元素釋放法於彈性動力之研究

A Study of Element Free Galerkin Method in Elastodynamics

李政達(Cheng-Da Lee)

指導教授 ： 莊清鏘

中原大學/工學院/土木工程研究所/碩士(2002年)

https://doi.org/10.6840/cycu200200307

摘要

本文主旨在運用元素釋放法分析彈性動力問題。元素釋放法為無網格法(Meshless)的一種，該法在節點形狀函數的建立上無需借助網格或元素的劃分，反而是運用到數值分析中曲線擬合(Curve Fitting)的概念，利用變動最小平方近似法(Moving Least Square Method，簡稱MLS)來近似建立節點的形狀函數。傳統上利用有限元素做動力模擬時可分為兩大部分，其一為空間域借助有限個節點所構成的元素做空間的離散，其二為時間域利用差分方程式或振態疊加法做動力模擬。
元素釋放法在處理彈性動力問題時基本上觀念和有限元素法類似但做法迴異，同樣可分為兩大部分，其中空間離散利用了變動式最小平方近似法來建立節點形狀函數，雖然該法在建立節點形狀函數會涉及許多的矩陣運算，但可避免元素生成和元素再細分的複雜性，因為無須劃分元素，所以在應力的模擬上將更具連續性。另外對於數值的積分基本上仍採用高斯積分法則且積分網格的劃分仍有其必須性。本論文的內容主要以元素釋放法來處理基本的彈性動力問題並討論數種分析模式所模擬的結果，並針對未來可能發展的方向提出看法與建議。

關鍵字

元素釋放法 ； 直接積分法 ； 變動最小平方近似法 ； 無網格法

並列摘要

The purpose of this thesis applies element free Galerkin method to analyse elastodynamic problems. Element free Galerkin method that needs not the division of meshes or elements to establish nodes shape functions and it is also one of meshless methods. However the idea is similar to curve fitting in numerical analysis and the notation uses moving least square method to construct the nodes shape functions. In analysis the elastodynamics problems typically divided into two major parts which one is space semi-discrete by finite nodes and the other part is time procedure that uses finite difference equations or model superposition to simulate dynamic motions.
In analysing elastodynamic problems the idea of Element Free Galerkin Method is similar to Finite Element Method, however the procedure is a large difference. The biggest one is Element Free Galerkin Method uses Moving Least Square Method for establishing node shape functions and introduce to the procedure of establishing space semi-discrete dynamic motion equations by energy variation principles. Although this approach involves large dimension matrixes operation but avoids the complication of forming and reforming elements. Because of no elements require so that the simulation of stress would be more continuous. As for numerical integration applies Gauss Integration Scheme and the divided integration cells is still necessary. The contents of this thesis applies Element Free Galerkin Method to analysis basic elasto dynamic problems and discusses the simulation results by considering several examples.

並列關鍵字

Element Free Method ； Element Free Galerkin Method ； Moving Least Square Method

參考文獻

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