FMM2D.jl

FMM2D.jl is a Julia interface for computing N-body interactions using the Flatiron Institute's FMM2D library.

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

$$u\left(x_t\right)=\sum _{s=1}^M{c}_s{H}_0^{(1)}(k|x_t-x_s|)-v_s{d}_s\cdot \mathrm{\nabla }{H}_0^{(1)}(k|x_t-x_s|),t=1,\dots ,N$$

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 $O((N+M)\text{log}({\epsilon }^{-1}))$ can be achieved with $\epsilon$ being the desired relative precision. Note that the library also includes the option for computing the gradient and Hessian of the potential.

Example

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

$$u(x)=\sum _{j=1}^N[c_j\text{log}(|x-x_j|)-d_j{v}_j\cdot \mathrm{\nabla }(\text{log}(|x-x_j|))],$$

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

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

$$u(x)=\sum _{j=1}^N{G}^{\text{stok}}(x,x_j)c_j+d_j\cdot T^{\text{stok}}(x,x_j)\cdot v_j$$

$$p(x)=\sum _{j=1}^N{P}^{\text{stok}}(x,x_j)c_j+d_j\cdot {\mathrm{\Pi }}^{\text{stok}}(x,x_j)\cdot v_j^{\mathrm{\top }}$$

Related Package

FMMLIB2D.jl interfaces the FMMLIB2D library which the FMM2D library improves on.