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研究生:江玉娟
研究生(外文):Yu-Chuan Chiang
論文名稱:以局部徑向基底函數佈點法求解壓電感應器及準晶體平板問題
論文名稱(外文):Piezoelectric Sensors and Quasicrystal Plate Problems by Localized Radial Basis Function Collocation Method
指導教授:楊德良楊德良引用關係
口試委員:廖清標陳正宗洪宏基徐國錦
口試日期:2015-07-25
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱:土木工程學研究所
學門:工程學門
學類:土木工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:英文
論文頁數:83
中文關鍵詞:局部徑向基底函數佈點法壓電感應器壓電效應聲子與相位子應變準晶體平板米德林定理
外文關鍵詞:localized radial basis functionpiezoelectric sensorpiezoelectric effectphonon and phason displacementsquasicrystal plateReissner-Mindlin theory
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局部徑向基底函數佈點無網格數值方法的優點在於能夠方便地利用局部徑向基底函數微分運算子近似控制方程式和諾伊曼型邊界條件中的空間導數,使研究者們更便利地求解複雜物理問題。以往局部徑向基底函數佈點無網格數值方法廣泛用於解決計算流體力學問題,為了將局部徑向基底函數佈點無網格法拓展至結構工程領域,本研究主旨為利用局部徑向基底函數佈點法求解壓電感應器及準晶體平板問題。
由於壓電材料本身的壓電特性,壓電材料被廣泛應用於智能裝置與感應器材。壓電感應器通常被製作成薄型圓板,因此本研究在三維圓柱模型多重維度計算域中,分析壓電感應器受壓於一均勻載重產生之位移與電勢。此外,本研究以有限元素法計算結果為基準,比較無網格局部Petrov-Galerkin法和局部徑向基底函數佈點法之計算結果差異。
在求解準晶體結構平板問題中,本研究根據米德林平板理論,將實際三為平板問題轉化為準三維問題。以二維控制方程式求解聲子與相位子變量,分別探討簡支平板及固支平板受於均勻載重下的物理行為。此外,本研究比較了傳統有限差分網格法和局部徑向基底函數佈點法的結果差異,以展示局部徑向基底函數佈點無網格法在求解本問題的優勢。最後提出了一項在正交均勻佈點計算域中,利用局部徑向基底函數運算子近似交互微分導數時所遇到的困難,待未來有更多研究進一步解決。


The advantage of the localized radial basis function collocation method (LRBFCM) is that we can easily utilize kinds of LRBFCM spatial differential operators for the approximation of the spatial derivatives from the governing equations and Neumann type boundary conditions. As a result, LRBFCM is a convenient strong form meshless method for researchers to conduct with complex physical problems numerically. In the past, LRBFCM are usually applied for solving computational fluid mechanics (CFD) problems. In order to extend LRBFCM into structure engineering problems, we focus on the two main problems: numerical studies of piezoelectric sensor and quasicrystal plate.
Due to the inherent piezoelectricity, piezoelectric electric materials are recognized as intelligent materials which play an important role on the development of various sensors and smart materials applications. In this thesis, we use LRBFCM to analyze a piezoelectric sensor under a uniform compressive load. Piezoelectric sensors are often manufactured as thin cylindrical plates, therefore a 3D cylindrical model with multi-scale nodal distribution domain is applied here. This thesis will demonstrate the results of mechanical displacement and induced electric potential by the LRBFCM. Furthermore, we also take the FEM-ANSYS solutions as a benchmark to compare the results with meshless local-Petrov-Galerkin method (MLPG) from Professor Sladek’s group.
For the second main problem, the LRBFCM is applied to analyze in a quasicrystal (QCs) plate under a uniform static loads. Due to the Reissner–Mindlin plate bending theory, the actual 3D plate problem can be reduced to a quasi-3D problem. Hence, we are allowed to simulate the phonon and phason displacements by 2D governing equations. The behavior of the simply supported and clamped quasicrystal plates will be discussed here. In addition, this study remakes this quasicrystal plate problem by a conventional mesh-dependent numerical method, finite difference method (FDM) and compare the FDM results and LRBFCM results in order to show the superiority of the LRBFCM. The last but not the least, this study points out the difficulties when we conduct with the cross term on the orthogonal uniform distribution domain in order to improve the stability and accuracy of the LRBFCM for further researches.

摘要 i
Abstract ii
Table of Contents iv
List of Figures vii
List of Tables xi
Chapter 1 Introduction 1
1.1 Motivations and Objectives 1
1.1.1 Mesh-dependent numerical methods 2
1.1.2 Meshless numerical methods 3
1.2 Organization of the thesis 4
Chapter 2 The Localized Radial Basis Function Collocation Method 6
2.1 Localized radial basis function collocation method 7
2.2 Radial basis function 10
2.3 The selection of local nodes 13
2.4 Shape parameter 17
2.5 Normalization technique 18
2.5.1 Normalized distance 18
2.5.2 Normalized shape parameter 18
Chapter 3 The Localized Radial Basis Function Collocation Method for Piezoelectric Sensor under Compressive Load 20
3.1 Introduction 20
3.1.1 Piezoelectric effect 21
3.1.2 Piezoelectric materials on civil engineering 24
3.1.3 Numerical methods for piezoelectric problems 25
3.2 Governing Equations 27
3.3 Boundary conditions 29
3.4 The derivations for LRBFCM 29
3.5 Numerical examples for the piezoelectric sensor 33
3.6 Numerical results for the static problem of the piezoelectric sensor under compressive load 39
Chapter 4 The Localized Radial Basis Function Collocation Method for the Quasicrystal Plate Problems 43
4.1 Introduction 43
4.2 Governing Equations 44
4.3 Boundary conditions 52
4.3.1 Simply-supported boundary conditions 52
4.3.2 Clamped boundary conditions 53
4.4 The derivations for LRBFCM 53
4.5 Numerical examples for the quasicrystal plate 56
4.6 Numerical results for the orthorhombic quasicrystal plate 57
4.6.1 Simply-supported QC plate 57
4.6.2 Clamped QC plate 61
4.7 Finite Difference Analysis 63
4.8 Investigation of sensitivity of shape parameter and selection of supported local nodes 67
Chapter 5 Conclusions and future work 70
5.1 Conclusions 70
5.2 Future Work 71
Acknowledgement 72
References 73
Appendix 82
A. Type of boundary conditions 82


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