Partial Differential Equations

Part 8 Chapter 29 1 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Part 8 Partial Differential Equations Table PT8.1 2 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Figure PT8.4 3

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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. 4 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Figure 29.1 5 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Figure 29.3 6 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The Laplacian Difference Equations/ 2T 2T 2 0 2

x y Laplace Equation 2T Ti 1, j 2Ti , j Ti 1, j O[(x)x)2] x 2 x 2 2T Ti , j 1 2Ti , j Ti , j 1 O[(x)y)2] 2 2 y y

Ti 1, j 2Ti , j Ti 1, j Ti , j 1 2Ti , j Ti , j 1 0 2 2 x y x y Ti 1, j Ti 1, j Ti , j 1 Ti , j 1 4Ti , j 0 Laplacian difference equation. Holds for all interior points 7

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Figure 29.4 8 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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: T21 T01 T12 T10 4T11 0 T01 75

T10 0 4T11 T12 T21 0 Similar equations can be developed for other interior points to result a set of simultaneous equations. 9 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The result is a set of nine simultaneous equations with nine unknowns: 4T11 T21 T11 4T21 T13 T21 T12 T22

4T31 T11 T21 T31 75 0 T32 4T12 T22 T13 T12 4T22 T32 T22 4T32

T12 T22 T32 T23 T33 50 75 0 50 4T13 T23 175 T13 4T23 T33 100

T23 4T33 150 10 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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 Liebmanns method. 11

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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. 12 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Figure 29.7

13 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. T1, j T 1, j T0, j 1 T0, j 1 4T0, j 0 T T1, j T 1, j x 2x T T 1, j T1, j 2x x T 2T1, j 2x T0, j 1 T0, j 1 4T0, j 0

x Thus, the derivative has been incorporated into the balance. Similar relationships can be developed for derivative boundary conditions at the other edges. 14 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Irregular Boundaries Many engineering problems exhibit

irregular boundaries. Figure 29.9 15 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. First derivatives in the x direction can be approximated as: Ti , j Ti 1, j T

1x x i 1,i Ti 1, j Ti , j T x 2 x i ,i 1 2T x 2

2T x 2 2T x 2 T T T x i ,i 1 x i 1,i 1x 2 x x x

2 Ti , j Ti 1, j Ti 1, j Ti , j 1x 2 x 2 1x 2 x 2 Ti 1, j Ti , j 2 Ti 1, j Ti , j 2 x 1 (1 2 ) 2 (1 2 ) Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

16 A similar equation can be developed in the y direction. Figure 29.12 Control-Volume Approach 17 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Figure 29.13 18

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 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. 19 Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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