Gauss-Legendre Quadrature Rule Research Articles

An Efficient Semianalytical Method for Hypersingularity Treatment Over Curved Patches

In this communication, we propose an efficient method to evaluate hypersingular integrals defined on curved surfaces. First an exact expression for hypersingular kernel is derived by projecting the integral on curvilinear element on a flat surface. Next singularity subtraction employed, where the singular core is hypersingular and the remaining part is weakly singular. The singular core is evaluated analytically using finite part interpretation and the remaining weakly singular part is evaluated numerically using Gauss–Legendre quadrature rules. By numerical experiments we have shown that the convergence rate of the purposed method is quite high even for few number of quadrature nodes. Accuracies over ten digits are obtained for relatively large and highly curved surfaces, which may cover entire domain of local corrections in Nystrom method.

The 35-Moment System with the Maximum-Entropy Closure for Rarefied Gas Flows

This paper presents a robust implementation of the maximum-entropy closure in the context of rarefied gas dynamics. Moment systems supplied with the maximum-entropy closure have attractive mathematical properties: They are hyperbolic in the interior of the domain of definition of the dual minimization problem and endowed with an entropy law. In contrast to Grad’s classical closure theory, the maximum-entropy closure allows for applications to strongly non-equilibrium gas flows. The 35-moment system studied in this paper includes as basis functions all monomials up to order four, so that evolution equations for important non-equilibrium quantities, such as the stress tensor and heat flux vector, are contained in the system. To remove the singularity in the maximum-entropy closure, we consider a bounded underlying velocity domain and approximate moments of the reconstructed maximum-entropy distribution with a fixed, block-wise Gauss–Legendre quadrature rule. The convex dual minimization problem is solved with a Newton type algorithm. We show that the Hessian matrix used in the Newton iteration can become ill-conditioned even for equilibrium states if monomial basis functions are used. To improve the robustness of the Newton iteration, we consider partially and fully adaptive basis algorithms and demonstrate that the 35-moment system allows for accurate and robust simulations of non-equilibrium rarefied gas flows in the transition regime by applying the model to one-dimensional gas processes, including a continuous shock-structure problem.

The numerical solution of nonlinear integral equations of the second kind using thin plate spline discrete collocation method

In this article, the thin plate splines are given to obtain the numerical solution of nonlinear Fredholm integral equations of the second kind. The scheme approximates the solution using the discrete collocation method based on the shape functions of thin plate splines constructed on a set of scattered points. The thin plate splines can be seen as a type of the free shape parameter radial basis functions which are used to establish an effective technique for the interpolation of an unknown function. The numerical method developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to compute its integrals. Since the proposed scheme does not require any background mesh for approximations and numerical integrations, it is meshless. This approach can be easily implemented and its algorithm is simple and effective to solve nonlinear integral equations. Moreover the error estimate and the convergence rate of the method are presented. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

A Mixed Quadrature Rule by Blending Clenshaw-Curtis and Gauss-Legendre Quadrature Rules for Approximate Evaluation of Real Definite Integrals in Two Dimensions

A Mixed Quadrature Rule by Blending Clenshaw-Curtis and Gauss-Legendre Quadrature Rules for Approximate Evaluation of Real Definite Integrals in Two Dimensions

Functional approximation and projection of stored energy functions in computational homogenization of hyperelastic materials: A probabilistic perspective

This work is concerned with the construction of a surrogate model for the homogenized stored energy functions defining the effective behavior of nonlinear elastic microstructures. Here, a probabilistic standpoint is adopted and allows for the definition of a nonlinear mapping between the macroscopic deformations and the homogenized potential. This functional approximation is specifically obtained by means of a polynomial chaos expansion, the coefficients of which are computed through a Gauss–Legendre quadrature rule. By invoking well-known results related to projections in Hilbert spaces, closest approximations of arbitrary potentials to standard (e.g. Ogden-type) models are subsequently defined and characterized by appropriate residuals. Numerical illustrations on various microstructures are finally provided in order to discuss the relevance of the proposed framework. In particular, it is shown that the surrogate model compares very well with reference results at both the macroscale and the structural scale, even for moderate-order expansions, and that the aforementioned closest approximations constitute very accurate approximation provided that the subset onto which the homogenized response is projected is properly chosen. This result readily allows for nonconcurrent coupling using standard constitutive models available in commercial codes and therefore makes the approach very attractive for engineering applications.

Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition

The element free Galerkin technique is a meshless method based on the variational weak form in which the test and trial functions are the shape functions of moving least squares approximation. Since the shape functions of moving least squares approximation do not have the Kronecker property thus the Dirichlet boundary condition can not be applied directly and also in this case obtaining an error estimate is not simple. The main aim of the current paper is to propose an error estimate for the extracted numerical scheme from the element free Galerkin method. To this end, we select the fractional cable equation with Dirichlet boundary condition. Firstly, we obtain a time-discrete scheme based on a finite difference formula with convergence order O(τ1+min⁡{α,β}), then we use the meshless element free Galerkin method, to discrete the space direction and obtain a full-discrete scheme. Also, for calculating the appeared integrals over the boundary and the domain of problem the Gauss–Legendre quadrature rule has been used. In the next, we change the main problem with Dirichlet boundary condition to a new problem with Robin boundary condition. Then, we show that the new technique is unconditionally stable and convergent using the energy method. We show convergence orders of the time discrete scheme and the full discrete scheme are O(τ1+min⁡{α,β}) and O(rp+1+τ1+min⁡{α,β}), respectively. So, we can say that the main aim of this paper is as follows, (1) Transferring the main problem with Dirichlet boundary condition (old problem) to a problem with Robin boundary condition (new problem), (2) Showing that with special condition (when σ→+∞) the solution of the new problem is convergent to the solution of the old problem, (3) Obtaining an error estimate for the new problem. Numerical examples confirm the efficiency and accuracy of the proposed scheme.

Quadrature algorithms for high dimensional singular integrands on simplices

Galerkin discretizations of integral operators in ?d$\mathbb {R}^{d}$ require an accurate numerical evaluation of integrals I=?S(1)?S(2)f(x,y)dydx$I={\int }_{\!\!S^{(1)}}{\int }_{\!\!S^{(2)}}f(x,y)dydx$ where S(1), S(2) are d-simplices and the integrand function f has a possibly nonintegrable singularity at x = y. In a previous paper (2011) we constructed several families of quadrature rules QN${Q}_{\mathcal {N}}$ for a class of functions f which allow algebraic singularities at x = y, including hypersingular kernels, and are Gevrey smooth for x ? y. This holds for kernel functions commonly occurring in integral equations. In this paper we address the implementation aspects for computing QN$Q_{\mathcal {N}}$ and give a detailed computation algorithm for arbitrary space dimension d and arbitrary mutual location of simplices S(1) and S(2). The algorithm consists of a "desingularizing" coordinate transformation, which reduces the singular support of the integrand to one variable while preserving Gevrey regularity in all 2d?1 remaining variables and simultaneously simplifies the integration domain to a unit cube. Due to the simple singularity structure after transformation, one can optionally use various combinations of quadrature rules in singular and regular directions. We report on comprehensive convergence studies for the full tensor product and Smolyak-type quadrature rules in the 2d?1 Gevrey-regular variables combined with either composite Gauss-Legendre or Gauss-Jacobi quadrature rules in the singular direction. A Matlab software implementing the algorithm complements this paper and is available from Netlib.

Error estimate for the numerical solution of fractional reaction–subdiffusion process based on a meshless method

In this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction–subdiffusion equation. Firstly, we obtain a time discrete scheme based on a finite difference scheme, then we use the meshless Galerkin method, to approximate the spatial derivatives and obtain a full discrete scheme. In the proposed scheme, some integrals appear over the boundary and the domain of problem which will be approximated using Gauss–Legendre quadrature rule. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method. We show convergence order of the time discrete scheme is O(τγ). The aim of this paper is to obtain an error estimate and to show convergence for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.

A seven-parameter spectral/hp finite element formulation for isotropic, laminated composite and functionally graded shell structures

In this paper we apply high-polynomial order spectral/hp basis functions in the numerical implementation of a novel 7-parameter continuum shell finite element formulation using general quadrilateral finite elements. The implementation is applicable to the analysis of isotropic, functionally graded and laminated composite shells undergoing fully geometrically nonlinear mechanical response. The shell finite element formulation is constructed in a purely displacement-based setting such that no mixed variational procedures are employed and full numerical integration of all quantities appearing in the virtual work statement is performed using high-order Gauss–Legendre quadrature rules. An efficient procedure for numerically integrating the discrete weak formulation through the shell thickness is implemented; hence no thin- or shallow-shell type restrictions are imposed on the element. For the case of laminated composites, we introduce a discrete tangent vector field, defined on the approximate shell mid-surface, that permits the use of skewed and/or arbitrarily curved elements in the numerical simulation of complex shell structures. To reduce computer memory requirements in the numerical implementation we adopt element-level static condensation, wherein, the interior degrees of freedom of each element are implicitly eliminated prior to assembly of the global sparse finite element coefficient matrix; an approach that results in storage requirements that are on par with traditional low-order finite element implementations. The accuracy and overall robustness of the developed shell element are illustrated through the solution of several nontrivial benchmark problems taken from the literature. These numerical studies provide evidence that the proposed shell element may be used to obtain accurate locking-free results and that large load increments can be employed (in the numerical simulation of shell structures undergoing very large deformations).

