Gaussian Rules Research Articles

An extension of composite bending strain κMITC3+ scheme in analysis of shell structures reinforced with rib stiffeners

In this paper, the composite bending strain κMITC3+ approach is extended for analyzing the shell structural domain in the rib-stiffened shell problems. Developed to achieve enhanced performance and emphasize a softer bending behavior, this novel element employs a polynomial projection technique to construct an assumed composite bending strain field. This technique originates from the Hu-Washizu three-filed principle and the orthogonality condition. In the membrane component context, an improved Allman-like triangular element is utilized to remove the spurious energy mode commonly associated with classical Allman's theory. To address the shear-locking phenomenon within Timoshenko's beam model for the rib stiffeners, a selective/reduced integration approach is adopted. This involves using a standard two-point Gaussian rule for accurate bending strain computation and a one-point Gaussian rule for shear strain computation. The numerical examples showcased in our study unequivocally highlight the distinct superiority of the κMITC3+ element.

Gaussian rule for integrals involving Bessel functions

Gaussian rule for integrals involving Bessel functions

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

Improved enrichments and numerical integrations in SGFEM for interface problems

Generalized/extended finite element methods (GFEM/XFEM) for an interface problem are generally enriched by a distance function to the interface to handle discontinuities across the interface. Evaluations of the distance function and its gradients are needed for assembling stiffness matrices, which requires certain computational geometry algorithms. The computational complexity grows if the number of integration points is increased. Besides, the mathematical analysis on the effect of numerical integration has not been established in the field of GFEM/XFEM yet. This study designs an improved enriched function for a stable GFEM (SGFEM, a stable version of GFEM) based on the distance function, which only requires evaluating the distance function at nodes of elements that interact the interface (the gradients of distance function are not needed). The optimal convergence order (in the energy norm) of the SGFEM with such an improved enriched function is proven. In turn, we propose a perturbed variational formula, which can be integrated exactly by simple Gaussian rules designed in this paper. We prove that the SGFEM based on the improved enriched function and the perturbed variational formula (exactly integrated by the numerical integration) achieves the optimal convergence order for the interface problem. Namely, we develop a numerical integration rule for the SGFEM such that the optimal convergence order of SGFEM is maintained for the interface problem. The theoretical achievements are verified by numerical experiments.

Radau- and Lobatto-type averaged Gauss rules

We describe numerical methods for the construction of interpolatory quadrature rules of Radau and Lobatto types. In particular, we are interested in deriving efficient algorithms for computing optimal averaged Gauss–Radau and Gauss–Lobatto type javascript:undefined;quadrature rules. These averaged rules allow us to estimate the quadrature error in Gauss–Radau and Gauss–Lobatto quadrature rules. This is important since the latter rules have higher algebraic degree of exactness than the corresponding Gauss rules, and this makes it possible to construct averaged quadrature rules of higher algebraic degree of exactness than the corresponding “standard” averaged Gauss rules available in the literature.

Solving boundary value problems via the Nyström method using spline Gauss rules

We propose to use spline Gauss quadrature rules for solving boundary value problems (BVPs) using the Nyström method. When solving BVPs, one converts the corresponding partial differential equation inside a domain into the Fredholm integral equation of the second kind on the boundary in the sense of boundary integral equation (BIE). The Fredholm integral equation is then solved using the Nyström method, which involves the use of a particular quadrature rule, thus, converting the BIE problem to a linear system. We demonstrate this concept on the 2D Laplace problem over domains with smooth boundary as well as domains containing corners. We validate our approach on benchmark examples and the results indicate that, for a fixed number of quadrature points (i.e., the same computational effort), the spline Gauss quadratures return an approximation that is by one to two orders of magnitude more accurate compared to the solution obtained by traditional polynomial Gauss counterparts.

Error estimates for a Gaussian rule involving Bessel functions

This paper deals with the estimation of the quadrature error of a Gaussian formula for weight functions involving powers, exponentials and Bessel functions of the first kind. For this purpose, in this work the averaged and generalized averaged Gaussian rules are employed, together with a tentative a priori approximation of the error. The numerical examples confirm the reliability of these approaches.

