Iterative Integration Research Articles

Buckling and post-buckling analysis of extensible beam–columns by using the differential quadrature method

Buckling and post-buckling analysis of extensible beam–columns is performed numerically in this paper. It was experienced earlier that in some cases the numerical integration would not produce the convergent post-buckling solutions, especially under high loads. Therefore, a new differential quadrature (DQ) based iterative numerical integration method is proposed to solve post-buckling differential equations of extensible beam–columns. Six cases, including five classical Euler buckling cases, are analyzed. Critical loads and convergent post-buckling solutions under different applied loads are obtained. The results are compared with the existing multiple scales solutions. It is found that under high applied loads, the small rotation assumption in obtaining multiple scales solutions is no longer valid. The proposed iterative DQ based numerical integration method can yield reliable and accurate post-buckling solutions even at high applied loads for the cases investigated.

An analytical solution for a low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate

An analytical solution for a low velocity impact between a thin transversely isotropic plate made of a strain-hardening material supported by a rigid substrate and a rigid sphere is presented. One of the novelties of this work is employing a linear strain-hardening model for investigating the indentation phenomenon in the plastic zone, rather than the traditional perfectly plastic model. Another novelty of this work is employing the homotopy perturbation method to derive analytical solutions for the highly nonlinear governing equations of contact. Since it is very important to accurately predict the contact force and its time history, the three stages of the indentation process, i.e., (1) the elastic indentation, (2) the plastic indentation, and (3) the elastic unloading stages, are investigated in detail. Comparison of the present results with results obtained from the iterative numerical time integration method confirms the accuracy and efficiency of the present solution.

Computer simulation of Fresnel diffraction from double rectangular apertures in one and two dimensions using the iterative Fresnel integrals method

We have applied the iterative Fresnel integrals method for the computation and simulation of the Fresnel diffraction images created by double apertures in one and two dimensions. Necessary formalism has been derived and the relevant computation algorithms have been developed for this application. The computer simulated images show an interesting combination of Fresnel diffraction images with mutual interference effects between the light diffracted from the two apertures. Transition to the expected Fraunhofer diffraction pattern for double apertures or slits is also observed by the simulations.

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

{Let $B=(B_1(t),...,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha<1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates, and call for an extension of Gaussian tools such as for instance the Malliavin calculus. After a first introductory paper \cite{MagUnt1}, this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as L\'evy area.

A robust and energy-conserving model of freezing variably-saturated soil

Abstract. Phenomena involving frozen soil or rock are important in many natural systems and, as a consequence, there is a great interest in the modeling of their behavior. Few models exist that describe this process for both saturated and unsaturated soil and in conditions of freezing and thawing, as the energy equation shows strongly non-linear characteristics and is often difficult to handle with normal methods of iterative integration. Therefore in this paper we propose a method for solving the energy equation in freezing soil. The solver is linked with the solution of Richards equation, and is able to approximate water movement in unsaturated soils and near the liquid-solid phase transition. A globally-convergent Newton method has been implemented to achieve robust convergence of this scheme. The method is tested by comparison with an analytical solution to the Stefan problem and by comparison with experimental data derived from the literature.

Guest Editor's Introduction: Special Section on the IEEE Symposium on Visual Analytics Science and Technology (VAST)

Guest Editor's Introduction: Special Section on the IEEE Symposium on Visual Analytics Science and Technology (VAST)

From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics

Let B = (B 1(t), . . . , B d (t)) be a d-dimensional fractional Brownian motion with Hurst index α ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (II) The rough path for \({\frac{1}{6} < \alpha < \frac{1}{4}}\): constructive proof of convergence, 2011, preprint) for the constructive proofs.

A Deflection of Nano-Cantilevers Using Monotone Solution

In this paper, a monotone positive solution is studied for deflection of nano cantilever beams which is subject to electrostatic and intermolecular forces in a distributed model. A positive monotone solution base on Green's function in the form of a nonlinear iterative integral is introduced, to obtain a solution for deflection of nano cantilever beams. In order to determine the accuracy of presented method, the results are compared with numerical results of a boundary value method. The results show the monotone iterative solution is stable and converged to numerical results with a few iterations. The results of present work are useful to prove the stability and convergence of Green's function to deal with deflection of nano cantilever switches in future works and simplifications.

Kinetic Analysis of the Lactate-dehydrogenase-coupled Reaction Process and Measurement of Alanine Transaminase by an Integration Strategy

Kinetic analyses of lactate-dehydrogenase (LD)-coupled alanine transaminase (ALT) reaction processes were investigated for measuring ALT by an integration strategy. For measuring ALT by a kinetic analysis of an LD-coupled ALT reaction curve, candidate reaction curves were calculated via iterative numerical integration of the differential velocity equations to execute a weighted nonlinear-least-square-fitting. To realize the integration strategy, the conventional initial-velocity method was used if the ALT activities were below 25 U/L; otherwise, kinetic analyses of the reaction curves were employed. Of the reaction curves recorded at 10-s intervals, kinetic analyses gave ALT activities resistant to deviations in the LD kinetic parameters. The integration strategy yielded a higher value of the lower limit, but an upper limit of over 100 U/L by simulations and over 75 U/L with purified ALT. Also, its intra-run relative standard deviations were below 9% for 0.50 U/L ALT and below 5% for final 1 to 65 U/L ALT. The integration strategy gave consistent ALT activities in clinical sera. Hence, this new approach for kinetic analyses of ALT reaction processes and the integration strategy were effective to measure ALT.

