31 Jul Numerical Technics In Engineering
Here is a typical steady-state heat
ow problem. Consider a thin steel plate to be a
10 20 (cm)2 rectangle. If one side of the 10 cm edge is held at 1000C and the other
three edges are held at 00C, what are the steady-state temperature at interior points?
We can state the problem mathematically in this way if we assume that heat
ows
only in the x and y directions:
Find u(x; y) (temperature) such that
@2u
@x2 +
@2u
@y2 = 0 (3)
with boundary conditions
u(x; 0) = 0
u(x; 10) = 0
u(0; y) = 0
u(20; y) = 100
We replace the di erential equation by a di erence equation
1
h2 [ui+1;j + ui????1;j + ui;j+1 + ui;j????1 ???? 4ui;j ] = 0 (4)
5
which relates the temperature at the point (xi; yj) to the temperature at four neigh-
bouring points, each the distance h away from (xi; yj ). An approximation of Equation
(3) results when we select a set of such points (these are often called as nodes) and
nd the solution to the set of di erence equations that result.
(a) If we choose h = 5 cm , nd the temperature at interior points.
(b) Write a program to calculate the temperature distribution on interior points with
h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Discuss your solutions and
examine the e ect of grid size h.
(c) Modi ed the di erence equation (4) so that it permits to solve the equation
@2u
@x2 +
@2u
@y2 = xy(x ???? 2)(y ???? 2)
on the region
0 x 2; 0 y 2
with boundary condition u = 0 on all boundaries except for y = 0, where u = 1:0.
Write and run the program with di erent grid sizes h and discuss your numerical
results.
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