An EEP Adaptive Strategy of the Galerkin FEM for Axially Forced Vibration of Bars

XING Qinyan; YANG Qinghao; LU Chenyu; YANG Xing

doi:10.21656/1000-0887.400051

Volume 40 Issue 9

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Article Navigation > Applied Mathematics and Mechanics > 2019 > 40(9): 945-956

XING Qinyan, YANG Qinghao, LU Chenyu, YANG Xing. An EEP Adaptive Strategy of the Galerkin FEM for Axially Forced Vibration of Bars[J]. Applied Mathematics and Mechanics, 2019, 40(9): 945-956. doi: 10.21656/1000-0887.400051

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An EEP Adaptive Strategy of the Galerkin FEM for Axially Forced Vibration of Bars

doi: 10.21656/1000-0887.400051

Department of Civil Engineering, Tsinghua University; Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Beijing 100084, P.R.China

Funds: The National Natural Science Foundation of China（51508305）

Received Date: 2019-02-03
Rev Recd Date: 2019-07-12
Publish Date: 2019-09-01

Abstract

Abstract

Based on the successful applications of the element energy projection (EEP) adaptive method for the static problems of bars and the dynamic equations for discrete systems, a strategy was proposed to adaptively solve the axially forced vibration problems of bars in both the time dimension and the space dimension. In this strategy, the continuous space-time Galerkin finite element method (FEM) was used. Based on the idea of semi-discretization, through discretization in space first, the governing partial differential equations of the model problem were transformed into a system of 2nd-order ordinary differential equations with initial boundary conditions, which were called dynamic equations for discrete systems hereinafter. These dynamic equations for discrete systems were then solved with the proposed EEP adaptive FEM in the time domain. After that, the EEP super-convergent formula for dynamic displacements in the space direction was derived, with which errors of the conventional Galerkin FEM solutions were estimated and the corresponding adaptive analysis method was established. Finally, the presented EEP adaptive strategy gave dynamic displacements with high accuracy point-wisely satisfying the pre-specified error tolerance, together with the automatically produced space-time mesh. The basic idea, the key technologies and the implementation strategy were elaborated. Representative numerical examples including seismic wave input demonstrate effectiveness and reliability of the method.
- forced vibration,
- space-time finite element,
- adaptivity,
- Galerkin method,
- element energy projection

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References

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