new block quadrature rules for the approximation of matrix functions abstract golub and meurant have shown how to use the symmetric block lanczos algorithm to compute block gauss quadrature rules for the approximation of certain matrix functions we describe new block quadrature rules that can be computed by the symmetric or nonsymmetric block lanczos algorithms and yield higher accuracy than standard block gauss rules after the same number of steps of the symmetric or nonsymmetric block lanczos algorithms the new rules are block generalizations of the generalized averaged gauss rules introduced by spalevic applications to network analysis are presented

ultraspherical gauss kronrod quadrature is not possible for lambda 3 with the help of a new representation of the stieltjes polynomial it is shown by using bessel functions that the stieltjes polynomial with respect to the ultraspherical weight function with parameter lambda has only few real zeros for lambda 3 and sufficiently large n since the nodes of the gauss kronrod quadrature formulae subdivide into the zeros of the stieltjes polynomial and the gaussian nodes it follows immediately that gauss kronrod quadrature is not possible for lambda 3 on the other hand for lambda 3 and sufficiently large n even partially positive gauss kronrod quadrature is possible

generalized anti gauss quadrature rules gauss quadrature is a popular approach to approximate the value of a desired integral determined by a measure with support on the real axis laurie proposed an n 1 point quadrature rule that gives an error of the same magnitude and of opposite sign as the associated n point gauss quadrature rule for all polynomials of degree up to 2 n 1 this rule is referred to as an anti gauss rule it is useful for the estimation of the error in the approximation of the desired integral furnished by the n point gauss rule this paper describes a modification of the n 1 point anti gauss rule that has n k nodes and gives an error of the same magnitude and of opposite sign as the associated n point gauss quadrature rule for all polynomials of degree up to 2 n 2 k 1 for some k 1 we refer to this rule as a generalized anti gauss rule an application to error estimation of matrix functionals is presented

suboptimal kronrod extension formulae for numerical quadrature we consider cases where the stieltjes polynomial associated with a gaussian quadrature formula has complex zeros in such cases a kronrod extension of the gaussian rule does not exist a method is described for modifying the stieltjes polynomial so that the resulting polynomial has no complex zeros the modification is performed in such a way that the kronrod type extension rule resulting from the addition of then 1 zeros of the modified stieltjes polynomial to the original knots of the gaussian rule has only slightly lower degree of precision than normally achieved when the kronrod extension rule exists as examples of the use of the method we present some new formulae extending the classical gauss hermite quadrature rules we comment on the limited success of the method in extending gauss laguerre rules and show that several modified extensions of the gauss gegenbauer formulae exist in cases where the standard kronrod extension does not

numerical integration of functions with a sharp peak at or near one boundary using mo bius transformations abstract a class of coordinate transformations depending on a single parameter is studied as quadrature tool working rules for the choice of the parameter are proposed numerical tests for the method are presented they show that these coordinate transformations when combined with gauss legendre quadrature rules are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of the interval of integration a method to combine the transformations with automatic quadrature routines is also proposed it seems to be useful for the numerical evaluation of integrals with the same kind of integrand behavior

kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients abstract we continue with analyzing quadrature formulas of high degree of precision for computing the fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials started recently by bojanov and petrova quadrature formulae for fourier coefficients j comput appl math 231 2009 378 391 and we extend their results construction of new gaussian quadrature formulas for the fourier coefficients of a function based on the values of the function and its derivatives is considered we prove the existence and uniqueness of kronrod extensions with multiple nodes of standard gaussian quadrature formulas with multiple nodes for several weight functions in order to construct some new generalizations of quadrature formulas for the fourier coefficients for the quadrature formulas for the fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct kronrod extensions with multiple nodes and highest algebraic degree of precision for this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas

an imbedded family of fully symmetric numerical integration rules we consider an imbedded family of m 1 generator 2m 1 degree fully symmetric rules for numerical integration over an n dimensional hypercube the rules are shown to exist in theory for arbitrary degree in practice the rules are most economical and useful for 3 leqq m leqq 6 3 leqq n leqq 3m

