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Natural and Cartesian Coordinates of Triangular Element-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Aligarh Muslim UniversityMathematical Methods for Numerical Analysis and Optimization

This assignment solution was submitted to Amar Sharma for Finite Element Method course at Aligarh Muslim University. It includes: Derive, Relationship, Natural, Area, Cartesian, Coordinates, Triangular, Element

Typology: Exercises

2011/2012

Uploaded on 07/08/2012

ramu
ramu 🇮🇳

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Problem 3.8: Derive the relationship between the natural ( area ) and Cartesian
coordinates of a triangular element.
Solution:
The natura
l coordinates are given as:
L1 =
L2 =
L3 =
Also
A = A1 + A2 + A3
We have
+ + = L1 + L2 + L3 = 1
Also
Ni = L1 Nj = L2 Nk = L3
Then the relation between the natural and Cartesian coordinates is given by
X = x1 L1 + x2 L2 + x3 L3
Y = y1 L1 + y2 L2 + y3 L3
These equations can be expressed as
=
This equation can be inverted to obtain
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Download Natural and Cartesian Coordinates of Triangular Element-Finite Element Method-Assignment Solution and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

Problem 3.8: Derive the relationship between the natural ( area ) and Cartesian coordinates of a triangular element.

Solution:

The natura l coordinates are given as:

L 1 =

L 2 =

L 3 =

Also

A = A 1 + A 2 + A 3

We have

    • = L 1 + L 2 + L 3 = 1

Also

Ni = L1 Nj = L2 Nk = L 3

Then the relation between the natural and Cartesian coordinates is given by

X = x 1 L 1 + x 2 L 2 + x 3 L 3

Y = y 1 L 1 + y 2 L 2 + y 3 L 3

These equations can be expressed as

This equation can be inverted to obtain

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Where A is given as

A =

By expanding, the relation is given as

= L 1 = (x2y3 – x3y2) + (y2 – y3)x + (x3 – x2)y

= L 2 = (x3y1 – x1y3) + (y3 – y1)x + (x1 – x3)y

= L 3 = (x1y2 – x2 y1) + (y1 – y2)x + (x2 – x3)y

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