讨论双层圆筒一维非稳态热传导解析解

udvd

本人最近解除了解析解，但是编程后计算结果和终态温度基本重合，
请高手指点，结果在附件

iomega

I think your 特征值 beta_m expression has problem...

tonnyw

If your goal is to check if your program is correct, I wouldn't suggest doing this way. The exact solution is involved Bessel function and eigen-value problems. It is too complicated.

Maybe you can do this way:

At the material interface, the continuity condition is: Heat flux component along r direction is continuous Lambda1*dT1/dr (t, r0)= Lambda2*dT2/dr(t, r0) at r = r0.
temperature is continuous at r=r0: T1(t, r0) = T2(t, r0)

We can assume that the functions T1 and T2 have the same time dependence. For convenience, let T1(t, r) = t*F1(r), T2(t,r)=t*F2(r)
From the continuity condition we have: Lambda1*dF1/dr(r0) = Lambda2*dF2/dr(r0)
                                                        F1(r0) = F2(r0).

We can further assume: F1(r) = C1*r + C2, F2(r) = r + 1, then we have
Lambda1*C1 = Lambda2
C1*r0  + C2 = r+1.

You can see we will have the explicit forms of F1(r) and F2(r)

Plug T1(t,r), T2(t,r) into the governing equation, you will have nonzero source term. At the other two boundary, just apply the heat flux boudary condition. At one end it should be Lambda1*dT1/dr and the other end Lambda2*dT2/dr

If you use linear element in space and backward or forward difference in time, your finite element solution should be exact, which means the error is zero. Hopefully I don't confuse you.

iomega

One simple way to check if your solution is correct or not is looking at the limiting case when time-> infinite, your transient solution should be the same as steady-state solution, which has closed-form format in this case.