def indicator_factory(dd, bt):
def indicator(X):
L = np.logical_and(float_cmp(X[:, 0], dd.domain[0]), dd.left == bt)
R = np.logical_and(float_cmp(X[:, 0], dd.domain[1]), dd.right == bt)
return np.logical_or(L, R)
inject_sid(indicator, indicator_id, dd, bt)
return indicator
def indicator_factory(dd, bt):
def indicator(X):
T = np.logical_and(float_cmp(X[:, 1], dd.domain[1, 1]), dd.top == bt)
B = np.logical_and(float_cmp(X[:, 1], dd.domain[0, 1]), dd.bottom == bt)
TB = np.logical_or(T, B)
return TB
inject_sid(indicator, indicator_id, dd, bt)
return indicator
def indicator_factory(dd, bt):
def indicator(X):
L = np.logical_and(float_cmp(X[:, 0], dd.domain[0, 0]), dd.left == bt)
R = np.logical_and(float_cmp(X[:, 0], dd.domain[1, 0]), dd.right == bt)
T = np.logical_and(float_cmp(X[:, 1], dd.domain[1, 1]), dd.top == bt)
B = np.logical_and(float_cmp(X[:, 1], dd.domain[0, 1]), dd.bottom == bt)
LR = np.logical_or(L, R)
TB = np.logical_or(T, B)
return np.logical_or(LR, TB)
inject_sid(indicator, indicator_id, dd, bt)
return indicator
def __init__(self, vx=1., vy=1., torus=True, initial_data_type='sin', parameter_range=(1., 2.)):
assert initial_data_type in ('sin', 'bump')
def burgers_flux(U, mu):
U = U.reshape(U.shape[:-1])
U_exp = np.sign(U) * np.power(np.abs(U), mu['exponent'])
R = np.empty(U.shape + (2,))
R[..., 0] = U_exp * vx
R[..., 1] = U_exp * vy
return R
inject_sid(burgers_flux, str(Burgers2DProblem) + '.burgers_flux', vx, vy)
def burgers_flux_derivative(U, mu):
U = U.reshape(U.shape[:-1])
U_exp = mu['exponent'] * (np.sign(U) * np.power(np.abs(U), mu['exponent']-1))
R = np.empty(U.shape + (2,))
R[..., 0] = U_exp * vx
R[..., 1] = U_exp * vy
return R
inject_sid(burgers_flux_derivative, str(Burgers2DProblem) + '.burgers_flux_derivative', vx, vy)
flux_function = GenericFunction(burgers_flux, dim_domain=1, shape_range=(2,),
parameter_type={'exponent': 0},
name='burgers_flux')
flux_function_derivative = GenericFunction(burgers_flux_derivative, dim_domain=1, shape_range=(2,),
parameter_type={'exponent': 0},
name='burgers_flux')
if initial_data_type == 'sin':
def initial_data(x):
return 0.5 * (np.sin(2 * np.pi * x[..., 0]) * np.sin(2 * np.pi * x[..., 1]) + 1.)
inject_sid(initial_data, str(Burgers2DProblem) + '.initial_data_sin')
dirichlet_data = ConstantFunction(dim_domain=2, value=0.5)
else:
def initial_data(x):
return (x[..., 0] >= 0.5) * (x[..., 0] <= 1) * 1
inject_sid(initial_data, str(Burgers2DProblem) + '.initial_data_bump')
dirichlet_data = ConstantFunction(dim_domain=2, value=0)
initial_data = GenericFunction(initial_data, dim_domain=2)
domain = TorusDomain([[0, 0], [2, 1]]) if torus else RectDomain([[0, 0], [2, 1]], right=None, top=None)
super(Burgers2DProblem, self).__init__(domain=domain,
rhs=None,
flux_function=flux_function,
flux_function_derivative=flux_function_derivative,
initial_data=initial_data,
dirichlet_data=dirichlet_data,
T=0.3, name='Burgers2DProblem')
self.parameter_space = CubicParameterSpace({'exponent': 0}, *parameter_range)
self.parameter_range = parameter_range
self.initial_data_type = initial_data_type
self.torus = torus
self.vx = vx
self.vy = vy
def discretize_nonlinear_instationary_advection_diffusion_fv(analytical_problem, diameter=None, nt=100, num_flux='lax_friedrichs',
lxf_lambda=1., domain_discretizer=None, grid=None, boundary_info=None):
assert isinstance(analytical_problem, InstationaryAdvectionDiffusionProblem)
assert grid is None or boundary_info is not None
assert boundary_info is None or grid is not None
assert grid is None or domain_discretizer is None
assert num_flux in ('lax_friedrichs', 'engquist_osher')
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if diameter is None:
grid, boundary_info = domain_discretizer(analytical_problem.domain)
else:
grid, boundary_info = domain_discretizer(analytical_problem.domain, diameter=diameter)
p = analytical_problem
if num_flux == 'lax_friedrichs':
L = nonlinear_advection_lax_friedrichs_operator(grid, boundary_info, p.flux_function, dirichlet_data=p.dirichlet_data,
