scholarly journals Improving the two-stage numerical integration in stability identification of oscillation with distributed delay

2019 ◽  
Vol 11 (1) ◽  
pp. 168781401881990
Author(s):  
Chigbogu Godwin Ozoegwu
Keyword(s):  
Numerical Integration ◽  
Distributed Delay ◽  
Original Method ◽  
Higher Order ◽  
Trapezoidal Rule ◽  
Point Of View ◽  
Engineering Systems ◽  
Two Stage ◽  
Integration Rule ◽  
Integration Rules

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order [Formula: see text] irrespective of the order of the numerical integration rule used to handle the distributed delay. But [Formula: see text] is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule).

Estimating stability boundaries of distributed-delay oscillators via two-stage numerical integration

2016 ◽  
Vol 24 (9) ◽  
pp. 1728-1740 ◽  
Author(s):  
Chigbogu Godwin Ozoegwu
Keyword(s):  
Numerical Integration ◽  
Distributed Delay ◽  
Computationally Efficient ◽  
Two Stage ◽  
Second Stage ◽  
Discrete Delays ◽  
The Stability ◽  
Spectral Finite Elements ◽  
Wheel Shimmy

A procedure based on the concept of full-discretization and numerical integration is established in this work for the stability analysis of periodic distributed-delay oscillators governed by delay integro-differential equations (DIDEs). DIDEs can be found as models of mechanical systems suffering from distributed-delay feedback, such as regenerative machine tool vibrations modeled by distributed force and wheel shimmy. Unstable vibrations in such systems are systematically avoided/controlled if the boundaries between the stable and unstable subspaces are established. The presented method involves the two-stage application of numerical integration to the governing DIDE. While the first-stage application discretizes and converts the distributed delay to fine series of short discrete delays, the second-stage application results in discrete solutions paving the way for a new method of constructing a finite monodromy operator. The error and convergence of the method are studied. It is found that the presented method is of the same convergence as that of the well-accepted first-order semi-discretization method, but more computationally efficient in terms of time savings. A number of case study DIDEs that have already been studied in the literature using methods of semi-discretization and spectral finite elements are studied with the presented method. It is seen that the presented method is valid as it produces stability results that compare well with those of the earlier works.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

scholarly journals Voigt Transform and Umbral Image

2020 ◽  
Vol 25 (3) ◽  
pp. 49
Author(s):  
Silvia Licciardi ◽  
Rosa Maria Pidatella ◽  
Marcello Artioli ◽  
Giuseppe Dattoli
Keyword(s):  
Spectroscopic Studies ◽  
Higher Order ◽  
Point Of View ◽  
The Other ◽  
Pivotal Role ◽  
Umbral Calculus ◽  
Laguerre Functions ◽  
Hermite And Laguerre Functions ◽  
Higher Order Functions ◽  
Voigt Function

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions.

DISTRIBUTED ADAPTIVE MULTIVARIATE FUNCTION VISUALIZATION

2006 ◽  
Vol 06 (02) ◽  
pp. 273-288 ◽  
Author(s):  
SHUJUN LI ◽  
KARLIS KAUGARS ◽  
ELISE DE DONCKER
Keyword(s):  
Numerical Integration ◽  
Time Series Data ◽  
Grid Cell ◽  
Adaptive Method ◽  
Series Data ◽  
Function Evaluation ◽  
Uniform Grid ◽  
Visualization Method ◽  
Remote Visualization ◽  
Integration Rule

In this article, we introduce a new function visualization method and demonstrate that numerical integration and visualization of multi-dimensional functions are closely related. Adaptive numerical integration is utilized to reduce the number of function evaluations, and generate time series data. The integration region is partitioned into a uniform grid. A grid cell can be sampled many times, or is not sampled at all, depending on the function properties and the integration rule. Function properties are extracted during the process of function evaluation. An aging technique helps visualize functions by retaining the most recently sampled areas and making the older ones transparent. This also results in giving the non-smooth areas more attention than the smooth areas. The new function visualization method gives a view of the whole function while elaborating on important areas such as ridges and troughs, which are critical in many fields, including numerical integration. A Grid service, called Integration Service, is used to solve computationally intensive integration problems. Remote visualization based on the adaptive method helps monitor the progress of a computation, and can be utilized for computational steering. The data are filtered by the server and transferred to the client, which is responsible for visualization mapping and rendering.

