[docs]
class ViscoResistiveMHD(StruphyModel):
"""Full (non-linear) visco-resistive MHD equations discretized with a variational method.
Parameters
----------
base_units: BaseUnits
Base units for normalization (default: BaseUnits())
mass_number: float
Mass number (in units of Proton mass) of the fluid species (default: 1.0)
with_viscosity: bool
Whether to include viscous dissipation (default: True)
with_resistivity: bool
Whether to include resistive dissipation (default: True)
"""
@classmethod
def model_type(cls) -> LiteralOptions.ModelTypes:
return "Fluid"
## species
class EMFields(FieldSpecies):
def __init__(self):
self.b_field = FEECVariable(space="Hdiv")
self.init_variables()
class MHD(FluidSpecies):
def __init__(self, mass_number: float = 1.0):
self.density = FEECVariable(space="L2")
self.velocity = FEECVariable(space="H1vec")
self.entropy = FEECVariable(space="L2")
self.init_variables(mass_number=mass_number)
## propagators
class Propagators:
def __init__(
self,
s: FEECVariable = None,
rho: FEECVariable = None,
with_viscosity: bool = True,
with_resistivity: bool = True,
):
self.variat_dens = VariationalDensityEvolve(s=s)
self.variat_mom = VariationalMomentumAdvection()
self.variat_ent = VariationalEntropyEvolve(rho=rho)
self.variat_mag = VariationalMagFieldEvolve()
if with_viscosity:
self.variat_viscous = VariationalViscosity(rho=rho)
if with_resistivity:
self.variat_resist = VariationalResistivity(rho=rho)
## abstract methods
def __init__(
self,
base_units: BaseUnits = BaseUnits(),
mass_number: float = 1.0,
with_viscosity: bool = True,
with_resistivity: bool = True,
):
# 0. store input parameters
self.params = copy.deepcopy(locals())
# 1. instantiate all species
self.em_fields = self.EMFields()
self.mhd = self.MHD(mass_number=mass_number)
# 2. derive units (must be done after instantiating species to access charge and mass numbers)
self.setup_equation_params(base_units=base_units)
# 3. instantiate all propagators
self.propagators = self.Propagators(
s=self.mhd.entropy,
rho=self.mhd.density,
with_viscosity=with_viscosity,
with_resistivity=with_resistivity,
)
# 4. assign variables to propagators
self.propagators.variat_dens.variables.rho = self.mhd.density
self.propagators.variat_dens.variables.u = self.mhd.velocity
self.propagators.variat_mom.variables.u = self.mhd.velocity
self.propagators.variat_ent.variables.s = self.mhd.entropy
self.propagators.variat_ent.variables.u = self.mhd.velocity
self.propagators.variat_mag.variables.u = self.mhd.velocity
self.propagators.variat_mag.variables.b = self.em_fields.b_field
if with_viscosity:
self.propagators.variat_viscous.variables.s = self.mhd.entropy
self.propagators.variat_viscous.variables.u = self.mhd.velocity
if with_resistivity:
self.propagators.variat_resist.variables.s = self.mhd.entropy
self.propagators.variat_resist.variables.b = self.em_fields.b_field
# 5. define scalars to be tracked during simulation
kinetic_energy = BilinearEnergyFEEC(self.mhd.velocity, bilinear_form_name="WMMnew")
magnetic_energy = BilinearEnergyFEEC(self.em_fields.b_field)
thermo_energy = FunctionScalarFEEC(self.update_thermo_energy)
total_energy = kinetic_energy + magnetic_energy + thermo_energy
self.scalars = Scalars(
en_U=kinetic_energy,
en_thermo=thermo_energy,
en_mag=magnetic_energy,
en_tot=total_energy,
dens_tot=VolumeFormEnergyFEEC(self.mhd.density),
entr_tot=VolumeFormEnergyFEEC(self.mhd.entropy),
tot_div_B=FunctionScalarFEEC(self._compute_tot_div_B),
)
@property
def bulk_species(self):
return self.mhd
@property
def velocity_scale(self):
return "alfvén"
def allocate_helpers(self):
projV3 = L2Projector("L2", Propagator.mass_ops)
def f(e1, e2, e3):
return 1
f = xp.vectorize(f)
self._integrator = projV3(f)
self._energy_evaluator = InternalEnergyEvaluator(Propagator.derham, self.propagators.variat_ent.options.gamma)
self._ones = Propagator.derham.V3pol.zeros()
if isinstance(self._ones, PolarVector):
self._ones.tp[:] = 1.0
else:
self._ones[:] = 1.0
self._tmp_div_B = Propagator.derham.V3pol.zeros()
def _compute_tot_div_B(self):
b = self.em_fields.b_field.spline.vector
div_B = Propagator.derham.div.dot(b, out=self._tmp_div_B)
return Propagator.mass_ops.M3.dot_inner(div_B, div_B)
def update_thermo_energy(self):
"""Reuse tmp used in VariationalEntropyEvolve to compute the thermodynamical energy.