anti gaussian quadrature formulas an anti gaussian quadrature formula is an n 1 point formula of degree 2n 1 which integrates polynomials of degree up to 2n 1 with an error equal in magnitude but of opposite sign to that of the n point gaussian formula its intended application is to estimate the error incurred in gaussian integration by halving the difference between the results obtained from the two formulas we show that an anti gaussian formula has positive weights and that its nodes are in the integration interval and are interlaced by those of the corresponding gaussian formula similar results for gaussian formulas with respect to a positive weight are given except that for some weight functions at most two of the nodes may be outside the integration interval the anti gaussian formula has only interior nodes in many cases when the kronrod extension does not and is as easy to compute as the n 1 point gaussian formula

on the computation of fourier transforms of singular functions abstract in this paper we consider a method for the computation of finite fourier transforms of functions having endpoint singularities of algebraic or logarithmic type this method may be considered as a modification of clenshaw curtis quadrature the nonoscillatory and nonsingular part of the integrand is replaced by a truncated chebyshev series approximation and the integral is then evaluated using recurrence relations the numerical stability of these recurrence relations is investigated

on the remainder term of gauss radau quadratures for analytic functions for analytic functions the remainder term of gauss radau quadrature formulae can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes r 1 for the chebyshev weight function of the second kind starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined the corresponding gautschi s conjecture from on the remainder term for analytic functions of gauss lobatto and gauss radau quadratures rocky mountain j math 21 1991 209 226 is proved

stopping functionals for gaussian quadrature formulas abstract gaussian formulas are among the most often used quadrature formulas in practice in this survey an overview is given on stopping functionals for gaussian formulas which are of the same type as quadrature formulas i e linear combinations of function evaluations in particular methods based on extended formulas like the important gauss kronrod and patterson schemes and methods which are based on gaussian nodes are presented and compared

polynomial approximation of cm functions by means of boundary values and applications a survey we collect classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only the problem is considered from the point of view of the existence of explicit formulas interpolation to boundary data bounds for the remainder and convergence of the polynomial series applications to some problems of numerical analysis are pointed out such as nonlinear equations numerical differentiation and integration formulas special associated differential boundary value problems some polynomial expansions for smooth enough functions defined in rectangles or in triangles of r 2 as well as in cuboids or in tetrahedrons in r 3 and their applications are also discussed

interfacing of fluid and structural models via innovative structural boundary element method an innovative structural boundary element method bem solver is developed for interfacing the computational fluid dynamics cfd and computational structural dynamics csd grids formulated as a solid mechanics problem with a minimum strain energy requirement the bem solver generates a truly three dimensional universal spline matrix the universal spline matrix is a vector operator that includes the coupling of displacements and forces along all axes based on a similar formulation an exterior bem solver is also developed to account for the flowfield grid deformation thus the bem solver allows a unified treatment of the displacement and force transformation for cfd csd interfacing as well as the computation of the flowfield grid deformation the solution procedure is fully automated and no additional model generation is required therefore is ideally suitable for computational aeroelasticity and multidisciplinary optimization applications

modified optimal quadrature extensions optimal extensions of quadrature rules are of importance in the construction of automatic integrators but many sequences fail to exist in usable form the paper considers some techniques for overcoming the problem of inextensibility with a minimal effect on integrating efficiency a modification to the extension procedure proposed recently by begumisa and robinson is shown to be just a special case of the standard theory for quadrature extension some illustrative examples are included

practical error estimation in numerical integration theoretical error estimates for quadrature rules involve quantities that are usually difficult if not impossible to obtain in practice various heuristics to obtain computable error estimates are compared by calculating their performance profiles on the lyness family of integrands two sets of tests are used corresponding to adaptive and single rule quadrature in the single rule case gaussian quadrature with error estimate provided by dropping one point from the formula performs best in the adaptive case the best heuristic is non linear extrapolation based on gaussian quadrature

on splitting the area under curves ii two area splitting problems involving real valued functions of a real variable are investigated the second of these is essentially equivalent to finding all functions a c1 0 r with 0 a x x which satisfy the functional differential equation a a x a x x for x 0 r all solutions analytic at x 0 and many which are not are exhibited in closed form

