julia

The julia interface has high-level subroutines for four interaction kernels:

  • Laplace wrappers: Fast multipole implementation (lfmm3d) and direct evaluation (l3ddir) for Laplace N-body interactions

  • Helmholtz wrappers: Fast multipole implementation (hfmm3d) and direct evaluation (h3ddir) for Helmholtz N-body interactions

  • Stokes wrappers: Fast multipole implementation (stfmm3d) and direct evaluation (st3ddir) for Stokes N-body interactions

  • Maxwell wrappers: Fast multipole implementation (emfmm3d) and direct evaluation (em3ddir) for Maxwell N-body interactions

Laplace wrappers

This subroutine computes the N-body Laplace interactions and its gradients in three dimensions where the interaction kernel is given by \(1/r\)

\[u(x) = \sum_{j=1}^{N} \frac{c_{j}}{\|x-x_{j}\|} - v_{j} \cdot \nabla \left( \frac{1}{\|x-x_{j}\|}\right)\]

where \(c_{j}\) are the charge densities \(v_{j}\) are the dipole orientation vectors, and \(x_{j}\) are the source locations. When \(x=x_{j}\), the term corresponding to \(x_{j}\) is dropped from the sum.

vals = lfmm3d(eps,sources;charges=nothing,dipvecs=nothing,
              targets=nothing,pg=0,pgt=0,nd=1)

Wrapper for fast multipole implementation for Laplace N-body interactions.

Args:

  • eps: double

    precision requested

  • sources: double(3,n)

    source locations, \(x_{j}\)

  • charges: double(n,) or double(nd,n)

    charge densities, \(c_{j}\)

  • dipvec: double(3,n) or double(nd,3,n)

    dipole orientation vectors, \(v_{j}\)

  • nd: integer

    number of charge/dipole vectors

  • pg: integer
    source eval flag
    potential at sources evaluated if pg = 1
    potenial and gradient at sources evaluated if pg=2
  • targets: double(3,nt)

    target locations (\(t_{i}\)) (optional)

  • pgt: integer
    target eval flag (optional)
    potential at targets evaluated if pgt = 1
    potenial and gradient at targets evaluated if pgt=2

Returns: The subroutine returns an object val of type FMMVals with the following variables

  • vals.pot: potential at source locations, if requested, \(u(x_{j})\)

  • vals.grad: gradient at source locations, if requested, \(\nabla u(x_{j})\)

  • vals.pottarg: potential at target locations, if requested, \(u(t_{i})\)

  • vals.gradtarg: gradient at target locations, if requested, \(\nabla u(t_{i})\)

  • vals.ier: error flag as returned by FMM3D library. A value of 0 indicates a successful call. Non-zero values may indicate insufficient memory available. See the documentation for the FMM3D library.

Wrapper for direct evaluation of Laplace N-body interactions. Note that this wrapper only returns potentials and gradients at the target locations.

vals = l3ddir(sources,targets;charges=nothing,
             dipvecs=nothing,pgt=0,nd=1,
             thresh=1e-16)

Example:

  • see lfmmexample.jl

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Helmholtz wrappers

This subroutine computes the N-body Helmholtz interactions and its gradients in three dimensions where the interaction kernel is given by \(e^{ikr}/r\)

\[u(x) = \sum_{j=1}^{N} c_{j} \frac{e^{ik\|x-x_{j}\|}}{\|x-x_{j}\|} - v_{j} \cdot \nabla \left( \frac{e^{ik\|x-x_{j}\|}}{\|x-x_{j}\|}\right)\]

where \(c_{j}\) are the charge densities \(v_{j}\) are the dipole orientation vectors, and \(x_{j}\) are the source locations. When \(x=x_{j}\), the term corresponding to \(x_{j}\) is dropped from the sum.

vals = hfmm3d(eps,zk,sources;charges=nothing,dipvecs=nothing,
              targets=nothing,pg=0,pgt=0,nd=1)

Wrapper for fast multipole implementation for Helmholtz N-body interactions.