A barrier evaluation framework for forest carbon sink project implementation in China using an integrated BWM-IT2F-PROMETHEE II method

Forest carbon sink project (FCSP) has received growing attention for its outstanding advantages in carbon emission reduction, environment protection, and achieving the carbon neutrality objective. Using a government-social capital cooperation model, FCSP helps to revitalize the ecological and economic benefits of the forest with standard and efficient management. However, the management and operation barriers of FCSP implementation are complex, multiple and indefinite in the long run, which requires a comprehensive barrier evaluation framework. Hence, this paper is devoted to constructing a barrier evaluation framework for FCSP implementation in China with an integrated multi-criteria decision-making (MCDM) method. To this end, critical barriers are identified and the criteria system is constructed with a two-stage systematic literature review and interaction with the experts. Later, an applicable and comprehensive barrier evaluation framework is built, wherein the interval type-2 fuzzy set (IT2FS) is used to process fuzziness and uncertain information, the best-worst method (BWM) is adopted to determine the criteria weight, and the Preference Ranking Organization Method for Enrichment Evaluations II (PROMETHEE II) is introduced for comprehensive barrier evaluation. The criteria weighting results indicate that FCSP implementation encounters challenges taking into account institutional and governance barriers (criteria weight: 0.3592), management barriers (0.2384), economic and market barriers (0.1637), knowledge barriers (0.0747), technical and infrastructure barriers (0.1639), C1 “Lack of clear leadership and policies” and C5 “Forest management performance” were determined the most significant criteria, A5 and A4 respectively performed the lowest and the highest barrier level. In sensitivity analysis, the ranking results were stable when the criteria changed, and Spearman’s rank correlations of B2, B4, and B5 are significant. The results were validated by a comparative analysis with other methods with the help of Spearman’s rank correlation coefficient, which reflects accepted high correlations (Spearman’s pro values of BWM-IT2F-PROMETHEE II (Gaussian rule), BWM-IT2F-TOPSIS, and BWM-TFN-VIKOR are 1.00, 1.00, 1.00). The findings of the paper help the decision makers to develop more rational strategies and successfully implement the FCSP.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

On a time-discrete convolution—space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains

Abstract For the numerical solution of a classical two-dimensional Dirichlet exterior problem for the wave equation in the time domain, we consider a space-time boundary integral equation approach, based on its discretization by means of the coupling of a time discrete convolution quadrature of order 2, with the classical continuous piecewise linear boundary element method. The latter is either the Galerkin method or the collocation one, this being computationally much cheaper. For the efficient evaluation of most of the integrals generated by these two boundary element methods, but also for the subsequent analysis, we first define a new Gaussian quadrature. After recalling that stability and convergence have been proved for the Galerkin method, while for the collocation one we only have numerical evidences of these properties, to (partially) fill this gap, we first note that collocation can be obtained directly from the Galerkin system, after discretizing its inner product integral by using the composite trapezoidal rule, which in this case turns out to be the Gaussian rule mentioned above with $1$ node. Then, we show that for all (positive) time step-sizes $\varDelta _t$, when we let $h\rightarrow 0$, independently from $\varDelta _t$, the matrix of the linear system produced by the collocation method, properly normalized, converges, in the $\infty $-norm, to the corresponding Galerkin matrix. The behavior of the associated absolute and relative errors in the $\infty $-norm, as $h\rightarrow 0$, are $O\big (h^{\frac {5}{3}}+\varDelta _t^{-(1+\varepsilon )}h^{\frac {5+\varepsilon }{2}}[|\!\ln \varDelta _t|+|\!\ln h|]\big )$ and $O\big (h^{\frac {2}{3}}/|\!\ln \varDelta _t|+\varDelta _t^{-(1+\varepsilon )}h^{\frac {3+\varepsilon }{2}}[1+|\!\ln h|/|\!\ln \varDelta _t|]\big )$, respectively, with $\varepsilon>0$ as close as one likes to $0$ and the $O(\cdot )$ constant independent from $\varDelta _t, h$. Some possible extensions of the investigation we carried out are also mentioned.

