Construction Of Cubature Formulae Research Articles

Constructing Steklov-type cubature formulas for a finite element in the shape of a bipyramid

This paper reports the construction of cubature formulas for a finite element in the form of a bipyramid, which have a second algebraic order of accuracy. The proposed formulas explicitly take into consideration the parameter of bipyramid deformation, which is important when using irregular grids. The cubature formulas were constructed by applying two schemes for the location of interpolation nodes along the polyhedron axes: symmetrical and asymmetrical. The intervals of change in the elongation (compression) parameter of a bipyramid semi-axis have been determined, within which interpolation nodes of the constructed formulas belong to the integration region, while the weight coefficients are positive, which warrants the stability of calculations based on these cubature formulas. If the deformation parameter of the bipyramid is equal to unity, then both cubature formulas hold for the octahedron and have a third algebraic order of accuracy. The resulting formulas make it possible to find elements of the local stiffness matrix on a finite element in the form of a bipyramid. When calculating with a finite number of digits, a rounding error occurs, which has the same order for each of the two cubature formulas. The intervals of change in the elongation (compression) parameter of the bipyramid semi-axis have been determined, which meet the requirements, which are employed in the ANSYS software package, for deviations in the volume of the bipyramid from the volume of the octahedron. Among the constructed cubature formulas for a bipyramid, the optimal formula in terms of the accuracy of calculations has been chosen, derived from applying a symmetrical scheme of the arrangement of nodes relative to the center of the bipyramid. This formula is invariant in relation to any affinity transformations of the local bipyramid coordinate system. The constructed cubature formulas could be included in libraries of methods for approximate integration used by those software suites that implement the finite element method.

Monte Carlo cubature construction

In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and existing examples only represent the particular domains of integrands, such as hypercubes and spheres. In this study, we show that cubature formulas can be constructed for probability measures provided that we have an i.i.d. sampler from the measure and the mean values of given test functions. Moreover, the proposed method also works as a means of data compression, even if sufficient prior information of the measure is not available.

Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

Construction of cubature formulas with fourth and fifth degrees by the reproducing kernel and Radon methods

Construction of cubature formulas with fourth and fifth degrees by the reproducing kernel and Radon methods

Justification of the Collocation Method for a Class of Surface Integral Equations

The application of the collocation method to the boundary integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation is justified. In addition, a new method for constructing cubature formulas for surface singular integrals is proposed.

Построение кубатурных формул на сфере, согласованных с правильной гексагональной решеткой

Построение кубатурных формул на сфере, согласованных с правильной гексагональной решеткой

Cubature and Quadrature Formulas of High Order of Approximation

This paper is concerned with the use of semilocal smoothing splines (or S-splines) for constructing cubature formulas. Such a spline is a piecewise-polynomial function. The first coefficients of each of the polynomials are determined by the smooth joint conditions, and the remaining ones, by the least-squares method. Previous studies were concerned with splines of degree 3 and 5. In the present paper, we consider S-splines of degree n (n = 9, 10). Of special importance for calculation of integrals are the S-splines of class C 0 (the continuous ones). Such splines are employed in building quadrature and cubature formulas of high order of approximation for calculation of one-, two-, and three-dimensional integrals in a simply connected domain to 10th and 11th orders of approximation. The integrable function is assumed to lie in the class C (n+1) (n = 9, 10) in a somewhat larger domain than the original one (in which the integration takes place). It is also assumed that the boundary of the domain is given parametrically. This makes it possible to take into account, with high order of accuracy, the boundary of the domain. The corresponding convergence rates are estimates. A similar approach is also capable of building formulas for integration of smooth functions in multidimensional domains.

Remarks on Hilbert identities, isometric embeddings, and invariant cubature

Victoir (2004) developed a method to construct cubature formulae with various combinatorial objects. Motivated by this, we generalize Victoir's method with one more combinatorial object, called regular t-wise balanced designs. Many cubature of small indices with few points are provided, which are used to update Shatalov's table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.

Weighted spherical semidesigns and cubature formulae for calculating integrals on a sphere

In this paper we study weighted spherical semidesigns, i.e., systems of points on a sphere of a specific type. We propose a new proof of the necessary and sufficient condition for a system of points on a sphere to be a weighted spherical semidesign. This criterion gives new approaches to the construction of cubature formulae for calculating integrals over a sphere with the degree of accuracy of 5 and 9.

