Thermal Bridge Heat Transfer & Vapour Diffusion Simulation Program AnTherm Version 6.115 - 10.137

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Basics and Some Theory of AnTherm

Numerical Solution

The calculation method of finite differences used by AnTherm is based on the sub-division of the heat conducting continuum into an orthogonal cell structure. Each cell of this calculation geometry is treated as a node of the thermal network, and is typically connected with six other such nodes. The conductance of each connection depends on cell size and thermal conductivity of the material "filling" the individual cells. Therefore, the cell structure must be defined such that each cell can be assigned a single, homogeneous material.

calculation geometry The calculation geometry is obtained by means of gridding the construction geometry of the model to be evaluated. A concrete grid is generated by dividing the model with a series of grid planes along material boundaries and at additional intervals (their number and distance depends on the precision of analysis required) parallel to the coordinate axes.

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.

relaxation factor The convergence process of the relaxation method used by AnTherm is entirely dependant on the value of the relaxation factor ω. Initially this factor is defined only within a given range:

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.

relaxation factor variation The range of ω variation,

ωmin ≤ ω < ωmax

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

ωmin = ω0 - k • (ω0 - 1) • (ω0 -2),

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

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

The upper limit of the range of ω variation, ωmax, 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 = ωmin. The relaxation factor is then increased incrementally with each iteration until either ωmax is reached or the process begins to diverge, whereupon ω is set back to ωmin 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 ωmin. The standard number of assumed steps is relatively small number set as OMEGA_WEIGHT=-3.

As already mentioned, ω is set back to ωmin 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, Δmax, i.e. the deviation of the results from one iteration to the next. Hereby a mean value of Δmax is calculated from the last n steps and compared continually with the current Δmax. 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 Δmax 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, Δmax, 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.

results precision In summary, three primary factors control the precision of calculation results:

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.

 Model, Calculate, Simulate and Analyse Thermal Heat Bridges in 2D and 3D with AnTherm®  

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