torsional springs for two dimensional dynamic unstructured fluid meshes abstract dynamic fluid grids are commonly used for the solution of flow problems with moving boundaries they are often represented by a network of fictitious lineal springs that can become unreliable when the fluid mesh undergoes large displacements and or deformations in this paper we propose to control the arbitrary motion of two dimensional dynamic unstructured fluid grids with additional torsional springs we show that such springs can be designed to prohibit the interpenetration of neighboring triangles and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications we illustrate our new dynamic mesh motion algorithm with several examples that highlight its advantages in terms of robustness quality and performance

an iterative method with error estimators abstract iterative methods for the solution of linear systems of equations produce a sequence of approximate solutions in many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small this paper presents a new iterative method based on the lanczos process for the solution of linear systems of equations with a symmetric matrix the method is designed to allow the computation of estimates of the euclidean norm of the error in the computed approximate solutions these estimates are determined by evaluating certain gauss anti gauss or gauss radau quadrature rules

on stratified extensions of gauss laguerre and gauss hermite quadrature formulas for the practical estimation of the error of gauss laguerre and gauss hermite quadrature formulas it is well known that real positive kronrod formulas do not exist among the alternatives which are available in the literature the stratified rules introduced by laurie are of particular interest in this note we determine the degree optimal stratified extensions for the gauss laguerre and gauss hermite formulas and we investigate their properties

on the construction of higher degree three dimensional embedded integration rules we want to approximate i f equiv int r 3 f x dx for a given function f x defined on a fully symmetric fs region r 3 in e 3 let x x 1 x 2 x n be a fully symmetric set of points in r 3 let q a f and q b f be two fs rules of degree 2m 1 and 2m 1 respectively approximating i f and defined as follows eqalign q a left f right equiv sum nolimits i 1 n a i f left x i right text degree 2 m text 1 cr q a left f right equiv sum nolimits i 1 n b i f left x i right text degree 2 m text 1 cr

anti szego quadrature rules szego quadrature rules are discretization methods for approximating integrals of the form f e t du t this paper presents a new class of discretization methods which we refer to as anti szego quadrature rules anti szego rules can be used to estimate the error in szego quadrature rules under suitable conditions pairs of associated szego and anti szego quadrature rules provide upper and lower bounds for the value of the given integral the construction of anti szego quadrature rules is almost identical to that of szego quadrature rules in that pairs of associated szego and anti szego rules differ only in the choice of a parameter of unit modulus several examples of szego and anti szego quadrature rule pairs are presented

efficient mesh motion using radial basis functions with data reduction algorithms mesh motion using radial basis functions has been demonstrated previously by the authors to produce high quality meshes suitable for use within unsteady and aeroelastic cfd codes in the aeroelastic case the structural mesh may be used as the set of control points governing the deformation which is efficient since the structural mesh is usually small however as a stand alone mesh motion tool where the surface mesh points control the motion radial basis functions may be restricted by the size of the surface mesh as an update of a single volume point depends on all surface points in this paper a method is presented that allows an arbitrary deformation to be represented to within a desired tolerance by using a significantly reduced set of surface points intelligently identified in a fashion that minimises the error in the interpolated surface this method may be used on much larger cases and is successfully demonstrated here for a 10 6 cell mesh where the initial solve phase cost reduces by a factor of eight with the new scheme and the mesh update by a factor of 55 it has also been shown that the number of surface points required to represent the surface is only geometry dependent i e grid size independent and so this reduction factor actually increases for larger meshes

when not to use an automatic quadrature routine numerical quadrature is required as a component in many large scale and small scale scientific programs frequently the scientist faces a choice either he may use one of the comparatively recent automatic quadrature routines aqr or he may use a standard rule evaluation quadrature routine rear based directly on traditional formulae the purpose of this article is to help the scientist to make an informed choice in the first half of this article we introduce some generic properties of aqrs the more straightforward of these are those concerned with reliability and economy less straightforward is the vital performance profile property the second half of this article is devoted principally to demonstrating how in some calculations the jagged performance profile associated with the aqr can cause other parts of the overall program to be far more expensive than necessary thus while in many programs the use of an aqr can save significant human time in many other programs its use can be a disaster

