利用瞬時非線性變化程度參數評估逐步積分法的非線性數值特性

Nonlinear Evaluation of Step-by-step Integration Method Using Instantaneous Degree of Nonlinearity

廖迪峰(Ti-Feng Liao)

指導教授 ： 張順益

國立臺北科技大學/工程學院/土木與防災研究所/碩士(2007年)

摘要

在結構系統的動態歷時分析中，使用逐步積分法早就已經成為非常普遍的方式，對於使用逐步積分法來求解線性結構系統的數值特性評估方法已經發展的非常完備，但是卻無法直接應用於非線性結構系統。在先前的論文之中已探討過使用一個定義名為單步非線性度(step degree of nonlinearity)的參數來描述結構勁度在積分時間步長內的變化情形，但其缺點是非線性變化的極值不易評估，因為此法的非線性變化程度與積分時間步長有密切的關係。而本論文則是使用定義名為瞬時非線性度(instantaneous degree of nonlinearity)的另一種新參數，這個方法的非線性變化極值是由材料的性質所控制，所以並不會因為積分時間步長的挑選而有太大的影響，故較易掌握結構的非線性變化程度。加入這個新參數之後，我們可以用線性系統的評估方法求解非線性結構系統的數值特性、進行非線性的數值特性分析，並探討此積分方法在結構系統下的數值消散與非線性變化程度的關係。在實際應用的時候也因為使用之前的方法來評估非線性變化程度的極值比較難，以至於挑選積分時間步長也較不易，而此法的非線性變化程度極值較易評估，所以挑選積分時間步長也會比較容易。

關鍵字

逐步積分法 ； HHT- 積分法 ； 瞬時非線性度 ； 收斂度

並列摘要

Since step-by-step integration methods have become widely used in the inelastic dynamic analysis, it is very important to assess their numerical characteristics in practical applications. Although the evaluation technique for evaluating the numerical properties of a step-by-step integration method in the solution of linear elastic systems has been well-developed it may not be directly applied to inelastic systems. The previous study has already probed into using step degree of nonlinearity parameter that describe the variation of stiffness in time step, but its shortcoming is that the extreme value of nonlinearity is difficult to assess, because the step degree of nonlinearity parameter and time step are close relations. In this paper, another kind of new parameter is used and defined as instantaneous degree of nonlinearity. The extreme value of this method is controlled by nature of material. After joining this new parameter, we can explore the numerical properties of the aforementioned dissipative integration methods for nonlinear elastic systems. Because the extreme value does not change with different time step, it will be easier to select the time step.

並列關鍵字

step-by-step integration method ； HHT- integral method ； instantaneous degree of nonlinearity ； step degree of convergence

參考文獻

[1] Bathe K.J. and Wilson E.L., “Stability and Accuracy Analysis of Direct Integration Methods,” Earthquake Engineering and Structural Dynamics, Vol. 1, pp. 283-291, 1973.

[2] Belytschko T. and Hughes T.J.R., Computational Methods for Transient Analysis, Elsevier Science Publishers B.V., North-Holland, 1983.

[3] Belytschko T. and Schoeberle D.F., “On the Unconditional Stability of An Implicit Algorithm for Nonlinear Structural Dynamics,” Journal of Applied Mechanics, Vol. 17, pp. 865-869, 1975.

[5] Chang S.Y., “A Series of Engergy Conserving Algorithms for Structural Dynamics,” Journal of the Chinese Institute of Engineers, Vol. 19, No. 2, pp. 219-230, 1996.

[6] Chang S.Y., “Explicit Pseudodynamic Algorithm with Unconditional Stability,” Journal of Engineering Mechanics, ASCE, Vol. 128, No.9, pp. 935-947, 2002.