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 n2+5n+3 points and for n-dimensional spherically symmetrical region contains only n2+3n+3 points. Moreover, the numbers can be reduced to n2+3n+1 and n2+n+1 if n=7 respectively, the latter of which is minimal.

Cubature formulas on a sphere invariant under the icosahedral rotation group

An algorithm is proposed for constructing cubature formulas on a sphere that are invariant under the icosahedral rotation group. The algorithm is applied to construct new cubature formulas of algebraic order of accuracy n = 19, 20, 21, 23, 24, 25. Parameters of these cubature formulas are given to 16 significant digits. A table, which containsmain characteristics of all optimal cubature formulas for the icosahedral rotation group exact to order 35, is included. A real version of F. Klein’s formula, which establishes a connection between basic invariant forms of the icosahedral rotation group, is presented.

Construction of cubature formulae with preassigned nodes§

In this article, a technique for developing cubature rules with preassigned nodes is presented to avoid wasting of information in scientific computation. The corresponding constructive method of the cubature rule is also given. As an application of the rules, a cubature formula on disk, which was derived via the method of reproducing kernel in (Xu, Y., 2000, Constructing cubature formulae by the method of reproducing kernel. Numerische Mathematik, 85, 155–173), is reconstructed by using our technique. When the preassigned nodes are selected as the nodes of a cubature formula of lower degree, an embedded cubature formula can be easily obtained by the presented method. Furthermore, some examples are included in the article. §This paper is dedicated to the 70th birthday of Prof. Zuhair Nashed.

Cubature formulas for a disk that are invariant under groups of transformations of regular polygons into themselves

A method for constructing cubature formulas of different degrees of accuracy that are invariant under the groups of transformations of regular polygons into themselves is proposed. The cubature sum can contain powers of the Laplace operator applied to the integrand at the origin.

Cubature formulas over the [formula omitted]-sphere

We examine the method of Cartesian product to construct cubature formulae on the unit sphere and obtain a kind of new Lobatto formulae. Furthermore, by constraining nodes to lie on orthogonal polynomials, an alternative method of producing new cubature formulae on the spherically symmetrical region is also presented. Finally, some new cubature formulae are derived.

Asymmetric Cubature Formulae with Few Points in High Dimension for Symmetric Measures

Let $\mu$ be a positive measure on $\mathbb{R}^d$ invariant under the group of reflections and permutations, and let m be a natural number. We describe a method to construct cubature formulae of degree m with respect to $\mu$, with n positive weights and n points in the support of $\mu$ and such that n grows at most like dm with the dimension d. We apply this method to classical measures to explicitly construct cubature formulae of degree 5 with the number of points growing at most like d3 .

On generalized invariant cubature formulae

An important quality criterion of cubature formulae is their algebraic or trigonometric degree of exactness. The invariant theory is a powerful tool to construct cubature formulae of a given degree. In this paper, a quantitative expression is established for the classical invariant cubature formulas (ICFs). Motivated by this expression (or structure), we generalize the concept of ICFs and extend the famous Sobolev's Theorem on ICFs. The transformations allowed are no longer just orthogonal transformations. We illustrate the concepts and the constructions of the generalized ICFs by several examples.

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.

Gaussian extended cubature formulae for polyharmonic functions

The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. This allows us to construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula that uses m values of linear functionals (integrals over hyperspheres) and is exact for all 2m-harmonic functions, and consequently, for all algebraic polynomials of n variables of degree 4m - 1.

Polynomial interpolation in several variables, cubature formulae, and ideals[*]Supported by the National Science Foundation under Grant DMS-9802265.

We discuss polynomial interpolation in several variables from a polynomial ideal point of view. One of the results states that if I is a real polynomial ideal with real variety and if its codimension is equal to the cardinality of its variety, then for each monomial order there is a unique polynomial that interpolates on the points in the variety. The result is motivated by the problem of constructing cubature formulae, and it leads to a theorem on cubature formulae which can be considered an extension of Gaussian quadrature formulae to several variables.

An interactive tool to visualize common zeros of two-dimensional polynomials

In this note an interactive program is presented which allows the visualization of the common roots of two-dimensional polynomials. The program is useful when constructing cubature formulas by use of orthogonal polynomials.