The Over-Relaxation

In general there is no explicit need for addition input during the calculation. The user can control the solver by choosing various parameters of the computation. Such most important control elements shall be described in short below.

Generating a temperature distribution is connected with the numerical solution of a very large linear system of equations. Such an equation system is solved through the relaxation method [2] . The relaxation factor ω must be between 1 and 2 .

Relaxation factor ω

The value of the optimal relaxation factor ω_opt , which leads to the most expedient convergence of the calculation method, is highly dependant on the size and conditioning of the equation system to be solved. This value differs from case to case and is initially unknown. During the initial step of calculation an analysis of the equations system is executed and an estimation of ω₀ for the optimal relaxation factor is calculated iteratively.

Due to the fact, that the optimal relaxation factor ω_opt, which ensures the fasted method convergence, can only be roughly estimated, the value calculated serves as an initial parameter for the relaxation factor ω which is then modified with each iteration in the course of the second stage of calculation. In particular, the relaxation factor grows from ω_min towards (asymptotically) the value of ω_max. If certain conditions are met it is reset to ω_min and the process continues.

General parameters responsible for the automated variation of ω are set to default values. Under normal condition (moderate number of cells calculated, absence of extremely high or low conductance values) the calculation will appropriately execute without any user intervention.

If the convergence might be slow in exceptional case the user can speed up the convergence by changing the default values. The major indication is provided if the number of cell is very high and the optimal value of ω is just below 2. Detailed description of ω–control parameters can be found in the separate section of this manual.

Solving the system of equations

At each iteration step the method determines the deviation of the value calculated for each network element (cell) compared to the previous iteration. The maximum of all absolute values of these differences is used to control the convergence and progress of the calculation. If that value continually is below some defined threshold ε during several successive iteration steps, the termination condition is considered fulfilled and the calculation finished.

During the evaluation the user shall check if all precision criteria are satisfied and, if required, he can reduce the termination condition parameter ε and enforce the calculation to continue.

The computation can be interrupted at any time (e.g. if the computing device is needed for some other purpose) to be resumed later. The calculation will resume starting from some intermediate results saved at the time of interruption.

Complete results of the calculation are written to several files, which are then required during the evaluation. There are no special output results shown are available for printout during the calculation stage.

See also: Theoretical background, Solver-Parameter, Application settings