應用無網格法於含有多領域耦合、大變形及不規則形狀三維問題之分析

Analysis of Three-dimensional Problems with Coupled Multi-physics, Large Deformatioin and Irregular-shaped Geometry by Meshless Method

李銘孝(Ming-Hsiao Lee)

指導教授 ： 陳文華

國立清華大學/工學院/動力機械工程學系/博士(2010年)

摘要

無網格法本質上具有一明顯優於其他數值方法的特點，即其分析模型只需要節點，而不需要像有限單元法一樣須建立複雜之單元網格。單元網格之設計與產生雖耗時費力，且其形狀亦常影響解之收斂性及精確度，但在另一方面，單元網格本身卻可以提供數值分析過程中所需的與幾何相關的資訊，如單元的邊或面即定義了分析場域的邊界形狀，而且其積分點原本就位於單元當中。
相對的，只有節點資料的無網格法分析模型卻缺少這些與幾何相關的資訊，因而在數值模擬過程中，通常會遇到一些困難，尤其是對於具三維不規則分析場域或多材料之問題。本論文係採用兩種幾何方式配合本論文發展之檢查機制來解決這些難題。此檢查機制可在分析過程中用來決定所需要的空間內外及相對位置。
此外，無網格法在分析劇烈大變形問題時節點分佈常會嚴重扭曲，嚴重扭曲的節點分佈將導致程式之準確度降低甚至發散而無法得到解答，本論文另提出「均佈背景格點法」來解決。在「均佈背景格點法」的分析過程中，將應用規則的節點分佈來進行分析，如此即可避免節點分佈扭曲所衍生的難題。此方法將以一個被廣泛採用於微機電系統中的靜電驅動元件之分析作為示範。靜電驅動元件往往因電極結構的位移而造成靜電分析場域劇烈大變形的問題，此類問題至今對於各種數值方法仍是一個難題。但可透過本論文提出之「均佈背景格點法」及本論文所建議的耦合分析程序，而順利獲得解決。
數個示範案例將用來驗證上述新開發技術的有效性及優越性。

關鍵字

無網格法 ； 幾何處理 ； 背景格點法 ； 耦合分析

並列摘要

The meshless method has a distinct advantage over other methods in that it requires only nodes without an element mesh which usually induces time-consuming work and inaccuracy when the elements are distorted during the analysis process. However, the element mesh can provide some geometry information for the numerical simulation, such as, the analysis domain is defined by the element’s edges or faces and the quadrature points are all inside the elements.
Because the analysis model with only nodes for the meshless method lacks these types of geometry-related information, some difficulties are usually encountered during numerical simulations, especially in the cases with three-dimensional irregular-shaped analysis domains. To overcome these difficulties, two geometry schemes and check mechanisms are proposed. The check mechanisams can be used for determining if certain points are inside or outside the anayslis domain, when the situations are often encoutered in the analysis processes of the meshless method.
Furthermore, on the analyses of extremely large deformation problems, the distortion of the distribution of nodes degrades the accuracy of the solution. A new three-dimensional meshless scheme with a uniform background grid is proposed herein. By this scheme, no matter how large the analysis domain deforms, the uniformly distributed nodes of the background grid will be selected to do the solution that the situation of the distortion of node distribution can be avoided and the accuracy of the solution can be maintained. An application to an electrostatic-structural problems encountered in many electrostatic driven MEMS devices is performed. In the type of cases, the electrostatic analysis domain is often extremely distorted due to the deflection of the structure of the electrode. This kind of problem is difficult to deal with by almost all kinds of available numerical methods. But, with the proposed background grid scheme and an iterative coupled-field analysis procedure, the electrostatic-structural problem can be solved without difficulties.
Several demonstrative cases have been conducted to prove the effectiveness and advantages of the proposed techniques.

並列關鍵字

meshless method ； Geometry-related treatment ； background grid ； coupled-field analysis

參考文獻

Atluri, S. N.; Zhu, T. (1998): A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, Vol. 22, pp. 117–127.

Atluri, S. N.; Shen. S. (2002): The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple & Less-costly Alternative to the Finite Element and Boundary Element Methods. Computational Mechanics, Vol. 3, no. 1, pp. 11-52.

Babuska, I.; Melenk, J. M. (1996): The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. Vol. 139, pp. 289-315.

Beissel, S.; Belytschko, T. (1996) : Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Engrg., Vol. 139, pp. 47-49.

Belytschko, T. and Fleming, M. (1999) : Smoothing enrichment and contact in the element-free Galerkin method. Comput. Methods Appl. Mech. Engrg. Vol. 71, pp. 173-195.