# 统计代写|主成分分析代写Principal Component Analysis代考|”Mathematical Formulation for Axisymmetric Finite Elements”

In axisymmetric problems, a three-dimensional domain can be analyzed using two-dimensional finite elements due to the inherent symmetry around an axis. The key aspects of axisymmetric problems include:

The domain has an axis of symmetry (usually the z-axis). Boundary conditions, loads, and material properties are symmetric about the axis. The field variable depends only on the radial (r) and axial (z) coordinates. The interpolation functions for the axisymmetric problem can be derived from their two-dimensional counterparts by replacing the Cartesian coordinates (x, y) with (r, z). This allows any two-dimensional element to be used in an axisymmetric context; however, in practice, the element is a physical three-dimensional entity—like a prism of revolution.

In the given examples, you’re asked to evaluate certain integrals involved in formulating the element characteristic matrices for axisymmetric elements. For the first example, the integral involves interpolation functions NiN_iN i ​ and NjN_jN j ​ integrated over the volume of an axisymmetric element based on a three-node triangle using area coordinates. By expressing rrr in terms of nodal coordinates and area coordinates, and applying the correct integration formula for the area AAA, the integral reduces to a simpler form involving the product of nodal radii and constants related to the area coordinates.

For the second example, the integral is evaluated for a six-node, quadratic triangular element. Here, although the interpolation functions are quadratic, the fact that the sides are straight means the radial coordinate rrr can still be expressed linearly in terms of the area coordinates. The integral is again transformed into a series of integrals over the element area, which can be computed using Equation 6.49, resulting in a final expression for III that involves the nodal radii and coefficients related to the element’s geometry and interpolation scheme. The exact numerical values are not provided but can be computed using the mentioned integration formula.

### MATLAB代写

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