臺灣博碩士論文加值系統

在有限單元分析中, 數值計算之準確度及可靠度對單元形狀的扭曲相當敏
感. 因此, 扭曲參數的定義以及單元敏感度的分析, 實為一重要研究課
題.由微分幾何中的測地線理論, 已成功地導出平面4節點, 平面8節點和
8節點六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, 而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值.本文基於上述之推廣, 對於6節點三角形和10節點
四面體選擇其形心位置為面積座標和體積座標之原點. 亦可導出其等參映
射的反函數關係由此, 則6節點三角形可得6個扭曲參數; 10節點四面體則
有18個扭曲參數.此扭曲參數可做為單元形狀扭曲之測度.本文定義扭曲參
數的方法具有數學理論上之一致性與一般性, 並可應用於其它有限單元之
相關研究.

In analyzing the finite element, The accurance and realibity of
numerical calculation are rather sensitive to the distortion
of element shape. so, defining distortion parameter and
analyzing sensitivity of element are an important research
course. By using the theory of geodesics in differential
geometry, inverse relations of the mapping for 4-node elements,
8-node quadrilaterals with curved boundaries and 8-node
hexahedra can be successfully derived and expressed in terms
of the element coordinates defined at the local origin. Such
inverse relations assume the form of infinite power series in
the element geodesic coordinates, which are shown to be the
skew Cartesion coordinates determined by the Jacobian of the
mapping evaluated at the origin By expressing the geodesic
coordinates in turn in terms of the isoparametric coordinates,
the coefficients in the resulted polynomials are suggested to
be the distortion parameters of the element. These distortion
parameters can be used to completely describe the inverse
relations and the determinant of the of the mapping. In this
study, for quadratic triangular element in two dimensions and
quadratic tetrahedral element in three dimensions, choose their
centroids as the origins of the area and volume coordinates.
Therefore, we can also derived the inverse relations of
mapping. For quadratic triangular element, there are 6
parameters; For quadratic tetrahedral element there are 18
parameters. From these distortion parameters, they can be the
measures of the distortion of element shape. These methods of
defining the distortion measures and deriving the inverse
relations of the mapping are completely general and can be
applied to any other two- or three- isoparametric elements.