MomentGauge.Models.ESBGK

Module Contents

Classes

ESBGK_Legendre_1D	The 1D Elliptic Statistical distribution.
ESBGK_Canonical_Legendre_1D	The 1D Elliptic Statistical distribution.
ESBGK_Gauged_Legendre_1D	The 1D Elliptic Statistical distribution with Gauge transformation.
ESBGK_Canonical_Gauged_Legendre_1D	The 1D Elliptic Statistical distribution with Gauge transformation.

Attributes

constant

class MomentGauge.Models.ESBGK.ESBGK_Legendre_1D(constant)

Bases: MomentGauge.Models.ExpFamilyBase.BaseExpFamilyModel

The 1D Elliptic Statistical distribution.

\begin{equation} \begin{split} f_0(\mathbf{u}) &= \exp\left( \beta_0 \phi_0(\mathbf{u}) + \beta_1 \phi_1(\mathbf{u}) + \beta_2 \phi_2(\mathbf{u} )+ \beta_3 \phi_3(\mathbf{u} ) \right)\\ &= \frac{\rho}{m} \exp\left( \frac{m v_x}{k_B T}\phi_1(\mathbf{u}) -\frac{m}{2 k_B T} \phi_2(\mathbf{u}) - \left(\frac{m v_x^2}{2k_B T}+ \log \left(\frac{2\pi k_B T}{m}\right)^{\frac{3}{2}}\right) \right), \end{split} \end{equation}

in which \(\rho\) is the number density, \(m\) is the molecule velocity, \(v_x\) is the flow velocity at the x direction, \(T\) is the temperature, and

\(\phi_0\) (u ) = 1.

\(\phi_1\) (u ) = \(u_x\)

\(\phi_2\) (u ) = \(u_x^2\)

\(\phi_3\) (u ) = \(u_y^2 + u_z^2\).

Parameters:

constant (dict) –

dictionary with the following keys

’m’ : float - the mass of particle considered

’kB’ : float - the Boltzmann constant

’n_x’ : int - the order of Gauss Legendre quadrature in x dimension. Required by MomentGauge.Sampler.QuadratureSampler.Gauss_Legendre_Sampler_Polar3D

’n_r’ : int - the order of Gauss Legendre quadrature in r dimension.

’B_x’ : int - how many blocks are splitted in the x dimension for Gauss Legendre quadrature.

’B_r’ : int - how many blocks are splitted in the r dimension for Gauss Legendre quadrature.

’alpha’: float - the initial step size used in backtracking line search, default = 1. Required by MomentGauge.Optim.NewtonOptimizer.Newton_Optimizer

’beta’: float - the decreasing factor of the step size used in backtracking line search, default = 0.5

’c’: float - the parameter used in the Armijo’s condiiton, must lies in (0,1), default = 5e-4

’atol’: float - the absolute error tolerance of the Armijo’s condiiton since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 5e-6.

’rtol’: float - the relative error tolerance of the Armijo’s condition since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 1e-5.

’max_iter’: int - the maximal iteration allowed for the Netwon’s method, default = 100

’max_back_tracking_iter’: int - the maximal iteration allowed for the backtracking line search, default = 25

’tol’: float - the tolerance for residual, below which the optimization stops.

’min_step_size’: float - the minimum step size given by back tracking, below which the optimization stops, default = 1e-6.

’reg_hessian’: bool - Regularize the Hessian if the Cholesky decomposition failed. Default = True

’debug’: bool - print debug information if True.

m

the mass of particle considered

Type:

float

kB

the Boltzmann constant

Type:

float

constant

dictionary with the keys containing ‘m’ and ‘kB’

Type:

dict

suff_statistics(u)

Compute the value of sufficient statistics at certain \(\mathbf{u}\)

Parameters:

u (float array of shape (3)) – The 3D sample vector

Returns:

The array \(\{ \phi_0(\mathbf{u}), \phi_1(\mathbf{u}), \phi_2(\mathbf{u}), \phi_3(\mathbf{u}) \}\)

Return type:

float array of shape (4)

natural_paras_to_moments(betas, domain)

Compute the moments of sufficient statistics given natural parameters`

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

The array \(\{ M_0, \cdots, M_M \}\)

Return type:

float array of shape (M+1)

moments_to_natural_paras(betas_ini, moments, domain)

Compute the natural parameters \(\boldsymbol{\beta}\) from the moments of sufficient statistics.

Parameters:

• betas_ini (float array of shape (4)) – A set of proposed natural parameter \(\boldsymbol{\beta}\)

• moments (float array of shape (4)) – the target moments that we expect exponential familty with the natural parameter \(\boldsymbol{\beta}\) to have.

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

A tuple containing

beta: float array of shape (M+1) - The natural parameters \(\boldsymbol{\beta}\)

opt_info: tuple - A tuple containing other information of the optimization process

values: float - the optimal value of target_function.

residuals: float - the residual of the optimization.

step: float - the total number of Newton’s step iteration.

bsteps: float - the total number of Backtracking step.

