From the trapezoidal rule to higher-order accurate and unconditionally stable time-integration method for structural dynamics

Kim, Seung Jo; Cho, Jin Yeon; Kim, Wie Dae

doi:10.1016/s0045-7825(97)00061-3

Cited by 25 publications

(11 citation statements)

References 6 publications

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“…Since the Newmark method does not need a distinct starting procedure, the method should be preferred in the combination with the DRBEM for analyzing this type of problems. This conclusion is in agreement with common practice in the FEM (Kim et al 1997).…”

Section: Discussionsupporting

confidence: 92%

A study on time schemes for DRBEM analysis of elastic impact wave

Chen¹,

Tanaka²

2002

Computational Mechanics

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The precise integration and differential quadrature methods are two new unconditionally stable numerical schemes to approximate time derivative with more than the second order accuracy. Recent studies showed that compared with the Houbolt and Newmark methods, they produced more accurate solutions with large time step for the problems where response is primarily dominated by low and intermediate frequency modes. This paper aims to investigate these time schemes in the context of the dual reciprocity BEM (DRBEM) formulation of various shockexcited scalar elastic wave problems, where high modes have important affect on traction response. The Houbolt method was widely recommended in many literatures for such DRBEM dynamic formulations. However, this study found that the damped Newmark algorithm was the most efficient and accurate for impact traction analysis in conjunction with the DRBEM. The precise integration and differential quadrature methods are shown inapplicable for such shock-excited problems due to the absence of numerical damping. On the other hand, we also found that to achieve the same order of accuracy, the differential quadrature method required much less computing effort than the precise integration method due to the use of the Bartels-Stewart algorithm solving the resulting Lyapunov matrix analogization equation.

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Section: Discussionsupporting

confidence: 92%

A study on time schemes for DRBEM analysis of elastic impact wave

Chen¹,

Tanaka²

2002

Computational Mechanics

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“…1). For a higher-order approximation, a higher-order interpolation function can be used, as by Kim et al 14 By the substitution of interpolating functions (5) into Eqs. (2) and (4), approximated relations for the time derivative are obtained.…”

Section: A Discrete Operator For Time Integrationmentioning

confidence: 99%

“…The detailed algorithm construction procedure is similar to that of Kim et al 14 The predictorcorrector algorithm based on displacement form can be written as follows:…”

Section: B Numerical Time-integration Algorithmmentioning

confidence: 99%

“…3) With the initial condition A n , V n , and U n obtained in the preceding time step, find A n +1, V n +1, and U n + \ of the next time step until the required time step, using the following procedure: Predict r(«) Calculate Correct (13) (14) (15) where A" + { and v" + 1 are predictors for acceleration and velocity, respectively. After the initialization and predictor stages, the displacements are obtained via Eq.…”

Section: B Numerical Time-integration Algorithmmentioning

confidence: 99%

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Discontinuous Time-Integration Method for Dynamic Contact/Impact Problems

Cho

Kim

1999

AIAA Journal

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A new time-integration method is proposed using the generalized derivative concept to simulate the dynamic phenomena with sudden constraint occurring in dynamic contact/impact problems. By the adoption of the generalized derivative concept and the jump assumption, discontinuity can be incorporated in time integration, and as a result, the algorithm does not need any other special consideration of jumps in dynamic field variables due to sudden constraint, such as dynamic contact-release conditions. To observe the characteristics of the proposed time-integration method, stability and convergence analyses are carried out. The analyses prove that the proposed method with linear approximation in the time domain yields unconditional stability and third-order accuracy. In numerical tests, several dynamic contact/impact problems are analyzed by straightforward application of the proposed time-integration method with the exterior penalty method. From the tests, it is shown that the proposed time-integration method can be used successfully with the penalty method in numerical simulations of dynamic contact/impact problems.

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“…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%

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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From the trapezoidal rule to higher-order accurate and unconditionally stable time-integration method for structural dynamics

Cited by 25 publications

References 6 publications

A study on time schemes for DRBEM analysis of elastic impact wave

A study on time schemes for DRBEM analysis of elastic impact wave

Discontinuous Time-Integration Method for Dynamic Contact/Impact Problems

Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

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