Numerical Integration Rules Research Articles

From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules

In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Corput sequence and generalizations thereof. This led to a huge number of results.On the occasion of the 125th birthday of J.G. van der Corput we survey many interesting results on van der Corput sequences and their generalizations. In this way we move from van der Corput’s ideas to the most modern constructions of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton sequences or Niederreiter’s (t,s)-sequences.

On a projection-corrected component-by-component construction

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm.

Computation of scattering kernels in radiative transfer

This note proposes rapidly convergent computational formulae for evaluating scattering kernels from radiative transfer theory. The approach used here does not rely on Legendre expansions, but rather uses exponentially convergent numerical integration rules. The relation between the domain of analyticity of a given phase function and the speed of convergence is studied in detail.

Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces

Quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences for smooth functions. A construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in Dick (2007), which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension.In this paper, we study the use of polynomial lattice point sets for interlaced components. We call quadrature rules using such point sets interlaced polynomial lattice rules. We consider weighted Walsh spaces containing smooth functions and derive two upper bounds on the worst-case error for interlaced polynomial lattice rules, both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules. We investigate the component-by-component construction and the Korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst-case error. Through this approach we are able to obtain a good dependence of the worst-case error bounds on the dimension under certain conditions on the weights, while significantly reducing the construction cost as compared to higher order polynomial lattice rules.

Application of Nyström method to a Fredholm integral equation describing induction heating

An induction heating problem can be described by a Fredholm Integral Equation of the second kind. The equation is used to compute the eddy current of density. One method for solving such an equation is the Nystrom method. It is based on the approximation of the integral in an equation by the numerical integration rule. This paper shows application of the Nystrom method to an induction heating problem. Results of the Nystrom method are compared with an alternative method.

A Mixed quadrature rule for numerical integration of analytic functions by using fejer and gaussian quadrature rules

In this paper 5-point Fejer’s second rule has been modified for computing integrals involving analytic functions on a line. This rule has been mixed with Gaussian 3-point rule to form a mixed quadrature rule. The relative efficiencies of this rule with respect to its constituent rules and other previously developed mixed rules have been numerically verified on test integrals. Asymptotic error estimate of this rule has been determined.

Algoritmos de integración numérica

En este artículo se desarrollan e implementan, los algoritmos de integración numérica, que permiten solucionar problemas de ciencias e ingeniería; tales como el cálculo de áreas, volúmenes, mecánica aplicada, y ecuaciones diferenciales (sistemas dinámicos). Dada la necesidad de contar con resultados de gran precisión, se analiza las diferentes reglas de integración numérica; basadas en los errores que ocurren cuando el integrando es reemplazado por un polinomio de interpolación P(x), conocida como las fórmulas de integración de Newton-Cotes. La regla de la extrapolación de Richardson, puede ser aplicada a cualquier fórmula de cuadratura de Newton-Cotes; o cualquier computación que se encuentra basada en una rejilla de ancho h y un error descrito como una potencia de h.

Stabilized finite‐element computation of compressible flow with linear and quadratic interpolation functions

SUMMARYThe unsteady compressible flow equations are solved using a stabilized finite‐element formulation with C0 elements. In 2D, the performance of three‐noded linear and six‐noded quadratic triangular elements is compared. In 3D, the relative performance is evaluated for 6‐noded linear and 18‐noded quadratic wedge elements. Results are compared for the solutions to Euler, laminar, and turbulent flows at different Mach numbers for several flow problems. The finite‐element meshes considered for comparison have same location of nodes for the linear and quadratic interpolations. For the turbulent flow, the Spalart–Allmaras model is used for closure. It is found that the quadratic elements yield better performance than the linear elements. This is attributed to accurate representation of the stabilization terms that involve second‐order derivatives in the formulation. When these terms are dropped from the formulation with quadratic interpolation, the numerical results are similar to those obtained with linear interpolation. The absence of these terms result in added numerical diffusion that seems to give the effect of a relatively reduced Reynolds number. For the same location of nodes, the computations with the linear triangular and wedge elements are approximately 20% and 100% faster than those with quadratic triangular and wedge elements, respectively. However, if the same quadrature rule for numerical integration is used for both interpolations, the computations with quadratic elements are approximately 20% and 45% faster in 2D and 3D, respectively. Copyright © 2014 John Wiley & Sons, Ltd.

