From 0f9ec7dfbad9098b2183f900bf38d98a74ac5825 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Mon, 17 Aug 2026 19:50:01 +1000 Subject: [PATCH] Figure: where the rotation lives under rotated boundary conditions Drawn for the boundary-conditions note (R1 in the technical-notes writing plan), which is not yet written. Parked here rather than left in a scratch directory: it follows the per-figure layout the cetz skill documents, and it moves to the note's own examples/ and figures/ when that article is created. Two panels. The left is a free surface with an inflection -- rising on the left, falling on the right -- so the outward normal swings through a wide range and the curvature changes sign. That is the case no global coordinate system straightens out, which is why rotating each node's frame is the general answer and solving in spherical components is not. The right is where the rotation actually lives, and the point is that it is contained. The velocity solve is rotated and carries its multigrid with it; the fieldsplit / Schur solve abuts it and never handles a rotated vector, because the pressure block carries no boundary condition of this kind. One un-rotation sits on the boundary between them and feeds both the Schur solve and everything downstream. Geometry is generated in Python and handed to Typst as JSON, per the skill: a real Delaunay triangulation, culled by centroid against the non-convex surface, with alternate rows staggered so the triangles are not degenerate. Underworld development team with AI support from Claude Code --- .../generate-rotated-basis.py | 102 ++ .../rotated-basis-data.json | 1011 +++++++++++++++++ .../rotated-basis.png | Bin 0 -> 199642 bytes .../rotated-basis.svg | 1 + .../rotated-basis.typ | 180 +++ 5 files changed, 1294 insertions(+) create mode 100644 publications/blog-posts/figures/rotated-boundary-conditions/generate-rotated-basis.py create mode 100644 publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis-data.json create mode 100644 publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis.png create mode 100644 publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis.svg create mode 100644 publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis.typ diff --git a/publications/blog-posts/figures/rotated-boundary-conditions/generate-rotated-basis.py b/publications/blog-posts/figures/rotated-boundary-conditions/generate-rotated-basis.py new file mode 100644 index 000000000..b203819ba --- /dev/null +++ b/publications/blog-posts/figures/rotated-boundary-conditions/generate-rotated-basis.py @@ -0,0 +1,102 @@ +"""Geometry for the rotated-boundary-conditions figure. + +The cetz skill's rule: geometry computation happens in Python and Typst just +draws. Doing the triangulation here rather than in the figure is what stops +nodes being left out of the mesh -- an earlier version connected nodes by a +distance threshold and silently missed several. + +Emits the schema in the skill's `underworld-bridge.md`: + + {"vertices": [[x, y], ...], + "triangles": [[i, j, k], ...], + "surface": [i, ...], indices of the constrained nodes + "frames": [{"p": [x, y], "n": [nx, ny], "t": [tx, ty]}, ...], + "curve": [[x, y], ...]