One-dimensional Quadrature Research Articles

Precision computation of one-dimensional quadratures

The calculation of quadratures arises in many physical and technical applications. In this paper, the best quadrature formulas are selected and their quantitative comparison is carried out on a number of representative examples.

High-order quadrature on multi-component domains implicitly defined by multivariate polynomials

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of h-refinement with q fixed, the quadrature schemes yield an order of accuracy of 2q, where q is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of q-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling q approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of numerical experiments demonstrates the quadrature algorithm on two- and three-dimensional problems, including: randomly generated geometry involving multiple high-curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyperrectangular constraint cells; and boolean operations to compute integrals on overlapping domains.

Generalized multiscale approximation of a mixed finite element method with velocity elimination for Darcy flow in fractured porous media

In this paper, we propose a multiscale method for solving the Darcy flow of a single-phase fluid in two-dimensional fractured porous media. We consider a discrete fracture-matrix (DFM) model that treats fractures as one-dimensional objects, and flows in both the matrix and fractures respect the Darcy’s law. A multipoint flux mixed finite element (MFMFE) method with the broken Raviart–Thomas (RT12) element and the trapezoidal quadrature rule is employed to approximate the matrix velocity and pressure, which results in a block diagonal, symmetric and positive definite mass matrix for the matrix velocity on general quadrilateral grids; the one-dimensional RT0 mixed finite element method with the one-dimensional trapezoidal quadrature rule is exploited to approximate the fracture velocity and pressure, which leads to a diagonal and positive definite mass matrix for the fracture velocity in each single fracture. All these features of the obtained mass matrices allow for velocity elimination. Multiscale basis functions are constructed for the two-dimensional matrix pressure following the generalized multiscale finite element method (GMsFEM) framework to capture the fine-scale information of heterogeneous fractured porous media and effectively reduce the degrees of freedom for the matrix pressure, while fine-grid basis functions are utilized for the one-dimensional fracture pressure in fractures. Various numerical tests with the oversampling technique for different fracture distributions are performed to show that the proposed multiscale method is effective and able to provide good approximations for the fine-grid solution.

DEVELOPMENT AND TESTING OF THE MODIFIED SIMPSON’S RULE FOR DISCRETE ORDINATES TRANSPORT APPLICATIONS

A new angular quadrature type termed Modified Simpson Trapezoidal (MST) is developed based on the conventional Simpson’s 1/3 rule where the angular pattern over polar levels has a trapezoid shape. An adaptive coefficient correction scheme is developed to enable our new quadrature to integrate the angular flux over subintervals separated by the interior jump irregularities. A two-dimensional test problem is employed to verify the angular discretization error in the uncollided SN scalar flux computed with our new quadrature sets, as well as conventional angular quadrature types. Numerical results show that the MST quadrature error in the point-wise scalar flux converges with second order against increasing number of discrete angles, while the error obtained with other conventional quadrature types converges slower than first order depending on the regularity of the exact point-wise uncollided angular flux. In order to reduce the number of discrete points needed, a variant of the MST quadrature, namely MSTP30, is developed by using the Quadruple Range [1] polar quadrature with fixed 30 polar angles and applying the MST quadrature to the azimuthal dependence in each polar level. The angular discretization error in the point-wise SN scalar flux obtained with MSTP30 sets converges with fourth order because the polar discretization error is sufficiently reduced that MSTP30 behaves like a one-dimensional quadrature. Furthermore, because MSTP30 computes the integral over subintervals that keep the true solution’s irregularity at the boundaries, this fourth order convergence rate is unaffected by such inevitable irregularities.

