Biomedical Engineering Reference
In-Depth Information
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2
2
2
µ
u
N
u
u u
w
1
p
w
ww
F
-
·
(5.45)
¦
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z
S
u
S
2
2
2
Dt
z
u u u
§
x
y
z
In another expression,
Dv
1
2
O
F r
-
p
v
(5.46)
S
Dt
5.2
Discretization Algorithm
Before solving a differential equation such as Navier-Stokes
equations, it is necessary to transform the equation into a machine-
readable equation. For it is impossible for computer to solve a
differential equation analytically, the procedure of writing an
algebra equation and relational expression for unknown value on
a point of grid within calculated region is needed. This operation
is called discretization. Based on the discrete equation, unknown
value is evaluated by solving algebra equation and operating four
fundamental rules of arithmetics. In the ield of numerical simulation
of low, there are three typical methods for discretization.
1
Finite-difference method:
Finite-difference method, which is
widely used in numerical analysis, expresses derivative term in
differential with Taylor series. The derivative term is approximated
by combination of relational equation expressed with Taylor series
on the point of grid and neighboring grids depending on degree of
precision for discretization. It is based on differential form of the
equation of low and the solution exists on the point of grid. Basically,
the space between dots is not considered. Structured grid is used in
FDM.
Finite-element method:
The basic equation is not a differential
form but a weak form derived from the integration of multiplied basic
function. By substituting this formula into the governing equation
and integrating after weighting, we obtain relational equation of
variable arranged at the vertex of cell. In many cases, calculating
area is composed of minim triangular elements. Therefore, it does
not matter whether it is structured grid or unstructured grid.
Finite-volume method:
Finite-volume method discretizes
differential equation after integration to satisfy the conservation
law of mass and momentum. It is based on the integral form of the
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