Thermal Bridge Heat Transfer and Vapour Diffusion Simulation Program AnTherm Version 6.106 

Once the network data has been sufficiently defined for calculation, the program sets up an appropriate linear system of equations automatically. This equation system then needs to be solved numerically with an iterative method.
 

1 ≤ ω < 2

The "optimal" ω-value is defined as the constant value of ω which leads to the most expedient convergence of the calculation method. This value differs from case to case and is initially unknown.

Once the system of equations describing the model has been determined, the first stage of calculation is therefore an analysis of this system, followed by an iterative calculation to approximate a preliminary "optimal" *-value, ω₀. This stage can be influenced by two parameters defined in the solver-parameters form:

•  The termination condition for determining ω₀ is satisfied when the absolute difference between last value of ω₀ just calculated and the "old" value from the previous iterative step falls below a limit for which ω₀ is considered solved. This limit is called OMEGAO_DELTA within the solver-parameter set.
•  A further termination condition is given by the maximum number of iterative steps which the program allows before stopping calculation and accepting the last value of ω₀ as optimal for the next stage of calculation. This step number limit is defined by OMEGAO_STOP.

A more precise approximation of ω₀ (e.g. by setting a more stringent termination condition or increasing calculation time) is usually unnecessary, since ω₀ just 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.
 

ω_min ≤ ω < ω_max

depends on ω₀ for the lower limit according to the following equation:

ω_min = ω₀ - k • (ω₀ - 1) • (ω₀ -2),

whereby k is a pre-defined parameter between -1 and 0.

Since k is negative (ω_min < ω₀), specifying a greater absolute value of k results in a larger difference between ω_min and ω₀, and thus a larger range. This parameter is set as OMEGA_MIN.

The upper limit of the range of ω variation, ω_max, is a set value between ω₀ and 2. This is defined as OMEGA_MAX.

In the second stage of calculation, the first step is performed with ω = ω_min. The relaxation factor is then increased incrementally with each iteration until either ω_max is reached or the process begins to diverge, whereupon ω is set back to ω_min and calculation continues.

The increments of ω variation also vary: low values of ω in the beginning are run through rapidly, but the increment approaches 0 asymptotically as ω approaches the value of 2. The increments are calculated such that the approximated optimal value of ω₀ is reached within a given number of iterations starting from ω_min. The standard number of assumed steps is relatively small number set as OMEGA_WEIGHT=-3.

As already mentioned, ω is set back to ω_min as soon as the iterative results begin to recognisably diverge. The criterium for discerning whether the solution process is converging or diverging is the absolute value, Δ_max, i.e. the deviation of the results from one iteration to the next. Hereby a mean value of Δ_max is calculated from the last n steps and compared continually with the current Δ_max. The number of steps included in the mean value for comparison, n, is also defined as a solver-parameter and called OMEGA_TESTNUM.

The calculation process is considered divergent when Δ_max equals or exceeds the comparative mean value. However, it would be impractical to set the relaxation factor back automatically every time this condition is satisfied before a certain minimum number of calculation steps has been performed. This quantity is a further criterium which must be met before an ω set-back occurs. The standard minimum number of iterations here is 23, defined as OMEGA_VETO.

An equation is ultimately considered solved when the deviation, Δ_max, remains smaller than a defined limit for a continuous series of a prescribed number of iterations. This quantity is defined as TERM_NUM.

Finally, in order to "smooth" the results of calculation, a post-run of iterations is performed with a constant relaxation factor, ω = 1 (defined as OMEGA_POSTRUN=1.0). An increase in this parameter should be avoided so as not to jeopardise the smoothing effect of post-calculation. The number of iterations of this stage is prescribed as POSTRUN=15.
 

•  fineness of the grid (calculation geometry).
•  stringency of the termination condition (calculation time).
•  precision of computation ("double precision").

The first two of these factors can be manipulated by the user, though the standard parameters used by AnTherm are adequate for the evaluation of most models to be considered.
 

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2012-01-01 19:53 +0100