Return type:

Tuple

natural_paras_to_custom_moments(betas, domain, statistics, stats_gauge_paras=())

Compute the moments of custom statistics given natural parameters.

Specific implementation of BaseExpFamilyModel should replace gauge_paras and base_args to explicit arguments.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

• statistics (function) –

  A float-valued or tensor-valued function \(\phi_{i_1,\cdots,i_k}\) ( u , \(*\) gauge_paras ) with

  Parameters:

  u : float array of shape (3) - The 3D sample vector \(\mathbf{u}\)

  \(*\) gauge_paras : - Arbitrary many extra parameters such as \(\mathbf{g}\). The \(*\) refers to the unpacking operator in python.

  Returns:

  float or array of arbitrary shape \((d_1,\cdots,d_k)\) – the value of the statistic \(\phi_{i_1,\cdots,i_k}(\mathbf{u},\mathbf{g})\)

• stats_gauge_paras (tuple) – A tuple ( para1, para2, … ) containing arbitrary many extra parameters the custom statistics function \(\phi_{i_1,\cdots,i_k}\) required. Defaut is (), an empty tuple.

Returns:

The moment value \(M_{i_1,\cdots,i_k}(\mathbf{g})\)

Return type:

float or array of arbitrary shape \((d_1,\cdots,d_k)\)

natural_paras_to_fluid_properties(betas, domain)

Compute the fluid properties including number density \(n\), density \(\rho\), flow velocities \(\mathbf{v} = \{v_\alpha, \alpha \in \{x,y,z\}\}\), temperature \(T\), pressure \(p\), stress \(\{\sigma_{\alpha \beta}, \alpha, \beta \in \{x,y,z\}\}\) and heat flux \(\{q_{\alpha}, \alpha \in \{x,y,z\}\}\).

\begin{equation} \begin{split} n &= \frac{\rho}{m}= \int f(\mathbf{u}) d^3 \mathbf{u} \\ v_\alpha &= \frac{1}{n}\int u_\alpha f(\mathbf{u}) d^3 \mathbf{u}\\ p &= n k_B T = \frac{1}{3} \int m c_\alpha c_\alpha f(\mathbf{u}) d^3 \mathbf{u} \\ \sigma_{\alpha\beta} &= \int m c_\alpha c_\beta f(\mathbf{u}) d^3 \mathbf{u} - p \delta_{\alpha\beta} \\ \epsilon &= \frac{3}{2} k_B T = \frac{1}{n}\int \frac{m}{2} \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u} \\ q_\alpha &= \int \frac{m}{2} c_\alpha \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u}; \quad \alpha, \beta \in \{x,y,z\} \end{split} \end{equation}

in which \(m\) is the mass of gas molecule.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

Array containing macroscopic quantities \(\{ \rho, n, v_x, v_y, v_z, T, p, \sigma_{xx}, \sigma_{xy}, \sigma_{xz}, \sigma_{yy}, \sigma_{yz}, \sigma_{zz}, q_x, q_y, q_z \}\)

Return type:

float array of shape (16)

rhoVTsigma_to_natural_paras(rhoVTsigma, domain, Pr=2 / 3)

Compute the natural parameters \(\boldsymbol{\beta}\) from the density \(\rho\), flow velocity \(v_x\) and temperature \(T\)

Parameters:

rhoVTsigma (float array of shape (4)) – A float array ( \(\rho\), \(v_x\), \(T\), \(\sigma_xx\) )

Returns:

The natural parameters \(\boldsymbol{\beta}\)

Return type:

float array of shape (3)

class MomentGauge.Models.ESBGK.ESBGK_Canonical_Legendre_1D(constant)

Bases: ESBGK_Legendre_1D, MomentGauge.Models.ExpFamilyBase.CanonicalExpFamilyModel

The 1D Elliptic Statistical distribution.

\begin{equation} \begin{split} f_0(\mathbf{u}) &= \exp\left( \beta_0 \phi_0(\mathbf{u}) + \beta_1 \phi_1(\mathbf{u}) + \beta_2 \phi_2(\mathbf{u} )+ \beta_3 \phi_3(\mathbf{u} ) \right)\\ &= \frac{\rho}{m} \exp\left( \frac{m v_x}{k_B T}\phi_1(\mathbf{u}) -\frac{m}{2 k_B T} \phi_2(\mathbf{u}) - \left(\frac{m v_x^2}{2k_B T}+ \log \left(\frac{2\pi k_B T}{m}\right)^{\frac{3}{2}}\right) \right), \end{split} \end{equation}

in which \(\rho\) is the number density, \(m\) is the molecule velocity, \(v_x\) is the flow velocity at the x direction, \(T\) is the temperature, and

\(\phi_0\) (u ) = 1.

\(\phi_1\) (u ) = \(u_x\)

\(\phi_2\) (u ) = \(u_x^2\)

\(\phi_3\) (u ) = \(u_y^2 + u_z^2\).

