on the construction of good fully symmetric integration rules in this paper we review some classical theory of moments for polynomials this theory turns out to be a useful tool when we want to construct good fully symmetric integration rules fsg rules for fully symmetric regions we also show how to apply this theory by constructing several new fsg rules of degree 7 9 and 11 for the 3 cube

economical symmetrical quadrature rules for complete polynomials over a square domain it is of interest in numerical analysis to develop symmetrical quadrature rules for integration of complete polynomial functions over a square domain with minimum computational effort gaussian product quadrature rules integrate such functions with maximum effort symmetrical quadrature rules are developed and presented for integration of complete polynomial functions up to 21st order with minimum computational effort

weighted minimum and maximun operations in fuzzy sets theory abstract this short note comments on the interpretation of weighted minimum and maximum operations introduced in a previous paper as medians and corrects a misleading statement about it

qualitative choice logic qualitative choice logic qcl is a propositional logic for representing alternative ranked options for problem solutions the logic adds to classical propositional logic a new connective called ordered disjunction a b intuitively means if possible a but if a is not possible then at least b the semantics of qualitative choice logic is based on a preference relation among models consequences of qcl theories can be computed through a compilation to stratified knowledge bases which in turn can be compiled to classical propositional theories we also discuss potential applications of the logic several variants of qcl based on alternative inference relations and their relation to existing nonmonotonic formalisms

on the identification of symmetric quadrature rules for finite element methods in this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals triangles tetrahedra prisms pyramids and hexahedra the methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points we also present polyquad which is an implementation of our methodology using polyquad v1 0 we proceed to derive a complete set of symmetric rules on the aforementioned domains all rules possess purely positive weights and have all points inside the domain many of the rules appear to be new and an improvement over those tabulated in the literature

on four noncommutative fuzzy connectives and their axiomatization in most of the query languages conjunctive and disjunctive combinations of conditions remain the usual way for aggregation fuzzy query languages also offer trade off operators such as means in order to compensate between elementary conditions in this paper we would like to introduce a new type of condition basically founded on the interaction between two predicates thus enriching the panoply of tools the user is provided with and the power of query languages

perfectly matched layer in three dimensions for the time domain finite element method applied to radiation problems a new perfectly matched layer pml formulation for the time domain finite element method is described and tested for problems in three dimensional space an integral part of this pml algorithm is a novel time integration scheme based on galerkin s method with a piecewise linear temporal expansion of the electric field our time integration procedure is constructed by forming a linear combination of exact and trapezoidal integration applied to the temporal weak form which reduces to the well known newmark scheme in the case where the pml is not present this technique allows for a consistent time discretization of the electric field vector wave equation augmented with the pml which includes terms that involve the electric field itself time derivatives thereof and exponential functions temporally convolved with the electric field the auxiliary variables that are associated with such convolutions are efficiently updated in an explicit manner by matrix vector products the new pml formulation is successfully tested on canonical problems the pml can very efficiently absorb propagating electromagnetic waves which is demonstrated by reflections coefficients as low as 100 db the pml formulation is capable of giving highly accurate results for very broad frequency bands which is confirmed by tests that show an error smaller than 2 over more than six octaves in frequency finally we apply the pml formulation to a bow tie antenna and comparisons with measurements show good agreement

adaptively weighted numerical integration in the finite cell method abstract with adaptively weighted aw numerical integration for a given set of quadrature nodes order and domain of integration the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense the moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain and then are corrected for the deviation from the original domain using shape sensitivity analysis in this paper we demonstrate the application of aw integration scheme in the context of the finite cell method which must perform numerical integration over arbitrary domains without meshing the standard integration technique employed in fcm is the characteristic function method that converts the continuous integrand over a complex domain into a discontinuous integrand over a simple box domain then well known integrand adaptivity techniques are employed to integrate the resulting discontinuous integrand over the box domain although this method is simple to implement it becomes computationally very expensive for realistic complex 3d domains such as sculptures bones and engines in contrast in aw scheme the quadrature weights directly adapt to the complex geometric domain without the need to making the integrand discontinuous leading to superior computational properties in this paper we demonstrate the computational efficiency of aw over the characteristic function method as it requires fewer subdivisions and less time to achieve a given accuracy in both two and three dimensions in addition aw offers a number of advantages including flexibility in the choice of quadrature points and basis functions

