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Improving the convergence rate of kinetic dynamic relaxation method with new R-point

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Abstract

This study deals with a new strategy for the restart phase in the kinetic dynamic relaxation (DR) method. First, the position of the restart point (R-point) is determined by the kinetic energy modeling as a quadratic function in successive DR iterations. Then, the displacement vector of the R-point is formulated based on the finite difference method. The proposed relation for the R-point is very simple and does not impose any additional calculations on the kinetic DR algorithm. For numerical evaluation, several truss, frame, and shell structures, with linear and nonlinear behaviors, are analyzed by different kinetic DR algorithms. The results show that the proposed R-point formulation increases the convergence rate of the kinetic DR method so that the average number of required iterations decreases by about 9% and 5% in linear and nonlinear analyzes, respectively.

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Abbreviations

\(A_{1} ,A_{2} ,A_{3}\) :

Some parameters

\({\varvec{D}}\) :

Structural displacement vector

\(\varvec{\ddot{D}}\) :

Fictitious acceleration vector

\({\varvec{F}}\) :

Structural internal force vector

\({\varvec{M}}\) :

Fictitious mass matrix

\(^{n}\) :

Number of DR iteration

\({\varvec{P}}\) :

Structural external load vector

\(q\) :

Number of degrees of freedom

\(R\) :

Residual force vector

\({\varvec{S}}\) :

Structural stiffness matrix

\(T\) :

Kinetic energy of fictitious dynamic system

\(\Delta T\) :

Relative kinetic energy function

\(\tau\) :

Fictitious time step

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Authors and Affiliations

  1. Civil Engineering Department, Mashhad Branch, Islamic Azad University, Mashhad, Iran

    Javad Alamatian

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Alamatian, J. Improving the convergence rate of kinetic dynamic relaxation method with new R-point. Engineering with Computers 38 (Suppl 3), 2555–2568 (2022). https://doi.org/10.1007/s00366-021-01410-x

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