Parameters:

constant (dict) –

dictionary with the following keys

’m’ : float - the mass of particle considered

’kB’ : float - the Boltzmann constant

’n_x’ : int - the order of Gauss Legendre quadrature in x dimension. Required by MomentGauge.Sampler.QuadratureSampler.Gauss_Legendre_Sampler_Polar3D

’n_r’ : int - the order of Gauss Legendre quadrature in r dimension.

’B_x’ : int - how many blocks are splitted in the x dimension for Gauss Legendre quadrature.

’B_r’ : int - how many blocks are splitted in the r dimension for Gauss Legendre quadrature.

’alpha’: float - the initial step size used in backtracking line search, default = 1. Required by MomentGauge.Optim.NewtonOptimizer.Newton_Optimizer

’beta’: float - the decreasing factor of the step size used in backtracking line search, default = 0.5

’c’: float - the parameter used in the Armijo’s condiiton, must lies in (0,1), default = 5e-4

’atol’: float - the absolute error tolerance of the Armijo’s condiiton since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 5e-6.

’rtol’: float - the relative error tolerance of the Armijo’s condition since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 1e-5.

’max_iter’: int - the maximal iteration allowed for the Netwon’s method, default = 100

’max_back_tracking_iter’: int - the maximal iteration allowed for the backtracking line search, default = 25

’tol’: float - the tolerance for residual, below which the optimization stops.

’min_step_size’: float - the minimum step size given by back tracking, below which the optimization stops, default = 1e-6.

’reg_hessian’: bool - Regularize the Hessian if the Cholesky decomposition failed. Default = True

’debug’: bool - print debug information if True.

m

the mass of particle considered

Type:

float

kB

the Boltzmann constant

Type:

float

constant

dictionary with the keys containing ‘m’ and ‘kB’

Type:

dict

rhoVTsigma_to_natural_paras(rhoVTsigma, domain, Pr=2 / 3)

Compute the natural parameters \(\boldsymbol{\beta}\) from the density \(\rho\), flow velocity \(v_x\) and temperature \(T\)

Parameters:

rhoVTsigma (float array of shape (4)) – A float array ( \(\rho\), \(v_x\), \(T\), \(\sigma_xx\) )

Returns:

The natural parameters \(\boldsymbol{\beta}\)

Return type:

float array of shape (3)

suff_statistics(u)

Compute the value of sufficient statistics at certain \(\mathbf{u}\)

Parameters:

u (float array of shape (3)) – The 3D sample vector

Returns:

The array \(\{ \phi_0(\mathbf{u}), \phi_1(\mathbf{u}), \phi_2(\mathbf{u}), \phi_3(\mathbf{u}) \}\)

Return type:

float array of shape (4)

natural_paras_to_moments(betas, domain)

Compute the moments of sufficient statistics given natural parameters`

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

The array \(\{ M_0, \cdots, M_M \}\)

Return type:

float array of shape (M+1)

moments_to_natural_paras(betas_ini, moments, domain)

Compute the natural parameters \(\boldsymbol{\beta}\) from the moments of sufficient statistics.

Parameters:

• betas_ini (float array of shape (4)) – A set of proposed natural parameter \(\boldsymbol{\beta}\)

• moments (float array of shape (4)) – the target moments that we expect exponential familty with the natural parameter \(\boldsymbol{\beta}\) to have.

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

A tuple containing

beta: float array of shape (M+1) - The natural parameters \(\boldsymbol{\beta}\)

opt_info: tuple - A tuple containing other information of the optimization process

values: float - the optimal value of target_function.

residuals: float - the residual of the optimization.

step: float - the total number of Newton’s step iteration.

bsteps: float - the total number of Backtracking step.

Return type:

Tuple

natural_paras_to_custom_moments(betas, domain, statistics, stats_gauge_paras=())

Compute the moments of custom statistics given natural parameters.

Specific implementation of BaseExpFamilyModel should replace gauge_paras and base_args to explicit arguments.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

• statistics (function) –

  A float-valued or tensor-valued function \(\phi_{i_1,\cdots,i_k}\) ( u , \(*\) gauge_paras ) with

  Parameters:

  u : float array of shape (3) - The 3D sample vector \(\mathbf{u}\)

  \(*\) gauge_paras : - Arbitrary many extra parameters such as \(\mathbf{g}\). The \(*\) refers to the unpacking operator in python.

  Returns:

  float or array of arbitrary shape \((d_1,\cdots,d_k)\) – the value of the statistic \(\phi_{i_1,\cdots,i_k}(\mathbf{u},\mathbf{g})\)

• stats_gauge_paras (tuple) – A tuple ( para1, para2, … ) containing arbitrary many extra parameters the custom statistics function \(\phi_{i_1,\cdots,i_k}\) required. Defaut is (), an empty tuple.