Composite Gauss-Legendre Formulas for Solving Fuzzy Integration

Two numerical integration rules based on composition of Gauss-Legendre formulas for solving integration of fuzzy numbers-valued functions are investigated in this paper. The methods' constructions are presented and the corresponding convergence theorems are shown in detail. Two numerical examples are given to illustrate the proposed algorithms finally.

Numerical implementation of constitutive model for arterial layers with distributed collagen fibre orientations

Several constitutive models have been proposed for the description of mechanical behaviour of soft tissues containing collagen fibres. Some of the commonly used approaches accounting for the dispersion of fibre orientations are based on the summation of (mechanical) contributions of differently oriented fibre families. This leads to the need of numerical integration on the sphere surface, and the related numerical consumption is the main disadvantage of this category of constitutive models. The paper is focused on the comparison of various numerical integration methods applied to a specific constitutive model applicable for arterial walls. Robustness and efficiency of several integration rules were tested with respect to application in finite element (FE) codes. Among all the analysed numerical integration rules, the best results were reached by Lebedev quadrature; the related parameters for the specific constitutive model are presented in the paper. The results were implemented into the commercial FE code ANSYS via user subroutines, and their applicability was demonstrated by an example of FE simulation with non-homogenous stress field.

Modeling of piezo-induced ultrasonic wave propagation in composite structures using layered solid spectral element

This article presents the development of a three-dimensional spectral element-based Piezo-Enabled Spectral Element Analysis code to simulate piezo-induced ultrasonic wave propagation in composite structures. Piezo-Enabled Spectral Element Analysis solves the coupled electromechanical governing equations for a given arbitrary voltage input to a piezoelectric actuator and outputs the voltage response of the piezoelectric sensors. In case of modeling laminated composites, one element per ply is computationally expensive, and smeared material properties may give inaccurate results; hence, a layered solid spectral element is introduced. Layered solid spectral element can model several plies per element and uses combined nodal quadrature and Simpson’s 1/3 integration rule for numerical integration of mechanical stiffness matrix. It is worth noting that Piezo-Enabled Spectral Element Analysis can interface with commercial finite element mesh generators such as Abaqus/CAE to model structures with complex configuration. Delaminations and debonds are modeled by separating the nodes in the damaged area to create a volume split. Experiments and simulations have been carried out to verify and validate Piezo-Enabled Spectral Element Analysis for composites. This article also presents simulations of ultrasonic wave propagation in a stiffened composite plate with delamination and debond.

Using New Hybrid Integrators to Design IIR Fractional-Order Integrators via Impulse Invariance Method

This paper presents a method for designing fullband and non-fullband IIR digital integrators with linear phase that has been used to approximate fractional-order integrators with IIR filters. Several numerical integration rules have been considered in this study, namely, Euler, Simpson, Schneider, bilinear, and F012. The main idea in our design is based on interpolating the well-known integration rules (Euler, Bilinear and Simpson) proposed by Al-Alaoui combined with deterministic signal modeling techniques. Numerical examples are presented to illustrate the performance of the proposed integrators. It was found that the Euler–Simpson integrator gives better approximation accuracy than existing integrators.

Quadrature for meshless Nitsche's method

AbstractIn this article, we study effect of numerical integration on Galerkin meshless method (GMM), applied to approximate solutions of elliptic partial differential equations with essential boundary conditions (EBC). It is well‐known that it is difficult to impose the EBC on the standard approximation space used in GMM. We have used the Nitsche's approach, which was introduced in context of finite element method, to impose the EBC. We refer to this approach as the meshless Nitsche's method (MNM). We require that the numerical integration rule satisfies (a) a “discrete Green's identity” on polynomial spaces, and (b) a “conforming condition” involving the additional integration terms introduced by the Nitsche's approach. Based on such numerical integration rules, we have obtained a convergence result for MNM with numerical integration, where the shape functions reproduce polynomials of degree k ≥ 1. Though we have presented the analysis for the nonsymmetric MNM, the analysis could be extended to the symmetric MNM similarly. Numerical results have been presented to illuminate the theoretical results and to demonstrate the efficiency of the algorithms.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 265–288, 2014

An iterative algorithm for polynomial approximation of rational triangular Bézier surfaces

This paper presents the weighted progressive iteration approximation (WPIA) property for the triangular Bernstein basis over a triangle domain with uniform parameters, which is extended from the PIA property for triangular Bernstein basis proposed by Chen and Wang in [J. Chen, G.J. Wang, Progressive-iterative approximation for triangular Bézier surfaces, Computer-Aided Design 43 (2011) 889–895]. We also provide how to choose an optimal value of the weight to own the fastest convergence rate for triangular Bernstein basis. Furthermore, a new and efficient iterative method is proposed for polynomial approximation of rational triangular Bézier surfaces. The algorithm is reiterated until a halting condition about approximation error is satisfied. And the approximation error in Lp-norm (p=1, 2, ∞) is calculated by the symmetric Gauss Legendre quadrature rule for composite numerical integration over a triangular surface. Finally, several numerical examples are presented to validate the effectiveness of this method.

