D-R-A-F-T
the model the chamber values are shifted back one chamber per time step and the
mixing process repeated. This procedure results in memory of the history of
concentrations crossing the boundary, delays full specification of the nominal
boundary concentration Cb, and generally provides more realistic boundary
conditions. Furthermore, the buffering also provides a buffer for the changes that
any plan alternatives to be tested may have on the boundary conditions.
Finite Element Formulation
Galerkin Finite Element Form
The governing equation (1) is then cast into the Galerkin finite element form using
quadratic shape functions, N ,
Equation 24
∂C ⎤
⎡
∂N j
∂C ∂N j
^
^
^
^
⎧
^⎫ +
∂C
∂C
NE
∑1 ∫∫ ⎢ N j ⎨Q + u ∂x + v ∂y -α 1 C ⎬ ∂x
Dx ∂x + ∂y
D y ∂y ⎥dxdy
⎥
Dne ⎢
⎩
⎭
⎦
⎣
ne=
NL
+∑∫N
s
q dζ
=0
j
i
i=1 ζ
⎛ ∧⎞
⎜∂C ⎟
⎜ ∂t ⎟ + α 2 for the transient problem
Where Q=
⎜
⎟
⎝
⎠
where
NE
=
total number of elements
N
=
the quadratic shape(or basis) functions
Q
=
See above
^
=
the approximate concentration in an element as evaluated from
C
shape functions and nodal point values of C
NL
=
total number of boundary segments
ζ
=
the local coordinate
=
flux from source on boundary i
s
q
i
The transient equation is expressed as
Equation 25
[T ]∂ {C} + [K ]{C}- {F } = 0
∂t
where each element in the computation mesh contributes the following terms to the
global matrix
Equation 26
Conceptual Program Design 17
Users Guide To SED2D-WES