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Commit 91a0cfad authored by Michael Kuron's avatar Michael Kuron
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Merge branch 'IncompressibleCumulants' into 'master' 

Fix minor issues and remove depricated usage of cumulant LB method

See merge request pycodegen/lbmpy!70
parents 847389c4 63b38380
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doc/notebooks/03_tutorial_lbm_formulation.ipynb
View file @ 91a0cfad
%% Cell type:code id: tags:
``` python
from lbmpy.session import *
```
%% Cell type:markdown id: tags:
# Tutorial 03: Defining LB methods in *lbmpy*
## A) General Form
%% Cell type:markdown id: tags:
The lattice Boltzmann equation in its most general form is:
$$f_q(\mathbf{x} + \mathbf{c}_q \delta t, t+\delta t) = K\left( f_q(\mathbf{x}, t) \right)$$
with a discrete velocity set $\mathbf{c}_q$ (stencil) and a generic collision operator $K$.
So a lattice Boltzmann method can be fully defined by picking a stencil and a collision operator.
The collision operator $K$ has the following structure:
- Transformation of particle distribution function $f$ into collision space. This transformation has to be invertible and may be nonlinear.
- The collision operation is an convex combination of the pdf representation in collision space $c$ and some equilibrium vector $c^{(eq)}$. This equilibrium can also be defined in physical space, then $c^{(eq)} = C( f^{(eq)} ) $. The convex combination is done elementwise using a diagonal matrix $S$ where the diagonal entries are the relaxation rates.
- After collision, the collided state $c'$ is transformed back into physical space
![](../img/collision.svg)
The full collision operator is:
$$K(f) = C^{-1}\left( (I-S)C(f) + SC(f^{(eq}) \right)$$
or
$$K(f) = C^{-1}\left( C(f) - S (C(f) - C(f^{(eq})) \right)$$
%% Cell type:markdown id: tags:
## B) Moment-based relaxation
The most commonly used LBM collision operator is the multi relaxation time (MRT) collision.
In MRT methods the collision space is spanned by moments of the distribution function. This is a very natural approach, since the pdf moments are the quantities that go into the Chapman Enskog analysis that is used to show that LB methods can solve the Navier Stokes equations. Also the lower order moments correspond to the macroscopic quantities of interest (density/pressure, velocity, shear rates, heat flux). Furthermore the transformation to collision space is linear in this case, simplifying the collision equations:
$$K(f) = C^{-1}\left( C(f) - S (C(f) - C(f^{(eq})) \right)$$
$$K(f) = f - \underbrace{ C^{-1}SC}_{A}(f - f^{(eq)})$$
in *lbmpy* the following formulation is used, since it is more natural to define the equilibrium in moment-space instead of physical space:
$$K(f) = f - C^{-1}S(Cf - c^{(eq)})$$
%% Cell type:markdown id: tags:
### Use a pre-defined method
Lets create a moment-based method in *lbmpy* and see how the moment transformation $C$ and the relaxation rates that comprise the diagonal matrix $S$ can be defined. We start with a function that creates a basic MRT model.
Don't use this for real simulations, there orthogonalized MRT methods should be used, as discussed in the next section.
%% Cell type:code id: tags:
``` python
from lbmpy.creationfunctions import create_lb_method
method = create_lb_method(stencil='D2Q9', method='mrt_raw')
# check also method='srt', 'trt', 'mrt'
method
```
%% Output
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7fc00786c3d0>
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7f1be00e30d0>
%% Cell type:markdown id: tags:
The first column labeled "Moment" defines the collision space and thus the transformation matrix $C$.
The remaining columns specify the equilibrium vector in moment space $c^{(eq)}$ and the corresponding relaxation rate.
Each row of the "Moment" column defines one row of $C$. In the next cells this matrix and the discrete velocity set (stencil) of our method are shown. Check for example the second last row of the table $x^2 y$: In the corresponding second last row of the moment matrix $C$ where each column stands for a lattice velocity (for ordering visualized stencil below) and each entry is the expression $x^2 y$ where $x$ and $y$ are the components of the lattice velocity.
