Reference no: EM132361406
Question - 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, sec the related topic of "finite clement 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.
(A) Write a system of (four by four) linear equations for the node values x1, x2, x3, and x4 and clearly identify the matrix A in the matrix equation Ax = b. Do this by averaging the values at each node x1, x2, x3, and x4 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 tour node values x1, x2, x3, and x4.
(C) Refine the mesh by setting up and solving the same system using a 10 x 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.