Numerical Integration Error Research Articles

Accurate Computation of Zernike Moments in Polar Coordinates

An algorithm for high-precision numerical computation of Zernike moments is presented. The algorithm, based on the introduced polar pixel tiling scheme, does not exhibit the geometric error and numerical integration error which are inherent in conventional methods based on Cartesian coordinates. This yields a dramatic improvement of the Zernike moments accuracy in terms of their reconstruction and invariance properties. The introduced image tiling requires an interpolation algorithm which turns out to be of the second order importance compared to the discretization error. Various comparisons are made between the accuracy of the proposed method and that of commonly used techniques. The results reveal the great advantage of our approach.

Simulation of springback: Through-thickness integration

The number of through-thickness integration points ( N IP) required for accurate springback analysis following sheet forming simulation using shell elements is a subject of confusion and controversy. Li and Wagoner recommended, in 1999, based on a finite element analysis (FEA) of draw-bending springback, the use of 25 integration points (IP), with up to 51 IP required to ensure accuracies of 1%. Several researchers have since reported that N IP between 5 and 11 are adequate, or even that 7 or 9 IP are optimal, with reduced accuracy for more IP. These apparent contradictions are addressed with an analytical model of elasto-plastic bending under tension, followed by elastic springback. The fractional error in the evaluated bending moment, which is equal to the fractional error in springback, was determined by comparing three numerical integration schemes, with various N IP, to the closed-form result. The results illustrate the oscillatory nature of numerical integration error with small parametric changes, such that fortuitous agreement can be obtained in isolated simulations where the number of integration points is inadequate. The concept of an assured error limit is introduced as well as a maximum error limit (for a range of generally unknown sheet tensions). The assured error limit varies with the integration scheme, N IP, bending ratio ( R/ t), and sheet tension. Guidelines for the number of integration points required for given error tolerances are reported to allow practitioners to choose numerical parameters appropriately.

Numerical Integration over Spheres of Arbitrary Dimension

In this paper, we study the worst-case error (of numerical integration) on the unit sphere $\\mathbb{S}^{d}$, $d\\geq 2$, for all functions in the unit ball of the Sobolev space $\\mathbb{H}^s(\\mathbb{S}^d)$, where $s>d/2$. More precisely, we consider infinite sequences $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of $m(n)$-point numerical integration rules $Q_{m(n)}$, where (i) $Q_{m(n)}$ is exact for all spherical polynomials of degree $\\leq n$, and (ii) $Q_{m(n)}$ has positive weights or, alternatively to (ii), the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration) $E(Q_{m(n)};\\mathbb{H}^s(\\matbb{S}^d))$ in $\\mathbb{H}^s(\\mathbb{S}^d)$ has the upper bound $c n^{-s}$, where the constant $c$ depends on $s$ and $d$ (and possibly the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$). This extends the recent results for the sphere $\\mathbb{S}^2$ by K.Hesse and I.H.Sloan to spheres $\\mathbb{S}^d$ of arbitrary dimension $d\\geq2$ by using an alternative representation of the worst-case error. If the sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of numerical integration rules satisfies $m(n)=\\mathcal{O}(n^d)$ an order-optimal rate of convergence is achieved.

Mitigating the integration error in numerical simulations of Newtonian systems

AbstractWe introduce a method for mitigating the numerical integration errors of linear, second‐order initial value problems. We propose a methodology for constructing an optimal state‐space representation that gives minimum numerical truncation error, and in this sense, is the optimal state‐space representation for modelling given phase‐space dynamics. To that end, we utilize a simple transformation of the state‐space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in electromagnetism. We then utilize the gauge function to reduce the numerical integration error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time‐step. Moreover, we derive general expressions yielding the optimal gauge functions given a Newtonian one degree‐of‐freedom ODE. For the n degrees‐of‐freedom case we describe MATLAB® code capable of finding the optimal gauge functions and integrating the given system using the gauge‐optimized integration algorithm. In all of our illustrating examples, the gauge‐optimized integration outperforms the integration using standard state variables by a few orders of magnitude. Copyright © 2006 John Wiley & Sons, Ltd.

A Petrov-Galerkin Natural Element Method Securing the Numerical Integration Accuracy

An improved meshfree method called the Petrov-Galerkin natural element (PG-NE) method is introduced in order to secure the numerical integration accuracy. As in the Bubnov-Galerkin natural element (BG-NE) method, we use Laplace interpolation function for the trial basis function and Delaunay triangles to define a regular integration background mesh. But, unlike the BG-NE method, the test basis function is differently chosen, based on the Petrov-Galerkin concept, such that its support coincides exactly with a regular integration region in background mesh. Illustrative numerical experiments verify that the present method successfully prevents the numerical accuracy deterioration stemming from the numerical integration error.