A Coarea Formulation for Grid-Based Evaluation of Volume Integrals

Abstract We present a new method for computing volume integrals based on data sampled on a regular Cartesian grid. We treat the case where the domain is defined implicitly by an inequality, and the input data include sampled values of the defining function and the integrand. The method employs Federer’s coarea formula (Federer, 1969, Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften, Springer) to convert the volume integral to a one-dimensional quadrature over level set values where the integrand is an integral over a level set surface. Application of any standard quadrature method produces an approximation of the integral over the continuous range as a weighted sum of integrals over level sets corresponding to a discrete set of values. The integral over each level set is evaluated using the grid-based approach presented by Yurtoglu et al. (2018, “Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains,” J. Comput. Inf. Sci. Eng., 18, p. 3). The new coarea method fills a need for computing volume integrals whose integrand cannot be written in terms of a vector potential. We present examples with known results, specifically integration of polynomials over the unit sphere. We also present Saye’s (2015, “High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles,” SIAM J. Sci. Comput., 37) example of integrating a logarithmic integrand over the intersection of a bounding box with an open domain implicitly defined by a trigonometric polynomial. For the final examples, the input data is a grid of mixture ratios from a direct numerical simulation of fluid mixing, and we demonstrate that the grid-based coarea method applies to computing volume integrals when no analytical form of the implicit defining function is given. The method is highly parallelizable, and the results presented are obtained using a parallel implementation capable of producing results at interactive rates.

An efficient method for estimating time-dependent global reliability sensitivity

Time-dependent global reliability sensitivity can quantify the effect of input variables in their whole distribution ranges on the time-dependent failure probability. To efficiently estimate it to help researchers control the time-dependent failure probability, a novel method is proposed. The proposed method transforms the estimation of unconditional-conditional time-dependent failure probabilities into that of the unconditional-conditional probability density functions (PDFs) of the minimum of time-dependent performance function. Firstly, the minimum of time-dependent performance function is evaluated by adaptive Kriging, and its unconditional-conditional fractional moments are estimated by multiplicative dimensional reduction method (M-DRM). Then, the maximum entropy (MaxEnt) constrained by these fractional moments is used to estimate the unconditional-conditional PDFs, on which the unconditional-conditional time-dependent failure probabilities can be obtained. Finally, the one-dimensional Gaussian quadrature is applied to estimate the time-dependent global reliability sensitivity indices. Due to the high efficiency of adaptive Kriging for estimating the minimum of time-dependent performance function, the avoidance of dimensional curse by M-DRM, and the high efficiency of MaxEnt constrained by fractional moments for estimating PDF, the proposed method can reduce the computational cost dramatically.

A Fast and Exact w-stacking and w-projection Hybrid Algorithm for Wide-field Interferometric Imaging

Abstract The standard wide-field imaging technique, the w-projection, allows correction for wide fields of view for non-coplanar radio interferometric arrays. However, calculating exact corrections for each measurement has not been possible due to the amount of computation required at high resolution and with the large number of visibilities from current interferometers. The required accuracy and computational cost of these corrections is one of the largest unsolved challenges facing next-generation radio interferometers such as the Square Kilometre Array. We show that the same calculation can be performed with a radially symmetric w-projection kernel, where we use one-dimensional adaptive quadrature to calculate the resulting Hankel transform, decreasing the computation required for kernel generation by several orders of magnitude, while preserving the accuracy. We confirm that the radial w-projection kernel is accurate to approximately 1% by imaging the zero-spacing with an added w-term. We demonstrate the potential of our radially symmetric w-projection kernel via sparse image reconstruction, using the software package PURIFY. We develop a distributed w-stacking and w-projection hybrid algorithm. We apply this algorithm to individually correct for non-coplanar effects in 17.5 million visibilities over a 25 by 25 degree FoV Murchison Widefield Array observation for image reconstruction. Such a level of accuracy and scalability is not possible with standard w-projection kernel generation methods. This demonstrates that we can scale to a large number of measurements with large image sizes while still maintaining both speed and accuracy.

Normal stress effects on Knudsen flow

Normal stress effects are investigated on tube flow of a single-component non-Newtonian fluid under a constant pressure gradient in a constant temperature field. The generalized hydrodynamic equations are employed, which are consistent with the laws of thermodynamics. In the cylindrical tube flow configuration, the solutions of generalized hydrodynamic equations are exactly solvable and the flow velocity is obtained in a simple one-dimensional integral quadrature. Unlike the case of flow in the absence of normal stresses, the flow develops an anomaly in that the flow in the boundary layer becomes stagnant and the thickness of such a stagnant velocity boundary layer depends on the pressure gradient, the aspect ratio of the radius to the length of the tube, and the pressure (or density and temperature) at the entrance of the tube. The volume flow rate formula through the tube is derived for the flow. It generalizes the Knudsen flow rate formula to the case of a non-Newtonian stress tensor in the presence of normal stress differences. It also reduces to the Navier–Stokes theory formula in the low shear rate limit near equilibrium.