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

efficient computation of rcs from scatterers of uncertain shapes we propose a way of accounting for the lack of detailed knowledge about material shapes in computational time domain electromagnetics we use legendre gauss lobatto stroud 2 and stroud 3 quadrature formulas to solve the resulting stochastic equation and we show the efficiency of the proposed method over statistical monte carlo simulations we also show how the radar cross section rcs in scattering is affected by the uncertainty in shape of the objects and by the direction of the incident field

adaptive approximation of higher order posterior statistics filtering is an approach for incorporating observed data into time evolving systems instead of a family of dirac delta masses that is widely used in monte carlo methods we here use the wiener chaos expansion for the parametrization of the conditioned probability distribution to solve the nonlinear filtering problem the wiener chaos expansion is not the best method for uncertainty propagation without observations nevertheless the projection of the system variables in a fixed polynomial basis spanning the probability space might be a competitive representation in the presence of relatively frequent observations because the wiener chaos approach not only leads to an accurate and efficient prediction for short time uncertainty quantification but it also allows to apply several data assimilation methods that can be used to yield a better approximate filtering solution the aim of the present paper is to investigate this hypothesis we answer in the affirmative for the stochastic lorenz 63 system based on numerical simulations in which the uncertainty quantification method and the data assimilation method are adaptively selected by whether the dynamics is driven by brownian motion and the near gaussianity of the measure to be updated respectively

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

asymmetric cubature formulas for polynomial integration in the triangle and square we present five new cubature formula in the triangle and square for exact integration of polynomials the points were computed numerically with a cardinal function algorithm which does not impose any symmetry requirements on the points cubature formula are presented which integrate degrees 10 11 and 12 in the triangle and degrees 10 and 12 in the square they have positive weights contain no points outside the domain and have fewer points than previously known results

a physically motivated sparse cubature scheme with applications to molecular density functional theory we present a novel approach for performing multi dimensional integration of arbitrary functions the method starts with smolyak type sparse grids as cubature formulae on the unit cube and uses a transformation of coordinates based on the conditional distribution method to adapt those formulae to real space our method is tested on integrals in one two three and six dimensions the three dimensional integration formulae are used to evaluate atomic interaction energies via the gordon kim model the six dimensional integration formulae are tested in conjunction with the nonlocal exchange correlation energy functional proposed by lee and parr this methodology is versatile and powerful we contemplate application to frozen density embedding next generation molecular mechanics force fields kernel type exchange correlation energy functionals and pair density functional theory

the construction of three dimensional invariant cubature formulae abstract a method of constructing three dimensional cubature formulae of high degree for three dimensional regions is presented the formulae possess full polyhedral symmetry this means that every knot of the formula belongs to an orbit invariant with respect to the symmetries of the polyhedron and that all points of an orbit are knots of the formula with the same weight

selective and reduced numerical integrations for nurbs based isogeometric analysis we propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on nurbs elements the notion of an approximation space that approximates the target space is introduced we explore the use of various approximation spaces associated with optimal patch wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points patch rules exploit the higher continuity of spline basis functions the tendency of smooth spline functions to exhibit numerical locking in nearly incompressible problems when using a full gauss legendre quadrature is alleviated with selective or reduced integration stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem we propose a local algorithm which is robust and computationally efficient to compute element by element the quadrature points and weights in patch rules the performance of the methods is assessed on several numerical examples in two dimensional elasticity and reissner mindlin shell structures

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

efficient symmetrical cubature rules for complete polynomials of high degree over the unit cube the fully symmetrical cubature rules developed and presented are at present the most economically available for numerical integration over the unit cube of complete polynomial functions up to and including nineteenth order conventional gaussian triple product integration formulae by comparison require more than twice as many integration points and weights

symmetric quadrature rules for tetrahedra based on a cubic close packed lattice arrangement a family of quadrature rules for integration over tetrahedral volumes is developed the underlying structure of the rules is based on the cubic close packed ccp lattice arrangement using 1 4 10 20 35 and 56 quadrature points the rules are characterized by rapid convergence positive weights and symmetry each rule is an optimal approximation in the sense that lower order terms have zero contribution to the truncation error and the leading order error term is minimized quadrature formulas up to order 9 are presented with relevant numerical examples

optimal spherical designs and numerical integration on the sphere so called spherical t designs in three dimensional euclidean space r3 are considered it is shown that the maximal possible value of t for designs which are permuted transitively under a finite group of euclidean transformations is 9 this means that we can find a set of 60 points on the surface of the unit sphere in r3 which form an orbit under the icosahedral group such that the average of the function values at the 60 points and the average over the spherical surface are identical for all spherical harmonics of degree less than or equal to t 9 while there is no such orbit for t 10 among all 9 designs which are orbits of the icosahedral group exactly one is optimal with respect to approximation of integrals of harmonics of tenth and eleventh order this set is given in section 2 of this paper it is very well suited for numerically evaluating integrals of continuous functions over the sphere a problem occurring very often in mathematics and science as e g physics astronomy etc

