本文選題：時程分析 + 顯式算法�。� 參考：《華僑大學》2017年碩士論文

【摘要】：在結構動力時程分析中,直接積分法常常被用來求解結構運動方程。直接積分法可以分為顯式算法和隱式算法。相比隱式算法,顯式算法的計算效率較高,但較差的穩(wěn)定性限制了其應用,尤其是結構進入非線性后,線性系統(tǒng)中無條件穩(wěn)定的顯式算法可能會退化為條件穩(wěn)定。另一方面,部分算法擁有的數(shù)值阻尼可以將數(shù)值結果中虛假的高頻震蕩成分迅速剔除。但這些數(shù)值阻尼特性不能方便地引入到其他算法之中。針對這兩個問題,本文開展了以下研究:(1)在狀態(tài)空間下,將結構動力方程改寫成了一階常微分方程的形式,并利用積分因子法導出了該問題含有隱式積分項的精確解。利用Pade近似,對上述積分項進行近似處理,提出了一類顯式單步算法,記為Pade-based算法。該算法對位移和速度都具有二階精度,且在線性系統(tǒng)和非線性系統(tǒng)中均無條件穩(wěn)定。(2)基于顯式Adams法并利用Pade近似與高斯數(shù)值積分法,對上述隱式積分項進行近似處理,構造出了一類顯式多步算法,記為Adams-based算法。該算法可以表達為具有任意高階精度的一般形式,且在線性系統(tǒng)和非線性系統(tǒng)中都保持無條件穩(wěn)定。通過控制Pade近似的形式,新算法的穩(wěn)定性可以在A穩(wěn)定和L穩(wěn)定之間轉換。(3)基于廣義Pade近似,提出了一種用于構造可控數(shù)值阻尼的一般方法。該方法通過調整單一參數(shù)?,達到控制數(shù)值阻尼大小的目的。合適的數(shù)值阻尼可以使計算結果中虛假的高頻震蕩成分被剔除,同時保留真實的低頻成分。
[Abstract]:In structural dynamic time history analysis, direct integration method is often used to solve structural equations of motion. Direct integration method can be divided into explicit algorithm and implicit algorithm. Compared with the implicit algorithm, the explicit algorithm is more efficient, but its application is limited by its poor stability. Especially when the structure is nonlinear, the unconditionally stable explicit algorithm in the linear system may degenerate into conditional stability. On the other hand, the partial numerical damping can quickly eliminate the false high frequency oscillation in the numerical results. However, these numerical damping characteristics can not be easily introduced into other algorithms. For these two problems, the following research is carried out: 1) in the state space, the structural dynamic equation is rewritten into the form of the first order ordinary differential equation, and the exact solution of the problem with implicit integral term is derived by using the integral factor method. By using Pade approximation, the above integral terms are approximated, and a class of explicit one-step algorithm is proposed, which is described as Pade-based algorithm. The algorithm has second-order accuracy for displacement and velocity, and is unconditionally stable in linear and nonlinear systems. Based on explicit Adams method and using Pade approximation and Gao Si numerical integration method, the implicit integral terms mentioned above are approximated. A class of explicit multistep algorithm is constructed, which is called Adams-based algorithm. The algorithm can be expressed as a general form with arbitrary higher order accuracy and is unconditionally stable in both linear and nonlinear systems. By controlling the form of Pade approximation, the stability of the new algorithm can be transformed between A stability and L stability. Based on the generalized Pade approximation, a general method for constructing controllable numerical damping is proposed. The numerical damping is controlled by adjusting a single parameter. With proper numerical damping, the false high frequency oscillation components can be eliminated and the true low frequency components can be retained.
【學位授予單位】：華僑大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：TU311.3

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