An asymptotic approach to solidification shrinkage-induced macrosegregation in the continuous casting of binary alloys

• Asymptotic analysis for continuous casting of a binary alloy using regular perturbation theory. • Micro-scale assumption for lever rule solidification feeds through to the leading order in the macroscale solution. • Boundary immobilization technique for determining the extent of the mushy zone and the level of macrosegregation. • Method implemented in Comsol Multiphysics. The modelling of macrosegregation in the continuous casting of alloys normally requires resource-intensive computational fluid dynamics (CFD). By contrast, here we develop an asymptotic framework for the case when macrosegregation is driven by solidification shrinkage; as a first step, a binary alloy is considered. Systematic asymptotic analysis of the steady-state two-dimensional mass, momentum, heat and solute conservation equations in terms of the shrinkage parameter indicates that the overall problem can be reduced to a hierarchy of decoupled problems: a leading-order problem that is non-linear, and a sequence of linear problems, with the actual macrosegregation of the solute then being determined by means of one-dimensional quadrature. A numerical method that solves this sequence is then developed and implemented, and yields realistic macrosegregation profiles at low computational cost.

Static analysis of non-uniform heterogeneous circular plate with porous material resting on a gradient hybrid foundation involving friction force

This paper is concerned with the static analysis of variable thickness of two directional functionally graded porous materials (FGPM) circular plate resting on a gradient hybrid foundation (Horvath-Colasanti type) with friction force and subjected to compound mechanical loads (e.g., transverse, in-plane shear traction and concentrated force at the center of the plate).The governing state equations are derived in terms of displacements based on the 3D theory of elasticity, assuming the elastic coefficients of the plate material except the Poisson‟s ratio varying continuously throughout the thickness and radial directions according to an exponential function. These equations are solved semi-analytically by employing the state space method (SSM) and one-dimensional differential quadrature (DQ) rule to obtain the displacements and stress components of the FGPM plate. The effect of concentrated force at the center of the plate is approximated with the shear force, uniformly distributed over the inner boundary of a FGPM annular plate. In addition to verification study and convergence analysis, numerical results are displayed to show the effect of material heterogeneity indices, foundation stiffness coefficients, foundation gradient indices, loads ratio, thickness to radius ratio, compressibility, porosity and friction coefficient of the foundation on the static behavior of the plate. Finally, the responses of FG and FG porous material circular plates to compound mechanical loads are compared.

Response and reliability analysis of nonlinear uncertain dynamical structures by the probability density evolution method

The paper deals with the response and reliability analysis of hysteretic or geometric nonlinear uncertain dynamical systems of arbitrary dimensionality driven by stochastic processes. The approach is based on the probability density evolution method proposed by Li and Chen (Stochastic dynamics of structures, 1st edn. Wiley, London, 2009; Probab Eng Mech 20(1):33–44, 2005), which circumvents the dimensional curse of traditional methods for the determination of non-stationary probability densities based on Markov process assumptions and the numerical solution of the related Fokker–Planck and Kolmogorov–Feller equations. The main obstacle of the method is that a multi-dimensional convolution integral needs to be carried out over the sample space of a set of basic random variables, for which reason the number of these need to be relatively low. In order to handle this problem an approach is suggested, which reduces the number of basic random variables to merely a single one. Correspondingly, the response and reliability problems reduce to the solution of one-dimensional quadratures.

High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

A high-order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle. By converting the implicitly defined geometry into the graph of an implicitly defined height function, the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one-dimensional root finding and one-dimensional Gaussian quadrature. The computed quadrature scheme yields strictly positive quadrature weights and inherits the high-order accuracy of Gaussian quadrature: a range of different convergence tests demonstrate orders of accuracy up to 20th order. Also presented is an application of the quadrature algorithm to a high-order embedded boundary discontinuous Galerkin method for solving partial differential equations on curved domains.

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.

An analytical formula for pricing m-th to default swaps

The computational effort of pricing an m-th to default swap depends highly on the size n of the underlying basket. Usually, n different default times are modeled, but in fact the valuation only depends on the m-th smallest default time of this tuple. In this paper we attain an analytical formula for the distribution of this m-th default time. With the help of this distribution we simplify the valuation problem from an n-dimensional quadrature to a one-dimensional quadrature and break the curse of dimensionality. Applications of this modification are efficient pricing of m-th to default swaps, estimation of sensitivities and pricing of European max/min options.