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

a posteriori error analysis for finite element solution of one dimensional elliptic differential equations using equidistributing meshes the paper is concerned with the adaptive linear finite element solution of linear one dimensional elliptic differential equations using equidistributing meshes a strategy is developed for defining such meshes based on a residual based a posteriori error estimate the mesh and the finite element solution are determined by the coupled system of the finite element equation and the equidistribution relation i e the mesh equation an iterative algorithm is proposed for solving this system for the mesh and the finite element solution the existence of an equidistributing mesh is proven for a given sufficiently large number of the points with help of a result on the continuous dependence of the finite element solution on the mesh which is also established in the current work error bounds for the finite element solution are obtained for the equidistributing and quasi equidistributing meshes they show that adaptive meshes can lead to more accurate solutions than a uniform mesh and it is unnecessary to compute the equidistribution relation accurately for the equidistributing meshes the departure from the equidistributing meshes has only a mild effect on the finite element error numerical examples are given to illustrate the convergence of the iterative algorithm and the theoretical findings

rotation invariant cubature formulas over the n dimensional unit cube we consider cubature formulas to approximate multivariate integrals that remain unchanged under the orientation preserving transformations of a cube the use of invariant theory molien series reynold operator for the construction of such cubature formulas is investigated some new cubature formulas for the unit cube that are obtained using this approach are presented

automatic computation of knots and weights of cubature formulae for circular symmetric planar regions abstract we are concerned with the construction of high degree formulae for circular symmetric planar regions such as a circle and the entire plane with the aid of invariant polynomials with respect to reflection groups we decompose the problem into relative small quadrature problems the cubature formulae can be constructed by an automatic procedure

scattering of flexural wave in a thin plate with multiple circular holes by using the multipole trefftz method the scattering of flexural wave by multiple circular holes in an infinite thin plate is analytically solved by using the multipole trefftz method the dynamic moment concentration factor dmcf along the edge of circular holes is determined based on the addition theorem the solution of the field represented by multiple coordinate systems centered at each circle can be transformed into one coordinate system centered at one circle where the boundary conditions are given in this way a coupled infinite system of simultaneous linear algebraic equations is derived as an analytical model for the scattering of flexural wave by multiple holes in an infinite plate subject to the incident flexural wave the formulation is general and is easily applicable to dealing with the problem containing multiple circular holes although the number of hole is not limited in our proposed method the numerical results of an infinite plate with three circular holes are presented in the truncated finite system the effects of both incident wave number and the central distance among circular holes on the dmcf are investigated numerical results show that the dmcf of three holes is larger than that of one when the space among holes is small and meanwhile the specified direction of incident wave is subjected to the plate

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

flexural wave scattering and dynamic stress concentration in a heterogeneous plate with multiple cylindrical patches by acoustical wave propagator technique this paper presents the acoustical wave propagator technique to describe the flexural wave propagation and scattering in an elastic plate with multiple cylindrical patches the wave propagation patterns in the presence of these patches are used to investigate dynamic stress concentration an improved combined scheme of chebyshev polynomial expansion and fast fourier transformation compared with the exact analytical solutions is found to be effective in predicting the distribution of dynamic stress concentration the dynamic stress concentration factors around the patches in the time domain are also discussed in detail

an isogeometric collocation method using superconvergent points abstract we develop an iga collocation method modified by collocating at points other than the standard greville abscissae the method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory therefore we refer to this method as super collocation iga sc by carefully choosing the collocation points it can be seen that the iga sc converges in the first derivative energy norms at rates similar to that of the galerkin solution this is different from the collocation at greville abscissae iga c where the convergence in energy norm for odd polynomial degrees is typically suboptimal the method is tested on 1d 2d and 3d numerical examples in which it is compared to iga c and galerkin s method iga g the comparison includes a detailed cost vs accuracy analysis which shows an improved efficiency of the proposed method in particular for odd polynomial degrees

