"""Fractional Kelvin-Voigt Zener (FKVZ) Model.
This model consists of a Fractional Kelvin-Voigt element in series with a spring,
providing retardation behavior with equilibrium compliance.
Theory
------
The FKVZ model consists of:
- Spring (G_e) in series with
- Fractional Kelvin-Voigt element (spring G_k in parallel with SpringPot)
Creep compliance:
J(t) = 1/G_e + (1/G_k) * (1 - E_α(-(t/τ)^α))
Complex compliance:
J*(ω) = 1/G_e + (1/G_k) / (1 + (iωτ)^α)
where E_α is the one-parameter Mittag-Leffler function.
Parameters
----------
Ge : float
Series spring modulus (Pa), bounds [1e-3, 1e9]
Gk : float
KV element modulus (Pa), bounds [1e-3, 1e9]
alpha : float
Fractional order, bounds [0.0, 1.0]
tau : float
Retardation time (s), bounds [1e-6, 1e6]
Limit Cases
-----------
- alpha → 0: Two springs in series (J = 1/G_e + 1/G_k)
- alpha → 1: Classical Zener solid in creep formulation
References
----------
- Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity
- Bagley, R.L. & Torvik, P.J. (1986). J. Rheol. 30, 133-155
"""
from __future__ import annotations
from rheojax.core.jax_config import safe_import_jax
from rheojax.logging import get_logger, log_fit
from rheojax.models.fractional.fractional_mixin import FRACTIONAL_ORDER_BOUNDS
jax, jnp = safe_import_jax()
from rheojax.core.base import BaseModel
from rheojax.core.inventory import Protocol
from rheojax.core.parameters import ParameterSet
from rheojax.core.registry import ModelRegistry
from rheojax.core.test_modes import DeformationMode
from rheojax.utils.mittag_leffler import mittag_leffler_e
# Module logger
logger = get_logger(__name__)
[docs]
@ModelRegistry.register(
"fractional_kv_zener",
protocols=[
Protocol.RELAXATION,
Protocol.CREEP,
Protocol.OSCILLATION,
],
deformation_modes=[
DeformationMode.SHEAR,
DeformationMode.TENSION,
DeformationMode.BENDING,
DeformationMode.COMPRESSION,
],
)
class FractionalKelvinVoigtZener(BaseModel):
"""Fractional Kelvin-Voigt Zener model.
A fractional viscoelastic model emphasizing retardation behavior
with finite equilibrium compliance.
Test Modes
----------
- Relaxation: Supported (via inversion)
- Creep: Supported (primary mode)
- Oscillation: Supported
- Rotation: Not supported (no steady-state flow)
Examples
--------
>>> import jax.numpy as jnp
>>> from rheojax.models import FractionalKelvinVoigtZener
>>>
>>> # Create model
>>> model = FractionalKelvinVoigtZener()
>>>
>>> # Set parameters
>>> model.set_params(Ge=1000.0, Gk=500.0, alpha=0.5, tau=1.0)
>>>
>>> # Predict creep compliance
>>> t = jnp.logspace(-2, 2, 50)
>>> J_t = model.predict(t)
"""
[docs]
def __init__(self):
"""Initialize Fractional Kelvin-Voigt Zener model."""
super().__init__()
# Define parameters with bounds and descriptions
self.parameters = ParameterSet()
self.parameters.add(
name="Ge",
value=1000.0,
bounds=(1e-3, 1e9),
units="Pa",
description="Series spring modulus",
)
self.parameters.add(
name="Gk",
value=500.0,
bounds=(1e-3, 1e9),
units="Pa",
description="KV element modulus",
)
self.parameters.add(
name="alpha",
value=0.5,
bounds=FRACTIONAL_ORDER_BOUNDS,
units="",
description="Fractional order",
)
self.parameters.add(
name="tau",
value=1.0,
bounds=(1e-6, 1e6),
units="s",
description="Retardation time",
)
@staticmethod
@jax.jit
def _predict_creep(
t: jnp.ndarray,
Ge: float,
Gk: float,
alpha: float,
tau: float,
) -> jnp.ndarray:
"""Predict creep compliance J(t).
