Kernels

greens3d!(integrand,r,k)

Green's function for the Helmholtz Equation in 3d:

\[\frac{e^{ikr_j}}{4\pi r_j}\]

freens3d!(integrand,r,interpolation,sources,normals,k)

Normal derivative of the 3D Helmholtz Green's function with respect to interpolation nodes:

\[\frac{e^{ikr_j}}{4\pi r_j^3}(ikr_j - 1) (x_j - y)\cdot n\]

freens3dk0!(integrand,r,interpolation,sources,normals)

freens3d! with k=0.

\[-\frac{(x_j - y)\cdot n}{4\pi r_j}\]

onefunction!(integrand,r,k)
onefunction!(integrand,r,interpolation,source,normals)
onefunction!(integrand,r,interpolation,source,normals,k)

Fills integrand with ones. Other inputs disregarded. Can be used to compute surface areas.

taylor_greens3d!(integrand,r,k,m)

$m$th derivative of the Green's function w.r.t. the wavenumber.

\[G^{(m)}(r_j,k_0) = \frac{(\mathrm{i}r)^m\exp(\mathrm{i}k_0r)}{4\pi r_j}\]

taylor_freens3d!(integrand,r,interpolation,sources,normals,k0,m)

$m$th derivative of the normal derivative of the Green's function w.r.t. the wavenumber.

\[n\cdot \nabla G^{(m)}(r_j,k_0)\]

taylor_greens_gradient3d!(integrand,r,interpolation,sources,normals,k0,m)

Gradient of the $m$th derivative of the Green's function w.r.t. the wavenumber.

\[\nabla G^{(m)}(r_j,k_0)\]

taylor_greens_tangential_gradient3d!(integrand,r,interpolation,sources,normals,k0,m)

Tangential gradient of the $m$th derivative of the Green's function w.r.t. the wavenumber.

\[(\mathbf{I} - \mathbf{n}\mathbf{n}^\top)\nabla G^{(m)}(r_j,k_0)\]