Abstract
在本論文研究中，我們使用由微分幾何理論所導出的正則座標系來解決一些傳統上在等參數有限元上所遇到的困難。由微分幾何中的測地線理論可導出平面或三維有限元的等參映射的反函數關係，在這種反函數關係裡假設了正則座標的無限級數，此座標可證明為由等參映射時在原點的Jacobian值所決定而得。若將正則座標以等參座標的多項式表示，則多項式中的係數可定義成元素的扭曲參數，這些扭曲參數能完全地描述等參映射的反函數關係及其Jacobian determinant。這種扭曲參數的定義方法及等參映射的反函數關係的推導是具普適性的而且可應用到二維和三維的等參元素上。本研究中，正則座標系也被引入為四節點混合型應力單元的參考座標，這樣一來假設應力可完全的滿足平衡條件，使的混合型應力單元的建立能根基於更合理的理論□石上。In this thesis study, a centroidal normal coordinate systembased on the theory in differential geometry is introduced toresolve some difficulties encountered in the formulation of thetraditional isoparametric elements. By using the theory ofgeodesics, the inverse relations of the isoparametric mappingfor the plane and solid elements are derived. Such inverserelations assume the form of infinite power series in theelement normal coordinates, which are shown to be the skewCartesian coordinates determined by the Jacobian of the mappingevaluated at the origin. By expressing the normal coordinatesin turn in terms of the isoparametric coordinates, thecoefficients in the resulting polynomials are suggested to bethe distortion parameters of the element. These distortionparameters can be used to completely describe the inverserelations and the determinant of the Jacobian of the mapping.The meanings of them can also be explained geometrically andmathematically. These methods of defining the distortionmeasures and deriving the inverse relations of the mapping arecompletely general, and can be applied to any other two- orthree- dimensional isoparametric elements.