Visualizing computational fluid dynamics on a car

In this example we visualize a mesh drawn from the CarCFDDataset.

Import dependencies

We first import our neuralop library and required dependencies.

import numpy as np
import torch
import matplotlib
import matplotlib.pyplot as plt
from neuralop.data.datasets import load_mini_car

font = {'size' : 12}
matplotlib.rc('font', **font)

torch.manual_seed(0)
np.random.seed(0)

Understanding the data

The data in a MeshDataModule is structured as a dictionary of tensors and important scalar values encoding a 3-d triangle mesh over the surface of a car. Each sample includes the coordinates of all triangle vertices and the centroids of each triangle face.

In this case, the creators used OpenFOAM to simulate the surface air pressure on car geometries in a wind tunnel. The 3-d Navier-Stokes equations were simulated for a variety of inlet velocities over each surface using the OpenFOAM computational solver to predict pressure at every vertex on the mesh. Each sample here also has an inlet velocity scalar and a pressure field that maps 1-to-1 with the vertices on the mesh. The full CarCFDDataset is stored in triangle mesh files for downstream processing. For the sake of simplicity, we’ve packaged a few examples of the data after processing in tensor form to visualize here:

data_list = load_mini_car()
sample = data_list[0]
print(f'{sample.keys()=}')

sample.keys()=dict_keys(['vertices', 'vertex_normals', 'triangle_normals', 'centroids', 'triangle_areas', 'distance', 'closest_points', 'normalized_triangle_areas', 'press', 'query_points'])

Visualizing the car

Let’s take a look at the vertices and pressure values.

fig = plt.figure()
ax = fig.add_subplot(projection='3d')
# By default the data is normalized into the unit cube. To get a
# better look at it, we scale the z-direction up.
scatter = ax.scatter(sample['vertices'][:,0],sample['vertices'][:,1],
                     sample['vertices'][:,2]*2, s=2, c=sample['press'])
ax.set_xlim(0,2)
ax.set_ylim(0,2)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.view_init(elev=20, azim=150, roll=0, vertical_axis='y')
ax.set_title("Pressure over car mesh vertices")
fig.colorbar(scatter, pad=0.2, label="normalized pressure", ax=ax)
plt.draw()

Query points

Each sample in the CarCFDDataset also includes a set of latent query points on which we learn a function to enable learning with an FNO in the middle of our geometry-informed models. Let’s visualize the queries on top of the car from before:

fig = plt.figure(figsize=(8,10))
ax = fig.add_subplot(projection='3d')
scatter = ax.scatter(sample['vertices'][:,0],sample['vertices'][:,1],
                     sample['vertices'][:,2]*2, s=4, label='Car surface')
queries = sample['query_points'].view(-1, 3) # unroll our cube tensor into a point cloud
ax.scatter(queries[:,0],queries[:,1],queries[:,2]*2,s=1, alpha=0.5, label='Latent queries')

ax.set_xlim(0,2)
ax.set_ylim(0,2)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.legend()
ax.view_init(elev=20, azim=150, roll=0, vertical_axis='y')
ax.set_title("Query points and vertices")

Text(0.5, 1.0, 'Query points and vertices')

Neighbor search between 3D point clouds In Visualizing neighbor search we demonstrate our neighbor search on a simple 2-d point cloud. Let’s try that again with our points here:

from neuralop.layers.neighbor_search import native_neighbor_search
verts = sample['vertices']
#query_point = queries[1000]
query_point = queries[3300] # 1550 and 0.4 is really good
#nbr_data = native_neighbor_search(data=verts, queries=query_point.unsqueeze(0), radius=0.15)
nbr_data = native_neighbor_search(data=verts, queries=query_point.unsqueeze(0), radius=0.5)

Let’s plot the new neighbors we just found on top of the car surface point cloud.

fig = plt.figure(figsize=(8,10))
ax1 = fig.add_subplot(2,1,1, projection='3d')
ax2 = fig.add_subplot(2,1,2, projection='3d')
neighbors = verts[nbr_data['neighbors_index']]