Iterative implicit integration procedure for hybrid simulation of large nonlinear structures

AbstractA fully implicit iterative integration procedure is presented for local and geographically distributed hybrid simulation of the seismic response of complex structural systems with distributed nonlinear behavior. The purpose of this procedure is to seamlessly incorporate experimental elements in simulations using existing fully implicit integration algorithms designed for pure numerical simulations. The difficulties of implementing implicit integrators in a hybrid simulation are addressed at the element level by introducing a safe iteration strategy and using an efficient procedure for online estimation of the experimental tangent stiffness matrix. In order to avoid physical application of iterative displacements, the required experimental restoring force at each iteration is estimated from polynomial curve fitting of recent experimental measurements. The experimental tangent stiffness matrix is estimated by using readily available experimental measurements and by a classical diagonalization approach that reduces the number of unknowns in the matrix. Numerical and hybrid simulations are used to demonstrate that the proposed procedure provides an efficient method for implementation of fully implicit numerical integration in hybrid simulations of complex nonlinear structures. The hybrid simulations presented include distributed nonlinear behavior in both the numerical and experimental substructures. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Iterative Monte Carlo path integral with optimal grids from whole-necklace sampling

The efficiency of the iterative Monte Carlo (IMC) path integral methodology for complex time correlation functions is increased through the use of optimal grids, which are sampled from paths that span the entire path integral necklace. The two-bead marginal distributions required in each step of the IMC iteration are obtained from a recursive procedure. Applications to one-dimensional and multi-dimensional model systems illustrate the enhancement in stability effected by the use of grids based on whole-necklace sampling.

Temperature‐induced transition between polyproline I and II helices: quantitative fitting of hysteresis effects

The conformational properties of the polyproline I (PPI) helix of oligoprolines toward heating were examined. Oligoproline H-Pro(12)-NH(2) served as a model which adopts in n-PrOH a pronounced PPI conformation with all cis amide bonds, whereas a polyproline II (PPII) conformation with all trans amide bonds is predominant in pure aqueous buffer. CD spectroscopic studies revealed that a conformational change from the PPI to the PPII helix takes place upon heating and back to the PPI helix upon cooling. This conformational transition cycle is characterized by a strong hysteresis. With a quantitative fitting of the experimentally observed hysteresis loops by a newly developed iterative integration with different starting conditions, kinetic and thermodynamic parameters for the transition from the PPI to the PPII helical conformation were determined. The transition is as expected for cis-trans isomerizations of amide bonds comparatively slow (k = 0.003 s(-1) at 80 °C) and characterized by an activation energy E(a) of 81.1 ± 3.6 kJ mol(-1). Thermodynamically, the transition from the PPI to the PPII helix is characterized by a positive standard enthalpy (ΔH(0) = 33.5 ± 2.1 kJ min(-1)) and a positive standard entropy (ΔS(0) = 102 ± 6.6 J mol(-1) K(-1)).

Dynamic analyses of gear pairs incorporating the effect of time-varying lubrication damping

This study presents a time-varying model to investigate dynamic responses of spur gear pairs. In particular, time-varying damping was directly used to account for the lubrication film of tooth pairs. At first, the governing equations of vibration for gear pairs were expressed in equivalent discrete models of mass, damping, and spring elements. The damping factor which incorporated the lubrication effect was derived using elastohydrodynamic-lubrication and squeezed-film theories. The time-varying property due to the moving meshing position was considered by immediately updating the meshing stiffness, damping factor, meshing force, and frictional force between the tooth pairs at each calculation step. After that, dynamic factors and fillet strains of the gear pairs under various operating conditions were obtained by performing iterative numerical integrations. The lubricant damping factor and its effects on gear dynamics was also discussed. Finally, the effects of the lubricant viscosity and applied torque on the gear dynamics were thoroughly investigated.

Non-abelian Reciprocity Laws on a Riemann Surface

On a Riemann surface, there are relations among the periods of holomorphic differential forms called Riemann’s relations. We prove them, using iterated integrals. In this paper, we generalize these relations to relations among generating series of iterated integrals. The generating series gives infinitely many relations—one for each coefficient of the generating series. The lower-order terms give the well-known classical relations. The new result is reciprocity for the higher degree terms, which give non-trivial relations among iterated integrals on a Riemann surface. As an application we refine the definition of Manin’s non-commutative modular symbol in order to include Eisenstein series. Finally, this paper contains some constructions needed for multi-dimensional reciprocity laws like a refinement of one of the Kato–Parshin reciprocity laws (Horozov, I. “A refinement of the Parshin symbol for surfaces”. Preprint, arXiv.org, AG, 1002.2698].

