Page 49 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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THE FINITE ELEMENT METHOD
3. Form element equations (Formulation)
Next, we have to determine the matrix equations that express the properties of the
individual elements by forming an element Left Hand Side (LHS) matrix and load vector.
For example, a typical LHS matrix and a load vector can be written as
Ak 1 −1
[K] e = (3.1)
l −1 1

Q i
{f} e = (3.2)
Q j
where the subscript e represents an element; Q is the total heat transferred; k is the thermal
conductivity; l is the length of a one-dimensional linear element and i and j represent the
nodes forming an element. The unknowns are the temperature values on the nodes.
4. Assemble the element equations to obtain a system of simultaneous equations
To find the properties of the overall system, we must assemble all the individual ele-
ment equations, that is, to combine the matrix equations of each element in an appropriate
way such that the resulting matrix represents the behaviour of the entire solution region
of the problem. The boundary conditions must be incorporated after the assemblage of the
individual element contributions (see Appendix C), that is,

[K]{T}={f} (3.3)

where [K] is the global LHS matrix, which is the assemblage of the individual element LHS
matrices, as given in Equation 3.1, {f} is the global load vector, which is the assemblage of
the individual element load vectors the Equation 3.2, and {T} is the global unknown vector.
5. Solve the system of equations
The resulting set of algebraic equations, Equation 3.3, may now be solved to obtain the
nodal values of the field variable, for example, temperature.
6. Calculate the secondary quantities
From the nodal values of the field variable, for example, temperatures, we can then
calculate the secondary quantities, for example, space heat fluxes.

3.2 Elements and Shape Functions

As shown in Figure 3.1, the finite element method involves the discretization of both the
domain and the governing equations. In this process, the variables are represented in a
piece-wise manner over the domain. By dividing the solution region into a number of
small regions, called elements, and approximating the solution over these regions by a
suitable known function, a relation between the differential equations and the elements is
established. The functions employed to represent the nature of the solution within each
element are called shape functions, or interpolating functions, or basis functions. They are
called interpolating functions as they are used to determine the value of the field variable
within an element by interpolating the nodal values. They are also known as basis functions
as they form the basis of the discretization method. Polynomial type functions have been