Gauss Rule Research Articles

Computation of pairs of related Gauss-type quadrature rules

The evaluation of Gauss-type quadrature rules is an important topic in scientific computing. To determine estimates or bounds for the quadrature error of a Gauss rule often another related quadrature rule is evaluated, such as an associated Gauss-Radau or Gauss-Lobatto rule, an anti-Gauss rule, an averaged rule, an optimal averaged rule, or a Gauss-Kronrod rule when the latter exists. We discuss how pairs of a Gauss rule and a related Gauss-type quadrature rule can be computed efficiently by a divide-and-conquer method.

Averaged Gauss quadrature formulas: Properties and applications

The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, Spalević derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. Numerical experiments reported in this paper show both kinds of averaged rules to often give much higher accuracy than can be expected from their degrees of exactness. This is important when estimating the error in a Gauss rule by an associated averaged rule. We use techniques similar to those employed by Trefethen in his investigation of Clenshaw–Curtis rules to shed light on the performance of the averaged rules. The averaged rules are not guaranteed to be internal, i.e., they may have nodes outside the convex hull of the support of the measure. This paper discusses three approaches to modify averaged rules to make them internal.

A new representation of generalized averaged Gauss quadrature rules

Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an ℓ-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2ℓ+1)-node Gauss–Kronrod rule. However, Gauss–Kronrod rules with 2ℓ+1 real nodes might not exist. The (2ℓ+1)-node generalized averaged Gauss formula associated with the ℓ-node Gauss rule described in Spalević (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2ℓ+1)-node Gauss–Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation.

Cálculo numérico de la matriz de flexibilidades de vigas de sección variable, con elementos finitos

In this work, the flexibility properties of variable cross section beams are derived, through the application of the second theorem of Castigliano; considering the complementary energy by bending and share forces. To perform the integration of the flexibility coefficients, a numerical method, which considers the discretization of the beam domain with first order rectangular finite elements, in conjunction with the Gauss rule, is proposed. At the end of the work, the proposed method is applied to a tapered beam that has been discretized with a maximum of five finite elements. It is shown that the method is general, and that it can be applied to beams of variable section in which the cross section can be complex. The results shown that no more than 3 finite elements are needed to discretize the domain of beams in which, the ratio height-length is of the order of ten.

Error estimates for certain cubature formulae

We estimate the errors of selected cubature formulae constructed by the product of Gauss quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gauss quadrature rules and cubature formula constructed by the product of corresponding Gauss-Kronrod or corresponding generalized averaged Gaussian quadrature rules. Generalized averaged Gaussian quadrature rule ?2l+1 is (2l + 1)-point quadrature formula. It has 2l + 1 nodes and the nodes of the corresponding Gauss rule Gl with l nodes form a subset, similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l+1 associated with Gl. The advantages of bG2l+1 are that it exists also when H2l+1 does not, and that the numerical construction of ?2l+1, based on recently proposed effective numerical procedure, is simpler than the construction of H2l+1.

Simplified anti-Gauss quadrature rules with applications in linear algebra

The need to compute inexpensive estimates of upper and lower bounds for matrix functions of the form w T f(A)v with $A\in {\mathbb {R}}^{n\times n}$ a large matrix, f a function, and $v,w\in {\mathbb {R}}^{n}$ arises in many applications such as network analysis and the solution of ill-posed problems. When A is symmetric, u = v, and derivatives of f do not change sign in the convex hull of the spectrum of A, a technique described by Golub and Meurant allows the computation of fairly inexpensive upper and lower bounds. This technique is based on approximating v T f(A)v by a pair of Gauss and Gauss-Radau quadrature rules. However, this approach is not guaranteed to provide upper and lower bounds when derivatives of the integrand f change sign, when the matrix A is nonsymmetric, or when the vectors v and w are replaced by “block vectors” with several columns. In the latter situations, estimates of upper and lower bounds can be computed quite inexpensively by evaluating pairs of Gauss and anti-Gauss quadrature rules. When the matrix A is large, the dominating computational effort for evaluating these estimates is the evaluation of matrix-vector products with A and possibly also with A T . The calculation of anti-Gauss rules requires one more matrix-vector product evaluation with A and maybe also with A T than the computation of the corresponding Gauss rule. The present paper describes a simplification of anti-Gauss quadrature rules that requires the evaluation of the same number of matrix-vector products as the corresponding Gauss rule. This simplification makes the computational effort for evaluating the simplified anti-Gauss rule negligible when the corresponding Gauss rule already has been computed.