In general the transformation matrix $C_{iq}$ is defined as;
$$c_i = C_{iq} f_q = \sum_q m_i(c_q)$$
where $m_i(c_q)$ is the $i$'th moment polynomial where $x$ and $y$ are substituted with the components of the $q$'th lattice velocity
%% Cell type:code id: tags:
``` python
# Transformation matrix C
method.moment_matrix
```
%% Output
$\displaystyle \left[\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\0 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1\\0 & 1 & -1 & 0 & 0 & 1 & 1 & -1 & -1\\0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1\\0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1\\0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1\\0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1\\0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\end{matrix}\right]$
⎡1 1 1 1 1 1 1 1 1 ⎤
⎢ ⎥
⎢0 0 0 -1 1 -1 1 -1 1 ⎥
⎢ ⎥
⎢0 1 -1 0 0 1 1 -1 -1⎥
⎢ ⎥
⎢0 0 0 1 1 1 1 1 1 ⎥
⎢ ⎥
⎢0 1 1 0 0 1 1 1 1 ⎥
⎢ ⎥
⎢0 0 0 0 0 -1 1 1 -1⎥
⎢ ⎥
⎢0 0 0 0 0 1 1 -1 -1⎥
⎢ ⎥
⎢0 0 0 0 0 -1 1 -1 1 ⎥
⎢ ⎥
⎣0 0 0 0 0 1 1 1 1 ⎦
%% Cell type:code id: tags:
``` python
ps.stencil.plot(method.stencil)
method.stencil
```
%% Output
$\displaystyle \left( \left( 0, \ 0\right), \ \left( 0, \ 1\right), \ \left( 0, \ -1\right), \ \left( -1, \ 0\right), \ \left( 1, \ 0\right), \ \left( -1, \ 1\right), \ \left( 1, \ 1\right), \ \left( -1, \ -1\right), \ \left( 1, \ -1\right)\right)$
((0, 0), (0, 1), (0, -1), (-1, 0), (1, 0), (-1, 1), (1, 1), (-1, -1), (1, -1))
%% Cell type:markdown id: tags:
### Orthogonal MRTs
For a real MRT method, the moments should be orthogonalized.
One can either orthogonalize using the standard scalar product or a scalar product that is weighted with the lattice weights. If unsure, use the weighted version.
The next cell shows how to get both orthogonalized MRT versions in lbmpy.
%% Cell type:code id: tags:
``` python
weighted_ortho_mrt = create_lb_method(stencil="D2Q9", method="mrt", weighted=True)
weighted_ortho_mrt
```
%% Output
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7fc00774f640>
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7f1b8f8e0400>
%% Cell type:code id: tags:
``` python
ortho_mrt = create_lb_method(stencil="D2Q9", method="mrt", weighted=False)
ortho_mrt
```
%% Output
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7fc0075d7b50>
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7f1b8763a4f0>
%% Cell type:markdown id: tags:
One can check if a method is orthogonalized:
%% Cell type:code id: tags:
``` python
ortho_mrt.is_orthogonal, weighted_ortho_mrt.is_weighted_orthogonal
```
%% Output
(True, True)
%% Cell type:markdown id: tags:
### Define custom MRT method
To choose custom values for the left moment column one can pass a nested list of moments.
Moments that should be relaxed with the same paramter are grouped together.
