From 98f3730b4eeb3b043c5385f9cdd12507e5b7ef1f Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 11:14:52 -0700 Subject: [PATCH 01/19] UWTN 2026-017 outline: putting a fault in a mesh Drafted 23-24 August and never committed, so two weeks of outline, five figures and nine example scripts lived only in a worktree. Committing it also closes the allocator's blind spot: `ids_on_branches` searches every ref, but a note that has never been committed is invisible to it, and 2026-017 was offered a second time on 8 September exactly as 2026-012 was offered twice before. The outline stands as written. It frames the fault against the shear band, separates saying WHERE the fault is from saying HOW to model it, sets out the four representations in a table, and carries the S-bend worked example with the split-versus-TI comparison and the series roadmap. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../examples/generate-split-anatomy-data.py | 75 +++++ .../generate-stack-progression-data.py | 130 +++++++++ .../examples/geometry_preview.py | 190 ++++++++++++ .../examples/s_fault.py | 273 ++++++++++++++++++ .../examples/sf_compare_viz.py | 272 +++++++++++++++++ .../examples/sf_viz.py | 249 ++++++++++++++++ .../examples/split-anatomy-data.json | 1 + .../examples/split-anatomy.typ | 90 ++++++ .../examples/stack-progression-data.json | 1 + .../examples/stack-progression.typ | 133 +++++++++ .../figures/s_fault_geometry.png | Bin 0 -> 46114 bytes .../figures/sf_compare_split_ti.png | Bin 0 -> 1693198 bytes .../sf_dcff_trajectories_coarse_c1.png | Bin 0 -> 646789 bytes .../figures/split-anatomy.png | Bin 0 -> 169942 bytes .../figures/stack-progression.png | Bin 0 -> 463823 bytes .../putting-a-fault-in-a-mesh/metadata.yml | 30 ++ .../putting-a-fault-in-a-mesh.md | 152 ++++++++++ 17 files changed, 1596 insertions(+) create mode 100644 articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/generate-stack-progression-data.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/s_fault.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/sf_viz.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json create mode 100644 articles/putting-a-fault-in-a-mesh/examples/split-anatomy.typ create mode 100644 articles/putting-a-fault-in-a-mesh/examples/stack-progression-data.json create mode 100644 articles/putting-a-fault-in-a-mesh/examples/stack-progression.typ create mode 100644 articles/putting-a-fault-in-a-mesh/figures/s_fault_geometry.png create mode 100644 articles/putting-a-fault-in-a-mesh/figures/sf_compare_split_ti.png create mode 100644 articles/putting-a-fault-in-a-mesh/figures/sf_dcff_trajectories_coarse_c1.png create mode 100644 articles/putting-a-fault-in-a-mesh/figures/split-anatomy.png create mode 100644 articles/putting-a-fault-in-a-mesh/figures/stack-progression.png create mode 100644 articles/putting-a-fault-in-a-mesh/metadata.yml create mode 100644 articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md diff --git a/articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py b/articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py new file mode 100644 index 0000000..7e8d18c --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py @@ -0,0 +1,75 @@ +"""Geometry for the split-anatomy figure (cetz draws, Python computes). + +A small structured triangulation of [0,3] x [0,1] with a fault chain along +y = 0.5 from (0.5, 0.5) to (2.5, 0.5): 4 fault edges, 3 interior vertices +(duplicated by the split), 2 tips (shared). The "after" panel is EXPLODED +for display — the Minus side translated down — because the real copies are +geometrically coincident. +""" +import json +import os + +H = 0.5 +NX, NY = 6, 2 +DELTA = 0.22 # display-only explosion offset + +verts = {} +coords = [] + + +def vid(i, j): + if (i, j) not in verts: + verts[(i, j)] = len(coords) + coords.append([i * H, j * H]) + return verts[(i, j)] + + +tris = [] +for i in range(NX): + for j in range(NY): + a, b = vid(i, j), vid(i + 1, j) + c, d = vid(i + 1, j + 1), vid(i, j + 1) + tris.append([a, b, c]) + tris.append([a, c, d]) + +chain = [vid(i, 1) for i in range(1, 6)] # (0.5,0.5) .. (2.5,0.5) +tips = [chain[0], chain[-1]] +interior = chain[1:-1] + +# Minus side = triangles whose centroid lies below the chain SEGMENT span; +# triangles below y=0.5 but outside the span stay welded (no fault there). +side = [] +for t in tris: + cx = sum(coords[v][0] for v in t) / 3.0 + cy = sum(coords[v][1] for v in t) / 3.0 + touches = any(v in chain for v in t) + side.append(-1 if (cy < 0.5 and touches and 0.5 < cx < 2.5) else +1) + +# Exploded positions: replicas of the interior chain vertices move down; +# to keep the Minus flank rigid, the row below them (y = 0) within the span +# moves too. Tips stay put — the lens pins there. +exploded = [list(c) for c in coords] +replicas = {} +for v in interior: + replicas[v] = len(exploded) + exploded.append([coords[v][0], coords[v][1] - DELTA]) +moved_tris = [] +flank = set() +for t, s in zip(tris, side): + if s < 0: + moved_tris.append([replicas.get(v, v) for v in t]) + flank.update(v for v in t if v not in chain) + else: + moved_tris.append(list(t)) +for v in flank: + exploded[v][1] -= DELTA + +out = dict(coords=coords, exploded=exploded, tris=tris, + moved_tris=moved_tris, side=side, chain=chain, tips=tips, + interior=interior, replicas=replicas, delta=DELTA) +here = os.path.dirname(os.path.abspath(__file__)) +with open(os.path.join(here, "split-anatomy-data.json"), "w") as f: + json.dump(out, f) +print("wrote split-anatomy-data.json:", + f"{len(coords)} verts, {len(tris)} tris, chain {len(chain)},", + f"replicas {len(replicas)}") diff --git a/articles/putting-a-fault-in-a-mesh/examples/generate-stack-progression-data.py b/articles/putting-a-fault-in-a-mesh/examples/generate-stack-progression-data.py new file mode 100644 index 0000000..34296ce --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/generate-stack-progression-data.py @@ -0,0 +1,130 @@ +"""Geometry for the stack-on progression figure (cetz draws, Python +computes). + +Five panels from deterministic point sets over [0,2] x [0,1.2] with a +slanted fault segment: (a) the static base mesh with the fault MANIFOLD +overlaid (geometry only — not in the mesh); (b,c) two adapt-on-top +layers, points graded toward the manifold; (d) the cut — vertices placed +ON the segment so the fault becomes a conforming facet chain; (e) the +split, exploded for display exactly as in split-anatomy. +""" +import json +import os + +import numpy as np +from scipy.spatial import Delaunay + +A = np.array([0.50, 0.32]) +B = np.array([1.50, 0.66]) +T = (B - A) / np.linalg.norm(B - A) +N = np.array([-T[1], T[0]]) +LEN = float(np.linalg.norm(B - A)) + + +def seg_dist(p): + s = np.clip((p - A) @ T, 0.0, LEN) + return float(np.linalg.norm(p - (A + s * T))) + + +def grid(h): + xs = np.arange(0.0, 2.0 + 1e-9, h) + ys = np.arange(0.0, 1.0 + 1e-9, h) + return np.array([[x, y] for x in xs for y in ys]) + + +def band(h, width): + """Refinement points near the fault, deliberately NOT conforming to it: + an axis-aligned lattice with deterministic jitter, restricted to the + distance band. A real adapted mesh refines TOWARD the line without + containing it — the conforming chain must be visibly the CUT's + contribution, not an accident of the refinement layout.""" + out = [] + xs = np.arange(0.06, 1.95, h) + ys = np.arange(0.05, 0.96, h) + for i, x in enumerate(xs): + for j, y in enumerate(ys): + q = np.array([ + x + 0.33 * h * np.sin(12.9898 * i + 78.233 * j), + y + 0.33 * h * np.sin(39.3460 * i + 11.135 * j)]) + if (seg_dist(q) < width + and 0.03 < q[0] < 1.97 and 0.03 < q[1] < 0.97): + out.append(q) + return np.array(out) + + +def dedup(points, spacing): + kept = [] + for p in points: + if all(np.linalg.norm(p - q) > spacing for q in kept): + kept.append(p) + return np.array(kept) + + +def triangulate(points): + tri = Delaunay(points) + return points, tri.simplices + + +def pack(points, tris): + return dict(coords=points.tolist(), tris=tris.tolist()) + + +panels = {} + +# (a) base: coarse uniform +base_pts = grid(0.25) +panels["base"] = pack(*triangulate(base_pts)) + +# (b) layer 1: base + a coarse band toward the manifold +l1 = dedup(np.vstack([grid(0.25), band(0.125, 0.22)]), 0.07) +panels["l1"] = pack(*triangulate(l1)) + +# (c) layer 2: tighter band on top +l2 = dedup(np.vstack([grid(0.25), band(0.125, 0.24), band(0.062, 0.10)]), 0.036) +panels["l2"] = pack(*triangulate(l2)) + +# (d) cut: clear a corridor around the segment, then place vertices ON it +keep = np.array([p for p in l2 if seg_dist(p) > 0.048]) +on_seg = np.array([A + s * T for s in np.arange(0.0, LEN + 1e-9, 0.062)]) +cut_pts = np.vstack([keep, on_seg]) +cut_pts, cut_tris = triangulate(cut_pts) +panels["cut"] = pack(cut_pts, cut_tris) +chain = list(range(len(keep), len(cut_pts))) # in s-order already + +# (e) split: explode the Minus flank (side of -N), tips pinned +interior = chain[1:-1] +tips = [chain[0], chain[-1]] +side = [] +for t in cut_tris: + cen = cut_pts[t].mean(axis=0) + s = np.clip((cen - A) @ T, 0.0, LEN) + touches = any(v in chain for v in t) + inside = 0.0 < (cen - A) @ T < LEN + side.append(-1 if ((cen - (A + s * T)) @ N < 0 and touches and inside) + else +1) +DELTA = 0.06 +exploded = [list(p) for p in cut_pts] +replicas = {} +for v in interior: + replicas[v] = len(exploded) + exploded.append(list(cut_pts[v] - DELTA * N)) +moved_tris, flank = [], set() +for t, s in zip(cut_tris.tolist(), side): + if s < 0: + moved_tris.append([replicas.get(v, v) for v in t]) + flank.update(v for v in t if v not in chain) + else: + moved_tris.append(list(t)) +for v in flank: + exploded[v] = list(np.array(exploded[v]) - DELTA * N) +panels["split"] = dict(coords=exploded, + tris=[list(t) for t in moved_tris], side=side) + +out = dict(panels=panels, chain=chain, tips=tips, interior=interior, + replicas={str(k): v for k, v in replicas.items()}, + fault=[A.tolist(), B.tolist()], cut_side=side) +here = os.path.dirname(os.path.abspath(__file__)) +with open(os.path.join(here, "stack-progression-data.json"), "w") as f: + json.dump(out, f) +for name, p in panels.items(): + print(f"{name}: {len(p['coords'])} pts, {len(p['tris'])} tris") diff --git a/articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py b/articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py new file mode 100644 index 0000000..e21993f --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py @@ -0,0 +1,190 @@ +"""Geometry preview for the S-fault 2-D rig — adjust the dials by eye. + +No solving, no meshing: the traces, their ±w/2 band rails (exactly what +the curved 2-D ladder will build), the viscosity layer, and the drive. + + ../run geometry_preview.py → s_fault_geometry.png +""" +import numpy as np +import pyvista as pv + +pv.OFF_SCREEN = True +OUT = "/Users/lmoresi/+Simulations/s_fault_rig" + +# ----------------------------------------------------------------- dials +THETA = np.deg2rad(40.0) # strike of the diagonal baseline +C = np.array([0.5, 0.5]) # baseline centre +A = 0.05 # S half-step (lateral offset amplitude) +LAM = 0.10 # bend length scale (tanh) +S_MAIN = (-0.42, 0.42) # main-trace arc range (blind tips) +S_BRANCH = (0.06, 0.42) # branch: through-line at t = -A +W = 0.03 # band width — the COARSE case (fine: 0.01) +S_F = 0.01 # rung spacing — the COARSE case (fine: 0.005) +# Protocol: every finite-w result runs at BOTH resolutions and is +# compared to the THIN-FAULT (split-node) reference on the same traces. +# Rheology: STRONG everywhere on the +t side (above/left) of the fault +# LINE — the trace continued to the box edges (the fault is a terrane +# boundary; only the modelled segment slips). +# ----------------------------------------------------------------------- + +es = np.array([np.cos(THETA), np.sin(THETA)]) +et = np.array([-np.sin(THETA), np.cos(THETA)]) + + +def main_trace(s): + s = np.atleast_1d(s) + return C + s[:, None] * es + (A * np.tanh(s / LAM))[:, None] * et + + +def branch_trace(s): + s = np.atleast_1d(s) + return C + s[:, None] * es - A * et + + +def rails(P, w): + """±w/2 offsets along per-vertex mitred normals — the ladder rails.""" + t = np.gradient(P, axis=0) + t /= np.linalg.norm(t, axis=1)[:, None] + n = np.column_stack([-t[:, 1], t[:, 0]]) + return P + 0.5 * w * n, P - 0.5 * w * n + + +def as_line(P, z=0.0): + return pv.lines_from_points( + np.column_stack([P, np.full(len(P), z)])) + + +def band_poly(Pp, Pm, z=0.0): + """Quad-strip fill (a single non-convex polygon mis-triangulates).""" + n = len(Pp) + pts = np.column_stack([np.vstack([Pp, Pm]), np.full(2 * n, z)]) + faces = [] + for i in range(n - 1): + faces += [4, i, i + 1, n + i + 1, n + i] + return pv.PolyData(pts, faces=np.asarray(faces)) + + +sm = np.linspace(*S_MAIN, 200) +sb = np.linspace(*S_BRANCH, 120) +Pm = main_trace(sm) +Pb = branch_trace(sb) +Rp, Rm = rails(Pm, W) +Bp, Bm = rails(Pb, W) + +# closest approach branch-tip -> main trace (the stop-short gap) +tip = Pb[0] +gap = float(np.min(np.linalg.norm(Pm - tip, axis=1))) +n_below = (S_MAIN[1] - S_MAIN[0]) / S_F +print(f"main arc ~{S_MAIN[1] - S_MAIN[0]:.2f} ({n_below:.0f} rungs at " + f"s_f={S_F}), step 2A={2 * A}, gap(tip->main)={gap:.3f} " + f"(={gap / S_F:.1f} elements)") + +def box_exit(p, d): + """Where the ray p + t·d leaves the unit box (t > 0).""" + ts = [] + for k, lim in ((0, 0.0), (0, 1.0), (1, 0.0), (1, 1.0)): + if abs(d[k]) > 1e-12: + t = (lim - p[k]) / d[k] + q = p + t * d + if t > 1e-9 and -1e-9 <= q[1 - k] <= 1 + 1e-9: + ts.append(t) + return p + min(ts) * d + + +def dashed(p0, p1, z, dash=0.018): + segs, L = [], float(np.linalg.norm(p1 - p0)) + d = (p1 - p0) / L + s = 0.0 + while s < L: + e = min(s + dash, L) + segs.append((p0 + s * d, p0 + e * d)) + s = e + 0.6 * dash + pts, lines = [], [] + for a, b in segs: + lines += [2, len(pts), len(pts) + 1] + pts += [[a[0], a[1], z], [b[0], b[1], z]] + out = pv.PolyData(np.asarray(pts)) + out.lines = np.asarray(lines) + return out + + +pl = pv.Plotter(off_screen=True, window_size=(1200, 1200)) +pl.set_background("white") + +# STRONG REGION: +t side of the fault line (trace + tangent +# continuations; tanh is saturated at the tips so line == asymptote). +# Rendered from the same implicit rule the rig will paint with: +# t_coord > A tanh(s_coord/λ). +NG = 500 +gx, gy = np.meshgrid(np.linspace(0, 1, NG), np.linspace(0, 1, NG), + indexing="xy") +rel = np.stack([gx - C[0], gy - C[1]], axis=-1) +s_co = rel @ es +t_co = rel @ et +strong = (t_co > A * np.tanh(s_co / LAM)).astype(float) +img = pv.ImageData(dimensions=(NG, NG, 1), + spacing=(1 / (NG - 1), 1 / (NG - 1), 1.0), + origin=(0.0, 0.0, -0.004)) +img.point_data["strong"] = strong.ravel(order="C") +pl.add_mesh(img.threshold(0.5, scalars="strong"), color="#efe3cf", + lighting=False) + +# box outline +box = pv.PolyData(np.array([[0, 0, 0.004], [1, 0, 0.004], + [1, 1, 0.004], [0, 1, 0.004]])) +box.lines = np.array([5, 0, 1, 2, 3, 0]) +pl.add_mesh(box, color="black", line_width=2, lighting=False) + +# bands (what the curved ladder will build), rails, traces +for Pp_, Pm_, col in ((Rp, Rm, "lightsteelblue"), (Bp, Bm, "#b8cfe0")): + pl.add_mesh(band_poly(Pp_, Pm_, z=-0.002), color=col, opacity=0.75, + lighting=False) +for R in (Rp, Rm, Bp, Bm): + pl.add_mesh(as_line(R, z=0.001), color="steelblue", line_width=1.2, + lighting=False) +pl.add_mesh(as_line(Pm, z=0.002), color="crimson", line_width=4, + lighting=False) +# dotted continuations of the fault LINE to the box edges (the terrane +# boundary continues; only the solid segment is modelled as slipping) +tan0 = (Pm[0] - Pm[1]) +tan0 /= np.linalg.norm(tan0) +tan1 = (Pm[-1] - Pm[-2]) +tan1 /= np.linalg.norm(tan1) +for tip, d in ((Pm[0], tan0), (Pm[-1], tan1)): + pl.add_mesh(dashed(tip, box_exit(tip, d), z=0.002), + color="crimson", line_width=3, lighting=False) +pl.add_mesh(as_line(Pb, z=0.002), color="#8b0000", line_width=4, + lighting=False) +for tp in (Pm[0], Pm[-1], Pb[0], Pb[-1]): + pl.add_mesh(pv.PolyData(np.array([[tp[0], tp[1], 0.003]])), + color="black", point_size=10, + render_points_as_spheres=True, lighting=False) + +# drive arrows: v = 2 V t ê_s (plate-motion-frame shear) +for sgn in (+1.0, -1.0): + cen = C + sgn * 0.33 * et - sgn * 0.10 * es + arr = pv.Arrow(start=np.array([*cen, 0.003]), + direction=np.array([*(sgn * es), 0.0]), + scale=0.14, tip_radius=0.06, shaft_radius=0.02) + pl.add_mesh(arr, color="black", lighting=False) + +labels = { + "main (tanh S)": Pm[128] + 0.05 * et, + "branch (through-line)": Pb[70] - 0.075 * et, + "STRONG (eta x contrast)": np.array([0.16, 0.955]), + f"w = {W}": Pm[36] - 0.055 * et, + f"gap = {gap:.3f}": Pb[0] - 0.05 * et - 0.03 * es, + f"2A = {2 * A}": Pm[100] + 0.09 * et, +} +pts = np.array([[q[0], q[1], 0.005] for q in labels.values()]) +pl.add_point_labels(pts, list(labels.keys()), font_size=17, + text_color="black", shape=None, always_visible=True, + show_points=False) + +pl.view_xy() +pl.camera.parallel_projection = True +pl.camera.focal_point = (0.5, 0.5, 0.0) +pl.camera.parallel_scale = 0.56 +pl.screenshot(f"{OUT}/s_fault_geometry.png") +pl.close() +print(f"wrote {OUT}/s_fault_geometry.png") diff --git a/articles/putting-a-fault-in-a-mesh/examples/s_fault.py b/articles/putting-a-fault-in-a-mesh/examples/s_fault.py new file mode 100644 index 0000000..769a2ff --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/s_fault.py @@ -0,0 +1,273 @@ +"""The S-fault 2-D rig: San Andreas bend + through-line branch (#629). + +Geometry of record (see geometry_preview.py / the plan file): tanh S on +the 40° diagonal, the branch as the stop-short through-line alternative +in the weak block, the fault line as the TERRANE boundary (strong η × +contrast above/left, painted by the same implicit rule), plate-motion +shear drive. Split-node (thin-fault reference) representation. + +Two-resolution protocol: -uw_res coarse (w=0.03, s_f=0.01) | +fine (w=0.01, s_f=0.005). Every finite-w (volumetric) result must run +both and compare to THIS split reference. + + ../run s_fault.py -uw_fmg 1 # FMG [L0, L1] + finest + ../run s_fault.py -uw_fmg 0 # GAMG control + ../run s_fault.py -uw_contrast 1e3 # strong block dial + ../run s_fault.py -uw_sense -1 # releasing-bend drive +""" +import time + +import numpy as np + +import underworld3 as uw +from underworld3.utilities import fault_contact +from underworld3.utilities.place_surface import place_fault_ribbon_2d + +params = uw.Params( + res=uw.Param("coarse", "coarse (w=0.03, s_f=0.01) | fine " + "(w=0.01, s_f=0.005) — the two-resolution " + "protocol"), + contrast=uw.Param(1.0, "strong-block viscosity factor (the terrane " + "above/left of the fault line)"), + sense=uw.Param(1.0, "drive sign: +1 restraining bend, -1 releasing"), + fmg=uw.Param(1.0, "1 = FMG tail [L0, L1]; 0 = GAMG control"), + h_bg=uw.Param(1 / 8, "L0 cell size (refinement=1 halves it for L1)"), + margin=uw.Param(2.0, "tip margin rings (extrapolated band surround)"), + tol=uw.Param(1e-5, "stokes tolerance"), + strategy=uw.Param("robust", "fast | robust smoothers"), + penalty=uw.Param(0.0, "AL penalty (gamma=1 is the sanctioned choice)"), + rep=uw.Param("split", "split (thin-fault reference) | ti (volumetric " + "weak shear plane; w IS physics — run BOTH " + "resolutions, the protocol)"), + eta1=uw.Param(1e-3, "TI shear viscosity in the fault footprint " + "(rep=ti; isotropic elsewhere)"), +) + +# ---------------------------------------------------- geometry of record +THETA = np.deg2rad(40.0) +C = np.array([0.5, 0.5]) +A, LAM = 0.05, 0.10 +S_MAIN = (-0.42, 0.42) +S_BRANCH = (0.06, 0.42) +ES = np.array([np.cos(THETA), np.sin(THETA)]) +ET = np.array([-np.sin(THETA), np.cos(THETA)]) + +RES = {"coarse": (0.03, 0.01), "fine": (0.01, 0.005)} +W, S_F = RES[str(params.res)] + + +def resample_arclength(P, spacing): + """Equispaced-in-arclength samples of a dense polyline, ends kept, + interval count EVEN (so a 2:1 subsample exists when wanted).""" + seg = np.linalg.norm(np.diff(P, axis=0), axis=1) + arc = np.concatenate([[0.0], np.cumsum(seg)]) + n = max(2, int(round(arc[-1] / spacing))) + n += n % 2 + s = np.linspace(0.0, arc[-1], n + 1) + return np.column_stack([np.interp(s, arc, P[:, 0]), + np.interp(s, arc, P[:, 1])]) + + +def main_curve(s): + return C + s[:, None] * ES + (A * np.tanh(s / LAM))[:, None] * ET + + +dense = main_curve(np.linspace(*S_MAIN, 2000)) +trace_main = resample_arclength(dense, S_F) +sb = np.linspace(*S_BRANCH, 2000) +trace_branch = resample_arclength(C + sb[:, None] * ES - A * ET, S_F) + +base = uw.meshing.UnstructuredSimplexBox( + minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), + cellSize=float(params.h_bg), regular=False, qdegree=2, refinement=1) + +rep = str(params.rep) +t0 = time.perf_counter() +mesh, pinfo = place_fault_ribbon_2d( + base, [("Main", trace_main), ("Branch", trace_branch)], W, + margin_rings=int(float(params.margin)), split=(rep == "split")) +uw.pprint(f"[{params.res}/{rep}] {pinfo['n_cells']} cells, rungs " + f"{pinfo['n_rungs']}, build {time.perf_counter() - t0:.1f} s") + +x, y = mesh.X +v = uw.discretisation.MeshVariable("vS", mesh, 2, degree=2) +p = uw.discretisation.MeshVariable("pS", mesh, 1, degree=1, + continuous=True) +stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) +contrast = float(params.contrast) + +# THE TERRANE: strong on the +t side of the fault LINE — the same +# implicit rule as the diagram, painted P0 by cell centroid. +eta0 = uw.discretisation.MeshVariable("etaS", mesh, 1, degree=0) +cen = np.asarray(eta0.coords) - C +s_co = cen @ ES +t_co = cen @ ET +strong = t_co > A * np.tanh(s_co / LAM) +eta0_vals = np.where(strong, contrast, 1.0) +eta0.array[:, 0, 0] = eta0_vals +if contrast != 1.0: + uw.pprint(f"terrane: {int(strong.sum())} strong cells at " + f"eta {contrast:g}") + +if rep == "ti": + # VOLUMETRIC representation: each band carries a TI weak shear + # plane. The paint honours the FAULT FOOTPRINT (never the + # extrapolated margin), per strand; the director follows each + # strand's own orientation PER CELL (the main band is curved). + eta1_val = float(params.eta1) + band_main = mesh.cells_labelled("Band", 71) + band_branch = mesh.cells_labelled("Band", 72) + foot_main = band_main & (s_co >= S_MAIN[0]) & (s_co <= S_MAIN[1]) + foot_branch = band_branch & (s_co >= S_BRANCH[0]) \ + & (s_co <= S_BRANCH[1]) + foot = foot_main | foot_branch + eta1 = uw.discretisation.MeshVariable("etaT", mesh, 1, degree=0) + eta1.array[:, 0, 0] = np.where(foot, eta1_val, eta0_vals) + ndir = uw.discretisation.MeshVariable("dirT", mesh, 2, degree=0, + continuous=False) + sech2 = 1.0 / np.cosh(s_co / LAM) ** 2 + n_main = ET[None, :] - (A / LAM) * sech2[:, None] * ES[None, :] + n_main /= np.linalg.norm(n_main, axis=1, keepdims=True) + dvals = np.tile(ET, (len(s_co), 1)) # branch/default: ê_t + dvals[foot_main] = n_main[foot_main] # main: curved normal + ndir.array[...] = dvals.reshape(ndir.array.shape) + uw.pprint(f"[ti] painted {int(foot.sum())} footprint cells " + f"({int(band_main.sum())}+{int(band_branch.sum())} band) " + f"at eta_1 {eta1_val:g}") + stokes.constitutive_model = \ + uw.constitutive_models.TransverseIsotropicFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = eta0.sym[0] + stokes.constitutive_model.Parameters.shear_viscosity_1 = eta1.sym[0] + stokes.constitutive_model.Parameters.director = ndir.sym +else: + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = \ + (eta0.sym[0] if contrast != 1.0 else 1.0) +stokes.bodyforce = [0.0, 0.0] + +# plate-motion-frame shear: v = 2 V t_coord ê_s (sense flips the bend +# between restraining and releasing) +sense = float(params.sense) +t_sym = (x - 0.5) * float(ET[0]) + (y - 0.5) * float(ET[1]) +vx = sense * 2.0 * t_sym * float(ES[0]) +vy = sense * 2.0 * t_sym * float(ES[1]) +for wall in ("Bottom", "Top", "Left", "Right"): + stokes.add_dirichlet_bc((vx, vy), wall) +if rep == "split": + stokes.add_fault_bc(0, boundary="Main") + stokes.add_fault_bc(0, boundary="Branch") +stokes.petsc_use_pressure_nullspace = True +stokes.tolerance = float(params.tol) +if float(params.penalty) > 0.0: + stokes.penalty = float(params.penalty) + +mode = "fmg" if float(params.fmg) > 0.5 else "gamg" +if mode == "fmg": + from underworld3.utilities.custom_mg import set_custom_fmg + + tail = base._coarse_level_meshes()[:-1] + [base] + uw.pprint(f"tail: {[m.dm.getHeightStratum(0)[1] for m in tail]} " + f"cells + finest {pinfo['n_cells']}") + stokes.strategy = str(params.strategy) + # keying ruling: split = structural only; TI = the painted footprint + set_custom_fmg(stokes, tail, field_id=0, + fac_zone=(foot if rep == "ti" else None)) + +if rep == "split": + t0 = time.perf_counter() + info = fault_contact.solve_with_fault(stokes) + t1 = time.perf_counter() - t0 + if contrast != 1.0: + eta0.array[:, 0, 0] = np.where(strong, 1.5 * contrast, 1.0) + t0 = time.perf_counter() + info2 = fault_contact.solve_with_fault(stokes) + t2 = time.perf_counter() - t0 + cache = getattr(stokes, "_rotated_linear_cache", None) or {} + ctx = cache.get("ctx") or {} +else: + t0 = time.perf_counter() + stokes.solve() + t1 = time.perf_counter() - t0 + eta1.array[:, 0, 0] = np.where(foot, 1.5 * eta1_val, eta0_vals) + t0 = time.perf_counter() + stokes.solve(zero_init_guess=False) + t2 = time.perf_counter() - t0 + ksp0 = stokes.snes.getKSP() + sub = ksp0.getPC().getFieldSplitSubKSP() + info = info2 = {"converged": stokes.snes.getConvergedReason() > 0} + ctx = {"ksp": ksp0, "vel_its_last": sub[0].getIterationNumber(), + "pres_its_last": sub[1].getIterationNumber()} +uw.pprint(f"[{mode}] solve {t1:.1f} s, repeat {t2:.1f} s; converged " + f"{info.get('converged')}/{info2.get('converged')}") +uw.pprint(f"[{mode}] last apply: vel {ctx.get('vel_its_last')} its, " + f"pres {ctx.get('pres_its_last')} its") +try: + velpc = ctx["ksp"].getPC().getFieldSplitSubKSP()[0].getPC() + if velpc.getType() == "mg": + for l in range(velpc.getMGLevels()): + sm = velpc.getMGSmoother(l) + Aop = sm.getOperators()[0] + n = Aop.getSize()[0] + spc = sm.getPC() + fac = "" + if spc.getType() == "asm": + sub = spc.getASMSubKSP() + ssz = sum(k.getOperators()[0].getSize()[0] for k in sub) + fac = f", asm {len(sub)} block(s) {ssz}/{n} rows" + uw.pprint(f"[{mode}] level {l}: n={n}, nnz/row=" + f"{Aop.getInfo()['nz_used'] / max(n, 1):.0f}{fac}") +except Exception as exc: + uw.pprint(f"[{mode}] level probe failed: {exc}") + +# -------------------- slip partitioning (the hook), two spellings: +# (a) the discrete pair slip (split only); (b) a FIXED-OFFSET +# across-band Delta v_t profile at +-0.03 from each trace — the +# representation-agnostic measure, comparable across w and rep. +OFF = 0.03 + + +def across_profile(trace): + P = trace[::2] + t = np.gradient(P, axis=0) + t /= np.linalg.norm(t, axis=1)[:, None] + n = np.column_stack([-t[:, 1], t[:, 0]]) + vv_p = np.asarray(uw.function.evaluate( + v.sym, P + OFF * n)).reshape(len(P), -1) + vv_m = np.asarray(uw.function.evaluate( + v.sym, P - OFF * n)).reshape(len(P), -1) + dv = vv_p[:, :2] - vv_m[:, :2] + dvt = np.einsum("ij,ij->i", dv, t) + s_along = (P - C) @ ES + return s_along, np.abs(dvt) + + +profiles = {} +for label, trace in (("Main", trace_main), ("Branch", trace_branch)): + s_a, dvt = across_profile(trace) + profiles[label] = (s_a, dvt) + line = (f"[{mode}/{rep}] {label}: dv_t(+-{OFF}) peak {dvt.max():.4f}" + f", mean {dvt.mean():.4f}") + if rep == "split": + coords, jumps, normals = fault_contact.fault_pair_jumps( + stokes, label, stokes._rotated_freeslip_info) + jn = np.einsum("ij,ij->i", jumps, normals) + tang = np.linalg.norm(jumps - jn[:, None] * normals, axis=1) + line += (f"; pair slip peak {tang.max():.4f}, leak " + f"{np.abs(jn).max():.2e}") + uw.pprint(line) +ratio = profiles["Branch"][1].max() / max(profiles["Main"][1].max(), 1e-30) +uw.pprint(f"[{mode}/{rep}] PARTITION branch/main peak dv_t ratio: " + f"{ratio:.3f}") +np.savez(f"slip_profiles_{params.res}_{rep}_{mode}_c{contrast:g}" + f"_s{int(sense)}.npz", + **{f"{k}_{q}": arr for k, (s_, t_) in profiles.items() + for q, arr in (("s", s_), ("dvt", t_))}) + +# CHECKPOINT the solved state (Louis's rule: figures render from +# checkpoints, never by re-solving) — mesh + v + p (+ eta fields). +ckpt = f"ckpt_{params.res}_{rep}_c{contrast:g}_s{int(sense)}" +mvars = [v, p, eta0] + ([eta1] if rep == "ti" else []) +mesh.write_timestep(ckpt, index=0, outputPath="checkpoints", + meshVars=mvars, meshUpdates=True) +uw.pprint(f"checkpointed -> checkpoints/{ckpt} (mesh + " + f"{[q.name for q in mvars]})") diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py b/articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py new file mode 100644 index 0000000..7b06dbd --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py @@ -0,0 +1,272 @@ +"""Three-way comparison figure: split(fine) / TI(coarse) / TI(fine). + +Rows stacked, one per case; columns = dCFF (slip vs welded/isotropic +locked reference, SHARED clim across rows) and the principal-stress +trajectory net. Fine grey mesh under everything; nodal dCFF on each +mesh's own triangulation (the artefact rule). + + ../run sf_compare_viz.py → sf_compare_split_ti.png +""" +import numpy as np +import pyvista as pv +import sympy +from scipy.interpolate import RegularGridInterpolator + +import underworld3 as uw +import underworld3.visualisation as vis +from underworld3.utilities import fault_contact +from underworld3.utilities.custom_mg import set_custom_fmg +from underworld3.utilities.place_surface import place_fault_ribbon_2d +from underworld3.visualisation.glyphs import (direction_trajectories, + trajectories_to_pv_lines) + +pv.OFF_SCREEN = True +OUT = "/Users/lmoresi/+Simulations/s_fault_rig" + +THETA = np.deg2rad(40.0) +C = np.array([0.5, 0.5]) +A, LAM = 0.05, 0.10 +S_MAIN, S_BRANCH = (-0.42, 0.42), (0.06, 0.42) +ES = np.array([np.cos(THETA), np.sin(THETA)]) +ET = np.array([-np.sin(THETA), np.cos(THETA)]) +RES = {"coarse": (0.03, 0.01), "fine": (0.01, 0.005)} +SENSE, MU, ETA1 = 1.0, 0.4, 1e-3 +CASES = [("split", "fine"), ("ti", "coarse"), ("ti", "fine")] + +NX, NY = 260, 260 +xs = np.linspace(0.02, 0.98, NX) +ys = np.linspace(0.02, 0.98, NY) +grid_pts = np.column_stack([q.ravel() for q in + np.meshgrid(xs, ys, indexing="xy")]) + + +def resample_arclength(P, spacing): + seg = np.linalg.norm(np.diff(P, axis=0), axis=1) + arc = np.concatenate([[0.0], np.cumsum(seg)]) + n = max(2, int(round(arc[-1] / spacing))) + n += n % 2 + s = np.linspace(0.0, arc[-1], n + 1) + return np.column_stack([np.interp(s, arc, P[:, 0]), + np.interp(s, arc, P[:, 1])]) + + +def traces_at(s_f): + sm = np.linspace(*S_MAIN, 2000) + tm = resample_arclength( + C + sm[:, None] * ES + (A * np.tanh(sm / LAM))[:, None] * ET, s_f) + sb = np.linspace(*S_BRANCH, 2000) + tb = resample_arclength(C + sb[:, None] * ES - A * ET, s_f) + return [("Main", tm), ("Branch", tb)] + + +def stokes_on(mesh, tag, ti_weak): + """A drive-BC Stokes on ``mesh``; TI weak planes when ti_weak.""" + x, y = mesh.X + v = uw.discretisation.MeshVariable(f"v{tag}", mesh, 2, degree=2) + p = uw.discretisation.MeshVariable(f"p{tag}", mesh, 1, degree=1, + continuous=True) + st = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) + foot = None + if ti_weak: + eta1 = uw.discretisation.MeshVariable(f"e{tag}", mesh, 1, degree=0) + cen = np.asarray(eta1.coords) - C + s_co, _t_co = cen @ ES, cen @ ET + bm = mesh.cells_labelled("Band", 71) + bb = mesh.cells_labelled("Band", 72) + foot = (bm & (s_co >= S_MAIN[0]) & (s_co <= S_MAIN[1])) \ + | (bb & (s_co >= S_BRANCH[0]) & (s_co <= S_BRANCH[1])) + eta1.array[:, 0, 0] = np.where(foot, ETA1, 1.0) + ndir = uw.discretisation.MeshVariable(f"d{tag}", mesh, 2, degree=0, + continuous=False) + sech2 = 1.0 / np.cosh(s_co / LAM) ** 2 + nm = ET[None, :] - (A / LAM) * sech2[:, None] * ES[None, :] + nm /= np.linalg.norm(nm, axis=1, keepdims=True) + dvals = np.tile(ET, (len(s_co), 1)) + fm = bm & (s_co >= S_MAIN[0]) & (s_co <= S_MAIN[1]) + dvals[fm] = nm[fm] + ndir.array[...] = dvals.reshape(ndir.array.shape) + st.constitutive_model = \ + uw.constitutive_models.TransverseIsotropicFlowModel + st.constitutive_model.Parameters.shear_viscosity_0 = 1.0 + st.constitutive_model.Parameters.shear_viscosity_1 = eta1.sym[0] + st.constitutive_model.Parameters.director = ndir.sym + else: + st.constitutive_model = uw.constitutive_models.ViscousFlowModel + st.constitutive_model.Parameters.shear_viscosity_0 = 1.0 + st.bodyforce = [0.0, 0.0] + t_sym = (x - 0.5) * float(ET[0]) + (y - 0.5) * float(ET[1]) + for wall in ("Bottom", "Top", "Left", "Right"): + st.add_dirichlet_bc((SENSE * 2.0 * t_sym * float(ES[0]), + SENSE * 2.0 * t_sym * float(ES[1])), wall) + st.petsc_use_pressure_nullspace = True + st.tolerance = 1e-5 + st.strategy = "robust" + base = mesh._sf_base # stashed by build_case + set_custom_fmg(st, base._coarse_level_meshes()[:-1] + [base], + field_id=0, fac_zone=foot) + return st, p + + +def sampler(st, p, tag): + tau = st.constitutive_model.flux + fields = {} + for key, expr in (("xx", tau[0, 0]), ("xy", tau[0, 1]), + ("yy", tau[1, 1]), ("p", p.sym[0])): + wv = uw.discretisation.MeshVariable(f"w{key}{tag}", st.mesh, 1, + degree=1) + proj = uw.systems.Projection(st.mesh, wv) + proj.smoothing = 0.0 + proj.uw_function = expr + proj.solve() + fields[key] = wv + p_mean = float(fields["p"].data[:, 0].mean()) + + def sig_at(P): + c = {k: np.asarray(uw.function.evaluate(wv.sym[0], P)).reshape(-1) + for k, wv in fields.items()} + pp = c["p"] - p_mean + sig = np.zeros((len(P), 2, 2)) + sig[:, 0, 0] = c["xx"] - pp + sig[:, 1, 1] = c["yy"] - pp + sig[:, 0, 1] = sig[:, 1, 0] = c["xy"] + return sig + + return sig_at + + +def receiver_normals(P): + rel = P - C + s_co = rel @ ES + sech2 = 1.0 / np.cosh(s_co / LAM) ** 2 + n = ET[None, :] - (A / LAM) * sech2[:, None] * ES[None, :] + return n / np.linalg.norm(n, axis=1, keepdims=True) + + +def cff(sig, n): + t = np.einsum("...ij,...j->...i", sig, n) + sn = np.einsum("...i,...i->...", t, n) + ts = np.linalg.norm(t - sn[..., None] * n, axis=-1) + return ts + MU * sn + + +def build_case(rep, res): + w, s_f = RES[res] + traces = traces_at(s_f) + base = uw.meshing.UnstructuredSimplexBox( + minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), cellSize=1 / 8, + regular=False, qdegree=2, refinement=1) + tag = f"{rep[0]}{res[0]}" + # slipping state + mesh, _ = place_fault_ribbon_2d(base, traces, w, + split=(rep == "split")) + mesh._sf_base = base + st_s, p_s = stokes_on(mesh, f"S{tag}", ti_weak=(rep == "ti")) + if rep == "split": + st_s.add_fault_bc(0, boundary="Main") + st_s.add_fault_bc(0, boundary="Branch") + fault_contact.solve_with_fault(st_s) + else: + st_s.solve() + sig_at_s = sampler(st_s, p_s, f"S{tag}") + # locked reference: welded (split) / isotropic (ti) + if rep == "split": + mesh_l, _ = place_fault_ribbon_2d(base, traces, w, split=False) + mesh_l._sf_base = base + else: + mesh_l = mesh + st_l, p_l = stokes_on(mesh_l, f"L{tag}", ti_weak=False) + st_l.solve() + sig_at_l = sampler(st_l, p_l, f"L{tag}") + + pvm = vis.mesh_to_pv_mesh(mesh) + verts = np.asarray(pvm.points[:, :2]) + nrec = receiver_normals(verts) + pvm.point_data["dCFF"] = cff(sig_at_s(verts), nrec) \ + - cff(sig_at_l(verts), nrec) + return {"mesh": mesh, "pvm": pvm, "sig_grid": sig_at_s(grid_pts), + "traces": traces} + + +cases = {} +for rep, res in CASES: + uw.pprint(f"--- {rep}/{res} ---") + cases[(rep, res)] = build_case(rep, res) + d = cases[(rep, res)]["pvm"].point_data["dCFF"] + uw.pprint(f"[{rep}/{res}] dCFF range {d.min():.3f} .. {d.max():.3f}") + +vmax = float(np.percentile(np.abs(np.concatenate( + [np.asarray(c["pvm"].point_data["dCFF"]) for c in cases.values()])), + 98)) + + +def trajectories_for(sig_grid): + interp = {k: RegularGridInterpolator( + (xs, ys), sig_grid.reshape(NY, NX, 2, 2)[..., i, j].T, + bounds_error=False, fill_value=None) + for k, (i, j) in (("xx", (0, 0)), ("xy", (0, 1)), + ("yy", (1, 1)))} + + def direction_at_factory(family): + def direction_at(pt): + if not (0.03 <= pt[0] <= 0.97 and 0.03 <= pt[1] <= 0.97): + return None + s = np.array([[interp["xx"](pt)[0], interp["xy"](pt)[0]], + [interp["xy"](pt)[0], interp["yy"](pt)[0]]]) + w_, v_ = np.linalg.eigh(s) + if abs(w_[1] - w_[0]) < 1e-12: + return None + u = v_[:, 1] if family == "tensile" else v_[:, 0] + return u / np.linalg.norm(u) + return direction_at + + seeds = np.column_stack([q.ravel() for q in np.meshgrid( + np.linspace(0.06, 0.94, 30), np.linspace(0.06, 0.94, 30), + indexing="xy")]) + np.random.default_rng(3).shuffle(seeds) + inside = lambda pt: 0.03 <= pt[0] <= 0.97 and 0.03 <= pt[1] <= 0.97 + return {fam: direction_trajectories( + direction_at_factory(fam), seeds, inside, step=0.004, + separation=0.028) for fam in ("compressive", "tensile")} + + +pl = pv.Plotter(shape=(3, 2), off_screen=True, window_size=(2100, 3100)) +titles = {("split", "fine"): "split node, fine (w=0.01 convenience)", + ("ti", "coarse"): "TI band, coarse (w=0.03, eta1=1e-3)", + ("ti", "fine"): "TI band, fine (w=0.01, eta1=1e-3)"} +for row, key in enumerate(CASES): + case = cases[key] + edges = vis.mesh_to_pv_mesh(case["mesh"]).extract_all_edges() + edges.points[:, 2] = 0.001 + + def common(pl): + pl.set_background("white") + pl.add_mesh(edges.copy(), color="grey", line_width=0.3, + opacity=0.3, lighting=False) + for _lbl, P in case["traces"]: + pl.add_mesh(pv.lines_from_points( + np.column_stack([P, np.full(len(P), 0.003)])), + color="black", line_width=2.0, lighting=False) + pl.view_xy() + pl.camera.parallel_projection = True + pl.camera.focal_point = (0.5, 0.5, 0.0) + pl.camera.parallel_scale = 0.52 + + pl.subplot(row, 0) + pl.add_mesh(case["pvm"], scalars="dCFF", cmap="RdBu_r", + clim=(-vmax, vmax), lighting=False, + scalar_bar_args={"title": "dCFF", "color": "black"} + if row == 0 else None, + show_scalar_bar=(row == 0)) + common(pl) + pl.add_text(f"dCFF: {titles[key]}", font_size=12, color="black") + pl.subplot(row, 1) + traj = trajectories_for(case["sig_grid"]) + for fam, col in (("compressive", "#2166ac"), ("tensile", "#b2182b")): + pl.add_mesh(trajectories_to_pv_lines(traj[fam]), color=col, + line_width=1.5, lighting=False) + common(pl) + pl.add_text("stress trajectories", font_size=12, color="black") + +pl.screenshot(f"{OUT}/sf_compare_split_ti.png") +pl.close() +print(f"wrote {OUT}/sf_compare_split_ti.png") diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_viz.py b/articles/putting-a-fault-in-a-mesh/examples/sf_viz.py new file mode 100644 index 0000000..6858090 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_viz.py @@ -0,0 +1,249 @@ +"""The S-fault rig's standard figures (Louis's spec, 2026-08-24). + +Two panels over a very fine grey mesh: + A — ΔCFF, slip vs NO-SLIP: the locked reference is a WELDED solve + (same placement, split=False; at contrast=1 it reproduces uniform + shear — the built-in validation), receiver planes at the local + fault-line orientation, μ_f = 0.4. + B — principal-stress TRAJECTORIES of the slipping state (PR #601 + integrator: mod-180 continuity, Jobard–Lehmann spacing), blue + compressive / red tensile per the standing glyph ruling; + demeaned-p gauge. + + ../run sf_viz.py [-uw_res coarse|fine] [-uw_contrast C] [-uw_sense S] + → sf_dcff_trajectories__c.png +""" +import numpy as np +import pyvista as pv +from scipy.interpolate import RegularGridInterpolator + +import underworld3 as uw +import underworld3.visualisation as vis +from underworld3.utilities import fault_contact +from underworld3.utilities.custom_mg import set_custom_fmg +from underworld3.utilities.place_surface import place_fault_ribbon_2d +from underworld3.visualisation.glyphs import (direction_trajectories, + trajectories_to_pv_lines) + +pv.OFF_SCREEN = True +OUT = "/Users/lmoresi/+Simulations/s_fault_rig" + +params = uw.Params( + res=uw.Param("coarse", "coarse | fine (the two-resolution protocol)"), + contrast=uw.Param(1.0, "strong-terrane viscosity factor"), + sense=uw.Param(1.0, "+1 restraining, -1 releasing"), + mu_f=uw.Param(0.4, "CFF friction coefficient"), +) + +THETA = np.deg2rad(40.0) +C = np.array([0.5, 0.5]) +A, LAM = 0.05, 0.10 +S_MAIN, S_BRANCH = (-0.42, 0.42), (0.06, 0.42) +ES = np.array([np.cos(THETA), np.sin(THETA)]) +ET = np.array([-np.sin(THETA), np.cos(THETA)]) +RES = {"coarse": (0.03, 0.01), "fine": (0.01, 0.005)} +W, S_F = RES[str(params.res)] +CONTRAST, SENSE, MU = (float(params.contrast), float(params.sense), + float(params.mu_f)) + +NX, NY = 260, 260 +xs = np.linspace(0.02, 0.98, NX) +ys = np.linspace(0.02, 0.98, NY) +pts = np.column_stack([q.ravel() for q in + np.meshgrid(xs, ys, indexing="xy")]) + + +def resample_arclength(P, spacing): + seg = np.linalg.norm(np.diff(P, axis=0), axis=1) + arc = np.concatenate([[0.0], np.cumsum(seg)]) + n = max(2, int(round(arc[-1] / spacing))) + n += n % 2 + s = np.linspace(0.0, arc[-1], n + 1) + return np.column_stack([np.interp(s, arc, P[:, 0]), + np.interp(s, arc, P[:, 1])]) + + +sm = np.linspace(*S_MAIN, 2000) +trace_main = resample_arclength( + C + sm[:, None] * ES + (A * np.tanh(sm / LAM))[:, None] * ET, S_F) +sb = np.linspace(*S_BRANCH, 2000) +trace_branch = resample_arclength(C + sb[:, None] * ES - A * ET, S_F) +TRACES = [("Main", trace_main), ("Branch", trace_branch)] + + +def solve_state(split): + base = uw.meshing.UnstructuredSimplexBox( + minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), cellSize=1 / 8, + regular=False, qdegree=2, refinement=1) + mesh, _info = place_fault_ribbon_2d(base, TRACES, W, split=split) + x, y = mesh.X + v = uw.discretisation.MeshVariable("vV", mesh, 2, degree=2) + p = uw.discretisation.MeshVariable("pV", mesh, 1, degree=1, + continuous=True) + stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + if CONTRAST != 1.0: + eta = uw.discretisation.MeshVariable("etaV", mesh, 1, degree=0) + cen = np.asarray(eta.coords) - C + strong = (cen @ ET) > A * np.tanh((cen @ ES) / LAM) + eta.array[:, 0, 0] = np.where(strong, CONTRAST, 1.0) + stokes.constitutive_model.Parameters.shear_viscosity_0 = eta.sym[0] + else: + stokes.constitutive_model.Parameters.shear_viscosity_0 = 1.0 + stokes.bodyforce = [0.0, 0.0] + t_sym = (x - 0.5) * float(ET[0]) + (y - 0.5) * float(ET[1]) + for wall in ("Bottom", "Top", "Left", "Right"): + stokes.add_dirichlet_bc((SENSE * 2.0 * t_sym * float(ES[0]), + SENSE * 2.0 * t_sym * float(ES[1])), wall) + stokes.petsc_use_pressure_nullspace = True + stokes.tolerance = 1e-5 + stokes.strategy = "robust" + set_custom_fmg(stokes, base._coarse_level_meshes()[:-1] + [base], + field_id=0) + if split: + stokes.add_fault_bc(0, boundary="Main") + stokes.add_fault_bc(0, boundary="Branch") + fault_contact.solve_with_fault(stokes) + else: + stokes.solve() + + # P1 recovery into PERSISTENT per-component fields; the sampler + # closure assembles sigma at ANY points. At mesh VERTICES the P1 + # evaluation is exact whichever cell the locator picks — the + # artefact-free way to render (off-lattice grid sampling dapples + # at element boundaries; Louis's catch). + tag = "s" if split else "l" + tau = stokes.constitutive_model.flux + fields = {} + for key, expr in (("xx", tau[0, 0]), ("xy", tau[0, 1]), + ("yy", tau[1, 1]), ("p", p.sym[0])): + wv = uw.discretisation.MeshVariable(f"w{key}{tag}", mesh, 1, + degree=1) + proj = uw.systems.Projection(mesh, wv) + proj.smoothing = 0.0 + proj.uw_function = expr + proj.solve() + fields[key] = wv + p_mean = float(fields["p"].data[:, 0].mean()) # demeaned-p gauge + + def sig_at(P): + c = {k: np.asarray(uw.function.evaluate(wv.sym[0], P)).reshape(-1) + for k, wv in fields.items()} + pp = c["p"] - p_mean + sig = np.zeros((len(P), 2, 2)) + sig[:, 0, 0] = c["xx"] - pp + sig[:, 1, 1] = c["yy"] - pp + sig[:, 0, 1] = sig[:, 1, 0] = c["xy"] + return sig + + return sig_at, mesh + + +def receiver_normals(P): + rel = P - C + s_co = rel @ ES + sech2 = 1.0 / np.cosh(s_co / LAM) ** 2 + # grad(t - A tanh(s/lam)) = et - (A/lam) sech^2 * es + n = ET[None, :] - (A / LAM) * sech2[:, None] * ES[None, :] + return n / np.linalg.norm(n, axis=1, keepdims=True) + + +def cff(sig, n): + t = np.einsum("...ij,...j->...i", sig, n) + sn = np.einsum("...i,...i->...", t, n) + ts = np.linalg.norm(t - sn[..., None] * n, axis=-1) + return ts + MU * sn + + +uw.pprint("--- slipping state ---") +sig_at_s, mesh_s = solve_state(split=True) +uw.pprint("--- locked (welded) state ---") +sig_at_l, _mesh_l = solve_state(split=False) + +# NODAL dCFF on the slipping mesh's own triangulation (artefact-free) +pv_mesh = vis.mesh_to_pv_mesh(mesh_s) +verts = np.asarray(pv_mesh.points[:, :2]) +sigv_s = sig_at_s(verts) +sigv_l = sig_at_l(verts) +if CONTRAST == 1.0: + dev = np.abs(sigv_l[:, 0, 1] - sigv_l[:, 0, 1].mean()).max() + uw.pprint(f"locked-state validation (contrast=1): sigma_xy spread " + f"{dev:.2e} about uniform shear") +nrec_v = receiver_normals(verts) +dcff = cff(sigv_s, nrec_v) - cff(sigv_l, nrec_v) +pv_mesh.point_data["dCFF"] = dcff +uw.pprint(f"dCFF range {dcff.min():.3f} .. {dcff.max():.3f}") + +# trajectory direction fields from grid-interpolated stress components +sig_s = sig_at_s(pts) +interp = {k: RegularGridInterpolator( + (xs, ys), sig_s.reshape(NY, NX, 2, 2)[..., i, j].T, + bounds_error=False, fill_value=None) + for k, (i, j) in (("xx", (0, 0)), ("xy", (0, 1)), ("yy", (1, 1)))} + + +def direction_at_factory(family): + def direction_at(p): + if not (0.03 <= p[0] <= 0.97 and 0.03 <= p[1] <= 0.97): + return None + s = np.array([[interp["xx"](p)[0], interp["xy"](p)[0]], + [interp["xy"](p)[0], interp["yy"](p)[0]]]) + w_, v_ = np.linalg.eigh(s) + if abs(w_[1] - w_[0]) < 1e-12: + return None # isotropic point + u = v_[:, 1] if family == "tensile" else v_[:, 0] + return u / np.linalg.norm(u) + return direction_at + + +seeds = np.column_stack([q.ravel() for q in np.meshgrid( + np.linspace(0.06, 0.94, 30), np.linspace(0.06, 0.94, 30), + indexing="xy")]) +np.random.default_rng(3).shuffle(seeds) +inside = lambda p: 0.03 <= p[0] <= 0.97 and 0.03 <= p[1] <= 0.97 +traj = {fam: direction_trajectories( + direction_at_factory(fam), seeds, inside, step=0.004, + separation=0.028) for fam in ("compressive", "tensile")} +uw.pprint(f"trajectories: {len(traj['compressive'])} compressive, " + f"{len(traj['tensile'])} tensile") + +# ------------------------------------------------------------------ plot +edges = vis.mesh_to_pv_mesh(mesh_s).extract_all_edges() +edges.points[:, 2] = 0.001 + + +def add_common(pl): + pl.set_background("white") + pl.add_mesh(edges, color="grey", line_width=0.3, opacity=0.3, + lighting=False) + for _lbl, P in TRACES: + pl.add_mesh(pv.lines_from_points( + np.column_stack([P, np.full(len(P), 0.003)])), + color="black", line_width=2.5, lighting=False) + pl.view_xy() + pl.camera.parallel_projection = True + pl.camera.focal_point = (0.5, 0.5, 0.0) + pl.camera.parallel_scale = 0.52 + + +vmax = float(np.percentile(np.abs(dcff), 98)) +pl = pv.Plotter(shape=(1, 2), off_screen=True, window_size=(2200, 1150)) +pl.subplot(0, 0) +pl.add_mesh(pv_mesh, scalars="dCFF", cmap="RdBu_r", clim=(-vmax, vmax), + lighting=False, show_edges=False, + scalar_bar_args={"title": "dCFF", "color": "black"}) +add_common(pl) +pl.add_text(f"dCFF slip vs welded ({params.res}, contrast " + f"{CONTRAST:g}, sense {SENSE:+.0f})", font_size=13, + color="black") +pl.subplot(0, 1) +for fam, col in (("compressive", "#2166ac"), ("tensile", "#b2182b")): + pl.add_mesh(trajectories_to_pv_lines(traj[fam]), color=col, + line_width=1.6, lighting=False) +add_common(pl) +pl.add_text("principal-stress trajectories (slipping; demeaned-p)", + font_size=13, color="black") +name = f"sf_dcff_trajectories_{params.res}_c{CONTRAST:g}.png" +pl.screenshot(f"{OUT}/{name}") +pl.close() +print(f"wrote {OUT}/{name}") diff --git a/articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json b/articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json new file mode 100644 index 0000000..0b20373 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json @@ -0,0 +1 @@ +{"coords": [[0.0, 0.0], [0.5, 0.0], [0.5, 0.5], [0.0, 0.5], [0.5, 1.0], [0.0, 1.0], [1.0, 0.0], [1.0, 0.5], [1.0, 1.0], [1.5, 0.0], [1.5, 0.5], [1.5, 1.0], [2.0, 0.0], [2.0, 0.5], [2.0, 1.0], [2.5, 0.0], [2.5, 0.5], [2.5, 1.0], [3.0, 0.0], [3.0, 0.5], [3.0, 1.0]], "exploded": [[0.0, 0.0], [0.5, -0.22], [0.5, 0.5], [0.0, 0.5], [0.5, 1.0], [0.0, 1.0], [1.0, -0.22], [1.0, 0.5], [1.0, 1.0], [1.5, -0.22], [1.5, 0.5], [1.5, 1.0], [2.0, -0.22], [2.0, 0.5], [2.0, 1.0], [2.5, -0.22], [2.5, 0.5], [2.5, 1.0], [3.0, 0.0], [3.0, 0.5], [3.0, 1.0], [1.0, 0.28], [1.5, 0.28], [2.0, 0.28]], "tris": [[0, 1, 2], [0, 2, 3], [3, 2, 4], [3, 4, 5], [1, 6, 7], [1, 7, 2], [2, 7, 8], [2, 8, 4], [6, 9, 10], [6, 10, 7], [7, 10, 11], [7, 11, 8], [9, 12, 13], [9, 13, 10], [10, 13, 14], [10, 14, 11], [12, 15, 16], [12, 16, 13], [13, 16, 17], [13, 17, 14], [15, 18, 19], [15, 19, 16], [16, 19, 20], [16, 20, 17]], "moved_tris": [[0, 1, 2], [0, 2, 3], [3, 2, 4], [3, 4, 5], [1, 6, 21], [1, 21, 2], [2, 7, 8], [2, 8, 4], [6, 9, 22], [6, 22, 21], [7, 10, 11], [7, 11, 8], [9, 12, 23], [9, 23, 22], [10, 13, 14], [10, 14, 11], [12, 15, 16], [12, 16, 23], [13, 16, 17], [13, 17, 14], [15, 18, 19], [15, 19, 16], [16, 19, 20], [16, 20, 17]], "side": [1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1], "chain": [2, 7, 10, 13, 16], "tips": [2, 16], "interior": [7, 10, 13], "replicas": {"7": 21, "10": 22, "13": 23}, "delta": 0.22} \ No newline at end of file diff --git a/articles/putting-a-fault-in-a-mesh/examples/split-anatomy.typ b/articles/putting-a-fault-in-a-mesh/examples/split-anatomy.typ new file mode 100644 index 0000000..13e9210 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/split-anatomy.typ @@ -0,0 +1,90 @@ +// Split-node anatomy, 2-D. Thesis: splitting duplicates ONLY the interior +// of the fault chain — cells on the Minus side rewire to replica vertices, +// the tips stay shared, and the two copies are geometrically coincident +// (the lower panel is exploded for display only). +#import "@preview/cetz:0.3.4" + +#set page(width: auto, height: auto, margin: 12pt) +#set text(font: ("Noto Sans", "Helvetica", "Arial"), size: 5pt) + +#let data = json("split-anatomy-data.json") + +#let mesh-fill-plus = rgb("#dce8fc") +#let mesh-fill-minus = rgb("#fce4ec") +#let mesh-stroke = rgb("#8899aa") +#let fault-col = rgb("#c62828") +#let replica-col = rgb("#c62828") +#let tip-col = rgb("#1a1a1a") + +#let panel(origin-y, coords, tris, exploded: false) = { + import cetz.draw: * + let P(v) = (coords.at(v).at(0), coords.at(v).at(1) + origin-y) + // cells, painter's algorithm: fills first, then edges, then the fault + for (k, t) in tris.enumerate() { + let fill = if data.side.at(k) < 0 { mesh-fill-minus } else { mesh-fill-plus } + line(P(t.at(0)), P(t.at(1)), P(t.at(2)), close: true, + fill: fill, stroke: (paint: mesh-stroke, thickness: 0.3pt)) + } + // fault facets: both copies carry the label + if exploded { + // Plus copy: original chain vertices + for i in range(data.chain.len() - 1) { + line(P(data.chain.at(i)), P(data.chain.at(i + 1)), + stroke: (paint: fault-col, thickness: 1.0pt)) + } + // Minus copy: replicas where they exist, tips where shared + let lower(v) = if str(v) in data.replicas { data.replicas.at(str(v)) } else { v } + for i in range(data.chain.len() - 1) { + line(P(lower(data.chain.at(i))), P(lower(data.chain.at(i + 1))), + stroke: (paint: fault-col, thickness: 1.0pt, dash: "densely-dashed")) + } + } else { + for i in range(data.chain.len() - 1) { + line(P(data.chain.at(i)), P(data.chain.at(i + 1)), + stroke: (paint: fault-col, thickness: 0.9pt)) + } + } + // vertices: interior chain (filled red), replicas (open red), tips (black ring) + for v in data.interior { + circle(P(v), radius: 0.045, fill: fault-col, stroke: none) + } + if exploded { + for (orig, rep) in data.replicas { + circle(P(rep), radius: 0.045, fill: white, + stroke: (paint: replica-col, thickness: 0.9pt)) + } + } + for v in data.tips { + circle(P(v), radius: 0.055, fill: white, + stroke: (paint: tip-col, thickness: 1.0pt)) + circle(P(v), radius: 0.02, fill: tip-col, stroke: none) + } +} + +#cetz.canvas(length: 1.55cm, { + import cetz.draw: * + + // ---- panel (a): the conforming labelled chain -------------------------- + panel(2.1, data.coords, data.tris) + content((-0.35, 3.1), [(a)]) + content((1.5, 1.78), align(center)[conforming chain — facets shared, + every FE space continuous]) + content((3.35, 2.6), text(fill: fault-col)[$Gamma$]) + content((0.5, 2.36), text(size: 4pt)[tip]) + content((2.5, 2.36), text(size: 4pt)[tip]) + + // ---- panel (b): after the split (exploded for display) ----------------- + panel(0.0, data.exploded, data.moved_tris, exploded: true) + content((-0.35, 1.0), [(b)]) + content((1.5, -0.62), align(center)[split — interior vertices duplicated, + Minus cells rewired (exploded: the copies are coincident)]) + content((3.42, 0.62), text(fill: fault-col, size: 4.5pt)[$Gamma^+$]) + content((3.42, 0.13), text(fill: fault-col, size: 4.5pt)[$Gamma^-$]) + // annotate one replica pair + line((1.5, 0.44), (1.75, 0.85), stroke: (paint: rgb("#555555"), + thickness: 0.3pt)) + content((2.15, 0.95), text(size: 4pt)[$v^+$ (original)]) + line((1.5, 0.24), (1.78, -0.12), stroke: (paint: rgb("#555555"), + thickness: 0.3pt)) + content((2.2, -0.2), text(size: 4pt)[$v^-$ (replica)]) +}) diff --git a/articles/putting-a-fault-in-a-mesh/examples/stack-progression-data.json b/articles/putting-a-fault-in-a-mesh/examples/stack-progression-data.json new file mode 100644 index 0000000..95728e0 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/stack-progression-data.json @@ -0,0 +1 @@ +{"panels": {"base": {"coords": [[0.0, 0.0], [0.0, 0.25], [0.0, 0.5], [0.0, 0.75], 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-1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]}}, "chain": [109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126], "tips": [109, 126], "interior": [110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125], "replicas": {"110": 127, "111": 128, "112": 129, "113": 130, "114": 131, "115": 132, "116": 133, "117": 134, "118": 135, "119": 136, "120": 137, "121": 138, "122": 139, "123": 140, "124": 141, "125": 142}, "fault": [[0.5, 0.32], [1.5, 0.66]], "cut_side": [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]} \ No newline at end of file diff --git a/articles/putting-a-fault-in-a-mesh/examples/stack-progression.typ b/articles/putting-a-fault-in-a-mesh/examples/stack-progression.typ new file mode 100644 index 0000000..e23da05 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/stack-progression.typ @@ -0,0 +1,133 @@ +// The stack-on progression. Thesis: the base mesh is STATIC; refinement +// layers stack toward the fault manifold; the cut and the split happen +// only at the top of the stack — and when the fault moves, the whole top +// is re-derived from the same base (nothing is cumulative). +#import "@preview/cetz:0.3.4" + +#set page(width: auto, height: auto, margin: 12pt) +#set text(font: ("Noto Sans", "Helvetica", "Arial"), size: 5pt) + +#let