## A 2D Backward-Facing Step case with Neural Concept Shape

**Backward-Facing Step **flows are a very common benchmark problem to assess the reliability of a software for industrial applications. Neural Concept explored such an application: **the generalisation to volumetric fields predictions**. The desired outcome of the project can be expressed in the following way:

We desire to predict the fields 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. As a proof-of-concept we will present here only the results around a **backward-facing-step** and a 2D airfoil, but are easily generalised to complex 3D flows as our applications will show.

The backward facing step is an important milestone for showing result 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 emphasise that the prediction was 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) does not satisfy the conservation of mass, as it has not been enforced by the model. To correct this behaviour, we add a penalisation 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:

One target application here is to use this prediction to initialise an iterative solver with a corrected prediction, hence drastically decreasing the number of iterations necessary to reach convergence, and with it the computational cost.