6eCh29-Part8

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1

Part 8 -

Chapter 29


2

Part 8

Partial Differential Equations

Table PT8.1


3

Figure PT8.4


4

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.


5

Figure 29.1


6

Figure 29.3


7

The Laplacian Difference Equations/

04

022

2

2

0

,1,1,,1,1

2

1,,1,

2

,1,,1

2

1,,1,

2

2

2

,1,,1

2

2

2

2

2

2

jijijijiji

jijijijijiji

jijiji

jijiji

TTTTT

yx

y

TTT

x

TTT

y

TTT

y

T

x

TTT

x

T

y

T

x

T

Laplacian difference

equation.

Holds for all interior points

Laplace Equation

O[(x)2]

O[(y)2]


8

Figure 29.4


9

• 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:

• Similar equations can be developed for other interior points

to result a set of simultaneous equations.

04

0

75

04

211211

10

01

1110120121

TTT

T

T

TTTTT


10

1504

1004

1754

504

04

754

504

04

754

332332

33231322

231312

33322231

2332221221

13221211

323121

22132111

122111

TTT

TTTT

TTT

TTTT

TTTTT

TTTT

TTT

TTTT

TTT

• The result is a set of nine simultaneous equations with nine

unknowns:


11

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.


12

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.


13

Figure 29.7


14

0422

2

2

04

,01,01,0,1

,1,1

,1,1

,01,01,0,1,1

jjjj

jj

jj

jjjjj

TTTx

TxT

x

TxTT

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.


15

Irregular

Boundaries

• Many

engineering

problems

exhibit

irregular

boundaries.

Figure 29.9


16

• First derivatives in the x direction can be

approximated as:

)()(

2

2

2

2

212

,,1

211

,,1

22

2

21

2

,,1

1

,1,

2

2

21

,11,

2

2

2

,,1

1,

1

,1,

,1

jijijiji

jijijiji

iiii

jiji

ii

jiji

ii

TTTT

xx

T

xx

x

TT

x

TT

x

T

xx

x

T

x

T

x

T

xx

T

x

TT

x

T

x

TT

x

T


17

• A similar equation can be developed in the y direction.

Control-Volume Approach Figure 29.12


18

Figure 29.13


19

• 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.

6eCh29-Part8

Documents

Transcript of 6eCh29-Part8