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82c1555 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 | """Mechanism forward equations shared by the NumPyro models (xp = jax.numpy), the agent tool and the
world generator (xp = numpy). No jax import here.
Density mechanisms (relative density of an SPS body; unchanged from v3/v3.1):
D1 diffusion-controlled (stress exponent n = 1), solid state
D2 dislocation-creep-controlled (n in [2.5, 6])
D3 liquid-phase-assisted (n = 1, rate x (1 + lam * sigmoid((T - TL)/wL)))
D0 'other': D1 with a broad low-dimensional discrepancy (T slope, pressure-mode offset)
Coercivity mechanisms H (ln Hci relative to RPI's E route: 850 C, degassed, load on cooling, dense). v4: the H cores
are load-timing-neutral; why load during heating lowers coercivity is a separate indicator LT (below).
H1 residual hydrogen (desorption kinetics with pores open through heating + dwell; no closure trapping here)
H2 grain-boundary liquid continuity (wetting above Tw; the v3 squeeze-out term moved to LT = load)
H3 alpha-Fe / decomposition (local overheating, fast ramps); parametrised by the penalty at sample A
H4 grain size / Kronmueller Neff = nB ln(D/dB): Hci(D)/Hci(D_E) = 1 - (nB Js / H_KE) ln(D/D_E)
H5 open porosity / oxidation of exposed grain boundaries
H0 'other': generic historical contrasts + out-of-hull discrepancy
Load-timing indicator LT (v4, decision D5): ln Hci = H core - b_LT * Pi_LT(route), with b_LT the penalty at RPI's D
route (850 C, degassed, load in heating) relative to E, shared by the three alternatives, and
Pi_k = [psi_k(x) - psi_k(x_E)] / [psi_k(x_D) - psi_k(x_E)], psi_k(x) = 1 - exp(-kappa_k x / x_D)
so Pi = 1 exactly at D and 0 exactly at E (saturating; kappa_k = how saturated the damage is at D's dose). Doses x:
trap hydrogen trapped by early pore closure in excess of the open-pore residual:
x = h0 [exp(-theta_closed) - exp(-theta_open)]; 0 for load on cooling and for complete degassing (h0 = 0)
load load acting while the grain-boundary phase is liquid/soft (squeeze-out to triple junctions):
x = integral (P / 50 MPa) sigmoid((T - Tsq) / wsq) dt over the whole schedule (persists when degassed)
current Joule overheating at necks when the loaded compact carries the pulsed current:
x = path * m * exp(-(QI/R)(1/T_peak - 1/1123 K)), m = (rate/10)^nI with load in heating, 1 with load from
the start of the dwell (holding current equated to a 10 C/min ramp's, declared), 0 for load on cooling or
path = 0 (insulated compact)
"""
from __future__ import annotations
import numpy as np
from . import physics as ph
DENSITY_MECHS = ("D1", "D2", "D3", "D0")
MAG_MECHS = ("H1", "H2", "H3", "H4", "H5", "H0")
LT_LEVELS = ("trap", "load", "current") # v4 load-timing indicator
E_RHO = 0.9738 # RPI E pellet relative density (7.42 / 7.62), used as the H5 reference
H_FLOOR = 1e-3 # kOe
def density_params(p: dict, mech: str) -> dict:
out = dict(lnk=p["lnk"], Q=p["Q"], s=p["s"], p=1.0 + p["pm1"], phi0=p["phi0"], cR=p["cR"],
lnkR=p["lnkR"], QR=p["QR"], phiR=p["phiR"],
n=p["n"] if mech == "D2" else 1.0, d1=p["d1"], d2=p["d2"], rho_max=p["rho_max"])
if mech == "D3":
out.update(lam=p["lam"], TL=p["TL"], wL=p["wL"])
else:
out.update(lam=0.0, TL=700.0, wL=25.0)
return out
def density(p: dict, mech: str, A: dict, lnk_offset=0.0, xp=np):
prm = density_params(p, mech)
prm["off"] = lnk_offset
