Interpolation Functions-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

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Assignment

FINITE ELEMENT

METHODS

P A G E | 1

Q.No.08 The interpolation functions corresponding to node i of a triangular element can be expressed in terms of natural coordinates L 1 , L 2 , and L 3 using the relationship

With,

Where i = 1, 2... n

n = total number of nodes in the element, and

m = order of the interpolation model (2 for quadratic, 3 for cubic, etc.),

Lj (i)^ = value of the coordinate L 2 at node i.

Using above equations, find the interpolation function corresponding to node 4 of a cubic triangular element.

Solution:

Given:

Element type: cubic triangular element

m = order of the interpolation model = 3

Find:

N 4 = the interpolation function corresponding to node 4

Schematic:

The cubic triangular element is shown in figure below.

Analysis:

P A G E | 3

Simplifying,

Hence

3. For, , realize that at 4th^ node

Therefore,

Hence we get,

This gives,