Implicit Higher-Order Accuracy Method for Numerical Integration in Dynamic Analysis

Rezaiee‐Pajand, Mohammad; Alamatian, Javad

doi:10.1061/(asce)0733-9445(2008)134:6(973)

Cited by 54 publications

(31 citation statements)

References 27 publications

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“…(5)). In a special case, if 0 = 1 = 2 = ::: = m 1 = 0 , the above relationships present IHOA integration [16]. Another interesting version of G-IHOA is obtained when 0 = 1 = 2 = ::: = m 1 = 0.…”

Section: The Generalized Implicit Higher Ordermentioning

confidence: 87%

“…The rst approach could be used in single step time integrations; however, it has some di culties, especially in the beginning of the process when higher-order derivatives should be estimated [19,20]. The multi-time-step integrations, which use information of several previous time increments to integrate the current step, are another way for satisfying the continuity of higherorder time derivatives [16,22,24,28]. In spite of more requirement memory, multi-time-step integrations are more accurate and e cient than single-step methods.…”

Section: The Generalized Implicit Higher Ordermentioning

confidence: 99%

“…as functions of the higherorder derivatives of displacement at the nth increment. For this propose, the inverse expansions of velocities and accelerations give [16]: From the mathematical point of view, the weighted factors should be determined so that a good number of derivatives/coe cients are equalized to their corresponding values in Taylor series expansion (Eqs. (12) and (13)).…”

Section: N+1mentioning

confidence: 99%

“…Running a statically analysis in each time step would be a di cult and time-consuming procedure. Newmark-method, Wilson-scheme, HHTprocedure [1], WBZ-integration [2], generalizedmethod [3], Newmark multi-time-step approach [4], third-order time step integration [5], the Newmark complex time step [6], the time weighted function procedure [7], the generalized single step integration [8], the N?rsett time integration [9], the composite time integration [10], the higher order acceleration function [11], the implicit integration based on conserving energy and momentum [12], the Green function approach [13,14], the precise integration methods [15], the IHOA [16], and the implicit integration combined with the nite-element method [17] are some of the implicit integrations.…”

Section: Introductionmentioning

confidence: 99%

“…It should be noted that by changing weighted factors, many of the previous implicit integrations have the ability to use as an explicit or predictorcorrector scheme [16,25,26]. For example, the implicit higher-order accuracy integration method (called IHOA) presents the P C m integration [24].…”

Section: Introductionmentioning

confidence: 99%

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Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

Alamatian

2017

Scientia Iranica

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Abstract. This paper deals with a generalized multi-time-step integration used for structural dynamic analysis. The proposed method presents three kinds of implicit schemes in which the accelerations and velocities of the previous steps are utilized to integrate the equations of motion. This procedure employs three groups of weighted factors calculated by minimizing the numerical errors of displacement and velocity in Taylor series expansion. Moreover, a comprehensive study on mathematical stability of the proposed technique, which is performed based on the ampli cation matrices, proves that the new method is more stable than existing schemes such as IHOA. For numerical veri cation, a wide range of dynamic systems, including linear and nonlinear, single and multi degrees of freedom, damped and undamped, as well as forced and free vibrations from nite-element and nitedi erence methods, are analyzed. These numerical studies demonstrate that e ciency and accuracy of the proposed method are higher than those of other techniques.

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Section: The Generalized Implicit Higher Ordermentioning

confidence: 87%

Section: The Generalized Implicit Higher Ordermentioning

confidence: 99%

Section: N+1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

Alamatian

2017

Scientia Iranica

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Improving stability domains of the implicit higher order accuracy method

Rezaiee‐Pajand

Sarafrazi

Hashemian

2011

Numerical Meth Engineering

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SUMMARYThe implicit higher order accuracy (IHOA) time integration family has a high order of accuracy. However, its stability domains are very restrictive. By improving the stability conditions, the larger time steps can be utilized. In this study, two parameters are introduced in the displacement and velocity extrapolations of IHOA. In order to find the optimum values of the proposed parameters, the dissipation and the dispersion are investigated. Three conditionally stable versions of the second-, third-, and forth-order algorithms are found. It is shown that these formulations have larger stability domains than the previous ones. Furthermore, the suggested strategies give responses that are more accurate.

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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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Implicit Higher-Order Accuracy Method for Numerical Integration in Dynamic Analysis

Cited by 54 publications

References 27 publications

Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

Improving stability domains of the implicit higher order accuracy method

Dynamic relaxation method based on Lanczos algorithm

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