optimal modeling parameters for higher order mom sie and fem mom electromagnetic simulations general guidelines and quantitative recipes for adoptions of optimal higher order parameters for computational electromagnetics cem modeling using the method of moments and the finite element method are established and validated based on an exhaustive series of numerical experiments and comprehensive case studies on higher order hierarchical cem models of metallic and dielectric scatterers the modeling parameters considered are electrical dimensions of elements h refinement polynomial orders of basis functions p refinement orders of gauss legendre integration formulas integration accuracy and geometrical curvature orders of elements in the model the goal of the study which is the first such study of higher order parameters in cem is to reduce the dilemmas and uncertainties associated with the great modeling flexibility of higher order elements basis and testing functions and integration procedures this flexibility is the principal advantage but also the greatest shortcoming of the higher order cem and to ease and facilitate the decisions to be made on how to actually use them by both cem developers and practitioners

constructing cubature formulae by the method of reproducing kernel we examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently several new cubature formulae are derived

on gauss lobatto integration on the triangle a recent result in b t helenbrook siam j numer anal 47 2009 pp 1304 1318 on the nonexistence of gauss lobatto cubature rules on the triangle is strengthened by establishing a lower bound for the number of nodes of such rules a method of constructing lobatto type cubature rules on the triangle is given and used to construct several examples

constructing cubature formulae for spheres and balls abstract it has been shown recently that cubature formulae for the unit sphere and for the unit ball are closely related in particular cubature formulae for the surface measure on the sphere correspond to formulae for the chebyshev weight function on the ball this provides a new method to generate cubature formulae on these regions in this paper we construct a number of cubature formulae for the chebyshev weight function on the unit ball in r 2 and use them to derive new formulae for the surface measure on the sphere s2

a discontinuous galerkin method for the shallow water equations in spherical triangular coordinates a global model of the atmosphere is presented governed by the shallow water equations and discretized by a runge kutta discontinuous galerkin method on an unstructured triangular grid the shallow water equations on the sphere a two dimensional surface in r 3 are locally represented in terms of spherical triangular coordinates the appropriate local coordinate mappings on triangles on every triangular grid element this leads to a two dimensional representation of tangential momentum and therefore only two discrete momentum equations the discontinuous galerkin method consists of an integral formulation which requires both area elements and line element faces integrals here we use a rusanov numerical flux to resolve the discontinuous fluxes at the element faces a strong stability preserving third order runge kutta method is applied for the time discretization the polynomial space of order k on each curved triangle of the grid is characterized by a lagrange basis and requires high order quadature rules for the integration over elements and element faces for the presented method no mass matrix inversion is necessary except in a preprocessing step the validation of the atmospheric model has been done considering standard tests from williamson et al d l williamson j b drake j j hack r jakob p n swarztrauber a standard test set for numerical approximations to the shallow water equations in spherical geometry j comput phys 102 1992 211 224 unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream a convergence rate of o dx k 1 was observed in the model experiments furthermore a numerical experiment is presented for which the third order time integration method limits the model error thus the time step dt is restricted by both the cfl condition and accuracy demands conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k

on a fuzzy bipolar relational algebra this paper presents an extension of relational algebra suitable for the handling of bipolar concepts the type of queries considered involves two parts a first one which expresses a possibly flexible constraint and a second one that corresponds to a possibly flexible wish the framework considered is that of bipolar fuzzy relations where each tuple is associated with a pair of satisfaction degrees

consistent structures of invariant quadrature rules for the n simplex in this paper we develop a technique to obtain in a systematic way the consistency conditions for the n dimensional simplex t n for any dimension n and degree of precision d the introduction of a convenient basis of invariant polynomials provides a powerful tool to analyze and obtain consistent structures we also present tables listing the optimal consistent structures for dimensions n 2 8 and degree of precision up to d 23 this paper is devoted only to structures no quadrature rules are presented here

numerical implementation of the edem for modified helmholtz bvps on annular domains in a recent paper by the current authors a new methodology called the extended domain eigenfunction method edem was proposed for solving elliptic boundary value problems on annular like domains in this paper we present and investigate one possible numerical algorithm to implement the edem this algorithm is used to solve modified helmholtz bvps on annular like domains two examples of annular like domains are studied the results and performance are compared with those of the well known boundary element method bem the high accuracy of the edem solutions and the superior efficiency of the edem over the bem make edem an excellent alternate candidate to use in the animation industry where speed is a predominant requirement and by the scientific community where accuracy is the paramount objective