Numerical Aspects on the Prediction of Stability Boundaries of Two-Phase Natural Circulation Circuits, Considering Flashing Evaluation

Abstract In this work, the stability of a two-phase, natural circulation circuit is analyzed, using a specially developed model. This thermohydraulic model results in a set of coupled, nonlinear, first-order partial differential equations, which are solved by means of the up-wind finite difference method, using combinations of explicit and implicit methods for the numerical integration of the different balance equations. An adaptive nodalization scheme is implemented, minimizing the error of the propagation of small perturbations through the discretized volumes and especially the ones having two-phase flow regime. A linearization method is implemented by means of numerical perturbations. Frequency domain calculations are carried out, allowing a rapid visualization of the stability of the linearized system. Two cases are analyzed: a test case, where the code is compared in a wide range of qualities with an analytical model, and an application case, where the model is used to analyze the stability of an integral reactor cooled by natural circulation. The CAREM prototype is taken as a reference. In both cases, the numerical diffusion and integration errors are analyzed in the stability limit prediction by means of a convergence analysis using different nodalization and numerical integration criteria.

Transverse Vibration of Mindlin Plates on Two-Parameter Foundations by Analytical Trapezoidal p-Elements

An analytical trapezoidal hierarchical element for the transverse vibration of Mindlin plates resting on two-parameter foundations is presented. Legendre orthogonal polynomials are used as enriching shape functions to avoid the shear-locking problem and to improve considerably the computational efficiency. Element matrices are integrated in closed form eliminating the numerical integration errors conventionally found. With the C0 continuity requirement, the element can be used to analyze any triangular and polygonal plates without difficulty, while the Kirchhoff p -version elements requiring C1 continuity are not as versatile. The computed natural frequencies for rectangular, skew, trapezoidal, triangular, annular, and polygonal plates on two-parameter foundations show that the convergence of the proposed element is very fast compared to the conventional linear finite elements with respect to the number of degrees of freedom used. Many numerical examples are given.

A modification of the numerical integration error method for the zero-finding problem of an analytic function

The numerical integration error method (NIEM) is a zero-finding method which is based on the numerical evaluation of integrals of ƒ′/ƒ NIEM is an iterative method of higher-order, and (therefore) a lot of computational complexities are required. Besides it is necessary to select one from a number of choices of NIEM correction. In this paper we introduce an algorithm that improves computational efficiency of NIEM, yielding better results for the case of multiple or clustered zeros. Some numerical examples demonstrating the efficiency of this algorithm are introduced.

Current accumulation for a self magnetic field calculation in a finite-element gun code

We describe a novel current accumulation algorithm for the three-dimensional self magnetic field calculation in a charged-particle beam optics code. The current source is a charged-particle beam represented by a collection of numerically-integrated current-carrying rays. We compute the magnetic vector potential using edge basis functions and the curl-curl formulation of the finite-element method. The current accumulation algorithm takes advantage of a novel particle tracker that happens to be ideal for this application. We show that our source vector is compatible with the singular finite-element matrix, even with modest numerical integration errors. Thus, a conjugate gradient matrix solver works well, with no need for additional gauge conditions. We confirmed this behavior in a series of numerical tests on a small problem with incomplete linear and quadratic basis functions on a variety of element shapes.

보간과 회귀를 위한 일반크리깅 모델

Kriging model is widely used as design analysis and computer experiment (DACE) model in the field of engineering design to accomplish computationally feasible design optimization. In general, kriging model has been applied to many engineering applications as an interpolation model because it is usually constructed from deterministic simulation responses. However, when the responses include not only global nonlinearity but also numerical error, it is not suitable to use Kriging model that can distort global behavior. In this research, generalized kriging model that can represent both interpolation and regression is proposed. The performances of generalized kriging model are compared with those of interpolating kriging model for numerical function with error of normal distribution type and trigonometric function type. As an application of the proposed approach, the response of a simple dynamic model with numerical integration error is predicted based on sampling data. It is verified that the generalized kriging model can predict a noisy response without distortion of its global behavior. In addition, the influences of maximum likelihood estimation to prediction performance are discussed for the dynamic model.