Returns:

The moment value \(M_{i_1,\cdots,i_k}(\mathbf{g})\)

Return type:

float or array of arbitrary shape \((d_1,\cdots,d_k)\)

natural_paras_to_fluid_properties(betas, domain)

Compute the fluid properties including number density \(n\), density \(\rho\), flow velocities \(\mathbf{v} = \{v_\alpha, \alpha \in \{x,y,z\}\}\), temperature \(T\), pressure \(p\), stress \(\{\sigma_{\alpha \beta}, \alpha, \beta \in \{x,y,z\}\}\) and heat flux \(\{q_{\alpha}, \alpha \in \{x,y,z\}\}\).

\begin{equation} \begin{split} n &= \frac{\rho}{m}= \int f(\mathbf{u}) d^3 \mathbf{u} \\ v_\alpha &= \frac{1}{n}\int u_\alpha f(\mathbf{u}) d^3 \mathbf{u}\\ p &= n k_B T = \frac{1}{3} \int m c_\alpha c_\alpha f(\mathbf{u}) d^3 \mathbf{u} \\ \sigma_{\alpha\beta} &= \int m c_\alpha c_\beta f(\mathbf{u}) d^3 \mathbf{u} - p \delta_{\alpha\beta} \\ \epsilon &= \frac{3}{2} k_B T = \frac{1}{n}\int \frac{m}{2} \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u} \\ q_\alpha &= \int \frac{m}{2} c_\alpha \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u}; \quad \alpha, \beta \in \{x,y,z\} \end{split} \end{equation}

in which \(m\) is the mass of gas molecule.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

Array containing macroscopic quantities \(\{ \rho, n, v_x, v_y, v_z, T, p, \sigma_{xx}, \sigma_{xy}, \sigma_{xz}, \sigma_{yy}, \sigma_{yz}, \sigma_{zz}, q_x, q_y, q_z \}\)

Return type:

float array of shape (16)

class MomentGauge.Models.ESBGK.ESBGK_Gauged_Legendre_1D(constant)

Bases: MomentGauge.Models.ExpFamilyBase.GaugedExpFamilyModel, MomentGauge.Models.ExpFamilyBase.BaseExpFamilyModel

The 1D Elliptic Statistical distribution with Gauge transformation.

\begin{equation} \begin{split} f_0(\mathbf{u}) &= \exp\left( \beta_0 \phi_0(\mathbf{u},s_r, s_x, w_x) + \beta_1 \phi_1(\mathbf{u}, s_r, s_x, w_x) + \beta_2 \phi_2(\mathbf{u} , s_r, s_x,, w_x) + \beta_3 \phi_3(\mathbf{u} , s_r, s_x,, w_x) \right)\\ \end{split} \end{equation}

in which \(\rho\) is the number density, \(m\) is the molecule velocity, \(v_x\) is the flow velocity at the x direction, \(T\) is the temperature, and

Specifically,

\(\phi_0\) (u, \(s_r, s_x, w_x\) ) = 1.

\(\phi_1\) (u, \(s_r, s_x, w_x\) ) = \(\bar{u}_x\)

\(\phi_2\) (u, \(s_r, s_x, w_x\) ) = \(\bar{u}_x^2\)

\(\phi_3\) (u, \(s_r, s_x, w_x\) ) = \(\bar{u}_y^2 + \bar{u}_z^2\).

in which \(\bar{u}_x = \frac{u_x - w_x}{s}\), \(\bar{u}_y = \frac{u_y}{s}\), \(\bar{u}_z = \frac{u_z}{s}\)

Parameters:

constant (dict) –

dictionary with the following keys

’m’ : float - the mass of particle considered

’kB’ : float - the Boltzmann constant

’n_x’ : int - the order of Gauss Legendre quadrature in x dimension. Required by MomentGauge.Sampler.QuadratureSampler.Gauss_Legendre_Sampler_Polar3D

’n_r’ : int - the order of Gauss Legendre quadrature in r dimension.

’B_x’ : int - how many blocks are splitted in the x dimension for Gauss Legendre quadrature.

’B_r’ : int - how many blocks are splitted in the r dimension for Gauss Legendre quadrature.

’alpha’: float - the initial step size used in backtracking line search, default = 1. Required by MomentGauge.Optim.NewtonOptimizer.Newton_Optimizer

’beta’: float - the decreasing factor of the step size used in backtracking line search, default = 0.5

’c’: float - the parameter used in the Armijo’s condiiton, must lies in (0,1), default = 5e-4

’atol’: float - the absolute error tolerance of the Armijo’s condiiton since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 5e-6.

’rtol’: float - the relative error tolerance of the Armijo’s condition since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 1e-5.

’max_iter’: int - the maximal iteration allowed for the Netwon’s method, default = 100

’max_back_tracking_iter’: int - the maximal iteration allowed for the backtracking line search, default = 25

’tol’: float - the tolerance for residual, below which the optimization stops.

’min_step_size’: float - the minimum step size given by back tracking, below which the optimization stops, default = 1e-6.

