# 统计代写|主成分分析代写Principal Component Analysis代考|Interpolation Functions for Triangular Finite Elements in 2D

This passage describes the formulation of interpolation functions for triangular finite elements in two dimensions and their application in various contexts, including axisymmetric 3-D cases and structural analyses involving plate and shell structures. The discussion starts with a presentation of three types of triangular elements: linear, quadratic, and cubic, noting that the cubic element has an internal node to ensure geometric isotropy.

For a simple three-node linear triangular element, the field variable is interpolated using a linear polynomial:

ϕ(x,y)=a0+a1x+a2y\phi(x, y) = a_0 + a_1x + a_2yϕ(x,y)=a 0 ​ +a 1 ​ x+a 2 ​ y

Applying the nodal conditions at the three vertices of the triangle allows us to solve for the polynomial coefficients a0,a1,a_0, a_1,a 0 ​ ,a 1 ​ , and a2a_2a 2 ​ using a system of equations derived from Equation 6.34, where the nodal displacements (ϕ1,ϕ2,ϕ3\phi_1, \phi_2, \phi_3ϕ 1 ​ ,ϕ 2 ​ ,ϕ 3 ​ ) are substituted into the matrix equation and solved for the coefficients through matrix inversion. The resulting expressions for the coefficients involve the area AAA of the triangle and combinations of the nodal displacements and vertex coordinates.

The interpolation functions Ni(x,y)N_i(x, y)N i ​ (x,y) are obtained by collecting the nodal variables in the expression for ϕ(x,y)\phi(x, y)ϕ(x,y). These functions are crucial because they determine how the field variable varies across the element.

When choosing a particular coordinate system for the element, the form of the interpolation functions can simplify significantly. However, this comes with the trade-off that if the local element coordinate system is not aligned with the global coordinate system, transformations must occur when assembling the global stiffness or other relevant matrices. Computational efficiency can be enhanced by orienting the element coordinate systems parallel to the global axes, thereby avoiding the need for these transformations.

Finally, the passage highlights that the linear interpolation scheme leads to constant strain components in structural applications and constant temperature gradients in heat transfer problems. Thus, in structural mechanics, the three-node linear triangular element is often referred to as a constant strain triangle (CST). This indicates that for small deformations, the strains do not vary within the element, which is an idealization that works well for linear elastic behavior over relatively small domains.

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