A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels

This paper describes a computational method for solving Fredholm integral equations of the second kind with logarithmic kernels. The method is based on the discrete Galerkin method with the shape functions of the moving least squares (MLS) approximation constructed on scattered points as basis. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least square polynomial fitting. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule for approximating logarithm-like singular integrals and so reduces the solution of the logarithmic integral equation to the solution of a linear system of algebraic equations. The proposed method is meshless, since it does not require any background mesh or domain elements. The error analysis of the method is provided. The scheme is also applied to a boundary integral equation which is a reformulation of a boundary value problem of Laplace’s equation with linear Robin boundary conditions. Finally, numerical examples are included to show the validity and efficiency of the new technique.

A Mixed Quadrature Rule by Blending Clenshaw-Curtis and Gauss-Legendre Quadrature Rules for Approximate Evaluation of Real Definite Integrals

A mixed quadrature rule blending Clenshaw-Curtis five point rule and Gauss-Legendre three point rule is formed. The mixed rule has been tested and found to be more effective than that of its constituent Clenshaw-Curtis five point rule.

Semi-analytic treatment of nearly-singular Galerkin surface integrals

On utilizing piecewise linear test and interpolation functions over flat triangles, three semi-analytic techniques for evaluating nearly-singular surface integrals of potential theory are developed and investigated in the context of a three-dimensional Galerkin approximation. In all procedures, the inner surface integral is calculated exactly via recursive expressions defined over the edges of the integration triangle. In addition, the proposed methods respectively employ a Duffy transformation, sinh transformations, and polynomial transformations to weaken or smooth out near singularities, followed by a standard product Gauss–Legendre quadrature rule to compute the outer boundary integral. Numerical experiments and comparisons of the proposed approaches are included to provide more insight into the performance of these techniques. Computational tests have demonstrated that the algorithm based on a Duffy transformation is the most efficient and effective method for dealing with nearly-singular as well as well-separated integrals.

On Stable, Complete, and Singularity-Free Boundary Integral Formulations of Exterior Stokes Flow

A new boundary integral formulation of the second kind for exterior Stokes flow is introduced. The formulation is stable, complete, singularity-free, and natural for bodies of complicated shape and topology. We prove an existence and uniqueness result for the formulation for arbitrary flows and illustrate its performance via several numerical examples using a Nyström method with Gauss–Legendre quadrature rules of different order.

Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities

Let /[/] = jh1f(x)dx, where / € C°°(-l,l), and let Gn[f] = SiLi wmf(xni) be the n-point Gauss-Legendre quadrature approximation to /[/]. In this paper, we derive an asymptotic expansion as n ► oo for the error En[f] = I[f] Gn[f] when f(x) has general algebraic-logarithmic singularities at one or both endpoints. We assume that f(x) has asymptotic expansions of the forms f(x)~jr,U8(\og(l-x))(l-xr° ass-1-, s=0

The kink phenomenon in Fejér and Clenshaw–Curtis quadrature

The Fejer and ClenshawCurtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like O(ρ-2N), where ρ is a constant greater than 1. For these values of N the accuracy of both the Fejer and ClenshawCurtis rules is almost indistinguishable from that of the more celebrated GaussLegendre quadrature rule. For larger N, however, the error decreases at the rate O(ρ-N), i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.