On generalized averaged Gaussian formulas. II

Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new $(2\ell +1)$-point quadrature rule $\widehat G_{2\ell +1}$, referred to as a generalized averaged Gaussian quadrature rule. This rule has $2\ell +1$ nodes and the nodes of the corresponding Gauss rule $G_\ell$ with $\ell$ nodes form a subset. This is similar to the situation for the $(2\ell +1)$-point Gauss-Kronrod rule $H_{2\ell +1}$ associated with $G_\ell$. An attractive feature of $\widehat G_{2\ell +1}$ is that it exists also when $H_{2\ell +1}$ does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of $\widehat G_{2\ell +1}$ is simpler than the construction of $H_{2\ell +1}$. A disadvantage might be that the algebraic degree of precision of $\widehat G_{2\ell +1}$ is $2\ell +2$, while the one of $H_{2\ell +1}$ is $3\ell +1$. Consider a (nonnegative) measure $d\sigma$ with support in the bounded interval $[a,b]$ such that the respective orthogonal polynomials, above a specific index $r$, satisfy a three-term recurrence relation with constant coefficients. For $\ell \ge 2r-1$, we show that $\widehat G_{2\ell +1}$ has algebraic degree of precision at least $3\ell +1$, and therefore it is in fact $H_{2\ell +1}$ associated with $G_\ell$. We derive some interesting equalities for the corresponding orthogonal polynomials.

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 2n+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 2n+2k−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.

Rational Gauss Quadrature

The existence of (standard) Gauss quadrature rules with respect to a nonnegative measure $d\mu$ with support on the real axis easily can be shown with the aid of orthogonal polynomials with respect to this measure. Efficient algorithms for computing the nodes and weights of an $n$-point Gauss rule use the $n\times n$ symmetric tridiagonal matrix determined by the recursion coefficients for the first $n$ orthonormal polynomials. Many rational functions that are orthogonal with respect to the measure $d\mu$ and have real or complex conjugate poles also satisfy a short recursion relations. This paper describes how banded matrices determined by the recursion coefficients for these orthonormal rational functions can be used to efficiently compute the nodes and weights of rational Gauss quadrature rules.

Using Green's functions and split Gaussian integration for two-point boundary value problems on the ray

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green's functions, building on the successful use of product integration to achieve the convergence expected of Gauss-type quadrature schemes. We introduce refinements such as the use of cardinal basis functions to eliminate the need for a transformation from the ordinates to the expansion coefficients. For problems on the ray, we algebraically map the ray to a segment, and there use (cardinal) Legendre polynomials for interpolation and Gauss's rule for quadrature. Numerical examples (the heat and Burgers equations) demonstrate the applicability of the method to problems on the ray, particularly for the sequence of two-point boundary value problems arising from constant time-stepping for parabolic evolution problems; nonlinear terms, as in the Burgers equation, present no special difficulty.

Degree optimal average quadrature rules for the generalized Hermite weight function

For the practical estimation of the error of Gauss quadrature rules Gauss-Kronrod rules are widely used; but, it is well known that for the generalized Hermite weight function, ωα(x )= |x| 2α exp(−x 2 ) over [−∞, ∞], real positive Gauss-Kronrod rules do not exist. Among the alternatives which are available in the literature, the anti-Gauss and average rules introduced by Laurie, and their modified versions, are of particular interest. In this paper, we investigate the properties of the modified anti-Gauss and average quadrature rules for ωα , and we determine the degree optimal average rules by proving that for each n-point Gauss rule for ωα there exists a unique average rule with the precise degree of exactness 2n+3. We also give some numerical examples to test the performance of the average rules obtained in this paper. Mathematics Subject Classification: 65D30, 65D32

Two-frequency-dependent Gauss quadrature rules

We construct two-frequency-dependent Gauss quadrature rules which can be applied for approximating the integration of the product of two oscillatory functions with different frequencies β 1 and β 2 of the forms, y i(x)=f i,1(x) cos(β ix)+f i,2(x) sin(β ix), i=1,2, where the functions f i, j ( x) are smooth. A regularization procedure is presented to avoid the singularity of the Jacobian matrix of nonlinear system of equations which is induced as one frequency approaches the other frequency. We provide numerical results to compare the accuracy of the classical Gauss rule and one- and two-frequency-dependent rules.

Gauss integration applied to a Green's function formulation for cylindrical fiber composites

The stress in an elastic two phase material is formulated in terms of eigenstrains and Green's functions leading to a singular integral equation. The singularity is dealt with by a subtraction technique and contour integrations, one of which corresponds to the Eshelby solution. The remaining non-singular integration is evaluated numerically using a Gauss rule. The calculated local stresses are within 1% of the analytical solution for an isolated inclusion problem. Also, the effective properties as a function of volume fraction are in good agreement with Christensen's analytical result. The method is then used to study rectangular versus square packing. The volume averaged stiffness for rectangular packing may be either higher or lower than that for square packing, depending on the loading direction. The stress concentration factor is shown to increase with fiber volume fraction. The stress concentration factor also increases at a decreasing rate as the fiber to matrix stiffness ratio is increased. All changes examined that an increase in the stiffness, whether by increased fiber volume fraction, increased fiber stiffness or an alternate packing arrangement, also increase the fiber stress concentration factor.

