Page 177 - Finite Element Analysis with ANSYS Workbench

168 Chapter 9 Heat Transfer Analysis

9.2 Finite Element Method

9.2.1 Finite Element Equations
Finite element equations can be derived by applying the
method of weighted residuals to the governing differential
equation. Details of the derivation can be found in many finite
element textbooks including the one written by the author. The
derived finite element equations in matrix form are,
T
   CT     K c  K h  K r  
  c Q   Q Q    Q q    h Q   r Q
C
where  is the capacitance matrix; K c  is the conduction
matrix;  h K is the convection matrix; K is the radiation matrix;

r

  is the vector containing rate of change of nodal temperatures;
T
T
Q
  is the vector containing nodal temperatures;   is the
c
Q
conduction load vector;   is the heat generation load vector;
Q
Q
  is the specified heating load vector;   is the convection
Q
q
h
Q
load vector; and   is the radiation load vector.
r
These element matrices and load vectors depend on
element types as described in the following section.

9.2.2 Element Types
The one-dimensional two-node rod element is shown in
the figure. The finite element matrices and load vectors can be
derived in closed form, such as,
Q q s (   ) hT T  ( 4 T  4 )T  p

1 2

x
L A