Fictitious Time Step for the Kinetic Dynamic Relaxation Method

Rezaiee‐Pajand, Mohammad; Rezaee, H.

doi:10.1080/15376494.2012.699603

Cited by 21 publications

(10 citation statements)

References 26 publications

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“…A nonlinear equations solver (e.g., the Newton-Raphson method) is used to determine the acceleration vector a +1 that satisfies (6). Subsequently, d +1 is calculated from (3), and the calculation of the k+1 step is completed.…”

Section: Summary Of Ckdrmentioning

confidence: 99%

“…The basic DR method was proposed in 1965 by A.S. Day [1], and, since then, there have been many applications in the field of structural dynamics [2], geomechanics [3], and biomechanics [4]. Research studies continue to develop faster and more efficient DR methods, and the state-of-the-art methods and examples are reported by Rezaiee-Pajand et al [5][6][7]. The DR method basically compels the dynamic system to converge to a static equilibrium through the application of fictitious damping.…”

Section: Introductionmentioning

confidence: 99%

“…The DR method basically compels the dynamic system to converge to a static equilibrium through the application of fictitious damping. The type of DR method depends on the type of fictitious damping, and many variants are introduced in [6]. Among them, dynamic relaxation with kinetic damping (KDR) is a widely used method proposed in a study by Cundall [8].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Adaptive Step-Size Control for Dynamic Relaxation Using Continuous Kinetic Damping

Jung

Kim

Yoo

2018

Mathematical Problems in Engineering

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Dynamic relaxation (DR) is a widely used numerical method to determine the static equilibrium of a dynamic system. However, it is difficult to apply conventional DR methods to nonlinear models because they require estimation of a stiffness matrix of the model. To resolve the forementioned problem, a new dynamic relaxation method using continuous kinetic damping (CKDR) was proposed in previous research. The CKDR method does not require any model parameters including the stiffness matrix, and it possesses absolute stability and a second-order convergence rate. However, the convergence rate is proportional to square of the step size, and this may result in a low convergence rate if the selected step size is excessively small. This problem leads to difficulties in the practical use of CKDR. Thus, an adaptive step-size method is proposed in this paper to control the convergence rate of CKDR. The proposed method estimates natural frequency of the model and determines adaptive step size. Static equilibrium simulations were performed for three different models to verify the method. The results revealed that the computational cost of CKDR with a variable step size was very efficient when compared to fixed step sizes and that the convergence rate was also controlled as intended. In addition, the lowest natural frequencies of models in static equilibrium were accurately estimated.

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Section: Summary Of Ckdrmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Adaptive Step-Size Control for Dynamic Relaxation Using Continuous Kinetic Damping

Jung

Kim

Yoo

2018

Mathematical Problems in Engineering

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“…A 15‐story moment‐resisting frame, as shown in Figure , is considered . The structure has 135 elements and total 225 degrees of freedom.…”

Section: Numerical Examplesmentioning

confidence: 99%

Dynamic relaxation method based on Lanczos algorithm

Namadchi

Alamatian

2017

Int. J. Numer. Meth. Engng

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This paper tries to accelerate the convergence rate of the general viscous dynamic relaxation method. For this purpose, a new automated procedure for estimating the critical damping factor is developed by employing a simple variant of the Lanczos algorithm, which does not require any re-orthogonalization process. All of the computational operations are performed by simple vector-matrix multiplication without requiring any matrix factorization or inversion. Some numerical examples with geometric nonlinear behavior are analyzed by the proposed algorithm. Results show that the suggested procedure could effectively decrease the total number of convergence iterations compared with the conventional dynamic relaxation algorithms.A. H. NAMADCHI AND J. ALAMATIAN time integration process. Large amount of research has been directed toward both improving the DR parameters and application of the DR method in various engineering problems. Otter employed DR to analyze axially symmetric pressure vessels and a three-dimensional elastic arch dam using central finite differences [5,6]. Cassel investigated the application of DR technique in analyze of cylindrical shells [7]. Rushton proposed a systematic procedure to tackle rectangular plate problems including small and large deflection of elastic plates as well as the buckling behavior [8][9][10]. In order to damp out the artificial oscillations as quick as possible, Rushton suggested that the critical damping of fundamental mode of vibration is selected as the fictitious damping factor, and an automated process for estimating this factor was also introduced [8]. Considering DR as a fictitious time-marching scheme, Brew and Brotton investigated DR characteristics based on the concept of amplification matrix and employed DR to analyze frame structures [11]. Pseudo-transient analysis of linear and geometric nonlinear Mindlin plates using finite element method was successfully carried out by adopting a specific form of DR. Some empirically based suggestions regarding time-step selection and fictitious mass matrix were also reported [12]. By assuming a load vector proportional to diagonal entries of the artificial mass matrix, which is formulated using Gershgorin's circle theorem, Bunce proposed a new technique to determine critical damping in DR algorithm based on the Rayleigh's principle [13]. In another study, Papadrakakis evaluated DR properties by transforming the fundamental difference equations into the error space, which leads to an eigenvalue problem. By examining the resultant characteristic equation, a new automated procedure for estimating DR fictitious parameters was proposed that improves the convergence rate of the method [3]. Zienkiewicz and Löhner studied the concept of viscous relaxation in which the inertial term is removed from the virtual dynamic system. They then utilized a line search technique to accelerate the convergence [14]. Instead of applying the external load in an incremental form, which is conventional in nonlinear quasi-static analyses, a novel techniq...

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“…They introduced the top five procedures by comparing their performance. Rezaiee-Pajand and Rezaiee proposed a new time step for kinetic dynamic relaxation in the latest study [47].…”

Section: Introductionmentioning

confidence: 99%