’reg_hessian’: bool - Regularize the Hessian if the Cholesky decomposition failed. Default = True

’debug’: bool - print debug information if True.

m

the mass of particle considered

Type:

float

kB

the Boltzmann constant

Type:

float

constant

dictionary with the keys containing ‘m’ and ‘kB’

Type:

dict

suff_statistics(u, gauge_paras)

Compute the value of sufficient statistics at certain \(\mathbf{u}\)

Parameters:

• u (float array of shape (3)) – The 3D sample vector

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

Returns:

The array \(\{ \phi_0(\mathbf{u}, s_r, s_x, w_x), \phi_1(\mathbf{u}, s_r, s_x, w_x), \phi_2(\mathbf{u}, s_r, s_x, w_x), \phi_3(\mathbf{u}, s_r, s_x, w_x) \}\)

Return type:

float array of shape (4)

natural_paras_to_moments(betas, gauge_paras, domain)

Compute the moments of sufficient statistics given natural parameters`

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\),:math:w_x).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

The array \(\{ M_0, \cdots, M_M \}\)

Return type:

float array of shape (4)

moments_to_natural_paras(betas_ini, moments, gauge_paras, domain)

Compute the natural parameters \(\boldsymbol{\beta}\) from the moments of sufficient statistics.

Parameters:

• betas_ini (float array of shape (4)) – A set of proposed natural parameter \(\boldsymbol{\beta}\)

• moments (float array of shape (4)) – the target moments that we expect exponential familty with the natural parameter \(\boldsymbol{\beta}\) to have.

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

A tuple containing

beta: float array of shape (4) - The natural parameters \(\boldsymbol{\beta}\)

opt_info: tuple - A tuple containing other information of the optimization process

values: float - the optimal value of target_function.

residuals: float - the residual of the optimization.

step: float - the total number of Newton’s step iteration.

bsteps: float - the total number of Backtracking step.

Return type:

Tuple

natural_paras_to_custom_moments(betas, gauge_paras, domain, statistics, stats_gauge_paras=())

Compute the moments of custom statistics given natural parameters.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

• statistics (function) –

  A float-valued or tensor-valued function \(\phi_{i_1,\cdots,i_k}\) ( u , \(*\) gauge_paras ) with

  Parameters:

  u : float array of shape (3) - The 3D sample vector \(\mathbf{u}\)

  \(*\) gauge_paras : - Arbitrary many extra parameters such as \(\mathbf{g}\). The \(*\) refers to the unpacking operator in python.

  Returns:

  float or array of arbitrary shape \((d_1,\cdots,d_k)\) – the value of the statistic \(\phi_{i_1,\cdots,i_k}(\mathbf{u},\mathbf{g})\)

• stats_gauge_paras (tuple) – A tuple ( para1, para2, … ) containing arbitrary many extra parameters the custom statistics function \(\phi_{i_1,\cdots,i_k}\) required. Defaut is (), an empty tuple.

Returns:

The moment value \(M_{i_1,\cdots,i_k}(\mathbf{g})\)

Return type:

float or array of arbitrary shape \((d_1,\cdots,d_k)\)

moments_gauge_transformation(moments, gauge_para2, gauge_para1, domain)

Gauge transformation for moments of sufficient statistics. The transformation is defined as

\begin{equation} M_i( s_r', s_x', w_x' )= T_{ij}(s_r', s_x', w_x', s_r, s_x, w_x) M_j(s_r, s_x, w_x); \quad i,j = 0, \cdots, 3 \end{equation}

which is induced from the gauge transformation \(T_{ij}(\mathbf{g}',\mathbf{g})\) between sufficient statistics \(\phi_i(\mathbf{u},(s, w_x))\) such that

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x',w_x') )= T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, 3 \end{equation}

with \(\phi_i(\mathbf{u},(s_r, s_x, w_x))\) is sufficient statistics parameterized by gauge parameters \((s_r, s_x, w_x)\).

Parameters:

• moments (float array of shape (4)) – the moments \(M_i(s_r, s_x, w_x)\) of sufficient statistics

• gauge_para2 (float array of shape (3)) – The array (\(s_r', s_x'\), \(w_x'\)).

• gauge_para1 (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

the moments \(M_i(s', w_x')\) of sufficient statistics

Return type:

float array of shape (3)

natural_paras_gauge_transformation(betas, gauge_para2, gauge_para1, domain)

Gauge transformation for natural parameters. The transformation is defined as

\begin{equation} \beta_i( s_r', s_x', w_x' )= T_{ij}^T(s_r, s_x, w_x, s_r', s_x', w_x') \beta_j(s_r, s_x, w_x); \quad i,j = 0, \cdots, 2 \end{equation}

which is induced from the gauge transformation \(T_{ij}(\mathbf{g}',\mathbf{g})\) between sufficient statistics \(\phi_i(\mathbf{u},\mathbf{g})\) such that

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x', w_x') )= T_{ij}(s_r', s_x', w_x',s, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, 2 \end{equation}

with \(\phi_i(\mathbf{u},(s_r, s_x, w_x))\) is sufficient statistics parameterized by gauge parameters \((s_r, s_x, w_x)\).

Parameters:

• betas (float array of shape (4)) – the moments \(\beta_i(s_r, s_x, w_x)\) of sufficient statistics

• gauge_para2 (float array of shape (3)) – The array (\(s_r', s_x'\), \(w_x'\)).