J(t) = 1/G_e + (1/G_k) * (1 - E_α(-(t/τ)^α))
Parameters
----------
t : jnp.ndarray
Time array (s)
Ge : float
Series spring modulus (Pa)
Gk : float
KV element modulus (Pa)
alpha : float
Fractional order
tau : float
Retardation time (s)
Returns
-------
jnp.ndarray
Creep compliance J(t) (1/Pa)
"""
# Add small epsilon to prevent issues
epsilon = 1e-12
# Clip alpha to safe range (works with JAX tracers)
alpha_safe = jnp.clip(alpha, epsilon, 1.0 - epsilon)
tau_safe = tau + epsilon
# Instantaneous compliance (elastic response)
J_inst = 1.0 / (Ge + epsilon)
# Retarded compliance amplitude
J_retard_amp = 1.0 / (Gk + epsilon)
# Compute argument: z = -(t/τ)^α
z = -jnp.power(t / tau_safe, alpha_safe)
# Mittag-Leffler function E_α(z) with concrete alpha
ml_term = mittag_leffler_e(z, alpha=alpha_safe)
# J(t) = 1/G_e + (1/G_k) * (1 - E_α(-(t/τ)^α))
J_t = J_inst + J_retard_amp * (1.0 - ml_term)
return J_t
@staticmethod
@jax.jit
def _predict_relaxation(
t: jnp.ndarray,
Ge: float,
Gk: float,
alpha: float,
tau: float,
) -> jnp.ndarray:
"""Predict relaxation modulus G(t).
Note: Analytical relaxation modulus requires numerical inversion.
This provides an approximation based on the creep-relaxation relationship.
Parameters
----------
t : jnp.ndarray
Time array (s)
Ge : float
Series spring modulus (Pa)
Gk : float
KV element modulus (Pa)
alpha : float
Fractional order
tau : float
Retardation time (s)
Returns
-------
jnp.ndarray
Relaxation modulus G(t) (Pa)
"""
# Add small epsilon to prevent issues
epsilon = 1e-12
# Clip alpha to safe range (works with JAX tracers)
alpha_safe = jnp.clip(alpha, epsilon, 1.0 - epsilon)
tau_safe = tau + epsilon
# Compute transition function
z = -jnp.power(t / tau_safe, alpha_safe)
ml_term = mittag_leffler_e(z, alpha=alpha_safe)
# Short time modulus
G_short = Ge
# Long time modulus (series combination)
G_long = (Ge * Gk) / (Ge + Gk + epsilon)
# Interpolate using Mittag-Leffler decay
G_t = G_long + (G_short - G_long) * ml_term
return G_t
@staticmethod
@jax.jit
def _predict_oscillation(
omega: jnp.ndarray,
Ge: float,
Gk: float,
alpha: float,
tau: float,
) -> jnp.ndarray:
"""Predict complex modulus G*(ω).
Convert from complex compliance:
J*(ω) = 1/G_e + (1/G_k) / (1 + (iωτ)^α)
G*(ω) = 1 / J*(ω)
Parameters
----------
omega : jnp.ndarray
Angular frequency array (rad/s)
Ge : float
Series spring modulus (Pa)
Gk : float
KV element modulus (Pa)
alpha : float
Fractional order
tau : float
Retardation time (s)
Returns
-------
jnp.ndarray
Complex modulus array with shape (..., 2) where [:, 0] is G' and [:, 1] is G''
"""
# Add small epsilon to prevent issues
epsilon = 1e-12
# Clip alpha to safe range (works with JAX tracers)
alpha_safe = jnp.clip(alpha, epsilon, 1.0 - epsilon)
tau_safe = tau + epsilon
# Compute (iωτ)^α
omega_tau_alpha = jnp.power(omega * tau_safe, alpha_safe)
phase = jnp.pi * alpha_safe / 2.0
i_omega_tau_alpha = omega_tau_alpha * (jnp.cos(phase) + 1j * jnp.sin(phase))
# Complex compliance
J_inst = 1.0 / (Ge + epsilon)
J_kv = (1.0 / (Gk + epsilon)) / (1.0 + i_omega_tau_alpha)
J_star = J_inst + J_kv
# Complex modulus (inverse of compliance)
G_star = 1.0 / (J_star + epsilon)
# Extract storage and loss moduli
G_prime = jnp.real(G_star)
G_double_prime = jnp.imag(G_star)
return jnp.stack([G_prime, G_double_prime], axis=-1)
def _fit(
self, X: jnp.ndarray, y: jnp.ndarray, **kwargs
) -> FractionalKelvinVoigtZener:
"""Fit model to data using NLSQ TRF optimization.
