MATLAB

The MATLAB interface has four callable subroutines:

  • 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.

function [U] = lfmm3d(eps,srcinfo,pg,targ,pgt)

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

Args:

  • eps: double

    precision requested

  • srcinfo: structure

    structure containing sourceinfo

    • srcinfo.sources: double(3,n)

      source locations, \(x_{j}\)

    • srcinfo.nd: integer

      number of charge/dipole vectors (optional, default - nd = 1)

    • srcinfo.charges: double(nd,n)

      charge densities, \(c_{j}\) (optional, default - term corresponding to charges dropped)

    • srcinfo.dipoles: double(nd,3,n)

      dipole orientation vectors, \(v_{j}\) (optional default - term corresponding to dipoles dropped)

  • pg: integer
    source eval flag
    potential at sources evaluated if pg = 1
    potenial and gradient at sources evaluated if pg=2
  • targ: 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:

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

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

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

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

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

function [U] = l3ddir(srcinfo,targ,pgt)

Example:

  • see lfmm3d_example.m

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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.

function [U] = hfmm3d(eps,zk,srcinfo,pg,targ,pgt)

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

Args:

  • eps: double

    precision requested

  • zk: complex

    Helmholtz parameter, k

  • srcinfo: structure

    structure containing sourceinfo

    • srcinfo.sources: double(3,n)

      source locations, \(x_{j}\)

    • srcinfo.nd: integer

      number of charge/dipole vectors (optional, default - nd = 1)

    • srcinfo.charges: complex(nd,n)

      charge densities, \(c_{j}\) (optional, default - term corresponding to charges dropped)

    • srcinfo.dipoles: complex(nd,3,n)

      dipole orientation vectors, \(v_{j}\) (optional default - term corresponding to dipoles dropped)

  • pg: integer
    source eval flag
    potential at sources evaluated if pg = 1
    potenial and gradient at sources evaluated if pg=2
  • targ: 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:

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

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

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

  • U.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.

function [U] = h3ddir(zk,srcinfo,targ,pgt)

Example:

  • see hfmm3d_example.m

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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.

function [U] = stfmm3d(eps,srcinfo,ifppreg,targ,ifppregtarg)

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

Args:

  • eps: double

    precision requested

  • srcinfo: structure

    structure containing sourceinfo

    • srcinfo.sources: double(3,n)

      source locations, \(x_{j}\)

    • srcinfo.nd: integer

      number of charge/dipole vectors (optional, default - nd = 1)

    • srcinfo.stoklet: double(nd,3,n)

      Stokeslet densities, \(\sigma_{j}\) (optional, default - term corresponding to Stokeslet dropped)

    • srcinfo.strslet: double(nd,3,n)

      Stresslet densities, \(\mu_{j}\) (optional default - term corresponding to stresslet dropped)

    • srcinfo.strsvec: double(nd,3,n)

      Stresslet orientiation vectors, \(\nu_{j}\) (optional default - term corresponding to stresslet dropped)

  • ifppreg: integer
    source eval flag
    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
  • targ: double(3,nt)

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

  • ifppregtarg: integer
    target eval flag (optional)
    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:

  • U.pot: velocity at source locations if requested

  • U.pre: pressure at source locations if requested

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

  • U.pottarg: velocity at target locations if requested

  • U.pretarg: pressure at target locations if requested

  • U.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.

function [U] = st3ddir(srcinfo,targ,ifppregtarg)

Example:

  • see stfmm3d_example.m

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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.

function [U] = emfmm3d(eps,zk,srcinfo,targ,ifE,ifcurlE,ifdivE)

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

  • srcinfo: structure

    structure containing sourceinfo

    • srcinfo.sources: double(3,n)

      source locations, \(x_{j}\)

    • srcinfo.nd: integer

      number of charge/dipole vectors (optional, default - nd = 1)

    • srcinfo.h_current: complex(nd,3,n)

      Magnetic current densities, \(M_{j}\) (optional, default - term corresponding to magnetic current dropped)

    • srcinfo.e_current: complex(nd,3,n)

      Electric current densities, \(J_{j}\) (optional, default - term corresponding to electric current dropped)

    • srcinfo.e_charge: complex(nd,n)

      Electric charge densities, \(\rho_{j}\) (optional, default - term corresponding to electric charge dropped)

  • targ: double(3,nt)

    target locations, \(t_{i}\)

  • ifE: integer

    E is returned at the target locations if ifE = 1

  • ifcurlE: integer

    curl E is returned at the target locations if ifcurlE = 1

  • ifdivE: integer

    div E is returned at the target locations if ifdivE = 1

Returns:

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

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

  • U.divE: 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.

function [U] = em3ddir(zk,srcinfo,targ,ifE,ifcurlE,ifdivE)

Example:

  • see emfmm3d_example.m

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