Abstract
等參單元之逆映射及扭曲參數最近已引起了許多研究學者的重視，雖有少數有關此方面的研究文獻，對於三維等參單元卻未曾導出其逆映射之解析表示式亦未曾建立其扭曲參數。由微分幾何中的測地線理論可導出8節點6面體單元等參映射的反函數關係，在這種反函數關係裡假設了測地線座標的無限級數，此座標可證明為由等參映射時在原點的Jacobian值所決定而得。若將測地線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。當等參座標表為測地線座標無窮級數之多項式時，各項之係數即為等參座標在原點之Christoffel symbol及其高階微分項的值，這些值皆可由所有的扭曲參數表示。由張量分析知，一座標系統之Christoffel symbol值反映了此座標系統基向量的變化。所以，扭曲參數反映了此座標系統在原點基向量的變化。當等參映射表示成卡氏座標為以係數向量為等參座標係數之多項式時，這些係數向量可完全建立等參單元之扭曲參數。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。The inverse relations of the isoparametric mapping for the8-node hexahedra are derived by using the theory ofgeodesics in differential geometry. Such inverse relationsassume the form of infinite power series in the elementgeodesic coordinates, which are shown to be the skewCartesian coordinates determined by the Jacobian of themapping evaluated at the origin. By expressing the geodesiccoordinates in turn in terms of the isoparametriccoordinates, the coefficients in the resulted polynomialsare suggested to be the distortion parameters of theelement. These distortion parameters can be used tocompletely describe the inverse relations and the determinantof the Jacobian of the mapping. The meanings of them canalso be explained geometrically and mathematically. Thesemethods of defining the distortion measures andderiving the inverse relations of the mapping arecompletely general, and can be applied to any other two- orthree- dimensional isoparametric elements.