Modeling aspects in linear stability analysis of a self-pressurized, natural circulation integral reactor

The stability of a self-pressurized natural circulation integral reactor is studied by means of a linear approach, taking the CAREM-25 reactor as reference. A thermohydraulic code has been improved for analysis of linear stability, great emphasis having been placed on the minimization of numerical diffusion and integration errors. A linearization method is implemented by means of numerical perturbations. The results are obtained within the frequency domain. The code is compared to a simpler analytical model, by contrasting stability maps obtained from both models for a test configuration, showing good agreement. In this type of reactor, oscillations are promoted by the two-phase regime in its long riser, and take place due to the counteraction between mass flow and buoyancy force. The stability of the system is strongly influenced by the steam-dome dynamics. Condensation in the steam zone, together with reactor power, determines the dynamical state of the system. The phase-lag introduced by the core dynamic regarding the riser timing, together with the sensitivity of the buoyancy force due to flow changes, determines the sustainability of the oscillation. A parametric study is carried out, gradually increasing the complexity of the model, to analyze the influence of different factors on the oscillation sustainability, concerning physical process and modeling approaches. The analysis includes the relative velocities between phases, the axial power profile along the core, the buoyancy force due to subcooled density changes, the flashing effect, the core dynamic and the pressure feedback due to self-pressurization. The steam-dome-pressure feedback is identified as a stabilizing effect, as long as it decreases the sensitivity of the buoyancy force.

Numerical integration in Natural Neighbour Galerkin methods

AbstractIn this paper, issues regarding numerical integration of the discrete system of equations arising from natural neighbour (natural element) Galerkin methods are addressed. The sources of error in the traditional Delaunay triangle‐based numerical integration are investigated. Two alternative numerical integration schemes are analysed. First, a ‘local’ approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results. Second, a stabilized nodal quadrature scheme is shown to render high levels of accuracy, while resulting specially appropriate in a Natural Neighbour Galerkin approximation method. The paper is completed with several examples showing the performance of the proposed techniques. Copyright © 2004 John Wiley & Sons, Ltd.

Solution of the radiative integral transfer equations in rectangular participating and isotropically scattering inhomogeneous medium

Radiative integral transfer equations for a rectangular participating and isotropically scattering inhomogeneous medium are solved numerically for the incident energy and the net partial heat fluxes using the method of “subtraction of singularity”. All the relevant single (surface integrals) and double integrals (volume integrals) are carried out analytically to reduce the computation time and numerical integration errors. The resulting system of linear equations are solved iteratively. A benchmark problem is chosen as a rectangular inhomogeneous cold participating medium which is subject to externally uniform diffuse radiation on the bottom surface. Solutions for linearly and quadratically varying scattering albedos are provided in tabular form.

A general paradigm to model reaction‐based biogeochemical processes in batch systems

This paper presents the development and illustration of a numerical model of reaction‐based geochemical and biochemical processes with mixed equilibrium and kinetic reactions. The objective is to provide a general paradigm for modeling reactive chemicals in batch systems, with expectations that it is applicable to reactive chemical transport problems. The unique aspects of the paradigm are to simultaneously (1) facilitate the segregation (isolation) of linearly independent kinetic reactions and thus enable the formulation and parameterization of individual rates one reaction by one reaction when linearly dependent kinetic reactions are absent, (2) enable the inclusion of virtually any type of equilibrium expressions and kinetic rates users want to specify, (3) reduce problem stiffness by eliminating all fast reactions from the set of ordinary differential equations governing the evolution of kinetic variables, (4) perform systematic operations to remove redundant fast reactions and irrelevant kinetic reactions, (5) systematically define chemical components and explicitly enforce mass conservation, (6) accomplish automation in decoupling fast reactions from slow reactions, and (7) increase the robustness of numerical integration of the governing equations with species switching schemes. None of the existing models to our knowledge has included these scopes simultaneously. This model (BIOGEOCHEM) is a general computer code to simulate biogeochemical processes in batch systems from a reaction‐based mechanistic standpoint, and is designed to be easily coupled with transport models. To make the model applicable to a wide range of problems, programmed reaction types include aqueous complexation, adsorption‐desorption, ion‐exchange, oxidation‐reduction, precipitation‐dissolution, acid‐base reactions, and microbial mediated reactions. In addition, user‐specified reaction types can be programmed into the model. Any reaction can be treated as fast/equilibrium or slow/kinetic reaction. An equilibrium reaction is modeled with an infinite rate governed by a mass action equilibrium equation or by a user‐specified algebraic equation. Programmed kinetic reaction rates include multiple Monod kinetics, nth order empirical, and elementary formulations. In addition, user‐specified rate formulations can be programmed into the model. No existing models to our knowledge offer these simultaneous features. Furthermore, most available reaction‐based models assume chemical components a priori so that reactions can be written in basic (canonical) forms and implicitly assume that fast equilibrium reactions occur only for homogeneous reactions. The decoupling of fast reactions from slow reactions lessens the stiffness typical of these systems. The explicit enforcement of mass conservation overcomes the mass conservation error due to numerical integration errors. The removal of redundant fast reactions alleviates the problem of singularity. The exclusion of irrelevant slow reactions eliminates the issue of exporting their problematic rate formulations/parameter estimations to different environment conditions. Taking the advantage of the nonuniqueness of components, a dynamic basis‐species switching strategy is employed to make the model numerically robust. Backward basis switching allows components to freely change in the simulation of the chemistry module, while being recovered for transport simulation. Three example problems were selected to demonstrate the versatility and robustness of the model.

