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(7) [12] A common problem encountered in thermodynamics is that of solving for the
equilibrium temperature distribution of a thin plate of metal. One way of solving this type
of problem is to solve a continuous-time differential equation that can be queried as a
function of (x,y) for any continuous-valued position on the plate of metal. In general, this
solution could be exact given certain assumptions, but this solution is somewhat difficult to
compute. A simpler way to approximately solve the problem is to discretize the plate and
solve a system of linear equations (for example, see the related topic of “finite element
methods"). The solution will be found by solving this system of linear equations using the
matrix inverse. To do this, consider the discretized square plate in Figure 1.
S"
10°
Figure 1 -Discretized Square Plate
(A) Write a system of (four by four) linear equations for the node values Xı. X2, X3, and
X4 and clearly identify the matrix A in the matrix equation Ax-5. Do this by averaging
the values at each node X1, X3, Xz, and x, by taking the (thermal) average of its (four)
adjacent nodes (with the given boundary conditions). [That is, write an equation for each
internal node where the thermal value for that node is the average of the four adjacent
nodes.)
(B) Compute the inverse of A and solve the system for the four node values X1, X2, X3.
and X.
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(C) Refine the mesh by setting up and solving the same system using a 10x 10 internal
grid of nodes (thus there will be 100 internal nodes). Describe the resulting 100 by 100
matrix (Hint: The matrix A will be sparce, that is, mostly zeros. What are its diagonal
terms?) For extra credit, use software to generate a solution of this system
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