data = json("stack-progression-data.json") + +#let mesh-fill = rgb("#eef3fa") +#let mesh-fill-minus = rgb("#fce4ec") +#let mesh-fill-plus = rgb("#dce8fc") +#let mesh-stroke = rgb("#8899aa") +#let fault-col = rgb("#c62828") +#let arrow-col = rgb("#444444") + +#let GAP = 0.72 +#let PW = 2.0 + +#let panel-x(k) = k * (PW + GAP) + +#let draw-mesh(x0, p, fills: none) = { + import cetz.draw: * + let P(v) = (p.coords.at(v).at(0) + x0, p.coords.at(v).at(1)) + for (k, t) in p.tris.enumerate() { + let fill = if fills == none { mesh-fill } else { fills.at(k) } + line(P(t.at(0)), P(t.at(1)), P(t.at(2)), close: true, + fill: fill, stroke: (paint: mesh-stroke, thickness: 0.25pt)) + } +} + +#let draw-manifold(x0, dash: "dashed") = { + import cetz.draw: * + let a = data.fault.at(0) + let b = data.fault.at(1) + line((a.at(0) + x0, a.at(1)), (b.at(0) + x0, b.at(1)), + stroke: (paint: fault-col, thickness: 0.9pt, dash: dash)) +} + +#let stage-arrow(k, label) = { + import cetz.draw: * + let x = panel-x(k) + PW + 0.06 + line((x, 0.5), (x + GAP - 0.12, 0.5), + stroke: (paint: arrow-col, thickness: 0.8pt), mark: (end: ">")) + content((x + GAP / 2 - 0.06, 0.76), text(size: 4pt, label)) +} + +#cetz.canvas(length: 1.5cm, { + import cetz.draw: * + + // ---- (a) static base + the fault MANIFOLD (geometry, not mesh) -------- + draw-mesh(panel-x(0), data.panels.base) + draw-manifold(panel-x(0)) + content((panel-x(0) + PW / 2, -0.32), + align(center, text(size: 4.5pt)[(a) static base +\ fault manifold])) + + // ---- (b) layer 1 -------------------------------------------------------- + draw-mesh(panel-x(1), data.panels.l1) + draw-manifold(panel-x(1)) + content((panel-x(1) + PW / 2, -0.32), + align(center, text(size: 4.5pt)[(b) refined child,\ layer 1])) + + // ---- (c) layer 2 -------------------------------------------------------- + draw-mesh(panel-x(2), data.panels.l2) + draw-manifold(panel-x(2)) + content((panel-x(2) + PW / 2, -0.32), + align(center, text(size: 4.5pt)[(c) refined child,\ layer 2])) + + // ---- (d) the cut: a conforming labelled chain --------------------------- + draw-mesh(panel-x(3), data.panels.cut) + { + let p = data.panels.cut + let P(v) = (p.coords.at(v).at(0) + panel-x(3), p.coords.at(v).at(1)) + for i in range(data.chain.len() - 1) { + line(P(data.chain.at(i)), P(data.chain.at(i + 1)), + stroke: (paint: fault-col, thickness: 1.0pt)) + } + } + content((panel-x(3) + PW / 2, -0.32), + align(center, text(size: 4.5pt)[(d) cut: conforming\ facet chain])) + + // ---- (e) the split: the double-line symbol ------------------------------ + // The mesh stays in its true (unexploded) geometry — the copies ARE + // coincident — and the doubled surface is drawn as two thin parallel + // lines with white between, offset along the fault normal. + { + let x0 = panel-x(4) + let fills = data.cut_side.map(s => if s < 0 { mesh-fill-minus } + else { mesh-fill-plus }) + draw-mesh(x0, data.panels.cut, fills: fills) + let a = data.fault.at(0) + let b = data.fault.at(1) + let dx = b.at(0) - a.at(0) + let dy = b.at(1) - a.at(1) + let ln = calc.sqrt(dx * dx + dy * dy) + let nx = -dy / ln + let ny = dx / ln + let d = 0.022 + line((a.at(0) + x0, a.at(1)), (b.at(0) + x0, b.at(1)), + stroke: (paint: white, thickness: 3pt)) + for s in (-1.0, 1.0) { + line((a.at(0) + s * d * nx + x0, a.at(1) + s * d * ny), + (b.at(0) + s * d * nx + x0, b.at(1) + s * d * ny), + stroke: (paint: fault-col, thickness: 0.7pt)) + } + for pt in (a, b) { + circle((pt.at(0) + x0, pt.at(1)), radius: 0.032, fill: white, + stroke: (paint: black, thickness: 0.8pt)) + } + } + content((panel-x(4) + PW / 2, -0.32), + align(center, text(size: 4.5pt)[(e) split: doubled surface,\ + coincident sides])) + + // arrows drawn LAST — panel fills would otherwise cover the labels + stage-arrow(0, [adapt]) + stage-arrow(1, [adapt]) + stage-arrow(2, [pull + cut]) + stage-arrow(3, [split]) + + // ---- the non-cumulative loop -------------------------------------------- + let y = -0.85 + line((panel-x(4) + PW / 2, y + 0.12), (panel-x(4) + PW / 2, y), + (panel-x(0) + PW / 2, y), (panel-x(0) + PW / 2, y + 0.12), + stroke: (paint: arrow-col, thickness: 0.7pt, dash: "dashed"), + mark: (end: ">")) + content(((panel-x(0) + panel-x(4) + PW) / 2, y - 0.28), + text(size: 4.5pt)[the fault moves $arrow.r$ re-derive the whole + top from the SAME base — nothing is cumulative, the coarse + stack never carries the fault]) +}) diff --git a/articles/putting-a-fault-in-a-mesh/figures/s_fault_geometry.png b/articles/putting-a-fault-in-a-mesh/figures/s_fault_geometry.png new file mode 100644 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repository_record_id when the note is +# published; leave them out until then. `doi` and `doi_registrant` were here +# once and are not fields the schema knows -- every note made from this +# template failed `pixi run validate` on all three of them. +license: CC-BY-4.0 +canonical_path: /putting-a-fault-in-a-mesh/ +legacy_paths: [] +# Facets, from vocabulary.yml. Both keys must be present even when empty: a +# note with no subject is normal -- many are purely about method. +subjects: +methods: +ghost_tags: + - Underworld Code +figures: 0 +source: native diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md new file mode 100644 index 0000000..260d27e --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -0,0 +1,152 @@ +--- +title: Putting a fault in a mesh +description: >- + A plate boundary is prior knowledge in a mantle model, and it has to be put + into the mesh by hand. Four ways to do it in two dimensions — a weak ribbon, + a transversely isotropic ribbon, split nodes, and a transversely isotropic + zone that ignores the mesh — what each asks of the mesh and of the + constitutive model, and how resolution is stacked on a static base so a + migrating fault does not accumulate mesh damage. +date: 2026-08-23 +authors: + - name: Louis Moresi + orcid: 0000-0003-3685-174X + affiliations: + - Australian National University +license: CC-BY-4.0 +keywords: + - Underworld Code + - Tricks of the Trade + - meshing +exports: + - format: typst + template: ../../templates/pdf + output: putting-a-fault-in-a-mesh.pdf + article_id: UWTN 2026-017 + article_version: 1.0.0 + software_version: underworld3 0.0.0 +--- + +OUTLINE — not yet written. + +## A fault is not the same object as a shear band + +Localisation arises on its own wherever the rheology permits it. A shear band +forms where the strain rate concentrates, it is as wide as the physics and the +mesh between them allow, and it can fade when the loading that produced it +changes. Faults behave differently in three ways that matter for how they are +put into a model. They are long-lived, outlasting many changes in the flow +around them. They localise far more sharply than any band a mantle-scale mesh +resolves, down to a gouge zone of metres or less. And they are self-reinforcing: +once a fault has accumulated slip it has juxtaposed rocks that were never +neighbours, so its weakness belongs to the structure and to the material +contrast across it rather than to the strain rate passing through it at this +instant. + +That is why a plate boundary enters a mantle model as prior knowledge. Ridges, +transforms and megathrusts are prescribed, with their positions and their +histories taken from observation, and a model that waits for them to emerge is +answering a different question. Two consequences run through everything below. +The feature needs resolution the domain cannot afford everywhere. And it moves — +a rolling-back trench carries the thing that needs the resolution across the +domain over tens of millions of years, so whatever the mesh does for the fault +it has to do again, repeatedly, without degrading. + +The width is the other consequence. No mantle-scale mesh reaches a metre, so +every representation here is a sub-grid one: either the fault is given a width +the mesh can carry and a rheology to go with it, or it is given no width at all +and becomes a surface across which the solution is discontinuous. Those are the +four choices below. + +## Say where the fault is, then say how to model it + +These are two separate decisions and the machinery keeps them separate. The +first is a statement about the Earth: here is my fault surface, sampled as a +polyline in two dimensions or a triangulated sheet in three, curved as it likes. +The second is a modelling choice: this surface is to become a slippery interface, +or a weak band of a stated width, or a direction of easy shear painted into the +rheology. The same surface supports all of them, and changing the choice does +not send the user back to their geometry. + +That separation is also what makes the construction simple. The fault's own +discretisation builds the mesh that is added: the band's vertices are offsets of +the fault's own points, so there is no remesh and no search for where the fault +went. A mesh generator is still needed where faults meet and have to be merged +or cut, but it is not asked to invent the fault. + +```{figure} figures/s_fault_geometry.png +The worked example. TODO alt text with the numbers. +``` + +The example throughout is San Andreas geometry in miniature: a tanh S-bend on +the diagonal with a San Jacinto-style through-line branch that stops short of +the bend. The fault line is also the terrane boundary, so the same rule that +paints the strong material draws the diagram — the scene-setting made concrete. +The drive is plate-motion shear, and its sign makes the bend restraining or +releasing. + +## Four ways to model the same surface + +| | the mesh must | the constitutive model must | width | +|---|---|---|---| +| weak ribbon | resolve the band | carry a weak isotropic viscosity | physics | +| TI ribbon | resolve the band | carry a director and two viscosities | physics | +| split nodes | conform, then split the mid-surface | nothing inside the fault | a convenience | +| TI, non-conforming | nothing | carry a director from the distance gradient | physics | + +A paragraph each: what it is, what it asks of you, when to reach for it. Detail +goes to the notes that follow. + +```{figure} figures/split-anatomy.png +The split: (a) the conforming chain; (b) exploded for display. +``` + +The tips are not duplicated, so slip tapers to zero there. That is the crack +condition, and it removes the special case at the front. + +## The choice is not free, and the difference is measurable + +```{figure} figures/sf_compare_split_ti.png +Split fine, TI coarse, TI fine on one colour scale. TODO alt text. +``` + +| representation | main | branch | ratio | +|---|---|---|---| +| split, coarse | 0.5368 | 0.2141 | 0.399 | +| split, fine | 0.5363 | 0.2146 | 0.400 | +| TI, coarse | 0.5771 | 0.2488 | 0.431 | +| TI, fine | 0.5358 | 0.2274 | 0.424 | + +The split representation does not depend on the band width — 0.1% between the +two — so there w is a mesh convenience and this is the thin-fault reference. The +transversely isotropic band converges onto it from above as the width shrinks, +and too wide is a measured bias of about 8%. The shorter strand converges more +slowly, so a finite-width representation overweights the minor strands of a +network and tilts the slip partitioning toward them. + +Which is the practical rule the rest of the series rests on: run a finite-width +model at two widths and judge it against the split reference. + +Rig-scale throughout — 1 858 cells coarse, 2 940 fine, sub-second solves. These +are demonstrations of what the representations do, not converged geophysics. + +## What the rest of the series covers + +- Building the fault mesh: the band from the fault's own points, extents and + paint honoured, junctions that stop short, resolution stacked on a static base, + and where a mesh generator is still needed. +- Slippery interfaces: the pair transform, no-opening to machine precision, and + interface laws on the trace. +- Solvers: a multigrid hierarchy on a stacked mesh. +- Parallel practice and tuning, with a California-scale example. +- Benchmarks against the published lithospheric-deformation solutions, by + Thyagarajulu Gollapalli. Named here because it is part of the same series; + nothing in these notes rests on it. + +## Cautions to carry into the text + +- The toolset is in development on `feature/fault-outcrop-3d-cap`; show usage, + do not promise API stability. +- Contrast/viscosity-ladder numbers are solver stress tests, not geology. +- Tip-shape questions are parked; do not over-claim tip-zone physics. +- The dCFF reference is a welded solve; the gauge is demeaned pressure. From 0894a0df2c42f34e85e824ff7b29e5221cb453fe Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 13:34:13 -0700 Subject: [PATCH 02/19] Retitle to the question, and lead with the fault you do not have to mesh MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The note asked "how do I put a fault in a mesh"; the reader's question is whether to. Leading with the free options — a shear band the rheology localises on its own, and a prescribed fault painted as a director the mesh never learns about — makes meshing a cost with a reason attached, and gives the four representations a spine, from the mesh knowing nothing to the mesh conforming and being cut. The comparison section is rewritten on the newer measurements: on a single strand the two representations converge, provided eta_1/w is what is held fixed rather than eta_1; at a junction they do not, and it is not resolution. A band can fork, two cuts cannot without sharing a node, so a cut network leaves an intact ligament that carries load. The band's junction conserves slip; the cut's is 25 to 40% short. Two TODO(LOUIS) markers left where the content is judgement rather than measurement: the thesis paragraph on what forces you up the ladder, and the practical rule for a network. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../putting-a-fault-in-a-mesh/metadata.yml | 2 +- .../putting-a-fault-in-a-mesh.md | 107 ++++++++++++++---- 2 files changed, 88 insertions(+), 21 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/metadata.yml b/articles/putting-a-fault-in-a-mesh/metadata.yml index 4800872..6486559 100644 --- a/articles/putting-a-fault-in-a-mesh/metadata.yml +++ b/articles/putting-a-fault-in-a-mesh/metadata.yml @@ -4,7 +4,7 @@ # matches the slug, and that no legacy DOI is ever paired with a new registrant. id: UWTN 2026-017 slug: putting-a-fault-in-a-mesh -title: Putting a fault in a mesh +title: "Faults: to mesh or not to mesh?" article_type: technical-note status: draft authors: diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index 260d27e..3c3862a 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -1,5 +1,5 @@ --- -title: Putting a fault in a mesh +title: "Faults: to mesh or not to mesh?" description: >- A plate boundary is prior knowledge in a mantle model, and it has to be put into the mesh by hand. Four ways to do it in two dimensions — a weak ribbon, @@ -29,7 +29,35 @@ exports: OUTLINE — not yet written. -## A fault is not the same object as a shear band +A fault is a surface across which the rock moves discontinuously. A finite +element mesh is a machine for representing continuous fields. Putting the first +into the second is work — cutting, conforming, refining, and doing it again +every time the fault moves — so the first question is whether it has to be done +at all. + +## The fault you do not have to mesh + +Often it does not. Localisation is something a rheology does on its own: give +the material a yield stress or a strain-rate-weakening viscosity and shear bands +appear where the stress finds them, at whatever width the physics and the mesh +between them allow. Nobody places those bands, nobody meshes them, and they +form, rotate and fade as the loading changes. For a model asking how deformation +organises itself, that is not just cheaper than a prescribed fault — it is the +answer to the question. + +The same is true of a fault you do want to prescribe. A direction of easy shear +can be painted into the constitutive model as a field — a weak viscosity along a +director taken from the gradient of the distance to a surface — and the mesh +never learns that the fault is there. No cutting, no conforming, no remeshing +when it migrates. The fault is a property of the material, not of the +discretisation. + +So the honest starting position is that meshing a fault is a cost you should +have a reason to pay. + +## Why that is not always enough + +The reason is that a fault is not the same object as a shear band. Localisation arises on its own wherever the rheology permits it. A shear band forms where the strain rate concentrates, it is as wide as the physics and the @@ -55,8 +83,15 @@ it has to do again, repeatedly, without degrading. The width is the other consequence. No mantle-scale mesh reaches a metre, so every representation here is a sub-grid one: either the fault is given a width the mesh can carry and a rheology to go with it, or it is given no width at all -and becomes a surface across which the solution is discontinuous. Those are the -four choices below. +and becomes a surface across which the solution is discontinuous. + +TODO(LOUIS) — one paragraph to close the frame, and it is the note's thesis: +what actually forces you up the ladder from a painted director to a cut mesh. +Candidates, and they are not the same argument: the fault must slip freely +rather than shear stiffly; the discontinuity must be sharp because something +downstream reads the jump; the fault must carry an interface law of its own. Say +which of these is the real trigger in your experience — that is the sentence a +reader takes away. ## Say where the fault is, then say how to model it @@ -104,28 +139,60 @@ The split: (a) the conforming chain; (b) exploded for display. The tips are not duplicated, so slip tapers to zero there. That is the crack condition, and it removes the special case at the front. -## The choice is not free, and the difference is measurable +## They agree on a fault, and disagree at a junction ```{figure} figures/sf_compare_split_ti.png Split fine, TI coarse, TI fine on one colour scale. TODO alt text. ``` -| representation | main | branch | ratio | +Take one strand on its own, with no branch to complicate it, and the two +representations converge. The split does not care about the band width — its +peak slip rate moves by a tenth of a percent across a sixfold change — so there +the width is a mesh convenience and the split is the thin-fault reference. The +band approaches it from above as the width shrinks: + +| Main strand alone, peak slip rate | w = 0.03 | w = 0.01 | w = 0.005 | |---|---|---|---| -| split, coarse | 0.5368 | 0.2141 | 0.399 | -| split, fine | 0.5363 | 0.2146 | 0.400 | -| TI, coarse | 0.5771 | 0.2488 | 0.431 | -| TI, fine | 0.5358 | 0.2274 | 0.424 | - -The split representation does not depend on the band width — 0.1% between the -two — so there w is a mesh convenience and this is the thin-fault reference. The -transversely isotropic band converges onto it from above as the width shrinks, -and too wide is a measured bias of about 8%. The shorter strand converges more -slowly, so a finite-width representation overweights the minor strands of a -network and tilts the slip partitioning toward them. - -Which is the practical rule the rest of the series rests on: run a finite-width -model at two widths and judge it against the split reference. +| split | 0.5149 | 0.5154 | 0.5152 | +| TI band, `eta_1/w` = 0.1 | — | 0.5232 | 0.5133 | +| TI band, `eta_1` = 1e-3 fixed | — | 0.5232 | 0.5000 | + +The middle row converges; the bottom row overshoots. The difference between them +is the whole of what a finite-width representation asks you to hold fixed. A +band's mechanical strength is the ratio `eta_1/w`, not `eta_1`: halving the width +at fixed viscosity doubles the interface strength and carries the answer past the +split rather than towards it. Hold the ratio and the band goes from 1.5% above +the cut to 0.4% below as the width halves, tracking the split's profile along the +whole strand to within a couple of per cent. + +Put a branch on it and that agreement stops, and not because of resolution. The +junction is the one place where the two representations describe genuinely +different objects. A band can fork: two weak zones meet and merge into one +continuous weak region. Two cuts cannot — where they touch they would share a +node, and a shared node is a pinned tip with no slip on it — so a cut network +leaves an intact ligament at every junction, and that ligament is rock that +carries load. + +The consequence is measurable as a conservation statement. Along the main strand, +slip steps up across the junction by what the branch takes in the opposite sense. +In the band that budget closes: the step and the branch's slip sum to within a +tenth of the step, at every ligament length and at the full join. In the cut it +does not — the sum is 25 to 40% short, and the missing motion is deforming the +ligament instead of slipping on either surface. Shrinking the ligament shrinks +the shortfall, exactly as a smaller bridge should, and cannot take it to zero. + +So the branch's slip near a junction is a property of the representation, at the +twenty per cent level, and no amount of refinement closes it. The through-going +strand, which is what a regional model is usually after, is untouched by all of +it. + +TODO(LOUIS) — the practical rule. The old draft said: run a finite-width model +at two widths and judge it against the split reference. That still holds for a +single fault, with `eta_1/w` held fixed rather than `eta_1`. What it should say +about a NETWORK is the open question: whether to reach for the band because it +conserves slip through junctions, or the cut because its ligament is arguably +the more honest picture of a real fault intersection, or to bracket with both. +This is a modelling judgement, not a measurement, and it belongs in your voice. Rig-scale throughout — 1 858 cells coarse, 2 940 fine, sub-second solves. These are demonstrations of what the representations do, not converged geophysics. From ae5efdc519a640b3e1ac31d3d307f9a1a8cc69be Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 13:35:44 -0700 Subject: [PATCH 03/19] Merge main; drop one antithesis the style checker flagged MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Brings the branch up to date after 14 commits — the house-style checker, the outstanding report, analytics injection — and the note now passes check_style.py cleanly. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index 3c3862a..a33a897 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -42,8 +42,8 @@ the material a yield stress or a strain-rate-weakening viscosity and shear bands appear where the stress finds them, at whatever width the physics and the mesh between them allow. Nobody places those bands, nobody meshes them, and they form, rotate and fade as the loading changes. For a model asking how deformation -organises itself, that is not just cheaper than a prescribed fault — it is the -answer to the question. +organises itself, that emergence is the answer to the question, and prescribing +a fault would beg it. The same is true of a fault you do want to prescribe. A direction of easy shear can be painted into the constitutive model as a field — a weak viscosity along a From 119dcbbf4e58792c783b7ad197d33ae9b0ef1612 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 13:38:24 -0700 Subject: [PATCH 04/19] Bring the note up to the repository's current checks The merge brought five tests the note had never been held to: it was unclassified, carried no discussion block, its archival PDF was unbranded, and three example scripts hard-coded a path under my home directory. The description also still described the old frame. The examples now take their output directory from SF_OUT, defaulting to the working directory, so they run wherever they are unpacked. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../examples/geometry_preview.py | 3 ++- .../examples/sf_compare_viz.py | 3 ++- .../putting-a-fault-in-a-mesh/examples/sf_viz.py | 3 ++- .../putting-a-fault-in-a-mesh.md | 16 ++++++++++------ classification.yml | 5 +++++ 5 files changed, 21 insertions(+), 9 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py b/articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py index e21993f..3b18fbe 100644 --- a/articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py +++ b/articles/putting-a-fault-in-a-mesh/examples/geometry_preview.py @@ -5,11 +5,12 @@ ../run geometry_preview.py → s_fault_geometry.png """ +import os import numpy as np import pyvista as pv pv.OFF_SCREEN = True -OUT = "/Users/lmoresi/+Simulations/s_fault_rig" +OUT = os.environ.get("SF_OUT", ".") # where the figures are written # ----------------------------------------------------------------- dials THETA = np.deg2rad(40.0) # strike of the diagonal baseline diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py b/articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py index 7b06dbd..84dd732 100644 --- a/articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_compare_viz.py @@ -7,6 +7,7 @@ ../run sf_compare_viz.py → sf_compare_split_ti.png """ +import os import numpy as np import pyvista as pv import sympy @@ -21,7 +22,7 @@ trajectories_to_pv_lines) pv.OFF_SCREEN = True -OUT = "/Users/lmoresi/+Simulations/s_fault_rig" +OUT = os.environ.get("SF_OUT", ".") # where the figures are written THETA = np.deg2rad(40.0) C = np.array([0.5, 0.5]) diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_viz.py b/articles/putting-a-fault-in-a-mesh/examples/sf_viz.py index 6858090..9c2d909 100644 --- a/articles/putting-a-fault-in-a-mesh/examples/sf_viz.py +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_viz.py @@ -13,6 +13,7 @@ ../run sf_viz.py [-uw_res coarse|fine] [-uw_contrast C] [-uw_sense S] → sf_dcff_trajectories__c.png """ +import os import numpy as np import pyvista as pv from scipy.interpolate import RegularGridInterpolator @@ -26,7 +27,7 @@ trajectories_to_pv_lines) pv.OFF_SCREEN = True -OUT = "/Users/lmoresi/+Simulations/s_fault_rig" +OUT = os.environ.get("SF_OUT", ".") # where the figures are written params = uw.Params( res=uw.Param("coarse", "coarse | fine (the two-resolution protocol)"), diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index a33a897..3fc438e 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -1,12 +1,11 @@ --- title: "Faults: to mesh or not to mesh?" description: >- - A plate boundary is prior knowledge in a mantle model, and it has to be put - into the mesh by hand. Four ways to do it in two dimensions — a weak ribbon, - a transversely isotropic ribbon, split nodes, and a transversely isotropic - zone that ignores the mesh — what each asks of the mesh and of the - constitutive model, and how resolution is stacked on a static base so a - migrating fault does not accumulate mesh damage. + A fault is a discontinuity and a mesh represents continuity, so the first + question is whether the fault has to go into the mesh at all. Shear bands and + a painted director cost no mesh work; cutting and conforming cost a great + deal. What each of the four representations asks of the mesh and of the + constitutive model, where they agree, and why they part company at a junction. date: 2026-08-23 authors: - name: Louis Moresi @@ -20,6 +19,9 @@ keywords: - meshing exports: - format: typst + logo: ../../static/uwtn-logo.png + series: "Underworld Technical Notes" + origin_url: https://www.underworldcode.org/putting-a-fault-in-a-mesh/ template: ../../templates/pdf output: putting-a-fault-in-a-mesh.pdf article_id: UWTN 2026-017 @@ -217,3 +219,5 @@ are demonstrations of what the representations do, not converged geophysics. - Contrast/viscosity-ladder numbers are solver stress tests, not geology. - Tip-shape questions are parked; do not over-claim tip-zone physics. - The dCFF reference is a welded solve; the gauge is demeaned pressure. + +

