Page 102 - Process Modelling and Simulation With Finite Element Methods

Partial Differential Equations and the Finite Element Method 89

2.1.5 Basic elements

Fundamental to the FEM is the concept that any domain can be implemented as a
collection of smaller subdomains of preferred shape. These subdomains are
called finite elements. Corners of an element are called nodes at which the
solutions to field variables are computed. There can be nodes in between corner
points that are commonly called edge nodes. In FEMLAB, when you generate
the mesh, it subdivides the computational domain in to a selected form of
elements and form of nodes accordingly. One can find more than a hundred
types of elements in use. If you are a beginner, it is natural to be puzzled over the
type of elements that should be used and the number of elements to be used.
The discretization process proscribes the type and the number of elements.
The number of elements is directly connected with the accuracy of the solution.
The higher the number of elements used, the lesser will be the error. However,
having a large number of elements would be computationally expensive,
demanding a large chunk of RAM and an extended runtime.
Defining an unnecessary number of elements is a very common practice.
There is no formula that allows you to choose optimally exact number of
elements. It is only by experience that you would be able to decide the right
amount of elements to pack in a domain. Though the accuracy increases with the
number of elements N, there will be a certain number N, beyond which the
sensitivity of accuracy becomes negligible.
Figure 2.9 shows the normalized error against the number of elements N.
The number of elements doubles in each iteration. One can see that the last three
points do not make any considerable improvement on the accuracy. However
one can perform a few short runs to find out the appropriate number of elements
to be used. There are instances where one is interested in a certain region of the
domain rather than the whole domain. For instance, take the case of the flow past
a cylinder. The boundary layer behavior around the cylinder is to be
investigated. In such cases one can and should pack more elements around the
cylinder having a lower density of elements in the far field. This way one can
attain the accuracy required without increasing the number of elements.
FEMLAB allows such grid stretching. Another point worth mentioning at this
juncture is the skewness of the elements in stretched grids. Skewed elements
make the formation of the Jacobian impossible. Therefore great care should be
taken in using stretched meshes.
The type of the elements to be used depends on the problem that has to be
solved. The dimensionality of the domain defines the dimensionality of the
elements. The most simple is the 1-D element that represents a line segment
between two nodes at each end. The most fundamental element in 2-D is a
triangle where as in 3-D it is a tetrahedron. Table 2.1 shows some of the basic
elements in use with respect to the dimensionality. Though we used straight