# 统计代写|主成分分析代写Principal Component Analysis代考|Development of Interpolation Functions for Four-Node Rectangular Finite Element

The text describes the development of interpolation functions for a four-node rectangular finite element. Here’s a summary of the main points:

Four-Node Rectangular Element:

The field variable ϕ(x,y)\phi(x, y)ϕ(x,y) within a four-node rectangular element is approximated by a two-dimensional, incomplete, symmetric polynomial of degree two (quadratic), given by Equation 6.50: ϕ(x,y)=a0+a1x+a2y+a3xy\phi(x, y) = a_0 + a_1x + a_2y + a_3xyϕ(x,y)=a 0 ​ +a 1 ​ x+a 2 ​ y+a 3 ​ xy To determine the coefficients a0,a1,a2,a_0, a_1, a_2,a 0 ​ ,a 1 ​ ,a 2 ​ , and a3a_3a 3 ​ , the polynomial is evaluated at each of the four nodes, and the nodal conditions are applied. This results in a system of equations (Equation 6.51 and 6.52), which when solved, gives the coefficients in terms of nodal values. Natural Coordinates (r, s):

Instead of using the standard Cartesian coordinates (x, y), natural coordinates rrr and sss are introduced, which are dimensionless and range from -1 to +1. They are defined as: r=x−xˉa,s=y−yˉbr = \frac{x – \bar{x}}{a}, \quad s = \frac{y – \bar{y}}{b}r= a x− x ˉ

​ ,s= b y− y ˉ ​

The centroid coordinates xˉ\bar{x} x ˉ and yˉ\bar{y} y ˉ ​ are calculated as the averages of the respective nodal coordinates across opposite sides of the rectangle. Interpolation Functions in Natural Coordinates:

The interpolation functions N1(r,s),N2(r,s),N3(r,s),N_1(r, s), N_2(r, s), N_3(r, s),N 1 ​ (r,s),N 2 ​ (r,s),N 3 ​ (r,s), and N4(r,s)N_4(r, s)N 4 ​ (r,s) are derived based on satisfying the conditions that each function equals 1 at its associated node and 0 at all other nodes. The explicit forms are provided in Equation 6.56a. Higher-Order Rectangular Elements:

For an eight-node rectangular element, the field variable would require an incomplete, symmetric cubic polynomial with eight terms. However, instead of directly choosing between two possible forms (Equations 6.57a and 6.57b), the interpolation functions are derived using the natural coordinates and nodal conditions. An example is given for the interpolation function N1N_1N 1 ​ associated with the first node, which is constructed such that it vanishes at all other nodes except node 1. The final form for N1N_1N 1 ​ after correction is presented in Equation 6.58. Similar procedures are followed to derive the interpolation functions for the other seven nodes, both for the corner nodes (Equations 6.59b-d) and midside nodes (Equations 6.59e-h). Advantages of Natural Coordinates:

Like area coordinates in triangular elements, using natural coordinates simplifies the algebraic expressions for interpolation functions and facilitates easier integration since the integration limits are fixed (-1 to +1). Internal Nodes in Higher-Order Elements:

For high-order elements, internal nodes may create complications during meshing and connectivity. Mechanically relevant effects from these internal nodes are often mathematically assigned to the external nodes through a process called ‘elimination’. This allows for a more practical implementation of higher-order elements without requiring direct connections between internal nodes and nodes of adjacent elements.

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