Part 8 Chapter 29 1 Copyright The Mc

Part 8 Partial Differential Equations Table PT 8. 1 2 Copyright © The Mc.

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Finite Difference: Elliptic Equations Chapter 29 Solution Technique • Elliptic equations in engineering are

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The Laplacian Difference Equations/ Laplace Equation O[D(x)2] O[D(y)2] Laplacian difference equation. Holds for all

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• In addition, boundary conditions along the edges must be specified to obtain

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

The Liebmann Method/ • Most numerical solutions of Laplace equation involve systems that are

Boundary Conditions • We will address problems that involve boundaries at which the derivative

Figure 29. 7 13 Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for

• Thus, the derivative has been incorporated into the balance. • Similar relationships

Irregular Boundaries • Many engineering problems exhibit irregular boundaries. Figure 29. 9 15 Copyright

• First derivatives in the x direction can be approximated as: 16 Copyright

• A similar equation can be developed in the y direction. Figure 29.

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• The control-volume approach resembles the point-wise approach in that points are determined

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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. 4 Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

The Laplacian Difference Equations/ Laplace Equation O[D(x)2] O[D(y)2] Laplacian difference equation. Holds for all interior points 7 Copyright © The Mc. Graw-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: • Similar equations can be developed for other interior points to result a set of simultaneous equations. 9 Copyright © The Mc. Graw-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 Liebmann’s method. 11 Copyright © The Mc. Graw-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 Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

• 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 Mc. Graw-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 Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.