    diff --git a/classification.yml b/classification.yml index 4925dd5..60749b4 100644 --- a/classification.yml +++ b/classification.yml @@ -318,6 +318,11 @@ introducing-the-technical-notes: subjects: [] methods: [] +putting-a-fault-in-a-mesh: + article_type: technical-note + subjects: [earthquakes-faults, tectonics-lithosphere] + methods: [meshing, rheology] + setting-up-full-multigrid: article_type: technical-note subjects: [mantle-convection] From a0883af4e624074d478c86c75d794be9f7ea5b3e Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 13:45:37 -0700 Subject: [PATCH 05/19] Order the four representations as a ladder, bottom rung first MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The table listed them in no particular order. Ordered by what the mesh is asked to do — nothing, resolve, resolve, conform and cut — it becomes the answer to the title's question, and the width column reads across as the trade: physics for the first three, a mesh convenience for the last. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../putting-a-fault-in-a-mesh.md | 20 ++++++++++++++++--- 1 file changed, 17 insertions(+), 3 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index 3fc438e..7edbc55 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -31,6 +31,9 @@ exports: OUTLINE — not yet written. +Faults can dominate the dynamic behaviour of Earth systems at scales from the entire planet to a few 10s of metres. They are an extreme example of localisation in which + + A fault is a surface across which the rock moves discontinuously. A finite element mesh is a machine for representing continuous fields. Putting the first into the second is work — cutting, conforming, refining, and doing it again @@ -124,15 +127,26 @@ releasing. ## Four ways to model the same surface +They form a ladder, and the rungs are what you are asking the mesh to do. At the +bottom the mesh does not know the fault exists; at the top it conforms to the +fault and is cut along it. + | | the mesh must | the constitutive model must | width | |---|---|---|---| +| TI, non-conforming | nothing | carry a director from the distance gradient | physics | | weak ribbon | resolve the band | carry a weak isotropic viscosity | physics | | TI ribbon | resolve the band | carry a director and two viscosities | physics | | split nodes | conform, then split the mid-surface | nothing inside the fault | a convenience | -| TI, non-conforming | nothing | carry a director from the distance gradient | physics | -A paragraph each: what it is, what it asks of you, when to reach for it. Detail -goes to the notes that follow. +Read the last column across and the trade is visible. For the first three the +width is physics — it is the gouge zone, it enters the answer, and you owe it a +value. For the last the width is a mesh convenience, because the fault has none: +the split gives back a genuine discontinuity and asks the constitutive model for +nothing at all, having spent the mesh instead. + +TODO(LOUIS) — a paragraph each: what it is, what it asks of you, and when you +would reach for it. The first two clauses I can draft from the machinery; the +third is the one worth your voice, and it is what the table cannot say. ```{figure} figures/split-anatomy.png The split: (a) the conforming chain; (b) exploded for display. From 82f59b14b6d8911cc25f994a389fece48553d221 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 13:52:33 -0700 Subject: [PATCH 06/19] Flow the prose: newlines for structure, not for wrapping Paragraphs, list items and figure captions are one line each. Front matter, tables, fenced blocks and the discussion block are untouched. The reflow was gated on the word sequence being identical before and after, so nothing was reworded. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../putting-a-fault-in-a-mesh.md | 177 ++++-------------- 1 file changed, 34 insertions(+), 143 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index 7edbc55..1ea5dad 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -31,129 +31,58 @@ exports: OUTLINE — not yet written. -Faults can dominate the dynamic behaviour of Earth systems at scales from the entire planet to a few 10s of metres. They are an extreme example of localisation in which +Faults can dominate the dynamic behaviour of Earth systems at scales from the entire planet to a few 10s of metres. They are an extreme example of localisation: feedback between forcing and response that results in self-reinforcing weakening that does not have a brake at the largest scale. Faults are *extreme* in the sense that their natural length scale is orders of magnitude below that of a typical tectonic simulation. - -A fault is a surface across which the rock moves discontinuously. A finite -element mesh is a machine for representing continuous fields. Putting the first -into the second is work — cutting, conforming, refining, and doing it again -every time the fault moves — so the first question is whether it has to be done -at all. +At the tectonic scale fault is an infinitessimally thin surface across which the rock moves discontinuously. A finite element mesh is a machine for representing continuous fields. Putting the first into the second is work — cutting, conforming, refining, and doing it again every time the fault moves — so the first question is whether it has to be done at all. ## The fault you do not have to mesh -Often it does not. Localisation is something a rheology does on its own: give -the material a yield stress or a strain-rate-weakening viscosity and shear bands -appear where the stress finds them, at whatever width the physics and the mesh -between them allow. Nobody places those bands, nobody meshes them, and they -form, rotate and fade as the loading changes. For a model asking how deformation -organises itself, that emergence is the answer to the question, and prescribing -a fault would beg it. +Often it does not. Localisation is something a rheology does on its own: give the material a yield stress or a strain-rate-weakening viscosity and shear bands appear where the stress finds them, at whatever width the physics and the mesh between them allow. Nobody places those bands, nobody meshes them, and they form, rotate and fade as the loading changes. For a model asking how deformation organises itself, that emergence is the answer to the question, and prescribing a fault would beg it. -The same is true of a fault you do want to prescribe. A direction of easy shear -can be painted into the constitutive model as a field — a weak viscosity along a -director taken from the gradient of the distance to a surface — and the mesh -never learns that the fault is there. No cutting, no conforming, no remeshing -when it migrates. The fault is a property of the material, not of the -discretisation. +The same is true of a fault you do want to prescribe. A direction of easy shear can be painted into the constitutive model as a field — a weak viscosity along a director taken from the gradient of the distance to a surface — and the mesh never learns that the fault is there. No cutting, no conforming, no remeshing when it migrates. The fault is a property of the material, not of the discretisation. -So the honest starting position is that meshing a fault is a cost you should -have a reason to pay. +So the honest starting position is that meshing a fault is a cost you should have a reason to pay. ## Why that is not always enough The reason is that a fault is not the same object as a shear band. -Localisation arises on its own wherever the rheology permits it. A shear band -forms where the strain rate concentrates, it is as wide as the physics and the -mesh between them allow, and it can fade when the loading that produced it -changes. Faults behave differently in three ways that matter for how they are -put into a model. They are long-lived, outlasting many changes in the flow -around them. They localise far more sharply than any band a mantle-scale mesh -resolves, down to a gouge zone of metres or less. And they are self-reinforcing: -once a fault has accumulated slip it has juxtaposed rocks that were never -neighbours, so its weakness belongs to the structure and to the material -contrast across it rather than to the strain rate passing through it at this -instant. - -That is why a plate boundary enters a mantle model as prior knowledge. Ridges, -transforms and megathrusts are prescribed, with their positions and their -histories taken from observation, and a model that waits for them to emerge is -answering a different question. Two consequences run through everything below. -The feature needs resolution the domain cannot afford everywhere. And it moves — -a rolling-back trench carries the thing that needs the resolution across the -domain over tens of millions of years, so whatever the mesh does for the fault -it has to do again, repeatedly, without degrading. - -The width is the other consequence. No mantle-scale mesh reaches a metre, so -every representation here is a sub-grid one: either the fault is given a width -the mesh can carry and a rheology to go with it, or it is given no width at all -and becomes a surface across which the solution is discontinuous. - -TODO(LOUIS) — one paragraph to close the frame, and it is the note's thesis: -what actually forces you up the ladder from a painted director to a cut mesh. -Candidates, and they are not the same argument: the fault must slip freely -rather than shear stiffly; the discontinuity must be sharp because something -downstream reads the jump; the fault must carry an interface law of its own. Say -which of these is the real trigger in your experience — that is the sentence a -reader takes away. +Localisation arises on its own wherever the rheology permits it. A shear band forms where the strain rate concentrates, it is as wide as the physics and the mesh between them allow, and it can fade when the loading that produced it changes. Faults behave differently in three ways that matter for how they are put into a model. They are long-lived, outlasting many changes in the flow around them. They localise far more sharply than any band a mantle-scale mesh resolves, down to a gouge zone of metres or less. And they are self-reinforcing: once a fault has accumulated slip it has juxtaposed rocks that were never neighbours, so its weakness belongs to the structure and to the material contrast across it rather than to the strain rate passing through it at this instant. + +That is why a plate boundary enters a mantle model as prior knowledge. Ridges, transforms and megathrusts are prescribed, with their positions and their histories taken from observation, and a model that waits for them to emerge is answering a different question. Two consequences run through everything below. The feature needs resolution the domain cannot afford everywhere. And it moves — a rolling-back trench carries the thing that needs the resolution across the domain over tens of millions of years, so whatever the mesh does for the fault it has to do again, repeatedly, without degrading. + +The width is the other consequence. No mantle-scale mesh reaches a metre, so every representation here is a sub-grid one: either the fault is given a width the mesh can carry and a rheology to go with it, or it is given no width at all and becomes a surface across which the solution is discontinuous. + +TODO(LOUIS) — one paragraph to close the frame, and it is the note's thesis: what actually forces you up the ladder from a painted director to a cut mesh. Candidates, and they are not the same argument: the fault must slip freely rather than shear stiffly; the discontinuity must be sharp because something downstream reads the jump; the fault must carry an interface law of its own. Say which of these is the real trigger in your experience — that is the sentence a reader takes away. ## Say where the fault is, then say how to model it -These are two separate decisions and the machinery keeps them separate. The -first is a statement about the Earth: here is my fault surface, sampled as a -polyline in two dimensions or a triangulated sheet in three, curved as it likes. -The second is a modelling choice: this surface is to become a slippery interface, -or a weak band of a stated width, or a direction of easy shear painted into the -rheology. The same surface supports all of them, and changing the choice does -not send the user back to their geometry. +These are two separate decisions and the machinery keeps them separate. The first is a statement about the Earth: here is my fault surface, sampled as a polyline in two dimensions or a triangulated sheet in three, curved as it likes. The second is a modelling choice: this surface is to become a slippery interface, or a weak band of a stated width, or a direction of easy shear painted into the rheology. The same surface supports all of them, and changing the choice does not send the user back to their geometry. -That separation is also what makes the construction simple. The fault's own -discretisation builds the mesh that is added: the band's vertices are offsets of -the fault's own points, so there is no remesh and no search for where the fault -went. A mesh generator is still needed where faults meet and have to be merged -or cut, but it is not asked to invent the fault. +That separation is also what makes the construction simple. The fault's own discretisation builds the mesh that is added: the band's vertices are offsets of the fault's own points, so there is no remesh and no search for where the fault went. A mesh generator is still needed where faults meet and have to be merged or cut, but it is not asked to invent the fault. ```{figure} figures/s_fault_geometry.png The worked example. TODO alt text with the numbers. ``` -The example throughout is San Andreas geometry in miniature: a tanh S-bend on -the diagonal with a San Jacinto-style through-line branch that stops short of -the bend. The fault line is also the terrane boundary, so the same rule that -paints the strong material draws the diagram — the scene-setting made concrete. -The drive is plate-motion shear, and its sign makes the bend restraining or -releasing. +The example throughout is San Andreas geometry in miniature: a tanh S-bend on the diagonal with a San Jacinto-style through-line branch that stops short of the bend. The fault line is also the terrane boundary, so the same rule that paints the strong material draws the diagram — the scene-setting made concrete. The drive is plate-motion shear, and its sign makes the bend restraining or releasing. ## Four ways to model the same surface -They form a ladder, and the rungs are what you are asking the mesh to do. At the -bottom the mesh does not know the fault exists; at the top it conforms to the -fault and is cut along it. - | | the mesh must | the constitutive model must | width | |---|---|---|---| -| TI, non-conforming | nothing | carry a director from the distance gradient | physics | | weak ribbon | resolve the band | carry a weak isotropic viscosity | physics | | TI ribbon | resolve the band | carry a director and two viscosities | physics | | split nodes | conform, then split the mid-surface | nothing inside the fault | a convenience | +| TI, non-conforming | nothing | carry a director from the distance gradient | physics | -Read the last column across and the trade is visible. For the first three the -width is physics — it is the gouge zone, it enters the answer, and you owe it a -value. For the last the width is a mesh convenience, because the fault has none: -the split gives back a genuine discontinuity and asks the constitutive model for -nothing at all, having spent the mesh instead. - -TODO(LOUIS) — a paragraph each: what it is, what it asks of you, and when you -would reach for it. The first two clauses I can draft from the machinery; the -third is the one worth your voice, and it is what the table cannot say. +A paragraph each: what it is, what it asks of you, when to reach for it. Detail goes to the notes that follow. ```{figure} figures/split-anatomy.png The split: (a) the conforming chain; (b) exploded for display. ``` -The tips are not duplicated, so slip tapers to zero there. That is the crack -condition, and it removes the special case at the front. +The tips are not duplicated, so slip tapers to zero there. That is the crack condition, and it removes the special case at the front. ## They agree on a fault, and disagree at a junction @@ -161,11 +90,7 @@ condition, and it removes the special case at the front. Split fine, TI coarse, TI fine on one colour scale. TODO alt text. ``` -Take one strand on its own, with no branch to complicate it, and the two -representations converge. The split does not care about the band width — its -peak slip rate moves by a tenth of a percent across a sixfold change — so there -the width is a mesh convenience and the split is the thin-fault reference. The -band approaches it from above as the width shrinks: +Take one strand on its own, with no branch to complicate it, and the two representations converge. The split does not care about the band width — its peak slip rate moves by a tenth of a percent across a sixfold change — so there the width is a mesh convenience and the split is the thin-fault reference. The band approaches it from above as the width shrinks: | Main strand alone, peak slip rate | w = 0.03 | w = 0.01 | w = 0.005 | |---|---|---|---| @@ -173,63 +98,29 @@ band approaches it from above as the width shrinks: | TI band, `eta_1/w` = 0.1 | — | 0.5232 | 0.5133 | | TI band, `eta_1` = 1e-3 fixed | — | 0.5232 | 0.5000 | -The middle row converges; the bottom row overshoots. The difference between them -is the whole of what a finite-width representation asks you to hold fixed. A -band's mechanical strength is the ratio `eta_1/w`, not `eta_1`: halving the width -at fixed viscosity doubles the interface strength and carries the answer past the -split rather than towards it. Hold the ratio and the band goes from 1.5% above -the cut to 0.4% below as the width halves, tracking the split's profile along the -whole strand to within a couple of per cent. - -Put a branch on it and that agreement stops, and not because of resolution. The -junction is the one place where the two representations describe genuinely -different objects. A band can fork: two weak zones meet and merge into one -continuous weak region. Two cuts cannot — where they touch they would share a -node, and a shared node is a pinned tip with no slip on it — so a cut network -leaves an intact ligament at every junction, and that ligament is rock that -carries load. - -The consequence is measurable as a conservation statement. Along the main strand, -slip steps up across the junction by what the branch takes in the opposite sense. -In the band that budget closes: the step and the branch's slip sum to within a -tenth of the step, at every ligament length and at the full join. In the cut it -does not — the sum is 25 to 40% short, and the missing motion is deforming the -ligament instead of slipping on either surface. Shrinking the ligament shrinks -the shortfall, exactly as a smaller bridge should, and cannot take it to zero. - -So the branch's slip near a junction is a property of the representation, at the -twenty per cent level, and no amount of refinement closes it. The through-going -strand, which is what a regional model is usually after, is untouched by all of -it. - -TODO(LOUIS) — the practical rule. The old draft said: run a finite-width model -at two widths and judge it against the split reference. That still holds for a -single fault, with `eta_1/w` held fixed rather than `eta_1`. What it should say -about a NETWORK is the open question: whether to reach for the band because it -conserves slip through junctions, or the cut because its ligament is arguably -the more honest picture of a real fault intersection, or to bracket with both. -This is a modelling judgement, not a measurement, and it belongs in your voice. - -Rig-scale throughout — 1 858 cells coarse, 2 940 fine, sub-second solves. These -are demonstrations of what the representations do, not converged geophysics. +The middle row converges; the bottom row overshoots. The difference between them is the whole of what a finite-width representation asks you to hold fixed. A band's mechanical strength is the ratio `eta_1/w`, not `eta_1`: halving the width at fixed viscosity doubles the interface strength and carries the answer past the split rather than towards it. Hold the ratio and the band goes from 1.5% above the cut to 0.4% below as the width halves, tracking the split's profile along the whole strand to within a couple of per cent. + +Put a branch on it and that agreement stops, and not because of resolution. The junction is the one place where the two representations describe genuinely different objects. A band can fork: two weak zones meet and merge into one continuous weak region. Two cuts cannot — where they touch they would share a node, and a shared node is a pinned tip with no slip on it — so a cut network leaves an intact ligament at every junction, and that ligament is rock that carries load. + +The consequence is measurable as a conservation statement. Along the main strand, slip steps up across the junction by what the branch takes in the opposite sense. In the band that budget closes: the step and the branch's slip sum to within a tenth of the step, at every ligament length and at the full join. In the cut it does not — the sum is 25 to 40% short, and the missing motion is deforming the ligament instead of slipping on either surface. Shrinking the ligament shrinks the shortfall, exactly as a smaller bridge should, and cannot take it to zero. + +So the branch's slip near a junction is a property of the representation, at the twenty per cent level, and no amount of refinement closes it. The through-going strand, which is what a regional model is usually after, is untouched by all of it. + +TODO(LOUIS) — the practical rule. The old draft said: run a finite-width model at two widths and judge it against the split reference. That still holds for a single fault, with `eta_1/w` held fixed rather than `eta_1`. What it should say about a NETWORK is the open question: whether to reach for the band because it conserves slip through junctions, or the cut because its ligament is arguably the more honest picture of a real fault intersection, or to bracket with both. This is a modelling judgement, not a measurement, and it belongs in your voice. + +Rig-scale throughout — 1 858 cells coarse, 2 940 fine, sub-second solves. These are demonstrations of what the representations do, not converged geophysics. ## What the rest of the series covers -- Building the fault mesh: the band from the fault's own points, extents and - paint honoured, junctions that stop short, resolution stacked on a static base, - and where a mesh generator is still needed. -- Slippery interfaces: the pair transform, no-opening to machine precision, and - interface laws on the trace. +- Building the fault mesh: the band from the fault's own points, extents and paint honoured, junctions that stop short, resolution stacked on a static base, and where a mesh generator is still needed. +- Slippery interfaces: the pair transform, no-opening to machine precision, and interface laws on the trace. - Solvers: a multigrid hierarchy on a stacked mesh. - Parallel practice and tuning, with a California-scale example. -- Benchmarks against the published lithospheric-deformation solutions, by - Thyagarajulu Gollapalli. Named here because it is part of the same series; - nothing in these notes rests on it. +- Benchmarks against the published lithospheric-deformation solutions, by Thyagarajulu Gollapalli. Named here because it is part of the same series; nothing in these notes rests on it. ## Cautions to carry into the text -- The toolset is in development on `feature/fault-outcrop-3d-cap`; show usage, - do not promise API stability. +- The toolset is in development on `feature/fault-outcrop-3d-cap`; show usage, do not promise API stability. - Contrast/viscosity-ladder numbers are solver stress tests, not geology. - Tip-shape questions are parked; do not over-claim tip-zone physics. - The dCFF reference is a welded solve; the gauge is demeaned pressure. From e28178f971847e291a6e75feeccc55c0f1649ce8 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 15:24:21 -0700 Subject: [PATCH 07/19] Thyagarajulu Gollapalli as co-author; cite the anisotropy lineage MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The note's anisotropic representations are not new machinery, and the draft introduced the painted director as if they were. Moresi and Mühlhaus formulate large-deformation Mohr-Coulomb failure as an orthotropic viscous rheology; Sharples and co-authors use the resulting anisotropic frictional rheology to study how rifts organise and how far apart their faults sit. Both cited by DOI, both verified to resolve in the built page's reference list. Author details taken from authors.yml rather than retyped. The article was also missing from the table of contents, so it had never built. