Clase 11 (SEMANA 12 ) Calculo Numerico II_PGRA_2024_I.pdf

CALCULO NUMEICO 2

1. Calculo Numérico II
IF-392
Semana 11
Dr. Pierre Giovanny Ramos Apestegui
Cálculo Numérico II-IF392

2. Elliptic Equations
 Neumann Problem (Mixed Conditions)
 Irregular Boundaries.
Cálculo Numérico II-IF392

3. Boundary Conditions
•Today, we will address problems that involve
boundaries at which the derivative is specified and
boundaries that are irregularly shaped.
Derivative Boundary Conditions/
•Known as a Neumann boundary condition.
•For the heated plate problem, heat flux is specified
at the boundary, rather than the temperature.
•If the edge is insulated, this derivative becomes zero.
Cálculo Numérico II-IF392

4. In the formulation of the Neumann problem, the values of the
normal derivatives of u are prescribed along the boundary. As
before, we are focusing on Laplace’s equation in a rectangular
region. In solving the Neumann problem, u is no longer available
on the boundary, hence the boundary points become part of the
unknown vector.
Neumann Problem
Cálculo Numérico II-IF392

5. Suppose the difference equation for Laplace’s
equation, is applied at the (3,3)meshpoint:
The three interior mesh points are part of the unknown vector.
Since boundary values u
43 and u
34 are not available, they must
also be included in the unknown vector. There are 25 unknowns:
9 in the interior and 16 on the boundary. But since Equation is
applied at the interior mesh points, there are as many equations as
there are interior mesh points. Therefore, several more equations
are needed to completely solve the ensuringsystemofequations.
Cálculo Numérico II-IF392
Fori=3,j=3

6. For a point on the edge:
Cálculo Numérico II-IF392

7. 0422
2
2
04
,01,01,0,1
,1,1
,1,1
,01,01,0,1,1
=−++


−


−=

−



=−+++
−+
−
−
−+−
jjjj
jj
jj
jjjjj
TTT
x
T
xT
x
T
xTT
x
TT
x
T
TTTTT •Thus, the derivative has been incorporated into the
balance.
•Similar relationships can be developed for derivative
boundary conditions at the other edges.
Cálculo Numérico II-IF392

8. Cálculo Numérico II-IF392
In order to generate these additional (auxiliary) equations, we
apply the following equation at each of the marked boundary
points. Since we are currently concentrating on �
43 and �
34, we
apply the equation at these two points
There are two quantities that are not part of the grid and need to
be eliminated: �
53 and �
35 . We will do this by extending the grid
beyond the boundary of the region, and using the information on
the vertical and horizontal boundary segments they are located
on. At the (3, 4) boundary point, we have access to ∂u/∂y = g(x).
Let us call it ??????
34 .

9. Cálculo Numérico II-IF392
Applying the two-point central difference formula at (3, 4), we find
Substitution of these two relations into the previous equation creates two new
equations that involve only the interior mesh points and the boundary points. In
order to proceed with this approach, we need to assume that Laplace’s
equation is valid beyond the rectangular region, at least in the exterior area
that contains the newly produced points such as ??????
�� and ??????
�� . If we continue
with this strategy, we will end up with a system containing as many equations as
the total number of interior mesh points and the boundary points. In the Figure,
forinstance, the system will consist of 25 equations and 25 unknowns.

10. Example 01 (Mixed Conditions)
Cálculo Numérico II-IF392
It is required to determine the steady state temperature at all
points of a heated sheet of metal. The edges of the sheet are
kept at a constant temperature: 100, 50, 75 degrees and de
bottom edge is insolated
50
100
75
The sheet is divided to 5X5 grids.

11. Example 01100
4,1
=T 100
4,2
=T 100
4,3
=T 50
3,4
=T 50
2,4
=T 50
1,4
=T 75
3,0
=T 75
2,0
=T 75
1,0
=T
??????
�,�??????
�,�
??????
�,�
Known
To be determined3,1
T 2,1
T 1,1
T 3,2
T 2,2
T 1,2
T 3,3
T 2,3
T 1,3
T
Cálculo Numérico II-IF392

12. First Equation100
4,1
=T 100
4,2
=T 75
3,0
=T 75
2,0
=T
Known
To be determined3,1
T 2,1
T 3,2
T 2,2
T 0410075
04
3,13,22,1
3,13,22,14,13,0
=−+++
=−+++
TTT
TTTTT
Cálculo Numérico II-IF392

13. The general equation that characterizes a derivative at the
lower end (i.e., at j = 0) on a hot plate is:
Cálculo Numérico II-IF392
At the isolated end, the derivative is zero and the equation
becomes
Therefore, the 3 additional equations due to the boundary are:
????????????�??????
10: ??????
00+??????
20+??????
11+??????
1−1−4??????
10=0
���,??????
1−1 = ??????
11−2∆�
????????????
??????�
=0

14. Cálculo Numérico II-IF392
The simultaneous equations for the temperature
distribution on the plate with an insulated bottom end
are written in matrix form as:
75+??????
20+2??????
11−4??????
10=0
Analogously we have:
????????????�??????
20: ??????
10+??????
30+??????
21+??????
2−1−4??????
20=0
??????
10+??????
30+2??????
21−4??????
20=0
????????????�??????
30: ??????
20+??????
31+??????
31+??????
3−1−4??????
30=0
??????
20+50+2??????
31−4??????
30=0