# Plotting just one query point vs. the car
ax1.scatter(verts[:, 0], verts[:, 1], verts[:, 2]*2, s=1, alpha=0.1)
ax1.scatter(query_point[0], query_point[1], query_point[2]*2, s=10, c='red', label='Single query')
ax1.view_init(elev=20, azim=-20, roll=0, vertical_axis='y')
ax1.legend()
ax1.set_xlim(0,2)
ax1.set_ylim(0,2)
ax1.set_xlabel("x")
ax1.set_ylabel("y")
ax1.set_zlabel("z")
ax1.view_init(elev=20, azim=-20, roll=0, vertical_axis='y')
ax1.grid(False)
ax1.set_title("One query point")

# Plotting all query points and neighbors
ax2.scatter(verts[:, 0], verts[:, 1], verts[:, 2]*2, s=0.5, alpha=0.4, label="Car surface")
ax2.scatter(queries[:, 0], queries[:, 1], queries[:, 2]*2, s=0.5, alpha=0.2, label="Latent queries")
ax2.scatter(neighbors[:, 0], neighbors[:, 1], neighbors[:, 2]*2, s=10, label="Neighbors on\ncar surface",)
ax2.legend()
ax2.set_xlim(0,2)
ax2.set_ylim(0,2)
ax2.set_xlabel("x")
ax2.set_ylabel("y")
ax2.set_zlabel("z")
ax2.view_init(elev=20, azim=-20, roll=0, vertical_axis='y')
ax2.set_title("Neighbor points from car for one query point")
ax2.grid(False)

for ax in ax1,ax2:
    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_zticks([])
plt.draw()

One query point, Neighbor points from car for one query point

Connecting neighbors to query

First, let’s make a simple utiltiy to add arrows to our 3D plot:

import numpy as np
from matplotlib import pyplot as plt
from matplotlib.patches import FancyArrowPatch
from mpl_toolkits.mplot3d import proj3d

class Arrow3D(FancyArrowPatch):
    def __init__(self, xs, ys, zs, *args, **kwargs):
        super().__init__((0,0), (0,0), *args, **kwargs)
        self._verts3d = xs, ys, zs

    def do_3d_projection(self, renderer=None):
        xs3d, ys3d, zs3d = self._verts3d
        xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, self.axes.M)
        self.set_positions((xs[0],ys[0]),(xs[1],ys[1]))

        return np.min(zs)

# Creating plots
fig = plt.figure(figsize=(8,10))
ax1 = fig.add_subplot(projection='3d')
neighbors = verts[nbr_data['neighbors_index']]

# Plotting just one query point vs. the car
ax1.scatter(verts[:, 0], verts[:, 1], verts[:, 2]*2, s=1, alpha=0.1)
ax1.scatter(query_point[0], query_point[1], query_point[2]*2, s=10, c='red', label='Single query')
ax1.scatter(neighbors[:, 0], neighbors[:, 1], neighbors[:, 2]*2, s=10, label="Neighbors on\ncar surface",)

ax1.view_init(elev=20, azim=-20, roll=0, vertical_axis='y')
ax1.legend()
ax1.set_xlim(0,2)
ax1.set_ylim(0,2)
ax1.set_xlabel("x")
ax1.set_ylabel("y")
ax1.set_zlabel("z")
ax1.view_init(elev=20, azim=-20, roll=0, vertical_axis='y')
ax1.grid(False)
ax1.set_title("One query point")


for ax in [ax1]:
    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_zticks([])

# Add arrows between neighbors and query
arrow_prop_dict = dict(mutation_scale=1, arrowstyle='-|>', color='red', shrinkA=1, shrinkB=1, alpha=0.1)
for nbr in neighbors:
    a = Arrow3D([query_point[0], nbr[0]],
                [query_point[1], nbr[1]],
                [query_point[2]*2, nbr[2]*2], **arrow_prop_dict)
    ax1.add_artist(a)

fig.tight_layout()
plt.draw()

Total running time of the script: (0 minutes 4.312 seconds)

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