order preserving mesh spacing for compound quadrature formulas and functions with endpoint singularities for compound quadrature formulas for the numerical evaluation of int a b w x f x dx where the function f as well as the weight function w are allowed to have singularities at the endpoints a and b we show how to preserve the order of convergence which is obtained for smooth w and f by an appropriate choice of the meshpoints the main emphasis is on singularities of algebraic and logarithmic type

internality of generalized averaged gaussian quadrature rules and truncated variants for modified chebyshev measures of the second kind abstract generalized averaged gaussian quadrature rules associated with some measure and truncated variants of these rules can be used to estimate the error in gaussian quadrature rules however the former quadrature rules may have nodes outside the interval of integration and therefore it may not be possible to apply them when the integrand is defined on the interval of integration only this paper investigates whether generalized averaged gaussian quadrature rules associated with modified chebyshev measures of the second kind and truncated variants of these rules are internal i e if all nodes of these quadrature rules are in the interval of integration

time efficient aeroelastic simulations based on radial basis functions aeroelasticity studies the interaction between aerodynamic forces and structural responses and is one of the fundamental problems to be considered in the design of modern aircraft the fluid structure interpolation fsi and mesh deformation are two key issues in the cfd csd coupling approach the partitioned approach which is the mainstream numerical strategy in aeroelastic simulations in this paper a time efficient coupling scheme is developed based on the radial basis function interpolations during the fsi process the positive definite system of linear equations is constructed with the introduction of pseudo structural forces the acting forces on the structural nodes can be calculated more efficiently via the solution of the linear system avoiding the costly computations of the aerodynamic structural coupling matrix the multi layer sequential mesh motion algorithm msm is proposed to improve the efficiency of the volume mesh deformations which is adequate for large scale time dependent applications with frequent mesh updates two dimensional mesh motion cases show that the msm algorithm can reduce the computing cost significantly compared to the standard rbf based method the computations of the agard 445 6 wing flutter and the static deflections of the three dimensional high aspect ratio aircraft demonstrate that the developed coupling scheme is applicable to both dynamic and static aeroelastic problems

adaptive change of basis in entropy based moment closures for linear kinetic equations entropy based m n moment closures for kinetic equations are defined by a constrained optimization problem that must be solved at every point in a space time mesh making it important to solve these optimization problems accurately and efficiently we present a complete and practical numerical algorithm for solving the dual problem in one dimensional slab geometries the closure is only well defined on the set of moments that are realizable from a positive underlying distribution and as the boundary of the realizable set is approached the dual problem becomes increasingly difficult to solve due to ill conditioning of the hessian matrix to improve the condition number of the hessian we advocate the use of a change of polynomial basis defined using a cholesky factorization of the hessian that permits solution of problems nearer to the boundary of the realizable set we also advocate a fixed quadrature scheme rather than adaptive quadrature since the latter introduces unnecessary expense and changes the computationally realizable set as the quadrature changes for very ill conditioned problems we use regularization to make the optimization algorithm robust we design a manufactured solution and demonstrate that the adaptive basis optimization algorithm reduces the need for regularization this is important since we also show that regularization slows and even stalls convergence of the numerical simulation when refining the space time mesh we also simulate two well known benchmark problems there we find that our adaptive basis fixed quadrature algorithm uses less regularization than alternatives although differences in the resulting numerical simulations are more sensitive to the regularization strategy than to the choice of basis

on generalized averaged gaussian formulas we present a simple numerical method for constructing the optimal generalized averaged gaussian quadrature formulas which are the optimal stratified extensions of gauss quadrature formulas these extensions exist in many cases in which real positive kronrod formulas do not exist for the jacobi weight functions w x w x 1 x 1 x 1 we give a necessary and sufficient condition on the parameters a and such that the optimal averaged gaussian quadrature formulas are internal

local analytic solution of a second order functional differential equation with a state derivative dependent delay abstract this paper is concerned with a second order functional differential equation x z 1 x az bx z with a state derivative dependent delay by reducing the equation to another functional differential equation with proportional delay an existence theorem is established for analytic solutions of the original equation and systematic methods for deriving explicit solutions are also given we not only prove the convergence of the formal solution under the diophantine condition i e eigenvalues is far from unit roots but also make progresses without the diophantine condition i e the convergence is equivalent to the well known small divisor problems