Args:

  • eps: double

    precision requested

  • zk: complex

    Helmholtz parameter, k

  • sources: double(3,n)

    source locations, \(x_{j}\)

  • charges: complex(n,) or complex(nd,n)

    charge densities, \(c_{j}\)

  • dipvec: complex(3,n) or complex(nd,3,n)

    dipole orientation vectors, \(v_{j}\)

  • nd: integer

    number of charge/dipole vectors

  • pg: integer
    source eval flag
    potential at sources evaluated if pg = 1
    potenial and gradient at sources evaluated if pg=2
  • targets: double(3,nt)

    target locations, \(t_{i}\) (optional)

  • pgt: integer
    target eval flag (optional)
    potential at targets evaluated if pgt = 1
    potenial and gradient at targets evaluated if pgt=2

Returns: The subroutine returns an object vals of type FMMVals with the following variables

  • vals.pot: potential at source locations, if requested, \(u(x_{j})\)

  • vals.grad: gradient at source locations, if requested, \(\nabla u(x_{j})\)

  • vals.pottarg: potential at target locations, if requested, \(u(t_{i})\)

  • vals.gradtarg: gradient at target locations, if requested, \(\nabla u(t_{i})\)

Wrapper for direct evaluation of Helmholtz N-body interactions. Note that this wrapper only returns potentials and gradients at the target locations.

vals = h3ddir(zk,sources,targets;charges=nothing,
                dipvecs=nothing,pgt=0,nd=1,
                thresh=1e-16)

Example:

  • see hfmmexample.jl

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Stokes wrappers

Let \(\mathcal{G}^{\textrm{stok}}(x,y)\) denote the Stokeslet given by

\[\begin{split}\mathcal{G}^{\textrm{stok}}(x,y)=\frac{1}{2 \|x-y\|^3} \begin{bmatrix} (x_{1}-y_{1})^2 + \|x-y \|^2 & (x_{1}-y_{1})(x_{2}-y_{2}) & (x_{1}-y_{1})(x_{3}-y_{3}) \\ (x_{2}-y_{2})(x_{1}-y_{1}) & (x_{2}-y_{2})^2 + \|x-y \|^2 & (x_{2}-y_{2})(x_{3}-y_{3}) \\ (x_{3}-y_{3})(x_{1}-y_{1}) & (x_{3}-y_{3})(x_{2}-y_{2}) & (x_{3}-y_{3})^2 + \|x-y \|^2 \end{bmatrix} \, ,\end{split}\]

and \(\mathcal{T}^{\textrm{stok}}(x,y)\) denote the Stresslet whose action on a vector \(v\) is given by

\[\begin{split}v\cdot \mathcal{T}^{\textrm{stok}}(x,y) = \frac{3 v \cdot (x-y)}{\|x-y \|^5} \begin{bmatrix} (x_{1}-y_{1})^2 & (x_{1}-y_{1})(x_{2}-y_{2}) & (x_{1}-y_{1})(x_{3}-y_{3}) \\ (x_{2}-y_{2})(x_{1}-y_{1}) & (x_{2}-y_{2})^2 & (x_{2}-y_{2})(x_{3}-y_{3}) \\ (x_{3}-y_{3})(x_{1}-y_{1}) & (x_{3}-y_{3})(x_{2}-y_{2}) & (x_{3}-y_{3})^2 \end{bmatrix} \, .\end{split}\]

This subroutine computes the N-body Stokes interactions, its gradients and the corresponding pressure in three dimensions given by

\[u(x) = \sum_{m=1}^{N} \mathcal{G}^{\textrm{stok}}(x,x_{j}) \sigma_{j} + \nu_{j} \cdot \mathcal{T}^{\textrm{stok}}(x,x_{j}) \cdot \mu_{j}\]

where \(\sigma_{j}\) are the Stokeslet densities, \(\nu_{j}\) are the stresslet orientation vectors, \(\mu_{j}\) are the stresslet densities, and \(x_{j}\) are the source locations. When \(x=x_{j}\), the term corresponding to \(x_{j}\) is dropped from the sum.

vals = stfmm3d(eps,sources;stoklet=nothing,strslet=nothing,
               strsvec=nothing,targets=nothing,ppreg=0,
               ppregt=0,nd=1)

Wrapper for fast multipole implementation for Stokes N-body interactions.