*lbmpy* also comes with a few templates for this list taken from literature:
%% Cell type:code id: tags:
``` python
from lbmpy.methods import mrt_orthogonal_modes_literature
from lbmpy.stencils import get_stencil
from lbmpy.moments import MOMENT_SYMBOLS
x, y, z = MOMENT_SYMBOLS
moments = mrt_orthogonal_modes_literature(get_stencil("D2Q9"), is_weighted=True, is_cumulant=False)
moments = mrt_orthogonal_modes_literature(get_stencil("D2Q9"), is_weighted=True)
moments
```
%% Output
$\displaystyle \left[ \left[ 1\right], \ \left[ x, \ y\right], \ \left[ 3 x^{2} + 3 y^{2} - 2, \ x^{2} - y^{2}, \ x y\right], \ \left[ x \left(3 x^{2} + 3 y^{2} - 4\right), \ y \left(3 x^{2} + 3 y^{2} - 4\right)\right], \ \left[ - 15 x^{2} - 15 y^{2} + 9 \left(x^{2} + y^{2}\right)^{2} + 2\right]\right]$
⎡
⎢ ⎡ 2 2 2 2 ⎤ ⎡ ⎛ 2 2 ⎞ ⎛ 2
⎣[1], [x, y], ⎣3⋅x + 3⋅y - 2, x - y , x⋅y⎦, ⎣x⋅⎝3⋅x + 3⋅y - 4⎠, y⋅⎝3⋅x +
⎡ 2 ⎤⎤
2 ⎞⎤ ⎢ 2 2 ⎛ 2 2⎞ ⎥⎥
3⋅y - 4⎠⎦, ⎣- 15⋅x - 15⋅y + 9⋅⎝x + y ⎠ + 2⎦⎦
%% Cell type:markdown id: tags:
This nested moment list can be passed to `create_lb_method`:
%% Cell type:code id: tags:
``` python
create_lb_method(stencil="D2Q9", method="mrt", nested_moments=moments)
```
%% Output
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7fc04815aa30>
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7f1b874f5e20>
%% Cell type:markdown id: tags:
If one needs to also specify custom equilibrium moments the following approach can be used
%% Cell type:code id: tags:
``` python
rho = sp.symbols("rho")
u = sp.symbols("u_:3")
omega = sp.symbols("omega_:4")
method_table = [
# Conserved moments
(1, rho, 0 ),
(x, u[0], 0 ),
(y, u[1], 0 ),
# Shear moments
(x*y, u[0]*u[1], omega[0]),
(x**2-y**2, u[0]**2 - u[1]**2, omega[0]),
(x**2+y**2, 2*rho/3 + u[0]**2 + u[1]**2, omega[1]),
# Higher order
(x * y**2, u[0]/3, omega[2]),
(x**2 * y, u[1]/3, omega[2]),
(x**2 * y**2, rho/9 + u[0]**2/3 + u[1]**2/3, omega[3]),
]
method = create_generic_mrt(get_stencil("D2Q9"), method_table)
method
```
%% Output
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7fc0075b2e50>
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7f1b873e9cd0>
%% Cell type:markdown id: tags:
Instead of manually defining all entries in the method table, *lbmpy* has functions to fill the table according to a specific pattern. For example:
- for a full stencil (D2Q9, D3Q27) there exist exactly 9 or 27 linearly independent moments. These can either be taken as they are, or orthogonalized using Gram-Schmidt, weighted Gram-Schmidt or a Hermite approach
- equilibrium values can be computed from the standard discrete equilibrium of order 1,2 or 3. Alternatively they can also be computed as continuous moments of a Maxwellian distribution
One option is to start with one of *lbmpy*'s built-in methods and modify it with `create_lb_method_from_existing`.
In the next cell we fix the fourth order relaxation rate to a constant, by writing a function that defines how to alter each row of the collision table. This is for demonstration only, of course we could have done it right away when passing in the collision table.
%% Cell type:code id: tags:
``` python
def modification_func(moment, eq, rate):
if rate == omega[3]:
return moment, eq, 1.0
return moment, eq, rate
method = create_lb_method_from_existing(method, modification_func)
method
```
%% Output
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7fc007825f70>
<lbmpy.methods.momentbased.momentbasedmethod.MomentBasedLbMethod at 0x7f1b8766a520>
%% Cell type:markdown id: tags:
Our customized method can be directly passed into one of the scenarios. We can for example set up a channel flow with it. Since we used symbols as relaxation rates, we have to pass them in as `kernel_params`.
