時間不連續葛勒金有限元素法於彈性動力學問題之研究

A Study on Time-Discontinuous Galerkin Finite Element Method for Elastodynamic Problems

吳東岳(Tong-Yue Wu)

指導教授 ： 簡秋記

中原大學/工學院/土木工程研究所/博士(2001年)

https://doi.org/10.6840/cycu200100074

摘要

本文主要目的在於以時間不連續葛勒金有限元素法（簡稱TDG法），分別配合空間域之有限元素法、邊界元素法與無網格法分析結構或彈性動力問題。首先，本文採用位移與速度雙場混合形式的TDG法，每個時階內的位移與速度皆以一階線性時域元素離散，並允許其在每個時階初的離散點上不連續，且具有三階的收斂率。為了降低TDG法每個時階所需的運算量，本文根據廣義迭代法的理論設計出無條件收斂穩定的區塊迭代演算方式，經理論的證明及數值執行結果顯示每個時階的計算成本可大幅降低。第二，本文將TDG法搭配有限元素法分析彈性動力問題，並分別與常用的直接積分法：HHT-a法、Park法與Houbolt法，做一數值上之比較。本文數值算例顯示，TDG法無論對假高頻反應的取捨或是精確度皆遠優於上述三者。第三，本文將TDG法搭配邊界元素法分析彈性動力問題。其中，邊界元素法乃基於雙重互換邊界元素法（簡稱DRBEM）或特解邊界元素法（簡稱PIBEM），此法使用近似特解的觀念將慣性域內積分項轉換至邊界，因此僅涉及彈性靜力之位移和曳引力基本解，不需用到一般較繁雜之彈性動力時間域或頻率域基本解。由於DRBEM或PIBEM所形成的矩陣不具帶寬且非對稱，因此在較高振態時易有複數振態或頻率產生，這些複數振態極易導致動力反應分析時的數值不穩定﹔因此，具有L-stable的Houbolt法通常被視為有較穩定的數值結果，然而Houbolt法具有嚴重的週期誤差之缺點。本文數值算例結果顯示，TDG法的穩定性、精確度與對假高頻反應的取捨皆遠優於Houbolt法。最後，本文使用無網格法分析彈性動力問題，其中，無網格法乃基於無網格局部彼得洛夫葛勒金法（簡稱MLPG法）。MLPG法係根據局部之對稱弱形式以及變動最小平方近似法，因此於物理量的近似與能量的積分，甚至必要邊界條件的引入，完全無須求助網格或有限元素之建立。由於MLPG法所形成的質量與勁度矩陣雖具帶寬但並不一定為對稱，因此動力領域的問題值得探討。除了自由震動分析之外，本文亦分別使用具L-stable的TDG法、Park法與Houbolt法以及有限元素法最常使用的Newmark法分析時域反應。本文數值算例結果顯示，MLPG法可成功地算出彈性問題之重要自然頻率，且唯有TDG法可以有效抑制時域反應之假高頻震盪現象。

關鍵字

時間不連續葛勒金法 ； 無網格法 ； 有限元素法 ； 無網格局部彼得洛夫葛勒金法 ； 彈性動力 ； 特解邊界元素法 ； 雙重互換邊界元素法

並列摘要

The main objective of this thesis is to deal with elastodynamic problems, in which the spatial domain is modeled by finite element method, boundary element method and meshless method, respectively, and then, using the time-discontinuous Galerkin method (TDG method), analyze the dynamic responses in the time domain. First, a two-fields and mixed formulated TDG method is studied, by which both displacements and velocities are interpolated independently as piecewise linear functions in time and may be discontinuous at beginning of each time step. In order to alleviate the computational cost on each time step, a block iteration algorithm based on generalized iteration is proposed. Theoretical proof and practical numerical implementations reveal that the proposed algorithm can effectively decrease the computational cost. Secondly, the TDG together with space finite element methods are applied to elastodynamic problems. As compared with commonly used time integration methods, HHT-a, Houbolt and Park methods, numerical results demonstrate that the TDG method considered here has superior performance on accuracy and is capable of filtering out the noise of spurious high modes. Then, the TDG and dual reciprocity boundary element method (DRBEM) or particular integral boundary element method (PIBEM) methodology are used for solving elastodynamic problems. Both of DRBEM and PIBEM use the technique of approximated particular solutions to circumvent the domain integral for inertial forces. Thus, the DRBEM and PIBEM depend only on elastostatic displacement and traction fundamental solutions without resorting to commonly used complex fundamental solutions for elastodynamic problems. However, the higher complex vibrating-modes or -frequencies may easily arise from asymmetric matrices formed by the DRBEM or PIBEM. Both of Houbolt and TDG methods are employed to deal with this main drawback of the DRBEM or PIBEM. The numerical results reveal that the TDG method not only has the same stability as Houbolt method but also adequately controls the spurious high-modes. Finally, the meshless local Petrov-Galerkin (MLPG) approach is adopted to analyze the elastodynamic problems. The MLPG approach considered here is based on local symmetric weak form and moving least square approximation. This approach is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation for the solution variables, or for the integration of the energy. The matrix properties of MLPG are banded but not always symmetric, thus here is a motivation to explore the dynamic problems using MLPG approach. Both of free vibration and transient dynamic responses analysis are explored via several benchmark problems. Numerical results support that MLPG approach is also able to extract the crucial frequencies and modes like finite element method. However, the spurious high-frequency oscillations could adversely affect the computed responses. Three L-stable techniques (TDG, Houbolt and Park methods) and Newmark method are surveyed. The numerical results reveal that the only TDG method provides very stable and accurate results in the sense that the crucial modes have been accurately integrated and the spurious ones successfully filtered out.

並列關鍵字

particular integral boundary element method ； meshless local Petrov-Galerkin approach. ； finite element method ； time-discontinuous Galerkin method ； elastodynamics ； meshless method ； dual reciprocity boundary element method

參考文獻

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