gauss quadrature rules for a generalized hermite weight function abstract in this paper we give an algorithm for finding the coefficients in the three term recurrence relation for the polynomials orthogonal with respect to the weight function w x x b 2 exp x c 2 on r where b c r 2 1 the algorithm is based on the discretization technique proposed by gautschi with the lanczos algorithm using the coefficients obtained by this algorithm we construct the gauss quadrature rules relative to the weight function w x to show the performance of the new algorithm we give some numerical examples

parallel efficient mesh motion using radial basis functions with application to multi bladed rotors radial basis functions are used to provide a solution to the problem of mesh motion for unsteady aerodynamic simulation the method is independent of connectivity and produces high quality meshes but is expensive for large meshes in its full form hence the efficiency of the technique has been greatly improved here by reducing the number of surface points used to define deformations of the surface and the minor error in position that this implies at other surface points is corrected with a simple decaying perturbation thus splitting the method into a primary basis function method and a secondary local correction method this means that the exact surface is retained but the mesh motion is significantly faster while splitting the motion into two stages allows both the methods to work on appropriate problems given their relative strengths an example deformation for a 5 x10 6 cell helicopter rotor mesh with an exaggerated cyclic pitch motion shows excellent mesh quality thus validating a scheme that is also simple robust and readily parallelized

modified stieltjes polynomials and gauss kronrod quadrature rules modified stieltjes polynomials are defined and used to construct suboptimal extensions of gaussian rules with one or two degrees less of polynomial exactness than the corresponding kronrod extension we prove that for wide classes of weight functions and a sufficiently large number of nodes the extended quadratures have positive weights and simple nodes on the interval 1 1 the classes of weight functions considered complement those for which the gauss kronrod rule is known to exist also strong asymptotic representations on the whole interval 1 1 are given for the modified stieltjes polynomials which prove that they behave asymptotically as orthogonal polynomials finally we provide some numerical examples

the functional differential equation x t x x t abstract a classification of the solutions of the functional differential equation x t x x t is given and it is proved that every solution either vanishes identically or is strictly monotonie for monotonically increasing solutions existence and uniqueness of the solution x are proved with the condition x t0 x0 where t0 x0 is any given pair of reals in some specified subset of r 2 every monotonically increasing solution is thus obtained it is analytic and depends analytically on t0 and x0 only for t0 x0 1 is the question of analyticity still open

computation of gauss type quadrature formulas gaussian formulas for a linear functional l such as a weighted integral are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to l in gauss type formulas one or more extraneous conditions such as pre assigning certain nodes replace some of the equations expressing exactness when applied to high order polynomials these extraneous conditions may be applied by modifying the same number of recursion coefficients we survey the methods of computing formulas from recursion coefficients methods of obtaining recursion coefficients and modifying them for gauss type formulas and questions of existence and numerical accuracy associated with those computations

analytic solutions of an iterative functional differential equation this paper is concerned with a functional differential equation x z 1 x az bx z where a b are two complex numbers by constructing a convergent power series solution y z of a auxiliary equation of the form b 2y z y m 2z ay mz my mz ay z analytic solutions of the form 1b y my 1 z az for the original differential equation are obtained

positive quadrature formulas iii asymptotics of weights first we discuss briefly our former characterization theorem for positive interpolation quadrature formulas abbreviated qf provide an equivalent characterization in terms of jacobi matrices and give links and applications to other qf in particular to gauss kronrod quadratures and recent rediscoveries then for any polynomial t n which generates a positive qf a weight function depending on n is given with respect to which t n is orthogonal to p n 1 with the help of this result an asymptotic representation of the quadrature weights is derived in general the asymptotic behaviour is different from that of the gaussian weights only under additional conditions do the quadrature weights satisfy the so called circle law corresponding results are obtained for positive qf of radau and lobatto type

smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients using the fixed point theorems of banach and schauder in this paper we discuss the existence uniqueness and stability of smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients

optimal addition of knots to cubature formulae for planar regions a method is described to add knots to a cubature formula of degree 2k 1 for an integral over a symmetric region to obtain a cubature formula of degree 2k 1 this method is used to construct cubature formulae for the square the circle the hexagon and the entire plane