%% Cell type:code id: tags:
``` python
ch = create_channel(domain_size=(100, 30), lb_method=method, u_max=0.05,
kernel_params={'omega_0': 1.8, 'omega_1': 1.4, 'omega_2': 1.5})
ch.run(500)
plt.figure(dpi=200)
plt.vector_field(ch.velocity[:, :]);
```
%% Output
%% Cell type:markdown id: tags:
#### Bonus: Automatic analysis
Above we have created a non-orthogonal MRT, where the shear viscosity and bulk viscosity can be independently controlled. For moment-based methods, *lbmpy* also offers an automatic Chapman Enskog analysis that can find the relation between viscosity and relaxation rate(s):
%% Cell type:code id: tags:
``` python
from lbmpy.chapman_enskog import ChapmanEnskogAnalysis
analysis = ChapmanEnskogAnalysis(method)
analysis.get_dynamic_viscosity()
```
%% Output
$\displaystyle - \frac{\omega_{0} - 2}{6 \omega_{0}}$
-(ω₀ - 2)
──────────
6⋅ω₀
%% Cell type:code id: tags:
``` python
analysis.get_bulk_viscosity()
```
%% Output
$\displaystyle - \frac{1}{9} - \frac{1}{3 \omega_{1}} + \frac{5}{9 \omega_{0}}$
1 1 5
- ─ - ──── + ────
9 3⋅ω₁ 9⋅ω₀
......
lbmpy/creationfunctions.py
View file @ 91a0cfad
......@@ -53,7 +53,6 @@ General:
- ``maxwellian_moments=True``: way to compute equilibrium moments/cumulants, if False the standard
discretized LBM equilibrium is used, otherwise the equilibrium moments are computed from the continuous Maxwellian.
This makes only a difference if sparse stencils are used e.g. D2Q9 and D3Q27 are not affected, D319 and DQ15 are
- ``cumulant=False``: use cumulants instead of moments (deprecated: use method=cumulant directly)
- ``initial_velocity=None``: initial velocity in domain, can either be a tuple (x,y,z) velocity to set a constant
velocity everywhere, or a numpy array with the same size of the domain, with a last coordinate of shape dim to set
velocities on cell level
......@@ -87,8 +86,6 @@ Cumulant methods:
- ``galilean_correction=False``: Special correction for D3Q27 cumulant LBMs. For Details see
:mod:`lbmpy.methods.centeredcumulant.galilean_correction`
- ``pre_simplification=True``: Simplifications applied during the derivaton of the collision rule for cumulant LBMs
For details see :mod:`lbmpy.methods.momentbased.moment_transforms`.
LES methods:
......@@ -112,7 +109,9 @@ Simplifications / Transformations:
load/store streams and thus speeds up the kernel on most architectures
- ``builtin_periodicity=(False,False,False)``: instead of handling periodicity by copying ghost layers, the periodicity
is built into the kernel. This parameters specifies if the domain is periodic in (x,y,z) direction. Even if the
periodicity is built into the kernel, the fields have one ghost layer to be consistent with other functions.
periodicity is built into the kernel, the fields have one ghost layer to be consistent with other functions.
- ``pre_simplification=True``: Simplifications applied during the derivaton of the collision rule for cumulant LBMs
For details see :mod:`lbmpy.methods.momentbased.moment_transforms`.
Field size information:
......@@ -190,6 +189,7 @@ For example, to modify the AST one can run::
func = create_lb_function(ast=ast, ...)