Parameters
----------
X : jnp.ndarray
Independent variable (time or frequency)
y : jnp.ndarray
Dependent variable (modulus or compliance)
**kwargs : dict
Additional fitting options
Returns
-------
self
Fitted model instance
"""
from rheojax.core.test_modes import TestMode
from rheojax.utils.optimization import (
create_least_squares_objective,
nlsq_optimize,
)
# Detect test mode
test_mode_str = kwargs.get("test_mode", "creep")
# Convert string to TestMode enum
if isinstance(test_mode_str, str):
test_mode_map = {
"relaxation": TestMode.RELAXATION,
"creep": TestMode.CREEP,
"oscillation": TestMode.OSCILLATION,
}
test_mode = test_mode_map.get(test_mode_str, TestMode.CREEP)
else:
test_mode = test_mode_str
# Store test mode for model_function
self._test_mode = test_mode
# Determine data shape for logging
data_shape = (len(X),) if hasattr(X, "__len__") else None
with log_fit(
logger,
model="FractionalKelvinVoigtZener",
data_shape=data_shape,
test_mode=(
test_mode_str if isinstance(test_mode_str, str) else str(test_mode)
),
) as ctx:
logger.debug(
"Starting FKVZ fit",
n_points=len(X) if hasattr(X, "__len__") else 1,
test_mode=str(test_mode),
initial_params=self.parameters.to_dict(),
)
# Smart initialization for oscillation mode (Issue #9)
if test_mode == TestMode.OSCILLATION:
try:
import numpy as np
from rheojax.utils.initialization import (
initialize_fractional_kv_zener,
)
success = initialize_fractional_kv_zener(
np.array(X), np.array(y), self.parameters
)
if success:
logger.debug(
"Smart initialization applied from frequency-domain features",
initialized_params=self.parameters.to_dict(),
)
except Exception as e:
logger.debug(
"Smart initialization failed, using defaults",
error=str(e),
)
# Create stateless model function for optimization
def model_fn(x, params):
"""Model function for optimization (stateless)."""
Ge, Gk, alpha, tau = params[0], params[1], params[2], params[3]
# Direct prediction based on test mode (stateless)
if test_mode == TestMode.RELAXATION:
return self._predict_relaxation(x, Ge, Gk, alpha, tau)
elif test_mode == TestMode.CREEP:
return self._predict_creep(x, Ge, Gk, alpha, tau)
elif test_mode == TestMode.OSCILLATION:
return self._predict_oscillation(x, Ge, Gk, alpha, tau)
else:
raise ValueError(f"Unsupported test mode: {test_mode}")
# Create objective function
logger.debug("Creating least squares objective", normalize=True)
objective = create_least_squares_objective(
model_fn, jnp.array(X), jnp.array(y), normalize=True
)
# Optimize using NLSQ TRF
logger.debug(
"Starting NLSQ optimization",
method=kwargs.get("method", "auto"),
max_iter=kwargs.get("max_iter", 1000),
)
try:
result = nlsq_optimize(
objective,
self.parameters,
use_jax=kwargs.get("use_jax", True),
method=kwargs.get("method", "auto"),
max_iter=kwargs.get("max_iter", 1000),
)
except Exception as e:
logger.error(
"NLSQ optimization raised exception",
error_type=type(e).__name__,
error=str(e),
exc_info=True,
)
raise
# Validate optimization succeeded
if not result.success:
logger.error(
"Optimization failed",
message=result.message,
final_params=self.parameters.to_dict(),
)
raise RuntimeError(
f"Optimization failed: {result.message}. "
f"Try adjusting initial values, bounds, or max_iter."