A Quadrature-Based Approach to Improving the Collocation Method for Splines of Even Degree

The "qualocation" method, a recently proposed quadrature-based extension of the collocation method, is here applied to a class of boundary integral equations, using an even degree spline trial space on a uniform partition. The problems handled are of the form (L + K)u = f where L is a convolutional operator with even symbol, and K is an operator with a greater smoothing effect than L . For a trial space of dimension n , it is shown that a certain 2n -point quadrature rule, which is a generalization of the repeated 2-point Gauss rule, gives a stable qualocation method, and yields an order of convergence, in suitable negative norms, two powers of h higher than achieved by the mid-point collocation method in the recent analysis of Saranen. The treatment of the smooth perturbation covers also the earlier analysis of the odd degree spline case by Sloan.

Rates of Convergence of Gauss, Lobatto, and Radau Integration Rules for Singular Integrands

Rates of convergence (or divergence) are obtained in the application of Gauss, Lobatto, and Radau integration rules to functions with an algebraic or logarithmic singularity inside, or at an endpoint of, the interval of integration. A typical result is the following: For a generalized Jacobi weight function on (-1,1), the error in applying an n-point rule to f(x) = Ix - VI-6 is O(n-2+28), if y = +1 and O(n-1+'6) if y E (-1,1), providedwe avoid the singularity. If we ignore the singularity y, the error is O( n-1 + 28(log n )6(log log n )?(1 e)) for almost all choices of y. These assertions are sharp with respect to order. 1. Introduction. This paper is a sequel to that of Lubinsky and myself (2) on the rates of convergence of Gauss integration rules for singular integrands. In this paper we extend the results in (2) in several directions. On the one hand, we extend the results for Gauss integration rules to results for Lobatto and Radau rules. This extension follows easily from the generalized Markov-Stieltjes inequality for Lobatto and Radau rules given in Lemma 3.2 and from a representation of the coefficients (or weights or Cotes numbers) in the Lobatto and Radau rules in terms of the coefficients in the Gauss rule with respect to a related weight function. A second extension is to the special case of Gauss-Jacobi integration rules as well as Lobatto- Jacobi and Radau-Jacobi rules for certain values of the parameters defining the Jacobi weight function

A stable Gauss-Kronrod algorithm for cauchy principal-value integrals

It is shown that the ratio of the precision of the stable Kronrod extension to the precision of the stable version of the Gauss rule for Cauchy principal-value integrals is approximately the same as the ratio of the precision of the Kronrod extension to that of the Gauss rule for ordinary integrals.

Numerically integrated finite elements as assemblies of bars

An eight node plane stress element is formulated using a system of bars. These bars are located at the Gauss points and their widths are equal to the weights of the Gauss rule considered. It is shown that an ( n × n) integration rule yields, for the stiffness matrix, exactly the same values as a grid of ( n × n) bars located at the corresponding Gauss points. This analogy is used to justify the existence of mechanisms with underintegration.

Viscoelastic analysis of nearly incompressible solids

The finite element method is well established for the analysis of structures and other field problems. However, its straightforward application for the analysis of nearly incompressible solids yields erratic results. In the present work, an efficient special purpose code for the Viscoelastic Analysis of Nearly Incompressible Solids (VANIS) is developed using isoparametric elements with selective integration procedure, which is a third order Gauss rule for deviatoric response and second order Gauss rule for volumetric response. The software can be effectively employed for the structures with lower Poisson's ratios. VANIS is based on the direct formulation using linear uncoupled thermoviscoelastic theory for the thermorheologically simple materials. The element library consists of 8-noded plane strain, 8-noded axisymmetric solid and 20-noded three dimensional quadratic isoparametric elements. These elements meet all the possible structural idealisation requirements of the solid continua. Experimentally obtained rigidity modulus can be used directly or expressing it in Prony series. The software is tested on a number of problems and gives very accurate results for all the permissible values of the Poisson's ratio.

Stream function finite element solution of Stokes flow with penalties

AbstractA finite element method for solution of the stream function formulation of Stokes flow is developed. The method involves complete cubic non‐conforming (C0) triangular Hermite elements. This element fails the patch test. To correct the element and produce a convergent method we employ a penalty method to weakly enforce the desired continuity constraint on the normal derivative across the inter‐element boundaries. Successful use of the method is demonstrated to require reduced integration of the inter‐element penalty with a 1‐point Gauss rule. Error estimates relate the optimal choice of penalty parameter to mesh size and are corroborated by numerical convergence studies. The need for reduced integration is interpreted using rank relations for an associated hybrid method.

An 8‐node brick finite element

AbstractAn 8‐node brick element based on the assumed stress hybrid formulation is described. With three additional stress fields, the element stiffness matrix now has the required rank of 18, and the ‘bending’ response is exact with rectangular elements. Surprisingly, the 2 × 2 × 2 Gauss rule suffices for all numerical integrations, to satisfy the constant stress patch test.