## BFS Fluids Prediction with Neural Concept Shape

** This use case demonstrates the application of Neural Concept Shape to the prediction of volumetric fields. **

In this project we want to predict the fields on the 3D volume around an object such as velocity and

pressure given its mesh, and thus getting a direct relation between the mesh and the

predicted fields.

Such a model has been built, leveraging geometric deep learning, i.e.

deep learning on irregular domains, on which standard convolutions would

fail.

The backward facing step is an important test-case for demonstrating results in

computational fluid dynamics as it has a simple geometry that hides a complex

phenomena: the re-circulation zone [see next paragraph]. Our model is able

to predict the internal flow around the backward-facing-step

as visually shown here below, with first prediction, then groundtruth and finally

the absolute point-wise error. We highlight the fact that the predictions were made

mesh-to-mesh, with as additional information only the boundary conditions and properties

of the flow (e.g. viscosity).

x-Velocity:

y-Velocity:

Pressure:

Once we got all x-velocity and y-velocity we can also predict the

re-circulation zone behind the step, as we illustrate below.

In the figure we show the simulation [left], and prediction [right].

As we see, the predicted solution (right) seems unfeasible with some

streamlines seemingly exiting the side of the step… And indeed, it is

infeasible! It does not satisfy the conservation of mass, as it has not been enforced

by the model. To correct this behavior, we can add a penalization term in the

loss of the network or iterate one step through the domain correcting the

conservation.

Of course, the same process can be repeated to other fields

that can be of interest for numerical simulations or other problems. Here below, we can see the turbulent viscosity

on the airfoil dataset:

Such a model can be used for automatic design of engineering shapes or for the initialization of an iterative solver with a corrected

prediction, hence drastically decreasing the number of iterations

necessary to reach convergence, and with it the computational cost.