Computational investigation of free convection flow inside inclined square cavities

The temperature and fluid profiles of flow inside tilted square cavities are analysed with two different cases of thermal boundary conditions, (1) Isothermally cold sidewalls of the cavity and the hot bottom wall kept parallel to the insulated top wall, (2) Hot left wall, cold right wall, insulated top and bottom walls. The Galerkin finite element method with penalty parameter is used to solve the nonlinear coupled system of partial differential equations governing the flow and thermal fields. The method is further used to solve the Poisson equation for stream function. The results are presented in terms of isotherms and streamlines. The Gaussian rule with the hybrid function formed from the block-pulse function and Lagrange polynomial is implemented for the evaluation of the definite integrals present in the residual equations. Attempting to affix the hybrid methods in the integration part for solving Finite Element Method (FEM) turned efficacious as the convergence is achieved for streamlines and isotherms with the existing results. The tilted square cavities with inclination angles Deg 0 to Deg 90 and Rayleigh number ranging between 10^3 and 10^5 for Pr=0.71 (air) are investigated. The source code for the finite element analysis is written in Mathematica. The results thus obtained are found to be competent with those of COMSOL, the Software for Multiphysics Simulation.

On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines

Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nyström method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for m→∞, m being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.

Explicit Formulas and Numerical Integral Equation of ARL for SARX(P,r)<sub>L</sub> Model Based on CUSUM Chart

The Cumulative Sum (CUSUM) chart is widely used and has many applications in different fields such as finance, medical, engineering, and other fields. In real applications, there are many situations in which the observations of random processes are serially correlated, such as a hospital admission in the medical field, a share price in the economic field, or a daily rainfall in the environmental field. The common characteristic of control charts that has been used to evaluate the performance of control charts is the Average Run Length (ARL). The primary goals of this paper are to derive the explicit formula and develop the numerical integral equation of the ARL for the CUSUM chart when observations are seasonal autoregressive models with exogenous variable, SARX(P,r)_L with exponential white noise. The Fredholm Integral Equation has been used for solving the explicit formula of ARL, and we used numerical methods including the Midpoint rule, the Trapezoidal rule, the Simpson's rule, and the Gaussian rule to approximate the numerical integral equation of ARL. The uniqueness of solutions is guaranteed by using Banach's Fixed Point Theorem. In addition, the proposed explicit formula was compared with their numerical methods in terms of the absolute percentage difference to verify the accuracy of the ARL results and the computational time (CPU). The results obtained indicate that the ARL from the explicit formula is close to the numerical integral equation with an absolute percentage difference of less than 1%. We found an excellent agreement between the explicit formulas and the numerical integral equation solutions. An important conclusion of this study was that the explicit formulas outperformed the numerical integral equation methods in terms of CPU time. Consequently, the proposed explicit formulas and the numerical integral equation have been the alternative methods for finding the ARL of the CUSUM control chart and would be of use in fields like biology, engineering, physics, medical, and social sciences, among others.

Nyström methods for approximating the solutions of an integral equation arising from a problem in mathematical biology

The paper deals with an integral equation arising from a problem in mathematical biology. We propose approximating its solution by Nyström methods based on Gaussian rules and on product integration rules according to the smoothness of the kernel function. In particular, when the latter is weakly singular we propose two Nyström methods constructed by means of different product formulas. The first one is based on the Lagrange interpolation while the second one is based on discrete spline quasi-interpolants. The stability and the convergence of the proposed methods are proved in uniform spaces of continuous functions. Finally, some numerical tests showing the effectiveness of the methods and the sharpness of the obtained error estimates are given.

Application of the Cauchy integral approach to singular and highly oscillatory integrals

This paper presents a method that is based on the sum of line integrals for fast computation of singular and highly oscillatory integrals , and . Where G and f are non-oscillatory sufficiently smooth functions on the interval of integration. is a product of singular factors and is an oscillatory parameter. The computation of these integrals requires f and G to be analytic in a large complex region C accommodating the interval of integration. The integrals are changed into a problem of integrals on ; which are later computed using the generalized Gauss-Laguerre rule or by the construction of Gauss rules relative to a Freud weights function with k positive. MATHEMATICA programming code, algorithms and illustrative numerical examples are provided to test the efficiency of the presented experiments.

Numerical Solution for Real Double Definite Integrals

The double integral is numerically evaluated in this paper using higher precision quadrature rules. With the combination of Newtonian and Gaussian rules of precision three each, a mixed quadrature rule of precision five is obtained. Three test problems are used to numerically validate the rule. The approximations are compared to analytical solutions, and error bounds are calculated.