S = {k: xp.asarray(v) for k, v in A["S"].items()}
return ph.densify(S, xp.asarray(A["cooling"]), xp.asarray(A["T_peak"]), xp.asarray(A["P"]), prm, xp=xp)
def _route_arrays(route):
S = ph.stack([ph.schedule(route)])
return dict(S=S, T_peak=np.array([route.T_peak_C]), rate=np.array([route.heating_rate_C_per_min]),
cooling=np.array([1.0 if route.pressure_mode == "cooling" else 0.0]),
mode_dwell=np.array([1.0 if route.pressure_mode == "dwell" else 0.0]),
path=np.array([0.0 if route.current_path == "insulated" else 1.0]),
degas=np.array([1.0]), hfac=np.array([1.0]), a900=np.array([0.0]), a520=np.array([0.0]))
# RPI's E (load on cooling) and D (load in heating) routes, slide 12 rows: 850 C, 10 C/min, 10 min, 50 MPa, degassed
E_ROUTE = ph.Route(T_peak_C=850.0, heating_rate_C_per_min=10.0, dwell_min=10.0, P_MPa=50.0, pressure_mode="cooling",
load_ramp_min=1.5, degas=1.0)
D_ROUTE = ph.Route(T_peak_C=850.0, heating_rate_C_per_min=10.0, dwell_min=10.0, P_MPa=50.0, pressure_mode="heating",
load_ramp_min=1.5, degas=1.0)
E_ARR = _route_arrays(E_ROUTE)
D_ARR = _route_arrays(D_ROUTE)
A_ROUTE = dict(T_peak=np.array([950.0]), rate=np.array([100.0])) # RPI sample A (alpha-Fe marked)
H_MAX = 37.5 # kOe: smooth physical ceiling ~0.5 * mu0HA (Bance 2014: alpha <= 0.48 for ideal Dy-free grains)
T_D_K = 1123.15 # D's peak temperature (current-path dose reference)
# Magnetic data hull: the four independent specimens (A, C, D, E) in (T_peak C, relative density, load on cooling,
# degas, log10 heating rate), standardised by (100 C, 0.1, 1, 1, 1 decade). H0's out-of-hull discrepancy scales with
# the nearest-neighbour distance; it is 0 on every fitted row. Shared by the tool and the twin candidate.
MAG_HULL_POINTS = np.array([[950.0, 0.9882, 0.0, 0.0, 2.0], # A: 950 C, 100 C/min, load in heating, no RPI degas
[850.0, 0.9777, 0.0, 0.0, 1.0], # C
[850.0, 0.9829, 0.0, 1.0, 1.0], # D
[850.0, 0.9738, 1.0, 1.0, 1.0]]) # E (and its anneals)
MAG_HULL_SCALES = np.array([100.0, 0.1, 1.0, 1.0, 1.0])
def mag_hull_distance(T_peak_C, rel_density, load_on_cooling, degas, rate_C_per_min):
x = np.array([T_peak_C, rel_density, load_on_cooling, degas, np.log10(rate_C_per_min)], float)
return float(np.sqrt((((MAG_HULL_POINTS - x) / MAG_HULL_SCALES) ** 2).sum(1)).min())
def cap_log_hci(lh, xp=np):
"""Smooth ceiling Hci <= H_MAX: Hci_c = Hci / (1 + (Hci/H_MAX)^4)^(1/4) (negligible below ~20 kOe)."""
return lh - 0.25 * xp.logaddexp(0.0, 4.0 * (lh - np.log(H_MAX)))
def grain(p: dict, A: dict, xp=np):
S = {k: xp.asarray(v) for k, v in A["S"].items()}
return ph.grain_size(S, xp.asarray(A["T_peak"]), xp.asarray(A["a900"]), p, xp=xp)
def decomp(p: dict, A: dict, xp=np):
return ph.decomposition(xp.asarray(A["T_peak"]), xp.asarray(A["rate"]), p, xp=xp)
def _arr(A, key, like, default, xp):
return xp.asarray(A[key]) if key in A else default * xp.ones_like(like)
# ------------------------------------------------------------------------------------------------------------------
# Load-timing doses and penalties (v4)
# ------------------------------------------------------------------------------------------------------------------
def _hydrogen(p, A, xp):
"""Initial hydrogen index h0 (1 = non-degassed powder, hdg = the fitted D/E degassed state, 0 = complete
degassing via hfac = 0) and the desorption integrals (theta_closed, theta_open) of the route."""
