A comparative study on application of Chebyshev and spline methods for geometrically non-linear analysis of truss structures

Mahdavi, Seyed Hossein; Razak, Hashim Abdul; Shojaee, Saeed; Mahdavi, Maedeh Sadat

doi:10.1016/j.ijmecsci.2015.08.001

Cited by 10 publications

(5 citation statements)

References 31 publications

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“…Сплайн широко используется в обработке сигналов, дискретных вычислениях, статистике и, в частности, сплайн сглаживания позволят получить гладкую кривую, которая наилучшим образом подходит для аппроксимации данных с шумом [1,2]. Теоретический сглаживающий сплайн в работе [3] является обобщением сглаживающего сплайна с использованием новых подходов управления, с помощью которых кривая сплайна определяется выходом линейной динамической системы.…”

Section: обзор исследованийunclassified

Investigation of Parameters of the Problem of Spline Approximation of Noisy Data by Numerical Methods of Optimal Control

Bolodurina¹,

Grishina²,

Antsiferova³

2021

CTAC&R

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Currently, the problems of distortion of measurement data by noise and the appearance of un-certainties in quality criteria have caused increased interest in research in the field of spline approx-imation. At the same time, existing methods of minimizing empirical risk, assuming that the noise is a uniform distribution with heavier tails than Gaussian, limit the scope of application of these studies. The problem of estimating noise-distorted data is usually based on solving an optimi-zation problem with a function containing uncertainty arising from the problem of finding optimal parameters. In this regard, the estimation of distorted noise cannot be solved by classical methods. Aim. This study is aimed at solving and analyzing the problem of spline approximation of data under uncertainty conditions based on the parametrization of control and the gradient projec-tion algorithm. Methods. The study of the problem of spline approximation of noisy data is carried out by the method of approximation of the piecewise constant control function. In this case, para-metrization of the control is possible only for a finite number of break points of the first kind. In the framework of the experimental study, the gradient projection algorithm is used for the numerical solution of the spline approximation problem. The proposed methods are used to study the parameters of the problem of spline approximation of data under conditions of uncertain-ty. Results. The numerical study of the control parametrization approach and the gradient projec-tion algorithm is based on the developed software and algorithmic tool for solving the problem of the spline approximation model under uncertainty. To evaluate the noise-distorted data, numerical experiments were conducted to study the model parameters and it was found that increasing the value of the parameter α leads to an increase in accuracy, but a loss of smoothness. In addition, the analysis showed that the considered distribution laws did not change the accuracy and convergence rate of the algorithm. Conclusion. The proposed approach for solving the problem of spline approx-imation under uncertainty conditions allows us to determine the problems of distortion of measure-ment data by noise and the appearance of uncertainties in the quality criteria. The study of the model parameters showed that the constructed system is stable to the error of the initial approxima-tion, and the distribution laws do not significantly affect the accuracy and convergence of the gra-dient projection method.

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Section: обзор исследованийunclassified

Investigation of Parameters of the Problem of Spline Approximation of Noisy Data by Numerical Methods of Optimal Control

Bolodurina¹,

Grishina²,

Antsiferova³

2021

CTAC&R

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“…Saffari;Mansouri (2011) suggested a two-point method with fourth-order convergence. Mahdavi et al (2015) proposed an iterative method free from second derivative originated from modified Chebyshev and cubic spline's schemes. Souza et al (2018) proposed new algorithms based on two-step methods with cubic convergence order and combined with the Linear Arc-Length path-following technique.…”

Section: Introductionmentioning

confidence: 99%

Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems

2021

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In this paper we adapt the Newton-Raphson and Potra-Pták algorithms by combining them with the modified Newton-Raphson method by inserting a condition. Problems of systems of sparse nonlinear equations are solved the algorithms implemented in Matlab® environment. In addition, the methods are adapted and applied to space trusses problems with geometric nonlinear behavior. Structures are discretized by the Finite Element Positional Method, and nonlinear responses are obtained in an incremental and iterative process using the Linear Arc-Length path-following technique. For the studied problems, the proposed algorithms had good computational performance reaching the solution with shorter processing time and fewer iterations until convergence to a given tolerance, when compared to the standard algorithms of the Newton-Raphson and Potra-Pták methods.

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“…There are many predictor – corrector algorithms based in the iterative scheme of Newton-Raphson that are widely used. 7…”

Section: Introductionmentioning

confidence: 99%

“…Saffari and Mansouri 5 suggested a two-point method with fourth-order convergence, and showed that this method can decrease the computational time and iterations. Mahdavi et al 7 proposed an iterative method free from second derivative originated from modified Chebyshev and cubic spline’s schemes. Souza et al 8 presented new algorithms based on two-step methods with cubic convergence order and combined with the Linear Arc-Length path-following technique.…”

Section: Introductionmentioning

confidence: 99%

New fourth-order convergent algorithm for analysis of trusses with material and geometric nonlinearities

Souza

Santos

Kawamoto

et al. 2021

The Journal of Strain Analysis for Engineering Design

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This paper presents a new algorithm to solve the system of nonlinear equations that describes the static equilibrium of trusses with material and geometric nonlinearities, adapting a three-step method with fourth-order convergence found in the literature. The co-rotational formulation of the Finite Element Method is used in the discretization of structures. The nonlinear behavior of the material is characterized by an elastoplastic constitutive model. The equilibrium paths with limit points of load and displacement are obtained using the linearized Arc-Length path-following technique. The numerical results obtained with the free program Scilab show that the new algorithm converges faster than standard procedures and modified Newton-Raphson, since the approximate solution of the problem is obtained with a smaller number of accumulated iterations and less CPU time. The equilibrium paths show that the structures exhibit a completely different behavior when the material nonlinearity is considered in the analysis with large displacements.

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A comparative study on application of Chebyshev and spline methods for geometrically non-linear analysis of truss structures

Cited by 10 publications

References 31 publications

Investigation of Parameters of the Problem of Spline Approximation of Noisy Data by Numerical Methods of Optimal Control

Investigation of Parameters of the Problem of Spline Approximation of Noisy Data by Numerical Methods of Optimal Control

Adaptation of the Newton-Raphson and Potra-Pták methods for the solution of nonlinear systems

New fourth-order convergent algorithm for analysis of trusses with material and geometric nonlinearities

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