Sensor optimization for urban wind estimation with cluster-based probabilistic framework

TL;DR

A physics-informed machine-learned framework for sensor-based flow estimation for drone trajectories in complex urban terrain that scales proportionally to the domain complexity, making it suitable for flows that are too complex for any monolithic reduced-order representation.

Abstract

We propose a physics-informed machine-learned framework for sensor-based flow estimation for drone trajectories in complex urban terrain. The input is a rich set of flow simulations at many wind conditions. The outputs are velocity and uncertainty estimates for a target domain and subsequent sensor optimization for minimal uncertainty. The framework has three innovations compared to traditional flow estimators. First, the algorithm scales proportionally to the domain complexity, making it suitable for flows that are too complex for any monolithic reduced-order representation. Second, the framework extrapolates beyond the training data, e.g., smaller and larger wind velocities. Last, and perhaps most importantly, the sensor location is a free input, significantly extending the vast majority of the literature. The key enablers are (1) a Reynolds number-based scaling of the flow variables, (2) a physics-based domain decomposition, (3) a cluster-based flow representation for each subdomain, (4) an information entropy correlating the subdomains, and (5) a multi-variate probability function relating sensor input and targeted velocity estimates. This framework is demonstrated using drone flight paths through a three-building cluster as a simple example. We anticipate adaptations and applications for estimating complete cities and incorporating weather input.

Figures (13)

• Figure 1: Methodology framework. In the offline stage, the simulation dataset of the building complex is generated. The dataset contains random wind speeds ranging from $7.9$ m/s to $20.7$ m/s (Beaufort wind levels 5 to 8), and incoming flow directions randomly distributed between $0^\circ$ and $360^\circ$. A physics-informed probabilistic model is built from the training data. First, the entire flow field is divided into several subdomains, and correlations between these subdomains are established by applying clustering within coarse-grained regions. Flow estimation is performed by inferring the correlations between the subdomains. The optimized sensor locations $\bm{x}^*_{s}$ are identified by selecting the most informative sensors. In the online stage, the velocity field along the drone trajectory is estimated by the sensor signal $\bm{s}^*$ from the optimized sensor locations $\bm{x}^*_s$.
• Figure 2: Sketch of the building complex. (a) The top view of the building complex in the $xy$-plane, and the side view in the $xz$-plane. The length used for normalization is $L=0.5$ m. The buildings within the complex are numbered from tallest to shortest as 1, 2, and 3. (b) The top view of the computational domains in the $XY$ plane, and the side view in the $xz$-plane. The entire computational domain consists of the inner domain and the outer domain. (c) Computational grid around the building complex.
• Figure 3: Results of the grid independence test conducted using 8 grid sets. The plot shows the mean streamwise velocity $\bar{U}$ across a $4L$ line parallel to the $z$-axis at location ($x=1.75L, y=0$).
• Figure 4: Sensor-based flow estimation exemplified for a single sensor. The training sensor signal $\bm{s}^m$ serves as input, from which the sensor’s cluster affiliation $\bm{c}_{1,i}(\bm{s})$ is inferred. This cluster affiliation $\bm{c}_{1,i}(\bm{s})$ is the input to the physics-informed probabilistic model, which then infers the cluster affiliation $\bm{c}_{2,j}(\bm{x}_d)$ of the subdomain containing the drone trajectory $\bm{x}_d$, ultimately estimating the velocity along the drone trajectory. The dashed box indicates the physics-informed probabilistic model, constructed by first decomposing the training flow field snapshots $\bm{u}^m$ into subdomains $\Omega_1, \Omega_2, \Omega_3$, then clustering each subdomain, and finally establishing the inference matrix $\mathrm{P}$ between them.
• Figure 5: Domain decomposition. (a) The oblique view of the three subdomains. For each snapshot, the entire flow field is decomposed into three subdomains $\Omega_1$, $\Omega_2$, and $\Omega_3$ around buildings 1, 2, and 3. From the tallest to the shortest, the heights of the subdomains are $5L$, $4L$ and $3L$, with $L=0.5$ m. (b) The top view of the three subdomains. Each subdomain has a square cross-section measuring $L \times L$.