An accelerated incremental algorithm to trace the nonlinear equilibrium path of structures
30 Jun 2012-Latin American Journal of Solids and Structures (LATIN AMERICAN JOURNAL OF SOLIDS AND STRUCTURES)-Vol. 9, Iss: 4, pp 425-442
TL;DR: In this paper, an effective iterative algorithm, named the three-point method, is applied to nonlinear analysis of structures, which can be used for structures with complex behavior, including: unloading, snap-through, elastic post-buckling and inelastic postbuckling analyses.
Abstract: This paper deals with the convergence acceleration of iterative nonlinear methods. An effective iterative algorithm, named the three-point method, is applied to nonlinear analysis of structures. In terms of computational cost, each iteration of the three-point method requires three evaluations of the function. In this study the effective functions have been proposed to accelerate the convergence process. The proposed method has a convergence order of eight, and it is important to note that its implementation does not require the computation of higher order derivatives compared to most other methods of the same order. To trace the equilibrium path beyond the limit point, a normal flow algorithm is implemented into a computer program. The three-point method is applied as an inner step in the normal flow algorithm. The procedure can be used for structures with complex behavior, including: unloading, snap-through, elastic post-buckling and inelastic post-buckling analyses. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results show that the new method is comparable with the well-known existing methods and gives better results in convergence speed.
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TL;DR: The basic conclusion from the study is that robust and efficient implementation of algorithms requires expert knowledge and considerable numerical experimentation.
Abstract: In this two part paper, the problem of implementation of computational algorithms for design optimization into a computer software is discussed. A recently developed algorithm that generates and incorporates approximate second order information about the problem is selected for detailed analyses and discussions. It is shown that numerical behaviour of the algorithm is influenced by variation of the key parameters and procedures. The concept of numerical experiments is introduced, and certain variations of the algorithm and parameters are selected and their influence on its performance is studied. It is shown that the numerical rate of convergence can be substantially improved with proper procedures and values of the parameters. The first part of the paper describes some preliminary analyses and investigations. The second part describes further numerical analyses and detailed procedures for evaluation of performance of various variations of an algorithm or different computer codes. The basic conclusion from the study is that robust and efficient implementation of algorithms requires expert knowledge and considerable numerical experimentation. A wide range of small scale and large scale problems of varying difficulty must be solved to evaluate performance of an algorithm. The study suggests development of knowledge-based systems for practical design optimization.
45 citations
TL;DR: In this paper, the improved perturbation algorithm is proposed to refine the classical methods in numerical computing techniques such as the Newton-Raphson method, and a nonlinear load control procedure is generated and implemented for structures.
Abstract: The objective of this study is to explore a noble application of the improved homotopy perturbation procedure bases in structural engineering by applying it to the geometrically nonlinear analysis of the space trusses. The improved perturbation algorithm is proposed to refine the classical methods in numerical computing techniques such as the Newton–Raphson method. A linear of sub-problems is generated by transferring the nonlinear problem with perturbation quantities and then approximated by summation of the solutions related to several sub-problems. In this study, a nonlinear load control procedure is generated and implemented for structures. Several numerical examples of known trusses are given to show the applicability of the proposed perturbation procedure without considering the passing limit points. The results reveal that perturbation modeling methodology for investigating the structural performance of various applications has high accuracy and low computational cost of convergence analysis, compared with the Newton–Raphson method.
10 citations
Journal Article•
TL;DR: In this article, the effect of angle between predictor and corrector surfaces on the structural analysis is investigated and two objective functions are formulated based on this angle and also the load factor Optimizing these functions, and using the structural equilibrium path's geometry, lead to two new constraints for the nonlinear solver.
Abstract: In this paper, the effect of angle between predictor and corrector surfaces on the structural analysis is investigated Two objective functions are formulated based on this angle and also the load factor Optimizing these functions, and using the structural equilibrium path’s geometry, lead to two new constraints for the nonlinear solver Besides, one more formula is achieved, which was previously found by other researchers, via a different mathematical process Several benchmark structures, which have geometric nonlinear behavior, are analyzed with the proposed methods The finite element method is utilized to analyze these problems The abilities of suggested schemes are evaluated in tracing the complex equilibrium paths Moreover, comparison study for the required number of increments and iterations is performed Results reflect the robustness of the authors’ formulations
6 citations
Cites background from "An accelerated incremental algorith..."
...From a strictly mathematical viewpoint, it could be implied to multi-point procedures with different convergence [20, 21]....
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01 Jan 2014
TL;DR: In this paper, a hybrid iterative algorithm is employed for solving nonlinear problems based on the homotopy perturbation method (HPM) and the Newton-Raphson method.
Abstract: The efficiency of the Newton-Raphson iteration method for solving nonlinear equations has made it popular, although the time required to achieve convergence inspires aspirations to find a more efficient alternative. In the current study a hybrid iterative algorithm is employed for solving nonlinear problems. To that effect, an alternative to the NewtonRaphson method, and related classical methods in numerical computing based on a Homotopy Perturbation Method (HPM) is introduced. In perturbation methods, perturbation quantities are used to replace a nonlinear problem by a number of manageable linear subproblems. Then, an approximate solution is reached by summing up the results of these subproblems. In this paper three global methods belonging to this family are discussed and then it is shown how to combine a global method with Newton-Raphson method into a hybrid algorithm as a possible way to reduce computational cost. Several well-known and difficult applications are considered for testing the performance of the new approach. The results reveal that using 2 HPM coupled with two-point method requires less time to achieve convergence and reduces the total number of iterations.
4 citations
Cites background from "An accelerated incremental algorith..."
...Recently, a new approach to accelerate the nonlinear analysis of structures with low computational cost has been proposed ([9,10,11])....
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References
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17 Aug 2012
TL;DR: De Borst et al. as mentioned in this paper present a condensed version of the original book with a focus on non-linear finite element technology, including nonlinear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity.
Abstract: Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist Rene de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature.
2,568 citations
TL;DR: In this paper, an incremental approach to the solution of buckling and snapping problems is explored, where the authors use the length of the equilibrium path as a control parameter, together with the second order iteration method of Newton.
1,821 citations
"An accelerated incremental algorith..." refers methods in this paper
...Normal flow GDC Arc-length MRD algorithm[21] method[6] method[20] method [7] Analysis N-R three-point N-R three-point N-R three-point N-R three-point Elastic 115....
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...The original idea behind the arc–length method was introduced by Riks[20] and Wempner[26]....
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...Normal flow GDC Arc-length MRD algorithm[21] method [6] method[20] method[7] Analysis N-R three-point N-R three-point N-R three-point N-R three-point Elastic 26....
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TL;DR: In this paper, a generalized arc-length is introduced in the configuration-load space in order to facilitate the incremental computations near limit points, and the arc length is used as the loading parameter in some illustrative problems.
487 citations
"An accelerated incremental algorith..." refers methods in this paper
...The original idea behind the arc–length method was introduced by Riks[20] and Wempner[26]....
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TL;DR: A generalized displacement control method has been proposed for geometrically nonlinear analysis and has been proven to be bery effective by several problems with multiple limit points and snap-back points as mentioned in this paper.
Abstract: A generalized displacement control method has been proposed for geometrically nonlinear analysis and has been proven to be bery effective by several problems with multiple limit points and snap-back points. This method can be implemented easily in a general-purpose finite-element program.
289 citations
"An accelerated incremental algorith..." refers methods in this paper
...This method is originally proposed by Yang and Shieh[27]; (b) the work control method which was proposed by Bathe and Dvorkin[5] to enforce a constant value of work done in each iteration....
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