some non linear problems and how the cm gives more accurate results compared with those of other methods copyright 2008 john wiley sons ltd

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

the generalized differential quadrature rule for fourth order differential equations the generalized differential quadrature rule gdqr proposed here is aimed at solving high order differential equations the improved approach is completely exempted from the use of the existing point technique by applying multiple conditions in a rigorous manner the gdqr is used here to static and dynamic analyses of bernoulli euler beams and classical rectangular plates numerical error analysis caused by the method itself is carried out in the beam analysis independent variables for the plate are first defined the explicit weighting coefficients are derived for a fourth order differential equation with two conditions at two different points it is quite evident that the gdqr expressions and weighting coefficients for two dimensional problems are not a direct application of those for one dimensional problems the gdqr are implemented through a number of examples good results are obtained in this work copyright 2001 john wiley sons ltd

a moving mesh finite difference method for equilibrium radiation diffusion equations an efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions the method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations the mesh adaptation is controlled through a hessian based monitor function and the so called equidistribution and alignment principles several challenging issues in the numerical solution are addressed particularly the radiation diffusion coefficient depends on the energy density highly nonlinearly this nonlinearity is treated using a predictor corrector and lagged diffusion strategy moreover the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations numerical examples with multi material multiple spot concentration situations are presented numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy it is also shown that a two level mesh movement strategy can significantly improve the efficiency of the computation

the preconditioned jacobian free newton krylov methods for nonequilibrium radiation diffusion equations in general it is difficult to use the newton krylov methods to solve the large scale multi variable nonequilibrium reaction diffusion systems in this paper by employing two new semi implicit discretization schemes to construct the preconditioners the preconditioned newton krylov methods are presented to solve the multidimensional problems these methods cannot only improve the number of iterations but also speed up the convergence of solutions numerical results are given to illustrate the effectiveness

numerical analysis of time integration errors for nonequilibrium radiation diffusion numerical analysis of time integration errors for nonequilibrium radiation diffusion is considered two first order implicit time integration methods are studied asymptotic analysis and modified equation analysis are applied to both time integration methods numerical experiments are used to highlight the results of the analysis asymptotic analysis is used to highlight the source of temperature spiking when a hot radiation wave propagates into a cold material modified equation analysis is used to provide insight into the thermal wave speed coming from the two different first order methods

on the solution of the thomas fermi equation by differential quadrature on utilise la methode de quadrature differentielle pour resoudre l equation de thomas fermi