统计代写|主成分分析代写Principal Component Analysis代考|Volume Coordinates and Higher-Order Elements in Finite Element Analysis

The text discusses the concept of volume coordinates for tetrahedral elements in finite element analysis. A four-node tetrahedral element is described, with its nodes numbered according to a counterclockwise convention. Volume coordinates L1,L2,L3,L_1, L_2, L_3,L 1 ​ ,L 2 ​ ,L 3 ​ , and L4L_4L 4 ​ are introduced as the ratios of the sub-volumes defined by each node and the total volume of the element. These coordinates vary linearly throughout the element and assume unity at the corresponding node and zero at the other nodes, thereby fulfilling the conditions for interpolation functions.

The field variable ϕ(x,y,z)\phi(x, y, z)ϕ(x,y,z) within the tetrahedral element is interpolated using the volume coordinates as a weighted sum of the nodal values (Equation 6.64). Integration using volume coordinates is relatively simple, and an integration formula (Equation 6.66) is provided for tetrahedral elements.

Analogous to the two-dimensional case, the tetrahedral element is suitable for modeling irregular three-dimensional geometries. However, it’s noted that tetrahedral elements are not easily compatible with other element types due to their nodal configurations.

The text then moves to discuss higher-order tetrahedral elements and introduces the eight-node brick element, also known as a hexahedral or cuboidal element. For this type of element, natural coordinates r,s,r, s,r,s, and ttt are employed, which are dimensionless and range from -1 to 1 over the element domain. The interpolation functions for the eight-node brick element are presented in Equation 6.69, and the field variable is expressed as a sum of these functions multiplied by the nodal values.

It’s emphasized that the field variable representation in the eight-node brick element (Equation 6.71) is an incomplete, symmetric polynomial ensuring geometric isotropy. The text demonstrates that, despite the apparent non-constant nature of the first partial derivatives, the interpolation functions satisfy the completeness condition, allowing the first derivatives to assume constant values (important for C0 continuity problems). This is shown by relating the gradient in the xxx direction to the nodal values and verifying that the partial derivative with respect to xxx remains constant throughout the element. Similar procedures can be followed to demonstrate the constancy of other partial derivatives, thus meeting the completeness condition for the eight-node brick element.

MATLAB代写

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