以局部徑向基底函數佈點法分析三維功能梯度壓電半導體問題

Abstract

本研究主旨為藉由局部徑向基底函數佈點法分析三維梯度功能壓電半導體問題。局部徑向基底函數無網格數值方法已經被廣泛運用在工程與科學領域上，由於在處理空間尺度相差甚異與無須數值積分的優勢下，因此將此數值方法運用在壓電材料的工程問題上。 壓電材料可以分為介電體與半導體兩類，不同於壓電絕緣體，由電子密度及電流所組成的電守恆式需被額外用以描述壓電半導體的現象，這也加深了彈性位移及電場間相互作用分析的複雜度。此研究以局部徑向基底函數佈點法來求解存在非定常數的偏微分方程，在物理場中的空間變化以多元二次曲面徑向基底函數近似；時變性的微分方程系統問題以Houbolt有限差分法求解。 以有限元素法之相對應結果來驗證局部徑向基底函數佈點法之結果，且分析在不同載重情形下分析的樑所產生的力學反應、電場、電流場間互相的關係。此外，此研究也分析梯度參數及初始電子密度所產生之影響。最後，暫態分析也在此研究的範疇中。

This thesis presents three-dimensional analysis of functionally graded piezoelectric semiconductor by the local radial basis function collocation method (LRBFCM). The LRBFCM is a commonly-used meshless numerical method in the field of engineering and sciences. On account of the advantages of addressing the problems with much different length scales in three dimensions and circumventing numerical quadrature, the LRBFCM is investigated and applied in the problems of piezoelectric materials. Piezoelectric materials can be divided by dielectrics and semiconductors. Unlike piezoelectric dielectric materials, the conservation of charge which is composed of electron density and electric current is additionally considered to depict the phenomenon for piezoelectric semiconductors. This will complicate our analyzing the mutual coupling of elastic displacements and electric fields. For the solution of the set of partial differential equations with non-constant coefficients the LRBFCM is proposed in this work. The spatial variations of all physical fields are approximated by the multiquadric radial basis function. For time dependent problems a resulting system of ordinary differential equations is solved by the Houbolt finite difference scheme as a time stepping method. The presented LRBFCM method is verified by using the corresponding results obtained by the finite element method. The effect of various loading scenarios is then considered in the numerical examples to analyze the mutual properties of the mechanical responses, electrical fields, and electrical current field. The influence of material parameter gradation and initial electron density is then investigated. The transient analysis is also analyzed.