# 统计代写|主成分分析代写Principal Component Analysis代考|Importance of Polynomial Representations in Finite Element Analysis

The passage explains the importance of using polynomial representations in finite element analysis that satisfy both compatibility and completeness requirements. For a one-dimensional element with M degrees of freedom, the polynomial representation must include all powers of the independent variable up to and including M-1. Geometric isotropy is a key property for these polynomials in two and three dimensions, meaning that the polynomial remains unchanged under translations or rotations of the coordinate system.

In two dimensions, a complete polynomial of order M can be constructed using a combination of powers of x and y up to xiyjx^i y^jx i y j where i+j≤Mi + j \leq Mi+j≤M, with the total number of terms given by N(2)tN(2)_tN(2) t ​ .

The Pascal triangle illustrates the structure of these complete polynomials, showing that for a polynomial of order M, all the terms lying on or below the Mth horizontal line must be included. For instance, a quadratic polynomial would require six nodal degrees of freedom in a finite element model.

Incomplete but symmetric polynomials can also satisfy geometric isotropy if the terms balance out the contributions from each independent variable. Such polynomials can be identified within the Pascal triangle by considering the terms along certain diagonals (indicated by dashed lines).

In three dimensions, the principle extends to a “Pascal pyramid” where the polynomial expression of the field variable should again be either complete or incomplete but symmetric to maintain geometric isotropy. This ensures that each independent variable (x, y, and z) contributes equally to the polynomial.

While geometric isotropy isn’t always strictly necessary, it is advantageous in ensuring convergence during mesh refinement (h-refinement). When using the p-refinement method, increasing the order of the interpolation functions approximates the power series expansion of the exact solution better, eventually approaching the true solution as the number of nodes goes to infinity.

### MATLAB代写

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