Geometries

Choose a named body, construct a meridian or elliptical-section loft, or supply a closed triangular surface. Geometry support and material support are separate: a mesh that works for a rigid scatterer does not automatically support elasticity. All lengths below are in meters. The generated bodies use x for their longitudinal or revolution axis, with y and z transverse.

Geometry gallery

Each gallery panel shows one geometry rendered from its package mesh. Click an image to enlarge it. Panels are independently scaled and preserve aspect ratios. The mesh resolution is for viewing, not a scattering convergence recommendation.

using AcousticScattering
using CairoMakie: Figure, Axis3, plot!, save, hidedecorations!, hidespines!

function save_geometries(entries; meridian=false)
    for (name, body) in entries
        fig = Figure(; size=(570, 360))
        ring = body isa Torus || (body isa RevolvedBody && body.closed)
        ax = Axis3(fig[1, 1];
            aspect=:data, azimuth=ring ? pi/6 : -0.35pi, elevation=0.12pi,
            viewmode=:fitzoom, perspectiveness=0.15)
        surface = meridian ? mesh(body; resolution=128) :
                  mesh(body; method=:full, resolution=0.16, mesh_order=2, qorder=2)
        plot!(ax, surface)
        hidedecorations!(ax)
        hidespines!(ax)
        slug = replace(lowercase(name), r"[^a-z0-9]+" => "_")
        save("geometry_$(slug).png", fig)
    end
    return nothing
end

canonical = [
    "Sphere" => Sphere(0.7),
    "Prolate spheroid" => Spheroid(1.0, 0.4),
    "Oblate spheroid" => Spheroid(0.35, 0.85),
    "Flat-ended cylinder" => Cylinder(0.4, 1.5),
    "Capsule" => Cylinder(0.4, 1.0; endcap_depth=0.4),
    "Bent, domed cylinder" => Cylinder(0.25, 2.0;
        radius_curvature=1.2, endcap_depth=0.25),
]
save_geometries(canonical)

primitives = [
    "Triaxial ellipsoid" => Ellipsoid(1.0, 0.6, 0.35),
    "Superellipsoid" => Superellipsoid(0.9, 0.6, 0.4;
        axial_exponent=0.6, equatorial_exponent=0.6),
    "Cone" => Cone(0.6, 1.5),
    "Bicone" => Bicone(0.6, 1.8; waist=0.4),
    "Circular frustum" => Frustum(0.65, 0.3, 1.4),
    "Elliptical frustum" => Frustum((0.65, 0.35), (0.3, 0.2), 1.4),
]
save_geometries(primitives)

Profiles, lofts and toroids

RevolvedBody accepts axial coordinates and radii describing a simple piecewise-linear meridian. Its ordinary form has two endpoints on the axis and positive interior radii. closed=true instead describes a loop away from the axis, allowing toroids with noncircular cross-sections. Crossing or touching profiles are rejected. Either traversal direction is accepted.

LoftedBody joins elliptical sections along a straight or bent centerline and closes both ends with flat caps. Supply centers as a $3\times N$ matrix and semi-axes as a $2\times N$ matrix. Section normals follow the centerline unless directions is supplied. twists rotates each section about its normal in radians. All semi-axes must be positive.

centers = [-1.0 0.0 1.0; 0.0 0.15 0.6; 0.0 0.0 0.15]
axes = [0.3 0.5 0.25; 0.2 0.3 0.15]
profiles = [
    "Circular frustum chain" => FrustumChain(
        [-1.0, -0.5, 0.2, 1.0], [0.0, 0.5, 0.35, 0.0]),
    "Elliptical frustum chain" => FrustumChain([-1.0, 0.0, 1.0], axes),
    "Bent elliptical loft" => LoftedBody(centers, axes; twists=[0.0, 0.2, 0.4]),
    "Ring torus" => Torus(0.75, 0.25),
    "Elliptical tube torus" => Torus(0.75, (0.15, 0.3)),
    "Rectangular-section toroid" => RevolvedBody(
        [-0.2, 0.2, 0.2, -0.2], [0.55, 0.55, 0.95, 0.95]; closed=true),
]
save_geometries(profiles)

Finite capped paraboloids, hyperboloids, ogives, waisted bodies, rounded noses and other surfaces of revolution can use sampled profiles without another solver. For example, the next paraboloid has $r^2=0.3(x+1)$ and a flat cap at $x=1$. The hyperboloid has $r=0.3\sqrt{1+(x/0.6)^2}$ between two flat caps.

parameter = collect(range(0.0, 1.0; length=33))
paraboloid = RevolvedBody(vcat(-1 .+ 2parameter.^2, 1.0),
    vcat(sqrt(0.6) .* parameter, 0.0))
x = collect(range(-1.0, 1.0; length=41))
hyperboloid = RevolvedBody(vcat(-1.0, x, 1.0),
    vcat(0.0, 0.3sqrt.(1 .+ (x ./ 0.6).^2), 0.0))
ogive = RevolvedBody(t -> (2t-1, sqrt(1.25^2-(2t-1)^2)-0.75); n=48)
save_geometries([
    "Capped paraboloid" => paraboloid,
    "Capped hyperboloid" => hyperboloid,
    "Double circular-arc ogive" => ogive,
]; meridian=true)

Increasing mesh resolution only subdivides the supplied profile segments. Increase profile sampling separately to converge a curved surface. Corners are preserved. The last gallery revolves meridian panels. Full triangular meshing of densely sampled profiles may require smaller elements or mesh_order=1 near tips and closely spaced corners. Loft section placement likewise controls geometry independently of surface element size. Superellipsoid nodes lie on the prescribed surface, while the triangular interpolants approximate it. Strong curvature and cusps may need finer meshes or linear elements.

Named constructors and solver routes

Here full mesh methods means BEM, MFS and mesh Kirchhoff. The table lists geometry routes, not every material combination. See Choosing a solver for canonical bodies and the material limits below for generated surfaces. Straight-cylinder meridian BEM/FEM use flat ends. Cylinder modal models are approximations to finite bodies, and bent-body lateral models omit endcaps.

Other supported geometry representations

See Geometry and incidence for mesh import, units, labels and coupled-region examples. A single supplied mesh describes one connected boundary. Separate region interfaces must be passed separately.

Solver and material limits

Generated surfaces support Rigid(), PressureRelease() and homogeneous FluidFilled(g,h) or GasFilled(g,h) through BEM and full surface MFS. Here g and h are the interior/exterior density and sound-speed ratios. Coupled fluid BEM additionally handles multiple lossless homogeneous regions.

body = Torus(0.025, 0.006)
solution = bem(body, PressureRelease(), 100.0; n=64, m_max=8)
surface = mesh(body; method=:full, resolution=0.003, qorder=4)
solution_3d = bem(surface, PressureRelease(), 100.0)

Axisymmetric BEM uses meridian panels and azimuthal modes. Full BEM uses the closed triangular boundary. MFS requires source offsets that remain on the correct side of the surface, which is more difficult near corners and narrow concavities. Kirchhoff is a high-frequency approximation whose local-normal illumination does not account for self-shadowing or multiple reflections.

FMM is an acceleration of boundary interactions, not an additional geometry class. There is currently no public FMM BEM path. Supported hierarchical-matrix compression can accelerate existing full BEM formulations without changing their geometry or material scope. See BEM and MFS.

Shapes and physics needing further support

A named constructor is a convenience, not a prerequisite for solving a valid closed acoustic surface. Conversely, adding a constructor alone cannot supply missing material physics or resolve a singular boundary.