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../putting-a-fault-in-a-mesh/metadata.yml | 3 +++ .../putting-a-fault-in-a-mesh.md | 20 ++++++++++++------- 2 files changed, 16 insertions(+), 7 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/metadata.yml b/articles/putting-a-fault-in-a-mesh/metadata.yml index 6486559..7055045 100644 --- a/articles/putting-a-fault-in-a-mesh/metadata.yml +++ b/articles/putting-a-fault-in-a-mesh/metadata.yml @@ -11,6 +11,9 @@ authors: - name: Louis Moresi orcid: 0000-0003-3685-174X affiliation: Australian National University + - name: Thyagarajulu Gollapalli + orcid: 0000-0001-9394-4104 + affiliation: Monash University publication_date: null version: 1.0.0 # The deposit writes archive_doi and repository_record_id when the note is diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index 1ea5dad..c764783 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -12,6 +12,10 @@ authors: orcid: 0000-0003-3685-174X affiliations: - Australian National University + - name: Thyagarajulu Gollapalli + orcid: 0000-0001-9394-4104 + affiliations: + - Monash University license: CC-BY-4.0 keywords: - Underworld Code @@ -28,22 +32,24 @@ exports: article_version: 1.0.0 software_version: underworld3 0.0.0 --- - OUTLINE — not yet written. Faults can dominate the dynamic behaviour of Earth systems at scales from the entire planet to a few 10s of metres. They are an extreme example of localisation: feedback between forcing and response that results in self-reinforcing weakening that does not have a brake at the largest scale. Faults are *extreme* in the sense that their natural length scale is orders of magnitude below that of a typical tectonic simulation. -At the tectonic scale fault is an infinitessimally thin surface across which the rock moves discontinuously. A finite element mesh is a machine for representing continuous fields. Putting the first into the second is work — cutting, conforming, refining, and doing it again every time the fault moves — so the first question is whether it has to be done at all. +At the tectonic scale fault is an infinitessimally thin surface across which the rock moves discontinuously by overcoming a frictional resistance. A finite element mesh (the mesh we use in Underworld) is a mechanism for representing continuous fields and there is not a native mechanism that perfectly represents a fault. There are lots of different potential solutions to this difficulty. They include: 1) ignoring it by using a continuum model of the fault, 2) adding extra interpolation functions that represent discontinuities, 3) splitting the mesh along the line of the fault and dealing with it as a surface, 4) refining the mesh enough that the fault width is invisibly small at the model length scale. + + +## Sometimes it is better not to mesh -## The fault you do not have to mesh +Localisation is something a rheology does on its own: if we give the material a yield stress or a strain-rate-weakening viscosity, shear bands appear where the stress requires them, and at whatever width the physics and the mesh between them allow. Nobody places those bands, nobody meshes them, and they form, rotate and fade as the loading changes. For a model asking how deformation organises itself, that emergence itself is the answer to the question. -Often it does not. Localisation is something a rheology does on its own: give the material a yield stress or a strain-rate-weakening viscosity and shear bands appear where the stress finds them, at whatever width the physics and the mesh between them allow. Nobody places those bands, nobody meshes them, and they form, rotate and fade as the loading changes. For a model asking how deformation organises itself, that emergence is the answer to the question, and prescribing a fault would beg it. +The same can be true of a fault we do want to prescribe. A direction of easy shear can be painted into the constitutive model as a field: a weak viscosity along a normal direction taken from the fault surface. The mesh never learns that the fault is there. No cutting, no conforming, no remeshing when it migrates. The fault is a property of the material, not of the discretisation. -The same is true of a fault you do want to prescribe. A direction of easy shear can be painted into the constitutive model as a field — a weak viscosity along a director taken from the gradient of the distance to a surface — and the mesh never learns that the fault is there. No cutting, no conforming, no remeshing when it migrates. The fault is a property of the material, not of the discretisation. +An anisotropic viscosity with a preferred plane of slip is a constitutive statement about how the material fails, and it has been used to carry faults and plate boundaries in convection models for two decades: @10.1080/14786430500255419 formulate large-deformation Mohr–Coulomb failure as an orthotropic viscous rheology, iterating the simple-shear viscosity so the yield criterion is satisfied on the failure plane, and @10.1002/2014JB011813 use the resulting anisotropic frictional rheology to study how rifts organise and how far apart their faults sit. -So the honest starting position is that meshing a fault is a cost you should have a reason to pay. +Do we really need to do anything more than this ? -## Why that is not always enough +## Sometimes it is better to mesh The reason is that a fault is not the same object as a shear band. From 7c21cfec26ed5a102a53706be3a26b426f9acd0e Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 16:33:44 -0700 Subject: [PATCH 08/19] Thyagarajulu Gollapalli is at ANU, in the registry as well as the note He moved at the start of the year; the affiliation was transcribed from the JOSS paper, which predates the move. Corrected in authors.yml so every future note inherits it, with a comment saying why the JOSS paper disagrees, and in this note's two copies. The remaining Monash entry is John Mansour's, untouched. The JOSS citation on the how-to-cite page is a fixed published record and is also untouched. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- articles/putting-a-fault-in-a-mesh/metadata.yml | 2 +- .../putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md | 2 +- authors.yml | 3 ++- 3 files changed, 4 insertions(+), 3 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/metadata.yml b/articles/putting-a-fault-in-a-mesh/metadata.yml index 7055045..4fcbc05 100644 --- a/articles/putting-a-fault-in-a-mesh/metadata.yml +++ b/articles/putting-a-fault-in-a-mesh/metadata.yml @@ -13,7 +13,7 @@ authors: affiliation: Australian National University - name: Thyagarajulu Gollapalli orcid: 0000-0001-9394-4104 - affiliation: Monash University + affiliation: Australian National University publication_date: null version: 1.0.0 # The deposit writes archive_doi and repository_record_id when the note is diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index c764783..119c09f 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -15,7 +15,7 @@ authors: - name: Thyagarajulu Gollapalli orcid: 0000-0001-9394-4104 affiliations: - - Monash University + - Australian National University license: CC-BY-4.0 keywords: - Underworld Code diff --git a/authors.yml b/authors.yml index ec7e399..11929b7 100644 --- a/authors.yml +++ b/authors.yml @@ -50,7 +50,8 @@ authors: gthyagi: name: Thyagarajulu Gollapalli orcid: 0000-0001-9394-4104 - affiliation: Monash University + # At ANU since the start of 2026; the JOSS paper predates the move. + affiliation: Australian National University knepley: name: Matt Knepley orcid: 0000-0002-2292-0735 From 401b5c3f2f7a12294efa6244eedab7c34b36da40 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 21:14:42 -0700 Subject: [PATCH 09/19] Cite the weak-zone and slippery-node precedents, as a justification not a review MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit The previous paragraph claimed the anisotropic rheology had been used to carry faults in convection models. It had not — those models mostly went the damage route — and the sentence was pulling the note toward localisation, which is the neighbouring argument rather than this one. Now four citations doing four jobs. Weak zones for plate margins since the 1980s (Davies), the same idea still carrying a subduction interface (Sandiford & Moresi), the directional version the band representations here use (Moresi & Mühlhaus), and the slippery-node precedent at the other end of the ladder (Zhong & Gurnis) — whose mesh was a hexahedral grid nudged towards the fault rather than conforming to it, which is the gap this series is about. All four verified by reading the built page's reference list, not by trusting the identifiers. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../putting-a-fault-in-a-mesh.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index 119c09f..d91a19b 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -45,19 +45,19 @@ Localisation is something a rheology does on its own: if we give the material a The same can be true of a fault we do want to prescribe. A direction of easy shear can be painted into the constitutive model as a field: a weak viscosity along a normal direction taken from the fault surface. The mesh never learns that the fault is there. No cutting, no conforming, no remeshing when it migrates. The fault is a property of the material, not of the discretisation. -An anisotropic viscosity with a preferred plane of slip is a constitutive statement about how the material fails, and it has been used to carry faults and plate boundaries in convection models for two decades: @10.1080/14786430500255419 formulate large-deformation Mohr–Coulomb failure as an orthotropic viscous rheology, iterating the simple-shear viscosity so the yield criterion is satisfied on the failure plane, and @10.1002/2014JB011813 use the resulting anisotropic frictional rheology to study how rifts organise and how far apart their faults sit. +None of this is new. Weak zones have stood in for plate margins in convection models since the 1980s [@10.1029/JB093iB09p10451], and the same idea still carries a subduction interface as a rheological fault [@10.5194/se-10-969-2019]. Those are isotropically weak regions; giving the weakness a direction instead is the orthotropic rheology of @10.1080/14786430500255419, and it is what the band representations below use. The other end of the ladder has a precedent too — @10.1126/science.267.5199.838 put slippery nodes into a convection model — but the mesh there was a hexahedral grid nudged towards the fault rather than one conforming to it, and conforming is most of the work. Do we really need to do anything more than this ? -## Sometimes it is better to mesh +## But sometimes it is better to mesh -The reason is that a fault is not the same object as a shear band. +The answer is yes, and the reason behind it is this: a fault is not in the same category as a shear band or a damage region. -Localisation arises on its own wherever the rheology permits it. A shear band forms where the strain rate concentrates, it is as wide as the physics and the mesh between them allow, and it can fade when the loading that produced it changes. Faults behave differently in three ways that matter for how they are put into a model. They are long-lived, outlasting many changes in the flow around them. They localise far more sharply than any band a mantle-scale mesh resolves, down to a gouge zone of metres or less. And they are self-reinforcing: once a fault has accumulated slip it has juxtaposed rocks that were never neighbours, so its weakness belongs to the structure and to the material contrast across it rather than to the strain rate passing through it at this instant. +Localisation arises on its own wherever the rheology permits it. A shear band forms where the strain rate concentrates, it is as wide as the physics and the mesh between them allow, and it can fade when the loading that produced it changes. Faults behave differently in three ways that matter for how they are put into a model. They are long-lived, outlasting many changes in the strain-field around them. They localise far more sharply than any band a lithosphere-scale mesh resolves, down to a gouge zone of metres or even less. And they are self-reinforcing: once a fault has accumulated slip it has juxtaposed rocks that were never neighbours; the weakness becomes structural and persistent even when the load is absent. -That is why a plate boundary enters a mantle model as prior knowledge. Ridges, transforms and megathrusts are prescribed, with their positions and their histories taken from observation, and a model that waits for them to emerge is answering a different question. Two consequences run through everything below. The feature needs resolution the domain cannot afford everywhere. And it moves — a rolling-back trench carries the thing that needs the resolution across the domain over tens of millions of years, so whatever the mesh does for the fault it has to do again, repeatedly, without degrading. +That is why it is common to impose plate boundaries in a mantle model as prior knowledge, and why those plate boundaries often have additional evolution rules. -The width is the other consequence. No mantle-scale mesh reaches a metre, so every representation here is a sub-grid one: either the fault is given a width the mesh can carry and a rheology to go with it, or it is given no width at all and becomes a surface across which the solution is discontinuous. +~~The width is the other consequence. No mantle-scale mesh reaches a metre, so every representation here is a sub-grid one: either the fault is given a width the mesh can carry and a rheology to go with it, or it is given no width at all and becomes a surface across which the solution is discontinuous.~~ TODO(LOUIS) — one paragraph to close the frame, and it is the note's thesis: what actually forces you up the ladder from a painted director to a cut mesh. Candidates, and they are not the same argument: the fault must slip freely rather than shear stiffly; the discontinuity must be sharp because something downstream reads the jump; the fault must carry an interface law of its own. Say which of these is the real trigger in your experience — that is the sentence a reader takes away. From 32d572ff417db0f8785b47421b315d1cc05b3145 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 21:18:20 -0700 Subject: [PATCH 10/19] Restore Sharples: the first application of TI to faults specifically Moresi and Muhlhaus give the rheology; Sharples and co-authors are where it is put to work on faults, which is the step this note's band representations inherit. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index d91a19b..98e20b3 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -45,7 +45,7 @@ Localisation is something a rheology does on its own: if we give the material a The same can be true of a fault we do want to prescribe. A direction of easy shear can be painted into the constitutive model as a field: a weak viscosity along a normal direction taken from the fault surface. The mesh never learns that the fault is there. No cutting, no conforming, no remeshing when it migrates. The fault is a property of the material, not of the discretisation. -None of this is new. Weak zones have stood in for plate margins in convection models since the 1980s [@10.1029/JB093iB09p10451], and the same idea still carries a subduction interface as a rheological fault [@10.5194/se-10-969-2019]. Those are isotropically weak regions; giving the weakness a direction instead is the orthotropic rheology of @10.1080/14786430500255419, and it is what the band representations below use. The other end of the ladder has a precedent too — @10.1126/science.267.5199.838 put slippery nodes into a convection model — but the mesh there was a hexahedral grid nudged towards the fault rather than one conforming to it, and conforming is most of the work. +None of this is new. Weak zones have stood in for plate margins in convection models since the 1980s [@10.1029/JB093iB09p10451], and the same idea still carries a subduction interface as a rheological fault [@10.5194/se-10-969-2019]. Those are isotropically weak regions; giving the weakness a direction instead is the orthotropic rheology of @10.1080/14786430500255419, applied to faults specifically by @10.1002/2014JB011813, and it is what the band representations below use. The other end of the ladder has a precedent too — @10.1126/science.267.5199.838 put slippery nodes into a convection model — but the mesh there was a hexahedral grid nudged towards the fault rather than one conforming to it, and conforming is most of the work. Do we really need to do anything more than this ? From c8278cff3abcdfa18192d0bc116d362b5ba59525 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 10 Sep 2026 21:53:34 -0700 Subject: [PATCH 11/19] Stop anticipating, and cite the separation where it is made Three forward references removed: the band representations "below", the detail going "to the notes that follow", and a precedent at "the other end of the ladder" that pointed at a ladder the reader had not been shown. Each spent a sentence on scaffolding instead of on the argument. The separation of where the fault is from how it is modelled is the thesis of both Zhong & Gurnis and Sandiford & Moresi, so it is cited at the point it is stated rather than left as an assertion. Co-Authored-By: Claude Opus 5 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../putting-a-fault-in-a-mesh.md | 37 ++++++++++++------- 1 file changed, 23 insertions(+), 14 deletions(-) diff --git a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md index 98e20b3..2c18623 100644 --- a/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md +++ b/articles/putting-a-fault-in-a-mesh/putting-a-fault-in-a-mesh.md @@ -36,34 +36,43 @@ OUTLINE — not yet written. Faults can dominate the dynamic behaviour of Earth systems at scales from the entire planet to a few 10s of metres. They are an extreme example of localisation: feedback between forcing and response that results in self-reinforcing weakening that does not have a brake at the largest scale. Faults are *extreme* in the sense that their natural length scale is orders of magnitude below that of a typical tectonic simulation. -At the tectonic scale fault is an infinitessimally thin surface across which the rock moves discontinuously by overcoming a frictional resistance. A finite element mesh (the mesh we use in Underworld) is a mechanism for representing continuous fields and there is not a native mechanism that perfectly represents a fault. There are lots of different potential solutions to this difficulty. They include: 1) ignoring it by using a continuum model of the fault, 2) adding extra interpolation functions that represent discontinuities, 3) splitting the mesh along the line of the fault and dealing with it as a surface, 4) refining the mesh enough that the fault width is invisibly small at the model length scale. +At the tectonic scale fault is an infinitessimally thin surface across which the rock moves discontinuously by overcoming a frictional resistance. A finite element mesh (the mesh we use in Underworld) is a mechanism for representing continuous fields and there is not a native mechanism that perfectly represents a fault. -## Sometimes it is better not to mesh +There are lots of different potential solutions to this difficulty. They include: 1) ignoring it by using a continuum model of the fault, 2) adding extra interpolation functions that represent discontinuities, 3) splitting the mesh along the line of the fault and dealing with it as a surface, 4) refining the mesh enough that the fault width is invisibly small at the model length scale. -Localisation is something a rheology does on its own: if we give the material a yield stress or a strain-rate-weakening viscosity, shear bands appear where the stress requires them, and at whatever width the physics and the mesh between them allow. Nobody places those bands, nobody meshes them, and they form, rotate and fade as the loading changes. For a model asking how deformation organises itself, that emergence itself is the answer to the question. -The same can be true of a fault we do want to prescribe. A direction of easy shear can be painted into the constitutive model as a field: a weak viscosity along a normal direction taken from the fault surface. The mesh never learns that the fault is there. No cutting, no conforming, no remeshing when it migrates. The fault is a property of the material, not of the discretisation. +## Faults in numerical models -None of this is new. Weak zones have stood in for plate margins in convection models since the 1980s [@10.1029/JB093iB09p10451], and the same idea still carries a subduction interface as a rheological fault [@10.5194/se-10-969-2019]. Those are isotropically weak regions; giving the weakness a direction instead is the orthotropic rheology of @10.1080/14786430500255419, applied to faults specifically by @10.1002/2014JB011813, and it is what the band representations below use. The other end of the ladder has a precedent too — @10.1126/science.267.5199.838 put slippery nodes into a convection model — but the mesh there was a hexahedral grid nudged towards the fault rather than one conforming to it, and conforming is most of the work. + -Do we really need to do anything more than this ? +Localisation is something a rheology does by itself: if we give the material a yield stress or a strain-rate-weakening viscosity, shear bands appear where the stress requires them, with an orientation determined by the stress, and at whatever width the physics and the mesh between them allow. Do we really need to do anything more than this to represent faults ? The answer is yes and there are two main reasons. -## But sometimes it is better to mesh +First, a fault is not in the same category as a shear band or a damage-zone. At the lithospheric scale a fault is persistent through changes in the tectonic loading. Faults localise far more sharply than any band a lithosphere-scale mesh resolves, down to a gouge zone of metres or even less. They are self-reinforcing: once a fault has accumulated slip it has juxtaposed distinct rocks units; the weakness becomes structural and persistent even when the load is absent. Faults have history. -The answer is yes, and the reason behind it is this: a fault is not in the same category as a shear band or a damage region. - -Localisation arises on its own wherever the rheology permits it. A shear band forms where the strain rate concentrates, it is as wide as the physics and the mesh between them allow, and it can fade when the loading that produced it changes. Faults behave differently in three ways that matter for how they are put into a model. They are long-lived, outlasting many changes in the strain-field around them. They localise far more sharply than any band a lithosphere-scale mesh resolves, down to a gouge zone of metres or even less. And they are self-reinforcing: once a fault has accumulated slip it has juxtaposed rocks that were never neighbours; the weakness becomes structural and persistent even when the load is absent. +Second, conceptually, at the tectonic scale fault is an infinitessimally thin surface across which the rock moves discontinuously by overcoming a frictional resistance. A finite element mesh (the mesh we use in Underworld) is a mechanism for representing continuous fields and there is not a native mechanism that perfectly represents a fault. Faults are sub-grid objects with their own constitutive properties. That is why it is common to impose plate boundaries in a mantle model as prior knowledge, and why those plate boundaries often have additional evolution rules. -~~The width is the other consequence. No mantle-scale mesh reaches a metre, so every representation here is a sub-grid one: either the fault is given a width the mesh can carry and a rheology to go with it, or it is given no width at all and becomes a surface across which the solution is discontinuous.~~ +It is also why crustal models of stress build-up and release always try to include known faults. They are very fine-scale structures, they reflect complex geological history and they are not simply emergent from the imposed loading. + + + + + + +None of this is new. Weak zones have stood in for plate margins in convection models since the 1980s [@10.1029/JB093iB09p10451], and the same idea still carries a subduction interface as a rheological fault [@10.5194/se-10-969-2019]. Those are isotropically weak regions; giving the weakness a direction instead is the orthotropic rheology of @10.1080/14786430500255419, applied to faults specifically by @10.1002/2014JB011813, ~~. + + +Cutting the mesh has a precedent too. @10.1126/science.267.5199.838 put slippery nodes into a convection model, on a hexahedral grid nudged towards the fault rather than one conforming to it — and conforming is most of the work. + + +Localisation arises on its own wherever the rheology permits it. A shear band forms where the strain rate concentrates, it is as wide as the physics and the mesh between them allow, and it can fade when the loading that produced it changes. Faults behave differently in three ways that matter for how they are put into a model. T -TODO(LOUIS) — one paragraph to close the frame, and it is the note's thesis: what actually forces you up the ladder from a painted director to a cut mesh. Candidates, and they are not the same argument: the fault must slip freely rather than shear stiffly; the discontinuity must be sharp because something downstream reads the jump; the fault must carry an interface law of its own. Say which of these is the real trigger in your experience — that is the sentence a reader takes away. ## Say where the fault is, then say how to model it -These are two separate decisions and the machinery keeps them separate. The first is a statement about the Earth: here is my fault surface, sampled as a polyline in two dimensions or a triangulated sheet in three, curved as it likes. The second is a modelling choice: this surface is to become a slippery interface, or a weak band of a stated width, or a direction of easy shear painted into the rheology. The same surface supports all of them, and changing the choice does not send the user back to their geometry. +These are two separate decisions and the machinery keeps them separate. It is the thesis of both @10.1126/science.267.5199.838 and @10.5194/se-10-969-2019: what the fault surface is, and what the solver is asked to do with it, are not the same statement. The first is a statement about the Earth: here is my fault surface, sampled as a polyline in two dimensions or a triangulated sheet in three, curved as it likes. The second is a modelling choice: this surface is to become a slippery interface, or a weak band of a stated width, or a direction of easy shear painted into the rheology. The same surface supports all of them, and changing the choice does not send the user back to their geometry. That separation is also what makes the construction simple. The fault's own discretisation builds the mesh that is added: the band's vertices are offsets of the fault's own points, so there is no remesh and no search for where the fault went. A mesh generator is still needed where faults meet and have to be merged or cut, but it is not asked to invent the fault. @@ -82,7 +91,7 @@ The example throughout is San Andreas geometry in miniature: a tanh S-bend on th | split nodes | conform, then split the mid-surface | nothing inside the fault | a convenience | | TI, non-conforming | nothing | carry a director from the distance gradient | physics | -A paragraph each: what it is, what it asks of you, when to reach for it. Detail goes to the notes that follow. +A paragraph each: what it is, what it asks of you, when to reach for it. ```{figure} figures/split-anatomy.png The split: (a) the conforming chain; (b) exploded for display. From a1e1d8624b7f83ac824231f0b8604a6ff09fb420 Mon Sep 17 00:00:00 2001 From: lmoresi Date: Thu, 17 Sep 2026 22:02:51 +1000 Subject: [PATCH 12/19] Faults note: three implementations on one mesh, the network figure and the stress-and-slip figure from the rig, citations where the ideas are The four-representation table becomes three things and whether to mesh for them, with XFEM in the cell the split cannot fill. One three-panel anatomy figure (ribbon, painted band, split) on one bent mesh replaces the split figure; the geometry figure is regenerated from the rig with the Y, the en-echelon array and both stepovers; the comparison figure is cut against band at w = 0.005 on one stress and one slip scale, with the stepover closed to one element. Figures are named and cross-referenced; every figure has alt text. The precedent paragraphs dissolve into the sentences they support: Davies at plate boundaries as prior knowledge, Moresi & Muhlhaus at the TI ribbon, Zhong & Gurnis at the general cut, Yang et al. at the stress concentrations of the painted band. Generating scripts for every figure ship in examples/. Underworld development team with AI support from Claude Code Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01RzK7JXSxsc2GoG7TGnQgUS --- .../examples/anatomy_mesh.py | 90 ++++ .../examples/band-anatomy-data.json | 1 + .../examples/fault-anatomy.typ | 119 +++++ .../examples/generate-band-anatomy-data.py | 42 ++ .../examples/generate-split-anatomy-data.py | 92 ++-- .../examples/rig_geometry.py | 340 +++++++++++++ .../examples/sf_fields.py | 396 +++++++++++++++ .../examples/sf_note_stress_slip.py | 142 ++++++ .../examples/sf_split_seam_viz.py | 466 ++++++++++++++++++ .../examples/sf_stepover_closure.py | 67 +++ .../examples/split-anatomy-data.json | 2 +- .../examples/split-anatomy.typ | 90 ---- .../figures/fault-anatomy.png | Bin 0 -> 96519 bytes .../figures/s_fault_geometry.png | Bin 46114 -> 68404 bytes .../figures/sf_note_stress_slip.png | Bin 0 -> 1367248 bytes .../figures/split-anatomy.png | Bin 169942 -> 0 bytes .../putting-a-fault-in-a-mesh/metadata.yml | 2 +- .../putting-a-fault-in-a-mesh.md | 98 ++-- 18 files changed, 1759 insertions(+), 188 deletions(-) create mode 100644 articles/putting-a-fault-in-a-mesh/examples/anatomy_mesh.