Increment formulations for rounding error reduction in the numerical solution of structured differential systems

Strategies for reducing the effect of cumulative rounding errors in geometric numerical integration are outlined. The focus is, in particular, on the solution of separable Hamiltonian systems using explicit symplectic integration methods and on solving orthogonal matrix differential systems using projection. Examples are given that demonstrate the advantages of an increment formulation over the standard implementation of conventional integrators. We describe how the aforementioned special purpose integration methods have been set up in a uniform, modular and extensible framework being developed in the problem solving environment Mathematica.

Application of Recursive Constructs for Numerical Integration and Determination of Invariant Errors

A general concept of a recursive construct is introduced to determine invariant errors in numerical integration of functions, which, in turn, may be useful in applying invariant constructs to more effective numerical integration from the viewpoint of computation time and accuracy.

Numerical integration error method for zeros of analytic functions

It is shown that a globally convergent zero finding method named the numerical integration error method (NIEM) can be applied to functions in a certain class of analytic functions including exponential type. Some results of numerical experiments are given.

Data continuation for the explicit solution of an inverse biomagnetic problem

This paper considers the problem of data continuation for an explicit identification method for electric dipoles situated in a conductor, and shows that the magnetic field measured over a part of a closed surface (which encloses the conductor) can be analytically continued to the rest of the surface. Such a continuation problem has a unique solution for an arbitrarily shaped conductor and observation surface. The paper proposes two straightforward methods for the continuation - the spherical harmonic expansion and the integral equation approach - and presents continuation examples for the case of a single dipole in a spherical conductor with the data measured on a part of a spherical observation surface. The spherical harmonic expansion method works well when the field is continued relatively far from the conductor and when the measured data are clean; however, the spherical harmonic expansion fails when the data are noisy. On the other hand, the integral equation approach, although it suffers from numerical integration errors, shows a much better continuation with quite noisy data. (A method to reduce the numerical integration errors is proposed.) The latter approach is also much more accurate than the spherical harmonic expansion for continuing the field in the proximity of the conductor. Finally, the paper discusses the applicability of both methods to an explicit identification method.

Relaxation oscillations in a kinetic model of catalytic hydrogen oxidation involving a chase on canards

A detailed study of two- and three-variable mathematical models of a heterogeneous catalytic system is presented with special attention to weakly stable dynamics, a type of complex irregular behavior frequently encountered in oscillating chemical reactions. One of the most important properties of the weakly stable dynamics is “a sensitive dependence on the initial conditions”. Our analysis of a global error in long-term numerical integration shows that a high sensitive dependence on the initial conditions appears in the three-variable system with fast, intermediate and slow variables due to existence of the canard cycles which occur close to Hopf bifurcation in the one-parameter family of two-variable subsystems.

Error and uncertainty in modeling and simulation

This article develops a general framework for identifying error and uncertainty in computational simulations that deal with the numerical solution of a set of partial differential equations (PDEs). A comprehensive, new view of the general phases of modeling and simulation is proposed, consisting of the following phases: conceptual modeling of the physical system, mathematical modeling of the conceptual model, discretization and algorithm selection for the mathematical model, computer programming of the discrete model, numerical solution of the computer program model, and representation of the numerical solution. Our view incorporates the modeling and simulation phases that are recognized in the systems engineering and operations research communities, but it adds phases that are specific to the numerical solution of PDEs. In each of these phases, general sources of uncertainty, both aleatory and epistemic, and error are identified. Our general framework is applicable to any numerical discretization procedure for solving ODEs or PDEs. To demonstrate this framework, we describe a system-level example: the flight of an unguided, rocket-boosted, aircraft-launched missile. This example is discussed in detail at each of the six phases of modeling and simulation. Two alternative models of the flight dynamics are considered, along with aleatory uncertainty of the initial mass of the missile and epistemic uncertainty in the thrust of the rocket motor. We also investigate the interaction of modeling uncertainties and numerical integration error in the solution of the ordinary differential equations for the flight dynamics.