biem-helmholtz-sphere

Documentation: https://biem-helmholtz-sphere.readthedocs.io

Source Code: https://github.com/ultrasphere-dev/biem-helmholtz-sphere

Acoustic scattering from multiple n-spheres in NumPy / PyTorch

Installation

Install this via pip (or your favourite package manager):

pip install biem-helmholtz-sphere

Usage (GUI)

uvx biem-helmholtz-sphere serve

Usage

Boundary Integral Equation Method (BIEM) for the Helmholtz equation.

Let $d \in \mathbb{N} \setminus \lbrace 1 \rbrace$ be the dimension of the space, $k$ be the wave number, and $\mathbb{S}^{d-1} = \lbrace x \in \mathbb{R}^d \mid |x| = 1 \rbrace$ be a unit sphere in $\mathbb{R}^d$. Let $B := {0, ..., }$ be the index set of spheres, $c_b \in \mathbb{R}^d$ be the center of sphere $b \in B$, and $\rho_b > 0$ be the radius of sphere $b \in B$. Assume that the closure of spheres do not overlap, i.e.,

$$ \forall b, b' \in B, b \neq b', |c_b - c_b'| > \rho_b + \rho_b' $$

Asuume that $u_\text{in}$ is an incident wave satisfying the Helmholtz equation

$$ \Delta u_\text{in} + k^2 u_\text{in} = 0 $$

and scattered wave $u$ satisfies the following:

$$ \begin{cases} \Delta u + k^2 u = 0 \quad &x \in \mathbb{R}^d \setminus \overline{\mathbb{S}^{d-1}} \ \alpha u + \beta \nabla u \cdot n_x = -\alpha u_\text{in} -\beta \nabla u_\text{in} \cdot n_x \quad &x \in \mathbb{S}^{d-1} \ \lim_{|x| \to \infty} |x|^{\frac{d-1}{2}} \left( \frac{\partial u}{\partial |x|} - i k u \right) = 0 \quad &\frac{x}{|x|} \in \mathbb{S}^{d-1} \end{cases} $$

The following code assumes

$$ d = 3, k = 1, u_\text{in} (x) = e^{i k x_0}, c_0 = (0, 2, 0), c_1 = (0, -2, 0), \rho_0 = 1, \rho_1 = 1, \alpha = 1, \beta = 0 \quad \text{(Sound-soft)} $$

and computes the scattered wave at $(0, 0, 0)$.

>>> from array_api_compat import numpy as xp
>>> from biem_helmholtz_sphere import BIEMResultCalculator, biem, plane_wave
>>> from ultrasphere import create_from_branching_types
>>> c = create_from_branching_types("ba")
>>> uin, uin_grad = plane_wave(k=xp.asarray(1.0), direction=xp.asarray((1.0, 0.0, 0.0)))
>>> calc = biem(c, uin=uin, uin_grad=uin_grad, k=xp.asarray(1.0), n_end=6, eta=xp.asarray(1.0), centers=xp.asarray(((0.0, 2.0, 0.0), (0.0, -2.0, 0.0))), radii=xp.asarray((1.0, 1.0)), kind="outer")
>>> complex(xp.round(calc.uscat(xp.asarray((0.0, 0.0, 0.0))), 6))
(-0.741333-0.669657j)

Accuracy

• The radius of spheres is fixed to 1.0.
• The incident wave is $u_\text{in} (x) = e^{i k x_0}$.
• For n_balls == 2, spheres are centered at (0, 2, 0, ...) and (0, -2, 0, ...).
• For n_balls == 4, ..., spheres are placed in 2D grid pattern with distance of 4.0 between adjacent spheres.

vs Wavenumber k

2D

3D

vs Number of Spheres n_balls

2D

Citation

Consider citing the following paper if you use this package in your research:

• doi.org/10.60443/jascome.25.0_143

Contributors ✨

Thanks goes to these wonderful people (emoji key):

This project follows the all-contributors specification. Contributions of any kind welcome!

Credits

This package was created with Copier and the browniebroke/pypackage-template project template.