• gauge_para1 (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

Returns:

the moments \(\beta_i(s_r', s_x', w_x')\) of sufficient statistics

Return type:

float array of shape (M+1)

natural_paras_to_fluid_properties(betas, gauge_paras, domain)

Compute the fluid properties including number density \(n\), density \(\rho\), flow velocities \(\mathbf{v} = \{v_\alpha, \alpha \in \{x,y,z\}\}\), temperature \(T\), pressure \(p\), stress \(\{\sigma_{\alpha \beta}, \alpha, \beta \in \{x,y,z\}\}\) and heat flux \(\{q_{\alpha}, \alpha \in \{x,y,z\}\}\).

\begin{equation} \begin{split} n &= \frac{\rho}{m}= \int f(\mathbf{u}) d^3 \mathbf{u} \\ v_\alpha &= \frac{1}{n}\int u_\alpha f(\mathbf{u}) d^3 \mathbf{u}\\ p &= n k_B T = \frac{1}{3} \int m c_\alpha c_\alpha f(\mathbf{u}) d^3 \mathbf{u} \\ \sigma_{\alpha\beta} &= \int m c_\alpha c_\beta f(\mathbf{u}) d^3 \mathbf{u} - p \delta_{\alpha\beta} \\ \epsilon &= \frac{3}{2} k_B T = \frac{1}{n}\int \frac{m}{2} \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u} \\ q_\alpha &= \int \frac{m}{2} c_\alpha \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u}; \quad \alpha, \beta \in \{x,y,z\} \end{split} \end{equation}

in which \(m\) is the mass of gas molecule.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

Array containing macroscopic quantities \(\{ \rho, n, v_x, v_y, v_z, T, p, \sigma_{xx}, \sigma_{xy}, \sigma_{xz}, \sigma_{yy}, \sigma_{yz}, \sigma_{zz}, q_x, q_y, q_z \}\)

Return type:

float array of shape (16)

rhoVTsigma_to_natural_paras(rhoVTsigma, gauge_paras, domain, Pr=2 / 3)

Compute the natural parameters \(\boldsymbol{\beta}\) from the density \(\rho\), flow velocity \(v_x\) and temperature \(T\)

Parameters:

• rhoVTsigma (float array of shape (4)) – A float array ( \(\rho\), \(v_x\), \(T\), \(\sigma_xx\) )

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

Returns:

The natural parameters \(\boldsymbol{\beta}\)

Return type:

float array of shape (3)

standard_gauge_para_from_moments(moments, gauge_paras)

the standard gauge parameters \(\mathbf{g}\) prefered among all possible gauges.

Parameters:

• moments (float array of shape (3)) – the moments \(M_i(s, w_x)\) of sufficient statistics

• gauge_paras (float array of shape (2)) – The array (\(s\), \(w_x\)).

Returns:

The gauge parameters (\(s\), \(w_x\)) in the Hermite gauge.

Return type:

float array of shape (2)

class MomentGauge.Models.ESBGK.ESBGK_Canonical_Gauged_Legendre_1D(constant)

Bases: ESBGK_Gauged_Legendre_1D, MomentGauge.Models.ExpFamilyBase.CanonicalExpFamilyModel

The 1D Elliptic Statistical distribution with Gauge transformation.

\begin{equation} \begin{split} f_0(\mathbf{u}) &= \exp\left( \beta_0 \phi_0(\mathbf{u},s_r, s_x, w_x) + \beta_1 \phi_1(\mathbf{u}, s_r, s_x, w_x) + \beta_2 \phi_2(\mathbf{u} , s_r, s_x,, w_x) + \beta_3 \phi_3(\mathbf{u} , s_r, s_x,, w_x) \right)\\ \end{split} \end{equation}

in which \(\rho\) is the number density, \(m\) is the molecule velocity, \(v_x\) is the flow velocity at the x direction, \(T\) is the temperature, and

Specifically,

\(\phi_0\) (u, \(s_r, s_x, w_x\) ) = 1.

\(\phi_1\) (u, \(s_r, s_x, w_x\) ) = \(\bar{u}_x\)

\(\phi_2\) (u, \(s_r, s_x, w_x\) ) = \(\bar{u}_x^2\)

\(\phi_3\) (u, \(s_r, s_x, w_x\) ) = \(\bar{u}_y^2 + \bar{u}_z^2\).

in which \(\bar{u}_x = \frac{u_x - w_x}{s}\), \(\bar{u}_y = \frac{u_y}{s}\), \(\bar{u}_z = \frac{u_z}{s}\)

Parameters:

constant (dict) –

dictionary with the following keys

’m’ : float - the mass of particle considered

’kB’ : float - the Boltzmann constant

’n_x’ : int - the order of Gauss Legendre quadrature in x dimension. Required by MomentGauge.Sampler.QuadratureSampler.Gauss_Legendre_Sampler_Polar3D

’n_r’ : int - the order of Gauss Legendre quadrature in r dimension.

’B_x’ : int - how many blocks are splitted in the x dimension for Gauss Legendre quadrature.

’B_r’ : int - how many blocks are splitted in the r dimension for Gauss Legendre quadrature.

