Geometries¶

A geometry says where the problem lives: its coordinates, which directions are periodic, where the walls are and what they are called.

import numpy as np
import conatus as cn

box = cn.Box(x=cn.periodic(2.0), z=cn.walls(-0.5, 0.5))   # periodic in x, walls at z = ±0.5
m = cn.Model(box, name="convection", Ra=2500, Pr=1)

A geometry holds no resolution and no basis. The spectral basis of each coordinate is fixed by the geometry (one choice per geometry, listed below and printed by explain()); the number of modes is given only when you solve, with N= (Simulations and time steppers).

Catalogue¶

Geometry	Coordinates (in order)	Named boundaries
`cn.Box(x=cn.periodic(Lx), z=cn.walls(a, b))`	the keyword order, 1 to 3 axes	`bottom`, `top`, `walls`, `z=a`, `z=b`
`cn.Channel(Lx, Lz=None, h=1)`	`x`, `y` (walls at \(y=\pm h\)), `z` in 3D	`bottom`, `top`, `walls`, `y=-1`, `y=1`
`cn.Disk(R)`	`phi`, `r`	`wall`, `r=R`
`cn.Annulus(ri, ro)`	`phi`, `r`	`inner`, `outer`, `walls`
`cn.Pipe(R, L)` (also `cn.Cylinder`)	`phi`, `r`, `z` (periodic, length \(L\))	`wall`
`cn.TaylorCouette(ri, ro, length)`	`phi`, `r`, `z` (periodic)	`inner`, `outer`, `walls`
`cn.Sphere(R)`	`phi`, `theta`	none (a closed surface)
`cn.Ball(R)`	`phi`, `theta`, `r`	`wall`
`cn.Shell(ri, ro)`	`phi`, `theta`, `r`	`inner`, `outer`, `walls`

walls names both ends of the bounded coordinate at once: m.at("walls", u=0) is a no-slip condition on both plates. bottom and top are the two ends of the bounded Cartesian axis, whatever its name. Spherical shells have their own page, Spherical shells.

import conatus as cn

print(cn.Annulus(1.0, 2.0).boundaries())
# {'inner': [('r', 'left')], 'outer': [('r', 'right')], 'walls': [...], 'r=1': [...], 'r=2': [...]}
print(cn.Annulus(1.0, 2.0).explain())
print(cn.TaylorCouette(1.0, 2.0, 2 * 3.14159).boundaries().keys())   # inner, outer, walls, ...

Geometry: Annulus(inner=1, outer=2) (annulus)
  bases (fixed per geometry, sizes given at discretization):
    phi: periodic on [0, 2 pi) -> RealFourier(N_phi)
    r: [1, 2] -> AnnulusBasis(N_r) (Chebyshev T)
  named boundaries: inner = r left; outer = r right; walls = r left, r right; r=1 = r left; r=2 = r right

What each geometry uses¶

Geometry	Expansion	Regularity
Box, Channel	real Fourier series (cos/sin) on periodic axes, Chebyshev polynomials on the wall axis	—
Disk, Pipe	Fourier in \(\varphi\) × Zernike-type polynomials \(r^{\lvert m\rvert} P(2r^2-1)\) in \(r\)	regular at the axis by construction: no axis condition, no \(1/r\) regularisation
Annulus, Taylor–Couette	Fourier in \(\varphi\) (and in \(z\)) × Chebyshev in \(r\)	—
Sphere	Fourier in \(\varphi\) × spin-weighted spherical harmonics	regular at the poles
Ball	spherical harmonics × one-sided Jacobi polynomials in \(r\)	regular at the centre
Shell	spherical harmonics × Chebyshev in \(r\)	—

cn.periodic(L, complex=True) uses complex exponentials instead of the real cos/sin series (for complex unknowns). For the sphere, the size given for theta is the number of spherical-harmonic degrees \(L_{\max}+1\).

Resolution¶

The resolution is a tuple in the coordinate order, a dict by coordinate name, or an integer when the geometry has one coordinate:

import conatus as cn

m = cn.problems.native.rayleigh_benard(Lx=2.0, Ra=2500, Pr=1)   # x periodic, z between walls
m.simulate(N=(64, 32), scheme="RK443", dt=1e-3)          # x: 64, z: 32
m.simulate(N={"x": 64, "z": 32}, scheme="RK443", dt=1e-3)

A missing size, or an integer for a geometry with several coordinates, is refused. The default dealiasing factor follows the degree p of the products of unknowns and given fields (a given coefficient counts as one factor, so U*u*u has p = 3): max(3/2, (p+1)/2) (3/2 for quadratic terms, 2 for cubic ones), and max(2, (p+1)/2) along Hermite/Laguerre axes, where no factor is alias-free; give dealias= (a number, or a dict by coordinate) to choose it. The value and its reason are printed by explain() and written in the Methods paragraph.

Limits¶

A Box has at most one bounded axis; two walled directions (a duct, a cavity) are refused with UnsupportedError.
Free-slip, no-slip and Navier partial-slip walls are named conditions of the native layer (m.at("top", u=cn.free_slip), cn.no_slip, cn.navier_slip(l), Boundary conditions); other conditions on single components of a vector unknown are written in the expert layer.
Every geometry runs on the CPU and on a GPU (backend="cuda"); the fused nonlinear kernels are for Cartesian geometries (GPU).