lxf_lambda=lxf_lambda)
else:
L = nonlinear_advection_engquist_osher_operator(grid, boundary_info, p.flux_function, p.flux_function_derivative,
dirichlet_data=p.dirichlet_data)
I = p.initial_data.evaluate(grid.quadrature_points(0, order=2)).squeeze()
I = np.sum(I * grid.reference_element.quadrature(order=2)[1], axis=1) * (1. / grid.reference_element.volume)
I = NumpyVectorArray(I)
inject_sid(I, __name__ + '.discretize_nonlinear_instationary_advection_diffusion_fv.initial_data', p.initial_data, grid)
D = DiffusionOperator(grid, boundary_info, p.diffusion)
D = type(D).lincomb(operators=[D], name='diffusion', coefficients_name='diffusion')
F = None if p.rhs is None else DiffusionRHSOperatorFunctional(grid, boundary_info, p.rhs,
p.neumann_data, p.dirichlet_data, p.diffusion)
F = type(F).lincomb(operators=[F], name='rhs', coefficients_name='diffusion')
products = {'l2': L2Product(grid, boundary_info)}
visualizer = PatchVisualizer(grid=grid, bounding_box=grid.domain, codim=0)
parameter_space = p.parameter_space if hasattr(p, 'parameter_space') else None
discretization = InstationaryImexDiscretization(explicit_operator=L, implicit_operator=D, rhs=F,
initial_data=I, T=p.T, nt=nt, products=products,
parameter_space=parameter_space, visualizer=visualizer,
name='{}_FV'.format(p.name))
return discretization, {'grid': grid, 'boundary_info': boundary_info}
def __init__(self, v=1., circle=True, initial_data_type='sin', parameter_range=(1., 2.)):
assert initial_data_type in ('sin', 'bump')
def burgers_flux(U, mu):
U_exp = np.sign(U) * np.power(np.abs(U), mu['exponent'])
R = U_exp * v
return R
inject_sid(burgers_flux, str(BurgersProblem) + '.burgers_flux', v)
def burgers_flux_derivative(U, mu):
U_exp = mu['exponent'] * (np.sign(U) * np.power(np.abs(U), mu['exponent']-1))
R = U_exp * v
return R
inject_sid(burgers_flux_derivative, str(BurgersProblem) + '.burgers_flux_derivative', v)
flux_function = GenericFunction(burgers_flux, dim_domain=1, shape_range=(1,),
parameter_type={'exponent': 0},
name='burgers_flux')
flux_function_derivative = GenericFunction(burgers_flux_derivative, dim_domain=1, shape_range=(1,),
parameter_type={'exponent': 0},
name='burgers_flux')
if initial_data_type == 'sin':
def initial_data(x):
return 0.5 * (np.sin(2 * np.pi * x[..., 0]) + 1.)
inject_sid(initial_data, str(BurgersProblem) + '.initial_data_sin')
dirichlet_data = ConstantFunction(dim_domain=1, value=0.5)
else:
def initial_data(x):
return (x[..., 0] >= 0.5) * (x[..., 0] <= 1) * 1
inject_sid(initial_data, str(BurgersProblem) + '.initial_data_bump')
dirichlet_data = ConstantFunction(dim_domain=1, value=0)
initial_data = GenericFunction(initial_data, dim_domain=1)
if circle:
domain = CircleDomain([0, 2])
else:
domain = LineDomain([0, 2], right=None)
super(BurgersProblem, self).__init__(domain=domain,
rhs=None,
flux_function=flux_function,
flux_function_derivative=flux_function_derivative,
initial_data=initial_data,
dirichlet_data=dirichlet_data,
T=0.3, name='BurgersProblem')
self.parameter_space = CubicParameterSpace({'exponent': 0}, *parameter_range)
self.parameter_range = parameter_range
self.initial_data_type = initial_data_type
self.circle = circle
self.v = v
def discretize_nonlinear_instationary_advection_fv(analytical_problem, diameter=None, nt=100, num_flux='lax_friedrichs',
lxf_lambda=1., eo_gausspoints=5, eo_intervals=1, num_values=None,
domain_discretizer=None, grid=None, boundary_info=None):
"""Discretizes an |InstationaryAdvectionProblem| using the finite volume method.
Simple explicit Euler time-stepping is used for time-discretization.
Parameters
----------
analytical_problem
The |InstationaryAdvectionProblem| to discretize.
diameter
If not None, is passed to the domain_discretizer.
nt
The number of time-steps.
num_flux
The numerical flux to use in the finite volume formulation. Allowed
values are `'lax_friedrichs'`, `'engquist_osher'`, `'simplified_engquist_osher'`.