’alpha’: float - the initial step size used in backtracking line search, default = 1. Required by MomentGauge.Optim.NewtonOptimizer.Newton_Optimizer

’beta’: float - the decreasing factor of the step size used in backtracking line search, default = 0.5

’c’: float - the parameter used in the Armijo’s condiiton, must lies in (0,1), default = 5e-4

’atol’: float - the absolute error tolerance of the Armijo’s condiiton since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 5e-6.

’rtol’: float - the relative error tolerance of the Armijo’s condition since we use -(atol + rtol*abs(next_value)) instead of 0 to handle single precision numerical error. default = 1e-5.

’max_iter’: int - the maximal iteration allowed for the Netwon’s method, default = 100

’max_back_tracking_iter’: int - the maximal iteration allowed for the backtracking line search, default = 25

’tol’: float - the tolerance for residual, below which the optimization stops.

’min_step_size’: float - the minimum step size given by back tracking, below which the optimization stops, default = 1e-6.

’reg_hessian’: bool - Regularize the Hessian if the Cholesky decomposition failed. Default = True

’debug’: bool - print debug information if True.

m

the mass of particle considered

Type:

float

kB

the Boltzmann constant

Type:

float

constant

dictionary with the keys containing ‘m’ and ‘kB’

Type:

dict

rhoVTsigma_to_natural_paras(rhoVTsigma, gauge_paras, domain, Pr=2 / 3)

Compute the natural parameters \(\boldsymbol{\beta}\) from the density \(\rho\), flow velocity \(v_x\) and temperature \(T\)

Parameters:

• rhoVTsigma (float array of shape (4)) – A float array ( \(\rho\), \(v_x\), \(T\), \(\sigma_xx\) )

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

Returns:

The natural parameters \(\boldsymbol{\beta}\)

Return type:

float array of shape (3)

suff_statistics(u, gauge_paras)

Compute the value of sufficient statistics at certain \(\mathbf{u}\)

Parameters:

• u (float array of shape (3)) – The 3D sample vector

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

Returns:

The array \(\{ \phi_0(\mathbf{u}, s_r, s_x, w_x), \phi_1(\mathbf{u}, s_r, s_x, w_x), \phi_2(\mathbf{u}, s_r, s_x, w_x), \phi_3(\mathbf{u}, s_r, s_x, w_x) \}\)

Return type:

float array of shape (4)

natural_paras_to_moments(betas, gauge_paras, domain)

Compute the moments of sufficient statistics given natural parameters`

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\),:math:w_x).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

The array \(\{ M_0, \cdots, M_M \}\)

Return type:

float array of shape (4)

moments_to_natural_paras(betas_ini, moments, gauge_paras, domain)

Compute the natural parameters \(\boldsymbol{\beta}\) from the moments of sufficient statistics.

Parameters:

• betas_ini (float array of shape (4)) – A set of proposed natural parameter \(\boldsymbol{\beta}\)

• moments (float array of shape (4)) – the target moments that we expect exponential familty with the natural parameter \(\boldsymbol{\beta}\) to have.

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

A tuple containing

beta: float array of shape (4) - The natural parameters \(\boldsymbol{\beta}\)

opt_info: tuple - A tuple containing other information of the optimization process

values: float - the optimal value of target_function.

residuals: float - the residual of the optimization.

step: float - the total number of Newton’s step iteration.

bsteps: float - the total number of Backtracking step.

Return type:

Tuple

natural_paras_to_custom_moments(betas, gauge_paras, domain, statistics, stats_gauge_paras=())

Compute the moments of custom statistics given natural parameters.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

• statistics (function) –

  A float-valued or tensor-valued function \(\phi_{i_1,\cdots,i_k}\) ( u , \(*\) gauge_paras ) with

  Parameters:

  u : float array of shape (3) - The 3D sample vector \(\mathbf{u}\)

  \(*\) gauge_paras : - Arbitrary many extra parameters such as \(\mathbf{g}\). The \(*\) refers to the unpacking operator in python.

  Returns:

  float or array of arbitrary shape \((d_1,\cdots,d_k)\) – the value of the statistic \(\phi_{i_1,\cdots,i_k}(\mathbf{u},\mathbf{g})\)

• stats_gauge_paras (tuple) – A tuple ( para1, para2, … ) containing arbitrary many extra parameters the custom statistics function \(\phi_{i_1,\cdots,i_k}\) required. Defaut is (), an empty tuple.

Returns:

The moment value \(M_{i_1,\cdots,i_k}(\mathbf{g})\)

Return type:

float or array of arbitrary shape \((d_1,\cdots,d_k)\)

moments_gauge_transformation(moments, gauge_para2, gauge_para1, domain)

Gauge transformation for moments of sufficient statistics. The transformation is defined as

\begin{equation} M_i( s_r', s_x', w_x' )= T_{ij}(s_r', s_x', w_x', s_r, s_x, w_x) M_j(s_r, s_x, w_x); \quad i,j = 0, \cdots, 3 \end{equation}

which is induced from the gauge transformation \(T_{ij}(\mathbf{g}',\mathbf{g})\) between sufficient statistics \(\phi_i(\mathbf{u},(s, w_x))\) such that

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x',w_x') )= T_{ij}(s_r', s_x', w_x',s_r, s_x, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, 3 \end{equation}

with \(\phi_i(\mathbf{u},(s_r, s_x, w_x))\) is sufficient statistics parameterized by gauge parameters \((s_r, s_x, w_x)\).

