This Easter weekend I finally found a few hours and decided to toy around with an idea I’ve had for ages: trying to create a physically-informed intepolation method for river bathymetries.
As you know survey campaigns are usually required to create bathymetric Digital Terrain Models (DTM). But very often your modeling domain is not fully surveyed, so you need to interpolate the available data somehow. Using general-purpose methods such as Inverse Distance Weighting (IDW), or Triangular Irregular Networks (TIN) can work, but usually, they take time and generate unphysical artifacts.
Instead of using these methods, in this experiment, I initially solve the flow field with a flat bathymetry and then use the resulting velocity field as an input to a custom Partial Equation Derivative (PDE) solver written in OpenFOAM. This solver uses the known data as constraints and computes bottom levels for the rest of the domain. The procedure can then be applied iteratively a few times, using the new bathymetries to update the flow field.
The trick here is to construct a sensible PDE that does the interpolation. For this experiment I used a very simple one, but with an important twist. Can you guess which? Would love to hear your guess in the comments!
I have several ideas to further improve this PDE, but those will have to wait for now.