Helmholtz dirichlet wrappers

Iterative solver

Solve the dirichlet boundary value problem for the Helmholtz equation using combined field integral equation

subroutine helm_comb_dir_solver(npatches,norders,ixyzs,iptype,npts,
srccoefs,srcvals,eps,zpars,numit,ifinout,rhs,eps_gmres,niter,errs,
rres,sigma)

This subroutine solves the interior or exterior Dirichlet boundary value problem using the combined field integral representation

\[u = \alpha \mathcal{S}_{k}[\sigma] + \beta D_{k}[\sigma]\]

The integral equation on the boundary is given by

\[f = \pm \frac{\beta \sigma}{2} + \alpha \mathcal{S}_{k}[\sigma] + \beta D_{k}[\sigma] \, ,\]

where f is the boundary data, and sign of the constant is positive for the exterior problem, and negative for the interior problem.

Input arguments:

  • npatches: integer

    number of patches

  • norders: integer(npatches)

    order of discretization on each patch

  • ixyzs: integer(npatches+1)

    ixyzs(i) denotes the starting location in srccoefs, and srcvals array where information for patch i begins

  • iptype: integer(npatches)

    type of patch

  • npts: integer

    total number of points on the boundary

  • srccoefs: double precision (9,npts)

    koornwinder exapansion coefficients for x, \(\partial_{u} x\), and \(\partial_{v} x\)

  • srcvals: double precision (12,npts)

    x,\(\partial_{u} x\), \(\partial_{v} x\), and \(n\) sampled at discretization nodes

  • eps: double precision

    precision requested for computing quadrature and fmm tolerance

  • zpars: double complex(3)

    kernel parameters, zpars(1)=k, zpars(2)=\(\alpha\), zpars(3)=\(\beta\)

  • ifinout: integer

    flag fpr interior or exterior problems (normals assumed to be pointing in the exertior of the region) - ifinout = 0, interior problem - ifinout = 1, exterior problem

  • rhs: double complex(npts)

    boundary data f

  • eps_gmres: double precision

    gmres tolerance requested

  • numit: integer

    max number of gmres iterations

Output arguments:
  • niter: integer

    number of gmres iterations required for relative residual

  • errs: double precision(1:niter)

    relative residual as a function of iteration number

  • rres: double precision

    relative residual of the computed solution

  • sigma: double complex(npts)

    solution of integral equation

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Dirichlet problem post-processor

Evaluate potential for the Helmholtz equation using combined field integral equation

subroutine lpcomp_helm_comb_dir(npatches,norders,ixyzs,iptype,npts,
srccoefs,srcvals,ndtarg,targs,ipatch_id,uvs_targ,eps,zpars,sigma,pot)

This subroutine evaluates the layer potential for the representation

\[u = \alpha \mathcal{S}_{k}[\sigma] + \beta \mathcal{D}_{k}[\sigma]\]

For targets on the boundary, this routine only computes the principal value part, the identity term corresponding to the jump in the layer potential is not included in the layer potential.

Input arguments:

  • npatches: integer

    number of patches

  • norders: integer(npatches)

    order of discretization on each patch

  • ixyzs: integer(npatches+1)

    ixyzs(i) denotes the starting location in srccoefs, and srcvals array where information for patch i begins

  • iptype: integer(npatches)

    type of patch

  • npts: integer

    total number of discretization points on the boundary

  • srccoefs: double precision (9,npts)

    koornwinder expansion coefficients of x, \(\partial_{u} x\), and \(\partial_{v} x\).

  • srcvals: double precision (12,npts)

    x, \(\partial_{u} x\), \(\partial_{v} x\), and \(n\) sampled at discretization nodes

  • ndtarg: integer

    leading dimension of target array

  • ntarg: integer

    number of targets

  • targs: double precision (ndtarg,ntarg)

    target information

  • ipatch_id: integer(ntarg)

    id of patch of target i, id = -1, if target is off-surface

  • uvs_targ: double precision (2,ntarg)

    local uv coordinates on patch if on surface, otherwise set to 0 by default

  • eps: double precision

    precision requested

  • zpars: double complex (3)

    kernel parameters (Referring to formula above) zpars(1) = k zpars(2) = \(\alpha\) zpars(3) = \(\beta\)

  • sigma: double complex(npts)

    density for layer potential

Output arguments
  • pot: double complex(ntarg)

    layer potential evaluated at the target points

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Advanced wrappers

We also the following advanced user interfaces

Near quadrature correction generation

Evaluate near quadrature correction and store in row sparse compressed format

subroutine getnearquad_helm_comb_dir(npatches,norders,ixyzs,iptype,npts,
srccoefs,srcvals,ndtarg,ntarg,targs,ipatch_id,uvs_targ,eps,zpars,
iquadtype,nnz,row_ptr,col_ind,iquad,rfac0,nquad,wnear)

This subroutine generates the near field quadrature for the representation

\[u = \alpha \mathcal{S}_{k}[\sigma] + \beta D_{k}[\sigma] \,,\]

where the near field is specified by the user in row sparse compressed format

Input arguments:

  • npatches: integer

    number of patches

  • norders: integer(npatches)

    order of discretization on each patch

  • ixyzs: integer(npatches+1)

    ixyzs(i) denotes the starting location in srccoefs, and srcvals array where information for patch i begins

