以強形式廣義位移控制法進行非線性分析

A Strong-Form Formulated Generalized Displacement Control Method for Nonlinear Analysis

陳建宇(Jian-Yu Chen)

指導教授 ： 楊子儀

國立交通大學/工學院/土木工程系所/碩士(2018年)

摘要

在傳統的幾何非線性分析中，最受歡迎的數值方法為基於弱形式表述的有限元素法，此方法由於元素的本質，限制了它的應用層面，例如在控制方程式中出現的數值積分項與網格變形之品質控制。自1990年代起，無網格法已逐漸發展成為計算力學領域中的首要研究主題，尤其是其中的強形式配置法，不需額外處理數值積分之運算，而且可以直接施加Dirichlet邊界條件，從而使配置法之計算效率提高。關於幾何非線性分析，如何精確地反應結構物之載重變形曲線的斜率變化，同時保持數值穩定性，一直是增量疊代過程中的主要課題。有鑑於此，我們提出一個以強形式表述的廣義位移控制法來分析幾何非線性問題，當中使用徑向基函數做近似並求解，並透過本論文中數個數值例題驗證所提出之方法在大變形之分析。

關鍵字

無網格法 ； 廣義位移控制法 ； 徑向基函數 ； 大變形 ； 增量疊代演算法

並列摘要

In the traditional analysis of geometric nonlinearity, the most popular method is formulated on the basis of weak-form such as the finite element method. Due to the element nature, its application is limited, for instance, by the numerical integration in the governing equation and the quality control of deformed mesh. The meshfree methods have been developed and become one leading research topic in the field of computational mechanics since 1990s. In particular, the strong form collocation methods require no additional efforts to deal with numerical integration and impose Dirichlet boundary conditions, thereby making the collocation methods computationally efficient. Concerning geometric nonlinearity, how to accurately reflect the change in the slope of the load-deflection curve of the structure and remain numerically stable are of major concerns in the incremental-iterative process. As a result, we propose a strong-form formulated generalized displacement control method to analyze geometrically nonlinear problems, where the radial basis collocation method is adopted. The numerical examples demonstrate the ability of the proposed method for large deformation analysis.

並列關鍵字

meshfree method ； generalized displacement control method ； radial basis function ； large deformation ； incremental-iterative algorithm

參考文獻

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[2] Jun, S., Liu, W. K. and Belytschko, T. [1998] “Explicit reproducing kernel particle methods for large deformation problems,” International Journal for Numerical Methods in Engineering 41, 137–166.

[3] ]Chen, J.S., Pan, C. and Wu, C. T. [1997] “Large deformation analysis of rubber based on a reproducing kernel particle method,” Computational Mechanics 19, 153–168.

[4] Chen, J.S., Pan, C. and Wu, C. T. [1998] “Application of reproducing kernel particle methods to large deformation and contact analysis of elastomers,” Rubber Chemistry and Technology 7, 191–213.

[5] Guan, P. C. and Sun, C. T. [2014] “The isoparametric reproducing kernel particle method for nonlinear deformation of plates,” Engineering Analysis with Boundary Elements 42, 67–76.