S = {k: xp.asarray(v) for k, v in A["S"].items()}
cool = xp.asarray(A["cooling"])
dw = _arr(A, "mode_dwell", cool, 0.0, xp)
d = xp.asarray(A["degas"])
h0 = (d * p["hdg"] + (1.0 - d)) * _arr(A, "hfac", cool, 1.0, xp)
th_c, th_o = ph.desorption_integrals(S, cool, p, xp=xp, mode_dwell=dw)
return h0, th_c, th_o
def hydrogen_open(p, A, xp=np):
"""Residual hydrogen index with the pores open through heating + dwell (load-timing-neutral; H1's core)."""
h0, _, th_o = _hydrogen(p, A, xp)
return h0 * xp.exp(-th_o)
LOG_TRAP_RATIO_MAX = 50.0 # the saturating penalty is flat long before r = e^50; the clamp only prevents overflow
def trap_ratio(p, A, xp=np):
"""x_trap(route) / x_trap(D), x_trap = h0 [exp(-theta_closed) - exp(-theta_open)] >= 0, computed in log space
and clamped at e^50 (where every penalty is already saturated), so extreme kinetics never overflow. Exactly 0
when nothing is trapped (load on cooling: theta_closed == theta_open; complete degassing: h0 = 0)."""
h0, c, o = _hydrogen(p, A, xp)
h0D, cD, oD = _hydrogen(p, D_ARR, xp)
d = o - c
pos = (d > 0) & (h0 > 0)
sd = xp.where(pos, d, 1.0) # safe values keep gradients finite in the branch that is not taken
sh = xp.where(pos, h0, 1.0)
lnum = xp.log(sh) - c + xp.log(-xp.expm1(-sd))
lden = xp.log(h0D) - cD + xp.log(-xp.expm1(-xp.maximum(oD - cD, 1e-300)))
return xp.where(pos, xp.exp(xp.minimum(lnum - lden, LOG_TRAP_RATIO_MAX)), 0.0)
def load_dose(p, A, xp=np):
"""Integral of (P / 50 MPa) x sigmoid((T - Tsq)/wsq) dt over the whole schedule (MPa-normalised minutes)."""
S = {k: xp.asarray(v) for k, v in A["S"].items()}
soft = ph.sigmoid((S["T_K"] - 273.15 - ph._g(p["Tsq"], xp)) / ph._g(p["wsq"], xp), xp)
return xp.sum(S["P"] / ph.P_REF * soft * S["dt"], axis=-1)
def current_dose(p, A, xp=np):
"""Current-path dose relative to D (D = 1): path x m x exp(-(QI/R)(1/T_peak - 1/T_D))."""
cool = xp.asarray(A["cooling"])
dw = _arr(A, "mode_dwell", cool, 0.0, xp)
path = _arr(A, "path", cool, 1.0, xp)
heat = 1.0 - cool - dw
m = heat * (xp.asarray(A["rate"]) / 10.0) ** p["nI"] + dw
Tk = xp.asarray(A["T_peak"]) + 273.15
return path * m * xp.exp(-(p["QI"] / ph.R_GAS) * (1.0 / Tk - 1.0 / T_D_K))
def _psi(r, kap, xp):
return -xp.expm1(-kap * r)
def lt_penalties(p, A, xp=np):
"""Normalised load-timing penalty Pi_k(route) for k in LT_LEVELS (1 at D, 0 at E). Returns a dict."""
out = {}
out["trap"] = _psi(trap_ratio(p, A, xp), p["kap_tr"], xp) / _psi(1.0, p["kap_tr"], xp)
xD = load_dose(p, D_ARR, xp)
rE = load_dose(p, E_ARR, xp) / xD
sE = _psi(rE, p["kap_ld"], xp)
out["load"] = (_psi(load_dose(p, A, xp) / xD, p["kap_ld"], xp) - sE) / (_psi(1.0, p["kap_ld"], xp) - sE)
out["current"] = _psi(current_dose(p, A, xp), p["kap_cu"], xp) / _psi(1.0, p["kap_cu"], xp)
return out
# ------------------------------------------------------------------------------------------------------------------
# ln Hci
# ------------------------------------------------------------------------------------------------------------------
def log_hci(p: dict, mech: str, A: dict, rho, u_spec=0.0, hull_dist=0.0, z_hull=0.0, xp=np, lt="load"):
"""Body-level ln Hci (kOe) at room temperature, isotropic, for one parameter draw and load-timing label `lt`,
with the smooth physical ceiling applied. u_spec: specimen effect (ln kOe). hull_dist/z_hull: H0 discrepancy."""
return cap_log_hci(_log_hci_raw(p, mech, A, rho, u_spec, hull_dist, z_hull, xp, lt), xp)
def log_hci_all(p: dict, mech: str, A: dict, rho, u_spec=0.0, hull_dist=0.0, z_hull=0.0, xp=np):
"""Capped ln Hci for every load-timing label, stacked along a new leading axis in LT_LEVELS order."""
