Page 100 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
92
where
K km =− k x ∂N k ∂N m + k y ∂N k ∂N m + k z ∂N k ∂N m d

∂x ∂x ∂y ∂y ∂z ∂z

+ hN k N m dS
S

f k = GN k d
− qN k dS + hT a N k dS (3.268)

S S
It may be observed that Equations 3.260 and 3.266 are identical, which substantiates
the fact that both the variational and Galerkin methods give the same result because there
exists a classical variational integral for the heat conduction equation.

3.5 Requirements for Interpolation Functions

The procedure for formulating the individual element equations from a variational princi-
ple and the assemblage of these equations relies on the assumption that the interpolation
functions satisfy the following requirements. This arises from the need to ensure that
Equation 3.248 holds and that our approximate solution converges to the correct solution
when we use an increasing number of elements, that is, when we refine the mesh.
a. Compatibility: At element interfaces, the field variable T and any of its partial derivatives
up to one order less than the highest-order derivative appearing in I(T ) must be continuous.
b. Completeness: All uniform states of T and its partial derivatives up to the highest order
appearing in I(T ) should have representation in T , when in the limit the element size
decreases to zero.
0
If the field variables are continuous at the element interfaces, then we have C conti-
1
nuity. If, in addition, the first derivatives are continuous, we have C continuity, and if the
2
second derivatives are continuous, then we have C continuity, and so on. If the functions
appearing in the integrals of the element equations contain derivatives up to the (r + 1)th
order, then to have a rigorous assurance of convergence as the element size decreases, we
must satisfy the following requirements.
r
For compatibility: At the element interfaces, we must have C continuity.
For completeness: Within an element, we must have C r+1 continuity.
These requirements will hold regardless of whether the element equations (integral
expressions) were derived using the variation method, the Galerkin method, the energy
balance methods or any other method yet to be devised. These requirements govern
the selection of proper interpolation functions depending on the order of the differential
equation. Thus, for a conduction heat transfer problem, the highest derivative in I is of the
first order. Thus, the shape function selected should provide for the continuity of temper-
ature at the interface between two elements and also ensure the continuity of temperature
and heat flux within each element.
In addition to the requirements of continuity of the field variable and convergence to the
correct solution as the element size reduces, we require that the field variable representation