Part 8 Chapter 29
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Part 8
Partial Differential Equations
Table PT8.1
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Figure PT8.4
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Finite Difference: Elliptic Equations
Chapter 29
Solution Technique
• Elliptic equations in engineering are typically used to
characterize steady-state, boundary value problems.
• For numerical solution of elliptic PDEs, the PDE is
transformed into an algebraic difference equation.
• Because of its simplicity and general relevance to
most areas of engineering, we will use a heated plate
as an example for solving elliptic PDEs.
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Figure 29.1
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Figure 29.3
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The Laplacian Difference Equations/
 T
2
x
2
 T
2

2
x
2
 T

2
y
2
 T

y
0
2
Laplace Equation
T i  1, j  2 T i , j  T i 1 , j
x
2
T i , j  1  2 T i , j  T i , j 1
y
2
T i  1, j  2 T i , j  T i 1 , j
x
2
O[(x)2]

O[(y)2]
T i , j  1  2 T i , j  T i , j 1
y
2
0
x  y
T i  1 , j  T i 1 , j  T i , j  1  T i , j 1  4 T i , j  0
Laplacian difference
equation.
Holds for all interior points
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Figure 29.4
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• In addition, boundary conditions along the edges must be
specified to obtain a unique solution.
• The simplest case is where the temperature at the boundary is
set at a fixed value, Dirichlet boundary condition.
• A balance for node (1,1) is:
T 21  T 01  T12  T10  4 T11  0
T 01  75
T10  0
 4 T11  T12  T 21  0
• Similar equations can be developed for other interior points
to result a set of simultaneous equations.
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• The result is a set of nine simultaneous equations with nine
unknowns:
4 T11
 T 21
 T11
 4 T 21
 T13
 T 21
 4 T 31
 T12
 T11
 T 21
 75
 T 22
0
 T 32
 4 T12
 T 22
 T12
 4 T 22
 T 32
 T 22
 4 T 32
 T 31
 50
 T13
 T12
 T 22
 T 32
 75
 T 23
0
 T 33
 50
 4 T13
 T 23
 T13
 4 T 23
 T 33
 100
 T 23
 4 T 33
 150
 175
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The Liebmann Method/
• Most numerical solutions of Laplace equation
involve systems that are very large.
• For larger size grids, a significant number of
terms will b e zero.
• For such sparse systems, most commonly
employed approach is Gauss-Seidel, which
when applied to PDEs is also referred as
Liebmann’s method.
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Boundary Conditions
• 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.
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Figure 29.7
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T1 , j  T 1 , j  T 0 , j  1  T 0 , j 1  4 T 0 , j  0
T
x

T1 , j  T 1 , j
2x
T 1 , j  T1 , j  2  x
2 T1 , j  2  x
T
x
T
x
 T 0 , j  1  T 0 , j 1  4 T 0 , j  0
•Thus, the derivative has been incorporated into the
balance.
•Similar relationships can be developed for derivative
boundary conditions at the other edges.
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Figure 29.9
Irregular
Boundaries
• Many
engineering
problems
exhibit
irregular
boundaries.
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• First derivatives in the x direction can be
approximated as:
T i , j  T i 1, j
 T 



 1 x
  x  i 1, i
T i 1, j  Ti , j
 T 




x
 2x

 i ,i 1
 T 
 T 





2
 T
   T    x  i , i  1   x  i 1 , i



2
 1 x   2  x
x
x  x 
2
T i , j  T i 1, j
 T
2
x
2
2

Ti 1, j  Ti , j
 1 x
 2x
 1 x   2  x
2
 T
2
x
2

Ti 1, j  Ti , j 
2  T i 1, j  T i , j


2 
 x   1 ( 1   2 )  2 ( 1   2 ) 
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• A similar equation can be developed in the y direction.
Control-Volume Approach
Figure 29.12
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Figure 29.13
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• The control-volume approach resembles the
point-wise approach in that points are
determined across the domain.
• In this case, rather than approximating the
PDE at a point, the approximation is applied to
a volume surrounding the point.
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