stieltjes polynomials and related quadrature formulae for a class of weight functions ii consider a nonnegative measure d sigma with support in the interval a b such that the respective orthogonal polynomials satisfy a three term recurrence relation with coefficients alpha n left begin array ll alpha e n hbox even alpha o n hbox odd end array right beta n beta hbox for n ge ell where alpha e alpha o beta and ell are specific constants we show that the corresponding stieltjes polynomials above the index 2 ell 1 have a very simple and useful representation in terms of the orthogonal polynomials as a result of this the gauss kronrod quadrature formula for d sigma has all the desirable properties namely the interlacing of nodes their inclusion in the closed interval a b under an additional assumption on d sigma and the positivity of all weights while the formula enjoys an elevated degree of exactness furthermore the interpolatory quadrature formula based on the zeros of the stieltjes polynomials has positive weights and also elevated degree of exactness it turns out that this formula is the anti gaussian formula for d sigma while the resulting averaged gaussian formula coincides with the gauss kronrod formula for this measure moreover we show that the only positive and even measure d sigma on a a for which the gauss kronrod formula is almost of chebyshev type i e it has almost all of its weights equal is the measure d sigma t a 2 t 2 1 2 dt

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

on numerical computation of integrals with integrands of the form f x sin w xr on 0 1 with existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the form 0 1f x sin wx r dxor 0 1f x cos wx r dx r 0 where f is a sufficiently smooth function on 0 1 gautschi has developed software as scripts in matlab for computing these integrals for the special case r w 1 in this paper an algorithm as a mathematica program is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range numerical examples are given of testing the performance of the algorithm program

mesh deformation using the biharmonic operator the use of the biharmonic operator for deforming a mesh in an arbitrary lagrangian eulerian simulation is investigated the biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh this allows both the position and the normal mesh spacing along a boundary to be controlled which is important for two fluid interfaces and periodic boundaries at these boundaries we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary in addition results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second order partial differential equations a final advantage is that with the biharmonic operator the integrity of a grid in a moving boundary layer can be preserved as the boundary moves the main disadvantage of the approach is its increased computational expense

on the construction of multi dimensional embedded cubature formulae in order to compute an integrali f one needs at least two cubature formulaeq j j 1 2 q 1 f q 2 f can be used as an error estimate for the less precise cubature formula in order to reduce the amount of work one can try to reuse some of the function evaluations needed forq 1 inq 2 the easiest way to construct embedded cubature formulae is start with a high degree formulaq 1 drop at least one knot and calculate the weights such that the new formulaq 2 is exact for as much monomials as possible we describe how such embedded formulae with positive weights can be found the disadvantage of such embedded cubature formulae is that there is in general a large difference in the degree of exactness of the two formulae in this paper we will explain how the high degree formula can be chosen to obtain an embedded pair of cubature formulae of degrees 2m 1 2m 1 the method works for all regions n n 2 we will also show the influence of structure on the cubature formulae

fast dynamic grid deformation based on delaunay graph mapping a simple and efficient dynamic grid deformation technique is proposed for computing unsteady flow problems with geometrical deformation relative body movement or shape changes due to aerodynamic optimisation and fluid structure interaction a delaunay graph of the solution domain is first generated which can be moved easily during the geometric dynamic deformation even for very large distortion a one to one mapping between the delaunay graph and the computational grid is maintained during the movement therefore the new computational grid after the dynamic movement can be generated efficiently through the mapping while maintaining the primary qualities of the grid while most dynamic grid deformation techniques are iterative based on the spring analogy the present method is non iterative and much more efficient on the other hand in comparison with dynamic grid techniques based on transfinite interpolation for structured grids it offers both geometric and cell topology flexibility which is crucial for many unsteady flow problems involving geometric deformation and relative motions the method is demonstrated through some typical unsteady flow test cases including a pitching aerofoil in a fixed domain boundary relative movement between multi element aerofoils due to flap deployment for high lift a deformable sphere in a fixed cube and a three dimensional flexible wing with large deformation