"""
from collections import OrderedDict
from copy import copy
import sympy as sp
......@@ -199,8 +199,9 @@ import lbmpy.methods.centeredcumulant.force_model as cumulant_force_model
from lbmpy.fieldaccess import CollideOnlyInplaceAccessor, PdfFieldAccessor, PeriodicTwoFieldsAccessor
from lbmpy.fluctuatinglb import add_fluctuations_to_collision_rule
from lbmpy.methods import (create_mrt_orthogonal, create_mrt_raw, create_srt, create_trt, create_trt_kbc)
from lbmpy.methods.abstractlbmethod import RelaxationInfo
from lbmpy.methods.centeredcumulant import CenteredCumulantBasedLbMethod
from lbmpy.methods.momentbased.moment_transforms import PdfsToCentralMomentsByShiftMatrix
from lbmpy.methods.momentbased.moment_transforms import FastCentralMomentTransform
from lbmpy.methods.centeredcumulant.cumulant_transform import CentralMomentsToCumulantsByGeneratingFunc
from lbmpy.methods.creationfunctions import (
create_with_monomial_cumulants, create_with_polynomial_cumulants, create_with_default_polynomial_cumulants)
......@@ -416,7 +417,6 @@ def create_lb_method(**params):
'equilibrium_order': params['equilibrium_order'],
'force_model': force_model,
'maxwellian_moments': params['maxwellian_moments'],
'cumulant': params['cumulant'],
'c_s_sq': params['c_s_sq'],
}
......@@ -443,10 +443,6 @@ def create_lb_method(**params):
if all(get_order(m) < 2 for m in moments):
if params['entropic']:
return relaxation_rates[0]
elif params['cumulant']:
result = relaxation_rates[next_relaxation_rate[0]]
next_relaxation_rate[0] += 1
return result
else:
return 0
res = relaxation_rates[next_relaxation_rate[0]]
......@@ -494,11 +490,27 @@ def create_lb_method_from_existing(method, modification_function):
modification_function: function receiving (moment, equilibrium_value, relaxation_rate) as arguments,
i.e. one row of the relaxation table, returning a modified version
"""
relaxation_table = (modification_function(m, eq, rr)
for m, eq, rr in zip(method.moments, method.moment_equilibrium_values, method.relaxation_rates))
compressible = method.conserved_quantity_computation.compressible
cumulant = isinstance(method, CenteredCumulantBasedLbMethod)
return create_generic_mrt(method.stencil, relaxation_table, compressible, method.force_model, cumulant)
if isinstance(method, CenteredCumulantBasedLbMethod):
rr_dict = OrderedDict()
relaxation_table = (modification_function(m, eq, rr)
for m, eq, rr in
zip(method.cumulants, method.cumulant_equilibrium_values, method.relaxation_rates))
for cumulant, eq_value, rr in relaxation_table:
cumulant = sp.sympify(cumulant)
rr_dict[cumulant] = RelaxationInfo(eq_value, rr)
return CenteredCumulantBasedLbMethod(method.stencil, rr_dict, method.conserved_quantity_computation,
method.force_model,
galilean_correction=method.galilean_correction,
central_moment_transform_class=method.central_moment_transform_class,
cumulant_transform_class=method.cumulant_transform_class)
else:
relaxation_table = (modification_function(m, eq, rr)
for m, eq, rr in
zip(method.moments, method.moment_equilibrium_values, method.relaxation_rates))
return create_generic_mrt(method.stencil, relaxation_table, compressible, method.force_model)
# ----------------------------------------------------------------------------------------------------------------------
......@@ -564,11 +576,10 @@ def update_with_default_parameters(params, opt_params=None, fail_on_unknown_para
'force_model': 'none',
'force': (0, 0, 0),
'maxwellian_moments': True,
'cumulant': False, # Depricated usage. Cumulant is now an own method
'initial_velocity': None,
'galilean_correction': False, # only available for D3Q27 cumulant methods
'central_moment_transform_class': PdfsToCentralMomentsByShiftMatrix,
'central_moment_transform_class': FastCentralMomentTransform,
'cumulant_transform_class': CentralMomentsToCumulantsByGeneratingFunc,
'entropic': False,
......