Args:

  • eps: double

    precision requested

  • sources: float(3,n)

    source locations

  • stoklet: float(nd,3,n) or float(3,n)

    Stokeslet charge strengths (\(\sigma_{j}\) above)

  • strslet: float(nd,3,n) or float(3,n)

    stresslet strengths (\(mu_{j}\) above)

  • strsvec: float(nd,3,n) or float(3,n)

    stresslet orientations (\(nu_{j}\) above)

  • targets: float(3,nt)

    target locations (x)

  • ifppreg: integer
    flag for evaluating potential, gradient, and pressure at sources
    potential at sources evaluated if ifppreg = 1
    potential and pressure at sources evaluated if ifppreg=2
    potential, pressure and gradient at sources evaluated if ifppreg=3
  • ifppregtarg: integer
    flag for evaluating potential, gradient, and pressure at targets
    potential at targets evaluated if ifppregtarg = 1
    potential and pressure at targets evaluated if ifppregtarg = 2
    potential, pressure and gradient at targets evaluated if ifppregtarg = 3

Returns:

  • vals.pot: velocity at source locations if requested

  • vals.pre: pressure at source locations if requested

  • vals.grad: gradient of velocity at source locations if requested

  • vals.pottarg: velocity at target locations if requested

  • vals.pretarg: pressure at target locations if requested

  • vals.gradtarg: gradient of velocity at target locations if requested

Wrapper for direct evaluation of Stokes N-body interactions. Note that this wrapper only returns potentials and gradients at the target locations.

vals = st3ddir(sources,targets;stoklet=nothing,strslet=nothing,
               strsvec=nothing,ppregt=0,nd=1,thresh=1e-16)

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Maxwell wrappers

This subroutine computes the N-body Maxwell interactions, its curl and its divergence in three dimensions given by

\[E(x) = \sum_{j=1}^{N} \nabla \times \frac{e^{ik\|x-x_{j}\|}}{\|x-x_{j}\|} M_{j} + \frac{e^{ik\|x-x_{j}\|}}{\|x-x_{j}\|} J_{j} + \nabla \frac{e^{ik\|x-x_{j}\|}}{\|x-x_{j}\|} \rho_{j}\]

where \(M_{j}\) are the magnetic current densities, \(J_{j}\) are the electric current densities, \(\rho_{j}\) are the electric charge densities, and \(x_{j}\) are the source locations. When \(x=x_{j}\), the term corresponding to \(x_{j}\) is dropped from the sum.

vals = emfmm3d(eps,zk,sources;h_current=nothing,e_current=nothing,e_charge=nothing,
            ifE=false,ifdivE=false,ifcurlE=false,
            ifEtarg=false,ifdivEtarg=false,ifcurlEtarg=false,
            nd=1,targets=nothing)

Wrapper for fast multipole implementation for Maxwell N-body interactions. Note that this wrapper only returns fields, divergences, and curls at the target locations.

Args:

  • eps: double

    precision requested

  • zk: complex

    Wavenumber, k

  • sources: float(3,n)

    source locations

  • h_current: complex(3,n) or complex(nd,3,n)

    Magnetic currents, \(M_{j}\)

  • e_current: complex(3,n) or complex(nd,3,n)

    Electric currents, \(J_{j}\)

  • e_charge: complex(n,) or complex(nd,n)

    Electric charges, \(\rho_{j}\)

  • targets: float(3,nt)

    target locations, \(t_{i}\)

  • ifE: boolean

    E is returned at the source locations if ifE = true

  • ifcurlE: boolean

    curl E is returned at the source locations if ifcurlE = true

  • ifdivE: boolean

    div E is returned at the source locations if ifdivE = true

  • ifEtarg: boolean

    E is returned at the target locations if ifE = true

  • ifcurlEtarg: boolean

    curl E is returned at the target locations if ifcurlE = true

  • ifdivEtarg: boolean

    div E is returned at the target locations if ifdivE = true

Returns:

  • vals.E: E field defined above at target locations if requested \((E(t_{j}))\)

  • vals.curlE: curl of E field at target locations if requested \((\nabla \times E(t_{j}))\)

  • vals.divE: divergence of E at target locations if requested \((\nabla \cdot E(t_{j}))\)

  • vals.Etarg: E field defined above at target locations if requested \((E(t_{j}))\)

  • vals.curlEtarg: curl of E field at target locations if requested \((\nabla \times E(t_{j}))\)

  • vals.divEtarg: divergence of E at target locations if requested \((\nabla \cdot E(t_{j}))\)

Wrapper for direct evaluation of Maxwell N-body interactions. Note that this wrapper only returns fields, divergences, and curls at the target locations.

vals = em3ddir(zk,sources,targets;h_current=nothing,e_current=nothing,e_charge=nothing,
            ifEtarg=false,ifdivEtarg=false,ifcurlEtarg=false,
            nd=1,thresh=1e-16)

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