高斯積分及參考點置點積分耦合於無元素法之研究

On the Coupling of Gaussian Quadrature and Reference Point Collocation Integration in Element Free Galerkin Method

黃致榮(chih-jung Huang)

指導教授 ： 莊清鏘

中原大學/工學院/土木工程學系/碩士(2006年)

https://doi.org/10.6840/cycu200600492

摘要

本研究主要目的是提高無元素法的外力邊界、位移邊界及材料區域的積分精確度。首先將會探討傳統高斯積分法配合不同積分網格處理不規則區域的適用性，同時建立節點無關取樣點積分法的處理程序並探討其特性，再嘗試耦合高斯積分法及節點無關取樣點置點積分法處理不規則區域的問題。

傳統無元素法以取樣點落在模型內外做為取捨的條件，而保留下的取樣點常因權重大小影響積分穩定程度。置點取樣點無權重特性，即控制面積是固定的，只要施加足夠的點數在邊界上就能得到較穩定精確的答案。本文將使用置點積分處理不規則區域的積分，同時保留高斯積分在規則區塊積分較佳的特性進行算例的分析，並和（1）傳統高斯積分法（2）節點無關取樣點置點積分法，將討論積分結果的精確性及穩定性。

關鍵字

不規則邊界 ； 無元素法 ； 變動式最小平方近似法 ； 高斯積分法耦合置點積分法

並列摘要

The main purpose of this research is to discuss the accuracy and stability of natural boundary, essential boundary and domain integration of the Element free Galerkin method. The feasibility of irregular domain by using the background integration cells combined with traditional Gaussian quadrature is evaluated first. Next, the procedure of reference point integration is established and properties are discussed. Finally, a coupling method to merge the Gaussian quadrature and reference point integration method is tested to handle irregular domain.
Traditionally, Gaussian quatrature method is used in Element free Galerkin method according to Gaussian quatrature points located in or out of the integration region. The weight of Gaussian quatrature points may influence the accuracy and stability of integration in irregular domain. If these integration are replaced by reference point integration method, the weight of each integration points will be 1, which means that the dominated area of each reference point is fixed. For that reason, precise results can be obtained if the number of reference points is large enough.
The advantages of the Gaussian quadrature and reference point integration method are combined in this research. Incomplete integration cell is dealt with by reference point integration method whereas complete integration cell dealt with Gaussian quadrature. The accuracy and stability of the coupled method are studied and compared with traditional Gaussian quadrature method and reference point integration method.

並列關鍵字

Moving Least Square ； Gaussian Quadrature ； Irregular Problem Domain ； Element free Galerkin Method ； Point Collocation Integration

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