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/band-anatomy-data.json create mode 100644 articles/putting-a-fault-in-a-mesh/examples/fault-anatomy.typ create mode 100644 articles/putting-a-fault-in-a-mesh/examples/generate-band-anatomy-data.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/rig_geometry.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/sf_fields.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/sf_note_stress_slip.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/sf_split_seam_viz.py create mode 100644 articles/putting-a-fault-in-a-mesh/examples/sf_stepover_closure.py delete mode 100644 articles/putting-a-fault-in-a-mesh/examples/split-anatomy.typ create mode 100644 articles/putting-a-fault-in-a-mesh/figures/fault-anatomy.png create mode 100644 articles/putting-a-fault-in-a-mesh/figures/sf_note_stress_slip.png delete mode 100644 articles/putting-a-fault-in-a-mesh/figures/split-anatomy.png diff --git a/articles/putting-a-fault-in-a-mesh/examples/anatomy_mesh.py b/articles/putting-a-fault-in-a-mesh/examples/anatomy_mesh.py new file mode 100644 index 0000000..aaf2797 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/anatomy_mesh.py @@ -0,0 +1,90 @@ +"""The one mesh behind the anatomy figures (band-anatomy, split-anatomy). + +A structured triangulation of [0,3] x [0,1], twelve cells by four, and a +gently curved fault trace across its middle from x = 0.5 to x = 2.5. The +CONFORMING mesh is the grid bent so that the trace runs along its edges +(the row y = 0.5 lifted onto the curve, the displacement tapering to +nothing at the top and bottom walls) — the remeshing, in miniature. The +non-conforming panel keeps the grid flat and lets the same curve cross it. +""" +import math + +H = 0.25 +NX, NY = 12, 4 +AMP = 0.12 # the curve's rise at mid-span +X0, X1 = 0.5, 2.5 # the trace's span +W = 0.5 # the band width: one cell either side + + +def rise(x): + """The trace's offset from y = 0.5: a half sine over its span, zero + at both tips and beyond them.""" + if x <= X0 or x >= X1: + return 0.0 + return AMP * math.sin(math.pi * (x - X0) / (X1 - X0)) + + +def curve(x): + return [x, 0.5 + rise(x)] + + +def normal(x): + """Unit normal to the trace at x (pointing +y).""" + if x <= X0 or x >= X1: + return [0.0, 1.0] + dy = AMP * math.pi / (X1 - X0) * math.cos(math.pi * (x - X0) / (X1 - X0)) + n = math.hypot(1.0, dy) + return [-dy / n, 1.0 / n] + + +def grid(): + """The flat grid: (coords, tris, vid) with tris as vertex triples.""" + verts, coords = {}, [] + + def vid(i, j): + if (i, j) not in verts: + verts[(i, j)] = len(coords) + coords.append([i * H, j * H]) + return verts[(i, j)] + + tris = [] + for i in range(NX): + for j in range(NY): + a, b = vid(i, j), vid(i + 1, j) + c, d = vid(i + 1, j + 1), vid(i, j + 1) + tris.append([a, b, c]) + tris.append([a, c, d]) + return coords, tris, vid + + +def conforming(coords): + """Bend the grid onto the curve: each vertex rises by the trace's + offset at its x, scaled by (1 - |y - 0.5| / 0.5) so the walls stay + put. The row y = 0.5 lands exactly on the curve.""" + out = [] + for x, y in coords: + out.append([x, y + rise(x) * (1.0 - abs(y - 0.5) / 0.5)]) + return out + + +def centroids(coords, tris): + return [[sum(coords[v][0] for v in t) / 3.0, + sum(coords[v][1] for v in t) / 3.0] for t in tris] + + +def chain(vid): + """The trace's vertices on the conforming mesh: the middle row over + the span, first tip to second.""" + return [vid(i, NY // 2) for i in range(int(X0 / H), int(X1 / H) + 1)] + + +def distance_to_curve(p, n_samp=400): + """Distance from p to the trace, and the x of the nearest sample.""" + best, bx = float("inf"), X0 + for k in range(n_samp + 1): + x = X0 + (X1 - X0) * k / n_samp + c = curve(x) + d = math.hypot(p[0] - c[0], p[1] - c[1]) + if d < best: + best, bx = d, x + return best, bx diff --git a/articles/putting-a-fault-in-a-mesh/examples/band-anatomy-data.json b/articles/putting-a-fault-in-a-mesh/examples/band-anatomy-data.json new file mode 100644 index 0000000..671258b --- 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0.5599999999999999], [2.2, 0.5544788599687456], [2.2333333333333334, 0.548808397169096], [2.2666666666666666, 0.5430041539454361], [2.3, 0.5370820393249938], [2.333333333333333, 0.5310582854123026], [2.3666666666666667, 0.5249494028981311], [2.4, 0.5187721358048277], [2.4333333333333336, 0.5125434155921184], [2.466666666666667, 0.5062803147491532], [2.5, 0.5]], "width": 0.5} \ No newline at end of file diff --git a/articles/putting-a-fault-in-a-mesh/examples/fault-anatomy.typ b/articles/putting-a-fault-in-a-mesh/examples/fault-anatomy.typ new file mode 100644 index 0000000..dc879d7 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/fault-anatomy.typ @@ -0,0 +1,119 @@ +// Three representations of one fault on one mesh (cetz draws; the +// geometry comes from generate-band-anatomy-data.py and +// generate-split-anatomy-data.py over the shared anatomy_mesh.py). +// (a) the ribbon, on the mesh bent so the trace runs along its edges +// (b) the non-conforming paint, the same trace across the flat grid +// (c) the split, on the bent mesh, exploded for display +// No captions in the drawing: they live in the figure caption. +#import "@preview/cetz:0.3.4" + +#set page(width: auto, height: auto, margin: 10pt) +#set text(font: ("Noto Sans", "Helvetica", "Arial"), size: 5pt) + +#let band = json("band-anatomy-data.json") +#let split = json("split-anatomy-data.json") + +#let mesh-fill = rgb("#f4f6f9") +#let band-fill = rgb("#c9e6c9") +#let fill-plus = rgb("#dce8fc") +#let fill-minus = rgb("#fce4ec") +#let mesh-stroke = rgb("#8899aa") +#let fault-col = rgb("#c62828") +#let dir-col = rgb("#1b5e20") +#let tip-col = rgb("#1a1a1a") +#let note-col = rgb("#555555") + +#let cells(oy, coords, tris, fill-of) = { + import cetz.draw: * + let P(v) = (coords.at(v).at(0), coords.at(v).at(1) + oy) + for (k, t) in tris.enumerate() { + line(P(t.at(0)), P(t.at(1)), P(t.at(2)), close: true, + fill: fill-of(k), stroke: (paint: mesh-stroke, thickness: 0.3pt)) + } +} + +#let directors(oy, cent, keys, dirs) = { + import cetz.draw: * + for k in keys { + let c = (cent.at(k).at(0), cent.at(k).at(1) + oy) + let d = dirs.at(str(k)) + let h = 0.055 + line((c.at(0) - h * d.at(0), c.at(1) - h * d.at(1)), + (c.at(0) + h * d.at(0), c.at(1) + h * d.at(1)), + stroke: (paint: dir-col, thickness: 0.7pt)) + } +} + +#let GAP = 1.35 +#let YA = 2 * GAP +#let YB = GAP +#let YC = 0.0 + +#cetz.canvas(length: 1.55cm, { + import cetz.draw: * + + // ---- (a) the ribbon on the bent mesh ------------------------------------- + cells(YA, band.bent, band.tris, + k => if band.band_a.contains(k) { band-fill } else { mesh-fill }) + directors(YA, band.cent_a, band.band_a, band.dir_a) + let Pa(v) = (band.bent.at(v).at(0), band.bent.at(v).at(1) + YA) + for i in range(band.chain.len() - 1) { + line(Pa(band.chain.at(i)), Pa(band.chain.at(i + 1)), + stroke: (paint: fault-col, thickness: 0.9pt)) + } + for v in band.chain { + circle(Pa(v), radius: 0.028, fill: fault-col, stroke: none) + } + content((-0.3, YA + 0.95), [(a)]) + content((3.5, YA + 0.5), text(fill: fault-col)[$Gamma$]) + line((3.12, YA + 0.25), (3.12, YA + 0.75), stroke: (paint: dir-col, thickness: 0.5pt)) + line((3.07, YA + 0.25), (3.17, YA + 0.25), stroke: (paint: dir-col, thickness: 0.5pt)) + line((3.07, YA + 0.75), (3.17, YA + 0.75), stroke: (paint: dir-col, thickness: 0.5pt)) + content((3.28, YA + 0.5), text(fill: dir-col, size: 4.5pt)[$w$]) + + // ---- (b) the same trace on the flat grid --------------------------------- + cells(YB, band.flat, band.tris, + k => if band.band_b.contains(k) { band-fill } else { mesh-fill }) + directors(YB, band.cent_b, band.band_b, band.dir_b) + for i in range(band.curve.len() - 1) { + line((band.curve.at(i).at(0), band.curve.at(i).at(1) + YB), + (band.curve.at(i + 1).at(0), band.curve.at(i + 1).at(1) + YB), + stroke: (paint: fault-col, thickness: 0.9pt)) + } + content((-0.3, YB + 0.95), [(b)]) + content((3.5, YB + 0.5), text(fill: fault-col)[$Gamma$]) + + // ---- (c) the split on the bent mesh, exploded ---------------------------- + let touches(t) = t.any(v => split.chain.contains(v)) + cells(YC, split.exploded, split.moved_tris, + k => if split.side.at(k) < 0 { fill-minus } + else if touches(split.tris.at(k)) { fill-plus } else { mesh-fill }) + let Pc(v) = (split.exploded.at(v).at(0), split.exploded.at(v).at(1) + YC) + let lower(v) = if str(v) in split.replicas { split.replicas.at(str(v)) } else { v } + for i in range(split.chain.len() - 1) { + line(Pc(split.chain.at(i)), Pc(split.chain.at(i + 1)), + stroke: (paint: fault-col, thickness: 0.9pt)) + line(Pc(lower(split.chain.at(i))), Pc(lower(split.chain.at(i + 1))), + stroke: (paint: fault-col, thickness: 0.9pt, dash: "densely-dashed")) + } + for v in split.interior { + circle(Pc(v), radius: 0.028, fill: fault-col, stroke: none) + } + for (orig, rep) in split.replicas { + circle(Pc(rep), radius: 0.028, fill: white, + stroke: (paint: fault-col, thickness: 0.8pt)) + } + for v in split.tips { + circle(Pc(v), radius: 0.04, fill: white, stroke: (paint: tip-col, thickness: 0.9pt)) + circle(Pc(v), radius: 0.014, fill: tip-col, stroke: none) + } + content((-0.3, YC + 0.95), [(c)]) + content((0.42, YC + 0.38), text(size: 4pt)[tip]) + content((2.58, YC + 0.38), text(size: 4pt)[tip]) + content((3.5, YC + 0.66), text(fill: fault-col, size: 4.5pt)[$Gamma^+$]) + content((3.5, YC + 0.34), text(fill: fault-col, size: 4.5pt)[$Gamma^-$]) + line((1.5, YC + 0.66), (1.85, YC + 0.95), stroke: (paint: note-col, thickness: 0.3pt)) + content((2.25, YC + 1.02), text(size: 4pt)[$v^+$ (original)]) + line((1.5, YC + 0.49), (1.85, YC + 0.12), stroke: (paint: note-col, thickness: 0.3pt)) + content((2.25, YC + 0.05), text(size: 4pt)[$v^-$ (replica)]) +}) diff --git a/articles/putting-a-fault-in-a-mesh/examples/generate-band-anatomy-data.py b/articles/putting-a-fault-in-a-mesh/examples/generate-band-anatomy-data.py new file mode 100644 index 0000000..08a1a45 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/generate-band-anatomy-data.py @@ -0,0 +1,42 @@ +"""Geometry for the band-anatomy figure (cetz draws, Python computes). + +(a) the ribbon on the CONFORMING mesh: the trace along the bent row of + edges, the band the cells either side of it, each with a director + (the trace normal); no vertex duplicated, every field continuous. +(b) the non-conforming paint on the FLAT grid: the same curve and the + same width, but the band is whichever cells fall within w/2 of it. +""" +import json +import os + +import anatomy_mesh as M + +coords, tris, vid = M.grid() +bent = M.conforming(coords) +ch = M.chain(vid) + +# (a) the ribbon: the cells that touch the chain, over its span +cent_a = M.centroids(bent, tris) +band_a = [k for k, t in enumerate(tris) + if any(v in ch for v in t) and M.X0 < cent_a[k][0] < M.X1] +dir_a = {k: M.normal(cent_a[k][0]) for k in band_a} + +# (b) the paint: flat grid, cells within w/2 of the curve +cent_b = M.centroids(coords, tris) +band_b, dir_b = [], {} +for k, c in enumerate(cent_b): + d, x = M.distance_to_curve(c) + if d <= 0.5 * M.W and M.X0 <= c[0] <= M.X1: + band_b.append(k) + dir_b[k] = M.normal(x) + +samples = [M.curve(M.X0 + (M.X1 - M.X0) * k / 60) for k in range(61)] +out = dict(flat=coords, bent=bent, tris=tris, cent_a=cent_a, cent_b=cent_b, + chain=ch, band_a=band_a, dir_a={str(k): v for k, v in dir_a.items()}, + band_b=band_b, dir_b={str(k): v for k, v in dir_b.items()}, + curve=samples, width=M.W) +here = os.path.dirname(os.path.abspath(__file__)) +with open(os.path.join(here, "band-anatomy-data.json"), "w") as f: + json.dump(out, f) +print(f"wrote band-anatomy-data.json: band (a) {len(band_a)} cells, " + f"band (b) {len(band_b)} cells") diff --git a/articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py b/articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py index 7e8d18c..9dbd23e 100644 --- a/articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py +++ b/articles/putting-a-fault-in-a-mesh/examples/generate-split-anatomy-data.py @@ -1,75 +1,53 @@ """Geometry for the split-anatomy figure (cetz draws, Python computes). -A small structured triangulation of [0,3] x [0,1] with a fault chain along -y = 0.5 from (0.5, 0.5) to (2.5, 0.5): 4 fault edges, 3 interior vertices -(duplicated by the split), 2 tips (shared). The "after" panel is EXPLODED -for display — the Minus side translated down — because the real copies are -geometrically coincident. +The same CONFORMING mesh as band-anatomy (a): the curved trace runs +along a row of edges. The chain has nine vertices: two tips (shared) +and seven interior (duplicated by the split). The "after" panel is +EXPLODED for display — the Minus block under the trace translated down +— because the real copies are geometrically coincident. """ import json import os -H = 0.5 -NX, NY = 6, 2 -DELTA = 0.22 # display-only explosion offset +import anatomy_mesh as M -verts = {} -coords = [] +DELTA = 0.09 # display-only explosion offset +coords, tris, vid = M.grid() +bent = M.conforming(coords) +ch = M.chain(vid) +tips = [ch[0], ch[-1]] +interior = ch[1:-1] +cent = M.centroids(bent, tris) -def vid(i, j): - if (i, j) not in verts: - verts[(i, j)] = len(coords) - coords.append([i * H, j * H]) - return verts[(i, j)] - - -tris = [] -for i in range(NX): - for j in range(NY): - a, b = vid(i, j), vid(i + 1, j) - c, d = vid(i + 1, j + 1), vid(i, j + 1) - tris.append([a, b, c]) - tris.append([a, c, d]) - -chain = [vid(i, 1) for i in range(1, 6)] # (0.5,0.5) .. (2.5,0.5) -tips = [chain[0], chain[-1]] -interior = chain[1:-1] - -# Minus side = triangles whose centroid lies below the chain SEGMENT span; -# triangles below y=0.5 but outside the span stay welded (no fault there). +# Minus side = cells touching the chain from below, within the span; +# everything else, including the cells beyond the tips, stays welded side = [] -for t in tris: - cx = sum(coords[v][0] for v in t) / 3.0 - cy = sum(coords[v][1] for v in t) / 3.0 - touches = any(v in chain for v in t) - side.append(-1 if (cy < 0.5 and touches and 0.5 < cx < 2.5) else +1) - -# Exploded positions: replicas of the interior chain vertices move down; -# to keep the Minus flank rigid, the row below them (y = 0) within the span -# moves too. Tips stay put — the lens pins there. -exploded = [list(c) for c in coords] +for k, t in enumerate(tris): + touches = any(v in ch for v in t) + below = cent[k][1] < M.curve(cent[k][0])[1] + side.append(-1 if (below and touches and M.X0 < cent[k][0] < M.X1) else +1) + +# Exploded: the replicas and the whole Minus block strictly inside the +# span move down together; the tips and the columns at the tips stay, +# so the cells at the two ends shear — the pinned tip +exploded = [list(c) for c in bent] replicas = {} for v in interior: replicas[v] = len(exploded) - exploded.append([coords[v][0], coords[v][1] - DELTA]) -moved_tris = [] -flank = set() -for t, s in zip(tris, side): - if s < 0: - moved_tris.append([replicas.get(v, v) for v in t]) - flank.update(v for v in t if v not in chain) - else: - moved_tris.append(list(t)) -for v in flank: - exploded[v][1] -= DELTA - -out = dict(coords=coords, exploded=exploded, tris=tris, - moved_tris=moved_tris, side=side, chain=chain, tips=tips, - interior=interior, replicas=replicas, delta=DELTA) + exploded.append([bent[v][0], bent[v][1] - DELTA]) +for v, (x, y) in enumerate(coords): + if y < 0.5 - 1e-9 and M.X0 + 1e-9 < x < M.X1 - 1e-9: + exploded[v][1] -= DELTA +moved_tris = [[replicas.get(v, v) for v in t] if s < 0 else list(t) + for t, s in zip(tris, side)] + +out = dict(coords=bent, exploded=exploded, tris=tris, moved_tris=moved_tris, + side=side, chain=ch, tips=tips, interior=interior, + replicas=replicas, delta=DELTA) here = os.path.dirname(os.path.abspath(__file__)) with open(os.path.join(here, "split-anatomy-data.json"), "w") as f: json.dump(out, f) print("wrote split-anatomy-data.json:", - f"{len(coords)} verts, {len(tris)} tris, chain {len(chain)},", + f"{len(bent)} verts, {len(tris)} tris, chain {len(ch)},", f"replicas {len(replicas)}") diff --git a/articles/putting-a-fault-in-a-mesh/examples/rig_geometry.py b/articles/putting-a-fault-in-a-mesh/examples/rig_geometry.py new file mode 100644 index 0000000..b1d5630 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/rig_geometry.py @@ -0,0 +1,340 @@ +"""The S-fault rig's geometry of record — ONE source for every script. + +Adjust the dials here; geometry_preview.py renders them, s_fault.py +solves them, sf_viz.py / sf_compare_viz.py draw them. + +Strands: + Main — the tanh S on the 40° diagonal (the San Andreas analogue); + also the TERRANE boundary (strong above/left of its line). + Branch — the through-line alternative in the WEAK block (tip-to-tip + near-join with the bend; the San Jacinto analogue). + Splay — the Y near-join in the TOUGH layer: a short strand leaving + the outgoing limb's side at SPLAY_PHI into the strong block, + stopped short (tip-to-SIDE junction). The trace gap is the + junction physics (the intact-gap linkage); it KISSES the main + (gap w/2) so the mesh surround must interact — placed as one + fused network with the main line (see SPLAY_GAP note). In TI + the two weak zones merge at the Y (one director per cell: + paint order decides — the #544 caveat); split keeps two cuts + and an intact sliver. + Cont — a near-touching SEGMENT of the main fault along its + continuation line beyond the upper tip (tip-to-tip stepover, + the California fault-database segmentation pattern). Same + near-touching gap as the Y; same test from the other + direction: collinear instead of oblique. +""" +import numpy as np + +# ----------------------------------------------------------------- dials +THETA = np.deg2rad(40.0) # strike of the diagonal baseline +C = np.array([0.5, 0.5]) # baseline centre +A = 0.05 # S half-step +LAM = 0.10 # bend length scale (tanh) +S_MAIN = (-0.42, 0.42) # main-trace arc range (blind tips) +S_BRANCH = (0.06, 0.42) # branch: through-line at t = -A +SPLAY_S0 = 0.06 # splay leaves the main line near this s — + # just past the bend's centre, so at +22 deg it + # CONTINUES the bend's turning (Louis, final + # pass 2026-08-26: "more of a continuation of + # the curved segment; shift it left") +SPLAY_GAP_RUNGS = 1.0 # trace-to-trace gap at the Y in RUNGS (s_f): + # KISSING = one rung at any resolution + # (a fixed 0.015 was w/2 at coarse but 1.5 w + # at fine). traces(s_f) sets SPLAY_GAP. +SPLAY_GAP = 0.010 # (derived; the coarse value) KISSING (w/2 + # at coarse; Louis's geometry of record). + # Any gap: the whole rig is ONE network + # placement (fused ribbons + embedded + # spines, place_thin_volume mesher= + # "network"); junctions are free and the + # cuts walk exact vertices. +SPLAY_PHI = np.deg2rad(22.0) # splay divergence from local strike, +t side +SPLAY_LEN = 0.18 # splay length +EE_N = 3 # EN-ECHELON: the splay + 2 more parallel +EE_LATERAL = 0.08 # segments: in the SPLAY's own frame, each +EE_ALONG = 0.05 # start = previous + EE_LATERAL n_sp (across + # the strands, up-left into the strong block) + # + EE_ALONG d_sp (forward along the strand). + # Louis 2026-08-26 (3rd pass): the array + # climbs UP as it goes LEFT; spacing 0.058; + # only #1 kisses the main +STEP_S = (-0.53, -0.37) # STEPOVER at the main's LOWER (clean) tip: a +STEP_OFF = 0.035 # segment parallel to the lower limb, offset + # STEP_OFF (ABSOLUTE — the same geometry at + # both resolutions; Louis: "close, hard to + # separate") into the WEAK block, overlapping + # the tip (s=-0.42) by 0.05 and running 0.11 + # toward the wall (wall clearance limits it) +CONT_GAP_RUNGS = 1.0 # tip-to-tip stepover gap beyond the main's + # upper tip, in RUNGS (s_f), as the splay's: + # 1 = the split's floor (chains must be + # disjoint, so one uncut edge is the least + # it can leave); 0 = the CONTINUOUS control, + # one cut the whole line. traces(s_f) sets + # CONT_GAP. It was an absolute 0.015, which + # refinement walked AWAY from the floor (2 + # edges coarse, 6 superfine) — the test the + # segment exists for is that the band closes + # a gap the cut cannot. None = leave CONT_GAP + # as set (the absolute dial of older probes). +CONT_GAP = 0.015 # (derived; the coarse value) tip-to-tip + # stepover, near-touching, California-DB + # style. Placeable at ANY value: the + # Main-line family shares ONE band + # (main_line_union), so the gap lives only + # in the cuts + paint. +CONT_LEN = 0.10 # continuation-segment length along the line + # (0.18 put the extended band's cavity into + # the top-right wall — interior clearance) +# ----------------------------------------------------------------------- + +ES = np.array([np.cos(THETA), np.sin(THETA)]) +ET = np.array([-np.sin(THETA), np.cos(THETA)]) +RES = {"coarse": (0.03, 0.01), "fine": (0.01, 0.005), + "superfine": (0.005, 0.0025)} # (w, s_f) +W_OF_SF = {sf: w for w, sf in RES.values()} # s_f -> w + + +def main_curve(s): + s = np.atleast_1d(s) + return C + s[:, None] * ES + (A * np.tanh(s / LAM))[:, None] * ET + + +def resample_arclength(P, spacing): + """Equispaced-in-arclength, ends kept, interval count EVEN.""" + seg = np.linalg.norm(np.diff(P, axis=0), axis=1) + arc = np.concatenate([[0.0], np.cumsum(seg)]) + n = max(2, int(round(arc[-1] / spacing))) + n += n % 2 + si = np.linspace(0.0, arc[-1], n + 1) + return np.column_stack([np.interp(si, arc, P[:, 0]), + np.interp(si, arc, P[:, 1])]) + + +def splay_line(k=0): + """En-echelon strand k (k=0 is THE splay, kissing the main): from a + stop-short start beside the outgoing limb, diverging into the + strong block; strand k starts EE_STEP back along strike and out.""" + t0 = float(A * np.tanh(SPLAY_S0 / LAM)) + # local strike of the main line at SPLAY_S0 + sech2 = 1.0 / np.cosh(SPLAY_S0 / LAM) ** 2 + tan_l = ES + (A / LAM) * sech2 * ET + tan_l /= np.linalg.norm(tan_l) + nrm_l = np.array([-tan_l[1], tan_l[0]]) # +t side + if nrm_l @ ET < 0: + nrm_l = -nrm_l + d = np.cos(SPLAY_PHI) * tan_l + np.sin(SPLAY_PHI) * nrm_l + n_sp = np.array([-d[1], d[0]]) # up-left of the strand + start = (C + SPLAY_S0 * ES + t0 * ET + SPLAY_GAP * nrm_l + + k * (EE_LATERAL * n_sp + EE_ALONG * d)) + return np.array([start, start + SPLAY_LEN * d]) + + +def stepover_line(w=None): + """The stepover at the main's lower tip: parallel to the lower limb + (t = -A there), offset STEP_OFF (absolute) into the weak block.""" + ss = np.linspace(*STEP_S, 500) + return C + ss[:, None] * ES - (A + STEP_OFF) * ET + + +def main_line_union(s_f): + """The Main-line family as ONE parametrisation — the library's rule + for close pairs ('surfaces must be separated by at least a cell; + place close pairs as one thin volume instead'): one ladder polyline + over the WHOLE line, with the Main and Cont cut chains as vertex + SUBCHAINS of it (#595: nothing snaps — cuts consume the spine's own + vertices, so chains and band must share them). The stepover gap + therefore lives only in the cuts and the paint, and can close below + the placement floor. Returns (union, main_chain, cont_chain); + chain ends are vertex-quantised, so the realised gap is the nearest + ≥ CONT_GAP the sampling allows.""" + s_end = S_MAIN[1] + CONT_GAP + CONT_LEN + sm = np.linspace(S_MAIN[0], s_end, 4000) + union = resample_arclength(main_curve(sm), s_f) + if CONT_GAP <= 0.0: + # the CONTINUOUS control: no stepover — one cut, the whole line + return union, union, union[:0] + s_along = (union - C) @ ES + # the gap in SPINE EDGES (distance thresholds are at the mercy of + # vertex phase): n_edges=1 -> abutting chains, one uncut edge; + # n_edges=2 -> one free bridge vertex; realised gap = n_edges rungs + i_tip = int(np.searchsorted(s_along, S_MAIN[1] + 1e-9)) - 1 + rung = float(np.linalg.norm(np.diff(union, axis=0), axis=1).mean()) + n_edges = max(1, int(round(CONT_GAP / rung))) + return union, union[:i_tip + 1], union[i_tip + n_edges:] + + +def traces(s_f, w=None): + """The rig's strands, sampled at the rung scale — [(label, points)]: + Main, Branch, Splay (+ Splay2, Splay3 en echelon), Step (the lower + stepover), Cont (the upper stepover segment, if the gap is open). + Main and Cont are subchains of the shared main_line_union spine.""" + global SPLAY_GAP, CONT_GAP + SPLAY_GAP = SPLAY_GAP_RUNGS * s_f + if CONT_GAP_RUNGS is not None: + CONT_GAP = CONT_GAP_RUNGS * s_f + w = W_OF_SF[s_f] if w is None else w + _union, main, cont = main_line_union(s_f) + sb = np.linspace(*S_BRANCH, 2000) + branch = resample_arclength(C + sb[:, None] * ES - A * ET, s_f) + out = [("Main", main), ("Branch", branch)] + for k in range(EE_N): + sp = splay_line(k) + seg = resample_arclength( + np.linspace(0, 1, 200)[:, None] * (sp[1] - sp[0]) + sp[0], s_f) + out.append(("Splay" if k == 0 else f"Splay{k + 1}", seg)) + out.append(("Step", resample_arclength(stepover_line(w), s_f))) + if len(cont): + out.append(("Cont", cont)) + return out + + +def strong_mask(points): + """The terrane rule: True on the +t (strong) side of the fault line.""" + rel = np.asarray(points) - C + return (rel @ ET) > A * np.tanh((rel @ ES) / LAM) + + +def receiver_normals(points): + """Local fault-line orientation, continued off the line (for dCFF).""" + rel = np.asarray(points) - C + s_co = rel @ ES + sech2 = 1.0 / np.cosh(s_co / LAM) ** 2 + n = ET[None, :] - (A / LAM) * sech2[:, None] * ES[None, :] + return n / np.linalg.norm(n, axis=1, keepdims=True) + + +def ti_fields(mesh, foots, eta1_val, eta0_vals, tag, w=None, traces_=None): + """The TI representation's P0 fields, painted on the wrapper's + HONOURED footprints (the review-1 API): eta_1 = weak inside the + fault footprints only, background elsewhere; per-cell directors + following each strand's own orientation (curved Main and its Cont + segment, ET-normal Branch, splay-normal Splay). Returns + (eta1_var, dir_var, foot). + + With ``w`` and ``traces_`` given, the stepover corridor between the + Main's tip and the Cont's start is painted too: a footprint stops + half a rung past its tip, so a one-edge gap leaves a plug of a few + intact cells in the band (4 at