Example: Heat Flow on a Square Plate - II
Solve for the steady-state temperature distribution of a square plate using the partial differential equation solver relax.
Solving Poisson's Equation
Solve the heat equation where values of the source function are known and the boundary conditions are non-zero.
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The relax function is based on an entirely different solving method, and hence requires a different set of arguments.
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1. Define five square matrices a, b, c, d, and e to contain the coefficients for the Laplacian approximation:
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These arrays can be of any size you specify. The larger they are, the finer the mesh in the solution.
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2. Define the dimension of the square plate:
3. Define the coefficients:
4. Define the strength and position of a constant source.
5. Define a square matrix f, of size equal to the size of the grid, to contain the known boundary values of function F(x,y) and guess values for the unknown interior values.
◦ Boundary condition along the top:
◦ Boundary condition along the bottom:
◦ Boundary condition along the edges:
6. Define the Jacobi spectral radius variable r, a real number between 0 and 1.
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This parameter controls the convergence of the algorithm. If you see the error message "too many iterations", then try reducing r.
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7. Call the relax function:
8. Create a 3D plot to show the heat distribution over the square plate.
9. Create a contour plot to show the lines of constant temperature.