Higher-Order Multilevel Solvers for the EHL Line and Point Contact Problem

1994 ◽  
Vol 116 (4) ◽  
pp. 741-750 ◽  
Author(s):  
C. H. Venner
Keyword(s):  
Contact Problem ◽  
Computing Time ◽  
Point Contact ◽  
Higher Order ◽  
Second Order ◽  
Point Of View ◽  
First Order ◽  
Fundamental Point ◽  
Numerical Solver ◽  
Circular Contact

This paper addresses the development of efficient numerical solvers for EHL problems from a rather fundamental point of view. A work-accuracy exchange criterion is derived, that can be interpreted as setting a limit to the price paid in terms of computing time for a solution of a given accuracy. The criterion can serve as a guideline when reviewing or selecting a numerical solver and a discretization. Earlier developed multilevel solvers for the EHL line and circular contact problem are tested against this criterion. This test shows that, to satisfy the criterion a second-order accurate solver is needed for the point contact problem whereas the solver developed earlier used a first-order discretization. This situation arises more often in numerical analysis, i.e., a higher order discretization is desired when a lower order solver already exists. It is explained how in such a case the multigrid methodology provides an easy and straightforward way to obtain the desired higher order of approximation. This higher order is obtained at almost negligible extra work and without loss of stability. The approach was tested out by raising an existing first order multilevel solver for the EHL line contact problem to second order. Subsequently, it was used to obtain a second-order solver for the EHL circular contact problem. Results for both the line and circular contact problem are presented.

scholarly journals An improved Trapezoidal rule for numerical integration

2021 ◽  
Vol 2090 (1) ◽  
pp. 012104
Author(s):  
A. F Abdulhameed ◽  
Q A Memon
Keyword(s):  
Numerical Integration ◽  
Integration Method ◽  
Research Community ◽  
Trapezoidal Rule ◽  
Engineering Software ◽  
Engineering Problems ◽  
Current Century ◽  
Single Interval ◽  
Improved Accuracy ◽  
Higher Order Functions

Abstract Numerical Methods have attracted of research community for solving engineering problems. This interest is due to its practicality and the improvement of highspeed calculations done on current century processors. The increase in numerical method tools in engineering software, such as Matlab, is an example of the increased interest. In this paper, we are present a new improved numerical integration method, that is based on the well-known trapezoidal rule. The proposed method gives a great enhancement to the trapezoidal rule and overcomes the issue of the error value when dealing with some higher order functions even when solving for a single interval. After literature review, the proposed system is mathematically explained along with error analysis. Few examples are illustrated to prove improved accuracy of the proposed method over traditional trapezoidal method.

scholarly journals Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

2015 ◽  
Vol 2015 ◽  
pp. 1-19 ◽  
Author(s):  
Krzysztof Gdawiec ◽  
Wiesław Kotarski ◽  
Agnieszka Lisowska
Keyword(s):  
Iteration Process ◽  
Higher Order ◽  
Point Of View ◽  
Complex Polynomials ◽  
Root Finding ◽  
Points Of View ◽  
Iteration Processes

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

Pionization at AGS and SPS energies ― degree of coherence and its target excitation dependence

2009 ◽  
Vol 87 (4) ◽  
pp. 311-319 ◽  
Author(s):  
Dipak Ghosh ◽  
Argha Deb ◽  
Srimonti Dutta
Keyword(s):  
Particle Production ◽  
Source Model ◽  
Higher Order ◽  
Point Of View ◽  
Projectile Energy ◽  
Data Set ◽  
Energy Scaling ◽  
Degree Of Coherence

The target excitation dependence of the higher order cumulant correlation of pions is studied in the present work in the framework of void probability scaling. The data set of pions produced for 16O–AgBr interactions at 2.1 AGeV and 32S–AgBr interactions at 200 AGeV was divided into three sets depending upon the number of grey tracks ng. The different sets of ng correspond to different degrees of target excitation. Analysis was carried out for each set in different pseudorapidity intervals. For the lower projectile energy the void probability scaling is observed only for low target excitation for all pseudorapidity intervals. For the comparatively higher projectile energy, scaling is observed for all pseudorapidity intervals and all target excitation. In other words, for higher projectile energy, scaling is almost independent of the target excitation. The results are interpreted from the point of view of the two source model of particle production.

Two-Stage Precession Type Gear: Design, Geometric and Kinematic Analysis

2017 ◽  
Author(s):  
Łukasz Macyszyn ◽  
Adam Myszkowski ◽  
Roman Staniek ◽  
Stanisław Pabiszczak
Keyword(s):  
Gear Ratio ◽  
Original Method ◽  
Two Stage ◽  
Precession Angle ◽  
Gear Design ◽  
The Face ◽  
Gear Efficiency ◽  
Rotary Speed ◽  
Gear Wheels ◽  
Neodymium Magnets

The paper presents the theoretical bases, design and the principle of operation of two-stage precession type transmission with face meshing. Description and the principle of forming the face meshing which is modified by the original method have been shown as well. Dimensional relations between particular components of the gears are established and the analysis of optimal gear ratio, depending on the number of teeth or magnets on the circumferences of meshing gear wheels is also provided in the paper. For further analysis four prototypes of mechanical precession transmission with face meshing were designed, built and investigated. Those prototypes present different sizes, reduction ratio and precession angle. Investigations, described in the paper, helped to determine the gear efficiency rate as well as the maximal torque that could be transferred for the given rotary speed. This paper presents also the conception of the design of a novel double stage precession magnetic gear with face neodymium magnets. The results of the initial studies are the background of the further research in the field of magnetic precession type transmission.

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