Numerical integration over the square in the presence of algebraic/logarithmic singularities with an application to aerodynamics

Tensor-product formulae based on one-dimensional Gaussian quadratures are developed for evaluating double integrals of the type indicated in the title. If the singularities occur only along the diagonal and the regular part of the integrand is a polynomial of total degree d, the formulae can be made exact by choosing the number of quadrature points larger than, or equal to, 1?+?d/2. Numerical examples are given as well as an application to a problem in aerodynamics.

Appropriate Gaussian quadrature formulae for triangles

This paper mainly presents higher order Gaussian quadrature formulae for numerical integration over the triangular surfaces. In order to show the exactness and efficiency of such derived quadrature formulae, it also shows first the effective use of available gaussian quadrature for square domain integrals to evaluate the triangular domain integrals. Finally, it presents nxn points and n(n+1)/2 - 1 points (for n >1) Gaussian quadrature formulae for triangle utilizing n-point one-dimensional Gaussian quadrature. By use of simple but straightforward algorithms, gaussian points and corresponding weights are calculated and presented for clarity and reference. The proposed n(n+1)/2 - 1 points formulae completely avoids the crowding of gaussian points and overcomes all the drawbacks in view of accuracy and efficiency for the numerical evaluation of the triangular domain integrals of any arbitrary functions encountered in the realm of science and engineering.

Interval Integration Revisited

We present an overview of approaches to selfvalidating one-dimensional integration quadrature formulas and a verified numerical integration algorithm with an adaptive strategy. The new interval integration adaptive algorithm delivers a desired integral enclosure with an error bounded by a specified error bound. The adaptive technique is usually much more efficient than Simpson’s rule and narrow interval results can be reached.

An Efficient Semianalytic Computation Method of Magnetic Field for a Circular Coil With Rectangular Cross Section

A semianalytic formula for calculating the magnetic field produced by a circular coil with rectangular cross section in air is proposed. In the formula, the coil is approximated as a series of circular rings with small cross sectional area, of which the current density can be considered as constant. In order to simplify the calculation of the magnetic field, the translation transformation and scaling transformation are introduced. Furthermore, all the possible singular cases, in the expressions, are treated properly. Therefore, with the help of one-dimensional Gauss-Legendre quadrature, the magnetic field at any points in space produced by current-carrying circular coil, especially inside the coil, can be calculated very easily.

Accurate Evaluation Schemes for Triangular Domain Integrals

This paper mainly presents higher order Gaussian quadrature formulae for numerical integration over the triangular surfaces. In order to show the exactness and efficiency of such derived quadrature formulae, it shows the effective use of available Gaussian quadrature for square domain integrals to evaluate the triangular domain integrals. Then it presents n × n points and n(n+1) 2 − 1 points (for n >1) Gaussian quadrature formulae for triangle utilizing n-point one-dimensional Gaussian quadrature. By use of simple but straight-forward algorithms, Gaussian points and corresponding weights are calculated and presented for clarity and reference. The proposed n(n+1) 2 − 1 point formula completely avoids the crowding of Gaussian points. Finally, it presents an error formula which can evaluate the error in monomial/polynomial integration using m×n point method successfully. The error calculated by the new formula and the error in calculation of integrals by the proposed methods are found to be in good agreement. Thus, it overcomes all the drawbacks in view of accuracy and efficiency for the numerical evaluation of the triangular domain integrals of any arbitrary functions encountered in the realm of science and engineering.

On a generalization of the Laplace approximation

Laplace approximation is one commonly used approach to the calculation of difficult integrals arising in Bayesian inference and the analysis of random effects models. Here we outline a procedure which is an extension of the Laplace approximation and which attempts to find changes of variable for which the integrand becomes approximately a product of one-dimensional functions. When the integrand is a product of one-dimensional functions, an approximation to the integral can be obtained using one-dimensional quadrature. The approximation is exact for a broader class of functions than the ordinary Laplace approximation and can be applied when the integrand is not smooth at the mode. As an illustration of this last point we consider calculation of marginal likelihoods for smoothing parameter selection in the lasso.