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| Title: | 調適無網格法解彈性問題之理論與應用 |
| Other Titles: | Adaptive Meshfree Methods for Elasticity Problems: Theory and Applications |
| Authors: | 胡馨云
Hu, Hsin-Yun |
| Contributors: | 行政院國家科學委員會
東海大學數學系 |
| Keywords: | 調適性無網格法；h-版調適性；p-版調適性；誤差估計；均質彈性材料；異質彈性材料；再生核心函數
Adaptive meshfree method；h-adaptivity；p-adaptivity；homogeneous elasticity,heterogeneous elasticity；error indicator；error estimation；reproducing kernel shape function,interface-enriched reproducing kernel shape function. |
| Date: | 2008 |
| Issue Date: | 2011-06-16T06:35:03Z (UTC)
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| Abstract: | 有限元素法在數值方法中居領導之地位已歷時數十年了，其數學的理論相當完備，應用相當的廣泛。而近十年來所謂無網格法漸漸受到工程及計算力學界的重視，成為數值方法的新趨勢。無網格法繼承了傳統有限元素法的優點，例如，局部化以及一些良好逼近的性質，同時亦克服了一項缺點，不需依賴網格。無網格法有一個共通的特性，就是無需建立網格，它的基底函數是由一組節點所建構出來，快速且簡單，大量減低網格生成所需耗費的時間。在數值方法中效率及精確度是兩個相當重要的議題，但往往需要從中作個取捨，然而調適法能同時具備好的精度及好的效率。有些問題如果不採用調適法是無法得到好的結果，例如，裂縫成長，以及衝擊波等。如果用有限元素法去解這類問題將不易得到精確的解，且耗費許多計算時間。傳統有限元調適法共有四類，分別為h-版，p-版， h-p 版以及r-版的調適法。在有限元的調適法中有所謂非協調之問題存在。此外，在前後兩組網格間數值之傳遞也是相當複雜的。然而，無網格調適法因為基底之建構是比較全面性的因此將不會出現所謂不協調性之問題，而且因為無網格法之基底具有可調整之平滑性及局部性，所以在前後步驟間它能提供較準之數值傳遞。所以調適性的想法可以成功與無網格法結合。有限元素法中相容性的要求在無網格法中是不需要的，這使得無網格法在作調適法時更簡單些了。此研究計劃擬採用無網格法中之 Reproducing kernel particle method 搭配h-版及p-版調適法來解均質及異質彈性材料問題。研究計劃分二部分，包括演算法之建立及誤差分析，透過數學的分析將能增進計算之效率及精確度。此預計二年完成，將與美國加州大學洛杉磯分校陳俊賢教授的研究團隊合作。在第一年我們僅考慮較單純之均質彈性材料問題，採用h-版調適性去尋找好的近似解，理論部分將參考相關文獻，作定理推導的工作，理論之結果希望能供材料科學界參考。第二年則是想解決較具挑戰性之異質彈性材料問題，因為有介面不連續的問題，所以採用p-版調適性，需額外加入interface enrichment function 預期有較好的結果。理論推導及數值驗證工作亦同時進行。往後再考慮用此方法去解裂縫成長及衝擊波等問題。
The finite element method has been a dominant numerical method for several decades. For a decade or so, a new family of methods, collectively called meshless methods or meshfree methods, has attracted much interest in the community of computational mechanics. This new family of numerical methods is designed to inherit the main advantages of the finite element method such as compact supports of shape functions and good approximation properties, while at the same time, overcome the main disadvantages of the finite element method caused by the mesh-dependence. The meshfree methods share a common feature that no mesh is needed and shape functions are constructed from sets of particles, thus eliminating the need for time-consuming mesh generation. However, efficiency and accuracy are the two most important issues for the computational methods. A trade-off often exists between these two aspects. To achieve better accuracy, as well as high efficiency, adaptive methods have been adopted. Further, there is a class of problems, for example, modeling of crack propagation (moving singularity at the crack tip) and shock waves (moving discontinuity) that can not be handled effectively without adaptive method. Four types of adaptive procedures have been proposed in finite element methods: h-, p-, h-p- and r-adaptive methods. Adaptive refinement under finite element framework requires the imposition of continuity requirement (compatibility) along the element boundaries that is tedious and time consuming. In finite element discretization of second order differential equations, a minimum of C0 continuity is needed. This requires that the approximation is continuous across element boundaries, which substantially restrains h- or p-adaptivity. Additional efforts must be devoted to the imposition of conforming constraints. A related issue is the meshing strategy in adaptive refinement and the associated information transfer and data management. On the other hand, due to flexibility in adjusting order of consistency, smoothness, locality and without restriction of element connectivity, adaptive methods can be formulated naturally under the framework of meshfree methods. The objective of this research is to develop an error estimation for adaptive analysis of boundary value problems under meshfree framework. By collaboration with Professor J. S. Chen, the developed error estimation will be employed in the meshfree code to perform adaptive meshfree analysis of some challenging boundary value problems. Under this work, we will develop error estimation of RKPM for homogeneous and heterogeneous linear elasticity and identify how the proposed error estimator and the adaptive RKPM formulation can achieve enhanced computational accuracy and efficiency compared to RKPM analysis without adaptive procedures. My group will take the main responsibility in developing mathematical analysis and error estimation of reproducing kernel approximation for adaptive analysis, and Prof. Chen』s group will be responsible for numerical implementation and testing of the adaptive reproducing kernel particle method. In particular, the error estimate for reproducing kernel adaptive analysis will be integrated into the computational code and be used as a decision maker as to where and how to perform adaptive refinement to achieve computational efficiency and accuracy simultaneously in the reproducing kernel particle method. |
| Relation: | 研究編號：NSC96-2115-M029-002-MY2
研究期間：2008-08~ 2009-07 |
| Appears in Collections: | [應用數學系所] 國科會研究報告
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