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Creating better dynamic relaxation methods

M. Rezaiee-Pajand (Civil Engineering Department, Ferdowsi University of Mashhad, Mashhad, Islamic Republic of Iran)
Hossein Estiri (Civil Engineering Department, Ferdowsi University of Mashhad, Mashhad, Islamic Republic of Iran)
Mohammad Mohammadi-Khatami (Civil Engineering Department, Ferdowsi University of Mashhad, Mashhad, Islamic Republic of Iran)

Engineering Computations

ISSN: 0264-4401

Article publication date: 19 July 2019

Issue publication date: 15 August 2019




The purpose of this study is to demonstrate that using appropriate values for fictitious parameters is very important in dynamic relation methods. It will be shown that a better scheme can be made by modifying these terms.


Former research studies have proposed diverse values for fictitious parameters. These factors are very essential and highly affect structural analyses’ abilities. In this paper, the fictitious masses in ten previous well-known schemes are replaced with each other. These formulations lead to the extra 41 different new procedures.


To compare the skills of the created processes with those of the ten previous ones, 14 benchmark problems with geometrical nonlinear behaviour are analysed. The performances’ evaluations are based on the number of iterations and analysis time. Considering these two criteria, the score of each technique is found for the ranking assessments.

Research limitations/implications

To solve a static problem by using a dynamic relaxation (DR) scheme, it should be first converted to a dynamic space. Using the appropriate values for fictitious terms is very important in this approach. The fictitious mass matrix and damping factor play the most effective role in the process stability. Besides, the fictitious time step is necessary for improving the method convergence rate.

Practical implications

Different famous DR procedures were compared with each other previously. These solvers used their original assumptions for the imaginary mass and damping. So far, no attempt has been made to change the fictitious parameters of the well-known DR methods. As these fictitious factors highly affect structural analyses’ efficiencies, these solvers are formulated again by using new parameters. In this study, the fictitious masses of ten previous famous methods are replaced with each other. These substitutions give 51 different procedures.


It is concluded that the present formulations lead to more effective and favourable methods than the solvers with previous assumptions.



Rezaiee-Pajand, M., Estiri, H. and Mohammadi-Khatami, M. (2019), "Creating better dynamic relaxation methods", Engineering Computations, Vol. 36 No. 5, pp. 1483-1521.



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