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

Abstract

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.