Lab #4 Non-Linear Temperature Distribution

with Newton Convergence

 

Problem Statement:  Using the jacobian and residual developed for problem P-07a, solve for the temperature distribution using a 32 element grid and a thermal conductivity of k(t) = 73.2 - 0.04T - 0.000003T².  Verify and report quadratic convergence for the iterative solver. Then remove one term from the jacobian, resolve, and comment on solution convergence/divergence.

 

Solution:  With help from page 6 of the Non-Linear Heat Conduction Handout, MatLab Code was generated to solve for the temperature distribution within the slab and also for the convergence of the iterative solution.  The solution was iterated until the maximum change in the temperature was with in 1e-6.  A plot of the 32 element Temperature Distribution can be seen here.

 

A graph of dQ Vs. # of Iterations was plotted to show the quadratic convergence of  Newton’s Iteration method. A second plot of log₁₀(dQ^p+1) Vs. log₁₀(dQ^p) was done to further verify quadratic convergence by having a slope of 2.  From the plot you can see that the slope is very close to 2, which shows the quadratic convergence of the Newton Iteration Method.

 

For the second part of the lab I removed the first b term in the Jacobian

asjac1d(b,[],Q ,-1,A3110L,[]) and ran the code again.  I found that it took 33 iterations compared to 6 iterations with all the jacobian terms in the code.  Convergence Graph with the first b term taken out of the Jacobian.  log₁₀(dQ^p+1) Vs. log₁₀(dQ^p) with the first term taken out of the Jacobian.

 

Conclusion: The graphs show that the Newton Iteration method converges quadratically.  When you take terms out of the jacobian the solution takes longer to converge, as was demonastrated when the first b term was taken out of the matlab code.