Introduction To Finite Element Analysis And Design
Introduction To Finite Element Analysis And Design
2nd Edition
ISBN: 9781119078722
Author: Kim, Nam H., Sankar, Bhavani V., KUMAR, Ashok V., Author.
Publisher: John Wiley & Sons,
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Chapter 2, Problem 2E

Use the Galerkin method to solve the following boundary-value problem using a: (a) one-term approximation and (b) two-term approximation. Compare your results with the exact solution by plotting them on the same graph.

d 2 u d x 2 + x 2 = 0 , 0 x 1 u ( 0 ) = 1 u ( 1 ) = 0 }  Boundary conditions

Hint: Use the following one- and two-term approximations

One-term approximation:

u ˜ ( x ) = ( 1 x ) + c 1 ϕ 1 ( x ) = ( 1 x ) + c 1 x ( 1 x ) .

Two-term approximation:

u ˜ ( x ) = ( 1 x ) + c 1 ϕ 1 ( x ) + c 2 ϕ 2 ( x ) = ( 1 x ) + c 1 x ( 1 x ) + c 2 x 2 ( 1 x ) .

The exact solution is u ( x ) = 1 x ( x 2 + 11 ) 12

The approximate solution is split into two parts. The first term satisfies the given essential boundary conditions exactly, i.e., u ( 0 ) = 1 and u ( 1 ) = 1 .The rest of the solution containing the unknown coefficients vanishes at the boundaries.

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