solution of second kind fredholm integral equations by means of gauss and anti gauss quadrature rules this paper is concerned with the numerical approximation of fredholm integral equations of the second kind a nystrom method based on the anti gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces the nystrom interpolants corresponding to the gauss and the anti gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation under suitable assumptions which are easily verified for a particular weight function hence an error estimate is available and the accuracy of the solution can be improved by approximating it by an averaged nystrom interpolant the effectiveness of the proposed approach is illustrated through different numerical tests

automatic integration using asymptotically optimal adaptive simpson quadrature we present a novel theoretical approach to the analysis of adaptive quadratures and adaptive simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration for a given function f in c 4 f c4 with f 4 ge 0 f 4 0 and possible endpoint singularities the algorithm produces an approximation to int a bf x mathrm d x abf x dx within a given varepsilon asymptotically as varepsilon rightarrow 0 0 moreover it is optimal among all adaptive simpson quadratures i e needs the minimal number n f varepsilon n f of function evaluations to obtain an varepsilon approximation and runs in time proportional to n f varepsilon n f

analytic solutions of an iterative functional differential equation abstract this paper is concerned with an iterative functional differential equation x z x m z where x m z x x x z is the m th iterate of the function x z by constructing a convergent power series solution y z of a companion equation of the form y z y z y m z analytic solutions of the form y y 1 z for the original differential equation are obtained

unified fluid structure interpolation and mesh motion using radial basis functions a multivariate interpolation scheme using radial basis functions is presented which results in a completely unified formulation for the fluid structure interpolation and mesh motion problems the method has several significant advantages primarily all volume mesh structural mesh and flow solver type dependence is removed and all operations are performed on totally arbitrary point clouds of any form hence all connectivity and user input requirements are removed from the computational fluid dynamics computational structural dynamics cfd csd coupling problem as only point clouds are required to determine the coupling also it may equally well be applied to structured and unstructured grids or structural and aerodynamic grids that intersect again because no connectivity information is required furthermore no expensive computations are required during an unsteady simulation just matrix vector multiplications since the required dependence relations are computed only once prior to any simulation and then remain constant this property means that the method is both perfectly parallel since only the data relevant to each structured block or unstructured partition are required to move those points and totally independent from the flow solver hence a completely generic black box tool can be developed which is ideal for use in an optimization approach aeroelastic behaviour of the brite euram mdo wing is analysed in terms of both static deflection and dynamic responses and it is demonstrated that responses are strongly dependent on the exact cfd csd interpolation used mesh quality is also examined during the motion resulting from a large surface deformation global grid quality is shown to be preserved well with local grid orthogonality also being maintained well particularly at and near the moving surface where the original orthogonality is retained copyright 2007 john wiley sons ltd

generalized block anti gauss quadrature rules golub and meurant describe how pairs of gauss and gauss radau quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real valued matrix functionals defined by a symmetric matrix however there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds in this situation it may be possible to determine upper and lower bounds by evaluating pairs of gauss and anti gauss rules unfortunately it is difficult to ascertain whether the values determined by gauss and anti gauss rules bracket the value of the given real valued matrix functional therefore generalizations of anti gauss rules have recently been described such that pairs of gauss and generalized anti gauss rules may determine upper and lower bounds for real valued matrix functionals also when pairs of gauss and standard anti gauss rules do not the available generalization requires the matrix that defines the functional to be real and symmetric the present paper reviews available anti gauss and generalized anti gauss rules and extends them in several ways that allow applications in new situations in particular the genarlized anti gauss rules for a real valued non negative measure described in pranic and reichel j comput appl math 284 235 243 2015 are extended to allow the estimation of the error in matrix functionals defined by a non symmetric matrix as well as to matrix valued matrix functions modifications that give simpler formulas and thereby make the application of the rules both easier and applicable to a larger class of problems also are described

stratified nested and related quadrature rules the stratified nested quadrature procedure due to laurie is discussed together with an alternative computational procedure which leads to the concept of hybrid gkp rules in the context of the approximation of stratified nested sequences the work of krogh and van snyder on the representation of the gkp rules is considered and a generalisation of this employing hybrid rules of special form is discussed

