Page 95 - Process Modelling and Simulation With Finite Element Methods

82 Process Modelling and Simulation with Finite Element Methods

Before moving in to explain the implementation of boundary conditions we
shall work through a simple ODE to highlight the concepts we discussed so far.
Following the worked example should guide you through the weak formulation
and finding approximate numerical solutions to a given second order ODE.

Exercise 2.4: A worked example offinite element calculations in detail
To elaborate the concepts described above a simple ODE is solved using the
variational principles that forms the core to FEM. The problem is simple. Solve
the second order ODE

d 'u 2
y+4u =8x (2.28)
dx
subjected to boundary conditions
u(0) = U(X /4) = 0

using the weak formulation.
This simple second order ODE has an analytic solution (Prove it!)

(2.29)

Since we know the analytic solution, a comparison would give the error of the
approximate solution we find.
The first step for the weak formulation is to assume a weight function and a
trial function. Take U(x) as the trial function and q5 (x) as the weight function.
We discuss the exact forms of these functions later. The trial function U(x) forms
a solution to the ODE. Therefore, if we substitute U(x) in (2.28), the resulting
equation gives the residual:

(2.30)

Subsequent steps really amount to the minimization of this residual. The
minimization process starts by evaluating the weighted residual. To evaluate the
weighted residual, multiply (2.30) by q5 (x) and integrate over the domain (i.e.
05 x 5 n/4).

(2.31)