A 3 cm-square plate has its edge temperatures maintained at...

A $3 \mathrm{~cm}$-square plate has its edge temperatures maintained at $30^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}, 10^{\circ} \mathrm{C}$, and $0^{\circ} \mathrm{C}$, as shown. The faces of the plate are well insulated. Use Gauss-Seidel iteration on a I cm-square mesh to obtain the steady-state temperature distribution. Give the interior node temperatures after the first four iterations.

Key Concepts

Steady-State Heat Conduction

This concept refers to a state in which the temperature distribution in a medium does not change over time because the heat entering any region of the medium equals the heat leaving it. In mathematical problems like these, the governing equation is typically Laplace’s equation, representing the balance of thermal diffusion in the absence of internal heat sources.

Laplace Equation

The Laplace equation (?²T = 0) is a partial differential equation that describes steady-state phenomena, such as temperature distribution in a plate when the surfaces are held at fixed temperatures. It arises naturally from the conservation of energy in heat conduction and requires that the solution be harmonic, meaning that the value at any point is the average of its surrounding values.

Finite Difference Method

The finite difference method is a numerical technique used to approximate solutions to differential equations by discretizing a continuous domain into a grid or mesh. For heat conduction problems, this involves replacing the continuous Laplace equation with a set of algebraic equations at the grid nodes, where each node's temperature is represented in terms of its neighboring nodes.

Gauss-Seidel Iteration

Gauss-Seidel iteration is an iterative method for solving a system of linear equations derived from discretized PDEs like the Laplace equation. In each iteration, the algorithm updates the value at each grid node using the most recent values of its neighbors, thus gradually converging to the steady-state solution. It is especially useful for problems with large systems or where direct solution methods are impractical.

Boundary Conditions

Boundary conditions specify the temperatures at the edges of the plate, crucial for determining the overall solution to the Laplace equation. They are typically imposed as fixed (Dirichlet) conditions, and they guide the finite difference method by providing known values at the boundaries against which the interior node temperatures are iterated.

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