Demo: Create lbmpy Method from Scratch

Defining transformation to collision space

from lbmpy.moments import moment_matrix, moments_up_to_component_order, exponents_to_polynomial_representations
moment_exponents = list(moments_up_to_component_order(2, 2))
moment_exponents

$\displaystyle \left[ \left( 0, \ 0\right), \ \left( 0, \ 1\right), \ \left( 0, \ 2\right), \ \left( 1, \ 0\right), \ \left( 1, \ 1\right), \ \left( 1, \ 2\right), \ \left( 2, \ 0\right), \ \left( 2, \ 1\right), \ \left( 2, \ 2\right)\right]$

moments = exponents_to_polynomial_representations(moment_exponents)
moments

$\displaystyle \left( 1, \ y, \ y^{2}, \ x, \ x y, \ x y^{2}, \ x^{2}, \ x^{2} y, \ x^{2} y^{2}\right)$

d2q9 = LBStencil(Stencil.D2Q9, ordering='walberla')
d2q9.plot()

M = moment_matrix(moments, stencil=d2q9)
M

$\displaystyle \left[\begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\0 & 1 & -1 & 0 & 0 & 1 & 1 & -1 & -1\\0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1\\0 & 0 & 0 & -1 & 1 & -1 & 1 & -1 & 1\\0 & 0 & 0 & 0 & 0 & -1 & 1 & 1 & -1\\0 & 0 & 0 & 0 & 0 & -1 & 1 & -1 & 1\\0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1\\0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1\\0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\end{matrix}\right]$

Defining the Equilibrium Distribution, and Computation of Conserved Quantities

from lbmpy.equilibrium import ContinuousHydrodynamicMaxwellian
from lbmpy.methods import DensityVelocityComputation

equilibrium = ContinuousHydrodynamicMaxwellian(dim=2, compressible=True, order=2)
cqc = DensityVelocityComputation(d2q9, True, False)

tuple(zip(moments, equilibrium.moments(moments)))

$\displaystyle \left( \left( 1, \ \rho\right), \ \left( y, \ \rho u_{1}\right), \ \left( y^{2}, \ c_{s}^{2} \rho + \rho u_{1}^{2}\right), \ \left( x, \ \rho u_{0}\right), \ \left( x y, \ \rho u_{0} u_{1}\right), \ \left( x y^{2}, \ c_{s}^{2} \rho u_{0}\right), \ \left( x^{2}, \ c_{s}^{2} \rho + \rho u_{0}^{2}\right), \ \left( x^{2} y, \ c_{s}^{2} \rho u_{1}\right), \ \left( x^{2} y^{2}, \ c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2}\right)\right)$

Defining Relaxation Behaviour and Creating the Method

from lbmpy.methods.creationfunctions import create_from_equilibrium

omega = sp.symbols("omega")
relaxation_rate_dict = {moment : omega for moment in moments}

force_model = forcemodels.Guo(sp.symbols("F_:2"))
method = create_from_equilibrium(d2q9, equilibrium, cqc, relaxation_rate_dict)
method

Moment-Based Method			Stencil: D2Q9	Zero-Centered Storage: ✗	Force Model: None

Continuous Hydrodynamic Maxwellian Equilibrium			$f (\rho, \left( u_{0}, \ u_{1}\right), \left( v_{0}, \ v_{1}\right)) = \frac{\rho e^{\frac{- \left(- u_{0} + v_{0}\right)^{2} - \left(- u_{1} + v_{1}\right)^{2}}{2 c_{s}^{2}}}}{2 \pi c_{s}^{2}}$
Compressible: ✓	Deviation Only: ✗	Order: 2

Relaxation Info
Moment	Eq. Value	Relaxation Rate
$1$	$\rho$	$\omega$
$y$	$\rho u_{1}$	$\omega$
$y^{2}$	$c_{s}^{2} \rho + \rho u_{1}^{2}$	$\omega$
$x$	$\rho u_{0}$	$\omega$
$x y$	$\rho u_{0} u_{1}$	$\omega$
$x y^{2}$	$c_{s}^{2} \rho u_{0}$	$\omega$
$x^{2}$	$c_{s}^{2} \rho + \rho u_{0}^{2}$	$\omega$
$x^{2} y$	$c_{s}^{2} \rho u_{1}$	$\omega$
$x^{2} y^{2}$	$c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2}$	$\omega$

Example of a update equation without simplifications

collision_rule = method.get_collision_rule()
collision_rule

Subexpressions:

$$vel0Term \leftarrow f_{4} + f_{6} + f_{8}$$

$$vel1Term \leftarrow f_{1} + f_{5}$$

$$\rho \leftarrow f_{0} + f_{2} + f_{3} + f_{7} + vel0Term + vel1Term$$

$$u_{0} \leftarrow \frac{- f_{3} - f_{5} - f_{7} + vel0Term}{\rho}$$

$$u_{1} \leftarrow \frac{- f_{2} + f_{6} - f_{7} - f_{8} + vel1Term}{\rho}$$

$$\delta_{\rho} \leftarrow \rho - 1$$

Main Assignments:

$$d_{0} \leftarrow f_{0} + \omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right) - \omega \left(c_{s}^{2} \rho - f_{1} - f_{2} - f_{5} - f_{6} - f_{7} - f_{8} + \rho u_{1}^{2}\right) - \omega \left(c_{s}^{2} \rho - f_{3} - f_{4} - f_{5} - f_{6} - f_{7} - f_{8} + \rho u_{0}^{2}\right) + \omega \left(- f_{0} - f_{1} - f_{2} - f_{3} - f_{4} - f_{5} - f_{6} - f_{7} - f_{8} + \rho\right)$$