extended domain eigenfunction method edem a study of ill posedness and regularization the extended domain eigenfunction method edem proposed for solving elliptic boundary value problems on annular like domains requires an inversion process the procedure thus represents an ill posed problem whose numerical solution involves an ill conditioned system of equations in this paper the ill posed nature of edem is studied and numerical solutions based on regularization schemes are considered it is shown that the edem solution methodology lends itself naturally to a formulation in terms of the well known iterative landweber method and the more general and faster converging semi iterative regularization schemes theoretical details and numerical results of the regularization schemes are presented for the case of the two dimensional laplace operator on annular domains

partially symmetric cubature formulas for even degrees of exactness the method presented consists of a reduction of the moment equation system and an a priori assumption of the structure of a formula it allows the determination of partially symmetric cubature formulas for regions with the same or higher symmetries this makes it specially suited for even degrees of exactness because these cubature formulas exhibit only partial symmetry the size of the resulting nonlinear system of equations calls for a numerical solution several new results are given for squares circles and hexagons the application to higher dimensional problems is straightforward however no systematic search has been conducted as yet

variational mesh adaptation isotropy and equidistribution we present a new approach for developing more robust and error oriented mesh adaptation methods specifically assuming that a regular in cell shape and uniform in cell size computational mesh is used as is commonly done in computation we develop a criterion for mesh adaptation based on an error function whose definition is motivated by the analysis of function variation and local error behavior for linear interpolation the criterion is then decomposed into two aspects the isotropy or conformity and uniformity or equidistribution requirements each of which can be easier to deal with the functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities a new functional is finally formulated by combining the functionals corresponding to the isotropy and uniformity requirements the features of the functional are analyzed and demonstrated by numerical results in particular unlike the existing mesh adaptation functionals the new functional has clear geometric meanings of minimization a mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters the analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions

a fast method of numerical quadrature for p version finite element matrices a new technique of numerical quadrature especially suited for quadrilateral and hexahedron p version finite element matrices is presented this new technique separates the integrand into two parts and numerically operates on each part separately the objective of this scheme is to minimize the computational cost of integrating the entire element matrix as opposed to minimizing the cost of integrating a single function the efficiency of the new technique is compared with gaussian quadrature and found to take a small fraction of the computational effort

natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method this paper presents a semi analytical approach to solve the eigenproblem of an acoustic cavity with multiple elliptical boundaries to satisfy the helmholtz equation in the elliptical coordinate system the multipole expansion for the acoustic pressure is formulated in terms of angular and radial mathieu functions the boundary conditions are satisfied by uniformly collocating points on the boundaries the acoustic pressure at each point is directly calculated in each elliptical coordinate system in different coordinate systems the normal derivative of the acoustic pressure is calculated by using the appropriate directional derivative an alternative to the addition theorem by truncating the multipole expansion a finite linear algebraic system is derived the direct searching approach is employed to determine the natural frequencies by using the singular value decomposition svd numerical results are widely discussed for several examples including an elliptical cavity a confocal elliptical annulus cavity and an elliptical cavity with two elliptical cylinders the accuracy and numerical convergence of the presented method is validated by comparison with available results from the analytical method and the commercial finite element code abaqus no spurious eigensolutions are found in the proposed formulation excellent accuracy and fast rate of convergence are the key features of the present method thanks to its semi analytical feature

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

symmetric gaussian quadrature formulae for tetrahedronal regions abstract quadrature formulae of degrees 2 to 6 are presented for the numerical integration of a function over tetrahedronal regions the formulae presented are of gaussian type and fully symmetric with respect to the four vertices of the tetrahedron