} the surface, finely sampled + +Run: python3 generate-rotated-basis.py +""" +import json +import pathlib + +import numpy as np +from scipy.spatial import Delaunay + +OUT = pathlib.Path(__file__).with_name("rotated-basis-data.json") + +X0, X1 = -3.5, 3.5 +BASE = -2.9 +NX = 9 # columns of nodes +NY = 4 # rows, surface included + + +def surface_y(x): + """A free surface: rises on the left, falls on the right, with an + inflection between, so the normal swings through a wide range and the + curvature changes sign. No global rotation straightens this out, which is + the reason the figure exists.""" + x = np.asarray(x, dtype=float) + return (0.95 * np.exp(-(((x + 1.75) / 1.30) ** 2)) + - 0.80 * np.exp(-(((x - 1.70) / 1.15) ** 2)) + + 0.75) + + +def surface_slope(x, eps=1.0e-4): + return (surface_y(x + eps) - surface_y(x - eps)) / (2 * eps) + + +# Nodes on a grid warped to sit under the surface. Columns are staggered on +# alternate rows so the Delaunay triangulation comes out as triangles rather +# than as near-degenerate right angles on a perfect lattice. +xs = np.linspace(X0, X1, NX) +pts = [] +surface_idx = [] +for row in range(NY): + frac = row / (NY - 1) # 0 at the base, 1 at the surface + offset = 0.0 if row % 2 == 0 else 0.5 * (xs[1] - xs[0]) + cols = xs + offset + if row == NY - 1: + cols = xs # surface row unstaggered + for x in cols: + if x < X0 - 1e-9 or x > X1 + 1e-9: + continue + top = float(surface_y(x)) + y = BASE + (top - BASE) * frac + if row == NY - 1: + surface_idx.append(len(pts)) + pts.append([float(x), float(y)]) + +pts = np.array(pts) +tri = Delaunay(pts) + +# Drop the slivers Delaunay leaves along a non-convex top edge: any triangle +# whose centroid sits above the surface is outside the domain. +keep = [] +for simplex in tri.simplices: + c = pts[simplex].mean(axis=0) + if c[1] <= float(surface_y(c[0])) + 1.0e-9: + keep.append([int(i) for i in simplex]) + +frames = [] +for i in surface_idx: + x = pts[i][0] + m = float(surface_slope(x)) + n = np.array([-m, 1.0]) + n /= np.linalg.norm(n) + t = np.array([1.0, m]) + t /= np.linalg.norm(t) + frames.append({"p": [float(pts[i][0]), float(pts[i][1])], + "n": [float(n[0]), float(n[1])], + "t": [float(t[0]), float(t[1])]}) + +curve_x = np.linspace(X0, X1, 121) +data = { + "vertices": [[float(a), float(b)] for a, b in pts], + "triangles": keep, + "surface": [int(i) for i in surface_idx], + "frames": frames, + "curve": [[float(a), float(b)] for a, b in zip(curve_x, surface_y(curve_x))], +} +OUT.write_text(json.dumps(data, indent=1)) +print("wrote %s: %d vertices, %d triangles, %d surface nodes" + % (OUT.name, len(data["vertices"]), len(data["triangles"]), + len(data["surface"]))) diff --git a/publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis-data.json b/publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis-data.json new file mode 100644 index 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diff --git a/publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis.typ b/publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis.typ new file mode 100644 index 000000000..0480ba4c8 --- /dev/null +++ b/publications/blog-posts/figures/rotated-boundary-conditions/rotated-basis.typ @@ -0,0 +1,180 @@ +// Rotated boundary conditions: where the basis changes, and where in the +// solver the rotation lives. +// +// Thesis: rotating the degrees of freedom leaves the discrete problem in a +// MIXED basis, but the obligation is CONTAINED -- it lives in the velocity +// block and its multigrid substructure, and the Schur/pressure machinery +// wrapping it never handles a rotated vector. +// +// House style follows publications/blog-posts/figures/arrays-sync-flow.typ +// (cetz 0.3.4, hex colours, helper-function pattern). + +#import "@preview/cetz:0.3.4" + +#set page(width: auto, height: auto, margin: 16pt) +#set text(size: 10pt) + +#cetz.canvas({ + import cetz.draw: * + + // Colours + let cart-fg = rgb("#4a7bf7") + let