The TAR's stacking volumetric design is the one more problem from the same area - How to model and design spatial heterogeneous structure which should posses few interconnected features, among them is one as like maximum permeability and/or maximum heat transfer, etc.?
Another application is regarding the study for formulation the problem of acoustical liners on the wall with the goal to achieve most of the attenuation possible with some assumptions - as, for example, having the thickness of a layer less or equal to assigned value, etc., then some results were achieved as following.
Among other issues the one particular was found that in the mathematical VAT acoustics governing equations there are the four additional (in the simplest forms of governing equations) terms and this fact and the closure analysis recognizes the great influence of those terms as well as of the phenomena they represent. Those are:
Also, the one more important note is that the transformation to the harmonic wave representation should be done only after the Heterogeneous Acoustics (VAT) equations obtained.
This is the technicality, which can have a great impact.
Note also, that these four phenomena above are completely missed from the homogeneous acoustics mathematical statements? They do not exist explicitly in the homogeneous acoustics?.
The complete closure of specific heterogeneous acoustics equations for cases of layer - damping medium - 2D grating layer - of straight vertical rib fins had been achieved recently.
The first observations gave the kind of understanding and knowledge why the damping can happen and how this simple damping design can be improved. For example, what is the difference in exact mathematical terms between the regular grating above and this one
Of coarse those observations are in addition to known homogeneous acoustics problems analysis summaries.