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

cubature for the sphere and the discrete spherical harmonic transform using a result of bannai and damerell it is shown that a cubature formula with n points of degree 2s 4 for the surface of the n dimensional sphere u n cannot achieve the classical lower bound of dim mathcal p s where mathcal p s is the space of all polynomials in n variables of at most degree s restricted to u n this implies that for n 2 there does not exist a cubature based discrete n dimensional spherical harmonic transform for degree s 2 with the same number of points as spectral coefficients

three dimensional acoustic scattering by multiple spheres using collocation multipole method abstract this paper presents a semi analytical approach to solve the three dimensional acoustic scattering problems with multiple spheres subjected to a plane sound wave to satisfy the three dimensional helmholtz equation in a spherical coordinate system the multipole expansion for the scattered acoustic field is formulated in terms of the associated legendre functions and the spherical hankel functions that also satisfy the radiation condition at infinity the multipole method the directional derivative and the collocation technique are combined to propose a collocation multipole method in which the acoustic field and its normal derivative with respect to the non local spherical coordinate system can be calculated without any truncated error frequently occurred when using the addition theorem the boundary conditions are satisfied by collocating points on the surface of each sphere by truncating the higher order terms of the multipole expansion a finite linear algebraic system is acquired the scattered field can then be determined according to the given incident sound wave the convergence analysis considering the specified error the separation of spheres and the wave number of an incident wave is first carried out to provide guide lines for the proposed method then the proposed results for acoustic scattering by one two and three spheres are validated by using the available analytical method and numerical methods such as boundary element method finally the effects of the separation between scatterers the incident wave number and the incident angle on the acoustic scattering are investigated extensively

lattice integration on the 7 sphere abstract recently the 7 sphere s7 has been the subject of increased interest for mathematicians and theoretical physicists in this paper we present integration algorithms of several different orders up to 19 which require no more than a few thousand points in this respect our formulae are highly superior to standard methods e g gaussian integration or the use of spherical t designs

the construction of algebraic cubature formulae by the distribution of nodes along selected lines a new technique for developing general cubature rules is discussed by distributing the nodes strategically along selected lines the solution is reduced to dealing with a series of one dimensional moment problems several analytical and numerical illustrative examples are provided the extension to higher dimensions is also considered

constructing cubature formulae of degree 5 with few points this paper is devoted to construct a family of fifth degree cubature formulae for n cube with symmetric measure and n dimensional spherically symmetrical region the formula forn cube contains at most n 2 5n 3 points and for n dimensional spherically symmetrical region contains only n 2 3n 3 points moreover the numbers can be reduced to n 2 3n 1 and n 2 n 1 if n 7 respectively the latter of which is minimal

symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements sphere close packed scp lattice arrangements of points are well suited for formulating symmetric quadrature rules on simplexes as they are symmetric under affine transformations of the simplex unto itself in 2d and 3d as a result scp lattice arrangements have been utilized to formulate symmetric quadrature rules with n p 1 4 10 20 35 and 56 points on the 3 simplex shunn and ham 2012 in what follows the work on the 3 simplex is extended and scp lattices are employed to identify symmetric quadrature rules with n p 1 3 6 10 15 21 28 36 45 55 and 66 points on the 2 simplex and n p 84 points on the 3 simplex these rules are found to be capable of exactly integrating polynomials of up to degree 17 in 2d and up to degree 9 in 3d

a nodal triangle based spectral element method for the shallow water equations on the sphere a nodal triangle based spectral element se method for the shallow water equations on the sphere is presented the original se method uses quadrilateral elements and high order nodal lagrange polynomials constructed from a tensor product of the legendre gauss lobatto points in this work we construct the high order lagrange polynomials directly on the triangle using nodal sets obtained from the electrostatics principle j s hesthaven from electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex siam journal on numerical analysis 35 1998 655 676 and fekete points m a taylor b a wingate r e vincent an algorithm for computing fekete points in the triangle siam journal on numerical analysis 38 2000 1707 1720 these points have good approximation properties and far better lebesgue constants than any other nodal set derived for the triangle by employing triangular elements as the basic building blocks of the se method and the cartesian coordinate form of the equations we can use any grid imaginable including adaptive unstructured grids results for six test cases are presented to confirm the accuracy and stability of the method the results show that the triangle based se method yields the expected exponential convergence and that it can be more accurate than the quadrilateral based se method even while using 30 60 fewer grid points especially when adaptive grids are used to align the grid with the flow direction however at the moment the quadrilateral based se method is twice as fast as the triangle based se method because the latter does not yield a diagonal mass matrix

construction of fully symmetric cubature formulae of degree 4 k 3 for fully symmetric planar regions abstract a new method is described for the construction of cubature formulae of degree 4k 3 for two dimensional symmetric regions this method is a generalisation of the t method some formulae of degree 5 9 13 and 17 for the square the circle and the entire plane are constructed

