Part 8 - Chapter 29

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Part 8 - Chapter 29

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Part 8Partial Differential Equations

Table PT8.1

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Figure PT8.4

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Finite Difference: Elliptic EquationsChapter 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/

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]

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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:

• 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:

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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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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.

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Irregular Boundaries

• Many engineering problems exhibit irregular boundaries.

Figure 29.9

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• 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

xxx

TT

x

TT

x

T

xxxT

xT

x

T

xx

T

x

TT

x

T

x

TT

x

T

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

Control-Volume ApproachFigure 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.

Part 8 - Chapter 29

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