Improved FFT approximations of probability functions based on modified quadrature rules

In the last ten years, the traditional Simpson quadrature rule for numerical integration has been improved to an optimal 3-point quadrature formula and a modified Simpson rule that takes additionally the first derivative of the approximated integrand at the two end-points of integration into account. The impact of the improved quadrature rules on the fast Fourier transform (FFT) approximation of probability density functions with known characteristic functions is discussed. The quality of the approximations is measured with a discrete version of the total variation distance. Numerical examples suggest that a discrete total variation distance of approximately 0.25 is the best possible attainable value across all the considered FFT approximations. Mathematics Subject Classification: 41A55, 62E17, 65C60, 65T50, 91G60

SEP Performance of Cross QAM Signaling with MRC over Fading Channels and its Arbitrarily Tight Approximation

The exact average symbol error probability (SEP) of the cross quadrature amplitude modulation signal in a single-input multiple-output system over independent but not necessarily identical fading channels is derived. The maximal-ratio combining (MRC) is considered as the diversity technique, and the average SEP is obtained by using the moment generating function (MGF) method. The obtained closed-form SEP expression is presented in terms of a finite sum of single integrals with finite limits and an integrand composed of a finite product of elementary functions. In addition, the arbitrarily tight approximations with the form of a sum of constant coefficient exponential functions for Gaussian Q-function and the generalization of its Craig's form are proposed by applying the composite rectangle and Simpson numerical integration rules, respectively. The proposed approximations are simple and accurate enough even with only a few terms of exponential functions, and they are particularly suitable for applications of averaging Q-function and the generalized Q-function over the fading distributions. As a result, the closed-form approximations of the SEP over the AWGN channel and fading multichannels are expressed as a finite sum of exponential functions and a finite sum of MGFs, respectively, such that it is convenient and rapid to evaluate the SEP performances. Both the simulation results and the approximations show excellent agreement with the exact analytical expressions.

Sparse pseudospectral approximation method

Multivariate global polynomial approximations – such as polynomial chaos or stochastic collocation methods – are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials.By reexamining Smolyak’s algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.

Variance bounds and existence results for randomly shifted lattice rules

We study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions. We consider integrands that are square integrable but whose Fourier series are not necessarily absolutely convergent. For such integrands, a bound on the variance is expressed through a certain type of weighted discrepancy. We prove existence and construction results for randomly shifted lattice rules such that the variance bounds are almost O(n−α), where n is the number of function evaluations and α>1 depends on our assumptions on the convergence speed of the Fourier coefficients. These results hold for general weights, arbitrary n, and any dimension. With additional conditions on the weights, we obtain a convergence that holds uniformly in the dimension, and this provides sufficient conditions for strong tractability of the integration problem. We also show that lattice rules that satisfy these bounds are not difficult to construct explicitly and we provide numerical illustrations of the behaviour of construction algorithms.

A comparative study of numerical integration based on block-pulse and sinc functions and Chebyshev wavelet

In this paper, numerical integration rules based on block-pulse functions and Chebyshev wavelet are proposed to find approximate values of definite integrals. Errors of these numerical integrations are given. These numerical integrations are compared by sinc functions numerical integration method. Some numerical examples are provided to illustrate the accuracy of proposed rules and comparison between them. The main advantage of proposed numerical integration methods are their efficiency and simple applicability.

Free Vibration Analysis of Cantilever Rectangular Plates with Variable Thickness

A new method-mixed boundary grid method (FBGM) is proposed to analyze the free vibration of cantilever plates with variable thickness. The fundamental differential equations of the plate are established for the bending problem. By transforming these equations into integral equations in a small area and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained and chosen as Green function to obtain the characteristic equation of the free vibration. The accuracy of the numerical results for the natural frequency parameter calculated by the proposed method is investigated.