:meta private:
"""
rho = self.mhd.density.spline.vector
s = self.mhd.entropy.spline.vector
en_prop = self.propagators.variat_dens
self._energy_evaluator.sf.vector = s
self._energy_evaluator.rhof.vector = rho
sf_values = self._energy_evaluator.sf.eval_tp_fixed_loc(
self._energy_evaluator.integration_grid_spans,
self._energy_evaluator.integration_grid_bd,
out=self._energy_evaluator._sf_values,
)
rhof_values = self._energy_evaluator.rhof.eval_tp_fixed_loc(
self._energy_evaluator.integration_grid_spans,
self._energy_evaluator.integration_grid_bd,
out=self._energy_evaluator._rhof_values,
)
e = self._energy_evaluator.ener
ener_values = en_prop._proj_rho2_metric_term * e(rhof_values, sf_values)
en_prop._get_L2dofs_V3(ener_values, dofs=en_prop._linear_form_dl_drho)
en_thermo = self._integrator.inner(en_prop._linear_form_dl_drho)
return en_thermo
# default parameters
def generate_default_parameter_file(self, path=None, prompt=True):
params_path = super().generate_default_parameter_file(path=path, prompt=prompt)
new_file = []
with open(params_path, "r") as f:
for line in f:
if "variat_dens.Options" in line:
new_file += [
"model.propagators.variat_dens.options = model.propagators.variat_dens.Options(model='full')\n",
]
elif "entropy.add_background" in line:
new_file += ["model.mhd.density.add_background(FieldsBackground())\n"]
new_file += [line]
else:
new_file += [line]
with open(params_path, "w") as f:
for line in new_file:
f.write(line)
[docs]
@classmethod
def doc_pde(cls):
r"""**PDEs solved by model:**
Continuity:
.. math::
\partial_t \rho + \nabla \cdot (\rho \mathbf{u}) = 0
Momentum:
.. math::
\partial_t (\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) + \rho \nabla \frac{(\rho \mathcal{U}(\rho, s))}{\partial \rho} + s \nabla \frac{(\rho \mathcal{U}(\rho, s))}{\partial s} + \mathbf{B} \times \nabla \times \mathbf{B} - \nabla \cdot \left( (\mu + \mu_a(\mathbf{x})) \nabla \mathbf{u} \right) = 0
Entropy:
.. math::
\partial_t s + \nabla \cdot (s \mathbf{u}) = \frac{1}{T} \left( (\mu + \mu_a(\mathbf{x})) |\nabla \mathbf{u}|^2 + (\eta + \eta_a(\mathbf{x})) |\nabla \times \mathbf{B}|^2 \right)
Induction:
.. math::
\partial_t \mathbf{B} + \nabla \times (\mathbf{B} \times \mathbf{u}) + \nabla \times (\eta + \eta_a(\mathbf{x})) \nabla \times \mathbf{B} = 0
where the internal energy per unit mass is :math:`\mathcal U(\rho) = \rho^{\gamma-1} \exp(s / \rho)`,
and :math:`\mu_a(\mathbf{x})` and :math:`\eta_a(\mathbf{x})` are artificial viscosity and resistivity coefficients.
"""
[docs]
@classmethod
def doc_normalization(cls):
r"""The model uses Alfvén-speed normalization together with entropy-based
thermodynamic units inherited from the chosen internal-energy
functional."""
[docs]
@classmethod
def doc_scalar_quantities(cls):
r"""**The following scalars are tracked during simulation:**
- Kinetic energy: ``en_U``
- Thermodynamic energy: ``en_thermo``
- Magnetic energy: ``en_mag``
- Total energy: ``en_tot``
- Total density / entropy: ``dens_tot``, ``entr_tot``
- Divergence diagnostic: ``tot_div_B``"""
[docs]
@classmethod
def doc_discretization(cls):
"""Time integration is performed by the following propagators (in sequence):
1. :class:`~struphy.propagators.variational_density_evolve.VariationalDensityEvolve`
2. :class:`~struphy.propagators.variational_momentum_advection.VariationalMomentumAdvection`
3. :class:`~struphy.propagators.variational_entropy_evolve.VariationalEntropyEvolve`
4. :class:`~struphy.propagators.variational_mag_field_evolve.VariationalMagFieldEvolve`
5. :class:`~struphy.propagators.variational_viscosity.VariationalViscosity` (if :attr:`with_viscosity` is True)
6. :class:`~struphy.propagators.variational_resistivity.VariationalResistivity` (if :attr:`with_resistivity` is True)
"""
doc = rf"""**1. VariationalDensityEvolve:**
{VariationalDensityEvolve.__doc__}
**2. VariationalMomentumAdvection:**
{VariationalMomentumAdvection.__doc__}
**3. VariationalEntropyEvolve:**
{VariationalEntropyEvolve.__doc__}
**4. VariationalMagFieldEvolve:**
{VariationalMagFieldEvolve.__doc__}
**5. VariationalViscosity:**
{VariationalViscosity.__doc__}
**6. VariationalResistivity:**
{VariationalResistivity.__doc__}
"""
return doc
[docs]
@classmethod
def doc_long_description(cls):
r"""ViscoResistiveMHD is the full nonlinear dissipative MHD model in the
entropy formulation. It is the most complete variational MHD model in
this family when both viscosity and resistivity are required."""
[docs]
@classmethod
def doc_examples(cls):
r"""Create and initialize the full visco-resistive MHD model:
.. code-block:: python
from struphy.models import ViscoResistiveMHD
model = ViscoResistiveMHD()
model.mhd.density
model.mhd.velocity
model.mhd.entropy
model.em_fields.b_field
"""
[docs]
@classmethod
def doc_use_cases(cls):
r"""This model is appropriate for:
- nonlinear dissipative MHD simulations
- entropy-based variational MHD benchmarks
- studies where both viscosity and resistivity matter"""
[docs]
@classmethod
def doc_cannot_be_used_for(cls):
r"""This model is not suitable for:
- ideal nondissipative MHD if strict conservation is required
- kinetic or hybrid particle effects
- reduced linear perturbation studies better served by the linear models"""
