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flow5 v7.13 introduces an option to generate once unit solutions to the panel problem and to reuse them to generate results across all type 1, 2, 3, 5, 7 and 8 analyses for a given plane and mesh configuration.
The goal of the option is to take advantage of the linear properties of the panel problem to reduce the computation times.
To illustrate the benefit, once a T1 or T2 analysis has been run
The option is activated by default and does not require any user action. It can be deactivated in the analysis settings.
This simplification has been an opportunity to introduce the more versatile type 8 analyses.
The solution to a panel analysis comes down to the resolution of a linear system with equal number of variables and equations. The different steps leading to the construction of the linear system are described for instance in the Technical Note NASA 4023.
The most computationally expensive tasks in the process are
On the other hand,
These observations form the basis of the solution method by linear combination of unit results.
It is referred to as the "influence" matrix since item aij contains the influence of a unit doublet density distributed on panel j at the location of panel i. It is a dense matrix, since each mesh panel has an influence on all the others. For comparison a finite element matrix is sparse, since a mesh element is only connected to its neighbours. This is the reason why finite element meshes use considerably less memory and may have larger sizes than boundary element meshes.
The influence matrix is constructed using only the information of the surface and wake meshes. As a consequence,
The right hand side vector (RHS) is constructed using the information of the surface and wake meshes and of the fluid's velocity field. The important point is that the information specific to each analysis ends up in the RHS of the linear system and not in the influence matrix.
In the typical case of a solid body movement such as the motion of a plane in a static atmosphere, each analysis defines a velocity field. This velocity field is usually specified by the angle of attack α, the sideslip angle β and the speed V∞. It can also include a rotational part which is typically of use in the estimation of the stability derivatives.
In all cases, this velocity field is a linear combination of the 6 unit velocity fields defined by elementary movements along the 6 degrees of freedom in motion, i.e.
Since type 1, 2, 3, 5, 7 and 8 analyses all use the same arrangement of surface and wake panels and therefore the same influence matrix, it makes sense to calculate once the unit elementary solutions and to store them for future use by these analyses. Once the unit solutions are available, the LU-factorized influence matrix is discarded.
Type 1, 2, 3, 5, 7 and 8 analyses are referred to as "linear analyses" herafter.
On the other hand in the case of type 6 analyses, the wake panels or particles depend on the velocity field. This implies that the influence matrix depends on the velocity field, and that the linear system needs to be built and solved for each operating point.
The generation of results by linear combinations is the standard procedure used by panel codes such as XFoil and AVL. It is also the method used in flow5 to calculate all the operating points in an analysis run.
The main change in v7.13 is that the unit solutions are kept in live memory instead of being discarded at the end of the run. This saves steps (1) to (3) of the linear system resolution for subsequent calculations of polars and operating points.
Whenever a linear analysis is requested, the application first checks whether the elementary solutions exist, and re-uses them if they do. If not, the problem is solved for the unit solutions which are then stored for future re-use.
It has been checked that the analysis results are the same whether the option is activated or not.
This verification can be performed at any time by activating and deactivating the option.
The option is activated in the interface of the analysis settings. This interface is accessed in the top menu "Analysis/3d analysis settings".
The menu option "Unit RHS vectors" enables the control of the live memory used by the RHS vectors.