精細時間積分法於結構問題之應用

An Application of High Precision Direct Integration Method on Structural Problems

蔡明謙(Ming-Chain Tsai)

指導教授 ： 莊清鏘

中原大學/工學院/土木工程研究所/碩士(2003年)

https://doi.org/10.6840/cycu200300016

摘要

本文主要利用有限元素法配合精細時間積分法處理彈性結構(連體)的靜、動力問題，其中靜力問題利用時間積分法配合運動阻尼模擬(每一個時間步程的末速度設為零，當成下一個時間步程的初速度)，動力問題則假設每一個時間步程的末速度為下一個時間步程的初速度。
利用精細時間積分法模擬問題時，首先將有限元素空間半離散的二階運動方程式降階變成一階聯立微分方程式，再用數學解析，配合矩陣的2N計算法求對應的解析解，因此該方法幾乎可以視為聯立微分方程的解析計算方法。
由於精細時間積分法為解析的計算法，所以沒有高頻消散能力，對於實際問題的模擬，為了能夠降低(或逐漸減少)物體因空間半離散所產生的不正確高頻反應，在模擬前先做分析模型基本振動頻率分析，再根據分析模型自然振動頻率的分佈，配合適當雷利阻尼(Rayleigh damping)的選取，使得精細時間積分法亦具有高頻消散的能力。同時根據精細時間積分法的穩定性和精確度分析，可知該法為無條件穩定、高精確度的計算方法(再配合雷利阻尼也可以有高頻消散的效果)，同時也是顯性的計算法。

關鍵字

精確解 ； 無條件穩定 ； 2N計算法 ； 直接積分法 ； 精細時間積分法

並列摘要

In this study a high precision direct integration method (HPD) is discussed and implemented on linear elastic structures. The kinetic damping, the initially velocity at each time step is set to zero, is used in static problems, and the initial velocity at each time step is the end velocity of the previous step in dynamic problems.
The equations of motion after space discretization must be transformed into first order differential equations in HPD. The solution in computation can be obtained from the analytical solution of first order simultaneous differential equations and 2N algorithm, so this method can be seen as a computing algorithmic method in differential equations.
Because HPD is an algorithmic method, it has not any frequency dissipation. In order to reduce the high frequency effects in the space discretization Rayleigh damping is used. After the frequency analysis of model, a suitable Rayleigh damping can be chosen to reduce the high frequency effects gradually.
From the discussion of stability and accuracy, HPD is an unconditional stability, high accuracy, explicit with high frequency dissipation method (combining with Rayleigh damping).

並列關鍵字

2N algorithm ； high precision direct integration method ； unconditional stability

參考文獻

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