向量式有限元時間積分法之研究

A Study in Time Integration Method of Vector Form Intrinsic Finite Element Method

李東奇(Dong-Chi Li)

指導教授 ： 莊清鏘

中原大學/工學院/土木工程學系/碩士(2008年)

https://doi.org/10.6840/CYCU.2008.00411

摘要

本文主要探討向量式有限元所使用的時間積分法，由於向量式有限元所採用的時間積分法為顯式的中值差分法，模擬問題時常因穩定性的關係，在時間步長的選取上常受限於一臨界值之下，模擬出來的結果才不會產生數值發散的現象，但時間步長選取的愈小，計算機所要花費的計算時間也就會相對增加。所以本文引入數值方法中常見的一些顯式時間積分法至向量式有限元，藉由討論各種時間積分法的穩定性與精確性，期能找出應用於向量式有限元較佳的數值積分法。
原本向量式有限元所使用之中值差分法為建立於動力平衡方程式的一種數值方法，系統所受之外力通常於離散的時間點上動力平衡，但離散點作用時中間之外力表現便無法描述。本論文引入動量平衡的概念，利用動量-衝量力平衡，如此便能使原本於離散點上之外力作面積的描述，另外介紹各種外力載荷積分方法並以數值算例探討其精度，還有依外力的不同作適合的載荷切割也可達到不錯的精度，結果利用動量平衡時間積分法在時間步長方面能做較大範圍的選取，如此便能在計算機的運算節省計算時間。
另外，本文引入各種數值方法於向量式有限元中，發現許多方法於內力部份會有耦合效應的發生，如：Runge-Kutta法…等。所以本文另一項重點為比較各種動力平衡與動量平衡數值方法在向量式有限元的適用性。

關鍵字

時間積分法 ； 向量式有限元 ； 動量平衡

並列摘要

This paper focuses primarily on time integration methods in VFIFE (Vector Form Intrinsic Finite Element). Because the method adopts explicit central difference method in VFIFE, and due to the stability problems during the simulation, it is restricted to a critical value in time step in order to avoid numerical dissipation; however, smaller the time steps are, more time a computer needs to calculate. As a result, in this paper some common explicit time integration methods are taken into VFIFE, hoping to find some better integration methods in VFIFE application by discussing the stability and accuracy of many kinds of time integration methods,
Originally, the central difference method used in VFIFE is one of the numerical methods to build a dynamic equilibrium equation. The external force received by the system usually equilibrates at the discrete time point. But in this way it the external force received between two discrete time points will not be able to be described. In this paper, momentum equilibrium equation is taken into VFIFE, utilizing momentum-impulse force equilibrium to describe the external force as mentioned above by superficial measure. Besides, in this paper some external force integration methods and ways to utilize numerical examples to confer accuracy are also introduced. Furthermore, different force segment in accordance with different external force can also improve accuracy. To sum up, by utilizing momentum equilibrium time integration method, we can choose time step in a wider range so that huge amount of time can be saved.
In addition, many kinds of numerical methods are taken into VFIFE in this paper and it is found that if we adopt certain methods (for example the Runge-Kutta method), couple effect occurs. So another key point of this paper is comparing the fitness of many kinds of numerical methods to VFIFE.

並列關鍵字

time integration methods ； VFIFE ； momentum equilibrium

參考文獻

[1] Chang, S. Y., Huang, Y. C., and Wang, C. H., “Analysis of Newmark Explicit Integration Method for Nonlinear Systems” , Journal of Mechanics, Vol.22, No.4, pp.321-329(2006).

[2] Chang, S. Y., “Improved Numerical Dissipation for Explicit Methods in Pseudodynamic Tests” , Earthquake Engineering and Structural Dynamics, 26, pp. 917-929(1997).

[3] Chang, S. Y., and Liao, W. I., “An Unconditionally Stable Explicit Method for Structural Dynamics” , Journal of Earthquake Engineering, 9(3), pp. 349{370 (2005).

[4] Dokainish, M. A., and Subbaraj, K., “A Survey of Direct Time-Integration Methods in Computational Structural Dynamics-I. Explicit Methods”, Comput.&Struct., Vol. 32, No.6, pp. 1371-1386 (1989).

[5] Dokainish, M. A., and Subbaraj, K., “A Survey of Direct Time-Integration Methods in Computational Structural Dynamics-II. Implicit Methods”, Comput.&Struct., Vol. 32, No.6, pp. 1371-1386 (1989).