Classes
class	Geo

Functions
	edge_frac (pa, pb)

	snap (a)

	level_op (ax, ay, unknown, h, periodic, W)

	transfers (n, periodic)

	build_hierarchy (G, mode, nmin=4)

	colour_sets (n)

	smooth (A, x, b, sets, order)

	vcycle (levels, l, b, singular)

	contraction (levels, singular, its=60)

	bicgstab_its (levels, singular)

	disc_union (centres, R, L)

Variables
int	SNAP = 1e-3

int	n = int(sys.argv[1]) if len(sys.argv) > 1 else 128

float	L = 1.0

float	off = 0.37 / n

dict	cases

	G = Geo(n, L, sdf, per)

str	line = f"{name:34s}"

	lv = build_hierarchy(G, mode)

	rho = contraction(lv, False)

	its = bicgstab_its(lv, False)

	flush

Detailed Description

2-D prototype (architect, Amendment A1 to doc/scalar_ibm_design.md): which coarse construction of
the lumped-probe surrogate's WALL term makes a sound V-cycle?

Surrogate (steady, Lam = 1): S = aperture 5-point Laplacian + diagonal wall sink W_i = alpha_i/s0_i,
alpha_i = |area vector|/h^2 (aperture-consistent), s0 = 1.1 * 1/2 (|n_x|+|n_y|) h (Dirichlet wall).
Coarse constructions compared (same smoother, transfers and exact bottom, so only A_c differs):
  redisc-sL : faces averaged + wall W_C = <alpha / s_L>, s_L = probe distance at the level's H (design §5.2)
  avg-s0    : faces averaged + wall W_C = <alpha / s0>   (= variational for piecewise-constant P)
  rap       : Galerkin R A P, R = 4-child average, P = bilinear (zero beyond Dirichlet box faces)
Smoother: 4-colour Gauss-Seidel (colour = (i%2, j%2): race-free for 5- and 9-point), 2 pre (colours
0..3) + 2 post (3..0); bilinear prolongation, pinned cells re-zeroed; exact bottom at 4x4.
Measure: V-cycle contraction rho(I - B S) by power iteration (mean projected out when singular).
Results (n = 64 -> 128) are in doc/scalar_ibm_design.md, Amendment A1.

Function Documentation

◆ edge_frac()

mg_coarse_wall.edge_frac	(	pa,
		pb
	)

Definition at line 20 of file mg_coarse_wall.py.

◆ snap()

mg_coarse_wall.snap ( a )

Definition at line 29 of file mg_coarse_wall.py.

◆ level_op()

mg_coarse_wall.level_op	(	ax,
		ay,
		unknown,
		h,
		periodic,
		W
	)

5-point per-volume operator on an n x n level, identity rows at pinned cells.

Definition at line 59 of file mg_coarse_wall.py.

Referenced by build_hierarchy().

◆ transfers()

mg_coarse_wall.transfers	(	n,
		periodic
	)

R: average of 4 children (nc^2 x n^2); P: cell-centred bilinear (n^2 x nc^2).

Definition at line 87 of file mg_coarse_wall.py.

Referenced by build_hierarchy().

◆ build_hierarchy()

mg_coarse_wall.build_hierarchy	(	G,
		mode,
		nmin = `4`
	)

Definition at line 108 of file mg_coarse_wall.py.

References level_op(), mean(), and transfers().

◆ colour_sets()

mg_coarse_wall.colour_sets ( n )

Definition at line 144 of file mg_coarse_wall.py.

Referenced by vcycle().

◆ smooth()

mg_coarse_wall.smooth	(	A,
		x,
		b,
		sets,
		order
	)

Definition at line 149 of file mg_coarse_wall.py.

Referenced by vcycle().

◆ vcycle()

mg_coarse_wall.vcycle	(	levels,
		l,
		b,
		singular
	)

Definition at line 157 of file mg_coarse_wall.py.

References colour_sets(), mean(), smooth(), and vcycle().

Referenced by bicgstab_its(), contraction(), and vcycle().

◆ contraction()

mg_coarse_wall.contraction	(	levels,
		singular,
		its = `60`
	)

Definition at line 180 of file mg_coarse_wall.py.

References mean(), and vcycle().

◆ bicgstab_its()

mg_coarse_wall.bicgstab_its	(	levels,
		singular
	)

Definition at line 199 of file mg_coarse_wall.py.

References mean(), and vcycle().

◆ disc_union()

mg_coarse_wall.disc_union	(	centres,
		R,
		L
	)

Definition at line 211 of file mg_coarse_wall.py.

Variable Documentation

◆ SNAP

int mg_coarse_wall.SNAP = 1e-3

Definition at line 17 of file mg_coarse_wall.py.

◆ n

int mg_coarse_wall.n = int(sys.argv[1]) if len(sys.argv) > 1 else 128

Definition at line 223 of file mg_coarse_wall.py.

◆ L

float mg_coarse_wall.L = 1.0

Definition at line 224 of file mg_coarse_wall.py.

◆ off

float mg_coarse_wall.off = 0.37 / n

Definition at line 225 of file mg_coarse_wall.py.

◆ cases

dict mg_coarse_wall.cases

Initial value:

=  {
        "box+disc R=L/4 (G1-like)": (disc_union([(0.5 + off, 0.5 + off)], 0.25, L), False, False),
        "periodic+disc R=L/8": (disc_union([(0.5 + off, 0.5 + off)], 0.125, L), True, False),
        "periodic+disc R=L/32": (disc_union([(0.5 + off, 0.5 + off)], 1 / 32, L), True, False),
        "periodic 4x4 dense (R=0.4 pitch)": (disc_union([((k + 0.5) / 4 + off, (m + 0.5) / 4 + off) for k in range(4) for m in range(4)], 0.1, L), True, False),
        "box, no solid": (lambda x, y: 1.0, False, False),
    }

Definition at line 226 of file mg_coarse_wall.py.

◆ G

mg_coarse_wall.G = Geo(n, L, sdf, per)

Definition at line 235 of file mg_coarse_wall.py.

◆ line

mg_coarse_wall.line = f"{name:34s}"

Definition at line 236 of file mg_coarse_wall.py.

◆ lv

mg_coarse_wall.lv = build_hierarchy(G, mode)

Definition at line 238 of file mg_coarse_wall.py.

◆ rho

mg_coarse_wall.rho = contraction(lv, False)

Definition at line 239 of file mg_coarse_wall.py.

◆ its

mg_coarse_wall.its = bicgstab_its(lv, False)

Definition at line 240 of file mg_coarse_wall.py.

◆ flush

mg_coarse_wall.flush

Definition at line 242 of file mg_coarse_wall.py.

Classes

Functions

Variables