variational mesh adaptation ii error estimates and monitor functions the key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation in this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work j comput phys 174 2001 924 two types of monitor functions scalar matrix and non scalar matrix ones are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods the choice of the adaptation intensity parameter is also discussed for each of these monitor functions asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions two dimensional numerical results are given to verify the theoretical findings

a note on a plate having a circular cavity excited by plane harmonic sh waves abstract a solution for the plate having a circular cavity subject to plane harmonic sh waves horizontally polarized shear waves is presented in this paper the method of solution involves series expansion of incident and reflected sh waves from the cavity and two free surfaces in terms of cylindrical wave functions and the image method respectively the image method proposed in this article is used to satisfy the boundary condition of the traction free surfaces on the plate then the boundary condition of the cavity is applied to the solution having unknown coefficients in order to simplify the problem only the lowest mode is taken into account in the plate the results obtained in this study provide important information about the behaviour of the plate near the discontinuity numerical results show that the distance between the upper surface and the nearest cavity boundary at 0 r a has primary effect if the radius of the cavity is relatively small compared with the thickness of the plate the solutions approach the full space s solutions for the cavity well known in literature also those solutions for the plate correspond to the solutions of the plate without cavity dynamic stress concentration factors around the cavity and displacements in the cross section of the plate are obtained near the cavity for various wave numbers it is concluded that the presented solutions are partially analytical solutions so they can be used for construction and verification of approximate numerical techniques such as the boundary element method bem finite element method fem finite difference method fdm etc

an imbedded family of cubature formulae for n dimensional product regions abstract we consider an imbedded family of cubature formulae for n dimensional fully symmetric product regions the formula of degree 2m 1 based on m 1 generators the n dimensional cube and the n dimensional space with a gaussian weight function are studied in detail

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

numerical integration on the sphere and its effect on the material symmetry of constitutive equations a comparative study in the present paper we consider a class of constitutive models based on numerical integration on the unit sphere the directional behaviour of the quadrature schemes and its effect on the symmetry properties of these constitutive models are studied by subjecting the set of integration points on the sphere to arbitrary rigid rotations we investigate a number of recently proposed integration schemes in application to a full network model of rubber elasticity and to an exponential model for soft tissues in order to assess and compare these schemes statistical methods are presented and applied the analysis discloses a number of integration schemes that offer a good compromise between the numerical error and the number of integration points however as a general result it turns out that numerical integration is prone to introduce strong anisotropy into originally isotropic constitutive equations in particular for highly non linear integrand functions the consequences for application of the investigated class of constitutive models in finite element calculations are highlighted in a benchmark like numerical example copyright 2009 john wiley sons ltd

high degree cubature kalman filter the cubature kalman filter ckf which is based on the third degree spherical radial cubature rule is numerically more stable than the unscented kalman filter ukf but less accurate than the gauss hermite quadrature filter ghqf to improve the performance of the ckf a new class of ckfs with arbitrary degrees of accuracy in computing the spherical and radial integrals is proposed the third degree ckf is a special case of the class the high degree ckfs of the class can achieve the accuracy and stability performances close to those of the ghqf but at lower computational cost a numerical integration problem and a target tracking problem are utilized to demonstrate the necessity of using the high degree cubature rules to improve the performance the target tracking simulation shows that the fifth degree ckf can achieve higher accuracy than the extended kalman filter the ukf the third degree ckf and the particle filter and is computationally much more efficient than the ghqf

computation of moderate degree fully symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving a novel method is presented for expressing the moment equations involved in computing fully symmetric cubature rules on the triangle by using symmetric polynomials to represent the two kinds of invariance inherent in these rules this method results in a system of polynomial equations that is amenable to solution using algebraic solving techniques using grobner bases rules of degree up to 15 are computed and presented some of them new and with all their points inside the triangle since all solutions to the polynomial system are computed it is for the first time possible to prove whether a given rule type results in specific rules of a given quality it is thus proved that for degrees up to 14 there are no non fortuitous rules that can improve on the presented results for degree 10 an example is also provided showing how the proposed method can be used to exclude the existence of better fortuitous rules as well

improved cubature formulae of high degrees of exactness for the square the method of constructing minimal cubature rules with high algebraic degrees of exactness is developed by adapting a powerful algorithm for solving the system of nonlinear equations as a result new cubature formulae of degrees 15 17 19 21 and 23 are derived for the square they lead to lower numbers of knots and or to better quality with respect to those known previously the formulae obtained should be considered as the most efficient for the calculation of two dimensional integrals with a high precision