cart-bg = rgb("#dce8fc") + let rot-fg = rgb("#e57373") + let rot-bg = rgb("#fce4ec") + let plain-bg = rgb("#f2f2f0") + let mg-bg = rgb("#fdf1f4") + let schur-fg = rgb("#49a87c") // the un-rotated half + let schur-bg = rgb("#e8f4e8") + let ink = rgb("#1a1a1a") + let muted = rgb("#7a7a7a") + let hair = rgb("#d0d0d0") + + // ====================================================================== + // LEFT: a free surface, meshed. Geometry (nodes, Delaunay triangles, + // surface frames) comes from generate-rotated-basis.py via JSON -- the + // skill's rule, and the reason every node is now in the triangulation. + // An earlier version joined nodes by a distance threshold and left some out. + // ====================================================================== + let data = json("rotated-basis-data.json") + let vtx = data.vertices + let at(i) = (vtx.at(i).at(0), vtx.at(i).at(1)) + let is-surface(i) = data.surface.contains(i) + + // triangles first, then the surface, then the nodes: painter's algorithm + for tri in data.triangles { + line(at(tri.at(0)), at(tri.at(1)), at(tri.at(2)), close: true, + stroke: (paint: hair, thickness: 0.55pt)) + } + + line(..data.curve.map(q => (q.at(0), q.at(1))), + stroke: (paint: ink, thickness: 1.2pt)) + + // A rotated frame at every constrained node: n the outward surface normal, + // t the tangent. Both follow the node, which is the whole point. + for (k, f) in data.frames.enumerate() { + let p = (f.p.at(0), f.p.at(1)) + let n = f.n + let tg = f.t + let ln = 0.72 + let lt = 0.46 + line(p, (p.at(0) + ln * n.at(0), p.at(1) + ln * n.at(1)), + mark: (end: ">", scale: 0.4), stroke: (paint: rot-fg, thickness: 1.1pt)) + line(p, (p.at(0) + lt * tg.at(0), p.at(1) + lt * tg.at(1)), + mark: (end: ">", scale: 0.4), stroke: (paint: rot-fg, thickness: 1.1pt)) + if k == 4 { + content((p.at(0) + 1.20 * ln * n.at(0), p.at(1) + 1.20 * ln * n.at(1)), + text(fill: rot-fg, size: 9pt, $n$)) + content((p.at(0) + 1.55 * lt * tg.at(0) + 0.24 * n.at(0), + p.at(1) + 1.55 * lt * tg.at(1) + 0.24 * n.at(1)), + text(fill: rot-fg, size: 9pt, $t$)) + } + } + + // the Cartesian frame at one interior node -- identical at every other one, + // which is what makes the surface the odd one out + let ip = at(12) // an interior node with room around it + line(ip, (ip.at(0) + 0.66, ip.at(1)), mark: (end: ">", scale: 0.4), + stroke: (paint: cart-fg, thickness: 1.1pt)) + line(ip, (ip.at(0), ip.at(1) + 0.66), mark: (end: ">", scale: 0.4), + stroke: (paint: cart-fg, thickness: 1.1pt)) + content((ip.at(0) + 0.88, ip.at(1)), text(fill: cart-fg, size: 9pt, $x$)) + content((ip.at(0), ip.at(1) + 0.88), text(fill: cart-fg, size: 9pt, $y$)) + + for i in range(vtx.len()) { + if is-surface(i) { + circle(at(i), radius: 0.13, fill: rot-fg, + stroke: (paint: rot-fg, thickness: 1pt)) + } else { + circle(at(i), radius: 0.11, fill: cart-bg, + stroke: (paint: cart-fg, thickness: 1pt)) + } + } + + content((0, 3.05), text(weight: "bold", size: 10pt, fill: ink, + "A free surface has no preferred direction")) + + circle((-3.3, -3.55), radius: 0.13, fill: rot-fg, stroke: (paint: rot-fg)) + content((-3.05, -3.55), anchor: "west", + text(size: 9pt, fill: ink, [surface node --- solve for $(v_n, v_t)$, hold $v_n$])) + circle((-3.3, -4.1), radius: 0.11, fill: cart-bg, + stroke: (paint: cart-fg, thickness: 1pt)) + content((-3.05, -4.1), anchor: "west", + text(size: 9pt, fill: ink, [interior node --- solve for $(v_x, v_y)$])) + + line((4.35, -4.9), (4.35, 3.3), stroke: (paint: hair, thickness: 0.8pt)) + + // ====================================================================== + // RIGHT: where the rotation lives. Two blocks ABUTTING, not nested: the + // velocity solve is rotated, the Schur/pressure solve is not, and the + // single un-rotation on the boundary between them feeds both the Schur + // solve and everything outside. + // ====================================================================== + let panel(tl, br, title, subtitle, bg, edge) = { + import cetz.draw: * + rect(tl, br, fill: bg, stroke: (paint: edge, thickness: 1pt), radius: 5pt) + content((tl.at(0) + 0.30, tl.at(1) - 0.36), anchor: "west", + text(weight: "bold", size: 9.5pt, fill: edge, title)) + if subtitle != none { + // cetz content() lays out at natural width and spills over the rect. + // Box it to the panel's own width so the text wraps inside the border. + // The canvas default is 1cm per unit, so the arithmetic is direct. + let tw = (br.at(0) - tl.at(0) - 0.60) * 1cm + content((tl.at(0) + 0.30, tl.at(1) - 0.90), anchor: "north-west", + box(width: tw, text(size: 8.5pt, fill: muted, subtitle))) + } + } + + content((10.3, 3.05), text(weight: "bold", size: 10pt, fill: ink, + "Where the rotation lives")) + + // -- the rotated half --------------------------------------------------- + panel((4.9, 1.65), (11.75, -3.70), [Velocity solve --- rotated], + [$hat(A) = Q^T A Q$, #h(0.7em) $hat(b) = Q^T b$, #h(0.7em) + $v_n$ held strongly at surface nodes], rot-bg, rot-fg) + + panel((5.35, -0.15), (11.50, -3.25), "Multigrid", + [the transfers are the only further obligation], mg-bg, rot-fg) + + let mgrow(y, lhs, rhs) = { + import cetz.draw: * + content((5.65, y), anchor: "west", text(size: 9pt, fill: ink, lhs)) + content((8.05, y), anchor: "west", text(size: 9pt, fill: rot-fg, rhs)) + } + mgrow(-1.85, [prolongation], [$P -> Q^T P$]) + mgrow(-2.40, [coarse operators], [inherit $Q$ via $R A P$]) + mgrow(-2.93, [coarse solve], [SVD (rigid rotations)]) + + // -- the un-rotated half, abutting -------------------------------------- + panel((13.35, 1.65), (17.85, -1.35), [Fieldsplit / Schur solve], + [pressure and constraints. Isotropic, and carrying no boundary + condition of this kind, so it never sees a rotated vector.], + schur-bg, schur-fg) + + // -- one un-rotation on the boundary, feeding both ---------------------- + // The junction sits in the gap BETWEEN the two blocks, low enough to clear + // the velocity block's own subtitle -- one un-rotation, two consumers. + let jx = 12.55 + let jy = -0.40 + line((11.75, jy), (jx, jy), stroke: (paint: rot-fg, thickness: 1pt)) + circle((jx, jy), radius: 0.075, fill: rot-fg, stroke: (paint: rot-fg)) + content((jx, jy + 0.45), text(size: 9pt, fill: rot-fg, $v = Q hat(v)$)) + + // branch 1: into the Schur solve, which needs it un-rotated + line((jx, jy), (13.29, jy), mark: (end: ">", scale: 0.4), + stroke: (paint: rot-fg, thickness: 1pt)) + + // branch 2: out to everything else + line((jx, jy), (jx, -2.70), stroke: (paint: rot-fg, thickness: 1pt)) + line((jx, -2.70), (13.20, -2.70), mark: (end: ">", scale: 0.4), + stroke: (paint: rot-fg, thickness: 1pt)) + content((13.33, -2.70), anchor: "west", text(size: 8.5pt, fill: rot-fg, + [and to everything outside ---\ output, advection, the surface update])) + + // convention, stated -- a reader who assumes the transpose reads the + // whole figure backwards + line((4.75, -5.05), (18.1, -5.05), stroke: (paint: hair, thickness: 0.8pt)) + content((4.75, -5.55), anchor: "west", text(size: 9pt, fill: ink, + [Convention: the columns of $Q$ are the nodal frame, so $hat(v) = Q^T v$, + and $Q = I$ at every unconstrained node.])) +})