lbmpy/methods/centeredcumulant/centeredcumulantmethod.py
View file @ 91a0cfad
......@@ -159,21 +159,45 @@ class CenteredCumulantBasedLbMethod(AbstractLbMethod):
self.set_first_moment_relaxation_rate(self._force_model.override_momentum_relaxation_rate)
self.force_model_rr_override = True
@property
def central_moment_transform_class(self):
self._central_moment_transform_class
@property
def cumulants(self):
return tuple(self._cumulant_to_relaxation_info_dict.keys())
@property
def cumulant_equilibrium_values(self):
return tuple([e.equilibrium_value for e in self._cumulant_to_relaxation_info_dict.values()])
@property
def cumulant_transform_class(self):
self._cumulant_transform_class
@property
def first_order_equilibrium_moment_symbols(self, ):
return self._conserved_quantity_computation.first_order_moment_symbols
@property
def force_model(self):
return self._force_model
@property
def galilean_correction(self):
return self._galilean_correction
@property
def relaxation_info_dict(self):
return self._cumulant_to_relaxation_info_dict
@property
def zeroth_order_equilibrium_moment_symbol(self, ):
return self._conserved_quantity_computation.zeroth_order_moment_symbol
def relaxation_rates(self):
return tuple([e.relaxation_rate for e in self._cumulant_to_relaxation_info_dict.values()])
@property
def first_order_equilibrium_moment_symbols(self, ):
return self._conserved_quantity_computation.first_order_moment_symbols
def zeroth_order_equilibrium_moment_symbol(self, ):
return self._conserved_quantity_computation.zeroth_order_moment_symbol
def set_zeroth_moment_relaxation_rate(self, relaxation_rate):
e = sp.Rational(1, 1)
......@@ -200,18 +224,6 @@ class CenteredCumulantBasedLbMethod(AbstractLbMethod):
def set_force_model(self, force_model):
self._force_model = force_model
@property
def cumulants(self):
return sorted(self._cumulant_to_relaxation_info_dict.keys(), key=moment_sort_key)
@property
def cumulant_equilibrium_values(self):
return tuple([e.equilibrium_value for e in self._cumulant_to_relaxation_info_dict.values()])
@property
def relaxation_rates(self):
return tuple([e.relaxation_rate for e in self._cumulant_to_relaxation_info_dict.values()])
def _repr_html_(self):
table = """
<table style="border:none; width: 100%">
......
lbmpy/methods/creationfunctions.py
View file @ 91a0cfad
This diff is collapsed.
lbmpy/methods/momentbased/moment_transforms.py
View file @ 91a0cfad
......@@ -142,7 +142,7 @@ class PdfsToCentralMomentsByMatrix(AbstractMomentTransform):
central_moments = self.forward_matrix * f_vec
main_assignments = [Assignment(sq_sym(moment_symbol_base, e), eq)
for e, eq in zip(self.moment_exponents, central_moments)]
symbol_gen = SymbolGen(subexpression_base, dtype=float)
symbol_gen = SymbolGen(subexpression_base)
ac = AssignmentCollection(main_assignments, subexpression_symbol_generator=symbol_gen)
if simplification:
......@@ -157,7 +157,7 @@ class PdfsToCentralMomentsByMatrix(AbstractMomentTransform):
moment_vec = sp.Matrix(moments)
pdfs_from_moments = self.backward_matrix * moment_vec
main_assignments = [Assignment(f, eq) for f, eq in zip(pdf_symbols, pdfs_from_moments)]
symbol_gen = SymbolGen(subexpression_base, dtype=float)
symbol_gen = SymbolGen(subexpression_base)
ac = AssignmentCollection(main_assignments, subexpression_symbol_generator=symbol_gen)
if simplification:
......@@ -228,7 +228,7 @@ class FastCentralMomentTransform(AbstractMomentTransform):
collect_partial_sums(e)
subexpressions = [Assignment(lhs, rhs) for lhs, rhs in subexpressions_dict.items()]
symbol_gen = SymbolGen(subexpression_base, dtype=float)
symbol_gen = SymbolGen(subexpression_base)
ac = AssignmentCollection(main_assignments, subexpressions=subexpressions,
subexpression_symbol_generator=symbol_gen)
if simplification:
......@@ -244,7 +244,7 @@ class FastCentralMomentTransform(AbstractMomentTransform):
pdf_symbols, moment_symbol_base=POST_COLLISION_CENTRAL_MOMENT, simplification=False)
raw_equations = raw_equations.new_without_subexpressions()
symbol_gen = SymbolGen(subexpression_base, dtype=float)