Finite Element Method (FEM)

1
Finite Element Method (FEM)
Different from the finite difference method (FDM)
described earlier, the FEM introduces
approximated solutions of the variables at every
nodal points, not their derivatives as has been
done in the FDM. The region of interest is
subdivided into small regions that are called
finite elements. It will then be assumed that
some predetermined function ( such as ?(x,y,z,t))
in terms of dependent variables (such as the
spatial and time coordinates, x,y,x,z t) can be
used to replace the dependent variable
(T(x,y,z,t)at the node (with unknown coefficients
(ai) to be determined). This function has
to satisfy both the governing equation (heat
diffusion equation for heat transfer problem, for
example) at every nodal points and the boundary
condition at every exterior nodal points. By
substitute this function into every points we can
obtain a system of algebraic equations in terms
of the unknown coefficients (ai). This system of
equations can then be solved using standard
numerical schemes described before.
2
Finite Element Example
Determine the temperature distribution of the
flat plate as shown below using finite element
analysis. Assume one-dimensional heat transfer,
steady state, no heat generation and constant
thermal conductivity. The two surfaces of the
plate are maintained at constant temperatures of
100C and 0C, respectively.
First, divide the plate into three elements (1,2
3). The temperatures of these three elements
are represented by their nodal temperatures T1,T2
T3, respectively. Next, assume the temperature
is a function of its coordinate T(x)Ax2BxC.
A,B C are three constants. Finally, determine
the constants using the governing equation and
all corresponding boundary conditions.
T100C
T0C
x23/2
x11/2
x35/2
2
3
1
L3
3
Example (cont.)
To simplify the solution, we can apply the
governing equation first
Therefore, A0 for all nodal temperature
functions. This is no surprise for us since we
know the steady state, no generation, 1-D heat
transfer should have a linear temperature
distribution. Therefore T(x)BxC and the three
nodal equations are T1Bx1C(B/2)C,
T2Bx2C(3B/2)C, T3 Bx3C(5B/3)C Therefore,
there are only two constants to be solved and
they can be determined using the two boundary
conditions. At the left-side surface, the
temperature is a constant 100C and there is a
constant heat transfer into the element 1 and the
same amount of the heat is transferred to the
element 2 since there can be no heat accumulation
inside the element to satisfy the steady state
condition. q(left surface to element 1)
q(element 1 to element 2)
4
Example (cont.)
zero
The second equation can be determined by using
the boundary condition on the other side of the
plate
5
Example (cont.)
This equation satisfies both boundary conditions
T(x0)100C and T(x3)0 C. For most finite
element problems, we have to use thousands or
even millions of elements in order to resolve as
much detailed information as possible.
Therefore, a fast numerical solver for the matrix
(system of equations) is necessary to obtain
satisfactory results. The use of numerical
scheme has been discussed previously when we
introduce the finite difference method.