Parameters:

• moments (float array of shape (4)) – the moments \(M_i(s_r, s_x, w_x)\) of sufficient statistics

• gauge_para2 (float array of shape (3)) – The array (\(s_r', s_x'\), \(w_x'\)).

• gauge_para1 (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

the moments \(M_i(s', w_x')\) of sufficient statistics

Return type:

float array of shape (3)

natural_paras_gauge_transformation(betas, gauge_para2, gauge_para1, domain)

Gauge transformation for natural parameters. The transformation is defined as

\begin{equation} \beta_i( s_r', s_x', w_x' )= T_{ij}^T(s_r, s_x, w_x, s_r', s_x', w_x') \beta_j(s_r, s_x, w_x); \quad i,j = 0, \cdots, 2 \end{equation}

which is induced from the gauge transformation \(T_{ij}(\mathbf{g}',\mathbf{g})\) between sufficient statistics \(\phi_i(\mathbf{u},\mathbf{g})\) such that

\begin{equation} \phi_i(\mathbf{u}, (s_r', s_x', w_x') )= T_{ij}(s_r', s_x', w_x',s, w_x) \phi_j(\mathbf{u}, (s_r, s_x, w_x)); \quad i,j = 0, \cdots, 2 \end{equation}

with \(\phi_i(\mathbf{u},(s_r, s_x, w_x))\) is sufficient statistics parameterized by gauge parameters \((s_r, s_x, w_x)\).

Parameters:

• betas (float array of shape (4)) – the moments \(\beta_i(s_r, s_x, w_x)\) of sufficient statistics

• gauge_para2 (float array of shape (3)) – The array (\(s_r', s_x'\), \(w_x'\)).

• gauge_para1 (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

Returns:

the moments \(\beta_i(s_r', s_x', w_x')\) of sufficient statistics

Return type:

float array of shape (M+1)

natural_paras_to_fluid_properties(betas, gauge_paras, domain)

Compute the fluid properties including number density \(n\), density \(\rho\), flow velocities \(\mathbf{v} = \{v_\alpha, \alpha \in \{x,y,z\}\}\), temperature \(T\), pressure \(p\), stress \(\{\sigma_{\alpha \beta}, \alpha, \beta \in \{x,y,z\}\}\) and heat flux \(\{q_{\alpha}, \alpha \in \{x,y,z\}\}\).

\begin{equation} \begin{split} n &= \frac{\rho}{m}= \int f(\mathbf{u}) d^3 \mathbf{u} \\ v_\alpha &= \frac{1}{n}\int u_\alpha f(\mathbf{u}) d^3 \mathbf{u}\\ p &= n k_B T = \frac{1}{3} \int m c_\alpha c_\alpha f(\mathbf{u}) d^3 \mathbf{u} \\ \sigma_{\alpha\beta} &= \int m c_\alpha c_\beta f(\mathbf{u}) d^3 \mathbf{u} - p \delta_{\alpha\beta} \\ \epsilon &= \frac{3}{2} k_B T = \frac{1}{n}\int \frac{m}{2} \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u} \\ q_\alpha &= \int \frac{m}{2} c_\alpha \mathbf{c}^2 f(\mathbf{u}) d^3 \mathbf{u}; \quad \alpha, \beta \in \{x,y,z\} \end{split} \end{equation}

in which \(m\) is the mass of gas molecule.

Parameters:

• betas (float array of shape (4)) – the natural parameter \(\boldsymbol{\beta}\) of the distribution

• gauge_paras (float array of shape (3)) – The array (\(s_r, s_x\), \(w_x\)).

• domain (float array of shape (3)) –

  an array (a_x, b_x, b_r), in which

  a_x : float - lower integration limit in x dimension for the Legendre quadrature

  b_x : float - upper integration limit in x dimension for the Legendre quadrature

  b_r : float - upper integration limit in r dimension for the Legendre quadrature

Returns:

Array containing macroscopic quantities \(\{ \rho, n, v_x, v_y, v_z, T, p, \sigma_{xx}, \sigma_{xy}, \sigma_{xz}, \sigma_{yy}, \sigma_{yz}, \sigma_{zz}, q_x, q_y, q_z \}\)

Return type:

float array of shape (16)

standard_gauge_para_from_moments(moments, gauge_paras)

the standard gauge parameters \(\mathbf{g}\) prefered among all possible gauges.

Parameters:

• moments (float array of shape (3)) – the moments \(M_i(s, w_x)\) of sufficient statistics

• gauge_paras (float array of shape (2)) – The array (\(s\), \(w_x\)).

Returns:

The gauge parameters (\(s\), \(w_x\)) in the Hermite gauge.

Return type:

float array of shape (2)

MomentGauge.Models.ESBGK.constant