Thermal noise inside a conducting cylinder

Compare numerical thermal magnetic noise inside a thin conducting cylinder with an analytical result.

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Static Scene

Interactive Scene

Static Scene

Interactive Scene

Static Scene

Interactive Scene

No component count given, computing all components.
Calculating surface harmonics expansion...
Computing the resistance matrix...
Computing the inductance matrix...
Computing self-inductance matrix using rough quadrature (degree=2).              For higher accuracy, set quad_degree to 4 or more.
Estimating 48300 MiB required for 3842 by 3842 vertices...
Computing inductance matrix in 80 chunks (13543 MiB memory free),              when approx_far=True using more chunks is faster...
Computing triangle-coupling matrix
Inductance matrix computation took 33.80 seconds.
Closed mesh or Neumann BC, leaving out the constant component
0 0
1 0
2 0
0 1
1 1
2 1
0 2
1 2
Computing magnetic field coupling matrix analytically, 3842 vertices by 30 target points... took 0.08 seconds.
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import pyvista as pv
from bfieldtools._example_plots import triangle_mesh

pv.set_plot_theme("document")

import numpy as np
import matplotlib.pyplot as plt
import trimesh
from bfieldtools.thermal_noise import (
    compute_current_modes,
    noise_covar,
    noise_var,
    visualize_current_modes,
)
from bfieldtools.mesh_magnetics import magnetic_field_coupling

from bfieldtools import resources


font = {"family": "normal", "weight": "normal", "size": 16}
plt.rc("font", **font)

# Fix the simulation parameters
d = 1e-3
sigma = 3.8e7
T = 293
kB = 1.38064852e-23
mu0 = 4 * np.pi * 1e-7
freqs = np.array((0,))


Nchunks = 8
quad_degree = 2


mesh = trimesh.load(
    resources.files("bfieldtools", "example_meshes/closed_cylinder.stl")
)
mesh.vertices, mesh.faces = trimesh.remesh.subdivide(mesh.vertices, mesh.faces)


vl = compute_current_modes(
    obj=mesh, T=T, resistivity=1 / sigma, thickness=d, mode="AC", return_eigenvals=False
)


vl[:, 0] = np.zeros(vl[:, 0].shape)  # fix DC-component

plotter = scene = pv.Plotter(window_size=(800, 800))

visualize_current_modes(mesh, vl[:, :, 0], 8, 1)


plotter = scene = pv.Plotter(window_size=(800, 800))
s = plotter.add_mesh(triangle_mesh(*mesh.vertices.T, mesh.faces), show_scalar_bar=False)
scene.view_xy(negative=True)
s.prop.style = "wireframe"
s.mapper.scalar_visibility = False
scene.view_isometric()
scene.render()


Np = 30

x = np.linspace(-0.95, 0.95, Np)
fp = np.array((x, np.zeros(x.shape), np.zeros(x.shape))).T

B_coupling = magnetic_field_coupling(mesh, fp, analytic=True)
B = noise_var(B_coupling, vl)

a = 0.5
L = 2
rat = L / (2 * a)
Gfact = (
    1
    / (8 * np.pi)
    * (
        (3 * rat**5 + 5 * rat**3 + 2) / (rat**2 * (1 + rat**2) ** 2)
        + 3 * np.arctan(rat)
    )
)
Ban = np.sqrt(Gfact) * mu0 * np.sqrt(kB * T * sigma * d) / a

plt.figure(figsize=(5, 5))
plt.plot(x, Ban * np.ones(x.shape) * 1e15, label="Analytic", linewidth=2)
plt.plot(
    x,
    np.sqrt(B[:, 0]) * 1e15,
    "x",
    label="Numerical",
    markersize=10,
    markeredgewidth=2,
)
plt.grid()
plt.gca().spines["right"].set_visible(False)
plt.gca().spines["top"].set_visible(False)
plt.legend(frameon=False)
plt.xlabel("z (m)")
plt.ylabel(r"$B_z$ noise at DC (fT/rHz)")
plt.tight_layout()

plt.figure()
plt.semilogy(x, np.sqrt(B[:, 0]), label="x")
plt.semilogy(x, np.sqrt(B[:, 1]), label="y")
plt.semilogy(x, np.sqrt(B[:, 2]), "--", label="z")
plt.legend()
plt.xlabel("z (m)")
plt.ylabel("DC noise (T/rHz)")

# Display the assembled scene (Sphinx-Gallery captures it during docs builds).
if "plotter" in globals() and not plotter._closed:
    plotter.show()