)
self.fitted_ = True
ctx["final_params"] = self.parameters.to_dict()
ctx["success"] = True
logger.debug(
"FKVZ fit completed successfully",
final_params=self.parameters.to_dict(),
)
return self
def _predict(self, X: jnp.ndarray, **kwargs) -> jnp.ndarray:
"""Predict response for given input.
Parameters
----------
X : jnp.ndarray
Independent variable
**kwargs
Additional arguments (test_mode handled via self._test_mode)
Returns
-------
jnp.ndarray
Predicted values
"""
from rheojax.core.test_modes import TestMode
# Get parameters
Ge = self.parameters.get_value("Ge")
Gk = self.parameters.get_value("Gk")
alpha = self.parameters.get_value("alpha")
tau = self.parameters.get_value("tau")
# Dispatch based on test_mode if set, otherwise auto-detect
_kw_mode = kwargs.get("test_mode")
test_mode = (
_kw_mode if _kw_mode is not None else getattr(self, "_test_mode", None)
)
if test_mode in ("oscillation", TestMode.OSCILLATION):
return self._predict_oscillation(X, Ge, Gk, alpha, tau)
elif test_mode in ("relaxation", TestMode.RELAXATION):
return self._predict_relaxation(X, Ge, Gk, alpha, tau)
elif test_mode in ("creep", TestMode.CREEP):
return self._predict_creep(X, Ge, Gk, alpha, tau)
# Auto-detect test mode (legacy fallback)
if jnp.all(X > 0) and len(X) > 1:
log_range = jnp.log10(jnp.max(X)) - jnp.log10(jnp.min(X) + 1e-12)
if log_range > 3:
return self._predict_oscillation(X, Ge, Gk, alpha, tau)
# Default to creep (primary mode for FKVZ)
return self._predict_creep(X, Ge, Gk, alpha, tau)
[docs]
def model_function(self, X, params, test_mode=None, **kwargs):
"""Model function for Bayesian inference.
This method is required by BayesianMixin for NumPyro NUTS sampling.
It computes predictions given input X and a parameter array.
Args:
X: Independent variable (time or frequency)
params: Array of parameter values [Ge, Gk, alpha, tau]
Returns:
Model predictions as JAX array
"""
from rheojax.core.test_modes import TestMode
# Extract parameters from array (in order they were added to ParameterSet)
Ge = params[0]
Gk = params[1]
alpha = params[2]
tau = params[3]
# Use test_mode from last fit if available, otherwise default to CREEP
# Use explicit test_mode parameter (closure-captured in fit_bayesian)
# Fall back to self._test_mode only for backward compatibility
if test_mode is None:
test_mode = getattr(self, "_test_mode", TestMode.CREEP)
# Normalize test_mode to handle both string and TestMode enum
if hasattr(test_mode, "value"):
test_mode = test_mode.value
# Call appropriate prediction function based on test mode
if test_mode == TestMode.RELAXATION:
return self._predict_relaxation(X, Ge, Gk, alpha, tau)
elif test_mode == TestMode.CREEP:
return self._predict_creep(X, Ge, Gk, alpha, tau)
elif test_mode == TestMode.OSCILLATION:
stacked = self._predict_oscillation(X, Ge, Gk, alpha, tau)
return stacked[..., 0] + 1j * stacked[..., 1]
else:
# Default to creep mode for FKVZ model
return self._predict_creep(X, Ge, Gk, alpha, tau)
# Convenience alias
FKVZ = FractionalKelvinVoigtZener
__all__ = ["FractionalKelvinVoigtZener", "FKVZ"]