Fourier matching

Fourier matching conformally maps an irregular axisymmetric body's meridian profile to a coordinate system in which the mapped surface is exactly circular, then matches boundary conditions there using spherical wave functions. It targets smooth bodies of revolution between the canonical (sphere/spheroid) and general numerical (BEM/MFS/FEM) solvers. See Reeder and Stanton (2004), extending DiPerna and Stanton (1994)'s two-dimensional method.

fourier(Irregular(profile), boundary, k) # Rigid, PressureRelease or FluidFilled
fourier(Sphere(a), boundary, k) # converts to an equivalent Irregular
incidence_angle_sweep(body::Irregular, boundary, k, angles)

fourier returns an FMSolution, the same calling convention as modal, bem and mfs. Converting a Sphere/Spheroid to Irregular works, but modal is exact and cheaper for those bodies. incidence_angle_sweep builds the conformal mapping and boundary-matching transition operator once and reuses it across all angles, since neither depends on incidence angle.

Conformal mapping

For a meridian profile $R(\theta)$ (radial distance from the origin, $\theta$ the polar angle), expand it in a Fourier series and solve for an angle-correction series $\theta(w)$ such that the mapped coordinate system becomes exactly circular at the body surface

\[R(\theta) = a + \sum_{n\ge1} r_n^c\cos(n\theta), \qquad \theta(w) = w + \sum_{l\ge1}\left[\delta_l^c\cos(lw) + \delta_l^s\sin(lw)\right].\]

Both poles lie on the axis, so only cosine terms appear. Odd cosine harmonics still allow fore-aft asymmetry. Irregular fits the mirrored profile and rejects nonzero sine coefficients.

The $\delta$ coefficients solve a nonlinear system with forward-mode automatic differentiation. A continuation homotopy handles bodies far from circular. The mapping is rejected (is_admissible) if its Jacobian vanishes anywhere outside the body.

Pressure-release boundary matching

The scattered field is expanded in spherical Bessel/Hankel and associated Legendre functions. For each azimuthal order $m$, matching the total pressure to zero on the mapped surface gives a dense linear system

\[\sum_n a_{nm} R_n^m + \sum_n b_{nm} Q_n^m = 0,\]

solved for the scattered coefficients $b_{nm}$, where $a_{nm}$ are the incident plane-wave coefficients and $R_n^m$, $Q_n^m$ are integrals of the mapped surface against spherical Bessel and Hankel functions. Incidence azimuth is fixed at $0$, and general azimuth is recovered by shifting the observation azimuth instead, matching the rest of this package's axisymmetric solvers.

Rigid boundary matching

The rigid condition instead matches the normal derivative of the total pressure to zero on the mapped surface, using the coordinate system's normal derivative

\[\hat{n}\cdot\nabla P(u,w,v) = \frac{1}{\sqrt{f_w^2+f_u^2}}\frac{\partial P}{\partial u}.\]

Differentiating the modal series w.r.t. $u$ through both the radial argument $kr(u,w)$ and the Legendre argument $g(u,w)/r(u,w)$ gives the same linear-system form as the soft case, with $R_n'^m$, $Q_n'^m$ in place of $R_n^m$, $Q_n^m$.

Fluid boundary matching

For a penetrable interior with the given density and sound-speed contrast (FluidFilled's own convention), pressure and velocity are both matched across the mapped surface, introducing an interior field expansion with wavenumber $k_1 = k/h$ (soundspeed_contrast is $h$) and coefficients $l_{nm}$, regular at the origin. Eliminating $l_{nm}$ from the two matched linear systems gives $b_{nm}$ in terms of the exterior matrices above and a third pair of interior matrices $S_n^m$, $S_n'^m$. Velocity continuity is weighted by density_contrast alone, since the sound-speed part of the impedance ratio is already carried by the $k$/$k_1$ factors in the normal-derivative matrices.

Numerical conditioning

R_n^m, Q_n^m and their primed and interior counterparts span many orders of magnitude across $n$ from spherical Hankel-function growth alone, not from genuine ill-conditioning. Each per-$m$ linear solve uses a row/column-equilibrated truncated-SVD pseudoinverse (pinv on the equilibrated matrix) rather than a plain solve, so truncation compares singular values on a properly-scaled basis instead of against raw Hankel-function magnitude.

Truncation guard

Fourier matching does not converge monotonically in n_max for elongated bodies. Each solve is repeated with n_max and m_max reduced by 2, and diagnostics(sol).convergence reports the largest relative change in the far-field amplitude. A warning is emitted above 1e-2.

Accuracy envelope

Worst complex-amplitude error against axisymmetric BEM over backscatter, forward and off-plane directions, at the default settings of fourier(Spheroid(aspect, 1), ...). Prolate spheroids at incidence $\pi/3$, fluid contrasts 1.05 and 1.02. ! marks a solve where the truncation guard warns.

At 5:1 the rigid boundary reaches only a few percent and the fluid boundary is marginal. Use bem for rigid bodies beyond 4:1.

The usable n_max is limited by mapping_order. Beyond about $\mathrm{mapping\_order}/(2\,\mathrm{aspect})+2$ the error grows rapidly, so elongated bodies (aspect above 2.5) default to mapping_order = 20*aspect (at most 96) and n_max = m_max set by that bound. The truncation guard can pass a result with an error of up to a few percent for a rigid 5:1 body. The quadrature tolerance rtol and node cap maxevals do not change the error at these settings.

Validation

Reconstruction and mapping convergence are checked against a sphere (exact) and prolate spheroids up to 10:1 aspect ratio (against the exact ellipse equation). All three boundary conditions are checked against the exact sphere modal solution and, for prolate spheroids at oblique incidence up to 5:1 aspect ratio, against independent axisymmetric BEM, at backscatter, forward scatter and off-plane bistatic angles. The fluid boundary is checked at both weak (density and sound-speed contrasts near $1$) and gas (strong) contrast. A fore-aft asymmetric noncanonical body ($R(\theta)=1+0.12\cos 3\theta+0.05\cos 2\theta$) agrees with axisymmetric BEM on the mapped surface within 3e-4 in complex amplitude for all three boundaries, including gas contrast across its resonance. Reeder et al. (2004) validate the method against measured fish morphology.