Transition-matrix solutions

tmatrix solves scattering by elastic prolate and oblate spheroids and elastic shells in spheroidal coordinates (Hackman, 1984). It returns a TMatrixSolution.

Field expansions

In prolate spheroidal coordinates $(\xi,\eta,\phi)$ with focal half-distance $f$ and size parameter $c=kf$, the exterior pressure is

\[p=\sum_{m,n}\left[a_{mn}R_{mn}^{(1)}(c,\xi)+f_{mn}R_{mn}^{(3)}(c,\xi)\right]S_{mn}(c,\eta)\,e^{im\phi},\]

with $R_{mn}^{(1)}$ the regular radial spheroidal wave function, $R_{mn}^{(3)}$ the outgoing one, and $S_{mn}$ the angular one. m_max and n_max truncate this sum.

Elastic displacement

Solid displacement satisfies the Navier equation of FEM and shell coupling. It is expanded in vector spheroidal wave functions built from the same $R_{mn}S_{mn}e^{im\phi}$ scalar potentials, one longitudinal family at the solid's compressional wavenumber $k_L$ and two transverse families at its shear wavenumber $k_T$:

\[\boldsymbol u=\frac{1}{k_L}\nabla\chi_L +\frac{1}{\sqrt\lambda}\nabla\chi_T\times\boldsymbol r +\frac{1}{k_T}\nabla\times\left(\nabla\chi_T\times\boldsymbol r\right),\]

where $\lambda$ is $\chi_T$'s own separation constant.

Transition matrix

Pressure, normal displacement and zero shear traction are matched at the interface, tested against Betti's reciprocal identity for two elastodynamic states of the same frequency:

\[\int_S\left(\boldsymbol u\cdot\boldsymbol t(\boldsymbol v)-\boldsymbol v\cdot\boldsymbol t(\boldsymbol u)\right)dS=0.\]

Every incident degree pairs with every test degree through this surface integral, so the resulting matrix $f=Ta$ is dense, unlike the diagonal series of modal.

Solid spheroid

tmatrix(Spheroid(a, b), SolidElastic(g, hl, ht), k; incidence_angle = pi / 3)

The solution reduces to the elastic sphere as the aspect ratio approaches one and to the rigid spheroid as the solid becomes stiff and dense. It agrees with a full-wave isogeometric reference and with the volume FEM of FEM and shell coupling.

Elastic shell

Shelled(ElasticLayer(...), FluidInterior(...) or VacuumInterior(), radius_ratio) is a shell with a confocal inner surface, whose equatorial semi-axis is radius_ratio times the outer one. An oblate shell needs a radius_ratio above the focal ratio so that this surface exists.

Elongated and thin shells converge slowly in n_max. Each shell solve is repeated with m_max and n_max reduced by 2 and warns when the amplitude changes by more than 1%. Use n_max near 30 for a 3:1 shell with radius_ratio = 0.8 and check against bem beyond that.

Sweeps

incidence_angle_sweep(body, boundary, k, angles) computes each azimuthal order once and reuses it for every angle. Pass method = :volume to sweep the volume FEM instead.