a monolithic approach to fluid structure interaction using space time finite elements abstract the paper presents a simultaneous solution procedure for fluid structure interaction problems the structural motion is described by geometrically nonlinear elastodynamics the fluid is modeled by the incompressible navier stokes equations the space time finite element method is applied to both continua leading to an almost uniform discretization in which velocity variables are used for fluid and structure in order to enforce momentum conservation and geometric continuity at the interface a weighted residual formulation of coupling conditions is introduced the discretized model equations for fluid structure and coupling conditions are formulated in a single equation system the nonlinear system and the fluid mesh movement are solved in a single iteration loop two dimensional examples of membrane flutter and vortex excited plate vibrations demonstrate the efficiency of the methodology

generalized averaged szegź quadrature rules szeg quadrature rules are commonly applied to integrate periodic functions on the unit circle in the complex plane however often it is difficult to determine the quadrature error recently spalevic introduced generalized averaged gauss quadrature rules for estimating the quadrature error obtained when applying gauss quadrature over an interval on the real axis we describe analogous quadrature rules for the unit circle that often yield higher accuracy than szeg rules using the same moment information and may be used to estimate the error in szeg quadrature rules

error estimates of anti gaussian quadrature formulae anti gauss quadrature formulae associated with four classical chebyshev weight functions are considered complex variable methods are used to obtain expansions of the error in anti gaussian quadrature formulae over the interval 1 1 the kernel of the remainder term in anti gaussian quadrature formulae is analyzed the location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated this leads to effective l error bounds of anti gauss quadratures moreover the effective l 1 error estimates are also derived the results obtained here are an analogue of some results of gautschi and varga 1983 11 gautschi et al 1990 9 and hunter 1995 10 concerning gaussian quadratures

unsteady euler algorithm with unstructured dynamic mesh for complex aircraft aerodynamic analysis an euler solution algorithm is presented for unsteady aerodynamic analysis of complex aircr aft configurations the flow solver involves a multistage runge kutta time stepping scheme that uses a finite volume spatial discretization on an unstructured grid made up of tetrahedra a significant contribution of the research is the development and implementation of a moving mesh algorithm that is employed for problems involving static or dynamic deformation of the aircraft the mesh algorithm is a general procedure that can treat realistic motions and deformations of complex aircraft configurations steady and unsteady results are presented for a supersonic fighter configuration to demonstrate applications of the euler solver and dynamic mesh algorithm the unsteady flow results were obtained for the aircraft oscillating harmonically in a complete vehicle bending mode effects of angle of attack and reduced frequency on instantaneous pressures and force responses were investigated the paper presents descriptions of the euler solver and dynamic mesh algorithm along with results that assess the capability

on neutral functional differential equations with proportional delays abstract in this paper we develop a comprehensive theory on the well posedness of the initial value problem for the neutral functional differential equation y t ay t i 1 b i y q i t i 1 c i y p i t t 0 y 0 y 0 and the asymptotic behaviour of its solutions we prove that the existence and uniqueness of solutions depend mainly on the coefficientsci i 1 2 and on the smoothness of functions in the solution space as far as the asymptotic behaviour of analytic solutions is concerned thecihave little effect we prove that if re a 0 then the solutiony t either grows exponentially or is polynomial the most interesting result is that if re a 0 anda 0 then the asymptotic behaviour of the solution depends mainly on the characteristic equation a i 1 b i q i 0 these results can be generalized to systems of equations finally we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters the main idea used in this paper is to express the solution in either dirichlet or dirichlet taylor series form

matlab program for quadrature in 2d abstract we discuss here the algorithms of twod a m atlab program for approximating integrals over generalized rectangles and sectors capabilities of the language are exploited to make twod very easy to use vectorization is exploited by a novel algorithm to make the program efficient and reliable twod provides for moderate singularities on boundaries of the region

calculation of gauss kronrod quadrature rules the jacobi matrix of the 2n 1 point gauss kronrod quadrature rule for a given measure is calculated efficiently by a five term recurrence relation the algorithm uses only rational operations and is therefore also useful for obtaining the jacobi kronrod matrix analytically the nodes and weights can then be computed directly by standard software for gaussian quadrature formulas