core = _log_hci_core(p, mech, A, rho, u_spec, hull_dist, z_hull, xp)
pen = lt_penalties(p, A, xp)
return xp.stack([cap_log_hci(core - p["b_LT"] * pen[k], xp) for k in LT_LEVELS])
def _log_hci_raw(p, mech, A, rho, u_spec, hull_dist, z_hull, xp, lt):
"""Uncapped ln Hci for one load-timing label (a string, or an array of labels aligned with the draws)."""
core = _log_hci_core(p, mech, A, rho, u_spec, hull_dist, z_hull, xp)
pen = lt_penalties(p, A, xp)
if isinstance(lt, str):
return core - p["b_LT"] * pen[lt]
lt = np.asarray(lt)
sel = sum(np.where(lt == k, 1.0, 0.0).reshape(-1, *([1] * (np.ndim(pen[k]) - 1))) * pen[k] for k in LT_LEVELS)
return core - p["b_LT"] * sel
def _log_hci_core(p, mech, A, rho, u_spec, hull_dist, z_hull, xp):
"""Load-timing-neutral ln Hci of mechanism `mech` (before the LT penalty and the ceiling)."""
d = xp.asarray(A["degas"])
hot = xp.where(xp.asarray(A["T_peak"]) >= 900.0, 1.0, 0.0)
a1 = xp.asarray(A["a900"])
a2 = xp.asarray(A["a520"])
ann = p["b_a1"] * a1 + p["b_a2"] * a2
if mech == "H4":
# Kronmueller grain-size law relative to the E route (v3 form; see METHOD.md section 9 item 5a)
D = grain(p, A, xp)
DE = grain(p, E_ARR, xp)
r = 1.0 - (p["nB"] * p["Js"] / p["HKE"]) * xp.log(D / DE)
fl = 0.02
ratio = fl * xp.logaddexp(0.0, r / fl)
return p["muE"] + p["b_deg"] * (1 - d) + ann + xp.log(xp.maximum(ratio, 1e-6)) + u_spec
if mech == "H1":
h = hydrogen_open(p, A, xp)
hE = hydrogen_open(p, E_ARR, xp)
return p["muE"] - p["gH"] * (h - hE) + p["b_hot"] * hot + ann + u_spec
if mech == "H2":
W = ph.sigmoid((xp.asarray(A["T_peak"]) - p["Tw"]) / p["ww"], xp)
WE = ph.sigmoid((850.0 - p["Tw"]) / p["ww"], xp)
return p["muE"] + p["b_deg"] * (1 - d) + p["gL"] * (W - WE) + p["b_hot"] * hot + ann + u_spec
if mech == "H3":
# parametrised by gA, the decomposition penalty at RPI sample A: gF = gA / (f_A - f_E)
f = decomp(p, A, xp)
fE = ph.decomposition(850.0, 10.0, p, xp=xp)
fA = decomp(p, A_ROUTE, xp)
gF = p["gA"] / xp.maximum(fA - fE, 1e-3)
return p["muE"] + p["b_deg"] * (1 - d) - gF * (f - fE) + ann + u_spec
generic = p["muE"] + p["b_deg"] * (1 - d) + p["b_hot"] * hot + ann + u_spec
if mech == "H5":
po = xp.maximum(1.0 - rho - p["phic"], 0.0)
poE = xp.maximum(1.0 - E_RHO - p["phic"], 0.0)
return generic - p["gphi"] * (po - poE)
if mech == "H0":
return generic + p["tau0"] * hull_dist * z_hull
raise ValueError(mech)
def remanence(p: dict, A: dict, rho, hci, xp=np):
"""Br (kG) at room temperature: Js * rho * <cos theta> * (1 + kappa) * R(Hci) * anneal factor.
R(H) = 1 / (1 + (HR/H)^kR): remanence retention collapses when Hci falls below ~HR (A, C)."""
R = 1.0 / (1.0 + (p["HR"] / xp.maximum(hci, H_FLOOR)) ** p["kR"])
ann = xp.exp(p["ba1"] * xp.asarray(A["a900"]) + p["ba2"] * xp.asarray(A["a520"]))
return p["Js"] * rho * xp.asarray(A["align"]) * (1.0 + p["kappa"]) * R * ann
def squareness_logit(p: dict, A: dict, xp=np):
return p["eta0"] + p["eta_a1"] * xp.asarray(A["a900"]) + p["eta_a2"] * xp.asarray(A["a520"])
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