  • iptype: integer(npatches)

    type of patch

  • npts: integer

    total number of points on the boundary

  • srccoefs: double precision (9,npts)

    koornwinder exapansion coefficients for x, \(\partial_{u} x\), and \(\partial_{v} x\)

  • srcvals: double precision (12,npts)

    x,\(\partial_{u} x\), \(\partial_{v} x\), and \(n\) sampled at discretization nodes

  • ndtarg: integer

    leading dimension of target array

  • ntarg: integer

    number of targets

  • targs: double precision (ndtarg,ntarg)

    target information. Note the first three entries must be the cartesian components of the target

  • ipatch_id: integer(ntarg)

    id of patch of target i, id= -1/0 if target is off-surface

  • uvs_targ: double precision (2,ntarg)

    local uv coordinates on patch, if target is on surface

  • eps: double precision

    precision requested for computing quadrature and fmm tolerance

  • zpars: double complex(3)

    kernel parameters, zpars(1)=k, zpars(2)=\(\alpha\), zpars(3)=\(\beta\)

  • iquadtype: integer

    quadrature type

    • iquadtype = 1: for generalized gaussian quadrature+adaptive integration

  • nnz: integer

    number of non-zero target-patch interactions

  • row_ptr: integer(ntarg+1)

    row_ptr(i) is the starting location in the col-ind list where the list of patches in the near field of target i start. If row_ptr(i) <= j < row_ptr(i+1), then target(i) and patch col_ind(j) are in the near-field of each other

  • col_ind: integer(nnz)

    list of patches which interact with the targets. See example

  • iquad: integer(nnz)

    iquad(i) is the location in the quadrature array where the matrix entries corresponding to the interaction of target i, and patch col_ind(j) start in wnear array

  • rfac0: double precision

    radius parameter for switiching between adaptive integration and oversampled smooth quadrature in quadrature generation routine (note this is different from the radius parameter used for identifying the near field)

  • nquad: integer

    size of wnear array

Output arguments:
  • wnear: double complex(nquad)

    near field quadrature correction

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Guru interface for layer potential evaluation

Guru interface for evaluating the potential for the Helmholtz equation using combined field integral equation

subroutine lpcomp_helm_comb_dir_addsub(npatches,norders,ixyzs,iptype,npts,
srccoefs,srcvals,ndtarg,targs,ipatch_id,uvs_targ,eps,zpars,
nnz,row_ptr,col_ind,iquad,nquad,wnear,sigma,novers,nptso,ixyzso,
srcover,wover,pot)

This subroutine evaluates the layer potential for the representation

\[u = \alpha \mathcal{S}_{k}[\sigma] + \beta D_{k}[\sigma]\]

For targets on the boundary, this routine only computes the principal value part, the identity term corresponding to the jump in the double layer is not included in the layer potential.

Input arguments:

  • npatches: integer

    number of patches

  • norders: integer(npatches)

    order of discretization on each patch

  • ixyzs: integer(npatches+1)

    ixyzs(i) denotes the starting location in srccoefs, and srcvals array where information for patch i begins

  • iptype: integer(npatches)

    type of patch

  • npts: integer

    total number of points on the boundary

  • srccoefs: double precision (9,npts)

    koornwinder exapansion coefficients for x, \(\partial_{u} x\), and \(\partial_{v} x\)

  • srcvals: double precision (12,npts)

    x,\(\partial_{u} x\), \(\partial_{v} x\), and \(n\) sampled at discretization nodes

  • ndtarg: integer

    leading dimension of target array

  • ntarg: integer

    number of targets

  • targs: double precision (ndtarg,ntarg)

    target information. Note the first three entries must be the cartesian components of the target

  • ipatch_id: integer(ntarg)

    id of patch of target i, id= -1/0 if target is off-surface

  • uvs_targ: double precision (2,ntarg)

    local uv coordinates on patch, if target is on surface

  • eps: double precision

    precision requested

  • zpars: double complex(3)

    kernel parameters, zpars(1)=k, zpars(2)=\(\alpha\), zpars(3)=\(\beta\)

  • nnz: integer

    number of non-zero target-patch interactions

  • row_ptr: integer(ntarg+1)

    row_ptr(i) is the starting location in the col-ind list where the list of patches in the near field of target i start. If row_ptr(i) <= j < row_ptr(i+1), then target(i) and patch col_ind(j) are in the near-field of each other

  • col_ind: integer(nnz)

    list of patches which interact with the targets. See example

  • iquad: integer(nnz)

    iquad(i) is the location in the quadrature array where the matrix entries corresponding to the interaction of target i, and patch col_ind(j) start in wnear array

  • nquad: integer

    size of wnear array

  • wnear: double complex(nquad)

    near field quadrature correction

  • sigma: double complex(npts)

    density for layer potential

  • novers: integer(npatches)

    order of oversampled discretization on each patch

  • nptso: integer

    number of oversampled discretization nodes

  • ixyzso: integer(npatches+1)

    ixyzso(i) denotes the starting location in srcover, and wover array where information for patch i begins

  • srcover: double precision (12,nptso)

    x,\(\partial_{u} x\), \(\partial_{v} x\), and \(n\) sampled at discretization nodes

  • wover: double precision(nptso)

    oversampled quadrature weights

Output arguments:
  • pot: double complex(ntarg)

    layer potential evaluated at target points

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