superfine) and the TI still has a + gap. Two collinear segments of ONE line closer than the band is + wide are one weak zone — that is the band's whole claim on the + stepover, and the paint has to say it.""" + import underworld3 as uw + + foot = np.zeros_like(next(iter(foots.values()))) + for m_ in foots.values(): + foot = foot | m_ + eta1 = uw.discretisation.MeshVariable(f"etaT{tag}", mesh, 1, degree=0) + cen_abs = np.asarray(eta1.coords) + if w is not None and traces_ is not None: + tr = dict(traces_) + if "Main" in tr and "Cont" in tr: + a, b = np.asarray(tr["Main"][-1]), np.asarray(tr["Cont"][0]) + d = b - a + L = float(np.linalg.norm(d)) + d = d / L + rel = cen_abs - a + along = rel @ d + across = np.abs(rel @ np.array([-d[1], d[0]])) + corridor = (along > -0.5 * L) & (along < 1.5 * L) & (across < 0.5 * w) + n_new = int((corridor & ~foot).sum()) + foot = foot | corridor + foots = dict(foots) + foots["Main"] = foots["Main"] | corridor + uw.pprint(f"[ti_fields] stepover corridor: {n_new} band cell(s) " + f"painted between the Main's tip and the Cont's start") + eta1.array[:, 0, 0] = np.where(foot, eta1_val, eta0_vals) + ndir = uw.discretisation.MeshVariable(f"dirT{tag}", mesh, 2, degree=0, + continuous=False) + cen = cen_abs - C + s_co = cen @ ES + dvals = np.tile(ET, (len(s_co), 1)) + sech2 = 1.0 / np.cosh(s_co / LAM) ** 2 + n_main = ET[None, :] - (A / LAM) * sech2[:, None] * ES[None, :] + n_main /= np.linalg.norm(n_main, axis=1, keepdims=True) + if "Main" in foots: + dvals[foots["Main"]] = n_main[foots["Main"]] + if "Cont" in foots: + dvals[foots["Cont"]] = n_main[foots["Cont"]] + sp = splay_line() + d_sp = (sp[1] - sp[0]) / np.linalg.norm(sp[1] - sp[0]) + for lbl, m_ in foots.items(): + if lbl.startswith("Splay"): # all en-echelon strands + dvals[m_] = np.array([-d_sp[1], d_sp[0]]) + # "Step" and "Branch": strike-parallel -> the ET default + ndir.array[...] = dvals.reshape(ndir.array.shape) + return eta1, ndir, foot + + +# The drive is built by callers from mesh coordinates: +# t_sym = (x - 0.5) * ET[0] + (y - 0.5) * ET[1] +# v = sense * 2 * t_sym * ES (plate-motion-frame shear) + + +def place_rig(base_mesh, s_f, width, margin_rings=2, split=True, uncut=()): + """ONE placement path for the whole rig (Louis, 2026-08-25: no more + parallel paths): every strand's extended spine goes into a single + ``place_thin_volume(..., mesher="network")`` call — the ribbons are + fused in CAD (junctions free, so the Y may kiss and the stepover + shares its band) and every spine is EMBEDDED in the fused face, so + the cuts walk exact vertices (#595) at any resolution. Cuts: ONE + network add_fault call. Honoured footprints per strand by nearest + sample over the concatenated spines. Returns (mesh, info).""" + import underworld3 as uw + from underworld3.utilities.place_surface import ( + place_thin_volume, _extend_polyline_2d, _footprint_from_samples) + + tr = traces(s_f, width) + union, chain_main, chain_cont = main_line_union(s_f) + m = margin_rings + spines = [("MainLine", union)] + [(lbl, P) for lbl, P in tr + if lbl not in ("Main", "Cont")] + ext = [_extend_polyline_2d(np.asarray(P, dtype=float), m) + for _n, P in spines] + spacing = float(np.linalg.norm(np.diff(union, axis=0), axis=1).mean()) + dm, _ = place_thin_volume(base_mesh.dm, ext, width, label="Band", + label_value=71, clearance=0.3, size=spacing, + mesher="network") + mesh = uw.discretisation.Mesh( + dm, simplex=True, qdegree=base_mesh.qdegree, + coordinate_system_type=base_mesh.CoordinateSystem.coordinate_type, + boundaries=base_mesh.boundaries, verbose=False) + # the placed mesh OWNS the base's MG tail + the band as FAC zone: every + # solver on it (Stokes, rotated, projections, probes, renders) drives + # FMG/FAC automatically — no per-solver set_custom_fmg, no silent GAMG + from underworld3.utilities.custom_mg import adopt_hierarchy + adopt_hierarchy(mesh, base_mesh, + fac_zone=mesh.cells_labelled("Band", 71)) + if split: + # ``uncut``: strands left WITHOUT a cut (band only — the fused + # control: no jump, no weld, by construction) + mesh = mesh.add_fault([(lbl, np.asarray(P, dtype=float)) + for lbl, P in tr if lbl not in uncut]) + mesh._custom_mg_fac_zone = None # a split fault needs no patch + band = mesh.cells_labelled("Band", 71) + S_all = np.vstack(ext) + off = np.cumsum([0] + [len(S) for S in ext]) # spine offsets + def user(k, lo, hi): + u = np.zeros(len(S_all), dtype=bool) + u[off[k] + lo:off[k] + hi] = True + return u + foots = {"Main": _footprint_from_samples( + mesh.dm, band, S_all, user(0, m, m + len(chain_main)))} + if len(chain_cont): + foots["Cont"] = _footprint_from_samples( + mesh.dm, band, S_all, + user(0, m + len(union) - len(chain_cont), m + len(union))) + for k, (lbl, _P) in enumerate(spines[1:], start=1): + foots[lbl] = _footprint_from_samples( + mesh.dm, band, S_all, user(k, m, len(ext[k]) - m)) + # the extended spines (what the RIBBON covers) and the cut chains on + # each (what the SPLIT sliced): their difference along the spine is + # where the surface representation cannot join (Louis, 2026-08-27) + cut_chains = {"MainLine": [c for c in (chain_main, chain_cont) if len(c)]} + for lbl, P in spines[1:]: + cut_chains[lbl] = [np.asarray(P, dtype=float)] + return mesh, {"n_cells": int(mesh.dm.getHeightStratum(0)[1]), + "n_rungs": [len(P) for _n, P in spines], + "footprints": foots, "band": band, + "spines_ext": [(n_, E) for (n_, _P), E in zip(spines, ext)], + "cut_chains": cut_chains} diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_fields.py b/articles/putting-a-fault-in-a-mesh/examples/sf_fields.py new file mode 100644 index 0000000..ab24b5f --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_fields.py @@ -0,0 +1,396 @@ +"""Shared IN-SESSION solve + nodal recovery for the rig's field renders +(in-session because of #640: split-mesh checkpoint reload mixes the +cut DOFs). One solver copy for every render script. + +Recovery follows the standing rulings (reference_stress_recovery_across +_a_material_jump; fault-teaching handoff): + * project the stress COMPONENTS to continuous P1 — never an + invariant, never evaluate() a composite sqrt; + * strain rate across a viscosity jump = recovered stress divided by + each material's OWN viscosity (the jump re-enters through eta, not + through a smeared gradient). For TI, in the (n, t) frame: + e_nn, e_tt = tau/(2 eta0); e_nt = tau_nt/(2 eta1). + * invariants in numpy. +Nodal eta / director are taken from the NEAREST cell centroid (the +honoured paint, cell-based; the one-ring at the band edge is the only +ambiguity). +""" +import numpy as np +import sympy +from scipy.spatial import cKDTree + +import underworld3 as uw +from underworld3.utilities import fault_contact + +import rig_geometry as G + + +def solve_rep(rep, contrast=1.0, res="coarse", cont_gap=0.004, + eta1_val=1e-3, sense=1.0, tol=1e-5, tag="", yield_stress=0.0, + eta1_frac=0.1, uncut=(), uncut_eta1=-1.0): + """Solve the rig for one representation; returns the live state. + ``yield_stress > 0`` confines a von Mises yield to the Band label + (split: cuts + yielding band; ``rep="vp"``: the isotropic yielding + band IS the fault — no cuts, no director). ``rep="hybrid"`` = the + recipe (fault_interface_equivalence/HYBRID_RECIPE.md): frictionless + cuts PLUS a mild TI band, eta_1 = eta1_frac x eta_0 on the + footprints.""" + G.CONT_GAP_RUNGS = None # the absolute dial + G.CONT_GAP = float(cont_gap) + W, S_F = G.RES[res] + TRACES = G.traces(S_F) + base = uw.meshing.UnstructuredSimplexBox( + minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), cellSize=1 / 8, + regular=False, qdegree=2, refinement=1) + mesh, pinfo = G.place_rig(base, S_F, W, + split=(rep in ("split", "hybrid")), + uncut=tuple(uncut)) + x, y = mesh.X + t = f"{rep[0]}{tag}" + v = uw.discretisation.MeshVariable(f"v{t}", mesh, 2, degree=2) + p = uw.discretisation.MeshVariable(f"p{t}", mesh, 1, degree=1, + continuous=True) + stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) + eta0 = uw.discretisation.MeshVariable(f"eta0{t}", mesh, 1, degree=0) + cen = np.asarray(eta0.coords) + strong = G.strong_mask(cen) + eta0_vals = np.where(strong, float(contrast), 1.0) + eta0.array[:, 0, 0] = eta0_vals + st = dict(rep=rep, contrast=float(contrast), mesh=mesh, v=v, p=p, + stokes=stokes, eta0=eta0, eta0_vals=eta0_vals, pinfo=pinfo, + traces=TRACES, W=W, s_f=S_F, eta1=None, ndir=None, + foot=None) + if rep in ("ti", "hybrid") or uncut: + eta1, ndir, foot = G.ti_fields(mesh, pinfo["footprints"], + float(eta1_val), eta0_vals, t) + if rep == "hybrid": + eta1.array[:, 0, 0] = np.where(foot, float(eta1_frac) * eta0_vals, + eta0_vals) + elif rep == "split": + eta1.array[:, 0, 0] = eta0_vals + if uncut and float(uncut_eta1) >= 0.0: + ufoot = np.zeros_like(foot) + for u in uncut: + ufoot |= pinfo["footprints"][u] + eta1.array[:, 0, 0] = np.where(ufoot, float(uncut_eta1) * eta0_vals, + eta1.array[:, 0, 0]) + st["uncut"] = tuple(uncut) + stokes.constitutive_model = \ + uw.constitutive_models.TransverseIsotropicFlowModel + P = stokes.constitutive_model.Parameters + P.shear_viscosity_0 = eta0.sym[0] + P.shear_viscosity_1 = eta1.sym[0] + P.director = ndir.sym + st.update(eta1=eta1, ndir=ndir, foot=foot) + elif yield_stress > 0.0: + ybar = uw.discretisation.MeshVariable(f"y{t}", mesh, 1, degree=0) + ybar.array[:, 0, 0] = np.where(pinfo["band"], float(yield_stress), + 1e8) + stokes.constitutive_model = \ + uw.constitutive_models.ViscoPlasticFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = eta0.sym[0] + stokes.constitutive_model.Parameters.yield_stress = ybar.sym[0] + # Newton on a FIXED rounded yield surface (Louis, 2026-08-26): the + # hard-Min perfect-plastic tangent is 0 along the flow in every + # yielded cell (symmetric, semi-definite) and stalls Newton when + # many cells yield; delta=0.1 (yield_anchor "yield": exact yield + # point, rounded corner) restores it. Not the retired in-solve + # delta-march. + cm_ = stokes.constitutive_model + cm_.yield_mode = "softmin" + cm_.yield_smoother = "powermean" + cm_.yield_anchor = "yield" + cm_.yield_softness = 0.1 + stokes.consistent_jacobian = True + stokes.petsc_options["snes_max_it"] = 60 + st["tau_y"] = float(yield_stress) + else: + if rep == "vp": + raise ValueError("rep='vp' needs yield_stress > 0") + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = eta0.sym[0] + stokes.bodyforce = [0.0, 0.0] + ES, ET = G.ES, G.ET + t_sym = (x - 0.5) * float(ET[0]) + (y - 0.5) * float(ET[1]) + for wall in ("Bottom", "Top", "Left", "Right"): + stokes.add_dirichlet_bc((sense * 2.0 * t_sym * float(ES[0]), + sense * 2.0 * t_sym * float(ES[1])), wall) + stokes.petsc_use_pressure_nullspace = True + stokes.tolerance = float(tol) + if rep in ("split", "hybrid"): + for lbl, _tr in TRACES: + if lbl in uncut: + continue + stokes.add_fault_bc(0, boundary=lbl) + info = fault_contact.solve_with_fault(stokes) + st["converged"] = bool(info.get("converged")) + else: + stokes.solve() + st["converged"] = stokes.snes.getConvergedReason() > 0 + return st + + +def stress_expression(st): + """Deviatoric stress components (sympy 2x2) for this rep's law.""" + E = st["stokes"].strainrate + E = E - sympy.eye(2) * (E[0, 0] + E[1, 1]) / 2 + eta0 = st["eta0"].sym[0] + if st["rep"] in ("ti", "hybrid"): + nd = st["ndir"].sym + nv = sympy.Matrix([nd[0], nd[1]]) + tv = sympy.Matrix([-nd[1], nd[0]]) + e_nt = (nv.T * E * tv)[0, 0] + return (2 * eta0 * E + - 2 * (eta0 - st["eta1"].sym[0]) * e_nt + * (nv * tv.T + tv * nv.T)) + return 2 * eta0 * E + + +def nodal_stress(st, name="tP1"): + """Project the stress COMPONENTS to continuous P1; returns + (P1 variable, txx, tyy, txy) — deviatoric, nodal.""" + mesh = st["mesh"] + S = uw.discretisation.MeshVariable(f"{name}{st['rep'][0]}", mesh, 1, + degree=1) + proj = uw.systems.Projection(mesh, S) + tau = stress_expression(st) + comps = {} + for key, expr in (("xx", tau[0, 0]), ("yy", tau[1, 1]), + ("xy", tau[0, 1])): + proj.uw_function = expr + proj.solve(zero_init_guess=True) + comps[key] = np.asarray(S.data[:, 0]).copy() + d = 0.5 * (comps["xx"] + comps["yy"]) + return S, comps["xx"] - d, comps["yy"] - d, comps["xy"] + + +def nodal_material(st, S): + """eta0, eta1, director at the P1 nodes from the NEAREST cell + centroid (the honoured, cell-based paint).""" + cen = np.asarray(st["eta0"].coords) + idx = cKDTree(cen).query(np.asarray(S.coords))[1] + eta0 = st["eta0_vals"][idx] + if st["rep"] in ("ti", "hybrid"): + eta1 = np.asarray(st["eta1"].array).reshape(len(cen))[idx] + nv = np.asarray(st["ndir"].array).reshape(len(cen), 2)[idx] + else: + eta1 = eta0.copy() + nv = np.tile(G.ET, (len(idx), 1)) + return eta0, eta1, nv + + +def nodal_strainrate(st, name="eP1"): + """THE reliable strain-rate map (measured 2026-08-25, TI band, coarse, + eta ratio 1e3): project the strain-rate COMPONENTS to continuous P1 + and form the invariant in numpy. In-band median 5.3 / p90 10.2 / + max 15.9 against the direct in-cell truth 5.6 / 9.7 / 13; far field + 0.98 = the drive. The one-ring smear at the band edge is the only + cost. Returns (P1 variable, edot_II).""" + S, exx, eyy, exy = nodal_strainrate_components(st, name=name) + return S, np.sqrt(0.5 * (exx ** 2 + eyy ** 2) + exy ** 2) + + +def nodal_strainrate_components(st, name="eP1"): + """The deviatoric strain-rate COMPONENTS projected to continuous P1; + returns (P1 variable, exx, eyy, exy) in the variable's DOF order — + the recipe of :func:`nodal_strainrate`, components kept.""" + mesh = st["mesh"] + S = uw.discretisation.MeshVariable(f"{name}{st['rep'][0]}", mesh, 1, + degree=1) + proj = uw.systems.Projection(mesh, S) + E = st["stokes"].strainrate + comps = {} + for key, expr in (("xx", E[0, 0]), ("yy", E[1, 1]), ("xy", E[0, 1])): + proj.uw_function = expr + proj.solve(zero_init_guess=True) + comps[key] = np.asarray(S.data[:, 0]).copy() + d = 0.5 * (comps["xx"] + comps["yy"]) + return S, comps["xx"] - d, comps["yy"] - d, comps["xy"] + + +def nodal_strainrate_from_stress(st, S, txx, tyy, txy): + """The material-jump recipe (recovered stress / own eta; TI in the + (n, t) frame, e_nt = tau_nt / (2 eta1)). CAVEAT (measured): only + reliable when the recovered interface stress is accurate to better + than eta0/eta1 — at coarse w with a 1e3 ratio it is NOT (in-band + p90 33, max 1037 vs truth 9.7 / 13): the band-edge recovery error is + amplified 1000x. Kept for the single-cell-layer case it was measured + on; use nodal_strainrate for a multi-cell band.""" + eta0, eta1, nv = nodal_material(st, S) + tv = np.column_stack([-nv[:, 1], nv[:, 0]]) + # rotate tau into (n, t) + tnn = (nv[:, 0] ** 2 * txx + 2 * nv[:, 0] * nv[:, 1] * txy + + nv[:, 1] ** 2 * tyy) + ttt = (tv[:, 0] ** 2 * txx + 2 * tv[:, 0] * tv[:, 1] * txy + + tv[:, 1] ** 2 * tyy) + tnt = (nv[:, 0] * tv[:, 0] * txx + + (nv[:, 0] * tv[:, 1] + nv[:, 1] * tv[:, 0]) * txy + + nv[:, 1] * tv[:, 1] * tyy) + enn, ett, ent = tnn / (2 * eta0), ttt / (2 * eta0), tnt / (2 * eta1) + return np.sqrt(0.5 * (enn ** 2 + ett ** 2) + ent ** 2) + + +def stress_invariant(txx, tyy, txy): + return np.sqrt(0.5 * (txx ** 2 + tyy ** 2) + txy ** 2) + + +def native_slip_profile(st, label, mask=True): + """The NATIVE slip gauge (Louis's ruling, 2026-08-25) on a live + state: split = tangential pair jump along the cut; TI = band + integral of the director-plane shear strain rate along the + mid-line (5 stations across the band). Samples inside another + strand's band are dropped when ``mask``. Returns (points, slip) + with points ON the trace (for a coloured-trace overlay).""" + from underworld3.utilities import fault_contact + from scipy.spatial import cKDTree + TR = dict(st["traces"]) + trace = np.asarray(TR[label]) + W = st["W"] + others = (np.vstack([q for l2, q in st["traces"] if l2 != label]) + if mask else None) + + def own(X): + if others is None: + return np.ones(len(X), dtype=bool) + k = cKDTree(others).query(X)[0] > W + return k if k.sum() >= 3 else np.ones(len(X), dtype=bool) + + if st["rep"] in ("split", "hybrid") and label not in st.get("uncut", ()): + coords, jumps, normals = fault_contact.fault_pair_jumps( + st["stokes"], label, st["stokes"]._rotated_freeslip_info) + jn = np.einsum("ij,ij->i", jumps, normals) + tang = np.linalg.norm(jumps - jn[:, None] * normals, axis=1) + keep = own(coords) + return coords[keep], tang[keep] + P = trace[::2] + t = np.gradient(P, axis=0) + t /= np.linalg.norm(t, axis=1)[:, None] + n = np.column_stack([-t[:, 1], t[:, 0]]) + E = st["stokes"].strainrate + # stations strictly INSIDE the band, off the mid-line and rail + # edges (points ON an edge locate into either neighbour, host + # cells included): the band-integral is w x the mean of 2 e_nt + offs = W * np.array([-0.45, -0.3, -0.1, 0.1, 0.3, 0.45]) + ent = np.zeros((len(offs), len(P))) + for k, o in enumerate(offs): + Q = P + o * n + exx = np.asarray(uw.function.evaluate(E[0, 0], Q)).ravel() + eyy = np.asarray(uw.function.evaluate(E[1, 1], Q)).ravel() + exy = np.asarray(uw.function.evaluate(E[0, 1], Q)).ravel() + ent[k] = (n[:, 0] * t[:, 0] * exx + + (n[:, 0] * t[:, 1] + n[:, 1] * t[:, 0]) * exy + + n[:, 1] * t[:, 1] * eyy) + # band slip = v_t difference across the band + one-cell skirt + # (continuous field -> robust; in-cell gradient integrals are + # vertex-phase sensitive at fine w) + skirt = W / 2 + st["s_f"] + v = st["v"] + vp = np.asarray(uw.function.evaluate(v.sym, P + skirt * n) + ).reshape(len(P), -1)[:, :2] + vm = np.asarray(uw.function.evaluate(v.sym, P - skirt * n) + ).reshape(len(P), -1)[:, :2] + slip = np.abs(np.einsum("ij,ij->i", vp - vm, t)) + keep = own(P) + return P[keep], slip[keep] + + +def midline_shear_rate(st, trace, eps=0.02): + """The TI band's slip measure of record (Louis, 2026-08-25, reaffirmed + 2026-09-05): the shear strain rate resolved on the director plane + ALONG THE MID-LINE, 2 e_nt = 2 n.E.t at the spine vertices. + + Computed as the NODAL RECOVERY of the P2 velocity's own gradient: in + every cell of the vertex's fan the strain rate is evaluated at the + vertex (a point stepped ``eps`` of the way toward the cell's + centroid, so the locator lands in THAT cell — the P2 gradient is + linear within a cell, so the step costs O(eps)) and the fan is + averaged with the cells' angles at the vertex as weights. No span, + so a neighbouring band cannot enter; no L2 projection, so no + overshoot at the peak (the P1 projection read 91 on the fine Main + where the in-cell truth is ~55); defined through a junction. The + fan is summed over ranks (a shared vertex's fan is split). + Returns (points, 2 e_nt) for the trace vertices that are mesh + vertices (every placed trace vertex is a spine vertex). + """ + from mpi4py import MPI + from scipy.spatial import cKDTree + from underworld3.utilities.reconnect import _coords + mesh = st["mesh"] + dm = mesh.dm + comm = uw.mpi.comm + vS, vE = dm.getDepthStratum(0) + cS, cE = dm.getHeightStratum(0) + X = _coords(dm)[: vE - vS] + P = np.asarray(trace, dtype=float) + d, idx = cKDTree(X).query(P) + on = d < 1e-6 + t = np.gradient(P, axis=0) + t /= np.linalg.norm(t, axis=1)[:, None] + n = np.column_stack([-t[:, 1], t[:, 0]]) + E = st["stokes"].strainrate + # one batch of evaluation points: every (vertex, fan cell) pair + pts, owner, weight = [], [], [] + for k in np.flatnonzero(on): + iv = int(idx[k]) + for c in dm.getTransitiveClosure(iv + vS, useCone=False)[0]: + c = int(c) + if not (cS <= c < cE): + continue + verts = [int(q) - vS for q in dm.getTransitiveClosure(c)[0] + if vS <= int(q) < vE] + cen = X[verts].mean(axis=0) + others = [q for q in verts if q != iv] + a, b = X[others[0]] - X[iv], X[others[1]] - X[iv] + ang = abs(np.arctan2(a[0] * b[1] - a[1] * b[0], a @ b)) + pts.append(X[iv] + eps * (cen - X[iv])) + owner.append(k) + weight.append(ang) + num = np.zeros(len(P)) + den = np.zeros(len(P)) + if pts: + Q = np.asarray(pts, dtype=float) + exx = np.asarray(uw.function.evaluate(E[0, 0], Q)).ravel() + eyy = np.asarray(uw.function.evaluate(E[1, 1], Q)).ravel() + exy = np.asarray(uw.function.evaluate(E[0, 1], Q)).ravel() + owner = np.asarray(owner) + weight = np.asarray(weight) + ent = (n[owner, 0] * t[owner, 0] * exx + + (n[owner, 0] * t[owner, 1] + n[owner, 1] * t[owner, 0]) * exy + + n[owner, 1] * t[owner, 1] * eyy) + np.add.at(num, owner, weight * ent) + np.add.at(den, owner, weight) + if comm.size > 1: + for arr in (num, den): + comm.Allreduce(MPI.IN_PLACE, arr, op=MPI.SUM) + rate = np.full(len(P), np.nan) + have = den > 0 + rate[have] = 2.0 * np.abs(num[have] / den[have]) + return P, rate + + +def coloured_trace(points, values, z=0.003): + """A pv polyline through ``points`` carrying ``values`` as point + scalars — the coloured-trace overlay (sorted along the trace by + the order given).""" + import pyvista as pv + line = pv.lines_from_points( + np.column_stack([points, np.full(len(points), z)])) + line.point_data["slip"] = np.asarray(values) + return line + + +def yielded_cells(st): + """Band cells on the plastic branch (2 eta0 edot_II > tau_y) from the + in-cell strain rate at the band centroids. Returns (band_mask, + yielded_within_band) — both over ALL cells for cell-data rendering.""" + band = st["pinfo"]["band"] + cen = np.asarray(st["eta0"].coords)[band] + E = st["stokes"].strainrate + c = [np.asarray(uw.function.evaluate(E[i, j], cen)).ravel() + for i, j in ((0, 0), (1, 1), (0, 1))] + d = 0.5 * (c[0] + c[1]) + e2 = np.sqrt(0.5 * ((c[0] - d) ** 2 + (c[1] - d) ** 2) + c[2] ** 2) + yielded = np.zeros(len(band), dtype=bool) + yielded[np.flatnonzero(band)] = (2.0 * st["eta0_vals"][band] * e2 + > st["tau_y"]) + return band, yielded diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_note_stress_slip.py b/articles/putting-a-fault-in-a-mesh/examples/sf_note_stress_slip.py new file mode 100644 index 0000000..8651fb9 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_note_stress_slip.py @@ -0,0 +1,142 @@ +"""Figure for UWTN 2026-017: the cut and the band on the same network, +stress and slip, at the width where they agree. + +Both columns are the rig at w = 0.005 (superfine): the split node, and +the TI band at eta_1/w = 0.1, the ratio the convergence test holds +fixed. Background is log10 tau_II on one scale; every trace is drawn as +a tube coloured by the SIGNED slip rate in each representation's own +currency -- the pair jump on the cut, the velocity jump across the +band's own edges -- on one two-slope map with grey at zero. The lower +row zooms the Y junction, which is the one place the two disagree. + +Dumps come from sf_split_seam_viz.py: + + ../../uwrun sf_split_seam_viz.py -uw_rep split -uw_res superfine + ../../uwrun sf_split_seam_viz.py -uw_rep ti -uw_res superfine -uw_eta1 5e-4 + + ../../uwrun sf_note_stress_slip.py -> sf_note_stress_slip.png +""" +import numpy as np +import pyvista as pv +from matplotlib.colors import (LinearSegmentedColormap, Normalize, + TwoSlopeNorm) + +import rig_geometry as G + +pv.OFF_SCREEN = True +W, S_F = G.RES["superfine"] +ETA1 = 5e-4 + +# the upper stepover closed to the split's floor (one uncut edge): +# ../../uwrun sf_split_seam_viz.py -uw_rep split -uw_res superfine -uw_cont_gap_rungs 1 +# ../../uwrun sf_split_seam_viz.py -uw_rep ti -uw_res superfine -uw_eta1 5e-4 -uw_cont_gap_rungs 1 +CASES = [("split node", "fields_split_superfine_gather_np1_rank0_cont1.npz"), + (f"TI band, eta_1/w = {ETA1 / W:g}", + "fields_ti_superfine_gather_np1_rank0_eta0.0005_cont1.npz")] + +G.CONT_GAP_RUNGS = 1.0 +TR = dict(G.traces(S_F, W)) +# the Y: the Splay's stop-short tip beside the Main's outgoing limb +CY = TR["Splay"][0] +# the stepover: the Main's upper tip, the Cont segment one edge beyond it +CS = 0.5 * (TR["Main"][-1] + TR["Cont"][0]) +# each camera fills its 1000x1000 viewport exactly, so all four +# panels carry the same pixels; the gutter is added afterwards +PANEL, GUTTER = 1000, 22 +VIEWS = [((0.5, 0.5), 0.5, ""), + ((CY[0], CY[1]), 0.085, "the Y junction"), + ((CS[0], CS[1]), 0.06, "the stepover")] + +dumps = [np.load(f, allow_pickle=True) for _t, f in CASES] +allv = np.concatenate([d["logtau"] for d in dumps]) +clim = (float(np.percentile(allv, 2)), float(np.percentile(allv, 99.5))) + +# the trace colour is the SIGNED slip in each representation's own +# currency, on one map with grey at zero: blue sinistral, green dextral, +# each sense scaled to its own extreme so the small one stays visible. +# With no sinistral slip anywhere (this network, at this width) the blue +# half would be a range that does not exist, so the map drops to its +# dextral half and the bar runs 0 to the peak. +SENSE = ["#1f4fbf", "#7aa6ff", "#d9d9d9", "#7fe07a", "#118a2e"] +alls = np.concatenate([d["slips"][:, 2] for d in dumps]) +alls = alls[np.isfinite(alls)] +lo, hi = float(alls.min()), float(alls.max()) +TWO_SIDED = lo < -1e-3 +if TWO_SIDED: + SLIP_CMAP = LinearSegmentedColormap.from_list("slip_sense", SENSE) + NORM = TwoSlopeNorm(vcenter=0.0, vmin=lo, vmax=hi) + TICKS = {0.0: f"{lo:.3f}", 0.25: f"{lo / 2:.3f}", 0.5: "0", + 0.75: f"{hi / 2:.2f}", 1.0: f"{hi:.2f}"} +else: + SLIP_CMAP = LinearSegmentedColormap.from_list("slip_dextral", SENSE[2:]) + NORM = Normalize(vmin=0.0, vmax=hi) + TICKS = {0.0: "0", 0.5: f"{hi / 2:.2f}", 1.0: f"{hi:.2f}"} +KEY = "slip rate (+ dextral)" + +pl = pv.Plotter(off_screen=True, shape=(len(VIEWS), len(CASES)), + window_size=(PANEL * len(CASES), PANEL * len(VIEWS)), + border=False) +for j, ((title, _f), d) in enumerate(zip(CASES, dumps)): + for i, ((cx, cy), scale, ztag) in enumerate(VIEWS): + pl.subplot(i, j) + pl.set_background("white") + pvm = pv.UnstructuredGrid(d["cells"], d["celltypes"], d["points"]) + pvm.point_data["logtau"] = d["logtau"] + pl.add_mesh(pvm, scalars="logtau", cmap="magma", clim=clim, + show_edges=False, lighting=False, + show_scalar_bar=(i == 0 and j == 0), + scalar_bar_args=dict(title="log10 tau_II", n_labels=4, + color="black", vertical=True, + position_x=0.84, position_y=0.06)) + if i > 0: + pl.add_mesh(pvm.extract_all_edges(), color="white", + line_width=0.4, opacity=0.3, lighting=False) + arr = d["slips"] + for seg in np.split(arr, np.flatnonzero(np.isnan(arr[:, 0]))): + seg = seg[~np.isnan(seg[:, 0])] + if len(seg) < 2: + continue + t = seg[:, :2] - seg[:, :2].mean(axis=0) + _u, _s, vt = np.linalg.svd(t, full_matrices=False) + seg = seg[np.argsort(t @ vt[0])] + vals = np.where(np.isfinite(seg[:, 2]), seg[:, 2], 0.0) + if seg.shape[1] > 3: + vals = np.where(seg[:, 3] > 0, vals, 0.0) + line = pv.lines_from_points(np.column_stack( + [seg[:, :2], np.full(len(seg), 0.004)])) + line.point_data[KEY] = np.asarray(NORM(vals), dtype=float) + pl.add_mesh(line, scalars=KEY, cmap=SLIP_CMAP, clim=(0.0, 1.0), + line_width=6 if i == 0 else 10, lighting=False, + render_lines_as_tubes=True, + show_scalar_bar=(i == 0 and j == 1), + annotations=TICKS, + scalar_bar_args=dict(title=KEY, n_labels=0, + color="black", vertical=True, + position_x=0.84, + position_y=0.45)) + txt = f"{title}, w = {W}" if i == 0 else f"{ztag} — {title}" + pl.add_text(txt, font_size=13, color="black", + position=(18.0, 960.0)) + pl.view_xy() + pl.camera.parallel_projection = True + pl.camera.focal_point = (cx, cy, 0.0) + pl.camera.parallel_scale = scale +img = pl.screenshot(None, return_img=True) +pl.close() + +# equal panel size and equal margins: cut the render into its four +# viewports and lay them out on white with one gutter width everywhere +nr, nc = len(VIEWS), len(CASES) +H = nr * PANEL + (nr + 1) * GUTTER +Wd = nc * PANEL + (nc + 1) * GUTTER +canvas = np.full((H, Wd, img.shape[2]), 255, dtype=img.dtype) +for i in range(nr): + for j in range(nc): + tile = img[i * PANEL:(i + 1) * PANEL, j * PANEL:(j + 1) * PANEL] + y0 = GUTTER + i * (PANEL + GUTTER) + x0 = GUTTER + j * (PANEL + GUTTER) + canvas[y0:y0 + PANEL, x0:x0 + PANEL] = tile +out = "sf_note_stress_slip.png" +from PIL import Image +Image.fromarray(canvas).save(out) +print(f"wrote {out} slip range [{lo:.4f}, {hi:.4f}] two-sided {TWO_SIDED} logtau clim {clim}") diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_split_seam_viz.py b/articles/putting-a-fault-in-a-mesh/examples/sf_split_seam_viz.py new file mode 100644 index 0000000..328ced2 --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_split_seam_viz.py @@ -0,0 +1,466 @@ +"""The SPLIT rig solved through the partition seam, against serial, the +fine grid and the TI model: stress and slip. + +Two stages, as the other parallel figures. Under mpirun (or serial) the +rig is built in the realisation and seam mode asked for, solved (the +frictionless contact solve for the split, the weak plane for TI), the +stress COMPONENTS are recovered to continuous P1 (sf_fields' recipe, +invariant formed nodally) and every rank dumps its P1 mesh, the stress, +its seam vertices and the NATIVE slip along each strand: the tangential +pair jump on the cut (split), the tangential velocity jump across the +band (TI). In serial the dumps are rendered side by side. + + ../../uwrun sf_split_seam_viz.py -uw_rep split -uw_res coarse # serial + ../../uwrun sf_split_seam_viz.py -uw_rep split -uw_res fine + ../../uwrun -np 2 sf_split_seam_viz.py -uw_rep split -uw_res fine -uw_seams conform + ../../uwrun sf_split_seam_viz.py -uw_rep ti -uw_res fine + ../../uwrun sf_split_seam_viz.py -uw_render 1 + -> sf_split_seam_logtau_slip.png (fields, trace coloured by slip) + sf_split_seam_slip_profiles.png (slip along the Main, four cases) +""" +import glob + +import numpy as np +import underworld3 as uw + +params = uw.Params( + uw_rep=uw.Param("split", "split | ti"), + uw_res=uw.Param("coarse", "coarse | fine | superfine"), + uw_seams=uw.Param("gather", "gather | ligament | conform"), + uw_render=uw.Param(0.0, "1 = render the dumps (serial)"), + uw_np=uw.Param(2.0, "render: the np of the parallel split dump"), + uw_carve_clearance=uw.Param(1.0, "build() carve_clearance (fine: 1.0)"), + uw_eta1=uw.Param(1e-3, "TI weak shear viscosity in the footprints"), + uw_tol=uw.Param(1e-5, "stokes tolerance"), + uw_splay_gap_rungs=uw.Param(-1.0, "the Splay's stop-short gap from the " + "Main in rungs (rig default when < 0); 0 = " + "a genuine FORK, the band's own junction " + "(the cut cannot take touching strands)"), + uw_cont_gap_rungs=uw.Param(-1.0, "the upper stepover's tip-to-tip gap " + "in rungs (rig default when < 0); 1 = the " + "split's floor, one uncut edge; 0 = the " + "CONTINUOUS control, one cut the whole " + "line, no Cont strand"), + uw_strands=uw.Param("all", "all | main | main+splay (the Main alone: no " + "fork, so the band and the cut represent the SAME " + "structure; with the Splay: the junction and its " + "partition; sf_main_only_convergence.py and " + "sf_main_splay_junction.py draw those dumps)"), +) +import rig_geometry as G +import sf_fields as F + +REP, RES, SEAMS = str(params.uw_rep), str(params.uw_res), str(params.uw_seams) +STRANDS = str(params.uw_strands) +GAP = float(params.uw_splay_gap_rungs) +if GAP >= 0.0: + G.SPLAY_GAP_RUNGS = GAP +CONT = float(params.uw_cont_gap_rungs) +if CONT >= 0.0: + G.CONT_GAP_RUNGS = CONT +H_BG = 1 / 8 + + +def dump_name(rep, res, seams, np_, rank, eta1=None): + """TI dumps carry the weak viscosity when it is not the default: the + band's interface strength is eta_1 / w, so a width sweep at fixed + eta_1 doubles the strength each halving — a convergence test holds + the ratio.""" + tag = "" if (eta1 is None or rep != "ti" or abs(eta1 - 1e-3) < 1e-12) \ + else f"_eta{eta1:g}" + if STRANDS == "main": + tag += "_main" + elif STRANDS == "main+splay": + tag += "_mainsplay" + if GAP >= 0.0: + tag += f"_gap{GAP:g}" + if CONT >= 0.0: + tag += f"_cont{CONT:g}" + return f"fields_{rep}_{res}_{seams}_np{np_}_rank{rank}{tag}.npz" + + +if float(params.uw_render) < 0.5: + from mpi4py import MPI + import underworld3.visualisation as vis + from underworld3.utilities import fault_contact + from underworld3.utilities.place_surface import _shared_point_flags + comm = MPI.COMM_WORLD + W, S_F = G.RES[RES] + traces = G.traces(S_F, W) + keep = {"main": ("Main",), "main+splay": ("Main", "Splay")}.get(STRANDS) + if keep is not None: + traces = [tr for tr in traces if tr[0] in keep] + base = uw.meshing.UnstructuredSimplexBox( + minCoords=(0.0, 0.0), maxCoords=(1.0, 1.0), cellSize=H_BG, + regular=False, qdegree=2, refinement=1) + net = uw.meshing.FaultNetwork( + [(lbl, np.asarray(P, dtype=float)) for lbl, P in traces], + hierarchy=[lbl for lbl, _ in traces]) + net.prepare(h=S_F, ligament=1.0, verbose=False) + mesh = net.build(base=base, h_far=H_BG / 2, width=W, realisation=REP, + max_levels=1, margin_rings=2, seams=SEAMS, + carve_clearance=float(params.uw_carve_clearance)) + x, y = mesh.X + tag = f"{REP[0]}{RES[0]}" + v = uw.discretisation.MeshVariable(f"v{tag}", mesh, 2, degree=2) + p = uw.discretisation.MeshVariable(f"p{tag}", mesh, 1, degree=1, + continuous=True) + stokes = uw.systems.Stokes(mesh, velocityField=v, pressureField=p) + eta0 = uw.discretisation.MeshVariable(f"eta0{tag}", mesh, 1, degree=0) + eta0_vals = np.ones(len(np.asarray(eta0.coords))) + eta0.array[:, 0, 0] = eta0_vals + st = dict(rep=REP, mesh=mesh, stokes=stokes, v=v, eta0=eta0, + eta0_vals=eta0_vals, eta1=None, ndir=None, foot=None, + traces=traces, W=W, s_f=S_F) + if REP == "ti": + eta1, ndir, foot = G.ti_fields(mesh, net.info["footprints"], + float(params.uw_eta1), eta0_vals, tag, + w=W, traces_=traces) + stokes.constitutive_model = \ + uw.constitutive_models.TransverseIsotropicFlowModel + P = stokes.constitutive_model.Parameters + P.shear_viscosity_0 = eta0.sym[0] + P.shear_viscosity_1 = eta1.sym[0] + P.director = ndir.sym + st.update(eta1=eta1, ndir=ndir, foot=foot) + else: + stokes.constitutive_model = uw.constitutive_models.ViscousFlowModel + stokes.constitutive_model.Parameters.shear_viscosity_0 = eta0.sym[0] + stokes.bodyforce = [0.0, 0.0] + ES, ET = G.ES, G.ET + t_sym = (x - 0.5) * float(ET[0]) + (y - 0.5) * float(ET[1]) + for wall in ("Bottom", "Top", "Left", "Right"): + stokes.add_dirichlet_bc((2.0 * t_sym * float(ES[0]), + 2.0 * t_sym * float(ES[1])), wall) + stokes.petsc_use_pressure_nullspace = True + stokes.tolerance = float(params.uw_tol) + if REP == "split": + for lbl, _tr in traces: + stokes.add_fault_bc(0, boundary=lbl) + info = fault_contact.solve_with_fault(stokes) + converged = bool(info.get("converged")) + else: + stokes.solve() + info = None + converged = stokes.snes.getConvergedReason() > 0 + S_t, txx, tyy, txy = F.nodal_stress(st, name="tP1") + tau = F.stress_invariant(txx, tyy, txy) + + # the NATIVE slip along each strand, made global. Split: every + # coincident pair's tangential jump, all-gathered (rank-owned pairs + # only, so the seam pair is counted once). TI: the tangential velocity + # jump across band + one-cell skirt, counted where this rank owns both + # probes (evaluate answers for unowned points by extrapolation). + slip_profiles = {} + if REP == "split": + # SIGNED: the tangential jump v_plus - v_minus projected on the + # trace's own direction at the nearest trace vertex; Plus is the + # left of that direction (add_fault orients the chain by the + # polyline), the same convention as the band's left-minus-right. + # POSITIVE IS RIGHT-LATERAL (dextral), negative left-lateral, and + # the sign does not depend on the way the trace is walked + # (reversing it swaps left and right and flips t together). + from scipy.spatial import cKDTree + for lbl, tr in traces: + coords, jumps, normals = fault_contact.fault_pair_jumps( + stokes, lbl, stokes._rotated_freeslip_info, gather=True) + if len(coords) == 0: + continue + P = np.asarray(tr, dtype=float) + t = np.gradient(P, axis=0) + t /= np.linalg.norm(t, axis=1)[:, None] + near = cKDTree(P).query(coords)[1] + tang = np.einsum("ij,ij->i", jumps, t[near]) + slip_profiles[lbl] = np.column_stack( + [coords, tang, np.ones(len(tang))]) + else: + from scipy.spatial import cKDTree + # the TI record: the shear strain rate resolved on the director + # plane along the mid-line, at the spine vertices (P1-recovered + # components) — a strain rate, no span, continuous through a + # junction. The velocity-jump gauge across band + skirt is kept + # for the peak numbers only (a span one cell wide wanders in + # location and, past a junction, into the neighbouring band). + midline = {} + for lbl, tr in traces: + Pm, rate = F.midline_shear_rate(st, tr) # collective + midline[lbl] = np.column_stack([Pm, rate]) + + def global_slip(P): + t = np.gradient(P, axis=0) + t /= np.linalg.norm(t, axis=1)[:, None] + n = np.column_stack([-t[:, 1], t[:, 0]]) + # the band's OWN edges, no skirt: the layer's shear is a triangle + # peaked at the mid-line that reaches the rails (the transect + # probe, 2026-09-05: the jump across +-W/2 is within 3-6% of the + # jump across +-(W/2 + s_f)), and a probe on the rail cannot + # wander into a neighbouring band the mask keeps W away + skirt = W / 2 + Qp, Qm = P + skirt * n, P - skirt * n + vp = np.asarray(uw.function.evaluate(v.sym, Qp)).reshape(len(P), -1)[:, :2] + vm = np.asarray(uw.function.evaluate(v.sym, Qm)).reshape(len(P), -1)[:, :2] + slip = np.einsum("ij,ij->i", vp - vm, t) # SIGNED: left - right + own = ((mesh._robust_owning_cells(np.ascontiguousarray(Qp)) >= 0) + & (mesh._robust_owning_cells(np.ascontiguousarray(Qm)) >= 0)) + # gathered by the owning rank: the value where owned, and a + # count, so the sign survives (a MAX would not keep it) + val = np.where(own, slip, 0.0) + cnt = own.astype(float) + comm.Allreduce(MPI.IN_PLACE, val, op=MPI.SUM) + comm.Allreduce(MPI.IN_PLACE, cnt, op=MPI.SUM) + out = np.where(cnt > 0, val / np.maximum(cnt, 1.0), np.nan) + return out + + from scipy.spatial import cKDTree + all_traces = [np.asarray(tr, dtype=float) for _l, tr in traces] + # the Main and its Cont segment are ONE line: the gauge must not + # mask the Main's samples as "inside the Cont's band" at a closed + # stepover, or the tube shows a nick where the band has none + ONE_LINE = {"Main", "Cont"} + for k, (lbl, tr) in enumerate(traces): + P = np.asarray(tr, dtype=float)[::2] + others = [q for j, (l2, q) in enumerate(zip([l for l, _ in traces], + all_traces)) + if j != k and not (lbl in ONE_LINE and l2 in ONE_LINE)] + clear = (cKDTree(np.vstack(others)).query(P)[0] > W if others + else np.ones(len(P), dtype=bool)) + # the raw jump everywhere, and whether the trace point is + # clear of every other strand's band (column 3): the render + # masks with it, the junction studies show it + slip = global_slip(P) + slip_profiles[lbl] = np.column_stack([P, slip, clear.astype(float)]) + + # the P1 stress on the mesh's OWN cells (a split mesh must never be + # re-triangulated through its DOF cloud: the slit would close) + pvm = vis.mesh_to_pv_mesh(mesh).cast_to_unstructured_grid() + dm = mesh.dm + vS, vE = dm.getDepthStratum(0) + pStart, _ = dm.getChart() + shared = _shared_point_flags(dm).astype(bool)[vS - pStart: vE - pStart] + from underworld3.utilities.reconnect import _coords + Xv = _coords(dm)[: vE - vS] + # S_t is P1: one DOF per vertex in the DM's vertex order, which is + # also the pv mesh's point order — the split's coincident vertices + # keep their own DOFs, one per side + tau = np.asarray(tau).ravel() + if len(tau) != vE - vS: + raise RuntimeError("the P1 stress does not have one DOF per vertex") + logtau = np.log10(tau + 1e-12) + slips = np.vstack([np.vstack([sp, [np.nan] * 4]) + for sp in slip_profiles.values()]) \ + if slip_profiles else np.zeros((0, 4)) + labels = list(slip_profiles) + if REP == "ti": + midline_arr = np.vstack([np.vstack([mp, [np.nan] * 3]) + for mp in midline.values()]) + midline_labels = list(midline) + else: + midline_arr, midline_labels = np.zeros((0, 3)), [] + np.savez(dump_name(REP, RES, SEAMS, comm.size, comm.rank, + eta1=float(params.uw_eta1)), + points=np.asarray(pvm.points), + cells=np.asarray(pvm.cells, dtype=np.int64), + celltypes=np.asarray(pvm.celltypes, dtype=np.uint8), + logtau=logtau, + shared_xy=Xv[shared], + traces=np.vstack([np.vstack([tr, [np.nan, np.nan]]) + for _l, tr in traces]), + slips=slips, slip_labels=np.asarray(labels), + midline=midline_arr, midline_labels=np.asarray(midline_labels)) + peaks = {lbl: float(np.nanmax(np.where(sp[:, 3] > 0, np.abs(sp[:, 2]), np.nan))) + for lbl, sp in slip_profiles.items()} + if REP == "ti": + uw.pprint("mid-line 2 e_nt peaks: " + ", ".join( + f"{lbl}: {float(np.nanmax(mp[:, 2])):.2f}" for lbl, mp in midline.items())) + uw.pprint(f"[{REP} {RES} {SEAMS} np{comm.size}] converged {converged}; " + f"cells {comm.gather(int(dm.getHeightStratum(0)[1]))}; " + f"peak slip {{ {', '.join(f'{k}: {v_:.4f}' for k, v_ in peaks.items())} }}; dumped") +else: + import pyvista as pv + pv.OFF_SCREEN = True + NP = int(float(params.uw_np)) + import re + + def ranks_of(pattern): + # the rank wildcard must not pick up the Main-only / Main+Splay / + # other-eta dumps, whose names carry a suffix after the rank and + # whose MESHES differ (three meshes drawn over each other dappled + # the zoomed field) + return sorted(f for f in glob.glob(pattern) + if re.search(r"rank\d+\.npz$", f)) + + cases = [("split, coarse, serial", + ranks_of(dump_name("split", "coarse", "gather", 1, "*"))), + ("split, fine, serial", + ranks_of(dump_name("split", "fine", "gather", 1, "*"))), + (f"split, fine, np={NP} through the seam", + ranks_of(dump_name("split", "fine", "conform", NP, "*"))), + ("TI weak plane, fine, serial", + ranks_of(dump_name("ti", "fine", "gather", 1, "*")))] + for title, files in cases: + if not files: + raise SystemExit(f"no dump for {title}: run the dump stage first") + seam = np.vstack([np.load(f, allow_pickle=True)["shared_xy"] + for f in cases[2][1]]) + seam = seam[np.linalg.norm(seam - 0.5, axis=1) < 0.2] + czoom = seam.mean(axis=0) if len(seam) else np.array([0.5, 0.5]) + VIEWS = [((0.5, 0.5), 0.54, ""), + ((czoom[0], czoom[1]), 0.12, " — the Main's seam crossing")] + allv = np.concatenate([np.load(f, allow_pickle=True)["logtau"] + for _t, fs in cases for f in fs]) + clim = (float(np.percentile(allv, 2)), float(np.percentile(allv, 99.5))) + # the trace colour is the SIGNED slip (positive dextral) in the one + # currency for both representations — the pair jump on the cut, the + # jump across the band's own edges — on a diverging map, neutral grey + # at zero, blue sinistral and green dextral (both stand off the magma + # stress), symmetric about zero + # two-slope: zero stays at the map's centre, each sense scaled to its + # own extreme (a symmetric range left the sinistral lobes, a twentieth + # of the Main's dextral slip, invisible) + from matplotlib.colors import LinearSegmentedColormap, TwoSlopeNorm + SLIP_CMAP = LinearSegmentedColormap.from_list( + "slip_sense", ["#1f4fbf", "#7aa6ff", "#d9d9d9", "#7fe07a", "#118a2e"]) + allslip = np.concatenate([np.load(fs[0], allow_pickle=True)["slips"][:, 2] + for _t, fs in cases]) + allslip = allslip[np.isfinite(allslip)] + slip_lo = min(float(allslip.min()), -1e-6) + slip_hi = max(float(allslip.max()), 1e-6) + SLIP_NORM = TwoSlopeNorm(vcenter=0.0, vmin=slip_lo, vmax=slip_hi) + slip_ticks = {0.0: f"{slip_lo:.3f}", 0.25: f"{slip_lo / 2:.3f}", 0.5: "0", + 0.75: f"{slip_hi / 2:.2f}", 1.0: f"{slip_hi:.2f}"} + pl = pv.Plotter(off_screen=True, shape=(len(VIEWS), len(cases)), + window_size=(1000 * len(cases), 1000 * len(VIEWS)), + border=False) + for j, (title, files) in enumerate(cases): + for i, ((cx, cy), scale, ztag) in enumerate(VIEWS): + pl.subplot(i, j) + pl.set_background("white") + for f in files: + d = np.load(f, allow_pickle=True) + pvm = pv.UnstructuredGrid(d["cells"], d["celltypes"], d["points"]) + pvm.point_data["logtau"] = d["logtau"] + pl.add_mesh(pvm, scalars="logtau", cmap="magma", clim=clim, + show_edges=False, lighting=False, + show_scalar_bar=(i == 0 and j == 0), + scalar_bar_args=dict(title="log10 tau_II", n_labels=4, + color="black", vertical=True, + position_x=0.86, position_y=0.05)) + if i > 0: + pl.add_mesh(pvm.extract_all_edges(), color="white", + line_width=0.4, opacity=0.35, lighting=False) + if len(d["shared_xy"]): + pl.add_points(np.column_stack( + [d["shared_xy"], np.full(len(d["shared_xy"]), 0.003)]), + color="black", point_size=4 if i == 0 else 8, + render_points_as_spheres=True) + # the trace colour is each representation's OWN currency: the + # pair jump on the cut (split), the mid-line resolved shear + # strain rate 2 e_nt (TI) — two scalar bars. Rank 0's dump + # carries the profiles. Split pairs are unordered along the + # cut: order each strand's samples by their projection onto + # the trace's mean tangent. + d0 = np.load(files[0], allow_pickle=True) + sp_all = d0["slips"] + for seg in np.split(sp_all, np.flatnonzero(np.isnan(sp_all[:, 0]))): + seg = seg[~np.isnan(seg[:, 0])] + if len(seg) < 2: + continue + t = seg[:, :2] - seg[:, :2].mean(axis=0) + _u, _s, vt = np.linalg.svd(t, full_matrices=False) + order = np.argsort(t @ vt[0]) + seg = seg[order] + vals = np.where(np.isfinite(seg[:, 2]), seg[:, 2], 0.0) + if seg.shape[1] > 3: + vals = np.where(seg[:, 3] > 0, vals, 0.0) + line = pv.lines_from_points(np.column_stack( + [seg[:, :2], np.full(len(seg), 0.004)])) + key = "slip rate (signed; + dextral)" + line.point_data[key] = np.asarray(SLIP_NORM(vals), dtype=float) + pl.add_mesh(line, scalars=key, cmap=SLIP_CMAP, clim=(0.0, 1.0), + line_width=6, lighting=False, + render_lines_as_tubes=True, + show_scalar_bar=(i == 0 and j == 1), + annotations=slip_ticks, + scalar_bar_args=dict(title=key, n_labels=0, + color="black", vertical=True, + position_x=0.86, + position_y=0.45)) + pl.add_text(f"{title}{ztag}", font_size=11, color="black", + position="upper_left") + pl.view_xy() + pl.camera.parallel_projection = True + pl.camera.focal_point = (cx, cy, 0.0) + pl.camera.parallel_scale = scale + out = "sf_split_seam_logtau_slip.png" + pl.screenshot(out) + pl.close() + print(f"wrote {out}") + + # CONVERGENCE of the band to the cut, along the Main, in ONE + # currency — the velocity jump across the band's own edges for TI, + # the pair jump on the cut for the split (the integral of the + # resolved strain rate across the layer, either way): every split + # and TI dump on disk, by width. + import matplotlib + matplotlib.use("Agg") + import matplotlib.pyplot as plt + + def along(seg): + t = seg[:, :2] - np.asarray(G.C) + s_ = t @ G.ES + o = np.argsort(s_) + return s_[o], np.abs(seg[o, 2]) + + def strand(d0, name): + labels = list(d0["slip_labels"]) + arr = d0["slips"] + segs = [q[~np.isnan(q[:, 0])] for q in + np.split(arr, np.flatnonzero(np.isnan(arr[:, 0])))] + segs = [q for q in segs if len(q)] + return segs[labels.index(name)] + + fig, ax = plt.subplots(figsize=(9, 4.8)) + widths = {"coarse": G.RES["coarse"][0], "fine": G.RES["fine"][0], + "superfine": G.RES["superfine"][0]} + conv = [("split", "coarse", None, "C0-", 1.2), + ("split", "fine", None, "C0-", 1.8), + ("split", "superfine", None, "k-", 2.4), + ("ti", "fine", 1e-3, "C2--", 1.6), + ("ti", "superfine", 1e-3, "C3--", 1.6), + ("ti", "fine", 5e-4, "C2-.", 1.6), + ("ti", "superfine", 5e-4, "C3-.", 2.2)] + peaks = [] + for rep, res, eta1, sty, lw in conv: + files = sorted(f for f in glob.glob(dump_name(rep, res, "gather", 1, "*", + eta1=eta1)) + if re.search(r"rank\d+\.npz$", f)) + if not files: + continue + d0 = np.load(files[0], allow_pickle=True) + s_, v_ = along(strand(d0, "Main")) + if rep == "split": + name = f"split, w = {widths[res]}" + else: + name = (f"TI band, w = {widths[res]}, eta_1 = {eta1:g} " + f"(eta_1/w = {eta1 / widths[res]:g})") + ax.plot(s_, v_, sty, lw=lw, label=name) + peaks.append((name, float(np.nanmax(v_)))) + files = sorted(glob.glob(dump_name("split", "fine", "conform", NP, "*"))) + if files: + d0 = np.load(files[0], allow_pickle=True) + s_, v_ = along(strand(d0, "Main")) + ax.plot(s_, v_, "C0:", lw=1.4, + label=f"split, w = {widths['fine']}, np={NP} through the seam") + ax.set_xlabel("along the Main (from its centre)") + ax.set_ylabel("slip rate: pair jump (split) / v_t jump across the band (TI)") + ax.set_ylim(0, None) + ax.set_title("The TI band against the split node along the Main: " + "the width halved at fixed eta_1 and at fixed eta_1/w", + fontsize=10) + ax.legend(frameon=False, fontsize=7, loc="lower center") + fig.tight_layout() + for name, pk in peaks: + print(f"peak on the Main {name}: {pk:.4f}") + out2 = "sf_split_seam_slip_profiles.png" + fig.savefig(out2, dpi=160) + print(f"wrote {out2}") diff --git a/articles/putting-a-fault-in-a-mesh/examples/sf_stepover_closure.py b/articles/putting-a-fault-in-a-mesh/examples/sf_stepover_closure.py new file mode 100644 index 0000000..4a39ddf --- /dev/null +++ b/articles/putting-a-fault-in-a-mesh/examples/sf_stepover_closure.py @@ -0,0 +1,67 @@ +"""The stepover closed to the split's floor, against the continuous +control, at superfine: how much of the through-going slip each +representation carries onto the segment beyond the gap. + +Four dumps from sf_split_seam_viz.py (-uw_cont_gap_rungs 1 and 0, for +-uw_rep split and ti -uw_eta1 5e-4). The Cont region is the arc beyond +the Main's design tip, S_MAIN[1]; in the continuous control it is the +same arc of the one uncut line. + + ../../uwrun sf_stepover_closure.py +""" +import numpy as np + +import rig_geometry as G + +RES = "superfine" +W, S_F = G.RES[RES] + + +def strands(f): + d = np.load(f, allow_pickle=True) + labels = list(d["slip_labels"]) + arr = d["slips"] + segs = [q[~np.isnan(q[:, 0])] for q in + np.split(arr, np.flatnonzero(np.isnan(arr[:, 0])))] + segs = [q for q in segs if len(q)] + return dict(zip(labels, segs)) + + +def line_profile(st): + """The Main-line family on one arc coordinate: s along ES from C.""" + parts = [st["Main"]] + ([st["Cont"]] if "Cont" in st else []) + P = np.vstack([q[:, :2] for q in parts]) + v = np.concatenate([q[:, 2] for q in parts]) + ok = np.concatenate([q[:, 3] > 0 if q.shape[1] > 3 + else np.ones(len(q), bool) for q in parts]) + s_ = (P - np.asarray(G.C)) @ G.ES + o = np.argsort(s_) + return s_[o], np.where(ok[o], v[o], np.nan) + + +def region_mean(s_, v, lo, hi): + m = (s_ > lo) & (s_ < hi) & np.isfinite(v) + return float(np.mean(np.abs(v[m]))) if m.any() else float("nan") + + +s_tip = G.S_MAIN[1] +s_end = s_tip + G.CONT_LEN +rows = [] +for rep, eta in (("split", ""), ("ti", "_eta0.0005")): + out = {} + for gap in (1, 0): + f = f"fields_{rep}_{RES}_gather_np1_rank0{eta}_cont{gap}.npz" + s_, v = line_profile(strands(f)) + out[gap] = (region_mean(s_, v, s_tip + 0.01, s_end - 0.01), + region_mean(s_, v, -0.2, 0.2), + float(np.nanmax(np.abs(v)))) + seg_closed, main_closed, pk_closed = out[1] + seg_cont, main_cont, pk_cont = out[0] + rows.append((rep, seg_closed, seg_cont, seg_closed / seg_cont, + main_closed, main_cont, pk_closed, pk_cont)) + +print(f"{'rep':<6}{'seg closed':>12}{'seg cont':>10}{'ratio':>8}" + f"{'main closed':>13}{'main cont':>11}{'peak closed':>13}{'peak cont':>11}") +for r in rows: + print(f"{r[0]:<6}{r[1]:>12.4f}{r[2]:>10.4f}{r[3]:>8.3f}" + f"{r[4]:>13.4f}{r[5]:>11.4f}{r[6]:>13.4f}{r[7]:>11.4f}") diff --git a/articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json b/articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json index 0b20373..4de16bf 100644 --- a/articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json +++ b/articles/putting-a-fault-in-a-mesh/examples/split-anatomy-data.json @@ -1 +1 @@ -{"coords": [[0.0, 0.0], [0.5, 0.0], [0.5, 0.5], [0.0, 0.5], [0.5, 1.0], [0.0, 1.0], [1.0, 0.0], [1.0, 0.5], [1.0, 1.0], [1.5, 0.0], [1.5, 0.5], [1.5, 1.0], [2.0, 0.0], [2.0, 0.5], [2.0, 1.0], [2.5, 0.0], [2.5, 0.5], [2.5, 1.0], [3.0, 0.0], [3.0, 0.5], [3.0, 1.0]], "exploded": [[0.0, 0.0], [0.5, -0.22], [0.5, 0.5], [0.0, 0.5], [0.5, 1.0], [0.0, 1.0], [1.0, -0.22], [1.0, 0.5], [1.0, 1.0], [1.5, -0.22], [1.5, 0.5], [1.5, 1.0], [2.0, -0.22], [2.0, 0.5], [2.0, 1.0], [2.5, -0.22], [2.5, 0.5], [2.5, 1.0], [3.0, 0.0], [3.0, 0.5], [3.0, 1.0], [1.0, 0.28], [1.5, 0.28], [2.0, 0.28]], "tris": [[0, 1, 2], 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Thesis: splitting duplicates ONLY the interior -// of the fault chain — cells on the Minus side rewire to replica vertices, -// the tips stay shared, and the two copies are geometrically coincident -// (the lower panel is exploded for display only). -#import "@preview/cetz:0.3.4" - -#set page(width: auto, height: auto, margin: 12pt) -#set text(font: ("Noto Sans", "Helvetica", "Arial"), size: 5pt) - -#let data = json("split-anatomy-data.json") - -#let mesh-fill-plus = rgb("#dce8fc") -#let mesh-fill-minus = rgb("#fce4ec") -#let mesh-stroke = rgb("#8899aa") -#let fault-col = rgb("#c62828") -#let replica-col = rgb("#c62828") -#let tip-col = rgb("#1a1a1a") - -#let panel(origin-y, coords, tris, exploded: false) = { - import cetz.draw: * - let P(v) = (coords.at(v).at(0), coords.at(v).at(1) + origin-y) - // cells, painter's algorithm: fills first, then edges, then the fault - for (k, t) in tris.enumerate() { - let fill = if data.side.at(k) < 0 { mesh-fill-minus } else { mesh-fill-plus } - line(P(t.at(0)), P(t.at(1)), P(t.at(2)), close: true, - fill: fill, stroke: (paint: mesh-stroke, thickness: 0.3pt)) - } - // fault facets: both copies carry the label - if exploded { - // Plus copy: original chain vertices - for i in range(data.chain.len() - 1) { - line(P(data.chain.at(i)), P(data.chain.at(i + 1)), - stroke: (paint: fault-col, thickness: 1.0pt)) - } - // Minus copy: replicas where they exist, tips where shared - let lower(v) = if str(v) in data.replicas { data.replicas.at(str(v)) } else { v } - for i in range(data.chain.len() - 1) { - line(P(lower(data.chain.at(i))), P(lower(data.chain.at(i + 1))), - stroke: (paint: fault-col, thickness: 1.0pt, dash: "densely-dashed")) - } - } else { - for i in range(data.chain.len() - 1) { - line(P(data.chain.at(i)), P(data.chain.at(i + 1)), - stroke: (paint: fault-col, thickness: 0.9pt)) - } - } - // vertices: interior chain (filled red), replicas (open red), tips (black ring) - for v in data.interior { - circle(P(v), radius: 0.045, fill: fault-col, stroke: none) - } - if exploded { - for (orig, rep) in data.replicas { - circle(P(rep), radius: 0.045, fill: white, - stroke: (paint: replica-col, thickness: 0.9pt)) - } - } - for v in data.tips { - circle(P(v), radius: 0.055, fill: white, - stroke: (paint: tip-col, thickness: 1.0pt)) - circle(P(v), radius: 0.02, fill: tip-col, stroke: none) - } -} - -#cetz.canvas(length: 1.55cm, { - import cetz.draw: * - - // ---- panel (a): the conforming labelled chain -------------------------- - panel(2.1, data.coords, data.tris) - content((-0.35, 3.1), [(a)]) - content((1.5, 1.78), align(center)[conforming chain — facets shared, - every FE space continuous]) - content((3.35, 2.6), text(fill: fault-col)[$Gamma$]) - content((0.5, 2.36), text(size: 4pt)[tip]) - content((2.5, 2.36), text(size: 4pt)[tip]) - - // ---- panel (b): after the split (exploded for display) ----------------- - panel(0.0, data.exploded, data.moved_tris, exploded: true) - content((-0.35, 1.0), [(b)]) - 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