# Demo: Create lbmpy Method from Scratch¶

## Defining transformation to collision space¶

[2]:

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

[2]:

$\displaystyle \left[ \left( 0, \ 0\right), \ \left( 0, \ 1\right), \ \left( 0, \ 2\right), \ \left( 1, \ 0\right), \ \left( 1, \ 1\right), \ \left( 1, \ 2\right), \ \left( 2, \ 0\right), \ \left( 2, \ 1\right), \ \left( 2, \ 2\right)\right]$
[3]:

moments = exponents_to_polynomial_representations(moment_exponents)
moments

[3]:

$\displaystyle \left( 1, \ y, \ y^{2}, \ x, \ x y, \ x y^{2}, \ x^{2}, \ x^{2} y, \ x^{2} y^{2}\right)$
[4]:

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

[5]:

M = moment_matrix(moments, stencil=d2q9)
M

[5]:

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

## Defining the Equilibrium Distribution, and Computation of Conserved Quantities¶

[6]:

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)))

[6]:

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

## Defining Relaxation Behaviour and Creating the Method¶

[7]:

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

[7]:

 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¶

[8]:

collision_rule = method.get_collision_rule()
collision_rule

[8]:

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¶

[9]:

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

[9]:

[10]:

simplification_strategy = create_simplification_strategy(method)