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Math · Advanced Math
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50 70 100 180 100 70 50 20 20 20 10mm 20 40 60 90 140 90 60 40 Figure 1: A rectangular plate with temperature defined on the boundaries 4 Figure 1 shows a rectangular plate made of an homogeneous isotropic material The temperature distribution in this plate satisfies the indicated boundary condi- tions (given in degrees centigrade) and has reached a steady-state condition so that the temperature is described by Laplaces equation a2T a2T (i) Draw a sketch of the solution domain showing clearly the line of symme- try for the temperature distribution and indicating which of the unknown temperatures are equal to each other (ii) Use the finite difference formulae on the formulae sheet to formulate the finite difference equations required to find estimates of the nodal temper- atures, TA. TB. Tc and TD

4(continued) (ii) Express these finite difference equations in the form AT B where A is a 4x4 matrix, T (TA. TB. Tc. To) and B -110,-130,-190,-320)T is a 4 x 1 column vector. Find matrix A, hence, given that -0.27-0.07-0.02-0.01 -1~ -0.07-0.29-0.08-0.02 0.020.08-0.31-0.08 -0.01-0.04-0.15-0.29 estimate TA, TB, Tc and TD correct to one degree.

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