Cluster model of the porosity of spongy titanium briquettes at the stage of pressing

The main factors of the formation of porosity of pressed products based on spongy titanium were studied. Three types of pores were studied and separated – cluster (in the place of particles), inter-cluster, and natural pores of the material. The cluster models of particles packing at the stages of pressing were developed (from bulk density, or the formation of temporary structures to the formation of stable structures). The number of cluster faces in the models depends on coordination number λ, which means tetrahedral (λ=4) clusters at the initial stage and cuboctahedral (λ=12) at the later ones. Based on the Gaussian rule, for spheres packing, it was found that the most correct form of clusters for later pressing stages is cuboctahedral, as the pores between the spheres at the maximum tight packing with the coordination number of 12 have the shape close to cuboctahedrons and octahedrons, but with concave faces. Based on the difference between the volume of spheres, for which particles and clusters in the model were accepted, based on calculated volumes of intercluster octahedrons and cuboctahedrons, the volume of pores in the shape of the Steiner octahedron or cuboctahedron was calculated. In calculating the strength of adhesion between the particles, the proper porosity of spongy titanium is determined through the assumption that a part of the powder is a conglomerate that is formed from hollow spheres of the regular shape at the stage of titanium reduction by the magnesium thermal method. Accordingly, in the formula for calculating the strength of adhesion, the force that influences a particle will consist of the difference between forces of elastic deformation and the destruction of hollow spheres contained in the deformed volume. The developed models were proved by the results of practical research. Actual measurements show the average exponential ratio of the porosity to pressing pressure, which makes it possible to calculate s maximum inter-cluster porosity at the maximum compaction of 66 % and the compression factor of the studied material of 0.15

A New Multivariable Grey Convolution Model Based on Simpson’s Rule and Its Applications

Accurate estimations can provide a solid basis for decision-making and policy-making that have experienced some kind of complication and uncertainty. Accordingly, a multivariable grey convolution model (GMC (1, n)) having correct solutions is put forward to deal with such complicated and uncertain issues, instead of the incorrect multivariable grey model (GM (1, n)). However, the conventional approach to computing background values of the GMC (1, n) model is inaccurate, and this model’s forecasting accuracy cannot be expected. Thereby, the drawback analysis of the GMC (1, n) model is conducted with mathematical reasoning, which can explain why this model is inaccurate in some applications. In order to eliminate the drawbacks, a new optimized GMC (1, n), shorted for OGMC (1, n), is proposed, whose background values are calculated based on Simpson’ rule that is able to efficiently approximate the integration of a function. Furthermore, its extended version that uses the Gaussian rule to discretize the convolution integral, abbreviated as OGMCG (1, n), is proposed to further enhance the model’s forecasting ability. In general, these two optimized models have such advantages as simplified structure, consistent forecasting performance, and satisfactory efficiency. Three empirical studies are carried out for verifying the above advantages of the optimized model, compared with the conventional GMC (1, n), GMCG (1, n), GM (1, n), and DGM (1, n) models. Results show that the new background values can effectively be calculated based on Simpson’s rule, and the optimized models significantly outperform other competing models in most cases.

A meshfree finite volume method with optimal numerical integration and direct imposition of essential boundary conditions

Meshfree methods (MMs) enjoy advantages in discretizing problem domains over mesh-based methods. Extensive progress has been made in the development of the MMs in the last three decades. The commonly used MMs, such as the reproducing kernel particle methods (RKP), the moving least-square methods (MLS), and the meshless Petrov-Galerkin methods, have main difficulties in numerical integration and in imposing essential boundary conditions (EBC). Motivated by conventional finite volume methods, we propose a meshfree finite volume method (MFVM), where the trial functions are constructed through the conventional RKP or MLS procedures, while the test functions are set to be piecewise constants on Voronoi diagrams built on scattered particles. The proposed method possesses three typical merits: (1) the standard Gaussian rules are proven to produce optimal approximation errors; (2) the EBC can be imposed directly on boundary particles; and (3) mass conservation is maintained locally due to its finite volume formulations. Inf-sup conditions for the MFVM are proven in a one-dimensional problem, and are demonstrated numerically using a generalized eigenvalue problem for higher dimensions. Numerical test results are reported to verify the theoretical findings.

Rational averaged gauss quadrature rules

It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules.