https://github.com/mipals/fmm2d.jl
A Julia interface to the FMM2D library from the Flatiron Institute
https://github.com/mipals/fmm2d.jl
2d fast-multipole-method fmm fmm2d
Last synced: 3 months ago
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A Julia interface to the FMM2D library from the Flatiron Institute
- Host: GitHub
- URL: https://github.com/mipals/fmm2d.jl
- Owner: mipals
- License: mpl-2.0
- Created: 2022-09-27T05:47:33.000Z (over 2 years ago)
- Default Branch: main
- Last Pushed: 2024-02-22T14:19:41.000Z (over 1 year ago)
- Last Synced: 2025-02-23T06:45:09.628Z (3 months ago)
- Topics: 2d, fast-multipole-method, fmm, fmm2d
- Language: Julia
- Homepage:
- Size: 62.5 KB
- Stars: 4
- Watchers: 1
- Forks: 1
- Open Issues: 1
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# FMM2D.jl
[](https://github.com/mipals/FMM2D.jl/actions/workflows/CI.yml?query=branch%3Amain)
[](https://codecov.io/gh/mipals/FMM2D.jl)
FMM2D.jl is a Julia interface for computing N-body interactions using the [Flatiron Institute's FMM2D library](https://github.com/flatironinstitute/fmm2d/).
Currently, the wrapper only wraps the Helmholtz and Laplace functionalities.
## Helmholtz FMM
Let $c_s \in \mathbb{C},\ s = 1,\dots,N$ denote a collection of charge strengths, $v_s \in \mathbb{C},\ s = 1,\dots,N$ denote a collection of dipole strengths, and $d_s\in\mathbb{R}^2,\ s = 1,\dots,N$ denote the corresponding dipole orientation vectors. Furthermore, $k \in \mathbb{C}$ denotes the wave number. The Helmholtz potential $u$ caused by the presence of a collection of $M$ sources ($x_s$) at $N$ target positions ($x_t$) is computed as
$$
u\left(x_t\right) = \sum_{s=1}^{M} c_sH_0^{(1)}(k\|x_t - x_s\|) - v_sd_s\cdot\nabla H_0^{(1)}(k\|x_t - x_s\|), \quad t = 1,\dots, N
$$
where $H_0^{(1)}$ is the Hankel function of the first kind of order 0. When $x = x_j$ the $j$ th term is dropped from the sum. Performing this summation would scale as $O(NM)$, but using the Flatiron Insitutes Fast Multipole Library a linear scaling of $O((N + M)\text{log}(\varepsilon^{-1}))$ can be achieved with $\varepsilon$ being the desired relative precision. Note that the library also includes the option for computing the gradient and Hessian of the potential.
### Example
```julia
using FMM2D
# Simple example for the FMM2D Library
thresh = 10.0^(-5) # Tolerance
zk = rand(ComplexF64) # Wavenumber
# Source-to-source
n = 200
sources = rand(2,n)
charges = rand(ComplexF64,n)
pg = 3 # Evaluate potential, gradient, and Hessian at the sources
vals = hfmm2d(eps=thresh,zk=zk,sources=sources,charges=charges,pg=pg)
vals.pot
vals.grad
vals.hess
# Source-to-target
m = 200
targets = rand(2,m)
pgt = 3 # Evaluate potential, gradient, and Hessian at the targets
vals = hfmm2d(targets=targets,eps=thresh,zk=zk,sources=sources,charges=charges,pgt=pgt)
vals.pottarg
vals.gradtarg
vals.hesstarg
```
## Laplace
The Laplace problem in 2D have the following form
$$
u(x) = \sum_{j=1}^{N} \left[c_{j} \text{log}\left(\|x-x_{j}\|\right) - d_{j}v_{j} \cdot \nabla( \text{log}(\|x-x_{j}\|) )\right],
$$
In the case of complex charges and dipole strengths ($c_j, v_j \in \mathbb{C}^n$) the function call `lfmm2d` has to be used. In the case of real charges and dipole strengths ($c_j, v_j \in \mathbb{R}^n$) the function call `rfmm2d` has to be used.
### Example
```julia
using FMM2D
# Simple example for the FMM2D Library
thresh = 10.0^(-5) # Tolerance
# Source-to-source
n = 200
sources = rand(2,n)
charges = rand(ComplexF64,n)
dipvecs = randn(2,n)
dipstr = rand(ComplexF64,n)
pg = 3 # Evaluate potential, gradient, and Hessian at the sources
vals = lfmm2d(eps=thresh,sources=sources,charges=charges,dipvecs=dipvecs,dipstr=dipstr,pg=pg)
vals.pot
vals.grad
vals.hess
# Source-to-target
m = 100
targets = rand(2,m)
pgt = 3 # Evaluate potential, gradient, and Hessian at the targets
vals = lfmm2d(targets=targets,eps=thresh,sources=sources,charges=charges,dipvecs=dipvecs,dipstr=dipstr,pgt=pgt)
vals.pottarg
vals.gradtarg
vals.hesstarg
```
## Stokes
$$
u(x) = \sum_{j=1}^NG^\text{stok}(x,x_j)c_j + d_j\cdot T^\text{stok}(x,x_j)\cdot v_j
$$
$$
p(x) = \sum_{j=1}^NP^\text{stok}(x,x_j)c_j + d_j\cdot \Pi^\text{stok}(x,x_j)\cdot v_j^\top
$$
## Related Package
[FMMLIB2D.jl](https://github.com/ludvigak/FMMLIB2D.jl) interfaces the [FMMLIB2D](https://github.com/zgimbutas/fmmlib2d) library which the [FMM2D library improves on](https://fmm2d.readthedocs.io/en/latest/).