SPACots

This paper presents a software tool for integrating a child-friendly computer system based on commercial off-the-shelf (COTS) components. The effective selection of COTS components, which meet a child’s requirements and expectations, is a non-trivial and challenging optimization problem. However, many published papers consider the functional requirements while ignoring usability requirements. The functional requirements are concerned with what the computer should be able to do, whereas the usability requirements are concerned with the extent to which the child is able to learn effectively and efficiently throughout the COTS based computer. In this paper, the authors propose an iterative five-task selection and integration of COTS process, including both hardware devices and software modules, to be automated. The core of the automated tool is employing Simulated Annealing (SA) to search the design space to match, select, and integrate COTS components with a maximal satisfaction while neither exceeding a given budget nor violating child and performance constraints. A Monte Carlo simulator was utilized to evaluate the goodness of the COTS based computer design. Computational results based on building a computer for a child handwriting e-learning application show feasibility of SPACots in finding a solution satisfying all constraints while reducing the cost by 58%.

Tree expansion in time-dependent perturbation theory

The computational complexity of time-dependent perturbation theory is well known to be largely combinatorial whatever the chosen expansion method and family of parameters (combinatorial sequences, Goldstone and other Feynman-type diagrams, etc.). We show that a very efficient perturbative expansion, both for theoretical and numerical purposes, can be obtained through an original parametrization by trees and generalized iterated integrals. We emphasize above all the simplicity and naturality of the new approach that links perturbation theory with classical and recent results in enumerative and algebraic combinatorics. These tools are applied to the adiabatic approximation and the effective Hamiltonian. We prove perturbatively and nonperturbatively the convergence of Morita’s generalization of the Gell–Mann and Low wave function. We show that summing all the terms associated with the same tree leads to an utter simplification where the sum is simpler than any of its terms. Finally, we recover the Rayleigh-Schrödinger time-independent equation for the wave operator and we give an explicit nonrecursive expression for the term corresponding to an arbitrary tree.

Iterative Integration of Visual Insights during Scalable Patent Search and Analysis

Patents are of growing importance in current economic markets. Analyzing patent information has, therefore, become a common task for many interest groups. As a prerequisite for patent analysis, extensive search for relevant patent information is essential. Unfortunately, the complexity of patent material inhibits a straightforward retrieval of all relevant patent documents and leads to iterative, time-consuming approaches in practice. Already the amount of patent data to be analyzed poses challenges with respect to scalability. Further scalability issues arise concerning the diversity of users and the large variety of analysis tasks. With "PatViz", a system for interactive analysis of patent information has been developed addressing scalability at various levels. PatViz provides a visual environment allowing for interactive reintegration of insights into subsequent search iterations, thereby bridging the gap between search and analytic processes. Because of its extensibility, we expect that the approach we have taken can be employed in different problem domains that require high quality of search results regarding their completeness.

Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems

In the present paper, the modified homotopy perturbation method (MHPM) is employed to investigate about both nonlinear swinging oscillation and the stability of circular sector oscillation systems. The sensitivity study performed for frequency analysis of the mentioned oscillatory circular sector body shows that frequency of nonlinear oscillation depends on some specific parameters and can be optimized. Furthermore onset of the instability is dependent to angle α and initial amplitude. Comparisons made among the results of the present closed-form analytical solution and the traditional numerical iterative time integration solution confirms the accuracy and efficiency of the presented analytical solution. In contrast to the available numerical methods, the present analytical method is free from the numerical damping and the time integration accumulated errors. Moreover, in comparison with the traditional multistep numerical iterative time integration methods, a much less computational time is required for the present analytical method. Responses of the dynamical systems to some extent are affected by the natural frequencies. Results reveal that for nonlinear systems, the natural frequency is remarkably affected by the initial conditions.

Gauss Hypergeometric Functions, Multiple Polylogarithms, and Multiple Zeta Values

In this article, we express solutions of the Gauss hypergeometric equation as a series of the multiple polylogarithms by using iterated integral. This representation is the most simple case of a semisimple representation of solutions of the formal KZ equation. Moreover, combining this representation with the connection relations of solutions of the Gauss hypergeometric equation, we obtain various relations of the multiple polylogarithms of one variable and the multiple zeta values.

On Nonmeasurable Functions of Two Variables and Iterated Integrals

Abstract Following the paper of Pkhakadze [Trudy Tbiliss. Mat. Inst. Razmadze 20: 167–209, 1954], we consider some properties of real-valued functions of two variables, which are not assumed to be measurable with respect to the two-dimensional Lebesgue measure on the plane 𝐑2, but for which the corresponding iterated integrals exist and are equal to each other. Close connections of these properties with certain set-theoretical axioms are emphasized.