On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region

In this paper we first present a Gauss-Legendre quadrature rule for the evaluation of I = f f T ff(x,y,z)dxdydz, where f(X,y,z) is an analytic function in x, y, z and Tis the standard tetrahedral region: {(x,y,z)|0 ≤ x,y,z ≤ I, x + y + z ≤ 1} in three space (x,y,z). We then use a transformation x = x(ξ,η,ζ), y = y(ξ,η,ζ) and z = z(ξ,η,ζ) to change the integral into an equivalent integral I =∫ 1 -1 ∫ 1 -1 ∫ 1 -1 f(x(ξ,η,ζ), y(ξ,η,ζ,)) ∂(x,y,z) ∂(ξ,η,ζ) dξdηdζ over the standard 2-cube in (ξ,η,ζ) space: {(ξ,η,ζ)|-1 ≤ ξ,η,ζ ≤ 1}. We then apply the one-dimensional Gauss-Legendre quadrature rules in ξ, η and ζ variables to arrive at an efficient quadrature rule with new weight coefficients and new sampling points. Then a second Gauss-Legendre quadrature rule of composite type is obtained. This rule is derived by discretising the tetrahedral region T into four new tetrahedra T c i (i= 1,2,3,4) of equal size which are obtained by joining the centroid of T, c = (1/4,1/4,1/4) to the four vertices of T. By use of the affine transformations defined over each T c i and the linearity property of integrals leads to the result: I = Σ4 i=1 ∫∫Tci∫f (x,y,z) dxdydz = 1 4 ∫∫T ∫ G(X,Y,Z)dXdYdZ, where G(X, Y,Z) = 1 p 3 Σ f(x (k) (X, Y,Z), y (k) (X, Y,Z),z (k) (X, Y, Z)), x (k) = x (k) (X,Y,Z), y (k) =y (k) (X,Y,Z) and z (k) = z (k) (X, Y,Z) refer to an affine transformations which map each T c i into the standard tetrahedral region T. We then write I = ∫∫ T ∫ G(X,Y,Z)dXdYdZ = ∫ 1 0 ∫ 1-ξ 0 ∫ 1-ξ-η 0 G(X(ξ,η,ζ), Y(ξ,η, ζ), Z(ξ,η,ζ)) |∂(X,Y,Z)| ∂(ξ,η,ζ) dξdηdζ and a composite rule of integration is thus obtained. We next propose the discretisation of the standard tetrahedral region T into p 3 tetrahedra T i (i = 1(1)p 3 ) each of which has volume equal to 1 /(6p 3 ) units. We have again shown that the use of affine transformations over each T, and the use of linearity property of integrals leads to the result: ∫∫ T ∫f(x,y,z)dxdydz= Σ p3 i=1 ∫∫ T c i ∫f(x,y,z)dxdydz = Σ p3 α=1 ∫ ∫ T[p]α ∫f(x (α,p) , y (α,p) , z (α,p) )dx (α,p) dy (α,p) dz (α,p) = 1 p3 ∫ ∫ T ∫H(X,Y,Z)dXdYdZ, where p3 H(X, Y,Z) = Σ α=1 f(x (α,p) (X Y,Z),y (α,p) (X, Y, Z), z (α,p) (X, Y,Z)), x (α,P) = x (α,P) (X,Y,Z), y (α,P) =y (α,P) (X,Y,Z) and z (α,p) = z (α,P) (X, Y, Z) refer to the affine transformations which map each T i in (x (α,P) ,y (α,P) , z (α,P) ) space into a standard tetrahedron T in the (X, Y,Z) space. We can now apply the two rules earlier derived to the integral ∫∫ T ∫H(X, Y,Z)dXdYdZ, this amounts to the application of composite numerical integration of T into p3 and 4p3 tetrahedra of equal sizes. We have demonstrated this aspect by applying the above composite integration method to some typical triple integrals.

THE TRAPEZOIDAL RULE WITH A NONLINEAR COORDINATE TRANSFORMATION FOR WEAKLY SINGULAR INTEGRALS

It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation <TEX>$\Omega$</TEX><TEX>$_{m}$</TEX>(b;<TEX>$\chi$</TEX>), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation <TEX>$\Omega$</TEX><TEX>$_{m}$</TEX>(b;<TEX>$\chi$</TEX>), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation <TEX>$\Omega$</TEX><TEX>$_{m}$</TEX>(b;<TEX>$\chi$</TEX>).TEX>).

A non‐linear co‐ordinate transformation for accurate numerical evaluation of weakly singular integrals

AbstractIn order to obtain accurate numerical evaluation of weakly singular integrals, we propose a new non‐linear co‐ordinate transformation of the sigmoidal type. The asymptotic behaviour of the present transformation near a singular point is governed by an additional parameter b≠0. Moreover, an analytical form of its inverse is derived, which is available for the case of interior‐point singularity. In the result the presented transformation can improve asymptotic truncation errors of the Gauss–Legendre quadrature rule for weakly singular integrals.By comparing the present transformation with existing sigmoidal transformations in numerical examples, we show that the former with a proper value of b can result in more accurate evaluation than the others. Copyright © 2004 John Wiley &amp; Sons, Ltd.

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule ∫ −1 1f(x) dx≃∑ i=1 nw if(x i) has the highest possible precision degree and is analytically exact for polynomials of degree at most 2 n−1, where nodes x i are zeros of Legendre polynomial P n ( x), and w i 's are corresponding weights. In this paper we are going to estimate numerical values of nodes x i and weights w i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ϵ 0, say ϵ 0=10 −8, for monomial functions f(x)=x j, j=0,1,…,2n+1. (Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.

Numerical stability analysis of steady state solutions of integral equations with distributed delays

This paper concerns the numerical stability analysis of scalar delay integral equations (DIEs). We discretize a DIE using a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate properties of this discretization in the context of stability analysis. Conditions are obtained under which the discretized equation retains certain stability properties of the original equation. Results of numerical experiments on computing stability of DIEs are presented.