在Arbitrary Lagrangian Eulerian架構下發展一具有守恆形式的有限元素法

Development of an Arbitrary Lagrangian Eulerian Finite Element Formulation in Conservative Form

伊菲利(Filip Ivancic)

指導教授 ： 許文翰

共同指導教授 ： 馬克沁(Maxim Solovchuk)

國立臺灣大學/工學院/工程科學及海洋工程學研究所/博士(2020年)

https://doi.org/10.6342/NTU202003676

摘要

本論文目的主要為發展一數值方法用以模擬在時變域上的多物理場系統。考慮此類問題的動機大部分來自於以時變域的偏微分方程式觀點所描述的生醫及生物流體力學問題。為此,我們將採用有限元素法來求解此類問題。此外,我們只考慮在演化過程中其場域拓樸不變的問題,此限制允許我們採用以對齊網格來顯式描述場域介面的任意拉格朗日-歐拉架構。因此,整個數值方法屬於顯式界面追蹤類別方法中的移動網格架構。論文的第一部分主要在守恆形式的ALE架構下推導出一個新穎的有限元數值方法,提供一個系統的方法用以消除由於移動網格下而產生的人工沉降及源項。即便此類人工數值沉降及源項已被眾所皆知,此問題仍是一個開放性且具挑戰性的主題。質量及離散空間律的守恆則為另外兩個需要被解決的問題,而所本論文的方法正是在結合此兩個特徵所發展出的。論文的第二部分將採用所提出的數值方法來解決真實的流體問題,將會著重在自由液面流跟流固耦合問題這兩類主題上,所選取的驗證問題中將會驗證所開發的方法具有良好靈活性及可信賴性。

關鍵字

有限元方法 ； 任意拉格朗日歐拉方法 ； 移動網格 ； 人工沉降源 ； 自由液面流 ； 流固耦合

並列摘要

The purpose of this thesis is to develop a numerical method for simulations of multiphysical systems on evolving domains. Motivation for the problems considered in this work comes largely from the field of bio-medicine and bio--fluid mechanics. These multiphysical systems are described in terms of systems of partial differential equations (PDEs) posed on time dependent domains. Finite element method (FEM) is employed for numerical approximation of such problems. Furthermore, only a special class of "domain-evolving" problems is considered - problems in which domain's topology does not change during its evolution. This restriction allows to work within the so-called arbitrary Lagrangian-Eulerian (ALE) framework in which the interface of domain is described explicitly by the aligned mesh. Thus, the complete numerical method employed falls under a moving mesh category within an explicit, so called interface tracking, approach.
The first part of the thesis deals with derivation of a novelty approach in finite element method within ALE framework focused on conservative formulations. This approach offers a systematic way to eliminate artificial sinks and sources arising from the moving mesh. Although the numerical origins of these artificial sinks and sources are well known, this problematics still remains to be an active and challenging topic. The mass conservation problem and the discrete space conservation law (SCL) are the two major issues resolved; actually, the novelty approach is integrated upon these two characteristics.
In the second part of the thesis, the newly proposed approach is applied to (academic) problems arising from the real world situations. The attention is on two particular class of problems: free-surface flows and fluid-structure interaction (FSI) problems. The flexibility and credibility of the methodology derived in the first part are demonstrated on selected examples.

並列關鍵字

finite element method ； arbitrary Langrangian-Eulerian ； moving mesh ； artificial sink/source ； free-surface flow ； fluid-structure interaction

參考文獻

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