A project for Case Studies in Advanced Computation, a course at the Australian National University. Compututational power was provided by the APAC National Facility.
The flow of heat through some conducting material is described by a parabolic partial differential equation. If the heat sources and sinks are constant then the temperature distribution will converge to a steady state. This is described more simply by an elliptic PDE.
It is possible to solve PDEs such as this numerically using finite differencing. This involves discretising onto a grid and approximating derivatives. In this case the problem is not time-dependent so we don't need to worry about accurately representing the convergence process -- we can use any scheme that converges to the steady state, as rapidly as possible.
There are several methods for solving the matrix problems arising from finite differencing of elliptic PDEs. The scheme used here was Successive Over-Relaxation (SOR) -- see Numerical Recipes chapter 19, for example.
This study was done in two parts. In the first part, methods were developed on a case for which analytic solutions were available. In the second part, these were applied to evaluating designs for a CPU heatsink.
Part 1: Analytic CasePart 2: Heatsink Design |
The SOR method is significantly more efficient than obvious schemes such as Gauss-Seidel. However, it still takes a long time to reach steady state in many problems -- many thousands of iterations. An attractive way forward would be to use Multigrid.
The heatsink problem gave mostly sensible results, although lower temperatures were found than realistically expected. It would need to better represent the air-aluminium interface and air flow to find an optimal design.