$$d_{1} \leftarrow f_{1} - \frac{\omega \left(c_{s}^{2} \rho u_{1} - f_{5} - f_{6} + f_{7} + f_{8}\right)}{2} + \frac{\omega \left(- f_{1} + f_{2} - f_{5} - f_{6} + f_{7} + f_{8} + \rho u_{1}\right)}{2} - \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{2} + \frac{\omega \left(c_{s}^{2} \rho - f_{1} - f_{2} - f_{5} - f_{6} - f_{7} - f_{8} + \rho u_{1}^{2}\right)}{2}$$

$$d_{2} \leftarrow f_{2} + \frac{\omega \left(c_{s}^{2} \rho u_{1} - f_{5} - f_{6} + f_{7} + f_{8}\right)}{2} - \frac{\omega \left(- f_{1} + f_{2} - f_{5} - f_{6} + f_{7} + f_{8} + \rho u_{1}\right)}{2} - \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{2} + \frac{\omega \left(c_{s}^{2} \rho - f_{1} - f_{2} - f_{5} - f_{6} - f_{7} - f_{8} + \rho u_{1}^{2}\right)}{2}$$

$$d_{3} \leftarrow f_{3} + \frac{\omega \left(c_{s}^{2} \rho u_{0} + f_{5} - f_{6} + f_{7} - f_{8}\right)}{2} - \frac{\omega \left(f_{3} - f_{4} + f_{5} - f_{6} + f_{7} - f_{8} + \rho u_{0}\right)}{2} - \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{2} + \frac{\omega \left(c_{s}^{2} \rho - f_{3} - f_{4} - f_{5} - f_{6} - f_{7} - f_{8} + \rho u_{0}^{2}\right)}{2}$$

$$d_{4} \leftarrow f_{4} - \frac{\omega \left(c_{s}^{2} \rho u_{0} + f_{5} - f_{6} + f_{7} - f_{8}\right)}{2} + \frac{\omega \left(f_{3} - f_{4} + f_{5} - f_{6} + f_{7} - f_{8} + \rho u_{0}\right)}{2} - \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{2} + \frac{\omega \left(c_{s}^{2} \rho - f_{3} - f_{4} - f_{5} - f_{6} - f_{7} - f_{8} + \rho u_{0}^{2}\right)}{2}$$

$$d_{5} \leftarrow f_{5} - \frac{\omega \left(f_{5} - f_{6} - f_{7} + f_{8} + \rho u_{0} u_{1}\right)}{4} - \frac{\omega \left(c_{s}^{2} \rho u_{0} + f_{5} - f_{6} + f_{7} - f_{8}\right)}{4} + \frac{\omega \left(c_{s}^{2} \rho u_{1} - f_{5} - f_{6} + f_{7} + f_{8}\right)}{4} + \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{4}$$

$$d_{6} \leftarrow f_{6} + \frac{\omega \left(f_{5} - f_{6} - f_{7} + f_{8} + \rho u_{0} u_{1}\right)}{4} + \frac{\omega \left(c_{s}^{2} \rho u_{0} + f_{5} - f_{6} + f_{7} - f_{8}\right)}{4} + \frac{\omega \left(c_{s}^{2} \rho u_{1} - f_{5} - f_{6} + f_{7} + f_{8}\right)}{4} + \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{4}$$

$$d_{7} \leftarrow f_{7} + \frac{\omega \left(f_{5} - f_{6} - f_{7} + f_{8} + \rho u_{0} u_{1}\right)}{4} - \frac{\omega \left(c_{s}^{2} \rho u_{0} + f_{5} - f_{6} + f_{7} - f_{8}\right)}{4} - \frac{\omega \left(c_{s}^{2} \rho u_{1} - f_{5} - f_{6} + f_{7} + f_{8}\right)}{4} + \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{4}$$

$$d_{8} \leftarrow f_{8} - \frac{\omega \left(f_{5} - f_{6} - f_{7} + f_{8} + \rho u_{0} u_{1}\right)}{4} + \frac{\omega \left(c_{s}^{2} \rho u_{0} + f_{5} - f_{6} + f_{7} - f_{8}\right)}{4} - \frac{\omega \left(c_{s}^{2} \rho u_{1} - f_{5} - f_{6} + f_{7} + f_{8}\right)}{4} + \frac{\omega \left(c_{s}^{4} \rho + c_{s}^{2} \rho u_{0}^{2} + c_{s}^{2} \rho u_{1}^{2} - f_{5} - f_{6} - f_{7} - f_{8}\right)}{4}$$

Generic simplification strategy - common subexpresssion elimination

generic_strategy = ps.simp.SimplificationStrategy()
generic_strategy.add(ps.simp.sympy_cse)
generic_strategy.create_simplification_report(collision_rule)

Name	Runtime	Adds	Muls	Divs	Total
OriginalTerm	-	245	248	2	495
sympy_cse	30.74 ms	82	36	1	119

A custom simplification strategy for moment-based methods

simplification_strategy = create_simplification_strategy(method)
simplification_strategy.create_simplification_report(collision_rule)
simplification_strategy.add(ps.simp.sympy_cse)