GRAPH NEURAL NETWORKS FOR AERODYNAMIC FLOW RECONSTRUCTION FROM SPARSE SENSING Anonymous authors Paper under double-blind review

Abstract

Sensing the fluid flow around an arbitrary geometry entails extrapolating from the physical quantities perceived at its surface in order to reconstruct the features of the surrounding fluid. This is a challenging inverse problem, yet one that if solved could have a significant impact on many engineering applications. The exploitation of such an inverse logic has gained interest in recent years with the advent of widely available cheap but capable MEMS-based sensors. When combined with novel data-driven methods, these sensors may allow for flow reconstruction around immersed structures, benefiting applications such as unmanned airborne/underwater vehicle path planning or control and structural health monitoring of wind turbine blades. In this work, we train deep reversible Graph Neural Networks (GNNs) to perform flow sensing (flow reconstruction) around two-dimensional aerodynamic shapes: airfoils. Motivated by recent work, which has shown that GNNs can be powerful alternatives to mesh-based forward physics simulators, we implement a Message-Passing Neural Network to simultaneously reconstruct both the pressure and velocity fields surrounding simulated airfoils based on their surface pressure distributions, whilst additionally gathering useful farfield properties in the form of context vectors. We generate a unique dataset of Computational Fluid Dynamics simulations by simulating random, yet meaningful combinations of input boundary conditions and airfoil shapes. We show that despite the challenges associated with reconstructing the flow around arbitrary airfoil geometries in high Reynolds turbulent inflow conditions, our framework is able to generalize well to unseen cases.

1. INTRODUCTION

Many engineering applications stand to benefit from the ability to sense and reconstruct fluid flow features from sparse measurements originating at a structure's surface. Flow sensing could be crucial for improvements in the accuracy and resilience of wind turbine and unmanned aircraft controllers. Another possible application is monitoring of wind loaded structures (Barber et al., 2022) , where the use of cheap micro-electromechanical systems (MEMS) in combination with novel methods for flow sensing could lead to robust structural health monitoring solutions. In this work, we focus on common aerodynamic structures: we aim to reconstruct the flow around 2-D airfoils. Traditionally, computing the flow around an airfoil requires approaches from Computational Fluid Dynamics (CFD), which are forward-physics simulators. In CFD, the inflow, outflow and wall boundary conditions are set, and over many iterations a solution for the discretized Navier-Stokes PDEs is reached, which then yields a pressure distribution at the airfoil surface. We aim to solve the inverse problem: given only the pressure distribution at the airfoil surface, a solution for the flow field and farfield boundary conditions is to be found. Moreover, our aim is to do so for any airfoil geometry subject to a wide variety of turbulent inflows. Adopting the notation of Erichson et al. (2020) , the problem can be described in the following manner. An airfoil equipped with p distributed barometric sensors is placed in a steady flow of air, providing surface pressure measurements s ∈ R p at multiple locations around its perimeter. The sensors sample from the surrounding flow field x ∈ R m through a measurement operator H: s = H(x) The goal is to construct an estimate of the flow field x surrounding the airfoil, by learning from training data a function F that approximates the highly nonlinear inverse measurement operator G such that: F(s) = x ≈ x = G(s) (2) Meshes are an extremely useful tool, indispensable in many engineering domains and especially in CFD. Contrary to Cartesian grid representations, mesh representations offer high flexibility for irregular geometries and allow for variable spatial density. This makes them ideal for discretizing complex physical problems, where one can balance the trade-off between numerical accuracy and computational efficiency in certain regions of interest. Furthermore, meshes can also be described in terms of nodes and edges, i.e. as a graph. In this context, the flow reconstruction problem can be described as follows. A graph G = (V, E, U) is constructed from an airfoil CFD mesh, with m fluid nodes V f and p airfoil boundary nodes V a . The flow features of all V f are unknown, whilst the features of V a are known. Our aim is then to learn a graph operator F that estimates the information at the fluid nodes using the information contained at the airfoil boundary nodes, the input graph-level attributes U in and the edges E: Vf = F (V a , E, U in ) (3) An ancillary goal is to estimate the global context U of the graph, as this contains relevant information for applications. Figure 1 provides an description of the flow reconstruction problem in terms of graph learning. From a geometric learning perspective, flow reconstruction is a challenging problem for several reasons. The first significant hurdle to overcome is the size of our graphs. We use meshes with high densities close to the airfoil in order to achieve good spatial resolution in these critical regions. Thus, our dataset contains graphs with a mean of around 55'000 nodes, which is an order of magnitude higher than previous mesh-graph learned simulation methods (Pfaff et al., 2020) . Moreover, the input information is concentrated in a very localized domain of the graph: the airfoil nodes. It is difficult to propagate the necessary information to reconstruct nodes far from the airfoil with a shallow Graph Neural Network (GNN), meaning that deep GNN architectures with a large number of messagepassing steps are required to push this 'information barrier' away from the airfoil nodes. However, deep GNNs go hand in hand with other issues such as large memory requirements, over-smoothing, and over-squashing. In this work, we combine a number of existing graph-learning methods to tackle the aforementioned challenges. Our contributions may be summarized as follows: • We combine unknown Feature Propagation (Rossi et al., 2021) with very deep Grouped Reversible GNNs (Li et al., 2021) to reconstruct flow features at the fluid nodes, whilst additionally gathering contextual farfield information. • Generalization of 2D aerodynamic flow field learning with GNNs, including (1) arbitrary airfoil geometries, (2) arbitrary turbulent inflow conditions (flow velocity, turbulence intensity, and angle of attack), and (3) simultaneous flow field reconstruction and inference of contextual farfield flow information based only on sparse pressure data on the surface of the airfoil.



Figure 1: Problem setup. We aim to learn the reconstruction operator F which estimates properties of the fluid nodes as well as the graph context. This amounts to reconstructing a solution to the Navier-Stokes equations which satisfies the boundary conditions perceived at the airfoil surface.

