# Scaling and nondimensionalizing parameterized geometry

Hi,
I followed the recommendations (Linear Elasticity) to non-dimensionalize my pdes using characteristic length and characteristic displacement. Nevertheless, the bracket example is for a non-parameterized geometry and the bounds are (-1,1) in all x,y and z directions. My geometry (I-section beam) has bounds ((0,8),(0,125),(0,10)) thus i set up characteristic length to be 125. If i parameterize my geometry along (width,length and height), how do then setup the characteristic length. Do I set to the maximum length of the length range parameter or dynamically change for every beam geometry?
The tutorial also recommends scaling the geometry. If i scale the parametrized geometry with the length dimension to be bounded in the range (0,1), do i need to correspondingly change the characteristic length now to 1 and thus the question remains how does characteristic displacement and other parameters change?
Hope I made my issue clear. Thanks

Scaling / Nondimensionalization is a critical part of learning physical systems and each case needs to be handled differently depending on the physics at play. Regarding your question about scaling the bounds of the geometry, your approach sounds correct. You should define you characteristic length such that the non-dimensionalized spatial coordinates fall into a good range.

However, this isn’t the end of the story. You need to make sure you adjust all other physical parameters for this new coordinate system (essentially anything that has a length unit will be effected. e.g. the force applied, stress tensor, etc.). There is some additional information on non-dimensionalizing the linear elastic equations here. We also have some support for using the Pint package which allows automatic scaling of variables. An example of this is the cylinder flow problem in the examples repo.

So for every parameterized geometry, its own characteristic length and displacement are computed to make sure the coordinates are in the good range. In the documentation, a good range is defined as [-1,1], I have bounded (only the length as the width and height are smaller than the length, their bounds are much smaller ) by geometry to [0,1] and set up the nondimensionalization. Would this work too? Thanks a lot.

Yes that should be fine. The key here is to scale the inputs for the machine learning model to improve convergence (similar to how you scale data in a traditional DL setting). What the absolute best scaling is a judgement call / empirical (again analogous to data-driven problems where you could min/max norm, guassian norm, etc.).