(See :mod:`pymor.operators.fv`.)
lxf_lambda
The stabilization parameter for the Lax-Friedrichs numerical flux.
(Ignored, if different flux is chosen.)
eo_gausspoints
Number of Gauss points for the Engquist-Osher numerical flux.
(Ignored, if different flux is chosen.)
eo_intervals
Number of sub-intervals to use for integration when using Engquist-Osher
numerical flux. (Ignored, if different flux is chosen.)
num_values
The number of returned vectors of the solution trajectory. If `None`, each
intermediate vector that is calculated is returned.
domain_discretizer
Discretizer to be used for discretizing the analytical domain. This has
to be a function `domain_discretizer(domain_description, diameter, ...)`.
If further arguments should be passed to the discretizer, use
:func:`functools.partial`. If `None`, |discretize_domain_default| is used.
grid
Instead of using a domain discretizer, the |Grid| can also be passed directly
using this parameter.
boundary_info
A |BoundaryInfo| specifying the boundary types of the grid boundary entities.
Must be provided if `grid` is provided.
Returns
-------
discretization
The discretization that has been generated.
data
Dictionary with the following entries:
:grid: The generated |Grid|.
:boundary_info: The generated |BoundaryInfo|.
"""
assert isinstance(analytical_problem, InstationaryAdvectionProblem)
assert grid is None or boundary_info is not None
assert boundary_info is None or grid is not None
assert grid is None or domain_discretizer is None
assert num_flux in ('lax_friedrichs', 'engquist_osher', 'simplified_engquist_osher')
if grid is None:
domain_discretizer = domain_discretizer or discretize_domain_default
if diameter is None:
grid, boundary_info = domain_discretizer(analytical_problem.domain)
else:
grid, boundary_info = domain_discretizer(analytical_problem.domain, diameter=diameter)
p = analytical_problem
if num_flux == 'lax_friedrichs':
L = nonlinear_advection_lax_friedrichs_operator(grid, boundary_info, p.flux_function,
dirichlet_data=p.dirichlet_data, lxf_lambda=lxf_lambda)
elif num_flux == 'engquist_osher':
L = nonlinear_advection_engquist_osher_operator(grid, boundary_info, p.flux_function,
p.flux_function_derivative,
gausspoints=eo_gausspoints, intervals=eo_intervals,
dirichlet_data=p.dirichlet_data)
else:
L = nonlinear_advection_simplified_engquist_osher_operator(grid, boundary_info, p.flux_function,
p.flux_function_derivative,
dirichlet_data=p.dirichlet_data)
F = None if p.rhs is None else L2ProductFunctional(grid, p.rhs)
if p.initial_data.parametric:
def initial_projection(U, mu):
I = p.initial_data.evaluate(grid.quadrature_points(0, order=2), mu).squeeze()
I = np.sum(I * grid.reference_element.quadrature(order=2)[1], axis=1) * (1. / grid.reference_element.volume)
I = NumpyVectorArray(I, copy=False)
return I.lincomb(U).data
inject_sid(initial_projection, __name__ + '.discretize_nonlinear_instationary_advection_fv.initial_data',
p.initial_data, grid)
I = NumpyGenericOperator(initial_projection, dim_range=grid.size(0), linear=True,
parameter_type=p.initial_data.parameter_type)
else:
I = p.initial_data.evaluate(grid.quadrature_points(0, order=2)).squeeze()
I = np.sum(I * grid.reference_element.quadrature(order=2)[1], axis=1) * (1. / grid.reference_element.volume)
I = NumpyVectorArray(I, copy=False)
inject_sid(I, __name__ + '.discretize_nonlinear_instationary_advection_fv.initial_data', p.initial_data, grid)
products = {'l2': L2Product(grid, boundary_info)}
if grid.dim == 2:
visualizer = PatchVisualizer(grid=grid, bounding_box=grid.domain, codim=0)
elif grid.dim == 1:
visualizer = Matplotlib1DVisualizer(grid, codim=0)
else:
visualizer = None
parameter_space = p.parameter_space if hasattr(p, 'parameter_space') else None
time_stepper = ExplicitEulerTimeStepper(nt=nt)
discretization = InstationaryDiscretization(operator=L, rhs=F, initial_data=I, T=p.T, products=products,
time_stepper=time_stepper,
parameter_space=parameter_space, visualizer=visualizer,
num_values=num_values, name='{}_FV'.format(p.name))
return discretization, {'grid': grid, 'boundary_info': boundary_info}
def diffusion_function_factory(x, y):
func = lambda X: (1. * (X[..., 0] >= x * dx) * (X[..., 0] < (x + 1) * dx)
* (X[..., 1] >= y * dy) * (X[..., 1] < (y + 1) * dy))
inject_sid(func, diffusion_function_id, x, y, dx, dy)
return GenericFunction(func, dim_domain=2, name='diffusion_function_{}_{}'.format(x, y))
