Page 91 - Process Modelling and Simulation With Finite Element Methods

78 Process Modelling and Simulation with Finite Element Methods

Note that at all times, these interesting wave dynamics remain “fixed” at the
boundaries, since the Dirichlet boundary condition ensures zero boundary
amplitude. So how do we combat this? Let’s establish periodic boundary
conditions, which are effected by a little knowledge of how FEMLAB keeps its
books. When two boundaries are equivalent, FEMLAB adds the Dirichlet
conditions on both boundaries to the constraint:

l2.p (0)+h2u (0) = 5 + r2 (2.16)

So the modest change to implement periodic boundaries is to set hz=-l.
Pull down the Boundary menu and select Boundary Settings.

0 Select domain 2
Check Dirichlet h=- 1 ; r=O
APPlY

Repeat the solution procedure. How does the final state (t=2) compare with the
state without periodic boundary conditions (Dirichlet)? Did you notice any
difference in the wave dynamics during the animation sequence?

Exercise 2.3
Try the initial conditions u(tO)=sin( lO*pi*x) and u-t(tO)= - lO*pi*cos( 1 O*pi*x).
What do you expect to see in the animation for u(t)? u-t(t)? Did anything
unexpected occur?
Note that MATLAB has a built-in constant pi.

The Finite Element Method
By now, you must be wondering how FEMLAB actually accomplishes this
magic of solving PDE systems. Finite element analysis has been around for
several decades, and has had commercial packages available since the 1980s. A
good introduction can be found in the book of Reddy [3]. It is not the intention
here to describe FEM in any great detail, nor to describe the full FEMLAB
implementation, but rather to give an impression of the type of calculations that
occur in FEM, and an understanding of why FEMLAB is a particularly
convenient tool for implementing FEM.
The essence of the finite element method is to state any constraints on the
field variables in weak form. To understand what a weak form is (and why
mathematicians termed it weak), it should be understood that the strong form of
a system of constraints is the partial differential equation system and appropriate