Geometry Reference

Class	What it samples	Example uses
Interval	1-D uniform or Sobol interior + boundary points	Burgers, wave, heat (1-D)
Rectangle	2-D interior (LHS / Sobol) + all four boundary edges	Helmholtz, LDC, diffusion inverse
NACAAirfoil	NACA 4-digit (symmetric & cambered) exterior domain, SDF-weighted near-surface, AoA via quarter-chord rotation	examples/airfoil/
Cylinder2D	Circular cylinder exterior cross-flow domain, analytic SDF, surface points	examples/cylinder/
Pipe	3-D cylindrical interior, lateral wall, circular inlet, circular outlet	examples/pipe_flow/, examples/pipe_flow_rheology/
BulgeGeometry	Axisymmetric AAA bulge R(x) (cosine²): interior, curved wall, inlet, outlet	examples/AAA/, examples/AAA_rheology/
Ramp	Trapezoidal domain above a wedge surface at angle θ	examples/ramp/
Composite	Boolean union / intersection / difference of any two geometry objects	LDC (cavity minus any obstacle)
ShapelyGeom	Arbitrary 2-D polygon backed by Shapely 2.x; rejection-samples interior	Custom geometries

Notable design details

Airfoil angle of attack via rotation, not inflow tilt

NACAAirfoil imposes angle of attack (AoA) by rotating the airfoil geometry about the quarter-chord point, rather than tilting the free-stream inflow direction. This keeps the inflow boundary condition horizontal and simple, while still producing the correct relative flow angle.

SDF-weighted near-surface sampling

Both NACAAirfoil and Cylinder2D expose analytic signed-distance functions (SDFs) that bias collocation-point density toward the body surface — where gradients are steepest — without hand-tuned refinement zones.

Every geometry implements a common sampler interface

class Pipe:
    def sample_interior(self, n, key): ...
    def sample_wall(self, n, key): ...
    def sample_inlet(self, n, key): ...
    def sample_outlet(self, n, key): ...

The exact method set varies slightly by dimensionality (e.g. Rectangle exposes four boundary-edge samplers instead of wall/inlet/outlet), but every geometry class follows the same sample_<region>(n, key) -> array convention, making it straightforward to swap geometries within an existing PDE/loss/solver stack.

See also

PDE Library Reference for the PDE residual classes each geometry pairs with, and Physics Examples for the full worked-example catalogue.