15. Cálculo Numérico II-IF392
Solution
The Rest of the Equations

16. Note that, due to the derivatives in the boundary conditions, the
matrix increased in size to 12 × 12, unlike the 9 × 9 system of the
Dirichlet problem seen in the previous lecture when considering the
three unknown temperatures at the lower end of the plate. From these
equations, we obtain:
Cálculo Numérico II-IF392

17. Temperature and flow distribution in a hot plate subject to
fixed boundary conditions, except at an isolated lower end.
Cálculo Numérico II-IF392
Isolated

18. The heat flow through the surface is calculated from Fourier's law:
Cálculo Numérico II-IF392
where �
??????= heat flow in the direction of dimension ?????? [????????????�/(??????�
2
· �)],� = thermal
diffusivity coefficient (??????�
2
/�), ?????? = density of the material (??????/??????�
3
), � = heat
capacity of the material [????????????�/(?????? · °�)] and ?????? = temperature°�.
Heat flow direction Heat flow direction

19. Using centered finite difference approximations for the first
derivatives are substituted into the Fourier equation to obtain the
following values of the heat flux in the x and y dimensions:
Cálculo Numérico II-IF392
The resulting heat flux is calculated from these two quantities by
where the direction of �
?????? is given by
for �
�>0

20. Cálculo Numérico II-IF392
Example 02 (Mixed Conditions)
Solve the mixed problem described in Figure using the grid with h = 1

21. Cálculo Numérico II-IF392
Example 02 (Mixed Conditions)
�
01=0;�
12=2;�
10=0;�
22=8;�
20=0;�
32=18;�
30=0

22. •Many engineering problems exhibit irregular boundaries.
Cálculo Numérico II-IF392
Irregular Boundaries
•�
2 = �
2 = 1. We will retain these parameters in the following
derivation so that the resulting equation applies to any irregular
boundary (and not just to the lower left corner of a hot plate).

23. Cálculo Numérico II-IF392
As an example, consider the problem of solving the 2D Laplace’s equation ൫
൯
�
��+�
��=
0 in the region shown in Figure. The region has an irregular boundary in that the curved
portion intersects the grid at points A and B, neither of which is a mesh point. The points
M and Q are treated as before because each has four adjacent mesh points that are located
on the grid. But a point such as P must be treated differently since two of its adjacent
points (A and B) are not on the grid. The objective therefore is to derive expressions for
�
��(P) and �
��(P), at mesh point P, to form a new difference equation for Laplace’s
equation. Assume that A is located a distance of αh to the right of P, and B is a distance of
βh aboveP. Writethe Taylor’s series expansion for u at the four points A, B, M, and N
about the pointP. Forexample, ??????(??????) and ??????(??????) are expressed as
(1)
(2)
Multiply Equation 1 by α and add the result to
Equation 2, while neglecting the terms involving ??????
�

and higher:

24. Cálculo Numérico II-IF392
(3)
(4)
Addition of Equations 3 and 4 gives
(5)

25. Cálculo Numérico II-IF392
If Laplace’s equation is
solved, then the left side of
Equation 5 is set to zero,
and wehave
(6)

26. Cálculo Numérico II-IF392
(7)
The case of Poisson’s equation can be handled in a similar
manner. Equation 7 is applied at any mesh point that has at least
one adjacent point not located on the grid. In the Figure, for
example, that would be points P and N. For the points M and Q,
we simply apply equation seen before, or equivalently Equation
7 with α = 1 = β.
With the usual notations involved in the Figure, the difference
equation is obtained by rewriting Equation 6, as

27. Cálculo Numérico II-IF392
Example 03 (Irregular Boundary)
Solve �
�� + �
�� = 0 in the region shown in the Figure subject to the given
boundary conditions. The slanting segment of the boundary obeys �=−
2
3
�+2

28. Cálculo Numérico II-IF392
Example 03 (Irregular Boundary)
Solution
Based on the grid shown in Figure, Equation seen previous can be
applied at mesh points (1, 1), (2, 1), and (1, 2) because all four
neighboring points at those mesh points are on thegrid. Using the
boundary conditions provided, the resulting difference equationsare
(8)

29. Cálculo Numérico II-IF392
Example 03 (Irregular Boundary)
However, at (2, 2), we need to use Equation 7. Using the equation of
the slanting segment, we find that the point at the (2, 3) location is a
vertical distance of Τ
1
3 from the (2,2) mesh point. But since ??????=
�
�
, we
have �=
�
�
. On the other hand, � = � since the neighboring point to
the right of (2, 2) is itself a mesh point. With these, and knowing ??????
��=??????
and ??????
��=� by the given boundary condition, Equation 7 yields
1
(1)(2)
�
32+
1
(
2
3
)(
5
3
)
�
23+
1
2
�
12+
1
(5/3)
�
21−
5
3
2
3
�
22=0
��.�+??????
��+�.�??????
��−�??????
��=�

30. Cálculo Numérico II-IF392
Example 03 (Irregular Boundary)
Combining this with Equations in 8, and simplifying, we find