An iterative scheme of flexibility‐based component mode synthesis with higher‐order residual modal compensation

2021 ◽  
Author(s):  
In Seob Chung ◽  
Jin‐Gyun Kim ◽  
Soo‐Won Chae ◽  
K. C. Park
Keyword(s):  
Iterative Scheme ◽  
Higher Order ◽  
Component Mode Synthesis

scholarly journals Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2570
Author(s):  
Alicia Cordero ◽  
Beny Neta ◽  
Juan R. Torregrosa
Keyword(s):  
Order Of Convergence ◽  
Iterative Scheme ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Multiple Roots ◽  
Higher Order Methods ◽  
Efficient Strategy ◽  
Schröder’S Method ◽  
With Memory

In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots.

VARIATIONAL ITERATION METHOD FOR SOLVING NONLINEAR HIGHER-ORDER INITIAL AND BOUNDARY VALUE PROBLEMS

2009 ◽  
Vol 06 (04) ◽  
pp. 521-555 ◽  
Author(s):  
SYED TAUSEEF MOHYUD-DIN ◽  
MUHAMMAD ASLAM NOOR ◽  
KHALIDA INAYAT NOOR
Keyword(s):  
Boundary Value Problems ◽  
Iteration Method ◽  
Nonlinear Boundary ◽  
Iterative Scheme ◽  
Boundary Value ◽  
Variational Iteration Method ◽  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Variational Iteration ◽  
Adomian’S Polynomials

In this paper, we apply variational iteration method (VIM) and variational iteration method using Adomian's polynomials for solving nonlinear boundary value problems. The proposed iterative scheme finds the solution without any discretization, linearization, perturbation, or restrictive assumptions. Several examples are given to verify the accuracy and efficiency of the method. We have also considered an example where the proposed VIM is not reliable.

A Hybrid Coordinates Component Mode Synthesis Method for Dynamic Analysis of Structures With Localized Nonlinearities

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Huan He ◽  
Tao Wang ◽  
Guoping Chen
Keyword(s):  
Dynamic Analysis ◽  
Nonlinear Dynamic ◽  
Synthesis Method ◽  
Nonlinear Dynamic Analysis ◽  
Higher Order ◽  
Component Mode Synthesis ◽  
Linear Component ◽  
Nonlinear Component ◽  
Order Matrix ◽  
Modal Coordinates

This paper reports on the development of the component mode synthesis (CMS) method using hybrid coordinates for a localized nonlinear dynamic system. As is well known, the CMS method is effective in reducing the degrees-of-freedom (DOF) of the system. In contrast to most existing CMS methods, which are usually developed for linear systems, a new CMS method using hybrid coordinates for nonlinear dynamic analysis has been developed in this paper. Generally, the system is divided into two parts, namely, a linear component and a nonlinear component. The equations of the linear component can be transformed into the modal coordinates using its linear vibration modes. To improve the accuracy, the equivalent higher-order matrix of the system is developed to capture the effects of the neglected higher-order modes. Quite different from early works, the flexibility attachment matrix can be obtained without using the inverse of the stiffness matrix by using an equivalent higher-order matrix, thus making it easier to deal with those components that have rigid-body freedom when formulating the residual flexibility attachment matrix. By introducing the residual flexibility attachment matrix and the retained lower-order modes, the dynamic governing equations of the linear component can be converted into the modal space and expressed by a few modal coordinates. To adopt the entire set of nonlinear terms into the final equations, the equations of the nonlinear component are kept in their original form. Compatibility conditions at the interface are used to combine the nonlinear component and the linear component to form the synthesis equations, which are expressed in hybrid coordinates. Finally, the computational efficiency and accuracy of the presented method is demonstrated using numerical examples.

scholarly journals An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1038 ◽  
Author(s):  
Sunil Kumar ◽  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano ◽  
Praveen Agarwal ◽  
...  
Keyword(s):  
Iterative Methods ◽  
Numerical Algorithm ◽  
Fourth Order ◽  
Iterative Scheme ◽  
Higher Order ◽  
New Technique ◽  
Multiple Roots ◽  
Order Convergence ◽  
Higher Order Methods ◽  
Derivative Free

A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques.

scholarly journals A Proposal of Mode Polynomials for Efficient Use of Component Mode Synthesis and Methodology to Simplify the Calculation of the Connecting Beams

2020 ◽  
Vol 9 (1) ◽  
pp. 20
Author(s):  
Jeong Hee Park ◽  
Duck Young Yoon
Keyword(s):  
Boundary Conditions ◽  
Practical Purpose ◽  
Ritz Method ◽  
Higher Order ◽  
Component Mode Synthesis ◽  
Element Analysis ◽  
Generalized Coordinates ◽  
Local Structures ◽  
Intensive Use ◽  
Geometrical Boundary

Analytical method using Rayleigh–Ritz method has not been widely used recently due to intensive use of finite element analysis (FEA). However as long as suitable mode functions together with component mode synthesis (CMS) can be provided, Rayleigh–Ritz method is still useful for the vibration analysis of many local structures in a ship such as tanks and supports for an equipment. In this study, polynomials which combines a simple and a fixed support have been proposed for the satisfaction of boundary conditions at a junction. Higher order polynomials have been generated using those suggested by Bhat. Since higher order polynomials used only satisfy geometrical boundary conditions, two ways are tried. One neglects moment continuity and the other satisfies moment continuity by sum of mode polynomials. Numerical analysis have been performed for typical shapes, which can generate easily more complicated structures. Comparison with FEA result shows good agreements enough to be used for practical purpose. Frequently dynamic behavior of one specific subcomponent is more concerned. In this case suitable way to estimate dynamic and static coupling of subcomponents connected to this specific subcomponent should be provided, which is not easy task. Elimination of generalized coordinates for subcomponents by mode by mode satisfaction of boundary conditions has been proposed. These results are still very useful for initial guidance.

scholarly journals On a Novel Fourth-Order Algorithm for Solving Systems of Nonlinear Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Diyashvir K. R. Babajee ◽  
Alicia Cordero ◽  
Fazlollah Soleymani ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Equations ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Iterative Scheme ◽  
Higher Order ◽  
Numerical Results ◽  
Systems Of Nonlinear Equations ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Order Algorithm

This paper focuses on solving systems of nonlinear equations numerically. We propose an efficient iterative scheme including two steps and fourth order of convergence. The proposed method does not require the evaluation of second or higher order Frechet derivatives per iteration to proceed and reach fourth order of convergence. Finally, numerical results illustrate the efficiency of the method.

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