Shape Functions 1-Finite Element Method-Assignment Solution, Exercises of Mathematical Methods for Numerical Analysis and Optimization

Download Shape Functions 1-Finite Element Method-Assignment Solution and more Exercises Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!

(SOLUTION OF PROBLEM # 4.6)

PROBLEM # 4.

Consider a triangular element with the corner nodes defined by the Cartesian coordinates (-2, - 1), (2, 4) and (4, 1). Using the expressions derived in Problem 4.5 for L 1 , L 2 , L 3 , evaluate the following in terms of the Cartesian coordinates x and y.

a) Shape function N 1 , N 2 , N 3 corresponding to a linear interpolation model. b) Shape function N 1 , N 2 ……………N 6 corresponding to a quadratic interpolation model. c) Shape function N 1 , N 2 , N 3 ……….……N 10 corresponding to a quadratic interpolation model.

GIVEN DATA

x 1 = -2, y 1 = -

x 2 = 4, y 2 = 1

x 3 = 2, y 3 = 4

SOLUTION

As we know that,

Write them in matrix form, as under

Therefore,

a) Shape function N 1 , N 2 , N 3 corresponding to a linear interpolation model.

As we know that, for linear interpolation model, the shape functions are

Therefore,

b) Shape function N 1 , N 2 ……………N 6 corresponding to a quadratic

interpolation model.

As we know that, for quadratic interpolation model, the shape functions are

for i = 1,2,

Now, replace corresponding values of L 1 , L 2 , L 3 in above equations and simplify them, as following

As we know that, for quadratic interpolation model, the shape functions are

for i = 1,2,

Now, replace corresponding values of L 1 , L 2 , L 3 in above equations and simplify them, as following